- Email: [email protected]
Metamagnetism and complex magnetic phase diagrams of rare earth intermetallics
Journol of
~YS
AND CONIPO~D5 ELSEVIER
J o u r n a l of Alloys a n d C o m p o u n d s 225 (1995) 423-431
Metamagnetism and complex magnetic phase diagrams of rare earth intermetallics D. Gignoux, D. Schmitt Laboratoire de Magndtisme Louis Ngel, C.N.R.S., BP 166, 38042 Grenoble Cedex 9, France
Abstract
A survey of the extreme variety of metamagnetic processes discovered during recent decades in rare earth intermetailics, where only the rare earth is magnetic, is presented together with macroscopic and microscopic aspects of the complex associated H - T magnetic phase diagrams. These advances were obtained owing to improvements in experimental techniques (in particular high-field magnetization measurements and neutron diffraction) and the increasing number of single crystals available of good quality. The role of basic interactions (crystal field, bilinear exchange interactions, quadrupolar interactions...) is discussed. The parallel improvements in theoretical models and quantitative analyses of these wealthy properties are reported briefly. Keywords: Metamagnetism; Magnetic phase diagrams; Rare earth intermetallics
I. Introduction
Initially limited to a first-order field-induced phase transition between a simple antiferromagnetic structure and the ferromagnetic state, the concept of metamagnetism has been extended during recent decades to many processes which break antiferromagnetic structures, and more generally to all types of field-induced magnetic phase transition [1,2]. The aim of this paper is to emphasize the specific role of the rare earth intermetallics in such an evolution by showing, through several examples, how each of the various types of metamagnetic process is associated with a particular physics depending on the material considered. The large increase in the number of rare earth intermetallics exhibiting metamagnetism has three main causes: (i) progress in the synthesis of single crystals; (ii) the increasing number of new rare earth intermetallics, in particular ternary compounds; (iii) the increase in the performance of experimental facilities, in particular high-field measurements and neutron diffraction experiments. We consider as metamagnetic behaviour an anomaly occurring during the magnetization process of a given compound, characterized by an upward curvature in a limited range of field. Two main classes of metamagnetic transition can be distinguished: (i) step-like behaviour where the magnetization jumps between two plateaux 0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 5 - 8 3 8 8 ( 9 4 ) 0 7 0 4 6 - 6
at a given critical field Hc (class (1)); (ii) S-shape behaviour where the magnetization increases more smoothly than in class (1) systems, showing only an inflexion point at the critical field (class (2)). Within this quite general definition of metamagnetism one can include the FOMPs (first order magnetization process) as described in Ref. [3] which are often associated with rotation of the magnetization vector in ferromagnets and ferrimagnets. In this paper we do not consider the following: (i) the FOMP occurring in ferromagnetic and ferrimagnetic rare earth-3d intermetallic compounds in which both 4f and 3d atoms are magnetic, since they are described elsewhere [4]; (ii) the antiferromagnetic rare earth-3d intermetallics, in which both 4f and 3d atoms are magnetic (for example RMn2), that exhibit the so-called antimetamagnetism [5]; (iii) itinerant electron metamagnetism associated with the onset of 3d magnetism [6]. The temperature dependence of the transition fields Hc allows boundaries of the H - T phase diagram to be defined, showing one or several distinct regions, each associated with a particular spin structure. The two main ingredients playing an essential role in metamagnetism and associated phase diagrams are the magnetocrystalline anisotropy and the interionic interactions. The former, i.e. the crystalline electric field (CEF) coupling, plays a fundamental role as it leads to Ising (one easy magnetization direction), Potts (several equiv-
424
D. Gignoux, D. Schmitt I Journal o f Alloys and Compounds 225 (1995) 423-431
alent non-collinear easy magnetization directions such as the (100) axes in hexagonal or tetragonal symmetry [7]) or X-Y (easy plane) systems. The one-ion magnetoelastic coupling must be included in this first category, being considered as the strain derivative of the CEF itself. The second fundamental ingredient includes all the types of interionic interaction, namely the isotropic or anisotropic bilinear exchange coupling and the two-ion quadrupolar coupling, as well as their possible strain dependence.
2. Spin-flop in weakly anisotropic metamagnetic systems In gadolinium (L = 0) based antiferromagnetic compounds the magnetocrystalline anisotropy is very weak and the metamagnetic behaviour characterizes the properties of the exchange interactions alone. However, a small anisotropy fixes the zero field orientation of the magnetic structure with regard to the lattice and is revealed under low field through a spin-flop metamagnetic transition. During this transition there is typically a rotation of the moments toward a direction perpendicular to the applied field. A good example is the orthorhombic GdCu6 compound [8], where the magnetization curves along the b and c axes exhibit small discontinuous steps before varying linearly up to the saturated value reached at Hs = 24 T (Fig. 1). In the case of a simple antiferromagnet, the spin-flop transition should occur along one single axis, namely parallel to the direction of the moments, at least in low symmetry systems. One can conclude that the magnetic structure of G d C u 6 is planar, for instance helical in the b--c plane.
./ .//
.//" ././//-
GdCu6 1.3K
c-a~is~///"
/2"./
-
:~ 2
"Z/
-
,f.-77"
,:,s,".;"
~." .J.I
nu
0
J¢" ,
I
,
I
48 Magnetic Field
,
I
12 [ T
,
]
Fig. 1. Magnetization curves for GdCu6 at 1.3 K (after Ref. [8]).
Once the perpendicular configuration is achieved, i.e. above the spin-flop transition, any anomaly on the magnetization curves will be associated with destruction of the antiferromagnetic structure itself. It is worth noting that, for a simple antiferromagnet or a helical structure, the magnetization process is, in principle, strictly linear. Therefore, any observed metamagnetic transition will suggest a more complex structure or will be the signature of anisotropic exchange interactions, as is the case in various gadolinium compounds. It should be pointed out that the absence of magnetocrystalline anisotropy in gadolinium compounds is not necessarily advantageous for understanding the magnetic behaviour under field. In fact, the situation is even more complicated because several types of coupling generally neglected should now be taken into consideration. Moreover, good determination of the magnetic structure is still important for describing the properties, and this is difficult to do for gadolinium compounds because of the huge absorption cross-section of natural gadolinium under normal conditions. Neutron diffraction experiments using a gadolinium isotope or at a very low wavelength (A = 0.5/~) are needed. This explains the small number of such experiments in the literature.
3. Modulated metamagnetic systems In the lanthanide intermetallic compounds, it is well known that the RKKY-type indirect coupling is long range and oscillatory, and its Fourier transform J(q) often exhibits a maximum for a vector Q not in a high symmetry position in the BriUouin zone, i.e. Q ~ 0 and Q ~ (1/2)K, where K is a reciprocal lattice vector. This leads, at least near TN, to incommensurate or long period commensurate structures. For rare earths with L ~ 0 and easy magnetization direction(s) (Ising or Potts systems), the corresponding structure is amplitude modulated, sinusoidal near TN. In Ising type systems, the metamagnetism associated with such modulated structures is simple when the magnetic field is applied along the moment direction, exhibiting a single smooth transition of class (2). In the case of modulated Potts systems, a difficulty arises concerning the magnetic domains because it is not obvious a priori whether the magnetic field favours the domain with moments parallel or perpendicular to itself. An increasing number of lanthanide intermetallic compounds showing an incommensurate modulated structure exhibit this type of transition. This is the case for numerous tetragonal compounds having the ThCr2Si2-type crystallographic structure. A good example is the Ising type TbNizSiz compound between 7",= 9 K and TN = 15 K, where the propagation vector is Q= (0.426, 0.574, 0); below Tt the structure changes towards a simple commensurate
D. Gignoux, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423-431 2
i
I
i
i
'4 :/ '~'
i
~
n. 0r
i
i
P r N i z S iz ..........
e
-~"r/I~
o
0
~ f"f"I
2
[001]
H = 5.8 T ,-"
4
6
8
MAGNETIC FIELD ( T ) Fig. 2. Magnetization processes in PrNi2Si2 at 1.5 K along and perpendicular to the c direction; note the cusp when reaching the induced ferromagnetic state along c. Lines were calculated using the PF model. Schematic representation of the magnetic structure at several fields along the c direction; note the residual modulation at 5.8 T, i.e. immediately below the cusp where it vanishes (after Ref. [11]).
antiferromagnetic structure with Q = (0.5, 0.5, 0) [9]. In the modulated phase there is one single metamagnetic transition ranging between 10 and 15 kOe according to the temperature. In the above examples as well as in the vast majority of modulated systems, the high temperature structure is not stable down to 0 K because of entropy effects, and the system transforms into a magnetic structure with a shorter period [10]. One exception may occur when the CEF ground state is a non-magnetic state, as in tetragonal PrNizSi2. In this compound, a sine-wave modulated structure occurs at TN = 20 K, the propagation vector being Q = (0, 0, 0.870) and the moments being aligned along the c-axis [11]. This structure persists down to 0 K owing to the existence of a non-magnetic CEF singlet as ground state well isolated from the excited levels. Under these conditions it is possible to study the modulated phase down to low temperatures, in particular the metamagnetic behaviour. In Fig. 2, this manifests itself very clearly as a positive curvature on the magnetization curve along the c-axis up to about Hc = 6 T where a cusp is observed. This curvature corresponds to the progressive disappearance of the modulation under the applied field, and the observed cusp indicates precisely the completion of this evolution, i.e. the entrance into the induced ferromagnetic state.
4. Spin-flip and spin-slip in anisotropic rare earth c o m p o u n d s with coUinear structures
In rare earth intermetallics with magnetocrystalline anisotropy (L=/=0) we are faced with Ising or Potts systems in which magnetic structures are often collinear
425
(at least when the symmetry is not too low). As quoted above, in a large number of materials the magnetic periodicity near TN is incommensurate or long period commensurate leading to amplitude modulated magnetic structures. Only a few compounds exhibit simple antiferromagnetic structures over the whole temperature range below TN (e.g. ErGaz). Except in the case of a non-magnetic CEF singlet ground state, amplitude modulated magnetic structures are not stable at low temperature, and transform toward a structure with equal moments in different ways: (i) the magnetic periodicity remains unchanged and there is a progressive squaring of the structure which manifests itself as the appearance in the neutron diffraction pattern of peaks associated with higher order harmonics of the ground propagation vector (e.g. TbCu2 and DyCu2 [12]); (ii) one or several transitions toward other, generally simpler, magnetic phases with different periodicity, i.e. where the period is shorter (devil's staircase). A very large number of systems belong to the latter category [10]. Magnetization processes of these collinear equal moment magnetic structures are step like, in most cases exhibiting multistep metamagnetic processes. This behaviour leads to complex magnetic H - T phase diagrams showing several intermediate magnetic structures. Note that in the second category of compounds quoted above, the resurgence of high temperature incommensurability is frequently observed during the low temperature metamagnetic process. In all these compounds, low temperature metamagnetism involves two types of process, namely spinflip and spin-slip. (i) A spin-flip transition corresponds to the reversal of moments antiparallel to the field in a given magnetic cell (one can find a common magnetic periodicity below and above the transition). Note that this process can be accompanied by the antiflipping of some moments. (ii) The concept of spin-slips or discommensurations was first introduced to explain the observed lock-in transitions in the magnetic spirals of rare earth metals such as pure holmium and dysprosium in terms of simple commensurate structures [13]. In both hexagonal metals, the magnetic propagation vector depends on the temperature through a succession of lock-in transitions on commensurate values close to each other. This is associated with the tendency for the basal plane component to lock close to the easy axes within the hexagonal plane, resulting in periodic defects (spinslips) instead of a regular rotation of the spiral order. More generally this term can be used to characterize structures which present periodic defaults in a simple sequence of magnetic moments. A spin-slip transition corresponds to the change from one discommensuration to another. It is generally associated with a metamagnetic transition of small amplitude in the magnetization process and a small shift of the propagation vector.
D. Gignoux, D. Schmitt / Journal of Alloys and Compounds 225 (1995) 423--431
426
To illustrate the variety of situations found in this type of material we present some characteristic examples. Note that, if the thermal dependence of the zero field magnetic structure has been determined in a large number of compounds, the field dependence of these structures has been investigated in a limited number of materials owing to the necessity of carrying out neutron diffraction experiments on single crystals under an applied magnetic field. Among the few examples of single-step spin-flip metamagnetism toward the induced ferromagnetic state one can quote the orthorhombic TbCu2 compound at low temperature when the field is applied along the easy a-axis [14] (Fig. 3). Under zero field, although the basic propagation vector is Q=(1/3, 0, 0), the structure is antiphase because there are four terbium atoms in the unit cell. At the critical field, all moments are antiparallel to the field flip. Two-step spin-flip metamagnetism, in which magnetization of the intermediate phase is exactly one half of the saturated value, has been observed in some simple antiferromagnets such as tetragonal DyCo2Si2 [15]. The intermediate field-induced phase corresponds, in agreement with neutron diffraction studies, to the spin-flip of half the moments initially antiparaUel to the field (Fig. 4). The tetragonal PrCo2Si2 compound has been particularly thoroughly investigated. It orders at T~= 30 K within a long period commensurate structure with Q1 = (0, 0, 7/9) [16]. As the temperature is lowered, a first-order transition occurs at T2 = 17 K, the new propagation vector being Q2 = (0, 0, 13/14), while the periodicity locks below T1 = 9 K into a simple antiferromagnetic structure with 0 3 = (0, 0, 1). The magnetization process at 1.3 K shows a four-step metamagnetic behaviour along the c-axis, the magnetization at each step being a fractional number of the saturated value, being approximately 1/14, 3/14, 2/9 and 1 of Mo = 3.2/zB/Pr (Fig. 5) [17]. These magnetization values are consistent with field-induced magnetic structures which have the same successive propagation vectors Q2 and Q1 as those stabilized at higher temperature. This 10
l
I
oOOO
o
I o
o
o o
TbCu2
z o
6
,< N .
Z.
Z ,<
2 0
4.2K
10
o
a
c .
20
30
40
H (kOe) Fig. 3. Magnetization curves for TbCu2 at 4.2 K (after Ref. [14]).
i
i
i
i
l
DyCo2Si2
[0011 12
IO
~o-o-O"O~
Co *Sl
•
...,~.:,o-o
8
4
IIII
I
I Foo,,.eo- o,- o,. -,o ,-o J
4
T:4.2~
f I Io
tI t I l I I / I j...~j~
2 O
~
"
e...l l - j
~
0
....e_.e-e --e-e'e*'e'e*e'er¢'e'~ HJ. '
20
~ 40
l
I 60
He (kOe) Fig. 4. Magnetization curves for DyCo2Si2 at 4.2 K (after Ref. [15]).
Pr C°2Si2
He4
I
T = 1.3 K Ha//C-axis
~2
H
Hc~
HaJ.c -axis
O0
10
2'0
3'0
Ha (T) Fig. 5. Magnetization curves for PrCo2Si~ at 1.3 K along a and c; the dashed line was calculated using a simple incommensurate m e a n field model (after Ref. [17]).
has been confirmed by a neutron diffraction study in pulsed high magnetic field [18]. The temperature variation of the critical field gives a phase diagram in which the high temperature phases extend toward the low temperature and high field region by turning around the origin (Fig. 6). Among the Ising systems with hexagonal symmetry, the HoA1Ga compound is one of the very few which has been studied extensively. It exhibits two different magnetic phases in zero field [19]: (i) an antiphase below Tt = 18.5 K, characterized by the basic vector Q1 = (1/3, 1/3, 1/2) and its third harmonics 3Q1 = (0, 0, 1/2); (ii) an amplitude modulated structure between T, and TN = 31 K with the single incommensurate vector Q2 = (1/3, 1/3, 0.481). At low temperature, a three-step metamagnetic process is observed along the c axis (Fig. 7), the two intermediate magnetizations reaching approximately 1/9 and 1/3 of the full moment. The magnetic phase diagram shows three ordered regions (Fig. 8)
427
D. Gignoux, D. Schrnitt / Journal o f Alloys and Compounds225 (1995) 423-431
PrCo2Si2
15
x X
B=
lO
\\
A
\ \x A
B=
5 -
r,
B¢I ~ -
•
°r~ - -o~ X
0
1
x
!
N
-- i
,
T1
0
-
j
m TN
T2
10
T(~
30
20
Fig. 6. Magnetic phase diagram of PrCo2Si 2 along c determined by high field n e u t r o n diffraction (after Ref. [18]).
9
i
]
6
~-~t -c
i
o
rt (~-'&4) 00
\[
~ ~///
~ _j
20
c axis
HoAIGa 40
60 H(kOe)
80
Fig. 7. Magnetization process at t.7 K in H o A I G a along the c-axis and associated magnetic structures (after Ref. [20]).
60
.-. •
0
50
..
: . . . ; . . . .
~
•
'
>~~,
"
~
! ....
i .... ,
II
.
,
. . . .
~.
,
. . . .
HoAIGa [001]
..~
~
The second field-induced state at 1.7 K is characterized by the flipping of all the remaining antiparallel moments of line B, the magnetic unit cell recovering the same size as at zero field. Hence there should exist a border line, not detected by magnetic measurements [19], between this commensurate high field-low temperature region and the incommensurate low field-high temperature region. Actually, such a border line (hatched area in Fig. 8) has been determined by neutron diffraction. However, broadening of the diffraction peaks at 1.7 K and 48 kOe is already present and suggests that a perfect arrangement with wavevector Q1 and its even harmonics is not fully achieved within this limited region, but presents defaults reminiscent of the high temperature propagation vector, illustrating the competition between both commensurate and incommensurate periodicities. PrGa2 (same hexagonal structure as HoA1Ga) is a wealthy Potts system in which the multistep metamagnetic behaviour and the associated complex phase diagram involve spin-flips, spin-slips and magnetic domain effects. At 1.5 K and in zero field, this compound exhibits a non-compensated long period commensurate antiphase structure, with the propagation vector Q = (4/ 27, 4/27, 0), the moments being aligned along the [100] axis [22]. Fig. 9 shows, for this axis, the second and subsequent magnetization processes in which the material is single domain, that with moments (and propagation vector) parallel to the field. The first magnetization process (in increasing field) exhibits a transition around 3.5 kOe associated with the change from a three-domain to a single-domain structure. Let us now discuss the phases induced during the second magnetization process. The first metamagnetic transition is of spin-flip type [23]; below a critical field Hc=2.3 kOe (phase II), the magnetic cell has 27 moments with the sequence (4 3)(3 4)(3 3)(4 3) along the [100] direction, i.e. it consists of four groups of seven moments (4 up, 3 down), with two defaults over 28 sites, namely
"~,--
, •
-
'
20 \ T1=18.5 K
TN=31 K
' 10
'
. 'L I (!]3,.113;112.'X)1 I . . . . 20 30
T(K)
I
I
[
1.SK 40
Fig. 8. Boundaries of the p h a s e diagram of H o A I G a along c (after Ref. [21]). Full circles are the experimental points. Full lines are calculations using the P F model. Boundaries of phase III are not accounted for by the model because this phase is described by several non-collinear propagation vectors.
,I
l it vi
=L
ii
0.5
.';:::::::;:'"
¢'4 .u
•'", " ,i:'L"" ,T
[20]. The first intermediate phase (phase III) is characterized, on one linie over three along c (line B), by the flipping of 2/3 of the initially antiparallel moments and the antiflipping of 1/3 of the parallel moments.
'
°:
,,oo, 2
G" 00 ~
I
P r G a
i
/ 2
II
t IV
t III ,
i
,
4
,
6 8 H (kOe)
,
,
,
i
10
i
12
Fig. 9. Low field magnetization process along [100] in PrGa2 at 1.5 K (after Ref. [23]).
428
D. Gignoux, D. Schmitt I Journal of Alloys and Compounds 225 (1995) 423-431
PrGa 2 T=l.8 K
H ffi4 koe- Plmm VI
a~ Spin-slip
++++
.
.
+++0-
.
H =3 kO¢- Plmm Hl
.
- I - ÷ -
~ Spin-flip
++++ H ffi0 ROe - P h i IT
+
.
-
-
÷+++---
"Jr"I
-
First default
_
++
- - - + + + + - - Second default
Fig. 10. Low field magnetic structure of PrGa2 at 1.8 K; note the successive spin-flip and spin-slip behaviours,
one inversion up--down in the second group and a missing up moment in the third group (Fig. 10). The induced magnetic structure above Hc (phase III) corresponds to the vanishing of the first fault of the zero field structure, i.e. to a single spin-flip, without changing the magnetic periodicity. Therefore, the resulting magnetization changes from (1.27)Mo to (3/27)Mo, where Mo = 3.1#B is the praseodymium magnetic moment. The other transitions occurring below 9 kOe are of spinslip type [23]. More precisely, the well defined plateau observed on decreasing the field between 6 and 2 kOe (phase VI), corresponds to a periodicity of exactly seven unit cells, i.e. Q=(4/28, 4/28, 0), where the second default in the sequence of magnetic blocks has disappeared (Fig. 10). Moreover, this periodicity appears under increasing field at around 4 kOe and subsists up to 8 kOe (phase IV), the width of the associated neutron diffraction peaks varying in the same field range and being related to the large superimposed susceptibility observed on the magnetization curve. This behaviour has been explained by the difficulty of establishing perfect long range periodicity of seven cells in the system, possibly owing to defects in the crystal.
5. Paramagnetic metamagnetic systems This behaviour is observed in systems where the CEF is the only interaction acting on the 4f shell. The most dramatic effect arises in the case of level crossing, as predicted in 1966 to occur in TmSb [24]. More recently this effect has been observed in praseodymium metal [25] which exhibits a single-step transition at low temperature. The crossing occurs between two CEF levels, namely a singlet which is the ground state under zero field and an excited level which is more magnetic. Actually the most typical example of this behaviour is provided by the hexagonal compound PrNi5 (Fig. 11) [26]. In this paramagnet, the F3 singlet ground state is followed by another singlet /'1 around 30 K. When
/~
t
r , ~
--
0
10
20
t ' [120] 30
' J
.o* e°*
i ...."" 1
,l.
"'"""
i
PrNi 5 1.4K 0
,
0
I
i
,,,~,~
I
,
~ ,
I
10 20 30 Applied Field (Tesla)
m
I
40
Fig. 11. High field magnetization curves of PrNi5 along the two main symmetry directions of the basal plane at 1.5 K (after Ref. [26]). Insets show the calculated field variations of the energy of the first two CEF levels along these directions.
the magnetic field is applied in the basal plane these two levels should cross for a given field. However, according to group theory considerations, the ground level F3 transforms into a F2 or F1 level if the applied field is along the [100] or [120] direction respectively, while the first excited level keeps the F1 symmetry. It follows that the two levels actually cross each other in the first case, because they belong to different representations, while in the second case an anticrossing occurs because they both belong to the same representation. This leads to a metamagnetic transition which becomes sharper and sharper in the first case when the temperature is lowered, finally reaching stepwise behaviour at 0 K, while in the second case the transition remains smooth even at 0 K. Both transitions have been observed experimentally for a magnetic field of 18.5 T and 12 T respectively.
D. Gignoux, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423-431
429
6. Ferromagnetic metamagnetic systems Under certain conditions, ferromagnets, where only the rare earth is magnetic, may exhibit metamagnetic processes. This is the case in low symmetry ferromagnets, such as the orthorhombic RNi compounds, where R atoms can be divided into sublattices with different local easy magnetization directions. The magnetic structures are then non-collinear with both ferromagnetic and antiferromagnetic components. When the field is applied along one of the latter, and if the magnetocrystalline anisotropy is large enough, a spin-flip occurs, corresponding to the reversal of one sublattice. In a large number of cases, the metamagnetic behaviour of rare earth ferromagnetic compounds is basically associated with crystal field properties, namely a crossing or an anticrossing of CEF levels. Examples, where such metamagnetic processes have been well studied, include cubic systems (e.g. RZn and HAl2) as well as lower symmetry compounds (e.g. RRu2, RGa, RNis). In non-cubic compounds, owing to the uniaxial character of the anisotropy, the transitions can be more spectacular than in cubic compounds. Transitions can be first or second order and have been widely studied and explained in the frame of a Hamiltonian which in most cases includes the CEF effects and an effective (applied plus exchange) field. However, care should be brought to distinguish between a true second-order transition and a smooth transition arising from thermal effects. The relation between the sharpness of the transition and the exact composition of the CEF levels is emphasized in the hexagonal compound ErNi5 [27]. In this compound, most magnetic properties are well described by two different sets of CEF parameters which differ only in the nature of the second excited state lying at about 70 K above the ground state. This apparently minor difference actually leads to drastically different behaviour on the magnetization curves along the [100] and [120] hard directions, the sharpness of the metamagnetic processes occurring in a high field (around 20 T) being strongly reduced for one set of parameters, in agreement with the experiment (Fig. 12). This underlines the need to use as much experimental information as possible for a reliable determination of the crystal field.
7. Quadrupolar metamagnetic systems Quadrupolar interactions in cubic systems can be at the origin of metamagnetic transitions. Two cases must be distinguished depending on the sign of this interaction. Negative two-ion interactions between 4f quadrupoles may favour multi-axial arrangements of the magnetic moments, as is the case for instance in TmGa3 [28].
. . . . . . . . . . . . . . .
8
~ --'T_T_T'-7-----'-*-~=
[o0,1
,~~_...~
6rl
/
0, ~ ,
............
0
100
T=,I~4,K, 200
300
400
H(kOe) Fig. 12. High field magnetization curves of ErNis at 1.4 K. Full and dashed lines are calculated curves for two different sets of CEF and exchange parameters (after Ref. [27]).
O oOO00
6
OOOO
O0
• • •
•
oAO
:
•
O
:
O
:
Oi~
'
O
T= 1.5 ° 4 , ' " 0 • O, 4
.% 8 0
• "5K •
•
t
./
"d
A
oO° °°:<° ° m D.
0 0
X
08
• • •
8. •
:oo •
TmGQ3 H#[111]
O
10
20
ab
I
20
H (kOe)
5'O
60
70
Fig. 13. Magnetization processes in TmGa3 along [111]; magnetic structures in zero field and in the intermediate phase (after Ref.
[28]). This compound orders at low temperature into an antiferromagnetic state described by the propagation vector Q= (1/2, 1/2, 0). A thorough analysis of all the magnetic properties of a single crystal led to the conclusion that the antiferroquadrupolar pair interactions within the trigonal symmetry favours a multi-axial (tripleQ) spin arrangement, the magnetic moments pointing along the four threefold axes. Two-step metamagnetic behaviour takes place along the three main symmetry directions (Fig. 13), leading to three magnetic phase diagrams similar to each other. The intermediate phase in each case is a biaxial or quadriaxial spin structure where only one fraction of the moments antiparallel to the field has rotated, either along its own direction or along another equivalent direction. This behaviour has been explained by strong competition between (i) a magnetocrystalline anisotropy favouring the <111>
430
D. Gignoux, D. Schmitt / Journal of Alloys and Compounds 225 (1995) 423-431
directions, (ii) a negative exchange interaction favouring an antiferromagnetic arrangement of the spins, and (iii) negative quadrupolar pair coupling favouring a mutually perpendicular arrangement of the quadrupoles. Ferroquadrupolar coupling may be at the origin of metamagnetic transitions in the paramagnetic state. This effect has been clearly evidenced in several quadrupolar compounds such as TmZn [29] where a singlestep transition occurs above the ordering temperature along the [001] easy magnetization direction. The transition arises when the internal field exceeds the critical field necessary to induce the ferromagnetic ordering. It has been shown that this behaviour may be associated with a positive value of third-order magnetic susceptibility. This receives a positive contribution from the ferroquadrupolar coupling which is able to over-compensate the negative contribution arising from the CEF effects.
8. Quantitative analysis of incommensurate magnetic systems During the last decade much attention has been paid to theoretical approaches to determination of the complex magnetic phase diagram. In particular, theoretical models focused on the transition between commensurate (C) and incommensurate (I) phases, and on the properties of the incommensurate phase. A model which has been widely used is the axial next nearest neighbour Ising (ANNNI) model [30,31]. It allowed the discovery of new quite exciting physical concepts and accounted qualitatively for experimental results but, because of its great simplifications, it failed to explain the experimental observations in any detail. In order to explain the complex phase diagram of CeSb [32], Date [33] considered an Ising spin chain immersed in a sinusoidally modulated exchange field. In spite of crude assumptions, the phase diagrams of some compounds such as CeSb and PrCo2Si2 [34] were outlined using this model. Much more realistic is the approach of Iwata [35]. Within the mean field approximation it considers the q dependence of J(q) and solves the problem selfconsistently in the same way. However, owing to a simplification of CEF effects, the model can only be applied to Ising systems. These conditions are fulfilled by PrCo2Si2, for which a remarkable quantitative account of the H-T phase diagram, in particular the multistep low temperature metamagnetic process (Fig. 5) and the zero applied field thermal dependence of the propagation vector, was obtained. Recently, a more realistic model which takes into account CEF effects has been applied successfully to describe the magnetic properties (e.g. magnetization processes, magnetic susceptibility, phase diagram, spe-
cific heat...) of several frustrated magnetic systems with any periodicity (commensurate as well as incommensurate) and any moment direction (collinear as well as non-collinear structure) [10,21,34]. This self-consistent periodic field (PF) model is based on an N-site Hamiltonian, N being the number of magnetic ions over one period of the magnetic structure. These magnetic ions are subjected to a modulated exchange field having the same periodicity and arising from the magnetic ions themselves in a self-consistent way. Quite satisfactory analyses of several uniaxial compounds such as PrNi2Si2 [11] and HoAIGa [21] were obtained, as illustrated in Figs. 2 and 8. The PF model applies not only for collinear structures but also for non-collinear arrangements such as those expected in easy axis systems when the field is applied perpendicular to the easy axis, for instance in PrNizSi2 [11]. In addition, the model allowed discussion of the magnetization processes observed in GdGa2, in particular the possibility of a helical to fan structure field induced transition [36]. Another remarkable result obtained by the PF model concerns the thermodynamic properties. Indeed, it has been shown that in the case of a single-Q magnetic structure, the specific heat discontinuity AC at TN of a system having an amplitude modulated (AM) structure is only two thirds of that in a compound with an equal moment (EM) structure, i.e. a ferromagnetic, a simple antiferromagnetic or a helimagnetic compound [37]. This unique feature, first derived in gadolinium based systems [37], has been extended to compounds in the presence of CEF effects [38]. In these latter compounds, however, the analysis is more complex and requires first a good knowledge of the CEF parameters. The second important consequence is that the slope of the specific heat at TN can be negative, leading to the original feature that a maximum of specific heat must occur at a temperature below TN. These original results throw a new light on the experimental temperature dependences of specific heat found in the literature, in particular for gadolinium based compounds [39]. They should be quite useful for choosing between a helical, i.e. EM, and an AM structure near TN, from specific heat data.
9. Summary and conclusion The above survey shows that in intermetallic compounds where only the rare earth is magnetic, a large variety of metamagnetic behaviours of quite different origins (crystal feld, quadrupolar interactions and/or frustration of bilinear exchange interactions) has been found and the often complex magnetic phase diagrams have been determined for some compounds. In most cases these metamagnetic processes are rather well understood and have been analysed quantitatively. At
D. Gignour, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423~131
the moment, a large number of studies are devoted to the determination of complex magnetic phase diagrams for a large number of compounds where incommensurate high temperature structures often compete with low temperature commensurate structures. The number of systems where the microscopic aspects of the phase diagrams have been investigated is rather limited. Accurate magnetization measurements and neutron diffraction under a magnetic field for single crystals would allow us in the future to obtain better knowledge of this type of systems, in particular the Potts systems which have more complex phase diagrams than the Ising systems. Special attention will also need to be paid to the weakly anisotropic gadolinium based systems, in particular with respect to their microscopic magnetic structures which have not, so far, been studied owing to the large neutron absorption of gadolinium.
References [1] M. Date, J. Magn. Magn. Mater., 90-91 (1990) 1. [2] D. Gignoux and D. Schmitt, Z Magn. Magn. Mater, 100 (1991) 99. [3] G. Asti, in K.H.J. Buschow and E.P. Wohlfarth (eds.), Ferromagnetic Materials, Vol. 5, North-Holland, Amsterdam, 1990, Chapter 5. [4] J.J.M. Franse and R.J. Radwanski, in K.H.J. Buschow (ed.), Handbook of Magnetic Materials, Vol. 7, North-Holland, Amsterdam, 1993, p. 307. [5] R. Ballou, B. Ouladdiaf, P.J. Brown, M.D. Nunez-Regueiro and C. Lacroix, Phys. Rev. B, 45 (1992) 3158. [6] E.P. Wohlfarth and P. Rhodes, Philos. Mag., 7 (1962) 1817. [7] F.Y. Wu, J. Appl. Phys., 55 (1984) 2421. [8] S. Takayanagi, Y. Onuki, K. Ina, T. Komatsubara, N. Wada, T. Watanabe, T. Sakakibara and T. Goto, J. Phys. Soc. Jpn., 58 (1989) 1031. [9] J.A. Blanco, D. Gignoux, D. Schmitt and C. Vettier, J. Magn. Magn. Mater., 97 (1991) 4. [10] D. Gignoux and D. Schmitt, Phys. Rev. B, 48 (1993) 12682. [11] J.A. Blanco, D. Schmitt and J.C. Gomez-Sal, J. Magn. Magn. Mater., 116 (1992) 128. [12] B. Lebech, Z. Smetana and V. Sima, J. Magn. Magn. Mater, 70 (1987) 97. [13] D. Gibbs, J. Bohr, J.D. Axe, D.E. Moncton and K.L. D'Amico, Phys. Rev. B, 34 (1986) 8182.
431
[14] N. Iwata, T. Kimura, T. Shigeoka and Y. Hashimoto, J. Magn. Magn. Mater, 70 (1987) 279. [15] N. Iwata, K. Honda, T. Shigeoka, Y. Hashimoto and H. Fujii, J. Magn. Magn. Mater., 90-91 (1990) 63. [16] T. Shigeoka, N. Iwata, H. Fujii, T. Okamoto and Y. Hashimoto, J. Magn. Magn. Mater., 70 (1987) 239. [17] H. Fujii and T. Shigeoka, J. Magn. Magn. Mater., 90-91 (1990) 115. [18] M. Motokawa, H. Nojiri and Y. Endoh, Physica B, 177 (1992) 279. [19] D. Gignoux, D. Schmitt, T. Takeuchi, F.Y. Zhang, C. Rouchon and E. Roudaut, J. Magn. Magn. Mater., 98 (1991) 333. [20] A.R. Ball, D. Gignoux, D. Schmitt, F.Y. Zhang and M. Reehuis, J. Magn. Magn. Mater., 110 (1992) 343. [21] A.R. Ball, D. Gignoux, D. Schmitt and F.Y. Zhang, Phys. Rev. B, 47 (1993) 11887. [22] A.R. Ball, D. Gignoux and D. Schmitt, J. Magn. Magn. Mater., 119 (1993) 96. [23] A.R. Ball, D. Gignoux, D. Schmitt, F.Y. Zhang and P. Burlet, J. Magn. Magn. Mater., 130 (1993) 317. [24] B.R. Cooper, Phys. Lett., 22 (1966) 244. [25] K.A. McEwen, G.J. Cock, L.W. Roeland and A.R. Mackintosh, Phys. Rev. Lett., 30 (1973) 287. [26] A.R. Ball, D. Gignoux, F.E. Kayzel, D. Schmitt and A. de Visser, J. Magn. Magn. Mater., 110 (1992) 337. [27] F.Y. Zhang, D. Gignoux, D. Schmitt, J.J.M. Franse, F.E. Kayzel, N.H. Kim-Ngan and R.J. Radwanski, J. Magn. Magn. Mater, 130 (1993) 108. [28] P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mater, 68 (1987) 107. [29] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B, 17 (1978) 3684. [30] P. Bak, Rep. Prog. Phys., 45 (1982) 587. [31] W. Selke and P.M. Duxbury, Z. Phys. 13, Corders. Matter, 57 (1984) 49. [32] J. Rossat-Mignod, P. Burlet, S. Quezel, J.M. Effantin, D. Delacote, H. Bartholin, O. Vogt and D. Ravot, J. Magn. Magn. Mater., 31-34 (1983) 398. [33] M. Date, J. Phys Soc. Jpn., 57 (1988) 3682. [34] T. Shigeoka, H. Fujii, K. Yonenobu, K. Sugiyama and M. Date, J. Phys. Soc. Jpn., 58 (1989) 394. [35] N. Iwata, J. Magn. Magn. Mater., 86 (1990) 225. [36] A.R. Ball, Thesis, University J. Fourier, Grenoble, 1993, unpublished. [37] J.A. Blanco, D. Gignoux and D. Schmitt, Phys. Rev. B, 43 (1991) 13145. [38] J.A. Blanco, D. Gignoux and D. Schmitt, Phys. Rev. B, 45 (1992) 2529. [39] M. Bouvier, P. Lethuillier and D. Schmitt, Phys. Rev. B, 43 (1991) 13137.
~YS
AND CONIPO~D5 ELSEVIER
J o u r n a l of Alloys a n d C o m p o u n d s 225 (1995) 423-431
Metamagnetism and complex magnetic phase diagrams of rare earth intermetallics D. Gignoux, D. Schmitt Laboratoire de Magndtisme Louis Ngel, C.N.R.S., BP 166, 38042 Grenoble Cedex 9, France
Abstract
A survey of the extreme variety of metamagnetic processes discovered during recent decades in rare earth intermetailics, where only the rare earth is magnetic, is presented together with macroscopic and microscopic aspects of the complex associated H - T magnetic phase diagrams. These advances were obtained owing to improvements in experimental techniques (in particular high-field magnetization measurements and neutron diffraction) and the increasing number of single crystals available of good quality. The role of basic interactions (crystal field, bilinear exchange interactions, quadrupolar interactions...) is discussed. The parallel improvements in theoretical models and quantitative analyses of these wealthy properties are reported briefly. Keywords: Metamagnetism; Magnetic phase diagrams; Rare earth intermetallics
I. Introduction
Initially limited to a first-order field-induced phase transition between a simple antiferromagnetic structure and the ferromagnetic state, the concept of metamagnetism has been extended during recent decades to many processes which break antiferromagnetic structures, and more generally to all types of field-induced magnetic phase transition [1,2]. The aim of this paper is to emphasize the specific role of the rare earth intermetallics in such an evolution by showing, through several examples, how each of the various types of metamagnetic process is associated with a particular physics depending on the material considered. The large increase in the number of rare earth intermetallics exhibiting metamagnetism has three main causes: (i) progress in the synthesis of single crystals; (ii) the increasing number of new rare earth intermetallics, in particular ternary compounds; (iii) the increase in the performance of experimental facilities, in particular high-field measurements and neutron diffraction experiments. We consider as metamagnetic behaviour an anomaly occurring during the magnetization process of a given compound, characterized by an upward curvature in a limited range of field. Two main classes of metamagnetic transition can be distinguished: (i) step-like behaviour where the magnetization jumps between two plateaux 0925-8388/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 5 - 8 3 8 8 ( 9 4 ) 0 7 0 4 6 - 6
at a given critical field Hc (class (1)); (ii) S-shape behaviour where the magnetization increases more smoothly than in class (1) systems, showing only an inflexion point at the critical field (class (2)). Within this quite general definition of metamagnetism one can include the FOMPs (first order magnetization process) as described in Ref. [3] which are often associated with rotation of the magnetization vector in ferromagnets and ferrimagnets. In this paper we do not consider the following: (i) the FOMP occurring in ferromagnetic and ferrimagnetic rare earth-3d intermetallic compounds in which both 4f and 3d atoms are magnetic, since they are described elsewhere [4]; (ii) the antiferromagnetic rare earth-3d intermetallics, in which both 4f and 3d atoms are magnetic (for example RMn2), that exhibit the so-called antimetamagnetism [5]; (iii) itinerant electron metamagnetism associated with the onset of 3d magnetism [6]. The temperature dependence of the transition fields Hc allows boundaries of the H - T phase diagram to be defined, showing one or several distinct regions, each associated with a particular spin structure. The two main ingredients playing an essential role in metamagnetism and associated phase diagrams are the magnetocrystalline anisotropy and the interionic interactions. The former, i.e. the crystalline electric field (CEF) coupling, plays a fundamental role as it leads to Ising (one easy magnetization direction), Potts (several equiv-
424
D. Gignoux, D. Schmitt I Journal o f Alloys and Compounds 225 (1995) 423-431
alent non-collinear easy magnetization directions such as the (100) axes in hexagonal or tetragonal symmetry [7]) or X-Y (easy plane) systems. The one-ion magnetoelastic coupling must be included in this first category, being considered as the strain derivative of the CEF itself. The second fundamental ingredient includes all the types of interionic interaction, namely the isotropic or anisotropic bilinear exchange coupling and the two-ion quadrupolar coupling, as well as their possible strain dependence.
2. Spin-flop in weakly anisotropic metamagnetic systems In gadolinium (L = 0) based antiferromagnetic compounds the magnetocrystalline anisotropy is very weak and the metamagnetic behaviour characterizes the properties of the exchange interactions alone. However, a small anisotropy fixes the zero field orientation of the magnetic structure with regard to the lattice and is revealed under low field through a spin-flop metamagnetic transition. During this transition there is typically a rotation of the moments toward a direction perpendicular to the applied field. A good example is the orthorhombic GdCu6 compound [8], where the magnetization curves along the b and c axes exhibit small discontinuous steps before varying linearly up to the saturated value reached at Hs = 24 T (Fig. 1). In the case of a simple antiferromagnet, the spin-flop transition should occur along one single axis, namely parallel to the direction of the moments, at least in low symmetry systems. One can conclude that the magnetic structure of G d C u 6 is planar, for instance helical in the b--c plane.
./ .//
.//" ././//-
GdCu6 1.3K
c-a~is~///"
/2"./
-
:~ 2
"Z/
-
,f.-77"
,:,s,".;"
~." .J.I
nu
0
J¢" ,
I
,
I
48 Magnetic Field
,
I
12 [ T
,
]
Fig. 1. Magnetization curves for GdCu6 at 1.3 K (after Ref. [8]).
Once the perpendicular configuration is achieved, i.e. above the spin-flop transition, any anomaly on the magnetization curves will be associated with destruction of the antiferromagnetic structure itself. It is worth noting that, for a simple antiferromagnet or a helical structure, the magnetization process is, in principle, strictly linear. Therefore, any observed metamagnetic transition will suggest a more complex structure or will be the signature of anisotropic exchange interactions, as is the case in various gadolinium compounds. It should be pointed out that the absence of magnetocrystalline anisotropy in gadolinium compounds is not necessarily advantageous for understanding the magnetic behaviour under field. In fact, the situation is even more complicated because several types of coupling generally neglected should now be taken into consideration. Moreover, good determination of the magnetic structure is still important for describing the properties, and this is difficult to do for gadolinium compounds because of the huge absorption cross-section of natural gadolinium under normal conditions. Neutron diffraction experiments using a gadolinium isotope or at a very low wavelength (A = 0.5/~) are needed. This explains the small number of such experiments in the literature.
3. Modulated metamagnetic systems In the lanthanide intermetallic compounds, it is well known that the RKKY-type indirect coupling is long range and oscillatory, and its Fourier transform J(q) often exhibits a maximum for a vector Q not in a high symmetry position in the BriUouin zone, i.e. Q ~ 0 and Q ~ (1/2)K, where K is a reciprocal lattice vector. This leads, at least near TN, to incommensurate or long period commensurate structures. For rare earths with L ~ 0 and easy magnetization direction(s) (Ising or Potts systems), the corresponding structure is amplitude modulated, sinusoidal near TN. In Ising type systems, the metamagnetism associated with such modulated structures is simple when the magnetic field is applied along the moment direction, exhibiting a single smooth transition of class (2). In the case of modulated Potts systems, a difficulty arises concerning the magnetic domains because it is not obvious a priori whether the magnetic field favours the domain with moments parallel or perpendicular to itself. An increasing number of lanthanide intermetallic compounds showing an incommensurate modulated structure exhibit this type of transition. This is the case for numerous tetragonal compounds having the ThCr2Si2-type crystallographic structure. A good example is the Ising type TbNizSiz compound between 7",= 9 K and TN = 15 K, where the propagation vector is Q= (0.426, 0.574, 0); below Tt the structure changes towards a simple commensurate
D. Gignoux, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423-431 2
i
I
i
i
'4 :/ '~'
i
~
n. 0r
i
i
P r N i z S iz ..........
e
-~"r/I~
o
0
~ f"f"I
2
[001]
H = 5.8 T ,-"
4
6
8
MAGNETIC FIELD ( T ) Fig. 2. Magnetization processes in PrNi2Si2 at 1.5 K along and perpendicular to the c direction; note the cusp when reaching the induced ferromagnetic state along c. Lines were calculated using the PF model. Schematic representation of the magnetic structure at several fields along the c direction; note the residual modulation at 5.8 T, i.e. immediately below the cusp where it vanishes (after Ref. [11]).
antiferromagnetic structure with Q = (0.5, 0.5, 0) [9]. In the modulated phase there is one single metamagnetic transition ranging between 10 and 15 kOe according to the temperature. In the above examples as well as in the vast majority of modulated systems, the high temperature structure is not stable down to 0 K because of entropy effects, and the system transforms into a magnetic structure with a shorter period [10]. One exception may occur when the CEF ground state is a non-magnetic state, as in tetragonal PrNizSi2. In this compound, a sine-wave modulated structure occurs at TN = 20 K, the propagation vector being Q = (0, 0, 0.870) and the moments being aligned along the c-axis [11]. This structure persists down to 0 K owing to the existence of a non-magnetic CEF singlet as ground state well isolated from the excited levels. Under these conditions it is possible to study the modulated phase down to low temperatures, in particular the metamagnetic behaviour. In Fig. 2, this manifests itself very clearly as a positive curvature on the magnetization curve along the c-axis up to about Hc = 6 T where a cusp is observed. This curvature corresponds to the progressive disappearance of the modulation under the applied field, and the observed cusp indicates precisely the completion of this evolution, i.e. the entrance into the induced ferromagnetic state.
4. Spin-flip and spin-slip in anisotropic rare earth c o m p o u n d s with coUinear structures
In rare earth intermetallics with magnetocrystalline anisotropy (L=/=0) we are faced with Ising or Potts systems in which magnetic structures are often collinear
425
(at least when the symmetry is not too low). As quoted above, in a large number of materials the magnetic periodicity near TN is incommensurate or long period commensurate leading to amplitude modulated magnetic structures. Only a few compounds exhibit simple antiferromagnetic structures over the whole temperature range below TN (e.g. ErGaz). Except in the case of a non-magnetic CEF singlet ground state, amplitude modulated magnetic structures are not stable at low temperature, and transform toward a structure with equal moments in different ways: (i) the magnetic periodicity remains unchanged and there is a progressive squaring of the structure which manifests itself as the appearance in the neutron diffraction pattern of peaks associated with higher order harmonics of the ground propagation vector (e.g. TbCu2 and DyCu2 [12]); (ii) one or several transitions toward other, generally simpler, magnetic phases with different periodicity, i.e. where the period is shorter (devil's staircase). A very large number of systems belong to the latter category [10]. Magnetization processes of these collinear equal moment magnetic structures are step like, in most cases exhibiting multistep metamagnetic processes. This behaviour leads to complex magnetic H - T phase diagrams showing several intermediate magnetic structures. Note that in the second category of compounds quoted above, the resurgence of high temperature incommensurability is frequently observed during the low temperature metamagnetic process. In all these compounds, low temperature metamagnetism involves two types of process, namely spinflip and spin-slip. (i) A spin-flip transition corresponds to the reversal of moments antiparallel to the field in a given magnetic cell (one can find a common magnetic periodicity below and above the transition). Note that this process can be accompanied by the antiflipping of some moments. (ii) The concept of spin-slips or discommensurations was first introduced to explain the observed lock-in transitions in the magnetic spirals of rare earth metals such as pure holmium and dysprosium in terms of simple commensurate structures [13]. In both hexagonal metals, the magnetic propagation vector depends on the temperature through a succession of lock-in transitions on commensurate values close to each other. This is associated with the tendency for the basal plane component to lock close to the easy axes within the hexagonal plane, resulting in periodic defects (spinslips) instead of a regular rotation of the spiral order. More generally this term can be used to characterize structures which present periodic defaults in a simple sequence of magnetic moments. A spin-slip transition corresponds to the change from one discommensuration to another. It is generally associated with a metamagnetic transition of small amplitude in the magnetization process and a small shift of the propagation vector.
D. Gignoux, D. Schmitt / Journal of Alloys and Compounds 225 (1995) 423--431
426
To illustrate the variety of situations found in this type of material we present some characteristic examples. Note that, if the thermal dependence of the zero field magnetic structure has been determined in a large number of compounds, the field dependence of these structures has been investigated in a limited number of materials owing to the necessity of carrying out neutron diffraction experiments on single crystals under an applied magnetic field. Among the few examples of single-step spin-flip metamagnetism toward the induced ferromagnetic state one can quote the orthorhombic TbCu2 compound at low temperature when the field is applied along the easy a-axis [14] (Fig. 3). Under zero field, although the basic propagation vector is Q=(1/3, 0, 0), the structure is antiphase because there are four terbium atoms in the unit cell. At the critical field, all moments are antiparallel to the field flip. Two-step spin-flip metamagnetism, in which magnetization of the intermediate phase is exactly one half of the saturated value, has been observed in some simple antiferromagnets such as tetragonal DyCo2Si2 [15]. The intermediate field-induced phase corresponds, in agreement with neutron diffraction studies, to the spin-flip of half the moments initially antiparaUel to the field (Fig. 4). The tetragonal PrCo2Si2 compound has been particularly thoroughly investigated. It orders at T~= 30 K within a long period commensurate structure with Q1 = (0, 0, 7/9) [16]. As the temperature is lowered, a first-order transition occurs at T2 = 17 K, the new propagation vector being Q2 = (0, 0, 13/14), while the periodicity locks below T1 = 9 K into a simple antiferromagnetic structure with 0 3 = (0, 0, 1). The magnetization process at 1.3 K shows a four-step metamagnetic behaviour along the c-axis, the magnetization at each step being a fractional number of the saturated value, being approximately 1/14, 3/14, 2/9 and 1 of Mo = 3.2/zB/Pr (Fig. 5) [17]. These magnetization values are consistent with field-induced magnetic structures which have the same successive propagation vectors Q2 and Q1 as those stabilized at higher temperature. This 10
l
I
oOOO
o
I o
o
o o
TbCu2
z o
6
,< N .
Z.
Z ,<
2 0
4.2K
10
o
a
c .
20
30
40
H (kOe) Fig. 3. Magnetization curves for TbCu2 at 4.2 K (after Ref. [14]).
i
i
i
i
l
DyCo2Si2
[0011 12
IO
~o-o-O"O~
Co *Sl
•
...,~.:,o-o
8
4
IIII
I
I Foo,,.eo- o,- o,. -,o ,-o J
4
T:4.2~
f I Io
tI t I l I I / I j...~j~
2 O
~
"
e...l l - j
~
0
....e_.e-e --e-e'e*'e'e*e'er¢'e'~ HJ.
20
~ 40
l
I 60
He (kOe) Fig. 4. Magnetization curves for DyCo2Si2 at 4.2 K (after Ref. [15]).
Pr C°2Si2
He4
I
T = 1.3 K Ha//C-axis
~2
H
Hc~
HaJ.c -axis
O0
10
2'0
3'0
Ha (T) Fig. 5. Magnetization curves for PrCo2Si~ at 1.3 K along a and c; the dashed line was calculated using a simple incommensurate m e a n field model (after Ref. [17]).
has been confirmed by a neutron diffraction study in pulsed high magnetic field [18]. The temperature variation of the critical field gives a phase diagram in which the high temperature phases extend toward the low temperature and high field region by turning around the origin (Fig. 6). Among the Ising systems with hexagonal symmetry, the HoA1Ga compound is one of the very few which has been studied extensively. It exhibits two different magnetic phases in zero field [19]: (i) an antiphase below Tt = 18.5 K, characterized by the basic vector Q1 = (1/3, 1/3, 1/2) and its third harmonics 3Q1 = (0, 0, 1/2); (ii) an amplitude modulated structure between T, and TN = 31 K with the single incommensurate vector Q2 = (1/3, 1/3, 0.481). At low temperature, a three-step metamagnetic process is observed along the c axis (Fig. 7), the two intermediate magnetizations reaching approximately 1/9 and 1/3 of the full moment. The magnetic phase diagram shows three ordered regions (Fig. 8)
427
D. Gignoux, D. Schrnitt / Journal o f Alloys and Compounds225 (1995) 423-431
PrCo2Si2
15
x X
B=
lO
\\
A
\ \x A
B=
5 -
r,
B¢I ~ -
•
°r~ - -o~ X
0
1
x
!
N
-- i
,
T1
0
-
j
m TN
T2
10
T(~
30
20
Fig. 6. Magnetic phase diagram of PrCo2Si 2 along c determined by high field n e u t r o n diffraction (after Ref. [18]).
9
i
]
6
~-~t -c
i
o
rt (~-'&4) 00
\[
~ ~///
~ _j
20
c axis
HoAIGa 40
60 H(kOe)
80
Fig. 7. Magnetization process at t.7 K in H o A I G a along the c-axis and associated magnetic structures (after Ref. [20]).
60
.-. •
0
50
..
: . . . ; . . . .
~
•
'
>~~,
"
~
! ....
i .... ,
II
.
,
. . . .
~.
,
. . . .
HoAIGa [001]
..~
~
The second field-induced state at 1.7 K is characterized by the flipping of all the remaining antiparallel moments of line B, the magnetic unit cell recovering the same size as at zero field. Hence there should exist a border line, not detected by magnetic measurements [19], between this commensurate high field-low temperature region and the incommensurate low field-high temperature region. Actually, such a border line (hatched area in Fig. 8) has been determined by neutron diffraction. However, broadening of the diffraction peaks at 1.7 K and 48 kOe is already present and suggests that a perfect arrangement with wavevector Q1 and its even harmonics is not fully achieved within this limited region, but presents defaults reminiscent of the high temperature propagation vector, illustrating the competition between both commensurate and incommensurate periodicities. PrGa2 (same hexagonal structure as HoA1Ga) is a wealthy Potts system in which the multistep metamagnetic behaviour and the associated complex phase diagram involve spin-flips, spin-slips and magnetic domain effects. At 1.5 K and in zero field, this compound exhibits a non-compensated long period commensurate antiphase structure, with the propagation vector Q = (4/ 27, 4/27, 0), the moments being aligned along the [100] axis [22]. Fig. 9 shows, for this axis, the second and subsequent magnetization processes in which the material is single domain, that with moments (and propagation vector) parallel to the field. The first magnetization process (in increasing field) exhibits a transition around 3.5 kOe associated with the change from a three-domain to a single-domain structure. Let us now discuss the phases induced during the second magnetization process. The first metamagnetic transition is of spin-flip type [23]; below a critical field Hc=2.3 kOe (phase II), the magnetic cell has 27 moments with the sequence (4 3)(3 4)(3 3)(4 3) along the [100] direction, i.e. it consists of four groups of seven moments (4 up, 3 down), with two defaults over 28 sites, namely
"~,--
, •
-
'
20 \ T1=18.5 K
TN=31 K
' 10
'
. 'L I (!]3,.113;112.'X)1 I . . . . 20 30
T(K)
I
I
[
1.SK 40
Fig. 8. Boundaries of the p h a s e diagram of H o A I G a along c (after Ref. [21]). Full circles are the experimental points. Full lines are calculations using the P F model. Boundaries of phase III are not accounted for by the model because this phase is described by several non-collinear propagation vectors.
,I
l it vi
=L
ii
0.5
.';:::::::;:'"
¢'4 .u
•'", " ,i:'L"" ,T
[20]. The first intermediate phase (phase III) is characterized, on one linie over three along c (line B), by the flipping of 2/3 of the initially antiparallel moments and the antiflipping of 1/3 of the parallel moments.
'
°:
,,oo, 2
G" 00 ~
I
P r G a
i
/ 2
II
t IV
t III ,
i
,
4
,
6 8 H (kOe)
,
,
,
i
10
i
12
Fig. 9. Low field magnetization process along [100] in PrGa2 at 1.5 K (after Ref. [23]).
428
D. Gignoux, D. Schmitt I Journal of Alloys and Compounds 225 (1995) 423-431
PrGa 2 T=l.8 K
H ffi4 koe- Plmm VI
a~ Spin-slip
++++
.
.
+++0-
.
H =3 kO¢- Plmm Hl
.
- I - ÷ -
~ Spin-flip
++++ H ffi0 ROe - P h i IT
+
.
-
-
÷+++---
"Jr"I
-
First default
_
++
- - - + + + + - - Second default
Fig. 10. Low field magnetic structure of PrGa2 at 1.8 K; note the successive spin-flip and spin-slip behaviours,
one inversion up--down in the second group and a missing up moment in the third group (Fig. 10). The induced magnetic structure above Hc (phase III) corresponds to the vanishing of the first fault of the zero field structure, i.e. to a single spin-flip, without changing the magnetic periodicity. Therefore, the resulting magnetization changes from (1.27)Mo to (3/27)Mo, where Mo = 3.1#B is the praseodymium magnetic moment. The other transitions occurring below 9 kOe are of spinslip type [23]. More precisely, the well defined plateau observed on decreasing the field between 6 and 2 kOe (phase VI), corresponds to a periodicity of exactly seven unit cells, i.e. Q=(4/28, 4/28, 0), where the second default in the sequence of magnetic blocks has disappeared (Fig. 10). Moreover, this periodicity appears under increasing field at around 4 kOe and subsists up to 8 kOe (phase IV), the width of the associated neutron diffraction peaks varying in the same field range and being related to the large superimposed susceptibility observed on the magnetization curve. This behaviour has been explained by the difficulty of establishing perfect long range periodicity of seven cells in the system, possibly owing to defects in the crystal.
5. Paramagnetic metamagnetic systems This behaviour is observed in systems where the CEF is the only interaction acting on the 4f shell. The most dramatic effect arises in the case of level crossing, as predicted in 1966 to occur in TmSb [24]. More recently this effect has been observed in praseodymium metal [25] which exhibits a single-step transition at low temperature. The crossing occurs between two CEF levels, namely a singlet which is the ground state under zero field and an excited level which is more magnetic. Actually the most typical example of this behaviour is provided by the hexagonal compound PrNi5 (Fig. 11) [26]. In this paramagnet, the F3 singlet ground state is followed by another singlet /'1 around 30 K. When
/~
t
r , ~
--
0
10
20
t ' [120] 30
' J
.o* e°*
i ...."" 1
,l.
"'"""
i
PrNi 5 1.4K 0
,
0
I
i
,,,~,~
I
,
~ ,
I
10 20 30 Applied Field (Tesla)
m
I
40
Fig. 11. High field magnetization curves of PrNi5 along the two main symmetry directions of the basal plane at 1.5 K (after Ref. [26]). Insets show the calculated field variations of the energy of the first two CEF levels along these directions.
the magnetic field is applied in the basal plane these two levels should cross for a given field. However, according to group theory considerations, the ground level F3 transforms into a F2 or F1 level if the applied field is along the [100] or [120] direction respectively, while the first excited level keeps the F1 symmetry. It follows that the two levels actually cross each other in the first case, because they belong to different representations, while in the second case an anticrossing occurs because they both belong to the same representation. This leads to a metamagnetic transition which becomes sharper and sharper in the first case when the temperature is lowered, finally reaching stepwise behaviour at 0 K, while in the second case the transition remains smooth even at 0 K. Both transitions have been observed experimentally for a magnetic field of 18.5 T and 12 T respectively.
D. Gignoux, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423-431
429
6. Ferromagnetic metamagnetic systems Under certain conditions, ferromagnets, where only the rare earth is magnetic, may exhibit metamagnetic processes. This is the case in low symmetry ferromagnets, such as the orthorhombic RNi compounds, where R atoms can be divided into sublattices with different local easy magnetization directions. The magnetic structures are then non-collinear with both ferromagnetic and antiferromagnetic components. When the field is applied along one of the latter, and if the magnetocrystalline anisotropy is large enough, a spin-flip occurs, corresponding to the reversal of one sublattice. In a large number of cases, the metamagnetic behaviour of rare earth ferromagnetic compounds is basically associated with crystal field properties, namely a crossing or an anticrossing of CEF levels. Examples, where such metamagnetic processes have been well studied, include cubic systems (e.g. RZn and HAl2) as well as lower symmetry compounds (e.g. RRu2, RGa, RNis). In non-cubic compounds, owing to the uniaxial character of the anisotropy, the transitions can be more spectacular than in cubic compounds. Transitions can be first or second order and have been widely studied and explained in the frame of a Hamiltonian which in most cases includes the CEF effects and an effective (applied plus exchange) field. However, care should be brought to distinguish between a true second-order transition and a smooth transition arising from thermal effects. The relation between the sharpness of the transition and the exact composition of the CEF levels is emphasized in the hexagonal compound ErNi5 [27]. In this compound, most magnetic properties are well described by two different sets of CEF parameters which differ only in the nature of the second excited state lying at about 70 K above the ground state. This apparently minor difference actually leads to drastically different behaviour on the magnetization curves along the [100] and [120] hard directions, the sharpness of the metamagnetic processes occurring in a high field (around 20 T) being strongly reduced for one set of parameters, in agreement with the experiment (Fig. 12). This underlines the need to use as much experimental information as possible for a reliable determination of the crystal field.
7. Quadrupolar metamagnetic systems Quadrupolar interactions in cubic systems can be at the origin of metamagnetic transitions. Two cases must be distinguished depending on the sign of this interaction. Negative two-ion interactions between 4f quadrupoles may favour multi-axial arrangements of the magnetic moments, as is the case for instance in TmGa3 [28].
. . . . . . . . . . . . . . .
8
~ --'T_T_T'-7-----'-*-~=
[o0,1
,~~_...~
6rl
/
0, ~ ,
............
0
100
T=,I~4,K, 200
300
400
H(kOe) Fig. 12. High field magnetization curves of ErNis at 1.4 K. Full and dashed lines are calculated curves for two different sets of CEF and exchange parameters (after Ref. [27]).
O oOO00
6
OOOO
O0
• • •
•
oAO
:
•
O
:
O
:
Oi~
'
O
T= 1.5 ° 4 , ' " 0 • O, 4
.% 8 0
• "5K •
•
t
./
"d
A
oO° °°:<° ° m D.
0 0
X
08
• • •
8. •
:oo •
TmGQ3 H#[111]
O
10
20
ab
I
20
H (kOe)
5'O
60
70
Fig. 13. Magnetization processes in TmGa3 along [111]; magnetic structures in zero field and in the intermediate phase (after Ref.
[28]). This compound orders at low temperature into an antiferromagnetic state described by the propagation vector Q= (1/2, 1/2, 0). A thorough analysis of all the magnetic properties of a single crystal led to the conclusion that the antiferroquadrupolar pair interactions within the trigonal symmetry favours a multi-axial (tripleQ) spin arrangement, the magnetic moments pointing along the four threefold axes. Two-step metamagnetic behaviour takes place along the three main symmetry directions (Fig. 13), leading to three magnetic phase diagrams similar to each other. The intermediate phase in each case is a biaxial or quadriaxial spin structure where only one fraction of the moments antiparallel to the field has rotated, either along its own direction or along another equivalent direction. This behaviour has been explained by strong competition between (i) a magnetocrystalline anisotropy favouring the <111>
430
D. Gignoux, D. Schmitt / Journal of Alloys and Compounds 225 (1995) 423-431
directions, (ii) a negative exchange interaction favouring an antiferromagnetic arrangement of the spins, and (iii) negative quadrupolar pair coupling favouring a mutually perpendicular arrangement of the quadrupoles. Ferroquadrupolar coupling may be at the origin of metamagnetic transitions in the paramagnetic state. This effect has been clearly evidenced in several quadrupolar compounds such as TmZn [29] where a singlestep transition occurs above the ordering temperature along the [001] easy magnetization direction. The transition arises when the internal field exceeds the critical field necessary to induce the ferromagnetic ordering. It has been shown that this behaviour may be associated with a positive value of third-order magnetic susceptibility. This receives a positive contribution from the ferroquadrupolar coupling which is able to over-compensate the negative contribution arising from the CEF effects.
8. Quantitative analysis of incommensurate magnetic systems During the last decade much attention has been paid to theoretical approaches to determination of the complex magnetic phase diagram. In particular, theoretical models focused on the transition between commensurate (C) and incommensurate (I) phases, and on the properties of the incommensurate phase. A model which has been widely used is the axial next nearest neighbour Ising (ANNNI) model [30,31]. It allowed the discovery of new quite exciting physical concepts and accounted qualitatively for experimental results but, because of its great simplifications, it failed to explain the experimental observations in any detail. In order to explain the complex phase diagram of CeSb [32], Date [33] considered an Ising spin chain immersed in a sinusoidally modulated exchange field. In spite of crude assumptions, the phase diagrams of some compounds such as CeSb and PrCo2Si2 [34] were outlined using this model. Much more realistic is the approach of Iwata [35]. Within the mean field approximation it considers the q dependence of J(q) and solves the problem selfconsistently in the same way. However, owing to a simplification of CEF effects, the model can only be applied to Ising systems. These conditions are fulfilled by PrCo2Si2, for which a remarkable quantitative account of the H-T phase diagram, in particular the multistep low temperature metamagnetic process (Fig. 5) and the zero applied field thermal dependence of the propagation vector, was obtained. Recently, a more realistic model which takes into account CEF effects has been applied successfully to describe the magnetic properties (e.g. magnetization processes, magnetic susceptibility, phase diagram, spe-
cific heat...) of several frustrated magnetic systems with any periodicity (commensurate as well as incommensurate) and any moment direction (collinear as well as non-collinear structure) [10,21,34]. This self-consistent periodic field (PF) model is based on an N-site Hamiltonian, N being the number of magnetic ions over one period of the magnetic structure. These magnetic ions are subjected to a modulated exchange field having the same periodicity and arising from the magnetic ions themselves in a self-consistent way. Quite satisfactory analyses of several uniaxial compounds such as PrNi2Si2 [11] and HoAIGa [21] were obtained, as illustrated in Figs. 2 and 8. The PF model applies not only for collinear structures but also for non-collinear arrangements such as those expected in easy axis systems when the field is applied perpendicular to the easy axis, for instance in PrNizSi2 [11]. In addition, the model allowed discussion of the magnetization processes observed in GdGa2, in particular the possibility of a helical to fan structure field induced transition [36]. Another remarkable result obtained by the PF model concerns the thermodynamic properties. Indeed, it has been shown that in the case of a single-Q magnetic structure, the specific heat discontinuity AC at TN of a system having an amplitude modulated (AM) structure is only two thirds of that in a compound with an equal moment (EM) structure, i.e. a ferromagnetic, a simple antiferromagnetic or a helimagnetic compound [37]. This unique feature, first derived in gadolinium based systems [37], has been extended to compounds in the presence of CEF effects [38]. In these latter compounds, however, the analysis is more complex and requires first a good knowledge of the CEF parameters. The second important consequence is that the slope of the specific heat at TN can be negative, leading to the original feature that a maximum of specific heat must occur at a temperature below TN. These original results throw a new light on the experimental temperature dependences of specific heat found in the literature, in particular for gadolinium based compounds [39]. They should be quite useful for choosing between a helical, i.e. EM, and an AM structure near TN, from specific heat data.
9. Summary and conclusion The above survey shows that in intermetallic compounds where only the rare earth is magnetic, a large variety of metamagnetic behaviours of quite different origins (crystal feld, quadrupolar interactions and/or frustration of bilinear exchange interactions) has been found and the often complex magnetic phase diagrams have been determined for some compounds. In most cases these metamagnetic processes are rather well understood and have been analysed quantitatively. At
D. Gignour, D. Schmitt /Journal of Alloys and Compounds225 (1995) 423~131
the moment, a large number of studies are devoted to the determination of complex magnetic phase diagrams for a large number of compounds where incommensurate high temperature structures often compete with low temperature commensurate structures. The number of systems where the microscopic aspects of the phase diagrams have been investigated is rather limited. Accurate magnetization measurements and neutron diffraction under a magnetic field for single crystals would allow us in the future to obtain better knowledge of this type of systems, in particular the Potts systems which have more complex phase diagrams than the Ising systems. Special attention will also need to be paid to the weakly anisotropic gadolinium based systems, in particular with respect to their microscopic magnetic structures which have not, so far, been studied owing to the large neutron absorption of gadolinium.
References [1] M. Date, J. Magn. Magn. Mater., 90-91 (1990) 1. [2] D. Gignoux and D. Schmitt, Z Magn. Magn. Mater, 100 (1991) 99. [3] G. Asti, in K.H.J. Buschow and E.P. Wohlfarth (eds.), Ferromagnetic Materials, Vol. 5, North-Holland, Amsterdam, 1990, Chapter 5. [4] J.J.M. Franse and R.J. Radwanski, in K.H.J. Buschow (ed.), Handbook of Magnetic Materials, Vol. 7, North-Holland, Amsterdam, 1993, p. 307. [5] R. Ballou, B. Ouladdiaf, P.J. Brown, M.D. Nunez-Regueiro and C. Lacroix, Phys. Rev. B, 45 (1992) 3158. [6] E.P. Wohlfarth and P. Rhodes, Philos. Mag., 7 (1962) 1817. [7] F.Y. Wu, J. Appl. Phys., 55 (1984) 2421. [8] S. Takayanagi, Y. Onuki, K. Ina, T. Komatsubara, N. Wada, T. Watanabe, T. Sakakibara and T. Goto, J. Phys. Soc. Jpn., 58 (1989) 1031. [9] J.A. Blanco, D. Gignoux, D. Schmitt and C. Vettier, J. Magn. Magn. Mater., 97 (1991) 4. [10] D. Gignoux and D. Schmitt, Phys. Rev. B, 48 (1993) 12682. [11] J.A. Blanco, D. Schmitt and J.C. Gomez-Sal, J. Magn. Magn. Mater., 116 (1992) 128. [12] B. Lebech, Z. Smetana and V. Sima, J. Magn. Magn. Mater, 70 (1987) 97. [13] D. Gibbs, J. Bohr, J.D. Axe, D.E. Moncton and K.L. D'Amico, Phys. Rev. B, 34 (1986) 8182.
431
[14] N. Iwata, T. Kimura, T. Shigeoka and Y. Hashimoto, J. Magn. Magn. Mater, 70 (1987) 279. [15] N. Iwata, K. Honda, T. Shigeoka, Y. Hashimoto and H. Fujii, J. Magn. Magn. Mater., 90-91 (1990) 63. [16] T. Shigeoka, N. Iwata, H. Fujii, T. Okamoto and Y. Hashimoto, J. Magn. Magn. Mater., 70 (1987) 239. [17] H. Fujii and T. Shigeoka, J. Magn. Magn. Mater., 90-91 (1990) 115. [18] M. Motokawa, H. Nojiri and Y. Endoh, Physica B, 177 (1992) 279. [19] D. Gignoux, D. Schmitt, T. Takeuchi, F.Y. Zhang, C. Rouchon and E. Roudaut, J. Magn. Magn. Mater., 98 (1991) 333. [20] A.R. Ball, D. Gignoux, D. Schmitt, F.Y. Zhang and M. Reehuis, J. Magn. Magn. Mater., 110 (1992) 343. [21] A.R. Ball, D. Gignoux, D. Schmitt and F.Y. Zhang, Phys. Rev. B, 47 (1993) 11887. [22] A.R. Ball, D. Gignoux and D. Schmitt, J. Magn. Magn. Mater., 119 (1993) 96. [23] A.R. Ball, D. Gignoux, D. Schmitt, F.Y. Zhang and P. Burlet, J. Magn. Magn. Mater., 130 (1993) 317. [24] B.R. Cooper, Phys. Lett., 22 (1966) 244. [25] K.A. McEwen, G.J. Cock, L.W. Roeland and A.R. Mackintosh, Phys. Rev. Lett., 30 (1973) 287. [26] A.R. Ball, D. Gignoux, F.E. Kayzel, D. Schmitt and A. de Visser, J. Magn. Magn. Mater., 110 (1992) 337. [27] F.Y. Zhang, D. Gignoux, D. Schmitt, J.J.M. Franse, F.E. Kayzel, N.H. Kim-Ngan and R.J. Radwanski, J. Magn. Magn. Mater, 130 (1993) 108. [28] P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mater, 68 (1987) 107. [29] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B, 17 (1978) 3684. [30] P. Bak, Rep. Prog. Phys., 45 (1982) 587. [31] W. Selke and P.M. Duxbury, Z. Phys. 13, Corders. Matter, 57 (1984) 49. [32] J. Rossat-Mignod, P. Burlet, S. Quezel, J.M. Effantin, D. Delacote, H. Bartholin, O. Vogt and D. Ravot, J. Magn. Magn. Mater., 31-34 (1983) 398. [33] M. Date, J. Phys Soc. Jpn., 57 (1988) 3682. [34] T. Shigeoka, H. Fujii, K. Yonenobu, K. Sugiyama and M. Date, J. Phys. Soc. Jpn., 58 (1989) 394. [35] N. Iwata, J. Magn. Magn. Mater., 86 (1990) 225. [36] A.R. Ball, Thesis, University J. Fourier, Grenoble, 1993, unpublished. [37] J.A. Blanco, D. Gignoux and D. Schmitt, Phys. Rev. B, 43 (1991) 13145. [38] J.A. Blanco, D. Gignoux and D. Schmitt, Phys. Rev. B, 45 (1992) 2529. [39] M. Bouvier, P. Lethuillier and D. Schmitt, Phys. Rev. B, 43 (1991) 13137.