The local exchange model of itinerant-electron ferromagnetism: Energy bands in nickel and iron

337 THE L O C A L E X C H A N G E M O D E L OF I T I N E R A N T - E L E C T R O N F E R R O M A G N E T I S M : ENERGY BANDS IN N I C K E L AND IRON* J. C A L L A W A Y and C.S. W A N G Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA

The local e x c h a n g e a p p r o x i m a t i o n is c o n s i d e r e d as a m o d e l for i t i n e r a n t - e l e c t r o n f e r r o m a g n e t i s m . F o r m a l p r o p e r t i e s are r e v i e w e d . The c u r r e n t s t a t u s of c a l c u l a t i o n s b a s e d on this m o d e l is s u m m a r i z e d , w i t h e m p h a s i s on r e c e n t s t u d i e s of e n e r g y b a n d s in iron and nickel.

I. Introduction

The essential idea of the local exchange approximation is the replacement of the non-local exchange operator of the H a r t r e e - F o c k equations by a local potential. In this form, the model was originated by Slater [1,2] 25 years ago. However, it is only within the past ten years that quantitative, self-consistent calculations for solids have become possible. A more fundamental basis for the model is provided by the H o h e n b e r g - K o h n theorem [3,4], and it is now apparent that some aspects of electron correlations can also be included. One considers a single-electron hamiltonian which has the form

H_P2 2m

~

Ze2

e2

f o(r') d~r, -

+ Vf(r)tr • n + H ........

V .... (r)

(1)

The first two terms represent the kinetic energy of an electron and its potential energy in the field of the nuclei; the third term is the average Coulomb repulsion of the electrons. Terms four and five represent exchange and correlation, which are described by local potentials (whence the name of the approximation), but these potentials are spin dependent. Thus, Slater proposed that the exchange potential for an electron of spin tr should be proportional to the cube root of the charge density of electrons of that spin. We write

V,(r) =-3e 2a(3p~/4~r)13,

(2)

in which a is a numerical factor. In the original work of Slater a = 1; while Kohn and Sham [5] and Gaspar [6] obtained a =32 (KSG potential). The essence of the currently popular X a method is to regard ~ as a variable parameter * S u p p o r t e d in part by the U S N a t i o n a l S c i e n c e F o u n d a t i o n .

Physica 91B (1976) 337-343 © North-Holland

[7]. We shall not discuss the rather extensive literature which has developed around the choice of a. It will be our contention in what follows that the limitations of eq. (2) are becoming apparent, at least in the ferromagnetism problem; recent work [8, 9] uses other functions of the charge density in an attempt to include additional correlation effects. We therefore define a spin averaged exchange-correlation potential I

V .... = ~ [Vxc., + Vx¢,~],

(3a)

(where 1" and { refer to spin directions with respect to some axis) and a difference potential 1 V f = 5 [Vxc, T -

Vx¢,,, ] ;

(3b)

however, a generalization to allow for arbitrary directions of spin is desirable, hence we introduce a t r • fi into (1). It is assumed here that V .... is an ordinary function of the charge and spin density. Finally H ...... is the spin orbit coupling, H~oc-

....

h

4m2c2~ - ( V V x p ) ,

(4)

in which V here includes the nuclear and electronic Coulomb terms in (1). Although the hamiltonian (1) refers explicitly to a single electron, it would be an oversimplification to suppose that (1) neglects electron interactions. In fact, these interactions are included through a self-consistent field (terms three through six). Bands calculated neglecting electron interactions entirely would be ridiculous. What is neglected is a detailed dynamic description of such interactions beyond the level of a self-consistent field. There are many formal problems associated with a procedure founded on the H o h e n b e r g Kohn theorem. In particular, there is no neces-

338 sarily s i m p l e r e l a t i o n b e t w e e n the e i g e n f u n c lions of (1) and a m a n y p a r t i c l e w a v e f u n c t i o n . e v e n t h o u g h the sum of the s q u a r e s of these f u n c t i o n s gives the c h a r g e d e n s i t y . T h e e n e r g i e s w h i c h are the e i g e n v a l u e s of (11 are. h o w e v e r . the d e r i v a t i v e s of the e n e r g y with r e s p e c t to the o c c u p a t i o n of the single p a l t i c l e s t a t e s [10l. This is w h a t is r e q u i r e d to d e t e r m i n e the F e r m i surface. T h e h a m i l t o n i a n (1) p e r m i t s the d e s c r i p t i o n of m a n y p h e n o m e n a c u s t o m a r i l y t r e a t e d in m o r e complex many body theories. The hamfltonian is s o l u b l e at T = 0 K, a l t h o u g h o n l y n u m e r i c a l l y . It is free of the t w o m o s t significant c a t a s t r o p h e s of H a r t r e e - F o c k t h e o r y : the i n c o r r e c t beh a v i o r at large i n t e r a t o m i c s e p a r a t i o n s and v a n i s h i n g d e n s i t y of s t a t e s at the F e r m i e n e r g y . Numerical calculations have yielded self-cons i s t e n t f e r r o m a g n e t i c s o l u t i o n s for iron and nickel with r e a s o n a b l e but not e x a c t l y c o r r e c t v a l u e s of the m a g n e t i c m o m e n t . If a w e a k osc i l l a t o r y e x t e r n a l m a g n e t i c field is a p p l i e d , the t r a n s v e r s e m a g n e t i c s u s c e p t i b i l i t y m a t r i x can be c a l c u l a t e d [11]. It has the f o r m (we neglect s p i n - o r b i t c o u p l i n g in the f o l l o w i n g ) : ,~(p, ~o) = ) t , ~ [ l

15i

- a)¢ <'~'] '

in w h i c h )¢"~' is the non s e l f - c o n s i s t e n t s u s c e p tibility i n v o l v i n g a sum o v e r b a n d s t a t e s ( r e f e r to [9] for details), and A is a m a t r i x w h i c h d e s c r i b e s the e x c h a n g e i n t e r a c t i o n within this m o d e l . T h e e s s e n t i a l f a c t is that a is c o m p l e t e l y d e t e r m i n e d f r o m the a s s u m e d h a m i l t o n i a n , and does not contain additional adjustable p a r a m e t e r s . Spin w a v e s tire o b t a i n e d f r o m the equation d e t iI

xx'"'l = 0.

~,~

it can be s h o w n that the spin w a v e s p e c t r u m c a l c u l a t e d f r o m (6) is g a p l e s s , w h i c h m e a n s thai if the spin w a v e e n e r g y is d e n o t e d b y w ( p ) , then o J ( 0 ) = 0 . F o r small p we h a v e ~o = Dp2: and r e c e n t l y E d w a r d s and R a h m a n [12], and we o u r s e l v e s [13] h a v e i n d e p e n d e n t l y o b t a i n e d e x p l i c i t f o r m u l a e f o r D. T h e g e n e r a l e x p r e s s i o n for D is ' D

#1 3

_(n.

~ rl. )

11 i

N,,(k)

_~l3 N ,,,'5] k

Ndk)

~O,,~.~k

<,,klS,

t#kb

17i

in w h i c h n , , . ~ is the n u m b e r of m a j o r i t , t m i n o r i t y ) spin e l e c t r o n s p e r a t o m : Nt(k) is the o c c u p a t i o n n u m b e r of a state in b a n d I with w a v e v e c t o r k Ispin is i n c l u d e d in the band index), S is the s p i n - r a i s i n g o p e r a t o r , p the nlomentunl

operator,

and

+o,,~.,s'

/:',,(kt

l~s(k)-

This result can also be d e d u c e d f r o m a g e n e r a l f o r m u l a of E d w a r d s and F i s h e r l l 4 i . E v i d e n t l y eq. (I) e n a b l e s a q u a l i t a t i v e l y satisf a c t o r y d e s c r i p t i o n of the g r o u n d s t a l e lind the l o w - l y i n g e x c i t e d states of s o m e i m p o r t a n t i t i n e r a n t - e l e c t r o n f e r r o m a g n e t s to be given. I! r e m a i n s to be s e e n how a d e q u a t e are the r e s u l t s m a q u a n t i t a t i v e s e n s e . T h e s i t u a t i o n lit finite t e m p e r a t u r e s , and p a r t i c u l a r l y in the critical region is, h o w e v e r , quite o b s c u r e . W e shall n o w turn to the main o b j e c t i v e of this p a p e r w h i c h is a d e t a i l e d c o n f r o n t a t i o n of the r e s u l t s of t i t with e x p e r i m e n t a l e v i d e n c e concerning band structure and magnetic p r o p e r t i e s of iron and nickel.

2. B a n d

calculations:

nickel

T h e r e h a v e b e e n s e v e r a l c a l c u l a t i o n s of the b a n d s t r u c t u r e s of iron 115-211 and nickel [17, 22--27] using e s s e n t i a l l y this f r a m e w o r k but different computational procedures (APW. K K R . L C A O . etc.). S e m i - e m p i r i c a l c a l c u l a t i o n s are not d i s c u s s e d here. A c l o s e l y r e l a t e d calc u l a t i o n c o n c e r n i n g c o p p e r s h o u l d also be cited [281. T h e L o u i s i a n a S t a t e U n i v e r s i t y g r o u p has r e c e n t l y r e p e a t e d the c a l c u l a t i o n s [20,241 with i m p r o v e d n u m e r i c a l t e c h n i q u e s , and has also c o m p l e t e d s e l f - c o n s i s t e n t c a l c u l a t i o n s using the e x c h a n g e - c o r r e l a t i o n p o t e n t i a l of von Barth and H e d i n [8] ( a b b r e v i a t e d v B H ) . T h e s e c a l c u l a t i o n s will be r e p o r t e d in m o r e detail e l s e w h e r e [29]. W e will begin with nickel, since the nickel F e r m i s u r f a c e is m u c h s i m p l e r than that of iron and it is c o r r e s p o n d i n g l y e a s i e r to c o m p a r e t h e o r y and e x p e r i m e n t . T h e g e n e r a l f e a t u r e s of the b a n d :structure as c a l c u l a t e d by the A P W m e t h o d [231 and the I . C A ( ) m e t h o d [24.251 are in s u b s t a n t i a l a g r e e m e n t : we note that C o n n o l l y ' s c a l c u l a t i o n was the first to i n d i c a t e that ¢~ = 2 was a g o o d c h o i c e for s e l f - c o n s i s t e n t calc u l a t i o n s in t r a n s i t i o n m e t a l s . S o m e e s s e n t i a l r e s u l t s are given in table 1. All c a l c u l a t i o n s w e r e m a d e for the o b s e r v e d lattice c o n s t a n t ext r a p o l a t e d to T = 0 . and e m p l o y e d a b a s i s of

339 thirteen s, ten p, five d and one f independent gaussian orbitals. Table I Band calculation results for nickel. Quantities given are the magneto n n umb er (ix*), the e x c h a n g e splitting of states at the top of the d band (A), the density of states at the Fermi energy (in states/(atom-Ry)), and the difference b e t w e e n majority and minority spin densities at the nuclear site e x p r e s s e d as an effective hyperfine field in kilogauss. The calculated values refer to the K o h n - S h a m - G a s p a r potential (line c~ = 2•3) and to the von B a r t h - H e d i n potential. The e x p e r i m e n t a l references are: a, ref. 29, including a g-factor of 2.18; b, refs. 24, 30; c, ref. 31; d, ref. 32

ct = 2/3 vBH Exp.

Ix*

A (eV)

N(E~)

[pr(O) ,o ~(O)](kG)

0.65 0.58 0.56"

0.88 0.63 0.5-0.3 h

22.92 25.45 40.41 ~

69.7 -57.9 7 6 -+ I d

It will be observed that the KSG potential leads to a significant overestimate of the magneton number and the exchange splitting (A). The calculated A values refer to the states X5 and W'~ at the top of the d band; the "experimental" values are crude averages. The discrepancy is substantially reduced when the vBH potential is employed, although the splitting may still be too large. A more precise experimental estimate of this quantity is urgently required. The optical transition between the filled and empty portions of the d band becomes weakly allowed when the hybridization of 1' and $ spin states due to spin-orbit coupling is considered, and there is an enormous joint density of states. Careful measurements of ordinary and magneto-optical properties around 0.5 eV might settle this question. The difference between theoretical and experimental (from the specific heat) results for the density of states at the Fermi energy probably tells us little more than that there is a large phonon-induced renormalization. The calculated effective hyperfine field includes the core polarization contribution, and allows for the modification of the core wave functions in the solid state potential, but does not include any relativistic effects. The agreement between the results from the KSG potential and experiment is quite satisfactory. It is not improved when the vBH potential is employed; however,

the omission of relativistic effects makes it impossible to draw definite conclusions. A cross section of the nickel Fermi surface in the (100) plane is shown in fig. 1. The solid lines are the results from the vBH potential and the dashed lines refer to the KSG potential. The open circles, triangles and squares come from the unpublished de Haas-van Alphen measurements of Stark [26]; the dotted line refers to measurements of Tsui [34]. The surfaces are as follows: (a) is the X~ d, light hole pocket. Its size is essentially unchanged between the two potentials, and it is significantly smaller than the experimentally observed hole pocket (dotted line). The contour (b) represents the d, heavy hole pocket associated with X2. This pocket has not been observed experimentally, and probably does not exist. However, although the calculated size of the pocket is decreased when the vBH potential is used, it does not disappear. A simple reduction of the exchange splitting is not enough to get rid of the extra pocket. The contours (c), (d), and (e) represent the major d ] hole surface, the majority spin s-p electron surface, and the minority spin s-p electron surface, respectively. Only in case (d) is the agreement between theory and experiment substantially improved by use of the F

×

W

Fig. I. Cross sections of the Fermi surface of nickel in a (100) plane. The solid lines are results obtained from the vBH potential, the dashed lines pertain to the KSG potential. The open circles, triangles, squares, and the dotted lines are e xpe ri me nt a l results. See text for further discussion.

340 vBH potential. The implication here is that it is quite difficult to obtain reliable information about the exchange splitting from the study of the Fermi surface. The general success of the calculation in describing the Fermi surface should not obscure the most significant error: the presence of the X2~ pocket, (b). In addition, the X~ pocket is too small. Note that Xzl and X~; have different symmetry: X, is % while Xs is t~. It would appear that we need additional non-spherical terms in the effective potential which will lower states of % symmetry and raise those of 1~ symmetry. Simply decreasing the exchange splitting will not be effective. A comparison of theoretical and experimental results for the anisotropy of the magnetic form factor (which we will not discuss in detail here) turns out to be consistent with this suggestion: we need to occupy more e~ states. The energy change required is not large, since the calculated X3, is only 0.008Ry above E~: with the vBH potential (0.01 Ry for the KSG potential). We argue that ;i significant defect of present density functional methods is that the effective potential is not sufficiently anisotropic. Possibly we need to consider gradient corrections to the exchangecorrelation potential. We have also used the wave functions and energies from the band calculation to compute the spin wave stiffness constant D [eq. (7)]. The formula is made suitable for computation by using the " f " sum rule, leading to the expression

D

N,,(k )V2E,,(k )

-

61n,

n ~)

N,,(k)-. N. , ( .k ) t.( n k. ] S. ~ p ilk) ~

+2

CO rtk,lk

t~I

k

(8) O) nk.l k

#~1

The laplacian needs to be evaluated only for occupied states in unfilled bands. Eq. (8) has been evaluated so far only for the KSG potential [13]. Wave functions and energies were considered at 505 points in 4'~ of the Brillouin zone. The derivatives of the energy bands were obtained by quadratic interpolation. Linear analytic tetrahedron methods were used for the k space integration. All the necessary

momentum matrix elements were calculated from the wave functions. Our result is D - 0 . 1 0 4 (atomic unitsl, which is in reasonable agreement with the experimental value of 0.113 [35]. 3. Iron

Some of the earlier attempts at self-consistent band calculations for iron experienced difficulty in obtaining reasonable values for the magnetic moment [16, 17]. Another apparently suffered from technical difficulties in evaluation of integrals and from incomplete convergence of O P W expansions {18]. In general, iron has been a more controversial subject for band theory than nickel. Because the Fermi level falls in the middle (roughly) of the d-band complex, the Fermi surface is more complicated [36, 37[, and it is possible that not all the pieces have been identified. We have made band calculations for iron which are similar to those for nickel. The essential results are given in table II. In this case there are three self-consistent calculations, including a = 0.64, the value used in [18] to obtain an improved fit to the Fermi surface, as well as the standard KSG and vBH potentials. Note that the percentage reduction in the magnetic moment and the exchange splitting when going from the KSG potential to the vBH potential is much smaller in iron than in nickel. Unfortunately we do not have a reasonable experimental estimate of the exchange splitting for iron. Some years ago, Wohlfarth 131] suggested 1.4-+0.2eV: however, most of his estimation procedures involve a grossly oversimplified model of the band structure or the density of states. One cannot apply the simple model of band ferromagnetism to a situation in which the exchange splitting is larger than the width of Table II B a n d c a l c u l a t i o n r e s u l t s for i r o n l ' h e e x p e r i m e n t a l ref e r e n c e s a r c : a, rcf. 29. i n c l u d i n g a e - f a c t o r of 2.1"19: b, ref. 31: c. rcf. 38

/J~

(cV)

,\~( bJt )

, :: 2/3 ,~ I).64

2.3o

2.6~

15.37

2.25

2.56

14.411

vBH }-ixp.

2.25 2.12'

2.2 I

.....................................................................

15.97

2".~v"

Ip-(0) t~. (0)](kG 1 343, 128. 237. 339.tl ~ 0. :~"

341

characteristic structure in density of states. Optical data may offer some information with regard to the actual exchange splitting, but experiments are sparse. There is a peak in the optical conductivity near 2.5 eV which may be associated with the transition across the exchange split bands; but it is not possible to be definite. If one assumes that the exchange splitting varies linearly with a, the exchange splitting obtained with the v B H potential corresponds to a reduction of a to about 0.55. It is interesting that in early work, Wakoh and Yamashita [15] found that c~ should be about 0.5 to obtain a good value of the magneton number. Of course, the d bands are narrower with the v B H potential than will be obtained with a = 0.5 (or even _~). There has been considerable interest in the density of states for iron. Our most recent results, based on the v B H potential, are shown in fig. 2. The high peak just below the Fermi level has been clearly observed by P e s s a et al. 640

I

i

BOTH

I

I

I

I

SPINS

I

I

I

EF

o 560 p-

480

t ,i

400

I

J

~2o

J

~ 24o ~

, i!

160

"/

"/L_~~

-

z

oo

.-4 -I0

-09

-08

I -07

I "06

I -05

I -04

I -05

l -02

-01

ENERGY (RY)

Fig. 2. Density of states for iron, including both spins, based on the v B H potential. The vertical line indicates the Fermi energy.

[39]. The v B H potential places this peak 0.8 eV below the Fermi energy while P e s s a et al. conclude that the displacement is 0.58eV. The second m a x i m u m is located 2 . 4 e V below EF which agrees rather well with their findings. In addition to the size of the magnetic moment, one is also concerned with its spatial distribution. It is important that the local exchange approximation should be able to describe this reasonably well since the explicit

H u n d ' s rule coupling of the H a r t r e e - F o c k equations for example is mutilated in the local exchange approximation. Results for the magnetic form factor of iron are given in table III. These quantities tabulated are the Fourier coefficients of the net spin distribution, normalized so that f(0) = 1.0. Corrections due to spin-orbit coupling (specifically, an orbital contribution) have been neglected. The general agreement with experiment is quite respectable, but the v B H results are perhaps slightly superior to those from the K S G potentials. On the other hand, the K S G potential gives a better result for the hyperfine field. Table I!I Normalized spin form factor for iron K

Exp.

c~ = 2/3

000 110 200 211 220 310 222 321 400 330 411 420 332 422 431 510 521 440 433 530 442 600 532 611 620 541 622

1.000 0.6347 0.4077 0.2520 0.1751 0.1383 0.0620 0.0461 0.0711 0.0132 0.0376 +0.0201 -0.0165 -0.0098 -0.0132 +0.0179 +0.0008 -0.0156 -0.0275 -0.0115 -0.0197 +0.0119 -0.0182 +0.0051 -0.0017 -0.0189 -0.0098

1.000 0.6583 0.4232 0.2648 0.1779 0.1396 0.0707 0.0532 0.0653 0.0185 0.0375 +0.0191 -0.0085 -0.0064 -0.0109 +0.0142 -0.0048 -0.0185 -0.0269 -0.0139 -0.0252 +0.0051 -0.0208 0.0011 -0.0056 0.0209 0.0122

a = 0.64 1.000 0.6573 0.4230 0.2629 0.1762 0.1400 0.0682 0.0520 0.0674 0.0175 0.0384 +0.0193 -0.0102 -0.0071 -0.0116 +0.0160 -0.0043 -0.0191 -0.0283 0.0138 -0.0262 +0.0071 -0.0213 +0.0004 -0.0045 -0.0213 -0.0118

vBH 1.000 0.6401 0.4136 0.2586 0.1735 0.1365 0.0679 0.0512 0.0645 0.0174 0.0367 +0.0186 -0.0092 0.0067 0.0110 +0.0147 -0.0045 0.0183 -0.0269 -0.0135 -0.0250 +0.0059 -0.0204 -0.0003 0.0049 -0.0205 -0.0116

aref. 40.

Cross sections of the Fermi surface in a (1,0, 0) plane according to the v B H potential are shown in fig. 3. Additional results will be published elsewhere [29]. The areas of these cross sections are listed in table IV, where they are c o m p a r e d with experimental results. The larger pieces (surfaces I, V, and VI) have calculated

342

N

I: i !;

E

H

Fig. 3. Cross sections of the Fermi surface of iron in a (I(X~) plane, based on the vBH potential. The solid lines are majority spin portions: the dashed lines pertain to minorit', spins. Areas of portions are given in table IV.

nickel and iron. In addition to those features discussed in detail here, reasonable success has been attained in describing the charge and momentum distribution functions, the latter through studies of the Complon profile 141,42]. We do not see any evidence for exceptionally large correlation effects. Inclusion of some electron correlation through use of the vBH potential produces significant improvements in the calculated magnetic moment and exchange splittings in comparison with the KSG potential. Additional experimental data are urgentl,~ required to ewduate the results of the band calculations. However, there are already some discrepancies involving small pieces of the

Table IX,' Cross sections of the iron Fermi surface in the 100 plane. Areas are given in tel m-, of de H a a s - v a n Alphen frequencies 51)5 × area in unils of (2rr/a)"

Surface

l)escription

1

l.arge I' centered maioritv spin electrons Majority spin hole surf:ace (armsl Intermediate hole pockel at H (majority spinl Small hole pocket at H (majorit~ spin) Minority spin H cenlered hole surface Minority spin 1" c e n t e r e d electrons Electron ball along A (minority spin) Ellipsoidal hole surface around N (minori'~ spin~

II 111 IV V

VI VII VIII

areas in reasonable agreement with experiment, but the smaller pieces (I11, IV), are too small. Surfaces II and VIII have not yet been observed. The situation is evidently not as satisfactory as in the case of nickel. It does not appear that the agreement with experiment would be improved much further simply by a further reduction of the exchange splitting. 4. C o n c l u s i o n s

The local exchange approximation has proved capable of accounting quantitatively for a wide variety of T = 0 K properties of ferromagnetic

in units

of

t0~G

,\rea IMG) (talc.)

(I MG):

area ( M ( ; I

Area tMG) ref. ~;~

~,9~

Area iM(;~ lef. ~," 4~,q

2711 t.8

2~ 8

2().~

~I

21.~}

I~,t!

2(52

1'~8

~9

-I

4.~

;s~

:~I

Fermi surface which indicate that, as one should expect, completely successful results at T - 0 K have not been obtained. Perhaps gradient terms must be inserted into the effective potential, as was proposed some years ago in another connexion by Herman et al. [43]. Additional nagging problems concerning the spin polarization of photoemitted electrons [44, 45] exist. Complicated calculations involving details of the photoemission process will be required to determine whether or not there is a discrepancy between theory and experiment. Beyond this there remain large unanswered questions, per-haps the most important of which is: how doe,

343

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