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Chapter 5 Magnetic properties of rare earth-Cu2 compounds
chapter 5 MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
Nguyen Hoang LUONG Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands (On leave from the Cryogenic Laboratory Faculty of Physics, University of Hanoi Vietnam)
and
J.J.M. FRANSE Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands
Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 415
CONTENTS 1. I n t r o d u c t i o n
.................................................................
417
2. B r i e f s u r v e y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. R a r e - e a r t h a n d r a r e - e a r t h - - C u c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418 418 ......
420
3. T h e o r e t i c a l a s p e c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Crystal-field i n t e r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. E x c h a n g e i n t e r a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. D e s c r i p t i o n o f m a g n e t i c p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
421 421 424 425
4. M a g n e t i c p r o p e r t i e s o f R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. C e C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432 432
4.2. 4.3. 4.4. 4.5.
PrCu 2 .................................................................. NdCu 2 .................................................................. SmCu 2 ................................................................. GdCu 2 ..................................................................
436 441 447 451
4.6. 4.7. 4.8. 4.9.
TbCu 2 DyCu2 HoCu 2 ErCu 2
.................................................................. .................................................................. .................................................................. ..................................................................
457 463 468 473
4.10.TmCu 2
.................................................................
480
5. C o m p a r i s o n o f i s o s t r u c t u r a l c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
485
6. A c k n o w l e d g e m e n t s
488
...........................................................
References .....................................................................
416
488
1. Introduction
Among the rare-earth intermetallic compounds, the RT2 (R = rare earth, T = transition metal) with the cubic Laves phase structure have been widely studied in the past decades. Initially, less attention has been paid to the RCu2 compounds after the early magnetic investigations performed by Sherwood et al. (1964). However, during the last decade substantial progress has been achieved in the study of the magnetic properties of the RCu 2 compounds. The RCu 2 compounds, except for LaCu2, crystallize in the orthorhombic CeCu2 structure, i.e. belong to the systems with low symmetry. The compounds show a large variety of magnetic behaviour. Most of the RCu 2 are antiferromagnetic. T b C u 2 has the highest N6el temperature with a value for TN near 50 K. The magnetic properties of the RCu 2 are largely affected by the crystal field (CF) effects. The magnetic ordering temperatures in the RCu 2 compounds are low and crystal-field and exchange interactions can be studied by varying temperature and/or magnetic field. Changes in the magnetic structure occur in most of the compounds below the N6el temperature and most of the compounds exhibit metamagnetic transitions below the magnetic ordering temperature. The metamagnetic transition in Z b C u 2 has been discussed by Gignoux and Schmitt (1991) as a typical example of a single-step transition in a metallic system. Unlike the RnTm compounds where T is a magnetic transition metal, 4f magnetism can be investigated in the RCu 2 without disturbing effects of the d magnetism. An extensive description of the magnetic properties of 4f-3d intermetallic compounds has been presented in the reviews by Buschow (1977), Kirchmayr and Poldy (1979) and Buschow (1980). The magnetic properties of binary rare-earth 3d-transition metals intermetallic compounds with the transition metals Mn, Fe, Co, and Ni have been reviewed by Franse and Radwanski (1993). It turns out to be of considerable interest to carry out in a systematic way investigations of the physical properties within isostructural series generated by the various rare-earth elements. As far as the binary rare-earth intermetallic compounds with non-magnetic partners are concerned, to our knowledge, only a review is available for the magnetic properties of the RA12 compounds (Purwins and Leson 1990). In the present work, the magnetic properties of the RCu 2 are reviewed. Transport properties of these compounds are not particularly considered in this review. For these properties we refer the reader to the review of Gratz and Zuckermann (1982). Thermal conductivity and electrical resistivity of some heavy RCu 2 compounds have been reported by Bartkowski et al. (1992). LaCu2, which crystallizes in the hexagonal A1B2 structure, is not included in this work. EuCu 2 and Y b C u 2 a r e both intermediate-valence compounds and are not considered in detail. We mention that EuCu 2 is known to be antiferromagnetic at 417
418
N.H. LUONGand J.J.M. FRANSE
4.2 K (Sherwood et al. 1964, Wickman et al. 1966). M6ssbauer studies have been performed on this compound by Nowik et al. (1973), Abd-Elmeguid and Kaindl (1978) and Abd-Elmeguid et al. (1981). YbCu2 is paramagnetic (Sherwood et al. 1964). The anomalous properties of Yb-Cu (as well as Ce-Cu) compounds have been reviewed by Bauer (1991). CeCu2 is a Kondo compound, and hence, is discussed not in detail. By lack of information we have also to omit the radioactive compound PmCu2. YCu2 and, occasionally, LuCu2, are treated as reference materials in connection with the other magnetically ordered compounds. Summarising, we will deal with a systematic discussion of the RCuz compounds where R = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Tm. A brief survey on all existing R-Cu compounds is given in section 2. In section 3 we present a short theoretical consideration of crystal-field and exchange interactions as well as of some important magnetic properties. Experimental results on RCu2 compounds are reviewed in section 4. A comparison of different RCu 2 compounds is given in section 5. 2. Brief survey 2.1. Rare-earth and rare-earth-Cu compounds In this chapter, the term lanthanides refers to the elements from La to Lu. Lanthanides to the left of Gd in the periodic table are referred to as the light rare earths, and the remaining ones as the heavy rare earths. The rare earth elements are of fundamental interest because most of them behave as well-defined ions in solids. Exceptions are known to be the elements Ce, Eu and Yb. The number of binary R - T (T = 3d transition metals Mn, Fe, Co, Ni) intermetallic compounds, available at present, amounts to nearly 200 (Franse and Radwanski 1993). As far as the R-C~ compounds are concerned, the nature of the compound formation is summarized in fig. 2.1. The light lanthanides RCu phases as well as YbCu crystallize in the orthorhombic FeB-type of structure. The remaining RCu compounds are formed within the cubic CsCl-type of structure. SmCu and EuCu can adopt both structures. ScCu and YCu have the CsCl-type of structure as well. C e C i l 2 crystallizes in an orthorhombic structure which is the prototype for the RCu2 series. I - ~ C u 2 and S c C u 2 a r e exceptions, forming the hexagonal A1BE-type and the tetragonal MoSi2-type of structure, respectively. The light rare earths RCu5 compounds have the hexagonal CaCus-type of structure, whereas the RCu5 compounds with the cubic AuBs-type of structure are formed for the heavy rare earths. An exception is YbCus, that crystallizes in the CaCus-type of structure. The RCu 6 compounds crystallize in the orthorhombic CeCu6-type of structure and have been reported to exist for the light rare earths from La to Sm. For the heavy rare earths, only Gd and Tb form phases with the RCu6 stoichiometry. The existence of the phases RzCu7, RCu4, R2Cu9 and RCu 7 has been reported for a limited number of rare earth elements (Subramanian and Laughlin 1988).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
I
I
I
I
i
I
I
I
I
I
I
I
I
I
I
I
CsCt
I',, \ \ \ \ \ \ ~ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N
RCu
419
Y////A
g//~
FeB MoSi
liB2 2 ~-
RCu2
I~,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " ,
3 CeCu2 ~-
E
k\\\\\\\\\\\\"
~RCu5
I
k\\\\\\\\\\\\\\"
I
C~Cu5
L~
V////////////A
V ~ AuBe5 .~
(/)
CeCu6
RCu6: I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Sc Y b Ce Pr ~ Pm SmEu Gd Tb Dy ~ Er Tm Yb Lu
Fig. 2.1. Compound formation in the R-Cu systems. After Subramanian and Laughlin (1988).
/
b
F,1 Fig. 2.2. Chemical unit cell of the orthorhombic CeCu2-type of structure. The open and closed circles indicate the rare-earth and copper atoms, respectively. After Hashimoto et al. (1979a).
420
N.H. LUONG and J.J.M. FRANSE TABLE 2.1 Crystallographic data for the RCu2 compounds.
R
Lattice parameters (/~) a b e
Ref.
Melting Reaction point (*C) type
Ref.
Sc Y La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
3.29 4.305 4.346 4.425 4.400 4.387 4.37 4.360 4.45 4.320 4.310 4.300 4.280 4.275 4.266 4.28 4.245
[1] [4] [4] [4] [4] [4] [2] [4] [4] [4] [4] [4] [4] [4] [4] [4] [4]
990 935 834 817 841 840 850 860 597 860 870 890 915 935 960 757 1000
[2, 3] [5] [4] [6] [7] [8] [2] (proposed) [2] [9] [10] [2] [11] [2] [12] [2] [13] [2]
6.800 7.057 7.024 7.059 6.96 6.925 7.25 6.858 6.825 6.792 6.759 6.726 6.697 6.76 6.627
8.388 7.315 3.807 7.475 7.435 7.420 7.40 7.375 7.54 7.330 7.320 7.300 7.290 7.265 7.247 7.40 7.220
congruent congruent congruent congruent congruent congruent congruent peritectic congruent congruent congruent congruent congruent congruent peritectic congruent
[1] Dwight et al. (1967). [2] Subramanian and Laughlin (1988), estimated on the basis of the systematics of crystallographic data in the R--Cu systems. [3] Markiv et al. (1978), quoted by Subramanian and Laughlin (1988). [4] Storm and Benson (1963). [5] Chakrabarti and Laughlin (1981). [6] Rhinehammer et ah (1964). [7] Canneri (1934), quoted by Subramanian and Laughlin (1988). [8] Carnasciali et ah (1983a). [9] Costa et al. (1985). [10] Carnasciali et ah (1983b). [11] Franceschi (1982). [12] Buschow (1970). [13] Iandelli and Palenzona (1971).
2.2. RCu2 compounds The orthorhombic CeCu2-type o f structure has first been determined by Larson and Cromer (1961). The chemistry o f the CeCuz-type structure was the subject o f considerable discussion (e.g., Debray 1973, Michel 1973, Bruzzone et al. 1973). This structure belongs to the space group Imma. The rare-earth atoms and the copper atoms o c c u p y the 4e and 8h sites, respectively, and form a double-layer structure along the c axis as s h o w n in fig. 2.2 (Hashimoto et ah 1979a). Crystallographic data for the RCu2 c o m p o u n d s are collected in table 2.1. As can be seen f r o m this table, the lattice parameters and the melting temperatures o f EuCu2 and YbCu2 are anomalous, indicating the divalent state o f Eu and Yb. For more detailed information we refer the reader to the review paper of Subramanian and Laughlin (1988), in which the R--C-~ phase diagrams have been compiled and evaluated. The Ce--Cu phase diagram is s h o w n in fig. 2.3, as an example.
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
421
Weight Percent Cerium
0 10 20 30 40
50
1200~ 1000
60
70
80
r
~
90 r
100
-~. ./~ 817 C
798"( 726°C
:'%111 `°
-2 r'~
•
E
tOO
II
-
2oo -
I
~I I~I
.~
= _.__L__--- . . . . . . . . . . . . . . . . . . . .
0 ~
0
10
20
Cu
/
72
30
40
50
60
70
~
80
61"C
90
AtomicPercentCerium
100
Ce
Fig. 2.3. The Ce-Cu phase diagram. After Subramanian and Laughlin (1988).
3. Theoretical aspects
3.1. Crystal-field interaction The 4f electrons of a rare-earth ion in a solid, being considered as well localized and separated from other charges, experience an electrostatic potential VcF(r) that originates from the surrounding charge distribution. The potential reflects the point symmetry of the site of the rare-earth ion. In case there is no overlap between this charge distribution and the wave functions of the 4f state, VCF satisfies Laplace's equation and can be expanded in terms of the spherical harmonics, Ynm. The value of n in this expansion is limited to 6, as higher multipoles cannot cause electronic transitions between states of the 4f ion. For details see Hutchings (1964), Fulde and Loewenhaupt (1985). It is an experimental fact that for the rare-earth ion, the spin-orbit interaction is much larger than the crystal-field and exchange interactions. A review of experimental data of the spin-orbit interactions based on the observation of intermultiplet transitions has been presented by Osborn et al. (1991). Owing to this fact, it is usually sufficient to consider the lowest multiplet J, given by Hund's rules. This
422
N.H. L U O N G and J.J.M. FRANSE
limitation results in a substantial simplification in computations. Eu +3 and Sm +3 are ions where the analysis requires the involvement of a higher multiplet since the higher multiplet is only 530 K and 1500 K above the ground multiplet, respectively. The crystal-field Hamiltonian that describes part of the electron-electron interactions in the solid due to the electrostatic interaction of the aspherical 4f charge distribution with the aspherical electrostatic field arising from the surrounding, can be written as: oo
n
AT n=0 ra=0
(3.1) i
where f,,m are Tesseral harmonics describing the spatial distribution of the charge associated with the 4f electrons; the summation over i is over all 4f electrons. Anm describes the spatial distribution of the charge surrounding the 4f electrons. Within the Stevens formalism the summation in eq. (3.1) over the 4f electrons leads to matrix elements of the total angular momentum: n rr~ Z fnra(ri) = On(7 ,4f)On (J'), i
(3.2)
where the x, y, z coordinates of a particular electron in the functions fnm(rl) are replaced by the components Jx, Jy, Jz of the multiplet J. The O F are the Stevens equivalent operators (Stevens 1952). On is the appropriate Stevens factor of order n which represents the proportionality between operator functions of x, y, z and operator functions of Jx, Jy, Jz. The parameter On is denoted as aj, flj, 7J for n = 2, 4, 6. The sign of On represents the type of asphericity associated with each Onm term describing the angular distribution of the 4f-electron shell. In particular, the factor ecj describes the ellipsoidal character of the 4f-electron distribution. For ~j > 0, the electron distribution associated with Jz = J is prolate, i.e. elongated along the moment direction whereas for cq < 0 the 4f-electron-charge distribution is oblate, i.e. expanded perpendicular to the moment direction. For ~j = 0 (which is the case of the Gd +3 ion) the charge density has spherical symmetry. (r~f) is the mean value of the n-th power of the 4f radius. Values for the average value (r~f) over the 4f wave function have been computed on the basis of Dirac-Fock studies of the electronic properties of the trivalent rare-earth ions by Freeman and Desclaux (1979) and are presented in table 3.1. Within the ground-state multiplet the crystal-field Hamiltonian (3.1) is written in the conventional form: oo
HCF = E
n
Z
BnmO~n(Y)'
(3.3)
n m=0
where Bnm are called the crystal-field parameters. Usually these parameters are evaluated from the analysis of experimental data. The parameters Bnm can be written as"
B m = On(r4r)A n n
gll
(3.4)
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
423
TABLE 3.1 The second-, fourth- and sixth-order multipole moments On(r~f) of the trivalent rare-earth ions. cq, flj and "Vl ate the Stevens factors of the second-, fourth- and sixth-order, respectively. The values for (r2), (r~f) and (r 6 ) have been taken after Freeman and Desclaux (1979). The multipole moments values enter into the relations between the CF parameters B m and the CF coefficients Anm. *-values for Pr have been deduced from interpolation. After Franse and Radwanski (1993).
R
S
I~> (~02)
I~> (%,)
I~> (~g)
Ce Pr Nd Sm Gd Tb Dy Ho Er Tm Yb
5/2 4 9/2 5/2 7/2 6 15/2 8 15/2 6 7/2
1.309 1.208" 1.114 0.9743 0.8671 0.8220 0.7814 0.7446 0.7111 0.6804 0.6522
3.964 3.396* 2.910 2.260 1.820 1.651 1.505 1.379 1.270 1.174 1.089
23.31 18.72" 15.03 10.55 7.831 6.852 6.048 5.379 4.816 4.340 3.932
~,l~> z,l~,'~> ,,/~4~,> (10-3%2) (10-,%,) (10-6:) -74.8 -25.54 -7.161 +40.209 0 -8.303 -5.020 - 1.676 +1.831 +6.976 +21.04
+251.7 -24.95 -8.471 +56.527 0 +2.021 -0.891 -0.459 +0.564 +1.916 -18.857
0 +1141.66 -570.96 0 0 -7.683 +6.260 -6.959 +9.969 -24.33 -581.94
in which expression terms related to the 4f ion, On(r'~f), and the term related to the surrounding charges, AT, are separated. The coefficients A T are known as the crystal-field coefficients. Values for On(r~f) have been collected by Franse and Radwanski (1993) and presented in table 3.1. The computation of the CF coefficients, AT, from microscopic, ab initio, calculations is a difficult problem. A full band-structure calculation of the charge distribution over the unit cell and, consequently, of the full set of CF coefficients, is lacking for almost all compounds. In some cases, the point-charge (PC) model, with electron charges centred at the ion positions in the lattice, can give the correct sign of the leading second-order crystal-field coefficients. However, this model is questioned, especially in metallic systems where the contribution of valence electrons is expected to be significant (Schmitt 1979a, b). This has been confirmed by band-structure calculations by Coehoorn (1991) who concluded that the second-order CF coefficient A ° is mainly determined by the asphericity of the valence-shell electron density of the rare earth under consideration. In band-structure calculations for GdCo 5 Coehoorn and Daalderop (1992) have found that, although the on-site contribution to A ° is the dominant effect, the lattice contribution cannot be neglected. The calculation predicts the correct sign and order of magnitude for A ° but the calculated value exceeds the experimental one by a factor of four. This discrepancy remains to be explained. For more detailed discussion we refer the reader to the reviews of Franse and Radwanski (1991, 1993), Givord and Nozieres (1991), Purwins and Leson (1990). There is only a limited number of compounds for which quantitative data for the CF interactions are available. Discussions are still continuing even for the bestknown systems like the cubic Laves-phase RT2 or the hexagonal RNi5 compounds. The situation becomes more complex for the systems with a lower crystal symmetry.
424
N.H. LUONG and J.J.M. FRANSE
The orthorhombic RCu2 compounds, which we are dealing with in this chapter, belong to these latter cases. For cubic symmetry (in case of the RA12 compounds, for instance) the crystal field is described by only two parameters B4 and B6: n c F - B 4 ( O ° -~- 5 0 4) -+ B 6 ( O ° - 21064),
(3.5)
whereas for the orthorhombic symmetry, nine crystal-field parameters B~ are needed: 00
22
00
22
H ~ = B202 + B202 + B404 + B404 + oo
+ B 6 06 +
22 B 606
44
66
44 B404+
(3.6)
+ B6 06 + B 606 •
3.2. E x c h a n g e interactions
Since the 4f electrons are strongly localized, direct exchange of the 4f wave functions of different rare-earth ions is precluded. In order to account for the observed ordering temperatures, one has to resort to an indirect exchange between the rare-earth ions via the conduction electrons. This interaction is known as the Ruderman-Kittel-KasuyaYosida (RKKY) interaction (Ruderman and Kittel 1954, Kasuya 1956, Yosida 1957). Within this approach, a spin of a rare-earth ion interacts with the conduction electrons. The effect of this interaction on the conduction electron density is not the same for the spin-up and for the spin-down electrons. Therefore, a spin polarization is created that interacts with the spin of a neighbouring rare-earth ion. The result is an indirect exchange interaction between the rare-earth ions. The RKKY model does not distinguish d- and s-conduction electrons, in contrast to the model of indirect exchange proposed by Campbell (1972) in which the role of the rare-earth 5d electrons is emphasized. Usually, in the RKKY model the simplifying assumption is made that the exchange integral is of the s-f type and constant and that the Fermi surface is spherical (free electrons). Efforts have been made to overcome the shortcomings of the RKKY model (Kirchmayr and Poldy 1979), but in many cases the improvements achieved are minimal. We list in the following some references in which the indirect exchange interaction in rare earths and rare-earth intermetallic compounds is described: Taylor (1971), Taylor and Darby (1972), Coqblin (1977), Kirchmayr and Poldy (1979), Bnschow (1980), Givord and Nozieres (1991), Franse and Radwanski (1993). The exchange interaction is assumed to be of the Heisenberg type: H e x = --
St,
(3.7)
i,j where the summation is over all the magnetic ions in the lattice and where J q is the exchange parameter between local spins at the sites i and j. Limiting ourself to the ground state, we can project the spin onto the total angular momentum and obtain Hex = - E i,j
JRR(gJ -- 1)zJ'J,
(3.8)
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
425
where J ~ is an effective exchange parameter between the 4f-spins of the rare-earth ions at sites i and j. In the RKKY model, the ordering temperature is expected to be proportional to the so-called De Gennes factor (g - 1)2j (J + 1). In the molecular-field approximation, the exchange interaction of a single R ion with other R ions is approximated by an effective molecular field B,~. Then one has:
Hm = -gj#BJ.Bm.
(3.9)
The field B,~ is related to the molecular-field coefficient naa by
B,,, = ~,,~R(M),
(3.10)
where "N = 2 ( g j - 1)/gj
(3.11)
and (M) is the thermal average of the magnetic moment. Taking into account the Zeeman magnetostatic energy term caused by an external magnetic field B, the Hamiltonian of the magnetic interactions is given by
Hm = --gJ#BJ" (Bin + B).
(3.12)
Thus, the ground state is described by the total Hamiltonian 6
n
HI~ = ~ , ~
B~O~ - g~u~J.(n,,, + n).
(3.13)
n=0 m=0
Eigenvalues and eigenfunctions are obtained from the diagonalization of this Hamiltonian, provided that the relevant CF and exchange field parameters are known.
3.3. Description of magnetic properties In the following we describe some important magnetic properties which are affected by the crystal-field and exchange interactions and mention some properties which are of help to get information on the energy splitting scheme of the rare-earth ion.
3.3.1. Magnetic moment For describing the magnetic moment of a rare-earth ion one has to calculate the thermal average of the magnetic moment. This quantity is given by: 2J+l
(u) = ( i / z ) ~
~ exp(-E, IkT),
(3.14)
i=1
where #i is the magnetic moment of the ith energy level,
#i = -gJ#B(i~]]i)
(3.15)
426
N.H. LUONG and J.J.M. FRANSE
and where Z is the partition function: 2J+l
Z = E exp(-Ei/kT).
(3.16)
i=1
Ei and Ii) are the eigenvalues and eigenfunctions of the Hamiltonian (3.13). Due to the complicated magnetic structure, the analysis of the magnetization process in case of the RCu2 compounds has been done by decomposing the structure into several sublattices (see, e.g., Iwata et al. 1987, 1988, 1989, Divis et al. 1987, Kimura 1985, 1987a, b). In this way the molecular field Bin(i) acting on the ith sublattice can be written as:
Bin(i) : E nq(Mj), J
(3.17)
where (Mj) is the sum of the thermal average of the magnetic moments per ion in the jth sublattice, nq the molecular field coefficients describing the interaction of ith and jth sublattices. The summation is over all sublattices.
3.3.2. Paramagnetic susceptibility In the paramagnetic region the susceptibility can be described by a Curie-Weiss law: C X - T - 0p'
(3.18)
where C is the Curie constant and Op the paramagnetic Curie temperature. Sometimes, the paramagnetic susceptibility can better be represented by a modified CurieWeiss law in the form: C X = X0 + T - 0p'
(3.19)
where the extra term Xo is a weakly temperature-dependent Pauli susceptibility. Using the eigenvalues Ei and appropriate eigenfunctions Ii) of the crystal-field Hamiltonian, HCF, one can calculate the paramagnetic susceptibility, X~F), along the principal axes (a = a, b, e) by the general Van Vleck formula (see, e.g., Zajac et al. 1987):
- ~
~ ( I(ils~li)12÷ (3.20)
+ 2kT ~j¢, ](JlJ~'li)12 -~j---~i j exp(-Ei/kT),
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
427
where J,~ are the components of the operator.J. For a polycrystalline sample the susceptibility arising from crystal-field Hamiltonian, X~, is obtained by the approximation: =
+
+
(3.21)
The crystal-field parameters can be derived from the paramagnetic Curie temperatures 9a, 8b and 8¢ along each principal axis. According to the modified molecular-field model given by Bowden et al. (1971), these paramagnetic Curie temperatures can be expressed in terms of the crystal-field parameters by the following relations (Shohata 1977): Oa = ep +
(2J
1)(2J + 3) (B ° + 10
0b = 0p + ( 2 J - 1)(2J + 3)BO,
5
(3.22)
0c = 0p + (2J - 1)(2J + 3) (B o _ Bo),
10 where B ° and B22 are measured in kelvin. On substituting the measured 0a, 0b and 0e values, the crystal-field parameters B ° and B22 can be derived. The influence of higher-order crystal-field parameters on the paramagnetic Curie temperatures has been studied by Zajac and Maczak (1985). 3.3.3. Specific heat The specific heat is written as the sum of electronic (ce), lattice (phonon) (Cph), magnetic (Cm), and nuclear (e,) contributions: C:
Ce ' + C p h + Cn - ] - c m.
(3.23)
The nuclear part of the specific heat is related to the hyperfine interactions of the 4f shell with the nuclear moment of the 4f ion and is only significant at low temperatures for most of the 4f ions. The largest nuclear contribution is observed in the ordered Ho compounds. This is also the case for the HoCu2 compound (Luong et al. 1985b). The electronic part is written as: ce = 7T
(3.24)
with the coefficient 7 yielding information on the density of states at the Fermi level. The experimental value for the electronic coefficient 7 from specific-heat measurements should be compared with -/-values from band-structure calculations. For the RCu2 compounds, band-structure calculations have been performed for YCu2 (Harima et al. 1990, Harima and Yanase 1992). The resulting value for the specific heat coefficient 7 is 3.47 mJ/K2 mol, which is about two times lower than the experimental value of 6.7 mJ/K2 mol (Luong et al. 1985a).
428
N.H. L U O N G and J.J.M. FRANSE
The phonon part of the specific heat (for a compound with r atoms per formula unit) is approximated by: Cph = 9 r R ( T / O D ) a
00D/T z4ez
(e~ _ 1)2 dz,
(3.25)
where OD is the Debye temperature, R the gas constant. In compounds with a magnetic R element the magnetic contribution, Cm, is associated with the increasing population of excited localized states. These localized states are due to CF and molecular field interactions of the 4f ion which lift the ( 2 J + 1)-fold degeneracy of the ground-state multiplet. Provided that the energy-level scheme is available, it is straightforward to evaluate this contribution by using the expression:
cm(T)
(3.26)
-02F = - - T OT2,
where F is the free energy of the R system. The reverse procedure is rather problematic. Contributions like the electronic and lattice contributions have to be evaluated and subtracted. However, in general it is difficult to separate the magnetic part from the other contributions. Isostructural compounds with non-magnetic R elements are usually employed to estimate the non-magnetic part of the specific heat. In the RCu 2 compounds, LaCu2 possesses a different crystallographic structure and crystallizes in the hexagonal AIB2-type of structure. Therefore, YCu2 is taken as the non-f reference material instead of LaCu 2 and data for YCu 2 are often used in the analysis of the experimental data on RCu 2 compounds. In the paramagnetic state the magnetic part of the specific heat is given by the Schottky contribution (Fulde 1979): CSeh = ~
Z
exp(-Ei/kr)
-
Ei e x p ( - E i / k T )
.
(3.27)
In the simple case of a two-level system, the Schottky specific heat attains a maximum value at an intermediate temperature Tm given by (Gopal 1966):
(g0/al) exp(6/Tm) = [(6/Tm) + 21/[(6/T.) - 21,
(3.28)
where 6 is the energy separation (measured in kelvin) between the ground state and the first excited state; go and gl denote the degeneracies of these two levels. Provided that the temperature Tm and the maximum value of the specific heat, cSch(Tm), are known, one can evaluate 6 according to the expression (Gopal 1966): 6 = [4CSch(Tm)/R + 411/2Tm.
(3.29)
In general, relevant information about the lower part of the energy-level scheme can be obtained from the specific-heat measurements. Experiments in external magnetic fields can give further information as the applied field additionally shifts the energy levels. The shift is not uniform for all levels and is largely dependent on the direction of the applied field. This effect can be detected by the measurements of the specific heat on single-crystalline samples.
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
429
3.3.4. Thermal expansion The thermal expansion contains electronic (fie), lattice (phonon) (flph), and magnetic, (tim) contributions:
= Ze + Zph + t - ,
(3.30)
where we neglect a nuclear contribution. The conduction electrons contribute to the thermal expansion through the volume dependence of their entropy. This entropy is related to the density of state at the Fermi surface. The electronic part of thermal expansion is a linear function of temperature, i.e.
fie = aT.
(3.31)
The phonon contribution to the thermal expansion, like the contribution to the specific heat, is approximated by: ~ph = b(T/OD)3
fOOD/T z4ex (e~'_~])2
dx,
(3.32)
where b is a numerical constant. In order to evaluate the magnetic contribution to the thermal expansion, nonmagnetic (electronic and lattice) contributions have to be subtracted from total thermal expansion as in the case of the specific heat. Isostructural compounds with nonmagnetic R elements are used to evaluate this non-magnetic part of the thermal expansion. For RCu 2 compounds, like in case of the specific heat, YCu 2 often serves as the reference material. In the paramagnetic region, the magnetic part of the thermal expansion arises from the thermal excitation of a series of energy levels E0, E b . . . with degeneracies go, g b . . . . This (Schottky) contribution is given by (Barron et al. 1980):
2
(Ei) (E¢/i) },
(3.33)
where 7i = - ( d In Ei/d In V) is the crystal-field Grtineisen parameter of the individual energy level Ei and where ~T is the isothermal compressibility. The brackets denote thermal averages of the form:
(x,) - E glz, e x p ( - E j k T ) / E g' exp(-EjkT). i
(3.34)
i
3.3.5. Graneisen analysis In the study of magnetic systems the specific heat and the thermal expansion are very important. The combined analysis of specific heat and thermal expansion can give valuable information on the system under consideration. Here we briefly describe a procedure that has successfully been applied to several different systems.
430
N.H. LUONG and J,J.M. FRANSE
An arbitrary contribution to the specific heat, el, is related to a corresponding contribution to the thermal expansion, ti, by a so-called Griineisen relation:
.l-'i = Vti/tgei,
(3.35)
where V is the molar volume, t¢ the compressibility and/~i the appropriate Griineisen parameter. For the electronic Griineisen parameter we have (see eqs (3.24) and (3.31)):
-re = V a / x T .
(3.36)
Using eqs (3.25) and (3.32), for the lattice Griineisen parameter we obtain:
l"ph = V b / 9 r Rt¢.
(3.37)
In treating the magnetic contributions to the specific heat and to the thermal expansion, a straightforward approach is to calculate an effective Griineisen parameter, Pen, by the relation: Feff = V t m / t C e m
(3.38)
and to follow its variation with temperature. A pronounced temperature dependence of Fee(T) indicates the presence of several contributions. ErCu2 can serve as an example, in which a change in sign in the parameter Feff is observed upon increasing the temperature (see section 4.9). In this case, at least two different contributions to em and tim can be distinguished. Therefore, we write: e m = err + Cef and
t m = t l r "lt- t e f
(3.39)
in which the Cm and tm are split up into two contributions, the 'long-range' magneticorder contributions % and tmr and the contributions eel and t~f associated with the crystal-field splitting of the energy levels, respectively. We have:
fiFi,
Feff= E
i = lr, cf,
(3.40)
i
with F~ and F~f, in first approximation, temperature-independent Grtineisen parameters and with fi = ci/em. The separated terms in Cm and tm can subsequently be written as: /"lr -- Fef cm'
(3.41)
ccf - /'lr - Fee era,
(3.42)
Clr =
1 - F~f/Fe~
ttr -- ~ - ~
tin,
F~f 1 - F e f d r t r
tef = Fe----ff~ -- ~
(3.43) tim.
(3.44)
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
431
Fransse et al. (1985) and Luong et al. (1985b) applied this method in the analysis of the ErCu2 compound (see section 4.9). The method of the combined analysis of specific heat and thermal expansion has been described in more detail by Brommer and Fransse (1990) and successfully applied to a variety of materials (see Brommer and Fransse 1990).
3.3.6. Inelastic neutron scattering Inelastic neutron scattering (INS) gives valuable information on the dynamic properties of the system. INS techniques enable to determine the crystal field scheme of the ground state multiplet of a R a÷ ion as the available neutron energies are of the order of this splitting. The scattering amplitude S(q, w) as function of the momentum transfer q and the energy transfer hw, depends on the matrix element and is given by:
S(q, w) ,~ I(il&LIf)[ z,
(3.45)
where Ja. is the component of the total angular momentum operator perpendicular to q. (i[ and If) denote the initial and final 4f-electron states (= CF levels), respectively. Within the dipole approximation, the intensities for the transitions are determined by the corresponding dipole matrix elements. For a review of the INS spectra in metallic 4f systems see, for instance, Fulde and Loewenhaupt (1985). The INS spectra of a system containing 4f ions depend very much on the relative strength of the CF and exchange interactions. The spectra belonging to only CF transitions contain a few well-defined peaks related to magnetic excitations between CF levels. The number of the transitions as well as the probability for these transitions is determined by matrix elements between the crystal-field eigenstates. In principle, these are governed by the local symmetry of the CF interactions. These transitions are typically single-site excitations. Including the exchange interactions, the transitions on neighbouring ions are coupled and lead to propagating modes. The spectra become more complex and the analysis is not straightforward. For RCu2 compounds, and NdCu 2 is a good example, at least two INS experiments (one above and one below antiferromagnetic ordering temperature) are needed to detect deviations from the CF-only case in the excitation spectrum (Loewenhaupt 1990). For orthorhombic and lower symmetries, the J ground-state multiplet splits into (2J + 1)/2 doublets for half integer values of J and into (2J + 1) singlets for integer d. In the simplest case (Ce 3+, J = 5/2) this gives a level scheme with 3 doublets and 3 possible inelastic transitions. In the most complicated case (Ho 3+, d = 8) a level scheme consists of 17 singlets and up to 136 inelastic transitions are possible (Loewenhaupt 1990). Loewenhaupt (1990) has also pointed out that not all transitions may have non-zero matrix elements but already the clear identification of 10 or 20 inelastic CF transitions is a great challenge for magnetic neutron spectroscopy.
3.3.7. MOssbauer spectroscopy Mrssbauer spectroscopy provides information on at least two important parameters: eft, and the quadrupolar splitting. the hyperfine field, Hhf
432
N.H. LUONG and J.J.M. FRANSE
The hyperfme field is used to evaluate the value for the magnetic moment of the unfilled 4f or 3d shell, usually by taking it directly proportional to the magnetic moment (as long as the additional contributions originating from core polarization and conduction electrons can be neglected). In that case, the temperature dependence of H ~ reflects the temperature dependence of the (4f or 3d) magnetic moment. The quadrupolar splitting gives information about interactions between the quadrupolar moment of the nuclei and the gradient of the electric field. The gradient of the electric field at the rare-earth nuclei results partly from the quadrupolar moment of the 4f shell and partly from that of the surrounding charges. Having an estimate of the 4f-shell contribution, the quadrupolar splitting arising from the lattice-charge arrangement can be evaluated and transformed into the crystal field coefficient A °. Higherorder CF coefficients cannot be determined by the Mrssbauer spectroscopy because of the much lower intensity of the higher-order multipolar interactions. Mrssbauerspectroscopy data for the RCu2 compounds have been presented by Gubbens et al. (1991, 1992).
4. Magnetic properties of RCn2 compounds 4.1. CeCu2 CeCu2 is a Kondo-lattice compound which shows antiferromagnetic order below 3.5 K (Gratz et al. 1985, Onuki et al. 1985). Earlier magnetic measurements have been reported by Sherwood et al. (1964), Olcese (1977) and Hashimoto et al. (1979b). Owing to the interesting magnetic behaviour, the compound has been extensively investigated. A characteristic feature of CeCu2 is the large value for the coefficient of the electronic contribution to the specific heat (between 80 and 105 mJ/K2 mol) (Gratz et al. 1985, Bredl 1987, Bauer et al. 1988, 1989 and Takayanagi et al. 1990). The specific heat of CeCu2 as a function of temperature is shown in fig. 4.1 (Gratz et al. 1985). Althouth magnetic ordering is observed in properties as specific heat and resistivity, a clear antiferromagnetic transition is not found in the low-temperature magnetic susceptibility (Onuki et al. 1990, Trump 1991 and Trump et al. 1991). Figure 4.2 shows the temperature dependence of the magnetic susceptibility in CeCu2 (Onuki et al. 1990). As can be seen from this figure, the susceptibility along the c axis shows a broad maximum near the Nrel temperature, indicating that the magnetic moments are along the e axis, in agreement with the neutron diffraction results of Trump et al. (1991) and Nunez et al. (1992) (see below). The magnetic susceptibility is highly anisotropic over the whole temperature range as revealed from the experiments (Onuki et al. 1985, Takayanagi et al. 1986, Onuki et al. 1990, Trump et al. 1991). However, the sequence of easy-intermediate-hard directions in CeCu2 is not quite clear. From magnetic measurements, Gratz et al. (1985) have determined an effective moment, #etI, of 2.45#B and a paramagnetic Curie temperature, 0p, of 30 K. The magnetization of CeCu2 has been measured on a polycrystalline sample by Hashimoto et al. (1979b) and by Gratz et al. (1985) and on a single-crystalline
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
ceCuz
_
0.5t-
_
433
Z
-,=
¢.i
2
0
0
1
2
3
/,
5
6
T (K) Fig. 4.1. Specific heat of CeCu 2 as a function of temperature. Inset: low-temperature data in a plot of e/T versus T2; the full curve represents e/T = "3'+ AT2exp(-AE/kT) with 7 = 82 mJ/K2 mol, A = 1.4J/K4 tool and AE/k = 1.2K. After Gratz et al. (1985).
0.3
..3 0
--
0.2
CeCu2
==
°~%,,o
QU
",o
0.1 c
0
i
i
J
5
10
T (K) Fig. 4.2. Temperature dependence of the magnetic susceptibility in CeCu 2. After Onuki et al. (1990).
434
N.H. LUONG and J.J.M. FRANSE
2.0 CeCuz
a
'6 ~
1.0
0
0
~
8
B(T) Fig. 4.3. Magnetization along each principal axis in CeCu 2 at 1.3 K. After Onuki et al. (1990).
sample by Onuki et al. (1988, 1990) and by Trump et al. (1991). Figure 4.3 shows the magnetization of single-crystalline CeCu2 at 1.3 K (Onuki et al. 1990). At a field of 1.7 T, the magnetization along the a axis exhibits a change of slope, indicating a metamagnetic phase transition. The magnetization is again highly anisotropic. Trump et al. (1991) have reported that a metamagnetic transition occurs along the a axis at a field of 1.1 T at T = 2 K. The metamagnetic behaviour of CeCu2 has received considerable attention (Gratz et al. 1985, Onuki et al. 1988, 1990, Satoh et al. 1990a, Takayanagi et al. 1990, 1991, Trump et al. 1991). Gratz et al. (1985) have reported neutron-diffraction studies on CeCu2. However, the exact nature of the magnetic ordering could not be determined. Later on, the magnetic structure of CeCu2 was studied by Trump et al. (1991) and Nunez et al. (1992). Figure 4.4 shows the magnetic structure of CeCu2 (Trump et al. 1991). In this structure, the magnetic moments of the Ce atoms are parallel and antiparallel to the c axis (Trump et al. 1991 and Nunez et al. 1992). The magnetic moment of the Ce atoms amounts to 0.33#n at 2.5 K (Nunez et al. 1992). Crystal-field effects in CeCu2 have been studied by different methods. From inelastic neutron-scattering experiments, Loewenhaupt et al. (1988) deduced excited levels at 9 meV (105 K) and 23.1 meV (268 K). Morin et al. (1992) have observed one CF transition at 23.4 meV. This value is very close to that obtained by Loewenhaupt et al. (1988). Morin et al. (1992) reported, however, that there does not exist another clear excitation from the ground state. Uwatoko et al. (1990a,b, 1991, 1992) have measured the thermal expansion of single-crystalline CeCu2. Assuming that the phonon contribution to the thermal expansion of CeCu2 equals that of YCu2, the
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
435
%c°I i! t.J
<---- a = 4.t.2 ~, ----> Fig. 4.4. Proposed magnetic structure of CeCu 2. After Trump et al. (1991).
magnetic contribution to the thermal-expansion coefficient of C e C u 2 , av(mag), has been estimated as: o t v ( m a g ) = c z v ( C e C u 2 ) - o~v(YCu2).
(4.1)
Figure 4.5 shows the temperature dependence of C~v(mag). In this figure, the calculated magnetic thermal-expansion coefficient of CeCu2 is also shown (see Uwatoko et al. 1991, 1992 and references therein). The level scheme used in these calculations is t/71 = 106 K and E2 = 334 K, which values are in reasonably good agreement with whose obtained by Loewenhaupt et al. (1988) from neutron scattering and by Loewenhaupt et al. (1990) from thermal-expansion measurements. However, the level scheme E1 = 10 K and t/72 = 800 K which has been used by Takayanagi et al. (1986) for describing the temperature dependence and the anisotropy of the magnetic susceptibility of CeCu2, disagrees with the neutron-scattering results. From de Haas-Van Alphen studies, Satoh et al. (1990b) have shown that the cyclotron effective masses of CeCu2 range from 0.5 to 5.3m0, which values are larger than those found in YCu2 (0.1--0.7m0). This underlines the heavy-fermion state of CeCu2. The light masses of Y C u 2 a r e consistent with the small specific-heat coefficient 7= 6.7 nd/K 2 mol in YCu 2 obtained by Luong et al. (1985a, b). The results of de Haas-Van Alphen measurements of Settai et al. (1990, 1992) agree with those obtained by Satoh et al. (1990b).
436
N.H. LUONG and J.J.M. FRANSE
12 lo 8
CeCuz
o ~
4
-2 0
I
I
100
I
I
200
Fig. 4.5. Temperaturedependenceof the magnetic part of the thermal-expansion coefficient,c~v(mag) = av(CeCu2)- txv(YCu2). The circles representthe measured thermal-expansioncoefficient. The solid line is a calculatedresult for the magnetic thermal-expansion coefficient. After Uwatokoet al. (1991). The effect of substituting Y for Ce in CeCu 2 has been studied by Bauer et al. (1989). Long-range magnetic order in CeCu2 rapidly disappears with increasing Y content. The replacement of Cu by Ag in CeCu2 leaves the orthorhombic structure unchanged (Iandelli and Palenzona 1968). Bauer et al. (1990) and Bauer (1991) have substituted Ag for Cu in CeCu2. On both the Cu-rich and Ag-rich sides longrange magnetic order is observed. In the middle of the concentration range some sort of spin-freezing phenomena is supposed to be present (Bauer 1991). The series Ce(Cu~,Gal_x)2 (0.5 x< z x< 1) has been studied by Bauer et al. (1988) and Bauer (1991). The transition temperature shifts from 3.5 K for z = 1 to about 1.9 K for z = 0.95. For higher Ga contents no indications of long-range magnetic order were resolved down to 1.5 K. The magnetic properties of Ce(Znl_~Cu~)2 have been investigated by Morin et al. (1992). When Cu is replaced by Ni in Ce(CUl_~Ni~)2 the orthorhombic phase is found in the range 0 ~< x ~< 0.5 (Olcese 1977).
4.2. PrCu2 Magnetic-susceptibility measurements performed on polycrystalline PrCu2 by Andres et al. (1972) (fig. 4.6) show a nearly temperature-independent Van Vleck paramagnetic behaviour below 4.2 K, confirming that the lowest crystal-field state is a singlet. The specific heat anomaly around 7 K (fig. 4.6) cannot be due to the onset of magnetic order, since there is no susceptibility anomaly around this temperature. This specific-heat anomaly will be discussed in more detail below. Andres et al. (1972) have shown that PrCu z exhibits cooperative nuclear antiferromagnetic order below 54 mK (see fig. 4.7). Using a molecular-field approximation, Andres (1973) has analyzed the nuclear magnetic order in PrCu2. The calculated ordering temperature is between 10 mK and 1 K. The magnetization at 4.2 K and the susceptibility along each principal axis for a single-crystalline sample has been measured by Hashimoto (1979) and Hashimoto
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
437
50 ~0
1.5
m: iJ
30
"-6 E
20
"3"
10
0
10
20
30
40
T(K) Fig. 4.6. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 above 2 K. After Andres et al. (1972).
.15
.14 T
1.0
w
.13 E
;~ .12
.11 '
~ .I
~' .2
21 .3
i
o--- 0
.4
T (K) Fig. 4.7. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 below 0.5 K. After Andres et al. (1972).
et al. (1979b) (figs 4.8 and 4.9). The results show a large anisotropy in the paramagnetic state. The sequence of the easy to hard magnetization direction at low temperatures is the a, b and e axes, which is different from that obtained at high temperatures: the a, c and b axes. The effective magnetic moment,/ten, amounts to 3.4#a and the paramagnetic Curie temperatures along the a, b and c axes are: 0a = 14 K, Ob = --107 K and Oe = - 3 2 K (Hashimoto et al. 1979b). From these values for the paramagnetic Curie temperatures Hashimoto et al. (1979b) have estimated the second-order crystal-field parameters for PrCu2: B ° -- 4.27 K, and B 2 = 2.97 K. By using a point-charge model, Hashimoto et al. (1979b) have also
438
N.H. LUONGand J.J.M. FRANSE
2
PrCuz Q
I
b c
O0
1
5
3 B (T)
Fig. 4.8. Magnetization curves along the principal axes for PrCu 2 at 4.2 IC After Hashimoto et al. (1979b).
10 x104" PrC u2
8
c
"T
~6 E
7o~4
,," ""
0
I
I
I
100
200
300
T (K) Fig. 4.9.
Temperature dependence of the inverse susceptibility along the principal axes of PrCu 2. After Hashimoto et al. (1979b).
calculated the values of B ° and B 2 and arrived at B ° c a l = 4.1 K and B E c a l = 3.48 K, in good agreement with experiment. Wun and Philips (1974) have also measured the specific heat of PrCu2 (fig. 4.10). They have shown that the 7 K anomaly is much sharper than expected for a twolevel Schottky anomaly. The shape of the anomaly cannot be produced by fixed crystal-field levels, but suggests, instead, that it arises from a cooperative transition. These authors attributed this anomaly to Jahn-Teller distortions. The microscopic picture of this class of structural transitions involves phonon-mediated quadrupole interactions between rare-earth ions (Kjems et al. 1978).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
439
12 10
* -
PrCuz MoteculnrFietd /
........ Schoffky
,7
Cuz
--.--Lo
1
oo07
0
I./
O
.-"'"
O O
... .,'"
•/"
5
10
TIK) Fig. 4.10. The zero-field specific heat of PrCu 2 between 0.3 and 10 IC After Wun and Philips (1974).
Thermal-expansion, susceptibility, resistivity and thermal-conductivity measurements performed by Andres et al. (1976), Ott et al. (1977) on single-crystalline PrCu 2 have confirmed that PrCu2 exhibits a cooperative Jahn-Teller effect at TD = 7.5 K. Figure 4.11 shows the temperature dependence of the linear thermal-expansion coefficients, ~i, along the three principal axes in PrCu2 (Ott et al. 1977). Distinct extremal values of oq are observed around 7.5 K giving evidence for an actual distortion of the crystal lattice. The structural distortion below TD has also been measured by neutron diffraction (Kjems et al. 1978). Inelastic neutron scattering in single-crystalline PrCu2 has been performed by Ott et al. (1978) at temperatures below and above the Jahn-Teller transition temperature. It has been shown that the most intense transition occurs at 1.3 meV (15 K). It is assigned as a transition from the ground state to an excited state. The effect of the crystal field on the paramagnetic susceptibility of PrCu2 has been studied by Zajac (1981). In neutron-diffraction measurements, Kawarazaki et al. (1984) and Nicklow et al. (1985) have observed magnetic order in PrCuz below 58 mK. These authors have also reported the magnetic structure of PrCu2 (fig. 4.12). Nicklow et al. (1985) have pointed out that the magnetic structure (electronic + nuclear) is incommensurate with the chemical lattice. Further neutron-diffraction experiments performed by Kawarazaki and Arthur (1988) (part of this work has been published earlier, see Kawarazaki and Arthur 1986) reveal that in the ordered state electronic and nuclear
440
N.H. LUONG and JJ.M. FRANSE
160 140
Pr Cuz
IZO 100
•
o//a
•
e//b
I~ •
o#c
$
80
•
6O
20
-20 -40 -60
I
I
I
2
/,
6 8 T(K}
I
I
I
I
10 12 1/+
Fig. 4.11. Linear thermal-expansion coefficient oti along the principal axes in PrCu2. After Ott et al. (1977).
o~o
\~
•
q
'
o o
\
\
'\
r o Pr • ru
oo
I~
C
---I
Fig. 4.12. Projection of the PrCu2 structure onto the (a, e) plane. The orthorhombic cell is indicated by the dashed lines. The vector ~ gives the direction of propagation and wavelength of the modulated magnetic structure. The indicated diagonal planes are parallel to (103) planes. After Kawarazaki et al. (1984).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
441
PrCu2 *+#~ ~,~ 0:0^.
200
÷4 o'0*°°
÷°
~..~0 ~ ~o ° x~ v
150
E
.. B:OT + B=3 T ° B: 5 T x B=TT
o
~'~x
,,~0 #
"...........,'
Yd lOO
50 NbTi
4
5
6
7
8
9
10
T (K)
Fig. 4.13. Specific heat (per unit volume) of PrCu2 in applied magnetic fields. For comparison the specific heat (per unit volume) of NbTi and Cu is also shown. After Kwasnitzaet al. (1983). moments of the praseodymium atoms are sinusoidally modulated in magnitude and oriented approximately parallel to the crystallographic a axis. PrCu2 may find its way to the applications, for instance, by stabilization of superconductors. Kwasnitza et al. (1983) and Barbisch and Kwasnitza (1984) have measured the specific heat of PrCu2 in magnetic fields up to 7 T (fig. 4.13). The specific heat has a broad maximum around 6 K with values (per unit volume) being about two orders of magnitude larger than those of NbTi (technical superconductor) and copper. An applied magnetic field does not destroy these very high specific heat values. Thus PrCu2, like some other materials (NdSn3, Nd0.9Pr0.1Sn3 and PrB6, see Kwasnitza et al. 1983), is a good candidate for superconductor stabilization. First attempts for application were made (Barbisch and Kwasnitza 1984). These authors have pointed out that PrCu2 powder would be advantageous as a filler in the epoxy used for coil impregnation.
4.3. NdCu2 The antiferromagnetic ordering temperature of NdCu 2 compound is 6 K (Hashimoto et al. 1979b, from ac susceptibility measurements) or 6.5 K (Gratz et al. 1991, from specific heat and resistivity measurements). From specific heat and resistivity measurements (see below), Gratz et al. (1991) have observed a spin reorientation at the temperature, Ts, equal to 4.1 K.
442
N.H. LUONG and J.J.M. FRANSE
The magnetic structure of NdCu2 is complex as revealed from recent neutron diffraction study in the temperature range from 1.4 K up to 8 K in zero external field (Arons et al. 1994). These authors show that two magnetic phases were observed between 1.4 K and TN = 6.5 K. In both phases the magnetic structure can be described by assuming a sinusoidal oscillation of the Nd moments which are oriented along the b direction. In the high-temperature phase between 5.2 K and TN, the magnetic structure is incommensurate with the lattice. In the low-temperature phase (below 4 K) the structure becomes commensurable. Around 4.4 K the neutrondiffraction results suggest a coexistence of the high- and low-temperature phases. The temperature dependence of the inverse susceptibility along each principal axis of NdCu2 is shown in fig. 4.14 (Hashimoto et al. 1979b). The effective magnetic moment,/~eff, amounts to 3.5/~B and the paramagnetic Curie temperatures along the a, b and c axes are: Oa = 17 K, Ob = - 1 6 K and 0e = - 4 K (Hashimoto et al. 1979b). The second-order crystal-field parameters B2° and B22 for NdCu2 were first estimated by Hashimoto et al. (1979b) from measurements of paramagnetic susceptibility in a single-crystalline sample: B ° = 0.8 K, B22 = 1.1 K. Hashimoto et al. (1979b) have also calculated the second-order crystal-field parameters using a point-charge model and obtained: B ° eal = 1.17 K and B 2 eat = 1.01 K. Inelastic neutron scattering investigations on polycrystalline samples have recently been reported by Loewenhaupt (1990) and Gratz et al. (1991). Figure 4.15 shows the inelastic neutron spectra in the paramagnetic state for Ndcu2 (Gratz et al. 1991). These authors observed four inelastic magnetic transitions which were interpreted as crystal-field transitions from the ground-state level to excited crystal-field levels at 34, 58, 84 and 164 K (see inset in fig. 4.15). This is expected for the Nd ion in an orthorhombic symmetry: a splitting of the J = 9/2 ground state into five doublets. Taking into account the values of B ° and B22 obtained by Hashimoto et al. (1979b) 10
"T
xl0/*
8
. ~ N2d C u
_~c
~-1 6 E
~g
"To~L~ 0
0
~
I
100
I
200 T (K)
I
300
Fig. 4.14. Temperature dependence of the inverse susceptibility along the principal axes of NdCu 2. After Hashimoto et al. (1979b).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
140
120
7
100
T=IOK
K
o
Eo=12 m e V
164
x
Eo=17 rneV
*
Eo=50 rrteV
443
meg
84 80
0®
7
®xx
,. 34
60
¢lr co
OLrue' Sehe'ra.e
x
40
_J
20 0 -5
® "
0
I
[
t
5
10
15
ENERGY
TRANSFER
.. .. .. ,
20
(rrte~
Fig. 4.15. Inelastic neutron spectra of NdCu 2 in the paramagnetic state (T = 10 K) obtained with different incident neutron energies E0. The deduced crystal-field transitions are given by the level scheme in the inset. The nine crystal-field parameters Bnm have been derived with the superposition model from these data (full curve). After Gratz et al. (1991). and using a superposition model ( N e w m a n and N g 1989, Divis 1991), Gratz et al. (1991) have derived the following set o f crystal-field parameters which best describe the inelastic neutron-scattering data:
B ° = (1.35 + 0.06) K, B g = (1.56 + 0.08) K, B ° = (2.23 + 0.12) K, B 2 = (1.01 + 0.2)K, B44 = (1.96 + 0.42) K, B6° = (4.89 + 0.43) K, B 2 = (1.35 + 0.45) K,
444
N.H. LUONGand J.J.M. FRANSE
B64 = (4.89 + 0.43) K, B66 = (4.25 + 0.19)K. Magnetization isotherms of single-crystalline NdCu2 measured at 2 K by Svoboda et al. (1992) are shown in fig. 4.16. These results as well as results of Hashimoto et al. (1979b) reveal that metamagnetic transitions are observed only in the b direction, whereas in the a and e directions no transitions are detectable. Bozukov et al. (1992) have performed magnetization measurements on polycrystalline NdCu2 in magnetic fields up to 28 T (see fig. 4.17). They have found a further transition around 12 T. Above 25 T a ferromagnetic spin arrangement seems to exist. At this magnetic-field value, the magnetic moment saturates to the value of 1.9/~B/ion which is about 60% of the gJ value for the trivalent Nd ion (Bozukov et al. 1992). Magnetization measurements of Bozukov et al. (1992) at 1.8 and 4.2 K have shown that the lowest lying field-induced transition disappears at temperatures above 4.2 K. This observation indicates the existence of a spin reorientation in the temperature range 1.8-4.2 K. As mentioned above, measurements without an external magnetic field (specific heat, resistivity) revealed a spin reorientation at the temperature Ts equal to 4.1 K (Gratz et al. 1991). Figure 4.18 shows the temperature dependence of the specific heat of NdCuz and of non-magnetic isostructural LuCu2 (Gratz et al. 1991). From these measurements the N6el temperature and the spin-reorientation temperature have been determined. The electronic specific-heat coefficient 7 is estimated to be around 8-12 mJ/K2 mol, comparable with 7(YCu2) = 6.7 mJ/K2 mol (Luong et al. 1985a) and ?(TmCu2) = 9 mJ/K2 mol (Sima et al. 1988). The magnetic contribution to the specific heat, era, is deduced as: em= c(NdCu2) - c(LuCu2)
(4.2)
2.0 b
Nd Cuz
1.5 i= 1.0 0.5
0.0 0
1
2
3
t~
5
B (T)
Fig. 4.16. Magnetizationisothermsalong the principal axes of NdCu2 at 2 K. After Svoboda et al. (1992).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
445
2.0 1.6
13K 16 K
-6 1.2 l::: .~
/
//
0.8 NdCuz 0.~
~" ""
,_ o., perma.nenf field puLsed field I
0.0
I
S
0
l
I
I
15 B (r)
25
Fig. 4.17. Magnetization curves of NdCu 2 at various temperatures in pulsed fields up to 28 T (solid curves) and in steady fields up to 14 T (dashed curves). After Bozukov et al. (1992).
50
ooO°~
NdCu2 o°°°° "*"t"
L,O
0 O0
O
o
,0°
oo
20 r L
*****
,
e"
e,~
0°1
eeeee
°oo°
30
/
tl°
LuCu2
_."
oO°°°~"/
10
0
10
20
30
,
,
~0
50
60
T (K) Fig. 4.1& Temperature dependence of the specific heat of NdCu 2 and L u C u 2. After Gratz et al. (1991).
and is given in fig. 4.19 as a function o f temperature, cm does not vanish above Try, but shows a broad and pronounced m a x i m u m around 20 K. In this figure, the calculated Schottky a n o m a l y as obtained by using the crystal-field eigenvalues Ei f r o m the neutron experiments is included (Gratz et al. 1991). The curvature of em above
446
N.H. LUONGand J.J.M. FRANSE
10
(experiment)
cm
i 0-~0"0-0-0.0 0
go 06
o
oI
E
/0
1.0
o/°"/ ./~CF ~
(theory)
t~ E t.I
/ ~i'
NdCu2
/ I
I
5
10
I
15 T (K)
I
20
25
Fig. 4.19. Temperaturedependenceof the magnetic contributionto the specific heat, era, of NdCu2, together with the calculated Sehottky specific heat, eCF, which is obtained by using the crystal-field eigenvaluesEi from neutron experiments. After Gratz et al. (1991). TN is clearly related to crystal-field effects, but there are also hints to correlations between the Nd 4f moments, because of the difference between the experimental data and the calculated Schottky peak. Gratz et al. (1991) have calculated the temperature variation of the magnetic entropy Sm. It is shown that Sm does not reach the theoretical value of R In 2 at TN, as one would expect for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet. They attributed this reduction of the entropy to short-range correlation effects in the paramagnetic state. The thermal expansion of NdCu2 has been measured by Gratz et al. (1991). The excess thermal expansion Ac~(T) was deduced by comparing the thermal expansion of magnetic NdCu2 and non-magnetic LuCu2 as: Ao~(T) = c~(NdCu2) - o~(LuCu2) ~-, Otsc(T) + ~cF(T),
(4.3)
where ase is the contribution due to spin correlations either in the ordered state or in the paramagnetic state and ~CF is the crystal-field induced contribution to the thermal expansion. Based on a model for magnetoelastic interactions described by Divis et al. (1990a), the temperature variation of ac~ was calculated and compared with Aa in the paramagnetic state (Gratz et al. 1991) (see fig. 4.20). The temperature variation of the lattice parameters a, b and c in NdCu2 in the temperature range from 4.2 K to 300 K has been reported by Gratz et al. (1993).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
__" Z~a
150 l "
T
447
100
°~CF
~
go -50 0
..~--.~.~. -~' I 50
l 100
~ 150
~ 200
) 250
300
T(K) Fig. 4.20. Temperaturevariationof the spin-dependentcontributionto the thermal expansion(symbols) in NdCu2. The full curve shows the calculatedinfluenceof the crystal field on the thermal expansion. After Gratz et al. (1991). 4.4. SmCu 2
The N6el temperature of SmCu2 compound is TN = 21.7 K as revealed from susceptibility, specific-heat and resistivity measurements performed by Isikawa et al. (1988) on single-crystalline samples. A value for TN of 23 K has been obtained by Gratz et al. (1990) from specific-heat and susceptibility measurements on a polycrystalline sample. Isikawa et al. (1988) have shown that the temperature dependence of the specific heat exhibits, apart from a peak at TN, two distinct anomalies at 9.4 and 16.5 K, while Gratz et al. (1990) did not observe the anomaly around 9 K. The effective magnetic moment, /ze~, amounts to 0.53/ZB; the paramagnetic Curie temperature, 0p, equals - 1 4 K (Gratz et al. 1990). In SmCu2, the Sm ion is in the trivalent state as is concluded from a lattice parameters comparison (Gratz et al. 1990) and from the experimental values of the susceptibility above TN, as obtained by Isikawa et al. (1988). Figure 4.21 shows the magnetization curves along the a and b axes, as obtained by Maezawa et al. (1989). It can be seen from this figure, that a spin-flop transition from the antiferromagnetic state to the ferromagnetic state occurs at 22.5 T. In fields above 23 T, the magnetic moment tends to saturate to the value of 0.3#B/ion, which is about 40% of the gJ value for the Sm 3+ ion. The temperature dependence of the susceptibility of single-crystalline SmCu 2 has been measured by Isikawa et al. (1988) and is shown in fig. 4.22. A sharp peak at TN has been found in the susceptibility curve along the b axis. It is suggested that the b axis is the easy direction and that the antiferromagnetic spin structure is collinear, with the spins aligned parallel to the b axis. Isikawa et al. (1988) did not
448
N.H. LUONG and J.J.M. FRANSE
f
SmCuz
b
/,.2K
EQJ tl
2
/
0
10
30
20
B(T) Fig. 4.21. Magnetization curves along the a and b axes of SmCu 2 at 4.2 K. After Maezawa et al. (1989).
2~L
SmCu2 000000 mOO000
1 lee
"~e~°
°eNeoeuee
° •
Io v--
0 0
I 100
I 200
300
T (K) Fig. 4.22. Temperature dependence of the susceptibility of single-crystalline SmCu 2 along the principal axes. After Isikawa et al. (1988).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
449
observe any other remarkable anomaly than that at TN in the temperature dependence of the susceptibility. These authors have also calculated the anisotropic temperature dependence of the susceptibility of SmCu2 on the basis of the crystal-field and exchange interactions. A good agreement between calculations and experiments is obtained but no explicit data for the crystal-field and exchange interaction parameters are given. Figure 4.23 shows the temperature dependence of the specific heat ep and the magnetic entropy Sm of SmCu2 (Gratz et al. 1990). Tr~ (= 23 K) and Ts (= 17 K) are the N4el and the spin-reorientation temperatures, respectively. The coefficient 7 is estimated to be about 20 mJ/K 2 mol, which value is about twice as high as the 7 values observed for the other RCuz compounds, e.g., 7(YCu2) = 6.7 mJ/K 2 mol (Luong et al. 1985a), 7(NdCu2) _~ 10 mJ/K 2 mol (Gratz et al. 1991) (see sections 4.1 and 4.3) and 7(TmCu2) = 9 mJ/K 2 mol (Sima et al. 1988). However, the value for 7(SmCuz) is four times smaller than that observed for the isostructural Kondo system CeCuz, 7(CeCu2) = 82 mJ/K z mol (Gratz et al. 1985) (see section 4.1). The spin-dependent specific heat, era, is obtained by comparing the specific heat of SmCu2 and that of non-magnetic LuCu2 as (Gratz et al. 1990): em(SmCu2) = e(SmCu2) - e(LuCu2).
(4.4)
Gratz et al. (1990) have obtained a value of 5.5 J/Kmol for the magnetic entropy Sin, which is very close to the value expected for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet (Sin = R In 2 = 5.76 J/K mol). These authors concluded that the ground state is a doublet. The magnetic entropy value estimated by Isakawa et al. (1988) is 88% of the theoretical value. The maximum temperature (45 K) reached in specific-heat measurements performed by Gratz et al. (1990) may be not high enough to observe the Schottky
SmCu2
ee
12
I
0
"6 12
,- ~
0
IIll ~
12
I
I
24 T (K]
I
I
36
Fig. 4.23. The specific heat c and the magnetic entropy Sm of SmCu2 as a function of temperature from 1.5 up to 45 K. After Gratz et al. (1990).
450
N.H. LUONG and J.J.M. FRANSE
anomaly caused by the crystal-field effect. Therefore, these authors have performed thermal-expansion measurements in the temperature range from 4 K up to room temperature. Figure 4.24 presents the coefficient of thermal expansion, a = l - 1 ( A I / A T ) , of SmCu 2 as a function of temperature. Figure 4.24 also shows of isostructural LuCu2. a(T) of SmCu2 shows a pronounced minimum near 40 K. Gratz et al. (1990) have deduced the spin-dependent contribution to the thermal expansion, am(T), for SmCu2 by subtracting a(T) of LuCu2. The results are shown in fig. 4.24. am(T) exhibits a minimum at 45 K, caused by the crystal-field effect. Assuming that the minimum in am(T) represents an effect comparable to the Schottky-type of anomaly in the specific heat (see, for instance, Franse et al. 1985, Sima et al. 1988), Gratz et al. (1990) have used the position of the temperature where the minimum occurs to estimate the splitting energy between the ground-state doublet and the first-excited doublet (see section 3). They have derived a value of about 110 K for this crystal-field splitting. Transport properties (resistivity, thermal conductivity and thermopower) of SmCu2 were measured by Gratz et al. (1990). The curvature of the spin-dependent contribution to the resistivity, Pro, indicates a crystal-field influence on the J = 5/2 ground state, although the effect of the J = 7/2 multiplet on Pm is still an open question. The high-field magnetoresistance in SmCu2 has been measured by Maezawa et al. (1986, 1989). Indications are found that SmCu2 has a complex Fermi surface. Based on the molecular-field theory and using the value of 22.5 T for the flopping field, Maezawa et al. (1989) have estimated the antiferromagnetic-exchangeinteraction parameter for SmCu2 to be 0.11 meV. This value is about half the value calculated from the NEd temperature. Based on the paramagnetic-susceptibility measurements of Gratz et al. (1990), Stewart (1992) has derived a value of 0.061 eV for the exchange interaction between the 4f and the itinerant electrons in SmCu2. 20 15 10
,:
...,,.
~5
~
,.,
. r, 4"
/
:--.
/
j mcu,_ ~,."
5
-10
',-"j
-15
0
_,-~" ~ .,-" ~ -,~ . 100
I
1'0
x.... '
I
~
--,-.-.-/ . '
/
5~/
200 T (K)
Fig. 4.24. Temperature dependence of the thermal expansion, c~, of S m C u 2 and L u C u 2 and the magnetic contribution to the thermal expansion, C~rn. Inset shows details of ~ versus T below 50 IC After Gratz et al. (1990).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
451
To our knowledge, up till now no data on the crystal-field interactions in SmCuz are available. Further experiments are needed to clarify the interesting magnetic properties of this compound. 4.5. G d C u 2
The compound GdCu2 is, by lack of an orbital moment of the 4f electrons, expected to be relatively simple as far as the magnetic anisotropy is concerned. Therefore, in the series RCu 2 this compound is the most convenient object for the study of RKKY exchange interactions. The magnetic properties of GdCu2, among other compounds, were first studied by Sherwood et al. (1964). These authors reported that GdCu2 is antiferromagnetic, with a value for the Nrel temperature equal to 41 K. Poldy and Kirchmayr (1974) have shown that the RKKY model can be successfully applied to GdCu2, i.e. the magnetic arrangement of the Gd a+ ions is due to the indirect exchange interaction via the conduction electrons. Further studies in polycrystalline samples were made on the crystallographic properties (Dwight 1980~ Borombaev et al. 1986a, b), magnetic properties (Gratz and Poldy 1977, Luong and Franse 1981, de Graaf et al. 1982, Borombaev et al. 1986a,b, Borombaev and Markosyan 1987), magnetostriction (Luong and Franse 1981), thermal expansion (Luong and Franse 1981, Luong et al. 1985a,b, Borombaev et al. 1986a,b, Borombaev et al. 1987), specific heat (Luong et al. 1985a, b), transport properties (Gratz 1981, Gratz and Zuckermann 1982) and MSssbauer effect (De Graaf et al. 1982). To our knowledge, only one work on single-crystalline GdCu2 is available (Borombaev et al. 1987). Figure 4.25 shows the results of magnetic measurements in low magnetic field of these authors. In the paramagnetic temperature region the magnetic susceptibility of single-crystalline GdCu2 does not depend on the direction of the external field and obeys the Curie-Weiss law with a paramagnetic Curie temperature, 0p, of (16 -4- 2) K and an effective magnetic moment,/~eff, of (8.14 -40.2)/~B, close to the theoretical value for the Gd 3+ ion (7.94#B). (De Graaf et al. (1982) have obtained the following values for a polycrystalline sample: 0p = 7 K a n d / ~ = 8.7/~B.) The Nrel temperature was derived as the temperature at which the magnetization exhibits a discontinuity and was found to be equal to 42 K, in good agreement with other literature data (Sherwood et al. 1964, Poldy and Kirchmayr 1974, Luong et al. 1985a, b). Interesting magnetization curves were obtained by Borombaev et al. (1987) below the Nrel temperature in high magnetic fields. Figure 4.26 shows the magnetization along the three principle axes of single-crystalline GdCu2 at 4.2 K. In the interval 6-12 T, a metamagnetic transition (in two stages) from antiferromagnetism to ferromagnetism occurs. At Bel = 6.8 T the magnetization sharply increases, reaching approximately 4#B, and then increases more slowly; at Bc2 = 9.5 T again a sharp increase of the magnetization is observed. The saturation magnetic moment is equal to (6.9 + 0.1)/~B, close to the Gd3+ moment (7.0/ZB), indicating that GdCu2 is in the ferromagnetic state in a field B > Be2. Figure 4.27 shows the magnetization curves of GdCu2 at different temperatures in relatively low fields up to 2.8 T. It can be seen from this figure that the magnetization along the b axis increases linearly with
452
N.H. LUONG and J.J.M. FRANSE
Xo(orb. unit) ~TN 200 1.5 150 m. 1.0~
TNI 0
ov -
b,0
I00 ~'a.
0.5 0
50
100
0
200
300
T {K) Fig. 4.25. Temperature dependence of the inverse susceptibility of GdCu 2 along the axes a (o), b (o) and e(o) and of the magnetization along the axis e(+) in a field of 0.15 T. The inset shows the temperature dependence of the initial susceptibility (in a field of 0.001 T). After Borombaev et al. (1987).
GdCu2 6
2
0
~"
0
i
I'"
I
I
10
5
I
I
15
B (T) Fig. 4.26. Magnetization curves along the axes a (solid line), b (dashed line) and c (dotted line) of GdCu 2 at 4.2 K. The inset shows the field dependence of the differential magnetic susceptibility along the a axis. After Borombaev et al. (1987).
field, and that the susceptibility does not depend on the temperature. However, the magnetization along the a and e axes displays a discontinuity near 1 T, in the vicinity of the N6el temperature, the magnetization in this region shows hysteresis. Above 2 T, the susceptibilities Xa and Xe are equal to the susceptibility XB. Borombaev
//JJ
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
// 0, / / / 0:////
/
b
o
0"20'~
I
I
0
I
2
I
0
453
2
1
0
I
I
1
2
B (T)
Fig. 4.27. Magnetization curves of GdCu2 along the principal crystallographic axes a (a), c (b) and b (e) at different temperatures: 1) 4.2 K; 2) 15 K; 3) 30 K; 4) 35 K. After Borombaev et al. (1987).
.o~.d'*~
0 1.0
GdCu 2
IE
04v'
~
~* ~ ~ ~ , ~*
~.~,.
___
I'---- 0.5
10
20
30
~0
50
T (K)
Fig. 4.28. The specific heat of GdCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of e/T versus T. The full curve represents the sum of the electronic and lattice contributions to the specific heat of GdCu2. The broken curve represents the specific heat of YCu2. After Luong et al. (1985a). et al. (1987) have explained the transition observed in GdCu 2 by comparing the energies o f the different magnetic phases in the R K K Y model. Their results s h o w that the presence o f a large magnetic anisotropy energy is not a necessary condition for the realization o f metamagnetic transitions. Such transitions are possible in the exchange approximation for ferrimagnetic structures which energies lie between the energies o f the initial antiferromagnetic state and the final ferromagnetic state.
454
N.H. LUONG and J.J.M. FRANSE
Figure 4.28 shows the specific heat of the compound GdCu2 at zero magnetic field and in a field of 5 T in a plot of c/T against T (Luong et al. 1985a). Non-magnetic YCu2 was used in order to separate out the magnetic contribution to the specific heat of GdCu2. A A-type of anomaly is observed around the N6el temperature TN. The value for TN has been determined from the peak in the plot of the specific heat against T and is found to be equal to 40 K, in good agreement with other literature data. Luong et al. (1985a) have also pointed out that their susceptibility experiments above 4.2 K confirm this value of TN. A second anomaly below the ordering temperature is visible in the plot of c/T against T in fig. 4.28 as a broad structure around 10 K. We attribute this anomaly, which is of a Schottky type, to the Zeeman splitting of eight-fold degenerate energy level below the N6el temperature. Finally, a sharp peak is found at 1.5 K that shifts to 1.7 K in a field of 5 T. This peak could indicate a change in magnetic structure. As mentioned before (see section 4.1), the electronic contribution to the specific heat, 7, was derived from the specific heat data of Y C u 2 and found to be equal to 6.7 mJ/K 2 mol. This number is close to the value of 8.2 mJ/K 2 mol reported for Y metal (Stewart 1983). The phonon part of the specific heat of YCu2 follows closely the Debye function, giving 0D = 236 K. Since the 7 value of 6.4 mJ/K 2 mol of Gd metal (Stewart 1983) is close to the above mentioned value for Y metal, Luong et al. (1985a) have taken the 7 value of Y C u 2 for GdCu2 and compared the specific-heat
15 o°
t +4+
~I0 E
4- 0 •
4•
w
4-
4-
E
5
X
g
C 0
g
+
o
4-
%
4-
•
O
I -°°3 %2.~, ° 10 20 30
:
% l z.O T (K)
50
Fig. 4.29. The magnetic contribution to the specific heat for G d = Y l _ x C u 2 as a function of temperature: z = 1 ( . ) , x = 0 . 8 ( + ) , z = 0.6(a), z = 0.4(X), z = 0.2(0). After Luong et al. (1985a).
M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S
455
data of YCu 2 and GdCu2 compounds well above the ordering temperature in order to derive the value for the Debye temperature for GdCu2. They obtained a value of 198 K for GdCu2. The electronic and lattice contributions to the specific heat of GdCu2 and YCu2 are represented by the full and broken curves in fig. 4.28, respectively. The magnetic part, Cm, which is obtained by subtracting the electronic and phonon parts from the observed specific heat, is shown in fig. 4.29. This figure also shows this magnetic contribution for the whole series of Gd=YI_=Cu2 compounds which have been studied by Luong et al. (1985a). An analysis of the specific heat data for G d ~ Y l _ x C u 2 has been performed in the same manner as for G d C u 2. We note that G d 0 . 2 Y 0 . 8 C I I 2 , although it has no long-range magnetic order down to 1.2 K, has a small but not negligible magnetic specific heat. From specific-heat measurements Luong et al. (1985a) derived the values of TN and 0D for the Gd=Yl_zCu 2 compounds. The thermal-expansion results for the GdzYl_~Cu 2 compounds obtained by Luong et al. (1985a) are shown in fig. 4.30 in a plot of ot (= I-1AI/AT) against T. The thermal-expansion results for non-magnetic YCu2 (fig. 4.30) are again used to separate out the magnetic contribution to the thermal expansion in the other compounds. The results are shown in fig. 4.31. The (am, T) curves behave similarly to the (Cm,T) curves (see fig. 4.29). The A-type of anomaly around TN as well as the Schottky anomaly below TN both can be observed. Luong et al. (1985a) have discussed the specific-heat and thermal-expansion results in terms of Griineisen parameters. These authors were able to determine the
3O
°o
~20
"7 Y to I o
-¢
:
•
,.k -4-+ o
++...+
el
=e ++;£
.I. .
j:#. ++ ,r ==
.
a~r ~ ' ~ ' - = " J:l~.,m.~ao"
0
-
2O
~
n
• :
.,. ~ ~,,~ t o . ~ "~ " "
+~ - + ÷ 69.~%" "r'l'++'r~ ~ ~LX~
4O
-
6O
80 T (K)
Fig. 4.30. The coefficient of thermal expansion, c~, for G d z Y I _ = C u 2 as a function of temperature: x = l ( o ) , x = 0 . 8 ( + ) , x = 0 . 6 ( a ) , x = 0 . 4 ( x ) , ar = 0.2(o), ~e = 0(--). After L u o n g et al. (1985a).
456
N.H. LUONG and JJ.M. FRANSE
30
@
~
2O
I Y
I 0
-CA: °
++ 4-+
Sg E
~I0
~°+ I
o -t~ ~ x
x+
++
oee
+
°
°
. + + +
Ot 0 a
0 0
+
o++ .:
°
•
•
~.:%N ++H-~%"°k" n
10
2O
30
40 T (K)
50
Fig. 4.31. The magnetic contribution to the coefficient of thermal expansion for GdxYI_=Cu2 as a function of temperature: :e = l(i), a: = 0.8(+), x = 0.6(o), z = 0.4(X), = = 0.2(0). After Luong et al. (1985a). electronic, lattice and magnetic Grtineisen parameters for the Gd~YI_=Cu 2 compounds. Taking the lattice Grtineisen parameter to be equal to 2, the electronic Griineisen parameter is found to be 1.7, quite close to the value Fe = 5/3 which has been obtained for d transition metals (Simizu 1974). The magnetic Grtineisen parameter is at least one order of magnitude larger than the electronic and lattice ones. The magnetic entropy, Sin, in the Gd~YI_~Cu2 compounds was found to be proportional to the number of Gd atoms and the value for S derived from the formula ,5'm -- mR In (2S + 1) is slightly smaller than 7/2 for a free Gd +3 ion (Luong et al.
1985a). Although GdCu2, as mentioned above, is a relatively simple compound, no experimental data about its magnetic structure are available. Because of the large cross section for scattering of neutrons by gadolinium, neutron-scattering studies of the magnetic structure of GdCu2 have not yet been performed. Poldy and Kirchmayr (1974) have calculated the magnetic structure of GdCu 2 in the RKKY model. According to their calculations, GdCuz has a spiral magnetic ordering: all spins in the be plane are parallel to one another, while the directions of the magnetic moments in neighbouring planes differ by 35 °. Studies on single-crystalline GdCu2 by Borombaev et al. (1987) have shown (fig. 4.27) that the curves of the magnetization of GdCu 2 along the a and c axes practically coincide with each other, but differ from
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
457
the curve of the magnetization along the b axis. These authors have explained the experimental results by assuming that at the lowest magnetic field (B < 1 T, see above) the spins of the Gd 3+ ions lie in the ac plane (the angle between the spins of neighbouring be layers is 35* (Poldy and Kirchmayr 1974)). Cu in GdCu2 has been partially replaced by Ni (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Smetana et al. 1985a, Borombaev et al. 1986a, Borombaev and Markosyan 1987), by Co and Fe (Borombaev et al. 1986b, Borombaev and Markosyan 1987) and by A1 (Borombaev et al. 1986a, Borombaev and Markosyan 1987). The aim of these replacements is to observe the effects of changing the conduction-electron concentration, since the exchange via the conduction electrons is sensitive to this parameter. In substituted GdCu2, the electron concentration is decreased with increasing Ni and Co content, whereas it is increased with increasing A1 concentration. It was assumed that Gd, Cu, Ni, Co and AI contribute 3, 1, 0, - 1 and 3 conduction electrons, respectively (Poldy and Kirchmayr 1974, Borombaev et al. 1986b, Borombaev and Markosyan 1987). The limits of structural stability were found to be 35% Ni, 10% Co, 1% Fe and 7% AI (Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). The experimental observations in GdCu2 substituted with Ni, Co and A1 have been discussed in terms of the RRKY model (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Kirchmayr and Poldy 1979, Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). 4.6. TbCu2
The compound TbCu2 has the highest value of the Nrel temperature in the series RCu 2. Early magnetic measurements of Sherwood et al. (1964) have yielded a value for TN of 54 K for this compound. A value for TN of 53.5 K, i.e. very close to that of Sherwood et al. (1964), has been obtained by Hashimoto et al. (1976) and Hashimoto et al. (1979a) on single-crystalline TbCu2. Somewhat lower values for T N of TbCu 2 have been obtained from magnetic measurements by Poldy and Gratz (1978) and by Smetana et al. (1985a) (TN = 48 K) and from specific-heat measurements by Luong et al. (1985b) (TN = 48.5 K). The early measurements of Sherwood et al. (1964) have shown metamagnetic behaviour of polycrystalline ThCu2. Hashimoto et al. (1976, 1979a) performed magnetization and magnetic-susceptibility studies on single-crystalline samples and observed anisotropies in the magnetic properties. Later, Iwata et al. (1988) have studied the magnetization process of single-crystalline TbCu 2 in the temperature range from 4.2 to 55 K. Their results are shown in fig. 4.32. For a magnetic field applied along the a axis, the magnetization is almost zero below Bc (,'~ 1.9 T) and jumps within one step to a nearly saturated value at higher field. The saturation moment per atom has been determined to be 8.8#B (Hashimoto et al. 1979a), very close to the theoretical value of 9/zB for the free Th3+ ion. Figure 4.33 shows the result of high-magnetic-field experiments on TbCu2 in a pulsed magnetic field performed by Luong and Frame (1981). Due to large magnetostrictive forces the sample was powdered and oriented in succeeding field runs. From these measurements we deduced a value of 1.9 T for the critical field Be.
458
N.H. LUONG and J.J.M. FRANSE
Observed
Calculated
°,,. . . . . . . . . . . .
10 0
2
Observed
10
/*
a
2
4.
a
6f.
::I
:
50.0K i /
..,........ *'I'°''''''''
2
4.
i
30.0K
00
2
4
81 2
c.. 2
4.
B (T)
0
'2
B (T)
o
z~
0
2
55.0K I
4-
=
:
0
50.0,K a
......c
0 L. ,.~:!:;:::': :::~': -''" '~'l 10 0 2 4,
'2
Calculated
o. . . .
4.
55.0K
..*"°'"" "'"""
.5...
::::::::::::::::::::::::::: 0 2 4 B IT)
2
4-
B (T)
Fig. 4.32. Observed and calculated magnetization of TbCu2. After Iwata et al. (1988).
]]I
f
150
100
f
"E
TbCu z
50
O I 0
I
I
10
20
30
B (T)
Fig. 4.33. High-magnetic-fieldexperiments on TbCu2 in a pulsed magnetic field. In succeeding field runs, the sample is powdered and oriented with the easy axis along the field direction. After Luong and Franse (1981).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
•
i
'r',
l!
,~i
"
II
II
ii
II
. . . . . . .
r,'-~
u
!"
. . . . . .
," A A ~
,,'~
C
"
I
Ii
V
I
,6
'
I
'
I
!
I
I
•
,J,
S"
P
J-
I iI
••
iS I
459
,4
:--
!
I
6
I
I
!
i
I
7
,
i
Fig. 4.34. Magnetic structure of TbCu 2 at 4.2 K. After Hashimoto et al. (1979a).
A first neutron-diffraction study on a powdered sample has been carded out by Brunet al. (1971). According to these authors, TbCu2 has a collinear antiferromagnetic structure in which the moments lie along the a axis. The magnetic unit cell is equal to the orthorhombic chemical unit cell along the b and c axes but tripled along the a axis. Further neutron-diffraction studies on a single-crystalline sample by Hashimoto et al. (1979a) and on a powdered sample by Smetana et al. (1983), Smetana and Sima (1985), Sima et al. (1986) and Lebech et al. (1987) have confirmed this magnetic structure. Figure 4.34 shows the magnetic structure of TbCu2 at 4.2 K. A fraction of 2/3 of the Tb magnetic moment in a given e plane is aligned along the +a direction (sublattices A, C) and a fraction of 1/3 along the - a direction (sublattice B). In the adjacent e planes, the magnetic moments of the Tb atoms are all reversed and these antiferromagnetically coupled double layers pile up in the c direction. The temperature dependence of the magnetic structure of TbCu2 has been studied by Brunet al. (1971) and Sima et al. (1986), the results of which are in good agreement with each other. Three distinct regions of magnetic order were identified (Sima et al. 1986, Divis et al. 1987). Between 47 K and the N6el temperature TN (55 K in the studied samples) (region I) the magnetic structure is longitudinally modulated with the propagation vector 1/3a*. At temperatures below 15 K (region III) all the magnetic moments are equal. Finally, for temperatures between 15 and 47 K (region II) there are Tb a+ ions with 'large' magnetic moments and with 'small' magnetic moments. The magnetic structure of TbCu2 in this intermediate temperature region is shown in fig. 4.35 (Divis et al. 1987). Figure 4.36 shows the temperature dependence of the inverse susceptibility along each principal axis of the TbCu2 compound (Hashimoto et al. 1979a). A large magnetic anisotropy in the paramagnetic region has been observed. The susceptibility along the different crystallographic axes obeys the Curie-Weiss law at high temperatures. The effective moment, #eft, amounts to 9.5#a, in good agreement with
460
N.H. LUONG and J.J.M. FRANSE
I°
i
i
iI ,
[
A
.vtl
Fig. 4.35. Magnetic structure of TbCu2 for the intermediate temperature region (15 K < T < 47 K). Numbers 1 to 12 enumerate the positions of moments in the magnetic unit cell and the arrows show their orientation and magnitude. After Divis et al. (1987).
the calculated value for the free T b 3+ ion. The paramagnetic Curie temperatures 0a, Ob and 0c along the a, b and e axes are remarkably anisotropic: 76 K, - 6 K and 36 K, respectively. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated the following values for the second-order crystal-field parameters: B ° = 1.23 K, B g = 1.23 K. Hashimoto et al. (1979a) have also calculated the second-order crystal-field parameters on the basis of the point-charge model. Their calculated values are: B ° -- 1.35 K and B 2 = 1.12 K. Experiments and calculations are in reasonable agreement with each other, indicating that the anisotropy of TbCu2 in the paramagnetic state can be explained mainly by the crystal-field effect. Measurements of the specific heat were performed by Luong et al. (1985b) at zero magnetic field and in a field of 5 T. A plot of e/T versus T at zero field is given in fig. 4.37. A A-type of anomaly is observed around the N6el temperature. Apart from this peak at Try, a broad hump was observed around 30 K and a sharp peak occurs at 2.2 K. Kimura et al. (1988) have calculated the specific heat for TbCu2 on the basis of the molecular-field theory including crystal-field interaction. In these calculations the authors have used two different exchange parameters and two crystal-field parameters, B ° and B22, as derived by Hashimoto et al. (1979a) (see above). The results are also shown in fig. 4.37. Based on the calculated temperature dependence of the magnetic contribution to the specific heat, Kimura et al. (1988) concluded that the broad anomaly observed around 30 K in TbCu2 must be attributed to the magnetic heat capacity. Luong and Fransse (1981) and Luong et al. (1985b) have measured the thermal expansion of TbCu2. The coefficient of the thermal expansion of TbCu2 is shown in fig. 4.38 as a function of temperature. Again a A-type of contribution is observed
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
10
461
x103 ~ c TbCu2
..
"7. 6
• coO°. °~
E4 X
2
100 i
•"
~"
I
I
l
200
300
T (K}
Fig. 4.36. Temperature dependence of the inverse susceptibility along the principal axes of TbCu2. After Hashimoto et al. (1979a).
1.5 TbCu2 o
1.0 0
==
,//
I--"
"3 0.5
S/
// 0
10
20
30
40
50
60
T (K) Fig. 4.37. The specific heat of TbCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T measured by Luong et al. (1985b). Lines are specific heat (at zero magnetic field) calculated by Kimura et al. (1988).
462
N.H. LUONG and J.J.M. FRANSE
15
TbCu 2
10 "7
o•
Qe•
'o
•o
•
e•
5 0
~'" I
I
I
10
20
30
I
40 r (K)
•
I
I
I
50
60
70
~ g . 4.38. The coemcientofthermalexpansion ~ r T b C u 2 as a ~nction of ~mpera~re. After Luong etal.(1985~.
around TN. Apart from this A-type of anomaly and apart from the lattice contributions, apparently a third contribution to c~ is present above 30 K. Discussions in terms of Grtineisen parameters (see section 3.3 and more details in section 4.9 for ErCuz) lead to the conclusion that for TbCu2 the same sign and approximately the same value of ~¢Fcfas for ErCu2 is observed (Luong et al. 1985b). This observation gives a natural explanation for the third contribution to the thermal expansion. Based on a molecular-field model taking into account crystal-field effects, Iwata et al. (1987, 1988) have performed a quantitative analysis of the magnetization process of single-crystalline "I'bCu2. In the analysis they used two second-order crystal-field parameters determined experimentally by Hashimoto et al. (1979a) (see above). The exchange parameters were chosen to obtain the best fit to the experimental data. The magnetization data are well explained, as shown in fig. 4.32. (We note that the Ngel temperature of their sample is Tr~ = 53.5 K.) Kimura (1985) has analyzed the magnetic structure and the magnetization process using a 12-sublattice model. The crystal-field effect on the paramagnetic susceptibility of TbCuz has been studied by Nowotny and Zajac (1985) using the general Van Vleck formula and by Kimura (1988) with the use of a high-temperature expansion method. Both these studies have shown that anisotropic behaviour of Xi (i = a, b, c) is cancelled in Xp (= (Xa + Xb + Xe)/3) for polycrystalline samples. Nowotny and Zajac (1985) have used only two lowest-order crystal-field parameters as obtained by Hashimoto et al. (1979a) (see above). The higher-order crystal-field parameters have also been taken into consideration but the effect on the paramagnetic susceptibility of RCuz was reported to be negligible (Zajac and Maczak 1985). Magnetic properties of the pseudobinary compounds Tb=YI_=Cu2 have been studied by Hashimoto (1979), Luong et al. (1982), Hien et al. (1983) and Zajac et al.
M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S
463
(1988b). When substituting Y for Tb, the antiferromagnetic interactions weaken but the magnetocrystalline anisotropy persists (Hashimoto 1979, Luong et al. 1982). A critical concentration, xc, where the magnetic ordering disappears, was taken by Hien et al. (1983) to be equal to 0.15, following data of Hien et al. (1983) and Hashimoto (1979). The latter author suggests that TN for TbxYI_~Cu2 is proportional to ~2/3 where G is an effective de Gennes factor equal to z(g - 1)2j(j + 1). However, the concentration dependence of TN expressed by the following equation: TN(
)/TN(1)
~
-
(4.5)
where TN(1) is TN(X) for x = 1, is in better agreement with the experimental data obtained by Hien et al. (1983). This concentration dependence of TN is also observed in Dy~YI_~Cu2 compounds (see also section 4.7). Tb0.rY0.4Cu2 has a collinear antiferromagnetic structure, as revealed from neutron-diffraction measurements (Smetana et al. 1983, Sima et al. 1986, Lebech et al. 1987). Hashimoto (1979) has derived values for the crystal-field parameters B ° and B22 in the TbxYl_xCu 2 compounds from values for the paramagnetic Curie temperatures obtained on single-crystalline samples. It was found that the values for B ° and B 2 change very little in the whole composition range. The magnetization process in the ordered Tb~YI_~Cu2 compounds is similar to that of TbCu2. It has been analyzed within the molecular and crystal-field model by Hashimoto (1979) in the paramagnetic state and by Zajac et al. (1988b) in the ordered state. For the crystal-field Hamiltonian only two parameters B ° and B 2 are considered in both studies. Partial replacement of Cu by Ni in TbCu2 leads to a lowering of the electron concentration and an increasing tendency to ferromagnetism (Poldy and Gratz 1978, Hashimoto 1979). This effect has been confirmed by neutron-diffraction studies (Smetana et al. 1983, Sima and Smetana 1984, Smetana and Sima 1985, Lebech et al. 1987). Smetana et al. (1985a) reported the temperature dependence of the ae susceptibility of a Tb(Cu0.77Ni0.23)2 sample. Hashimoto (1979) has derived values for B2° and B22 in Tb(cuxNil_x)2 compounds from values of the paramagnetic Curie temperatures. The magnetization process in single-crystalline Tb(Cu0.7Ni0.3)2 in high magnetic fields (up to 24 T) has been studied by Mamaguzhin et al. (1985) and Divis et al. (1989a, 1990b). In the analysis, the latter authors have taken into account higher-order crystal-field terms. 4.7. DyCu2
Measurements of the temperature dependence of the magnetization have yielded a value for TN of 24 K for the compound DyCu 2 (Sherwood et al. 1964). A value of 31.4 K for TN has been obtained from susceptibility measurements (Hashimoto et al. 1979a). The specific-heat measurements by Luong et al. (1985b) show a sharp A-type of peak at a value for TTq of 26.7 K. A value for the Nrel temperature found with Mbssbauer spectroscopy is (27 4- 0.5) K (Gubbens et al. 1991), in good agreement with that obtained by Luong et al. Lebech et al. (1987) have determined the magnetic structure of DyCu2 at 5 and 15 K by neutron diffraction measurements. The structure is similar to that of TbCu2 as
464
N.H. LUONG and JJ.M. FRANSE
C
!
r" .,
--tJi
.
.
.
.
~ ,~'Ws +~,.'f, !
.
.
t
/
I
b
p,s a°s~/'/' Fig. 4.39. Magnetic structure of DyCu 2. The structure is decomposed into four sublattices A, A', B and B ~. The Dy atoms labeled 4, 5, 9, 12 are on the A sublattice; atoms 3, 6, 10, 11 are on A~; atoms 1 and 8 are on B; atoms 2 and 7 are on B f. After lwata et al. (1989), data from Lebeeh et al. (1987).
10 r 8 t~.2K i f ............. a" :? 6 ', b r
I
I01
8L~5 K
+~
~'
............~
2
2
4
6
8 - 9.5K ,,~ . ....................... Q. •
--
n
2
#
6
~ ................
............. ii"
~.
.
..
2
~- 0
0
2
4
10 8 150K ~ 6
6
I bl
~
0
2
4 B iT}
0
2
6
t,
10 8130"5K
II
6 4.
{ ...,,.•,"
Fig. 4.40.
0
8
':
:~'4"
O t . ~- - ' ~ ,
.............. ...............
O.l~,~-f, 0 2
b
!
, 4 B (T)
Magnetization curves of DyCu2. Solid curves represent the calculated magnetization. After lwata et al. (1989)•
s h o w n in fig. 4.39. H o w e v e r , d u e to the fact that D y has a h i g h n e u t r o n a b s o r p t i o n c r o s s s e c t i o n , the t e m p e r a t u r e d e p e n d e n c e o f the m a g n e t i c s t r u c t u r e has not yet b e e n analyzed.
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
10
465
(ii)
0 ~i) c - axis
t~.2K 2 0
0
5
I
I
I
I
10
15
20
25
B (T)
Fig. 4.41. Magnetizationcurves of DyCu2along the c axis in applied field up to 25 T. Curve (i) shows the magnetization process in an initial state. Curve(ii) is the magnetizationcurve obtainedjust after the measurement of (i). After Hashimotoet al. (1990). Measurements of the magnetization at 4.2 K (Hashimoto et al. 1979a) and of the temperature dependence of isothermal magnetization curves (Iwata et al. 1989) have been performed on single-crystalline samples. Figure 4.40 presents the isothermal magnetization curves for DyCu2 (Iwata et al. 1989). The magnetization curve along the a axis has two critical fields, Bel and Be2, and the magnetization increases to about 1/3 of the saturation value at Bel and to nearly its saturation value at Be2. At high temperatures, the two-step process is smeared out. On the other hand, the magnetization along the b and c axes increases monotonically. Magnetization curves have been measured in high magnetic fields up to 30 T by Hashimoto et al. (1990) and by Date (1992). Figure 4.41 shows the isothermal magnetization curves measured along the c axis at 4.2 K (Hashimoto et al. 1990). The magnetization increases abruptly at about 14 T and shows a hysteresis at decreasing magnetic fields (i). The magnetization curve measured just after the measurement of curve (i) is indicated as curve (ii) in this figure. This is quite similar to the curve along the a axis (see fig. 4.40). It turns out that a switching of the magnetic axis is observed. The recovery of the virgin state is obtained either by warming the crystal above about 100 K or by applying a field higher than 5 T along the a axis. Figure 4.42 shows the magnetization curves along the b axis at 4.2 K (Hashimoto et al. 1990). A little jump of the magnetization with a small hysteresis is observed. The critical field decreases by repeating the high-field cycles. The sample was broken in two pieces when a high field was applied along this b axis due to magnetostriction effects. This phenomenon is similar to that observed in the polycrystalline samples of TbCu2 (Luong and Fransse 1981, see fig. 4.33). The results indicate the important role of large magnetocrystalline anisotropy and magnetoelastic energies in DyCu2. Kimura (1987a, b) has made a theoretical study of the magnetization process in RCu2 compounds using the Ising model with four exchange parameters. Necessary conditions for the exchange constants for which DyCu2 exhibits a two-step magnetization process are given (Kimura 1987b).
466
N.H. LUONG and J.J.M. FRANSE
I
2 0~ e-
= .6 0
3
b -axis 4.2 K
0
4
0
I
0
I
I
10
I
I
20
30
B (T) Fig. 4.42. Magnetization curve of DyCu2 along the b axis in applied fields up to 30 T. The number in the figure indicates the order of measurement. The critical field, at which a small jump of the magnetization occurs, decreases in subsequent measurements. After Hashimoto et al. (1990).
10 x103
b
8
DyCu2
"T
a
~6 i=
2
100
200 T (K)
300
Fig. 4.43. Temperature dependence of the inverse susceptibility along the principal axes of DyCu2. After Hashimoto et al. (1979a). The t e m p e r a t u r e d e p e n d e n c e o f the inverse susceptibility along each p r i n c i p a l axis o f D y C u 2 is s h o w n in fig. 4.43 ( H a s h i m o t o et al. 1979a). T h e effective m a g n e t i c m o m e n t , #en, a m o u n t s to 10.3#a. T h e p a r a m a g n e t i c Curie temperatures a l o n g the
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
467
a, b and c axes amount to: 0a = 35 K, 0b = --17 K, 0e = 0 K. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated experimental results for the two lowest-order crystal-field parameters: B ° = 0.43 K and B 2 = 0.72 K. A point-charge calculation by the same authors results in B ° ~l = 0.89 K and B 2 ~! = 0.71 K, in satisfactory agreement with the experimental values. Nowotny and Zajac (1985) have used the experimental results for the two crystalfield parameters to calculate the paramagnetic susceptibility of DyCu2. They could explain the observed curvature in Xi(T)- 1 CUrveS (i = a, b, C) at low temperatures. The isothermal magnetization curves up to TN have been analyzed using a molecular field model (Iwata et al. 1989). The calculated magnetization curves are presented in fig. 4.40 as the solid lines. A satisfactory agreement between experiments and calculations is obtained. Apart from the experimental results for B2° and B22, a small positive value of 2.47 x 10 - 3 K for B ° was taken into account because its term gives a large contribution to the b axis magnetization at low temperatures. On the other hand, in high field studies Date (1992) has pointed out that a switching of the magnetic axis (see above) implies that this phenomenon cannot be explained by a simple point-charge crystal-field model but should be explained in terms of a Jahn-Teller effect. Specific heat and thermal expansion of DyCu2 have been measured by Luong et al. (1985b). Figure 4.44 shows the specific heat of DyCu2 at zero magnetic field and in a field of 5 T. The A-type of peak at TN disappears in a magnetic field of 5 T. A broad anomaly around 6 K, which is field dependent, is observed. Kimura et al. (1988) have calculated the specific heat of DyCu2 in terms of the molecular-field model including the crystal-field interaction. They have obtained the temperature dependence of the specific heat which is similar in trend with the experimental one. They have used three crystal-field parameters: B °, B22 experimentally obtained by Hashimoto et al. (1979a) and B ° equal to 2.37 x 10 -3 K. According to Kimura et al. (1988), the specific heat hump of DyCu2 near 6 K is associated with the fact that the spacing of the energy between the lowest and the next lowest levels is not large: 10 K. The thermal expansion coefficient of DyCu2 as a function of temperature is shown in fig. 4.45 (Luong et al. 1985b). Apart from the A-type of contribution around TN, an additional peak at about 19.5 K is observed. Up till now no prove has been given whether this peak is due to a change of magnetic structure or not. This should be connected with the difficulty of neutron diffraction studies in Dycu2 as mentioned above. Magnetization measurements for Dy~YI-~Cu2 compounds have been performed by Hien et al. (1983). The magnetization curves obtained for these compounds have a metamagnetic behaviour similar to that for Tb~YI_~Cu2. The Nrel temperatures for Dy,,Yl_rCu 2 with z = 0.2, 0.4, 0.6, 0.8 and 1 are 5, 13, 18, 23 and 26.7 K, respectively. The composition dependence of TN for Dy~YI-~CU2 is similar to that for Tb~YI_~Cu2, i.e. obeys eq. (4.5). For DyrYl_rCu2 the critical concentration, zc, has been evaluated to be equal to 0.15, the same as for the TbrYI-rCu2 compounds (see section 4.6).
468
N.H, LUONG and J.J.M. FRANSE
OyCuz
+/°
1.0 -- ~
E
÷.t14,~
C4 v
0
++%2°°
~- 0.5
...... 0
I
I
I
lO
20
30
T (K) Fig. 4.44. The specific heat of DyCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of e/T versus T. After Luong et al. (1985b).
15
DyCu2
I
I,
10 "7x,,
to
~o
5
I
•
o #
~0
•
V
" 5 1P
0
o
° • ~oo
•
•
11
e
°
•
I
I
I
I
I
I
I
10
20
30
40
50
60
70
T (K) Fig.
4.45.
The coefficient of thermal expansion for DyCu 2 as a function of temperature. Luong et al. (1985b).
After
4.8. HoCu2 For HoCu2, a value for TN of 9 K was obtained by Sherwood et al. (1964) from magnetic measurements. From the experimental data on single-crystalline HoCuz, Hashimoto et al. (1979a) and Hashimoto (1979) deduced a value for TN of 9.8 K. Lord and McEven (1980) and Birss et al. (1980) performed neutron-diffraction, resistivity and magnetoresistance measurements on HoCu2. They showed that in
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
469
HoCuz, apart from an antiferromagnetic-order transition at 11.4 K, a change of the antiferromagnetic structure occurs at about 8 K. Gratz et al. (1982) carded out measurements of magnetization, resistivity, thermopower and neutron diffraction in HoCu2. They deduced values for TN of 9.7 K and for Ts of 7 K (order-to-order transition). From measurements of the magnetization, Hien et al. (1983) have confLrmed these results. Values for TN of 9.6 K and for Ts of 7 K were derived from specific-heat (Luong et al. 1985b, c) and thermal-expansion (Luong et al. 1985b) measurements. The at-susceptibility measurements by Smetana et al. (1985a) gave values for TN of 9.7 K and for Ts of 6.7 K. Neutron-diffraction studies of the magnetic structure of HoCu2 have been performed on a single-crystalline sample (Hashimoto et al. 1979a) and on powder material (Lord and McEven 1980, Gratz et al. 1982, Smetana et al. 1983, 1985b, Smetana and Sima 1985, Smetana et al. 1986b, Lebech et al. 1987). These studies reveal that the magnetic structure of HoCuz below TN is a collinear commensurable longitudinally modulated one along the a axis (see fig. 4.46). As temperature decreases, a second transition occurs at Ts ~-, 7 K. Below this transition, the magnetic structure has two components. One component is the above mentioned a axis modulated structure and the other component is a c axis incommensurable transversely modulated structure with moments along the b direction. The resulting magnetic structure of HoCu2 below 7 K is then a non-collinear incommensurably modulated structure with magnetic moments lying in the a b plane (Lebech et al. 1987), see fig. 4.47. The magnetic structure of HoCuz has been analyzed by Kimura (1990a) in a molecular-field approximation taking second-order crystal-field effects into account. He has shown that the spin arrangement in the a e plane for 7 K < T < TN can be realized by a competition of three kinds of exchange interactions between neighbouring ions in the same a c plane. He has also shown that the origin of the spin canting at temperatures below Ts is the biquadratic exchange interaction between the nearest Ho ions in the neighbouring a c planes. In later work Kimura (1990b) has investigated the effect of higher-order components of the crystal-field on the spin arrangement of HoCu2. He has shown that one of two possible magnetic structures at 0 K is stabilized by the 6th order component of the crystal field acting on the Ho
--,
JA '
11'
I
'
'
Fig. 4.46. The high-temperature (7 K < T < 10.4 K) magnetic structure of HoCu 2. After Smetana et al. (1986b).
470
N.H. LUONG and J.J.M. FRANSE
2
I
1
I
i
I
I
I I I
I I
I
I
r-"
J
I
I
I
The Mb component is modu[ated o[ong
I
Fig. 4.47. Projection of the magnetic structure of HoCu2 in the ab plane at T < 7 K. After Lebech et al. (1987).
10 xl0 3 b
c
"T
6
E ~u
4
2 0
.,~;" 0
l 100
i 200 T (K)
i 300
Fig. 4.48. Temperature dependence of the inverse susceptibility along the principal axes of HoCu 2. After Hasbimoto et al. (1979a).
ions. Thus the interplay of the biquadratic exchange interaction and the higher-order components of crystal-field has a large effect on the spin arrangement below Ts. Figure 4.48 shows the temperature dependence of the inverse susceptibility along the principal axes for HoCuz in the paramagnetic region (Hashimoto et al. 1979a). The effective magnetic moment, /-teir, amounts to 9.6#n. The paramagnetic Curie temperatures along the a, b and e axes are: 0a = 45 K, t9b = 30 K and 0c = 38 K. From these values for the paramagnetic Curie temperature Hashimoto et al. have estimated the following values for the second-order crystal-field parameters: B ° = 0.14 K, B22 = 0.12 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have calculated the second-order crystal-field parameters and arrived at: B2° ~l = 0.28 K and B 2 c~l = 0,23 K. Using experimentally derived crystal-field parameters of Hashimoto et al., Nowotny and Zajac (1985) have calculated the paramagnetic
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
471
200 H°Cu2
o
150 b 100 c
50
0
1
2
3
t~
5
B (T) Fig. 4.49. Magnetization curves along the principal axes of HoCu 2 at 4.2 K. After Hashimoto et al. (1979a).
susceptibility of HoCu 2. They have obtained a total splitting energy of 33 K, which is lowest compared with those of PrCu2, NdCu2, TbCu2, DyCu2 and ErCu 2. The magnetization curves for single-crystalline HoCu 2 at 4.2 K are shown in fig. 4.49 (Hashimoto et al. 1979a). The anisotropy of this compound is not so large compared with T b C u 2 and DyCu z. The specific heat of HoCu2 has been measured by Luong et al. (1985b, c). Figure 4.50 presents the specific heat of HoCuz at zero magnetic field and in a field of 5 T. A A-type of anomaly is observed around TN. Luong et al. (1985b) have obtained the value of 19.9 J/Kmol f.u. for the magnetic entropy. This value points to a multiplicity of at least 11, indicating that the J = 8 (for Ho 3+) multiplet is splitted by crystal-field effects. A sharp peak, which occurs at 7 K and which is strongly field dependent, points to changes in the magnetic structure. Apart from this peak and the A-type of peak around TN, there are two other anomalies at 4.5 K and at temperatures below 1.3 K. The first one is rather broad and is not much influenced by the applied magnetic field. The anomaly below 1.3 K is ascribed to the nuclear contribution of holmium to the specific heat (Luong et al. 1985b). The specific heat in the temperature range from 2 to 20 K and in magnetic fields up to 4 T has also been measured by Bischof et al. (1989). The results at zero magnetic field are in good agreement with the specific-heat data of Luong et al. (1985b, c). Bischof et al. determined the change of specific heat due to the magnetic field, ACCF, by writting ACCF = c c F ( B ) - CCF(0), where CCF is obtained by subtracting contributions of electrons and phonons to the specific heat. Using the values for the CF parameters B2° and B 2 of Hashimoto et al. (1979a) (see above), Bischof et al. have calculated ACCF and compared it with experiments. They observed discrepancies between the calculated and experimental specific-heat curves, which demonstrate the importance of higher-order crystal-field terms. Measurements of the specific heat at higher temperatures (up to 100 K, for instance) would be very useful,
472
N.H. LUONG and J.J.M. FRANSE
2.5
/ f 2.0 0
o
E
HoCu 2
* ~
o
°o ~o
1.5
o~ o
1.0
t3
~o++÷++~+
+ +
°°°°~°°°°°°°o
0.5
0
o o
0 0
o
O O0
0
I
I
I
I
I
I
5
10
15
20
25
30
T (K) Fig. 4.50. The specific heat of HoCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T. After Luong et al. (1985b).
among other experiments, for the determination of the crystal-field scheme of the HoCu2 compound. The coefficient of the thermal expansion of HoCu2 as a function of temperature is shown in fig. 4.51 (Luong et al. 1985b). Two sharp peaks were observed, reflecting 12 HoCu 2 go
~
"7%,"
8
:',,.
'0 o
4.
eo° •
oo
~IL ~
• •
o • o° %oe%eoeeNooeoeoo • •
I
I
I
10
20
30
I
40 T (K)
I
I
I
i
50
60
70
80
Fig. 4.51. The coefficient of thermal expansion for HoCu2 as a function o f temperature. After Luong et al. (1985b).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
473
specific anomalies around Tr~ and Ts. The relative volume change due to magnetic ordering in HoCu2 is -0.25 x 10 -3, which is almost four times smaller than in GdCu2. This fact possibly indicates a reduced polarization of conduction electrons by the holmium moments (Luong et al. 1985b). Neutron-diffraction studies on a powder sample of Ho(Cu0.9Ni0a)2 have been performed by Smetana et al. (1983) and Smetana and Sima (1985). The effect of Ni substitution in the Ho(Cu, Ni)2 system is qualitatively similar to that observed in Tb(Cu, Ni)2.
4°9. ErCu2 The Nrel temperature of ErCu2 has been found to lie in the range 11-13.5 K; 11 K, 13.5 K and 11.8 K, according to the magnetic measurements of Sherwood et al. (1964), Hashimoto et al. (1979a) and Smetana et al. (1985a), respectively and 11.5 K according to the specific-heat measurements of Luong et al. (1985b). In magnetization measurements Hien et al. (1983) have observed, apart from the anomaly that is characteristic for disordering the antiferromagnetic state, a second transition at about Tsl = 6 K. These authors ascribed this transition to the change of the antiferromagnetic structure. This conclusion is supported by ac-susceptibility measurements (Smetana et al. 1985a) and specific-heat measurements (Luong et al. 1985b, c). From specific-heat and thermal-expansion measurements, Luong et al. (1985b) have observed two other anomalies at Ts2 = 4.1 K and Ts3 = 3.3 K. The anomaly around 4 K has also been reported by Smetana et al. (1984, 1985a). These additional peaks at Tse and Ts3 could also point to changes in the magnetic structure. Neutron-diffraction experiments on ErCu 2 have been performed by Smetana et al. (1984), Smetana and Sima (1985) and Lebech et al. (1987). These latter authors 10
x103
°~~/~c
ErCu '7
~6 -i
E aJ
2 J"
¢
,,"
I
I
I
100
200
300
T (K) Fig. 4.52. Temperature dependence of the inverse susceptibility along the principal axes of ErCu 2. After Hashimoto et al. (1979a).
474
N.H. LUONG and J.J.M. FRANSE
have reported that neutron-diffraction powder patterns obtained between 1.5 and 40 K revealed a rather complicated antiferromagnetic structure. In spite of many attempts to interpret the powder patterns, they were not able to describe the magnetic structure of ErCu2 satisfactorily. Single-crystalline data would be helpful. Figure 4.52 shows the temperature dependence of the inverse susceptibility along each principal axis of ErCu2 (Hashimoto et al. 1979a). The effective magnetic moment,/~e~, amounts to 8.9/ZB. The paramagnetic Curie temperatures along the a, b and c axes are: 0a = 18 K, 0b = 53 K and 0c = 36 K. From these data, Hashimoto et al. have estimated the following values for the second-order crystal-field parameters for ErCu2: B ° = -0.35 K, B 2 = -0.36 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have also calculated these second-order parameters and obtained: B ° l = -0.31 K and B2~al = -0.26 K. The magnetization curves for single-crystalline ErCu2 are shown in fig. 4.53. The magnetization along the b axis increases abruptly at about 0.6 T and saturates magnetically at a higher field. From fig. 4.53 it is also clear that ErCu2 has a large magnetocrystalline anisotropy. The high-field magnetization curve of polycrystalline ErCu2 at 4.2 K is shown in fig. 4.54 (Hien et al., unpublished result). A two-step metamagnetic process was observed. In a magnetic field of about 25 T the magnetization is still not saturated. The specific heat and the thermal expansion of ErCu2 have been measured by Luong et al. (1985b, c). Figure 4.55 shows the specific heat of ErCu2 at zero magnetic field and in a field of 5 T. The thermal-expansion coefficient of this compound is presented in fig. 4.56. The anomalies at TN, Tsl, T~2 and Ts3 were observed in the specific-heat and thermal-expansion-coefficient curves. Apart from these anomalies, broad anomalies occurred above TN. Inspecting figs 4.55 and 4.56 Luong et al. (1985b) have assumed that in ErCu2, apart from the 'long-range' magnetic contributions to the specific heat and thermal expansion, crystal-field effects were present. 200
150 Er Cu2
El00
50 -__~.~-~ ~
0
1
2
3
4
5
B (T) Fig. 4.53. Magnetization curves along the principal axes of ErCu2 at 4.2 K. After Hashimoto et al. (1979a).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
150
475
ErCu2
100 E
50
0
0
I
I
I
I
l
5
10
15
20
25
B (T) Fig. 4.54. High-field magnetization curve of ErCu2 at 4.2 IC After Hien et al. (unpublished result).
1.0 -
0
*
O E
ErCu2
#
0
0~
+
o°
÷÷%o
0 00
,,. F-
÷+
00
o°
+
o
+*
÷
4- ÷ 0
4. 0
"l"-I. 0 o
.,I0
0
0
°
0 0 0
o
÷ ÷=-o o
0.5
,I 0
20
10 T
30
t~0
(KI
Fig. 4.55. The specific heat of ErCu2 at zero magnetic field (.) and in a field of 5 T (÷) in plots of e/T versus T. The full curve represents the non-magnetic contribution to the specific heat of ErCu2. The dashed curve represents the specific heat of YCu 2. After Luong et al. (1985b). Luong et al. ( 1 9 8 5 b ) and Franse et al. (1985) have analyzed the e x c e s s contributions to the specific heat and thermal expansion in ErCuz by applying Grtlneisen relations (see section 3.3), s h o w i n g that they consist of a 'long-range' magnetic and a crystal-
476
N.H. LUONGand J.J.M. FRANSE
ErCu 2
10
l
-5 m
-1(3
0
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
T (K) Fig. 4.56. Thermal-expansioncoefficientof ErCu2. The non-magneticcontribution to the thermal expansion is indicatedby the dashed line. AfterLuonget al. (1985b). field part. The excess specific heat, era, was obtained by subtracting the electronic and phonon parts (non-magnetic) from the observed specific heat. The excess thermal expansion, O~m,is also obtained by subtracting the non-magnetic parts from the observed thermal-expansion coefficient (see fig. 4.57). Cm and am are related by the effective Grtineisen parameter, Fen [see eq. (3.38)]. From the temperature dependence of the Griineisen parameter, ~Fe~, for ErCu2, a change in sign in this parameter was observed. This change in sign indicates that at least two different contributions to am and am have to be distinguished. Values for ~Flr (6×10 -11 me/N) and ~/~ef (-15 x 10 -11 m2fN) have been derived by comparing em and am well below and well above TN, respectively. With these values, Cm and am have been separated into two different contributions (see fig. 4.58). The term Clr is A-type, whereas ccf is Schottky-type. The energy difference between the two lowest doublets was derived to be 76 K. We note that the temperature dependence of cef above and below Tm (at which cef exhibits a maximum, see section 3.3), as reported by Luong et al. (1985b), has not properly been calculated. Franse and Luong (private communication) have calculated ccf as a function of temperature taking into account only two lowest doublets with the experimentally derived energy difference of 76 K between these two doublets. They obtained a similar behaviour of the eel(T) curve compared with the experimental one but the calculated value of ecf is lower (about half the experimental value, at Tin). Using values for the two second-order crystal-field parameters reported by Hashimoto et al. (1979a) (see above), Franse et al. (1985) have calculated the energy levels in the ErCu2 compound. These authors derived a splitting of
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
477
10
!s
;, 10
20
30
40
50
T (K)
.I,I.
~s
t,
/.
o
.
i,-. I
i
I
I"
1
20
10 ~t.
t
i
i
I
30
I
40
50
T (K)
44-
4"4" -5 ÷
4"
-10
4" 4"
4" 4" 4÷
4" 4"
4.
4"
~fl.+ + +
Fig. 4.57. The excess specific heat, cm, and the excess thermal expansion, am, of the ErCu2 compound. After Franse et al. (1985).
13 K between the lowest doublets. Apparently, higher-order terms have to be taken into account in order to bring the splitting closer to the experimental value of 76 IC The relative volume dependence of this energy splitting was compared with the volume dependence of the crystal-field parameters as calculated in a point-charge model (Franse et al. 1985). The two results obtained in this way differed in sign. As discussed by Franse et al. (1985), this unexpected volume dependence could originate either from an anisotropic compressibility or from an anisotropic thermal expansion due to preferential orientations in the sample on which the experiments were performed. To clarify this question from the experimental point of view, the thermal expansion in three mutually perpendicular directions of a ErCuz polycrystalline sample has been measured (Duc et al. 1988), see fig. 4.59. Data in this figure clearly prove that the thermal expansion is almost isotropic in the studied sample. These experiments once more show that a crystal-field calculation based on
478
N.H. LUONG and J.J.M. FRANSE
15
0
0
10
oc
(o lr
.°**'m°°
0
.£ u
0
*OOZe :of
0
I I I I
5
0
0
o
o
•
o
0
10
o
20
~
I
I
30
~0
50
I 60
T (K)
t=tr w
~ 0
t
.
.
I
;'0" .
.
t~0 T (K)
o CXrf
-5 0 0 0
-10
O o oo
o o oo
o
O
o
Fig. 4.58. Splitting of cm and C~m of ErCu 2 according to eqs (3.41-3.44) into a long-range magnetic (eir, air) and a crystal-field (c~, c~ce) part with values for n_Fir and ~_r'ef given in the text.
a point-charge model is inadequate to describe the experimental observations for this compound. The ErCu 2 compound has been measured with 166Er M6ssbauer spectroscopy (Gubbens et al. 1991). These authors also reported the results of inelastic neutron scattering, which show a not-yet definitively determined level sequence of doublets above TN at 0, 61, 78, 88, 124, 142, 148 and 160 K. Using a superposition model (Newman 1983, Divis 1991) and by fitting the results of inelastic-neutron-scattering and M6ssbauer spectroscopy, Gubbens et al. (1991) have determined a tentative set
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
479
15I ErCu2
10 A
•- !
5
v'
! Q
O
•--
0
~o ~o
-5
o
o°j/
oOJ ,,,/
x x
×
"10
-1_=
0
t
~
10
20
30
40
50
60
TtK)
Fig. 4.59. The linear thermal-expansion coefficient measured in three mutually perpendicular directions of a polycrystalline sample of ErCu2 with a cubic shape (5 x 5 × 5 mm3); the full curve represents the mean value of c~ along the three directions. After Due et al. (1988).
o f all the nine crystal-field parameters. This set is: B ° = - 0 . 2 8 K, B22 = - 0 . 2 2 K, B 0 -- - 0 . 3 0 X 10 - 2 K, B 2 = - 0 . 1 4 x 10 - 2 K, B44 = 0.30 x 10 - 2 K, B ° = - 0 . 2 0 x 10 - 4 K, B62 = - 0 . 4 7 x 10 - 4 K, B64 = - 0 . 9 7 x 10 - 4 K, B 6 = - 2 . 9 6 x 10 - 4 K. The value o f B ° of this set is very close to a value o f - 0 . 2 6 K determined f r o m the quadrupole splitting (Gubbens et al. 1991). One also can see that there is agreement
480
N.H. LUONG and J.J.M. FRANSE
between the values of B ° and B 2 obtained by Hashimoto et al. (1979a) and those obtained by Gubbens et al. (1991). However, the latter authors have pointed out that their set of nine crystal-field parameters is not unique at the present stage of the investigation. Gratz et al. (1993) have measured the anisotropic thermal expansion of polycrystalline Ercu2 sample from 4.2 K to 400 K by means of X-ray powder diffraction. They have described theoretically the obtained data using the above listed set of crystal-field parameters deduced by Gubbens et al. (1991).
4.10. TmCu2 Magnetic measurements on TmCu 2 have been reported by Sherwood et al. (1964) but no value for the ordering temperature has been given. The ae susceptibility (Smetana et al. 1985a), magnetization, susceptibility, resistivity and specific heat (Smetana et al. 1986a) have been measured on polycrystalline TmCu2. These measurements reveal a Nrel temperature value of 6.3 K and magnetic transitions at Tsl = 4.3 K, Ts2 = 3.9 K and Ts3 = 3.3 K which reflect changes of the magnetic structure. The paramagnetic Curie temperatures along the a, b and c axes are reported as: 0a = - 1 5 K, 0b = 49 K and 0c = 18 K (Sima et al. 1989). The effective magnetic moment, #eft, is equal to (7.2 4- 0.2) #a (Smetana et al. 1986a). Neutron-diffraction investigations have been performed by Lebech et al. (1987), Sima et al. (1989) and, later, by Heidelmann et al. (1992). According to the latter authors, TmCuz shows at least three different antiferromagnetic structures. The lowtemperature structure (AFI) (T < 3.1 K) is composed by the first two odd harmonics of a propagation vector (21r/8)a*. In a second temperature region, 3.1 K < T < 4.7 K, the phase is a mixture of two structures (AFII). A third phase, (AFIII), appears in the temperature region 4.7 K < T < TN ~, 8 K. All these structures may be described as sinusoidally modulated with magnetic moments parallel to the b direction, except for one of the components of AFII, where it is necessary to invoke a component perpendicular to the b axis. The result that only one magnetic moment component in the b direction is necessary to describe the observed structure in the case of AFI and AFIII is in agreement with the crystal-field anisotropy studies (Sima et al. 1989). A discrepancy between the Nrel temperature obtained from bulk data mentioned above (TN = 6.3 K) and that obtained from the neutron-diffraction study of Heidelmann et al. (1992) (TN ~ 8 K) is still unexplained. Heidelmann et al. (1992) also pointed out that from the neutrondiffraction data there is no evidence for the existence of the transition at 3.9 K which has been reported from bulk data (see above). The temperature dependencies of the inverse magnetic susceptibility of TmCu2 along each principal crystallographic axis are shown in fig. 4.60 (Sima et al. 1989). From these results the following experimental values for the second-order crystalfield parameters have been derived: B ° = -0.94 K, B 2 = -1.01 K. Figure 4.61 presents the magnetization curves of single-crystalline TmCu2 (in comparison with a polycrystalline sample) at 4.2 K (Svoboda et al. 1989). A large anisotropy is observed. The b axis is the easy magnetization direction. I n this direction, a two-step metamagnetic transition (of possible spin flip type) occurs
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
TmCu2
481
a o
3 C
% %
0
100
200
300
T (K) Fig. 4.60. Temperature dependence of the inverse susceptibility along the principal axes of TmCu 2. Symbols are the experimental data. Lines represent the calculated susceptibility according to the set of crystal-field parameters No. 6 (table 4.1). After Sima et al. (1989).
b "R.
6
::3
,oOf,* .
~
o
oe
°
e
po/yory.~tcthne
e
•
•
°
°
°
°
°
°ee e z
2
,see
// •
0
e
C
1
2 B (T)
3
Fig. 4.61. Magnetization curves of TmCu2 along the principal axes (full line) in comparison with the polycrystalline one (dotted line) obtained at 4.2 K. After Svoboda et al. (1989).
with two critical fields Bet = 0.06 T and Be2 = 0.36 T. Above the second step, the magnetization reaches the saturated value of (6.9 4- 0.2)/~B per T m atom which is in good agreement with the value for a free T m 3+ ion. The magnetization along a and e axes is an order of magnitude lower than that along the b axis.
482
N.H. LUONG and J.J.M. FRANSE
Crystal-field effects in TmCu 2 have extensively been studied (Zajac et al. 1987, 1988a, Sima et al. 1988, 1989, Divis et al. 1989b, and Gubbens et al. 1992). Zajac et al. (1987) have calculated the crystal-field splitting and the crystal-field susceptibility along the principal crystallographic axes. In their calculations, they used only the second-order crystal-field parameters B ° = -2.17 K and B 2 = -1.88 K. Their calculations show a total splitting of about 290 K. By comparing the calculations with the experimental results reported by Smetana et al. (1986a), Zajac et al. (1987) have pointed out the need to take higher-order terms into account. Based on the analysis of the specific-heat and thermal-expansion data, Sima et al. (1988, 1989) have studied crystal-field effects in TmCu 2 in more detail. Figure 4.62 shows the experimental and calculated temperature dependencies of the specific heat of TmCu2 in the paramagnetic region. The linear thermal-expansion coefficient of TmCu2 as a function of temperature is shown in fig. 4.63 (Sima et al. 1989). These authors have taken the 7 value equal to 9 mJ/K 2 mol, comparable with the values for the other RCu2 compounds (Luong et al. 1985b). The phonon part of the specific heat of the RCu 2 compounds follows a Debye function rather well (Luong et al. 1985b). From the best fit to the RCu2 data, Sima et al. (1989) obtained 0 D = 194 + 2 K for TmCu2. The values of m and l'ph for TmCu2 are unknown. In case of isostructural YCu2 and (Gd,Y)Cuz, Luong et al. (1985a) have found that x/~ph and 8i) are temperature-independent up to 100 K. Similarly, for TmCu2, Sima et al. (1989) assumed that n_r'ph is also independent of temperature and they have used a ~-r'ph value of 11.9 x 10- 12 m 2 /N, as obtained for YCu2 by Luong et al. (1985a). The problem of analyzing c(T) and ~(T) above TN has been reduced to the determination of parameters OD, Ei and 7i, where 7/[= - d ( l n Ei)/d(ln V)] is the crystal-field Grtineisen parameter of the individual energy
&E{K) 60
• 40
~ "4" % "~
~
~ -
-
////"
~ too
"
~
//
~
/11/
O =/17
//
J,
m
........
0
20
CO T(K)
60
, .......
80
Fig. 4.62. Temperaturedependences of the specific heat of TmCu2 in the paramagneticregion: o, experimental data; - - , calculatedtotal specificheat; - - -, phonon specificheat; •.., electronicspecific heat; - - -, Sehottky specific heat. The arrow indicates the N6el temperature. The inset shows the energies of the CF levels used in calculations. After Sima et al. (1989).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
483
10
T
-10
".......'
-20
i 20
i /40
i 60
t 80
100
T[K~ Fig. 4.63. Temperaturedependences of the linear thermal-expansioncoefficientof TmCu2 in the paramagnetie region: o, experimental data; q , calculated total thermal-expansioncoefficient; - - -, phonon thermal-expansioncoefficient;..., Schottkythermal-expansioncoefficient.The arrow indicates the Nrel temperature. After Sima et al. (1989). level Ei (i = 0, 1, . . . , 1 2 , E0 = 0). To solve this many-parameter (25) problem, Sima et al. (1989) used a Monte Carlo simulation of all parameters in order to find the intervals of appropriate possible values. The final solution for Ei has to be conform to the nine-parameter orthorhombic CF Hamiltonian. This leads to a reduction of the number of parameters to 22 (8D, B ,rr~, 7i). These authors treated the Bnm as adjustable parameters to be determined from the proposed CF level scheme. The number of possible solutions is quite large. About 100 fits were done and the sets of parameters which are in agreement with experimental observations have been selected (see table 4.1). The values of the second-order terms B ° and B 2 are very close to those obtained from susceptibility measurements (see above). From the analysis it appears that the ground state of TmCu2 is an isolated quasi-doublet (El = 5 K, E2 = 68 K). The total splitting is 210 K, i.e. less than that reported by Zajac et al. (1987) (see above). Sima et al. (1989) pointed out that a major contribution to the CF splitting comes from the second-order terms B2° and B 2. However, these terms cannot satisfactorily describe the observed splitting of the quasi-doublet ground state and the inverse susceptibility Xa I (see fig. 4.60) between 20 and 70 K. All nine parameters, Bnm, must be taken into account in order to obtain, at least qualitative, agreement with experiments. The value for E1 agrees with preliminary results of inelastic neutron spectroscopy obtained by Loewenhaupt and Gratz (1989). Gubbens et al. (1992) have studied crystal-field effects in TmCu2 by 169Tm MSssbauer spectroscopy. By combining the results of magnetization (Svoboda et al. 1989, Divis et al. 1989b), specific heat and thermal expansion (Sima et al. 1989)
484
N.H. LUONG and J.J.M. FRANSE TABLE 4.1 Crystal-field parameters, in units of K, for the compound TmCu 2. The sets Nos. 1-6 are the selected by Sima et al. (1989) from an analysis of specific heat and thermal expansion data. Set (i) gives the calculated CF parameters using a modification of the point-charge model (Sima et al. 1989). Set (ii) gives the values obtained by Zajac et al. (1988a) in point-charge model. Set (iii) gives the parameters derived by Gubbens et al. (1992) by combining the results of different experiments.
No
st
B0,
s:,
(10 -3 ) (10 -2 ) 1 2 3 4 5 6 i ii iii
-1.16 -1.30 -1.30 -1.01 -1.01 -0.94 -1.45 -1.13 -0.94
-0.80 -0.87 -1.09 -1.23 -1.23 -1.30 -0.29 0.87 -1.23
-1.81 -0.07 -4.35 0.36 -0.51 0.29 -5.58 0.15 -9.0
-2.90 -1.45 -0.43 -3.26 -3.40 -3.69 -0.36 -0.28 -0.39
B',
s06
(10 -2 )
(10-~
2.17 0.58 -0.07 2.82 3.62 2.90 0.87 0.77 -0.36
0.72 1.45 1.52 -2.32 -0.80 -2.54 3.26 0.43 5.8
(10 -4) -6.52 -9.63 0.72 -7.53 -7.46 -7.82 2.46 0.15 2.47
(10-') 5.79 2.90 -11.66 3.19 2.39 4.06 1.59 0.20 -0.48
(10 -4 ) -7.97 6.59 0.87 4.49 3.19 3.91 -7.17 0.07 6.31
with those of inelastic neutron scattering (Divis, Heidelmann and Loewenhaupt, unpublished results) and 169Tm M6ssbauer spectroscopy, Gubbens et al. (1992) have derived the set of crystal-field parameters for TmCu 2 shown in the last row of table 4.1. However, Gubbens et al. (1992) pointed out that this set of crystal-field parameters still has a preliminary character. By studying the relaxation in TmCu2, these authors have indicated that the proposed crystal-field parameters might need future reconsideration. Javorsky et al. (1992) have measured the specific heat of polycrystalline samples Tm~YI_~Cu2 (z = 0.1, 0.2, 0.3, 0.4, 0.6 and 0.8) at temperatures up to 30 K. The antiferromagnetic ordering has been observed for z = 0.6 and 0.8. For z ~< 0.4, no magnetic ordering has been found and the specific-heat curves turned out to show only a Shottky anomaly due to the crystal-field effect. Using the molecular-field model in a two-level approximation (Fulde 1979), Javorsky et al. (1992) calculated the temperature dependence of the Nrel temperature according to the relation: tanh (E1/2Ttq) = (1/z) tanh (E1/2T~),
(4.6)
where T~ is the NEel temperature for TmCu2 and E1 is the energy separation (measured in kelvin) between the two lowest levels. They have obtained good agreement with experiment. By comparing the experimental data in the paramagnetic region with eq. (3.23), Javorsky et al. derived the best-fit values for El, E2 and E3. They concluded that the value for E1 changes minimally with concentration, (El = (4 4- 2) K). and that the quasidoublet ground state is well separated from the higher excited states for all Tm concentrations, in very good agreement with the preliminary results of inelastic neutron spectroscopy obtained by Heidelmann and Loewenhaupt (private communication), who reported for x = 0.05 the values for El and E2 of 2.1 K and 45 K, respectively.
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
485
Specific-heat and magnetization measurements on Tm(Cu~Nil_~)2 have been performed by Divis et al. (1990c). A change from antiferromagnetic (for z --- 0.95 and 0.9) to ferromagnetic ordering (for z = 0.8, 0.75 and 0.7) is observed. For x < 0.7, the CeCu 2 structure becomes unstable. From the analysis of the data, Divis et al. conclude that the crystal-field ground state of the Tm(Cu~Nil_~:)2 compounds (z ~< 0.7) is a quasidoublet.
5. Comparison of isostructural compounds In the preceding section we have discussed the properties of various RCu 2 compounds. It is of interest to make an attempt to see some systematic behaviour, comparing different compounds with the same crystallographic structure. It can be inferred from section 4 that information about the crystal-field interaction ill RCu 2 is not complete. Due to the orthorhombic structure, nine CF parameters are needed to describe the CF Hamiltonian. Up till now, the full set of CF parameters is available only for NdCu2, ErCu2 and TmCu2. Nevertheless, the lowest-order CF parameters Bg and B 2 have been derived for most of the RCu2 compounds. In table 5.1, we collect the second-order coefficients A ° and A2z for the RCu2 compounds. These coefficients are related to the CF parameters B 20 and B 22 by (see section 3): Ao =
B°loej(r2e),
(5.1)
A 2 m B 22/ a J ( r 4 2f ) ,
where the values for the quantity otj(rlf ) are taken from table 3.1. As pointed out in section 4, to our knowledge, CF parameters for SmCu2 are not available. Rather low TABLE 5.1 Crystal-field coefficients, in units of Kao 2, for the RCu 2 compounds. R
A~
A~
ReL
Pr Nd
-168 -112 -188
-117 -154 -218
[1] [1] [2]
Tb
-148
-148
[31
Dy Ho Er
-85.7 -83.5 -190 -153 -134.7 - 134.7
-143.4 -71.6 -194 -120 -186.4 - 176.3
[3] [3] [3] 14] [5] [6]
Tm [1] [2] [3] [4] [5] [6]
Hashimoto (1979). Gratz et al. (1991). Hashimoto et al. (1979a). Gubbens et al. (1991). Sima et al. (1989). Gubbens et al. (1992).
486
N.H. LUONG and J.J.M. FRANSE
values of A ° and A 2 have been obtained by Trump (1991) for the Kondo compound CeCu2, in which anomalous properties are observed. Except for CeCu2, as can be seen from table 5.1, the coefficients A ° and A22 have the same sign and are of the same order of magnitude. From the similarity of the lowest-order CF parameters it seems that the crystal-field model can be used for describing the behaviour of isostructural RCu 2 compounds. At the same time, there is no reason why higherorder CF parameters should be neglected. The necessity to take these higher-order terms into account is indicated in section 4. The importance of the higher-order CF terms is also revealed from the studies on substituted RCu2 compounds. Analyzing the data obtained on a Tb(Cu0.7Ni0.3)2 sample, Divis et al. (1990b) have shown that the step-like appearance in the magnetization curves along the b axis in this compound cannot be explained using second-order terms in CF Hamiltonian only. These authors have shown that in order to account for all features of the magnetization data, the higher-order terms should be included into the Hamiltonian. Divis et al. (1990c) have also used nine CF parameters for describing the specific-heat data on Tm(Cul_xNix)2. Even for simple cubic systems such as RA12, where only two CF parameters are needed, the CF model fails to give a rigorous systematic and quantitative description of the magnetic properties (Purwins and Leson 1990). Little can be said about a systematic analysis of exchange interactions in RCu2. This is probably due to the complicated magnetic structure of these compounds. In some cases, the temperature dependence of the sublattice magnetizations and the magnetization process can be described in the molecular-field and crystal-field model in which, apart from the crystal-field parameters, a limited number of the molecular-field constants is considered. TbCu2 is an example. The magnetic unit cell of this compound consists of twelve magnetic moments but different temperature dependencies of the magnetization are observed in two sublattices only (Sima et al. 1986, Divis et al. 1987). Then the effective molecular fields at these two sublattices can be expressed as (Divis et al. 1987): Bin1 - -
LI(M1) + 2L2(M2),
(5.2)
Bin2 = (51 - L2)(M2) + L2(M1), where M1 and M2 are the sublattice magnetic moments and where LI and L2 are effective molecular-field constants. The observed temperature dependence of the sublattice magnetizations can be well reproduced by a calculation using these two effective molecular-field constants together with two experimentally determined crystal-field parameters (Divis et al. 1987, Iwata et al. 1987). Values for these molecular-field constants are derived from the best fit to the experimental data. The effective molecular-field constants for TbCu2 determined by Divis et al. (1987) and Zajac et al. (1988b) (L1 = 5.77 x 105 A/m/.tB and L2 = 5.42 x 105 A/m#B) are only slightly different from those obtained by Iwata et al. (1987, 1988). Divis et al. (1987) have pointed out that the observed temperature dependence of the sublattice magnetization in TbCu2 can be considered as a result of comparable contributions of the molecular-field and the crystal-field contributions to the total energy in this
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
487
compound. Magnetization processes in YbCu 2 compound are satisfactorily explained by Iwata et al. (1988) using the molecular-field model with two crystal-field parameters and five exchange parameters. From the best fit to the magnetization processes in DyCu2, Iwata et al. (1989) have determined the exchange parameters for this compound. Values for these parameters are comparable in magnitude with those of the TbCu2 compound. One of the features of the RCu2 compounds is that the values for the N6el temperature of these compounds are not simply proportional to the de Gelmes factor G = (gj - 1)2j(j + 1) and takes a maximum for TbCu2 (see fig. 5.1). This fact suggests that the RKKY interaction is not sufficient for fully understanding the exchange interactions in the RCu2 compounds. In the discussions of the ground-state spin configurations in the CeCu2-type crystal structure, Kimura (1986) considered four kinds of bilinear exchange interactions. In later work, Kimura et al. (1991), apart from three kinds of bilinear exchange interactions, have introduced the four-spin exchange interaction. These authors show that the four-spin exchange interaction gives rise to new complex states as a possible ground-state spin configuration in the CeCu 2type crystal structure and that, within the 12-sublattice model, 24 types of spin configurations are possible for various values of exchange interactions. Analyzing magnetization process in the TbrYl_=Cu2 compounds in the same molecular-field and crystal-field model as that used by Divis et al. (1987), Zajac et al. (1988b) pointed out that better agreement between calculated and measured magnetization curves can be expected if taking into account higher-order crystal- and molecularfield parameters. In some of the RCu2 compounds, it may be necessary to consider the quadrupole and magnetoelastic interactions for better explanation of the magnetic
16
5O
~0
,-i---
0
12 "7
0
3O
..-i ¢q
a~ 0
{71
2O 0
10
O
0 0
$ O
0
0 0 n ~ I i J i r T r I i ~9 Pm Eu Tb Ho Tm Lu Lo Pr Ce Nd Srn Od Dy Er Yb
Fig. 5.1. Values for the N6el temperature in the R C u 2 series and values for the de Gennes factor (9 - 1)23"(,/+ 1) across the 4f series.
488
N.H. LUONG and J.J.M. FRANSE
behaviour. The role of these interactions in rare-earth intermetallic compounds has been discussed, e.g., by Morin and Schmitt (1978, 1990).
6. Acknowledgements The authors wish to thank Prof. T.D. Hien of the Cryogenic Laboratory of the University of Hanoi for his interest, help and encouragement. The cooperations with Prof. N.P. Thuy, Dr. N.H. Duc and other colleaques of the Cryogenic Laboratory and with Dr. E E Bekker of the University of Amsterdam are kindly acknowledged. The authors are grateful to Dr. Y. Hashimoto, Prof. I. Kimura and Prof. M. Loewenhaupt for providing their work, to Dr. D. Givord for comments on the manuscript, to Dr. N.M. Hang for his comments and assistance in collecting some references and to Dr. R.R. Arons et al. for sending a preprint on the magnetic structure of NdCu2. This work has been supported by the Commission of the European Community (BRITE/EURAM Programme) in the scope of the project 'Basic Interactions in RareEarth Magnets' (BIREM).
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Onuki, Y., A. Fukada, I. Ukon, I. Umehara, K. Satoh and Y. Kurosawa, 1990, Physica B 163, 600. Onuki, Y., Y. Machii, Y. Shimizu, T. Komatsubara and T. Fujita, 1985, J. Phys. Soc. Jpn 54, 3562. Onuki, Y., Y. Nakai, T. Omi, T. Yamazaki and T. Komatsubara, 1988, J. Magn. Magn. Mater. 76-77, 119. Osborn, R., S.W. Lovesey, A.D. Taylor and E. Balcar, 1991, in: Handbook on the Physics and Chemistry of Rare Earths, M~1.14, eds K.A. Gschneidner, Jr. and L. Eyring (North-Holland, Amsterdam) p. 1. Ott, H.R., K. Andres, P.S. Wang, Y.FL Wong and B. Liithi, 1977, in: Crystal Field Effects in Metals and Alloys, ed. A. Furrer (Plenum, New York) p. 84. Ott, H.R., J.K. Kjems and K. Andres, 1978, Inst. Phys. Conf. Ser. 37, 149. Poldy, C.A. and E. Gratz, 1978, J. Magn. Magn. Mater. 8, 223. Poldy, C.A. and H.R. Kirchmayr, 1974, Phys. Status Solidi B: 65, 553. Purwins, H.-G. and A. Leson, 1990, Adv. Phys. 39, 309. Rhinehammer, T.B., D.E. Etter, J.E. Selle and P.A. Tucker, 1964, Trans. Metall. Soc. AIME 230, 1193. Ruderman, M.A. and C. Kittel, 1954, Phys. Rev. 96, 99. Satoh, K., I. Umehara, A. Fukada, Y. Kurosawa, Y. Onuki and K. Maezawa, 1990a, J. Magu. Magn. Mater. 90-91, 141. Satoh, K., I. Umehara, Y. Kurosawa and Y. Onuki, 1990b, Physica B 165-166, 329. Schmitt, D., 1979a, J. Phys. F: 9, 1745. Schmitt, D., 1979b, J. Phys. F: 9, 1759. Settai, R., T. Goto and Y. Onuki, 1992, J. Phys. Soc. Jpn 61, 609. Settai, R., FL Matsui, Y. Ohe, T. Goto, T. Fujimura, S. Sakatsume, Y. Onuki, F. Iga and M. Kasaya, 1990, J. Magn. Magn. Mater. 90-91, 446. Sherwood, R.C., H.J. Williams and J.H. Wernick, 1964, J. Appl. Phys. 35, 1049. Shohata, N., 1977, J. Phys. Soc. Jpn 42, 1873. Sima, V., M. Divis, P. Svoboda, Z. Smetana, S. Zajac and J. Bischof, 1989, J. Phys.: Condens. Matter 1, 10153. Sima, V. and Z. Smetana, 1984, Solid State Commun. 49, 981. Sima, V., Z. Smetana, M. Divis, P. Svoboda, S. Zajac, J. Bischof, B. Lebeeh and E Kayzel, 1988. J. Phys. (Paris) 49, C8-415.
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Sima, V., Z. Smetana, B. Lebech and E. Gratz, 1986, J. Magn. Magn. Mater. 54-57, 1357. Simizu, M., 1974, Phys. Lett. A 50, 95. Smetana, Z. and V. Sima, 1985, Czech. J. Phys. B 35, 1232. Smetana, Z., V. Sima and AN. Andreev, 1984, Phys. Status Solidi B: 121, K131. Smetana, Z., V. Sima, J. Bischof, E Svoboda, S. Zajac, L. Havela and A.V. Andreev, 1986a, J. Phys. F: 16, L201. Smetana, Z., V. Sima, E. Gratz and J.J.M. Franse, 1983, J. Magn. Magn. Mater. 31-34, 633. Smetana, Z., V. Sima, M.K. Borombajev, A.S. Markosyan and R.Z. Levitin, 1985a, Acta Phys. Slov. 35, 300. Smetana, Z., V. Sima and B. Lebech, 1986b, J. Magn. Magn. Mater. 59, 145. Smetana, Z., V. Sima, B. Lebeck and E. Gratz, 1985b, Physica 130 B, 385. Stevens, K.W.H., 1952, Proc. Phys. Soc. A 65, 209. Stewart, A.M., 1992, J. Phys.: Condens. Matter 4, 6859. Stewart, G.R., 1983, Rev. Sci. Instrum. 54, 1. Storm, A.R. and K.E. Benson, 1963, Acta Cryst. 16, 701. Svoboda, E, M. Divis, A.V. Andreev, N.V. Baranov, M.I. Bartashevich and EE. Markin, 1992, J. Magn. Magn. Mater. 104-107, 1329. Svoboda, E, Z. Smetana and A.V. Andreev, 1989, Phys. Status Solidi Pc 111, 285. Subramanian, ER. and D.E. Laughlin, 1988, Bull. Alloys Phase Diagrams 9, 316. Takayanagi, S., Y. Onuki and T. Komatsubara, 1986, J. Phys. Soc. Jpn 55, 2384. Takayanagi, S., I. Umehara, K. Satoh and Y. Onuki, 1991, Physica C 185-189, 2641. Takayanagi, S., I. Umehara, K. Satoh, Y. Onuki and N. Wada, 1990, J. Magn. Magn. Mater. 90-91,479.
Taylor, K.N.R., 1971, Adv. Phys. 20, 551. Taylor, K.N.R. and M.I. Darby, 1972, Physics of Rare Earth Solids (Chapman and Hall, London). Trump; R., 1991, PhD Thesis, KSln University. Trump, R., S. Thierfeldt, M. Loewenhaupt and T. Chattopadhyay, 1991, J. Appl. Phys. 69, 4699. Uwatoko, Y., H. Okita, G. Oomi, I. Umehara and Y. Onuld, 1992, J. Alloys and Compounds 181,293. Uwatoko, Y., H. Okita, G. Oomi, I. Umehara and Y. Onuki, 1990a, J. Magn. Magn. Mater. 90-91, 485. Uwatoko, Y., H. Okita, G. Oomi, I. Umehara and Y. Onuki, 1990b, J. Phys. Soc. Jpn 59, 3443. Uwatoko, Y., G. Oomi, I. Umehara and Y. Onuki, 1991, J. Phys. Soc. Jpn 60, 3823. Wickman, H.H., I. Nowik, J.H. Wernick, D.A. Shirley and R.B. Frankel, 1966, J. Appl. Phys. 37, 1246. Wun, M. and N.E. Philips, 1974, Phys. Lett. A 50, 195. Yosida, K., 1957, Phys. Rev. 106, 893. Yumaguzhin, R.Yu., R.Z. Levitin and Yu.E Popov, 1985, Fiz. "lVerd. Tela 27, 579; Eng. Transl., Soy. Phys. Solid State 27, 361. Zajac, S., 1981, Acta Phys. Slov. 31, 185. Zajac, S., M. Divis, V. Sima and Z Smetana, 1988a, J. Magn. Magn. Mater. 76-77, 197. Zajac, S., M. Divis, V. Sima and Z Smetana, 1988b, J. Phys. (Paris) 49, C8-413. Zajac, S. and S. Maczak, 1985, Acta Phys. Slov. 35, 312. Zajac, S., V. Sima and Z~ Smetana, 1987, Czech. J. Phys. B 37, 1.
Nguyen Hoang LUONG Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands (On leave from the Cryogenic Laboratory Faculty of Physics, University of Hanoi Vietnam)
and
J.J.M. FRANSE Van der Waals-Zeeman Laboratorium University of Amsterdam The Netherlands
Handbook of Magnetic Materials, Vol. 8 Edited by K. H.J. Buschow ©1995 Elsevier Science B.V. All rights reserved 415
CONTENTS 1. I n t r o d u c t i o n
.................................................................
417
2. B r i e f s u r v e y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. R a r e - e a r t h a n d r a r e - e a r t h - - C u c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
418 418 ......
420
3. T h e o r e t i c a l a s p e c t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Crystal-field i n t e r a c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. E x c h a n g e i n t e r a c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. D e s c r i p t i o n o f m a g n e t i c p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
421 421 424 425
4. M a g n e t i c p r o p e r t i e s o f R C u 2 c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. C e C u 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432 432
4.2. 4.3. 4.4. 4.5.
PrCu 2 .................................................................. NdCu 2 .................................................................. SmCu 2 ................................................................. GdCu 2 ..................................................................
436 441 447 451
4.6. 4.7. 4.8. 4.9.
TbCu 2 DyCu2 HoCu 2 ErCu 2
.................................................................. .................................................................. .................................................................. ..................................................................
457 463 468 473
4.10.TmCu 2
.................................................................
480
5. C o m p a r i s o n o f i s o s t r u c t u r a l c o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
485
6. A c k n o w l e d g e m e n t s
488
...........................................................
References .....................................................................
416
488
1. Introduction
Among the rare-earth intermetallic compounds, the RT2 (R = rare earth, T = transition metal) with the cubic Laves phase structure have been widely studied in the past decades. Initially, less attention has been paid to the RCu2 compounds after the early magnetic investigations performed by Sherwood et al. (1964). However, during the last decade substantial progress has been achieved in the study of the magnetic properties of the RCu 2 compounds. The RCu 2 compounds, except for LaCu2, crystallize in the orthorhombic CeCu2 structure, i.e. belong to the systems with low symmetry. The compounds show a large variety of magnetic behaviour. Most of the RCu 2 are antiferromagnetic. T b C u 2 has the highest N6el temperature with a value for TN near 50 K. The magnetic properties of the RCu 2 are largely affected by the crystal field (CF) effects. The magnetic ordering temperatures in the RCu 2 compounds are low and crystal-field and exchange interactions can be studied by varying temperature and/or magnetic field. Changes in the magnetic structure occur in most of the compounds below the N6el temperature and most of the compounds exhibit metamagnetic transitions below the magnetic ordering temperature. The metamagnetic transition in Z b C u 2 has been discussed by Gignoux and Schmitt (1991) as a typical example of a single-step transition in a metallic system. Unlike the RnTm compounds where T is a magnetic transition metal, 4f magnetism can be investigated in the RCu 2 without disturbing effects of the d magnetism. An extensive description of the magnetic properties of 4f-3d intermetallic compounds has been presented in the reviews by Buschow (1977), Kirchmayr and Poldy (1979) and Buschow (1980). The magnetic properties of binary rare-earth 3d-transition metals intermetallic compounds with the transition metals Mn, Fe, Co, and Ni have been reviewed by Franse and Radwanski (1993). It turns out to be of considerable interest to carry out in a systematic way investigations of the physical properties within isostructural series generated by the various rare-earth elements. As far as the binary rare-earth intermetallic compounds with non-magnetic partners are concerned, to our knowledge, only a review is available for the magnetic properties of the RA12 compounds (Purwins and Leson 1990). In the present work, the magnetic properties of the RCu 2 are reviewed. Transport properties of these compounds are not particularly considered in this review. For these properties we refer the reader to the review of Gratz and Zuckermann (1982). Thermal conductivity and electrical resistivity of some heavy RCu 2 compounds have been reported by Bartkowski et al. (1992). LaCu2, which crystallizes in the hexagonal A1B2 structure, is not included in this work. EuCu 2 and Y b C u 2 a r e both intermediate-valence compounds and are not considered in detail. We mention that EuCu 2 is known to be antiferromagnetic at 417
418
N.H. LUONGand J.J.M. FRANSE
4.2 K (Sherwood et al. 1964, Wickman et al. 1966). M6ssbauer studies have been performed on this compound by Nowik et al. (1973), Abd-Elmeguid and Kaindl (1978) and Abd-Elmeguid et al. (1981). YbCu2 is paramagnetic (Sherwood et al. 1964). The anomalous properties of Yb-Cu (as well as Ce-Cu) compounds have been reviewed by Bauer (1991). CeCu2 is a Kondo compound, and hence, is discussed not in detail. By lack of information we have also to omit the radioactive compound PmCu2. YCu2 and, occasionally, LuCu2, are treated as reference materials in connection with the other magnetically ordered compounds. Summarising, we will deal with a systematic discussion of the RCuz compounds where R = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Tm. A brief survey on all existing R-Cu compounds is given in section 2. In section 3 we present a short theoretical consideration of crystal-field and exchange interactions as well as of some important magnetic properties. Experimental results on RCu2 compounds are reviewed in section 4. A comparison of different RCu 2 compounds is given in section 5. 2. Brief survey 2.1. Rare-earth and rare-earth-Cu compounds In this chapter, the term lanthanides refers to the elements from La to Lu. Lanthanides to the left of Gd in the periodic table are referred to as the light rare earths, and the remaining ones as the heavy rare earths. The rare earth elements are of fundamental interest because most of them behave as well-defined ions in solids. Exceptions are known to be the elements Ce, Eu and Yb. The number of binary R - T (T = 3d transition metals Mn, Fe, Co, Ni) intermetallic compounds, available at present, amounts to nearly 200 (Franse and Radwanski 1993). As far as the R-C~ compounds are concerned, the nature of the compound formation is summarized in fig. 2.1. The light lanthanides RCu phases as well as YbCu crystallize in the orthorhombic FeB-type of structure. The remaining RCu compounds are formed within the cubic CsCl-type of structure. SmCu and EuCu can adopt both structures. ScCu and YCu have the CsCl-type of structure as well. C e C i l 2 crystallizes in an orthorhombic structure which is the prototype for the RCu2 series. I - ~ C u 2 and S c C u 2 a r e exceptions, forming the hexagonal A1BE-type and the tetragonal MoSi2-type of structure, respectively. The light rare earths RCu5 compounds have the hexagonal CaCus-type of structure, whereas the RCu5 compounds with the cubic AuBs-type of structure are formed for the heavy rare earths. An exception is YbCus, that crystallizes in the CaCus-type of structure. The RCu 6 compounds crystallize in the orthorhombic CeCu6-type of structure and have been reported to exist for the light rare earths from La to Sm. For the heavy rare earths, only Gd and Tb form phases with the RCu6 stoichiometry. The existence of the phases RzCu7, RCu4, R2Cu9 and RCu 7 has been reported for a limited number of rare earth elements (Subramanian and Laughlin 1988).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
I
I
I
I
i
I
I
I
I
I
I
I
I
I
I
I
CsCt
I',, \ \ \ \ \ \ ~ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N
RCu
419
Y////A
g//~
FeB MoSi
liB2 2 ~-
RCu2
I~,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " ,
3 CeCu2 ~-
E
k\\\\\\\\\\\\"
~RCu5
I
k\\\\\\\\\\\\\\"
I
C~Cu5
L~
V////////////A
V ~ AuBe5 .~
(/)
CeCu6
RCu6: I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Sc Y b Ce Pr ~ Pm SmEu Gd Tb Dy ~ Er Tm Yb Lu
Fig. 2.1. Compound formation in the R-Cu systems. After Subramanian and Laughlin (1988).
/
b
F,1 Fig. 2.2. Chemical unit cell of the orthorhombic CeCu2-type of structure. The open and closed circles indicate the rare-earth and copper atoms, respectively. After Hashimoto et al. (1979a).
420
N.H. LUONG and J.J.M. FRANSE TABLE 2.1 Crystallographic data for the RCu2 compounds.
R
Lattice parameters (/~) a b e
Ref.
Melting Reaction point (*C) type
Ref.
Sc Y La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
3.29 4.305 4.346 4.425 4.400 4.387 4.37 4.360 4.45 4.320 4.310 4.300 4.280 4.275 4.266 4.28 4.245
[1] [4] [4] [4] [4] [4] [2] [4] [4] [4] [4] [4] [4] [4] [4] [4] [4]
990 935 834 817 841 840 850 860 597 860 870 890 915 935 960 757 1000
[2, 3] [5] [4] [6] [7] [8] [2] (proposed) [2] [9] [10] [2] [11] [2] [12] [2] [13] [2]
6.800 7.057 7.024 7.059 6.96 6.925 7.25 6.858 6.825 6.792 6.759 6.726 6.697 6.76 6.627
8.388 7.315 3.807 7.475 7.435 7.420 7.40 7.375 7.54 7.330 7.320 7.300 7.290 7.265 7.247 7.40 7.220
congruent congruent congruent congruent congruent congruent congruent peritectic congruent congruent congruent congruent congruent congruent peritectic congruent
[1] Dwight et al. (1967). [2] Subramanian and Laughlin (1988), estimated on the basis of the systematics of crystallographic data in the R--Cu systems. [3] Markiv et al. (1978), quoted by Subramanian and Laughlin (1988). [4] Storm and Benson (1963). [5] Chakrabarti and Laughlin (1981). [6] Rhinehammer et ah (1964). [7] Canneri (1934), quoted by Subramanian and Laughlin (1988). [8] Carnasciali et ah (1983a). [9] Costa et al. (1985). [10] Carnasciali et ah (1983b). [11] Franceschi (1982). [12] Buschow (1970). [13] Iandelli and Palenzona (1971).
2.2. RCu2 compounds The orthorhombic CeCu2-type o f structure has first been determined by Larson and Cromer (1961). The chemistry o f the CeCuz-type structure was the subject o f considerable discussion (e.g., Debray 1973, Michel 1973, Bruzzone et al. 1973). This structure belongs to the space group Imma. The rare-earth atoms and the copper atoms o c c u p y the 4e and 8h sites, respectively, and form a double-layer structure along the c axis as s h o w n in fig. 2.2 (Hashimoto et ah 1979a). Crystallographic data for the RCu2 c o m p o u n d s are collected in table 2.1. As can be seen f r o m this table, the lattice parameters and the melting temperatures o f EuCu2 and YbCu2 are anomalous, indicating the divalent state o f Eu and Yb. For more detailed information we refer the reader to the review paper of Subramanian and Laughlin (1988), in which the R--C-~ phase diagrams have been compiled and evaluated. The Ce--Cu phase diagram is s h o w n in fig. 2.3, as an example.
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
421
Weight Percent Cerium
0 10 20 30 40
50
1200~ 1000
60
70
80
r
~
90 r
100
-~. ./~ 817 C
798"( 726°C
:'%111 `°
-2 r'~
•
E
tOO
II
-
2oo -
I
~I I~I
.~
= _.__L__--- . . . . . . . . . . . . . . . . . . . .
0 ~
0
10
20
Cu
/
72
30
40
50
60
70
~
80
61"C
90
AtomicPercentCerium
100
Ce
Fig. 2.3. The Ce-Cu phase diagram. After Subramanian and Laughlin (1988).
3. Theoretical aspects
3.1. Crystal-field interaction The 4f electrons of a rare-earth ion in a solid, being considered as well localized and separated from other charges, experience an electrostatic potential VcF(r) that originates from the surrounding charge distribution. The potential reflects the point symmetry of the site of the rare-earth ion. In case there is no overlap between this charge distribution and the wave functions of the 4f state, VCF satisfies Laplace's equation and can be expanded in terms of the spherical harmonics, Ynm. The value of n in this expansion is limited to 6, as higher multipoles cannot cause electronic transitions between states of the 4f ion. For details see Hutchings (1964), Fulde and Loewenhaupt (1985). It is an experimental fact that for the rare-earth ion, the spin-orbit interaction is much larger than the crystal-field and exchange interactions. A review of experimental data of the spin-orbit interactions based on the observation of intermultiplet transitions has been presented by Osborn et al. (1991). Owing to this fact, it is usually sufficient to consider the lowest multiplet J, given by Hund's rules. This
422
N.H. L U O N G and J.J.M. FRANSE
limitation results in a substantial simplification in computations. Eu +3 and Sm +3 are ions where the analysis requires the involvement of a higher multiplet since the higher multiplet is only 530 K and 1500 K above the ground multiplet, respectively. The crystal-field Hamiltonian that describes part of the electron-electron interactions in the solid due to the electrostatic interaction of the aspherical 4f charge distribution with the aspherical electrostatic field arising from the surrounding, can be written as: oo
n
AT n=0 ra=0
(3.1) i
where f,,m are Tesseral harmonics describing the spatial distribution of the charge associated with the 4f electrons; the summation over i is over all 4f electrons. Anm describes the spatial distribution of the charge surrounding the 4f electrons. Within the Stevens formalism the summation in eq. (3.1) over the 4f electrons leads to matrix elements of the total angular momentum: n rr~ Z fnra(ri) = On(7 ,4f)On (J'), i
(3.2)
where the x, y, z coordinates of a particular electron in the functions fnm(rl) are replaced by the components Jx, Jy, Jz of the multiplet J. The O F are the Stevens equivalent operators (Stevens 1952). On is the appropriate Stevens factor of order n which represents the proportionality between operator functions of x, y, z and operator functions of Jx, Jy, Jz. The parameter On is denoted as aj, flj, 7J for n = 2, 4, 6. The sign of On represents the type of asphericity associated with each Onm term describing the angular distribution of the 4f-electron shell. In particular, the factor ecj describes the ellipsoidal character of the 4f-electron distribution. For ~j > 0, the electron distribution associated with Jz = J is prolate, i.e. elongated along the moment direction whereas for cq < 0 the 4f-electron-charge distribution is oblate, i.e. expanded perpendicular to the moment direction. For ~j = 0 (which is the case of the Gd +3 ion) the charge density has spherical symmetry. (r~f) is the mean value of the n-th power of the 4f radius. Values for the average value (r~f) over the 4f wave function have been computed on the basis of Dirac-Fock studies of the electronic properties of the trivalent rare-earth ions by Freeman and Desclaux (1979) and are presented in table 3.1. Within the ground-state multiplet the crystal-field Hamiltonian (3.1) is written in the conventional form: oo
HCF = E
n
Z
BnmO~n(Y)'
(3.3)
n m=0
where Bnm are called the crystal-field parameters. Usually these parameters are evaluated from the analysis of experimental data. The parameters Bnm can be written as"
B m = On(r4r)A n n
gll
(3.4)
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
423
TABLE 3.1 The second-, fourth- and sixth-order multipole moments On(r~f) of the trivalent rare-earth ions. cq, flj and "Vl ate the Stevens factors of the second-, fourth- and sixth-order, respectively. The values for (r2), (r~f) and (r 6 ) have been taken after Freeman and Desclaux (1979). The multipole moments values enter into the relations between the CF parameters B m and the CF coefficients Anm. *-values for Pr have been deduced from interpolation. After Franse and Radwanski (1993).
R
S
I~> (~02)
I~> (%,)
I~> (~g)
Ce Pr Nd Sm Gd Tb Dy Ho Er Tm Yb
5/2 4 9/2 5/2 7/2 6 15/2 8 15/2 6 7/2
1.309 1.208" 1.114 0.9743 0.8671 0.8220 0.7814 0.7446 0.7111 0.6804 0.6522
3.964 3.396* 2.910 2.260 1.820 1.651 1.505 1.379 1.270 1.174 1.089
23.31 18.72" 15.03 10.55 7.831 6.852 6.048 5.379 4.816 4.340 3.932
~,l~> z,l~,'~> ,,/~4~,> (10-3%2) (10-,%,) (10-6:) -74.8 -25.54 -7.161 +40.209 0 -8.303 -5.020 - 1.676 +1.831 +6.976 +21.04
+251.7 -24.95 -8.471 +56.527 0 +2.021 -0.891 -0.459 +0.564 +1.916 -18.857
0 +1141.66 -570.96 0 0 -7.683 +6.260 -6.959 +9.969 -24.33 -581.94
in which expression terms related to the 4f ion, On(r'~f), and the term related to the surrounding charges, AT, are separated. The coefficients A T are known as the crystal-field coefficients. Values for On(r~f) have been collected by Franse and Radwanski (1993) and presented in table 3.1. The computation of the CF coefficients, AT, from microscopic, ab initio, calculations is a difficult problem. A full band-structure calculation of the charge distribution over the unit cell and, consequently, of the full set of CF coefficients, is lacking for almost all compounds. In some cases, the point-charge (PC) model, with electron charges centred at the ion positions in the lattice, can give the correct sign of the leading second-order crystal-field coefficients. However, this model is questioned, especially in metallic systems where the contribution of valence electrons is expected to be significant (Schmitt 1979a, b). This has been confirmed by band-structure calculations by Coehoorn (1991) who concluded that the second-order CF coefficient A ° is mainly determined by the asphericity of the valence-shell electron density of the rare earth under consideration. In band-structure calculations for GdCo 5 Coehoorn and Daalderop (1992) have found that, although the on-site contribution to A ° is the dominant effect, the lattice contribution cannot be neglected. The calculation predicts the correct sign and order of magnitude for A ° but the calculated value exceeds the experimental one by a factor of four. This discrepancy remains to be explained. For more detailed discussion we refer the reader to the reviews of Franse and Radwanski (1991, 1993), Givord and Nozieres (1991), Purwins and Leson (1990). There is only a limited number of compounds for which quantitative data for the CF interactions are available. Discussions are still continuing even for the bestknown systems like the cubic Laves-phase RT2 or the hexagonal RNi5 compounds. The situation becomes more complex for the systems with a lower crystal symmetry.
424
N.H. LUONG and J.J.M. FRANSE
The orthorhombic RCu2 compounds, which we are dealing with in this chapter, belong to these latter cases. For cubic symmetry (in case of the RA12 compounds, for instance) the crystal field is described by only two parameters B4 and B6: n c F - B 4 ( O ° -~- 5 0 4) -+ B 6 ( O ° - 21064),
(3.5)
whereas for the orthorhombic symmetry, nine crystal-field parameters B~ are needed: 00
22
00
22
H ~ = B202 + B202 + B404 + B404 + oo
+ B 6 06 +
22 B 606
44
66
44 B404+
(3.6)
+ B6 06 + B 606 •
3.2. E x c h a n g e interactions
Since the 4f electrons are strongly localized, direct exchange of the 4f wave functions of different rare-earth ions is precluded. In order to account for the observed ordering temperatures, one has to resort to an indirect exchange between the rare-earth ions via the conduction electrons. This interaction is known as the Ruderman-Kittel-KasuyaYosida (RKKY) interaction (Ruderman and Kittel 1954, Kasuya 1956, Yosida 1957). Within this approach, a spin of a rare-earth ion interacts with the conduction electrons. The effect of this interaction on the conduction electron density is not the same for the spin-up and for the spin-down electrons. Therefore, a spin polarization is created that interacts with the spin of a neighbouring rare-earth ion. The result is an indirect exchange interaction between the rare-earth ions. The RKKY model does not distinguish d- and s-conduction electrons, in contrast to the model of indirect exchange proposed by Campbell (1972) in which the role of the rare-earth 5d electrons is emphasized. Usually, in the RKKY model the simplifying assumption is made that the exchange integral is of the s-f type and constant and that the Fermi surface is spherical (free electrons). Efforts have been made to overcome the shortcomings of the RKKY model (Kirchmayr and Poldy 1979), but in many cases the improvements achieved are minimal. We list in the following some references in which the indirect exchange interaction in rare earths and rare-earth intermetallic compounds is described: Taylor (1971), Taylor and Darby (1972), Coqblin (1977), Kirchmayr and Poldy (1979), Bnschow (1980), Givord and Nozieres (1991), Franse and Radwanski (1993). The exchange interaction is assumed to be of the Heisenberg type: H e x = --
St,
(3.7)
i,j where the summation is over all the magnetic ions in the lattice and where J q is the exchange parameter between local spins at the sites i and j. Limiting ourself to the ground state, we can project the spin onto the total angular momentum and obtain Hex = - E i,j
JRR(gJ -- 1)zJ'J,
(3.8)
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
425
where J ~ is an effective exchange parameter between the 4f-spins of the rare-earth ions at sites i and j. In the RKKY model, the ordering temperature is expected to be proportional to the so-called De Gennes factor (g - 1)2j (J + 1). In the molecular-field approximation, the exchange interaction of a single R ion with other R ions is approximated by an effective molecular field B,~. Then one has:
Hm = -gj#BJ.Bm.
(3.9)
The field B,~ is related to the molecular-field coefficient naa by
B,,, = ~,,~R(M),
(3.10)
where "N = 2 ( g j - 1)/gj
(3.11)
and (M) is the thermal average of the magnetic moment. Taking into account the Zeeman magnetostatic energy term caused by an external magnetic field B, the Hamiltonian of the magnetic interactions is given by
Hm = --gJ#BJ" (Bin + B).
(3.12)
Thus, the ground state is described by the total Hamiltonian 6
n
HI~ = ~ , ~
B~O~ - g~u~J.(n,,, + n).
(3.13)
n=0 m=0
Eigenvalues and eigenfunctions are obtained from the diagonalization of this Hamiltonian, provided that the relevant CF and exchange field parameters are known.
3.3. Description of magnetic properties In the following we describe some important magnetic properties which are affected by the crystal-field and exchange interactions and mention some properties which are of help to get information on the energy splitting scheme of the rare-earth ion.
3.3.1. Magnetic moment For describing the magnetic moment of a rare-earth ion one has to calculate the thermal average of the magnetic moment. This quantity is given by: 2J+l
(u) = ( i / z ) ~
~ exp(-E, IkT),
(3.14)
i=1
where #i is the magnetic moment of the ith energy level,
#i = -gJ#B(i~]]i)
(3.15)
426
N.H. LUONG and J.J.M. FRANSE
and where Z is the partition function: 2J+l
Z = E exp(-Ei/kT).
(3.16)
i=1
Ei and Ii) are the eigenvalues and eigenfunctions of the Hamiltonian (3.13). Due to the complicated magnetic structure, the analysis of the magnetization process in case of the RCu2 compounds has been done by decomposing the structure into several sublattices (see, e.g., Iwata et al. 1987, 1988, 1989, Divis et al. 1987, Kimura 1985, 1987a, b). In this way the molecular field Bin(i) acting on the ith sublattice can be written as:
Bin(i) : E nq(Mj), J
(3.17)
where (Mj) is the sum of the thermal average of the magnetic moments per ion in the jth sublattice, nq the molecular field coefficients describing the interaction of ith and jth sublattices. The summation is over all sublattices.
3.3.2. Paramagnetic susceptibility In the paramagnetic region the susceptibility can be described by a Curie-Weiss law: C X - T - 0p'
(3.18)
where C is the Curie constant and Op the paramagnetic Curie temperature. Sometimes, the paramagnetic susceptibility can better be represented by a modified CurieWeiss law in the form: C X = X0 + T - 0p'
(3.19)
where the extra term Xo is a weakly temperature-dependent Pauli susceptibility. Using the eigenvalues Ei and appropriate eigenfunctions Ii) of the crystal-field Hamiltonian, HCF, one can calculate the paramagnetic susceptibility, X~F), along the principal axes (a = a, b, e) by the general Van Vleck formula (see, e.g., Zajac et al. 1987):
- ~
~ ( I(ils~li)12÷ (3.20)
+ 2kT ~j¢, ](JlJ~'li)12 -~j---~i j exp(-Ei/kT),
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
427
where J,~ are the components of the operator.J. For a polycrystalline sample the susceptibility arising from crystal-field Hamiltonian, X~, is obtained by the approximation: =
+
+
(3.21)
The crystal-field parameters can be derived from the paramagnetic Curie temperatures 9a, 8b and 8¢ along each principal axis. According to the modified molecular-field model given by Bowden et al. (1971), these paramagnetic Curie temperatures can be expressed in terms of the crystal-field parameters by the following relations (Shohata 1977): Oa = ep +
(2J
1)(2J + 3) (B ° + 10
0b = 0p + ( 2 J - 1)(2J + 3)BO,
5
(3.22)
0c = 0p + (2J - 1)(2J + 3) (B o _ Bo),
10 where B ° and B22 are measured in kelvin. On substituting the measured 0a, 0b and 0e values, the crystal-field parameters B ° and B22 can be derived. The influence of higher-order crystal-field parameters on the paramagnetic Curie temperatures has been studied by Zajac and Maczak (1985). 3.3.3. Specific heat The specific heat is written as the sum of electronic (ce), lattice (phonon) (Cph), magnetic (Cm), and nuclear (e,) contributions: C:
Ce ' + C p h + Cn - ] - c m.
(3.23)
The nuclear part of the specific heat is related to the hyperfine interactions of the 4f shell with the nuclear moment of the 4f ion and is only significant at low temperatures for most of the 4f ions. The largest nuclear contribution is observed in the ordered Ho compounds. This is also the case for the HoCu2 compound (Luong et al. 1985b). The electronic part is written as: ce = 7T
(3.24)
with the coefficient 7 yielding information on the density of states at the Fermi level. The experimental value for the electronic coefficient 7 from specific-heat measurements should be compared with -/-values from band-structure calculations. For the RCu2 compounds, band-structure calculations have been performed for YCu2 (Harima et al. 1990, Harima and Yanase 1992). The resulting value for the specific heat coefficient 7 is 3.47 mJ/K2 mol, which is about two times lower than the experimental value of 6.7 mJ/K2 mol (Luong et al. 1985a).
428
N.H. L U O N G and J.J.M. FRANSE
The phonon part of the specific heat (for a compound with r atoms per formula unit) is approximated by: Cph = 9 r R ( T / O D ) a
00D/T z4ez
(e~ _ 1)2 dz,
(3.25)
where OD is the Debye temperature, R the gas constant. In compounds with a magnetic R element the magnetic contribution, Cm, is associated with the increasing population of excited localized states. These localized states are due to CF and molecular field interactions of the 4f ion which lift the ( 2 J + 1)-fold degeneracy of the ground-state multiplet. Provided that the energy-level scheme is available, it is straightforward to evaluate this contribution by using the expression:
cm(T)
(3.26)
-02F = - - T OT2,
where F is the free energy of the R system. The reverse procedure is rather problematic. Contributions like the electronic and lattice contributions have to be evaluated and subtracted. However, in general it is difficult to separate the magnetic part from the other contributions. Isostructural compounds with non-magnetic R elements are usually employed to estimate the non-magnetic part of the specific heat. In the RCu 2 compounds, LaCu2 possesses a different crystallographic structure and crystallizes in the hexagonal AIB2-type of structure. Therefore, YCu2 is taken as the non-f reference material instead of LaCu 2 and data for YCu 2 are often used in the analysis of the experimental data on RCu 2 compounds. In the paramagnetic state the magnetic part of the specific heat is given by the Schottky contribution (Fulde 1979): CSeh = ~
Z
exp(-Ei/kr)
-
Ei e x p ( - E i / k T )
.
(3.27)
In the simple case of a two-level system, the Schottky specific heat attains a maximum value at an intermediate temperature Tm given by (Gopal 1966):
(g0/al) exp(6/Tm) = [(6/Tm) + 21/[(6/T.) - 21,
(3.28)
where 6 is the energy separation (measured in kelvin) between the ground state and the first excited state; go and gl denote the degeneracies of these two levels. Provided that the temperature Tm and the maximum value of the specific heat, cSch(Tm), are known, one can evaluate 6 according to the expression (Gopal 1966): 6 = [4CSch(Tm)/R + 411/2Tm.
(3.29)
In general, relevant information about the lower part of the energy-level scheme can be obtained from the specific-heat measurements. Experiments in external magnetic fields can give further information as the applied field additionally shifts the energy levels. The shift is not uniform for all levels and is largely dependent on the direction of the applied field. This effect can be detected by the measurements of the specific heat on single-crystalline samples.
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
429
3.3.4. Thermal expansion The thermal expansion contains electronic (fie), lattice (phonon) (flph), and magnetic, (tim) contributions:
= Ze + Zph + t - ,
(3.30)
where we neglect a nuclear contribution. The conduction electrons contribute to the thermal expansion through the volume dependence of their entropy. This entropy is related to the density of state at the Fermi surface. The electronic part of thermal expansion is a linear function of temperature, i.e.
fie = aT.
(3.31)
The phonon contribution to the thermal expansion, like the contribution to the specific heat, is approximated by: ~ph = b(T/OD)3
fOOD/T z4ex (e~'_~])2
dx,
(3.32)
where b is a numerical constant. In order to evaluate the magnetic contribution to the thermal expansion, nonmagnetic (electronic and lattice) contributions have to be subtracted from total thermal expansion as in the case of the specific heat. Isostructural compounds with nonmagnetic R elements are used to evaluate this non-magnetic part of the thermal expansion. For RCu 2 compounds, like in case of the specific heat, YCu 2 often serves as the reference material. In the paramagnetic region, the magnetic part of the thermal expansion arises from the thermal excitation of a series of energy levels E0, E b . . . with degeneracies go, g b . . . . This (Schottky) contribution is given by (Barron et al. 1980):
2
(Ei) (E¢/i) },
(3.33)
where 7i = - ( d In Ei/d In V) is the crystal-field Grtineisen parameter of the individual energy level Ei and where ~T is the isothermal compressibility. The brackets denote thermal averages of the form:
(x,) - E glz, e x p ( - E j k T ) / E g' exp(-EjkT). i
(3.34)
i
3.3.5. Graneisen analysis In the study of magnetic systems the specific heat and the thermal expansion are very important. The combined analysis of specific heat and thermal expansion can give valuable information on the system under consideration. Here we briefly describe a procedure that has successfully been applied to several different systems.
430
N.H. LUONG and J,J.M. FRANSE
An arbitrary contribution to the specific heat, el, is related to a corresponding contribution to the thermal expansion, ti, by a so-called Griineisen relation:
.l-'i = Vti/tgei,
(3.35)
where V is the molar volume, t¢ the compressibility and/~i the appropriate Griineisen parameter. For the electronic Griineisen parameter we have (see eqs (3.24) and (3.31)):
-re = V a / x T .
(3.36)
Using eqs (3.25) and (3.32), for the lattice Griineisen parameter we obtain:
l"ph = V b / 9 r Rt¢.
(3.37)
In treating the magnetic contributions to the specific heat and to the thermal expansion, a straightforward approach is to calculate an effective Griineisen parameter, Pen, by the relation: Feff = V t m / t C e m
(3.38)
and to follow its variation with temperature. A pronounced temperature dependence of Fee(T) indicates the presence of several contributions. ErCu2 can serve as an example, in which a change in sign in the parameter Feff is observed upon increasing the temperature (see section 4.9). In this case, at least two different contributions to em and tim can be distinguished. Therefore, we write: e m = err + Cef and
t m = t l r "lt- t e f
(3.39)
in which the Cm and tm are split up into two contributions, the 'long-range' magneticorder contributions % and tmr and the contributions eel and t~f associated with the crystal-field splitting of the energy levels, respectively. We have:
fiFi,
Feff= E
i = lr, cf,
(3.40)
i
with F~ and F~f, in first approximation, temperature-independent Grtineisen parameters and with fi = ci/em. The separated terms in Cm and tm can subsequently be written as: /"lr -- Fef cm'
(3.41)
ccf - /'lr - Fee era,
(3.42)
Clr =
1 - F~f/Fe~
ttr -- ~ - ~
tin,
F~f 1 - F e f d r t r
tef = Fe----ff~ -- ~
(3.43) tim.
(3.44)
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
431
Fransse et al. (1985) and Luong et al. (1985b) applied this method in the analysis of the ErCu2 compound (see section 4.9). The method of the combined analysis of specific heat and thermal expansion has been described in more detail by Brommer and Fransse (1990) and successfully applied to a variety of materials (see Brommer and Fransse 1990).
3.3.6. Inelastic neutron scattering Inelastic neutron scattering (INS) gives valuable information on the dynamic properties of the system. INS techniques enable to determine the crystal field scheme of the ground state multiplet of a R a÷ ion as the available neutron energies are of the order of this splitting. The scattering amplitude S(q, w) as function of the momentum transfer q and the energy transfer hw, depends on the matrix element and is given by:
S(q, w) ,~ I(il&LIf)[ z,
(3.45)
where Ja. is the component of the total angular momentum operator perpendicular to q. (i[ and If) denote the initial and final 4f-electron states (= CF levels), respectively. Within the dipole approximation, the intensities for the transitions are determined by the corresponding dipole matrix elements. For a review of the INS spectra in metallic 4f systems see, for instance, Fulde and Loewenhaupt (1985). The INS spectra of a system containing 4f ions depend very much on the relative strength of the CF and exchange interactions. The spectra belonging to only CF transitions contain a few well-defined peaks related to magnetic excitations between CF levels. The number of the transitions as well as the probability for these transitions is determined by matrix elements between the crystal-field eigenstates. In principle, these are governed by the local symmetry of the CF interactions. These transitions are typically single-site excitations. Including the exchange interactions, the transitions on neighbouring ions are coupled and lead to propagating modes. The spectra become more complex and the analysis is not straightforward. For RCu2 compounds, and NdCu 2 is a good example, at least two INS experiments (one above and one below antiferromagnetic ordering temperature) are needed to detect deviations from the CF-only case in the excitation spectrum (Loewenhaupt 1990). For orthorhombic and lower symmetries, the J ground-state multiplet splits into (2J + 1)/2 doublets for half integer values of J and into (2J + 1) singlets for integer d. In the simplest case (Ce 3+, J = 5/2) this gives a level scheme with 3 doublets and 3 possible inelastic transitions. In the most complicated case (Ho 3+, d = 8) a level scheme consists of 17 singlets and up to 136 inelastic transitions are possible (Loewenhaupt 1990). Loewenhaupt (1990) has also pointed out that not all transitions may have non-zero matrix elements but already the clear identification of 10 or 20 inelastic CF transitions is a great challenge for magnetic neutron spectroscopy.
3.3.7. MOssbauer spectroscopy Mrssbauer spectroscopy provides information on at least two important parameters: eft, and the quadrupolar splitting. the hyperfine field, Hhf
432
N.H. LUONG and J.J.M. FRANSE
The hyperfme field is used to evaluate the value for the magnetic moment of the unfilled 4f or 3d shell, usually by taking it directly proportional to the magnetic moment (as long as the additional contributions originating from core polarization and conduction electrons can be neglected). In that case, the temperature dependence of H ~ reflects the temperature dependence of the (4f or 3d) magnetic moment. The quadrupolar splitting gives information about interactions between the quadrupolar moment of the nuclei and the gradient of the electric field. The gradient of the electric field at the rare-earth nuclei results partly from the quadrupolar moment of the 4f shell and partly from that of the surrounding charges. Having an estimate of the 4f-shell contribution, the quadrupolar splitting arising from the lattice-charge arrangement can be evaluated and transformed into the crystal field coefficient A °. Higherorder CF coefficients cannot be determined by the Mrssbauer spectroscopy because of the much lower intensity of the higher-order multipolar interactions. Mrssbauerspectroscopy data for the RCu2 compounds have been presented by Gubbens et al. (1991, 1992).
4. Magnetic properties of RCn2 compounds 4.1. CeCu2 CeCu2 is a Kondo-lattice compound which shows antiferromagnetic order below 3.5 K (Gratz et al. 1985, Onuki et al. 1985). Earlier magnetic measurements have been reported by Sherwood et al. (1964), Olcese (1977) and Hashimoto et al. (1979b). Owing to the interesting magnetic behaviour, the compound has been extensively investigated. A characteristic feature of CeCu2 is the large value for the coefficient of the electronic contribution to the specific heat (between 80 and 105 mJ/K2 mol) (Gratz et al. 1985, Bredl 1987, Bauer et al. 1988, 1989 and Takayanagi et al. 1990). The specific heat of CeCu2 as a function of temperature is shown in fig. 4.1 (Gratz et al. 1985). Althouth magnetic ordering is observed in properties as specific heat and resistivity, a clear antiferromagnetic transition is not found in the low-temperature magnetic susceptibility (Onuki et al. 1990, Trump 1991 and Trump et al. 1991). Figure 4.2 shows the temperature dependence of the magnetic susceptibility in CeCu2 (Onuki et al. 1990). As can be seen from this figure, the susceptibility along the c axis shows a broad maximum near the Nrel temperature, indicating that the magnetic moments are along the e axis, in agreement with the neutron diffraction results of Trump et al. (1991) and Nunez et al. (1992) (see below). The magnetic susceptibility is highly anisotropic over the whole temperature range as revealed from the experiments (Onuki et al. 1985, Takayanagi et al. 1986, Onuki et al. 1990, Trump et al. 1991). However, the sequence of easy-intermediate-hard directions in CeCu2 is not quite clear. From magnetic measurements, Gratz et al. (1985) have determined an effective moment, #etI, of 2.45#B and a paramagnetic Curie temperature, 0p, of 30 K. The magnetization of CeCu2 has been measured on a polycrystalline sample by Hashimoto et al. (1979b) and by Gratz et al. (1985) and on a single-crystalline
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
ceCuz
_
0.5t-
_
433
Z
-,=
¢.i
2
0
0
1
2
3
/,
5
6
T (K) Fig. 4.1. Specific heat of CeCu 2 as a function of temperature. Inset: low-temperature data in a plot of e/T versus T2; the full curve represents e/T = "3'+ AT2exp(-AE/kT) with 7 = 82 mJ/K2 mol, A = 1.4J/K4 tool and AE/k = 1.2K. After Gratz et al. (1985).
0.3
..3 0
--
0.2
CeCu2
==
°~%,,o
QU
",o
0.1 c
0
i
i
J
5
10
T (K) Fig. 4.2. Temperature dependence of the magnetic susceptibility in CeCu 2. After Onuki et al. (1990).
434
N.H. LUONG and J.J.M. FRANSE
2.0 CeCuz
a
'6 ~
1.0
0
0
~
8
B(T) Fig. 4.3. Magnetization along each principal axis in CeCu 2 at 1.3 K. After Onuki et al. (1990).
sample by Onuki et al. (1988, 1990) and by Trump et al. (1991). Figure 4.3 shows the magnetization of single-crystalline CeCu2 at 1.3 K (Onuki et al. 1990). At a field of 1.7 T, the magnetization along the a axis exhibits a change of slope, indicating a metamagnetic phase transition. The magnetization is again highly anisotropic. Trump et al. (1991) have reported that a metamagnetic transition occurs along the a axis at a field of 1.1 T at T = 2 K. The metamagnetic behaviour of CeCu2 has received considerable attention (Gratz et al. 1985, Onuki et al. 1988, 1990, Satoh et al. 1990a, Takayanagi et al. 1990, 1991, Trump et al. 1991). Gratz et al. (1985) have reported neutron-diffraction studies on CeCu2. However, the exact nature of the magnetic ordering could not be determined. Later on, the magnetic structure of CeCu2 was studied by Trump et al. (1991) and Nunez et al. (1992). Figure 4.4 shows the magnetic structure of CeCu2 (Trump et al. 1991). In this structure, the magnetic moments of the Ce atoms are parallel and antiparallel to the c axis (Trump et al. 1991 and Nunez et al. 1992). The magnetic moment of the Ce atoms amounts to 0.33#n at 2.5 K (Nunez et al. 1992). Crystal-field effects in CeCu2 have been studied by different methods. From inelastic neutron-scattering experiments, Loewenhaupt et al. (1988) deduced excited levels at 9 meV (105 K) and 23.1 meV (268 K). Morin et al. (1992) have observed one CF transition at 23.4 meV. This value is very close to that obtained by Loewenhaupt et al. (1988). Morin et al. (1992) reported, however, that there does not exist another clear excitation from the ground state. Uwatoko et al. (1990a,b, 1991, 1992) have measured the thermal expansion of single-crystalline CeCu2. Assuming that the phonon contribution to the thermal expansion of CeCu2 equals that of YCu2, the
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
435
%c°I i! t.J
<---- a = 4.t.2 ~, ----> Fig. 4.4. Proposed magnetic structure of CeCu 2. After Trump et al. (1991).
magnetic contribution to the thermal-expansion coefficient of C e C u 2 , av(mag), has been estimated as: o t v ( m a g ) = c z v ( C e C u 2 ) - o~v(YCu2).
(4.1)
Figure 4.5 shows the temperature dependence of C~v(mag). In this figure, the calculated magnetic thermal-expansion coefficient of CeCu2 is also shown (see Uwatoko et al. 1991, 1992 and references therein). The level scheme used in these calculations is t/71 = 106 K and E2 = 334 K, which values are in reasonably good agreement with whose obtained by Loewenhaupt et al. (1988) from neutron scattering and by Loewenhaupt et al. (1990) from thermal-expansion measurements. However, the level scheme E1 = 10 K and t/72 = 800 K which has been used by Takayanagi et al. (1986) for describing the temperature dependence and the anisotropy of the magnetic susceptibility of CeCu2, disagrees with the neutron-scattering results. From de Haas-Van Alphen studies, Satoh et al. (1990b) have shown that the cyclotron effective masses of CeCu2 range from 0.5 to 5.3m0, which values are larger than those found in YCu2 (0.1--0.7m0). This underlines the heavy-fermion state of CeCu2. The light masses of Y C u 2 a r e consistent with the small specific-heat coefficient 7= 6.7 nd/K 2 mol in YCu 2 obtained by Luong et al. (1985a, b). The results of de Haas-Van Alphen measurements of Settai et al. (1990, 1992) agree with those obtained by Satoh et al. (1990b).
436
N.H. LUONG and J.J.M. FRANSE
12 lo 8
CeCuz
o ~
4
-2 0
I
I
100
I
I
200
Fig. 4.5. Temperaturedependenceof the magnetic part of the thermal-expansion coefficient,c~v(mag) = av(CeCu2)- txv(YCu2). The circles representthe measured thermal-expansioncoefficient. The solid line is a calculatedresult for the magnetic thermal-expansion coefficient. After Uwatokoet al. (1991). The effect of substituting Y for Ce in CeCu 2 has been studied by Bauer et al. (1989). Long-range magnetic order in CeCu2 rapidly disappears with increasing Y content. The replacement of Cu by Ag in CeCu2 leaves the orthorhombic structure unchanged (Iandelli and Palenzona 1968). Bauer et al. (1990) and Bauer (1991) have substituted Ag for Cu in CeCu2. On both the Cu-rich and Ag-rich sides longrange magnetic order is observed. In the middle of the concentration range some sort of spin-freezing phenomena is supposed to be present (Bauer 1991). The series Ce(Cu~,Gal_x)2 (0.5 x< z x< 1) has been studied by Bauer et al. (1988) and Bauer (1991). The transition temperature shifts from 3.5 K for z = 1 to about 1.9 K for z = 0.95. For higher Ga contents no indications of long-range magnetic order were resolved down to 1.5 K. The magnetic properties of Ce(Znl_~Cu~)2 have been investigated by Morin et al. (1992). When Cu is replaced by Ni in Ce(CUl_~Ni~)2 the orthorhombic phase is found in the range 0 ~< x ~< 0.5 (Olcese 1977).
4.2. PrCu2 Magnetic-susceptibility measurements performed on polycrystalline PrCu2 by Andres et al. (1972) (fig. 4.6) show a nearly temperature-independent Van Vleck paramagnetic behaviour below 4.2 K, confirming that the lowest crystal-field state is a singlet. The specific heat anomaly around 7 K (fig. 4.6) cannot be due to the onset of magnetic order, since there is no susceptibility anomaly around this temperature. This specific-heat anomaly will be discussed in more detail below. Andres et al. (1972) have shown that PrCu z exhibits cooperative nuclear antiferromagnetic order below 54 mK (see fig. 4.7). Using a molecular-field approximation, Andres (1973) has analyzed the nuclear magnetic order in PrCu2. The calculated ordering temperature is between 10 mK and 1 K. The magnetization at 4.2 K and the susceptibility along each principal axis for a single-crystalline sample has been measured by Hashimoto (1979) and Hashimoto
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
437
50 ~0
1.5
m: iJ
30
"-6 E
20
"3"
10
0
10
20
30
40
T(K) Fig. 4.6. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 above 2 K. After Andres et al. (1972).
.15
.14 T
1.0
w
.13 E
;~ .12
.11 '
~ .I
~' .2
21 .3
i
o--- 0
.4
T (K) Fig. 4.7. Molar susceptibility (closed circles) and specific heat (open circles) of PrCu2 below 0.5 K. After Andres et al. (1972).
et al. (1979b) (figs 4.8 and 4.9). The results show a large anisotropy in the paramagnetic state. The sequence of the easy to hard magnetization direction at low temperatures is the a, b and e axes, which is different from that obtained at high temperatures: the a, c and b axes. The effective magnetic moment,/ten, amounts to 3.4#a and the paramagnetic Curie temperatures along the a, b and c axes are: 0a = 14 K, Ob = --107 K and Oe = - 3 2 K (Hashimoto et al. 1979b). From these values for the paramagnetic Curie temperatures Hashimoto et al. (1979b) have estimated the second-order crystal-field parameters for PrCu2: B ° -- 4.27 K, and B 2 = 2.97 K. By using a point-charge model, Hashimoto et al. (1979b) have also
438
N.H. LUONGand J.J.M. FRANSE
2
PrCuz Q
I
b c
O0
1
5
3 B (T)
Fig. 4.8. Magnetization curves along the principal axes for PrCu 2 at 4.2 IC After Hashimoto et al. (1979b).
10 x104" PrC u2
8
c
"T
~6 E
7o~4
,," ""
0
I
I
I
100
200
300
T (K) Fig. 4.9.
Temperature dependence of the inverse susceptibility along the principal axes of PrCu 2. After Hashimoto et al. (1979b).
calculated the values of B ° and B 2 and arrived at B ° c a l = 4.1 K and B E c a l = 3.48 K, in good agreement with experiment. Wun and Philips (1974) have also measured the specific heat of PrCu2 (fig. 4.10). They have shown that the 7 K anomaly is much sharper than expected for a twolevel Schottky anomaly. The shape of the anomaly cannot be produced by fixed crystal-field levels, but suggests, instead, that it arises from a cooperative transition. These authors attributed this anomaly to Jahn-Teller distortions. The microscopic picture of this class of structural transitions involves phonon-mediated quadrupole interactions between rare-earth ions (Kjems et al. 1978).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
439
12 10
* -
PrCuz MoteculnrFietd /
........ Schoffky
,7
Cuz
--.--Lo
1
oo07
0
I./
O
.-"'"
O O
... .,'"
•/"
5
10
TIK) Fig. 4.10. The zero-field specific heat of PrCu 2 between 0.3 and 10 IC After Wun and Philips (1974).
Thermal-expansion, susceptibility, resistivity and thermal-conductivity measurements performed by Andres et al. (1976), Ott et al. (1977) on single-crystalline PrCu 2 have confirmed that PrCu2 exhibits a cooperative Jahn-Teller effect at TD = 7.5 K. Figure 4.11 shows the temperature dependence of the linear thermal-expansion coefficients, ~i, along the three principal axes in PrCu2 (Ott et al. 1977). Distinct extremal values of oq are observed around 7.5 K giving evidence for an actual distortion of the crystal lattice. The structural distortion below TD has also been measured by neutron diffraction (Kjems et al. 1978). Inelastic neutron scattering in single-crystalline PrCu2 has been performed by Ott et al. (1978) at temperatures below and above the Jahn-Teller transition temperature. It has been shown that the most intense transition occurs at 1.3 meV (15 K). It is assigned as a transition from the ground state to an excited state. The effect of the crystal field on the paramagnetic susceptibility of PrCu2 has been studied by Zajac (1981). In neutron-diffraction measurements, Kawarazaki et al. (1984) and Nicklow et al. (1985) have observed magnetic order in PrCuz below 58 mK. These authors have also reported the magnetic structure of PrCu2 (fig. 4.12). Nicklow et al. (1985) have pointed out that the magnetic structure (electronic + nuclear) is incommensurate with the chemical lattice. Further neutron-diffraction experiments performed by Kawarazaki and Arthur (1988) (part of this work has been published earlier, see Kawarazaki and Arthur 1986) reveal that in the ordered state electronic and nuclear
440
N.H. LUONG and JJ.M. FRANSE
160 140
Pr Cuz
IZO 100
•
o//a
•
e//b
I~ •
o#c
$
80
•
6O
20
-20 -40 -60
I
I
I
2
/,
6 8 T(K}
I
I
I
I
10 12 1/+
Fig. 4.11. Linear thermal-expansion coefficient oti along the principal axes in PrCu2. After Ott et al. (1977).
o~o
\~
•
q
'
o o
\
\
'\
r o Pr • ru
oo
I~
C
---I
Fig. 4.12. Projection of the PrCu2 structure onto the (a, e) plane. The orthorhombic cell is indicated by the dashed lines. The vector ~ gives the direction of propagation and wavelength of the modulated magnetic structure. The indicated diagonal planes are parallel to (103) planes. After Kawarazaki et al. (1984).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
441
PrCu2 *+#~ ~,~ 0:0^.
200
÷4 o'0*°°
÷°
~..~0 ~ ~o ° x~ v
150
E
.. B:OT + B=3 T ° B: 5 T x B=TT
o
~'~x
,,~0 #
"...........,'
Yd lOO
50 NbTi
4
5
6
7
8
9
10
T (K)
Fig. 4.13. Specific heat (per unit volume) of PrCu2 in applied magnetic fields. For comparison the specific heat (per unit volume) of NbTi and Cu is also shown. After Kwasnitzaet al. (1983). moments of the praseodymium atoms are sinusoidally modulated in magnitude and oriented approximately parallel to the crystallographic a axis. PrCu2 may find its way to the applications, for instance, by stabilization of superconductors. Kwasnitza et al. (1983) and Barbisch and Kwasnitza (1984) have measured the specific heat of PrCu2 in magnetic fields up to 7 T (fig. 4.13). The specific heat has a broad maximum around 6 K with values (per unit volume) being about two orders of magnitude larger than those of NbTi (technical superconductor) and copper. An applied magnetic field does not destroy these very high specific heat values. Thus PrCu2, like some other materials (NdSn3, Nd0.9Pr0.1Sn3 and PrB6, see Kwasnitza et al. 1983), is a good candidate for superconductor stabilization. First attempts for application were made (Barbisch and Kwasnitza 1984). These authors have pointed out that PrCu2 powder would be advantageous as a filler in the epoxy used for coil impregnation.
4.3. NdCu2 The antiferromagnetic ordering temperature of NdCu 2 compound is 6 K (Hashimoto et al. 1979b, from ac susceptibility measurements) or 6.5 K (Gratz et al. 1991, from specific heat and resistivity measurements). From specific heat and resistivity measurements (see below), Gratz et al. (1991) have observed a spin reorientation at the temperature, Ts, equal to 4.1 K.
442
N.H. LUONG and J.J.M. FRANSE
The magnetic structure of NdCu2 is complex as revealed from recent neutron diffraction study in the temperature range from 1.4 K up to 8 K in zero external field (Arons et al. 1994). These authors show that two magnetic phases were observed between 1.4 K and TN = 6.5 K. In both phases the magnetic structure can be described by assuming a sinusoidal oscillation of the Nd moments which are oriented along the b direction. In the high-temperature phase between 5.2 K and TN, the magnetic structure is incommensurate with the lattice. In the low-temperature phase (below 4 K) the structure becomes commensurable. Around 4.4 K the neutrondiffraction results suggest a coexistence of the high- and low-temperature phases. The temperature dependence of the inverse susceptibility along each principal axis of NdCu2 is shown in fig. 4.14 (Hashimoto et al. 1979b). The effective magnetic moment,/~eff, amounts to 3.5/~B and the paramagnetic Curie temperatures along the a, b and c axes are: Oa = 17 K, Ob = - 1 6 K and 0e = - 4 K (Hashimoto et al. 1979b). The second-order crystal-field parameters B2° and B22 for NdCu2 were first estimated by Hashimoto et al. (1979b) from measurements of paramagnetic susceptibility in a single-crystalline sample: B ° = 0.8 K, B22 = 1.1 K. Hashimoto et al. (1979b) have also calculated the second-order crystal-field parameters using a point-charge model and obtained: B ° eal = 1.17 K and B 2 eat = 1.01 K. Inelastic neutron scattering investigations on polycrystalline samples have recently been reported by Loewenhaupt (1990) and Gratz et al. (1991). Figure 4.15 shows the inelastic neutron spectra in the paramagnetic state for Ndcu2 (Gratz et al. 1991). These authors observed four inelastic magnetic transitions which were interpreted as crystal-field transitions from the ground-state level to excited crystal-field levels at 34, 58, 84 and 164 K (see inset in fig. 4.15). This is expected for the Nd ion in an orthorhombic symmetry: a splitting of the J = 9/2 ground state into five doublets. Taking into account the values of B ° and B22 obtained by Hashimoto et al. (1979b) 10
"T
xl0/*
8
. ~ N2d C u
_~c
~-1 6 E
~g
"To~L~ 0
0
~
I
100
I
200 T (K)
I
300
Fig. 4.14. Temperature dependence of the inverse susceptibility along the principal axes of NdCu 2. After Hashimoto et al. (1979b).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
140
120
7
100
T=IOK
K
o
Eo=12 m e V
164
x
Eo=17 rneV
*
Eo=50 rrteV
443
meg
84 80
0®
7
®xx
,. 34
60
¢lr co
OLrue' Sehe'ra.e
x
40
_J
20 0 -5
® "
0
I
[
t
5
10
15
ENERGY
TRANSFER
.. .. .. ,
20
(rrte~
Fig. 4.15. Inelastic neutron spectra of NdCu 2 in the paramagnetic state (T = 10 K) obtained with different incident neutron energies E0. The deduced crystal-field transitions are given by the level scheme in the inset. The nine crystal-field parameters Bnm have been derived with the superposition model from these data (full curve). After Gratz et al. (1991). and using a superposition model ( N e w m a n and N g 1989, Divis 1991), Gratz et al. (1991) have derived the following set o f crystal-field parameters which best describe the inelastic neutron-scattering data:
B ° = (1.35 + 0.06) K, B g = (1.56 + 0.08) K, B ° = (2.23 + 0.12) K, B 2 = (1.01 + 0.2)K, B44 = (1.96 + 0.42) K, B6° = (4.89 + 0.43) K, B 2 = (1.35 + 0.45) K,
444
N.H. LUONGand J.J.M. FRANSE
B64 = (4.89 + 0.43) K, B66 = (4.25 + 0.19)K. Magnetization isotherms of single-crystalline NdCu2 measured at 2 K by Svoboda et al. (1992) are shown in fig. 4.16. These results as well as results of Hashimoto et al. (1979b) reveal that metamagnetic transitions are observed only in the b direction, whereas in the a and e directions no transitions are detectable. Bozukov et al. (1992) have performed magnetization measurements on polycrystalline NdCu2 in magnetic fields up to 28 T (see fig. 4.17). They have found a further transition around 12 T. Above 25 T a ferromagnetic spin arrangement seems to exist. At this magnetic-field value, the magnetic moment saturates to the value of 1.9/~B/ion which is about 60% of the gJ value for the trivalent Nd ion (Bozukov et al. 1992). Magnetization measurements of Bozukov et al. (1992) at 1.8 and 4.2 K have shown that the lowest lying field-induced transition disappears at temperatures above 4.2 K. This observation indicates the existence of a spin reorientation in the temperature range 1.8-4.2 K. As mentioned above, measurements without an external magnetic field (specific heat, resistivity) revealed a spin reorientation at the temperature Ts equal to 4.1 K (Gratz et al. 1991). Figure 4.18 shows the temperature dependence of the specific heat of NdCuz and of non-magnetic isostructural LuCu2 (Gratz et al. 1991). From these measurements the N6el temperature and the spin-reorientation temperature have been determined. The electronic specific-heat coefficient 7 is estimated to be around 8-12 mJ/K2 mol, comparable with 7(YCu2) = 6.7 mJ/K2 mol (Luong et al. 1985a) and ?(TmCu2) = 9 mJ/K2 mol (Sima et al. 1988). The magnetic contribution to the specific heat, era, is deduced as: em= c(NdCu2) - c(LuCu2)
(4.2)
2.0 b
Nd Cuz
1.5 i= 1.0 0.5
0.0 0
1
2
3
t~
5
B (T)
Fig. 4.16. Magnetizationisothermsalong the principal axes of NdCu2 at 2 K. After Svoboda et al. (1992).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
445
2.0 1.6
13K 16 K
-6 1.2 l::: .~
/
//
0.8 NdCuz 0.~
~" ""
,_ o., perma.nenf field puLsed field I
0.0
I
S
0
l
I
I
15 B (r)
25
Fig. 4.17. Magnetization curves of NdCu 2 at various temperatures in pulsed fields up to 28 T (solid curves) and in steady fields up to 14 T (dashed curves). After Bozukov et al. (1992).
50
ooO°~
NdCu2 o°°°° "*"t"
L,O
0 O0
O
o
,0°
oo
20 r L
*****
,
e"
e,~
0°1
eeeee
°oo°
30
/
tl°
LuCu2
_."
oO°°°~"/
10
0
10
20
30
,
,
~0
50
60
T (K) Fig. 4.1& Temperature dependence of the specific heat of NdCu 2 and L u C u 2. After Gratz et al. (1991).
and is given in fig. 4.19 as a function o f temperature, cm does not vanish above Try, but shows a broad and pronounced m a x i m u m around 20 K. In this figure, the calculated Schottky a n o m a l y as obtained by using the crystal-field eigenvalues Ei f r o m the neutron experiments is included (Gratz et al. 1991). The curvature of em above
446
N.H. LUONGand J.J.M. FRANSE
10
(experiment)
cm
i 0-~0"0-0-0.0 0
go 06
o
oI
E
/0
1.0
o/°"/ ./~CF ~
(theory)
t~ E t.I
/ ~i'
NdCu2
/ I
I
5
10
I
15 T (K)
I
20
25
Fig. 4.19. Temperaturedependenceof the magnetic contributionto the specific heat, era, of NdCu2, together with the calculated Sehottky specific heat, eCF, which is obtained by using the crystal-field eigenvaluesEi from neutron experiments. After Gratz et al. (1991). TN is clearly related to crystal-field effects, but there are also hints to correlations between the Nd 4f moments, because of the difference between the experimental data and the calculated Schottky peak. Gratz et al. (1991) have calculated the temperature variation of the magnetic entropy Sm. It is shown that Sm does not reach the theoretical value of R In 2 at TN, as one would expect for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet. They attributed this reduction of the entropy to short-range correlation effects in the paramagnetic state. The thermal expansion of NdCu2 has been measured by Gratz et al. (1991). The excess thermal expansion Ac~(T) was deduced by comparing the thermal expansion of magnetic NdCu2 and non-magnetic LuCu2 as: Ao~(T) = c~(NdCu2) - o~(LuCu2) ~-, Otsc(T) + ~cF(T),
(4.3)
where ase is the contribution due to spin correlations either in the ordered state or in the paramagnetic state and ~CF is the crystal-field induced contribution to the thermal expansion. Based on a model for magnetoelastic interactions described by Divis et al. (1990a), the temperature variation of ac~ was calculated and compared with Aa in the paramagnetic state (Gratz et al. 1991) (see fig. 4.20). The temperature variation of the lattice parameters a, b and c in NdCu2 in the temperature range from 4.2 K to 300 K has been reported by Gratz et al. (1993).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
__" Z~a
150 l "
T
447
100
°~CF
~
go -50 0
..~--.~.~. -~' I 50
l 100
~ 150
~ 200
) 250
300
T(K) Fig. 4.20. Temperaturevariationof the spin-dependentcontributionto the thermal expansion(symbols) in NdCu2. The full curve shows the calculatedinfluenceof the crystal field on the thermal expansion. After Gratz et al. (1991). 4.4. SmCu 2
The N6el temperature of SmCu2 compound is TN = 21.7 K as revealed from susceptibility, specific-heat and resistivity measurements performed by Isikawa et al. (1988) on single-crystalline samples. A value for TN of 23 K has been obtained by Gratz et al. (1990) from specific-heat and susceptibility measurements on a polycrystalline sample. Isikawa et al. (1988) have shown that the temperature dependence of the specific heat exhibits, apart from a peak at TN, two distinct anomalies at 9.4 and 16.5 K, while Gratz et al. (1990) did not observe the anomaly around 9 K. The effective magnetic moment, /ze~, amounts to 0.53/ZB; the paramagnetic Curie temperature, 0p, equals - 1 4 K (Gratz et al. 1990). In SmCu2, the Sm ion is in the trivalent state as is concluded from a lattice parameters comparison (Gratz et al. 1990) and from the experimental values of the susceptibility above TN, as obtained by Isikawa et al. (1988). Figure 4.21 shows the magnetization curves along the a and b axes, as obtained by Maezawa et al. (1989). It can be seen from this figure, that a spin-flop transition from the antiferromagnetic state to the ferromagnetic state occurs at 22.5 T. In fields above 23 T, the magnetic moment tends to saturate to the value of 0.3#B/ion, which is about 40% of the gJ value for the Sm 3+ ion. The temperature dependence of the susceptibility of single-crystalline SmCu 2 has been measured by Isikawa et al. (1988) and is shown in fig. 4.22. A sharp peak at TN has been found in the susceptibility curve along the b axis. It is suggested that the b axis is the easy direction and that the antiferromagnetic spin structure is collinear, with the spins aligned parallel to the b axis. Isikawa et al. (1988) did not
448
N.H. LUONG and J.J.M. FRANSE
f
SmCuz
b
/,.2K
EQJ tl
2
/
0
10
30
20
B(T) Fig. 4.21. Magnetization curves along the a and b axes of SmCu 2 at 4.2 K. After Maezawa et al. (1989).
2~L
SmCu2 000000 mOO000
1 lee
"~e~°
°eNeoeuee
° •
Io v--
0 0
I 100
I 200
300
T (K) Fig. 4.22. Temperature dependence of the susceptibility of single-crystalline SmCu 2 along the principal axes. After Isikawa et al. (1988).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
449
observe any other remarkable anomaly than that at TN in the temperature dependence of the susceptibility. These authors have also calculated the anisotropic temperature dependence of the susceptibility of SmCu2 on the basis of the crystal-field and exchange interactions. A good agreement between calculations and experiments is obtained but no explicit data for the crystal-field and exchange interaction parameters are given. Figure 4.23 shows the temperature dependence of the specific heat ep and the magnetic entropy Sm of SmCu2 (Gratz et al. 1990). Tr~ (= 23 K) and Ts (= 17 K) are the N4el and the spin-reorientation temperatures, respectively. The coefficient 7 is estimated to be about 20 mJ/K 2 mol, which value is about twice as high as the 7 values observed for the other RCuz compounds, e.g., 7(YCu2) = 6.7 mJ/K 2 mol (Luong et al. 1985a), 7(NdCu2) _~ 10 mJ/K 2 mol (Gratz et al. 1991) (see sections 4.1 and 4.3) and 7(TmCu2) = 9 mJ/K 2 mol (Sima et al. 1988). However, the value for 7(SmCuz) is four times smaller than that observed for the isostructural Kondo system CeCuz, 7(CeCu2) = 82 mJ/K z mol (Gratz et al. 1985) (see section 4.1). The spin-dependent specific heat, era, is obtained by comparing the specific heat of SmCu2 and that of non-magnetic LuCu2 as (Gratz et al. 1990): em(SmCu2) = e(SmCu2) - e(LuCu2).
(4.4)
Gratz et al. (1990) have obtained a value of 5.5 J/Kmol for the magnetic entropy Sin, which is very close to the value expected for a complete removal of the two-fold spin degeneracy of the crystal-field ground-state doublet (Sin = R In 2 = 5.76 J/K mol). These authors concluded that the ground state is a doublet. The magnetic entropy value estimated by Isakawa et al. (1988) is 88% of the theoretical value. The maximum temperature (45 K) reached in specific-heat measurements performed by Gratz et al. (1990) may be not high enough to observe the Schottky
SmCu2
ee
12
I
0
"6 12
,- ~
0
IIll ~
12
I
I
24 T (K]
I
I
36
Fig. 4.23. The specific heat c and the magnetic entropy Sm of SmCu2 as a function of temperature from 1.5 up to 45 K. After Gratz et al. (1990).
450
N.H. LUONG and J.J.M. FRANSE
anomaly caused by the crystal-field effect. Therefore, these authors have performed thermal-expansion measurements in the temperature range from 4 K up to room temperature. Figure 4.24 presents the coefficient of thermal expansion, a = l - 1 ( A I / A T ) , of SmCu 2 as a function of temperature. Figure 4.24 also shows of isostructural LuCu2. a(T) of SmCu2 shows a pronounced minimum near 40 K. Gratz et al. (1990) have deduced the spin-dependent contribution to the thermal expansion, am(T), for SmCu2 by subtracting a(T) of LuCu2. The results are shown in fig. 4.24. am(T) exhibits a minimum at 45 K, caused by the crystal-field effect. Assuming that the minimum in am(T) represents an effect comparable to the Schottky-type of anomaly in the specific heat (see, for instance, Franse et al. 1985, Sima et al. 1988), Gratz et al. (1990) have used the position of the temperature where the minimum occurs to estimate the splitting energy between the ground-state doublet and the first-excited doublet (see section 3). They have derived a value of about 110 K for this crystal-field splitting. Transport properties (resistivity, thermal conductivity and thermopower) of SmCu2 were measured by Gratz et al. (1990). The curvature of the spin-dependent contribution to the resistivity, Pro, indicates a crystal-field influence on the J = 5/2 ground state, although the effect of the J = 7/2 multiplet on Pm is still an open question. The high-field magnetoresistance in SmCu2 has been measured by Maezawa et al. (1986, 1989). Indications are found that SmCu2 has a complex Fermi surface. Based on the molecular-field theory and using the value of 22.5 T for the flopping field, Maezawa et al. (1989) have estimated the antiferromagnetic-exchangeinteraction parameter for SmCu2 to be 0.11 meV. This value is about half the value calculated from the NEd temperature. Based on the paramagnetic-susceptibility measurements of Gratz et al. (1990), Stewart (1992) has derived a value of 0.061 eV for the exchange interaction between the 4f and the itinerant electrons in SmCu2. 20 15 10
,:
...,,.
~5
~
,.,
. r, 4"
/
:--.
/
j mcu,_ ~,."
5
-10
',-"j
-15
0
_,-~" ~ .,-" ~ -,~ . 100
I
1'0
x.... '
I
~
--,-.-.-/ . '
/
5~/
200 T (K)
Fig. 4.24. Temperature dependence of the thermal expansion, c~, of S m C u 2 and L u C u 2 and the magnetic contribution to the thermal expansion, C~rn. Inset shows details of ~ versus T below 50 IC After Gratz et al. (1990).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
451
To our knowledge, up till now no data on the crystal-field interactions in SmCuz are available. Further experiments are needed to clarify the interesting magnetic properties of this compound. 4.5. G d C u 2
The compound GdCu2 is, by lack of an orbital moment of the 4f electrons, expected to be relatively simple as far as the magnetic anisotropy is concerned. Therefore, in the series RCu 2 this compound is the most convenient object for the study of RKKY exchange interactions. The magnetic properties of GdCu2, among other compounds, were first studied by Sherwood et al. (1964). These authors reported that GdCu2 is antiferromagnetic, with a value for the Nrel temperature equal to 41 K. Poldy and Kirchmayr (1974) have shown that the RKKY model can be successfully applied to GdCu2, i.e. the magnetic arrangement of the Gd a+ ions is due to the indirect exchange interaction via the conduction electrons. Further studies in polycrystalline samples were made on the crystallographic properties (Dwight 1980~ Borombaev et al. 1986a, b), magnetic properties (Gratz and Poldy 1977, Luong and Franse 1981, de Graaf et al. 1982, Borombaev et al. 1986a,b, Borombaev and Markosyan 1987), magnetostriction (Luong and Franse 1981), thermal expansion (Luong and Franse 1981, Luong et al. 1985a,b, Borombaev et al. 1986a,b, Borombaev et al. 1987), specific heat (Luong et al. 1985a, b), transport properties (Gratz 1981, Gratz and Zuckermann 1982) and MSssbauer effect (De Graaf et al. 1982). To our knowledge, only one work on single-crystalline GdCu2 is available (Borombaev et al. 1987). Figure 4.25 shows the results of magnetic measurements in low magnetic field of these authors. In the paramagnetic temperature region the magnetic susceptibility of single-crystalline GdCu2 does not depend on the direction of the external field and obeys the Curie-Weiss law with a paramagnetic Curie temperature, 0p, of (16 -4- 2) K and an effective magnetic moment,/~eff, of (8.14 -40.2)/~B, close to the theoretical value for the Gd 3+ ion (7.94#B). (De Graaf et al. (1982) have obtained the following values for a polycrystalline sample: 0p = 7 K a n d / ~ = 8.7/~B.) The Nrel temperature was derived as the temperature at which the magnetization exhibits a discontinuity and was found to be equal to 42 K, in good agreement with other literature data (Sherwood et al. 1964, Poldy and Kirchmayr 1974, Luong et al. 1985a, b). Interesting magnetization curves were obtained by Borombaev et al. (1987) below the Nrel temperature in high magnetic fields. Figure 4.26 shows the magnetization along the three principle axes of single-crystalline GdCu2 at 4.2 K. In the interval 6-12 T, a metamagnetic transition (in two stages) from antiferromagnetism to ferromagnetism occurs. At Bel = 6.8 T the magnetization sharply increases, reaching approximately 4#B, and then increases more slowly; at Bc2 = 9.5 T again a sharp increase of the magnetization is observed. The saturation magnetic moment is equal to (6.9 + 0.1)/~B, close to the Gd3+ moment (7.0/ZB), indicating that GdCu2 is in the ferromagnetic state in a field B > Be2. Figure 4.27 shows the magnetization curves of GdCu2 at different temperatures in relatively low fields up to 2.8 T. It can be seen from this figure that the magnetization along the b axis increases linearly with
452
N.H. LUONG and J.J.M. FRANSE
Xo(orb. unit) ~TN 200 1.5 150 m. 1.0~
TNI 0
ov -
b,0
I00 ~'a.
0.5 0
50
100
0
200
300
T {K) Fig. 4.25. Temperature dependence of the inverse susceptibility of GdCu 2 along the axes a (o), b (o) and e(o) and of the magnetization along the axis e(+) in a field of 0.15 T. The inset shows the temperature dependence of the initial susceptibility (in a field of 0.001 T). After Borombaev et al. (1987).
GdCu2 6
2
0
~"
0
i
I'"
I
I
10
5
I
I
15
B (T) Fig. 4.26. Magnetization curves along the axes a (solid line), b (dashed line) and c (dotted line) of GdCu 2 at 4.2 K. The inset shows the field dependence of the differential magnetic susceptibility along the a axis. After Borombaev et al. (1987).
field, and that the susceptibility does not depend on the temperature. However, the magnetization along the a and e axes displays a discontinuity near 1 T, in the vicinity of the N6el temperature, the magnetization in this region shows hysteresis. Above 2 T, the susceptibilities Xa and Xe are equal to the susceptibility XB. Borombaev
//JJ
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
// 0, / / / 0:////
/
b
o
0"20'~
I
I
0
I
2
I
0
453
2
1
0
I
I
1
2
B (T)
Fig. 4.27. Magnetization curves of GdCu2 along the principal crystallographic axes a (a), c (b) and b (e) at different temperatures: 1) 4.2 K; 2) 15 K; 3) 30 K; 4) 35 K. After Borombaev et al. (1987).
.o~.d'*~
0 1.0
GdCu 2
IE
04v'
~
~* ~ ~ ~ , ~*
~.~,.
___
I'---- 0.5
10
20
30
~0
50
T (K)
Fig. 4.28. The specific heat of GdCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of e/T versus T. The full curve represents the sum of the electronic and lattice contributions to the specific heat of GdCu2. The broken curve represents the specific heat of YCu2. After Luong et al. (1985a). et al. (1987) have explained the transition observed in GdCu 2 by comparing the energies o f the different magnetic phases in the R K K Y model. Their results s h o w that the presence o f a large magnetic anisotropy energy is not a necessary condition for the realization o f metamagnetic transitions. Such transitions are possible in the exchange approximation for ferrimagnetic structures which energies lie between the energies o f the initial antiferromagnetic state and the final ferromagnetic state.
454
N.H. LUONG and J.J.M. FRANSE
Figure 4.28 shows the specific heat of the compound GdCu2 at zero magnetic field and in a field of 5 T in a plot of c/T against T (Luong et al. 1985a). Non-magnetic YCu2 was used in order to separate out the magnetic contribution to the specific heat of GdCu2. A A-type of anomaly is observed around the N6el temperature TN. The value for TN has been determined from the peak in the plot of the specific heat against T and is found to be equal to 40 K, in good agreement with other literature data. Luong et al. (1985a) have also pointed out that their susceptibility experiments above 4.2 K confirm this value of TN. A second anomaly below the ordering temperature is visible in the plot of c/T against T in fig. 4.28 as a broad structure around 10 K. We attribute this anomaly, which is of a Schottky type, to the Zeeman splitting of eight-fold degenerate energy level below the N6el temperature. Finally, a sharp peak is found at 1.5 K that shifts to 1.7 K in a field of 5 T. This peak could indicate a change in magnetic structure. As mentioned before (see section 4.1), the electronic contribution to the specific heat, 7, was derived from the specific heat data of Y C u 2 and found to be equal to 6.7 mJ/K 2 mol. This number is close to the value of 8.2 mJ/K 2 mol reported for Y metal (Stewart 1983). The phonon part of the specific heat of YCu2 follows closely the Debye function, giving 0D = 236 K. Since the 7 value of 6.4 mJ/K 2 mol of Gd metal (Stewart 1983) is close to the above mentioned value for Y metal, Luong et al. (1985a) have taken the 7 value of Y C u 2 for GdCu2 and compared the specific-heat
15 o°
t +4+
~I0 E
4- 0 •
4•
w
4-
4-
E
5
X
g
C 0
g
+
o
4-
%
4-
•
O
I -°°3 %2.~, ° 10 20 30
:
% l z.O T (K)
50
Fig. 4.29. The magnetic contribution to the specific heat for G d = Y l _ x C u 2 as a function of temperature: z = 1 ( . ) , x = 0 . 8 ( + ) , z = 0.6(a), z = 0.4(X), z = 0.2(0). After Luong et al. (1985a).
M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S
455
data of YCu 2 and GdCu2 compounds well above the ordering temperature in order to derive the value for the Debye temperature for GdCu2. They obtained a value of 198 K for GdCu2. The electronic and lattice contributions to the specific heat of GdCu2 and YCu2 are represented by the full and broken curves in fig. 4.28, respectively. The magnetic part, Cm, which is obtained by subtracting the electronic and phonon parts from the observed specific heat, is shown in fig. 4.29. This figure also shows this magnetic contribution for the whole series of Gd=YI_=Cu2 compounds which have been studied by Luong et al. (1985a). An analysis of the specific heat data for G d ~ Y l _ x C u 2 has been performed in the same manner as for G d C u 2. We note that G d 0 . 2 Y 0 . 8 C I I 2 , although it has no long-range magnetic order down to 1.2 K, has a small but not negligible magnetic specific heat. From specific-heat measurements Luong et al. (1985a) derived the values of TN and 0D for the Gd=Yl_zCu 2 compounds. The thermal-expansion results for the GdzYl_~Cu 2 compounds obtained by Luong et al. (1985a) are shown in fig. 4.30 in a plot of ot (= I-1AI/AT) against T. The thermal-expansion results for non-magnetic YCu2 (fig. 4.30) are again used to separate out the magnetic contribution to the thermal expansion in the other compounds. The results are shown in fig. 4.31. The (am, T) curves behave similarly to the (Cm,T) curves (see fig. 4.29). The A-type of anomaly around TN as well as the Schottky anomaly below TN both can be observed. Luong et al. (1985a) have discussed the specific-heat and thermal-expansion results in terms of Griineisen parameters. These authors were able to determine the
3O
°o
~20
"7 Y to I o
-¢
:
•
,.k -4-+ o
++...+
el
=e ++;£
.I. .
j:#. ++ ,r ==
.
a~r ~ ' ~ ' - = " J:l~.,m.~ao"
0
-
2O
~
n
• :
.,. ~ ~,,~ t o . ~ "~ " "
+~ - + ÷ 69.~%" "r'l'++'r~ ~ ~LX~
4O
-
6O
80 T (K)
Fig. 4.30. The coefficient of thermal expansion, c~, for G d z Y I _ = C u 2 as a function of temperature: x = l ( o ) , x = 0 . 8 ( + ) , x = 0 . 6 ( a ) , x = 0 . 4 ( x ) , ar = 0.2(o), ~e = 0(--). After L u o n g et al. (1985a).
456
N.H. LUONG and JJ.M. FRANSE
30
@
~
2O
I Y
I 0
-CA: °
++ 4-+
Sg E
~I0
~°+ I
o -t~ ~ x
x+
++
oee
+
°
°
. + + +
Ot 0 a
0 0
+
o++ .:
°
•
•
~.:%N ++H-~%"°k" n
10
2O
30
40 T (K)
50
Fig. 4.31. The magnetic contribution to the coefficient of thermal expansion for GdxYI_=Cu2 as a function of temperature: :e = l(i), a: = 0.8(+), x = 0.6(o), z = 0.4(X), = = 0.2(0). After Luong et al. (1985a). electronic, lattice and magnetic Grtineisen parameters for the Gd~YI_=Cu 2 compounds. Taking the lattice Grtineisen parameter to be equal to 2, the electronic Griineisen parameter is found to be 1.7, quite close to the value Fe = 5/3 which has been obtained for d transition metals (Simizu 1974). The magnetic Grtineisen parameter is at least one order of magnitude larger than the electronic and lattice ones. The magnetic entropy, Sin, in the Gd~YI_~Cu2 compounds was found to be proportional to the number of Gd atoms and the value for S derived from the formula ,5'm -- mR In (2S + 1) is slightly smaller than 7/2 for a free Gd +3 ion (Luong et al.
1985a). Although GdCu2, as mentioned above, is a relatively simple compound, no experimental data about its magnetic structure are available. Because of the large cross section for scattering of neutrons by gadolinium, neutron-scattering studies of the magnetic structure of GdCu2 have not yet been performed. Poldy and Kirchmayr (1974) have calculated the magnetic structure of GdCu 2 in the RKKY model. According to their calculations, GdCuz has a spiral magnetic ordering: all spins in the be plane are parallel to one another, while the directions of the magnetic moments in neighbouring planes differ by 35 °. Studies on single-crystalline GdCu2 by Borombaev et al. (1987) have shown (fig. 4.27) that the curves of the magnetization of GdCu 2 along the a and c axes practically coincide with each other, but differ from
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
457
the curve of the magnetization along the b axis. These authors have explained the experimental results by assuming that at the lowest magnetic field (B < 1 T, see above) the spins of the Gd 3+ ions lie in the ac plane (the angle between the spins of neighbouring be layers is 35* (Poldy and Kirchmayr 1974)). Cu in GdCu2 has been partially replaced by Ni (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Smetana et al. 1985a, Borombaev et al. 1986a, Borombaev and Markosyan 1987), by Co and Fe (Borombaev et al. 1986b, Borombaev and Markosyan 1987) and by A1 (Borombaev et al. 1986a, Borombaev and Markosyan 1987). The aim of these replacements is to observe the effects of changing the conduction-electron concentration, since the exchange via the conduction electrons is sensitive to this parameter. In substituted GdCu2, the electron concentration is decreased with increasing Ni and Co content, whereas it is increased with increasing A1 concentration. It was assumed that Gd, Cu, Ni, Co and AI contribute 3, 1, 0, - 1 and 3 conduction electrons, respectively (Poldy and Kirchmayr 1974, Borombaev et al. 1986b, Borombaev and Markosyan 1987). The limits of structural stability were found to be 35% Ni, 10% Co, 1% Fe and 7% AI (Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). The experimental observations in GdCu2 substituted with Ni, Co and A1 have been discussed in terms of the RRKY model (Poldy and Kirchmayr 1974, Gratz and Poldy 1977, Kirchmayr and Poldy 1979, Borombaev et al. 1986a, b, Borombaev and Markosyan 1987). 4.6. TbCu2
The compound TbCu2 has the highest value of the Nrel temperature in the series RCu 2. Early magnetic measurements of Sherwood et al. (1964) have yielded a value for TN of 54 K for this compound. A value for TN of 53.5 K, i.e. very close to that of Sherwood et al. (1964), has been obtained by Hashimoto et al. (1976) and Hashimoto et al. (1979a) on single-crystalline TbCu2. Somewhat lower values for T N of TbCu 2 have been obtained from magnetic measurements by Poldy and Gratz (1978) and by Smetana et al. (1985a) (TN = 48 K) and from specific-heat measurements by Luong et al. (1985b) (TN = 48.5 K). The early measurements of Sherwood et al. (1964) have shown metamagnetic behaviour of polycrystalline ThCu2. Hashimoto et al. (1976, 1979a) performed magnetization and magnetic-susceptibility studies on single-crystalline samples and observed anisotropies in the magnetic properties. Later, Iwata et al. (1988) have studied the magnetization process of single-crystalline TbCu 2 in the temperature range from 4.2 to 55 K. Their results are shown in fig. 4.32. For a magnetic field applied along the a axis, the magnetization is almost zero below Bc (,'~ 1.9 T) and jumps within one step to a nearly saturated value at higher field. The saturation moment per atom has been determined to be 8.8#B (Hashimoto et al. 1979a), very close to the theoretical value of 9/zB for the free Th3+ ion. Figure 4.33 shows the result of high-magnetic-field experiments on TbCu2 in a pulsed magnetic field performed by Luong and Frame (1981). Due to large magnetostrictive forces the sample was powdered and oriented in succeeding field runs. From these measurements we deduced a value of 1.9 T for the critical field Be.
458
N.H. LUONG and J.J.M. FRANSE
Observed
Calculated
°,,. . . . . . . . . . . .
10 0
2
Observed
10
/*
a
2
4.
a
6f.
::I
:
50.0K i /
..,........ *'I'°''''''''
2
4.
i
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00
2
4
81 2
c.. 2
4.
B (T)
0
'2
B (T)
o
z~
0
2
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:
0
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'2
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o. . . .
4.
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::::::::::::::::::::::::::: 0 2 4 B IT)
2
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B (T)
Fig. 4.32. Observed and calculated magnetization of TbCu2. After Iwata et al. (1988).
]]I
f
150
100
f
"E
TbCu z
50
O I 0
I
I
10
20
30
B (T)
Fig. 4.33. High-magnetic-fieldexperiments on TbCu2 in a pulsed magnetic field. In succeeding field runs, the sample is powdered and oriented with the easy axis along the field direction. After Luong and Franse (1981).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
•
i
'r',
l!
,~i
"
II
II
ii
II
. . . . . . .
r,'-~
u
!"
. . . . . .
," A A ~
,,'~
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I
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I iI
••
iS I
459
,4
:--
!
I
6
I
I
!
i
I
7
,
i
Fig. 4.34. Magnetic structure of TbCu 2 at 4.2 K. After Hashimoto et al. (1979a).
A first neutron-diffraction study on a powdered sample has been carded out by Brunet al. (1971). According to these authors, TbCu2 has a collinear antiferromagnetic structure in which the moments lie along the a axis. The magnetic unit cell is equal to the orthorhombic chemical unit cell along the b and c axes but tripled along the a axis. Further neutron-diffraction studies on a single-crystalline sample by Hashimoto et al. (1979a) and on a powdered sample by Smetana et al. (1983), Smetana and Sima (1985), Sima et al. (1986) and Lebech et al. (1987) have confirmed this magnetic structure. Figure 4.34 shows the magnetic structure of TbCu2 at 4.2 K. A fraction of 2/3 of the Tb magnetic moment in a given e plane is aligned along the +a direction (sublattices A, C) and a fraction of 1/3 along the - a direction (sublattice B). In the adjacent e planes, the magnetic moments of the Tb atoms are all reversed and these antiferromagnetically coupled double layers pile up in the c direction. The temperature dependence of the magnetic structure of TbCu2 has been studied by Brunet al. (1971) and Sima et al. (1986), the results of which are in good agreement with each other. Three distinct regions of magnetic order were identified (Sima et al. 1986, Divis et al. 1987). Between 47 K and the N6el temperature TN (55 K in the studied samples) (region I) the magnetic structure is longitudinally modulated with the propagation vector 1/3a*. At temperatures below 15 K (region III) all the magnetic moments are equal. Finally, for temperatures between 15 and 47 K (region II) there are Tb a+ ions with 'large' magnetic moments and with 'small' magnetic moments. The magnetic structure of TbCu2 in this intermediate temperature region is shown in fig. 4.35 (Divis et al. 1987). Figure 4.36 shows the temperature dependence of the inverse susceptibility along each principal axis of the TbCu2 compound (Hashimoto et al. 1979a). A large magnetic anisotropy in the paramagnetic region has been observed. The susceptibility along the different crystallographic axes obeys the Curie-Weiss law at high temperatures. The effective moment, #eft, amounts to 9.5#a, in good agreement with
460
N.H. LUONG and J.J.M. FRANSE
I°
i
i
iI ,
[
A
.vtl
Fig. 4.35. Magnetic structure of TbCu2 for the intermediate temperature region (15 K < T < 47 K). Numbers 1 to 12 enumerate the positions of moments in the magnetic unit cell and the arrows show their orientation and magnitude. After Divis et al. (1987).
the calculated value for the free T b 3+ ion. The paramagnetic Curie temperatures 0a, Ob and 0c along the a, b and e axes are remarkably anisotropic: 76 K, - 6 K and 36 K, respectively. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated the following values for the second-order crystal-field parameters: B ° = 1.23 K, B g = 1.23 K. Hashimoto et al. (1979a) have also calculated the second-order crystal-field parameters on the basis of the point-charge model. Their calculated values are: B ° -- 1.35 K and B 2 = 1.12 K. Experiments and calculations are in reasonable agreement with each other, indicating that the anisotropy of TbCu2 in the paramagnetic state can be explained mainly by the crystal-field effect. Measurements of the specific heat were performed by Luong et al. (1985b) at zero magnetic field and in a field of 5 T. A plot of e/T versus T at zero field is given in fig. 4.37. A A-type of anomaly is observed around the N6el temperature. Apart from this peak at Try, a broad hump was observed around 30 K and a sharp peak occurs at 2.2 K. Kimura et al. (1988) have calculated the specific heat for TbCu2 on the basis of the molecular-field theory including crystal-field interaction. In these calculations the authors have used two different exchange parameters and two crystal-field parameters, B ° and B22, as derived by Hashimoto et al. (1979a) (see above). The results are also shown in fig. 4.37. Based on the calculated temperature dependence of the magnetic contribution to the specific heat, Kimura et al. (1988) concluded that the broad anomaly observed around 30 K in TbCu2 must be attributed to the magnetic heat capacity. Luong and Fransse (1981) and Luong et al. (1985b) have measured the thermal expansion of TbCu2. The coefficient of the thermal expansion of TbCu2 is shown in fig. 4.38 as a function of temperature. Again a A-type of contribution is observed
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
10
461
x103 ~ c TbCu2
..
"7. 6
• coO°. °~
E4 X
2
100 i
•"
~"
I
I
l
200
300
T (K}
Fig. 4.36. Temperature dependence of the inverse susceptibility along the principal axes of TbCu2. After Hashimoto et al. (1979a).
1.5 TbCu2 o
1.0 0
==
,//
I--"
"3 0.5
S/
// 0
10
20
30
40
50
60
T (K) Fig. 4.37. The specific heat of TbCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T measured by Luong et al. (1985b). Lines are specific heat (at zero magnetic field) calculated by Kimura et al. (1988).
462
N.H. LUONG and J.J.M. FRANSE
15
TbCu 2
10 "7
o•
Qe•
'o
•o
•
e•
5 0
~'" I
I
I
10
20
30
I
40 r (K)
•
I
I
I
50
60
70
~ g . 4.38. The coemcientofthermalexpansion ~ r T b C u 2 as a ~nction of ~mpera~re. After Luong etal.(1985~.
around TN. Apart from this A-type of anomaly and apart from the lattice contributions, apparently a third contribution to c~ is present above 30 K. Discussions in terms of Grtineisen parameters (see section 3.3 and more details in section 4.9 for ErCuz) lead to the conclusion that for TbCu2 the same sign and approximately the same value of ~¢Fcfas for ErCu2 is observed (Luong et al. 1985b). This observation gives a natural explanation for the third contribution to the thermal expansion. Based on a molecular-field model taking into account crystal-field effects, Iwata et al. (1987, 1988) have performed a quantitative analysis of the magnetization process of single-crystalline "I'bCu2. In the analysis they used two second-order crystal-field parameters determined experimentally by Hashimoto et al. (1979a) (see above). The exchange parameters were chosen to obtain the best fit to the experimental data. The magnetization data are well explained, as shown in fig. 4.32. (We note that the Ngel temperature of their sample is Tr~ = 53.5 K.) Kimura (1985) has analyzed the magnetic structure and the magnetization process using a 12-sublattice model. The crystal-field effect on the paramagnetic susceptibility of TbCuz has been studied by Nowotny and Zajac (1985) using the general Van Vleck formula and by Kimura (1988) with the use of a high-temperature expansion method. Both these studies have shown that anisotropic behaviour of Xi (i = a, b, c) is cancelled in Xp (= (Xa + Xb + Xe)/3) for polycrystalline samples. Nowotny and Zajac (1985) have used only two lowest-order crystal-field parameters as obtained by Hashimoto et al. (1979a) (see above). The higher-order crystal-field parameters have also been taken into consideration but the effect on the paramagnetic susceptibility of RCuz was reported to be negligible (Zajac and Maczak 1985). Magnetic properties of the pseudobinary compounds Tb=YI_=Cu2 have been studied by Hashimoto (1979), Luong et al. (1982), Hien et al. (1983) and Zajac et al.
M A G N E T I C PROPERTIES OF R A R E EARTH-Cu 2 C O M P O U N D S
463
(1988b). When substituting Y for Tb, the antiferromagnetic interactions weaken but the magnetocrystalline anisotropy persists (Hashimoto 1979, Luong et al. 1982). A critical concentration, xc, where the magnetic ordering disappears, was taken by Hien et al. (1983) to be equal to 0.15, following data of Hien et al. (1983) and Hashimoto (1979). The latter author suggests that TN for TbxYI_~Cu2 is proportional to ~2/3 where G is an effective de Gennes factor equal to z(g - 1)2j(j + 1). However, the concentration dependence of TN expressed by the following equation: TN(
)/TN(1)
~
-
(4.5)
where TN(1) is TN(X) for x = 1, is in better agreement with the experimental data obtained by Hien et al. (1983). This concentration dependence of TN is also observed in Dy~YI_~Cu2 compounds (see also section 4.7). Tb0.rY0.4Cu2 has a collinear antiferromagnetic structure, as revealed from neutron-diffraction measurements (Smetana et al. 1983, Sima et al. 1986, Lebech et al. 1987). Hashimoto (1979) has derived values for the crystal-field parameters B ° and B22 in the TbxYl_xCu 2 compounds from values for the paramagnetic Curie temperatures obtained on single-crystalline samples. It was found that the values for B ° and B 2 change very little in the whole composition range. The magnetization process in the ordered Tb~YI_~Cu2 compounds is similar to that of TbCu2. It has been analyzed within the molecular and crystal-field model by Hashimoto (1979) in the paramagnetic state and by Zajac et al. (1988b) in the ordered state. For the crystal-field Hamiltonian only two parameters B ° and B 2 are considered in both studies. Partial replacement of Cu by Ni in TbCu2 leads to a lowering of the electron concentration and an increasing tendency to ferromagnetism (Poldy and Gratz 1978, Hashimoto 1979). This effect has been confirmed by neutron-diffraction studies (Smetana et al. 1983, Sima and Smetana 1984, Smetana and Sima 1985, Lebech et al. 1987). Smetana et al. (1985a) reported the temperature dependence of the ae susceptibility of a Tb(Cu0.77Ni0.23)2 sample. Hashimoto (1979) has derived values for B2° and B22 in Tb(cuxNil_x)2 compounds from values of the paramagnetic Curie temperatures. The magnetization process in single-crystalline Tb(Cu0.7Ni0.3)2 in high magnetic fields (up to 24 T) has been studied by Mamaguzhin et al. (1985) and Divis et al. (1989a, 1990b). In the analysis, the latter authors have taken into account higher-order crystal-field terms. 4.7. DyCu2
Measurements of the temperature dependence of the magnetization have yielded a value for TN of 24 K for the compound DyCu 2 (Sherwood et al. 1964). A value of 31.4 K for TN has been obtained from susceptibility measurements (Hashimoto et al. 1979a). The specific-heat measurements by Luong et al. (1985b) show a sharp A-type of peak at a value for TTq of 26.7 K. A value for the Nrel temperature found with Mbssbauer spectroscopy is (27 4- 0.5) K (Gubbens et al. 1991), in good agreement with that obtained by Luong et al. Lebech et al. (1987) have determined the magnetic structure of DyCu2 at 5 and 15 K by neutron diffraction measurements. The structure is similar to that of TbCu2 as
464
N.H. LUONG and JJ.M. FRANSE
C
!
r" .,
--tJi
.
.
.
.
~ ,~'Ws +~,.'f, !
.
.
t
/
I
b
p,s a°s~/'/' Fig. 4.39. Magnetic structure of DyCu 2. The structure is decomposed into four sublattices A, A', B and B ~. The Dy atoms labeled 4, 5, 9, 12 are on the A sublattice; atoms 3, 6, 10, 11 are on A~; atoms 1 and 8 are on B; atoms 2 and 7 are on B f. After lwata et al. (1989), data from Lebeeh et al. (1987).
10 r 8 t~.2K i f ............. a" :? 6 ', b r
I
I01
8L~5 K
+~
~'
............~
2
2
4
6
8 - 9.5K ,,~ . ....................... Q. •
--
n
2
#
6
~ ................
............. ii"
~.
.
..
2
~- 0
0
2
4
10 8 150K ~ 6
6
I bl
~
0
2
4 B iT}
0
2
6
t,
10 8130"5K
II
6 4.
{ ...,,.•,"
Fig. 4.40.
0
8
':
:~'4"
O t . ~- - ' ~ ,
.............. ...............
O.l~,~-f, 0 2
b
!
, 4 B (T)
Magnetization curves of DyCu2. Solid curves represent the calculated magnetization. After lwata et al. (1989)•
s h o w n in fig. 4.39. H o w e v e r , d u e to the fact that D y has a h i g h n e u t r o n a b s o r p t i o n c r o s s s e c t i o n , the t e m p e r a t u r e d e p e n d e n c e o f the m a g n e t i c s t r u c t u r e has not yet b e e n analyzed.
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
10
465
(ii)
0 ~i) c - axis
t~.2K 2 0
0
5
I
I
I
I
10
15
20
25
B (T)
Fig. 4.41. Magnetizationcurves of DyCu2along the c axis in applied field up to 25 T. Curve (i) shows the magnetization process in an initial state. Curve(ii) is the magnetizationcurve obtainedjust after the measurement of (i). After Hashimotoet al. (1990). Measurements of the magnetization at 4.2 K (Hashimoto et al. 1979a) and of the temperature dependence of isothermal magnetization curves (Iwata et al. 1989) have been performed on single-crystalline samples. Figure 4.40 presents the isothermal magnetization curves for DyCu2 (Iwata et al. 1989). The magnetization curve along the a axis has two critical fields, Bel and Be2, and the magnetization increases to about 1/3 of the saturation value at Bel and to nearly its saturation value at Be2. At high temperatures, the two-step process is smeared out. On the other hand, the magnetization along the b and c axes increases monotonically. Magnetization curves have been measured in high magnetic fields up to 30 T by Hashimoto et al. (1990) and by Date (1992). Figure 4.41 shows the isothermal magnetization curves measured along the c axis at 4.2 K (Hashimoto et al. 1990). The magnetization increases abruptly at about 14 T and shows a hysteresis at decreasing magnetic fields (i). The magnetization curve measured just after the measurement of curve (i) is indicated as curve (ii) in this figure. This is quite similar to the curve along the a axis (see fig. 4.40). It turns out that a switching of the magnetic axis is observed. The recovery of the virgin state is obtained either by warming the crystal above about 100 K or by applying a field higher than 5 T along the a axis. Figure 4.42 shows the magnetization curves along the b axis at 4.2 K (Hashimoto et al. 1990). A little jump of the magnetization with a small hysteresis is observed. The critical field decreases by repeating the high-field cycles. The sample was broken in two pieces when a high field was applied along this b axis due to magnetostriction effects. This phenomenon is similar to that observed in the polycrystalline samples of TbCu2 (Luong and Fransse 1981, see fig. 4.33). The results indicate the important role of large magnetocrystalline anisotropy and magnetoelastic energies in DyCu2. Kimura (1987a, b) has made a theoretical study of the magnetization process in RCu2 compounds using the Ising model with four exchange parameters. Necessary conditions for the exchange constants for which DyCu2 exhibits a two-step magnetization process are given (Kimura 1987b).
466
N.H. LUONG and J.J.M. FRANSE
I
2 0~ e-
= .6 0
3
b -axis 4.2 K
0
4
0
I
0
I
I
10
I
I
20
30
B (T) Fig. 4.42. Magnetization curve of DyCu2 along the b axis in applied fields up to 30 T. The number in the figure indicates the order of measurement. The critical field, at which a small jump of the magnetization occurs, decreases in subsequent measurements. After Hashimoto et al. (1990).
10 x103
b
8
DyCu2
"T
a
~6 i=
2
100
200 T (K)
300
Fig. 4.43. Temperature dependence of the inverse susceptibility along the principal axes of DyCu2. After Hashimoto et al. (1979a). The t e m p e r a t u r e d e p e n d e n c e o f the inverse susceptibility along each p r i n c i p a l axis o f D y C u 2 is s h o w n in fig. 4.43 ( H a s h i m o t o et al. 1979a). T h e effective m a g n e t i c m o m e n t , #en, a m o u n t s to 10.3#a. T h e p a r a m a g n e t i c Curie temperatures a l o n g the
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
467
a, b and c axes amount to: 0a = 35 K, 0b = --17 K, 0e = 0 K. From these values of the paramagnetic Curie temperatures, Hashimoto et al. (1979a) have estimated experimental results for the two lowest-order crystal-field parameters: B ° = 0.43 K and B 2 = 0.72 K. A point-charge calculation by the same authors results in B ° ~l = 0.89 K and B 2 ~! = 0.71 K, in satisfactory agreement with the experimental values. Nowotny and Zajac (1985) have used the experimental results for the two crystalfield parameters to calculate the paramagnetic susceptibility of DyCu2. They could explain the observed curvature in Xi(T)- 1 CUrveS (i = a, b, C) at low temperatures. The isothermal magnetization curves up to TN have been analyzed using a molecular field model (Iwata et al. 1989). The calculated magnetization curves are presented in fig. 4.40 as the solid lines. A satisfactory agreement between experiments and calculations is obtained. Apart from the experimental results for B2° and B22, a small positive value of 2.47 x 10 - 3 K for B ° was taken into account because its term gives a large contribution to the b axis magnetization at low temperatures. On the other hand, in high field studies Date (1992) has pointed out that a switching of the magnetic axis (see above) implies that this phenomenon cannot be explained by a simple point-charge crystal-field model but should be explained in terms of a Jahn-Teller effect. Specific heat and thermal expansion of DyCu2 have been measured by Luong et al. (1985b). Figure 4.44 shows the specific heat of DyCu2 at zero magnetic field and in a field of 5 T. The A-type of peak at TN disappears in a magnetic field of 5 T. A broad anomaly around 6 K, which is field dependent, is observed. Kimura et al. (1988) have calculated the specific heat of DyCu2 in terms of the molecular-field model including the crystal-field interaction. They have obtained the temperature dependence of the specific heat which is similar in trend with the experimental one. They have used three crystal-field parameters: B °, B22 experimentally obtained by Hashimoto et al. (1979a) and B ° equal to 2.37 x 10 -3 K. According to Kimura et al. (1988), the specific heat hump of DyCu2 near 6 K is associated with the fact that the spacing of the energy between the lowest and the next lowest levels is not large: 10 K. The thermal expansion coefficient of DyCu2 as a function of temperature is shown in fig. 4.45 (Luong et al. 1985b). Apart from the A-type of contribution around TN, an additional peak at about 19.5 K is observed. Up till now no prove has been given whether this peak is due to a change of magnetic structure or not. This should be connected with the difficulty of neutron diffraction studies in Dycu2 as mentioned above. Magnetization measurements for Dy~YI-~Cu2 compounds have been performed by Hien et al. (1983). The magnetization curves obtained for these compounds have a metamagnetic behaviour similar to that for Tb~YI_~Cu2. The Nrel temperatures for Dy,,Yl_rCu 2 with z = 0.2, 0.4, 0.6, 0.8 and 1 are 5, 13, 18, 23 and 26.7 K, respectively. The composition dependence of TN for Dy~YI-~CU2 is similar to that for Tb~YI_~Cu2, i.e. obeys eq. (4.5). For DyrYl_rCu2 the critical concentration, zc, has been evaluated to be equal to 0.15, the same as for the TbrYI-rCu2 compounds (see section 4.6).
468
N.H, LUONG and J.J.M. FRANSE
OyCuz
+/°
1.0 -- ~
E
÷.t14,~
C4 v
0
++%2°°
~- 0.5
...... 0
I
I
I
lO
20
30
T (K) Fig. 4.44. The specific heat of DyCu2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of e/T versus T. After Luong et al. (1985b).
15
DyCu2
I
I,
10 "7x,,
to
~o
5
I
•
o #
~0
•
V
" 5 1P
0
o
° • ~oo
•
•
11
e
°
•
I
I
I
I
I
I
I
10
20
30
40
50
60
70
T (K) Fig.
4.45.
The coefficient of thermal expansion for DyCu 2 as a function of temperature. Luong et al. (1985b).
After
4.8. HoCu2 For HoCu2, a value for TN of 9 K was obtained by Sherwood et al. (1964) from magnetic measurements. From the experimental data on single-crystalline HoCuz, Hashimoto et al. (1979a) and Hashimoto (1979) deduced a value for TN of 9.8 K. Lord and McEven (1980) and Birss et al. (1980) performed neutron-diffraction, resistivity and magnetoresistance measurements on HoCu2. They showed that in
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
469
HoCuz, apart from an antiferromagnetic-order transition at 11.4 K, a change of the antiferromagnetic structure occurs at about 8 K. Gratz et al. (1982) carded out measurements of magnetization, resistivity, thermopower and neutron diffraction in HoCu2. They deduced values for TN of 9.7 K and for Ts of 7 K (order-to-order transition). From measurements of the magnetization, Hien et al. (1983) have confLrmed these results. Values for TN of 9.6 K and for Ts of 7 K were derived from specific-heat (Luong et al. 1985b, c) and thermal-expansion (Luong et al. 1985b) measurements. The at-susceptibility measurements by Smetana et al. (1985a) gave values for TN of 9.7 K and for Ts of 6.7 K. Neutron-diffraction studies of the magnetic structure of HoCu2 have been performed on a single-crystalline sample (Hashimoto et al. 1979a) and on powder material (Lord and McEven 1980, Gratz et al. 1982, Smetana et al. 1983, 1985b, Smetana and Sima 1985, Smetana et al. 1986b, Lebech et al. 1987). These studies reveal that the magnetic structure of HoCuz below TN is a collinear commensurable longitudinally modulated one along the a axis (see fig. 4.46). As temperature decreases, a second transition occurs at Ts ~-, 7 K. Below this transition, the magnetic structure has two components. One component is the above mentioned a axis modulated structure and the other component is a c axis incommensurable transversely modulated structure with moments along the b direction. The resulting magnetic structure of HoCu2 below 7 K is then a non-collinear incommensurably modulated structure with magnetic moments lying in the a b plane (Lebech et al. 1987), see fig. 4.47. The magnetic structure of HoCuz has been analyzed by Kimura (1990a) in a molecular-field approximation taking second-order crystal-field effects into account. He has shown that the spin arrangement in the a e plane for 7 K < T < TN can be realized by a competition of three kinds of exchange interactions between neighbouring ions in the same a c plane. He has also shown that the origin of the spin canting at temperatures below Ts is the biquadratic exchange interaction between the nearest Ho ions in the neighbouring a c planes. In later work Kimura (1990b) has investigated the effect of higher-order components of the crystal-field on the spin arrangement of HoCu2. He has shown that one of two possible magnetic structures at 0 K is stabilized by the 6th order component of the crystal field acting on the Ho
--,
JA '
11'
I
'
'
Fig. 4.46. The high-temperature (7 K < T < 10.4 K) magnetic structure of HoCu 2. After Smetana et al. (1986b).
470
N.H. LUONG and J.J.M. FRANSE
2
I
1
I
i
I
I
I I I
I I
I
I
r-"
J
I
I
I
The Mb component is modu[ated o[ong
I
Fig. 4.47. Projection of the magnetic structure of HoCu2 in the ab plane at T < 7 K. After Lebech et al. (1987).
10 xl0 3 b
c
"T
6
E ~u
4
2 0
.,~;" 0
l 100
i 200 T (K)
i 300
Fig. 4.48. Temperature dependence of the inverse susceptibility along the principal axes of HoCu 2. After Hasbimoto et al. (1979a).
ions. Thus the interplay of the biquadratic exchange interaction and the higher-order components of crystal-field has a large effect on the spin arrangement below Ts. Figure 4.48 shows the temperature dependence of the inverse susceptibility along the principal axes for HoCuz in the paramagnetic region (Hashimoto et al. 1979a). The effective magnetic moment, /-teir, amounts to 9.6#n. The paramagnetic Curie temperatures along the a, b and e axes are: 0a = 45 K, t9b = 30 K and 0c = 38 K. From these values for the paramagnetic Curie temperature Hashimoto et al. have estimated the following values for the second-order crystal-field parameters: B ° = 0.14 K, B22 = 0.12 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have calculated the second-order crystal-field parameters and arrived at: B2° ~l = 0.28 K and B 2 c~l = 0,23 K. Using experimentally derived crystal-field parameters of Hashimoto et al., Nowotny and Zajac (1985) have calculated the paramagnetic
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
471
200 H°Cu2
o
150 b 100 c
50
0
1
2
3
t~
5
B (T) Fig. 4.49. Magnetization curves along the principal axes of HoCu 2 at 4.2 K. After Hashimoto et al. (1979a).
susceptibility of HoCu 2. They have obtained a total splitting energy of 33 K, which is lowest compared with those of PrCu2, NdCu2, TbCu2, DyCu2 and ErCu 2. The magnetization curves for single-crystalline HoCu 2 at 4.2 K are shown in fig. 4.49 (Hashimoto et al. 1979a). The anisotropy of this compound is not so large compared with T b C u 2 and DyCu z. The specific heat of HoCu2 has been measured by Luong et al. (1985b, c). Figure 4.50 presents the specific heat of HoCuz at zero magnetic field and in a field of 5 T. A A-type of anomaly is observed around TN. Luong et al. (1985b) have obtained the value of 19.9 J/Kmol f.u. for the magnetic entropy. This value points to a multiplicity of at least 11, indicating that the J = 8 (for Ho 3+) multiplet is splitted by crystal-field effects. A sharp peak, which occurs at 7 K and which is strongly field dependent, points to changes in the magnetic structure. Apart from this peak and the A-type of peak around TN, there are two other anomalies at 4.5 K and at temperatures below 1.3 K. The first one is rather broad and is not much influenced by the applied magnetic field. The anomaly below 1.3 K is ascribed to the nuclear contribution of holmium to the specific heat (Luong et al. 1985b). The specific heat in the temperature range from 2 to 20 K and in magnetic fields up to 4 T has also been measured by Bischof et al. (1989). The results at zero magnetic field are in good agreement with the specific-heat data of Luong et al. (1985b, c). Bischof et al. determined the change of specific heat due to the magnetic field, ACCF, by writting ACCF = c c F ( B ) - CCF(0), where CCF is obtained by subtracting contributions of electrons and phonons to the specific heat. Using the values for the CF parameters B2° and B 2 of Hashimoto et al. (1979a) (see above), Bischof et al. have calculated ACCF and compared it with experiments. They observed discrepancies between the calculated and experimental specific-heat curves, which demonstrate the importance of higher-order crystal-field terms. Measurements of the specific heat at higher temperatures (up to 100 K, for instance) would be very useful,
472
N.H. LUONG and J.J.M. FRANSE
2.5
/ f 2.0 0
o
E
HoCu 2
* ~
o
°o ~o
1.5
o~ o
1.0
t3
~o++÷++~+
+ +
°°°°~°°°°°°°o
0.5
0
o o
0 0
o
O O0
0
I
I
I
I
I
I
5
10
15
20
25
30
T (K) Fig. 4.50. The specific heat of HoCu 2 at zero magnetic field (o) and in a field of 5 T (+) in a plot of c/T versus T. After Luong et al. (1985b).
among other experiments, for the determination of the crystal-field scheme of the HoCu2 compound. The coefficient of the thermal expansion of HoCu2 as a function of temperature is shown in fig. 4.51 (Luong et al. 1985b). Two sharp peaks were observed, reflecting 12 HoCu 2 go
~
"7%,"
8
:',,.
'0 o
4.
eo° •
oo
~IL ~
• •
o • o° %oe%eoeeNooeoeoo • •
I
I
I
10
20
30
I
40 T (K)
I
I
I
i
50
60
70
80
Fig. 4.51. The coefficient of thermal expansion for HoCu2 as a function o f temperature. After Luong et al. (1985b).
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
473
specific anomalies around Tr~ and Ts. The relative volume change due to magnetic ordering in HoCu2 is -0.25 x 10 -3, which is almost four times smaller than in GdCu2. This fact possibly indicates a reduced polarization of conduction electrons by the holmium moments (Luong et al. 1985b). Neutron-diffraction studies on a powder sample of Ho(Cu0.9Ni0a)2 have been performed by Smetana et al. (1983) and Smetana and Sima (1985). The effect of Ni substitution in the Ho(Cu, Ni)2 system is qualitatively similar to that observed in Tb(Cu, Ni)2.
4°9. ErCu2 The Nrel temperature of ErCu2 has been found to lie in the range 11-13.5 K; 11 K, 13.5 K and 11.8 K, according to the magnetic measurements of Sherwood et al. (1964), Hashimoto et al. (1979a) and Smetana et al. (1985a), respectively and 11.5 K according to the specific-heat measurements of Luong et al. (1985b). In magnetization measurements Hien et al. (1983) have observed, apart from the anomaly that is characteristic for disordering the antiferromagnetic state, a second transition at about Tsl = 6 K. These authors ascribed this transition to the change of the antiferromagnetic structure. This conclusion is supported by ac-susceptibility measurements (Smetana et al. 1985a) and specific-heat measurements (Luong et al. 1985b, c). From specific-heat and thermal-expansion measurements, Luong et al. (1985b) have observed two other anomalies at Ts2 = 4.1 K and Ts3 = 3.3 K. The anomaly around 4 K has also been reported by Smetana et al. (1984, 1985a). These additional peaks at Tse and Ts3 could also point to changes in the magnetic structure. Neutron-diffraction experiments on ErCu 2 have been performed by Smetana et al. (1984), Smetana and Sima (1985) and Lebech et al. (1987). These latter authors 10
x103
°~~/~c
ErCu '7
~6 -i
E aJ
2 J"
¢
,,"
I
I
I
100
200
300
T (K) Fig. 4.52. Temperature dependence of the inverse susceptibility along the principal axes of ErCu 2. After Hashimoto et al. (1979a).
474
N.H. LUONG and J.J.M. FRANSE
have reported that neutron-diffraction powder patterns obtained between 1.5 and 40 K revealed a rather complicated antiferromagnetic structure. In spite of many attempts to interpret the powder patterns, they were not able to describe the magnetic structure of ErCu2 satisfactorily. Single-crystalline data would be helpful. Figure 4.52 shows the temperature dependence of the inverse susceptibility along each principal axis of ErCu2 (Hashimoto et al. 1979a). The effective magnetic moment,/~e~, amounts to 8.9/ZB. The paramagnetic Curie temperatures along the a, b and c axes are: 0a = 18 K, 0b = 53 K and 0c = 36 K. From these data, Hashimoto et al. have estimated the following values for the second-order crystal-field parameters for ErCu2: B ° = -0.35 K, B 2 = -0.36 K. On the basis of the point-charge model, Hashimoto et al. (1979a) have also calculated these second-order parameters and obtained: B ° l = -0.31 K and B2~al = -0.26 K. The magnetization curves for single-crystalline ErCu2 are shown in fig. 4.53. The magnetization along the b axis increases abruptly at about 0.6 T and saturates magnetically at a higher field. From fig. 4.53 it is also clear that ErCu2 has a large magnetocrystalline anisotropy. The high-field magnetization curve of polycrystalline ErCu2 at 4.2 K is shown in fig. 4.54 (Hien et al., unpublished result). A two-step metamagnetic process was observed. In a magnetic field of about 25 T the magnetization is still not saturated. The specific heat and the thermal expansion of ErCu2 have been measured by Luong et al. (1985b, c). Figure 4.55 shows the specific heat of ErCu2 at zero magnetic field and in a field of 5 T. The thermal-expansion coefficient of this compound is presented in fig. 4.56. The anomalies at TN, Tsl, T~2 and Ts3 were observed in the specific-heat and thermal-expansion-coefficient curves. Apart from these anomalies, broad anomalies occurred above TN. Inspecting figs 4.55 and 4.56 Luong et al. (1985b) have assumed that in ErCu2, apart from the 'long-range' magnetic contributions to the specific heat and thermal expansion, crystal-field effects were present. 200
150 Er Cu2
El00
50 -__~.~-~ ~
0
1
2
3
4
5
B (T) Fig. 4.53. Magnetization curves along the principal axes of ErCu2 at 4.2 K. After Hashimoto et al. (1979a).
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
150
475
ErCu2
100 E
50
0
0
I
I
I
I
l
5
10
15
20
25
B (T) Fig. 4.54. High-field magnetization curve of ErCu2 at 4.2 IC After Hien et al. (unpublished result).
1.0 -
0
*
O E
ErCu2
#
0
0~
+
o°
÷÷%o
0 00
,,. F-
÷+
00
o°
+
o
+*
÷
4- ÷ 0
4. 0
"l"-I. 0 o
.,I0
0
0
°
0 0 0
o
÷ ÷=-o o
0.5
,I 0
20
10 T
30
t~0
(KI
Fig. 4.55. The specific heat of ErCu2 at zero magnetic field (.) and in a field of 5 T (÷) in plots of e/T versus T. The full curve represents the non-magnetic contribution to the specific heat of ErCu2. The dashed curve represents the specific heat of YCu 2. After Luong et al. (1985b). Luong et al. ( 1 9 8 5 b ) and Franse et al. (1985) have analyzed the e x c e s s contributions to the specific heat and thermal expansion in ErCuz by applying Grtlneisen relations (see section 3.3), s h o w i n g that they consist of a 'long-range' magnetic and a crystal-
476
N.H. LUONGand J.J.M. FRANSE
ErCu 2
10
l
-5 m
-1(3
0
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
T (K) Fig. 4.56. Thermal-expansioncoefficientof ErCu2. The non-magneticcontribution to the thermal expansion is indicatedby the dashed line. AfterLuonget al. (1985b). field part. The excess specific heat, era, was obtained by subtracting the electronic and phonon parts (non-magnetic) from the observed specific heat. The excess thermal expansion, O~m,is also obtained by subtracting the non-magnetic parts from the observed thermal-expansion coefficient (see fig. 4.57). Cm and am are related by the effective Grtineisen parameter, Fen [see eq. (3.38)]. From the temperature dependence of the Griineisen parameter, ~Fe~, for ErCu2, a change in sign in this parameter was observed. This change in sign indicates that at least two different contributions to am and am have to be distinguished. Values for ~Flr (6×10 -11 me/N) and ~/~ef (-15 x 10 -11 m2fN) have been derived by comparing em and am well below and well above TN, respectively. With these values, Cm and am have been separated into two different contributions (see fig. 4.58). The term Clr is A-type, whereas ccf is Schottky-type. The energy difference between the two lowest doublets was derived to be 76 K. We note that the temperature dependence of cef above and below Tm (at which cef exhibits a maximum, see section 3.3), as reported by Luong et al. (1985b), has not properly been calculated. Franse and Luong (private communication) have calculated ccf as a function of temperature taking into account only two lowest doublets with the experimentally derived energy difference of 76 K between these two doublets. They obtained a similar behaviour of the eel(T) curve compared with the experimental one but the calculated value of ecf is lower (about half the experimental value, at Tin). Using values for the two second-order crystal-field parameters reported by Hashimoto et al. (1979a) (see above), Franse et al. (1985) have calculated the energy levels in the ErCu2 compound. These authors derived a splitting of
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
477
10
!s
;, 10
20
30
40
50
T (K)
.I,I.
~s
t,
/.
o
.
i,-. I
i
I
I"
1
20
10 ~t.
t
i
i
I
30
I
40
50
T (K)
44-
4"4" -5 ÷
4"
-10
4" 4"
4" 4" 4÷
4" 4"
4.
4"
~fl.+ + +
Fig. 4.57. The excess specific heat, cm, and the excess thermal expansion, am, of the ErCu2 compound. After Franse et al. (1985).
13 K between the lowest doublets. Apparently, higher-order terms have to be taken into account in order to bring the splitting closer to the experimental value of 76 IC The relative volume dependence of this energy splitting was compared with the volume dependence of the crystal-field parameters as calculated in a point-charge model (Franse et al. 1985). The two results obtained in this way differed in sign. As discussed by Franse et al. (1985), this unexpected volume dependence could originate either from an anisotropic compressibility or from an anisotropic thermal expansion due to preferential orientations in the sample on which the experiments were performed. To clarify this question from the experimental point of view, the thermal expansion in three mutually perpendicular directions of a ErCuz polycrystalline sample has been measured (Duc et al. 1988), see fig. 4.59. Data in this figure clearly prove that the thermal expansion is almost isotropic in the studied sample. These experiments once more show that a crystal-field calculation based on
478
N.H. LUONG and J.J.M. FRANSE
15
0
0
10
oc
(o lr
.°**'m°°
0
.£ u
0
*OOZe :of
0
I I I I
5
0
0
o
o
•
o
0
10
o
20
~
I
I
30
~0
50
I 60
T (K)
t=tr w
~ 0
t
.
.
I
;'0" .
.
t~0 T (K)
o CXrf
-5 0 0 0
-10
O o oo
o o oo
o
O
o
Fig. 4.58. Splitting of cm and C~m of ErCu 2 according to eqs (3.41-3.44) into a long-range magnetic (eir, air) and a crystal-field (c~, c~ce) part with values for n_Fir and ~_r'ef given in the text.
a point-charge model is inadequate to describe the experimental observations for this compound. The ErCu 2 compound has been measured with 166Er M6ssbauer spectroscopy (Gubbens et al. 1991). These authors also reported the results of inelastic neutron scattering, which show a not-yet definitively determined level sequence of doublets above TN at 0, 61, 78, 88, 124, 142, 148 and 160 K. Using a superposition model (Newman 1983, Divis 1991) and by fitting the results of inelastic-neutron-scattering and M6ssbauer spectroscopy, Gubbens et al. (1991) have determined a tentative set
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
479
15I ErCu2
10 A
•- !
5
v'
! Q
O
•--
0
~o ~o
-5
o
o°j/
oOJ ,,,/
x x
×
"10
-1_=
0
t
~
10
20
30
40
50
60
TtK)
Fig. 4.59. The linear thermal-expansion coefficient measured in three mutually perpendicular directions of a polycrystalline sample of ErCu2 with a cubic shape (5 x 5 × 5 mm3); the full curve represents the mean value of c~ along the three directions. After Due et al. (1988).
o f all the nine crystal-field parameters. This set is: B ° = - 0 . 2 8 K, B22 = - 0 . 2 2 K, B 0 -- - 0 . 3 0 X 10 - 2 K, B 2 = - 0 . 1 4 x 10 - 2 K, B44 = 0.30 x 10 - 2 K, B ° = - 0 . 2 0 x 10 - 4 K, B62 = - 0 . 4 7 x 10 - 4 K, B64 = - 0 . 9 7 x 10 - 4 K, B 6 = - 2 . 9 6 x 10 - 4 K. The value o f B ° of this set is very close to a value o f - 0 . 2 6 K determined f r o m the quadrupole splitting (Gubbens et al. 1991). One also can see that there is agreement
480
N.H. LUONG and J.J.M. FRANSE
between the values of B ° and B 2 obtained by Hashimoto et al. (1979a) and those obtained by Gubbens et al. (1991). However, the latter authors have pointed out that their set of nine crystal-field parameters is not unique at the present stage of the investigation. Gratz et al. (1993) have measured the anisotropic thermal expansion of polycrystalline Ercu2 sample from 4.2 K to 400 K by means of X-ray powder diffraction. They have described theoretically the obtained data using the above listed set of crystal-field parameters deduced by Gubbens et al. (1991).
4.10. TmCu2 Magnetic measurements on TmCu 2 have been reported by Sherwood et al. (1964) but no value for the ordering temperature has been given. The ae susceptibility (Smetana et al. 1985a), magnetization, susceptibility, resistivity and specific heat (Smetana et al. 1986a) have been measured on polycrystalline TmCu2. These measurements reveal a Nrel temperature value of 6.3 K and magnetic transitions at Tsl = 4.3 K, Ts2 = 3.9 K and Ts3 = 3.3 K which reflect changes of the magnetic structure. The paramagnetic Curie temperatures along the a, b and c axes are reported as: 0a = - 1 5 K, 0b = 49 K and 0c = 18 K (Sima et al. 1989). The effective magnetic moment, #eft, is equal to (7.2 4- 0.2) #a (Smetana et al. 1986a). Neutron-diffraction investigations have been performed by Lebech et al. (1987), Sima et al. (1989) and, later, by Heidelmann et al. (1992). According to the latter authors, TmCuz shows at least three different antiferromagnetic structures. The lowtemperature structure (AFI) (T < 3.1 K) is composed by the first two odd harmonics of a propagation vector (21r/8)a*. In a second temperature region, 3.1 K < T < 4.7 K, the phase is a mixture of two structures (AFII). A third phase, (AFIII), appears in the temperature region 4.7 K < T < TN ~, 8 K. All these structures may be described as sinusoidally modulated with magnetic moments parallel to the b direction, except for one of the components of AFII, where it is necessary to invoke a component perpendicular to the b axis. The result that only one magnetic moment component in the b direction is necessary to describe the observed structure in the case of AFI and AFIII is in agreement with the crystal-field anisotropy studies (Sima et al. 1989). A discrepancy between the Nrel temperature obtained from bulk data mentioned above (TN = 6.3 K) and that obtained from the neutron-diffraction study of Heidelmann et al. (1992) (TN ~ 8 K) is still unexplained. Heidelmann et al. (1992) also pointed out that from the neutrondiffraction data there is no evidence for the existence of the transition at 3.9 K which has been reported from bulk data (see above). The temperature dependencies of the inverse magnetic susceptibility of TmCu2 along each principal crystallographic axis are shown in fig. 4.60 (Sima et al. 1989). From these results the following experimental values for the second-order crystalfield parameters have been derived: B ° = -0.94 K, B 2 = -1.01 K. Figure 4.61 presents the magnetization curves of single-crystalline TmCu2 (in comparison with a polycrystalline sample) at 4.2 K (Svoboda et al. 1989). A large anisotropy is observed. The b axis is the easy magnetization direction. I n this direction, a two-step metamagnetic transition (of possible spin flip type) occurs
MAGNETIC PROPERTIES OF RARE EARTH-Cu 2 COMPOUNDS
TmCu2
481
a o
3 C
% %
0
100
200
300
T (K) Fig. 4.60. Temperature dependence of the inverse susceptibility along the principal axes of TmCu 2. Symbols are the experimental data. Lines represent the calculated susceptibility according to the set of crystal-field parameters No. 6 (table 4.1). After Sima et al. (1989).
b "R.
6
::3
,oOf,* .
~
o
oe
°
e
po/yory.~tcthne
e
•
•
°
°
°
°
°
°ee e z
2
,see
// •
0
e
C
1
2 B (T)
3
Fig. 4.61. Magnetization curves of TmCu2 along the principal axes (full line) in comparison with the polycrystalline one (dotted line) obtained at 4.2 K. After Svoboda et al. (1989).
with two critical fields Bet = 0.06 T and Be2 = 0.36 T. Above the second step, the magnetization reaches the saturated value of (6.9 4- 0.2)/~B per T m atom which is in good agreement with the value for a free T m 3+ ion. The magnetization along a and e axes is an order of magnitude lower than that along the b axis.
482
N.H. LUONG and J.J.M. FRANSE
Crystal-field effects in TmCu 2 have extensively been studied (Zajac et al. 1987, 1988a, Sima et al. 1988, 1989, Divis et al. 1989b, and Gubbens et al. 1992). Zajac et al. (1987) have calculated the crystal-field splitting and the crystal-field susceptibility along the principal crystallographic axes. In their calculations, they used only the second-order crystal-field parameters B ° = -2.17 K and B 2 = -1.88 K. Their calculations show a total splitting of about 290 K. By comparing the calculations with the experimental results reported by Smetana et al. (1986a), Zajac et al. (1987) have pointed out the need to take higher-order terms into account. Based on the analysis of the specific-heat and thermal-expansion data, Sima et al. (1988, 1989) have studied crystal-field effects in TmCu 2 in more detail. Figure 4.62 shows the experimental and calculated temperature dependencies of the specific heat of TmCu2 in the paramagnetic region. The linear thermal-expansion coefficient of TmCu2 as a function of temperature is shown in fig. 4.63 (Sima et al. 1989). These authors have taken the 7 value equal to 9 mJ/K 2 mol, comparable with the values for the other RCu2 compounds (Luong et al. 1985b). The phonon part of the specific heat of the RCu 2 compounds follows a Debye function rather well (Luong et al. 1985b). From the best fit to the RCu2 data, Sima et al. (1989) obtained 0 D = 194 + 2 K for TmCu2. The values of m and l'ph for TmCu2 are unknown. In case of isostructural YCu2 and (Gd,Y)Cuz, Luong et al. (1985a) have found that x/~ph and 8i) are temperature-independent up to 100 K. Similarly, for TmCu2, Sima et al. (1989) assumed that n_r'ph is also independent of temperature and they have used a ~-r'ph value of 11.9 x 10- 12 m 2 /N, as obtained for YCu2 by Luong et al. (1985a). The problem of analyzing c(T) and ~(T) above TN has been reduced to the determination of parameters OD, Ei and 7i, where 7/[= - d ( l n Ei)/d(ln V)] is the crystal-field Grtineisen parameter of the individual energy
&E{K) 60
• 40
~ "4" % "~
~
~ -
-
////"
~ too
"
~
//
~
/11/
O =/17
//
J,
m
........
0
20
CO T(K)
60
, .......
80
Fig. 4.62. Temperaturedependences of the specific heat of TmCu2 in the paramagneticregion: o, experimental data; - - , calculatedtotal specificheat; - - -, phonon specificheat; •.., electronicspecific heat; - - -, Sehottky specific heat. The arrow indicates the N6el temperature. The inset shows the energies of the CF levels used in calculations. After Sima et al. (1989).
MAGNETIC PROPERTIESOF RARE EARTH-Cu2 COMPOUNDS
483
10
T
-10
".......'
-20
i 20
i /40
i 60
t 80
100
T[K~ Fig. 4.63. Temperaturedependences of the linear thermal-expansioncoefficientof TmCu2 in the paramagnetie region: o, experimental data; q , calculated total thermal-expansioncoefficient; - - -, phonon thermal-expansioncoefficient;..., Schottkythermal-expansioncoefficient.The arrow indicates the Nrel temperature. After Sima et al. (1989). level Ei (i = 0, 1, . . . , 1 2 , E0 = 0). To solve this many-parameter (25) problem, Sima et al. (1989) used a Monte Carlo simulation of all parameters in order to find the intervals of appropriate possible values. The final solution for Ei has to be conform to the nine-parameter orthorhombic CF Hamiltonian. This leads to a reduction of the number of parameters to 22 (8D, B ,rr~, 7i). These authors treated the Bnm as adjustable parameters to be determined from the proposed CF level scheme. The number of possible solutions is quite large. About 100 fits were done and the sets of parameters which are in agreement with experimental observations have been selected (see table 4.1). The values of the second-order terms B ° and B 2 are very close to those obtained from susceptibility measurements (see above). From the analysis it appears that the ground state of TmCu2 is an isolated quasi-doublet (El = 5 K, E2 = 68 K). The total splitting is 210 K, i.e. less than that reported by Zajac et al. (1987) (see above). Sima et al. (1989) pointed out that a major contribution to the CF splitting comes from the second-order terms B2° and B 2. However, these terms cannot satisfactorily describe the observed splitting of the quasi-doublet ground state and the inverse susceptibility Xa I (see fig. 4.60) between 20 and 70 K. All nine parameters, Bnm, must be taken into account in order to obtain, at least qualitative, agreement with experiments. The value for E1 agrees with preliminary results of inelastic neutron spectroscopy obtained by Loewenhaupt and Gratz (1989). Gubbens et al. (1992) have studied crystal-field effects in TmCu2 by 169Tm MSssbauer spectroscopy. By combining the results of magnetization (Svoboda et al. 1989, Divis et al. 1989b), specific heat and thermal expansion (Sima et al. 1989)
484
N.H. LUONG and J.J.M. FRANSE TABLE 4.1 Crystal-field parameters, in units of K, for the compound TmCu 2. The sets Nos. 1-6 are the selected by Sima et al. (1989) from an analysis of specific heat and thermal expansion data. Set (i) gives the calculated CF parameters using a modification of the point-charge model (Sima et al. 1989). Set (ii) gives the values obtained by Zajac et al. (1988a) in point-charge model. Set (iii) gives the parameters derived by Gubbens et al. (1992) by combining the results of different experiments.
No
st
B0,
s:,
(10 -3 ) (10 -2 ) 1 2 3 4 5 6 i ii iii
-1.16 -1.30 -1.30 -1.01 -1.01 -0.94 -1.45 -1.13 -0.94
-0.80 -0.87 -1.09 -1.23 -1.23 -1.30 -0.29 0.87 -1.23
-1.81 -0.07 -4.35 0.36 -0.51 0.29 -5.58 0.15 -9.0
-2.90 -1.45 -0.43 -3.26 -3.40 -3.69 -0.36 -0.28 -0.39
B',
s06
(10 -2 )
(10-~
2.17 0.58 -0.07 2.82 3.62 2.90 0.87 0.77 -0.36
0.72 1.45 1.52 -2.32 -0.80 -2.54 3.26 0.43 5.8
(10 -4) -6.52 -9.63 0.72 -7.53 -7.46 -7.82 2.46 0.15 2.47
(10-') 5.79 2.90 -11.66 3.19 2.39 4.06 1.59 0.20 -0.48
(10 -4 ) -7.97 6.59 0.87 4.49 3.19 3.91 -7.17 0.07 6.31
with those of inelastic neutron scattering (Divis, Heidelmann and Loewenhaupt, unpublished results) and 169Tm M6ssbauer spectroscopy, Gubbens et al. (1992) have derived the set of crystal-field parameters for TmCu 2 shown in the last row of table 4.1. However, Gubbens et al. (1992) pointed out that this set of crystal-field parameters still has a preliminary character. By studying the relaxation in TmCu2, these authors have indicated that the proposed crystal-field parameters might need future reconsideration. Javorsky et al. (1992) have measured the specific heat of polycrystalline samples Tm~YI_~Cu2 (z = 0.1, 0.2, 0.3, 0.4, 0.6 and 0.8) at temperatures up to 30 K. The antiferromagnetic ordering has been observed for z = 0.6 and 0.8. For z ~< 0.4, no magnetic ordering has been found and the specific-heat curves turned out to show only a Shottky anomaly due to the crystal-field effect. Using the molecular-field model in a two-level approximation (Fulde 1979), Javorsky et al. (1992) calculated the temperature dependence of the Nrel temperature according to the relation: tanh (E1/2Ttq) = (1/z) tanh (E1/2T~),
(4.6)
where T~ is the NEel temperature for TmCu2 and E1 is the energy separation (measured in kelvin) between the two lowest levels. They have obtained good agreement with experiment. By comparing the experimental data in the paramagnetic region with eq. (3.23), Javorsky et al. derived the best-fit values for El, E2 and E3. They concluded that the value for E1 changes minimally with concentration, (El = (4 4- 2) K). and that the quasidoublet ground state is well separated from the higher excited states for all Tm concentrations, in very good agreement with the preliminary results of inelastic neutron spectroscopy obtained by Heidelmann and Loewenhaupt (private communication), who reported for x = 0.05 the values for El and E2 of 2.1 K and 45 K, respectively.
MAGNETIC PROPERTIES OF RARE EARTH-Cu2 COMPOUNDS
485
Specific-heat and magnetization measurements on Tm(Cu~Nil_~)2 have been performed by Divis et al. (1990c). A change from antiferromagnetic (for z --- 0.95 and 0.9) to ferromagnetic ordering (for z = 0.8, 0.75 and 0.7) is observed. For x < 0.7, the CeCu 2 structure becomes unstable. From the analysis of the data, Divis et al. conclude that the crystal-field ground state of the Tm(Cu~Nil_~:)2 compounds (z ~< 0.7) is a quasidoublet.
5. Comparison of isostructural compounds In the preceding section we have discussed the properties of various RCu 2 compounds. It is of interest to make an attempt to see some systematic behaviour, comparing different compounds with the same crystallographic structure. It can be inferred from section 4 that information about the crystal-field interaction ill RCu 2 is not complete. Due to the orthorhombic structure, nine CF parameters are needed to describe the CF Hamiltonian. Up till now, the full set of CF parameters is available only for NdCu2, ErCu2 and TmCu2. Nevertheless, the lowest-order CF parameters Bg and B 2 have been derived for most of the RCu2 compounds. In table 5.1, we collect the second-order coefficients A ° and A2z for the RCu2 compounds. These coefficients are related to the CF parameters B 20 and B 22 by (see section 3): Ao =
B°loej(r2e),
(5.1)
A 2 m B 22/ a J ( r 4 2f ) ,
where the values for the quantity otj(rlf ) are taken from table 3.1. As pointed out in section 4, to our knowledge, CF parameters for SmCu2 are not available. Rather low TABLE 5.1 Crystal-field coefficients, in units of Kao 2, for the RCu 2 compounds. R
A~
A~
ReL
Pr Nd
-168 -112 -188
-117 -154 -218
[1] [1] [2]
Tb
-148
-148
[31
Dy Ho Er
-85.7 -83.5 -190 -153 -134.7 - 134.7
-143.4 -71.6 -194 -120 -186.4 - 176.3
[3] [3] [3] 14] [5] [6]
Tm [1] [2] [3] [4] [5] [6]
Hashimoto (1979). Gratz et al. (1991). Hashimoto et al. (1979a). Gubbens et al. (1991). Sima et al. (1989). Gubbens et al. (1992).
486
N.H. LUONG and J.J.M. FRANSE
values of A ° and A 2 have been obtained by Trump (1991) for the Kondo compound CeCu2, in which anomalous properties are observed. Except for CeCu2, as can be seen from table 5.1, the coefficients A ° and A22 have the same sign and are of the same order of magnitude. From the similarity of the lowest-order CF parameters it seems that the crystal-field model can be used for describing the behaviour of isostructural RCu 2 compounds. At the same time, there is no reason why higherorder CF parameters should be neglected. The necessity to take these higher-order terms into account is indicated in section 4. The importance of the higher-order CF terms is also revealed from the studies on substituted RCu2 compounds. Analyzing the data obtained on a Tb(Cu0.7Ni0.3)2 sample, Divis et al. (1990b) have shown that the step-like appearance in the magnetization curves along the b axis in this compound cannot be explained using second-order terms in CF Hamiltonian only. These authors have shown that in order to account for all features of the magnetization data, the higher-order terms should be included into the Hamiltonian. Divis et al. (1990c) have also used nine CF parameters for describing the specific-heat data on Tm(Cul_xNix)2. Even for simple cubic systems such as RA12, where only two CF parameters are needed, the CF model fails to give a rigorous systematic and quantitative description of the magnetic properties (Purwins and Leson 1990). Little can be said about a systematic analysis of exchange interactions in RCu2. This is probably due to the complicated magnetic structure of these compounds. In some cases, the temperature dependence of the sublattice magnetizations and the magnetization process can be described in the molecular-field and crystal-field model in which, apart from the crystal-field parameters, a limited number of the molecular-field constants is considered. TbCu2 is an example. The magnetic unit cell of this compound consists of twelve magnetic moments but different temperature dependencies of the magnetization are observed in two sublattices only (Sima et al. 1986, Divis et al. 1987). Then the effective molecular fields at these two sublattices can be expressed as (Divis et al. 1987): Bin1 - -
LI(M1) + 2L2(M2),
(5.2)
Bin2 = (51 - L2)(M2) + L2(M1), where M1 and M2 are the sublattice magnetic moments and where LI and L2 are effective molecular-field constants. The observed temperature dependence of the sublattice magnetizations can be well reproduced by a calculation using these two effective molecular-field constants together with two experimentally determined crystal-field parameters (Divis et al. 1987, Iwata et al. 1987). Values for these molecular-field constants are derived from the best fit to the experimental data. The effective molecular-field constants for TbCu2 determined by Divis et al. (1987) and Zajac et al. (1988b) (L1 = 5.77 x 105 A/m/.tB and L2 = 5.42 x 105 A/m#B) are only slightly different from those obtained by Iwata et al. (1987, 1988). Divis et al. (1987) have pointed out that the observed temperature dependence of the sublattice magnetization in TbCu2 can be considered as a result of comparable contributions of the molecular-field and the crystal-field contributions to the total energy in this
MAGNETIC PROPERTIESOF RARE EARTH-Cu2COMPOUNDS
487
compound. Magnetization processes in YbCu 2 compound are satisfactorily explained by Iwata et al. (1988) using the molecular-field model with two crystal-field parameters and five exchange parameters. From the best fit to the magnetization processes in DyCu2, Iwata et al. (1989) have determined the exchange parameters for this compound. Values for these parameters are comparable in magnitude with those of the TbCu2 compound. One of the features of the RCu2 compounds is that the values for the N6el temperature of these compounds are not simply proportional to the de Gelmes factor G = (gj - 1)2j(j + 1) and takes a maximum for TbCu2 (see fig. 5.1). This fact suggests that the RKKY interaction is not sufficient for fully understanding the exchange interactions in the RCu2 compounds. In the discussions of the ground-state spin configurations in the CeCu2-type crystal structure, Kimura (1986) considered four kinds of bilinear exchange interactions. In later work, Kimura et al. (1991), apart from three kinds of bilinear exchange interactions, have introduced the four-spin exchange interaction. These authors show that the four-spin exchange interaction gives rise to new complex states as a possible ground-state spin configuration in the CeCu 2type crystal structure and that, within the 12-sublattice model, 24 types of spin configurations are possible for various values of exchange interactions. Analyzing magnetization process in the TbrYl_=Cu2 compounds in the same molecular-field and crystal-field model as that used by Divis et al. (1987), Zajac et al. (1988b) pointed out that better agreement between calculated and measured magnetization curves can be expected if taking into account higher-order crystal- and molecularfield parameters. In some of the RCu2 compounds, it may be necessary to consider the quadrupole and magnetoelastic interactions for better explanation of the magnetic
16
5O
~0
,-i---
0
12 "7
0
3O
..-i ¢q
a~ 0
{71
2O 0
10
O
0 0
$ O
0
0 0 n ~ I i J i r T r I i ~9 Pm Eu Tb Ho Tm Lu Lo Pr Ce Nd Srn Od Dy Er Yb
Fig. 5.1. Values for the N6el temperature in the R C u 2 series and values for the de Gennes factor (9 - 1)23"(,/+ 1) across the 4f series.
488
N.H. LUONG and J.J.M. FRANSE
behaviour. The role of these interactions in rare-earth intermetallic compounds has been discussed, e.g., by Morin and Schmitt (1978, 1990).
6. Acknowledgements The authors wish to thank Prof. T.D. Hien of the Cryogenic Laboratory of the University of Hanoi for his interest, help and encouragement. The cooperations with Prof. N.P. Thuy, Dr. N.H. Duc and other colleaques of the Cryogenic Laboratory and with Dr. E E Bekker of the University of Amsterdam are kindly acknowledged. The authors are grateful to Dr. Y. Hashimoto, Prof. I. Kimura and Prof. M. Loewenhaupt for providing their work, to Dr. D. Givord for comments on the manuscript, to Dr. N.M. Hang for his comments and assistance in collecting some references and to Dr. R.R. Arons et al. for sending a preprint on the magnetic structure of NdCu2. This work has been supported by the Commission of the European Community (BRITE/EURAM Programme) in the scope of the project 'Basic Interactions in RareEarth Magnets' (BIREM).
References Abd-Elmeguid, M.M. and G. Kaindl, 1978, Hyperfine Interact. 4, 420. Abd-Elmeguid, M.M., H. Micklitz and G. Kaindl, 1981, Phys. Rev. B 23, 75. Andres, K., 1973, Phys. Rev. B 7, 4295. Andres, K., E. Bucher, J.P. Malta and A.S. Cooper, 1972, Phys. Rev. Lett. 28, 1652. Andres, K., P.S. Wang, Y.H. Wang and B. Ltithi, 1976, AlP Conf. Proe. 34, 222. Arons, R.R., M. Loewenhaupt, Th. Reif and E. Gratz, 1994, to be published. Barbisch, B.J. and K. Kwasnitza, 1984, J. Phys. (Paris) 45, C1-561. Barron, T.H.IC, J.G. Collins and G.IC White, 1980, Adv. Phys. 29, 609. Bartkowski, K., E. Gratz, J. Rafalowicz and T. Tyc, 1992, Phys. Status Solidi B: 170, 255. Bauer, E., 1991, Adv. Phys. 40, 417. Bauer, E., E. Gratz, N. Pillmayr, D. Gignoux and D. Sehmitt, 1988, J. Magn. Magn. Mater. 76-77, 131. Bauer, E., E. Gratz, N. Pillmayr, D. Gignoux and D. Schmitt, 1989, J. Phys.: Condens. Matter 1, 5463. Bauer, E., E. Gratz, N. Pillmayr, T. H~iufler, M. Lees, D. Gignoux and D. Schmitt, 1990, Physica B 163, 375. Birss, R.R., R.V. Houldsworth and D.G. Lord, 1980, J. Magn. Magn. Mater. 15-15, 917.
Bischof, J., M. Divis, P. Svoboda and Z~ Smetana, 1989, Phys. Status Solidi A: 114, K229. Borombaev, M.K., R.Z. Levitin, A.S. Markosyan, V.A. Reimer, E.V. Sinitsyn and Z. Smetana, 1987, Zh. Eksp. Teor. Fiz. 93, 1517; Eng. Transl. Sov. Phys. JETP 66, 866. Borombaev, M.K., R.Z. Levitin, A.S. Markosyan, Z. Smetana, V.V. Snegirev and P. Svoboda, 1986a, Phys. Status Solidi A: 97, 501. Borombaev, M.K., R.Z. Levitin, A.S. Markosyan, Z. Smetana, P. Svoboda and V. Sima, 1986b, Phys. Status Solidi A: 98, 221. Borombaev, M.K. and A.S. Markosyan, 1987, Fiz. Metallov i Metalloved. 63, 714 (in Russian). Bowden, G.J., D.P. Bunbury and M.A.H. McCausland, 1971, J. Phys. C: 4, 1870. Bozukov, L., A. Gilewski, E. Gratz, A. Apostolov and IC Kamenov, 1992, Physica B 177, 299. Bredl, C.D., 1987, J. Magn. Magn. Mater. 63-64, 355. Brommer, P.E. and J.J.M. Franse, 1990, in: Ferromagnetic Materials, Vol. 5, eds K.H.J. Buschow and E.E Wohlfarth (North-Holland, Amsterdam) p. 323. Bruzzone, G., M.L. Fornasini and E Merlo, 1973, J. Less-Common Met. 32, 317.
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