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Chapter 1 Quadrupolar interactions and magneto-elastic effects in rare earth intermetallic compounds
chapter 1 QUADRUPOLAR INTERACTIONS AND MAGNETO-ELASTIC EFFECTS IN RARE EARTH INTERMETALLIC COMPOUNDS
P. MORIN and D. SCHMITT Laboratoire Louis N@el* C.N.R.S. B.P. 166 X 38042 Grenoble Cedex France
*Associated with Universit6 Joseph Fourier, Grenoble
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. I~ t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Fc rl~ alL, m . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. T h e c t b i c H a n n i l t c n i a n . . . . . . . . . . . . . . . . . . . . . 2.1.1. G e n e r a l e ~ p l e s s i o n . . . . . . . . . . . . . . . . . . . . 2.1.2. H a r m o n i c d e s c r i p t i o n of m a g n e t o s t r i c t i o n . . . . . . . . . . . . . 2.1.3. T h e f e t a l q ~ a & u p o l a r H a m i l ( o n i a n . . . . . . . . . . . . . . . 2.2. T r e a t m e n t of the c u b i c H a m i l t c n i a n . . . . . . . . . . . . . . . . . 2.2.1. F e r t m b a t f o n t h e o r y . . . . . . . . . . . . . . . . . . . . 2.2.2. C e n e r a l i z a t i o n of l h e s u s c e p t i b i l i t y f o r m a l i s m . . . . . . . . . . . . 2.2.3. B e h a v i o u r of l h e ~ a r i c u s s u s c e p t i b i l i t i e s . . . . . . . . . . . . . 2.2.4. M a g n e t i c a n d q u a & u p o l a r t r a n s i l i o n s . . . . . . . . . . . . . . 2.3. E x t e n s i o n to l o w e r s y m m e t r i e s ( h e x a g o n a l a n d t e t r a g o n a l ) . . . . . . . . . 2.3.1. T h e H ~ m i l t o n i ~ n . . . . . . . . . . . . . . . . . . . . . . 2.3.2. H a r m o n : ' c d e s c r i p t i o n of 1Tagnetostr c t i c n . . . . . . . . . . . . . 2.3.3. P e l t u r b a t i o n tl;eo~y . . . . . . . . . . . . . . . . . . . . 2.3.4. C o m p a r i s o n w i t h c u b i c s y m m e t r y . . . . . . . . . . . . . . . . 2.4. A p e c u l i a r c a . e : the semi-classical t r e a l m e n t . . . . . . . . . . . . . . 2.4.1. T h e p a r a m ~ g n e l i c lzhase . . . . . . . . . . . . . . . . . . . 2.4.2. T h e c r d e r e d p h a s e . . . . . . . . . . . . . . . . . . . . . 3. E x p e r i m e n t a l e v i d e n c e . . . . . . . . . . . . . . . . . . . . . . . 3.1. Q u a d l u p o l a r o r d e r i n g s . . . . . . . . . . . . . . . . . . . . . 3.1.1. F e r i c q u a d r ~ p o l a r older:'ng . . . . . . . . . . . . . . . . . . 3.1.1.1. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2. T m C d . . . . . . . . . . . . . . . . . . . . . . 3.1.1.3. C e A g . . . . . . . . . . . . . . . . . . . . . . 3.1.2. A n t : f e r l c q u a d ~ p o / ~ r o l d e r i n g . . . . . . . . . . . . . . . . 3.1.2.1. C e B 6 . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2. T m G a 3 . . . . . . . . . . . . . . . . . . . . . 3.1.2.3. P I P b 3 . . . . . . . . . . . . . . . . . . . . . . 3.1.3. S t r u c t u r a l l r a n s i t i o n s p o s s i b l y of q u a d r u p o l a r o r i g i n . . . . . . . . . 3.1.3.1. P r C u z . . . . . . . . . . . . . . . . . . . . . . 3.1.3.2. U P d 3 . . . . . . . . . . . . . . . . . . . . . . 3.2. N a t u r e of q u a d r u p o l a r a n d m a g n e t i c t r a n s i t i o n s . . . . . . . . . . . . . 3.2.1. Q u a d r u p o l a r t l ~ n s t i o n s . . . . . . . . . . . . . . . . . . . 3.2.2. M a g n e t i c o r d e r i n g in t h e p r e s e n c e of q u a d r u p o l a r i n t e r a c t i o n s . . . . . . . . . . . . . . . . . 3.2.2.1. F i r s t - o r d e r t r a n s i t i o n s a n d tricriti.cality
5 6 7 7 11 12 12 13 16 17 20 24 24 27 28 30 30 30 32 34 34 34 35 36 36 38 38 39 40 41 41 42 43 43 43 44
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
3.2.2.2. Second-order transition in PrMg 2 due to negative qua drupol a r interactions . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Magnetic ordering in the q u a d r u p o l a r phase . . . . . . . . . . . . 3.2.3.1. C e A g , CeB 6 . . . . . . . . . . . . . . . . . . . 3.2.3.2. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Presence of additional couplings . . . . . . . . . . . . . . . . 3.2.4.1. DySb and rare earth monopnictides . . . . . . . . . . . . 3.2.4.2. The RA12 series . . . . . . . . . . . . . . . . . . 3.3. D e t e r m i n a t i o n of the q u a d r u p o l a r p a r a m e t e r s from susceptibility techniques . 3.3.1, Elastic constants . . . . . . . . . . . . . . . . . . . . . 3.3.2, Parastriction . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Third-order p a r a m a g n e t i c susceptibility . . . . . . . . . . . . . 3.4. Effects on the magnetostriction . . . . . . . . . . . . . . . . . . 3.4.1. Magnetostriction of dilute compounds . . . . . . . . . . . . . . 3.4.2. Magnetostriction in the ordered phase . . . . . . . . . . . . . . 3.5. Effects on the magnetization processes . . . . . . . . . . . . . . . . 3.5.1. P a r a m a g n e t i c phase . . . . . . . . . . . . . . . . . . . . 3.5.2. O r d e r e d phase . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Effect on the direction of easy magnetization . . . . . . . . . . . 3.6. Effects on the magnetic structures . . . . . . . . . . . . . . . . . 3.6.1. Multi-axial structures . . . . . . . . . . . . . . . . . . . 3.6.2. I n c o m m e n s u r a t e magnetic structures . . . . . . . . . . . . . . 3.7. Effects on magnetic excitations . . . . . . . . . . . . . . . . . . 3.8. Two-ion anisotropic magneto-elasticity . . . . . . . . . . . . . . . . 3.9. Isotropic magneto-elasticity and pressure effects . . . . . . . . . . . . . 3.9.1. Two-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.2. Single-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.3. Pressure effects . . . . . . . . . . . . . . . . . . . . . 3.10. Magneto-elasticity in the presence of lattice instability . . . . . . . . . . . 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series . . . . 4A. CsCl-type structure compounds . . . . . . . . . . . . . . . . . . . 4.1.1. R Z n series . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. R C d series . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. R A g series . . . . . . . . . . . . . . . . . . . . . . . 4.1.4, R C u series . . . . . . . . . . . . . . . . . . . . . . . 4.1.5, R M g series . . . . . . . . . . . . . . . . . . . . . . . 4.2. NaCl-type structure compouods . . . . . . . . . . . . . . . . . . . 4.3. AuCu3-type structure compounds . . . . . . . . . . . . . . . . . . 4.4. Cubic Laves phase compounds . . . . . . . . . . . . . . . . . . . 4.4.1. RA12 compounds . . . . . . . . . . . . . . . . . . . . . 4.4.2. R M 2 compounds (M = Mg, Ni, Co) . . . . . . . . . . . . . . . 4.4.3. R F e 2 compounds . . . . . . . . . . . . . . . . . . . . . 4.5. Hexagonal CaCus-type structure compounds . . . . . . . . . . . . . . 4.5.1. RNi 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.5.2. RCo 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.6. Dilute rare earth systems . . . . . . . . . . . . . . . . . . . . . 4.6.1. R a r e earths diluted in cubic noble-metal hosts . . . . . . . . . . . 4.6.2. R a r e earths diluted in cubic pnictides . . . . . . . . . . . . . . 4.6.3. Other dilute systems . . . . . . . . . . . . . . . . . . . . 4.7. Miscellaneous compounds . . . . . . . . . . . . . . . . . . . . 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. G e n e r a l analysis . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Coherency of the various determinations . . . . . . . . . . . . .
3
49 49 50 50 50 50 52 53 54 56 57 61 61 64 67 67 69 70 72 72 75 77 79 79 80 80 81 82 83 83 84 85 85 90 92 93 95 95 95 98 100 102 102 102 106 106 107 110 111 113 113 113
4
P. M O R I N and D. SCHMITI"
5.1.2. Comparison of the magneto-elastic coefficients . . . . . . . . . . . 5.1.3. Comparison of the quadrupolar pair-interaction coefficients . . . . . . . 5.2. Origin of the quadrupolar interactions . . . . . . . . . . . . . . . . 5.2.1. The one-ion magneto-elastic coupling . . . . . . . . . . . . . . 5.2.2. Quadrupolar pair interactions . . . . . . . . . . . . . . . . . 5.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Symmetrized Stevens operators for the cubic symmetry . . . . . . . . . . A.2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~ME~ . . . . . . . A.3. Perturbation theory: the tetragonal mode 3' . . . . . . . . . . . . . . A.4. Perturbation theory: the trigonal mode e . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114 116 117 117 119 120 122 122 123 124 125 126
1. Introduction
The study of multipolar interactions concerning the unfilled 4f shell has known a great development in rare earth (R) intermetallic compounds. It involves the complex problems of both the crystalline electric field (CEF) acting on a 4f ion and the pair interactions between different ions. The electron-lattice coupling can drive very characteristic magneto-elastic features, in particular in elastic constants and in magnetostriction, through symmetry lowering modes which depend on the 4f electronic states. According to the Jahn-Teller theorem, the magneto-elastic modulation of the CEF corresponds to a gain of energy larger than the elastic energy lost at the thermodynamic equilibrium. These properties have been extensively studied in the past in many systems exhibiting an orbital degeneracy as reported for cubic spinels (Englman and Halperin 1970, Englman 1972) in particular mixed chromites (Kataoka and Kanamori 1972). Gehring and Gehring (1975) and Melcher (1976) have provided the literature with very complete reviews of the cooperative Jahn-TeUer transitions occurring in rare earth insulators within the tetragonal zircon structure. All these reviews show the fundamental importance of studying elastic constants. Indeed they couple, as thermodynamical variables, the strain to the ultrasonic stress in the presence of a quadrupolar moment, describing the orbital character of the 4f shell. Another important feature is the large success met by the mean-field approximation for describing thoroughly the structural transitions. More recently, very large symmetry lowerings, which are mainly of one-ion origin have been observed to occur at the magnetic ordering in rare earth intermetallics (L6vy 1969, Morin et al. 1979). In the example of RFe 2 compounds which are ordered at room temperature, they are large enough to motivate industrial applications as magnetostrictive transducers which are able to compete with ceramics (Clark 1980). In all the series, they are associated with large softenings of elastic constants in the non-ordered phase (Mullen et al. 1974, Moran et al. 1973). From a theoretical point of view, they have required the development of a quantum treatment which takes into account all the characteristics of the 4f wave functions, as will be discussed in the present chapter. However, in R intermetaUics an additional coupling has been revealed, mediated from one site to the others by the conduction electrons. This quadrupolar pair interaction may dominate the magneto-elastic coupling and drive a
6
P. MORIN and D. SCHMITI"
quadrupolar ordering in the paramagnetic phase, as occurs in TmZn and TmCd (Morin et al. 1978b). According to the sign of the quadrupolar pair interactions, ferro- or antiferroquadrupolar orderings have been observed. The symmetry lowering is then, when it exists, a consequence of the quadrupolar symmetry through the magneto-elastic coupling. Such compounds are not stricto sensu Jahn-Teller compounds, because then the magneto-elastic coupling is the driving mechanism. In the presence of dominating bilinear interactions, which is the majority of cases, quadrupolar interactions act on the minimization of the total free energy, and magnetic properties cannot be understood without them in several series (Giraud et al. 1985). For different experimental techniques (Ltithi 1980a), the magneto-elasticity remains the most appropriate probe for determining the quadrupolar pair interactions. This chapter is organized as follows. First, the quantum Hamiltonian is built, which describes both the one- and two-ion, spin and quadrupolar couplings in systems of cubic, tetragonal and hexagonal symmetries. The power of the susceptibility formalism, which allows independent determinations of the various coefficients, is emphasized (section 2). The main experimental evidence of quadrupolar orderings is presented in section 3, before giving the quadrupolar consequences on the magnetic properties. The experimental techniques which are based on the susceptibility formalism of section 2 are illustrated. Section 4 gives a review of the magneto-elastic and pair interaction coefficients which have been determined in the rare earth intermetaUics. Due to the anisotropic character of the magneto-elasticity, studies on single crystals are of fundamental importance and these are the only ones considered here. Section 5 gives an analysis of all these results from both a macroscopic and a microscopic point of view.
2. Formalism
The existence of non-quenched orbital moments in rare earth ions is the source of their anisotropic magnetic properties. A classical description of these properties is no longer valid. In a quantum formalism, the 4f shell is represented by wavefunctions expanded in the IJ, M:) basis. The appropriate Hamiltonian, acting on these wave functions, depends on the point symmetry of the rare earth site under consideration. In addition to the crystalline electric field (CEF) Hamiltonian, any kind of interactions acting on the rare earths or between them (Heisenberg coupling, magneto-elasticity, quadrupolar pair interactions . . . . ) must be considered. In the literature, most of the experimental results reported on the magnetoelastic properties concerns rare earth compounds with cubic symmetry, because for these compounds the number of parameters required for analyzing these properties is the smallest. In the following two sections, we extensively develop the formalism and the treatment of magneto-elasticity and quadrupolar interactions for cubic symmetry. The case of lower symmetries (hexagonal and tetragon-
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
7
al) will be discussed more briefly in section 2.3, while the connection with the classical formalism will be made in section 2.4. 2.1. The cubic Hamiltonian 2.1. i. General expression
The Hamiltonian appropriate for cubic symmetry includes several terms. By using the operator-equivalent method (Stevens 1952) they are written as: ffb°= ~CEF + ~°Z + ~B + ~OQ+ '~ME1 + ~ME2 "l- (Eel + E B + EQ q- EME2) .
(I)
The different terms represent successively the crystal field term, the Zeeman coupling, the Heisenberg-type bilinear interaction, the two-ion quadrupolar term, the magneto-elastic coupling, the elastic energy, and bilinear and quadrupolar corrective energy terms. Each term is described in detail below. The crystalline electric field Hamiltonian: Y(CEF According to different authors, this Hamiltonian takes one of the following forms. The x, y and z axes have been chosen as the four-fold axes in the cubic crystal: ~ C E F = B4(O°4 + 5 0 4 )
-]- B6(0°6 - 2104)
= A4 ( r 4 ) ~ 1 ( 0 40+ 5 0 4 ) + A 6 ( r 6 ) y 1 ( O ~ - 2 1 0 4 = __
Wx (004 -~- 5 0 ~ ) +
F4
W(1 - - I X l )
F6
0
)
(2)
4
( 0 6 -- 2 1 0 6 ) .
The O~'s are the Stevens operators in the formulation of Hutchings (1964). 13I and 7i are the 4th- and 6th-order Stevens coefficients, the value of which depends on the rare earth. The CEF parameters A4 and A6(r 6> originate in the electric field acting on the 4f shell, which arises from the surrounding point charges (Hutchings 1964) or from the conduction electrons (Schmitt 1979). In the last formulation, W represents a scale factor, and x the relative proportion of 4th and 6th order CEF terms (Lea et al. 1962). The Zeeman Hamiltonian: Y(z ~ z = --gJtzB H" J ,
(3)
represents the Zeeman coupling between the 4f magnetic moment M = gj/x B( J ) and the internal field H (applied magnetic field corrected for demagnetization effects). The bilinear Heisenberg-type Hamiltonian: ~ ~fB = -- g J IxBnM " J ,
(4)
8
P. MORIN and D. SCHMITT
represents the Heisenberg-type bilinear interaction, written within the molecular field approximation (MFA), n being the bilinear exchange parameter. An alternative definition for this parameter is 0", with: O* n
=
-C
3kBO* =
(5)
g :2t z B2 J ( J + 1)
where C is the Curie constant. The two-ion quadrupolar Hamiltonian: ~ o In a form analogous to the Heisenberg-type bilinear coupling between magnetic dipoles, a two-ion coupling between 4f quadrupoles is possible (see Schmitt and Levy 1985 and references therein), for which evidence has been found in several rare earth intermetallic compounds (Levy 1973, Levy et al. 1979). According to symmetry considerations, the corresponding Hamiltonian, again in the MFA, may be written as: ~Q = -KY((O°)
O0 + 3 < 0 2 ) O 2 ) - Ke(Pxy + < P y z > G z + ( P z x ) P z x ) •
(6) In this expression, 0~, 022 and the Pi:'s are the second-order (quadrupolar) Stevens operators (Hutchings 1964, Sivardi6re 1975) (see table 1), and K ~ and K ~ are two-ion quadrupolar parameters corresponding to the two linear combinations of quadrupolar operators, which are invariant to all the cubic symmetry operations. As will be discussed below, they are associated with the two anisotropic normal strain modes in cases with cubic symmetry, namely the tetragonal (3/) and trigonal (e) modes. TABLE 1 Expression of the quadrupolar operators. Operator
Expression
0°2
3JZz - J(J + 1)
o~
J~ - s,2 = ~(s~++ :_)
exy
l(JxJy + JyJx) = ~- (J+ - j2_)
--i
Pyz
e zx ~_
1
0 2
2
~(Y~L + L :,) ½(LL+LL)
e
The magneto-elastic Hamiltonian: ggME The magneto-elastic properties of rare earth compounds have been extensively studied since the 1960s (Callen and Callen 1965, Du Tr6molet de Lacheisserie 1970, Du Tr6molet de Lacheisserie et al. 1978). The analysis of these properties employs one-ion and/or two-ion magneto-elastic Hamiltonians, it is usually
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
9
limited to the first order in the strain (harmonic approximation) and contains terms up to the sixth rank for the Stevens operators. Few studies have been made in connection with second-order terms in the strain (anharmonic coupling) (Du Tr6molet de Lacheisserie et al. 1978, Rouchy and Du Tr6molet de Lacheisserie 1979, Wang and Lfithi 1977b) and this second-order magneto-elasticity will be not considered in this chapter. The one-ion magneto-elastic Hamiltonian corresponds to the direct coupling between the deformations of the lattice and the 4f shell. It can be considered as the strain derivative of the crystal field Hamiltonian. The two-ion magnetoelasticity is related to the modification of the two-ion magnetic interactions by the strains. The one-ion magneto-elastic H a m i l t o n i a n : ~fME1
Group theory gives us the correct procedure for constructing this Hamiltonian. First, one writes linear combinations of the first-order strain components eq which transform according to the three irreducible representations F1, F3 and F5 or a, y, e (see table 2) (Du Tr6molet de Lacheisserie 1970). Second, one considers the linear combinations of the Stevens operators of rank I (l = 2, 4 or 6) which transform also according to the irreducible representations of the cubic group (see appendix 1). Finally, one obtains all the possible invariants by summing the direct products of the different combinations which belong to the same representation. A magneto-elastic coefficient then is associated with each invariant. For l = 2 , two magneto-elastic terms are obtained, coming from the two representations /"3(7) and Fs(e), WMEa(I = 2) = Y ~ m ( l = 2) + aq~°hEl(l 2 ) , -----
(7)
with ~IEI(/
= 2)
e
=
--
~, 2 X/3e202),
B~(e~O ° +
Y(MEI(I = 2) = - B
8
e
e
(e,Pxy + e2Py ~ + e3Pzx ) .
(8)
(9)
Thus there are only two normal strain modes associated with symmetry lowerings of the cubic cell, one for a tetragonal and the other for a trigonal strain. TABLE 2 Normalized, syrnmetrizedstrains for the cubic symmetry. Representation
Straincomponents
r~
E° = ~
1"5
E1 = %¢/2 x y '
1
(Ex~+ ~, + ~zz)
E2 = " ~ Eyz'
10
P. MORIN and D. SCHMITT
The l = 4 and l = 6 one-ion magneto-elastic Hamiltonians have also been given (Creuzet and Campbell 1981, Niksch et al. 1982). Their full expressions are given in appendix 2.
The two-ion magneto-elastic Hamiltonian: ~ME2 Although any kind of two-ion interaction can a priori depend on the strains, only the bilinear coupling will be considered here. The same theoretical considerations as above allow us to write the corresponding Hamiltonian as the sum of an isotropic term and two anisotropic ones: ~ME2
=
o~ ~ME2
+
ffb'O~vlE2 "[- ~ M E 2
(10)
"
The first term, (11)
~ME2 = -D~e~ ( J ) " J ,
is related to the volume dependence of the isotropic bilinear exchange interactions. On the contrary, the two other terms,
Yg~E2 =
-Dr[e~'(2(Jz)J~
)~ME2
-De[eel( ( Jx)Jy + ( Jy) Jx) + 1=:2(( J,)Jz + ( Jz)Jy)
-
(L)L
- (4)4)
+ v3e~((L)L
- (4)4)],
(12) =
+ e3((Jz)Jx + (Jx)Jz)],
(13)
are associated with the appearance of anisotropic bilinear interactions under a strain. As for the bilinear coupling itself [eq. (4)] the MFA has been used for this two-ion magneto-elasticity.
The elastic energy: Fez The elastic contribution to the free energy may be expressed as a function of strains and symmetrized elastic constants. In cubic symmetry, the three normal strain modes a, y, e appear in the expression of the elastic energy 1 .,-,al a x 2 1 3' Y 2 Eo, = ~,~ot~ ) + ~c0[(~,) + ( ~ D 21+ 1 Co[(e~) ~ ~ 2 + (e~) 2 + (e~)2].
(14)
The C~ are background elastic constants without magnetic interactions; they are related to the conventional constants by a 0 C O ~-- C 1 1 +
0
.
2C12 ,
C~
:
0 C11 -
0 C12
. ,
s
0
C O-= 2C44
.
(15)
The corrective energy terms E~, EQ, EMe 2 These terms arise from the fact that each rare earth pair is counted twice in the MFA treatment of two-ion magnetic couplings. This occurs for the bilinear [eq. (4)], the quadrupolar [eq. (6)] interactions, and for the two-ion magneto-elastic
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
11
terms as well [eqs. (11)-(13)]: E B = ½nM 2 ,
(16)
Eo=½K'((o°)Z+3(OZ2>z)+½K~(
(17)
EME2 = ½D"e~2 + ½DV[e~(22_ 2_ 2)
+ V~E~(2__ 2)]_~_De[E~q_E; ~- e;] .
(18)
2.1.2. Harmonic description of magnetostriction Starting from the above Hamiltonian, the actual magnetostriction, i.e. the equilibrium values for the various strains e~'s, may be obtained by minimizing the free energy F with respect to each of them: OF/Oe ~= O. That leads to the following general expressions for the 6 symmetrized strains for cubic symmetry ~, 0 + 5 0 4 > + B(6) Ea = ~1 [B(4)
./
1
+ ½V,~2]
y,1
+ ½D~(2(jz)2_2_ (jr>2)], =
(19)
(20)
'Y T,2 1 [~/-~B.V+ B~4)+ B(6)(O 6 >
- -
+ ½V~D~((j~>2_
(21)
~. l~l [()el,l\ ~2 81 = ~1 [Be+ B(4) + u(6)\~ 6 / -}-B(6)
+ m~(Jx)(Jy>], 1
E2 : ~ 0
(22)
[Be(PYz>+ B (~4 ) < O 4 ' 2 >
l:?a:l / /-'}el,2 ~ e2 062,2> + u(6)\"6 / -~- B(6)<
+D~(J,)(Jz>],
(23)
1 ~ el O;1,3 ]~,52 [ /~e2,3 k e 3 = ~ g [ B ~ ( P ~ > + B(4) -4- B(6)< > + u(6)\,j6 / Co
+ D~(L)
(24)
The macroscopic change of length of a magnetic lattice with cubic symmetry may then be written as a-
8l 1 1 l - V~ e~ + ~
v
2
2
e1(2/3z -/3x - / 3 2 ) + ~
+ V~(e~/3~/3y + e~/3,/3 z + 23/3~/3x).
1
~
2
e2(/3" - / 3 2 ) (25)
12
P. MORIN and D. SCHMITF
To obtain a reliable description of the various magnetic and elastic properties, the expectation values of the Stevens operators must be treated in a self-consistent manner. In particular, this yields the actual temperature dependence of the various strains [eqs. (19)-(25)] in the paramagnetic as well as ordered phases.
2.1.3. The total quadrupolar Hamiltonian By including the preceding equilibrium values of the strains in the starting Hamiltonian, one can write it in a form which does not explicitly depend on the e~'s, but only on the magneto-elastic coefficients and on the expectation values of the various Stevens operators. In most of the cases reported in the literature, one-ion magneto-elasticity involving the quadrupolar operators appears as the dominant contribution. This justifies focusing our attention on it in the following. The relative importance of other terms from an experimental point of view will be discussed later, in particular in sections 3.8 and 3.9. Keeping only the B " contribution leads to an expression for NMEI(1 = 2) which appears to be formally undistinguishable from the two-ion quadrupolar Hamiltonian No; both terms can be grouped together in the total quadrupolar Hamiltonian NOT fft°QT= -G'Y[(O~)O°2 + 3 ( 0 2 2 ) 0 2] - G [( p xy)Pxy ~- (Pyz)Pyz -I- (P~x)P~x] (26) where the new total quadrupolar parameters G ~ are defined as G ~ - (B~) 2 -
-
cg
- - -
C;
~IE ~- K ~ '
,
(27)
+ K ~ = GME + K ~ .
(28)
+
K "~ =
G
E
Equation (26) provides an alternative expression for the quadrupolar couplings, which may be used instead of eqs. (6), (8) and (9), every time that the e "'s are not explicitly needed.
2.2. Treatment of the cubic Hamiltonian The treatment of the Hamiltonian described above can be carried out in two ways: i) First, by diagonalizing the full Hamiltonian in a self-consistent manner. This allows the description of all the magnetization and magnetostriction processes, in particular in the ordered phase or under large external stresses (magnetic field or uniaxial stresses) as shown in section 3. ii) The second possibility is to apply perturbation theory to the above Hamiltonian in order to describe the magnetic and magneto-elastic properties in the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
13
non-ordered phase and under small external stress. There are two advantages working in the cubic paramagnetic phase. First, analytical expressions can be obtained for the various susceptibilities connecting the various variables. Second, any strain mode can be systematically studied by applying external stress along the appropriate direction, whilst in the ordered phase only the spontaneous state may be usually achieved because of the strong anisotropy effects. In this section, the susceptibility formalism is developed by applying perturbation theory to the above Hamiltonian which is limited to second-rank quadrupolar terms. An analytical expression will be derived for the free energy, defining the various susceptibilities which appear in the analysis of the magnetization, parastriction and elastic constants measurements. Finally, the behaviour of these susceptibilities and their role in describing phase transitions will be described. 2.2.1. Perturbation theory
The zeroth-order Hamiltonian Y(0 consists of the CEF term [eq. (2)], the diagonalization of which provides the zeroth-order energies and CEF wave functions. According to the direction of external stress, the two normal cubic strain modes may be fully decoupled. Therefore they are considered separately. Tetragonal m o d e y
For a magnetic field applied along a four-fold axis, e.g. z, the only non-zero terms are M = g H a a 3 ( J z ) , Q = ( 0 2°) and e~. The perturbating Hamiltonian is then written as: YU1 = - g j l - % ( H + n M ) J z - (B3,e~ + K3,Q)O°2 + ~-~'0~,,'~3,to3,,= )~ + ½ n M : + -½K3,Q2 .
(29) Perturbation theory has to be carried out up to the second order for e~ and Q and up to the fourth order for H and M in order to derive analytical expression of the generalized Gibbs free energy F = - k B T l n Z. Here Z = T r e x p ( - / 3 Y ( ) is the partition function, k B the Boltzman constant and/3 = 1/(k B T ) . The details of the calculation may be found in appendix 3. The free energy expansion takes the form F3,= FVo - ½Xo(H + n M ) : -
~X3,tt~ 1 ,,,3, ea3, + K3,Q) 2
1. (3)(H + n M ) 4 - (2)tB3,e3,, + K 3 , Q ) ( H + n M ) 2 - zx3, \ •
- - , , I [ 3,
+ ~a"-'o,,ele~'t^3,'2)+ l n M 2 + 1K3,Q2 + . . . .
(30)
This expression includes four CEF single-ion susceptibilities which are defined without any magnetic or quadrupolar interactions, and depend only on the non-perturbed CEF state, i.e. on the cubic CEF level spacings and on the matrix elements of Jz and O ° between cubic CEF wavefunctions (see appendix 3). X0 is the usual (first-order) magnetic susceptibility, X3, is a quadrupolar strain susceptibility, X3, - (2) is a quadrupolar field-susceptibility which couples the quadrupolar
14
P. M O R I N
a n d D. S C H M I T T
operator O ° to the magnetic field, xz, - (3) is the third-order magnetic susceptibility which is related to the initial curvature of the magnetization curves and, finally, F 0 is the zeroth-order free energy corresponding to ~fcEF alone. The equilibrium values for M, e~ and Q are found by minimizing the generalized free energy F3, OF v
.
.
.
OM
OF3,
.
Oe~
OF3,
.
.
0.
oO
(31)
This yields a system of three coupled equations, the solution of which leads to the following relations M=
(3)11 u3 , XMH + XM
(32)
.
B ~'
~=Q ~ C~
(33)
'
Q = xoH 2 ,
(34)
XO
(35)
with XM -- (1 -- nXo) '
["~
1
X(M 3)=
(1
- nx0) 4
+ 203,
xv
(x~) ~ ] 1 - GrX3,J '
(36)
X(2) 3'
Xo
9
(1 - ~ X o ) 2 ( 1 - 6 % )
(37)
Note that we find for the equilibrium value of e~ the same expression as that found without recourse to perturbation theory [eq. (20)]. In the limit of zero field, OF3,/OQ = 0 leads to the strain dependence of the quadrupolar operator: B3,X3, -
Q = 1-
-
?
(38)
K:'X3, e l '
and the expression for the corresponding elastic constant can be written as: d2F 3, C3,= d(e~)2 = C~
( B y) 2Xv (1 - K~'X~,) '
(39)
or
Cy
C--~ = (1 - Grxr)/(1 - K~Xr).
(40)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
15
Trigonal m o d e e
When the magnetic field is applied along a three-fold axis, e.g. [111], the new symmetry (trigonal symmetry) implies that the only non-zero values involved are 1 M ( J x ) = : - V ~ -gJ~£B
,
(41)
( P x y ) = (Pyz) = ( P z x ) = P ,
(42)
and e 2~ =
E 1 ~
e 3~ ~
e"
(43)
The perturbating Hamiltonian takes the following form 1
~1 =
V ~ gJld"B(H "[- lq'm)(Jx -[- Jy -Jr Jz) 3 ~-~e[
ex2
- (B~e ~ + K P)(Pxy + Pyz + Pzx) + ~ C o t e ) + ½nM2 + 3 K ' p 2
.
(44)
For the calculations it is more convenient to rotate the coordinate axes so that [111] is the new z-axis (see appendix 4). In this new axes system, the same perturbation theory may be carried out to obtain the trigonal free energy F ". Then applying an inverse rotation allows us to express the latter in the initial coordinate system with four-fold symmetry F ~ = Fo _ ½Xo( H + n M ) 2 - ~ x9~ ( B
~e + K ~ P ) 2
- 3X~2)(Bee" + K ~ P ) ( H + r t M ) 2 - 1X(e3)(n + n M ) 4 3~e," + ~t~ol, e 8 \)2 + ½ n M 2 + 3K~p2
(45)
Here X0 is the same as in F r [eq. (30)] because of the isotropy of the (first-order) magnetic susceptibility, but the other CEF susceptibilities X~, )~z) and x~" (3) differ from the corresponding X~i) 's (see appendix 4). This yields the anisotropy of the properties related to the quadrupolar variables. The equilibrium conditions for M, e ~ and P are OF ~
OF ~
OF t
OM
Oe ~
3P
-0,
(46)
whence the following expressions .
(3) Lr3
M = XMH + X M I,
e = ~BP ,E Co
p
=
XpH 2
,
(47) (48) (49)
16
P. MORIN and D. SCHMITT
with XM as in eq. (35), F
1
X(M3 ) -
[)(~3)
4
+6
(1 - n X o )
1 - 3G~x, ] '
X(2) Xe = (1
-
nx0)2(1
(51)
3G~x~)
-
(50)
In the limit of zero field, OF~/OP leads to the strain dependence of the trigonal quadrupolar operators, 3B~x~ e ~ P = 1 - 3K~x, '
(52)
and the corresponding elastic constant, 1 d2F"
c . . 3 . d(e~) . .e
Co
C~ CO
•
3(B~)ZX~ 1 - 3K~x, '
(53)
or
1 - 3G~x~ ~ 1 - 3K X,
(54)
2.2.2. Generalization of the susceptibility formalism Symmetry considerations allow us to easily extend the preceding expressions for a magnetic field in any direction. For the third-order magnetic susceptibility, symmetry considerations show that there are two independent parameters in the expansion of the magnetization to order H 3 (Birss 1964), the general expressions are
Mx
x (3))I-I (I-Iy 2 + I-I z) ,
.~ (3) /At3 + . (3) Llr3
(3)
2
(3)
2
/ I ( M y I I Y ._[._ 1
(3) - XM~,)H,(H~ + H~Z), (3XM~
(3)/_/3 m z = X M , , / I I z .Jr- i
. (3) -- XMr)Hz(Hx + H y2) . (3XM~
.
.
(55)
. (3) and XM~ - (3) are the total third-order magnetic susceptibilities associated Here xa~:, with the four-fold and three-fold axes, respectively [eqs. (36) and (50)]. This allows us to predict the H 3 dependence of the magnetization for any direction; e.g., applying the magnetic field along any two-fold direction leads to the following relation
X~)l = ZXM~I-(3)+ ~,~,,(3) M~3 .
(56)
In a similar way, symmetry considerations lead to only two independent
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
17
parameters in the expansion of the quadrupolar susceptibilities to order H 2 ( 0 2 0 ) ~ ~XQ(2H~ 1 2 (O~)__
- H~2 - H y2 ) ,
1 2 ~XQ(Hx-H2y),
(57)
= 3XeI-I~t-IJ" (ij = xy, yz, zx) , where X~ and Xe are given by eqs. (37) and (51). These latter relations are useful in the expression for the parastriction, i.e. for the relative change of length A [eq. (25)] for a paramagnetic crystal in an applied magnetic field (see section 3.3.3). 2.2.3. Behaviour o f the various susceptibilities From the expressions of the CEF susceptibilities defined above, it is obvious that the crystal field plays an important role in their temperature dependence, in particular at low temperature where only a few CEF levels are populated. First-order magnetic susceptibility: Xo It is well-known that X0 varies as 1 / T at high temperatures, and that deviations from this law may occur at low temperatures, according to two typical behaviours: (i) with a magnetic CEF ground state, X0 diverges at 0 K; (ii) with a non-magnetic CEF level as ground state, a van Vleck-type behaviour is observed (see inset of fig. la). Third-order magnetic susceptibilities: X~ " (3) Because of the numerous terms in the expression of x (3)~ (/x = y, e), a more complex behaviour may be expected. While the high-temperature variation always follows a 1 / T 3 law, the behaviour of X- ~(3) at low temperature depends on the nature of the ground state. With a magnetic CEF ground state, X- ~(3) diverges at 0 K (fig. la). In the case of a non-magnetic singlet ground state, a van Vleck-type behaviour is observed (fig. lb). A very particular situation arises with a non-magnetic doublet ground state (fig. lc): if the van Vleck-type behaviour is observed for the trigonal susceptibility (H II[111]), a positive divergence occurs for the tetragonal one (HII[001]), due to the presence of mixed Curie-van Vleck terms in the expression of A~y - (3) • Thus, in addition to their anisotropy, the different (3) . (3) low-temperature behaviours oi Xr and x~ give valuable information about the CEF, in particular on the nature of the ground state. Strain susceptibilities: X~ The quadrupolar strain susceptibilities X~ give information about the quadrupolar character of the lowest CEF levels in the same way as X0 gives information about the magnetic character. Thus, for a singlet (fig. 2b) or a Kramers doublet (fig. 2d) ground state, there is no intrinsic quadrupolar moment and the low-temperature behaviour is of van Vleck-type. On the other hand, for a magnetic level as a
18
P. M O R I N and D. SCHMITI"
10
c° "~ I0
0
."
¢ .. /" --
0
20
40
0
q
20
TEMPERATURE
2ooi
-10
[.;t,)
--
40
(K)
Fig. 1. Typical temperature behaviour of the third-order magnetic susceptibility x~ - (3) for the two modes 3' (tctragonal) and e (trigonal) of cubic symmetry, and with a magnetic triplet (a)~ a singlet (b) and a non magnetic doublet (c) as ground state (CEF levels spacings are indicated). Inset of (a): corresponding temperature behaviour of the reciprocal susceptibility 1/Xo (dashed line: behaviour without CEF). (a)
'
--
G
- -
F,
'
r;'~ 1
80
J=6
~0
10
v
i
!
(e)
-
-
r,,
i
(d)
~ --
~5
J=6
20C
"
\
j =~
r;' r;" 100
10(
i
100
TEMPERATURE
'
lb0
(K)
Fig. 2. Typical temperature behaviour of the strain susceptibility X~ for both modes y and e, with a magnetic triplet (a), a singlet (b), a nonmagnetic, quadrupolar doublet (c) and a magnetic, nonquadrupolar Kramers doublet (d) as ground state. C E F levels spacings are indicated.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
19
triplet/,~t) (fig. 2a) or a quartet Fs, there is an intrinsic quadrupolar moment and the X,'S diverge at 0 K. Finally, the non-magnetic doublet F 3 (fig. 2c) is quadrupolarly active for the 3' mode but inactive for the e one, leading to a Curie and a van Vleck behaviour for X~ and X~, respectively. In all cases, the high-temperature variation of X, follows a 1/T law, as does X0-
Quadrupolar field-susceptibilities: X ~) The low-temperature variation of quadrupolar field susceptibilities depends on both the magnetic and quadrupolar character of the ground state. Thus a pure van Vleck-type behaviour is expected only for a non-magnetic and non-quadrupolar level, i.e. a singlet (fig. 3b). For a Kramers doublet, the presence of an intrinsic magnetic moment leads to the divergence of x~, - (2) (fig- 3d). For the non-magnetic doublet F 3 (fig. 3c), the van Vleck-type behaviour is observed only for the e mode for which there is no corresponding intrinsic quadrupolar moment. For the other cases, a divergence occurs at low temperatures. A very special case arises for the triplet F~ t) (fig. 3a), where the competition between the various components of the level leads to a change of sign of x- r(2) as a function of temperature: this leads to the so-called "reverse magnetostriction effect" (see section 3.4.1). Also, at high temperature, a 1/T 2 law is expected.
(a) 20
- -
r+
_ _
F'~ 1
8
100
\
--->
I --
r,,+++
--r]
10
1
----[-i" J=B
--Q J= 6
P,i
\ (b)
0
T
0 v
i
-1(
i
(c)
- -
r,
--15.
~>~
- -
F~2)
J=6
2°°~T
F~
40
t/ I\
\\+
100
TEMPERATURE
'++q
,.,
L = ,-+,,
20
100
( K)
Fig. 3. Typical t e m p e r a t u r e behaviour of the quadrupolar field-susceptibility X~ ) for both m o d e s 3' and e, with the same ground states as in fig. 2.
20
P. MORIN and D. SCHMITT
2.2.4. Magnetic and q u a d r u p o l a r transitions
In the presence of quadrupolar interactions, rare earth systems are characterized by up to five quadrupolar order parameters (Sivardirre et al. 1973) in addition to the usual magnetic order parameter M = g J P ~ ( J z ) . For cubic symmetry, there are only two independent order parameters, namely Q = (O °) and P = ( P i i ) , associated with the two quadrupolar parameters G r and G ~. In most cases, the bilinear interactions are dominant and drive one single magnetic phase transition, since the quadrupoles necessarily follow the dipoles as they magnetically order. Nevertheless, the quadrupolar interactions may influence the nature of this transition (see section 3.2). In a few cases, the quadrupolar interactions are dominant and drive a quadrupolar ordering without any magnetic dipolar ordering (see section 3.1). Several authors have investigated the phase diagrams associated with the presence of magnetic and quadrupolar interactions, with the frequent restriction of considering effective spins, i.e. by taking into account only the low lying levels. The cases S = 1 (Chen and Levy 1973), S = 3 (Sivardirre and Blume 1972) and S = ~ (Chen and Levy 1971) have been treated. Also the case of the F3-F5 magnetic system (S = 2) has been considered, explicitly including the related crystal field effects (Ray and Sivardi~re 1978). These calculations were all performed within the MFA. Numerous phase transitions have been found to occur, depending on the strength and the sign of the various parameters. Uniaxial ferro- (FQ) and biaxial antiferroquadrupolar (AFQ) orderings have been described (Sivardirre et al. 1973). More complex situations have been calculated by simultaneously considering the quadrupolar interactions belonging to both modes y and e, e.g., a ferriquadrupolar phase associated with the Q component followed by a FQ phase related to the P component (Chen and Levy 1973). In all the cases, the nature of the phase transitions was discussed and triple or tricritical points were found. While the existence of the various magnetic and quadrupolar phase transitions can be qualitatively described within the effective spin formalism, accounting for the real situations found experimentally requires one to consider the actual CEF level schemes, since the precise nature of the ground state as well as possible effects of excited levels via van Vleck-type matrix elements may noticeably change the physical properties. This can be done by using the susceptibility formalism developed above (Morin and Schmitt 1983a). Indeed, the expression for the free energy derived in section 2.2.1 can be used to investigate the possible phase transitions by taking H = 0. Equations (30) and (45) are then written as F v = F~ + ½n(1 - n X o ) M 2 - in4-,~,(3),~ar4 ~,~ + ½G~'(1- G r x r ) Q 2 - n 2r,'y t 1 X r(2) Q M 2 ,
F ~ = F 0 + ½n(1 - n x o ) M 2 - zin4"a t(3)hAr4 1,1 + 3G~(1-3G~x~)P 2 - 3n2G~2)pM
z"
(58)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
21
According to Landau theory, magnetic (tD) and quadrupolar (to_ and te) critical temperatures are defined by canceling the coefficient of the M 2, Qz or p2 terms in the expansion of F 1 - nXo(to) = 0,
1 -- G~Xr(tQ) = O,
1 - 3G~x~(tp) = 0.
(59)
The actual phase diagram as well as the nature of the transitions is determined by the relative magnitudes of tD, tQ and t e and by the signs and relative magnitudes of the other coefficients. The occurrence of a quadrupolar phase transition is a particularly interesting situation in cases with cubic symmetry where no quadrupolar component can spontaneously exist. In such a case (to_> t D for example) the transition is generally first order because the next term in the Q expansion of the free energy is of odd parity (Q3). The ordered value of Q is positive or negative depending on the sign of the corresponding coefficient. This coefficient is a second-order strain susceptibility [X(s2) in the notation of Morin and Schmitt (1983a)] and involves quadrupolar matrix elements between the various CEF levels. Note that the quadrupolar transition may be second order if Xs" (2) = 0 and the next term in the Q expansion of the free energy is positive. This occurs when the ground state is a quadrupolar doublet (F3 doublet, /"4, F5 triplets or F8 quartet) (Kanamori 1960). Examples of pure quadrupolar phases are shown in fig. 4 for both modes 3' and e (phases a 0 and ae, respectively). In the presence of dominant bilinear interactions (t D > tQ and tp), the magnetic transition is first- or second order, according to the sign of the next term in the M expansion of the free energy, namely x~ -(3) (/x = y, e). As seen in a previous section, xg -(3) is usually negative, whence a second-order magnetic transition. However, with a F3 doublet as ground state, xz, -(3) becomes positive at low temperature (see fig. lc), according to the characteristics of the low-lying CEF levels, leading to a first-order transition (see fig. 5). The limit between the two situations, i.e. xv - ( 3 ) = 0 , corresponds to the tricritical point C in the phase diagram. In the presence of competing bilinear and quadrupolar interactions, the situa-tion is more complex. First, various magnetic and Q or P quadrupolar phases may exist according to the relative strength of the associated coupling parameters n, G ~, G ~ (see fig. 4). Note in particular the possibility for two types of magnetic phases: (i) the phases b 0 or b e correspond to magnetic order where the ground state, coming from the F~ 1) CEF triplet, has mainly a Mj = 5 component; (ii) the phase co_ may be considered as a magnetic phase induced on the non-magnetic level, which is the ground state in the quadrupolar phase ao, and which has a dominant M : = 6 component. Another interesting feature of competing bilinear and quadrupolar couplings is the modification of the nature of the magnetic transition by the quadrupolar interactions. Indeed, the character of the phase transition is now determined by the sign o f -X M,'O (3) = X~)In=0 (see Morin and Schmitt 1983a). According to eqs. (36)
22
P. M O R I N and D. S C H M I T T
a) 5, / ~ ~ '
b) (]*=0 ~ aKp ~ 8'=2K
5 •
O'=4K/
W 0..
5"
,
)-
0'~=6K
o'=gK
bQ ,fQ 0
10
G I (mY,)
bp 0
100
G2(m K)
Fig. 4. Magnetic and quadrupolar phase diagrams for various bilinear coefficients 0* as a function of the quadrupolar parameters G ~ (=-G 1, left part) or G ~ (-=G 2, right part) for a T m 3+ ion (J = 6). a o, ap = pure quadrupolar phases; bo, bp = magnetic phases; Co: induced magnetic phase (from Morin et al. 1987c).
and (50), a positive quadrupolar parameter may drive the transition to be . (3) is larger than ]X~)[ (see fig. 6). Also first-order if its contribution to xM,0 tricritical points may be observed. The opposite situation may also occur, where a .(3) leads to a first-order transition, but a negative quadrupolar positive value of x~ . (3) changes it into a second-order one. Several examples of this contribution to XM,O will be discussed below (see section 3.2.2). Note that most of the experimental first-order transitions may be explained through the present model and do not require us to invoke critical fluctuations (see K6tzler 1984). In conclusion, the existence of bilinear and 3'- and e-type quadrupolar interactions gives rise to a variety of situations. Analyzing the complex related phase diagrams as well as the nature of the magnetic and quadrupolar transitions is qualitatively achieved within effective spin models, but requires a more realistic formalism in order to take the exact nature of the CEF levels involved into account correctly, and to better understand the experimental situations. This can easily be done through the present susceptibility formalism.
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
J=4 W=-5.SK x=.6B
(El 10
23
I-1 ~, ~ ? [ _ _ F 3
[001]
0 4c; 3O -
~
" -TD=tD
2C 1C I
0
/
I
20 TEMPERATURE(K)
40
Fig. 5. Temperature variation of xr - (a) (___X~3)) calculated for a Pr 3+ ion for the C E F level scheme indicated (upper part). Magnetic phase diagram (lower part); the dot-dashed line is the second-order magnetic transition line if neglecting X~3); the dashed line corresponds to a second-order magnetic transition, the full line (TD) to a first-order one; C is a tricritical point (from Morin and Schmitt 1983b).
~"
10 - -
:~
r 2
f2)
5
G1=13mK
--
r3
I 0 .-.
/
_'
10~-
/ I," 0
.
." T.-t IT.,
/
.~.-~_" . -
,
/
J=6 vv = u . ~ 3
n
x=- .34!
, , [oo ] 5 10 15 TEMPERATURE ( K )
20
Fig. 6. Temperature variation of X~)0 calculated for a Tm 3+ ion for the CEF level scheme indicated, for two G r (~G1) values (upper part). Lower part: magnetic phase diagram for G r = 13 inK; the solid (dashed) line represents a first- (second-) order transition; the tricritical point C occurs when XM,O - (3) vanishes; the dot-dashed line corresponds to a second-order transition which would occur if G ~= 0; T o is the quadrupolar ordering temperature, and t r the magnetic ordering temperature within the quadrupolar phase; T o and t o correspond to the first- and second-order magnetic transition, respectively (from Morin and Schmitt 1983a).
24
P. M O R I N
and D. SCHMITT
2.3. Extension to lower symmetries (hexagonal and tetragonal) The magneto-elastic and, more generally, quadrupolar properties have been studied less thoroughly in rare earth intermetallic compounds with a symmetry lower than cubic, e.g. in hexagonal and tetragonal compounds, because of the greater number of parameters involved in the CEF Hamiltonian as well as in the quadrupolar couplings. Only a few systems have been investigated such as metallic Pr (Hendy et al. 1979), dilute yttrium based rare earth compounds (Pureur et al. 1985), and more recently the compound PrNi 5 (Barthem et al. 1988). In this section, the formalism describing the magneto-elastic and quadrupolar properties of the rare earth compounds with hexagonal and tetragonal symmetries is developed. The full Hamiltonian is given, including all the relevant terms for both the one-ion magneto-elastic coupling and the two-ion quadrupolar interactions. Then analogies and differences with respect to the cubic symmetry are emphasized. 2.3.1. The Hamiltonian The general Hamiltonian appropriate for hexagonal and tetragonal symmetries takes the same form as eq. (1). However, the detailed expressions for most terms are different from those of cubic symmetry and are given below (see Morin et al. 1988). The CEF Hamiltonian The CEF Hamiltonian may be written, in terms of the Stevens operators (Hutchings 1964) and in the axes system where the z-axis is parallel to the c-axis of the hexagonal or tetragonal unit cell, as, ~Hex CEF =
~a'et
CEF =
0
0
0
0
0
0
6
(60)
6
B2 0 2 + B404 + B606 + B606 ,
0 0
0 0
4 4
0 0
4 4
B2 02 + B 404 + B404 + B606 q- B606 •
(61)
The bilinear Hamiltonian For hexagonal and tetragonal symmetries, the existence of one privileged direction, namely [001], dictates that there are two invariants in the full bilinear exchange coupling. Their expression is the same in both symmetries and takes the following form within the molecular field approximation ~ 1 = _ ( gjlJ.B)2nC~l ( j ) j , 2
~ B 2= --(g, lxB) n
~2
( 2 ( J z ) J z - (Jx)Jx - ( J y ) J y ) .
(62) (63)
The first term corresponds to the usual isotropic Heisenberg-type bilinear coupling, while the second term is an anisotropic bilinear coupling, forbidden in cubic symmetry, which leads to effective n coefficients which are different parallel and perpendicular to the c-axis in hexagonal and tetragonal symmetry. An experimental way to estimate the magnitude of this a2-term, would be to investigate the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
25
anisotropy of the magnetic susceptibility in the gadolinium compounds where there is, in principle, no other source of anisotropy. The two-ion quadrupolar Hamiltonian In hexagonal and tetragonal compounds, the O ° quadrupolar component is already ordered (i.e. (O2°) ¢ 0) by the crystal field, but that does not exclude the possible existence of quadrupolar pair interactions, as in cubic symmetry. However, due to the lower symmetry, more than two quadrupolar parameters are involved, and the corresponding Hamiltonian may be written, in the MFA, as a function of the linear combinations of products of second-order Stevens operators, which are invariant under the symmetry operations of the hexagonal or tetragonal point groups: ~Hex
o
=-K
a
0
0
2
(O2)O 2-KÈ[(O2)O2+4(Pxy)Pxy ]
- K~[(Pyz)Py~ + ( P z ~ ) P ~ ] , ~o Tet
~o
0
=
-
K~(02)02
0
7 -
-- K ~ [ ( P y z ) P y z
K
+
2
(64)
2
( 0 2 ) 0 z - K (Pxy)Pxy
(P~)P~].
(65)
The magneto-elastic Hamiltonian As in cubic symmetry, the one-ion magneto-elastic couplings may be considered as the strain derivatives of the crystal field term itself. Within the same harmonic approximation, and limiting ourselves to the second-rank (quadrupolar) terms, the full one-ion magneto-elastic Hamiltonian is written as:
~Hex ME1 = __(B,~le~l
~-
B,~2e,~2)O0 _ B ~(elO ~ 2z + 2e2Pxy ~ ) _ B¢(e~pyz + eIpzx) ,
(66) ~ TME1 et =_(B,~lal
+ B.aa2)
0 2 o --
B,s,O~_Baeapxy
--
B~(s~pyz+e2Pzx).
(67) In these expressions the e~'s are the symmetrized strains (Du Tr6molet de Lacheisserie 1970) (see table 3) and the B"'s the magneto-elastic coefficients associated with the corresponding normal strain modes. Note that there are two independent fully symmetric a-terms, related to the two deformations which maintain the initial symmetry, namely the volume al-strain and the axial a2-strain, the latter one is associated only with a change of the ratio c/a. For the other terms, a symmetry lowering towards orthorhombic or even lower symmetry Occurs.
The two-ion magneto-elasticity is related to the modification of the two-ion magnetic interactions by the strains. In this section, we restrict ourselves to the two a-type bilinear couplings. For each one, two strain modes must be considered; they are associated with volume and axial strain dependences of bilinear interactions. For hexagonal and tetragonal symmetries, the corresponding two-ion
26
P. M O R I N and D. SCHMITI" TABLE 3
Normalized, symmetrized strains for the hexagonal and tetragonal symmetry. Hexagonal
Tetragonal
Expression
Representation
Strain
Representation
Strain
5
~°~
/'1
~°~
-~v~(~ + ~. + ~z~)
5
~°~
5
~°~
~ ( ~ - ½(% + ~x))
5
4
r4
d
v~ %
F6
eg
F5
s;
V ~ G~
magneto-elastic Hamiltonians take an identical form, and may be written in the M F A as o~1
~ME2
~--- t'r~otl otl D ~ 2 e ~2 --~LIalE At- al
gME2a2 = --IZDaXa2 E ~
+
)(j)j
(68)
D~2 J e~2)(2( z a 2L )
-
(L)L
- (Jy)Jy).
(69)
The elastic energy The elastic contribution to the free energy may be expressed as a function of the strains s" and of the appropriate symmetrized elastic constants C~" ( D u Tr6molet de Lacheisserie 1970). For hexagonal and tetragonal symmetries, they are given by:
EHeX : 1 C 0 1 ( G a l ) 2 q_ CO12Eo~IEa2 + 1 C 0 2 ( E a 2 ) 2 el
+ ~Co[(~1)2 + (~)21 + ½Coq(~)2 + (~)21, ETet e, = 1
Co,(eal)2 +
(70)
C012EotlEo~2 + ~ 1t ~..~ao 2 ,ks -ot2\2 ) l z~/
6\2
1
e
e 2
+ ½G(~) 2+ : ~ o ~ ) + ~Co[(<) + ( 4 ) q -
(71)
The background elastic constants without magnetic interactions Co~ are given in table 4.
TABLE 4
Symmetrized elastic constants in hexagonal and tetragonal symmetry.
Hexagonal
Tetragonal
Expression
C al
C ~1
½(2Cn +2C~2 +4Ca3 + C33)
C~2 C~2
C~12 Co2
C~ C~
C~ C~
_
C ~
-~V~(G~ + c,~ - c . - G~) 3'(C n + C ~ 2 - 4 C ~ + 2 G 3 ) C n - C12 2G~ 2(2"66
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
27
The two-ion corrective energy terms As in cubic symmetry, the molecular field treatment of the two-ion interactions requires the usage of corrective terms in the deduced single-ion free energy. According to the two-ion couplings considered above, three corrective terms have to be taken into account, corresponding to the bilinear, the quadrupolar and the two-ion magneto-elastic Hamiltonian, respectively.
Hexagonal symmetry EB'= ln~lM2,
(72)
]IA2 2 = ~1 n,~2[,-) t .... z - Mx2 - My) ,
E~2
. 1KO: < o O > 2
EHeX
o
2 2
+ ½K"[
(73) 1 ' ~K [ 2 "~ 2 ],
+4
(74)
j~?Hex l,~cd ~1 _}_'rja2Ea2)2_}_ l z r ~ l ~1 a2Ea2 ) I,/.)28 + O,~ 2 ~ME2 = 2[,Ual 8 ~ctl x
[2(L) 2- (L) 2 -
<4>2]
(75)
.
Tetragonal symmetry E~ 1 and E~ ~ are the same as for hexagonal symmetry (eqs. 72 and 73), and 1 y 2\2 rTset=½K~(Oz°)Z+~K (O2/ + ~1K 8 (Pxy) 2 + ½ K * [ ( P y z ) 2 + ( P z x ) 2] (76)
ESet ME2 is the same as for hexagonal symmetry [eq. (75)].
2.3.2. Harmonic description of magnetostriction The equilibrium values of the strains eU's are obtained by minimizing the total free energy with respect to the strains. This leads to relations analogous to those for cubic symmetry, but they involve appropriately symmetrized coefficients.
Hexagonal symmetry E~I
=
[(BalCo2
+
- B
l g FI ot l['~ae2
X [ColCo
2 --
a2
~12
0
l_/rlalt,'~2
oe2 ,~12
C O ) ( O 2 ) + 2\~.-al~0 -- D~Co ) ( J ) a2
:Co~12 )(22 - < L
-D
>2
2
- <4)2)]
(Co'2)21-1 ,
(77)
e ~2 =[(B~2Co I _ B~1C012)( O ° ) + !gD,~2C,~1 2, =1 o - D ~ul I C ou12) ( J ) 2 1 //-,i ~ 2 / - ~ 1 + 2~,"~2~o -X [ColCo
B~
ea = ~
B ~
(022),
e~ = C'-~o(Pyz>,
al 412 Da2Co )(2(L)2 - (L) 2 -
2 - (Co12)2] -1 ,
~
2B ~
e 2= Co" (P~y),
(jy)2)] (78)
(79)
B ~
e~2= C---~o(Vzx)"
(80)
28
P. M O R I N and D. SCHMITT
Tetragonal symmetry e a l and e oe2 are the same as for hexagonal symmetry [eqs. (77) and (78)], and BT
e ~=
-
C~ (O~)
(81)
-
B ~
8 = ~g ( P x y ) , Co
(82)
B *
B ~
e~ = -~o (PYz)'
e2 = ~
(Pzx).
(83)
In the same way as for cubic symmetry, including these equilibrium values in the starting Hamiltonian allows us to express this Hamiltonian in a form which depends only on magneto-elastic coefficients. For example, considering only one-ion magneto-elasticity leads to t h e following total quadrupolar Hamiltonian ff(Hex QT = _ Goe(O2°) O ° - G~[(O~) 0 2 + 4(Pxy)Pxy]
-- G¢[(Pyz)Pyz + (Pzx)Pzx], ~fTet a 0 ~ Q T : -- G < 0 2 ) 0 2
0
2 2 __ G T ( 0 2 ) 0 2
(84) a a
-
-
(Pxy)exy
-- Ge[(Pyz)Pyz + (Pzx)Pzx],
(85)
where the total quadrupolar coefficients G ~ have contributions from both the one-ion magneto-elasticity and the quadrupolar pair interaction:
Hexagonal a2 (B a l ) 2 C O - 2B'~IB~2Co 12 + (B~2)2C01
G~ =
G ~, _
C01C02
(B~)
2
C----~"- +
K ~,
_
~
(C012) 2
: GME +
K~
;
+ R °e = G M E
(/*
=
e, ~'),
"{- K a ,
(86)
(87)
Tetragonal G oe is the same as above (eq. 86), and G~-
(B'~) 2
+ K ~ = C M~E + K ~ ;
=
(88)
2.3.3. Perturbation theory An analysis of the magnetic and magneto-elastic properties of hexagonal and tetragonal compounds within the paramagnetic phases may be done by using perturbation theory. The full development of the susceptibility formalism is treated elsewhere (Morin et al. 1988) and only the main results are reported below.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
29
Among the different modes, only the a- and e-modes in hexagonal symmetry (a-, y- and 6-modes in tetragonal symmetry) can easily be investigated by applying a magnetic field along the main symmetry directions, namely the [001], [100] and possibly the [110] directions. The quadrupolar components involved are ( O ° ) , ( 0 2) and possibly (Pxy)" Various associated susceptibilities may then be defined, namely strain susceptibilities X, and quadrupolar field-susceptibilities X~ - (2) (/z = a, e . . . . ). The only difference with respect to cubic symmetry is the explicit presence of a spontaneous quadrupolar moment (O°)0 in the expressions for X~, and Xtz -(2) • Also, the field dependence of the magnetization and quadrupole moments may be determined. For the quadrupolar components, relations analogous to those for cubic symmetry are found for hexagonal symmetry =
-
(89)
(9o)
2 ,
with ((2)
X~ = (1 - nx0)2(1 - G " X g ) '
(91)
and similar relations for tetragonal symmetry. For the magnetization, the following expressions are found M=
XMH--
4- . (3) L/3 X M 11
,
(92)
with X0 XM
--
,
(93)
1 - nXo
and X~ ) -
G a (x~(2),~2 ) . (2)-~2 q] [ (/l(e I |X~ 3) + 2 +2G ~ . (1 -- rtX0) 4 t_ 1 ~ G-~Xa 1 --- G- ~ 1
(94)
In these relations, all the susceptibilities assume different numerical values according to the direction of the applied magnetic field, although their expressions are formally equivalent. This is due to the different CEF wave functions determined by the field direction. This behaviour is well-known for the first-order magnetic susceptibility which is different for fields parallel and perpendicular to the c-axis in hexagonal and tetragonal compounds. For the third-order magnetic susceptibility, it turns out that only the G" contribution is present when the field is applied along the c-axis, while both G ~ and G ~ terms must be considered when the field is within the basal plane. Note that the y- and 6-modes in cases with
30
P. MORIN and D. SCHMITT
tetragonal symmetry may be separately investigated by applying a magnetic field along the [100] and [110] directions. Finally, only the isotropic bilinear exchange coefficient n ~1 has been considered here and is denoted n. 2.3.4. Comparison with cubic symmetry The main difference between hexagonal and tetragonal symmetries compared to cubic ones comes from the fact that the quadrupolar component (O~) is already ordered (i.e. ( 0 2°) # 0 ) by the crystal field. This leads to several consequences for the magnetic and magneto-elastic properties. First of all, from eqs. (77) and (78), there exists a spontaneous quadrupolar contribution to the volume (through e ~1) and to the ratio c/a (through e ~2) of the hexagonal and tetragonal unit cell. This feature manifests itself through an anomalous thermal expansion with regard to that of the corresponding compound with non-magnetic lanthanum or yttrium. This has been observed, e.g., in Pr (L/ithi and Ott 1980) and PrNi 5 (Barthem et al. 1988). Second, the a-term in YgOT[eqs. (84) and (85)] gives an additional quadrupolar contribution to the pure CEF second-order Hamiltonian as soon as either B ~1, B ~2 or K ~ is present. As for (O2°), this contribution is temperature dependent; this is a possible explanation for a thermal variation for the total effective second-order crystal field parameter B°2. As a consequence, the CEF level spacing will be temperature dependent, and it is worth taking this contribution explicitly into account besides the pure CEF one, at least in the case of strong quadrupolar coupling. The presence of a quadrupolar moment (020) does not exclude the possible existence of the corresponding quadrupolar pair coupling K ~. However, a quadrupolar phase transition involving this coupling is no longer allowed. This behaviour can be compared to the effect of high rank (fourth- and sixth-rank) pair interactions in cubic compounds which lead to apparent temperature-dependent fourth- and sixth-order CEF parameters (Morin and Williamson 1984). On the other hand, quadrupolar transitions involving the other couplings as K ~, K ~ (K r, K s, K" in tetragonal compounds) are quite possible and have been observed in several rare earth insulators (Gehring and Gehring 1975). 2.4. A peculiar case: the semi-classical treatment
In the case when bilinear interactions overwhelm the CEF coupling, the magnetic ordering temperature is large in comparison with the total CEF spacing. In the paramagnetic phase, the CEF levels may be considered as equally populated, while in the ordered phase, each of the CEF wave functions is "purified" by the exchange field to its dominant IJz) component, at least in the limit of low temperatures. Both situations may then be discussed within a semi-classical analysis. 2.4.1. The paramagnetic phase For vanishing CEF effect, the 4f multiplet is (2J + 1)-fold degenerate and the expressions of the various one-ion susceptibilities previously introduced become
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS C Xo = T
(95)
CQ T'
X~X(2)
(96)
2
2
CQ
(97)
= gjtx B 6 T 2 , C (3)
X(3) =
X(21 s
31
~--
r 3 ,
(98)
C~ ) T2 ,
(99)
where C is the usual Curie constant [see eq. (5)], and CQ, C ~ ) are quadrupolar constants and C (3) is the third-order Curie constant
CQ =
J(J + 1)(2J - 1)(2J + 3) 5k B ,
(lOO)
C(o2) = J(J + 1)(2J - 1)(2J + 3 ) ( 2 J - 3 ) ( 2 J + 5) 70k 2 C(3)
4 4 J(J+l)(2Jz+2J+1) = -gjtx B 90k 3
(101)
(102)
In the same way as C and C (3) are the coefficients of the first two terms in the H~ T expansion of the Brillouin function, CQ and C ~ ) are the coefficients of the first two terms in the s / T expansion of the response function of Q in the Brillouin-Boltzmann statistics 2
e C~ ) e Q ~ Ca -~ + ~5 +'".
(103)
Note that all the C's are isotropic. After renormalization by the bilinear and quadrupolar interactions, the actual susceptibilities . are
C XM -- T - O *
'
2 2 Co.T XQ = gllXB 6 ( T - 0 * ) 2 ( T - 0~) ' C(3)T
(i-U)
[1-
4
L
(lO4)
(2J-1)(ZJ+3)
?-2
75;-T
-T - -' O ~ J '
32
P. M O R I N and D. S C H M I T F
and the actual elastic constant is given by
C
T-O~
C0
T - 0~: "
(lo5)
This equation is very close to that given by Kataoka and Kanamori (1972). O~ = CQG and 0~ = CQK are defined by analogy with 0", as functions of the total and two-ion isotropic quadrupolar parameters G and K. The temperature dependences in eq. (104) have been extensively described by Morin and Schmitt (1981b) and Morin et al. (1980b) at high temperature as well as close to the magnetic ordering temperature. Less attention has been paid to the nature of the quadrupolar transition. It is easy to verify that for J = ½there is no quadrupole moment. For J = ~, C~2) is zero and the next term in the e expansion of Q is negative, the transition is then second-order. In all the other cases, C~ ) is positive and the transition is first order.
2.4.2. The ordered phase Several discussions of the energy in the ordered phase have been given in the past in particular by Clark in volume 1 of these handbooks (Clark 1980). Only those features that are necessary for describing properties encountered in the following sections will be derived here from our basic Hamiltonian. Following Callen and Callen (1963, 1965), the temperature dependences of the average value of the different operators O~' present in eq. (1) are expressed as functions of the normalized 4f magnetic moment, m, through modified hyperbolic Bessel functions. Their angular dependences are written as functions of the 4f magnetic moment cosines, ai, by using the harmonic polynomials. ( 0 O) = j ( j _
l ~
~)Is/2(~
-1
2
(m))(3a 3 - 1),
( 02> = J(J - 1)Is/2(~/~-1(m))(ot
2 -
ce2),
(Pij) = J ( J - ~) 1 ~1s / 2 f(~( -mi ) ) ( a i % ) (O04+504)=_40J(j -- ½ ) ( J _ l ) ( j _ 3
(i,j=1,2,3) 2 ~ 22 + ) I"9 / 2 ( ~ ' - 1 ( m ) ) ( O l l O
{O~ + 2 1 0 6) = 1 8 4 8 J ( J - ½ ) ( J - 1 ) ( J ^
-1
(106)
2
2
3)(j_2)(j_ 2
2
2
cycl. - ½)
~) 2
2
2
2
x I13,2(5g (m))[c~,c~2a 3 _ 1(12ct3 + c~3a a + a,t~2) +
~1
.
Du Tr6molet de Lacheisserie et al. (1978) have studied in detail the classical free energy of a cubic ferromagnet. The classical magnetostriction constants and anisotropy constants are found to be 0 --- 0 J ( J - ½)I5/2(5¢-1(rn)),
A100(T) = 2
Cll
-- C12
B ~
AHa(T ) - 3v~cO4 J ( J - ½)_15/2(~ ~(m)),
(107)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
33
K I ( T ) = - 4 J ( J - ½) ( J - 1)(J - 23-)[10B419/2(Gt?-'(m))
+ 42(J - 2)(J -
G(T) =
5 )B6~13/2(~ '
-'(m))],
~)B6?~l~/Z(iLf-l(m)).
1848Y(Y- 1 ) ( j _ 1 ) ( J - k ) ( J - 2 ) ( J -
(108)
The K 1 anisotropy constant receives two contributions from 4th and 6th rank polynomials. Owing to their different temperature dependences (fig. 7) they may be the causes of a change in the easy magnetization direction in the ordered phase. It is also well-known that the magnetostriction contributes to the magnetocrystalline energy, e.g. AK 1
=
9
-- ~(Cll
0
0 2 _ C,2)Aioo(T )
-I-
0 2 79 C44~,,11(T)
(109)
.
This magnetostriction contribution may be an appreciable part of the intrinsic K 1 constant: it reaches ~ of K 1 in TbFe 2 at room temperature [see table 8 in Clark (1980)]. In addition, due to different temperature variations, K 1 may change of
I
I
I
I
I
I
I
I
Is/2 10"1 19/2
10-2
113/2 10"3
1.0
.8
.6
.4
.2
0
m
Fig. 7. Normalized hyperbolic Bessel functions J5~2, 19/2 and I13/2 versus the normalized magnetization (from Clark 1980).
34
P. MORIN and D. SCHMITI"
sign. This situation occurs, e.g. in hexagonal RC% compounds as revealed by the variation of the lattice parameters and elastic constants (se e section 4). This classical description of magnetostriction and magnetocrystalline anisotropy gives a phenomenological connection between their temperature variation and that of the 4f magnetic moment. The details of the magnetic couplings leading to this observed magnetization itself are not of too great importance. It is useful in complex situations, as e.g., for rare earths dissolved in a magnetic metal (Fe, Co, M n , . . . ) . Indeed, at least at low temperatures, the magneto-elastic coefficient, in particular for the symmetry associated with the spontaneous state, may be determined by using the assumption that the 3d magneto-elastic contributions are negligible. However, both the necessary separation of the 4f and 3d magnetic moment and the existence of many sublattices limit the insight in the 4f magnetism. Obviously it would be unrealistic to search for quadrupolar pair interactions, when they are overwhelmed by bilinear ones. 3. Experimental evidence
3.1. Quadrupolar orderings To observe quadrupolar orderings it is necessary to find rare earth systems where the bilinear interactions are weak compared to the quadrupolar interactions. Favourable situations exist at each end of the lanthanides series. Indeed, first, the bilinear interactions are at their minimum for the series of compounds under consideration and, second, both the orbital moment L (Tm a+ : J = 6, L = 5; Ce 3+ : J = ~, L = 3; pr3+: J = 4 , L = 5 ) and the Stevens coefficient as, which deeply control the quadrupolar coefficients (Morin and Schmitt 1981a, Schmitt and Levy 1985), are large. In other cases, magnetic and structural transitions coincide (section 3.2.2). Quadrupolar orderings have been reported to occur in TmZn (Morin et al. 1978b), TmCd (Al6onard and Morin 1979), CeAg (Ushizaka et al. 1984, Morin 1988), PrPb a (Bucher et al. 1972a), TmGa 3 (Czopnik et al. 1985), CeB 6 (Effantin et al. 1985) and PrCu2 (Ott et al. 1977a). The first three of them are clearly ferroquadrupolar (FQ), the next three antiferroquadrupolar (AFQ) and the actual situation in the last one is not yet fully elucidated. UPd a which exhibits a structural transition may be relevant to the present review owing to its localized 5f electrons (3H 4 multiplet) (Andres et al. 1978). Structural transitions have been also observed in the RB 4 series, however, their origin has not been fully understood (Will et al. 1986).
3.1.1. Ferroquadrupolar ordering As for the ferromagnetic ordering (FM), the FQ ordering is characterized by a q = 0 propagation vector and is expected to be closely described by the singlesublattice formalism (see section 2.2).
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
35
3.1.1.1. TmZn. Cubic compounds within the CsCl-type structure are characterized by large quadrupolar interactions, TmZn being the archetype compound (Giraud et al. 1985). The temperature variation of the specific heat (fig. 8) reveals two well-defined transitions, close to each other (Morin et al. 1978b). TmZn undergoes a FQ ordering at T e = 8.55K, where ( O °) is the order-parameter. The transition is clearly first-order. An associated tetragonal strain takes place simultaneously due to the B y magneto-elastic coefficient. A FM ordering occurs in the quadrupolar phase at T c = 8.12 K, with the spins lying along the four-fold axis. The spontaneous tetragonal strain is reinforced by the bilinear interactions and reaches c/a - 1 = -8%0 at 1.5 K. From the jumps in the electrical resistivity at the two transitions, the electronic scatterings of quadrupolar and magnetic origins are of similar magnitude. Numerous properties investigated in the paramagnetic phase of TmZn have been consistently analyzed with a total quadrupolar coefficient G v= 25 mK; the dominant contribution to G v comes from the pair interactions (Givord et al. 1983). This G ~ value describes the temperature T e and no additional coupling (anharmonicity, higher order terms . . . . ) is needed. The FQ ordering lifts the degeneracy of the F~ 1) cubic ground state and splits it into a doublet and a singlet; the non-magnetic singlet is the ground state in the quadrupolar phase (structure a o in fig. 4 of section 2.2.3). Bilinear interactions are strong enough to induce a magnetic moment in this phase. Substituting L u 3+ for Tm 3+ in Tml_xLuxZn or Cu for Zn in TmCuxZnl_ x decreases the bilinear interactions quicker than the quadrupolar ones and the spin system remains paramagnetic: only the FQ ordering is observed (fig. 9) (Morin and Schmitt 1986). Note that in this case the splitting of the F~ 1) cubic triplet in the tetragonal paramagnetic phase has been observed by inelastic neutron scattering (Morin et al. 1981). 60
!
1
i
50 Tm Zn
o
E 40
30
20
l
T
r,, sss
T 8.12K= .
K , , ~ .
I
1
I
7
8
9
TEMPERATURE
( K )
Fig. 8. Specific heat of TmZn; T c is the magnetic ordering temperature, T O the quadrupolar ordering temperature (from Morin et al. 1978b).
36
P. MORIN and D. SCHMITT
10
% TmCu
v
Zn 1-c c
p
LU mr
2 ¢rUJ O.
;* 2 5 ~
:E
--.
hi
-_
.
,
/
f--
-5
20
/
oG 1 //
0.5
,,
15
)
C
Fig. 9. Phase diagram for TmCUl_cZnc; P: paramagnetic, M: modulated, AF: antiferromagnetic, F: ferromagnetic, Q: quadrupolar phases; lines are guides for the eye. Variation of the quadrupolar parameter G r (=G~) and bilinear exchange parameter 0* with c (from Morin and Schmitt 1986).
3.1.1.2. TmCd. Historically, the first FQ ordering in a rare earth intermetallic was observed in this compound at T o = 3.16 K. It was initially analyzed with a F3 cubic ground state, and the first-order character of the transition was assumed to be driven by anharmonic elastic effects (L/ithi et al. 1973b). However, RZn and RCd isomorphous compounds exhibit very similar properties, in particular the CEF (Alronard and Morin 1985). As a matter of fact, there is a close parallel between TmCd and TmZn and all the TmCd properties are fully explained with a G ~= 12 rnK total quadrupolar coefficient without the need of any other coupling (Alronard and Morin 1979). As in TmZn, pair interactions are the dominant contribution to the quadrupolar coupling and the ground state is also the non-magnetic singlet coming from the F~ 1) cubic triplet. Here, however, the spin system remains paramagnetic owing to the weakness of the bilinear interactions. As in TmZn, electrical resistivity measurements give clear evidence for the vanishing of the quadrupolar scattering below T o (fig. 10). 3.1.1.3. CeAg. CeAg is the only example of a FQ ordering for a Kramers ion. It
also belongs to the family of CsCl-type compounds. The cubic ground state is the F8 quartet, well-isolated from the F7 excited level (A = 266 K) (Schmitt et al. 1978). It exhibits a cubic-tetragonal transition at T o = 15.85 K with (O2°) as the order parameter (fig. 11). A FM transition occurs at T c = 5.2 K, the c axis of the tetragonal cell is the easy magnetization direction (Morin 1988). The c / a - 1 strain is very large (1.7% at 1.5 K). A strong dependence of the low-temperature properties on the metallurgical conditions is observed, which is related to the very large magneto-elastic coupling.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
37
5.6 I
,
,p..."
E 5.4
de_' dT
1
°
I.-te
E
W °
5.0
•"°
0
°
°¢,°°
""
I
i
2
½
t
~
n
I
4 i
4
3
TEM PE RATURE(K) Fig. 10. Low-temperature variation of the resistivity and its derivative (inset) in TmCd (from A16onard and Morin 1985).
u
30 CeAg
_~2o E
0
5
10
1~i
20
T (K) Fig. 11. Specific heat of CeAg. Anomalies at T c = 5.5 K and TQ = 15.5 K correspond to the ferromagnetic and the quadrupolar orderings, respectively (from Ushizaka et al. 1984).
The study of the quadrupolar properties in the cubic phase leads to G r = 150 inK. However, this value would drive a FQ ordering at only 10 K instead of T o = 15.85 K (Morin 1988). This difference is consistent with the weak softening of the Cll - C12 elastic mode (Takke et al. 1981) and reveals the existence of an additional term, i.e. the anharmonic elastic coupling of the Cll - C12 mode with a zone-boundary phonon, which softens at the M point of the Brillouin zone (Knorr et al. 1980). This last mechanism is associated with the incipient instability of the CsCl-type lattice for light rare earth-silver compounds which is of electronic origin (see section 3.10).
38
E MORIN and D. SCHMITI"
3.1.2. Antiferroquadrupolar ordering Several types of AFQ structures may exist. First, the lifting of the degeneracy of the cubic ground state may lead to wave functions different from one site to the other (Sivardi~re and Blume 1972). If the absolute values of the quadrupolar moments are different for the sublattices, that corresponds to a ferriquadrupolar ordering. Second, the ground state can be the same on each site, but the local z-axis may change, although belonging to a given star of crystallographic directions: the corresponding quadrupolar arrangement may be called multi-axial (Sivardi6re 1975). Lastly, more than one propagation vector may be necessary to describe the AFQ structure. In all the cases, magnetic structures with many sublattices may be favoured (see section 3.6). Whatever the actual structure may be, a multi-sublattice formalism is needed. The previous single-ion treatment may be used only for describing the occurrence of a magnetic or quadrupolar moment within a given sublattice. To describe the AFQ as well as AFM properties, no reliable information can be expected from static measurements in these intermetallics. In particular, the macroscopic symmetry may remain cubic. Only microscopic probes, such as neutron diffraction, are effective in revealing the existence of more than one magnetic sublattice.
3.1.2.1.
C e B 6. This cubic compound exhibits very complex properties due to the interplay of localized and delocalized 4f behaviours. The temperature variation of the specific heat reveals two anomalies at T o = 3.3 K and TN = 2.4 K (Peysson et al. 1986). Neutron diffraction experiments have shown that the compound remains paramagnetic down to 2.4 K (Horn et al. 1981). Magneto-elastic effects are present above TQ, although the C44 and Cll - C1~ elastic modes softens by only 2% (L/ithi et al. 1984). As the F8 ground state is well isolated (A = 540 K) (Zirngiebl et al. 1984), a quasi-full softening is expected in the case of a FQ ordering. On the other hand, the temperature variation of C44 and Cll - C12 may be analyzed with negative quadrupolar pair interactions. An AFQ ordering is then probable below TQ. This is confirmed by a neutron diffraction experiment on a monocrystalline sample in a magnetic field (Effantin et al. 1985). Indeed, two kinds of Ce ions have been observed which differ in zero field only by their magnetic susceptibility. The corresponding superstructure lines are described by a [½ ½ ½] propagation vector. The AFQ ordering then consists of an alternating sequence of (111) planes with different quadrupolar moments (fig. 12). Below TN, the spontaneous biaxial spin structure reflects the existence of the two sublattices driven by AFQ ordering. The magnetic field phase diagram reveals the interplay between localized and delocalized 4f magnetism as demonstrated by the anomalous increase of the AFQ phase under a magnetic field (fig. 13). In zero field, T O and the AFQ phase are reduced by the Kondo coupling but with an increasing magnetic field, the Kondo state is progressively suppressed and the AFQ ordering can manifest itself at a higher temperature.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
Magnetic ph<,.,,,,,
modulated
Anti ferro
~:[~/+ ,/, ,/2]
,
pho,.,,
:
39
quadrupolar
"~:[~]
z
(b)
(a)
Fig. 12. (a) Biaxial magnetic structure in CeB6 described by the propagation vectors indicated. (b) Antiferroquadrupolar structure described by the k o propagation vector; open and full circles correspond to the two different Ce-quadrupole sublattices (from Effantin et al. 1985).
' A- ,~ ' 'o ~,;too~j "-= ~6 *-I~//[111:]
20
t,....~ It ",
140
/
"iT
160
J/
120 100 80
0
+oIl,.,,,..=,,,,,,.+o,. q o!
60
pl ,+,,°,.o., V ,oI+,+..,.+.°,.o/,.
II
+~"'<""'i
0 1 " " @ 0 2
4
.
17 I~L_.
i
TernperaLure(K)
I 8
l
,o
=
0 10
Fig. 13. Magnetic phase diagram of CeB 6. Phases III and II are drawn in fig. 12 (from Effantin et al. 1985).
T m G a 3. This AuCu3-type cubic compound also exhibits two transitions in the temperature dependence of the specific heat, but very close to each other (TQ = 4.29 and Tr~ = 4.26 K) (fig. 14) (Czopnik et al. 1985). Here again, studies in the paramagnetic phase lead us to introduce quadrupolar interactions, but they 3.1.2.2.
40
P. MORIN and D. S C H M I T r
rN 60
50
I 40 30 20 10 o 4.1
4.2
4.3 T(K)
4.4 -
Fig. 14. Specific heat anomalies in TmGa 3. Arrows indicate the antiferromagnetic (TN) and antiferroquadrupolar (TQ) transitions (from Czopnik et al. 1985).
are too weak to induce a FQ ordering at T e (Morin et al. 1987c). In addition the spontaneous spin structure determined by neutron diffraction on a single crystal is multi-axial (see fig. 53 in section 3.6). This indicates there are different (111) directions as local z-axes at the 4f sites stabilized by AFQ interactions of trigonal symmetry (Morin et al. 1987d). Thus the quadrupolar ordering is of AFQ-type. As for the ferromagnetism in TmZn, the conditions of induced antiferromagnetism on the singlet ground state cause the temperature range of the paramagnetic AFQ phase to be very narrow. P r P b 3 . Another AuCu3-type compound, PrPb3, exhibits a non-magnetic transition at TQ = 0.35 K corresponding to the lifting of the F3 ground state degeneracy (fig. 15) (Bucher et al. 1972a). No spontaneous macroscopic strain has been detected in the quadrupolar phase. This may be due either to its weakness in the case of a FQ ordering or to its nullity in the case of an AFQ ordering. The softening of the C44 and Cll - C12 ultrasonic is far from complete and influenced by high-order magneto-elastic terms (Niksch et al. 1982). They may be analyzed by positive as well as negative quadrupolar interactions. However, both the parastriction and the third-order magnetic susceptibility need negative quadrupolar coefficients to be described (Morin et al. 1982). An AFQ ordering is then highly probable, and ( O °) is the order parameter. In conclusion, it appears that the case of AFQ ordering is not as well understood as the FQ ordering. This is due to experimental difficulties as well as theoretical complications. Indeed, none of the previously mentioned three compounds is a completely favourable candidate for investigating AFQ properties. In CeB 6 the quantitative analysis is complicated by the Kondo behaviour of the 3.1.2.3.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
41
0.55"K 0.7
0.6
PrPb 3
0.5
~
0.4
0.~
0.2
0.1
0
I
I
0.5
1.0
T(*K)
Fig. 15. Molar specific heat CJ R of PrPb 3 between 0.04 and 1.0 K (from Bucher et al. 1972a).
cerium ions. In TmGa 3 the temperature range of the AFQ phase is much too narrow. PrPb 3 orders at 0.35 K, a temperature characteristic of very weak AFQ interactions, which may easily be broken by the application of a magnetic field. From a theoretical point of view, the existence of many inter- and intralattice pair coefficients complicates the analysis of the data.
3.1.3. Structural transitions possibly of quadrupolar origin A few other structural transitions have been reported in the literature, which may be evidence for quadrupolar interactions. However, the actual situation is not yet fully determined, often because the crystal electric field is not known.
3.1.3.1. PrCu 2. This orthorhombic compound exhibits a second order structural transition at 7.5 K which has been observed in many different experiments, in particular by specific heat (Wun and Phillips 1974). From elastic constant measurements, the primary order parameter was found to be the ez, strain (Ott et al. 1977a). This agrees with neutron diffraction data which have shown that below 7.5 K there is a sizeable splitting of the (101) nuclear reflections corresponding to an angle/3 between the a- and c-axes of the orthorhombic cell that is no longer equal to 90° (Kjems et al. 1978) (fig. 16). A FQ ordering involving the (Pz~) quadrupolar component is likely. Unfortunately, there does not exist a determination of the CEF; thus a quantitative analysis of the observed properties is not possible.
42
P. MORIN and D. SCHMITI"
PrCu z
T=B.OK
T=4.75 g 1.0
0.5 ,
,
-0.01
0 i
~
~-I
0.01-0.01
0
0.01
k" ="
~ ~OR'¢
0.0
i-
4.0
L
~ I.v
=
A
6.0 8.0 TEMPERATURE (K)
10.0
Fig. 16. The upper part shows high resolution scans through the (101) Bragg peak below and above the structural transition temperature in PrCu2 at T = 7.5 K. In the lower part the observed splitting (A) is plotted versus the temperature. The full line represents a mean field calculation (from Kjems et al. 1978). 3.1.3.2. U P d 3. Contrarily to most of the metallic actinides, the hexagonal compound U P d 3 (P63/mmc) exhibits properties characteristic of localized magnetism associated with a 5f 2 configuration (3H 4 ground-state multiplet) (Andres et al. 1978). In a way reminiscent of Pr metal, two U sites exist with hexagonal and nearly cubic symmetries. T w o C E F transitions have been observed by inelastic neutron scattering on a single crystal. They have been attributed to an excitation process from a singlet ground state for both sites (Murray and Buyers 1980). In addition, U P d 3 exhibits a well-defined transition at 7.5 K, strongly depending on the metallurgical status of the sample (fig. 17). N o change of the hexagonal
4
t2
3 B~
-
ou -~-
,
0
2
4
6
4
8
10
12
14
o~_I
0
16
T(K)
Fig. 17. Specific heat of a UPd 3 monocrystalline sample minus that of ThPd 3 used as background
(open circles). The solid line (right-hand scale) is a plot of dp/dT. The slight difference in peak position of the two curves is caused by inaccuracies in the temperature scale of the resistivity measurement (from Andres et al. 1978).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
43
symmetry was observed from X-ray diffraction experiments on polycrystalline samples (Andres et al. 1978, Zaplinski et al. 1980). However, the increase of the c/a ratio observed at low temperatures might be associated with U atoms becoming more prolate ( ( O ° 2 ) < 0 ) at the quasi-cubic sites, according to an internal ferroquadrupolar process. This assumption might be checked by a full determination of the crystallographic structure on both sides of the transition by using a single crystal.
3.2. Nature of quadrupolar and magnetic transitions 3.2.1. Quadrupolar transitions As discussed in section 2.2.4, in cubic symmetry the transition at the quadrupolar ordering is usually first-order due to the existence of van Vleck matrix elements from the ground state. Indeed for the 6 cases (TmZn, TmCd, CeAg, CeB6, TmGa3, PrPb3) presented in section 3.1, the so called Xs(2~ susceptibility (Morin and Schmitt 1983a), which determines the order of the quadrupolar transition is calculated to be non zero. Thus the observed first-order quadrupolar transitions in the Tm compounds are well predicted by the present formalism. However a quantitative description of the specific heat anomalies is difficult in TmGa 3 and TmZn due to the vicinity of the magnetic and quadrupolar transitions. In TmCd, the measurements are not accurate enough, in particular in the low temperature phase. In PrPb3, the quadrupolar ordering temperature, T~ = 0 . 3 5 K , is small in comparison with the F3 - F4 spacing. The corresponding van Vleck terms do not affect the ordering process too drastically and the ( O °) jump is calculated to be weak in agreement with the temperature dependence of the specific heat (fig. 15). The same feature is also observed in CeB 6 and CeAg where the F8 - F7 splitting reaches 540 and 260 K respectively. For instance in CeAg the jump of (O2°) is calculated to reach only about 10% of its maximum value. The existence of anharmonic elastic terms in this compound makes it impossible to obtain a quantitative analysis of the T~ transition within the present formalism.
3.2.2. Magnetic ordering in the presence of quadrupolar interactions The nature of the magnetic transition in ferro- and antiferromagnetic cubic compounds has motivated a great number of studies, which can be classified according to whether they are based on an MFA analysis or renormalization group (RG) theory (K6tzler 1984). In both cases, the (isotropic) Heisenberg exchange is the starting point. The two approaches differ in the additional couplings which also have to be considered. The MFA model presented above, is in the line of models previously developed for transition metals as MnO, MnS (Lines and Jones 1965). However, instead of exchange striction, it considers for rare earth compounds quadrupolar pair and magneto-elastic interactions. The resulting coefficients of the free energy expansion are expressed in terms of single-ion susceptibilities and determining the phenomenological parameters for the different couplings then leads to the description of the magnetic transition (see section 2.2.4).
44
P. MORIN and D. SCHMITI"
On the other hand, the RG approach is based on symmetry considerations and includes critical fluctuations to predict the order of the magnetic transition. Within this hypothesis, continuous magnetic transitions become discontinuous ones according to the sign of the M4-term in the Landau expansion of the free energy. Note that the RG theory usually considers an incompressible lattice and nonquantized magnetic moments. The hypothesis of an incompressible lattice is in strong opposition with the evidences of large symmetry lowerings in cubic rare earth compounds. Due to the assumption of classical moments, the RG theory cannot predict the occurrence of a first-order transition associated with the positive X:, - (3) value in the case of a F3 non-magnetic ground state (see section 2.2.4 and Blume 1966). The aim of this section is to show through examples from the literature that the nature as well as the temperature of the magnetic transition may be predicted as soon as the quadrupolar magneto-elastic and pair interactions are reliably known (sections 3.2.2.1 and 3.2.2.2). Of course more complex situations may occur due to additional contributions from spin (and quadrupolar) fluctuations (section 3.2.3).
3.2.2.1. First-order transitions and tricriticality TmCu and TmAg A good example of the effect of the quadrupolar interactions on the magnetic transition is given by TmCu. This compound orders at TN = 7 . 7 K in a modulated antiferromagnetic structure, with all the moments parallel to the [001] axis. The transition is first-order (fig. 18). Various experiments performed in the paramagnetic state reliably provide the tetragonal quadrupolar parameter G r = 11 mK
1oo TmCu
-~ so
5 i
+I
i
10
,~-'"'"
.......
i
" "
.-
S
5
i
(b)
f" .,.." ,,, ~ ." " I 6
I
I
+
7
8
g
TEMPERATURE
(K)
Fig. 18. Temperature variation of the specific heat Cp (a) and of the entropy S (b) in TmCu. A n incommensurate antiferromagnetic structure is established at T N = 7.7 K and transforms at 6.7 K into a commensurate one (from Morin and Schmitt 1980).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
45
(Jaussaud et al. 1980). Just above TN the CEF third-order paramagnetic susceptibility Xz, - (3) is negative; this would lead to a second-order magnetic transition. However, the quadrupolar contribution to the total third-order magnetic susceptibility is positive and large enough to change the sign of X~)0, in the vicinity of TN, as seen experimentally. Consequently, the magnetic dipolar transition is expected to have a first-order character, as effectively observed (fig. 19a).
2
'
I'0.5
'
I
'
i -.
i
I
_m°!
,/ 3.5,,"
-~
T
ii..................
W =1~2K x -=-.42
'
1~o
'
f J" j/J
W=i.05K x=-.4.7
:.:" I
0
"
/ (~
.-"'' ( /"" T~
0/
0
I
I
10
TEMPERATURE(K) Fig. 19. (Top): temperature dependence of X~ ) observed for the tetragonal symmetry in TmCu (a) and TmAg (b). Full and dotted lines describe, for the CEF parameters W and x indicated, the cases with and without quadrupolar interactions, as indicated by the numerical values of the G v coefficient. (Bottom): the deduced magnetic phase diagram for both compounds, n is the bilinear coefficient (n N for an antiferromagnet). Dotted lines: second-order magnetic transition without quadrupolar interactions; dashed lines and full lines denote respectively: second- and first-order magnetic transition in the presence of quadrupolar transitions. C is a tricritical point (from Giraud and Morin 1986).
The isomorphous compound TmAg exhibits very similar CEF and bilinear interactions. It orders antiferromagnetically at T N =- 9.5 K, the transition is second order (Morin and Schmitt 1982a). Quadrupolar interactions within the tetragonal symmetry are weaker than in TmCu. They are characterized by G r = 4 m K (Giraud and Morin 1986). X~ ) is observed to be negative above 10 K, but is calculated to change sign at 6.2 K (fig. 19b). This defines a tricritical point between second-order transition (T > 6.2 K) and first-order ones (T < 6.2 K): the magnetic phase diagram predicts a second-order transition for a temperature of 9.5 K in agreement with the observation. An interesting feature is the large dependence of the tricritical temperature on quadrupolar interactions. Indeed, if G ~ reaches 5 mK it would shift the tricritical point up to 12.5 K, and the magnetic transition would be first-order for the same bilinear coefficient. Owing to the quadrupolar interactions, TmAg appears to be very close to tricriticality.
46
P. MOR1N and D. S C H M I T r
CeZn and CeMg Among other CsCl-type compounds, the CeX present a large set of different properties. Indeed whereas CeAg undergoes two well-separated magnetic and quadrupolar transitions (section 3.1.1.3), CeZn and CeMg order antiferromagnetically at T N = 30 and 19.5 K, respectively, the transition being first-order (Pierre et al. 1981, 1984). Contrary to the case of CeAg, the lattice stability is large; as in CeAg the Kondo-type behaviour for the Ce ion is not important. The tetragonal symmetry lowering below T N reaches record values ( c / a - 1 = 1.7 × 10 -2 and 1.3 × 10 -2 in CeZn and CeMg, respectively). Fitting magnetic properties in the ordered phase leads to G r = 220 and 50 mK in CeZn and CeMg, respectively. Note that these values have not been confirmed by measurements in the cubic paramagnetic phase. In CeZn the corresponding third-order magnetic susceptibility has been calculated to be positive in the vicinity of the N6el temperature, in agreement with the first-order character observed (fig. 20). In CeMg, the value G r = 50mK is a little bit too small: a value of about 75 mK would be large enough and more relevant.
/~-~=264 K
i
I'-~- ~=190KI
~
G.~=5OrnKI
CeZn
'75mK
1, ~ OOmK
\
0
I
/
20 TEM
I
PERATURE(K)
I
40
Fig. 20. Temperature dependence of the third-order magnetic susceptibility /~M,O - (3~ for CeZn and CeMg in the presence of quadrupolar interactions characterized by the G ~ values indicated.
erB 6 The same features are found again in P r B 6 , a compound isomorphous t o C e B 6. Here bilinear interactions are larger than the quadrupolar ones and a first-order antiferromagnetic transition is observed at 6.9 K (McCarthy et al. 1980). This first-order character seems to be stabilized by the same type of AFQ interactions as in the quadrupolar phase of C e B 6 ; indeed at low temperature the magnetic structure is described by the same propagation vectors (Burlet et al. 1988) and at high temperature large softenings of elastic constants are observed (Tamaki et al. 1985). By using quadrupolar coefficients deduced from f e B 6 (L/ithi et al. 1984) and NdB 6 (Tamaki et al. 1985) calculations within the present formalism show that the third-order magnetic susceptibility associated with the trigonal symmetry is positive. Measurements would allow to confirm this analysis.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
47
Tb pnictides Equi-atomic compounds within the NaCl-type structure have been extensively studied (Hulliger 1978). They are the earliest intermetallics observed to exhibit large one-ion magneto-elastic couplings (L6vy 1969, Mullen et al. 1974). In addition, from numerous studies, they have been shown to exhibit a more complex behaviour, they are at the meeting point of different couplings (section 3.2.4). However, a careful analysis of the antiferromagnetic TbX has shown that quadrupolar interactions unambiguously play a dominant role (K6tzler 1984). Indeed the AF transition is first order in TbP (T N = 7.1 K) and TbAs (T N = 12.5K) and second-order in TbSb (TN= 15.5K), TbBi (TN=17.5K), TbS (TN = 45 K), TbSe (TN = 49 K) and TbTe (T N = 51 K). All the compounds exhibit trigonal spontaneous distortions (Hulliger and Stucki 1978b). Figure 21 shows the resulting jumps of the magnetization and the susceptibility at the first-order transition in TbP compared to the behaviours observed in TbBi. In the t
__°_,--°_
i
TbP
MS(r) Ms(O)
i
i
°~Ooo~_-~ "~
(TN=7.1K )
.8 ! f
1.0
°~,
,
TbBi
\
IT, =17.5K)
~,
\
.6 .4
0.0
\
.2
o
0.4
Xo(T) Xo(T~)
0.6
0.8 T/TN
1.0
1.0
"'~ TbBi (TN :l?.SK)
~.0
0.5
.6
°
.8
a
1.0
3.5
..
0 (TN =7.1 K)
b o,
I
z
~
I
z,
~
I
6
~
T/TN
I
8
Fig. 21. Comparison between: (a) the spontaneous sublattice moments, and (b) the magnetic susceptibility for TbP and TbBi. Full lines are calculated within a formalism similar to that of section 2
(from Koetzler 1984).
48
P. MORIN and D. SCHMIT]"
paramagnetic phase, there are large magneto-elastic contributions to the elastic constants in TbP (Bucher et al. 1976) and TbSb (K6tzler 1984). The quadrupolar interactions are characterized by G" = 61, 32, 22 and 22 mK in TbP, TbAs, TbSb and TbBi, respectively (K6tzler 1984). These G ~ values lead to the temperature variations of X~u,o -(3) shown in fig. 22. Obviously the antiferromagnetic ordering occurs in TbP and TbAs at temperatures TN, for which xM,0 - (3) is positive; this explains the first-order transitions. From an experimental point of view, positive values of X}~) have been observed in TbP for the trigonal symmetry (fig. 23). For TbSb and TbBi, a second-order transition is predicted, as is experimentally
-/4
i
~32mK
I m
,
,~N
22m½
-10-4O
=
10 TEMPERATURE(K)
20
Fig. 22. Temperature dependence of the third-order magnetic susceptibility XM,O - (3) calculated in the presence of quadrupolar interactions in TbP (G ~= 61 mK), TbAs (G " = 32 mK) and TbSb, TbBi (G ~= 22 mK). W and x define the level scheme common to the four compounds. The first-order (second-order) transition at T N in TbP and TbAs (TbSb and TbBi) are driven by quadrupolar interactions.
0.00 B
,
,
X2)
~X =6inK
,
,
TbP ~ l~[I.]
o.oo/-, i i ~ 0.002 0.0
8
10
12
lt,,
T[K]
16
Fig. 23. Temperature dependence of the third-order magnetic susceptibility in TbP. Full lines are calculated with the indicated values for the trigonal quadrupolar coefficient Y( = 12G" (from Raffius and Koetzler 1983).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
49
observed. Note, however, that the critical point C strongly depends on G ~ and that TbSb and TbBi are not far from tricriticality of a quadrupolar origin.
3.2.2.2. Second-order transition in PrMge due to negative quadrupolar interactions. This compound orders ferromagnetically at T c = 10 K through a secondorder transition (Loidl et al. 1981). The ground state is a F 3 doublet. As mentioned in section 2.2.4, this CEF configuration gives rise to a positive value for Xz, - (3) in the vicinity of T c. This should produce by itself a first-order magnetic transition. On the other hand, the G ~ value suggested by Loidl et al. (1981) for explaining the temperature dependence of the spontaneous magnetic moment, i.e., G ~= -13.3 mK, leads to a negative total third-order magnetic susceptibility )¢(3) M,0" This now, accounts for the second-order character of the transition (fig. 24). I
.J.lo..
3
I
I
i
l
I
%%0 "
#2 PrMg21 0
5
T[K]
t~t 10
Fig. 24. Spontaneous magnetic moment, versus temperature in PrMg z. The dotted line is the Brillouin curve for J = 4. The full and broken lines give the result of a mean-field calculation with and without the quadrupolar interactions G ~= -13.3 mK (from Loidl et al. 1981).
Note that in the ferromagnetic phase, the ( O ° ) quadrupolar components are ferroquadrupolarly aligned by the dominant ferromagnetic interactions, whereas the negative quadrupolar interactions would prefer an AFQ arrangement and are thus frustrated.
3.2.3. Magnetic ordering in the quadrupolar phase The last case corresponds to a dipolar phase transition inside the quadrupolar phase (T < To). Below T c the quadrupoles are ordered and the crystal field is no longer cubic, e.g. it may have tetragonal symmetry. Therefore the various susceptibilities (X0, x~ " (3) , • • .) have new values (X~, x"~ T(3) . . . . ) according to the new tetragonal CEF level scheme. Note that this level scheme is now temperature dependent, like the ordered quadrupolar moment. The discussion of the nature of the dipolar transition is the same as in section 2.2.4, where /1-C(3) M,0 is replaced by )(T(3) 7 . Depending on the sign of X~ (3), the dipolar phase transition has a first- or second-order character.
50
P. MORIN and D. SCHMIqT
3.2.3.1. CeAg, f e B 6. In the case of CeAg, the ferromagnetic transition is second-order in the tetragonal phase (fig. 11) in accordance with theory. Indeed, . T(3) the x~ susceptibility is calculated to be negative as expected for a Kramers ion through the divergence of negative Curie-type terms associated with the magnetic doublet ground state. The same conclusion is also valid for CeB 6. 3.2.3.2. TmZn. In TmCd, bilinear interactions in the tetragonal phase are insufficient to induce a magnetic moment on the singlet ground state. On the contrary, in TmZn bilinear interactions generate a ferromagnetic ordering (T c = 8.12 K) slightly below the quadrupolar ordering (T o = 8.55 K). Calculations lead to a critical situation for Tc: the nature of the magnetic transition seems to be very sensitive to the values of the various parameters. For instance, T c is calculated to be of first order for G ~ = 25 inK, but would be of second-order with G v = 28 mK, as it experimentally appears. Consequently, the situation of TmZn with regard to the ferromagnetic transition seems to be not far from a tricritical point. 3.2.4. Presence of additional couplings In the compounds presented in section 3.2.2.1, the quadrupolar and Heisenberg bilinear interactions are dominant; any other interaction remains undetected; nonetheless, the competition between various couplings may be more balanced in other series such as rare earth pnictides and dialuminides. Two-ion magnetoelasticity and/or fluctuations may play a larger role in the minimization of the free energy. 3.2.4.1. DySb and rare earth monopnictides. This compound appeared in the past as an archetype for a first-order magnetic transition driven by quadrupolar interactions (Bucher et al. 1972b). It antiferromagnetically orders at TN = 9.5 K with a four-fold axis of easy magnetization. Associated with the magnetic structure, the lattice undergoes a structural symmetry lowering, mainly tetragonal, but with a small monoclinic component (L6vy 1969, Felcher et al. 1973) (fig. 25). This implies a magnetic moment not strictly parallel to the [001] axis. The existence of quadrupolar interactions has been unambiguously proved by the strong softening of C l l - C12 (Moran et al. 1973, Levy 1973); the value of G ~= 1 mK deduced is large enough to explain the first-order character of the joint spins and quadrupoles orderings, the third-order magnetic susceptibility is calculated to be positive (fig. 26). However, the analysis of the magnetization processes (Kouvel and Brun 1980, Everett and Streit 1979) and of the third-order magnetic susceptibility (A16onard et al. 1984b) in terms of only quadrupolar interactions fails (fig. 26). Indeed data are positive only in a very short range above TN and cannot be described by a unique G ~ value. Thus, additional spin interactions are present especially for the tetragonal symmetry. The most commonly proposed coupling is based on anisotropic bilinear interactions (Trammell 1963, Kim and Levy 1982, Jensen et al. 1980). It was in
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
.~;
,.y-e-V--L~c~-
Oa
h
....+, + , .
-I
l
~
-2-c
(t~tragonal)
-3-E
51
-4-__~n
n
,',
(cubique)
l° c/a
Oy P,
0.997 0.996
= Dy S b
0.995
-5--
Temperature
T° K
Fig. 25. Temperature dependence of the relative change of the tetragonal cell parameters in the ordered phase of DyP (two samples), DyAs and DySb; a 0 is the cubic cell parameter (from L6vy 1969).
a =
i
i
"
\
i
W=.2BK
i
~
x =-.82
GI~0"lmK 0
\.,~e3 1 ,,p o
=b
I
t
~
'
DI(kO~PB/ 3 0 ~
10
I
~
'
[0 0 1]
20 IEMPERATURE(K)
' W=.28K x =-.82 1~=-6.5K GI=1mK
30
Fig. 26. Temperature variation of the third-order magnetic susceptibility for H[I[001 ] in DySb. (a) tentative fits are drawn with the quadrupolar model alone, G 1 = G r = I mK being the value obtained from the softening of the Cu - C12 normal mode. (b) Fits here include, in addition to G: = i mK, the two-ion magneto-elastic coupling characterized by D1 = D r (from A16onard et al. 1984b).
52
P. MORIN and D. SCHMITT
particular postulated that rare earth pnictides, where the nearest-neighbour exchange is dominated by direct exchange and superexchange, are excellent candidates for large two-ion magneto-elastic effects (Lacueva and Levy 1985). From an experimental point of view, anisotropic bilinear couplings have been observed in the magnetic excitation spectrum of TbP (Loidl et al. 1979) and PrSb (Vettier et al. 1977). They have been also used for explaining the different antiferromagnetic structures found according to the temperature in GdX pnictides. Associated with these structures, different two-ion magneto-elastic strains have been also detected (Hulliger and Stucki 1978a, Hulliger and Siegrist 1979). For the experimental conditions which are used to study the third-order magnetic susceptibility , anisotropic bilinear interactions occur through their strain derivative, i.e., the two-ion magneto-elastic coupling, which may be described within the MFA formalism (Aldonard et al. 1984b). For DySb this coupling, characterized by a D 1 coefficient in fig. 26, is opposite to the quadrupolar interactions and improves the fit down to 12 K. However, a second-order transition would be expected. The same short-comings are also observed on the temperature dependence of the reciprocal susceptibility under an uniaxial [001] stress (Morin et al. 1985a). Even if present, two-ion magneto-elasticity, considered in the MFA, is not able to describe the features observed close to T~. On the other hand, a large number of studies has shown that NaCl-type compounds exhibit fluctuations through their critical behaviours close to the N6el temperature (Taub and Parente 1975, Hfilg et al. 1985). These critical fluctuations appear to be favoured in a deformable lattice (Taub and Williamson 1973). In DySb, as in HoSb, neutron diffraction measurements have shown an anomalously large magnetic scattering above T N. In the same type of experiments, the paramagnetic correlations in ErSb are characterized by a strong two-ion anisotropy (Knorr et al. 1983). In conclusion, the nature of the magnetic transition in numerous monopnictides seems to result from a well-balanced competition between different couplings. For many cases, it is difficult to determine the driving mechanism as e.g., in HoSb, which is very close to tricriticality (Jensen et al. 1980). The determination of quadrupolar interactions, coherent in CsCl-type compounds, cannot be achieved in these series. In addition the large dependence of anisotropic magnetic interactions on metallurgical stresses may change the nature of the transition in a given sample as shown in Er compounds (Hulliger and Natterer 1973). As in DySb the difficulty in properly considering fluctuations and anisotropic bilinear couplings may prevent one to fully understand the first-order transitions in DyBi (Hulliger 1980), DyP, DyAs (L6vy 1969), ErBi and ErSb (Hulliger and Natterer 1973) even if quadrupolar interactions are obviously present as proved by the temperature dependence of elastic modes. The R A l 2 series. RA12 compounds (C14 structure) appear to pose a similar problem as was observed in pnictides (del Moral et al. 1987). DyA12 orders ferromagnetically at T c --61.4K through a second-order process. The third-order magnetic susceptibility along the [001] easy magnetization direction is 3.2.4.2.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
-3 x 10"5~ /oB/KO¢3
I Gt=OrnK
I
70
80
53
mK i
-2 <3 -1
0
I 60
90
T(K]
Fig. 27. Temperature dependence of the third-order magnetic susceptibility A.... = X~ ) for HI[[001 ] in
DyAI2. Full lines are calculated with a bilinear coefficient n = 17.5 kOe//xB and the values indicated for the quadrupolar parameter G1 = G ~ (from Del Moral et al. 1987). negative in the entire paramagnetic range and cannot be described with a unique G ~ coefficient (fig. 27). However, the data close to T c do not indicate critical behaviour. The present range (8-10 mK) of G ~ is different from values deduced from magnetization measurements in the ordered phase, i.e. - 2 m K (Rossignol 1980). The same failure for describing the third-order susceptibility is found again in ErA12 (del Moral et al. 1987). Data go from positive values at high temperature to negatiye ones close to T c which countermands a first-order transition. The 2 possible G ~ values deduced are larger than in DyA12 in violation of the o~j variation law across the rare earth series (Schmitt and Levy 1985) (see section 5). All these features may be the signature of additional couplings and an improved knowledge would need systematic studies in particular of ultrasonic properties.
3.3. Determination of the quadrupolar parameters from susceptibility techniques Different ways of determining quadrupolar parameters are directly related to the various susceptibilities described in section 2. Indeed, these susceptibilities connect the magnetic variables on one hand and external stresses on the other hand. Thus, (i) the strain susceptibility represents the response of the quadrupolar moments to the corresponding stress; (ii) the quadrupolar field-susceptibility connects the quadrupolar moments to the magnetic field; (iii) the third-order magnetic susceptibility couples the magnetic moment, more exactly the initial curvature of the magnetization curves, and the magnetic field. Each of these three
54
P. M O R I N and D. S C H M I T T
susceptibilities may be acquired by a different experimental technique, namely measurement of the elastic constants, the parastriction and the low-field magnetization, respectively; these techniques are described below. 3.3.1. Elastic constants
As seen in section 2.2, the existence of a magneto-elastic coupling leads to the softening of the elastic constants of appropriate symmetry when the temperature is lowered, and this softening is enhanced by positive two-ion quadrupolar interactions. The experimental technique generally used is the ultrasonic velocity measurement, and several geometrical configurations are considered in order to observe the various ultrasonic propagation modes. In general, the normal (symmetrized) elastic constants are not directly accessible; they have to be obtained from a linear combination of those which are measured. The observation of a softening of elastic constants due to magneto-elastic coupling in the intermetallic rare earth compounds was realized in the early 1970s, and extensively studied in particular in cubic rare-earth antimonides (see figs. 28 and 29). Several behaviours are observed, according to the strength of the quadrupolar (magneto-elastic) coupling as well as the nature of the CEF ground state. The temperature variation of the elastic constants in LaSb is taken as the background for the other compounds. Large anomalies are often detected at the magnetic ordering temperatures and below; they are due to domain-wall stress effects. However, the most spectacular effect remains the softening above the ordering
I.C
0.8
0.6
O~
°il I
'
8b
'
I
J~o
'
z~,o
TEMPERATURE(°K)
'
3~o
Fig. 28. Change in C o = C ~ = (Cal - C12) as a function of temperature in DySb. A fit of the data leads to (g0) 2 = GEE = 1 . 4 m K and I o = K v = - 0 . 4 m K and thus G ~ = 1 mK (from Moran et al. 1973).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
14.0~t3.9~
55
TmSb
13.8 ~:
13.6 r ~
13.5 Ci1-C12 • I
1
5
.
2
el
~
~2.70 ee o
- • • ee~ e
~:__.z6.sz6.6z6.4z6.~
o
°
C44 ee
"
I
I
Ioo
200
300
T (°K)
Fig. 29. Temperature dependence of the symmetrized modes Clt - C12 = C 7 and C44 = ½ C e for TmSb. Full lines are calculated with only magneto-elastic couplings in the presence of CEF (from Mullen et al. 1974).
temperature, e.g., this softening reaches about 60% of the room-temperature value of C r at 10 K in DySb and occurs over a wide range of temperature (see fig. 28). A good description of the softening of the C ~ elastic constant requires the introduction of a two-ion quadrupolar interaction in addition to the one-ion magneto-elastic coupling (Levy 1973, Moran et al. 1973). The cubic TmX series (X = Zn, Cd, Cu, Ag, Mg) is another series which is particularly spectacular with regard to the softening of the C v elastic constant. For example in TmZn and TmCu (fig. 30) the softening reaches more than 50% of the room temperature value just above the ordering temperature, indicating very strong quadrupolar interactions (Liithi et al. 1979, Jaussaud et al. 1980). In the opposite limit, the softening is much less pronounced in TmAg and TmMg (Giraud et al. 1986, Giraud and Morin 1986). It is worth noting that the analysis of the temperature variation of the elastic constants may give information about the crystal field in addition to the magnitude of the quadrupolar parameters. Indeed, according to the nature and spacing of the low-lying CEF levels, anomalies may occur at low temperature in the strain susceptibilities, thus in the elastic constants (see section 2.2.3). This is particularly emphasized in singlet ground-state systems as in PrSb (Liithi et al. 1973a) and TmSb (see fig. 29) for both C ~ and C ~ modes.
56
P. M O R I N and D. S C H M I T T
I
I
I
l
l
C~(YCu) •/ ' 1 / " .~.I
/
,t~
/.
E
c" :½(c,-c,2
?
o
I
O,l 0
T
t..j
i
- -
experiment fit with G1 = l l . 5 m K R1 = 7,0 mK
J
I
l
100
I
I
2OO
300
T(K)
Fig. 30. T e m p e r a t u r e d e p e n d e n c e of the C' = ½C r = ½(C n - Ct2 ) m o d e in T m C u ; the full line is a theoretical fit with G 1 = G ~ and K 1 = K ~ ( f r o m J a u s s a u d et al. 1980).
3.3.2. Parastriction Parastriction constitutes a very suitable technique for investigating magnetoelastic coupling. It consists of measuring changes in the length of a sample in an applied magnetic field in the paramagnetic phase (Morin et al. 1978c, 1980b). It differs from the usual magnetostriction by the fact that only the low-field variation of the latter is analyzed, namely its H2-dependence. Indeed, as shown above (section 2.2.1), within the limits of validity of the perturbation theory, the coefficient of this HZ-variation is related to the quadrupolar field-susceptibility Xo (or Xp) which depends only on the crystal field and on the magnetic and magneto-elastic parameters. In addition, in agreement with the classical hightemperature approximation (Callen and Callen 1965), a linearization of this susceptibility may be achieved by plotting ) ( Q 1 / 2 a s a function of the temperature. Deviations from this linear behaviour may occur at low temperature due to CEF effects in the same way it does for the magnetic susceptibility. Experimentally the measurement of the relative change of length A = 8l/l is performed successively along and perpendicular to the direction of the applied magnetic fields; one eliminates the volume effects e ~ by considering the difference All - A~. As a consequence, for a magnetic field along a four-fold or three-fold direction, and according to the expression for 81/l [eq. (25)], one obtains respectively
(All- Al)v =
~93_B~ 2 C---~oXQH2'
3 B+ (All- A±)+ = V'2 C O x e H 2 '
(110)
for the two tetragonal and trigonal strain modes. It is worth noting that the same
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
57
relations are found, by considering a fixed measurement direction and by applying successively the magnetic field parallel and perpendicular to that direction. This latter geometry is habitually used by experimentalists for the sake of simplicity. As mentioned above it is useful to plot HI(I All - A± I) 1/2 in order to observe the linear high-temperature b~haviour. A good example is provided by the compound TmZn where the quadrupolar effects are particularly strong (fig. 31). According to eq. 110, the high-temperature slopes give the magneto-elastic coefficients B v and B', as the background elastic constants are known from other measurements. ~I~
~
I
0
,
v
o, o / / '
I
O•
."
I
I
1
Fig. 31. Parastriction of TmZn, measured for the tetragonal and trigonal symmetry lowering modes. Curves are calculated within the susceptibility formalism (G ~= Ga) (from Morin et al. 1980a).
In addition, large deviations from this linear behaviour are observed for the tetragonal mode below 100 K due to CEF effects and to a strong G v parameter. On the other hand the anisotropy of the parastriction is particularly obvious in this compound, showing that measuring the magnetostriction on polycrystals is quite meaningless in these anisotropic systems. Note also that the parastriction gives the sign of the magneto-elastic coefficients, while only its absolute value may be obtained from elastic constant measurements. This technique has been successfully used in several series of rare earth compounds, in particular in RZn compounds where the coefficients associated with both the tetragonal and trigonal strain modes have been obtained for the heavy rare earths (figs. 32 and 33).
3.3.3. Third-order paramagnetic susceptibility The third-order paramagnetic susceptibility has been introduced as a new method for studying quadrupolar interactions in rare-earth compounds (Morin and Schmitt 1979, 1981b). It consists of a detailed analysis of the magnetization induced by an external field in the paramagnetic phase. More precisely it
58
P. MORIN and D. SCHMITF =
i
1S
Er Zn/y
/.
H . [ oo]
.
..~
.
•
,/ TmZn
%s DyZn
"r
20o
too
300
(K)
TEMPERATURE
Fig. 32. Temperature variations of g/I;,, All 1/2 for tetragonal strain in RZn. The full lines are calculated dependencies (from Morin et al. 1980b). -
H //
8 ~o
1o
v
[111]
TmZn DyZn
5
TbZn
"I-
i
100
TEMPERATURE
200
300
(K)
Fig. 33. Temperature variation of H/l All- A±I 1/2 (rhombohedral strain) in RZn. The full lines are calculated with the susceptibility formalism (from Morin et al. 1980b).
corresponds to the second term (in H 3) of th e development of the magnetization as a function of the magnetic field, this term depends on the total quadrupolar interactions in addition to the crystal field (see section 2.2.1). An experimental way to investigate the magnetization curves is to plot M / H versus H2; this provides both the first-order magnetic susceptibility XM, i.e. the
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
59
isotropic null field value, and the third-order one X~ ) , i.e. the anisotropic slope of the linear low-field range. This is well-illustrated by the compound TmCd, where the anisotropic character of X~ ) is particularly obvious (see fig. 34). It is worth noting that the linear part of these curves diminishes as the temperature is decreased, since the higher-order terms (in H s , . . . ) become relatively more and more important. An alternative experimental way, although less direct, is to consider the Arrott's plots M 2 versus H/M. In low field, the variation is linear; the intersection with the x-axis yields (X~)) --1 while the third-order magnetic susceptibility is related to the slope dM2/d(H/M) through the relation
X(~ )
-(XM)4/(dM2/d(H/M)) .
=
The variation of the third-order magnetic susceptibility may then be drawn as a function of the temperature and analyzed with the above formalism (see fig. 35). As an other example, in fig. 36 we show the temperature dependence of X~ ) for the two main symmetry directions in PrPb 3. The anisotropic character of X~ ) is obvious from these curves again, as well as its dependence on the bilinear and quadrupolar interactions. The third-order magnetic susceptibility also explains the paramagnetic anisotropy of the magnetization observed in TbxLal_xA12 for a constant magnetic field within a given plane (Hoenig 1980). The technique of third-order magnetic susceptibility has also been applied to the tetragonal compound TmPO 4 which presents noticeable non-linear terms in the low-field magnetization within the basal plane (Andronenko et al. 1983). Another experimental way to extract X~ ) is to perform torque anisotropy 120
i
"
~
"- "
115
[001]
6 11o ~,
"
"
"
"
,
[111]
TmCd
' ~ 105 EI 77
"
~ - - - - ' -
"" ~ ~
""
76
K
[OO]l
75 74 .....
0
i 100
2
Hi
I ~ 200
I 300
400
(k(De2 )
Fig. 34. Experimentalvalues of M / H versus H 2 in TmCd (from Morin and Schmitt 1982c).
60
P, MORIN and D. SCHMITT
//// ///"
&"
!// II
-4
:o,5 K
,:o
/
×
~
5
,
=-o.34
i
10 15 20 TEMPERATURE ( K )
i
25
30
Fig. 35. Experimental and calculated variations with temperature of X~ ) along the [001] axis in TmCd system. Note the dependence of the coefficient G1 = G ~ on dilution (from Morin and Schmitt 1981b).
1 e*=-9K GI=O '. -
tI
i
l
H//[001]
~o O ~.~:E ~<
I
I
H//[111]
'
]
\", i L
i
PrPb3 W=-.65K x=.6
\
1¢---9K JG .3K
i . . . . . . . . . ;_:.=_._.....
K ............2 : 2 : ~ _-'r='- .....
'G =, --.9. -15mK -/ , //--GI="0' ' - I . IG2= ~ O*---09K \
,e*-gK ;.2K O,=G2__O// //
//
/
~_ /
// /
-1
I
0
5
I
I
10 5 TEMPERATURE (K)
/I
10
15
Fig. 36. Temperature variation of the third-order susceptibility along [001] and [111] in PrPb 3. Dots are experimental points. The dashed lines correspond to behaviours without quadrupolar interactions, but show the effects of the bilinear interactions (0" = - 9 K). The full lines show quadrupolar effects (G v= G1, G ~ = G2) (from Morin et al. 1982). m e a s u r e m e n t s which, in p r i n c i p l e , allow us to d e t e r m i n e p a r a m a g n e t i c susceptibility t e n s o r s o f f o u r t h (~- X ~ )) a n d e v e n sixth r a n k s a n d has b e e n a p p l i e d to R A I 2 c o m p o u n d s ( D e l M o r a l et al. 1987). A n a l t e r n a t e a p p r o a c h to a s c e r t a i n i n g t h e q u a d r u p o l a r effects o n t h e m a g n e t i z a t i o n is b a s e d o n t h e n o n - l i n e a r i t y o f t h e e x c h a n g e field Hexch as a f u n c t i o n o f M
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
61
(Kouvel and Brun 1980) Hex~h = AM
+ A'M 3 .
(111)
However, this procedure provides a )t' value which is temperature dependent, since it is related to the true quadrupolar parameter G" through the relation )t, =
2G u'(X~)) 2 (X0)4( 1 _ G~X# )
(112)
This ,V parameter is therefore connected with the quadrupolar part of X}~); its temperature variation has indeed been observed (Abu-Aljarayesh et al. 1986a)
3.4. Effects on the magnetostriction While parastriction allows us to describe the initial magnetostriction in the paramagnetic phase, i.e. the first term in its field development, a full diagonalization of the Hamiltonian describes the whole magnetostriction curve, in the ordered or non-ordered state, provided the magneto-elastic couplings and all the two-ion interactions are included.
3.4.1. Magnetostriction of dilute compounds The main advantage in investigating dilute magnetic compounds is to be free, at least to a large degree, of two-ion interactions. Indeed it is these interactions that make the analysis difficult. If the effect of the dilution on the macroscopic measured strain is correctly taken into account, one can determine the microscopic one-ion magneto-elastic coupling. Usually authors simply assume that the macroscopic deformation is proportional to the magnetic impurities concentration (Nieuwenhuys et al. 1982b, Pureur et al. 1985). A systematic study of the forced magnetostriction has been carried out for the cubic noble metals compounds (silver, gold) containing magnetic rare earth impurities (Creuzet and Campbell 1981, Campbell and Creuzet 1985). The magnetostriction follows the second-order Stevens coefficient a I (see figs. 37 and 38), indicating that a single-ion model is pertinent, and that the second-rank magneto-elasticity is preponderant in these systems. Nevertheless, a fourth-rank magneto-elastic coupling has also been determined, its contribution is about ten times smaller than the second-rank one. In addition to the magneto-elastic coefficients, the magnetostriction curves are strongly affected by the nature of the low-lying CEF levels. This allows one to choose between possible levels schemes through the shape as well as the temperature variation of the magnetostriction curves (fig. 39). Another spectacular effect related to the crystal field concerns the so-called "reverse" magnetostriction, i.e. the sign reversal of the magnetostriction between the low and high temperatures. This occurs in CexLal_xSb, SmxLaa_xSb (Nieuwenhuys et al. 1982a), and CexLal_~Bi (H~ifner et al. 1983) compounds.
62
P. MORIN and D. SCHMITF 14,
y
lu-,,-~nt
12
10' 8 6
~
14. •106x6L "~Ag Tb
Ag Tb
t
12.
10, 6
A
4 2 0
2
~
2 -4
--6 -8
kG
(a)
0
AgEr
-2
X,~Tm \
-t, -6 -e
~
o
H.kG 0AgEr AgTm (b)
Fig. 37. Parallel magnetostriction data for a field applied along the [001] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental data, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
6, ' 106 x"~t
.
/
6-
/AgTb
4.
AgDy
4 / - -
A_gHo
4u , 4u , 6,0 ,H.kG 0[ ~
,
H,kG
-2 -4. •
(a)
-q
fo)
Fig. 38. Parallel magnetostriction data for a field applied along the [111] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
For example, the change of sign of the measured strain takes place at 2 K in LaBiCe and at 8 K in LaSbCe (see fig. 40). This can be explained by a simple crystal field theory, which, in some particular cases, leads to an expectation value of the quadrupolar operator O ° which is not always positive. For example, in Ce0.15La0.85Sb, the quadrupolar m o m e n t induced by an external magnetic field on the F 7 doublet ground state is negative at low temperatures. It changes its sign either for very large magnetic fields (about 40 T at 1.5 K) or above 8 K for any field, namely when the influence of the excited F 8 quartet becomes important. This change of sign of the initial magnetostriction curvature is accounted for by the formalism developed in section 2 and corresponds to the change of sign of the quadrupolar field susceptibility x~ - (2) (see fig. 41). Similar reverse magnetostriction
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
~-110-6) 20
40
60
t
t
i
63
H(kG)
_1
_2 ~2.8 t
_3
Fig. 39. [001] parallel magnetostriction in ErA g (concentration 0.5%) corrected for the volume contribution; circles are experimental data, solid lines calculated. Note the anomalous variation with temperature which is characteristic of the CEF level scheme (from Creuzet and Campbell 1982b).
LaBi:Ce (lO°/o)
6 4
-5T
2
/I 3T
O -2
_1 ~-
-4
o
~ ~\
3 2/
O
LaSb:Ce (15%)
4.8T k,,..~ 4 T
5
10 15 T , (K)
20
Fig. 40. Comparison of the magnetostriction of (a) LaBi : Ce (upper panel) and (b) LaSb : Ce (lower panel) as a function of temperature at various fields. As clearly seen the "reverse magnetostriction" occurs at a lower temperature for (a) than for (b). Solid lines represent a fit to theory (from H/ifner et al. 1983).
64
P. M O R I N and D. SCHMITT r
50
-50
o
LaSb:Ce
LoBi:Ce
/ /
40o
•>• 0
-400
lo
T (K)
20
Fig. 41. Temperature dependences of Xy(~) , the quadrupolar field-susceptibility, calculated in the presence of only the CEF for LaBi : Ce (a), LaSb : Ce (b), LuCd : Tm (c) and LuZn : Tm (d). In these four cases the CEF level scheme drives the so-called reverse magnetostriction.
effects may be expected in other situations, e.g., with a F5 triplet as ground state, as in TmCd or TmZn, for an appropriate dilution.
3.4.2. Magnetostriction in the ordered phase The magnetostriction observed in the ordered phase of magnetic compounds is generally much larger than that in the paramagnetic phase. This is due to the large influence of the exchange field which tends to saturate both the magnetic and quadrupolar moments. The symmetry of the cell distortion is directly related to the direction of the magnetic moments, e.g., in ferromagnetic compounds. In cubic antiferromagnets, the observation of a spontaneous strain may be very helpful in determining the actual magnetic structure (see section 3.6). A quantitative analysis remains more difficult than for a non-ordered state, because of two-ion effects. Depending on the strength of the bilinear exchange coupling, it has been done using either a semi-classical (see section 2.4) or a quantum treatment (see section 2.1.2). In the case where the bilinear interactions are big, large magnetostrictions have been measured at room temP6erature, as e.g. in TbFe 2 where a record magnetostriction of Am = 2400 × 10- was reported (see fig. 42). This motivated numerous investigations because of the interest of technological applications. Still higher strains have been measured at low temperature in the RFe 2 series (see section 4). Huge magnetostriction has also been observed at low temperature in numerous series where the alloyed metal is nonmagnetic, in particular within the NaC1- and CsCl-type structures (L6vy 1969, Morin et al. 1977). Large tetragonal strains of the order of 10 - 2 have been measured; e.g., c / a - 1 =--1.4% in PrZn at 1.5K (Morin and Pierre 1975). In TbZn a change of easy direction within the ordered
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
I
I
I Tb Fe2
'
L
]
65
|
2,000I
~.poly [hi
1,000 t Y
0
V
I
I
I
I
I
5
10
15 H [kOel
20
25
F
Fig. 42. Magnetostriction of TbFe z at room temperature for a single crystal (Aiq curve) and a polycrystal taken for two different orientations (a and b) (from Clark et al. 1974).
10 -2
10- 3
IO-~
Lu
TbZn
1
io-s
10 -6
u~o.s 10" ;' 0 10 - 8
.....
i
100 i
100
200 )
I
200 TEMPERATURE OQ
300
Fig. 43. Thermal variations of the spontaneous strain e 3 = e~ above the temperature at which there is a change in direction of the easy axis and the e 3 strain induced by a 1 kOe field in the paramagnetic state for TbZn ((3, strain-gauge data; Q, capacitance dilatometer data; full curves are calculated variations). Inset: experimental variation of the volume anomaly e v = X/3 e" (from Morin et al. 1977).
66
P. MORIN and D. S C H M I T r -2 10
10.3
10-4
10~ 5 U3
TbZn
10-6
TC
H: lkCe 10- 7
\ ,
10- 8
,
~"
q.
100 200 TEMPERATURE (K)
300
Fig. 44. V a r i a t i o n s w i t h t e m p e r a t u r e o f the spontaneous strain sxz = e~/~/2 b e l o w the temperature where the easy axis changes, and of the exz strain induced by a 1 kOe field in the paramagnetic range for ThZn ( 0 , strain-gauge data; @, capacitance dilatometer data; full lines are calculated variations) (from Morin et al. 1977). ,
Tm Zn
2
~
,
T - 107 K 1
0
-- '?1 _..
01i°, E
9 10 11 TEMPERATURE (K)
-4
o
2
,
6
;
APPLIED FIELD
;o
12
(kOe)
Fig. 45. Experimental variations of the strains parallel (All) and perpendicular ( A l ) to the four-fold direction of the magnetic field at 10.7 K in TmZn. Full and dashed lines are relative to fast and slow rates of variation of the field. The inset shows the temperature dependence of the critical field H c (from Morin et al. 1978b).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
67
range occurs; this allows one to follow the temperature variation of the strain associated with the [001] and [101] moment axes (figs. 43 and 44). Finally, in TmZn one transition appears on the magnetostriction curves at temperatures above the Curie temperature (see fig. 45). This behaviour, also observed on the magnetization curves (see section 3.5), can be explained by the appearance of a strained ferromagnetic state above a critical field within the paramagnetic phase, arising from strong quadrupolar interactions.
3.5. Effects on the magnetization processes 3.5.1. Paramagnetic phase Perturbation theory has shown that quadrupolar interactions strongly influence the initial curvature of the magnetization as a function of field in the paramagnetic phase, i.e., the third-order paramagnetic susceptibility (section 2.2). In the same way, quadrupolar couplings influence the higher-order terms, i.e. they have an effect on the whole magnetization process. A diagonalization of the full Hamiltonian allows us to take these couplings into account particularly at high fields. As an example, the magnetization is predicted to be largest along a three-fold axis in [111]
(ooi] I
6
G,=9mK
.... +.:;,~ I
.0 ~+~+- +.+.+-16K
,
/ . +] ~o+- ~o+2OK I
4
-/'/+/4,/*/+°I o/'/+ 4,/ /0 I / /4,/ / ] 0 I/I /4" 4,0 /
d'i
2
o~ +~
•..'+25K,
/n
-
W=1.4 K x =-.42
•
• 4,o011/ +4,0 • 1+4,0 • o+~Oll u
+.,.
I
~'."
jEP=_ 3K
J
"'"
x =-42
~ ,
4 I-
W=I.4K E)w:I 3K ,
" 9K
+ ~...
..-'.~"
G.=gr~--I "11 -----
•
• +~/@/4,/0/
..,+0o., •
--
4~~
e~ e 0 i / @ .:+...-..
04"@0
0
.-1;:;. / 4+
|
e/o/../•
i+tOll t ~+ ll . .
~.-
I
G2=-80 mK
//+."
/."
/
.di.".
-
®// m II
,g .'"
0
I;::"
,
0
SO
/ 100
0 H(k~)
W=I.4K
? ." .e • •
x =-.42
!
50
it~=. 3 K 100
Fig. 46. Magnetization curves along the directions [001] and [111] for TmCu at various temperatures; lines are theoretical fits with the p a r a m e t e r s indicated ( G 1 = G ~, G 2 = G ~) ( f r o m J a u s s a u d et al. 1980).
68
P. MORIN and D. SCHMITF
TmCu in the absence of quadrupolar interactions; by using appropriate values of the parameters G ~ and G ~ one can restore the four-fold axis as the direction where the magnetization is maximum, as observed experimentally (see fig. 46). The magnetization curves show a more spectacular behaviour in case of huge quadrupolar couplings, as in TmZn (Morin et al. 1978b). In this compound, the quadrupolar coupling is strong enough to induce a transition from the paramagnetic to a ferromagnetic state in a wide range of temperatures above the quadrupolar ordering (see fig. 47). This transition occurs for a critical field which depends on the strength of the quadrupolar interactions and is well-described within the MFA. The critical field is also present along the directions of hard
.~
1K 111
0
20
(b
40
20
60
INTERNAL FIELD
80
100
120
(kO~)
Fig. 47. Magnetization curves of TmZn along the [001] axis at different temperatures; the insets show the temperature dependence of the critical field He; (a) experimental, (b) calculated (from Morin et al. 1978b).
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
69
magnetization, although it is larger than along the easy direction, i.e. [001]. For example, along the [101] direction, the magnetic moment is first aligned with the external field, then rotates towards the [001] axis when the component of the field along the [001] direction reaches the associated critical value; finally it comes back toward the initial [101] axis when the field is large enough.
3.5.2. Ordered phase In the ordered state, quadrupolar interactions play an important role in the free energy balance between the various crystallographic axes. In particular, when the field is applied along a direction of hard magnetization in a ferromagnetic compound, the magnetic moment rotates from the easy axis towards the field. The area between the corresponding curve and that extrapolated from the high field behaviour is related to the anisotropy of energy between both directions. This energy difference is strongly influenced by the quadrupolar couplings, as for example in ErZn (see fig. 48). This behaviour is enhanced when the free energies corresponding to two moment directions cross each other at a given temperature T R within the ordered phase. This leads to a first-order phase transition in which the direction of easy magnetization changes. This phenomenon is driven by entropy contributions induced by the anisotropic splitting of the CEF level scheme under the exchange field (see fig. 49). There are many examples in rare earth compounds, especially
6
5F//
b >
, 1mK-
~_o4~.4
* [101]
~I Z
=(I 1 I)
3 I
'
'
'
l
O N FLU Z L~ :E
,
Z/
~ 10
i:I \ 1 TEMIPERATUREI(K) .
4 0
;o
~o
INTERNAL
30 FIELD
40
50
(kOe)
Fig. 48. Magnetization curves in E r Z n along the three principal cubic directions at (a) 4.2 K and (b) 10 K; the inset shows the temperature dependence of the critical field; the points are the experimental data along [101] (O), [111] (zX) and [001] ( 0 ) ; the lines are the theoretical fits (broken curves: without quadrupolar terms; full curves: with G ~ = G 1 = 1.5 m K and G ~ = G 2 = 11 m K terms (from Morin and Schmitt 1978).
70
P. MORIN and D. SCHMITT - 250
-50P
///
HoZn 12
-
// /- ~" " i l l
v
>.-
L9
"Tl
-15OF
-26G
0~ W
Z W
/
W W
_.__~[i01]
//
----
// . . . .
G2:0
/
rr
b_
.... -270 0
G1 : - 4 mK G2= 10 mK
/ / ~[101] i 20
GI=0
i 40
TEMPERATURE
-2sot-
0
(K)
80
[lOl]
[111]
Fig. 49. (Left-hand part) calculated temperature variation of the free energy along [101] and [ l l l ] for HoZn, without quadrupolar terms (broken curves) and with G ~ and G ~ contributions (full curves); (right-hand part) arrangement of the lowest CEF levels in HoZn at T R = 25 K according to whether the moment lies along the [101] or [1ll] direction and taking into account the quadrupolar coefficients G ~= - 4 inK, G ~= 10 mK (from Morin and Schmitt 1978).
with Nd, Tb and Ho (see, e.g., Gignoux et al. 1975, Sankar et al. 1977, A16onard and Morin 1985). In each compound, the two-fold axis is the direction of easy magnetization at low temperature. In HoZn for example, the critical temperature T R is calculated to be 48K with tile CEF alone (Morin and Schmitt 1978). Introducing appropriate quadrupolar parameters G r and G ~ shifts the free energy curves and allows to reduce T R to 25 K in agreement with the experimental data (fig. 49). The effects of quadrupolar interactions on the magnetization processes in antiferromagnets are more difficult to analyze due to the existence of several sublattices (Jensen et al. 1980). On the other hand, they are more relevant for the stability of the actual magnetic structures. This will be discussed in the next section (3.6).
3.5.3. Effect on the direction of easy magnetization In the previous section, the dominant CEF determined the direction of easy magnetization and the quadrupolar interactions determined the exact temperature TR, for the change of the easy axis. In this section, we discuss some cases where the quadrupolar energies are larger than the CEF anisotropy. Good examples are provided by the ferromagnets DyCd and DyZn which have the CsCl-type structure (T c - - 8 0 and 140K, respectively). In both cases, a four-fold axis is observed as the direction of easy magnetization and a large tetragonal spontaneous strain occurs in the ordered phase (0.8% at 4.2K in DyZn) (Morin et al. 1977). Their cubic level schemes are given in fig. 50a together with those of the antiferromagnets DyCu and DyAg. Their anisotropic splitting by bilinear interactions always favours the three-fold axis: the CEF
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
zOO
71
100
I
r(3) DyAgi~--~. <..8 ~
!~-Y~.-~-~.~ /
200
c11"~i
®
.....
o
I
@easyaxis [I11]~k~O011
-200
2s~
W=-.57K \ 6
~
(-.}7)
-400
¢_-60K
',
(.9
\
T:4.2K J
(-5)
0 0
-.5
X
Fig. 50. (a) CEF level scheme in dysprosium compounds having the CsCl-type structure. E is the energy in Kelvin, W is a scale factor, the value of which is given in parentheses for each compound, x is the ratio between fourth- and sixth-order CEF terms; (b) x-dependence of the strain susceptibilities, X2 (=Xr) and X~ (=X~), for a given W at low temperature; (c) easy axis at low temperature according to x and G1 ( = G ~) for given values of W and 0". The full line is mainly related to the case of DyAg and DyCu; for DyZn and DyCd, the critical line would lie lower (from Al6onard et al. 1984a).
anisotropy between three-fold and four-fold axes is too large in DyAg and DyCu to be reversed by the tetragonal quadrupolar energy; the three-fold axes remain the easy ones and the (negative) trigonal quadrupolar interactions stabilize the muti-axial spin arrangement (section 3.6.1). In DyZn and DyCd, tetragonal quadrupolar interactions have the same order
8
-J - ~ m ] ~ ~ ~ _ _ .1 t~o~. . . . . . . . . . . -o-
7 "
i
~
" o
.
o
n
o
o[~OO1~
-
- -
2
GI=ImK ,G2=-22inK
N
ErNi 2
Z
. . . . . GI=O ,G2=0 T=1.5 K
:E'
o
2'o
go
go
8'0
'60
1~,o
INTERNAL MAGNETIC FIELD (kOe)
Fig. 51. ErNi2: magnetization variation at 1.5 K as a function of internal field. ©, • and [] are experimental values. Full lines are the variations calculated with 0* = 7 K, W = - 1.0 K, x = - 0.35, G1 (=G ~) = 1 mK and G 2 ( = G ~) = - 2 2 m K . Dashed lines are the variations calculated without quadrupolar terms (from Gignoux and Givord 1983).
72
E MORIN and D. SCHMITr
of magnitude as in DyCu and DyAg, but are more efficient due to the groundstate wave functions (see fig. 50b). They are then able to make the four-fold axis the easy direction above a G r critical value depending on the exact level scheme. This is realized for D y Z n as well as for DyCd (A16onard and Morin 1985). These two compounds represent the border-line case of quadrupolar effects on the temperature T R for the change of the axis of easy magnetization in the ordered phase (section 3.5.2); also in the paramagnetic state the quadrupolar effects are noticed through anisotropic magnetization processes. Similar features may be found also in other series specially when the C E F anisotropy is weak. This is, e.g., the case in ferromagnetic ErNi 2 which exhibits a [001] easy magnetization axis (fig. 51). The magnetization processes along the three main cubic directions are characterized by a crossing between the [001] magnetization curve and the [101] and [111] ones. Their description, as in D y Z n and DyCd, is impossible without quadrupolar interactions stabilizing the [001] axis as the easy one, instead of the [111] axis. 3.6. Effects on the magnetic structures 3. 6.1. Multi-axial structures In addition to their effects on the magnetic transition, the quadrupolar interactions also contribute to the determination of the magnetic structure through the minimization of the free energy. It is well-known that in cubic symmetry the spin structures that are described by one or more propagation vectors, q, which belong to the same star have the same bilinear energy and lead to identical powder neutron diffraction spectra (Wintemberger and Chamard-Bois 1972). The presence of several propagation vectors determine a multi-axial arrangement for spins (see, e.g., fig. 52). The knowledge of the C E F allows one to partly remove this ambiguity.
(17 FI O )
(a)
(b)
@=90 °
(c)
(d) Fig. 52. Possible configurationsfor the ( ½, 1,0)-type antiferromagneticstructure with 0 = 90°; (a) and (d) collinear; (b), (c) and (e) bi-axis; (f) multi-axis. All these structures lead to the same powder neutron diffraction spectrum as that observed in PrAg (from Morin and Schmitt 1982b).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
73
The magneto-elasticity can determine the actual structure. For instance with a four-fold easy axis, a large ( c / a - 1) magnetostriction may be a clear signature for a collinear arrangement as observed in many CsCl-type compounds (Morin et al. 1978a, Schmitt et al. 1978). On the contrary, the existence of collinear structures with three-fold easy axes has not been demonstrated for CsCl-type rare earth intermetallics. However, there is plentiful evidence for this in the NaCl-type series, in particular in Tb compounds (L6vy 1969, Hulliger and Stucki 1978b). In the presence of negative quadrupolar interactions, multi-axial spin structures are stabilized. This occurs within the trigonal symmetry in CsCl-type compounds as DyAg, DyCu (A16onard et al. 1984a) and NdZn (Morin and de Combarieu 1975) and in the AuCu3-type compound TmGa 3. In the latter compound for instance, powder neutron diffraction spectra have been indexed by using a [ ½ ½0] propagation vector and a [001] spin direction and the hypothesis of a collinear spin arrangement (Morin et al. 1987a). In spite of the sizeable magneto-elastic coupling found in the non-ordered phase (Morin et al. 1987c), no spontaneous strain was observed, making this collinear structure unlikely. The actual spin structure was determined by neutron diffraction under a magnetic field applied along the three main axes of a single crystal (Morin et al. 1987d). No change of magnetic intensities was detected when cooling the sample in different applied fields. This indicates the sample is spontaneously single domain; consequently, the actual structure is a triple-q structure which preserves the cubic symmetry (see fig. 53). This solution was confirmed by determining the different structures revealed by magnetization processes. In particular, along the Jill] direction, the quadrupolar structure and energy are unchanged at the lower critical field, He1, only the spins being reversed (fig. 54), the Zeeman energy corresponds mainly to this change of bilinear energy. At the high critical field, He2, the Zeeman energy has
•
• HC2
-,-10
I
I
2
3 4 5 TEMPERATURE (K)
I
TI
I
I
6
Fig. 53. Magnetic phase diagram for a magnetic field along a three-fold axis in TmGa 3 (from Morin et al. 1987d).
74
P. MORIN and D. SCHMITT
oooo0O0,°~° o°°° O
T=
1,5°4, " o •
o: .%
•
• •
A~A~
•
°"
°~'°t
0"£
•
"5K
Ao*
"d •
oOO oO4°,°° o • o Jo o ~Ae
rn ,~1.¸ v
oJ
•
.%" 2.o ,e •
o
8° o ~ oo
,0
2'o
3'0
20
H (kOe)
10
6'0
70
Fig. 54. Magnetization processes for a magnetic field along [111] in TmG% (from Morin et al. 1987d).
to compensate both the bilinear and quadrupolar energies as the AFQ arrangement is broken at this field. The existence of multi-axial spin structures in rare earth antiferromagnets with high symmetry appears to be a common consequence of AFQ interactions. In series other than the CsC1- and AuCu3-type ones, multi-axial structures also exist, e.g., as in rare earth hexaborides. In the AFQ phase of CeB6, the local anisotropy drives the spins to order in a multi-axial arrangement described by two propagation vectors [¼ 1 ½] and [1 _ ] ½] whereas the AFQ structure itself is described by [½ ½ ½] (fig. 12). The magnetic moments of the two cerium AFQ sublattices order along two mutually perpendicular directions (Effantin et al. 1985). The low-temperature magnetic structure of PrB 6 was initially described by a single [I ~ ½] propagation vector from powder neutron diffraction spectra (McCarthy et al. 1980). Recent experiments on a monocrystalline sample have shown that the structure is in fact double q, described, as in CeB6, by [¼ ¼ ½] and [ ~ - ¼ ½] and thus probably results from the effects of AFQ interactions (Burlet et al. 1988). Another complex situation is observed in the NaCl-type compound, HoP (Fischer et al. 1985). At Tc = 5.4 K it undergoes a second-order transition to a ferromagnetic state. At TF = 4.8 K there is a first-order transition to a flopside state with two Ho sublattices consisting of alternate ferromagnetic (111) planes. From one plane to the next one, the 4f moments change from one four-fold axis to another. Due to the antiferro- and ferromagnetic components, this lowtemperature structure is biaxial (fig. 55). The spontaneous and field-dependent properties were analyzed using the cubic model with two sublattices (Kim and Levy 1982). This is justified by the presence of six low-lying CEF levels nearly degenerate and well-separated from the excited ones. Bilinear and quadrupolar pair interactions were limited to nearest and next-nearest neighbours. The
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
75
lO-3Jn
10-
5-
10
5.OK I
I
I
I
~1 i
i
5
i
o
71 ""-------
3'0
eo
'
9'0
i
1~o
~'ec*~
Fig. 55. Temperature dependence of neutron diffraction pattern of a HoP powder sample. At 4.2 K the indexing corresponds to the magnetic unit cell with lattice constant a m = 2a. Insets indicate corresponding magnetic structures (from Fischer et al. 1985).
corresponding parameters were deduced from the magnetic excitations measured in the flopside phase by inelastic neutron scattering. Bilinear interactions are positive, in accordance with the ferromagnetic tendency, tetragonal quadrupolar ones are negative in accordance with the change of the z-axis from one plane to another. Thus the negative quadrupolar interactions stabilize the flopside structure. 3.6.2. Incommensurate magnetic structures The nature of bilinear interactions in rare earth intermetallic systems very often leads to modulated magnetic structures (Koehler 1972, Rossat-Mignod 1979). This corresponds to a maximum value of the Fourier transform of the bilinear interactions for an incommensurate wave vector. These structures are either
76
P. MORIN and D. SCHMITT
phase modulated or, when the magnetic moment is confined along a given direction by the anisotropy, amplitude modulated. When considering only bilinear interactions, other couplings being equal, the modulated structures can change when decreasing the temperature below the N6el transition in three ways: (a) For a non-magnetic ground-state (non-Kramers ions, Kondo systems) the modulated structure can remain stable down to 0 K as observed in HoNi0.sCu0. 5 (Gignoux et al. 1977) and in CeA12 (Barbara et al. 1980). (b) For a magnetic ground state (Kramers ions) entropy effects drive the structure to move progressively towards an antiphase structure with constant magnetic moments as in metallic Tm (Koehler et al. 1962). (c) A first-order transition to a commensurate structure (constant magnetic moments) occurs at a relatively high reduced temperature T]TN; e.g., T I T N = 0.87 in TmCu (Morin and Schmitt 1980), 0.82 in DyAg (Kaneko et al. 1987). In the presence of quadrupolar interactions, some modifications are expected. In particular for the first case (a), the nonmagnetic ground state can be quadrupolar (F3 doublet) and the same entropy effects exist for quadrupoles as for spins: consequently, the magnetic structure also has to change as in cases (b) or (c). For case (b), the tendency to a commensurate quadrupolar structure facilitates the squaring-up of a modulated spin arrangement. Unfortunately there is no clear evidence of these two situations up to now. In case (c) the temperature of the first-order commensurate transition Tt appears to be strongly dependent on the large quadrupolar interactions. In TmCu, Tt = 6.7K is close to the temperature To_= 5 K, at which it would ferroquadrupolarly order in the absence of bilinear interactions. Below Tt, both the magnetic and quadrupolar structures are collinear with tetragonal symmetry (Morin and Schmitt 1980). In DyAg, a modulated structure is also observed close to TN (Kaneko et al. 1987). However, the commensurate triple-q structure with spins along the three-fold axes is rapidly stabilized by the trigonal AFQ interactions through a first-order transition at Tt = 46.5 K. The influence of quadrupolar interactions on the commensurate-incommensurate transition appears to be a rather general feature and can be observed in other series such as RB 6 (Effantin 1985, McCarthy et al. 1980) and in RGa 3 (Morin et al. 1987a,d). From a purely magneto-elastic point of view, it is worth noting that a static strain wave is associated with the incommensurate spin wave. Its modulation is one half that of the spins because of the time-reversibility of the quadrupoles. It manifests itself through lattice satellite reflections at wave vectors ~'m~= 2~'n~, where r m is the magnetic propagation vector. They have been observed in metallic Cr (Tsunoda et al. 1974, Pynn et al. 1976), and in rare earth metals such as Ho (Keating 1969, Bohr et al. 1986) and Er (Gibbs et al. 1986). In the incommensurate helicoidal phase of these metals, 2%-satellites have been observed with X-ray diffraction and their intensity was related to the aspherical 4f charge density changing its orientation from plane to plane (fig. 56). In addition, % varies with temperature, the spin structure locks-in at commensurate values of % because it gains some magneto-elastic energy. These lock-ins are also observed in Dy
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS I
I
I
j/~
I
77
I
ERBIUM(OOQ) 2 Tm
47.5K ,;'/ .~".~..
20O o I00 C
49 K.,..-~
~[l~-
2~ ~o× p__. z o
_
,95K..~
505K
---
.... J
51.6K~ 52K .--
k__
k____
i
4/71 200 55K I00 I I I I ~] 0 2.55 2.56 2.57 2.58 2.59 (unitsof c*) m
Fig. 56. Temperature variation of the nuclear scattering at 2Tm in Er near the % = ~ lock-in transition (% is the magnetic propagation vector) (from Gibbs et al. 1986). ( G r e e n o u g h et al. 1981), they appear to be m o r e frequent in phase m o d u l a t e d structfires (existence of a quasi-isotropic plane) than in amplitude m o d u l a t e d structures (easy axis).
3.7. Effects on magnetic excitations T h e effects of quadrupolar interactions o n the e l e m e n t a r y excitations of m a g n e t i c systems w e r e first studied, by using effective spin m o d e l s , at the s a m e time as the
78
P. MORIN and D. SCHMITT
quadrupolar Hamiltonian itself. Cases with S = 1 (Chiu-Tsao et al. 1975, Sivardi~re 1975), S = 3 (Chiu-Tsao and Levy 1976), and S => 3 systems (Sivardibre 1976) were successively investigated. Two types of effects were considered in these calculations, namely: (i) the modification of the usual (spin waves) magnetic excitations by the quadrupolar couplings, and (ii) the occurrence of pure quadrupolar excitations corresponding to two-spin (AM = 2) deviations. The first effect consists simply of a variation of the size of the spin waves dispersion, at least in the ferromagnetic state. In the quadrupolar phase, however, an additional change of shape occurs, the dispersion curves becoming linear in q when q--~ 0. More exciting is the possibility of two-spin (AM = 2) excitations (quadrons) with the same type of dispersion as the AM = 1 magnetic excitations, but for different energies. Such quadrupolar excitations can be measured by inelastic neutron scattering experiments (Chiu-Tsao and Levy 1976, Levy and Trammell 1977). However, to our knowledge, they have never been observed experimentally. In more realistic calculations, the influence of the CEF must be taken into account by considering the whole level scheme, i.e., the actual CEF wave functions. Elementary excitations have then been obtained through a Green function (Sablik and Wang 1978, 1979) or a generalized susceptibility formalism (Morin et al. 1980c) with similar results. Attention was mainly focused on the influence of quadrupolar couplings on the magnetic (spin wave) excitations, which led to the same general conclusions as the effective spin models. In the magnetic phase, quadrupolar interactions modify the average energy of the dispersion curves, through the single-ion Hamiltonian, as well as their shape, through the Fourier transform of the two-ion quadrupolar couplings. The excitations are then dipolar and quadrupolar mixed excitations. The first effect has been widely used, as described in the literature, for adjusting the energy of the excited CEF levels with the average value of the dispersion curves (Furrer et al. 1977, Pierre et al. 1984). The second effect is difficult to observe; it has been invoked in NdSb in order to explain the shape of the experimentally observed excitation spectra (Sablik and Wang 1985). Surprisingly in the quadrupolar phase, the quadrupolar coupling which induces the static quadrupolar ordering does not produce any specific dispersion in addition to tha(arising from the bilinear coupling; it only modifies the overall amplitude of the dispersion curves. This has been clearly demonstrated in TmxLul_xZn, in which a 10% dilution of Tm by Lu suppresses the ferromagnetic ordering existing in TmZn and provides a pure quadrupolar phase below TQ = 5.8 K (Morin et al. 1981). In that phase, the total amplitude of the longitudinal excitation reaches 16 K, while the corresponding value is only 10K in the ferromagnetic phase of TmZn. The only expected effect in the quadrupolar phase is the modification of the transverse excitations by the second (trigonal) quadrupolar interaction. Unfortunately the latter is too weak in Tm0.9Lu0.1Zn to give rise to a noticeable effect.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
79
3.8. Two-ion anisotropic magneto-elasticity The magneto-elastic modulation of the pair interactions has been neglected in all the preceding sections. This may be justified by analyzing Gd compounds in the absence of any single-ion terms. Usually two-ion magneto-elasticity is only associated with the bilinear interactions. Its modification occurs through either the isotropic exchange striction (see section 3.9) or symmetry lowering modes which introduce anisotropic interactions. The corresponding equilibrium strains have been given in section 2 [eqs. (19)-(24)]. In the CsCl-type compound GdZn, a spontaneous tetragonal strain has been observed below T c = 270 K (Morin et al. 1977, Rouchy et al. 1981). Its zeroKelvin value is e~ = -4.5 x 10 -4, a value one order of magnitude smaller than observed in other RZn (see section 4). The two-ion strain is expected to vary roughly as the De Gennes factor across a series and to take its maximum value for the Gd compound. Thus its contribution to the tetragonal strain rapidly decreases towards each end of the rare earth sequence; e.g., it is only half a percent of the observed value in TmZn. The e-strain has been estimated to be two orders of magnitude smaller than the y-strain. In the Laves phase compound GdAI2, the opposite situation is encountered: e~ is negligible and the e-strain reaches e ~= 1.5 x 10 -5 at 0 K (Du Tr6molet de Lacheisserie 1988). As for RZn, the spontaneous strain is one order of magnitude smaller in GdZn than in other compounds (section 4). The ratio reaches 40 between~strains in TbFe 2 and GdFe 2 polycrystalline samples (Abbundi and Clark 1978). In hexagonal RCo 5 also anisotropic strains are noticeable in the presence of a rare earth orbital moment, but weak in YCo s (Andreev et al. 1985a). In NaCl-type Gd compounds, a spontaneous trigonal strain is observed (Hulliger and Stucki 1978a). The e ~ values observed at low temperatures are 8 x 10 -4, 12 x 10 -4 and 17 x 10 -4 in GdAs, GdSb and GdBi, respectively. For the same trigonal symmetry they are -37, -28 and -33 x 10 -4 in TbAs, TbSb and TbBi (Hulliger and Stucki 1978b). The ratio is then not as large as in the preceding examples, in particular for antimonides and bismuthides. This noticeable two-ion anisotropic magneto-elasticity seems to be a general feature of NaCl-type compounds as observed experimentally and expected theoretically (section 3.2.4). Although the two-ion symmetry lowering modes are usually negligible in comparison with the one-ion modes, the latter example shows that it is necessary to study the Gd compound in a given series systematically.
3.9. lsotropic magneto-elasticity and pressure effects The e ~ volume strain may be obtained from a combination of parallel and perpendicular changes-of length Ix/3 e ~ = AlL+ 2h. from eq. (25)]. Equation (19) shows that one- and two-ion magneto-elastic terms coexist in a given compound. Only the two-ion term is present in Gd compounds. As they are expected to
80
P. MORIN and D. SCHMI'IT
decrease from the Gd value according to the De Gennes law, this allows to evaluate the one-ion contribution in other compounds. 3.9.1. Two-ion contribution In addition to the change of length method, the volume strain may be deduced from measurements of the dependence of the magnetic transition on hydrostatic pressure. For vanishing CEF effects, a MFA treatment proposes an equivalent expression for e ~ as a function of the reduced magnetization o- (Bean and Rodbell 1962): e~-
3 J 2 J+l
ATe 2 NkB -Aff ~r '
where o" is the normalized magnetization. Both methods usually give compatible determinations in Gd compounds. In GdAI2, e.g., the isotropic strain at low temperatures is found to reach - 1 . 7 5 x 10 -3 from change of length measurements (Du Tr6molet de Lacheisserie 1988) and -1.85 × 10 -3 from pressure data (Jaakkola and Hanninen 1980). In GdZn, at T = 0 K a value of +5 x 10 -4 is obtained (Morin et al. 1977, Hiraoka 1974). In GdAg, at T = 0 K a large value, e ~ = -3.8 × 10 -3 may be deduced from ATN/AP = 1.17 K/kbar (Yoshida et al. 1987). This is characteristic of a strong dependence of the magnetic properties on the actual electronic structure, which is close to instability (see section 3.10). 3.9.2. Single-ion contribution For rare earths other than Gd, the isotropic strain also has a single-ion term describing the bulk modulation of the CEF. As the 4th and 6th order CEF terms are of comparable magnitude in rare earth (R) intermetallics, the two magnetoelastic coefficients B(4 ) and B(6 ) have to be considered [eq. (19)]. By comparing results on GdZn with those on TbZn and DyZn, e ~ has been found to be negative in the latter two ( - 3 × 10 -4 and -13 × 10 -4, respectively). This indicates that the CEF part is opposite to that of the exchange and at least twice as large in magnitude (Morin et al. 1977). With regard to the single-ion contribution, thermal expansion appears as the best experimental probe. It may be discussed in the paramagnetic phase within a susceptibility formalism (Morin and Williamson 1984) as well as in terms of Gr/ineisen parameters (Ott and Lfithi 1977). Thus, thanks to thermal expansion results, the pressure dependence of the magnetic susceptibility has been explained in TmSb by a modification of the single-ion susceptibility rather than by a change of the bilinear interactions (Ott and L/ithi 1977). In another singlet ground state system, PrSb, inelastic neutron scattering under pressure (Vettier et al. 1977) and thermal expansion results (Ott 1977) have consistently shown the CEF spacing to decrease when the volume decreases.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
81
3.9.3. Pressure effects Except for the previous examples of the magnetic susceptibility or the ordering temperature, the literature is rather scarce about pressure effects in rare earth intermetallics. Indeed the energies associated with pressure remain weak in comparison with the intrinsic couplings. Hydrostatic or uniaxial stress seems to be able to have dramatic effects only in the case of magnetic systems which are subcritical. For instance, hydrostatic pressure has been shown to drive an antiferromagnetic ordering in PrSb through a softening of the X point magnetic exciton, the zero-pressure value of which is determined by the intrinsic anisotropic bilinear interactions (McWhan et al. 1979). More relevant to the present review is the case of another single ground state system, Pr (Jensen 1979). The bilinear interactions between rare earths at hexagonal sites are just below the critical value for an antiferromagnetic ordering. This makes the spin system very sensitive to various influences, in particular to magneto-elastic effects with e-symmetry. They have been show to influence deeply the magnetic excitons ( H o u m a n n et al. 1979). As the B ~ magneto-elastic coefficient was deduced from ultrasonic measurements (Palmer and Jensen 1978), it was possible to calculate the e" critical value that is needed to modify the level spacing enough so as to drive the spin system to be critical; an uniaxial stress of 770 bar applied along the a-axis of the hexagonal cell was found to be necessary. Experimentally antiferromagnetic ordering was observed below 7.5 K just under about 800 bar (see fig. 57), thereby verifying the above prediction.
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Fig. 57. Dispersion relations for magnetic excitons propagating on the hexagonal sites in Pr, at 5.2 K, at room pressure (circles) and under an uniaxial pressure of 800 bar (squares). TO and LO branches correspond to transverse and longitudinal optical modes; the TA and LA transverse and longitudinal acoustic modes were not examined under stress. An asterisk (,) denotes the magnetic satellite position in the ordered phase (from McEwen et al. 1978).
82
P. MORIN and D. SCHMITT
3.10. Magneto-elasticity in the presence of lattice instability In all of the preceding sections, magneto-elasticity was acting on an otherwise stable lattice: As for the lattice parameter, the background elastic constants were assumed to vary weakly across a series as well as to be weakly dependent on the temperature. This situation is the norm, but there are some exceptions. For instance, the CsCl-type structure appears less stable than other crystallographic arrangements such as the MgCu 2- or NaCl-type ones. Indeed some series are incomplete (Iandelli and Palenzona 1979) with compounds exhibiting martensitic transformations (Gefen and Rosen 1981, Al6onard and Morin 1984, Sekizawa et al. 1981). For example, RAg compounds are cubic at all temperatures, and an incipient lattice instability is easily evidenced by hydrostatic pressures at temperatures which drastically depend on the pressure (Kurisu 1987). Chemical pressure obtained by substituting Ag by In produces analogous effects (Ihrig et al. 1975, Fujiwara et al. 1983). The martensitic transformation has been shown to occur mainly through a phonon softening at the M point of the Brillouin zone in LaAgxIn~_ x (Knorr et al. 1980, Abu-Aljarayesh et al. 1986b) rather than through a band Jahn-Teller coupling developing at the F point. Through an anharmonic coupling with the F-point acoustic phonon, this M-point phonon softening drives a partial softening of the lattice elastic constant. A full understanding of this lattice instability requires, in the presence of a strong electron-phonon coupling, dynamical models considering the full dielectric response function of the electronic system. The starting point must be the determination of the actual band structure as has been done for transition metals (Dacorogna et al. 1982). The coexistence of magneto-elasticity and lattice instability makes it difficult to determine the predominant coupling from the temperature dependence of the elastic constants. For instance, in light rare earth-silver compounds, the tempera-
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Q U A D R U P O E A R EFFECTS IN RARE EARTH INTERMETALLICS
83
~.5
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ture variation of Cll - C12 in the LaAg compounds is obviously anomalous and cannot be used directly as background in a magnetic compound (fig. 58). Across the series, the instability tendency progressively vanishes when proceeding to heavier rare earth compounds (fig. 59). As discussed from a theoretical point of view for the orthochromites (Kataoka and Kanamori 1972), the anharmonicity involved with the lattice instability manifests itself in the occurrence of the quadrupolar ordering. In CeAg, where the quadrupolar interactions have been determined by means of parastriction and third-order magnetic susceptibility, both the temperature and the discontinuity of the order parameter are increased, e.g., the quadrupolar ordering would occur around 10 K with only quadrupolar interactions, but this temperature is increased to 15.85 K due to the presence of the incipient lattice instability (Morin 1988). 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series
4.1. CsCl-type structure compounds CsCl-type rare earth intermetallics RX have been extensively studied due to the availability of their single crystals. Their very simple structure allows a deep insight in the 4f magnetism; analyses are possible according to the alloyed non-magnetic metal (X = Mg, Zn, Cd, Cu, A g , . . . ) . They have revealed large quadrupolar interactions which usually prevent any analysis which would only consider CEF and bilinear interactions of the magnetic data. In particular large spontaneous strains exist with tetragonal symmetry; they reach record values ( c / a - 1 = 1.4% in PrZn and -0.8% in TmZn). Important elastic constant softenings are observed in the paramagnetic range. FQ orderings were discovered in TmZn and TmCd (section 3). The following subsections outline the numerous effects of quadrupolar interactions on the magnetic properties in each series. Quadrupolar interactions are largest in tetragonal symmetry. They are of a ferroquadrupolar type. In trigonal symmetry, they are negative and very often stabilize multi-axial spin structures when the three-fold axis is the direction of easy magnetization. Systematic studies of the TmX have shown how very relevant the (spin and) quadrupolar Hamiltonian is within the MFA. In particular the
84
E MORIN and D. SCHMITT
absence (or the weak value) of fluctuations leads to determinations of G r and G" from third-order magnetic susceptibility measurements in close agreement with those obtained from magneto-elasticity data. However, there are particular situations where the quadrupolar coupling may exist with additional terms (as in the case of the lattice instability for CeAg).
4.1.1. RZn series Light rare earth compounds are antiferromagnetic with a (½,0, 0) propagation vector. From SmZn to TmZn, a ferromagnetic behaviour is observed (table 5). The CEF parameters A4(r4), A6(r 6) are found to be negative by inelastic neutron spectroscopy, without exhibiting an erratic variation (Giraud et al. 1986). When the four-fold axis is the easy magnetization direction, a large spontaneous tetragonal strain is observed. The various magneto-elastic coefficients were determined by different experimental techniques. Note that when the temperature range investigated by parastriction is reduced (high Tc values), only a relationship between G ~ and B ~ is obtained. Some differences between B ~ values deduced from parastriction and spontaneous strain measurements may be noticed, in particular in TbZn, DyZn and HoZn. They are partly explained by an overestimation of the two-ion contribution to the spontaneous strain. Across the series, the coefficient B ~, normalized by the Stevens coefficient exhibits an average value of about -2050 K with the ErZn value slightly remote (see section 5.1). In trigonal symmetry, reliable values of B ~ are achieved in TbZn and HoZn from the spontaneous trigonal strain. In other compounds, only the parastriction may be efficient. However, as the tetragonal symmetry lowering mode is highly favoured in CsCl-type compounds, a very accurate orientation of the magnetic field along [111] is necessary in order to avoid the presence of any tetragonal contribution to the change of length measured. Here again the parastriction leads only to a relation between values of B e and G ~. B e values obtained with both techniques are in good agreement in TbZn and HoZn. The ratio B ~/aj obviously varies across the heavy rare earth compounds; in TbZn, it even Changes sign. The magneto-elastic contribution to G ~ is less than 1 mK for the trigonal symmetry, but reaches 5.2, 9 and 120 mK for tetragonal symmetry in TmZn, PrZn and CeZn, respectively. However, in RZn, the quadrupolar interactions originate mainly from the pair coupling, as proved by the ratio K~/G~jE of 1.6 and 4 which is found in ErZn and TmZn, respectively. They are systematically positive in tetragonal symmetry, negative in trigonal symmetry. In TmZn, they dominate the bilinear interactions, and induce a quadrupolar ordering in the paramagnetic phase. Depending on their sign, they may stabilize either collinear antiferromagnetic structures as in CeZn and PrZn or multi-axial ones as in NdZn. In other compounds, none of the magnetic properties can be described without the quadrupolar interactions as shown for the four-fold axis of easy magnetization in DyZn, the magnetization processes in HoZn and ErZn, the actual temperature for the change of easy axis in NdZn and HoZn and the first-order transition in CeZn (see section 3).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
85
4.1.2. RCd series This series has been less extensively studied than the RZn one, due to the large neutron absorption cross section of cadmium, which in particular forbids a direct determination of the CEF level scheme by neutron spectroscopy. However, the properties appear to be very close to those of the RZn series from a magnetic point of view. One of its peculiarities rests in the properties of the lattice: La, Ce and Pr compounds exhibit a well-defined martensitic-type transition, which is revealed by electrical resistivity or susceptibility measurements, at 61, 107 and 125 K, respectively (A16onard and Morin 1984). At 216 K CeCd undergoes an additional transition which is as well-defined as the previous one (Fujii et al. 1985). Compounds from Nd to Tm are ferromagnetic, CeCd and PrCd being antiferromagnetic (A16onard and Morin 1985). CeCd was also found to be ferromagnetic (Fujii et al. 1985). The easy magnetization axes and the magnetic moments observed in ferromagnets are closely reminiscent of the corresponding results in RZn. For instance, all of the properties studied in TmCd, in particular the quadrupolar properties, can be described with a CEF level scheme very close to that in TmZn. Assuming the CEF parameters as constant, quadrupolar coefficients may be obtained through the full description of the magnetization properties. This is true in particular for the four-fold easy direction in DyCd as well as in DyZn, and for the temperature of change of easy direction in NdCd as well as in NdZn. On the contrary, observing [111] axes as easy direction in CeCd and PrCd clearly reveals a deep modification of all the one- and two-ion terms driven by the lattice instability. In RCd ferromagnets, quadrupolar interactions are present having large analogies with the RZn case as proved by the comparison for a given rare earth (tables 5 and 6). However, from the thorough study of TmCd, both the pair-interaction coupling and the magneto-elasticity are smaller than in TmZn in tetragonal symmetry, the ratio Kr/G~E, however, is twice as large. This is emphasized by the T o value in both Tm compounds. In trigonal symmetry, the coefficients G ~ are negative as in RZn. 4.1.3. RAg series This series exhibits some specific behaviours. Bilinear interactions favour incommensurate magnetic structures. One notes, however, a tendency to ferromagnetism at the beginning of the series through a ferromagnetic component in PrAg and a ferromagnetic moment in CeAg. The observed properties strongly depend on the preparation. In an unannealed sample, the magnetic structure is systematically commensurate with the lattice periodicity. Removing the metallurgical stresses by annealing the sample restores the intrinsic nature of the bilinear interactions as it was demonstrated in HoAg, ErAg and TmAg (Nereson 1973a,b, Morin and Schmitt 1982a). Another important peculiarity rests in the incipient tendency to lattice instability, although all the RAg are stable within the CsCl-type structure in the absence of external stress (section 3.10). Under hydrostatic or chemical pressure, the instability towards a tetragonal-symmetry based structure (Maetz et al. 1980) is
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P. MORIN and D. SCHMITF
revealed by a martensitic transition at a strongly pressure-dependent temperature (Kurisu 1987). Its driving mechanism consists of a softening of the TA phonon at the M point on the Brillouin zone (Knorr et al. 1980) and induces anharmonic contributions to the elastic constant. In the absence of an external stress, the incipient lattice instability takes the opportunity of the quadrupolar ordering in CeAg to manifest itself through both an increase of the quadrupolar ordering temperature (see section 3.1.1) and a spontaneous tetragonal strain ( c / a - 1 = 1.9%) which is larger than expected from the one-ion magneto-elasticity only. In RAg, the tendency to instability is particularly clear in LaAg, CeAg and, to a smaller extent, in PrAg (see section 3.10). The tetragonal magneto-elastic parameters, normalized by the second-order Stevens coefficient, are of the same order of magnitude as in RZn. Their average value is around -2000 K/at., the CeAg value being somewhat smaller (table 7). The trigonal symmetry coefficient observed in PrAg and TmAg leads to positive BE/aj ratios (4400 and 2200 K/at., respectively). As in RZn and RCd, the pair interactions are positive in tetragonal symmetry, negative in trigonal symmetry. For example, in TmAg, they seem to be smaller than in RZn. Their effects are, however, numerous on the magnetic properties. The commensurate-incommensurate transition at 9.5 K in ErAg may be influenced by quadrupolar interactions as discussed for DyAg (section 3.6.2). The N6el temperature in TbAg is the highest one known with a first-order character, which agrees with the G v value determined in the paramagnetic phase. In PrAg, a neutron diffraction experiment with polarization analysis on a single crystal has, at least partly, removed the ambiguities among the possible structures of fig. 52 and has shown that a double-q structure (structure b or c) is very likely in agreement with the existence of rather negative quadrupolar interactions (Givord et al. 1985). 4.1.4. RCu series CsCl-type structure compounds range from GdCu to LuCu; equiatomic alloys with light rare earth crystallize within the FeB structure (Buschow 1980). Lattice instability develops itself in boundary compounds: GdCu undergoes a martensitictype transition with a very large hysteresis (Gefen and Rosen 1981) (table 8). In TbCu the martensitic transition seems to be triggered by the antiferromagnetic ordering (Pierre and Hennion 1982). All the other heavy rare earth compounds are stable down to 0 K. Antiferromagnetic interactions are observed across the series, they are characterized by a (1, ½,0) propagation vector. However, ( ½ - r, ½,0) modulated structures have been observed in TmCu and ErCu close to the N6el temperature in a way reminiscent of RAg compounds. Magneto-elastic coefficients are available for TmCu and DyCu in tetragonal symmetry. In TbCu, a B ~ value may be achieved from the spontaneous strain by assuming a maximum value for ( O °) due to the large exchange field at 4.2K. The resulting ratios Br/aj are -2300, -2450 and -2800 K/at. in Tb-, Dy- and TmCu, respectively: the same sign and order of magnitude are observed as in other CsCl-type series. In trigonal symmetry, B ~ is found to be positive in TmCu as in other TmX.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
91
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P. MORIN and D. SCHMITT
Quadrupolar pair interactions are also present. They are positive in tetragonal symmetry although they are a little weaker than in RZn; they dominate the magneto-elastic coupling. The total quadrupolar coefficient G r would drive a quadrupolar ordering at only T o = 5 K in TmCu. Its value in ErCu, deduced from the magnetic excitation spectrum, is lower than in ErZn. The tendency to negative quadrupolar pair interactions in trigonal symmetry is also established in RCu. As discussed in section 3.6.1, they determine the multi-axial structure of DyCu, the CEF being responsible of the three-fold direction of easy magnetization as in DyAg and TmGa 3 (section 3.6). 4.1.5. R M g series
Light rare earth compounds are antiferromagnets. In heavy rare earth compounds, the spin structure is either canted, surprisingly also in GdMg (Kim and Levy 1982), or more complex, because they mix ferro- and antiferromagnetic tendencies (A16onard et al. 1976). From a quadrupolar point of view, spontaneous strains are observed in CeMg and PrMg (table 9). From the extensive study of TmMg (Giraud et al. 1986), it was found that the magneto-elastic and pair quadrupolar interactions are weak in heavy RMg compounds (section 5.1). Then an analogy appears between quadrupolar and antiferromagnetic interactions: they are strong and well-established at the beginning of the series and they rapidly decrease in heavy rare earth compounds.
TABLE 9 Quadrupolar coefficients in the RMg series. Also indicated are the N6el temperature, the antiferromagnetic propagation vector Q, and the spontaneous strain at low temperature. RMg compound
Tr~ Q Easy axis c/a - 1 (%) B" (K) G ~ (mK) B ~ (K) a " (mK)
CeMg
PrMg
TmMg [5]
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45 [3]
<1.3
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-5.4 (0, 3) (14, 21) (-100,-200)
References: [1] Pierre et al. (1984), from magnetic excitation studies. [2] Schmitt et al. (1978). [3] A16onard et al. (1976). [4] Morin et al. (1978a). [5] Giraud et al. (1986) from measurements of parastriction, elastic constants and X~ )[6] From c / a - 1 data, using (O °) ='8 (CeMg) and 28 (PrMg).
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
93
4.2. NaCl-type structure compounds Rare earth pnictides (rare earth compounds with group V-A elements) crystallize within the cubic NaCl-type structure. They have been extensively studied from a magnetic point of view (Hulliger 1978). Their semi-metallic behaviour leads to RKKY-type bilinear interactions driving rather low magnetic transition temperatures (T < 60 K). The low-temperature phase is antiferromagnetic of type-I or -II with some exceptions for Ce compounds. The small size of nitrogen ions is probably the origin of ferromagnetism in some nitrides. Associated with the semi-metallic behaviour, the strength of the magnetic interactions strongly depends on the stoichiometric conditions as may be deduced from the spread in the T s values listed in tables 10 and 11. The CEF parameters are very similar for compounds of a given rare earth ion and the dependence across the rare earth series follows, more or less fortuitously, point-charge estimates. However, a clear exception is the case of Ce compounds (Birgeneau et al. 1973). One of the consequences of the positive A4(r 4) parameter and of the weak A6(r 6) parameters, is the great number of nonmagnetic singlet ground states for Pr, Tb and Tm ions; over- (sub-) critical conditions are usually observed in Tb (Pr and Tm) compounds. The magnetoelastic coupling provides clear evidence for the actual magnetic structure, through the tetragonal or trigonal spontaneous strain. The number of multi-axial structures appears to be very small contrarily to the situation in CsCl-type compounds. Numerous compounds order at a first-order transition, in particular those with Tb, Dy and Er ions. In connection with the low-temperature range, this is explained by positive values for the third-order susceptibility driven by quadrupolar interactions in both tetragonal symmetry (Levy 1973) and trigonal symmetry (Koetzler 1984). In tetragonal symmetry, the magneto-elastic quadrupolar term has been observed to be dominant (Mullen et al. 1974, L6vy 1969). The B ~ values have signs opposite to those in the CsCl-type series and they roughly follow the aj variation with a magnitude slightly larger than in the CsCl-type structure (Liithi 1980b). This seems also to be valid for the coefficient in trigonal symmetry. As discussed in section 3.2.4, the anisotropic character of the bilinear interactions as well as the existence of an associated two-ion magneto-elastic coupling has been shown theoretically and experimentally. The occurrence of a sizeable trigonal spontaneous strain in Gd compounds, in particular in GdBi (HuUiger and Stucki 1978a), complicates the analysis of the magnetic properties, such as, e.g., the magnetic transition. Anisotropic bilinear interactions seem to have a dominant role in Ce pnictides due to hybridization effects (Cooper 1982, Takahashi et al. 1982). According to the De Gennes law, two-ion magneto-elastic effects are expected to be noticeable in the middle of the rare earth sequence, e.g., in Tb and Dy compounds. In DySb below TN, the actual direction for the magnetic moment is tilted from the c-axis of the tetragonal cell; the resulting monoclinic cell underlines the presence of the trigonal two-ion magneto-elasticity (Felcher et al. 1973). Above TN, in tetragonal symmetry, the parastriction also reveals the competition between one- and two-ion magneto-elastic terms (Al6onard et al.
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1984b). As for Ho compounds, HoBi exhibits a positive tetragonal strain (Hulliger et al. 1983) which is opposite to the negative one expected on the basis of a dominant one-ion contribution as effectively observed in HoAs; this may be also a signature of this competition.
4.3. AuCu3-type structure compounds Numerous RX 3 intermetallics crystallize with the cubic AuCu3-type structure, in particular for X = Pb, In, T1, Pd, Sn and Ga. Only a small number of studies has been devoted to magneto-elastic and quadrupolar interactions. The non-magnetic transition observed at 0.37 K in PrPb 3 has been analyzed to be driven by an AFQ ordering (see section 3.1.2). This behaviour agrees with a study of TmGa 3 which shows that AFQ interactions drive quadrupoles to order slightly above TN and stabilize the multi-axial spin structure below T N (sections 3.1.2 and 3.6.1). A quadrupolar ordering was also claimed to occur in the paramagnetic phase of Stain 3 and SmSn3; a noticeable softening of the C44elastic constant is observed at high temperatures (Kasaya et al. 1985). The tendency to negative quadrupolar interactions seems to be general in AuCu3-type compounds and responsible for the multi-axial magnetic structures observed. From a purely magneto-elastic point of view, PrPb3, TmGa 3 and CePb 3 (Nikl et al. 1987, Morin et al. 1987b) are the only AuCu3-type compounds studied. The spontaneous strain is null in TmGa3, as the magnetic structure is cubic. It is unobserved in CePb3, owing to the low N6el temperature (TN = 1.1 K) and the non-quadrupolar F7 ground state, and in PrPb3, due to the low value of TQ and the AFQ structure. The magneto-elastic coefficients and their ratios to aj are given in table 12. In both symmetries, the ratios B"/aj are positive; a good coherency is achieved for the tetragonal mode with an average magnitude which is coherent with values in preceding series. Some scattering is observed for the trigonal mode; however, values for PrPb 3 and TmGa 3 are noticeably stronger than in CsC1- and NaCl-type compounds.
4.4. Cubic Laves phase compounds 4.4.1. RAI 2 compounds The RAI 2 compounds have been thoroughly studied for many years with regard to, first, the crystal field and the magnetic properties and, second, the magnetoelastic coupling. All the available experimental results are reported in table 13. Most of them are based on magnetostriction measurements in the ferromagnetic range. That explains why some values seem out of range, as e.g., B" for PrA12 or B r for TbAI2, because they have been obtained from measurements along an axis of hard magnetization in compounds with a large anisotropy. The high value of B ~ for CeA12 has been related to a coupling with optical phonons at the F-point (Liithi and Lingner 1979). All the other experimental values are rather consistent with each other, in particular in NdA12 where the B r value has been obtained by elastic constants, X-ray and magnetostriction measurements. Recently, some
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P. MORIN and D. SCHMITF TABLE 12 Magnetic and quadrupolar ordering temperatures and magnetoelastic coefficients from elastic constant, parastriction and g~ ) experiments in AuCu3-type compounds. AuCu3-type compound CePb3 [1] TN (K) To (K) B r (K) B~/oq (K) K" (inK) G"MZ(mK) a" (inK) B" (K) B~/a~ (K) K" (mK) GME (mK) G ~ (mK)
PrPb3 [2]
TmGa3 [31
0.35 [5] -41 1950 -13 8 -5 -190 [11 9050 -200 7 -200
4.26 [6] 4.29 17 1700 (8,14) 0.5 (8,14) 71 7000 20 10 30
1.1 [4] -80 1400
-190 3350
References: [1] Morin et al. (1987b). [2] Morin et al. (1982). [3] Morin et al. (1987c). [4] Lin et al. (1985). [5] Bucher et al. (1972a). [6] Czopnik et al. (1985). quadrupolar p a r a m e t e r s G v have been obtained by the torque method, which allows to extract the third-order paramagnetic susceptibility x- e(3) in a less direct way than from magnetization m e a s u r e m e n t s (Del Moral et al. 1987). As for the sign of magneto-elastic coefficients, they appear to follow well that of the Stevens factor aj. The ratios BY/aj and BE/aj range around 1000 K and - 3 0 0 0 K, respectively, with a few exceptions (see section 5.1). F r o m values of G r and G rME, the existence of two-ion quadrupolar coupling is suggested. This is in disagreement with elastic constants fits, which did not require considering this coupling.
4.4.2. R M 2 compounds (M = Mg, Ni, Co) For the R M 2 cubic Laves phase compounds where M is not A1 (section 4.4.1) or Fe (section 4.4.3), quadrupolar data are rather scarce. The PrM 2 compounds (M = Pt, Rh, Ru, It) have been investigated by magnetization, neutron diffraction, and specific heat m e a s u r e m e n t s (Greidanus et al. 1983a,b). A systematic overestimation of the ferromagnetic m o m e n t by a calculation including C E F and bilinear exchange interactions, with regard to the observed spontaneous magnetization, was explained by the presence of negative quadrupolar interactions. The same is true in PrMg2, where a quantitative analysis of the t e m p e r a t u r e variation of the ferromagnetic m o m e n t provided the quadrupolar p a r a m e t e r G r = - 1 3 . 3 m K (see table 14). F u r t h e r m o r e , this negative quadrupolar coupling leads
.i
+I
+I
e~
=
Z c~
~
i ~
I
¢J
8
c~c~ I i
~
~°~'~
Z ~
~
"~.~o~ °°
~ ~ ~oo 8
< •
e.
o Z
I
I
I
÷I
S
~
~4~
0
+I I
4~ ~ +I
~:~o~
4
0
i ~°
+I
.o e.
-g X
X
~
99
~
100
P. M O R I N and D. SCHMITT T A B L E 14 Quadrupolar coefficients in some R M 2 compounds. R M 2 compound PrMg 2 [1]
ErNi 2 [2]
B ~ (K) B ~ (K) G ~' (inK) G ~ (mK)
TbCo 2 [3]
HoCo 2 [4]
~0 79 -13.3
1 -22
-2.7 -11.7
References: [1] Loidl et al. (1981). [2] Gignoux and Givord (1983), from magnetization measurements. [3] Gignoux et al. (1979), from X-ray measurements. [4] Castets et al. (1982), from magnetic excitation studies.
to the disappearance of the first-order transition which is expected without it (see section 3.2.2). Some magneto-elastic results are also available in the RNi 2 series, where nickel is not magnetic. The compounds order ferromagnetically at temperatures roughly three times smaller than the corresponding temperatures in the RAI 2 series. At low temperatures, the directions of easy magnetization are the same in RNi 2 and RAI 2 compounds (R = Tb, Dy, Ho); the spontaneous distortions are comparable in both series, except for HoNi 2 where the strain is noticeably larger than in HoA12 (e ~ = -1.1 x 10 -3 in TbNi2, c/a - 1 = - 2 x 10 - 3 and -1.5 x 10 -3 in DyNi 2 and HoNi2, respectively) (Markosyan 1980, 1981). The behaviour of ErNi 2 differs considerably from that of ERA12, since its direction of easy magnetization is [100] instead of [111]. This has been well accounted for by introducing the parameters G ~ and G ~ (see table 14). In the RCo 2 series, cobalt is magnetic and contributes mainly to the volume magnetostriction. A positive trigonal magneto-elastic coefficient B ~ associated with the rare earth has been determined in TbCo 2 (see table 14). This leads to a positive trigonal magnetostriction e ~= 3.1 x 10 .3 at 4.2K which is opposite in sign to that measured in TbNi 2 and TbAI 2. This feature has been related to environmental effects in this type of cubic structure (Gignoux et al. 1979). In HoCo 2, a consistent description of the magnetic excitations as well as the spin rotation temperature requires introducing negative quadrupolar terms in the Hamiltonian (table 14), indicating the probable existence of a contribution of two-ion origin.
4.4.3.
RFe 2
compounds
The magneto-elastic properties of cubic ferromagnetic Laves phase compounds RFe 2 are relevant for the classical analysis, owing to their high magnetic transition temperature (around 700 K in TbFe2). Indeed, the Fe magnetization induces an exchange field large enough to purify fully the 4f magnetic moment towards its maximum value at 0 K. Their magnetostriction has been studied, in particular at
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
101
room temperature, for industrial applications. The large set of available data has been collected in the first volume of this Handbook by Clark (1980). Here, only characteristic features and analyses are considered again. Comparing the strength of the tetragonal and trigonal symmetry-lowering modes leads to a conclusion opposite to that in CsCl-compounds. Indeed, in RF% the trigonal mode is many times larger than the tetragonal mode (fig. 60). TbFe2, the basic compound for applications, is the paradigm of the series. The temperature variation of the Aln trigonal spontaneous strain has been described using eq. (107). As pedagogically explained by Clark (1980), an important step is to determine the 4f magnetization at any temperature by means of neutron diffraction or magnetization measurements. In this latter case, the Fe magnetization (about 1.64/~) is deduced from M6ssbauer spectroscopic results. Noticeable uncertainties may be present at high temperatures; however, the saturation of the 4f moment observed at low temperatures allows one to derive reliable values of B ~ through the relation Be=
3v~ o%,,(o)
J(2J-
1)
5000
4000
-
'
'
'
'
I
.
.
.
.
~ j I ~ Tb Fez
3000
I
'
^
'
'
'
I
(£l(m))
-
~'111
(a)
2000
40"
7
0 -40 -80 -120 -160 -200
L
100
i
i
]
200
n
i
u
I
300
T (*K) Fig. 60. Temperature dependence of the spontaneous strain in TbF% (trigonal symmetry) and in DyF% (tetragonal symmetry). The An1 variation is classically described according to the Tb magnetic moment. For the DyF% data, the full line is only a guide for the eye (from Clark et al. 1977).
102
P. M O R I N and D. SCHMITI" T A B L E 15 Curie temperature, easy magnetization axis, trigonal spontaneous strain at 300 and OK, trigonal magneto-elastic coefficient B e and its ratio to the Stevens coefficient c~j in RFe 2 compounds [data are from tables 5 and 6 of Clark (1980)]. RFe z compound
T c (K) Easy axis hal 1 (300 K) × 1 0 6 )tll 1 (0 K) x 106 B" (K) B'/a: (K)
TbFe 2
DyFe 2
HoFez
ErFe 2
TmFe 2
SmFe 2
697 [111] 2460 4400 94 -9400
635 [001] 1260 3320 a 44 -7070
606 [001] 200 770 a 9 -4100
590 ' [1111 -300 -1760" -24 -9300
560 [111] -210 -3500 -75 -7500
676 -2100
Note: "estimated from the room-temperature value.
Across the series, B" is observed to vary as the second-order Stevens coefficient a: in sign and, with some erratic fluctuations as in other series, also in order of magnitude (table 15). The same technique, based on eq. (107), fails to describe the tetragonal spontaneous strain in DyFe 2 (fig. 60) and HoFe 2 (Abbundi et al. 1979). Indeed, the change of sign of hl00(T) in DyFe 2 and the general temperature variation in HoFe 2 are both unexpected within this model. Whereas the hi00 sign at 4.2 K follows the sign of aj in both compounds, its absolute value is about ten times smaller in DyFe 2 than in HoFe 2, in full disagreement with the ratio of the aj's. Cullen and Clark (1977) have developed an atomic model that shows that internal distortions drive the large external trigonal strain mode whereas the hi00 strain may be sorted out because of the high tetrahedral symmetry of the rare earth site.
4.5. Hexagonal CaCus-type structure compounds Among the series of hexagonal 3d-4f intermetallic compounds, only few RNi 5 and RCo 5 compounds have been investigated from a magneto-elastic point of view; nickel is not magnetic in the first series, but cobalt is in the second one.
4.5.1. RNi 5 compounds Whereas the RNi 5 compounds have been studied for many years with regard to their magnetic properties and CEF effects, only few results are available about their magneto-elastic properties. A first study concerned the thermal expansion of PrNi 5 (L/ithi and Ott 1980) which evidenced crystal field effects, and showed the existence of the two a-mode magneto-elastic coefficients. However, the latter were not determined because the elastic constants were not known at that time. Using2the values of C" from Barthem et al. (1988), we are able to calculate B ~1 and B from these early measurements (see table 16). These values have the same sign, but are 2-3 times smaller than more recent values determined by magnetostriction measurements. In addition, the latter provide an estimation of
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
103
TABLE 16 Quadrupolar coefficientsin RNi5 compounds. RNis compounds B'~1 (K) B ~z (K) B ~ (K) G ~ (inK) G ~ (mK)
PrNi5 [11
PrNi5 [21
TmNi5 [3]
-7.1 28.6
-22 68.6 <15 -10 -20
(-500,-100) (-300,-150) (-1500,-1000)
References: [1] L/ithi and Ott (1980), from thermal expansion measurements. [2] Barthem et al. (1988), from magnetostriction measurements. [3] Barthem et al. (1989), from magnetostriction measurements. the total quadrupolar parameters G ~ and GL By contrast, Barthem et al. (1988) deduce antiferroquadrupolar (negative) interactions between Pr ions in PrNi 5 (K ~ - - 1 5 mK, K ~ - - 2 0 i n K ) . Note that a magneto-elastic coupling was also deduced from inelastic neutron scattering in PrNi 5 (Aksenov et al. 1983), where it should contribute through the hybridization of magnetic excitons and phonons. However, no accurate determination could be obtained. A recent magnetostriction study of TmNi 5 also provided one- and two-ion magneto-elastic coefficients (Barthem et al. 1989). Particularly due to the CEF configuration of this compound, the one-ion and two-ion magneto-elasticities could be separated. More precisely, the axial strain dependence of the isotropic exchange interaction ( a 2 - m o d e ) leads to a strongly anisotropic anomaly in the thermal expansion along a and c-axes below the ferromagnetic ordering temperature T c = 4.5 K, while the magnetostriction is more sensitive to the one-ion magneto-elasticity. However, the deduced B " values are quite large with regard to those in PrNi 5 and do not scale with aj. This behaviour is still to be explained. 4.5.2. R C o 5 c o m p o u n d s
Just as the cubic RFe 2 compounds, RCo 5 compounds are typical cases for applying the classical formalism, as the Co-sublattice induces very large bilinear interactions: the Curie temperatures are around 1000 K (e.g., 940 K in YCos). It is therefore difficult to study paramagnetic properties or to determine the CEF for the rare earth. A main feature is the huge magnetocrystalline anisotropy even in the absence of a magnetic rare earth as observed in YCo 5 (Alameda et al. 1983). From a magneto-elastic point of view, large anomalies have been observed in the temperature variation of the lattice parameters (Andreev et al. 1983, 1985b). It is difficult to observe magneto-elastic properties other than the spontaneous ones except in several compounds which exhibit a change in the direction of easy magnetization, and thus a temperature range of reduced anisotropy. Indeed, as classically explained in hexagonal Co (Barnier et al. 1962) and Gd (Klimker and Rosen 1973), this p h e n o m e n o n is driven by the change of sign at a given temperature of the K 1 anisotropy constant, which phenomenologically includes
104
P. MORIN and D. SCHMITI"
magneto-elastic contributions. This occurs for compounds with a negative Stevens coefficient aj for the rare earth (PrCo 5, NdCo 5, TbCo 5, DyCo 5 and HoCos). The corresponding temperature regions range from 13 K (in TbCos) to 130 K (in HoCos); the rotation of the magnetization induces clear anomalies in both the lattice parameters (Andreev et al. 1983) (fig. 61) and the ultrasonic modes (Deryagin et al. 1985) (fig. 62). It has been shown by X-ray studies that the magneto-elastic coefficient B ~ is small. On the other hand, the relative changes Aa/a and Ac/c of the lattice
v.gs~ --
1*~t~f,
,, -4 v, A3
.!'""I
q.g#Z~
~
85 1
"
Oq. g
q.960
~ I ~ J'l'~+"~
390
350
i
l-I
qJo T K
Fig. 61. Temperature dependence of the lattice parameters a and c, and of the volume V of the unit-cell in the temperature range of the change of easy axis in TbCo 5 (from Andreev et al. 1983).
z+
r,~r~r2
"E i
t
;'¢.2 I0.8
5.2 ~ 0
r
i
200
t r ~"w
I
o00T.~
Fig. 62. Temperature dependence of the elastic constants in the temperature range of the change of easy axis in DyCo 5 (from Deryagin et al. 1985).
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
105
parameters at the average rotation temperature between the high-temperature (magnetization along the c-axis) and low-temperature (magnetization in the basal plane) phases indicate contributions of the two a-modes (see section 2.3). Temperature dependent magneto-elastic coefficients have been introduced (Deryagin et al. 1985),
V~
2(Aa 11 + 3 a
AC'~c~,
B12-
.
2 ( Aa
AC~c .
B~2=
AC'~c ~ 2 ( A a 2 (Aa X/3 2 - - + a c ! t 2 + 3 a
AC'}c ~
2 --a + c /
c /
c/
12 ,
22"
Experimentally, the values of (2Aa/a + Ac/c) have been observed to be very small. The 0 K values of B12 and B22 have been deduced assuming a temperature variation according to the Callen and Callen function is/z[sg-l(m)]. Note that some uncertainties may arise from inaccurate knowledge of the rare earth's magnetic moment, especially for the light rare earths (Andreev et al. 1983). The B~2(0 ) and B~2(0 ) are related to the B ~ and B "2 magneto-elastic coefficients of the present formalism through, BI:(O ) = 2B '~1
B~:(O) = 2V'-6 B "e
It is then possible to follow their variations through the entire series of rare earths. Table 17 collects the values for the different parameters in the series. The T A B L E 17 Magnetostriction constants at OK, h~2(0), h~2(0), associated to the relative change of lattice parameters at the rotation temperature, aa/a and Ac/c, elastic constants C~2 = 2X/2-7-gC~12, C~z = 2C ~2, B ~ and B ~2 magneto-elastic coefficients and their ratio to the Stevens coefficient a~ in RCo 5 compounds. RCo s compounds
h~2(0) x 103 [11 A~2(0) x 103 C~2 (101° Pa) C22 (1010Pa) B "1 (K) B "2 (K)
B~l/aj (K)
B"Z/a: (K)
PrCo s
NdCo s
SmCo 5
TbCo 5
DyCo 5
HoCo5
ErCo 5
TmCo 5
-1.1 2,2 1,6 a 20 a -6.6 -39 314 1860
-0.8 1.7 1.6 [2] 21.2 [2] -5.1 -25 793 3890
1.3 -2.6 1.6 = 22 a 24.6 143 597 3460
-1.8 3.5 1.7 [3] 23.3 [3] -5.5 -31.2 544 3090
-1.7 3.4 1.9 [31 25 [3] -3.7 -20.5 583 3230
-0.7 1.2 2a 27 ~ -1.2 -6.9 540 3100
0.6 -1.2 2" 27 a 1.4 7.6 551 3000
1.5 -3 2a 27" 5.5 30.8 544 3050
Note: aExtrapolated values. References: [1] Andreev et al. (1983). [2] Deryagin et al. (1984), [3] Deryagin et al. (1985).
106
R MORIN and D. SCHMITr
a f d e p e n d e n c e of B ~1 and B ~2 is very closely observed. The result indicates that the a - m o d e s depend on the rare earth and not on the cobalt. This conclusion is confirmed by the study of Y C % by X-ray diffraction. In this c o m p o u n d , the cobalt sublattice appears as contributing only through a two-ion magneto-elastic coupling, modifying the c/a ratio (Andreev et al. 1985a). The partial softening observed on the different elastic constants m a y be expected to be described using the coefficients B ~1, B ~a previously determined. Note that the softening observed on C44 during the rotation also indicates a sizeable magneto-elastic coefficient B ;. Some significant results might be provided by magnetostriction m e a s u r e m e n t s in high magnetic fields for the various symmetries in the rotation t e m p e r a t u r e range.
4.6. Dilute rare earth systems The magnetization of concentrated rare earth compounds generally includes oneand two-ion terms and m a y be strongly influenced by magnetic interactions. Dilute systems should, in principle, be easier to understand by limiting the magneto-elastic effects to one-ion contributions only. A dilution in the 1% range seems to be a satisfactory compromise between experimental sensitivity and residual two-ion effects. Most of the available studies concern rare earths diluted in noble-metal hosts, as Ag, A u and Cu.
4.6.1. Rare earths diluted in cubic noble-metal hosts The noble metals crystallize within the face centered cubic structure, and, because of the low solubility of rare earths in them, concentrations of rare earth impurities smaller than 1% were usually taken for the experiments. Silver-rare earth dilute compounds. A systematic study of heavy rare earths diluted in silver was carried out by magnetostriction m e a s u r e m e n t s on single crystals (Creuzet and Campbell 1981) (see table 18). Varying the cubic C E F TABLE 18 Magneto-elastic coefficients in the AgR series. All results were taken from Creuzet and Campbell (1982a) and from Creuzet (1982) and refer to magnetostriction measurements (rare earth concentration c = 0.5 at.%), except where indicated otherwise. AgR (R =) B r (K) B ~ (K) B(r4) (mK) B~4) (raN)
Tb
Dy
Ho
Er
Tm
2.72 3.9 [21 31.1 -6.3 -67
1.71 0.9 [21 12.1 1.9 45
0.60 3.5 [21 1.18 5.3 9
-1.24 <-0.39 [1] -7 -3.5 -29
-2.47 -3.3 [21 -16.1 -6.0 -200
References: [1] Garifullin et al. (1985) by EPR bulk, c = 0.03%. [2] Nieuwenhuys et al. (1982b) [alternate analysis of results of Creuzet and Campbell (1982a) with another set of CEF parameters].
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
107
parameters may affect the result of the magneto-elastic coefficients. However, the best agreement was achieved by using a CEF level scheme which also describes other results such as susceptibility measurements. They correspond to the same CEF parameters Aa(r 4) =--18 K and A6(r 6) = 7.6K across the series (except for Tm where A6(r 6) -= 6.5 K). These results have also been compared with estimates of the magneto-elastic coefficients obtained by EPR measurements (Pela et al. 1981). The EPR results appear to be systematically underestimated in experiments performed on strained thin films and, to a lesser extent, on strained bulk samples (Campbell and Creuzet 1985). Magnetostriction measurements were analyzed by considering second- and fourth-order one-ion magneto-elastic terms, the latter always contributing for about 10% to the total magnetostriction. The experimental values appear consistent along the series, and follow roughly the appropriate ~j or /3j Stevens coefficients. Typical values are, By oq
_
U~4)_
V~
3
V~
3
B ~
2V2 V2 ~ -270 K ,
U~4)
/3j - 16----~ V4 ~ - 4 0 K ,
3
5
O/j - W~ V 2 - -2100 K ~
5
/3j - 4V2 V4 ~ - 7 5 0 K ,
for all the dilute AgR compounds, the V~'s are defined by Campbell and Creuzet (1985). This coherency is reminiscent of the constant CEF parameters across the series. The isotropic one-ion magneto-elasticity was also investigated from these magnetostriction measurements (Creuzet 1982). The uncertainty in the coefficients is largely due to the weakness of the volume effects. Typical values for the AgR series are,
n(4)_ /3j
Weft
1
16V2 V4 - -120 K ,
n(6)
- -yj -
V3
1
64 V6~ -t- 135 K
Gold- and copper-rare earth dilute compounds. Compared to the AgR-series, rather less results are available on the A..__~uR and CuR dilute compounds. Systematic comparisons are therefore difficult. Nevertheless, the main result is the change of sign of B e and/or B y in AuR and C__uuRcompounds (see table 19), compared to AgR compounds. On the other hand, the volume magnetostriction appears to be still weaker in these compounds than in A__ggRones (Creuzet 1982). 4.6.2. Rare earths diluted in cubic pnictides A systematic study of the tetragonal magneto-elasticity was performed in the cubic NaCl-type LaSbR series by [001] magnetostriction measurements (Nieuwenhuys et al. 1982a). As in the A___ggRsystem, B y follows the sign of the Stevens factor % well (table 20). In comparison to corresponding concentrate
108
P. M O R I N and D. SCHMITT
T A B L E 19 Magneto-elastic coefficients in the A__._quRand C__._~uRseries. All the results are from Creuzet and Campbell (1982b,c) and Creuzet (1982); they were obtained by magnetostriction measurements (rare earth concentration e = 0.5 at.%), except when indicated otherwise. The C E F parameters used are A 4 ( r 4) = - 3 0 K, A 6 ( r 6) = 6 K for the A._.._uuRseries, A 4 ( r 4) = - 8 6 K, A 6 ( r 6) = 12 K for CuEr compound. Compound A..._~uTb B y (K)
1.7
B" (K)
-20
B~4~ (mK) B~4 ~ (mK)
12 -33
A..._~uEr [1]
AuYb
-0.8
-8.3 - 8 . 7 [11 89 44 [1] -17 1200
4.5
C__._~uEr[2] 0.44 -6.5
References: [I] Pela et al. (1981), from E P R measurements on films (c = 1%). [2] Garifullin et al. (1985), from E P R measurements on bulk samples (c = 0.03%).
compounds (section 4.2), these parameters appear to be quite comparable in magnitude, except in LaSbSm. The value obtained by magnetostriction in LaSbDy is consistent with that deduced from magnetization measurements. However, the latter may be considered to be more reliable than the former, because the magnetization reflects the lattice distortion in the immediate vicinity of the rare earth ions, while the magnetostriction cumulates the distortion effects far away from the rare earths. Indeed, it appears closer to the value in DySb than that obtained by magnetostriction. The most original feature in the series is the observation of a "reverse" magnetostriction in LaSbCe and LaSbSm, i.e., a change of sign of the distortion as a function of temperature, which is explained by the sign reversal of the expectation value ( O ° ) (see section 3.4.1). Magnetostriction was also observed for S-state Gd ions diluted in LaSb and LaBi (Hfifner et al. 1983). This has been explained within a phenomenological model which includes admixture effects of the first-excited 6p multiplet. However, corresponding magnetostriction values reach ~ l / l - - 8 x 10 -6 at 1.5 K in a 5 T field for 10%-dilute compounds. These values, scaled for the concentration, are only 1/10 of that observed in GdZn, where the distortion was interpreted as the manifestation of a two-ion magneto-elasticity (Morin et al. 1977). According to section 3.2.4, it could then be appropriate to consider the latter term explicitly. The magnetostriction measurements on LaSbR compounds were extended to the LaBiR series in order to check, in particular, the reverse magnetostriction in Ce and Sm based compounds (H/ifner et al. 1983). Although the CEF itself is smaller in bismuthides than in antimonides, the magneto-elastic coupling seems to be larger (see table 21).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
i ~ ,...,
< II
~
~
~9
.~ o=
o C, t"q
Z
I
~
.I
cq
O
r.)
109
110
P. MORIN and D. SCHMITT TABLE 21 Crystal field parameters W, x and magneto-elastic coefficient B y in the diluted LaBiR series (c is the rare earth atomic concentration), from H~fner et al. (1983), from magnetostriction measurements. Rare-earth
W (K) x B y (K) c (%)
Ce
Tb
Er
1.91 1 -65.3 10
-0.45 -0.96 -43.7 3
0.20 0.79 10.0 3
4.6.3. Other dilute systems Few other cubic dilute rare earth systems have been studied with regard to magneto-elasticity. Only on Tb based compounds has there been a determination of magneto-elastic coupling parameters (see table 22). Creuzet et al. (1982) determined both coefficients B y and B e by magnetostriction measurements in cubic CsCl-type Tb0.01La0.99Ag compound. The B y value is roughly eight times smaller than in the concentrated TbAg compound (see table 7), which could be related to the incipient instability of the LaAg matrix (section 3.10). The same is valid for the value of G ~. On the other hand, the B e value seems to be doubtful, since it would lead to an unreasonable value of G~E, i.e., 1 K. Some studies have also been carried out on dilute systems having a lower symmetry, namely the hexagonal YR and MgR compounds. The three secondorder magneto-elastic coefficients B or1, Ba2 and B ~ were determined in YTb, __YDy and __YErcompounds (see table 23). A change of sign of all coefficients occurs for TABLE 22 Crystal field parameters W, x and magneto-elastic coefficients in some diluted rare-earth systems. Compound LaAgTb
w (K) x B ~ (K) B E (K) G y (mK) G" (mK)
0.5 -1 3.3 [1] - - 3 2 0 [1] - 1 [2]
LaA12Tb
0.27 0.9
0 [31 - 3 0 [31
References: [1] Creuzet et al. (1982), from magnetostriction measurements (c = 1%). [2] Hoenig et al. (1980), from magnetization (SQUID) measurements (c = 2%). [3] Hoenig (1980), from magnetization (SQUID) measurements (c = 1 to 6%).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
111
TABLE 23 Magneto-elastic coefficientsin hexagonal YR system "(fromPureur et al. 1985), using magnetostriction measurements (c = 1 to 2%). Compound B "' (K)
B~2 (K) Be (K)
YTb
YDy
YEr
6.1 18.9 14.8
9.4 8.5 20.4
-1.9
-4.4 -8.4
the compound YEr, due to the Stevens factor aj. In addition, the ratio B ~ / a j is roughly constant throughout the series, B ' ~ / a j -~ V'3 B'~2/a, ~ ~
V2(/'a,1)/Olj
~
--850 K ,
V2(F~,2)/a s ~ -1600 K ,
B~/aj ==-~X/-2 Vz(F~,) / a j ~ -3200 K .
These values lead to distortions which are quite consistent, after correction for the concentration, with those observed in pure rare earth metals (Pureur et al. 1985). In a few cases, the introduction of a fourth-order magneto-elastic coefficient slightly improved the fits. In spite of the low concentration, spin-glass effects have been observed at low temperatures, in particular with Tb. Lastly, magnetostriction for YGd compound (c = 1%) reaches ~c/c ~ 10 -6 for a magnetic field of 6 T along the c axis, which is again of the same order of magnitude as in LaSbGd and LaBiGd (see above). Magnetostriction and thermal expansion data are also available for the heavy rare earths dilute in a single-crystal Mg host (Bijvoet et al. 1980, De Jong et al. 1980, 1982) (concentration about 0.1%). Here, also the magnetostriction agrees with the sign of aj. Unfortunately, no complete analysis was performed in order to determine the actual magneto-elastic coefficients. 4.7. Miscellaneous c o m p o u n d s
The universal character of the magneto-elasticity is emphasized in the literature through accurately conceived data on individual compounds, however, without thorough and systematic analyses across a given series. One can find, e.g., magneto-elastic effects in RCo 3 (Pourarian and Tajabor 1980) for which magnetostriction data on single crystals are available. In other series, like rare earth metals and rare earth borides few experimental results have been quantitatively analyzed. Rare earth metals. Among the rare earth metals, dhcp Pr has been widely studied over the last decade, because of the magneto-elastic effects on the criticality of antiferromagnetic interactions; the strain mode involved in these effects is the e mode (section 3.9). A magneto-elastic coefficient B ~ = 72 K acting
112
P. MORIN and D. SCHMITr
on the hexagonal Pr sites, i.e., G ~ = 76 mK, leads to a consistent description of various experiments, such as elastic constant measurements, magnetostriction, magnetic excitations and the occurrence of an antiferromagnetic ordering under an uniaxial stress (Houmann et al. 1979). This e-mode magneto-elastic coupling was also suggested to explain the field dependence of the thermal expansion (Liithi and Ott 1980). Without any magnetic field, these thermal expansion experiments have been described with the a-mode magneto-elastic coefficients of the hexagonal symmetry B al (-----~V/-3Bv) = - 4 6 K and B a2 (--=V~ B3) -~ 35.5 K. On the other hand, a detailed study of the magneto-elasticity was carried out by magnetostriction measurements (Hendy et al. 1979). A complete analysis was made in terms of one-ion and two-ion magneto-elastic coupling. It was found that one-ion a2-strain coupling is the dominant mechanism between the lattice and the 4f ion. Unfortunately no value for B ~ or D ~ was extracted from the experiments; only the h ~ coefficients relative to both cubic and hexagonal sites of Pr have been obtained. When taking the values of elastic constants from Greiner et al. (1973), the following evaluations can be made for the Pr ions on hexagonal sites: B "2 = 2 2 . 6 K , B ~ = 4 2 K , nal = o a~1 l = D~I a2 ~ 0. These values have the same order of magnitude as those from Houmann et al. (1979) in which the contribution arising from the cubic sites was included. Rare earth borides. Among the six families of binary rare earth borides (Etourneau 1985), RB 4 and R B 6 have been the ones most studied. Their crystallographic and magnetic properties have been shown to depend drastically on the nature and rate of impurities and on the metallurgical conditions. R B 6 compounds. RB 6 compounds all crystallize within a CsCl-type cubic arrangement of B octahedra and R ions. The main part of the literature on this series is devoted to light R compounds. Magneto-elastic and quadrupolar properties have been studied mainly in CeB 6 and negative quadrupolar interactions appear to be the origin of the properties observed (section 3.1.2). The tetragonal magneto-elastic coefficient [gr3[ = 50 K leads to [B~/,~j[ -- 875 K and the trigonal one = 135K to IB'/a,I = 3300K (L/ithi et al. 1984). PrB 6 exhibits properties reminiscent of those of CeB 6 in particular regarding its magnetic structure (section 3.6). The first-order transition (section 3.2.2) may result from bilinear and quadrupolar couplings of equivalent strength, as the bilinear exchange is larger than in CeB 6. Sizeable effects on the elastic constants are observed in PrB 6 and in NdB 6 (Tamaki et al. 1985). Unfortunately no quantitative analysis is available at the present time. In addition, the first-order Nrel transition in GdB 6 sets a puzzling problem which cannot be analyzed in terms of pure quadrupolar interactions. RB 4 compounds. The rare earth tetraborides have tetragonal crystal symmetry (Will et al. 1981). The rare earth atoms are confined in boron cages with an mm local symmetry (4 g). The compounds are antiferromagnetic with the exception of PrB 4 (Buschow and Creighton 1972). They exhibit some magneto-elastic features, especially ErB 4 which undergoes a tetragonal-orthorhombic transition at the Nrel temperature. At low temperature, the lattice symmetry (Pbam) agrees with the
[grsI
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
113
magnetic one (Pb'am). However, the most interesting compound might be TbB 4 for which the structural transition occurs in the paramagnetic range at 80 K, far above the N6el temperature (T N = 43 K). Here again, lattice (Pbam) and magnetic (Pbam') symmetries are closely related. In both compounds the structural symmetry lowerings have been observed by X-ray diffraction on polycrystalline samples and are rather weak (Will et al. 1986). To the author's knowledge there are, at present, neither experimental determinations for the CEF nor for the magneto-elastic coupling. It is worth noting that the mm rare earth symmetry induces non-zero (O~) and (022) quadrupolar values. In addition, the easy magnetization directions do not vary across the series in the same way as the second-order Stevens coefficient, revealing that the anisotropy is not only of single-ion origin (Etourneau 1985). 5. Discussion
5.1. General analysis 5.1.1. Coherency of the various determinations To prove the relevancy of the basic Hamiltonian, coherent determinations of the quadrupolar coefficients in independent experimental ways are needed. For example, table 24 collects the various values obtained for the quadrupolar parameters in tetragonal symmetry for TmX compounds within the CsCl-type structure. A very good agreement is observed. In particular, the parastriction removes the ambiguity about the sign of the magneto-elastic coefficient which remains after the elastic constant analysis. The consistency between determinations obtained by direct (elastic constants) and indirect (parastriction and third-order susceptibility) ways, i.e., without and with the presence of an applied magnetic field, demonstrates the relevancy of the quadrupolar Hamiltonian and the weakness of the two-ion magneto-elasticity in this family of TmX compounds. However, we have to keep in mind that the latter contribution may be more significant for large Land6 factors (gj). Higher order one-ion magneto-elasticity as well as anharmonic elastic terms appear to be negligible here. From table 24, the one-ion magneto-elasticity of compounds with 4d elements (Ag, Cd) is observed to be weaker than that of compounds with 3d elements (Cu, Zn). The eccentric case of TmMg is consistent with the absence of d electrons on the Mg-ions (see section 5.2). The same behaviour is observed for the pairinteraction coefficient, which is at its maximum for II-B elements, in particular for zinc. In most cases, the pair interaction contribution to the total coefficient G ~ dominates the magneto-elastic one. TmMg is the only TmX compound within the CsCl-type structure not to be strongly influenced by tetragonal quadrupolar interactions. The microscopic origin of these different coefficients is analyzed in section 5.2. In the presence of cubic level schemes close to each other, this series provides us with clear experimental evidence of quadrupolar effects. Large softenings of
P. MORIN and D. SCHMITT
114
TABLE 24 Quadrupolar coefficients in TmX compounds within the CsCl-type structure. Compound
Coefficient
TmZn
TQ [1] Cll - C12 [2] Parastriction [2] X~ ~ [31
-+29 -31
20
r e [41 c n - c12 [4] Parastriction [5] x~ ~ [6]
-+12.7 -14
11.2
TmCd
B ~ (K)
K s (mK)
G ~ (mK)
Kr/~ ~ ~ M E
25 25 27 25 12.5 12.5 12 13
TmCu
Cn - C12 [7] Parastriction [7] X~ ) [31
-+30 -27
7
11.5 11 10.5
TmAg(8~
Cll - CI2
---20 - 20.3
2
4.4 4 3.5
-+5.4 -5.4
(0, 3) (0, 3) (0, 3)
(0, 3) (0, 3) (0, 3)
1.75 Parastriction X~ ) TmMg(9)
CI1 C12 Parastriction X~ ) --
References: [1] Morin et al. (1978b). [2] Givord et al. (1983). [3] Morin and Schmitt (1981b). [4] A16onard and Morin (1979). [5] Morin et al. (1978c). [6] Morin and Schmitt (1982c). [7] Jaussaud et al. (1980). [8] Giraud and Morin (1986). [9] Giraud et al. (1986). elastic constants exist, with t h e r m a l variations strongly d e p e n d i n g on the competition b e t w e e n the two q u a d r u p o l a r couplings, as shown in T m Z n and T m C d [fig. 1 in Liithi et al. (1979)]. Similar selective characteristics are o b s e r v e d in parastriction ( M o r i n et al. 1978c) and third-order m a g n e t i c susceptibility ( M o r i n and Schmitt 1981b) experiments.
5.1.2. Comparison o f the magneto-elastic coefficients T h e various magneto-elastic coefficients p r e s e n t e d in section 4 for cubic c o m p o u n d s are collected in figs. 63 and 64, after normalization to aj. T h e first c o m m e n t c o n c e r n s the Order of m a g n i t u d e which is relatively i n d e p e n d e n t o f the rare earth across a given series. This is reminiscent o f the b e h a v i o u r of 4th and 6th o r d e r C E F p a r a m e t e r s within a series. A relatively g o o d a g r e e m e n t is o b s e r v e d for the B~/aj ratio b e t w e e n concentrated R S b and dilute L a S b R c o m p o u n d s . This indicates, in particular, that the
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS i
I
I
5
I
i
i
i
I
RLoSb/ '~
\ /~-~Sb
,,
o. /..o
0
ePb3,,.," • C-3
"
o0
#0%
115
+
~
~
~
.0.-'%o. TmGo3I
÷
RA[2~
+
F
RMg
l'\
.
.
.
el
-5
RZn__
.
RAg
....... •
-
-
TmCd
RC-u". . . . . . . . ">'~,~~.>,...-:~~"
i
I
I
I
i
t
i
i
Ce
Pr
Nd
Tb
Dy
Ho
Er
Tm
Fig. 63. Magneto-elastic coefficient B y for the tetragonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
I
]
:PrPb 3 ,
L
TmGg3
.~ • PrAg lCePb3
TmMg.
,,t
0
Ce
o/-
Pr
Nd
Tb
Oy
Ho
Er
Tm
Fig. 64. Magneto-elastic coefficient B ~ for the trigonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
macroscopic strain measured in dilute samples is, in a first approximation, linearly related to the local deformation through the concentration of active sites. For a given series, the two magneto-elastic coefficients are often of opposite sign, as expected from point charge estimates as will be discussed in section 5.2. The only exceptions are the cases of AuCu3-type compounds and RAg dilute alloys. For a given type of structure, values have been observed to be similar in a first approximation. This is in particular true for the CsCl-type structure. However, the positive sign of B ~/as for TbZn is in opposition with the other determinations in the RZn series. This might be the signature of a two-ion contribution although it has been observed nowhere else for this symmetry. Note in addition the small average value of B~/a: in RZn.
116
P. M O R I N and D. SCHMITT
Among the different series, the values observed for the B~/aj's are of the same order of magnitude. This is reminiscent of the CEF parameters which usually correspond to total level spacings of a few hundred Kelvin. In the same way as peculiar surroundings or densities of electronic states drive small or large CEF spacings, as in RBe13 (Vigneron et al. 1980) and RB 6 (Zirngiebl et al. 1984), respectively, the magneto-elastic coefficients may also vary from one series to another. The actual spontaneous strain depends not only on the B" values, but also on other parameters. The CEF usually determines the axis of easy magnetization, then the symmetry lowering mode. The pair interactions state the purification of the CEF wave functions, then the various expectation values. For this reason, RFe 2 compounds develop "giant" strains at room temperature although their low-temperature values are in the same range as for other series. On the other hand, the spontaneous strain, which would be large in the case of a collinear spin structure, fully vanishes in the case of a multi-axial spin structure, preserving cubic symmetry as in DyCu. The magneto-elastic effects are of similar magnitude in rare earth intermetallics and insulators. For instance, in TbVO4, DyVO 4 and TmVO4, the vanadates which undergo a Jahn-Teller tetragonal-orthorhombic transition (Gehring and Gehring 1975), the low-temperature spontaneous strains are e ~= 12.7 × 10 -3, e ~ ' = 5.9 × 10 -3 and e ~= 3 × 10 -3, respectively. These strains are reminiscent of those in intermetallics. Unfortunately, whereas determinations of magneto-elastic coefficients in metals are numerous in cubic symmetry and scarce in tetragonal symmetry, the situation is the opposite in insulators; thus comparing the magneto-elastic coefficients is difficult. However, from the magneto-elastic contributions G~aE = 6 /[~1 and GME = /X2 determined from the ultrasonic velocity softenings (Melcher 1976), the following values IB~I--S7K, In l--80 and 22K are deduced in DyWO4, TbVO4, TmWO4, respectively. These magnitudes are similar to those observed in intermetallics.
5.1.3. Comparison of the quadrupolar pair-interaction coefficients There are fewer systematic studies for the pair interactions than there are for the magneto-elastic coupling. However, from table 24, their existence in TmX is well established and a very good coherency is observed in tetragonal symmetry. Within the CsCl-type structure, the tetragonal pair interactions dominate the magnetoelastic coupling and are the main origin of the ferroquadrupolar ordering. They are always positive and particularly important as soon as the aj coefficient is large. On the contrary, from the determinations in the paramagnetic phase, the trigonal pair interactions are negative and overwhelm the weak magneto-elasticity. For NaCl-type compounds, the tendency is less clear (fig. 65): K ~ is negative in DySb, but seems to be positive in HoSb and null in TmSb, in which no pair interactions seem to be present (Wang and Liithi 1977b). Tetragonal pair interactions are weaker than the magneto-elastic coupling in RSb pnictides. For the trigonal symmetry, the study of the TbX system indicates K ~ to be positive
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
20
-t
80
-Ag
117
.Ag -Go 3.~. '-Cd
-Cd
II
-'Cu
-Cu°
-Sb E 0 X)
- cd "zo ;Ag
"-P%
I
Ce
I
Pr
-Sb
-sbD
I
I
I
Tb Dy Ho Er
I
Tm
Fig. 65. Quadrupolar pair coefficient K ~ for the tetragonal symmetry in some rare earth cubic intermetallics (0: CsCl-type, +: AuCu3-type, [~: NaCl-type compounds).
and larger than G ~E. However, the total quadrupolar interactions remain smaller than the bilinear coupling and only modify the nature of the N6el transition. In insulators, such as the vanadates, the pair-interaction coefficients are fundamentally different. From the analysis of elastic constants (Melcher 1976) the coefficients K ~= h 1 and K ~= A2 are found to be negative and their absolute values are smaller than the magneto-elastic contribution (K ~= -0.35 mK in D y g O 4 , K ~ = -0.77 and - 1 1 mK in TmVO 4 and TbVO4, respectively). In these Jahn-Teller compounds, the structural transition is thus driven by the magnetoelastic coupling, whereas in the intermetallics the pair interactions drive the quadrupolar ordering and then, through the magneto-elastic coupling, the structural transition (Levy et al. 1979). In insulators the pair interactions are mediated by phonons (Gehring and Gehring 1975), and the h coefficient is a so-called "phonon exchange" parameter. It integrates the effective coupling of all the phonons to the Jahn-Teller ions for a given symmetry. The K/GME ratios are very close to --1/3, the theoretical expectation in the absence of any optic phonon mode contribution (Harley et al. 1972). On the other hand, the positive K values observed in intermetallics can originate neither from a k = 0 optical phonon in the NaC1- and CsCl-type structures, nor from the negative phonon self-energy term (Levy et al. 1979). In section 5.2 it will be shown that pair interactions in intermetallics are mainly mediated by conduction electrons.
5.2. Origin of the quadrupolar interactions 5.2.1. The one-ion magneto-elastic coupling From the comparison between the expressions for the cubic crystal field Hamiltonian [eq. (2)] and those for the one-ion magneto-elastic coupling [eq. (7)], it becomes obvious that the magneto-elastic coefficients are the strain derivatives of
118
P. MORIN and D. SCHMITT
the CEF parameters. For example, the second-order CEF parameter B ° vanishes in cubic symmetry, but its derivative with regard to a tetragonal distortion does not, B ~ , =-=
OB °
0~
"
The same is valid for a trigonal deformation, as well as for the higher-rank magneto-elastic coefficients, or if we consider crystals with lower symmetry, e.g., hexagonal or tetragonal ones. Therefore, the question of the origin of the magneto-elastic coupling is identical to the problem of the origin of the crystal field itself. In rare earth intermetallic compounds, it has been shown that the CEF parameters originate mainly from two contributions, namely the point charges and the contributions of the conduction electrons (Schmitt 1979). In cubic systems, these latter have been found to be preponderant, due to the strong anisotropic orbital character of the conduction electrons, as shown, e.g., by augmented-plane-wave band calculations (Belakhovsky and Ray 1975). In systems with hexagonal or tetragonal symmetries, the point-charge contribution seems to play a role which is relatively more important than in cubic systems, at least for the second-order CEF parameters. Then, calculating the magneto-elastic coefficients requires a precise knowledge of how these two CEF contributions depend on a given strain. First, in the presence of a weak strain, the localized charges are displaced from their previous non-strained arrangement, generating a new potential having the new symmetry of the lattice and proportional to the strain. This potential gives the point-charge contribution to the magneto-elastic coefficients. On the other hand, and to the first order of perturbation, this new potential modifies the conduction band. In particular, it lifts the orbital degeneracy of the conduction electron states in the vicinity of the Fermi level. According to Fermi statistics, this leads to a redistribution of the conduction electrons in the band which gives rise to a new CEF contribution, i.e., the band contribution to the magneto-elastic coefficients. Both contributions have been thoroughly investigated in cubic rare earth intermetallic compounds (Morin and Schmitt 1981a). In particular, the expressions for the direct and exchange Coulomb contributions of each type of conduction electrons (p, d, or f character) have been derived. It has been found that the magneto-elastic modifications of the cubic fourth- and sixth-order CEF parameters are negligible and that the second-order magneto-elastic coefficients result from the competition between numerous antagonistic contributions, which are of the same order of magnitude as the experimental values. Within the hypothesis of constant effective charges and a rigid-band model, all contributions are expected to vary as aj, ~j, yj for the second-, fourth- and sixth-order magneto-elastic coefficients, respectively. Furthermore, the pointcharge estimation leads to the constant ratio B ~ / B ~ = -4/V'-3 between the trigonal
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
119
and tetragonal second-order coefficients (see Creuzet 1982); this is not true for the band contribution. As shown in the previous section, this relation is far from being satisfied experimentally for a given compound, and this failure clearly indicates the importance of the conduction electrons. Nevertheless, the difference in sign between both parameters in most of the series as well as their overall magnitude (figs. 63 and 64) suggests a driving role for the point-charge contribution, contrarily to the case of the cubic CEF parameters. In realistic systems, because of the various contributions involved, the magnetoelastic coefficients are thus liable to vary significantly from one compound to another, especially if they belong to different series. This variation may occur via a shift of the values of the effective point charges. For the conduction-band contribution, the variation may be related to the modification of the partial densities of states at the Fermi level, which can be important when the Fermi level lies near a peak of the density of states. The importance of the d electrons on the rare earth site has also been suggested in dilute rare earth alloys (Campbell and Creuzet 1985). In these systems, the magneto-elastic coupling constants were evaluated within an indirect electronlattice interaction model, and using the nearly free electron approximation (Kozarzewski et al. 1985). The general features of the results obtained in this calculation are similar to those described above. The effect of screening of the magneto-elastic coupling by the conduction electrons has been found to significantly reduce the point-charge estimate of the parameters in Laves-phase rare earth intermetallics (Del Moral 1983). As a conclusion, ab initio evaluations of the one-ion magneto-elastic coefficients appear to be difficult to perform. As for the CEF parameters, they originate from numerous antagonistic contributions, and obtaining realistic values seems rather difficult. Qualitatively, their order of magnitude is well accounted for by point-charge and/or conduction electron contributions, and they follow the Stevens factor throughout a given series.
5.2.2. Quadrupolar pair interactions In metallic systems, conduction electrons have been shown to play an essential role in the coupling between magnetic ions, by propagating the local interactions between them and the localized magnetic electrons. This indirect exchange mechanism has been widely invoked in magnetic materials and is known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Most of the studies has been focused on bilinear-type indirect coupling, which, in general, is the dominant term. Much less attention was paid to the origin of quadrupolar-type interactions. In rare earth and transition metal insulators, direct coupling between electric multipole moments of the two atoms and indirect coupling through the lattice-ion interaction have both been investigated (Birgeneau et al. 1969). However, in rare earth intermetallic compounds with the NaC1- and CsCl-type structures, both couplings have been proved to be negligible and therefore cannot be the dominant mechanism for the quadrupolar pair coupling (Levy et al. 1979): the
120
P. MORIN and D. SCHMITF
primary source of such a coupling is the indirect Coulomb and exchange interaction via the conduction electrons. The coupling of quadrupoles via the conduction band is analogous to the RKKY interaction quoted above, which involves the spins at different sites through the local s - f exchange interaction. For quadrupoles, it is the direct and exchange part of the full 4f-conduction electron (k-f) Coulomb interaction that contributes to the couplings K ". The corresponding effective couplings were calculated in second-order perturbation theory and by using plane-wave states for the conduction electrons (Teitelbaum and Levy 1976). A more realistic ab initio calculation of bilinear and quadrupolar pair coupling was performed later in the cubic rare earth intermetallic compound DyZn, by using conduction-electron wave-functions and energy bands obtained by a selfconsistent augmented-plane-wave method (Schmitt and Levy 1984, 1985). Spin and orbital effects in the complete k - f interaction were taken into account. The results underline the dominant role played by the d conduction electrons in the bilinear and quadrupolar couplings. In this calculation, the Fourier transform of the bilinear 0*(q) and quadrupolar Kr(q) and K~(q) interactions were found to be of the correct order of magnitude. Their different q-dependence was explained by the behaviour of specific matrix elements of the k - f interaction: the magnitude of these matrix elements is directly related to the symmetry of the two k-states involved. In particular, the ferromagnetism and the ferroquadrupolar tendencies (y-mode) of DyZn are associated with the strong eg-character of the conduction electrons near the Fermi level. On the contrary, the antiferroquadrupolar tendency of the trigonal coupling K~(q) is related to the small amount of t2g-type conduction electrons near the Fermi energy. The dependence of the bilinear and quadrupolar couplings on the Fermi energy was also investigated. Only part of the coupling, i.e. the intraband terms at q = 0, follows the partial densities of states at the Fermi level, which are associated with the type of electrons (% o r t2g ) from which they arise. The other terms, in particular the interband ones, are not directly correlated with the Fermi energy but depend on details of the entire energy band structure, including energy bands well above the Fermi level, which contribute for a noticeable amount. At last, the orbital interactions are the main contributions to the quadrupolar couplings K" so that the latter are expected to vary as otj. 2 That explains the importance of quadrupolar effects at the beginning and end of the rare earth series, where weak spin contributions (small De Gennes factor) allow the quadrupolar coupling to be better revealed. As for the one-ion magneto-elastic coefficients, ab initio evaluations of the two-ion quadrupolar parameters are very difficult to perform and would be necessary for each compound under consideration. 5.3. Conclusion
The aim of this chapter has been to present an extensive overview of the effect of the magneto-elastic coupling and of the quadrupolar interactions in rare earth intermetallic compounds. A complete Hamiltonian has been detailed, which
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
121
includes all the relevant terms describing the one-ion magneto-elasticity as well as the two-ion quadrupolar coupling, for all the possible normal strain modes in cubic, hexagonal and tetragonal symmetries. The formalism for using this Hamiltonian has been developed, in particular that related to a perturbation theory. The various effects of the magneto-elastic coupling and of the two-ion quadrupolar interactions on the magnetic and elastic properties have been discussed, with a particular emphasis on the properties related to the perturbation methods: elastic constants, parastriction and third-order magnetic susceptibility. All these experimental manifestations appear as reliable ways to determine the quadrupolar parameters. Due to the strong anisotropic character of the magneto-elastic properties, using single crystals for experiments is quite essential. The magneto-elastic and quadrupolar data available in the literature have been summarized for a great number of rare earth series, within the present formalism, in order to make a comparison between them easier. Some series, especially those having the cubic CsC1- and NaCl-type structures, have been extensively studied in the last decades. Several others have been investigated much less, and some of them would be worth to be studied more thoroughly, in particular when single crystals become available. In most of the experimental results, the second-order one-ion magneto-elastic coefficients are concerned. The ratios B~'/%, in general, present a regular variation throughout a given series, but their magnitude and their sign can drastically change from one series to another. Nevertheless, their absolute value always remains in the 0-4000 K range. These differences show that, even if the point-charge contribution to the magneto-elastic coefficient is important, the role of the conduction electron is far from being negligible in these metallic compounds. An accurate ab initio estimate appears, however, difficult to obtain, due to the great number of different antagonistic contributions. The two-ion quadrupolar interactions have been studied less thoroughly in the rare earth intermetallic compounds. A consistent determination of the corresponding parameters has been successfully carried out in those series where the effects are strong, in particular those where a quadrupolar ordering occurs. Through their contribution to the free energy, the quadrupolar pair-interactions may strongly influence the character of the magnetic transitions, as well as, when they are negative, the stability of multi-axial spin structures. At last the dominant role of the conduction electrons has been emphasized in these systems, contrarily to the case of the rare earth insulators where the coupling between 4f quadrupoles is mainly due to the phonons. The existence of magneto-elastic and quadrupolar couplings then appear quite general in rare earth interrnetallic compounds. They need to be included in order to quantitatively understand the magnetic and elastic properties of these systems.
Acknowledgement It is a great pleasure to thank Professor P.M. L~vy from New York University for very fruitful discussions and a critical reading of the manuscript.
122
P. MORIN and D. SCHMITI"
Appendix I. Symmetrized Stevens operators for the cubic symmetry To completely express the magneto-elasticity of rank l (l = 2, 4, 6), all 21 + 1 Stevens operators are needed. In the present chapter we use the operators defined by Hutchings (1964), namely the quadrupolar operators of table 1 (l = 2) and the O / (m = 0 to l) and O'S(s) (m = 1 to l) for l = 4 and 6 (see table VIII of Hutchings 1964). The linear combinations of these operators, which transform according to the irreducible representations of the cubic symmetry group, read as, Second-rank 5"
O~ ' 1 = 002'
o l '2 = v ~ o i .
5"
Oi '1 = P~y,
° 2 '2 = Pr= ,
0 2 '3 = Pz=.
Fourth-rank ot 0 4 = 0 0 "~- 5 0 ~
FI:
r3:0~4
.
0~4'2 = -4X/-3 0 ] .
'1 = 004 -- 7 0 4 '
6,2
1
3
5:
064 '1 = O ~ ( S ) ,
0 4 = 0 4 -- 0 4 ,
5:
o i '1= o~4(s),
01,2 = --lOi(s
8,3
0 4 =
__ 0 1 ( S ) __ O I ( S )
) q- 7: O 43( s ) ,
0 4 , 3 = - - ~1 1O 4 -- 1 0 43.
Sixth-rank 5:O6
a
= O1--210~"
r~: 06~-- o l - 06 5:
O~ '1~- O 1 71-3 0 ~ 01,1 _-- 0 4 ( S ) ,
5:
01,2=
=
~5 r ~ 0 6 _2[ _
- - ~10 6I
~/'~
06 6 .
-- 5
~ 0 6 "31 - T11/',15 v6,
Y11~.5/ IJ6(S)", .
011,2 = ~ 01 16 ( s ) - 9
3 )-{- ~'~ O I ( s ) , ~06(S
1 9 3 33 [-15 0 6 1 3 = ~1 O 6-[~O 6 q- T t / 6 •
0621 = OI(S),
5:
O~,2
0~6,3 = 1o~6(s)_ ~5 O 63( s ) 06~, 1 = O I ( S ) ,
5:
,
3 1 0 6 ~ 3 = -~0 6-
5 3 ) + IOI(s), O12,2 = I O ~ ( s ) ~'- ~06(S 5 3 3 5 ~0 6 + ~0 6 •
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
123
Appendix 2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~Mm Starting from the expressions in table 2 and appendix 1, three terms may be derived for l = 4: one isotropic term which keeps the cubic symmetry, and two anisotropic terms as for l = 2, O~MEI(I = 4) = O~MEI(I = 4) + Yg~m(l = 4) + Y(MEI(I = 4 ) , with ~x ~MEI(I
m --
4) =
~ -B(4)8
a
0 4 ,
--~(4)\e, ltJ4
~/~MEI(I e = 4)
At- /:,2tJ4
) ,
= - - /nJ ( 4. .). . ['Ell . . J 4 1 ..}_ E 2e,,~e I J 4 , 2 + E 3. .[.J. 4 3,) .
The case l = 6 is identical to l = 4, except that, for the e mode, there exist two independent sets of linear combinations of the Stevens operators. Therefore, ~MEI(I = 6) = ffffMEl(l=6) + ffff~ciEl(l=6)+ ffffMEX(/ el e2 a = 6) + ~MEI(/ = 6), with oz
o~
~1;
~MEI(I = 6)
= -B(6)8
F3:
Y(~E,(I
:
F5:
fft~MEI(/ el = 6):
F5:
g Me2E I ( I = 6)
:
6)
ot
o~
0 6 ,
- - D, (~6r) ~, E Ir~r,1 I J 6 + ~_r,-~r,2x 2~6 J , ... 1,1 ..[_ 8 2 0 ; 1 , 2 _]_ E 3e,-~el,3x, - - Dh e( 61 ) .I'EIIA'6 IJ 6 ) ,
= - - / ~ ( 6 ) .j E . .l.l J 6
2,1 + 406
,2
+ E;O;2,3)
,
All the above expressions are referred to axes with four-fold symmetry, i.e., with z-axis parallel to the [001] direction. In some cases, however, it may be useful to consider the z-axis along a [111] direction, e.g., when a magnetic field is applied along that direction. In that case, symmetry considerations imply that the tetragonal strain e r should vanish and that the trigonal strains e" are all identical, e 7 = e 2 = e; = e'. Therefore, the corresponding trigonal magneto-elastic Hamiltonians each involve a single linear combination of symmetrized operators which may be rewritten in the new axes system (according to the convention of Hutchings 1964) as, I=2:
t'xy+ e,z+ ezx-, o ,
1=4:
0 4,1+ 0~,2 + 0 4,3~ ~(O 1 4o +7V'2 O ] ) ,
l=6:
O ; 1 , 1 + O ; 1 , 2 _[_ O;1,3..._> _ _ ~ ( 0 6 1 0 __ ~ / ~ 0 6 3
O62,1 + 062,2 + O6Z,3__> -- T~ 1 ( O 60 --
_ 110~)
2 5 X / 2 O 63 +
130~)
124
P. MORIN and D. SCHMITr
Appendix 3. Perturbation theory: the tetragonal mode Y First we have to define the zeroth-order energies E~ and the eigenvectors [ik) corresponding to the CEF Hamiltonian ~fCEF alone, ~CEF [ik) = E i lik)
(k
= 1,...,
m,).
(Cl)
In each degenerate subspace i, the m~ independent eigenfunctions lik) have to be adapted to the perturbation Y('~ [eq. (29)]. This can be achieved easily by choosing the following expansion in the IJ, M) basis [for the complete expressions, see Lea et al. (1962)], [ik) = aiklJ,1
M ) + a~2lJ,
M
- 4) +
a~klJ, M
- 8) + " "
.
(C2)
A perturbation theory carried out to second-order in e~ and fourth order in H allows one to obtain analytic expressions of the perturbed energies Ezk,
4 Eik = E i + 2
n=l
~ikl~(n)+ ' ' "
(C3)
"
Then, the partition function Z, Z = Z e -t~E'k ,
(C4)
i,k
can be calculated, where/3 = 1 / ( k a T ) , where k B is the Boltzmann constant and T the temperature. One obtains, Z=
ZcEFZ
1 r~ ,BVeV1 + KVQ): .... [I + ½/3Xo(H+ nM)2 + :PXvt
+ PXv'~(2)'B~ert1 + K r Q ) ( H + n M ) 2
-1-1/3(X(3) -t- 1/3(X0)2)(n--}-/,/M) 4-1-.. "l,
(C5)
where ZCE v is relative to the Ei's and Z .... to the elastic energy and to the corrective energy terms [see eqs. (14), (16), (17)]. Finally, this leads to the expression of the total free energy F ~ = - k BT In Z given in eq. (30). . (2) and x~ - (3) are given The expressions of the four CEF susceptibilities X0, X~, x~ by,
x0 = g . . B
i z.12
Il Ok,j, 2 i,k
l
+ 1 ~B T
I/ k,,kl 2 , \ I ;~.i~l )
'
(c6)
(C7)
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS t,(2)
,
2 2
= gylxB
125
Jik,jlQjl,J'l'JY'l',ik q- 2Qik,jlJjl,j'l'Jy'l',ik fi jei,l £ (U i -- E i ) ( E i -- Ej,) j'~i,l' (
Ei,k
I4~,~,l O,~.,~ + 2Oik,jl41,ikJik,ik (
- ]¢i,I 2
7--V
1
1
)
(c8)
+ 2(kBT)2 [J~k,ikl Q,k,ik , __ ,y ,•(3)
1 4 4 ( Jik,jlJjl,j'l'Jj'l',j'T'Jj'T',ik 2k~r (x°)= + g'~"£i,k f' - 4 /#i,, £ (E,--~---E-7')--~i----Ef) j'#i,l' j"¢i,l" 2
2
IJ,k,j,I IJ,k,/','~ + 2J,'k,/,J/t,/','J/',',ikJik,ik(
+2 ~
/~,,,
(E,- g)(E,- E/,)
2
E,
j' #i,l'
2
IJ....~12l.l,,.,jll 2
-2 ,,i,,E
((E i - Ej)2
~ii:-~])
i
+ 6k3T 3 IJik,ik
i4)
2 q- ( E i _
E/)kuT
,
+©I (C9)
where
Jik,/, =
(C10)
a.~,/, = ,
(Cll)
and
are the matrix elements of Jz and O ~ in the CEF levels. For each CEF sublevel, 1 exp(-/3Ei) f / - ZCEF exp(-/3E~) = ~ exp(-/3E~)
(C12)
i,k
is the Boltzmann population factor. Appendix 4. Perturbation theory: the trigonal mode e
When the magnetic field H is applied along a three-fold axis, e.g., [111], it is more convenient to first rotate the coordinate axes so that [111] is the new z-axis. Then the two combinations of operators in eq. (44) may be rewritten as:
L+~+L--.V~L,
126
P. MORIN and D. SCHMITT
and P x y + P y z q-
1
0
P~x --> ~02 "
The perturbation Y(] takes the new form:
Y({'= --g,p.B(H + n M ) ] z - ½(B~e ~ + K~P)O ° 3 ,.-~e; e.,2 + ~t..oke ) + ½nM2 + ~K*P 2.
(D1)
In each subspace i, the new zeroth-order eigenfunctions Ilk }' (corresponding to ! the appropriate YgCEF) have to be adapted to the above Hamiltonian. This condition is always fulfilled if these basis functions are chosen so that AM = 3 in the IY, M) expansion: 2 [ik )' = bill J, M ) + b~klJ, M - 3) + b~klj, M _ 6 )
+ ..- .
(D2)
The same perturbation theory as in appendix 3 may then be carried out, leading to the following expression for the total free energy F ~, 1 ! F ~ = Foe -- ~Xo( H + nM)2 -- ~XvkD 1 ..... e +K~P) 2 1 (2)' [n~ -~X~, ~D e e + K ~ P ) ( H + n M ) 2 - - ~i a, . y (3)'(H + nM)4
+ ~3 o,,.~e,, te
e,,2
) + l n M 2 + ~K~P 2,
(D3)
¢
where the new CEF susceptibilities X~ are related to the operators Jz and O ° in the new system of axes. Their expressions are identical to those given in appendix 3, but their values differ from the corresponding unprimed susceptibilities, due to the new CEF eigenfunctions lik)'. Returning to the initial system of four-fold axes and defining new trigonal susceptibilities we are led to the expression of eq. (45) for the free energy, where: Xo
:
,
Xo
X~
=
1
,
~X~,
. (2)
X~
1.
(2)'
= ga~,
/~3)
. (3)'
= ,'(v
"
(D4)
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P. MORIN and D. SCHMITT Laboratoire Louis N@el* C.N.R.S. B.P. 166 X 38042 Grenoble Cedex France
*Associated with Universit6 Joseph Fourier, Grenoble
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. I~ t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Fc rl~ alL, m . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. T h e c t b i c H a n n i l t c n i a n . . . . . . . . . . . . . . . . . . . . . 2.1.1. G e n e r a l e ~ p l e s s i o n . . . . . . . . . . . . . . . . . . . . 2.1.2. H a r m o n i c d e s c r i p t i o n of m a g n e t o s t r i c t i o n . . . . . . . . . . . . . 2.1.3. T h e f e t a l q ~ a & u p o l a r H a m i l ( o n i a n . . . . . . . . . . . . . . . 2.2. T r e a t m e n t of the c u b i c H a m i l t c n i a n . . . . . . . . . . . . . . . . . 2.2.1. F e r t m b a t f o n t h e o r y . . . . . . . . . . . . . . . . . . . . 2.2.2. C e n e r a l i z a t i o n of l h e s u s c e p t i b i l i t y f o r m a l i s m . . . . . . . . . . . . 2.2.3. B e h a v i o u r of l h e ~ a r i c u s s u s c e p t i b i l i t i e s . . . . . . . . . . . . . 2.2.4. M a g n e t i c a n d q u a & u p o l a r t r a n s i l i o n s . . . . . . . . . . . . . . 2.3. E x t e n s i o n to l o w e r s y m m e t r i e s ( h e x a g o n a l a n d t e t r a g o n a l ) . . . . . . . . . 2.3.1. T h e H ~ m i l t o n i ~ n . . . . . . . . . . . . . . . . . . . . . . 2.3.2. H a r m o n : ' c d e s c r i p t i o n of 1Tagnetostr c t i c n . . . . . . . . . . . . . 2.3.3. P e l t u r b a t i o n tl;eo~y . . . . . . . . . . . . . . . . . . . . 2.3.4. C o m p a r i s o n w i t h c u b i c s y m m e t r y . . . . . . . . . . . . . . . . 2.4. A p e c u l i a r c a . e : the semi-classical t r e a l m e n t . . . . . . . . . . . . . . 2.4.1. T h e p a r a m ~ g n e l i c lzhase . . . . . . . . . . . . . . . . . . . 2.4.2. T h e c r d e r e d p h a s e . . . . . . . . . . . . . . . . . . . . . 3. E x p e r i m e n t a l e v i d e n c e . . . . . . . . . . . . . . . . . . . . . . . 3.1. Q u a d l u p o l a r o r d e r i n g s . . . . . . . . . . . . . . . . . . . . . 3.1.1. F e r i c q u a d r ~ p o l a r older:'ng . . . . . . . . . . . . . . . . . . 3.1.1.1. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2. T m C d . . . . . . . . . . . . . . . . . . . . . . 3.1.1.3. C e A g . . . . . . . . . . . . . . . . . . . . . . 3.1.2. A n t : f e r l c q u a d ~ p o / ~ r o l d e r i n g . . . . . . . . . . . . . . . . 3.1.2.1. C e B 6 . . . . . . . . . . . . . . . . . . . . . . 3.1.2.2. T m G a 3 . . . . . . . . . . . . . . . . . . . . . 3.1.2.3. P I P b 3 . . . . . . . . . . . . . . . . . . . . . . 3.1.3. S t r u c t u r a l l r a n s i t i o n s p o s s i b l y of q u a d r u p o l a r o r i g i n . . . . . . . . . 3.1.3.1. P r C u z . . . . . . . . . . . . . . . . . . . . . . 3.1.3.2. U P d 3 . . . . . . . . . . . . . . . . . . . . . . 3.2. N a t u r e of q u a d r u p o l a r a n d m a g n e t i c t r a n s i t i o n s . . . . . . . . . . . . . 3.2.1. Q u a d r u p o l a r t l ~ n s t i o n s . . . . . . . . . . . . . . . . . . . 3.2.2. M a g n e t i c o r d e r i n g in t h e p r e s e n c e of q u a d r u p o l a r i n t e r a c t i o n s . . . . . . . . . . . . . . . . . 3.2.2.1. F i r s t - o r d e r t r a n s i t i o n s a n d tricriti.cality
5 6 7 7 11 12 12 13 16 17 20 24 24 27 28 30 30 30 32 34 34 34 35 36 36 38 38 39 40 41 41 42 43 43 43 44
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
3.2.2.2. Second-order transition in PrMg 2 due to negative qua drupol a r interactions . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Magnetic ordering in the q u a d r u p o l a r phase . . . . . . . . . . . . 3.2.3.1. C e A g , CeB 6 . . . . . . . . . . . . . . . . . . . 3.2.3.2. T m Z n . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Presence of additional couplings . . . . . . . . . . . . . . . . 3.2.4.1. DySb and rare earth monopnictides . . . . . . . . . . . . 3.2.4.2. The RA12 series . . . . . . . . . . . . . . . . . . 3.3. D e t e r m i n a t i o n of the q u a d r u p o l a r p a r a m e t e r s from susceptibility techniques . 3.3.1, Elastic constants . . . . . . . . . . . . . . . . . . . . . 3.3.2, Parastriction . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Third-order p a r a m a g n e t i c susceptibility . . . . . . . . . . . . . 3.4. Effects on the magnetostriction . . . . . . . . . . . . . . . . . . 3.4.1. Magnetostriction of dilute compounds . . . . . . . . . . . . . . 3.4.2. Magnetostriction in the ordered phase . . . . . . . . . . . . . . 3.5. Effects on the magnetization processes . . . . . . . . . . . . . . . . 3.5.1. P a r a m a g n e t i c phase . . . . . . . . . . . . . . . . . . . . 3.5.2. O r d e r e d phase . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Effect on the direction of easy magnetization . . . . . . . . . . . 3.6. Effects on the magnetic structures . . . . . . . . . . . . . . . . . 3.6.1. Multi-axial structures . . . . . . . . . . . . . . . . . . . 3.6.2. I n c o m m e n s u r a t e magnetic structures . . . . . . . . . . . . . . 3.7. Effects on magnetic excitations . . . . . . . . . . . . . . . . . . 3.8. Two-ion anisotropic magneto-elasticity . . . . . . . . . . . . . . . . 3.9. Isotropic magneto-elasticity and pressure effects . . . . . . . . . . . . . 3.9.1. Two-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.2. Single-ion contribution . . . . . . . . . . . . . . . . . . . 3.9.3. Pressure effects . . . . . . . . . . . . . . . . . . . . . 3.10. Magneto-elasticity in the presence of lattice instability . . . . . . . . . . . 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series . . . . 4A. CsCl-type structure compounds . . . . . . . . . . . . . . . . . . . 4.1.1. R Z n series . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. R C d series . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. R A g series . . . . . . . . . . . . . . . . . . . . . . . 4.1.4, R C u series . . . . . . . . . . . . . . . . . . . . . . . 4.1.5, R M g series . . . . . . . . . . . . . . . . . . . . . . . 4.2. NaCl-type structure compouods . . . . . . . . . . . . . . . . . . . 4.3. AuCu3-type structure compounds . . . . . . . . . . . . . . . . . . 4.4. Cubic Laves phase compounds . . . . . . . . . . . . . . . . . . . 4.4.1. RA12 compounds . . . . . . . . . . . . . . . . . . . . . 4.4.2. R M 2 compounds (M = Mg, Ni, Co) . . . . . . . . . . . . . . . 4.4.3. R F e 2 compounds . . . . . . . . . . . . . . . . . . . . . 4.5. Hexagonal CaCus-type structure compounds . . . . . . . . . . . . . . 4.5.1. RNi 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.5.2. RCo 5 compounds . . . . . . . . . . . . . . . . . . . . . 4.6. Dilute rare earth systems . . . . . . . . . . . . . . . . . . . . . 4.6.1. R a r e earths diluted in cubic noble-metal hosts . . . . . . . . . . . 4.6.2. R a r e earths diluted in cubic pnictides . . . . . . . . . . . . . . 4.6.3. Other dilute systems . . . . . . . . . . . . . . . . . . . . 4.7. Miscellaneous compounds . . . . . . . . . . . . . . . . . . . . 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. G e n e r a l analysis . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Coherency of the various determinations . . . . . . . . . . . . .
3
49 49 50 50 50 50 52 53 54 56 57 61 61 64 67 67 69 70 72 72 75 77 79 79 80 80 81 82 83 83 84 85 85 90 92 93 95 95 95 98 100 102 102 102 106 106 107 110 111 113 113 113
4
P. M O R I N and D. SCHMITI"
5.1.2. Comparison of the magneto-elastic coefficients . . . . . . . . . . . 5.1.3. Comparison of the quadrupolar pair-interaction coefficients . . . . . . . 5.2. Origin of the quadrupolar interactions . . . . . . . . . . . . . . . . 5.2.1. The one-ion magneto-elastic coupling . . . . . . . . . . . . . . 5.2.2. Quadrupolar pair interactions . . . . . . . . . . . . . . . . . 5.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Symmetrized Stevens operators for the cubic symmetry . . . . . . . . . . A.2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~ME~ . . . . . . . A.3. Perturbation theory: the tetragonal mode 3' . . . . . . . . . . . . . . A.4. Perturbation theory: the trigonal mode e . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114 116 117 117 119 120 122 122 123 124 125 126
1. Introduction
The study of multipolar interactions concerning the unfilled 4f shell has known a great development in rare earth (R) intermetallic compounds. It involves the complex problems of both the crystalline electric field (CEF) acting on a 4f ion and the pair interactions between different ions. The electron-lattice coupling can drive very characteristic magneto-elastic features, in particular in elastic constants and in magnetostriction, through symmetry lowering modes which depend on the 4f electronic states. According to the Jahn-Teller theorem, the magneto-elastic modulation of the CEF corresponds to a gain of energy larger than the elastic energy lost at the thermodynamic equilibrium. These properties have been extensively studied in the past in many systems exhibiting an orbital degeneracy as reported for cubic spinels (Englman and Halperin 1970, Englman 1972) in particular mixed chromites (Kataoka and Kanamori 1972). Gehring and Gehring (1975) and Melcher (1976) have provided the literature with very complete reviews of the cooperative Jahn-TeUer transitions occurring in rare earth insulators within the tetragonal zircon structure. All these reviews show the fundamental importance of studying elastic constants. Indeed they couple, as thermodynamical variables, the strain to the ultrasonic stress in the presence of a quadrupolar moment, describing the orbital character of the 4f shell. Another important feature is the large success met by the mean-field approximation for describing thoroughly the structural transitions. More recently, very large symmetry lowerings, which are mainly of one-ion origin have been observed to occur at the magnetic ordering in rare earth intermetallics (L6vy 1969, Morin et al. 1979). In the example of RFe 2 compounds which are ordered at room temperature, they are large enough to motivate industrial applications as magnetostrictive transducers which are able to compete with ceramics (Clark 1980). In all the series, they are associated with large softenings of elastic constants in the non-ordered phase (Mullen et al. 1974, Moran et al. 1973). From a theoretical point of view, they have required the development of a quantum treatment which takes into account all the characteristics of the 4f wave functions, as will be discussed in the present chapter. However, in R intermetaUics an additional coupling has been revealed, mediated from one site to the others by the conduction electrons. This quadrupolar pair interaction may dominate the magneto-elastic coupling and drive a
6
P. MORIN and D. SCHMITI"
quadrupolar ordering in the paramagnetic phase, as occurs in TmZn and TmCd (Morin et al. 1978b). According to the sign of the quadrupolar pair interactions, ferro- or antiferroquadrupolar orderings have been observed. The symmetry lowering is then, when it exists, a consequence of the quadrupolar symmetry through the magneto-elastic coupling. Such compounds are not stricto sensu Jahn-Teller compounds, because then the magneto-elastic coupling is the driving mechanism. In the presence of dominating bilinear interactions, which is the majority of cases, quadrupolar interactions act on the minimization of the total free energy, and magnetic properties cannot be understood without them in several series (Giraud et al. 1985). For different experimental techniques (Ltithi 1980a), the magneto-elasticity remains the most appropriate probe for determining the quadrupolar pair interactions. This chapter is organized as follows. First, the quantum Hamiltonian is built, which describes both the one- and two-ion, spin and quadrupolar couplings in systems of cubic, tetragonal and hexagonal symmetries. The power of the susceptibility formalism, which allows independent determinations of the various coefficients, is emphasized (section 2). The main experimental evidence of quadrupolar orderings is presented in section 3, before giving the quadrupolar consequences on the magnetic properties. The experimental techniques which are based on the susceptibility formalism of section 2 are illustrated. Section 4 gives a review of the magneto-elastic and pair interaction coefficients which have been determined in the rare earth intermetaUics. Due to the anisotropic character of the magneto-elasticity, studies on single crystals are of fundamental importance and these are the only ones considered here. Section 5 gives an analysis of all these results from both a macroscopic and a microscopic point of view.
2. Formalism
The existence of non-quenched orbital moments in rare earth ions is the source of their anisotropic magnetic properties. A classical description of these properties is no longer valid. In a quantum formalism, the 4f shell is represented by wavefunctions expanded in the IJ, M:) basis. The appropriate Hamiltonian, acting on these wave functions, depends on the point symmetry of the rare earth site under consideration. In addition to the crystalline electric field (CEF) Hamiltonian, any kind of interactions acting on the rare earths or between them (Heisenberg coupling, magneto-elasticity, quadrupolar pair interactions . . . . ) must be considered. In the literature, most of the experimental results reported on the magnetoelastic properties concerns rare earth compounds with cubic symmetry, because for these compounds the number of parameters required for analyzing these properties is the smallest. In the following two sections, we extensively develop the formalism and the treatment of magneto-elasticity and quadrupolar interactions for cubic symmetry. The case of lower symmetries (hexagonal and tetragon-
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
7
al) will be discussed more briefly in section 2.3, while the connection with the classical formalism will be made in section 2.4. 2.1. The cubic Hamiltonian 2.1. i. General expression
The Hamiltonian appropriate for cubic symmetry includes several terms. By using the operator-equivalent method (Stevens 1952) they are written as: ffb°= ~CEF + ~°Z + ~B + ~OQ+ '~ME1 + ~ME2 "l- (Eel + E B + EQ q- EME2) .
(I)
The different terms represent successively the crystal field term, the Zeeman coupling, the Heisenberg-type bilinear interaction, the two-ion quadrupolar term, the magneto-elastic coupling, the elastic energy, and bilinear and quadrupolar corrective energy terms. Each term is described in detail below. The crystalline electric field Hamiltonian: Y(CEF According to different authors, this Hamiltonian takes one of the following forms. The x, y and z axes have been chosen as the four-fold axes in the cubic crystal: ~ C E F = B4(O°4 + 5 0 4 )
-]- B6(0°6 - 2104)
= A4 ( r 4 ) ~ 1 ( 0 40+ 5 0 4 ) + A 6 ( r 6 ) y 1 ( O ~ - 2 1 0 4 = __
Wx (004 -~- 5 0 ~ ) +
F4
W(1 - - I X l )
F6
0
)
(2)
4
( 0 6 -- 2 1 0 6 ) .
The O~'s are the Stevens operators in the formulation of Hutchings (1964). 13I and 7i are the 4th- and 6th-order Stevens coefficients, the value of which depends on the rare earth. The CEF parameters A4
(3)
represents the Zeeman coupling between the 4f magnetic moment M = gj/x B( J ) and the internal field H (applied magnetic field corrected for demagnetization effects). The bilinear Heisenberg-type Hamiltonian: ~ ~fB = -- g J IxBnM " J ,
(4)
8
P. MORIN and D. SCHMITT
represents the Heisenberg-type bilinear interaction, written within the molecular field approximation (MFA), n being the bilinear exchange parameter. An alternative definition for this parameter is 0", with: O* n
=
-C
3kBO* =
(5)
g :2t z B2 J ( J + 1)
where C is the Curie constant. The two-ion quadrupolar Hamiltonian: ~ o In a form analogous to the Heisenberg-type bilinear coupling between magnetic dipoles, a two-ion coupling between 4f quadrupoles is possible (see Schmitt and Levy 1985 and references therein), for which evidence has been found in several rare earth intermetallic compounds (Levy 1973, Levy et al. 1979). According to symmetry considerations, the corresponding Hamiltonian, again in the MFA, may be written as: ~Q = -KY((O°)
O0 + 3 < 0 2 ) O 2 ) - Ke(
(6) In this expression, 0~, 022 and the Pi:'s are the second-order (quadrupolar) Stevens operators (Hutchings 1964, Sivardi6re 1975) (see table 1), and K ~ and K ~ are two-ion quadrupolar parameters corresponding to the two linear combinations of quadrupolar operators, which are invariant to all the cubic symmetry operations. As will be discussed below, they are associated with the two anisotropic normal strain modes in cases with cubic symmetry, namely the tetragonal (3/) and trigonal (e) modes. TABLE 1 Expression of the quadrupolar operators. Operator
Expression
0°2
3JZz - J(J + 1)
o~
J~ - s,2 = ~(s~++ :_)
exy
l(JxJy + JyJx) = ~- (J+ - j2_)
--i
Pyz
e zx ~_
1
0 2
2
~(Y~L + L :,) ½(LL+LL)
e
The magneto-elastic Hamiltonian: ggME The magneto-elastic properties of rare earth compounds have been extensively studied since the 1960s (Callen and Callen 1965, Du Tr6molet de Lacheisserie 1970, Du Tr6molet de Lacheisserie et al. 1978). The analysis of these properties employs one-ion and/or two-ion magneto-elastic Hamiltonians, it is usually
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
9
limited to the first order in the strain (harmonic approximation) and contains terms up to the sixth rank for the Stevens operators. Few studies have been made in connection with second-order terms in the strain (anharmonic coupling) (Du Tr6molet de Lacheisserie et al. 1978, Rouchy and Du Tr6molet de Lacheisserie 1979, Wang and Lfithi 1977b) and this second-order magneto-elasticity will be not considered in this chapter. The one-ion magneto-elastic Hamiltonian corresponds to the direct coupling between the deformations of the lattice and the 4f shell. It can be considered as the strain derivative of the crystal field Hamiltonian. The two-ion magnetoelasticity is related to the modification of the two-ion magnetic interactions by the strains. The one-ion magneto-elastic H a m i l t o n i a n : ~fME1
Group theory gives us the correct procedure for constructing this Hamiltonian. First, one writes linear combinations of the first-order strain components eq which transform according to the three irreducible representations F1, F3 and F5 or a, y, e (see table 2) (Du Tr6molet de Lacheisserie 1970). Second, one considers the linear combinations of the Stevens operators of rank I (l = 2, 4 or 6) which transform also according to the irreducible representations of the cubic group (see appendix 1). Finally, one obtains all the possible invariants by summing the direct products of the different combinations which belong to the same representation. A magneto-elastic coefficient then is associated with each invariant. For l = 2 , two magneto-elastic terms are obtained, coming from the two representations /"3(7) and Fs(e), WMEa(I = 2) = Y ~ m ( l = 2) + aq~°hEl(l 2 ) , -----
(7)
with ~IEI(/
= 2)
e
=
--
~, 2 X/3e202),
B~(e~O ° +
Y(MEI(I = 2) = - B
8
e
e
(e,Pxy + e2Py ~ + e3Pzx ) .
(8)
(9)
Thus there are only two normal strain modes associated with symmetry lowerings of the cubic cell, one for a tetragonal and the other for a trigonal strain. TABLE 2 Normalized, syrnmetrizedstrains for the cubic symmetry. Representation
Straincomponents
r~
E° = ~
1"5
E1 = %¢/2 x y '
1
(Ex~+ ~, + ~zz)
E2 = " ~ Eyz'
10
P. MORIN and D. SCHMITT
The l = 4 and l = 6 one-ion magneto-elastic Hamiltonians have also been given (Creuzet and Campbell 1981, Niksch et al. 1982). Their full expressions are given in appendix 2.
The two-ion magneto-elastic Hamiltonian: ~ME2 Although any kind of two-ion interaction can a priori depend on the strains, only the bilinear coupling will be considered here. The same theoretical considerations as above allow us to write the corresponding Hamiltonian as the sum of an isotropic term and two anisotropic ones: ~ME2
=
o~ ~ME2
+
ffb'O~vlE2 "[- ~ M E 2
(10)
"
The first term, (11)
~ME2 = -D~e~ ( J ) " J ,
is related to the volume dependence of the isotropic bilinear exchange interactions. On the contrary, the two other terms,
Yg~E2 =
-Dr[e~'(2(Jz)J~
)~ME2
-De[eel( ( Jx)Jy + ( Jy) Jx) + 1=:2(( J,)Jz + ( Jz)Jy)
-
(L)L
- (4)4)
+ v3e~((L)L
- (4)4)],
(12) =
+ e3((Jz)Jx + (Jx)Jz)],
(13)
are associated with the appearance of anisotropic bilinear interactions under a strain. As for the bilinear coupling itself [eq. (4)] the MFA has been used for this two-ion magneto-elasticity.
The elastic energy: Fez The elastic contribution to the free energy may be expressed as a function of strains and symmetrized elastic constants. In cubic symmetry, the three normal strain modes a, y, e appear in the expression of the elastic energy 1 .,-,al a x 2 1 3' Y 2 Eo, = ~,~ot~ ) + ~c0[(~,) + ( ~ D 21+ 1 Co[(e~) ~ ~ 2 + (e~) 2 + (e~)2].
(14)
The C~ are background elastic constants without magnetic interactions; they are related to the conventional constants by a 0 C O ~-- C 1 1 +
0
.
2C12 ,
C~
:
0 C11 -
0 C12
. ,
s
0
C O-= 2C44
.
(15)
The corrective energy terms E~, EQ, EMe 2 These terms arise from the fact that each rare earth pair is counted twice in the MFA treatment of two-ion magnetic couplings. This occurs for the bilinear [eq. (4)], the quadrupolar [eq. (6)] interactions, and for the two-ion magneto-elastic
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
11
terms as well [eqs. (11)-(13)]: E B = ½nM 2 ,
(16)
Eo=½K'((o°)Z+3(OZ2>z)+½K~(
(17)
EME2 = ½D"e~
+ V~E~(
(18)
2.1.2. Harmonic description of magnetostriction Starting from the above Hamiltonian, the actual magnetostriction, i.e. the equilibrium values for the various strains e~'s, may be obtained by minimizing the free energy F with respect to each of them: OF/Oe ~= O. That leads to the following general expressions for the 6 symmetrized strains for cubic symmetry ~, 0 + 5 0 4 > + B(6) Ea = ~1 [B(4)
./
1
+ ½V,~
y,1
+ ½D~(2(jz)2_
(19)
(20)
'Y T,2 1 [~/-~B.V
- -
+ ½V~D~((j~>2_
(21)
~. l~l [()el,l\ ~2 81 = ~1 [Be
+ m~(Jx)(Jy>], 1
E2 : ~ 0
(22)
[Be(PYz>+ B (~4 ) < O 4 ' 2 >
l:?a:l / /-'}el,2 ~ e2 062,2> + u(6)\"6 / -~- B(6)<
+D~(J,)(Jz>],
(23)
1 ~ el O;1,3 ]~,52 [ /~e2,3 k e 3 = ~ g [ B ~ ( P ~ > + B(4)
+ D~(L)
(24)
The macroscopic change of length of a magnetic lattice with cubic symmetry may then be written as a-
8l 1 1 l - V~ e~ + ~
v
2
2
e1(2/3z -/3x - / 3 2 ) + ~
+ V~(e~/3~/3y + e~/3,/3 z + 23/3~/3x).
1
~
2
e2(/3" - / 3 2 ) (25)
12
P. MORIN and D. SCHMITF
To obtain a reliable description of the various magnetic and elastic properties, the expectation values of the Stevens operators must be treated in a self-consistent manner. In particular, this yields the actual temperature dependence of the various strains [eqs. (19)-(25)] in the paramagnetic as well as ordered phases.
2.1.3. The total quadrupolar Hamiltonian By including the preceding equilibrium values of the strains in the starting Hamiltonian, one can write it in a form which does not explicitly depend on the e~'s, but only on the magneto-elastic coefficients and on the expectation values of the various Stevens operators. In most of the cases reported in the literature, one-ion magneto-elasticity involving the quadrupolar operators appears as the dominant contribution. This justifies focusing our attention on it in the following. The relative importance of other terms from an experimental point of view will be discussed later, in particular in sections 3.8 and 3.9. Keeping only the B " contribution leads to an expression for NMEI(1 = 2) which appears to be formally undistinguishable from the two-ion quadrupolar Hamiltonian No; both terms can be grouped together in the total quadrupolar Hamiltonian NOT fft°QT= -G'Y[(O~)O°2 + 3 ( 0 2 2 ) 0 2] - G [( p xy)Pxy ~- (Pyz)Pyz -I- (P~x)P~x] (26) where the new total quadrupolar parameters G ~ are defined as G ~ - (B~) 2 -
-
cg
- - -
C;
~IE ~- K ~ '
,
(27)
+ K ~ = GME + K ~ .
(28)
+
K "~ =
G
E
Equation (26) provides an alternative expression for the quadrupolar couplings, which may be used instead of eqs. (6), (8) and (9), every time that the e "'s are not explicitly needed.
2.2. Treatment of the cubic Hamiltonian The treatment of the Hamiltonian described above can be carried out in two ways: i) First, by diagonalizing the full Hamiltonian in a self-consistent manner. This allows the description of all the magnetization and magnetostriction processes, in particular in the ordered phase or under large external stresses (magnetic field or uniaxial stresses) as shown in section 3. ii) The second possibility is to apply perturbation theory to the above Hamiltonian in order to describe the magnetic and magneto-elastic properties in the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
13
non-ordered phase and under small external stress. There are two advantages working in the cubic paramagnetic phase. First, analytical expressions can be obtained for the various susceptibilities connecting the various variables. Second, any strain mode can be systematically studied by applying external stress along the appropriate direction, whilst in the ordered phase only the spontaneous state may be usually achieved because of the strong anisotropy effects. In this section, the susceptibility formalism is developed by applying perturbation theory to the above Hamiltonian which is limited to second-rank quadrupolar terms. An analytical expression will be derived for the free energy, defining the various susceptibilities which appear in the analysis of the magnetization, parastriction and elastic constants measurements. Finally, the behaviour of these susceptibilities and their role in describing phase transitions will be described. 2.2.1. Perturbation theory
The zeroth-order Hamiltonian Y(0 consists of the CEF term [eq. (2)], the diagonalization of which provides the zeroth-order energies and CEF wave functions. According to the direction of external stress, the two normal cubic strain modes may be fully decoupled. Therefore they are considered separately. Tetragonal m o d e y
For a magnetic field applied along a four-fold axis, e.g. z, the only non-zero terms are M = g H a a 3 ( J z ) , Q = ( 0 2°) and e~. The perturbating Hamiltonian is then written as: YU1 = - g j l - % ( H + n M ) J z - (B3,e~ + K3,Q)O°2 + ~-~'0~,,'~3,to3,,= )~ + ½ n M : + -½K3,Q2 .
(29) Perturbation theory has to be carried out up to the second order for e~ and Q and up to the fourth order for H and M in order to derive analytical expression of the generalized Gibbs free energy F = - k B T l n Z. Here Z = T r e x p ( - / 3 Y ( ) is the partition function, k B the Boltzman constant and/3 = 1/(k B T ) . The details of the calculation may be found in appendix 3. The free energy expansion takes the form F3,= FVo - ½Xo(H + n M ) : -
~X3,tt~ 1 ,,,3, ea3, + K3,Q) 2
1. (3)(H + n M ) 4 - (2)tB3,e3,, + K 3 , Q ) ( H + n M ) 2 - zx3, \ •
- - , , I [ 3,
+ ~a"-'o,,ele~'t^3,'2)+ l n M 2 + 1K3,Q2 + . . . .
(30)
This expression includes four CEF single-ion susceptibilities which are defined without any magnetic or quadrupolar interactions, and depend only on the non-perturbed CEF state, i.e. on the cubic CEF level spacings and on the matrix elements of Jz and O ° between cubic CEF wavefunctions (see appendix 3). X0 is the usual (first-order) magnetic susceptibility, X3, is a quadrupolar strain susceptibility, X3, - (2) is a quadrupolar field-susceptibility which couples the quadrupolar
14
P. M O R I N
a n d D. S C H M I T T
operator O ° to the magnetic field, xz, - (3) is the third-order magnetic susceptibility which is related to the initial curvature of the magnetization curves and, finally, F 0 is the zeroth-order free energy corresponding to ~fcEF alone. The equilibrium values for M, e~ and Q are found by minimizing the generalized free energy F3, OF v
.
.
.
OM
OF3,
.
Oe~
OF3,
.
.
0.
oO
(31)
This yields a system of three coupled equations, the solution of which leads to the following relations M=
(3)11 u3 , XMH + XM
(32)
.
B ~'
~=Q ~ C~
(33)
'
Q = xoH 2 ,
(34)
XO
(35)
with XM -- (1 -- nXo) '
["~
1
X(M 3)=
(1
- nx0) 4
+ 203,
xv
(x~) ~ ] 1 - GrX3,J '
(36)
X(2) 3'
Xo
9
(1 - ~ X o ) 2 ( 1 - 6 % )
(37)
Note that we find for the equilibrium value of e~ the same expression as that found without recourse to perturbation theory [eq. (20)]. In the limit of zero field, OF3,/OQ = 0 leads to the strain dependence of the quadrupolar operator: B3,X3, -
Q = 1-
-
?
(38)
K:'X3, e l '
and the expression for the corresponding elastic constant can be written as: d2F 3, C3,= d(e~)2 = C~
( B y) 2Xv (1 - K~'X~,) '
(39)
or
Cy
C--~ = (1 - Grxr)/(1 - K~Xr).
(40)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
15
Trigonal m o d e e
When the magnetic field is applied along a three-fold axis, e.g. [111], the new symmetry (trigonal symmetry) implies that the only non-zero values involved are 1 M ( J x ) =
,
(41)
( P x y ) = (Pyz) = ( P z x ) = P ,
(42)
and e 2~ =
E 1 ~
e 3~ ~
e"
(43)
The perturbating Hamiltonian takes the following form 1
~1 =
V ~ gJld"B(H "[- lq'm)(Jx -[- Jy -Jr Jz) 3 ~-~e[
ex2
- (B~e ~ + K P)(Pxy + Pyz + Pzx) + ~ C o t e ) + ½nM2 + 3 K ' p 2
.
(44)
For the calculations it is more convenient to rotate the coordinate axes so that [111] is the new z-axis (see appendix 4). In this new axes system, the same perturbation theory may be carried out to obtain the trigonal free energy F ". Then applying an inverse rotation allows us to express the latter in the initial coordinate system with four-fold symmetry F ~ = Fo _ ½Xo( H + n M ) 2 - ~ x9~ ( B
~e + K ~ P ) 2
- 3X~2)(Bee" + K ~ P ) ( H + r t M ) 2 - 1X(e3)(n + n M ) 4 3~e," + ~t~ol, e 8 \)2 + ½ n M 2 + 3K~p2
(45)
Here X0 is the same as in F r [eq. (30)] because of the isotropy of the (first-order) magnetic susceptibility, but the other CEF susceptibilities X~, )~z) and x~" (3) differ from the corresponding X~i) 's (see appendix 4). This yields the anisotropy of the properties related to the quadrupolar variables. The equilibrium conditions for M, e ~ and P are OF ~
OF ~
OF t
OM
Oe ~
3P
-0,
(46)
whence the following expressions .
(3) Lr3
M = XMH + X M I,
e = ~BP ,E Co
p
=
XpH 2
,
(47) (48) (49)
16
P. MORIN and D. SCHMITT
with XM as in eq. (35), F
1
X(M3 ) -
[)(~3)
4
+6
(1 - n X o )
1 - 3G~x, ] '
X(2) Xe = (1
-
nx0)2(1
(51)
3G~x~)
-
(50)
In the limit of zero field, OF~/OP leads to the strain dependence of the trigonal quadrupolar operators, 3B~x~ e ~ P = 1 - 3K~x, '
(52)
and the corresponding elastic constant, 1 d2F"
c . . 3 . d(e~) . .e
Co
C~ CO
•
3(B~)ZX~ 1 - 3K~x, '
(53)
or
1 - 3G~x~ ~ 1 - 3K X,
(54)
2.2.2. Generalization of the susceptibility formalism Symmetry considerations allow us to easily extend the preceding expressions for a magnetic field in any direction. For the third-order magnetic susceptibility, symmetry considerations show that there are two independent parameters in the expansion of the magnetization to order H 3 (Birss 1964), the general expressions are
Mx
x (3))I-I (I-Iy 2 + I-I z) ,
.~ (3) /At3 + . (3) Llr3
(3)
2
(3)
2
/ I ( M y I I Y ._[._ 1
(3) - XM~,)H,(H~ + H~Z), (3XM~
(3)/_/3 m z = X M , , / I I z .Jr- i
. (3) -- XMr)Hz(Hx + H y2) . (3XM~
.
.
(55)
. (3) and XM~ - (3) are the total third-order magnetic susceptibilities associated Here xa~:, with the four-fold and three-fold axes, respectively [eqs. (36) and (50)]. This allows us to predict the H 3 dependence of the magnetization for any direction; e.g., applying the magnetic field along any two-fold direction leads to the following relation
X~)l
(56)
In a similar way, symmetry considerations lead to only two independent
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
17
parameters in the expansion of the quadrupolar susceptibilities to order H 2 ( 0 2 0 ) ~ ~XQ(2H~ 1 2 (O~)__
- H~2 - H y2 ) ,
1 2 ~XQ(Hx-H2y),
(57)
18
P. M O R I N and D. SCHMITI"
10
c° "~ I0
0
."
¢ .. /" --
0
20
40
0
q
20
TEMPERATURE
2ooi
-10
[.;t,)
--
40
(K)
Fig. 1. Typical temperature behaviour of the third-order magnetic susceptibility x~ - (3) for the two modes 3' (tctragonal) and e (trigonal) of cubic symmetry, and with a magnetic triplet (a)~ a singlet (b) and a non magnetic doublet (c) as ground state (CEF levels spacings are indicated). Inset of (a): corresponding temperature behaviour of the reciprocal susceptibility 1/Xo (dashed line: behaviour without CEF). (a)
'
--
G
- -
F,
'
r;'~ 1
80
J=6
~0
10
v
i
!
(e)
-
-
r,,
i
(d)
~ --
~5
J=6
20C
"
\
j =~
r;' r;" 100
10(
i
100
TEMPERATURE
'
lb0
(K)
Fig. 2. Typical temperature behaviour of the strain susceptibility X~ for both modes y and e, with a magnetic triplet (a), a singlet (b), a nonmagnetic, quadrupolar doublet (c) and a magnetic, nonquadrupolar Kramers doublet (d) as ground state. C E F levels spacings are indicated.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
19
triplet/,~t) (fig. 2a) or a quartet Fs, there is an intrinsic quadrupolar moment and the X,'S diverge at 0 K. Finally, the non-magnetic doublet F 3 (fig. 2c) is quadrupolarly active for the 3' mode but inactive for the e one, leading to a Curie and a van Vleck behaviour for X~ and X~, respectively. In all cases, the high-temperature variation of X, follows a 1/T law, as does X0-
Quadrupolar field-susceptibilities: X ~) The low-temperature variation of quadrupolar field susceptibilities depends on both the magnetic and quadrupolar character of the ground state. Thus a pure van Vleck-type behaviour is expected only for a non-magnetic and non-quadrupolar level, i.e. a singlet (fig. 3b). For a Kramers doublet, the presence of an intrinsic magnetic moment leads to the divergence of x~, - (2) (fig- 3d). For the non-magnetic doublet F 3 (fig. 3c), the van Vleck-type behaviour is observed only for the e mode for which there is no corresponding intrinsic quadrupolar moment. For the other cases, a divergence occurs at low temperatures. A very special case arises for the triplet F~ t) (fig. 3a), where the competition between the various components of the level leads to a change of sign of x- r(2) as a function of temperature: this leads to the so-called "reverse magnetostriction effect" (see section 3.4.1). Also, at high temperature, a 1/T 2 law is expected.
(a) 20
- -
r+
_ _
F'~ 1
8
100
\
--->
I --
r,,+++
--r]
10
1
----[-i" J=B
--Q J= 6
P,i
\ (b)
0
T
0 v
i
-1(
i
(c)
- -
r,
--15.
~>~
- -
F~2)
J=6
2°°~T
F~
40
t/ I\
\\+
100
TEMPERATURE
'++q
,.,
L = ,-+,,
20
100
( K)
Fig. 3. Typical t e m p e r a t u r e behaviour of the quadrupolar field-susceptibility X~ ) for both m o d e s 3' and e, with the same ground states as in fig. 2.
20
P. MORIN and D. SCHMITT
2.2.4. Magnetic and q u a d r u p o l a r transitions
In the presence of quadrupolar interactions, rare earth systems are characterized by up to five quadrupolar order parameters (Sivardirre et al. 1973) in addition to the usual magnetic order parameter M = g J P ~ ( J z ) . For cubic symmetry, there are only two independent order parameters, namely Q = (O °) and P = ( P i i ) , associated with the two quadrupolar parameters G r and G ~. In most cases, the bilinear interactions are dominant and drive one single magnetic phase transition, since the quadrupoles necessarily follow the dipoles as they magnetically order. Nevertheless, the quadrupolar interactions may influence the nature of this transition (see section 3.2). In a few cases, the quadrupolar interactions are dominant and drive a quadrupolar ordering without any magnetic dipolar ordering (see section 3.1). Several authors have investigated the phase diagrams associated with the presence of magnetic and quadrupolar interactions, with the frequent restriction of considering effective spins, i.e. by taking into account only the low lying levels. The cases S = 1 (Chen and Levy 1973), S = 3 (Sivardirre and Blume 1972) and S = ~ (Chen and Levy 1971) have been treated. Also the case of the F3-F5 magnetic system (S = 2) has been considered, explicitly including the related crystal field effects (Ray and Sivardi~re 1978). These calculations were all performed within the MFA. Numerous phase transitions have been found to occur, depending on the strength and the sign of the various parameters. Uniaxial ferro- (FQ) and biaxial antiferroquadrupolar (AFQ) orderings have been described (Sivardirre et al. 1973). More complex situations have been calculated by simultaneously considering the quadrupolar interactions belonging to both modes y and e, e.g., a ferriquadrupolar phase associated with the Q component followed by a FQ phase related to the P component (Chen and Levy 1973). In all the cases, the nature of the phase transitions was discussed and triple or tricritical points were found. While the existence of the various magnetic and quadrupolar phase transitions can be qualitatively described within the effective spin formalism, accounting for the real situations found experimentally requires one to consider the actual CEF level schemes, since the precise nature of the ground state as well as possible effects of excited levels via van Vleck-type matrix elements may noticeably change the physical properties. This can be done by using the susceptibility formalism developed above (Morin and Schmitt 1983a). Indeed, the expression for the free energy derived in section 2.2.1 can be used to investigate the possible phase transitions by taking H = 0. Equations (30) and (45) are then written as F v = F~ + ½n(1 - n X o ) M 2 - in4-,~,(3),~ar4 ~,~ + ½G~'(1- G r x r ) Q 2 - n 2r,'y t 1 X r(2) Q M 2 ,
F ~ = F 0 + ½n(1 - n x o ) M 2 - zin4"a t(3)hAr4 1,1 + 3G~(1-3G~x~)P 2 - 3n2G~2)pM
z"
(58)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
21
According to Landau theory, magnetic (tD) and quadrupolar (to_ and te) critical temperatures are defined by canceling the coefficient of the M 2, Qz or p2 terms in the expansion of F 1 - nXo(to) = 0,
1 -- G~Xr(tQ) = O,
1 - 3G~x~(tp) = 0.
(59)
The actual phase diagram as well as the nature of the transitions is determined by the relative magnitudes of tD, tQ and t e and by the signs and relative magnitudes of the other coefficients. The occurrence of a quadrupolar phase transition is a particularly interesting situation in cases with cubic symmetry where no quadrupolar component can spontaneously exist. In such a case (to_> t D for example) the transition is generally first order because the next term in the Q expansion of the free energy is of odd parity (Q3). The ordered value of Q is positive or negative depending on the sign of the corresponding coefficient. This coefficient is a second-order strain susceptibility [X(s2) in the notation of Morin and Schmitt (1983a)] and involves quadrupolar matrix elements between the various CEF levels. Note that the quadrupolar transition may be second order if Xs" (2) = 0 and the next term in the Q expansion of the free energy is positive. This occurs when the ground state is a quadrupolar doublet (F3 doublet, /"4, F5 triplets or F8 quartet) (Kanamori 1960). Examples of pure quadrupolar phases are shown in fig. 4 for both modes 3' and e (phases a 0 and ae, respectively). In the presence of dominant bilinear interactions (t D > tQ and tp), the magnetic transition is first- or second order, according to the sign of the next term in the M expansion of the free energy, namely x~ -(3) (/x = y, e). As seen in a previous section, xg -(3) is usually negative, whence a second-order magnetic transition. However, with a F3 doublet as ground state, xz, -(3) becomes positive at low temperature (see fig. lc), according to the characteristics of the low-lying CEF levels, leading to a first-order transition (see fig. 5). The limit between the two situations, i.e. xv - ( 3 ) = 0 , corresponds to the tricritical point C in the phase diagram. In the presence of competing bilinear and quadrupolar interactions, the situa-tion is more complex. First, various magnetic and Q or P quadrupolar phases may exist according to the relative strength of the associated coupling parameters n, G ~, G ~ (see fig. 4). Note in particular the possibility for two types of magnetic phases: (i) the phases b 0 or b e correspond to magnetic order where the ground state, coming from the F~ 1) CEF triplet, has mainly a Mj = 5 component; (ii) the phase co_ may be considered as a magnetic phase induced on the non-magnetic level, which is the ground state in the quadrupolar phase ao, and which has a dominant M : = 6 component. Another interesting feature of competing bilinear and quadrupolar couplings is the modification of the nature of the magnetic transition by the quadrupolar interactions. Indeed, the character of the phase transition is now determined by the sign o f -X M,'O (3) = X~)In=0 (see Morin and Schmitt 1983a). According to eqs. (36)
22
P. M O R I N and D. S C H M I T T
a) 5, / ~ ~ '
b) (]*=0 ~ aKp ~ 8'=2K
5 •
O'=4K/
W 0..
5"
,
)-
0'~=6K
o'=gK
bQ ,fQ 0
10
G I (mY,)
bp 0
100
G2(m K)
Fig. 4. Magnetic and quadrupolar phase diagrams for various bilinear coefficients 0* as a function of the quadrupolar parameters G ~ (=-G 1, left part) or G ~ (-=G 2, right part) for a T m 3+ ion (J = 6). a o, ap = pure quadrupolar phases; bo, bp = magnetic phases; Co: induced magnetic phase (from Morin et al. 1987c).
and (50), a positive quadrupolar parameter may drive the transition to be . (3) is larger than ]X~)[ (see fig. 6). Also first-order if its contribution to xM,0 tricritical points may be observed. The opposite situation may also occur, where a .(3) leads to a first-order transition, but a negative quadrupolar positive value of x~ . (3) changes it into a second-order one. Several examples of this contribution to XM,O will be discussed below (see section 3.2.2). Note that most of the experimental first-order transitions may be explained through the present model and do not require us to invoke critical fluctuations (see K6tzler 1984). In conclusion, the existence of bilinear and 3'- and e-type quadrupolar interactions gives rise to a variety of situations. Analyzing the complex related phase diagrams as well as the nature of the magnetic and quadrupolar transitions is qualitatively achieved within effective spin models, but requires a more realistic formalism in order to take the exact nature of the CEF levels involved into account correctly, and to better understand the experimental situations. This can easily be done through the present susceptibility formalism.
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
J=4 W=-5.SK x=.6B
(El 10
23
I-1 ~, ~ ? [ _ _ F 3
[001]
0 4c; 3O -
~
" -TD=tD
2C 1C I
0
/
I
20 TEMPERATURE(K)
40
Fig. 5. Temperature variation of xr - (a) (___X~3)) calculated for a Pr 3+ ion for the C E F level scheme indicated (upper part). Magnetic phase diagram (lower part); the dot-dashed line is the second-order magnetic transition line if neglecting X~3); the dashed line corresponds to a second-order magnetic transition, the full line (TD) to a first-order one; C is a tricritical point (from Morin and Schmitt 1983b).
~"
10 - -
:~
r 2
f2)
5
G1=13mK
--
r3
I 0 .-.
/
_'
10~-
/ I," 0
.
." T.-t IT.,
/
.~.-~_" . -
,
/
J=6 vv = u . ~ 3
n
x=- .34!
, , [oo ] 5 10 15 TEMPERATURE ( K )
20
Fig. 6. Temperature variation of X~)0 calculated for a Tm 3+ ion for the CEF level scheme indicated, for two G r (~G1) values (upper part). Lower part: magnetic phase diagram for G r = 13 inK; the solid (dashed) line represents a first- (second-) order transition; the tricritical point C occurs when XM,O - (3) vanishes; the dot-dashed line corresponds to a second-order transition which would occur if G ~= 0; T o is the quadrupolar ordering temperature, and t r the magnetic ordering temperature within the quadrupolar phase; T o and t o correspond to the first- and second-order magnetic transition, respectively (from Morin and Schmitt 1983a).
24
P. M O R I N
and D. SCHMITT
2.3. Extension to lower symmetries (hexagonal and tetragonal) The magneto-elastic and, more generally, quadrupolar properties have been studied less thoroughly in rare earth intermetallic compounds with a symmetry lower than cubic, e.g. in hexagonal and tetragonal compounds, because of the greater number of parameters involved in the CEF Hamiltonian as well as in the quadrupolar couplings. Only a few systems have been investigated such as metallic Pr (Hendy et al. 1979), dilute yttrium based rare earth compounds (Pureur et al. 1985), and more recently the compound PrNi 5 (Barthem et al. 1988). In this section, the formalism describing the magneto-elastic and quadrupolar properties of the rare earth compounds with hexagonal and tetragonal symmetries is developed. The full Hamiltonian is given, including all the relevant terms for both the one-ion magneto-elastic coupling and the two-ion quadrupolar interactions. Then analogies and differences with respect to the cubic symmetry are emphasized. 2.3.1. The Hamiltonian The general Hamiltonian appropriate for hexagonal and tetragonal symmetries takes the same form as eq. (1). However, the detailed expressions for most terms are different from those of cubic symmetry and are given below (see Morin et al. 1988). The CEF Hamiltonian The CEF Hamiltonian may be written, in terms of the Stevens operators (Hutchings 1964) and in the axes system where the z-axis is parallel to the c-axis of the hexagonal or tetragonal unit cell, as, ~Hex CEF =
~a'et
CEF =
0
0
0
0
0
0
6
(60)
6
B2 0 2 + B404 + B606 + B606 ,
0 0
0 0
4 4
0 0
4 4
B2 02 + B 404 + B404 + B606 q- B606 •
(61)
The bilinear Hamiltonian For hexagonal and tetragonal symmetries, the existence of one privileged direction, namely [001], dictates that there are two invariants in the full bilinear exchange coupling. Their expression is the same in both symmetries and takes the following form within the molecular field approximation ~ 1 = _ ( gjlJ.B)2nC~l ( j ) j , 2
~ B 2= --(g, lxB) n
~2
( 2 ( J z ) J z - (Jx)Jx - ( J y ) J y ) .
(62) (63)
The first term corresponds to the usual isotropic Heisenberg-type bilinear coupling, while the second term is an anisotropic bilinear coupling, forbidden in cubic symmetry, which leads to effective n coefficients which are different parallel and perpendicular to the c-axis in hexagonal and tetragonal symmetry. An experimental way to estimate the magnitude of this a2-term, would be to investigate the
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
25
anisotropy of the magnetic susceptibility in the gadolinium compounds where there is, in principle, no other source of anisotropy. The two-ion quadrupolar Hamiltonian In hexagonal and tetragonal compounds, the O ° quadrupolar component is already ordered (i.e. (O2°) ¢ 0) by the crystal field, but that does not exclude the possible existence of quadrupolar pair interactions, as in cubic symmetry. However, due to the lower symmetry, more than two quadrupolar parameters are involved, and the corresponding Hamiltonian may be written, in the MFA, as a function of the linear combinations of products of second-order Stevens operators, which are invariant under the symmetry operations of the hexagonal or tetragonal point groups: ~Hex
o
=-K
a
0
0
2
(O2)O 2-KÈ[(O2)O2+4(Pxy)Pxy ]
- K~[(Pyz)Py~ + ( P z ~ ) P ~ ] , ~o Tet
~o
0
=
-
K~(02)02
0
7 -
-- K ~ [ ( P y z ) P y z
K
+
2
(64)
2
( 0 2 ) 0 z - K (Pxy)Pxy
(P~)P~].
(65)
The magneto-elastic Hamiltonian As in cubic symmetry, the one-ion magneto-elastic couplings may be considered as the strain derivatives of the crystal field term itself. Within the same harmonic approximation, and limiting ourselves to the second-rank (quadrupolar) terms, the full one-ion magneto-elastic Hamiltonian is written as:
~Hex ME1 = __(B,~le~l
~-
B,~2e,~2)O0 _ B ~(elO ~ 2z + 2e2Pxy ~ ) _ B¢(e~pyz + eIpzx) ,
(66) ~ TME1 et =_(B,~lal
+ B.aa2)
0 2 o --
B,s,O~_Baeapxy
--
B~(s~pyz+e2Pzx).
(67) In these expressions the e~'s are the symmetrized strains (Du Tr6molet de Lacheisserie 1970) (see table 3) and the B"'s the magneto-elastic coefficients associated with the corresponding normal strain modes. Note that there are two independent fully symmetric a-terms, related to the two deformations which maintain the initial symmetry, namely the volume al-strain and the axial a2-strain, the latter one is associated only with a change of the ratio c/a. For the other terms, a symmetry lowering towards orthorhombic or even lower symmetry Occurs.
The two-ion magneto-elasticity is related to the modification of the two-ion magnetic interactions by the strains. In this section, we restrict ourselves to the two a-type bilinear couplings. For each one, two strain modes must be considered; they are associated with volume and axial strain dependences of bilinear interactions. For hexagonal and tetragonal symmetries, the corresponding two-ion
26
P. M O R I N and D. SCHMITI" TABLE 3
Normalized, symmetrized strains for the hexagonal and tetragonal symmetry. Hexagonal
Tetragonal
Expression
Representation
Strain
Representation
Strain
5
~°~
/'1
~°~
-~v~(~ + ~. + ~z~)
5
~°~
5
~°~
~ ( ~ - ½(% + ~x))
5
4
r4
d
v~ %
F6
eg
F5
s;
V ~ G~
magneto-elastic Hamiltonians take an identical form, and may be written in the M F A as o~1
~ME2
~--- t'r~otl otl D ~ 2 e ~2 --~LIalE At- al
gME2a2 = --IZDaXa2 E ~
+
)(j)j
(68)
D~2 J e~2)(2( z a 2L )
-
(L)L
- (Jy)Jy).
(69)
The elastic energy The elastic contribution to the free energy may be expressed as a function of the strains s" and of the appropriate symmetrized elastic constants C~" ( D u Tr6molet de Lacheisserie 1970). For hexagonal and tetragonal symmetries, they are given by:
EHeX : 1 C 0 1 ( G a l ) 2 q_ CO12Eo~IEa2 + 1 C 0 2 ( E a 2 ) 2 el
+ ~Co[(~1)2 + (~)21 + ½Coq(~)2 + (~)21, ETet e, = 1
Co,(eal)2 +
(70)
C012EotlEo~2 + ~ 1t ~..~ao 2 ,ks -ot2\2 ) l z~/
6\2
1
e
e 2
+ ½G(~) 2+ : ~ o ~ ) + ~Co[(<) + ( 4 ) q -
(71)
The background elastic constants without magnetic interactions Co~ are given in table 4.
TABLE 4
Symmetrized elastic constants in hexagonal and tetragonal symmetry.
Hexagonal
Tetragonal
Expression
C al
C ~1
½(2Cn +2C~2 +4Ca3 + C33)
C~2 C~2
C~12 Co2
C~ C~
C~ C~
_
C ~
-~V~(G~ + c,~ - c . - G~) 3'(C n + C ~ 2 - 4 C ~ + 2 G 3 ) C n - C12 2G~ 2(2"66
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
27
The two-ion corrective energy terms As in cubic symmetry, the molecular field treatment of the two-ion interactions requires the usage of corrective terms in the deduced single-ion free energy. According to the two-ion couplings considered above, three corrective terms have to be taken into account, corresponding to the bilinear, the quadrupolar and the two-ion magneto-elastic Hamiltonian, respectively.
Hexagonal symmetry EB'= ln~lM2,
(72)
]IA2 2 = ~1 n,~2[,-) t .... z - Mx2 - My) ,
E~2
. 1KO: < o O > 2
EHeX
o
2 2
+ ½K"[
(73) 1 ' ~K [
+4
(74)
j~?Hex l,~cd ~1 _}_'rja2Ea2)
[2(L) 2- (L) 2 -
<4>2]
(75)
.
Tetragonal symmetry E~ 1 and E~ ~ are the same as for hexagonal symmetry (eqs. 72 and 73), and 1 y 2\2 rTset=½K~(Oz°)Z+~K (O2/ + ~1K 8 (Pxy) 2 + ½ K * [ ( P y z ) 2 + ( P z x ) 2] (76)
ESet ME2 is the same as for hexagonal symmetry [eq. (75)].
2.3.2. Harmonic description of magnetostriction The equilibrium values of the strains eU's are obtained by minimizing the total free energy with respect to the strains. This leads to relations analogous to those for cubic symmetry, but they involve appropriately symmetrized coefficients.
Hexagonal symmetry E~I
=
[(BalCo2
+
- B
l g FI ot l['~ae2
X [ColCo
2 --
a2
~12
0
l_/rlalt,'~2
oe2 ,~12
C O ) ( O 2 ) + 2\~.-al~0 -- D~Co ) ( J ) a2
:Co~12 )(2
-D
>2
2
- <4)2)]
(Co'2)21-1 ,
(77)
e ~2 =[(B~2Co I _ B~1C012)( O ° ) + !gD,~2C,~1 2, =1 o - D ~ul I C ou12) ( J ) 2 1 //-,i ~ 2 / - ~ 1 + 2~,"~2~o -X [ColCo
B~
ea = ~
B ~
(022),
e~ = C'-~o(Pyz>,
al 412 Da2Co )(2(L)2 - (L) 2 -
2 - (Co12)2] -1 ,
~
2B ~
e 2= Co" (P~y),
(jy)2)] (78)
(79)
B ~
e~2= C---~o(Vzx)"
(80)
28
P. M O R I N and D. SCHMITT
Tetragonal symmetry e a l and e oe2 are the same as for hexagonal symmetry [eqs. (77) and (78)], and BT
e ~=
-
C~ (O~)
(81)
-
B ~
8 = ~g ( P x y ) , Co
(82)
B *
B ~
e~ = -~o (PYz)'
e2 = ~
(Pzx).
(83)
In the same way as for cubic symmetry, including these equilibrium values in the starting Hamiltonian allows us to express this Hamiltonian in a form which depends only on magneto-elastic coefficients. For example, considering only one-ion magneto-elasticity leads to t h e following total quadrupolar Hamiltonian ff(Hex QT = _ Goe(O2°) O ° - G~[(O~) 0 2 + 4(Pxy)Pxy]
-- G¢[(Pyz)Pyz + (Pzx)Pzx], ~fTet a 0 ~ Q T : -- G < 0 2 ) 0 2
0
2 2 __ G T ( 0 2 ) 0 2
(84) a a
-
-
(Pxy)exy
-- Ge[(Pyz)Pyz + (Pzx)Pzx],
(85)
where the total quadrupolar coefficients G ~ have contributions from both the one-ion magneto-elasticity and the quadrupolar pair interaction:
Hexagonal a2 (B a l ) 2 C O - 2B'~IB~2Co 12 + (B~2)2C01
G~ =
G ~, _
C01C02
(B~)
2
C----~"- +
K ~,
_
~
(C012) 2
: GME +
K~
;
+ R °e = G M E
(/*
=
e, ~'),
"{- K a ,
(86)
(87)
Tetragonal G oe is the same as above (eq. 86), and G~-
(B'~) 2
+ K ~ = C M~E + K ~ ;
=
(88)
2.3.3. Perturbation theory An analysis of the magnetic and magneto-elastic properties of hexagonal and tetragonal compounds within the paramagnetic phases may be done by using perturbation theory. The full development of the susceptibility formalism is treated elsewhere (Morin et al. 1988) and only the main results are reported below.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
29
Among the different modes, only the a- and e-modes in hexagonal symmetry (a-, y- and 6-modes in tetragonal symmetry) can easily be investigated by applying a magnetic field along the main symmetry directions, namely the [001], [100] and possibly the [110] directions. The quadrupolar components involved are ( O ° ) , ( 0 2) and possibly (Pxy)" Various associated susceptibilities may then be defined, namely strain susceptibilities X, and quadrupolar field-susceptibilities X~ - (2) (/z = a, e . . . . ). The only difference with respect to cubic symmetry is the explicit presence of a spontaneous quadrupolar moment (O°)0 in the expressions for X~, and Xtz -(2) • Also, the field dependence of the magnetization and quadrupole moments may be determined. For the quadrupolar components, relations analogous to those for cubic symmetry are found for hexagonal symmetry =
(89)
(9o)
2 ,
with ((2)
X~ = (1 - nx0)2(1 - G " X g ) '
(91)
and similar relations for tetragonal symmetry. For the magnetization, the following expressions are found M=
XMH--
4- . (3) L/3 X M 11
,
(92)
with X0 XM
--
,
(93)
1 - nXo
and X~ ) -
G a (x~(2),~2 ) . (2)-~2 q] [ (/l(e I |X~ 3) + 2 +2G ~ . (1 -- rtX0) 4 t_ 1 ~ G-~Xa 1 --- G- ~ 1
(94)
In these relations, all the susceptibilities assume different numerical values according to the direction of the applied magnetic field, although their expressions are formally equivalent. This is due to the different CEF wave functions determined by the field direction. This behaviour is well-known for the first-order magnetic susceptibility which is different for fields parallel and perpendicular to the c-axis in hexagonal and tetragonal compounds. For the third-order magnetic susceptibility, it turns out that only the G" contribution is present when the field is applied along the c-axis, while both G ~ and G ~ terms must be considered when the field is within the basal plane. Note that the y- and 6-modes in cases with
30
P. MORIN and D. SCHMITT
tetragonal symmetry may be separately investigated by applying a magnetic field along the [100] and [110] directions. Finally, only the isotropic bilinear exchange coefficient n ~1 has been considered here and is denoted n. 2.3.4. Comparison with cubic symmetry The main difference between hexagonal and tetragonal symmetries compared to cubic ones comes from the fact that the quadrupolar component (O~) is already ordered (i.e. ( 0 2°) # 0 ) by the crystal field. This leads to several consequences for the magnetic and magneto-elastic properties. First of all, from eqs. (77) and (78), there exists a spontaneous quadrupolar contribution to the volume (through e ~1) and to the ratio c/a (through e ~2) of the hexagonal and tetragonal unit cell. This feature manifests itself through an anomalous thermal expansion with regard to that of the corresponding compound with non-magnetic lanthanum or yttrium. This has been observed, e.g., in Pr (L/ithi and Ott 1980) and PrNi 5 (Barthem et al. 1988). Second, the a-term in YgOT[eqs. (84) and (85)] gives an additional quadrupolar contribution to the pure CEF second-order Hamiltonian as soon as either B ~1, B ~2 or K ~ is present. As for (O2°), this contribution is temperature dependent; this is a possible explanation for a thermal variation for the total effective second-order crystal field parameter B°2. As a consequence, the CEF level spacing will be temperature dependent, and it is worth taking this contribution explicitly into account besides the pure CEF one, at least in the case of strong quadrupolar coupling. The presence of a quadrupolar moment (020) does not exclude the possible existence of the corresponding quadrupolar pair coupling K ~. However, a quadrupolar phase transition involving this coupling is no longer allowed. This behaviour can be compared to the effect of high rank (fourth- and sixth-rank) pair interactions in cubic compounds which lead to apparent temperature-dependent fourth- and sixth-order CEF parameters (Morin and Williamson 1984). On the other hand, quadrupolar transitions involving the other couplings as K ~, K ~ (K r, K s, K" in tetragonal compounds) are quite possible and have been observed in several rare earth insulators (Gehring and Gehring 1975). 2.4. A peculiar case: the semi-classical treatment
In the case when bilinear interactions overwhelm the CEF coupling, the magnetic ordering temperature is large in comparison with the total CEF spacing. In the paramagnetic phase, the CEF levels may be considered as equally populated, while in the ordered phase, each of the CEF wave functions is "purified" by the exchange field to its dominant IJz) component, at least in the limit of low temperatures. Both situations may then be discussed within a semi-classical analysis. 2.4.1. The paramagnetic phase For vanishing CEF effect, the 4f multiplet is (2J + 1)-fold degenerate and the expressions of the various one-ion susceptibilities previously introduced become
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS C Xo = T
(95)
CQ T'
X~X(2)
(96)
2
2
CQ
(97)
= gjtx B 6 T 2 , C (3)
X(3) =
X(21 s
31
~--
r 3 ,
(98)
C~ ) T2 ,
(99)
where C is the usual Curie constant [see eq. (5)], and CQ, C ~ ) are quadrupolar constants and C (3) is the third-order Curie constant
CQ =
J(J + 1)(2J - 1)(2J + 3) 5k B ,
(lOO)
C(o2) = J(J + 1)(2J - 1)(2J + 3 ) ( 2 J - 3 ) ( 2 J + 5) 70k 2 C(3)
4 4 J(J+l)(2Jz+2J+1) = -gjtx B 90k 3
(101)
(102)
In the same way as C and C (3) are the coefficients of the first two terms in the H~ T expansion of the Brillouin function, CQ and C ~ ) are the coefficients of the first two terms in the s / T expansion of the response function of Q in the Brillouin-Boltzmann statistics 2
e C~ ) e Q ~ Ca -~ + ~5 +'".
(103)
Note that all the C's are isotropic. After renormalization by the bilinear and quadrupolar interactions, the actual susceptibilities . are
C XM -- T - O *
'
2 2 Co.T XQ = gllXB 6 ( T - 0 * ) 2 ( T - 0~) ' C(3)T
(i-U)
[1-
4
L
(lO4)
(2J-1)(ZJ+3)
?-2
75;-T
-T - -' O ~ J '
32
P. M O R I N and D. S C H M I T F
and the actual elastic constant is given by
C
T-O~
C0
T - 0~: "
(lo5)
This equation is very close to that given by Kataoka and Kanamori (1972). O~ = CQG and 0~ = CQK are defined by analogy with 0", as functions of the total and two-ion isotropic quadrupolar parameters G and K. The temperature dependences in eq. (104) have been extensively described by Morin and Schmitt (1981b) and Morin et al. (1980b) at high temperature as well as close to the magnetic ordering temperature. Less attention has been paid to the nature of the quadrupolar transition. It is easy to verify that for J = ½there is no quadrupole moment. For J = ~, C~2) is zero and the next term in the e expansion of Q is negative, the transition is then second-order. In all the other cases, C~ ) is positive and the transition is first order.
2.4.2. The ordered phase Several discussions of the energy in the ordered phase have been given in the past in particular by Clark in volume 1 of these handbooks (Clark 1980). Only those features that are necessary for describing properties encountered in the following sections will be derived here from our basic Hamiltonian. Following Callen and Callen (1963, 1965), the temperature dependences of the average value of the different operators O~' present in eq. (1) are expressed as functions of the normalized 4f magnetic moment, m, through modified hyperbolic Bessel functions. Their angular dependences are written as functions of the 4f magnetic moment cosines, ai, by using the harmonic polynomials. ( 0 O) = j ( j _
l ~
~)Is/2(~
-1
2
(m))(3a 3 - 1),
( 02> = J(J - 1)Is/2(~/~-1(m))(ot
2 -
ce2),
(Pij) = J ( J - ~) 1 ~1s / 2 f(~( -mi ) ) ( a i % ) (O04+504)=_40J(j -- ½ ) ( J _ l ) ( j _ 3
(i,j=1,2,3) 2 ~ 22 + ) I"9 / 2 ( ~ ' - 1 ( m ) ) ( O l l O
{O~ + 2 1 0 6) = 1 8 4 8 J ( J - ½ ) ( J - 1 ) ( J ^
-1
(106)
2
2
3)(j_2)(j_ 2
2
2
cycl. - ½)
~) 2
2
2
2
x I13,2(5g (m))[c~,c~2a 3 _ 1(12ct3 + c~3a a + a,t~2) +
~1
.
Du Tr6molet de Lacheisserie et al. (1978) have studied in detail the classical free energy of a cubic ferromagnet. The classical magnetostriction constants and anisotropy constants are found to be 0 --- 0 J ( J - ½)I5/2(5¢-1(rn)),
A100(T) = 2
Cll
-- C12
B ~
AHa(T ) - 3v~cO4 J ( J - ½)_15/2(~ ~(m)),
(107)
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
33
K I ( T ) = - 4 J ( J - ½) ( J - 1)(J - 23-)[10B419/2(Gt?-'(m))
+ 42(J - 2)(J -
G(T) =
5 )B6~13/2(~ '
-'(m))],
~)B6?~l~/Z(iLf-l(m)).
1848Y(Y- 1 ) ( j _ 1 ) ( J - k ) ( J - 2 ) ( J -
(108)
The K 1 anisotropy constant receives two contributions from 4th and 6th rank polynomials. Owing to their different temperature dependences (fig. 7) they may be the causes of a change in the easy magnetization direction in the ordered phase. It is also well-known that the magnetostriction contributes to the magnetocrystalline energy, e.g. AK 1
=
9
-- ~(Cll
0
0 2 _ C,2)Aioo(T )
-I-
0 2 79 C44~,,11(T)
(109)
.
This magnetostriction contribution may be an appreciable part of the intrinsic K 1 constant: it reaches ~ of K 1 in TbFe 2 at room temperature [see table 8 in Clark (1980)]. In addition, due to different temperature variations, K 1 may change of
I
I
I
I
I
I
I
I
Is/2 10"1 19/2
10-2
113/2 10"3
1.0
.8
.6
.4
.2
0
m
Fig. 7. Normalized hyperbolic Bessel functions J5~2, 19/2 and I13/2 versus the normalized magnetization (from Clark 1980).
34
P. MORIN and D. SCHMITI"
sign. This situation occurs, e.g. in hexagonal RC% compounds as revealed by the variation of the lattice parameters and elastic constants (se e section 4). This classical description of magnetostriction and magnetocrystalline anisotropy gives a phenomenological connection between their temperature variation and that of the 4f magnetic moment. The details of the magnetic couplings leading to this observed magnetization itself are not of too great importance. It is useful in complex situations, as e.g., for rare earths dissolved in a magnetic metal (Fe, Co, M n , . . . ) . Indeed, at least at low temperatures, the magneto-elastic coefficient, in particular for the symmetry associated with the spontaneous state, may be determined by using the assumption that the 3d magneto-elastic contributions are negligible. However, both the necessary separation of the 4f and 3d magnetic moment and the existence of many sublattices limit the insight in the 4f magnetism. Obviously it would be unrealistic to search for quadrupolar pair interactions, when they are overwhelmed by bilinear ones. 3. Experimental evidence
3.1. Quadrupolar orderings To observe quadrupolar orderings it is necessary to find rare earth systems where the bilinear interactions are weak compared to the quadrupolar interactions. Favourable situations exist at each end of the lanthanides series. Indeed, first, the bilinear interactions are at their minimum for the series of compounds under consideration and, second, both the orbital moment L (Tm a+ : J = 6, L = 5; Ce 3+ : J = ~, L = 3; pr3+: J = 4 , L = 5 ) and the Stevens coefficient as, which deeply control the quadrupolar coefficients (Morin and Schmitt 1981a, Schmitt and Levy 1985), are large. In other cases, magnetic and structural transitions coincide (section 3.2.2). Quadrupolar orderings have been reported to occur in TmZn (Morin et al. 1978b), TmCd (Al6onard and Morin 1979), CeAg (Ushizaka et al. 1984, Morin 1988), PrPb a (Bucher et al. 1972a), TmGa 3 (Czopnik et al. 1985), CeB 6 (Effantin et al. 1985) and PrCu2 (Ott et al. 1977a). The first three of them are clearly ferroquadrupolar (FQ), the next three antiferroquadrupolar (AFQ) and the actual situation in the last one is not yet fully elucidated. UPd a which exhibits a structural transition may be relevant to the present review owing to its localized 5f electrons (3H 4 multiplet) (Andres et al. 1978). Structural transitions have been also observed in the RB 4 series, however, their origin has not been fully understood (Will et al. 1986).
3.1.1. Ferroquadrupolar ordering As for the ferromagnetic ordering (FM), the FQ ordering is characterized by a q = 0 propagation vector and is expected to be closely described by the singlesublattice formalism (see section 2.2).
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
35
3.1.1.1. TmZn. Cubic compounds within the CsCl-type structure are characterized by large quadrupolar interactions, TmZn being the archetype compound (Giraud et al. 1985). The temperature variation of the specific heat (fig. 8) reveals two well-defined transitions, close to each other (Morin et al. 1978b). TmZn undergoes a FQ ordering at T e = 8.55K, where ( O °) is the order-parameter. The transition is clearly first-order. An associated tetragonal strain takes place simultaneously due to the B y magneto-elastic coefficient. A FM ordering occurs in the quadrupolar phase at T c = 8.12 K, with the spins lying along the four-fold axis. The spontaneous tetragonal strain is reinforced by the bilinear interactions and reaches c/a - 1 = -8%0 at 1.5 K. From the jumps in the electrical resistivity at the two transitions, the electronic scatterings of quadrupolar and magnetic origins are of similar magnitude. Numerous properties investigated in the paramagnetic phase of TmZn have been consistently analyzed with a total quadrupolar coefficient G v= 25 mK; the dominant contribution to G v comes from the pair interactions (Givord et al. 1983). This G ~ value describes the temperature T e and no additional coupling (anharmonicity, higher order terms . . . . ) is needed. The FQ ordering lifts the degeneracy of the F~ 1) cubic ground state and splits it into a doublet and a singlet; the non-magnetic singlet is the ground state in the quadrupolar phase (structure a o in fig. 4 of section 2.2.3). Bilinear interactions are strong enough to induce a magnetic moment in this phase. Substituting L u 3+ for Tm 3+ in Tml_xLuxZn or Cu for Zn in TmCuxZnl_ x decreases the bilinear interactions quicker than the quadrupolar ones and the spin system remains paramagnetic: only the FQ ordering is observed (fig. 9) (Morin and Schmitt 1986). Note that in this case the splitting of the F~ 1) cubic triplet in the tetragonal paramagnetic phase has been observed by inelastic neutron scattering (Morin et al. 1981). 60
!
1
i
50 Tm Zn
o
E 40
30
20
l
T
r,, sss
T 8.12K= .
K , , ~ .
I
1
I
7
8
9
TEMPERATURE
( K )
Fig. 8. Specific heat of TmZn; T c is the magnetic ordering temperature, T O the quadrupolar ordering temperature (from Morin et al. 1978b).
36
P. MORIN and D. SCHMITT
10
% TmCu
v
Zn 1-c c
p
LU mr
2 ¢rUJ O.
;* 2 5 ~
:E
--.
hi
-_
.
,
/
f--
-5
20
/
oG 1 //
0.5
,,
15
)
C
Fig. 9. Phase diagram for TmCUl_cZnc; P: paramagnetic, M: modulated, AF: antiferromagnetic, F: ferromagnetic, Q: quadrupolar phases; lines are guides for the eye. Variation of the quadrupolar parameter G r (=G~) and bilinear exchange parameter 0* with c (from Morin and Schmitt 1986).
3.1.1.2. TmCd. Historically, the first FQ ordering in a rare earth intermetallic was observed in this compound at T o = 3.16 K. It was initially analyzed with a F3 cubic ground state, and the first-order character of the transition was assumed to be driven by anharmonic elastic effects (L/ithi et al. 1973b). However, RZn and RCd isomorphous compounds exhibit very similar properties, in particular the CEF (Alronard and Morin 1985). As a matter of fact, there is a close parallel between TmCd and TmZn and all the TmCd properties are fully explained with a G ~= 12 rnK total quadrupolar coefficient without the need of any other coupling (Alronard and Morin 1979). As in TmZn, pair interactions are the dominant contribution to the quadrupolar coupling and the ground state is also the non-magnetic singlet coming from the F~ 1) cubic triplet. Here, however, the spin system remains paramagnetic owing to the weakness of the bilinear interactions. As in TmZn, electrical resistivity measurements give clear evidence for the vanishing of the quadrupolar scattering below T o (fig. 10). 3.1.1.3. CeAg. CeAg is the only example of a FQ ordering for a Kramers ion. It
also belongs to the family of CsCl-type compounds. The cubic ground state is the F8 quartet, well-isolated from the F7 excited level (A = 266 K) (Schmitt et al. 1978). It exhibits a cubic-tetragonal transition at T o = 15.85 K with (O2°) as the order parameter (fig. 11). A FM transition occurs at T c = 5.2 K, the c axis of the tetragonal cell is the easy magnetization direction (Morin 1988). The c / a - 1 strain is very large (1.7% at 1.5 K). A strong dependence of the low-temperature properties on the metallurgical conditions is observed, which is related to the very large magneto-elastic coupling.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
37
5.6 I
,
,p..."
E 5.4
de_' dT
1
°
I.-te
E
W °
5.0
•"°
0
°
°¢,°°
""
I
i
2
½
t
~
n
I
4 i
4
3
TEM PE RATURE(K) Fig. 10. Low-temperature variation of the resistivity and its derivative (inset) in TmCd (from A16onard and Morin 1985).
u
30 CeAg
_~2o E
0
5
10
1~i
20
T (K) Fig. 11. Specific heat of CeAg. Anomalies at T c = 5.5 K and TQ = 15.5 K correspond to the ferromagnetic and the quadrupolar orderings, respectively (from Ushizaka et al. 1984).
The study of the quadrupolar properties in the cubic phase leads to G r = 150 inK. However, this value would drive a FQ ordering at only 10 K instead of T o = 15.85 K (Morin 1988). This difference is consistent with the weak softening of the Cll - C12 elastic mode (Takke et al. 1981) and reveals the existence of an additional term, i.e. the anharmonic elastic coupling of the Cll - C12 mode with a zone-boundary phonon, which softens at the M point of the Brillouin zone (Knorr et al. 1980). This last mechanism is associated with the incipient instability of the CsCl-type lattice for light rare earth-silver compounds which is of electronic origin (see section 3.10).
38
E MORIN and D. SCHMITI"
3.1.2. Antiferroquadrupolar ordering Several types of AFQ structures may exist. First, the lifting of the degeneracy of the cubic ground state may lead to wave functions different from one site to the other (Sivardi~re and Blume 1972). If the absolute values of the quadrupolar moments are different for the sublattices, that corresponds to a ferriquadrupolar ordering. Second, the ground state can be the same on each site, but the local z-axis may change, although belonging to a given star of crystallographic directions: the corresponding quadrupolar arrangement may be called multi-axial (Sivardi6re 1975). Lastly, more than one propagation vector may be necessary to describe the AFQ structure. In all the cases, magnetic structures with many sublattices may be favoured (see section 3.6). Whatever the actual structure may be, a multi-sublattice formalism is needed. The previous single-ion treatment may be used only for describing the occurrence of a magnetic or quadrupolar moment within a given sublattice. To describe the AFQ as well as AFM properties, no reliable information can be expected from static measurements in these intermetallics. In particular, the macroscopic symmetry may remain cubic. Only microscopic probes, such as neutron diffraction, are effective in revealing the existence of more than one magnetic sublattice.
3.1.2.1.
C e B 6. This cubic compound exhibits very complex properties due to the interplay of localized and delocalized 4f behaviours. The temperature variation of the specific heat reveals two anomalies at T o = 3.3 K and TN = 2.4 K (Peysson et al. 1986). Neutron diffraction experiments have shown that the compound remains paramagnetic down to 2.4 K (Horn et al. 1981). Magneto-elastic effects are present above TQ, although the C44 and Cll - C1~ elastic modes softens by only 2% (L/ithi et al. 1984). As the F8 ground state is well isolated (A = 540 K) (Zirngiebl et al. 1984), a quasi-full softening is expected in the case of a FQ ordering. On the other hand, the temperature variation of C44 and Cll - C12 may be analyzed with negative quadrupolar pair interactions. An AFQ ordering is then probable below TQ. This is confirmed by a neutron diffraction experiment on a monocrystalline sample in a magnetic field (Effantin et al. 1985). Indeed, two kinds of Ce ions have been observed which differ in zero field only by their magnetic susceptibility. The corresponding superstructure lines are described by a [½ ½ ½] propagation vector. The AFQ ordering then consists of an alternating sequence of (111) planes with different quadrupolar moments (fig. 12). Below TN, the spontaneous biaxial spin structure reflects the existence of the two sublattices driven by AFQ ordering. The magnetic field phase diagram reveals the interplay between localized and delocalized 4f magnetism as demonstrated by the anomalous increase of the AFQ phase under a magnetic field (fig. 13). In zero field, T O and the AFQ phase are reduced by the Kondo coupling but with an increasing magnetic field, the Kondo state is progressively suppressed and the AFQ ordering can manifest itself at a higher temperature.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
Magnetic ph<,.,,,,,
modulated
Anti ferro
~:[~/+ ,/, ,/2]
,
pho,.,,
:
39
quadrupolar
"~:[~]
z
(b)
(a)
Fig. 12. (a) Biaxial magnetic structure in CeB6 described by the propagation vectors indicated. (b) Antiferroquadrupolar structure described by the k o propagation vector; open and full circles correspond to the two different Ce-quadrupole sublattices (from Effantin et al. 1985).
' A- ,~ ' 'o ~,;too~j "-= ~6 *-I~//[111:]
20
t,....~ It ",
140
/
"iT
160
J/
120 100 80
0
+oIl,.,,,..=,,,,,,.+o,. q o!
60
pl ,+,,°,.o., V ,oI+,+..,.+.°,.o/,.
II
+~"'<""'i
0 1 " " @ 0 2
4
.
17 I~L_.
i
TernperaLure(K)
I 8
l
,o
=
0 10
Fig. 13. Magnetic phase diagram of CeB 6. Phases III and II are drawn in fig. 12 (from Effantin et al. 1985).
T m G a 3. This AuCu3-type cubic compound also exhibits two transitions in the temperature dependence of the specific heat, but very close to each other (TQ = 4.29 and Tr~ = 4.26 K) (fig. 14) (Czopnik et al. 1985). Here again, studies in the paramagnetic phase lead us to introduce quadrupolar interactions, but they 3.1.2.2.
40
P. MORIN and D. S C H M I T r
rN 60
50
I 40 30 20 10 o 4.1
4.2
4.3 T(K)
4.4 -
Fig. 14. Specific heat anomalies in TmGa 3. Arrows indicate the antiferromagnetic (TN) and antiferroquadrupolar (TQ) transitions (from Czopnik et al. 1985).
are too weak to induce a FQ ordering at T e (Morin et al. 1987c). In addition the spontaneous spin structure determined by neutron diffraction on a single crystal is multi-axial (see fig. 53 in section 3.6). This indicates there are different (111) directions as local z-axes at the 4f sites stabilized by AFQ interactions of trigonal symmetry (Morin et al. 1987d). Thus the quadrupolar ordering is of AFQ-type. As for the ferromagnetism in TmZn, the conditions of induced antiferromagnetism on the singlet ground state cause the temperature range of the paramagnetic AFQ phase to be very narrow. P r P b 3 . Another AuCu3-type compound, PrPb3, exhibits a non-magnetic transition at TQ = 0.35 K corresponding to the lifting of the F3 ground state degeneracy (fig. 15) (Bucher et al. 1972a). No spontaneous macroscopic strain has been detected in the quadrupolar phase. This may be due either to its weakness in the case of a FQ ordering or to its nullity in the case of an AFQ ordering. The softening of the C44 and Cll - C12 ultrasonic is far from complete and influenced by high-order magneto-elastic terms (Niksch et al. 1982). They may be analyzed by positive as well as negative quadrupolar interactions. However, both the parastriction and the third-order magnetic susceptibility need negative quadrupolar coefficients to be described (Morin et al. 1982). An AFQ ordering is then highly probable, and ( O °) is the order parameter. In conclusion, it appears that the case of AFQ ordering is not as well understood as the FQ ordering. This is due to experimental difficulties as well as theoretical complications. Indeed, none of the previously mentioned three compounds is a completely favourable candidate for investigating AFQ properties. In CeB 6 the quantitative analysis is complicated by the Kondo behaviour of the 3.1.2.3.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
41
0.55"K 0.7
0.6
PrPb 3
0.5
~
0.4
0.~
0.2
0.1
0
I
I
0.5
1.0
T(*K)
Fig. 15. Molar specific heat CJ R of PrPb 3 between 0.04 and 1.0 K (from Bucher et al. 1972a).
cerium ions. In TmGa 3 the temperature range of the AFQ phase is much too narrow. PrPb 3 orders at 0.35 K, a temperature characteristic of very weak AFQ interactions, which may easily be broken by the application of a magnetic field. From a theoretical point of view, the existence of many inter- and intralattice pair coefficients complicates the analysis of the data.
3.1.3. Structural transitions possibly of quadrupolar origin A few other structural transitions have been reported in the literature, which may be evidence for quadrupolar interactions. However, the actual situation is not yet fully determined, often because the crystal electric field is not known.
3.1.3.1. PrCu 2. This orthorhombic compound exhibits a second order structural transition at 7.5 K which has been observed in many different experiments, in particular by specific heat (Wun and Phillips 1974). From elastic constant measurements, the primary order parameter was found to be the ez, strain (Ott et al. 1977a). This agrees with neutron diffraction data which have shown that below 7.5 K there is a sizeable splitting of the (101) nuclear reflections corresponding to an angle/3 between the a- and c-axes of the orthorhombic cell that is no longer equal to 90° (Kjems et al. 1978) (fig. 16). A FQ ordering involving the (Pz~) quadrupolar component is likely. Unfortunately, there does not exist a determination of the CEF; thus a quantitative analysis of the observed properties is not possible.
42
P. MORIN and D. SCHMITI"
PrCu z
T=B.OK
T=4.75 g 1.0
0.5 ,
,
-0.01
0 i
~
~-I
0.01-0.01
0
0.01
k" ="
~ ~OR'¢
0.0
i-
4.0
L
~ I.v
=
A
6.0 8.0 TEMPERATURE (K)
10.0
Fig. 16. The upper part shows high resolution scans through the (101) Bragg peak below and above the structural transition temperature in PrCu2 at T = 7.5 K. In the lower part the observed splitting (A) is plotted versus the temperature. The full line represents a mean field calculation (from Kjems et al. 1978). 3.1.3.2. U P d 3. Contrarily to most of the metallic actinides, the hexagonal compound U P d 3 (P63/mmc) exhibits properties characteristic of localized magnetism associated with a 5f 2 configuration (3H 4 ground-state multiplet) (Andres et al. 1978). In a way reminiscent of Pr metal, two U sites exist with hexagonal and nearly cubic symmetries. T w o C E F transitions have been observed by inelastic neutron scattering on a single crystal. They have been attributed to an excitation process from a singlet ground state for both sites (Murray and Buyers 1980). In addition, U P d 3 exhibits a well-defined transition at 7.5 K, strongly depending on the metallurgical status of the sample (fig. 17). N o change of the hexagonal
4
t2
3 B~
-
ou -~-
,
0
2
4
6
4
8
10
12
14
o~_I
0
16
T(K)
Fig. 17. Specific heat of a UPd 3 monocrystalline sample minus that of ThPd 3 used as background
(open circles). The solid line (right-hand scale) is a plot of dp/dT. The slight difference in peak position of the two curves is caused by inaccuracies in the temperature scale of the resistivity measurement (from Andres et al. 1978).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
43
symmetry was observed from X-ray diffraction experiments on polycrystalline samples (Andres et al. 1978, Zaplinski et al. 1980). However, the increase of the c/a ratio observed at low temperatures might be associated with U atoms becoming more prolate ( ( O ° 2 ) < 0 ) at the quasi-cubic sites, according to an internal ferroquadrupolar process. This assumption might be checked by a full determination of the crystallographic structure on both sides of the transition by using a single crystal.
3.2. Nature of quadrupolar and magnetic transitions 3.2.1. Quadrupolar transitions As discussed in section 2.2.4, in cubic symmetry the transition at the quadrupolar ordering is usually first-order due to the existence of van Vleck matrix elements from the ground state. Indeed for the 6 cases (TmZn, TmCd, CeAg, CeB6, TmGa3, PrPb3) presented in section 3.1, the so called Xs(2~ susceptibility (Morin and Schmitt 1983a), which determines the order of the quadrupolar transition is calculated to be non zero. Thus the observed first-order quadrupolar transitions in the Tm compounds are well predicted by the present formalism. However a quantitative description of the specific heat anomalies is difficult in TmGa 3 and TmZn due to the vicinity of the magnetic and quadrupolar transitions. In TmCd, the measurements are not accurate enough, in particular in the low temperature phase. In PrPb3, the quadrupolar ordering temperature, T~ = 0 . 3 5 K , is small in comparison with the F3 - F4 spacing. The corresponding van Vleck terms do not affect the ordering process too drastically and the ( O °) jump is calculated to be weak in agreement with the temperature dependence of the specific heat (fig. 15). The same feature is also observed in CeB 6 and CeAg where the F8 - F7 splitting reaches 540 and 260 K respectively. For instance in CeAg the jump of (O2°) is calculated to reach only about 10% of its maximum value. The existence of anharmonic elastic terms in this compound makes it impossible to obtain a quantitative analysis of the T~ transition within the present formalism.
3.2.2. Magnetic ordering in the presence of quadrupolar interactions The nature of the magnetic transition in ferro- and antiferromagnetic cubic compounds has motivated a great number of studies, which can be classified according to whether they are based on an MFA analysis or renormalization group (RG) theory (K6tzler 1984). In both cases, the (isotropic) Heisenberg exchange is the starting point. The two approaches differ in the additional couplings which also have to be considered. The MFA model presented above, is in the line of models previously developed for transition metals as MnO, MnS (Lines and Jones 1965). However, instead of exchange striction, it considers for rare earth compounds quadrupolar pair and magneto-elastic interactions. The resulting coefficients of the free energy expansion are expressed in terms of single-ion susceptibilities and determining the phenomenological parameters for the different couplings then leads to the description of the magnetic transition (see section 2.2.4).
44
P. MORIN and D. SCHMITI"
On the other hand, the RG approach is based on symmetry considerations and includes critical fluctuations to predict the order of the magnetic transition. Within this hypothesis, continuous magnetic transitions become discontinuous ones according to the sign of the M4-term in the Landau expansion of the free energy. Note that the RG theory usually considers an incompressible lattice and nonquantized magnetic moments. The hypothesis of an incompressible lattice is in strong opposition with the evidences of large symmetry lowerings in cubic rare earth compounds. Due to the assumption of classical moments, the RG theory cannot predict the occurrence of a first-order transition associated with the positive X:, - (3) value in the case of a F3 non-magnetic ground state (see section 2.2.4 and Blume 1966). The aim of this section is to show through examples from the literature that the nature as well as the temperature of the magnetic transition may be predicted as soon as the quadrupolar magneto-elastic and pair interactions are reliably known (sections 3.2.2.1 and 3.2.2.2). Of course more complex situations may occur due to additional contributions from spin (and quadrupolar) fluctuations (section 3.2.3).
3.2.2.1. First-order transitions and tricriticality TmCu and TmAg A good example of the effect of the quadrupolar interactions on the magnetic transition is given by TmCu. This compound orders at TN = 7 . 7 K in a modulated antiferromagnetic structure, with all the moments parallel to the [001] axis. The transition is first-order (fig. 18). Various experiments performed in the paramagnetic state reliably provide the tetragonal quadrupolar parameter G r = 11 mK
1oo TmCu
-~ so
5 i
+I
i
10
,~-'"'"
.......
i
" "
.-
S
5
i
(b)
f" .,.." ,,, ~ ." " I 6
I
I
+
7
8
g
TEMPERATURE
(K)
Fig. 18. Temperature variation of the specific heat Cp (a) and of the entropy S (b) in TmCu. A n incommensurate antiferromagnetic structure is established at T N = 7.7 K and transforms at 6.7 K into a commensurate one (from Morin and Schmitt 1980).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
45
(Jaussaud et al. 1980). Just above TN the CEF third-order paramagnetic susceptibility Xz, - (3) is negative; this would lead to a second-order magnetic transition. However, the quadrupolar contribution to the total third-order magnetic susceptibility is positive and large enough to change the sign of X~)0, in the vicinity of TN, as seen experimentally. Consequently, the magnetic dipolar transition is expected to have a first-order character, as effectively observed (fig. 19a).
2
'
I'0.5
'
I
'
i -.
i
I
_m°!
,/ 3.5,,"
-~
T
ii..................
W =1~2K x -=-.42
'
1~o
'
f J" j/J
W=i.05K x=-.4.7
:.:" I
0
"
/ (~
.-"'' ( /"" T~
0/
0
I
I
10
TEMPERATURE(K) Fig. 19. (Top): temperature dependence of X~ ) observed for the tetragonal symmetry in TmCu (a) and TmAg (b). Full and dotted lines describe, for the CEF parameters W and x indicated, the cases with and without quadrupolar interactions, as indicated by the numerical values of the G v coefficient. (Bottom): the deduced magnetic phase diagram for both compounds, n is the bilinear coefficient (n N for an antiferromagnet). Dotted lines: second-order magnetic transition without quadrupolar interactions; dashed lines and full lines denote respectively: second- and first-order magnetic transition in the presence of quadrupolar transitions. C is a tricritical point (from Giraud and Morin 1986).
The isomorphous compound TmAg exhibits very similar CEF and bilinear interactions. It orders antiferromagnetically at T N =- 9.5 K, the transition is second order (Morin and Schmitt 1982a). Quadrupolar interactions within the tetragonal symmetry are weaker than in TmCu. They are characterized by G r = 4 m K (Giraud and Morin 1986). X~ ) is observed to be negative above 10 K, but is calculated to change sign at 6.2 K (fig. 19b). This defines a tricritical point between second-order transition (T > 6.2 K) and first-order ones (T < 6.2 K): the magnetic phase diagram predicts a second-order transition for a temperature of 9.5 K in agreement with the observation. An interesting feature is the large dependence of the tricritical temperature on quadrupolar interactions. Indeed, if G ~ reaches 5 mK it would shift the tricritical point up to 12.5 K, and the magnetic transition would be first-order for the same bilinear coefficient. Owing to the quadrupolar interactions, TmAg appears to be very close to tricriticality.
46
P. MOR1N and D. S C H M I T r
CeZn and CeMg Among other CsCl-type compounds, the CeX present a large set of different properties. Indeed whereas CeAg undergoes two well-separated magnetic and quadrupolar transitions (section 3.1.1.3), CeZn and CeMg order antiferromagnetically at T N = 30 and 19.5 K, respectively, the transition being first-order (Pierre et al. 1981, 1984). Contrary to the case of CeAg, the lattice stability is large; as in CeAg the Kondo-type behaviour for the Ce ion is not important. The tetragonal symmetry lowering below T N reaches record values ( c / a - 1 = 1.7 × 10 -2 and 1.3 × 10 -2 in CeZn and CeMg, respectively). Fitting magnetic properties in the ordered phase leads to G r = 220 and 50 mK in CeZn and CeMg, respectively. Note that these values have not been confirmed by measurements in the cubic paramagnetic phase. In CeZn the corresponding third-order magnetic susceptibility has been calculated to be positive in the vicinity of the N6el temperature, in agreement with the first-order character observed (fig. 20). In CeMg, the value G r = 50mK is a little bit too small: a value of about 75 mK would be large enough and more relevant.
/~-~=264 K
i
I'-~- ~=190KI
~
G.~=5OrnKI
CeZn
'75mK
1, ~ OOmK
\
0
I
/
20 TEM
I
PERATURE(K)
I
40
Fig. 20. Temperature dependence of the third-order magnetic susceptibility /~M,O - (3~ for CeZn and CeMg in the presence of quadrupolar interactions characterized by the G ~ values indicated.
erB 6 The same features are found again in P r B 6 , a compound isomorphous t o C e B 6. Here bilinear interactions are larger than the quadrupolar ones and a first-order antiferromagnetic transition is observed at 6.9 K (McCarthy et al. 1980). This first-order character seems to be stabilized by the same type of AFQ interactions as in the quadrupolar phase of C e B 6 ; indeed at low temperature the magnetic structure is described by the same propagation vectors (Burlet et al. 1988) and at high temperature large softenings of elastic constants are observed (Tamaki et al. 1985). By using quadrupolar coefficients deduced from f e B 6 (L/ithi et al. 1984) and NdB 6 (Tamaki et al. 1985) calculations within the present formalism show that the third-order magnetic susceptibility associated with the trigonal symmetry is positive. Measurements would allow to confirm this analysis.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
47
Tb pnictides Equi-atomic compounds within the NaCl-type structure have been extensively studied (Hulliger 1978). They are the earliest intermetallics observed to exhibit large one-ion magneto-elastic couplings (L6vy 1969, Mullen et al. 1974). In addition, from numerous studies, they have been shown to exhibit a more complex behaviour, they are at the meeting point of different couplings (section 3.2.4). However, a careful analysis of the antiferromagnetic TbX has shown that quadrupolar interactions unambiguously play a dominant role (K6tzler 1984). Indeed the AF transition is first order in TbP (T N = 7.1 K) and TbAs (T N = 12.5K) and second-order in TbSb (TN= 15.5K), TbBi (TN=17.5K), TbS (TN = 45 K), TbSe (TN = 49 K) and TbTe (T N = 51 K). All the compounds exhibit trigonal spontaneous distortions (Hulliger and Stucki 1978b). Figure 21 shows the resulting jumps of the magnetization and the susceptibility at the first-order transition in TbP compared to the behaviours observed in TbBi. In the t
__°_,--°_
i
TbP
MS(r) Ms(O)
i
i
°~Ooo~_-~ "~
(TN=7.1K )
.8 ! f
1.0
°~,
,
TbBi
\
IT, =17.5K)
~,
\
.6 .4
0.0
\
.2
o
0.4
Xo(T) Xo(T~)
0.6
0.8 T/TN
1.0
1.0
"'~ TbBi (TN :l?.SK)
~.0
0.5
.6
°
.8
a
1.0
3.5
..
0 (TN =7.1 K)
b o,
I
z
~
I
z,
~
I
6
~
T/TN
I
8
Fig. 21. Comparison between: (a) the spontaneous sublattice moments, and (b) the magnetic susceptibility for TbP and TbBi. Full lines are calculated within a formalism similar to that of section 2
(from Koetzler 1984).
48
P. MORIN and D. SCHMIT]"
paramagnetic phase, there are large magneto-elastic contributions to the elastic constants in TbP (Bucher et al. 1976) and TbSb (K6tzler 1984). The quadrupolar interactions are characterized by G" = 61, 32, 22 and 22 mK in TbP, TbAs, TbSb and TbBi, respectively (K6tzler 1984). These G ~ values lead to the temperature variations of X~u,o -(3) shown in fig. 22. Obviously the antiferromagnetic ordering occurs in TbP and TbAs at temperatures TN, for which xM,0 - (3) is positive; this explains the first-order transitions. From an experimental point of view, positive values of X}~) have been observed in TbP for the trigonal symmetry (fig. 23). For TbSb and TbBi, a second-order transition is predicted, as is experimentally
-/4
i
~32mK
I m
,
,~N
22m½
-10-4O
=
10 TEMPERATURE(K)
20
Fig. 22. Temperature dependence of the third-order magnetic susceptibility XM,O - (3) calculated in the presence of quadrupolar interactions in TbP (G ~= 61 mK), TbAs (G " = 32 mK) and TbSb, TbBi (G ~= 22 mK). W and x define the level scheme common to the four compounds. The first-order (second-order) transition at T N in TbP and TbAs (TbSb and TbBi) are driven by quadrupolar interactions.
0.00 B
,
,
X2)
~X =6inK
,
,
TbP ~ l~[I.]
o.oo/-, i i ~ 0.002 0.0
8
10
12
lt,,
T[K]
16
Fig. 23. Temperature dependence of the third-order magnetic susceptibility in TbP. Full lines are calculated with the indicated values for the trigonal quadrupolar coefficient Y( = 12G" (from Raffius and Koetzler 1983).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
49
observed. Note, however, that the critical point C strongly depends on G ~ and that TbSb and TbBi are not far from tricriticality of a quadrupolar origin.
3.2.2.2. Second-order transition in PrMge due to negative quadrupolar interactions. This compound orders ferromagnetically at T c = 10 K through a secondorder transition (Loidl et al. 1981). The ground state is a F 3 doublet. As mentioned in section 2.2.4, this CEF configuration gives rise to a positive value for Xz, - (3) in the vicinity of T c. This should produce by itself a first-order magnetic transition. On the other hand, the G ~ value suggested by Loidl et al. (1981) for explaining the temperature dependence of the spontaneous magnetic moment, i.e., G ~= -13.3 mK, leads to a negative total third-order magnetic susceptibility )¢(3) M,0" This now, accounts for the second-order character of the transition (fig. 24). I
.J.lo..
3
I
I
i
l
I
%%0 "
#2 PrMg21 0
5
T[K]
t~t 10
Fig. 24. Spontaneous magnetic moment, versus temperature in PrMg z. The dotted line is the Brillouin curve for J = 4. The full and broken lines give the result of a mean-field calculation with and without the quadrupolar interactions G ~= -13.3 mK (from Loidl et al. 1981).
Note that in the ferromagnetic phase, the ( O ° ) quadrupolar components are ferroquadrupolarly aligned by the dominant ferromagnetic interactions, whereas the negative quadrupolar interactions would prefer an AFQ arrangement and are thus frustrated.
3.2.3. Magnetic ordering in the quadrupolar phase The last case corresponds to a dipolar phase transition inside the quadrupolar phase (T < To). Below T c the quadrupoles are ordered and the crystal field is no longer cubic, e.g. it may have tetragonal symmetry. Therefore the various susceptibilities (X0, x~ " (3) , • • .) have new values (X~, x"~ T(3) . . . . ) according to the new tetragonal CEF level scheme. Note that this level scheme is now temperature dependent, like the ordered quadrupolar moment. The discussion of the nature of the dipolar transition is the same as in section 2.2.4, where /1-C(3) M,0 is replaced by )(T(3) 7 . Depending on the sign of X~ (3), the dipolar phase transition has a first- or second-order character.
50
P. MORIN and D. SCHMIqT
3.2.3.1. CeAg, f e B 6. In the case of CeAg, the ferromagnetic transition is second-order in the tetragonal phase (fig. 11) in accordance with theory. Indeed, . T(3) the x~ susceptibility is calculated to be negative as expected for a Kramers ion through the divergence of negative Curie-type terms associated with the magnetic doublet ground state. The same conclusion is also valid for CeB 6. 3.2.3.2. TmZn. In TmCd, bilinear interactions in the tetragonal phase are insufficient to induce a magnetic moment on the singlet ground state. On the contrary, in TmZn bilinear interactions generate a ferromagnetic ordering (T c = 8.12 K) slightly below the quadrupolar ordering (T o = 8.55 K). Calculations lead to a critical situation for Tc: the nature of the magnetic transition seems to be very sensitive to the values of the various parameters. For instance, T c is calculated to be of first order for G ~ = 25 inK, but would be of second-order with G v = 28 mK, as it experimentally appears. Consequently, the situation of TmZn with regard to the ferromagnetic transition seems to be not far from a tricritical point. 3.2.4. Presence of additional couplings In the compounds presented in section 3.2.2.1, the quadrupolar and Heisenberg bilinear interactions are dominant; any other interaction remains undetected; nonetheless, the competition between various couplings may be more balanced in other series such as rare earth pnictides and dialuminides. Two-ion magnetoelasticity and/or fluctuations may play a larger role in the minimization of the free energy. 3.2.4.1. DySb and rare earth monopnictides. This compound appeared in the past as an archetype for a first-order magnetic transition driven by quadrupolar interactions (Bucher et al. 1972b). It antiferromagnetically orders at TN = 9.5 K with a four-fold axis of easy magnetization. Associated with the magnetic structure, the lattice undergoes a structural symmetry lowering, mainly tetragonal, but with a small monoclinic component (L6vy 1969, Felcher et al. 1973) (fig. 25). This implies a magnetic moment not strictly parallel to the [001] axis. The existence of quadrupolar interactions has been unambiguously proved by the strong softening of C l l - C12 (Moran et al. 1973, Levy 1973); the value of G ~= 1 mK deduced is large enough to explain the first-order character of the joint spins and quadrupoles orderings, the third-order magnetic susceptibility is calculated to be positive (fig. 26). However, the analysis of the magnetization processes (Kouvel and Brun 1980, Everett and Streit 1979) and of the third-order magnetic susceptibility (A16onard et al. 1984b) in terms of only quadrupolar interactions fails (fig. 26). Indeed data are positive only in a very short range above TN and cannot be described by a unique G ~ value. Thus, additional spin interactions are present especially for the tetragonal symmetry. The most commonly proposed coupling is based on anisotropic bilinear interactions (Trammell 1963, Kim and Levy 1982, Jensen et al. 1980). It was in
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
.~;
,.y-e-V--L~c~-
Oa
h
....+, + , .
-I
l
~
-2-c
(t~tragonal)
-3-E
51
-4-__~n
n
,',
(cubique)
l° c/a
Oy P,
0.997 0.996
= Dy S b
0.995
-5--
Temperature
T° K
Fig. 25. Temperature dependence of the relative change of the tetragonal cell parameters in the ordered phase of DyP (two samples), DyAs and DySb; a 0 is the cubic cell parameter (from L6vy 1969).
a =
i
i
"
\
i
W=.2BK
i
~
x =-.82
GI~0"lmK 0
\.,~e3 1 ,,p o
=b
I
t
~
'
DI(kO~PB/ 3 0 ~
10
I
~
'
[0 0 1]
20 IEMPERATURE(K)
' W=.28K x =-.82 1~=-6.5K GI=1mK
30
Fig. 26. Temperature variation of the third-order magnetic susceptibility for H[I[001 ] in DySb. (a) tentative fits are drawn with the quadrupolar model alone, G 1 = G r = I mK being the value obtained from the softening of the Cu - C12 normal mode. (b) Fits here include, in addition to G: = i mK, the two-ion magneto-elastic coupling characterized by D1 = D r (from A16onard et al. 1984b).
52
P. MORIN and D. SCHMITT
particular postulated that rare earth pnictides, where the nearest-neighbour exchange is dominated by direct exchange and superexchange, are excellent candidates for large two-ion magneto-elastic effects (Lacueva and Levy 1985). From an experimental point of view, anisotropic bilinear couplings have been observed in the magnetic excitation spectrum of TbP (Loidl et al. 1979) and PrSb (Vettier et al. 1977). They have been also used for explaining the different antiferromagnetic structures found according to the temperature in GdX pnictides. Associated with these structures, different two-ion magneto-elastic strains have been also detected (Hulliger and Stucki 1978a, Hulliger and Siegrist 1979). For the experimental conditions which are used to study the third-order magnetic susceptibility , anisotropic bilinear interactions occur through their strain derivative, i.e., the two-ion magneto-elastic coupling, which may be described within the MFA formalism (Aldonard et al. 1984b). For DySb this coupling, characterized by a D 1 coefficient in fig. 26, is opposite to the quadrupolar interactions and improves the fit down to 12 K. However, a second-order transition would be expected. The same short-comings are also observed on the temperature dependence of the reciprocal susceptibility under an uniaxial [001] stress (Morin et al. 1985a). Even if present, two-ion magneto-elasticity, considered in the MFA, is not able to describe the features observed close to T~. On the other hand, a large number of studies has shown that NaCl-type compounds exhibit fluctuations through their critical behaviours close to the N6el temperature (Taub and Parente 1975, Hfilg et al. 1985). These critical fluctuations appear to be favoured in a deformable lattice (Taub and Williamson 1973). In DySb, as in HoSb, neutron diffraction measurements have shown an anomalously large magnetic scattering above T N. In the same type of experiments, the paramagnetic correlations in ErSb are characterized by a strong two-ion anisotropy (Knorr et al. 1983). In conclusion, the nature of the magnetic transition in numerous monopnictides seems to result from a well-balanced competition between different couplings. For many cases, it is difficult to determine the driving mechanism as e.g., in HoSb, which is very close to tricriticality (Jensen et al. 1980). The determination of quadrupolar interactions, coherent in CsCl-type compounds, cannot be achieved in these series. In addition the large dependence of anisotropic magnetic interactions on metallurgical stresses may change the nature of the transition in a given sample as shown in Er compounds (Hulliger and Natterer 1973). As in DySb the difficulty in properly considering fluctuations and anisotropic bilinear couplings may prevent one to fully understand the first-order transitions in DyBi (Hulliger 1980), DyP, DyAs (L6vy 1969), ErBi and ErSb (Hulliger and Natterer 1973) even if quadrupolar interactions are obviously present as proved by the temperature dependence of elastic modes. The R A l 2 series. RA12 compounds (C14 structure) appear to pose a similar problem as was observed in pnictides (del Moral et al. 1987). DyA12 orders ferromagnetically at T c --61.4K through a second-order process. The third-order magnetic susceptibility along the [001] easy magnetization direction is 3.2.4.2.
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
-3 x 10"5~ /oB/KO¢3
I Gt=OrnK
I
70
80
53
mK i
-2 <3 -1
0
I 60
90
T(K]
Fig. 27. Temperature dependence of the third-order magnetic susceptibility A.... = X~ ) for HI[[001 ] in
DyAI2. Full lines are calculated with a bilinear coefficient n = 17.5 kOe//xB and the values indicated for the quadrupolar parameter G1 = G ~ (from Del Moral et al. 1987). negative in the entire paramagnetic range and cannot be described with a unique G ~ coefficient (fig. 27). However, the data close to T c do not indicate critical behaviour. The present range (8-10 mK) of G ~ is different from values deduced from magnetization measurements in the ordered phase, i.e. - 2 m K (Rossignol 1980). The same failure for describing the third-order susceptibility is found again in ErA12 (del Moral et al. 1987). Data go from positive values at high temperature to negatiye ones close to T c which countermands a first-order transition. The 2 possible G ~ values deduced are larger than in DyA12 in violation of the o~j variation law across the rare earth series (Schmitt and Levy 1985) (see section 5). All these features may be the signature of additional couplings and an improved knowledge would need systematic studies in particular of ultrasonic properties.
3.3. Determination of the quadrupolar parameters from susceptibility techniques Different ways of determining quadrupolar parameters are directly related to the various susceptibilities described in section 2. Indeed, these susceptibilities connect the magnetic variables on one hand and external stresses on the other hand. Thus, (i) the strain susceptibility represents the response of the quadrupolar moments to the corresponding stress; (ii) the quadrupolar field-susceptibility connects the quadrupolar moments to the magnetic field; (iii) the third-order magnetic susceptibility couples the magnetic moment, more exactly the initial curvature of the magnetization curves, and the magnetic field. Each of these three
54
P. M O R I N and D. S C H M I T T
susceptibilities may be acquired by a different experimental technique, namely measurement of the elastic constants, the parastriction and the low-field magnetization, respectively; these techniques are described below. 3.3.1. Elastic constants
As seen in section 2.2, the existence of a magneto-elastic coupling leads to the softening of the elastic constants of appropriate symmetry when the temperature is lowered, and this softening is enhanced by positive two-ion quadrupolar interactions. The experimental technique generally used is the ultrasonic velocity measurement, and several geometrical configurations are considered in order to observe the various ultrasonic propagation modes. In general, the normal (symmetrized) elastic constants are not directly accessible; they have to be obtained from a linear combination of those which are measured. The observation of a softening of elastic constants due to magneto-elastic coupling in the intermetallic rare earth compounds was realized in the early 1970s, and extensively studied in particular in cubic rare-earth antimonides (see figs. 28 and 29). Several behaviours are observed, according to the strength of the quadrupolar (magneto-elastic) coupling as well as the nature of the CEF ground state. The temperature variation of the elastic constants in LaSb is taken as the background for the other compounds. Large anomalies are often detected at the magnetic ordering temperatures and below; they are due to domain-wall stress effects. However, the most spectacular effect remains the softening above the ordering
I.C
0.8
0.6
O~
°il I
'
8b
'
I
J~o
'
z~,o
TEMPERATURE(°K)
'
3~o
Fig. 28. Change in C o = C ~ = (Cal - C12) as a function of temperature in DySb. A fit of the data leads to (g0) 2 = GEE = 1 . 4 m K and I o = K v = - 0 . 4 m K and thus G ~ = 1 mK (from Moran et al. 1973).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
14.0~t3.9~
55
TmSb
13.8 ~:
13.6 r ~
13.5 Ci1-C12 • I
1
5
.
2
el
~
~2.70 ee o
- • • ee~ e
~:__.z6.sz6.6z6.4z6.~
o
°
C44 ee
"
I
I
Ioo
200
300
T (°K)
Fig. 29. Temperature dependence of the symmetrized modes Clt - C12 = C 7 and C44 = ½ C e for TmSb. Full lines are calculated with only magneto-elastic couplings in the presence of CEF (from Mullen et al. 1974).
temperature, e.g., this softening reaches about 60% of the room-temperature value of C r at 10 K in DySb and occurs over a wide range of temperature (see fig. 28). A good description of the softening of the C ~ elastic constant requires the introduction of a two-ion quadrupolar interaction in addition to the one-ion magneto-elastic coupling (Levy 1973, Moran et al. 1973). The cubic TmX series (X = Zn, Cd, Cu, Ag, Mg) is another series which is particularly spectacular with regard to the softening of the C v elastic constant. For example in TmZn and TmCu (fig. 30) the softening reaches more than 50% of the room temperature value just above the ordering temperature, indicating very strong quadrupolar interactions (Liithi et al. 1979, Jaussaud et al. 1980). In the opposite limit, the softening is much less pronounced in TmAg and TmMg (Giraud et al. 1986, Giraud and Morin 1986). It is worth noting that the analysis of the temperature variation of the elastic constants may give information about the crystal field in addition to the magnitude of the quadrupolar parameters. Indeed, according to the nature and spacing of the low-lying CEF levels, anomalies may occur at low temperature in the strain susceptibilities, thus in the elastic constants (see section 2.2.3). This is particularly emphasized in singlet ground-state systems as in PrSb (Liithi et al. 1973a) and TmSb (see fig. 29) for both C ~ and C ~ modes.
56
P. M O R I N and D. S C H M I T T
I
I
I
l
l
C~(YCu) •/ ' 1 / " .~.I
/
,t~
/.
E
c" :½(c,-c,2
?
o
I
O,l 0
T
t..j
i
- -
experiment fit with G1 = l l . 5 m K R1 = 7,0 mK
J
I
l
100
I
I
2OO
300
T(K)
Fig. 30. T e m p e r a t u r e d e p e n d e n c e of the C' = ½C r = ½(C n - Ct2 ) m o d e in T m C u ; the full line is a theoretical fit with G 1 = G ~ and K 1 = K ~ ( f r o m J a u s s a u d et al. 1980).
3.3.2. Parastriction Parastriction constitutes a very suitable technique for investigating magnetoelastic coupling. It consists of measuring changes in the length of a sample in an applied magnetic field in the paramagnetic phase (Morin et al. 1978c, 1980b). It differs from the usual magnetostriction by the fact that only the low-field variation of the latter is analyzed, namely its H2-dependence. Indeed, as shown above (section 2.2.1), within the limits of validity of the perturbation theory, the coefficient of this HZ-variation is related to the quadrupolar field-susceptibility Xo (or Xp) which depends only on the crystal field and on the magnetic and magneto-elastic parameters. In addition, in agreement with the classical hightemperature approximation (Callen and Callen 1965), a linearization of this susceptibility may be achieved by plotting ) ( Q 1 / 2 a s a function of the temperature. Deviations from this linear behaviour may occur at low temperature due to CEF effects in the same way it does for the magnetic susceptibility. Experimentally the measurement of the relative change of length A = 8l/l is performed successively along and perpendicular to the direction of the applied magnetic fields; one eliminates the volume effects e ~ by considering the difference All - A~. As a consequence, for a magnetic field along a four-fold or three-fold direction, and according to the expression for 81/l [eq. (25)], one obtains respectively
(All- Al)v =
~93_B~ 2 C---~oXQH2'
3 B+ (All- A±)+ = V'2 C O x e H 2 '
(110)
for the two tetragonal and trigonal strain modes. It is worth noting that the same
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
57
relations are found, by considering a fixed measurement direction and by applying successively the magnetic field parallel and perpendicular to that direction. This latter geometry is habitually used by experimentalists for the sake of simplicity. As mentioned above it is useful to plot HI(I All - A± I) 1/2 in order to observe the linear high-temperature b~haviour. A good example is provided by the compound TmZn where the quadrupolar effects are particularly strong (fig. 31). According to eq. 110, the high-temperature slopes give the magneto-elastic coefficients B v and B', as the background elastic constants are known from other measurements. ~I~
~
I
0
,
v
o, o / / '
I
O•
."
I
I
1
Fig. 31. Parastriction of TmZn, measured for the tetragonal and trigonal symmetry lowering modes. Curves are calculated within the susceptibility formalism (G ~= Ga) (from Morin et al. 1980a).
In addition, large deviations from this linear behaviour are observed for the tetragonal mode below 100 K due to CEF effects and to a strong G v parameter. On the other hand the anisotropy of the parastriction is particularly obvious in this compound, showing that measuring the magnetostriction on polycrystals is quite meaningless in these anisotropic systems. Note also that the parastriction gives the sign of the magneto-elastic coefficients, while only its absolute value may be obtained from elastic constant measurements. This technique has been successfully used in several series of rare earth compounds, in particular in RZn compounds where the coefficients associated with both the tetragonal and trigonal strain modes have been obtained for the heavy rare earths (figs. 32 and 33).
3.3.3. Third-order paramagnetic susceptibility The third-order paramagnetic susceptibility has been introduced as a new method for studying quadrupolar interactions in rare-earth compounds (Morin and Schmitt 1979, 1981b). It consists of a detailed analysis of the magnetization induced by an external field in the paramagnetic phase. More precisely it
58
P. MORIN and D. SCHMITF =
i
1S
Er Zn/y
/.
H . [ oo]
.
..~
.
•
,/ TmZn
%s DyZn
"r
20o
too
300
(K)
TEMPERATURE
Fig. 32. Temperature variations of g/I;,, All 1/2 for tetragonal strain in RZn. The full lines are calculated dependencies (from Morin et al. 1980b). -
H //
8 ~o
1o
v
[111]
TmZn DyZn
5
TbZn
"I-
i
100
TEMPERATURE
200
300
(K)
Fig. 33. Temperature variation of H/l All- A±I 1/2 (rhombohedral strain) in RZn. The full lines are calculated with the susceptibility formalism (from Morin et al. 1980b).
corresponds to the second term (in H 3) of th e development of the magnetization as a function of the magnetic field, this term depends on the total quadrupolar interactions in addition to the crystal field (see section 2.2.1). An experimental way to investigate the magnetization curves is to plot M / H versus H2; this provides both the first-order magnetic susceptibility XM, i.e. the
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
59
isotropic null field value, and the third-order one X~ ) , i.e. the anisotropic slope of the linear low-field range. This is well-illustrated by the compound TmCd, where the anisotropic character of X~ ) is particularly obvious (see fig. 34). It is worth noting that the linear part of these curves diminishes as the temperature is decreased, since the higher-order terms (in H s , . . . ) become relatively more and more important. An alternative experimental way, although less direct, is to consider the Arrott's plots M 2 versus H/M. In low field, the variation is linear; the intersection with the x-axis yields (X~)) --1 while the third-order magnetic susceptibility is related to the slope dM2/d(H/M) through the relation
X(~ )
-(XM)4/(dM2/d(H/M)) .
=
The variation of the third-order magnetic susceptibility may then be drawn as a function of the temperature and analyzed with the above formalism (see fig. 35). As an other example, in fig. 36 we show the temperature dependence of X~ ) for the two main symmetry directions in PrPb 3. The anisotropic character of X~ ) is obvious from these curves again, as well as its dependence on the bilinear and quadrupolar interactions. The third-order magnetic susceptibility also explains the paramagnetic anisotropy of the magnetization observed in TbxLal_xA12 for a constant magnetic field within a given plane (Hoenig 1980). The technique of third-order magnetic susceptibility has also been applied to the tetragonal compound TmPO 4 which presents noticeable non-linear terms in the low-field magnetization within the basal plane (Andronenko et al. 1983). Another experimental way to extract X~ ) is to perform torque anisotropy 120
i
"
~
"- "
115
[001]
6 11o ~,
"
"
"
"
,
[111]
TmCd
' ~ 105 EI 77
"
~ - - - - ' -
"" ~ ~
""
76
K
[OO]l
75 74 .....
0
i 100
2
Hi
I ~ 200
I 300
400
(k(De2 )
Fig. 34. Experimentalvalues of M / H versus H 2 in TmCd (from Morin and Schmitt 1982c).
60
P, MORIN and D. SCHMITT
//// ///"
&"
!// II
-4
:o,5 K
,:o
/
×
~
5
,
=-o.34
i
10 15 20 TEMPERATURE ( K )
i
25
30
Fig. 35. Experimental and calculated variations with temperature of X~ ) along the [001] axis in TmCd system. Note the dependence of the coefficient G1 = G ~ on dilution (from Morin and Schmitt 1981b).
1 e*=-9K GI=O '. -
tI
i
l
H//[001]
~o O ~.~:E ~<
I
I
H//[111]
'
]
\", i L
i
PrPb3 W=-.65K x=.6
\
1¢---9K JG .3K
i . . . . . . . . . ;_:.=_._.....
K ............2 : 2 : ~ _-'r='- .....
'G =, --.9. -15mK -/ , //--GI="0' ' - I . IG2= ~ O*---09K \
,e*-gK ;.2K O,=G2__O// //
//
/
~_ /
// /
-1
I
0
5
I
I
10 5 TEMPERATURE (K)
/I
10
15
Fig. 36. Temperature variation of the third-order susceptibility along [001] and [111] in PrPb 3. Dots are experimental points. The dashed lines correspond to behaviours without quadrupolar interactions, but show the effects of the bilinear interactions (0" = - 9 K). The full lines show quadrupolar effects (G v= G1, G ~ = G2) (from Morin et al. 1982). m e a s u r e m e n t s which, in p r i n c i p l e , allow us to d e t e r m i n e p a r a m a g n e t i c susceptibility t e n s o r s o f f o u r t h (~- X ~ )) a n d e v e n sixth r a n k s a n d has b e e n a p p l i e d to R A I 2 c o m p o u n d s ( D e l M o r a l et al. 1987). A n a l t e r n a t e a p p r o a c h to a s c e r t a i n i n g t h e q u a d r u p o l a r effects o n t h e m a g n e t i z a t i o n is b a s e d o n t h e n o n - l i n e a r i t y o f t h e e x c h a n g e field Hexch as a f u n c t i o n o f M
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
61
(Kouvel and Brun 1980) Hex~h = AM
+ A'M 3 .
(111)
However, this procedure provides a )t' value which is temperature dependent, since it is related to the true quadrupolar parameter G" through the relation )t, =
2G u'(X~)) 2 (X0)4( 1 _ G~X# )
(112)
This ,V parameter is therefore connected with the quadrupolar part of X}~); its temperature variation has indeed been observed (Abu-Aljarayesh et al. 1986a)
3.4. Effects on the magnetostriction While parastriction allows us to describe the initial magnetostriction in the paramagnetic phase, i.e. the first term in its field development, a full diagonalization of the Hamiltonian describes the whole magnetostriction curve, in the ordered or non-ordered state, provided the magneto-elastic couplings and all the two-ion interactions are included.
3.4.1. Magnetostriction of dilute compounds The main advantage in investigating dilute magnetic compounds is to be free, at least to a large degree, of two-ion interactions. Indeed it is these interactions that make the analysis difficult. If the effect of the dilution on the macroscopic measured strain is correctly taken into account, one can determine the microscopic one-ion magneto-elastic coupling. Usually authors simply assume that the macroscopic deformation is proportional to the magnetic impurities concentration (Nieuwenhuys et al. 1982b, Pureur et al. 1985). A systematic study of the forced magnetostriction has been carried out for the cubic noble metals compounds (silver, gold) containing magnetic rare earth impurities (Creuzet and Campbell 1981, Campbell and Creuzet 1985). The magnetostriction follows the second-order Stevens coefficient a I (see figs. 37 and 38), indicating that a single-ion model is pertinent, and that the second-rank magneto-elasticity is preponderant in these systems. Nevertheless, a fourth-rank magneto-elastic coupling has also been determined, its contribution is about ten times smaller than the second-rank one. In addition to the magneto-elastic coefficients, the magnetostriction curves are strongly affected by the nature of the low-lying CEF levels. This allows one to choose between possible levels schemes through the shape as well as the temperature variation of the magnetostriction curves (fig. 39). Another spectacular effect related to the crystal field concerns the so-called "reverse" magnetostriction, i.e. the sign reversal of the magnetostriction between the low and high temperatures. This occurs in CexLal_xSb, SmxLaa_xSb (Nieuwenhuys et al. 1982a), and CexLal_~Bi (H~ifner et al. 1983) compounds.
62
P. MORIN and D. SCHMITF 14,
y
lu-,,-~nt
12
10' 8 6
~
14. •106x6L "~Ag Tb
Ag Tb
t
12.
10, 6
A
4 2 0
2
~
2 -4
--6 -8
kG
(a)
0
AgEr
-2
X,~Tm \
-t, -6 -e
~
o
H.kG 0AgEr AgTm (b)
Fig. 37. Parallel magnetostriction data for a field applied along the [001] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental data, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
6, ' 106 x"~t
.
/
6-
/AgTb
4.
AgDy
4 / - -
A_gHo
4u , 4u , 6,0 ,H.kG 0[ ~
,
H,kG
-2 -4. •
(a)
-q
fo)
Fig. 38. Parallel magnetostriction data for a field applied along the [111] axis of silver-0.5% rare earth single crystals at 1.4 K: (a) experimental, (b) calculated neglecting the e~ contribution (from Campbell et al. 1979).
For example, the change of sign of the measured strain takes place at 2 K in LaBiCe and at 8 K in LaSbCe (see fig. 40). This can be explained by a simple crystal field theory, which, in some particular cases, leads to an expectation value of the quadrupolar operator O ° which is not always positive. For example, in Ce0.15La0.85Sb, the quadrupolar m o m e n t induced by an external magnetic field on the F 7 doublet ground state is negative at low temperatures. It changes its sign either for very large magnetic fields (about 40 T at 1.5 K) or above 8 K for any field, namely when the influence of the excited F 8 quartet becomes important. This change of sign of the initial magnetostriction curvature is accounted for by the formalism developed in section 2 and corresponds to the change of sign of the quadrupolar field susceptibility x~ - (2) (see fig. 41). Similar reverse magnetostriction
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
~-110-6) 20
40
60
t
t
i
63
H(kG)
_1
_2 ~2.8 t
_3
Fig. 39. [001] parallel magnetostriction in ErA g (concentration 0.5%) corrected for the volume contribution; circles are experimental data, solid lines calculated. Note the anomalous variation with temperature which is characteristic of the CEF level scheme (from Creuzet and Campbell 1982b).
LaBi:Ce (lO°/o)
6 4
-5T
2
/I 3T
O -2
_1 ~-
-4
o
~ ~\
3 2/
O
LaSb:Ce (15%)
4.8T k,,..~ 4 T
5
10 15 T , (K)
20
Fig. 40. Comparison of the magnetostriction of (a) LaBi : Ce (upper panel) and (b) LaSb : Ce (lower panel) as a function of temperature at various fields. As clearly seen the "reverse magnetostriction" occurs at a lower temperature for (a) than for (b). Solid lines represent a fit to theory (from H/ifner et al. 1983).
64
P. M O R I N and D. SCHMITT r
50
-50
o
LaSb:Ce
LoBi:Ce
/ /
40o
•>• 0
-400
lo
T (K)
20
Fig. 41. Temperature dependences of Xy(~) , the quadrupolar field-susceptibility, calculated in the presence of only the CEF for LaBi : Ce (a), LaSb : Ce (b), LuCd : Tm (c) and LuZn : Tm (d). In these four cases the CEF level scheme drives the so-called reverse magnetostriction.
effects may be expected in other situations, e.g., with a F5 triplet as ground state, as in TmCd or TmZn, for an appropriate dilution.
3.4.2. Magnetostriction in the ordered phase The magnetostriction observed in the ordered phase of magnetic compounds is generally much larger than that in the paramagnetic phase. This is due to the large influence of the exchange field which tends to saturate both the magnetic and quadrupolar moments. The symmetry of the cell distortion is directly related to the direction of the magnetic moments, e.g., in ferromagnetic compounds. In cubic antiferromagnets, the observation of a spontaneous strain may be very helpful in determining the actual magnetic structure (see section 3.6). A quantitative analysis remains more difficult than for a non-ordered state, because of two-ion effects. Depending on the strength of the bilinear exchange coupling, it has been done using either a semi-classical (see section 2.4) or a quantum treatment (see section 2.1.2). In the case where the bilinear interactions are big, large magnetostrictions have been measured at room temP6erature, as e.g. in TbFe 2 where a record magnetostriction of Am = 2400 × 10- was reported (see fig. 42). This motivated numerous investigations because of the interest of technological applications. Still higher strains have been measured at low temperature in the RFe 2 series (see section 4). Huge magnetostriction has also been observed at low temperature in numerous series where the alloyed metal is nonmagnetic, in particular within the NaC1- and CsCl-type structures (L6vy 1969, Morin et al. 1977). Large tetragonal strains of the order of 10 - 2 have been measured; e.g., c / a - 1 =--1.4% in PrZn at 1.5K (Morin and Pierre 1975). In TbZn a change of easy direction within the ordered
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
I
I
I Tb Fe2
'
L
]
65
|
2,000I
~.poly [hi
1,000 t Y
0
V
I
I
I
I
I
5
10
15 H [kOel
20
25
F
Fig. 42. Magnetostriction of TbFe z at room temperature for a single crystal (Aiq curve) and a polycrystal taken for two different orientations (a and b) (from Clark et al. 1974).
10 -2
10- 3
IO-~
Lu
TbZn
1
io-s
10 -6
u~o.s 10" ;' 0 10 - 8
.....
i
100 i
100
200 )
I
200 TEMPERATURE OQ
300
Fig. 43. Thermal variations of the spontaneous strain e 3 = e~ above the temperature at which there is a change in direction of the easy axis and the e 3 strain induced by a 1 kOe field in the paramagnetic state for TbZn ((3, strain-gauge data; Q, capacitance dilatometer data; full curves are calculated variations). Inset: experimental variation of the volume anomaly e v = X/3 e" (from Morin et al. 1977).
66
P. MORIN and D. S C H M I T r -2 10
10.3
10-4
10~ 5 U3
TbZn
10-6
TC
H: lkCe 10- 7
\ ,
10- 8
,
~"
q.
100 200 TEMPERATURE (K)
300
Fig. 44. V a r i a t i o n s w i t h t e m p e r a t u r e o f the spontaneous strain sxz = e~/~/2 b e l o w the temperature where the easy axis changes, and of the exz strain induced by a 1 kOe field in the paramagnetic range for ThZn ( 0 , strain-gauge data; @, capacitance dilatometer data; full lines are calculated variations) (from Morin et al. 1977). ,
Tm Zn
2
~
,
T - 107 K 1
0
-- '?1 _..
01i°, E
9 10 11 TEMPERATURE (K)
-4
o
2
,
6
;
APPLIED FIELD
;o
12
(kOe)
Fig. 45. Experimental variations of the strains parallel (All) and perpendicular ( A l ) to the four-fold direction of the magnetic field at 10.7 K in TmZn. Full and dashed lines are relative to fast and slow rates of variation of the field. The inset shows the temperature dependence of the critical field H c (from Morin et al. 1978b).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
67
range occurs; this allows one to follow the temperature variation of the strain associated with the [001] and [101] moment axes (figs. 43 and 44). Finally, in TmZn one transition appears on the magnetostriction curves at temperatures above the Curie temperature (see fig. 45). This behaviour, also observed on the magnetization curves (see section 3.5), can be explained by the appearance of a strained ferromagnetic state above a critical field within the paramagnetic phase, arising from strong quadrupolar interactions.
3.5. Effects on the magnetization processes 3.5.1. Paramagnetic phase Perturbation theory has shown that quadrupolar interactions strongly influence the initial curvature of the magnetization as a function of field in the paramagnetic phase, i.e., the third-order paramagnetic susceptibility (section 2.2). In the same way, quadrupolar couplings influence the higher-order terms, i.e. they have an effect on the whole magnetization process. A diagonalization of the full Hamiltonian allows us to take these couplings into account particularly at high fields. As an example, the magnetization is predicted to be largest along a three-fold axis in [111]
(ooi] I
6
G,=9mK
.... +.:;,~ I
.0 ~+~+- +.+.+-16K
,
/ . +] ~o+- ~o+2OK I
4
-/'/+/4,/*/+°I o/'/+ 4,/ /0 I / /4,/ / ] 0 I/I /4" 4,0 /
d'i
2
o~ +~
•..'+25K,
/n
-
W=1.4 K x =-.42
•
• 4,o011/ +4,0 • 1+4,0 • o+~Oll u
+.,.
I
~'."
jEP=_ 3K
J
"'"
x =-42
~ ,
4 I-
W=I.4K E)w:I 3K ,
" 9K
+ ~...
..-'.~"
G.=gr~--I "11 -----
•
• +~/@/4,/0/
..,+0o., •
--
4~~
e~ e 0 i / @ .:+...-..
04"@0
0
.-1;:;. / 4+
|
e/o/../•
i+tOll t ~+ ll . .
~.-
I
G2=-80 mK
//+."
/."
/
.di.".
-
®// m II
,g .'"
0
I;::"
,
0
SO
/ 100
0 H(k~)
W=I.4K
? ." .e • •
x =-.42
!
50
it~=. 3 K 100
Fig. 46. Magnetization curves along the directions [001] and [111] for TmCu at various temperatures; lines are theoretical fits with the p a r a m e t e r s indicated ( G 1 = G ~, G 2 = G ~) ( f r o m J a u s s a u d et al. 1980).
68
P. MORIN and D. SCHMITF
TmCu in the absence of quadrupolar interactions; by using appropriate values of the parameters G ~ and G ~ one can restore the four-fold axis as the direction where the magnetization is maximum, as observed experimentally (see fig. 46). The magnetization curves show a more spectacular behaviour in case of huge quadrupolar couplings, as in TmZn (Morin et al. 1978b). In this compound, the quadrupolar coupling is strong enough to induce a transition from the paramagnetic to a ferromagnetic state in a wide range of temperatures above the quadrupolar ordering (see fig. 47). This transition occurs for a critical field which depends on the strength of the quadrupolar interactions and is well-described within the MFA. The critical field is also present along the directions of hard
.~
1K 111
0
20
(b
40
20
60
INTERNAL FIELD
80
100
120
(kO~)
Fig. 47. Magnetization curves of TmZn along the [001] axis at different temperatures; the insets show the temperature dependence of the critical field He; (a) experimental, (b) calculated (from Morin et al. 1978b).
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
69
magnetization, although it is larger than along the easy direction, i.e. [001]. For example, along the [101] direction, the magnetic moment is first aligned with the external field, then rotates towards the [001] axis when the component of the field along the [001] direction reaches the associated critical value; finally it comes back toward the initial [101] axis when the field is large enough.
3.5.2. Ordered phase In the ordered state, quadrupolar interactions play an important role in the free energy balance between the various crystallographic axes. In particular, when the field is applied along a direction of hard magnetization in a ferromagnetic compound, the magnetic moment rotates from the easy axis towards the field. The area between the corresponding curve and that extrapolated from the high field behaviour is related to the anisotropy of energy between both directions. This energy difference is strongly influenced by the quadrupolar couplings, as for example in ErZn (see fig. 48). This behaviour is enhanced when the free energies corresponding to two moment directions cross each other at a given temperature T R within the ordered phase. This leads to a first-order phase transition in which the direction of easy magnetization changes. This phenomenon is driven by entropy contributions induced by the anisotropic splitting of the CEF level scheme under the exchange field (see fig. 49). There are many examples in rare earth compounds, especially
6
5F//
b >
, 1mK-
~_o4~.4
* [101]
~I Z
=(I 1 I)
3 I
'
'
'
l
O N FLU Z L~ :E
,
Z/
~ 10
i:I \ 1 TEMIPERATUREI(K) .
4 0
;o
~o
INTERNAL
30 FIELD
40
50
(kOe)
Fig. 48. Magnetization curves in E r Z n along the three principal cubic directions at (a) 4.2 K and (b) 10 K; the inset shows the temperature dependence of the critical field; the points are the experimental data along [101] (O), [111] (zX) and [001] ( 0 ) ; the lines are the theoretical fits (broken curves: without quadrupolar terms; full curves: with G ~ = G 1 = 1.5 m K and G ~ = G 2 = 11 m K terms (from Morin and Schmitt 1978).
70
P. MORIN and D. SCHMITT - 250
-50P
///
HoZn 12
-
// /- ~" " i l l
v
>.-
L9
"Tl
-15OF
-26G
0~ W
Z W
/
W W
_.__~[i01]
//
----
// . . . .
G2:0
/
rr
b_
.... -270 0
G1 : - 4 mK G2= 10 mK
/ / ~[101] i 20
GI=0
i 40
TEMPERATURE
-2sot-
0
(K)
80
[lOl]
[111]
Fig. 49. (Left-hand part) calculated temperature variation of the free energy along [101] and [ l l l ] for HoZn, without quadrupolar terms (broken curves) and with G ~ and G ~ contributions (full curves); (right-hand part) arrangement of the lowest CEF levels in HoZn at T R = 25 K according to whether the moment lies along the [101] or [1ll] direction and taking into account the quadrupolar coefficients G ~= - 4 inK, G ~= 10 mK (from Morin and Schmitt 1978).
with Nd, Tb and Ho (see, e.g., Gignoux et al. 1975, Sankar et al. 1977, A16onard and Morin 1985). In each compound, the two-fold axis is the direction of easy magnetization at low temperature. In HoZn for example, the critical temperature T R is calculated to be 48K with tile CEF alone (Morin and Schmitt 1978). Introducing appropriate quadrupolar parameters G r and G ~ shifts the free energy curves and allows to reduce T R to 25 K in agreement with the experimental data (fig. 49). The effects of quadrupolar interactions on the magnetization processes in antiferromagnets are more difficult to analyze due to the existence of several sublattices (Jensen et al. 1980). On the other hand, they are more relevant for the stability of the actual magnetic structures. This will be discussed in the next section (3.6).
3.5.3. Effect on the direction of easy magnetization In the previous section, the dominant CEF determined the direction of easy magnetization and the quadrupolar interactions determined the exact temperature TR, for the change of the easy axis. In this section, we discuss some cases where the quadrupolar energies are larger than the CEF anisotropy. Good examples are provided by the ferromagnets DyCd and DyZn which have the CsCl-type structure (T c - - 8 0 and 140K, respectively). In both cases, a four-fold axis is observed as the direction of easy magnetization and a large tetragonal spontaneous strain occurs in the ordered phase (0.8% at 4.2K in DyZn) (Morin et al. 1977). Their cubic level schemes are given in fig. 50a together with those of the antiferromagnets DyCu and DyAg. Their anisotropic splitting by bilinear interactions always favours the three-fold axis: the CEF
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
zOO
71
100
I
r(3) DyAgi~--~. <..8 ~
!~-Y~.-~-~.~ /
200
c11"~i
®
.....
o
I
@easyaxis [I11]~k~O011
-200
2s~
W=-.57K \ 6
~
(-.}7)
-400
¢_-60K
',
(.9
\
T:4.2K J
(-5)
0 0
-.5
X
Fig. 50. (a) CEF level scheme in dysprosium compounds having the CsCl-type structure. E is the energy in Kelvin, W is a scale factor, the value of which is given in parentheses for each compound, x is the ratio between fourth- and sixth-order CEF terms; (b) x-dependence of the strain susceptibilities, X2 (=Xr) and X~ (=X~), for a given W at low temperature; (c) easy axis at low temperature according to x and G1 ( = G ~) for given values of W and 0". The full line is mainly related to the case of DyAg and DyCu; for DyZn and DyCd, the critical line would lie lower (from Al6onard et al. 1984a).
anisotropy between three-fold and four-fold axes is too large in DyAg and DyCu to be reversed by the tetragonal quadrupolar energy; the three-fold axes remain the easy ones and the (negative) trigonal quadrupolar interactions stabilize the muti-axial spin arrangement (section 3.6.1). In DyZn and DyCd, tetragonal quadrupolar interactions have the same order
8
-J - ~ m ] ~ ~ ~ _ _ .1 t~o~. . . . . . . . . . . -o-
7 "
i
~
" o
.
o
n
o
o[~OO1~
-
- -
2
GI=ImK ,G2=-22inK
N
ErNi 2
Z
. . . . . GI=O ,G2=0 T=1.5 K
:E'
o
2'o
go
go
8'0
'60
1~,o
INTERNAL MAGNETIC FIELD (kOe)
Fig. 51. ErNi2: magnetization variation at 1.5 K as a function of internal field. ©, • and [] are experimental values. Full lines are the variations calculated with 0* = 7 K, W = - 1.0 K, x = - 0.35, G1 (=G ~) = 1 mK and G 2 ( = G ~) = - 2 2 m K . Dashed lines are the variations calculated without quadrupolar terms (from Gignoux and Givord 1983).
72
E MORIN and D. SCHMITr
of magnitude as in DyCu and DyAg, but are more efficient due to the groundstate wave functions (see fig. 50b). They are then able to make the four-fold axis the easy direction above a G r critical value depending on the exact level scheme. This is realized for D y Z n as well as for DyCd (A16onard and Morin 1985). These two compounds represent the border-line case of quadrupolar effects on the temperature T R for the change of the axis of easy magnetization in the ordered phase (section 3.5.2); also in the paramagnetic state the quadrupolar effects are noticed through anisotropic magnetization processes. Similar features may be found also in other series specially when the C E F anisotropy is weak. This is, e.g., the case in ferromagnetic ErNi 2 which exhibits a [001] easy magnetization axis (fig. 51). The magnetization processes along the three main cubic directions are characterized by a crossing between the [001] magnetization curve and the [101] and [111] ones. Their description, as in D y Z n and DyCd, is impossible without quadrupolar interactions stabilizing the [001] axis as the easy one, instead of the [111] axis. 3.6. Effects on the magnetic structures 3. 6.1. Multi-axial structures In addition to their effects on the magnetic transition, the quadrupolar interactions also contribute to the determination of the magnetic structure through the minimization of the free energy. It is well-known that in cubic symmetry the spin structures that are described by one or more propagation vectors, q, which belong to the same star have the same bilinear energy and lead to identical powder neutron diffraction spectra (Wintemberger and Chamard-Bois 1972). The presence of several propagation vectors determine a multi-axial arrangement for spins (see, e.g., fig. 52). The knowledge of the C E F allows one to partly remove this ambiguity.
(17 FI O )
(a)
(b)
@=90 °
(c)
(d) Fig. 52. Possible configurationsfor the ( ½, 1,0)-type antiferromagneticstructure with 0 = 90°; (a) and (d) collinear; (b), (c) and (e) bi-axis; (f) multi-axis. All these structures lead to the same powder neutron diffraction spectrum as that observed in PrAg (from Morin and Schmitt 1982b).
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
73
The magneto-elasticity can determine the actual structure. For instance with a four-fold easy axis, a large ( c / a - 1) magnetostriction may be a clear signature for a collinear arrangement as observed in many CsCl-type compounds (Morin et al. 1978a, Schmitt et al. 1978). On the contrary, the existence of collinear structures with three-fold easy axes has not been demonstrated for CsCl-type rare earth intermetallics. However, there is plentiful evidence for this in the NaCl-type series, in particular in Tb compounds (L6vy 1969, Hulliger and Stucki 1978b). In the presence of negative quadrupolar interactions, multi-axial spin structures are stabilized. This occurs within the trigonal symmetry in CsCl-type compounds as DyAg, DyCu (A16onard et al. 1984a) and NdZn (Morin and de Combarieu 1975) and in the AuCu3-type compound TmGa 3. In the latter compound for instance, powder neutron diffraction spectra have been indexed by using a [ ½ ½0] propagation vector and a [001] spin direction and the hypothesis of a collinear spin arrangement (Morin et al. 1987a). In spite of the sizeable magneto-elastic coupling found in the non-ordered phase (Morin et al. 1987c), no spontaneous strain was observed, making this collinear structure unlikely. The actual spin structure was determined by neutron diffraction under a magnetic field applied along the three main axes of a single crystal (Morin et al. 1987d). No change of magnetic intensities was detected when cooling the sample in different applied fields. This indicates the sample is spontaneously single domain; consequently, the actual structure is a triple-q structure which preserves the cubic symmetry (see fig. 53). This solution was confirmed by determining the different structures revealed by magnetization processes. In particular, along the Jill] direction, the quadrupolar structure and energy are unchanged at the lower critical field, He1, only the spins being reversed (fig. 54), the Zeeman energy corresponds mainly to this change of bilinear energy. At the high critical field, He2, the Zeeman energy has
•
• HC2
-,-10
I
I
2
3 4 5 TEMPERATURE (K)
I
TI
I
I
6
Fig. 53. Magnetic phase diagram for a magnetic field along a three-fold axis in TmGa 3 (from Morin et al. 1987d).
74
P. MORIN and D. SCHMITT
oooo0O0,°~° o°°° O
T=
1,5°4, " o •
o: .%
•
• •
A~A~
•
°"
°~'°t
0"£
•
"5K
Ao*
"d •
oOO oO4°,°° o • o Jo o ~Ae
rn ,~1.¸ v
oJ
•
.%" 2.o ,e •
o
8° o ~ oo
,0
2'o
3'0
20
H (kOe)
10
6'0
70
Fig. 54. Magnetization processes for a magnetic field along [111] in TmG% (from Morin et al. 1987d).
to compensate both the bilinear and quadrupolar energies as the AFQ arrangement is broken at this field. The existence of multi-axial spin structures in rare earth antiferromagnets with high symmetry appears to be a common consequence of AFQ interactions. In series other than the CsC1- and AuCu3-type ones, multi-axial structures also exist, e.g., as in rare earth hexaborides. In the AFQ phase of CeB6, the local anisotropy drives the spins to order in a multi-axial arrangement described by two propagation vectors [¼ 1 ½] and [1 _ ] ½] whereas the AFQ structure itself is described by [½ ½ ½] (fig. 12). The magnetic moments of the two cerium AFQ sublattices order along two mutually perpendicular directions (Effantin et al. 1985). The low-temperature magnetic structure of PrB 6 was initially described by a single [I ~ ½] propagation vector from powder neutron diffraction spectra (McCarthy et al. 1980). Recent experiments on a monocrystalline sample have shown that the structure is in fact double q, described, as in CeB6, by [¼ ¼ ½] and [ ~ - ¼ ½] and thus probably results from the effects of AFQ interactions (Burlet et al. 1988). Another complex situation is observed in the NaCl-type compound, HoP (Fischer et al. 1985). At Tc = 5.4 K it undergoes a second-order transition to a ferromagnetic state. At TF = 4.8 K there is a first-order transition to a flopside state with two Ho sublattices consisting of alternate ferromagnetic (111) planes. From one plane to the next one, the 4f moments change from one four-fold axis to another. Due to the antiferro- and ferromagnetic components, this lowtemperature structure is biaxial (fig. 55). The spontaneous and field-dependent properties were analyzed using the cubic model with two sublattices (Kim and Levy 1982). This is justified by the presence of six low-lying CEF levels nearly degenerate and well-separated from the excited ones. Bilinear and quadrupolar pair interactions were limited to nearest and next-nearest neighbours. The
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS
75
lO-3Jn
10-
5-
10
5.OK I
I
I
I
~1 i
i
5
i
o
71 ""-------
3'0
eo
'
9'0
i
1~o
~'ec*~
Fig. 55. Temperature dependence of neutron diffraction pattern of a HoP powder sample. At 4.2 K the indexing corresponds to the magnetic unit cell with lattice constant a m = 2a. Insets indicate corresponding magnetic structures (from Fischer et al. 1985).
corresponding parameters were deduced from the magnetic excitations measured in the flopside phase by inelastic neutron scattering. Bilinear interactions are positive, in accordance with the ferromagnetic tendency, tetragonal quadrupolar ones are negative in accordance with the change of the z-axis from one plane to another. Thus the negative quadrupolar interactions stabilize the flopside structure. 3.6.2. Incommensurate magnetic structures The nature of bilinear interactions in rare earth intermetallic systems very often leads to modulated magnetic structures (Koehler 1972, Rossat-Mignod 1979). This corresponds to a maximum value of the Fourier transform of the bilinear interactions for an incommensurate wave vector. These structures are either
76
P. MORIN and D. SCHMITT
phase modulated or, when the magnetic moment is confined along a given direction by the anisotropy, amplitude modulated. When considering only bilinear interactions, other couplings being equal, the modulated structures can change when decreasing the temperature below the N6el transition in three ways: (a) For a non-magnetic ground-state (non-Kramers ions, Kondo systems) the modulated structure can remain stable down to 0 K as observed in HoNi0.sCu0. 5 (Gignoux et al. 1977) and in CeA12 (Barbara et al. 1980). (b) For a magnetic ground state (Kramers ions) entropy effects drive the structure to move progressively towards an antiphase structure with constant magnetic moments as in metallic Tm (Koehler et al. 1962). (c) A first-order transition to a commensurate structure (constant magnetic moments) occurs at a relatively high reduced temperature T]TN; e.g., T I T N = 0.87 in TmCu (Morin and Schmitt 1980), 0.82 in DyAg (Kaneko et al. 1987). In the presence of quadrupolar interactions, some modifications are expected. In particular for the first case (a), the nonmagnetic ground state can be quadrupolar (F3 doublet) and the same entropy effects exist for quadrupoles as for spins: consequently, the magnetic structure also has to change as in cases (b) or (c). For case (b), the tendency to a commensurate quadrupolar structure facilitates the squaring-up of a modulated spin arrangement. Unfortunately there is no clear evidence of these two situations up to now. In case (c) the temperature of the first-order commensurate transition Tt appears to be strongly dependent on the large quadrupolar interactions. In TmCu, Tt = 6.7K is close to the temperature To_= 5 K, at which it would ferroquadrupolarly order in the absence of bilinear interactions. Below Tt, both the magnetic and quadrupolar structures are collinear with tetragonal symmetry (Morin and Schmitt 1980). In DyAg, a modulated structure is also observed close to TN (Kaneko et al. 1987). However, the commensurate triple-q structure with spins along the three-fold axes is rapidly stabilized by the trigonal AFQ interactions through a first-order transition at Tt = 46.5 K. The influence of quadrupolar interactions on the commensurate-incommensurate transition appears to be a rather general feature and can be observed in other series such as RB 6 (Effantin 1985, McCarthy et al. 1980) and in RGa 3 (Morin et al. 1987a,d). From a purely magneto-elastic point of view, it is worth noting that a static strain wave is associated with the incommensurate spin wave. Its modulation is one half that of the spins because of the time-reversibility of the quadrupoles. It manifests itself through lattice satellite reflections at wave vectors ~'m~= 2~'n~, where r m is the magnetic propagation vector. They have been observed in metallic Cr (Tsunoda et al. 1974, Pynn et al. 1976), and in rare earth metals such as Ho (Keating 1969, Bohr et al. 1986) and Er (Gibbs et al. 1986). In the incommensurate helicoidal phase of these metals, 2%-satellites have been observed with X-ray diffraction and their intensity was related to the aspherical 4f charge density changing its orientation from plane to plane (fig. 56). In addition, % varies with temperature, the spin structure locks-in at commensurate values of % because it gains some magneto-elastic energy. These lock-ins are also observed in Dy
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS I
I
I
j/~
I
77
I
ERBIUM(OOQ) 2 Tm
47.5K ,;'/ .~".~..
20O o I00 C
49 K.,..-~
~[l~-
2~ ~o× p__. z o
_
,95K..~
505K
---
.... J
51.6K~ 52K .--
k__
k____
i
4/71 200 55K I00 I I I I ~] 0 2.55 2.56 2.57 2.58 2.59 (unitsof c*) m
Fig. 56. Temperature variation of the nuclear scattering at 2Tm in Er near the % = ~ lock-in transition (% is the magnetic propagation vector) (from Gibbs et al. 1986). ( G r e e n o u g h et al. 1981), they appear to be m o r e frequent in phase m o d u l a t e d structfires (existence of a quasi-isotropic plane) than in amplitude m o d u l a t e d structures (easy axis).
3.7. Effects on magnetic excitations T h e effects of quadrupolar interactions o n the e l e m e n t a r y excitations of m a g n e t i c systems w e r e first studied, by using effective spin m o d e l s , at the s a m e time as the
78
P. MORIN and D. SCHMITT
quadrupolar Hamiltonian itself. Cases with S = 1 (Chiu-Tsao et al. 1975, Sivardi~re 1975), S = 3 (Chiu-Tsao and Levy 1976), and S => 3 systems (Sivardibre 1976) were successively investigated. Two types of effects were considered in these calculations, namely: (i) the modification of the usual (spin waves) magnetic excitations by the quadrupolar couplings, and (ii) the occurrence of pure quadrupolar excitations corresponding to two-spin (AM = 2) deviations. The first effect consists simply of a variation of the size of the spin waves dispersion, at least in the ferromagnetic state. In the quadrupolar phase, however, an additional change of shape occurs, the dispersion curves becoming linear in q when q--~ 0. More exciting is the possibility of two-spin (AM = 2) excitations (quadrons) with the same type of dispersion as the AM = 1 magnetic excitations, but for different energies. Such quadrupolar excitations can be measured by inelastic neutron scattering experiments (Chiu-Tsao and Levy 1976, Levy and Trammell 1977). However, to our knowledge, they have never been observed experimentally. In more realistic calculations, the influence of the CEF must be taken into account by considering the whole level scheme, i.e., the actual CEF wave functions. Elementary excitations have then been obtained through a Green function (Sablik and Wang 1978, 1979) or a generalized susceptibility formalism (Morin et al. 1980c) with similar results. Attention was mainly focused on the influence of quadrupolar couplings on the magnetic (spin wave) excitations, which led to the same general conclusions as the effective spin models. In the magnetic phase, quadrupolar interactions modify the average energy of the dispersion curves, through the single-ion Hamiltonian, as well as their shape, through the Fourier transform of the two-ion quadrupolar couplings. The excitations are then dipolar and quadrupolar mixed excitations. The first effect has been widely used, as described in the literature, for adjusting the energy of the excited CEF levels with the average value of the dispersion curves (Furrer et al. 1977, Pierre et al. 1984). The second effect is difficult to observe; it has been invoked in NdSb in order to explain the shape of the experimentally observed excitation spectra (Sablik and Wang 1985). Surprisingly in the quadrupolar phase, the quadrupolar coupling which induces the static quadrupolar ordering does not produce any specific dispersion in addition to tha(arising from the bilinear coupling; it only modifies the overall amplitude of the dispersion curves. This has been clearly demonstrated in TmxLul_xZn, in which a 10% dilution of Tm by Lu suppresses the ferromagnetic ordering existing in TmZn and provides a pure quadrupolar phase below TQ = 5.8 K (Morin et al. 1981). In that phase, the total amplitude of the longitudinal excitation reaches 16 K, while the corresponding value is only 10K in the ferromagnetic phase of TmZn. The only expected effect in the quadrupolar phase is the modification of the transverse excitations by the second (trigonal) quadrupolar interaction. Unfortunately the latter is too weak in Tm0.9Lu0.1Zn to give rise to a noticeable effect.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
79
3.8. Two-ion anisotropic magneto-elasticity The magneto-elastic modulation of the pair interactions has been neglected in all the preceding sections. This may be justified by analyzing Gd compounds in the absence of any single-ion terms. Usually two-ion magneto-elasticity is only associated with the bilinear interactions. Its modification occurs through either the isotropic exchange striction (see section 3.9) or symmetry lowering modes which introduce anisotropic interactions. The corresponding equilibrium strains have been given in section 2 [eqs. (19)-(24)]. In the CsCl-type compound GdZn, a spontaneous tetragonal strain has been observed below T c = 270 K (Morin et al. 1977, Rouchy et al. 1981). Its zeroKelvin value is e~ = -4.5 x 10 -4, a value one order of magnitude smaller than observed in other RZn (see section 4). The two-ion strain is expected to vary roughly as the De Gennes factor across a series and to take its maximum value for the Gd compound. Thus its contribution to the tetragonal strain rapidly decreases towards each end of the rare earth sequence; e.g., it is only half a percent of the observed value in TmZn. The e-strain has been estimated to be two orders of magnitude smaller than the y-strain. In the Laves phase compound GdAI2, the opposite situation is encountered: e~ is negligible and the e-strain reaches e ~= 1.5 x 10 -5 at 0 K (Du Tr6molet de Lacheisserie 1988). As for RZn, the spontaneous strain is one order of magnitude smaller in GdZn than in other compounds (section 4). The ratio reaches 40 between~strains in TbFe 2 and GdFe 2 polycrystalline samples (Abbundi and Clark 1978). In hexagonal RCo 5 also anisotropic strains are noticeable in the presence of a rare earth orbital moment, but weak in YCo s (Andreev et al. 1985a). In NaCl-type Gd compounds, a spontaneous trigonal strain is observed (Hulliger and Stucki 1978a). The e ~ values observed at low temperatures are 8 x 10 -4, 12 x 10 -4 and 17 x 10 -4 in GdAs, GdSb and GdBi, respectively. For the same trigonal symmetry they are -37, -28 and -33 x 10 -4 in TbAs, TbSb and TbBi (Hulliger and Stucki 1978b). The ratio is then not as large as in the preceding examples, in particular for antimonides and bismuthides. This noticeable two-ion anisotropic magneto-elasticity seems to be a general feature of NaCl-type compounds as observed experimentally and expected theoretically (section 3.2.4). Although the two-ion symmetry lowering modes are usually negligible in comparison with the one-ion modes, the latter example shows that it is necessary to study the Gd compound in a given series systematically.
3.9. lsotropic magneto-elasticity and pressure effects The e ~ volume strain may be obtained from a combination of parallel and perpendicular changes-of length Ix/3 e ~ = AlL+ 2h. from eq. (25)]. Equation (19) shows that one- and two-ion magneto-elastic terms coexist in a given compound. Only the two-ion term is present in Gd compounds. As they are expected to
80
P. MORIN and D. SCHMI'IT
decrease from the Gd value according to the De Gennes law, this allows to evaluate the one-ion contribution in other compounds. 3.9.1. Two-ion contribution In addition to the change of length method, the volume strain may be deduced from measurements of the dependence of the magnetic transition on hydrostatic pressure. For vanishing CEF effects, a MFA treatment proposes an equivalent expression for e ~ as a function of the reduced magnetization o- (Bean and Rodbell 1962): e~-
3 J 2 J+l
ATe 2 NkB -Aff ~r '
where o" is the normalized magnetization. Both methods usually give compatible determinations in Gd compounds. In GdAI2, e.g., the isotropic strain at low temperatures is found to reach - 1 . 7 5 x 10 -3 from change of length measurements (Du Tr6molet de Lacheisserie 1988) and -1.85 × 10 -3 from pressure data (Jaakkola and Hanninen 1980). In GdZn, at T = 0 K a value of +5 x 10 -4 is obtained (Morin et al. 1977, Hiraoka 1974). In GdAg, at T = 0 K a large value, e ~ = -3.8 × 10 -3 may be deduced from ATN/AP = 1.17 K/kbar (Yoshida et al. 1987). This is characteristic of a strong dependence of the magnetic properties on the actual electronic structure, which is close to instability (see section 3.10). 3.9.2. Single-ion contribution For rare earths other than Gd, the isotropic strain also has a single-ion term describing the bulk modulation of the CEF. As the 4th and 6th order CEF terms are of comparable magnitude in rare earth (R) intermetallics, the two magnetoelastic coefficients B(4 ) and B(6 ) have to be considered [eq. (19)]. By comparing results on GdZn with those on TbZn and DyZn, e ~ has been found to be negative in the latter two ( - 3 × 10 -4 and -13 × 10 -4, respectively). This indicates that the CEF part is opposite to that of the exchange and at least twice as large in magnitude (Morin et al. 1977). With regard to the single-ion contribution, thermal expansion appears as the best experimental probe. It may be discussed in the paramagnetic phase within a susceptibility formalism (Morin and Williamson 1984) as well as in terms of Gr/ineisen parameters (Ott and Lfithi 1977). Thus, thanks to thermal expansion results, the pressure dependence of the magnetic susceptibility has been explained in TmSb by a modification of the single-ion susceptibility rather than by a change of the bilinear interactions (Ott and L/ithi 1977). In another singlet ground state system, PrSb, inelastic neutron scattering under pressure (Vettier et al. 1977) and thermal expansion results (Ott 1977) have consistently shown the CEF spacing to decrease when the volume decreases.
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
81
3.9.3. Pressure effects Except for the previous examples of the magnetic susceptibility or the ordering temperature, the literature is rather scarce about pressure effects in rare earth intermetallics. Indeed the energies associated with pressure remain weak in comparison with the intrinsic couplings. Hydrostatic or uniaxial stress seems to be able to have dramatic effects only in the case of magnetic systems which are subcritical. For instance, hydrostatic pressure has been shown to drive an antiferromagnetic ordering in PrSb through a softening of the X point magnetic exciton, the zero-pressure value of which is determined by the intrinsic anisotropic bilinear interactions (McWhan et al. 1979). More relevant to the present review is the case of another single ground state system, Pr (Jensen 1979). The bilinear interactions between rare earths at hexagonal sites are just below the critical value for an antiferromagnetic ordering. This makes the spin system very sensitive to various influences, in particular to magneto-elastic effects with e-symmetry. They have been show to influence deeply the magnetic excitons ( H o u m a n n et al. 1979). As the B ~ magneto-elastic coefficient was deduced from ultrasonic measurements (Palmer and Jensen 1978), it was possible to calculate the e" critical value that is needed to modify the level spacing enough so as to drive the spin system to be critical; an uniaxial stress of 770 bar applied along the a-axis of the hexagonal cell was found to be necessary. Experimentally antiferromagnetic ordering was observed below 7.5 K just under about 800 bar (see fig. 57), thereby verifying the above prediction.
1.0 -
/---r-X
N
"1" l->-
,,
n.I..iJ Z W
/
~,.¢/ t2<
,'
v
......
S-TO
.
0.5 ['-- ~.a~""',v" ~ l r . ~ '''r
0
0 F
0.1
0.2
4/ ,,p"
,'/
,'/ ,,,,//
EL/
0.3
0.4
WAVEVECTOR
0.5 M
Fig. 57. Dispersion relations for magnetic excitons propagating on the hexagonal sites in Pr, at 5.2 K, at room pressure (circles) and under an uniaxial pressure of 800 bar (squares). TO and LO branches correspond to transverse and longitudinal optical modes; the TA and LA transverse and longitudinal acoustic modes were not examined under stress. An asterisk (,) denotes the magnetic satellite position in the ordered phase (from McEwen et al. 1978).
82
P. MORIN and D. SCHMITT
3.10. Magneto-elasticity in the presence of lattice instability In all of the preceding sections, magneto-elasticity was acting on an otherwise stable lattice: As for the lattice parameter, the background elastic constants were assumed to vary weakly across a series as well as to be weakly dependent on the temperature. This situation is the norm, but there are some exceptions. For instance, the CsCl-type structure appears less stable than other crystallographic arrangements such as the MgCu 2- or NaCl-type ones. Indeed some series are incomplete (Iandelli and Palenzona 1979) with compounds exhibiting martensitic transformations (Gefen and Rosen 1981, Al6onard and Morin 1984, Sekizawa et al. 1981). For example, RAg compounds are cubic at all temperatures, and an incipient lattice instability is easily evidenced by hydrostatic pressures at temperatures which drastically depend on the pressure (Kurisu 1987). Chemical pressure obtained by substituting Ag by In produces analogous effects (Ihrig et al. 1975, Fujiwara et al. 1983). The martensitic transformation has been shown to occur mainly through a phonon softening at the M point of the Brillouin zone in LaAgxIn~_ x (Knorr et al. 1980, Abu-Aljarayesh et al. 1986b) rather than through a band Jahn-Teller coupling developing at the F point. Through an anharmonic coupling with the F-point acoustic phonon, this M-point phonon softening drives a partial softening of the lattice elastic constant. A full understanding of this lattice instability requires, in the presence of a strong electron-phonon coupling, dynamical models considering the full dielectric response function of the electronic system. The starting point must be the determination of the actual band structure as has been done for transition metals (Dacorogna et al. 1982). The coexistence of magneto-elasticity and lattice instability makes it difficult to determine the predominant coupling from the temperature dependence of the elastic constants. For instance, in light rare earth-silver compounds, the tempera-
1,4
=
=
I
~" E
(PrAg)o ...................
- ................................................
~c, 1 . 0 ~
.....
* ............
L
o
v
a ,..~ -6 -
---~~--CeAg v/'
,~''~ I
I
I
I
lOO 200 TEMPERATURE (K)
I
300
Fig. 58. Temperature variation of ½Cv= ½(Cll - C1~) for LaAg, CeAg and PrAg. (PrAg)o is the lattice behaviour after correction for the magneto-elastic contribution (from Giraud et al. 1983).
Q U A D R U P O E A R EFFECTS IN RARE EARTH INTERMETALLICS
83
~.5
t t
/,--"
\:~ci,_~
R=Tm Dy
.9
L
Pr Ce La
411
1.2
ionic radius {~,)
Fig. 59. Variation of the three normal elastic constants at room temperature as a function of the ionic radius in the RAg series, normalized to their value in TmAg (from Morin 1988).
ture variation of Cll - C12 in the LaAg compounds is obviously anomalous and cannot be used directly as background in a magnetic compound (fig. 58). Across the series, the instability tendency progressively vanishes when proceeding to heavier rare earth compounds (fig. 59). As discussed from a theoretical point of view for the orthochromites (Kataoka and Kanamori 1972), the anharmonicity involved with the lattice instability manifests itself in the occurrence of the quadrupolar ordering. In CeAg, where the quadrupolar interactions have been determined by means of parastriction and third-order magnetic susceptibility, both the temperature and the discontinuity of the order parameter are increased, e.g., the quadrupolar ordering would occur around 10 K with only quadrupolar interactions, but this temperature is increased to 15.85 K due to the presence of the incipient lattice instability (Morin 1988). 4. Magneto-elastic and pair interaction coefficients in rare earth intermetallic series
4.1. CsCl-type structure compounds CsCl-type rare earth intermetallics RX have been extensively studied due to the availability of their single crystals. Their very simple structure allows a deep insight in the 4f magnetism; analyses are possible according to the alloyed non-magnetic metal (X = Mg, Zn, Cd, Cu, A g , . . . ) . They have revealed large quadrupolar interactions which usually prevent any analysis which would only consider CEF and bilinear interactions of the magnetic data. In particular large spontaneous strains exist with tetragonal symmetry; they reach record values ( c / a - 1 = 1.4% in PrZn and -0.8% in TmZn). Important elastic constant softenings are observed in the paramagnetic range. FQ orderings were discovered in TmZn and TmCd (section 3). The following subsections outline the numerous effects of quadrupolar interactions on the magnetic properties in each series. Quadrupolar interactions are largest in tetragonal symmetry. They are of a ferroquadrupolar type. In trigonal symmetry, they are negative and very often stabilize multi-axial spin structures when the three-fold axis is the direction of easy magnetization. Systematic studies of the TmX have shown how very relevant the (spin and) quadrupolar Hamiltonian is within the MFA. In particular the
84
E MORIN and D. SCHMITT
absence (or the weak value) of fluctuations leads to determinations of G r and G" from third-order magnetic susceptibility measurements in close agreement with those obtained from magneto-elasticity data. However, there are particular situations where the quadrupolar coupling may exist with additional terms (as in the case of the lattice instability for CeAg).
4.1.1. RZn series Light rare earth compounds are antiferromagnetic with a (½,0, 0) propagation vector. From SmZn to TmZn, a ferromagnetic behaviour is observed (table 5). The CEF parameters A4(r4), A6(r 6) are found to be negative by inelastic neutron spectroscopy, without exhibiting an erratic variation (Giraud et al. 1986). When the four-fold axis is the easy magnetization direction, a large spontaneous tetragonal strain is observed. The various magneto-elastic coefficients were determined by different experimental techniques. Note that when the temperature range investigated by parastriction is reduced (high Tc values), only a relationship between G ~ and B ~ is obtained. Some differences between B ~ values deduced from parastriction and spontaneous strain measurements may be noticed, in particular in TbZn, DyZn and HoZn. They are partly explained by an overestimation of the two-ion contribution to the spontaneous strain. Across the series, the coefficient B ~, normalized by the Stevens coefficient exhibits an average value of about -2050 K with the ErZn value slightly remote (see section 5.1). In trigonal symmetry, reliable values of B ~ are achieved in TbZn and HoZn from the spontaneous trigonal strain. In other compounds, only the parastriction may be efficient. However, as the tetragonal symmetry lowering mode is highly favoured in CsCl-type compounds, a very accurate orientation of the magnetic field along [111] is necessary in order to avoid the presence of any tetragonal contribution to the change of length measured. Here again the parastriction leads only to a relation between values of B e and G ~. B e values obtained with both techniques are in good agreement in TbZn and HoZn. The ratio B ~/aj obviously varies across the heavy rare earth compounds; in TbZn, it even Changes sign. The magneto-elastic contribution to G ~ is less than 1 mK for the trigonal symmetry, but reaches 5.2, 9 and 120 mK for tetragonal symmetry in TmZn, PrZn and CeZn, respectively. However, in RZn, the quadrupolar interactions originate mainly from the pair coupling, as proved by the ratio K~/G~jE of 1.6 and 4 which is found in ErZn and TmZn, respectively. They are systematically positive in tetragonal symmetry, negative in trigonal symmetry. In TmZn, they dominate the bilinear interactions, and induce a quadrupolar ordering in the paramagnetic phase. Depending on their sign, they may stabilize either collinear antiferromagnetic structures as in CeZn and PrZn or multi-axial ones as in NdZn. In other compounds, none of the magnetic properties can be described without the quadrupolar interactions as shown for the four-fold axis of easy magnetization in DyZn, the magnetization processes in HoZn and ErZn, the actual temperature for the change of easy axis in NdZn and HoZn and the first-order transition in CeZn (see section 3).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
85
4.1.2. RCd series This series has been less extensively studied than the RZn one, due to the large neutron absorption cross section of cadmium, which in particular forbids a direct determination of the CEF level scheme by neutron spectroscopy. However, the properties appear to be very close to those of the RZn series from a magnetic point of view. One of its peculiarities rests in the properties of the lattice: La, Ce and Pr compounds exhibit a well-defined martensitic-type transition, which is revealed by electrical resistivity or susceptibility measurements, at 61, 107 and 125 K, respectively (A16onard and Morin 1984). At 216 K CeCd undergoes an additional transition which is as well-defined as the previous one (Fujii et al. 1985). Compounds from Nd to Tm are ferromagnetic, CeCd and PrCd being antiferromagnetic (A16onard and Morin 1985). CeCd was also found to be ferromagnetic (Fujii et al. 1985). The easy magnetization axes and the magnetic moments observed in ferromagnets are closely reminiscent of the corresponding results in RZn. For instance, all of the properties studied in TmCd, in particular the quadrupolar properties, can be described with a CEF level scheme very close to that in TmZn. Assuming the CEF parameters as constant, quadrupolar coefficients may be obtained through the full description of the magnetization properties. This is true in particular for the four-fold easy direction in DyCd as well as in DyZn, and for the temperature of change of easy direction in NdCd as well as in NdZn. On the contrary, observing [111] axes as easy direction in CeCd and PrCd clearly reveals a deep modification of all the one- and two-ion terms driven by the lattice instability. In RCd ferromagnets, quadrupolar interactions are present having large analogies with the RZn case as proved by the comparison for a given rare earth (tables 5 and 6). However, from the thorough study of TmCd, both the pair-interaction coupling and the magneto-elasticity are smaller than in TmZn in tetragonal symmetry, the ratio Kr/G~E, however, is twice as large. This is emphasized by the T o value in both Tm compounds. In trigonal symmetry, the coefficients G ~ are negative as in RZn. 4.1.3. RAg series This series exhibits some specific behaviours. Bilinear interactions favour incommensurate magnetic structures. One notes, however, a tendency to ferromagnetism at the beginning of the series through a ferromagnetic component in PrAg and a ferromagnetic moment in CeAg. The observed properties strongly depend on the preparation. In an unannealed sample, the magnetic structure is systematically commensurate with the lattice periodicity. Removing the metallurgical stresses by annealing the sample restores the intrinsic nature of the bilinear interactions as it was demonstrated in HoAg, ErAg and TmAg (Nereson 1973a,b, Morin and Schmitt 1982a). Another important peculiarity rests in the incipient tendency to lattice instability, although all the RAg are stable within the CsCl-type structure in the absence of external stress (section 3.10). Under hydrostatic or chemical pressure, the instability towards a tetragonal-symmetry based structure (Maetz et al. 1980) is
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revealed by a martensitic transition at a strongly pressure-dependent temperature (Kurisu 1987). Its driving mechanism consists of a softening of the TA phonon at the M point on the Brillouin zone (Knorr et al. 1980) and induces anharmonic contributions to the elastic constant. In the absence of an external stress, the incipient lattice instability takes the opportunity of the quadrupolar ordering in CeAg to manifest itself through both an increase of the quadrupolar ordering temperature (see section 3.1.1) and a spontaneous tetragonal strain ( c / a - 1 = 1.9%) which is larger than expected from the one-ion magneto-elasticity only. In RAg, the tendency to instability is particularly clear in LaAg, CeAg and, to a smaller extent, in PrAg (see section 3.10). The tetragonal magneto-elastic parameters, normalized by the second-order Stevens coefficient, are of the same order of magnitude as in RZn. Their average value is around -2000 K/at., the CeAg value being somewhat smaller (table 7). The trigonal symmetry coefficient observed in PrAg and TmAg leads to positive BE/aj ratios (4400 and 2200 K/at., respectively). As in RZn and RCd, the pair interactions are positive in tetragonal symmetry, negative in trigonal symmetry. For example, in TmAg, they seem to be smaller than in RZn. Their effects are, however, numerous on the magnetic properties. The commensurate-incommensurate transition at 9.5 K in ErAg may be influenced by quadrupolar interactions as discussed for DyAg (section 3.6.2). The N6el temperature in TbAg is the highest one known with a first-order character, which agrees with the G v value determined in the paramagnetic phase. In PrAg, a neutron diffraction experiment with polarization analysis on a single crystal has, at least partly, removed the ambiguities among the possible structures of fig. 52 and has shown that a double-q structure (structure b or c) is very likely in agreement with the existence of rather negative quadrupolar interactions (Givord et al. 1985). 4.1.4. RCu series CsCl-type structure compounds range from GdCu to LuCu; equiatomic alloys with light rare earth crystallize within the FeB structure (Buschow 1980). Lattice instability develops itself in boundary compounds: GdCu undergoes a martensitictype transition with a very large hysteresis (Gefen and Rosen 1981) (table 8). In TbCu the martensitic transition seems to be triggered by the antiferromagnetic ordering (Pierre and Hennion 1982). All the other heavy rare earth compounds are stable down to 0 K. Antiferromagnetic interactions are observed across the series, they are characterized by a (1, ½,0) propagation vector. However, ( ½ - r, ½,0) modulated structures have been observed in TmCu and ErCu close to the N6el temperature in a way reminiscent of RAg compounds. Magneto-elastic coefficients are available for TmCu and DyCu in tetragonal symmetry. In TbCu, a B ~ value may be achieved from the spontaneous strain by assuming a maximum value for ( O °) due to the large exchange field at 4.2K. The resulting ratios Br/aj are -2300, -2450 and -2800 K/at. in Tb-, Dy- and TmCu, respectively: the same sign and order of magnitude are observed as in other CsCl-type series. In trigonal symmetry, B ~ is found to be positive in TmCu as in other TmX.
QUADRUPOLAR
E F F E C T S IN R A R E E A R T H I N T E R M E T A L L I C S
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Quadrupolar pair interactions are also present. They are positive in tetragonal symmetry although they are a little weaker than in RZn; they dominate the magneto-elastic coupling. The total quadrupolar coefficient G r would drive a quadrupolar ordering at only T o = 5 K in TmCu. Its value in ErCu, deduced from the magnetic excitation spectrum, is lower than in ErZn. The tendency to negative quadrupolar pair interactions in trigonal symmetry is also established in RCu. As discussed in section 3.6.1, they determine the multi-axial structure of DyCu, the CEF being responsible of the three-fold direction of easy magnetization as in DyAg and TmGa 3 (section 3.6). 4.1.5. R M g series
Light rare earth compounds are antiferromagnets. In heavy rare earth compounds, the spin structure is either canted, surprisingly also in GdMg (Kim and Levy 1982), or more complex, because they mix ferro- and antiferromagnetic tendencies (A16onard et al. 1976). From a quadrupolar point of view, spontaneous strains are observed in CeMg and PrMg (table 9). From the extensive study of TmMg (Giraud et al. 1986), it was found that the magneto-elastic and pair quadrupolar interactions are weak in heavy RMg compounds (section 5.1). Then an analogy appears between quadrupolar and antiferromagnetic interactions: they are strong and well-established at the beginning of the series and they rapidly decrease in heavy rare earth compounds.
TABLE 9 Quadrupolar coefficients in the RMg series. Also indicated are the N6el temperature, the antiferromagnetic propagation vector Q, and the spontaneous strain at low temperature. RMg compound
Tr~ Q Easy axis c/a - 1 (%) B" (K) G ~ (mK) B ~ (K) a " (mK)
CeMg
PrMg
TmMg [5]
19.5 [11 1st order (~,0,0) [001] [2] 1.3 -+ 0.2 [2] 98 [6] (50, 150) [11
45 [3]
<1.3
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References: [1] Pierre et al. (1984), from magnetic excitation studies. [2] Schmitt et al. (1978). [3] A16onard et al. (1976). [4] Morin et al. (1978a). [5] Giraud et al. (1986) from measurements of parastriction, elastic constants and X~ )[6] From c / a - 1 data, using (O °) ='8 (CeMg) and 28 (PrMg).
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
93
4.2. NaCl-type structure compounds Rare earth pnictides (rare earth compounds with group V-A elements) crystallize within the cubic NaCl-type structure. They have been extensively studied from a magnetic point of view (Hulliger 1978). Their semi-metallic behaviour leads to RKKY-type bilinear interactions driving rather low magnetic transition temperatures (T < 60 K). The low-temperature phase is antiferromagnetic of type-I or -II with some exceptions for Ce compounds. The small size of nitrogen ions is probably the origin of ferromagnetism in some nitrides. Associated with the semi-metallic behaviour, the strength of the magnetic interactions strongly depends on the stoichiometric conditions as may be deduced from the spread in the T s values listed in tables 10 and 11. The CEF parameters are very similar for compounds of a given rare earth ion and the dependence across the rare earth series follows, more or less fortuitously, point-charge estimates. However, a clear exception is the case of Ce compounds (Birgeneau et al. 1973). One of the consequences of the positive A4(r 4) parameter and of the weak A6(r 6) parameters, is the great number of nonmagnetic singlet ground states for Pr, Tb and Tm ions; over- (sub-) critical conditions are usually observed in Tb (Pr and Tm) compounds. The magnetoelastic coupling provides clear evidence for the actual magnetic structure, through the tetragonal or trigonal spontaneous strain. The number of multi-axial structures appears to be very small contrarily to the situation in CsCl-type compounds. Numerous compounds order at a first-order transition, in particular those with Tb, Dy and Er ions. In connection with the low-temperature range, this is explained by positive values for the third-order susceptibility driven by quadrupolar interactions in both tetragonal symmetry (Levy 1973) and trigonal symmetry (Koetzler 1984). In tetragonal symmetry, the magneto-elastic quadrupolar term has been observed to be dominant (Mullen et al. 1974, L6vy 1969). The B ~ values have signs opposite to those in the CsCl-type series and they roughly follow the aj variation with a magnitude slightly larger than in the CsCl-type structure (Liithi 1980b). This seems also to be valid for the coefficient in trigonal symmetry. As discussed in section 3.2.4, the anisotropic character of the bilinear interactions as well as the existence of an associated two-ion magneto-elastic coupling has been shown theoretically and experimentally. The occurrence of a sizeable trigonal spontaneous strain in Gd compounds, in particular in GdBi (HuUiger and Stucki 1978a), complicates the analysis of the magnetic properties, such as, e.g., the magnetic transition. Anisotropic bilinear interactions seem to have a dominant role in Ce pnictides due to hybridization effects (Cooper 1982, Takahashi et al. 1982). According to the De Gennes law, two-ion magneto-elastic effects are expected to be noticeable in the middle of the rare earth sequence, e.g., in Tb and Dy compounds. In DySb below TN, the actual direction for the magnetic moment is tilted from the c-axis of the tetragonal cell; the resulting monoclinic cell underlines the presence of the trigonal two-ion magneto-elasticity (Felcher et al. 1973). Above TN, in tetragonal symmetry, the parastriction also reveals the competition between one- and two-ion magneto-elastic terms (Al6onard et al.
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1984b). As for Ho compounds, HoBi exhibits a positive tetragonal strain (Hulliger et al. 1983) which is opposite to the negative one expected on the basis of a dominant one-ion contribution as effectively observed in HoAs; this may be also a signature of this competition.
4.3. AuCu3-type structure compounds Numerous RX 3 intermetallics crystallize with the cubic AuCu3-type structure, in particular for X = Pb, In, T1, Pd, Sn and Ga. Only a small number of studies has been devoted to magneto-elastic and quadrupolar interactions. The non-magnetic transition observed at 0.37 K in PrPb 3 has been analyzed to be driven by an AFQ ordering (see section 3.1.2). This behaviour agrees with a study of TmGa 3 which shows that AFQ interactions drive quadrupoles to order slightly above TN and stabilize the multi-axial spin structure below T N (sections 3.1.2 and 3.6.1). A quadrupolar ordering was also claimed to occur in the paramagnetic phase of Stain 3 and SmSn3; a noticeable softening of the C44elastic constant is observed at high temperatures (Kasaya et al. 1985). The tendency to negative quadrupolar interactions seems to be general in AuCu3-type compounds and responsible for the multi-axial magnetic structures observed. From a purely magneto-elastic point of view, PrPb3, TmGa 3 and CePb 3 (Nikl et al. 1987, Morin et al. 1987b) are the only AuCu3-type compounds studied. The spontaneous strain is null in TmGa3, as the magnetic structure is cubic. It is unobserved in CePb3, owing to the low N6el temperature (TN = 1.1 K) and the non-quadrupolar F7 ground state, and in PrPb3, due to the low value of TQ and the AFQ structure. The magneto-elastic coefficients and their ratios to aj are given in table 12. In both symmetries, the ratios B"/aj are positive; a good coherency is achieved for the tetragonal mode with an average magnitude which is coherent with values in preceding series. Some scattering is observed for the trigonal mode; however, values for PrPb 3 and TmGa 3 are noticeably stronger than in CsC1- and NaCl-type compounds.
4.4. Cubic Laves phase compounds 4.4.1. RAI 2 compounds The RAI 2 compounds have been thoroughly studied for many years with regard to, first, the crystal field and the magnetic properties and, second, the magnetoelastic coupling. All the available experimental results are reported in table 13. Most of them are based on magnetostriction measurements in the ferromagnetic range. That explains why some values seem out of range, as e.g., B" for PrA12 or B r for TbAI2, because they have been obtained from measurements along an axis of hard magnetization in compounds with a large anisotropy. The high value of B ~ for CeA12 has been related to a coupling with optical phonons at the F-point (Liithi and Lingner 1979). All the other experimental values are rather consistent with each other, in particular in NdA12 where the B r value has been obtained by elastic constants, X-ray and magnetostriction measurements. Recently, some
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P. MORIN and D. SCHMITF TABLE 12 Magnetic and quadrupolar ordering temperatures and magnetoelastic coefficients from elastic constant, parastriction and g~ ) experiments in AuCu3-type compounds. AuCu3-type compound CePb3 [1] TN (K) To (K) B r (K) B~/oq (K) K" (inK) G"MZ(mK) a" (inK) B" (K) B~/a~ (K) K" (mK) GME (mK) G ~ (mK)
PrPb3 [2]
TmGa3 [31
0.35 [5] -41 1950 -13 8 -5 -190 [11 9050 -200 7 -200
4.26 [6] 4.29 17 1700 (8,14) 0.5 (8,14) 71 7000 20 10 30
1.1 [4] -80 1400
-190 3350
References: [1] Morin et al. (1987b). [2] Morin et al. (1982). [3] Morin et al. (1987c). [4] Lin et al. (1985). [5] Bucher et al. (1972a). [6] Czopnik et al. (1985). quadrupolar p a r a m e t e r s G v have been obtained by the torque method, which allows to extract the third-order paramagnetic susceptibility x- e(3) in a less direct way than from magnetization m e a s u r e m e n t s (Del Moral et al. 1987). As for the sign of magneto-elastic coefficients, they appear to follow well that of the Stevens factor aj. The ratios BY/aj and BE/aj range around 1000 K and - 3 0 0 0 K, respectively, with a few exceptions (see section 5.1). F r o m values of G r and G rME, the existence of two-ion quadrupolar coupling is suggested. This is in disagreement with elastic constants fits, which did not require considering this coupling.
4.4.2. R M 2 compounds (M = Mg, Ni, Co) For the R M 2 cubic Laves phase compounds where M is not A1 (section 4.4.1) or Fe (section 4.4.3), quadrupolar data are rather scarce. The PrM 2 compounds (M = Pt, Rh, Ru, It) have been investigated by magnetization, neutron diffraction, and specific heat m e a s u r e m e n t s (Greidanus et al. 1983a,b). A systematic overestimation of the ferromagnetic m o m e n t by a calculation including C E F and bilinear exchange interactions, with regard to the observed spontaneous magnetization, was explained by the presence of negative quadrupolar interactions. The same is true in PrMg2, where a quantitative analysis of the t e m p e r a t u r e variation of the ferromagnetic m o m e n t provided the quadrupolar p a r a m e t e r G r = - 1 3 . 3 m K (see table 14). F u r t h e r m o r e , this negative quadrupolar coupling leads
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ErNi 2 [2]
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TbCo 2 [3]
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References: [1] Loidl et al. (1981). [2] Gignoux and Givord (1983), from magnetization measurements. [3] Gignoux et al. (1979), from X-ray measurements. [4] Castets et al. (1982), from magnetic excitation studies.
to the disappearance of the first-order transition which is expected without it (see section 3.2.2). Some magneto-elastic results are also available in the RNi 2 series, where nickel is not magnetic. The compounds order ferromagnetically at temperatures roughly three times smaller than the corresponding temperatures in the RAI 2 series. At low temperatures, the directions of easy magnetization are the same in RNi 2 and RAI 2 compounds (R = Tb, Dy, Ho); the spontaneous distortions are comparable in both series, except for HoNi 2 where the strain is noticeably larger than in HoA12 (e ~ = -1.1 x 10 -3 in TbNi2, c/a - 1 = - 2 x 10 - 3 and -1.5 x 10 -3 in DyNi 2 and HoNi2, respectively) (Markosyan 1980, 1981). The behaviour of ErNi 2 differs considerably from that of ERA12, since its direction of easy magnetization is [100] instead of [111]. This has been well accounted for by introducing the parameters G ~ and G ~ (see table 14). In the RCo 2 series, cobalt is magnetic and contributes mainly to the volume magnetostriction. A positive trigonal magneto-elastic coefficient B ~ associated with the rare earth has been determined in TbCo 2 (see table 14). This leads to a positive trigonal magnetostriction e ~= 3.1 x 10 .3 at 4.2K which is opposite in sign to that measured in TbNi 2 and TbAI 2. This feature has been related to environmental effects in this type of cubic structure (Gignoux et al. 1979). In HoCo 2, a consistent description of the magnetic excitations as well as the spin rotation temperature requires introducing negative quadrupolar terms in the Hamiltonian (table 14), indicating the probable existence of a contribution of two-ion origin.
4.4.3.
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compounds
The magneto-elastic properties of cubic ferromagnetic Laves phase compounds RFe 2 are relevant for the classical analysis, owing to their high magnetic transition temperature (around 700 K in TbFe2). Indeed, the Fe magnetization induces an exchange field large enough to purify fully the 4f magnetic moment towards its maximum value at 0 K. Their magnetostriction has been studied, in particular at
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
101
room temperature, for industrial applications. The large set of available data has been collected in the first volume of this Handbook by Clark (1980). Here, only characteristic features and analyses are considered again. Comparing the strength of the tetragonal and trigonal symmetry-lowering modes leads to a conclusion opposite to that in CsCl-compounds. Indeed, in RF% the trigonal mode is many times larger than the tetragonal mode (fig. 60). TbFe2, the basic compound for applications, is the paradigm of the series. The temperature variation of the Aln trigonal spontaneous strain has been described using eq. (107). As pedagogically explained by Clark (1980), an important step is to determine the 4f magnetization at any temperature by means of neutron diffraction or magnetization measurements. In this latter case, the Fe magnetization (about 1.64/~) is deduced from M6ssbauer spectroscopic results. Noticeable uncertainties may be present at high temperatures; however, the saturation of the 4f moment observed at low temperatures allows one to derive reliable values of B ~ through the relation Be=
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102
P. M O R I N and D. SCHMITI" T A B L E 15 Curie temperature, easy magnetization axis, trigonal spontaneous strain at 300 and OK, trigonal magneto-elastic coefficient B e and its ratio to the Stevens coefficient c~j in RFe 2 compounds [data are from tables 5 and 6 of Clark (1980)]. RFe z compound
T c (K) Easy axis hal 1 (300 K) × 1 0 6 )tll 1 (0 K) x 106 B" (K) B'/a: (K)
TbFe 2
DyFe 2
HoFez
ErFe 2
TmFe 2
SmFe 2
697 [111] 2460 4400 94 -9400
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606 [001] 200 770 a 9 -4100
590 ' [1111 -300 -1760" -24 -9300
560 [111] -210 -3500 -75 -7500
676 -2100
Note: "estimated from the room-temperature value.
Across the series, B" is observed to vary as the second-order Stevens coefficient a: in sign and, with some erratic fluctuations as in other series, also in order of magnitude (table 15). The same technique, based on eq. (107), fails to describe the tetragonal spontaneous strain in DyFe 2 (fig. 60) and HoFe 2 (Abbundi et al. 1979). Indeed, the change of sign of hl00(T) in DyFe 2 and the general temperature variation in HoFe 2 are both unexpected within this model. Whereas the hi00 sign at 4.2 K follows the sign of aj in both compounds, its absolute value is about ten times smaller in DyFe 2 than in HoFe 2, in full disagreement with the ratio of the aj's. Cullen and Clark (1977) have developed an atomic model that shows that internal distortions drive the large external trigonal strain mode whereas the hi00 strain may be sorted out because of the high tetrahedral symmetry of the rare earth site.
4.5. Hexagonal CaCus-type structure compounds Among the series of hexagonal 3d-4f intermetallic compounds, only few RNi 5 and RCo 5 compounds have been investigated from a magneto-elastic point of view; nickel is not magnetic in the first series, but cobalt is in the second one.
4.5.1. RNi 5 compounds Whereas the RNi 5 compounds have been studied for many years with regard to their magnetic properties and CEF effects, only few results are available about their magneto-elastic properties. A first study concerned the thermal expansion of PrNi 5 (L/ithi and Ott 1980) which evidenced crystal field effects, and showed the existence of the two a-mode magneto-elastic coefficients. However, the latter were not determined because the elastic constants were not known at that time. Using2the values of C" from Barthem et al. (1988), we are able to calculate B ~1 and B from these early measurements (see table 16). These values have the same sign, but are 2-3 times smaller than more recent values determined by magnetostriction measurements. In addition, the latter provide an estimation of
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
103
TABLE 16 Quadrupolar coefficientsin RNi5 compounds. RNis compounds B'~1 (K) B ~z (K) B ~ (K) G ~ (inK) G ~ (mK)
PrNi5 [11
PrNi5 [21
TmNi5 [3]
-7.1 28.6
-22 68.6 <15 -10 -20
(-500,-100) (-300,-150) (-1500,-1000)
References: [1] L/ithi and Ott (1980), from thermal expansion measurements. [2] Barthem et al. (1988), from magnetostriction measurements. [3] Barthem et al. (1989), from magnetostriction measurements. the total quadrupolar parameters G ~ and GL By contrast, Barthem et al. (1988) deduce antiferroquadrupolar (negative) interactions between Pr ions in PrNi 5 (K ~ - - 1 5 mK, K ~ - - 2 0 i n K ) . Note that a magneto-elastic coupling was also deduced from inelastic neutron scattering in PrNi 5 (Aksenov et al. 1983), where it should contribute through the hybridization of magnetic excitons and phonons. However, no accurate determination could be obtained. A recent magnetostriction study of TmNi 5 also provided one- and two-ion magneto-elastic coefficients (Barthem et al. 1989). Particularly due to the CEF configuration of this compound, the one-ion and two-ion magneto-elasticities could be separated. More precisely, the axial strain dependence of the isotropic exchange interaction ( a 2 - m o d e ) leads to a strongly anisotropic anomaly in the thermal expansion along a and c-axes below the ferromagnetic ordering temperature T c = 4.5 K, while the magnetostriction is more sensitive to the one-ion magneto-elasticity. However, the deduced B " values are quite large with regard to those in PrNi 5 and do not scale with aj. This behaviour is still to be explained. 4.5.2. R C o 5 c o m p o u n d s
Just as the cubic RFe 2 compounds, RCo 5 compounds are typical cases for applying the classical formalism, as the Co-sublattice induces very large bilinear interactions: the Curie temperatures are around 1000 K (e.g., 940 K in YCos). It is therefore difficult to study paramagnetic properties or to determine the CEF for the rare earth. A main feature is the huge magnetocrystalline anisotropy even in the absence of a magnetic rare earth as observed in YCo 5 (Alameda et al. 1983). From a magneto-elastic point of view, large anomalies have been observed in the temperature variation of the lattice parameters (Andreev et al. 1983, 1985b). It is difficult to observe magneto-elastic properties other than the spontaneous ones except in several compounds which exhibit a change in the direction of easy magnetization, and thus a temperature range of reduced anisotropy. Indeed, as classically explained in hexagonal Co (Barnier et al. 1962) and Gd (Klimker and Rosen 1973), this p h e n o m e n o n is driven by the change of sign at a given temperature of the K 1 anisotropy constant, which phenomenologically includes
104
P. MORIN and D. SCHMITI"
magneto-elastic contributions. This occurs for compounds with a negative Stevens coefficient aj for the rare earth (PrCo 5, NdCo 5, TbCo 5, DyCo 5 and HoCos). The corresponding temperature regions range from 13 K (in TbCos) to 130 K (in HoCos); the rotation of the magnetization induces clear anomalies in both the lattice parameters (Andreev et al. 1983) (fig. 61) and the ultrasonic modes (Deryagin et al. 1985) (fig. 62). It has been shown by X-ray studies that the magneto-elastic coefficient B ~ is small. On the other hand, the relative changes Aa/a and Ac/c of the lattice
v.gs~ --
1*~t~f,
,, -4 v, A3
.!'""I
q.g#Z~
~
85 1
"
Oq. g
q.960
~ I ~ J'l'~+"~
390
350
i
l-I
qJo T K
Fig. 61. Temperature dependence of the lattice parameters a and c, and of the volume V of the unit-cell in the temperature range of the change of easy axis in TbCo 5 (from Andreev et al. 1983).
z+
r,~r~r2
"E i
t
;'¢.2 I0.8
5.2 ~ 0
r
i
200
t r ~"w
I
o00T.~
Fig. 62. Temperature dependence of the elastic constants in the temperature range of the change of easy axis in DyCo 5 (from Deryagin et al. 1985).
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H I N T E R M E T A L L I C S
105
parameters at the average rotation temperature between the high-temperature (magnetization along the c-axis) and low-temperature (magnetization in the basal plane) phases indicate contributions of the two a-modes (see section 2.3). Temperature dependent magneto-elastic coefficients have been introduced (Deryagin et al. 1985),
V~
2(Aa 11 + 3 a
AC'~c~,
B12-
.
2 ( Aa
AC~c .
B~2=
AC'~c ~ 2 ( A a 2 (Aa X/3 2 - - + a c ! t 2 + 3 a
AC'}c ~
2 --a + c /
c /
c/
12 ,
22"
Experimentally, the values of (2Aa/a + Ac/c) have been observed to be very small. The 0 K values of B12 and B22 have been deduced assuming a temperature variation according to the Callen and Callen function is/z[sg-l(m)]. Note that some uncertainties may arise from inaccurate knowledge of the rare earth's magnetic moment, especially for the light rare earths (Andreev et al. 1983). The B~2(0 ) and B~2(0 ) are related to the B ~ and B "2 magneto-elastic coefficients of the present formalism through, BI:(O ) = 2B '~1
B~:(O) = 2V'-6 B "e
It is then possible to follow their variations through the entire series of rare earths. Table 17 collects the values for the different parameters in the series. The T A B L E 17 Magnetostriction constants at OK, h~2(0), h~2(0), associated to the relative change of lattice parameters at the rotation temperature, aa/a and Ac/c, elastic constants C~2 = 2X/2-7-gC~12, C~z = 2C ~2, B ~ and B ~2 magneto-elastic coefficients and their ratio to the Stevens coefficient a~ in RCo 5 compounds. RCo s compounds
h~2(0) x 103 [11 A~2(0) x 103 C~2 (101° Pa) C22 (1010Pa) B "1 (K) B "2 (K)
B~l/aj (K)
B"Z/a: (K)
PrCo s
NdCo s
SmCo 5
TbCo 5
DyCo 5
HoCo5
ErCo 5
TmCo 5
-1.1 2,2 1,6 a 20 a -6.6 -39 314 1860
-0.8 1.7 1.6 [2] 21.2 [2] -5.1 -25 793 3890
1.3 -2.6 1.6 = 22 a 24.6 143 597 3460
-1.8 3.5 1.7 [3] 23.3 [3] -5.5 -31.2 544 3090
-1.7 3.4 1.9 [31 25 [3] -3.7 -20.5 583 3230
-0.7 1.2 2a 27 ~ -1.2 -6.9 540 3100
0.6 -1.2 2" 27 a 1.4 7.6 551 3000
1.5 -3 2a 27" 5.5 30.8 544 3050
Note: aExtrapolated values. References: [1] Andreev et al. (1983). [2] Deryagin et al. (1984), [3] Deryagin et al. (1985).
106
R MORIN and D. SCHMITr
a f d e p e n d e n c e of B ~1 and B ~2 is very closely observed. The result indicates that the a - m o d e s depend on the rare earth and not on the cobalt. This conclusion is confirmed by the study of Y C % by X-ray diffraction. In this c o m p o u n d , the cobalt sublattice appears as contributing only through a two-ion magneto-elastic coupling, modifying the c/a ratio (Andreev et al. 1985a). The partial softening observed on the different elastic constants m a y be expected to be described using the coefficients B ~1, B ~a previously determined. Note that the softening observed on C44 during the rotation also indicates a sizeable magneto-elastic coefficient B ;. Some significant results might be provided by magnetostriction m e a s u r e m e n t s in high magnetic fields for the various symmetries in the rotation t e m p e r a t u r e range.
4.6. Dilute rare earth systems The magnetization of concentrated rare earth compounds generally includes oneand two-ion terms and m a y be strongly influenced by magnetic interactions. Dilute systems should, in principle, be easier to understand by limiting the magneto-elastic effects to one-ion contributions only. A dilution in the 1% range seems to be a satisfactory compromise between experimental sensitivity and residual two-ion effects. Most of the available studies concern rare earths diluted in noble-metal hosts, as Ag, A u and Cu.
4.6.1. Rare earths diluted in cubic noble-metal hosts The noble metals crystallize within the face centered cubic structure, and, because of the low solubility of rare earths in them, concentrations of rare earth impurities smaller than 1% were usually taken for the experiments. Silver-rare earth dilute compounds. A systematic study of heavy rare earths diluted in silver was carried out by magnetostriction m e a s u r e m e n t s on single crystals (Creuzet and Campbell 1981) (see table 18). Varying the cubic C E F TABLE 18 Magneto-elastic coefficients in the AgR series. All results were taken from Creuzet and Campbell (1982a) and from Creuzet (1982) and refer to magnetostriction measurements (rare earth concentration c = 0.5 at.%), except where indicated otherwise. AgR (R =) B r (K) B ~ (K) B(r4) (mK) B~4) (raN)
Tb
Dy
Ho
Er
Tm
2.72 3.9 [21 31.1 -6.3 -67
1.71 0.9 [21 12.1 1.9 45
0.60 3.5 [21 1.18 5.3 9
-1.24 <-0.39 [1] -7 -3.5 -29
-2.47 -3.3 [21 -16.1 -6.0 -200
References: [1] Garifullin et al. (1985) by EPR bulk, c = 0.03%. [2] Nieuwenhuys et al. (1982b) [alternate analysis of results of Creuzet and Campbell (1982a) with another set of CEF parameters].
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
107
parameters may affect the result of the magneto-elastic coefficients. However, the best agreement was achieved by using a CEF level scheme which also describes other results such as susceptibility measurements. They correspond to the same CEF parameters Aa(r 4) =--18 K and A6(r 6) = 7.6K across the series (except for Tm where A6(r 6) -= 6.5 K). These results have also been compared with estimates of the magneto-elastic coefficients obtained by EPR measurements (Pela et al. 1981). The EPR results appear to be systematically underestimated in experiments performed on strained thin films and, to a lesser extent, on strained bulk samples (Campbell and Creuzet 1985). Magnetostriction measurements were analyzed by considering second- and fourth-order one-ion magneto-elastic terms, the latter always contributing for about 10% to the total magnetostriction. The experimental values appear consistent along the series, and follow roughly the appropriate ~j or /3j Stevens coefficients. Typical values are, By oq
_
U~4)_
V~
3
V~
3
B ~
2V2 V2 ~ -270 K ,
U~4)
/3j - 16----~ V4 ~ - 4 0 K ,
3
5
O/j - W~ V 2 - -2100 K ~
5
/3j - 4V2 V4 ~ - 7 5 0 K ,
for all the dilute AgR compounds, the V~'s are defined by Campbell and Creuzet (1985). This coherency is reminiscent of the constant CEF parameters across the series. The isotropic one-ion magneto-elasticity was also investigated from these magnetostriction measurements (Creuzet 1982). The uncertainty in the coefficients is largely due to the weakness of the volume effects. Typical values for the AgR series are,
n(4)_ /3j
Weft
1
16V2 V4 - -120 K ,
n(6)
- -yj -
V3
1
64 V6~ -t- 135 K
Gold- and copper-rare earth dilute compounds. Compared to the AgR-series, rather less results are available on the A..__~uR and CuR dilute compounds. Systematic comparisons are therefore difficult. Nevertheless, the main result is the change of sign of B e and/or B y in AuR and C__uuRcompounds (see table 19), compared to AgR compounds. On the other hand, the volume magnetostriction appears to be still weaker in these compounds than in A__ggRones (Creuzet 1982). 4.6.2. Rare earths diluted in cubic pnictides A systematic study of the tetragonal magneto-elasticity was performed in the cubic NaCl-type LaSbR series by [001] magnetostriction measurements (Nieuwenhuys et al. 1982a). As in the A___ggRsystem, B y follows the sign of the Stevens factor % well (table 20). In comparison to corresponding concentrate
108
P. M O R I N and D. SCHMITT
T A B L E 19 Magneto-elastic coefficients in the A__._quRand C__._~uRseries. All the results are from Creuzet and Campbell (1982b,c) and Creuzet (1982); they were obtained by magnetostriction measurements (rare earth concentration e = 0.5 at.%), except when indicated otherwise. The C E F parameters used are A 4 ( r 4) = - 3 0 K, A 6 ( r 6) = 6 K for the A._.._uuRseries, A 4 ( r 4) = - 8 6 K, A 6 ( r 6) = 12 K for CuEr compound. Compound A..._~uTb B y (K)
1.7
B" (K)
-20
B~4~ (mK) B~4 ~ (mK)
12 -33
A..._~uEr [1]
AuYb
-0.8
-8.3 - 8 . 7 [11 89 44 [1] -17 1200
4.5
C__._~uEr[2] 0.44 -6.5
References: [I] Pela et al. (1981), from E P R measurements on films (c = 1%). [2] Garifullin et al. (1985), from E P R measurements on bulk samples (c = 0.03%).
compounds (section 4.2), these parameters appear to be quite comparable in magnitude, except in LaSbSm. The value obtained by magnetostriction in LaSbDy is consistent with that deduced from magnetization measurements. However, the latter may be considered to be more reliable than the former, because the magnetization reflects the lattice distortion in the immediate vicinity of the rare earth ions, while the magnetostriction cumulates the distortion effects far away from the rare earths. Indeed, it appears closer to the value in DySb than that obtained by magnetostriction. The most original feature in the series is the observation of a "reverse" magnetostriction in LaSbCe and LaSbSm, i.e., a change of sign of the distortion as a function of temperature, which is explained by the sign reversal of the expectation value ( O ° ) (see section 3.4.1). Magnetostriction was also observed for S-state Gd ions diluted in LaSb and LaBi (Hfifner et al. 1983). This has been explained within a phenomenological model which includes admixture effects of the first-excited 6p multiplet. However, corresponding magnetostriction values reach ~ l / l - - 8 x 10 -6 at 1.5 K in a 5 T field for 10%-dilute compounds. These values, scaled for the concentration, are only 1/10 of that observed in GdZn, where the distortion was interpreted as the manifestation of a two-ion magneto-elasticity (Morin et al. 1977). According to section 3.2.4, it could then be appropriate to consider the latter term explicitly. The magnetostriction measurements on LaSbR compounds were extended to the LaBiR series in order to check, in particular, the reverse magnetostriction in Ce and Sm based compounds (H/ifner et al. 1983). Although the CEF itself is smaller in bismuthides than in antimonides, the magneto-elastic coupling seems to be larger (see table 21).
QUADRUPOLAR
EFFECTS IN RARE EARTH INTERMETALLICS
i ~ ,...,
< II
~
~
~9
.~ o=
o C, t"q
Z
I
~
.I
cq
O
r.)
109
110
P. MORIN and D. SCHMITT TABLE 21 Crystal field parameters W, x and magneto-elastic coefficient B y in the diluted LaBiR series (c is the rare earth atomic concentration), from H~fner et al. (1983), from magnetostriction measurements. Rare-earth
W (K) x B y (K) c (%)
Ce
Tb
Er
1.91 1 -65.3 10
-0.45 -0.96 -43.7 3
0.20 0.79 10.0 3
4.6.3. Other dilute systems Few other cubic dilute rare earth systems have been studied with regard to magneto-elasticity. Only on Tb based compounds has there been a determination of magneto-elastic coupling parameters (see table 22). Creuzet et al. (1982) determined both coefficients B y and B e by magnetostriction measurements in cubic CsCl-type Tb0.01La0.99Ag compound. The B y value is roughly eight times smaller than in the concentrated TbAg compound (see table 7), which could be related to the incipient instability of the LaAg matrix (section 3.10). The same is valid for the value of G ~. On the other hand, the B e value seems to be doubtful, since it would lead to an unreasonable value of G~E, i.e., 1 K. Some studies have also been carried out on dilute systems having a lower symmetry, namely the hexagonal YR and MgR compounds. The three secondorder magneto-elastic coefficients B or1, Ba2 and B ~ were determined in YTb, __YDy and __YErcompounds (see table 23). A change of sign of all coefficients occurs for TABLE 22 Crystal field parameters W, x and magneto-elastic coefficients in some diluted rare-earth systems. Compound LaAgTb
w (K) x B ~ (K) B E (K) G y (mK) G" (mK)
0.5 -1 3.3 [1] - - 3 2 0 [1] - 1 [2]
LaA12Tb
0.27 0.9
0 [31 - 3 0 [31
References: [1] Creuzet et al. (1982), from magnetostriction measurements (c = 1%). [2] Hoenig et al. (1980), from magnetization (SQUID) measurements (c = 2%). [3] Hoenig (1980), from magnetization (SQUID) measurements (c = 1 to 6%).
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
111
TABLE 23 Magneto-elastic coefficientsin hexagonal YR system "(fromPureur et al. 1985), using magnetostriction measurements (c = 1 to 2%). Compound B "' (K)
B~2 (K) Be (K)
YTb
YDy
YEr
6.1 18.9 14.8
9.4 8.5 20.4
-1.9
-4.4 -8.4
the compound YEr, due to the Stevens factor aj. In addition, the ratio B ~ / a j is roughly constant throughout the series, B ' ~ / a j -~ V'3 B'~2/a, ~ ~
V2(/'a,1)/Olj
~
--850 K ,
V2(F~,2)/a s ~ -1600 K ,
B~/aj ==-~X/-2 Vz(F~,) / a j ~ -3200 K .
These values lead to distortions which are quite consistent, after correction for the concentration, with those observed in pure rare earth metals (Pureur et al. 1985). In a few cases, the introduction of a fourth-order magneto-elastic coefficient slightly improved the fits. In spite of the low concentration, spin-glass effects have been observed at low temperatures, in particular with Tb. Lastly, magnetostriction for YGd compound (c = 1%) reaches ~c/c ~ 10 -6 for a magnetic field of 6 T along the c axis, which is again of the same order of magnitude as in LaSbGd and LaBiGd (see above). Magnetostriction and thermal expansion data are also available for the heavy rare earths dilute in a single-crystal Mg host (Bijvoet et al. 1980, De Jong et al. 1980, 1982) (concentration about 0.1%). Here, also the magnetostriction agrees with the sign of aj. Unfortunately, no complete analysis was performed in order to determine the actual magneto-elastic coefficients. 4.7. Miscellaneous c o m p o u n d s
The universal character of the magneto-elasticity is emphasized in the literature through accurately conceived data on individual compounds, however, without thorough and systematic analyses across a given series. One can find, e.g., magneto-elastic effects in RCo 3 (Pourarian and Tajabor 1980) for which magnetostriction data on single crystals are available. In other series, like rare earth metals and rare earth borides few experimental results have been quantitatively analyzed. Rare earth metals. Among the rare earth metals, dhcp Pr has been widely studied over the last decade, because of the magneto-elastic effects on the criticality of antiferromagnetic interactions; the strain mode involved in these effects is the e mode (section 3.9). A magneto-elastic coefficient B ~ = 72 K acting
112
P. MORIN and D. SCHMITr
on the hexagonal Pr sites, i.e., G ~ = 76 mK, leads to a consistent description of various experiments, such as elastic constant measurements, magnetostriction, magnetic excitations and the occurrence of an antiferromagnetic ordering under an uniaxial stress (Houmann et al. 1979). This e-mode magneto-elastic coupling was also suggested to explain the field dependence of the thermal expansion (Liithi and Ott 1980). Without any magnetic field, these thermal expansion experiments have been described with the a-mode magneto-elastic coefficients of the hexagonal symmetry B al (-----~V/-3Bv) = - 4 6 K and B a2 (--=V~ B3) -~ 35.5 K. On the other hand, a detailed study of the magneto-elasticity was carried out by magnetostriction measurements (Hendy et al. 1979). A complete analysis was made in terms of one-ion and two-ion magneto-elastic coupling. It was found that one-ion a2-strain coupling is the dominant mechanism between the lattice and the 4f ion. Unfortunately no value for B ~ or D ~ was extracted from the experiments; only the h ~ coefficients relative to both cubic and hexagonal sites of Pr have been obtained. When taking the values of elastic constants from Greiner et al. (1973), the following evaluations can be made for the Pr ions on hexagonal sites: B "2 = 2 2 . 6 K , B ~ = 4 2 K , nal = o a~1 l = D~I a2 ~ 0. These values have the same order of magnitude as those from Houmann et al. (1979) in which the contribution arising from the cubic sites was included. Rare earth borides. Among the six families of binary rare earth borides (Etourneau 1985), RB 4 and R B 6 have been the ones most studied. Their crystallographic and magnetic properties have been shown to depend drastically on the nature and rate of impurities and on the metallurgical conditions. R B 6 compounds. RB 6 compounds all crystallize within a CsCl-type cubic arrangement of B octahedra and R ions. The main part of the literature on this series is devoted to light R compounds. Magneto-elastic and quadrupolar properties have been studied mainly in CeB 6 and negative quadrupolar interactions appear to be the origin of the properties observed (section 3.1.2). The tetragonal magneto-elastic coefficient [gr3[ = 50 K leads to [B~/,~j[ -- 875 K and the trigonal one = 135K to IB'/a,I = 3300K (L/ithi et al. 1984). PrB 6 exhibits properties reminiscent of those of CeB 6 in particular regarding its magnetic structure (section 3.6). The first-order transition (section 3.2.2) may result from bilinear and quadrupolar couplings of equivalent strength, as the bilinear exchange is larger than in CeB 6. Sizeable effects on the elastic constants are observed in PrB 6 and in NdB 6 (Tamaki et al. 1985). Unfortunately no quantitative analysis is available at the present time. In addition, the first-order Nrel transition in GdB 6 sets a puzzling problem which cannot be analyzed in terms of pure quadrupolar interactions. RB 4 compounds. The rare earth tetraborides have tetragonal crystal symmetry (Will et al. 1981). The rare earth atoms are confined in boron cages with an mm local symmetry (4 g). The compounds are antiferromagnetic with the exception of PrB 4 (Buschow and Creighton 1972). They exhibit some magneto-elastic features, especially ErB 4 which undergoes a tetragonal-orthorhombic transition at the Nrel temperature. At low temperature, the lattice symmetry (Pbam) agrees with the
[grsI
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
113
magnetic one (Pb'am). However, the most interesting compound might be TbB 4 for which the structural transition occurs in the paramagnetic range at 80 K, far above the N6el temperature (T N = 43 K). Here again, lattice (Pbam) and magnetic (Pbam') symmetries are closely related. In both compounds the structural symmetry lowerings have been observed by X-ray diffraction on polycrystalline samples and are rather weak (Will et al. 1986). To the author's knowledge there are, at present, neither experimental determinations for the CEF nor for the magneto-elastic coupling. It is worth noting that the mm rare earth symmetry induces non-zero (O~) and (022) quadrupolar values. In addition, the easy magnetization directions do not vary across the series in the same way as the second-order Stevens coefficient, revealing that the anisotropy is not only of single-ion origin (Etourneau 1985). 5. Discussion
5.1. General analysis 5.1.1. Coherency of the various determinations To prove the relevancy of the basic Hamiltonian, coherent determinations of the quadrupolar coefficients in independent experimental ways are needed. For example, table 24 collects the various values obtained for the quadrupolar parameters in tetragonal symmetry for TmX compounds within the CsCl-type structure. A very good agreement is observed. In particular, the parastriction removes the ambiguity about the sign of the magneto-elastic coefficient which remains after the elastic constant analysis. The consistency between determinations obtained by direct (elastic constants) and indirect (parastriction and third-order susceptibility) ways, i.e., without and with the presence of an applied magnetic field, demonstrates the relevancy of the quadrupolar Hamiltonian and the weakness of the two-ion magneto-elasticity in this family of TmX compounds. However, we have to keep in mind that the latter contribution may be more significant for large Land6 factors (gj). Higher order one-ion magneto-elasticity as well as anharmonic elastic terms appear to be negligible here. From table 24, the one-ion magneto-elasticity of compounds with 4d elements (Ag, Cd) is observed to be weaker than that of compounds with 3d elements (Cu, Zn). The eccentric case of TmMg is consistent with the absence of d electrons on the Mg-ions (see section 5.2). The same behaviour is observed for the pairinteraction coefficient, which is at its maximum for II-B elements, in particular for zinc. In most cases, the pair interaction contribution to the total coefficient G ~ dominates the magneto-elastic one. TmMg is the only TmX compound within the CsCl-type structure not to be strongly influenced by tetragonal quadrupolar interactions. The microscopic origin of these different coefficients is analyzed in section 5.2. In the presence of cubic level schemes close to each other, this series provides us with clear experimental evidence of quadrupolar effects. Large softenings of
P. MORIN and D. SCHMITT
114
TABLE 24 Quadrupolar coefficients in TmX compounds within the CsCl-type structure. Compound
Coefficient
TmZn
TQ [1] Cll - C12 [2] Parastriction [2] X~ ~ [31
-+29 -31
20
r e [41 c n - c12 [4] Parastriction [5] x~ ~ [6]
-+12.7 -14
11.2
TmCd
B ~ (K)
K s (mK)
G ~ (mK)
Kr/~ ~ ~ M E
25 25 27 25 12.5 12.5 12 13
TmCu
Cn - C12 [7] Parastriction [7] X~ ) [31
-+30 -27
7
11.5 11 10.5
TmAg(8~
Cll - CI2
---20 - 20.3
2
4.4 4 3.5
-+5.4 -5.4
(0, 3) (0, 3) (0, 3)
(0, 3) (0, 3) (0, 3)
1.75 Parastriction X~ ) TmMg(9)
CI1 C12 Parastriction X~ ) --
References: [1] Morin et al. (1978b). [2] Givord et al. (1983). [3] Morin and Schmitt (1981b). [4] A16onard and Morin (1979). [5] Morin et al. (1978c). [6] Morin and Schmitt (1982c). [7] Jaussaud et al. (1980). [8] Giraud and Morin (1986). [9] Giraud et al. (1986). elastic constants exist, with t h e r m a l variations strongly d e p e n d i n g on the competition b e t w e e n the two q u a d r u p o l a r couplings, as shown in T m Z n and T m C d [fig. 1 in Liithi et al. (1979)]. Similar selective characteristics are o b s e r v e d in parastriction ( M o r i n et al. 1978c) and third-order m a g n e t i c susceptibility ( M o r i n and Schmitt 1981b) experiments.
5.1.2. Comparison o f the magneto-elastic coefficients T h e various magneto-elastic coefficients p r e s e n t e d in section 4 for cubic c o m p o u n d s are collected in figs. 63 and 64, after normalization to aj. T h e first c o m m e n t c o n c e r n s the Order of m a g n i t u d e which is relatively i n d e p e n d e n t o f the rare earth across a given series. This is reminiscent o f the b e h a v i o u r of 4th and 6th o r d e r C E F p a r a m e t e r s within a series. A relatively g o o d a g r e e m e n t is o b s e r v e d for the B~/aj ratio b e t w e e n concentrated R S b and dilute L a S b R c o m p o u n d s . This indicates, in particular, that the
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS i
I
I
5
I
i
i
i
I
RLoSb/ '~
\ /~-~Sb
,,
o. /..o
0
ePb3,,.," • C-3
"
o0
#0%
115
+
~
~
~
.0.-'%o. TmGo3I
÷
RA[2~
+
F
RMg
l'\
.
.
.
el
-5
RZn__
.
RAg
....... •
-
-
TmCd
RC-u". . . . . . . . ">'~,~~.>,...-:~~"
i
I
I
I
i
t
i
i
Ce
Pr
Nd
Tb
Dy
Ho
Er
Tm
Fig. 63. Magneto-elastic coefficient B y for the tetragonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
I
]
:PrPb 3 ,
L
TmGg3
.~ • PrAg lCePb3
TmMg.
,,t
0
Ce
o/-
Pr
Nd
Tb
Oy
Ho
Er
Tm
Fig. 64. Magneto-elastic coefficient B ~ for the trigonal symmetry normalized by the second-order Stevens coefficient in various series of rare earth cubic intermetallic compounds.
macroscopic strain measured in dilute samples is, in a first approximation, linearly related to the local deformation through the concentration of active sites. For a given series, the two magneto-elastic coefficients are often of opposite sign, as expected from point charge estimates as will be discussed in section 5.2. The only exceptions are the cases of AuCu3-type compounds and RAg dilute alloys. For a given type of structure, values have been observed to be similar in a first approximation. This is in particular true for the CsCl-type structure. However, the positive sign of B ~/as for TbZn is in opposition with the other determinations in the RZn series. This might be the signature of a two-ion contribution although it has been observed nowhere else for this symmetry. Note in addition the small average value of B~/a: in RZn.
116
P. M O R I N and D. SCHMITT
Among the different series, the values observed for the B~/aj's are of the same order of magnitude. This is reminiscent of the CEF parameters which usually correspond to total level spacings of a few hundred Kelvin. In the same way as peculiar surroundings or densities of electronic states drive small or large CEF spacings, as in RBe13 (Vigneron et al. 1980) and RB 6 (Zirngiebl et al. 1984), respectively, the magneto-elastic coefficients may also vary from one series to another. The actual spontaneous strain depends not only on the B" values, but also on other parameters. The CEF usually determines the axis of easy magnetization, then the symmetry lowering mode. The pair interactions state the purification of the CEF wave functions, then the various expectation values. For this reason, RFe 2 compounds develop "giant" strains at room temperature although their low-temperature values are in the same range as for other series. On the other hand, the spontaneous strain, which would be large in the case of a collinear spin structure, fully vanishes in the case of a multi-axial spin structure, preserving cubic symmetry as in DyCu. The magneto-elastic effects are of similar magnitude in rare earth intermetallics and insulators. For instance, in TbVO4, DyVO 4 and TmVO4, the vanadates which undergo a Jahn-Teller tetragonal-orthorhombic transition (Gehring and Gehring 1975), the low-temperature spontaneous strains are e ~= 12.7 × 10 -3, e ~ ' = 5.9 × 10 -3 and e ~= 3 × 10 -3, respectively. These strains are reminiscent of those in intermetallics. Unfortunately, whereas determinations of magneto-elastic coefficients in metals are numerous in cubic symmetry and scarce in tetragonal symmetry, the situation is the opposite in insulators; thus comparing the magneto-elastic coefficients is difficult. However, from the magneto-elastic contributions G~aE = 6 /[~1 and GME = /X2 determined from the ultrasonic velocity softenings (Melcher 1976), the following values IB~I--S7K, In l--80 and 22K are deduced in DyWO4, TbVO4, TmWO4, respectively. These magnitudes are similar to those observed in intermetallics.
5.1.3. Comparison of the quadrupolar pair-interaction coefficients There are fewer systematic studies for the pair interactions than there are for the magneto-elastic coupling. However, from table 24, their existence in TmX is well established and a very good coherency is observed in tetragonal symmetry. Within the CsCl-type structure, the tetragonal pair interactions dominate the magnetoelastic coupling and are the main origin of the ferroquadrupolar ordering. They are always positive and particularly important as soon as the aj coefficient is large. On the contrary, from the determinations in the paramagnetic phase, the trigonal pair interactions are negative and overwhelm the weak magneto-elasticity. For NaCl-type compounds, the tendency is less clear (fig. 65): K ~ is negative in DySb, but seems to be positive in HoSb and null in TmSb, in which no pair interactions seem to be present (Wang and Liithi 1977b). Tetragonal pair interactions are weaker than the magneto-elastic coupling in RSb pnictides. For the trigonal symmetry, the study of the TbX system indicates K ~ to be positive
Q U A D R U P O L A R EFFECTS IN R A R E EARTH INTERMETALLICS
20
-t
80
-Ag
117
.Ag -Go 3.~. '-Cd
-Cd
II
-'Cu
-Cu°
-Sb E 0 X)
- cd "zo ;Ag
"-P%
I
Ce
I
Pr
-Sb
-sbD
I
I
I
Tb Dy Ho Er
I
Tm
Fig. 65. Quadrupolar pair coefficient K ~ for the tetragonal symmetry in some rare earth cubic intermetallics (0: CsCl-type, +: AuCu3-type, [~: NaCl-type compounds).
and larger than G ~E. However, the total quadrupolar interactions remain smaller than the bilinear coupling and only modify the nature of the N6el transition. In insulators, such as the vanadates, the pair-interaction coefficients are fundamentally different. From the analysis of elastic constants (Melcher 1976) the coefficients K ~= h 1 and K ~= A2 are found to be negative and their absolute values are smaller than the magneto-elastic contribution (K ~= -0.35 mK in D y g O 4 , K ~ = -0.77 and - 1 1 mK in TmVO 4 and TbVO4, respectively). In these Jahn-Teller compounds, the structural transition is thus driven by the magnetoelastic coupling, whereas in the intermetallics the pair interactions drive the quadrupolar ordering and then, through the magneto-elastic coupling, the structural transition (Levy et al. 1979). In insulators the pair interactions are mediated by phonons (Gehring and Gehring 1975), and the h coefficient is a so-called "phonon exchange" parameter. It integrates the effective coupling of all the phonons to the Jahn-Teller ions for a given symmetry. The K/GME ratios are very close to --1/3, the theoretical expectation in the absence of any optic phonon mode contribution (Harley et al. 1972). On the other hand, the positive K values observed in intermetallics can originate neither from a k = 0 optical phonon in the NaC1- and CsCl-type structures, nor from the negative phonon self-energy term (Levy et al. 1979). In section 5.2 it will be shown that pair interactions in intermetallics are mainly mediated by conduction electrons.
5.2. Origin of the quadrupolar interactions 5.2.1. The one-ion magneto-elastic coupling From the comparison between the expressions for the cubic crystal field Hamiltonian [eq. (2)] and those for the one-ion magneto-elastic coupling [eq. (7)], it becomes obvious that the magneto-elastic coefficients are the strain derivatives of
118
P. MORIN and D. SCHMITT
the CEF parameters. For example, the second-order CEF parameter B ° vanishes in cubic symmetry, but its derivative with regard to a tetragonal distortion does not, B ~ , =-=
OB °
0~
"
The same is valid for a trigonal deformation, as well as for the higher-rank magneto-elastic coefficients, or if we consider crystals with lower symmetry, e.g., hexagonal or tetragonal ones. Therefore, the question of the origin of the magneto-elastic coupling is identical to the problem of the origin of the crystal field itself. In rare earth intermetallic compounds, it has been shown that the CEF parameters originate mainly from two contributions, namely the point charges and the contributions of the conduction electrons (Schmitt 1979). In cubic systems, these latter have been found to be preponderant, due to the strong anisotropic orbital character of the conduction electrons, as shown, e.g., by augmented-plane-wave band calculations (Belakhovsky and Ray 1975). In systems with hexagonal or tetragonal symmetries, the point-charge contribution seems to play a role which is relatively more important than in cubic systems, at least for the second-order CEF parameters. Then, calculating the magneto-elastic coefficients requires a precise knowledge of how these two CEF contributions depend on a given strain. First, in the presence of a weak strain, the localized charges are displaced from their previous non-strained arrangement, generating a new potential having the new symmetry of the lattice and proportional to the strain. This potential gives the point-charge contribution to the magneto-elastic coefficients. On the other hand, and to the first order of perturbation, this new potential modifies the conduction band. In particular, it lifts the orbital degeneracy of the conduction electron states in the vicinity of the Fermi level. According to Fermi statistics, this leads to a redistribution of the conduction electrons in the band which gives rise to a new CEF contribution, i.e., the band contribution to the magneto-elastic coefficients. Both contributions have been thoroughly investigated in cubic rare earth intermetallic compounds (Morin and Schmitt 1981a). In particular, the expressions for the direct and exchange Coulomb contributions of each type of conduction electrons (p, d, or f character) have been derived. It has been found that the magneto-elastic modifications of the cubic fourth- and sixth-order CEF parameters are negligible and that the second-order magneto-elastic coefficients result from the competition between numerous antagonistic contributions, which are of the same order of magnitude as the experimental values. Within the hypothesis of constant effective charges and a rigid-band model, all contributions are expected to vary as aj, ~j, yj for the second-, fourth- and sixth-order magneto-elastic coefficients, respectively. Furthermore, the pointcharge estimation leads to the constant ratio B ~ / B ~ = -4/V'-3 between the trigonal
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
119
and tetragonal second-order coefficients (see Creuzet 1982); this is not true for the band contribution. As shown in the previous section, this relation is far from being satisfied experimentally for a given compound, and this failure clearly indicates the importance of the conduction electrons. Nevertheless, the difference in sign between both parameters in most of the series as well as their overall magnitude (figs. 63 and 64) suggests a driving role for the point-charge contribution, contrarily to the case of the cubic CEF parameters. In realistic systems, because of the various contributions involved, the magnetoelastic coefficients are thus liable to vary significantly from one compound to another, especially if they belong to different series. This variation may occur via a shift of the values of the effective point charges. For the conduction-band contribution, the variation may be related to the modification of the partial densities of states at the Fermi level, which can be important when the Fermi level lies near a peak of the density of states. The importance of the d electrons on the rare earth site has also been suggested in dilute rare earth alloys (Campbell and Creuzet 1985). In these systems, the magneto-elastic coupling constants were evaluated within an indirect electronlattice interaction model, and using the nearly free electron approximation (Kozarzewski et al. 1985). The general features of the results obtained in this calculation are similar to those described above. The effect of screening of the magneto-elastic coupling by the conduction electrons has been found to significantly reduce the point-charge estimate of the parameters in Laves-phase rare earth intermetallics (Del Moral 1983). As a conclusion, ab initio evaluations of the one-ion magneto-elastic coefficients appear to be difficult to perform. As for the CEF parameters, they originate from numerous antagonistic contributions, and obtaining realistic values seems rather difficult. Qualitatively, their order of magnitude is well accounted for by point-charge and/or conduction electron contributions, and they follow the Stevens factor throughout a given series.
5.2.2. Quadrupolar pair interactions In metallic systems, conduction electrons have been shown to play an essential role in the coupling between magnetic ions, by propagating the local interactions between them and the localized magnetic electrons. This indirect exchange mechanism has been widely invoked in magnetic materials and is known as the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Most of the studies has been focused on bilinear-type indirect coupling, which, in general, is the dominant term. Much less attention was paid to the origin of quadrupolar-type interactions. In rare earth and transition metal insulators, direct coupling between electric multipole moments of the two atoms and indirect coupling through the lattice-ion interaction have both been investigated (Birgeneau et al. 1969). However, in rare earth intermetallic compounds with the NaC1- and CsCl-type structures, both couplings have been proved to be negligible and therefore cannot be the dominant mechanism for the quadrupolar pair coupling (Levy et al. 1979): the
120
P. MORIN and D. SCHMITF
primary source of such a coupling is the indirect Coulomb and exchange interaction via the conduction electrons. The coupling of quadrupoles via the conduction band is analogous to the RKKY interaction quoted above, which involves the spins at different sites through the local s - f exchange interaction. For quadrupoles, it is the direct and exchange part of the full 4f-conduction electron (k-f) Coulomb interaction that contributes to the couplings K ". The corresponding effective couplings were calculated in second-order perturbation theory and by using plane-wave states for the conduction electrons (Teitelbaum and Levy 1976). A more realistic ab initio calculation of bilinear and quadrupolar pair coupling was performed later in the cubic rare earth intermetallic compound DyZn, by using conduction-electron wave-functions and energy bands obtained by a selfconsistent augmented-plane-wave method (Schmitt and Levy 1984, 1985). Spin and orbital effects in the complete k - f interaction were taken into account. The results underline the dominant role played by the d conduction electrons in the bilinear and quadrupolar couplings. In this calculation, the Fourier transform of the bilinear 0*(q) and quadrupolar Kr(q) and K~(q) interactions were found to be of the correct order of magnitude. Their different q-dependence was explained by the behaviour of specific matrix elements of the k - f interaction: the magnitude of these matrix elements is directly related to the symmetry of the two k-states involved. In particular, the ferromagnetism and the ferroquadrupolar tendencies (y-mode) of DyZn are associated with the strong eg-character of the conduction electrons near the Fermi level. On the contrary, the antiferroquadrupolar tendency of the trigonal coupling K~(q) is related to the small amount of t2g-type conduction electrons near the Fermi energy. The dependence of the bilinear and quadrupolar couplings on the Fermi energy was also investigated. Only part of the coupling, i.e. the intraband terms at q = 0, follows the partial densities of states at the Fermi level, which are associated with the type of electrons (% o r t2g ) from which they arise. The other terms, in particular the interband ones, are not directly correlated with the Fermi energy but depend on details of the entire energy band structure, including energy bands well above the Fermi level, which contribute for a noticeable amount. At last, the orbital interactions are the main contributions to the quadrupolar couplings K" so that the latter are expected to vary as otj. 2 That explains the importance of quadrupolar effects at the beginning and end of the rare earth series, where weak spin contributions (small De Gennes factor) allow the quadrupolar coupling to be better revealed. As for the one-ion magneto-elastic coefficients, ab initio evaluations of the two-ion quadrupolar parameters are very difficult to perform and would be necessary for each compound under consideration. 5.3. Conclusion
The aim of this chapter has been to present an extensive overview of the effect of the magneto-elastic coupling and of the quadrupolar interactions in rare earth intermetallic compounds. A complete Hamiltonian has been detailed, which
Q U A D R U P O L A R EFFECTS IN RARE EARTH INTERMETALLICS
121
includes all the relevant terms describing the one-ion magneto-elasticity as well as the two-ion quadrupolar coupling, for all the possible normal strain modes in cubic, hexagonal and tetragonal symmetries. The formalism for using this Hamiltonian has been developed, in particular that related to a perturbation theory. The various effects of the magneto-elastic coupling and of the two-ion quadrupolar interactions on the magnetic and elastic properties have been discussed, with a particular emphasis on the properties related to the perturbation methods: elastic constants, parastriction and third-order magnetic susceptibility. All these experimental manifestations appear as reliable ways to determine the quadrupolar parameters. Due to the strong anisotropic character of the magneto-elastic properties, using single crystals for experiments is quite essential. The magneto-elastic and quadrupolar data available in the literature have been summarized for a great number of rare earth series, within the present formalism, in order to make a comparison between them easier. Some series, especially those having the cubic CsC1- and NaCl-type structures, have been extensively studied in the last decades. Several others have been investigated much less, and some of them would be worth to be studied more thoroughly, in particular when single crystals become available. In most of the experimental results, the second-order one-ion magneto-elastic coefficients are concerned. The ratios B~'/%, in general, present a regular variation throughout a given series, but their magnitude and their sign can drastically change from one series to another. Nevertheless, their absolute value always remains in the 0-4000 K range. These differences show that, even if the point-charge contribution to the magneto-elastic coefficient is important, the role of the conduction electron is far from being negligible in these metallic compounds. An accurate ab initio estimate appears, however, difficult to obtain, due to the great number of different antagonistic contributions. The two-ion quadrupolar interactions have been studied less thoroughly in the rare earth intermetallic compounds. A consistent determination of the corresponding parameters has been successfully carried out in those series where the effects are strong, in particular those where a quadrupolar ordering occurs. Through their contribution to the free energy, the quadrupolar pair-interactions may strongly influence the character of the magnetic transitions, as well as, when they are negative, the stability of multi-axial spin structures. At last the dominant role of the conduction electrons has been emphasized in these systems, contrarily to the case of the rare earth insulators where the coupling between 4f quadrupoles is mainly due to the phonons. The existence of magneto-elastic and quadrupolar couplings then appear quite general in rare earth interrnetallic compounds. They need to be included in order to quantitatively understand the magnetic and elastic properties of these systems.
Acknowledgement It is a great pleasure to thank Professor P.M. L~vy from New York University for very fruitful discussions and a critical reading of the manuscript.
122
P. MORIN and D. SCHMITI"
Appendix I. Symmetrized Stevens operators for the cubic symmetry To completely express the magneto-elasticity of rank l (l = 2, 4, 6), all 21 + 1 Stevens operators are needed. In the present chapter we use the operators defined by Hutchings (1964), namely the quadrupolar operators of table 1 (l = 2) and the O / (m = 0 to l) and O'S(s) (m = 1 to l) for l = 4 and 6 (see table VIII of Hutchings 1964). The linear combinations of these operators, which transform according to the irreducible representations of the cubic symmetry group, read as, Second-rank 5"
O~ ' 1 = 002'
o l '2 = v ~ o i .
5"
Oi '1 = P~y,
° 2 '2 = Pr= ,
0 2 '3 = Pz=.
Fourth-rank ot 0 4 = 0 0 "~- 5 0 ~
FI:
r3:0~4
.
0~4'2 = -4X/-3 0 ] .
'1 = 004 -- 7 0 4 '
6,2
1
3
5:
064 '1 = O ~ ( S ) ,
0 4 = 0 4 -- 0 4 ,
5:
o i '1= o~4(s),
01,2 = --lOi(s
8,3
0 4 =
__ 0 1 ( S ) __ O I ( S )
) q- 7: O 43( s ) ,
0 4 , 3 = - - ~1 1O 4 -- 1 0 43.
Sixth-rank 5:O6
a
= O1--210~"
r~: 06~-- o l - 06 5:
O~ '1~- O 1 71-3 0 ~ 01,1 _-- 0 4 ( S ) ,
5:
01,2=
=
~5 r ~ 0 6 _2[ _
- - ~10 6I
~/'~
06 6 .
-- 5
~ 0 6 "31 - T11/',15 v6,
Y11~.5/ IJ6(S)", .
011,2 = ~ 01 16 ( s ) - 9
3 )-{- ~'~ O I ( s ) , ~06(S
1 9 3 33 [-15 0 6 1 3 = ~1 O 6-[~O 6 q- T t / 6 •
0621 = OI(S),
5:
O~,2
0~6,3 = 1o~6(s)_ ~5 O 63( s ) 06~, 1 = O I ( S ) ,
5:
,
3 1 0 6 ~ 3 = -~0 6-
5 3 ) + IOI(s), O12,2 = I O ~ ( s ) ~'- ~06(S 5 3 3 5 ~0 6 + ~0 6 •
QUADRUPOLAR EFFECTS IN RARE EARTH INTERMETALLICS
123
Appendix 2. Fourth- and sixth-rank one-ion magneto-elastic Hamiltonian ~Mm Starting from the expressions in table 2 and appendix 1, three terms may be derived for l = 4: one isotropic term which keeps the cubic symmetry, and two anisotropic terms as for l = 2, O~MEI(I = 4) = O~MEI(I = 4) + Yg~m(l = 4) + Y(MEI(I = 4 ) , with ~x ~MEI(I
m --
4) =
~ -B(4)8
a
0 4 ,
--~(4)\e, ltJ4
~/~MEI(I e = 4)
At- /:,2tJ4
) ,
= - - /nJ ( 4. .). . ['Ell . . J 4 1 ..}_ E 2e,,~e I J 4 , 2 + E 3. .[.J. 4 3,) .
The case l = 6 is identical to l = 4, except that, for the e mode, there exist two independent sets of linear combinations of the Stevens operators. Therefore, ~MEI(I = 6) = ffffMEl(l=6) + ffff~ciEl(l=6)+ ffffMEX(/ el e2 a = 6) + ~MEI(/ = 6), with oz
o~
~1;
~MEI(I = 6)
= -B(6)8
F3:
Y(~E,(I
:
F5:
fft~MEI(/ el = 6):
F5:
g Me2E I ( I = 6)
:
6)
ot
o~
0 6 ,
- - D, (~6r) ~, E Ir~r,1 I J 6 + ~_r,-~r,2x 2~6 J , ... 1,1 ..[_ 8 2 0 ; 1 , 2 _]_ E 3e,-~el,3x, - - Dh e( 61 ) .I'EIIA'6 IJ 6 ) ,
= - - / ~ ( 6 ) .j E . .l.l J 6
2,1 + 406
,2
+ E;O;2,3)
,
All the above expressions are referred to axes with four-fold symmetry, i.e., with z-axis parallel to the [001] direction. In some cases, however, it may be useful to consider the z-axis along a [111] direction, e.g., when a magnetic field is applied along that direction. In that case, symmetry considerations imply that the tetragonal strain e r should vanish and that the trigonal strains e" are all identical, e 7 = e 2 = e; = e'. Therefore, the corresponding trigonal magneto-elastic Hamiltonians each involve a single linear combination of symmetrized operators which may be rewritten in the new axes system (according to the convention of Hutchings 1964) as, I=2:
t'xy+ e,z+ ezx-, o ,
1=4:
0 4,1+ 0~,2 + 0 4,3~ ~(O 1 4o +7V'2 O ] ) ,
l=6:
O ; 1 , 1 + O ; 1 , 2 _[_ O;1,3..._> _ _ ~ ( 0 6 1 0 __ ~ / ~ 0 6 3
O62,1 + 062,2 + O6Z,3__> -- T~ 1 ( O 60 --
_ 110~)
2 5 X / 2 O 63 +
130~)
124
P. MORIN and D. SCHMITr
Appendix 3. Perturbation theory: the tetragonal mode Y First we have to define the zeroth-order energies E~ and the eigenvectors [ik) corresponding to the CEF Hamiltonian ~fCEF alone, ~CEF [ik) = E i lik)
(k
= 1,...,
m,).
(Cl)
In each degenerate subspace i, the m~ independent eigenfunctions lik) have to be adapted to the perturbation Y('~ [eq. (29)]. This can be achieved easily by choosing the following expansion in the IJ, M) basis [for the complete expressions, see Lea et al. (1962)], [ik) = aiklJ,1
M ) + a~2lJ,
M
- 4) +
a~klJ, M
- 8) + " "
.
(C2)
A perturbation theory carried out to second-order in e~ and fourth order in H allows one to obtain analytic expressions of the perturbed energies Ezk,
4 Eik = E i + 2
n=l
~ikl~(n)+ ' ' "
(C3)
"
Then, the partition function Z, Z = Z e -t~E'k ,
(C4)
i,k
can be calculated, where/3 = 1 / ( k a T ) , where k B is the Boltzmann constant and T the temperature. One obtains, Z=
ZcEFZ
1 r~ ,BVeV1 + KVQ): .... [I + ½/3Xo(H+ nM)2 + :PXvt
+ PXv'~(2)'B~ert1 + K r Q ) ( H + n M ) 2
-1-1/3(X(3) -t- 1/3(X0)2)(n--}-/,/M) 4-1-.. "l,
(C5)
where ZCE v is relative to the Ei's and Z .... to the elastic energy and to the corrective energy terms [see eqs. (14), (16), (17)]. Finally, this leads to the expression of the total free energy F ~ = - k BT In Z given in eq. (30). . (2) and x~ - (3) are given The expressions of the four CEF susceptibilities X0, X~, x~ by,
x0 = g . . B
i z.12
Il Ok,j, 2 i,k
l
+ 1 ~B T
I/ k,,kl 2 , \ I ;~.i~l )
'
(c6)
(C7)
Q U A D R U P O L A R EFFECTS IN R A R E E A R T H INTERMETALLICS t,(2)
,
2 2
= gylxB
125
Jik,jlQjl,J'l'JY'l',ik q- 2Qik,jlJjl,j'l'Jy'l',ik fi jei,l £ (U i -- E i ) ( E i -- Ej,) j'~i,l' (
Ei,k
I4~,~,l O,~.,~ + 2Oik,jl41,ikJik,ik (
- ]¢i,I 2
7--V
1
1
)
(c8)
+ 2(kBT)2 [J~k,ikl Q,k,ik , __ ,y ,•(3)
1 4 4 ( Jik,jlJjl,j'l'Jj'l',j'T'Jj'T',ik 2k~r (x°)= + g'~"£i,k f' - 4 /#i,, £ (E,--~---E-7')--~i----Ef) j'#i,l' j"¢i,l" 2
2
IJ,k,j,I IJ,k,/','~ + 2J,'k,/,J/t,/','J/',',ikJik,ik(
+2 ~
/~,,,
(E,- g)(E,- E/,)
2
E,
j' #i,l'
2
IJ....~12l.l,,.,jll 2
-2 ,,i,,E
((E i - Ej)2
~ii:-~])
i
+ 6k3T 3 IJik,ik
i4)
2 q- ( E i _
E/)kuT
,
+©I (C9)
where
Jik,/, =
(C10)
a.~,/, =
(Cll)
and
are the matrix elements of Jz and O ~ in the CEF levels. For each CEF sublevel, 1 exp(-/3Ei) f / - ZCEF exp(-/3E~) = ~ exp(-/3E~)
(C12)
i,k
is the Boltzmann population factor. Appendix 4. Perturbation theory: the trigonal mode e
When the magnetic field H is applied along a three-fold axis, e.g., [111], it is more convenient to first rotate the coordinate axes so that [111] is the new z-axis. Then the two combinations of operators in eq. (44) may be rewritten as:
L+~+L--.V~L,
126
P. MORIN and D. SCHMITT
and P x y + P y z q-
1
0
P~x --> ~02 "
The perturbation Y(] takes the new form:
Y({'= --g,p.B(H + n M ) ] z - ½(B~e ~ + K~P)O ° 3 ,.-~e; e.,2 + ~t..oke ) + ½nM2 + ~K*P 2.
(D1)
In each subspace i, the new zeroth-order eigenfunctions Ilk }' (corresponding to ! the appropriate YgCEF) have to be adapted to the above Hamiltonian. This condition is always fulfilled if these basis functions are chosen so that AM = 3 in the IY, M) expansion: 2 [ik )' = bill J, M ) + b~klJ, M - 3) + b~klj, M _ 6 )
+ ..- .
(D2)
The same perturbation theory as in appendix 3 may then be carried out, leading to the following expression for the total free energy F ~, 1 ! F ~ = Foe -- ~Xo( H + nM)2 -- ~XvkD 1 ..... e +K~P) 2 1 (2)' [n~ -~X~, ~D e e + K ~ P ) ( H + n M ) 2 - - ~i a, . y (3)'(H + nM)4
+ ~3 o,,.~e,, te
e,,2
) + l n M 2 + ~K~P 2,
(D3)
¢
where the new CEF susceptibilities X~ are related to the operators Jz and O ° in the new system of axes. Their expressions are identical to those given in appendix 3, but their values differ from the corresponding unprimed susceptibilities, due to the new CEF eigenfunctions lik)'. Returning to the initial system of four-fold axes and defining new trigonal susceptibilities we are led to the expression of eq. (45) for the free energy, where: Xo
:
,
Xo
X~
=
1
,
~X~,
. (2)
X~
1.
(2)'
= ga~,
/~3)
. (3)'
= ,'(v
"
(D4)
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