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Magnetic phases of europium triarsenide
Journal of Magnetism and Magnetic Materials 58 (1986) 207-215 North-Holland, Amsterdam
MAGVETIC
PHASES OF EUROPIUM
207
TRIARSENIDE
P. THALMEIER Institut
ftir Festkijrperforschung der KFA Jiilich, D-51 70 Jiilich, Fed. Rep. Germany
Received 2 August 1985
Europium triarsenide is a new semi-metallic antiferromagnet which develops an incommensurate structure below TN = 11.3 K and shows a lock-in transition to a commensurate structure at T,_ = 10.26 K. A model for a realistic exchange Hamiltonian is proposed and shown to lead to a proper incommensurate modulation if the exchange interactions are sufficiently anisotropic. A phase diagram of possible magnetic phases closely below TN is obtained in the exchange parameter plane by calculating exchange bands and the mean-field susceptibility.
1. Introduction Modulated magnetic structures are commonly divided into two empirical classes. Commensurate (C) modulated structures where the modulation wave vector qm changes in a step-like behavior as temperature varies, taking only values qm = (P/q)K where p, q are integers and K is a reciprocal lattice vector. Examples are the well studied Ce-pnictides, e.g. CeSb, CeBi [l] and also some actinide compounds [2]. In the second class of genuinely incommensurate (IC) structures the modulation qm changes smoothly with temperature and ultimately shows a lock-in to a C-value q. at low temperatures. The heavy Rare Earth metals (RE) are classic examples of IC-structures. Their theoretical description [3] is complicated by the fact that not only q, but also the direction of magnetic moments with respect to the modulation axis change with temperature. The existence of modulated structures is due to competing exchange interactions of nearest and next-nearest neighbors of possible anisotropic character. In most cases the RKKY-mechanism is thought to be responsible for the different signs of competing interactions. The most thoroughly discussed theoretical model is the ANNNI (axial next nearest neighbor Ising) model [4,5], where mean field calculations predict the existence of various Cphases in finite temperature intervals. This leads 0304~8853/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
to a ‘staircase’ behavior of q,,,(T) which has a qualitative similarity to the behavior of the modulation vector in the Ce-pnictides. For the ICstructures with smooth q,,,(T) the ‘soliton lattice’ model based on a Landau expansion of the free energy in terms of an order parameter with inhomogeneous phase is more appropriate. Recently a new IC-antiferromagnet, EuAs,, was discovered by neutron diffraction [6] This semimetallic compound has a monoclinic structure [7] as shown in fig. 1. From specific heat measurements [7] it was already known to have two magnetic phase transitions at TN = 11.3 K and TL = 10.26 K. Subsequently, the phase below TL was shown to be a C-structure of the AFl-type [8]. The intermediate phase (T,_ -c T-c TN) has now been identified as a IC-modulated antiferromagnet with from the 4mlle * (see fig. 1). This was concluded existence of satellite reflexes at [i,0, 3 f a]positions in the reciprocal chemical unit cell whose basis is denoted by (I*, b*, c*. The satellite position 6(T) varies smoothly with temperature from 6(T,) = 0.15 down to 6 = 0 at TL where a lock-in transition to the C-AFl-phase takes place. No higher order satellites were found indicating a modulation of purely sinusoidal type . The continuous S(T), the small value of (TN - TL)/TN = 0.1 and the fact that the moment direction (lb does not change with temperature suggest that EuAs, should be well described by a Landau B.V.
208
P. Thalmeier / Magnetic phases of europium triarsenide exchange mn.
Fig. 1. Crystal structure and magnetic low temperature AFlphase of EuAs, (As-atoms are not shown). Chemical unit cell (atoms 1-4, basis vectors a, b, c) is indicated by dotted lines, b is perpendicular to the plane. Reciprocal lattice vectors a*, b*, c* are also shown. I is the inversion center. Magnetic moments are parallel (open circles) and antiparallel (full circles) to b. (a = 9.43 A, b = 1.50 A, c = 5.75 A, /3 = 61.5”.)
theory. As a first step, however, it is necessary to obtain a clear picture of possible exchange interactions in EuAs, and about the conditions of existence for an IC-structure of the observed type. Therefore the topic of this paper is an investigation of the possible magnetic phases of EuAs, immediately below TN and the resulting phase diagram as function of the exchange parameters. The results of a Landau theory for the C-lock-in transition will be published elsewhere.
2. Crystal structure and exchange model Europium triarsenide has a monoclinic structure which can be viewed as folded layers of As atoms forming tubes for the chains of divalent Euzf-ions (fig. 1). The space group is (2/m) which includes a twofold rotation C, around b, a reflection ur, in a plane perpendicular to b and consequently an inversion Ci = C, . u,,. The coordination of Eu-neighbors is shown in fig. 2, they can be associated with different (101) planes and fall into two groups: Next neighbors (n.n.) with distances ranging from 3.91 to 4.29 A and next nearest neighbors (n.n.n.) with distances of 5.76 to 5.99 A. Starting point for understanding the IC-
‘c--@
I0
coupling nnn.
C--+1,
I1
Fig. 2. Coordination of nearest and next nearest Eu2+ neighbors with respect to center ion C (d = 4.24 A). All Eu2+-ions are lying in (501) planes. Exchange coupling: IO, I, 10, 1, < 0.
structure is a suitable exchange Hamiltonian. Bestate without cause Eu2+ is in a 8S ,,,-ground orbital moment, a Heisenberg exchange
is assumed. Here 1, 1’ = unit cell and OL,(Y’= basis atom. In principle, J$ could also depend on spin directions in a way compatible with monoclinic symmetry. The exchange tensor in (2.1), however, is assumed to be isotropic in spin space and it is characterized by three phenomenological constants whose signs are chosen in a way to satisfy obvious constraints given by the low temperature AFl-phase (see fig. 1). For the in-plane n.n. coupling Z, > 0 is assumed, and Z, < 0 for the n.n.n. coupling to adjacent planes (fig. 2). There is a single out-of-plane n.n. coupling Z, left which should have the same sign (Z,, > 0) as the other n.n. couplings but may have different magnitude since their directions are perpendicular. This means that Eu2+-ions lying in adjacent (201)planes are coupled ferromagnetically (Z, > 0) as well as antiferromagnetically ( Z2 < 0). This is quite different from the ANNNI-model where moments lying in (OOl)-cubic planes have interplane-couplings with only one sign for a given pair of planes. In the model for EuAs,, couplings to overnext (201)-planes are neglected in a first approximation because distances involved would be
P. Thalmeier / Magnetic phases
of europium triarsenide
209
larger than 8 A, twice as large as nn. distances. The out-of-plane n.n. coupling I,, is frustrated in the low temperature phase. Immediately below TN this frustration could lead to a transition to an energetically more favorable incommensurate phase.
3. Symmetry coordinates and ‘exchange bands’ Whether an IC-phase exists as first magnetic phase (FMP) below T,,, can be seen by looking at the paramagnetic susceptibility (SC) and its singular behavior. In RPA-approximation the SC-tensor is given by
xi&)
= [T-
W&+‘C~ (3.1)
with C= fS(S+ 1). Here (Y= 1...4 denotes the sites of the unit cell, not the space directions. This has been suppressed because x(q) is isotropic in the Heisenberg case. The FMP is defined by the singularity of the SC-tensor which according to (3.1) is determined by the exchange tensor Z,,(q). This important quantity is given in the appendix as obtained from the exchange model of fig. 2. To find the singularity of x(q) it is necessary to diagonalize 1(q) and x(q) which requires a transformation to normal spin coordinates. For reasons of better understanding, this is done in two steps. First a block-diagonalisation by transforming to unit cell collective coordinates S:(q) is performed where
(3.2) Here a, = (S,Y) are the expectation values of individual unit cell spins pointing along b. The q-independent transformation U” is given by
Fig. 3. Collective unit cell spin coordinates and closed circles correspond to spins parallel to b, respectively.
Sj(q = 0). Open and antiparallel
For q = 0 the collective coordinates S.j are shown in fig. 3. They transform as irreducible representations of the (2/m) symmetry group according to the rules given in table 1. A similar classification holds for a zone boundary vectorq = & ic*. For a general q = qc* along the relevant modulation direction, a transformation from a, to Sz leads to a block-diagonalisation of the exchange tensor Za = U”ZUo+ in the (Si’, S,‘) (A) and (S,‘, S,“) (B) subspaces. For example, in the A-subspace one has / (21, - 31, - I,)
- i( Zi - Z2) sin 2aq \
+ (31, - Zi) cos 2lTq PA(Q) =
i( Zi - I,)
sin 2nq
’
(z,-2z,-Z*) +(&+z,)
\
cos2aq (3.4)
where q = qc*, 1q 1 = 2Tq/c sin p. An equivalent expression holds for G(q). For q = 0, f i, i.e. zone center and zone boundary, the nondiagonal elements of P vanish and the symmetry coordinates Si are already the normal coordinates. For a general q, these elements are - sin 2+nq and lead
Table 1 Symmetry coordinates sion)
(2/m) repres.
properties (character table) of the S.” collective for q = 0 under the operations C, and Ci (inver-
Coil. word.
c2
ci
P. Tholmeier / Magnetic phases ofeuropium triarsenide
210
S,(q) (K = 1.. .4) through of (St”, S,“) and (S,“, S,“),
to normal coordinates pair-wise hybridisation i.e.
1
E, S,(q)
= %(‘#%I)
+ %(&%I),
K= 1, 2,
= a:(q)S,o(q)
+ u:(q)S:(q),
K = 394,
I
,
I
%-
4-
or S,(q)
_______
(3.5) where the sequence of K within each set is according to decreasing energy of the ‘exchange bands’ r,(q) which are the eigenvalues of f’(q) corresponding to S,(q). They are given by (q’ = 2nq) r,,,
=
-2Z,(l &- ((z,
- cos q’) - z,)’
+ (I,
+ z, -
0
0.1
0.2
0.3 9
0.4
0.5
Fig. 5. Comparison of exchange bands E,” (broken lines) and E, (full lines) for K = 1, 2 which correspond to collective S.“(q) and normal S,.(q)spin coordinates. (q = q c*.)
2zJ2
+ 2( I, - I,)( z, + I, - 21,) cos ql}1’2, &+=2Z,(l
+cos
For further troduce the
q’)
+{(2z,+z2+z0)2+(z,+z2)2 +2( I, + Z,)(2Zt
‘competition
+ I, + z,) cos q’}“*. (3.6)
and ‘anisotropy
discussions,
ratio’
r=
it is convenient
to in-
-Z2/Z,
(3.7) ratio’
r’ = IO/Z,.
Fig. 4. Exchange bands E, = r,.(q)/Z, for q = qc* and exchange parameters (r, r’) = (1.6,5.7). Absolute maximum (arrow) correspond to a modulation q,,,= 0.38 (6 = 0.12). Symmetry at q = 0, ; is indicated for some bands. The insert shows the hybridization coefficients for the top-most band according to (3.5).
P. Thalmeier / Magneiic phases of europium triarsenide
They determine the shape of the exchange bands E, = r,/Z, and Z, defines the energy scale. The FMP is a modulated IC-phase if the topmost exchange band has an absolute maximum for a wave number 0 < q, < t in the interior of Brillouin zone; this is equivalent to a (T - TN)-’ singularity_ in the diagonalized SC-tensor x,(q) = CATCl,(q)1 at a,, with a transition temperature TN = CZ,(q,). Such a case is shown in fig. 4 for a special (r, r’) parameter set. If the absolute maximum exists for 0 < q, < 4, the modulation q, and the corresponding TN are given by q, =
& aces -
(1 + r)(r’
- r - 2)
8r2
i
r’-r-2
l+r 2( r’ - r - 2) -
2(1 + r)
,
(3.8a)
2r+(I+r)tr’-r-2)
TNzcz
1
4r
i
r(1
+
+ (r’-r-2) = CZ,F(r,r’).
r)
r(r’-r-2) +
(l+r) (3.8b)
It is interesting to look for the origin of the maximum at q,. If one neglects the nondiagonal mixing elements in (3.4) one obtains the ‘ unhybridized’ exchange bands E,” corresponding to the S:(q) collective coordinates. For K = 1,2 they are shown in fig. 5 and they have no maximum within the Brillouin zone but rather at the zone boundary. Inclusion of the mixing terms in (3.4) results in a strong hybridization at the crossing point of the E:(q) leading to a repulsion such that the upper exchange band E,(q) develops an absolute maximum at an incommensurate wave number q,. At q = 0, i there is no hybridization and E, = Ef, SK = S,” because the SK0 show different behavior under the inversion operation ( Ci Sy = Sp, Ci S: = -S,“). For 0 < q -c 4, however, the inversion is not contained in the q-vector group and mixing of Sp, S: occurs according to (3.5). The mixing coefficients u*(q), u,(q) are shown in fig. 4 as an insert which leads to the conclusion that the normal coordinates S,(q), S,(q) exchange their symmetries as q varies from 0 to *.
211
4. Phase diagram of first magnetic phases (T ,< TN) Depending on the (r, r’) parameter set, the exchange bands E,(q) can look very different from fig. 4. A selection of possible cases is shown in figs. 6a-d. In each case the q-vector of the absolute maximum and the corresponding normal coordinate characterize the FMP below TN. For example, fig. 6a shows a situation with q,,, = l/2 and S,(q,) = SF. This phase corresponds directly to the antiferromagnetic AFl phase observed at low temperatures. Another case is shown in fig. 6b where one band, a (SF, $)-mixture, still has a relative maximum at an incommensurate q,,,, but the absolute maximum has shifted to q,,, = l/2 with a different symmetry S,” corresponding to a different antiferromagnetic phase AF2. For parameters with small competition ratio r = - Z,/Z,, one obtains an absolute maximum at q = 0 with S: symmetry, i.e. a ferromagnetic (FM) phase. In fig. 6d finally, the special case for isotropic n.n. coupling IO = Z, is shown which again leads to an AFl phase for the competition ratio r chosen. In this way, identifying the absolute maximum, its q,-wave number and symmetry of the normal coordinate, a complete phase diagram of FMP’s in the (r, r’)-parameter plane was obtained. The IC-region of the phase diagram is defined by the requirement that the absolute value of the argument in (3.8a) is less than one. This leads to the condition (r+2)+b_(r)
= (l/a)[l l+r --i 4r2
* (1+ a(1 + r))]“‘, 1 l+r
(4.1)
i’
necessary for the existence of a relative maximum E,(q,). In addition, if E,(q,) = T,/CZ, (3.8b) is larger than any E,(O), E,(i) (table 2) one has an absolute maximum and the parameters (r, r’) lie in the IC-region. The resulting phase diagram is shown in fig. 7. To each of the sectors in this diagram the typical exchange bands are shown in figs. 4, 6. One can see that a large anisotropy ratio r’ = IO/Z, > 4 is necessary for the existence of an IC-phase. In the isotropic case r’ = 1 only FM or AFl commensurate phases would be possible.
212
of europiumtriarsenide
P. Thnlmeier / Magnetic phases
r’=6
16 */
ICI
6-
S;b):FM L
_
L. z-
L
O-
0
-2 r’=4
-L -
-0
E,lZ a
01
02
03
04
-L
05
0
01
02
9
03
04
05
9
Fig. 6. Exchange bands for various parameters (r, r’). Symmetry of normal coordinate at the absolute maximum and corresponding type of magnetic phase is indicated (compare with fig. 7). (d) Special isotropic case with I, = I, has degeneracies.
Starting from the IC-region a reduction of r’ leads to a continuous transition to the AFl region, i.e. the q,-value in fig. 4 moves to the zone boundary continuously. If r is reduced instead, one crosses into the AF2 region with a discontinuous change of the order parameter symmetry; it corresponds to fig. 6b where the absolute maximum has jumped to q = l/2 with a St normal coordinate although
6
r’
q=;
0
04
08
12
16 r
Fig. 7. Phase diagram for T 5 TN in the (r, r’)-parameter plane. AFl, AF2 are different antiferromagnetic C-phases and FM is the ferromagnetic phase. IC is the sector of the incommensurate antiferromagnetic phase. Crossing from the IC-region into AFl or AF2 leads to a continuous or a discontinuous change of order parameter symmetry, respectively.
Table 2 Exchange band energies E&( 4) at zone center (4 = 0) and zone boundary (4 = i) as function of r, r’ K
K(0)
E,(:)
1 2 3 4
II- r’l -(lr’l (-4r+ (-4r-
(4r+ 13+2r-r’l) (4r - 13+2r - r’l) II+ r’l -II+ r’l
13-2r+r’l) 13-2r+r’l)
P. Thalmeier / Magnetic phases
ofeuropium
triarsenide
213
was adopted, leading to 6 = 0.12. These values were also chosen in fig. 4. Using (3.8b) with TN = 11.3 K one obtains the exchange constants I, = 1.66 K, Z, = 0.29 K, Z, = -0.47
3L
I 1.0
I 1.4
I 1.8
I I 2.2 ,. 2.6
(4.2)
They are comparable to magnitude to those found in the Eu-monochalcogenides [9]. The out-of-plane n.n. coupling is rather large because in this model it is the only coupling which is ‘frustrated’ in the AFl C-phase and tends to establish the IC-structure. In an extension of this model one might consider additional ferromagnetic coupling to overnext (201)-planes with distances > 8 A (figs. 1, 2) and also an anisotropy of the antiferromagnetic n.n.n. coupling. Both modifications would lead to a smaller I,, for a given IC-wave number, so that I, in (4.2) has to be considered as an effective exchange coupling. For a more precise determination of exchange constants, one must await the investigation of spin wave excitations in EuAs,. On the other hand, the present model is in reasonable agreement with paramagnetic sc-measurements [lo]. In mean field theory the SCis given
J 3.0
Fig. 8. Enlarged IC-region of the phase diagram with contours for 6(r, r’) = f - q,,, according to (3.8a). Curves (a)-(i) respectively: S = 0.140, 0.135, 0.130, 0.125, 0.120, 0.115, 0.110, 0.10, 0.08.
there still remains a relative maximum at an IC wave vector. The variation of 6(r, r’) = + - q, in the IC-region is shown in fig. 8 as a contour plot and in fig. 9 as function of r for constant anisotropy. It shows that 6 5 0.145 which is close to the extrapolated experimental value 8, = S(T,) = 0.15. However, this is not sufficient to determine the ratios r, r’ unambiguously. One would prefer the anisotropy r’ as small as possible but compatible with 6 = S,. As a compromise (r, r’) = (1.6, 5.7)
0.16
K.
by X0’ = C/(T-t To), T,=$Cx&(O). I(
(4.3)
Using table 2, one obtains To= - 2CZ,. Furthermore, TN = CZ,F(r,r')according to (3.8b), there-
C’
0
1.4
1.6
1.8
20
2.2
21
r
2.6
28
30
Fig. 9. Variation of 6(r) for constant r’. This corresponds to a continuous transition from IC to AFl in fig. 7 with the absolute maximum of EX moving to the zone boundary (see fig. 4).
Fig. 10. Inverse paramagnetic susceptibility of EuAsr. Data are taken from Bauhofer et al. [lo]. TN = 11.3 K, TO = 5.4 K, x0 = 1 Sl umt.
P. Thalmeier / Magnetic phases
214
momentum transfer, tor and the structure
fore TN/TO=-&F(r,
r’).
(4.4)
This relation connects (TN/TO) which is a purely thermodynamic ratio to a r.h.s. which is in principle independently determined by the IC-modulation wave number 6. With TN = 11.3 K and TO= 5.4 K (see fig. 10) one has TN/TO = 2.1. Using (r, r’) = (1.6, 5.7) the r.h.s. of (3.4) is equal to 2.3 which is in reasonable agreement with TN/TO. Therefore, the exchange model used is compatible with both the magnitude of 6 and the thermodynamic ratio TN/TO.
5. Incommensurate modulation scattering cross section
wave and neutron
It is interesting to investigate the spin structure in real space u,(r) = (S:(r)) which corresponds to the IC order parameter S,(q,) for a given set (r, r’). According to (3.3)-(3.5) it is given by u,(r)
=A{cos
$3~~ - isin +4$}
T(Z) = - sin(2aqz p(z)
= sin(2Tqz
I f(Q)
I2
xis;Q-4,-K)cS(Q-,,-K)), (5.3) f(Q)
= ~eiQ”~.sa, a
s, = 0,/A.
Here K = ha* + kb* + Ic* is a reciprocal lattice vector and t, (a = 1 . . .4) are position vectors of unit cell atoms with the inversion center as origin. Eq. (5.3) shows that IC-magnetic satellite reflexes occur at K+ q,,, and have an intensity proportional to the square of the structure factor f(K + qm). It depends on the unit cell spin configuration, explicitly one has for q, = qmc*: f(K+q,)=e”*
[
Z,cos:(h+k) -iZ,
sint(h
+ k)]
(5.4)
za=(Si+s2+S~+s4);
Zb=(S1-S2-S~+SJ,
(5.1)
where cos + = u,(q,,,), sin + = 1u,(q,) 1 and w, = f( -1, 1, l,- 1); rv, = i(l, -1, 1, -1) are the first two row vectors in (3.3). In the present model the modulation vector q, corresponds nearly to the crossing point of the unhybridized exchange bands in fig. 5 where 1u1 1 = I u1 I = l/&f, i.e. + = a/4. In this approximation (5.1) can be written as =‘+A(-r(z),
S(Q) = iA2C
F(Q) the magnetic form facfunction S(Q) is given by
with
exp(iq;r)
(cr=1...4:basisatom),
u(r)
of europium triarsenide
r(z),
p(z),
-P(Z)),
(5.2)
- n/4), + ~/4),
A - (T,., - T)‘/2 is the amplitude of the IC modulation wave. Thus, the spin structure in the ICphase can be viewed as AF pairs (a = 1, 2) and ((Y = 3, 4) modulated sinusoidally along c* with a phase difference 71/2. The coherent neutron scattering cross-section due to the IC spin structure with wave vector qm is proportional to I F(Q) I 2S(Q) where Q is the
s, = e,/A, and f9* is an overall phase. Inserting (5.1) with @J= IT/~ into (5.4) one observes that because Z,(wt,) = Z,(rv,) = 0 only the first part of (5.1) contributes to the cross section. From Z,( wi) = 0, Z,( wi) = 4 one concludes that only Bragg points with odd (h + k) = 2m + 1 have IC-satellites. Therefore experimentally [0,0, 4 + 61 reflexes cannot be observed but [i, 0, f + 61 has finite intensity. For temperatures below 10.26 K, q, = ic* and one has a magnetic supercell with doubling in the c-direction which leads to the additional condition I = 2n + 1 for the commensurate magnetic reflexes.
6. Conclusion The existence of an incommensurate antiferromagnetic phase in EuAs, could be explained within an anisotropic exchange-model including next and next nearest neighbors. A considerable anisotropy
P. Thnlmeier / Magnetic phases
of the ferromagnetic n.n. interaction must be assumed. The IC modulation is due to a hybridization of exchange bands with different symmetry. The magnitude of the modulation wave vector agrees with the experimental value and the implied exchange parameters are compatible with susceptibility measurements. The phase diagram in the exchange parameter plane shows that the IC-phase is lying next to the experimentally observed low temperature antiferromagnetic C-phase. It is possible to describe the lock-in transition to the C-phase within the context of a Landau theory [ll]. It remains to explain the H-T phase diagram which shows many interesting IC- and C-phases for nonzero fields [12] within the present model.
Acknowledgements The author would like to thank T. Chattopadhyay and W. Bauhofer for helpful discussions. Financial support by the Deutsche Forschungsgemeinschaft is also acknowledged.
Appendix The exchange tensor Z,,(q) (a, j3 = 1.. .4 = basis atom) of (2.2) is given in terms of the exchange constants Z,, Z,, Z, by the following matrix:
of europiumtriarsenide
With
+,
215
+2 = q.
= q . a,
A, = 21, cos +3,
A, = IO + I, ele3 + Z, em’+‘,
B, = Z,(l + e-‘+I), B, = Z, e-‘+l(I
b, $J~= q ac one has
+ e&)
B, = Z,(l + ePiG2) + I, ei93, + Z~e-i(+1-&).
(A-2)
References [l] J. Rossat-Mignod, P. Burlet, J. Quezel, J.M. Effantin, D. DelacBte, H. Bartholin, 0. Vogt and D. Ravot, J. Magn. Magn. Mat. 31-34 (1983) 398. [2] M. Kuznietz. P. Burlet, J. Rossat-Mignod and 0. Vogt, to be published. [3] B. Coqblin, The Electronic Structure of Rare Earth Metals and Alloys: the Magnetic Heavy Rare-Earths (Academic Press, London, 1977). [4] P. Bak and J. v. Boehm, Phys. Rev. B 21 (1980) 5297. [5] W. Selke and P.M. Duxbury, Z. Phys. B 57 (1984) 49. [6] T. Chattopadhyay, P.J. Brown, P. Thalmeier and H.G. v. Schnering, to be published. [7] W. Bauhofer, M. Wittman and H.G. v. Schnering, J. Phys. Chem. Solids 42 (1981) 687. [8] T. Cbattopadhyay, H.G. v. Schnering and P.J. Brown, J. Magn. Magn. Mat. 28 (1982) 247. [9] V.C. Lee and L. Lin, Phys. Rev. B 30 (1984) 2026. [lo] W. Bauhofer, E. Gmelin, M. Mollendorf, R. Nesper and H.G. v. Schnering, J. Phys. C 18 (1985) 3017. [ll] P. Thalmeier, to be published. [12] W. Bauhofer et al., to be published.
MAGVETIC
PHASES OF EUROPIUM
207
TRIARSENIDE
P. THALMEIER Institut
ftir Festkijrperforschung der KFA Jiilich, D-51 70 Jiilich, Fed. Rep. Germany
Received 2 August 1985
Europium triarsenide is a new semi-metallic antiferromagnet which develops an incommensurate structure below TN = 11.3 K and shows a lock-in transition to a commensurate structure at T,_ = 10.26 K. A model for a realistic exchange Hamiltonian is proposed and shown to lead to a proper incommensurate modulation if the exchange interactions are sufficiently anisotropic. A phase diagram of possible magnetic phases closely below TN is obtained in the exchange parameter plane by calculating exchange bands and the mean-field susceptibility.
1. Introduction Modulated magnetic structures are commonly divided into two empirical classes. Commensurate (C) modulated structures where the modulation wave vector qm changes in a step-like behavior as temperature varies, taking only values qm = (P/q)K where p, q are integers and K is a reciprocal lattice vector. Examples are the well studied Ce-pnictides, e.g. CeSb, CeBi [l] and also some actinide compounds [2]. In the second class of genuinely incommensurate (IC) structures the modulation qm changes smoothly with temperature and ultimately shows a lock-in to a C-value q. at low temperatures. The heavy Rare Earth metals (RE) are classic examples of IC-structures. Their theoretical description [3] is complicated by the fact that not only q, but also the direction of magnetic moments with respect to the modulation axis change with temperature. The existence of modulated structures is due to competing exchange interactions of nearest and next-nearest neighbors of possible anisotropic character. In most cases the RKKY-mechanism is thought to be responsible for the different signs of competing interactions. The most thoroughly discussed theoretical model is the ANNNI (axial next nearest neighbor Ising) model [4,5], where mean field calculations predict the existence of various Cphases in finite temperature intervals. This leads 0304~8853/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
to a ‘staircase’ behavior of q,,,(T) which has a qualitative similarity to the behavior of the modulation vector in the Ce-pnictides. For the ICstructures with smooth q,,,(T) the ‘soliton lattice’ model based on a Landau expansion of the free energy in terms of an order parameter with inhomogeneous phase is more appropriate. Recently a new IC-antiferromagnet, EuAs,, was discovered by neutron diffraction [6] This semimetallic compound has a monoclinic structure [7] as shown in fig. 1. From specific heat measurements [7] it was already known to have two magnetic phase transitions at TN = 11.3 K and TL = 10.26 K. Subsequently, the phase below TL was shown to be a C-structure of the AFl-type [8]. The intermediate phase (T,_ -c T-c TN) has now been identified as a IC-modulated antiferromagnet with from the 4mlle * (see fig. 1). This was concluded existence of satellite reflexes at [i,0, 3 f a]positions in the reciprocal chemical unit cell whose basis is denoted by (I*, b*, c*. The satellite position 6(T) varies smoothly with temperature from 6(T,) = 0.15 down to 6 = 0 at TL where a lock-in transition to the C-AFl-phase takes place. No higher order satellites were found indicating a modulation of purely sinusoidal type . The continuous S(T), the small value of (TN - TL)/TN = 0.1 and the fact that the moment direction (lb does not change with temperature suggest that EuAs, should be well described by a Landau B.V.
208
P. Thalmeier / Magnetic phases of europium triarsenide exchange mn.
Fig. 1. Crystal structure and magnetic low temperature AFlphase of EuAs, (As-atoms are not shown). Chemical unit cell (atoms 1-4, basis vectors a, b, c) is indicated by dotted lines, b is perpendicular to the plane. Reciprocal lattice vectors a*, b*, c* are also shown. I is the inversion center. Magnetic moments are parallel (open circles) and antiparallel (full circles) to b. (a = 9.43 A, b = 1.50 A, c = 5.75 A, /3 = 61.5”.)
theory. As a first step, however, it is necessary to obtain a clear picture of possible exchange interactions in EuAs, and about the conditions of existence for an IC-structure of the observed type. Therefore the topic of this paper is an investigation of the possible magnetic phases of EuAs, immediately below TN and the resulting phase diagram as function of the exchange parameters. The results of a Landau theory for the C-lock-in transition will be published elsewhere.
2. Crystal structure and exchange model Europium triarsenide has a monoclinic structure which can be viewed as folded layers of As atoms forming tubes for the chains of divalent Euzf-ions (fig. 1). The space group is (2/m) which includes a twofold rotation C, around b, a reflection ur, in a plane perpendicular to b and consequently an inversion Ci = C, . u,,. The coordination of Eu-neighbors is shown in fig. 2, they can be associated with different (101) planes and fall into two groups: Next neighbors (n.n.) with distances ranging from 3.91 to 4.29 A and next nearest neighbors (n.n.n.) with distances of 5.76 to 5.99 A. Starting point for understanding the IC-
‘c--@
I0
coupling nnn.
C--+1,
I1
Fig. 2. Coordination of nearest and next nearest Eu2+ neighbors with respect to center ion C (d = 4.24 A). All Eu2+-ions are lying in (501) planes. Exchange coupling: IO, I, 10, 1, < 0.
structure is a suitable exchange Hamiltonian. Bestate without cause Eu2+ is in a 8S ,,,-ground orbital moment, a Heisenberg exchange
is assumed. Here 1, 1’ = unit cell and OL,(Y’= basis atom. In principle, J$ could also depend on spin directions in a way compatible with monoclinic symmetry. The exchange tensor in (2.1), however, is assumed to be isotropic in spin space and it is characterized by three phenomenological constants whose signs are chosen in a way to satisfy obvious constraints given by the low temperature AFl-phase (see fig. 1). For the in-plane n.n. coupling Z, > 0 is assumed, and Z, < 0 for the n.n.n. coupling to adjacent planes (fig. 2). There is a single out-of-plane n.n. coupling Z, left which should have the same sign (Z,, > 0) as the other n.n. couplings but may have different magnitude since their directions are perpendicular. This means that Eu2+-ions lying in adjacent (201)planes are coupled ferromagnetically (Z, > 0) as well as antiferromagnetically ( Z2 < 0). This is quite different from the ANNNI-model where moments lying in (OOl)-cubic planes have interplane-couplings with only one sign for a given pair of planes. In the model for EuAs,, couplings to overnext (201)-planes are neglected in a first approximation because distances involved would be
P. Thalmeier / Magnetic phases
of europium triarsenide
209
larger than 8 A, twice as large as nn. distances. The out-of-plane n.n. coupling I,, is frustrated in the low temperature phase. Immediately below TN this frustration could lead to a transition to an energetically more favorable incommensurate phase.
3. Symmetry coordinates and ‘exchange bands’ Whether an IC-phase exists as first magnetic phase (FMP) below T,,, can be seen by looking at the paramagnetic susceptibility (SC) and its singular behavior. In RPA-approximation the SC-tensor is given by
xi&)
= [T-
W&+‘C~ (3.1)
with C= fS(S+ 1). Here (Y= 1...4 denotes the sites of the unit cell, not the space directions. This has been suppressed because x(q) is isotropic in the Heisenberg case. The FMP is defined by the singularity of the SC-tensor which according to (3.1) is determined by the exchange tensor Z,,(q). This important quantity is given in the appendix as obtained from the exchange model of fig. 2. To find the singularity of x(q) it is necessary to diagonalize 1(q) and x(q) which requires a transformation to normal spin coordinates. For reasons of better understanding, this is done in two steps. First a block-diagonalisation by transforming to unit cell collective coordinates S:(q) is performed where
(3.2) Here a, = (S,Y) are the expectation values of individual unit cell spins pointing along b. The q-independent transformation U” is given by
Fig. 3. Collective unit cell spin coordinates and closed circles correspond to spins parallel to b, respectively.
Sj(q = 0). Open and antiparallel
For q = 0 the collective coordinates S.j are shown in fig. 3. They transform as irreducible representations of the (2/m) symmetry group according to the rules given in table 1. A similar classification holds for a zone boundary vectorq = & ic*. For a general q = qc* along the relevant modulation direction, a transformation from a, to Sz leads to a block-diagonalisation of the exchange tensor Za = U”ZUo+ in the (Si’, S,‘) (A) and (S,‘, S,“) (B) subspaces. For example, in the A-subspace one has / (21, - 31, - I,)
- i( Zi - Z2) sin 2aq \
+ (31, - Zi) cos 2lTq PA(Q) =
i( Zi - I,)
sin 2nq
’
(z,-2z,-Z*) +(&+z,)
\
cos2aq (3.4)
where q = qc*, 1q 1 = 2Tq/c sin p. An equivalent expression holds for G(q). For q = 0, f i, i.e. zone center and zone boundary, the nondiagonal elements of P vanish and the symmetry coordinates Si are already the normal coordinates. For a general q, these elements are - sin 2+nq and lead
Table 1 Symmetry coordinates sion)
(2/m) repres.
properties (character table) of the S.” collective for q = 0 under the operations C, and Ci (inver-
Coil. word.
c2
ci
P. Tholmeier / Magnetic phases ofeuropium triarsenide
210
S,(q) (K = 1.. .4) through of (St”, S,“) and (S,“, S,“),
to normal coordinates pair-wise hybridisation i.e.
1
E, S,(q)
= %(‘#%I)
+ %(&%I),
K= 1, 2,
= a:(q)S,o(q)
+ u:(q)S:(q),
K = 394,
I
,
I
%-
4-
or S,(q)
_______
(3.5) where the sequence of K within each set is according to decreasing energy of the ‘exchange bands’ r,(q) which are the eigenvalues of f’(q) corresponding to S,(q). They are given by (q’ = 2nq) r,,,
=
-2Z,(l &- ((z,
- cos q’) - z,)’
+ (I,
+ z, -
0
0.1
0.2
0.3 9
0.4
0.5
Fig. 5. Comparison of exchange bands E,” (broken lines) and E, (full lines) for K = 1, 2 which correspond to collective S.“(q) and normal S,.(q)spin coordinates. (q = q c*.)
2zJ2
+ 2( I, - I,)( z, + I, - 21,) cos ql}1’2, &+=2Z,(l
+cos
For further troduce the
q’)
+{(2z,+z2+z0)2+(z,+z2)2 +2( I, + Z,)(2Zt
‘competition
+ I, + z,) cos q’}“*. (3.6)
and ‘anisotropy
discussions,
ratio’
r=
it is convenient
to in-
-Z2/Z,
(3.7) ratio’
r’ = IO/Z,.
Fig. 4. Exchange bands E, = r,.(q)/Z, for q = qc* and exchange parameters (r, r’) = (1.6,5.7). Absolute maximum (arrow) correspond to a modulation q,,,= 0.38 (6 = 0.12). Symmetry at q = 0, ; is indicated for some bands. The insert shows the hybridization coefficients for the top-most band according to (3.5).
P. Thalmeier / Magneiic phases of europium triarsenide
They determine the shape of the exchange bands E, = r,/Z, and Z, defines the energy scale. The FMP is a modulated IC-phase if the topmost exchange band has an absolute maximum for a wave number 0 < q, < t in the interior of Brillouin zone; this is equivalent to a (T - TN)-’ singularity_ in the diagonalized SC-tensor x,(q) = CATCl,(q)1 at a,, with a transition temperature TN = CZ,(q,). Such a case is shown in fig. 4 for a special (r, r’) parameter set. If the absolute maximum exists for 0 < q, < 4, the modulation q, and the corresponding TN are given by q, =
& aces -
(1 + r)(r’
- r - 2)
8r2
i
r’-r-2
l+r 2( r’ - r - 2) -
2(1 + r)
,
(3.8a)
2r+(I+r)tr’-r-2)
TNzcz
1
4r
i
r(1
+
+ (r’-r-2) = CZ,F(r,r’).
r)
r(r’-r-2) +
(l+r) (3.8b)
It is interesting to look for the origin of the maximum at q,. If one neglects the nondiagonal mixing elements in (3.4) one obtains the ‘ unhybridized’ exchange bands E,” corresponding to the S:(q) collective coordinates. For K = 1,2 they are shown in fig. 5 and they have no maximum within the Brillouin zone but rather at the zone boundary. Inclusion of the mixing terms in (3.4) results in a strong hybridization at the crossing point of the E:(q) leading to a repulsion such that the upper exchange band E,(q) develops an absolute maximum at an incommensurate wave number q,. At q = 0, i there is no hybridization and E, = Ef, SK = S,” because the SK0 show different behavior under the inversion operation ( Ci Sy = Sp, Ci S: = -S,“). For 0 < q -c 4, however, the inversion is not contained in the q-vector group and mixing of Sp, S: occurs according to (3.5). The mixing coefficients u*(q), u,(q) are shown in fig. 4 as an insert which leads to the conclusion that the normal coordinates S,(q), S,(q) exchange their symmetries as q varies from 0 to *.
211
4. Phase diagram of first magnetic phases (T ,< TN) Depending on the (r, r’) parameter set, the exchange bands E,(q) can look very different from fig. 4. A selection of possible cases is shown in figs. 6a-d. In each case the q-vector of the absolute maximum and the corresponding normal coordinate characterize the FMP below TN. For example, fig. 6a shows a situation with q,,, = l/2 and S,(q,) = SF. This phase corresponds directly to the antiferromagnetic AFl phase observed at low temperatures. Another case is shown in fig. 6b where one band, a (SF, $)-mixture, still has a relative maximum at an incommensurate q,,,, but the absolute maximum has shifted to q,,, = l/2 with a different symmetry S,” corresponding to a different antiferromagnetic phase AF2. For parameters with small competition ratio r = - Z,/Z,, one obtains an absolute maximum at q = 0 with S: symmetry, i.e. a ferromagnetic (FM) phase. In fig. 6d finally, the special case for isotropic n.n. coupling IO = Z, is shown which again leads to an AFl phase for the competition ratio r chosen. In this way, identifying the absolute maximum, its q,-wave number and symmetry of the normal coordinate, a complete phase diagram of FMP’s in the (r, r’)-parameter plane was obtained. The IC-region of the phase diagram is defined by the requirement that the absolute value of the argument in (3.8a) is less than one. This leads to the condition (r+2)+b_(r)
= (l/a)[l l+r --i 4r2
* (1+ a(1 + r))]“‘, 1 l+r
(4.1)
i’
necessary for the existence of a relative maximum E,(q,). In addition, if E,(q,) = T,/CZ, (3.8b) is larger than any E,(O), E,(i) (table 2) one has an absolute maximum and the parameters (r, r’) lie in the IC-region. The resulting phase diagram is shown in fig. 7. To each of the sectors in this diagram the typical exchange bands are shown in figs. 4, 6. One can see that a large anisotropy ratio r’ = IO/Z, > 4 is necessary for the existence of an IC-phase. In the isotropic case r’ = 1 only FM or AFl commensurate phases would be possible.
212
of europiumtriarsenide
P. Thnlmeier / Magnetic phases
r’=6
16 */
ICI
6-
S;b):FM L
_
L. z-
L
O-
0
-2 r’=4
-L -
-0
E,lZ a
01
02
03
04
-L
05
0
01
02
9
03
04
05
9
Fig. 6. Exchange bands for various parameters (r, r’). Symmetry of normal coordinate at the absolute maximum and corresponding type of magnetic phase is indicated (compare with fig. 7). (d) Special isotropic case with I, = I, has degeneracies.
Starting from the IC-region a reduction of r’ leads to a continuous transition to the AFl region, i.e. the q,-value in fig. 4 moves to the zone boundary continuously. If r is reduced instead, one crosses into the AF2 region with a discontinuous change of the order parameter symmetry; it corresponds to fig. 6b where the absolute maximum has jumped to q = l/2 with a St normal coordinate although
6
r’
q=;
0
04
08
12
16 r
Fig. 7. Phase diagram for T 5 TN in the (r, r’)-parameter plane. AFl, AF2 are different antiferromagnetic C-phases and FM is the ferromagnetic phase. IC is the sector of the incommensurate antiferromagnetic phase. Crossing from the IC-region into AFl or AF2 leads to a continuous or a discontinuous change of order parameter symmetry, respectively.
Table 2 Exchange band energies E&( 4) at zone center (4 = 0) and zone boundary (4 = i) as function of r, r’ K
K(0)
E,(:)
1 2 3 4
II- r’l -(lr’l (-4r+ (-4r-
(4r+ 13+2r-r’l) (4r - 13+2r - r’l) II+ r’l -II+ r’l
13-2r+r’l) 13-2r+r’l)
P. Thalmeier / Magnetic phases
ofeuropium
triarsenide
213
was adopted, leading to 6 = 0.12. These values were also chosen in fig. 4. Using (3.8b) with TN = 11.3 K one obtains the exchange constants I, = 1.66 K, Z, = 0.29 K, Z, = -0.47
3L
I 1.0
I 1.4
I 1.8
I I 2.2 ,. 2.6
(4.2)
They are comparable to magnitude to those found in the Eu-monochalcogenides [9]. The out-of-plane n.n. coupling is rather large because in this model it is the only coupling which is ‘frustrated’ in the AFl C-phase and tends to establish the IC-structure. In an extension of this model one might consider additional ferromagnetic coupling to overnext (201)-planes with distances > 8 A (figs. 1, 2) and also an anisotropy of the antiferromagnetic n.n.n. coupling. Both modifications would lead to a smaller I,, for a given IC-wave number, so that I, in (4.2) has to be considered as an effective exchange coupling. For a more precise determination of exchange constants, one must await the investigation of spin wave excitations in EuAs,. On the other hand, the present model is in reasonable agreement with paramagnetic sc-measurements [lo]. In mean field theory the SCis given
J 3.0
Fig. 8. Enlarged IC-region of the phase diagram with contours for 6(r, r’) = f - q,,, according to (3.8a). Curves (a)-(i) respectively: S = 0.140, 0.135, 0.130, 0.125, 0.120, 0.115, 0.110, 0.10, 0.08.
there still remains a relative maximum at an IC wave vector. The variation of 6(r, r’) = + - q, in the IC-region is shown in fig. 8 as a contour plot and in fig. 9 as function of r for constant anisotropy. It shows that 6 5 0.145 which is close to the extrapolated experimental value 8, = S(T,) = 0.15. However, this is not sufficient to determine the ratios r, r’ unambiguously. One would prefer the anisotropy r’ as small as possible but compatible with 6 = S,. As a compromise (r, r’) = (1.6, 5.7)
0.16
K.
by X0’ = C/(T-t To), T,=$Cx&(O). I(
(4.3)
Using table 2, one obtains To= - 2CZ,. Furthermore, TN = CZ,F(r,r')according to (3.8b), there-
C’
0
1.4
1.6
1.8
20
2.2
21
r
2.6
28
30
Fig. 9. Variation of 6(r) for constant r’. This corresponds to a continuous transition from IC to AFl in fig. 7 with the absolute maximum of EX moving to the zone boundary (see fig. 4).
Fig. 10. Inverse paramagnetic susceptibility of EuAsr. Data are taken from Bauhofer et al. [lo]. TN = 11.3 K, TO = 5.4 K, x0 = 1 Sl umt.
P. Thalmeier / Magnetic phases
214
momentum transfer, tor and the structure
fore TN/TO=-&F(r,
r’).
(4.4)
This relation connects (TN/TO) which is a purely thermodynamic ratio to a r.h.s. which is in principle independently determined by the IC-modulation wave number 6. With TN = 11.3 K and TO= 5.4 K (see fig. 10) one has TN/TO = 2.1. Using (r, r’) = (1.6, 5.7) the r.h.s. of (3.4) is equal to 2.3 which is in reasonable agreement with TN/TO. Therefore, the exchange model used is compatible with both the magnitude of 6 and the thermodynamic ratio TN/TO.
5. Incommensurate modulation scattering cross section
wave and neutron
It is interesting to investigate the spin structure in real space u,(r) = (S:(r)) which corresponds to the IC order parameter S,(q,) for a given set (r, r’). According to (3.3)-(3.5) it is given by u,(r)
=A{cos
$3~~ - isin +4$}
T(Z) = - sin(2aqz p(z)
= sin(2Tqz
I f(Q)
I2
xis;Q-4,-K)cS(Q-,,-K)), (5.3) f(Q)
= ~eiQ”~.sa, a
s, = 0,/A.
Here K = ha* + kb* + Ic* is a reciprocal lattice vector and t, (a = 1 . . .4) are position vectors of unit cell atoms with the inversion center as origin. Eq. (5.3) shows that IC-magnetic satellite reflexes occur at K+ q,,, and have an intensity proportional to the square of the structure factor f(K + qm). It depends on the unit cell spin configuration, explicitly one has for q, = qmc*: f(K+q,)=e”*
[
Z,cos:(h+k) -iZ,
sint(h
+ k)]
(5.4)
za=(Si+s2+S~+s4);
Zb=(S1-S2-S~+SJ,
(5.1)
where cos + = u,(q,,,), sin + = 1u,(q,) 1 and w, = f( -1, 1, l,- 1); rv, = i(l, -1, 1, -1) are the first two row vectors in (3.3). In the present model the modulation vector q, corresponds nearly to the crossing point of the unhybridized exchange bands in fig. 5 where 1u1 1 = I u1 I = l/&f, i.e. + = a/4. In this approximation (5.1) can be written as =‘+A(-r(z),
S(Q) = iA2C
F(Q) the magnetic form facfunction S(Q) is given by
with
exp(iq;r)
(cr=1...4:basisatom),
u(r)
of europium triarsenide
r(z),
p(z),
-P(Z)),
(5.2)
- n/4), + ~/4),
A - (T,., - T)‘/2 is the amplitude of the IC modulation wave. Thus, the spin structure in the ICphase can be viewed as AF pairs (a = 1, 2) and ((Y = 3, 4) modulated sinusoidally along c* with a phase difference 71/2. The coherent neutron scattering cross-section due to the IC spin structure with wave vector qm is proportional to I F(Q) I 2S(Q) where Q is the
s, = e,/A, and f9* is an overall phase. Inserting (5.1) with @J= IT/~ into (5.4) one observes that because Z,(wt,) = Z,(rv,) = 0 only the first part of (5.1) contributes to the cross section. From Z,( wi) = 0, Z,( wi) = 4 one concludes that only Bragg points with odd (h + k) = 2m + 1 have IC-satellites. Therefore experimentally [0,0, 4 + 61 reflexes cannot be observed but [i, 0, f + 61 has finite intensity. For temperatures below 10.26 K, q, = ic* and one has a magnetic supercell with doubling in the c-direction which leads to the additional condition I = 2n + 1 for the commensurate magnetic reflexes.
6. Conclusion The existence of an incommensurate antiferromagnetic phase in EuAs, could be explained within an anisotropic exchange-model including next and next nearest neighbors. A considerable anisotropy
P. Thnlmeier / Magnetic phases
of the ferromagnetic n.n. interaction must be assumed. The IC modulation is due to a hybridization of exchange bands with different symmetry. The magnitude of the modulation wave vector agrees with the experimental value and the implied exchange parameters are compatible with susceptibility measurements. The phase diagram in the exchange parameter plane shows that the IC-phase is lying next to the experimentally observed low temperature antiferromagnetic C-phase. It is possible to describe the lock-in transition to the C-phase within the context of a Landau theory [ll]. It remains to explain the H-T phase diagram which shows many interesting IC- and C-phases for nonzero fields [12] within the present model.
Acknowledgements The author would like to thank T. Chattopadhyay and W. Bauhofer for helpful discussions. Financial support by the Deutsche Forschungsgemeinschaft is also acknowledged.
Appendix The exchange tensor Z,,(q) (a, j3 = 1.. .4 = basis atom) of (2.2) is given in terms of the exchange constants Z,, Z,, Z, by the following matrix:
of europiumtriarsenide
With
+,
215
+2 = q.
= q . a,
A, = 21, cos +3,
A, = IO + I, ele3 + Z, em’+‘,
B, = Z,(l + e-‘+I), B, = Z, e-‘+l(I
b, $J~= q ac one has
+ e&)
B, = Z,(l + ePiG2) + I, ei93, + Z~e-i(+1-&).
(A-2)
References [l] J. Rossat-Mignod, P. Burlet, J. Quezel, J.M. Effantin, D. DelacBte, H. Bartholin, 0. Vogt and D. Ravot, J. Magn. Magn. Mat. 31-34 (1983) 398. [2] M. Kuznietz. P. Burlet, J. Rossat-Mignod and 0. Vogt, to be published. [3] B. Coqblin, The Electronic Structure of Rare Earth Metals and Alloys: the Magnetic Heavy Rare-Earths (Academic Press, London, 1977). [4] P. Bak and J. v. Boehm, Phys. Rev. B 21 (1980) 5297. [5] W. Selke and P.M. Duxbury, Z. Phys. B 57 (1984) 49. [6] T. Chattopadhyay, P.J. Brown, P. Thalmeier and H.G. v. Schnering, to be published. [7] W. Bauhofer, M. Wittman and H.G. v. Schnering, J. Phys. Chem. Solids 42 (1981) 687. [8] T. Cbattopadhyay, H.G. v. Schnering and P.J. Brown, J. Magn. Magn. Mat. 28 (1982) 247. [9] V.C. Lee and L. Lin, Phys. Rev. B 30 (1984) 2026. [lo] W. Bauhofer, E. Gmelin, M. Mollendorf, R. Nesper and H.G. v. Schnering, J. Phys. C 18 (1985) 3017. [ll] P. Thalmeier, to be published. [12] W. Bauhofer et al., to be published.