a study of structurally disordered single-element magnetic materials beyond the homogeneous...

J. SLECHTA: Study of Structurdly Disordered Magnetic Materials 5 95

phys. stat. sol. (b) 67, 595 (1975)

Subject, classification: 2; 18.2; 18.3; 18.4

School of Mathematics and Physics, University of East Anglia, Norwich

A Study of Structurally Disordered Single-Element Magnetic Materials beyond the Homogeneous

Molecular Field Approximation

BY J. SLECHTA

A usual assumption in the theory of amorphous magnetic materials is CS,) = S. It is shown that in amorphous materials this IS a very crude approximation. The magnetic properties of small random clusters of localised spins were studied beyond the above simplification. It is found that in this case the fluctuations of the exchange interaction around its crystalline mean value cause an increase of the Curie temperature. In the light of this conclusion most of the experimental results in this direction should be reinterpreted. It is argued that the magnetic properties inherrntly depend on higher spatial correlations than pair-pair ones.

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1. Introduction

Recently there has been considerable interest in the theory of random magnetic systems which niay be divided into two groups: 1. substitutional alloys and 2. topologically disordered single element materials (amorphous). I n this paper we shall concentrate on the latter case and study the properties of arnor- phous materials in which localised spins interact with each other by a Heisen- berg interaction [ 1 to 71.

Usually the averaging procedure in the above mentioned papers is valid for linear systems. However cooperative phenomena are nonlinear and so its appli- cations must be taken with caution. Moreover the assumption (S , ) = S may he a poor one in the real amorphous systems and may cause a n important mis- interpretation of the relation between the structure of those materials and their magnetic properties.

I n amorphous materials a profound change of magnetic properties, in comparison with those for the related crystalline ones, may occur due to the three separate phenomena : 1. change of the average nuniber of nearest neighbours, 2. change of the average exchange interaction, 3. local fluctuations of the exchange interaction around their niem value which can be equal to the related 39 phy<ic a (11) 67/2

596 J. ~ E C H T A

crystalline one. The change of the average number of nearest neighbours, in comparison with a related crystal, is very often connected with the change of the average distance between pairs of spins and this is connected with a change of the average exchange interaction through its dependence on the pair-pair distance. So in the real amorphous magnetic materials the change of the Curie (NBel) temperature is caused by all three factors. To understand the experimental results in more details it is necessary to separate the effect of those three phenomena on the magnetic phenomena.

As the result of the first phenomenon is most simple to evaluate, and is inten- sively studied in the case of substitutional alloys, we shall concentrate on the last two, For this reason we studied two separate cases.

1. A random distribution of positions of spins around a fixed mean value, with a simple model of the exchange interaction (see Section 3.1).

2. A random distribution of exchange interactions around a fixed mean value (see Section 3.2).

We found that: a) the ferromagnetic state does not necessarily collapse with the appearance of exchange integrals of the opposite sign, in contradiction to e.g. [ 2 , 4, 131. b) The Curie teiiiperature increases and not decreases, as widely believed, with increasing fluctuations of the exchange integrals. This fact has an important consequence for the interpretation of experimental data (see Sec- tion 4). c) Magnetic properties of amorphous materials inherently depend on higher spatial correlations of spins and can possibly be directly measured.

2. Mathematical Model

For our porpose we modify the niatheinatical fornialism presented in the book

We shall describe our magnetic system by the Heisenberg Hamiltonian by Tyablikov ([B]).

A

ix = -p 2 (H> Sf) - a 2 I(fl> fi) (Sfl . Sf2) > (1)

where fl, fi, f3 are indexes of the positions of the spins in the random lattice, Sf the vector of the spin a t the place f, H i s the external magnetic field, ,LL is the magnetic moment connected with the spin, I(fi, fi) is the exchange interaction between spins a t places fl and fi.

f flf,

The basic idea of the procedure adopted here is as follows: First it is assumed that S, are classical vectors. Then we can write them as

Sf = yrSf where yf are unit vectors. The Hamiltonian of the system is now

We may now find the directions of the vectors y f by minimizing the energy E ~ .

The equations for them are

-2 I(f - fl) SLY: - pfHa = 2AfY.; > (3)

( ~ 7 ) ~ = 1 for all f , (4) f ,

a

where a denotes the coordinates indexes of vectors yf, 2, are Lagrange niulti- pliers, Xf is the magnitude of spin a t the place f.

Study of Structurally Disordered Single-Element Magnetic Materials 597

We call the array of yf vectors, obtained by minimizing ( 2 ) in the case of H -+ 0, the magnetic ground state of the systems of interacting spins. It char- acterises the magnetisation of the system at T = 0. To include the teniper- ature dependence of the total magnetisation we have to consider spins as dynani- ical variables. The vectors yI are then taken as directions of quantisat on of z-component of the spin operator X,Z. If we define operators of elementary excitations as (see [8])

S; = K b , , 8, = J@ b; , S, - Si' = nj = b;bf

then the Haniiltonian (l) , including deliations from the magnetic ground state, may be written in the form ,.

= ~o + Z: Bfnj - f L' IU'i, fi) (yilj yi,) ni,ni, + H m a g 9 (5) f Id,

where Hmag is the Haniiltonian of the magnon excitations of the system and will be discussed in the end of this paragraph.

Following [8] the generalised molecular field equations for the case Sf = f for all f are

of = tanh (?),

Here 19 = k,T and of is the average magnetisation at the place f , projected on the oriented directions defined by yf-vectors. k, is the Boltzniann constant. I n the case of 1' = 0 the solution of (6) is of = 1 for all f. Tyablikov discussed equations similar to (6) for crystalline materials with one type of ferromagnetic exchange interaction only, in which case the system of the coupled nonlinear algebraic equations (6) degenerates into the Weiss molecular field equation. I n the case of crystalline antiferroniagnetics with two different interactions (6) becomes a coupled system of two equations and is equivalent t o the usual two lattice molecular field theory. Our system of equations is similar to that used previously e.g. in [4] and [9]. Hou7evc.r in their treatment there are two large differences which will prove important : a) we shall discuss in detail the magnetic ground state, b) we shall not assume that a, = o for all f.

terise the properties of the systems under study: At this place me shall define additional quantities by which we shall charac-

(i) Average effective exchange interaction

where N , is the number of exchange interactions. (ii) The local teinperature of magnetic ordering a t site f

yrn1 - I - 2 Jff1(YjYf,) .

f, near ncighb. (9)

598 J . SLECRTA

(iii) The average temperature of magnetic ordering

2 TY f T E =---

N ’ where N is the number of spins.

(iv) The relative mean square deviation of the fluctuations of TF from T E

(11) -

T E f T x =

(v) The relative mean square deviation of the fluctuations of of from 0

z of (12)

f - a=--- N ’

(vi) The total niagnetisation M = 2 of sign ( y r . e ) ,

f where e is the unit vector in the direction of Z-axis.

(vii) The exact temperature of magnetic ordering T , which is obtained from the numerical solution of (6).

An important property of (6) is that, if the proper magnetic ground state has been found, it has a positive solution for all f . To show this we have to realize that af given by (7) include yf which are obtained from minimizing of .so. The minimum of .so requires that 2 I ( f - fi) (yryf,) are positive for all f which

is a sufficient condition for the positive solution of (6). This fact is of a crucial importance for the numerical solution of (6) as there exists a unically defined zero aproximation (otherwise it would be very difficult to solve it a t all).

The procedure described here physically means that the mutual orientations of S,Z are given by the minimisation of the energy for T = 0. These orientations are then maintained throughout 0 < T < T,, only the magnitudes of the time average projections of the spins on those directions change with temperature. In the crystalline ferromagnets they vary with temperature in synchronism, but in the amorphous case they vary in different ways from site to site. This is described in detail by the system (6). T , is the Curie temperature in the case of a ferromagnetic ground state, the Nee1 temperature for an antiferromagnetic ground state. However in the amorphous case the ground state may have a more complex structure than for either ferro- ot antiferroniagnetic limits (mixed ferromagnets or mixed antiferromagnets - those names introduced in

The magnetic ground state is obtained by solving (3). In the case of crystalline solids it was shown (see [ll, 121) that conditions (4) may be substituted by a weaker one of the form 2 2 ( ~ 7 ) ~ = N . In this case il, = ii for all f and

(3) becomes a standard eigenvalue problem. The ground state is then given by the eigenvectors which correspond to the lowest eigenvalue. However, it may be shown that for the amorphous systems the full formulation gives a different result than the simplified one and therefore the simplification is not allowed.

il

[101).

f a


It is not possible a t the present time to solve (3) and (6) for a very large system of spins. We carried the calculation for two-dimensional clusters of spins of the size nine and twenty five spins and three-dimensional clusters of the size 27 spins. We also investigated a selfconsistent solution of (6) for two-dimensional clusters of spins imbedded in an effective medium (on the line similar to MCPn approximation (see [13] and [14])) and the results will be published elsewhere [15]. From the convergence of the results for magnetic clusters in vacuum (see Section 3) and the selfconsistent calculation [15] follows that the main qualitative results presented in this paper do not depend on the size of the clusters involved in the calculation (see Section 3) and so they are representative also for the bulk.

For the sake of clarity and simplicity we shall demonstrate them on two- dimensional clusters of nine spins. In the end of Section 3 we shall give a sum- mary of the main results for large clusters.

Furthermore we shall limit ourselves to the nearest neighbours isotropic exchange interaction only. In this cade all directions y, lay on parallel lines. The energy of the system does not change when the direction of those lines is changed and so in the case of the two dimensional clusters we shall choose them perpendicular to the plane of the clusters.

The problem of the stability of the magnetic ground state obtained by the niinimization of (2) can be investigated by studying spin wave excitations described by the term Sfmag. This ternh can be written in the form

_ _ _ ~ where

Rf,i2 = - ifliA% J’(fi - fi) ( A x 9 >

flfA = - iSf?S;% I(f1 - I-2) (AZAI,) * For the definition of A, see [8]. It may be diagonalised by the Tyablikov cano- nical transformation (see [S]). The result of the transformation is

x m , , = A&, + 2 EuL7:Ev > V

where the niagnon energies E, are obtained by the solution of the system of the equations

E,pu,, = 2 s , g t q Y 4- z: RaB% >

-E& = 2 s33, f 2 R$Upv *

B B

B P

This system gives a spectrum of h’, symmetrical around the zero energy. The physical excitations are those with positive energy and so the ground state is stable. This is consistent with the fact that the ground state is obtained as the absolute minimum of E,,.

3. Results of the Numerical Solution 3.1 Random distribiction of spin positions

The clusters discussed in this section consist of nine spins and their configura- tion is projected on a square lattice with the lattice spacing 4 (as in all our dis- cussions in this paper we are interested predominantly in qualitative compara-

600 J. SLECHTA

Fig. 1. R-dependence of the function J ( R )

1 2 3 4 5 R--

tive study, the absolute magnitudes of the parameters involved are of a less importance). The lattice is used as a reference system and its nodes represent the positions of spins in a “crystalline” cluster. The results for the disordered systems are then compared with those for the reference one.

We used the model exchange interaction J ( R ) = l (R)/k, , used also in [4], where

with parameters a = 0.2, y = 20 and 6 = 0.05. The R dependence of the interaction is plotted in Fig. 1.

The disordered clusters were obtained from the reference crystalline system by using a generator of random numbers. The centre of the intervals, from which the random numbers were generated, are in the nodes of the reference lattice and their half width is Ax. We shall discuss here results for represen- tatives of clusters with Ax = 0, 0.5 and 2 .

For Ax = 0 all exchange interactions are equal -2.6. However, as the number of the nearest neighbours is not the same for every spin, the local temperature of magnetic ordering TF’ is not the same for every spin. From this point of view the cluster is also amorphous. Because of that T” for this system is different from 10.4, the NBel temperature of an inifinite two dimensional crystalline lattice with the given interaction. For this reason we used the case Ax = 0 as a reference “crystalline” system with which we compare systems with random- ness in the exchange interaction (i.e. Ax = 0.5 and 2 ) .

Solution of (3) gives an antiferromagnetic ground state. We chose the basic orientations of vectors yr as shown on Fig. 2 , where + ineans the direction perpendicular to the plane of the cluster and pointing upwards and - means the direction pointing downwards. Projected on this ground state, all are positive and (6) are the same as for the system with exchange interaction -I,? for all i, j (ferromagnetic interaction) which would be projected on a ferromagnetic ground state. This fact can be generalised into the statement: we may often study simultaneously a given system together with its inverse one (the system derived from the original one by changing signs of all interactions involved) by solving the same form of (6). Only the interpretation of the results in the end of the calculation is different. For both systems wc get the same behaviour of 0,. However, the definition of the total magnetisation (13) is different.

The characteristic parameters for all three cases are given in the Table 1, where X is the average spin-spin distance.

I ( R ) = y( -R2 + 6R4) exp (-&A2)


Fig. 2. The orientation of yf vectors and the distribution of 5";' for Ax=O

Fig. 3. The case of Ax = 0.5. a) The orientation of yf vectors and the distribution of J t l ( y q y l ) . b) The distribution of TF

T a b l e 1

Ax ~ Jav I x 1 Ti'v 1 T h 1 ~ ~ T Y

0 1 -2.6 1 4 1 6.92 1 (7'7f" ~ 0.25 0.5 -3.46 1 3.98 9.23 0.507 2 0.652

For Ax = 0.5 the magnetic ground state is also antiferromagnetic despite that one of the exchange interactions is positive. The ordering of the vectors yr and the distributions of the effective exchange interactions projected on the ground state is given in Fig. 3a. The distribution of TfMis given in Fig. 3b. The system of (6) is the same as for the ferroniagnetic ground state with the distribution of the exchange interactions given in Fig. 3b, but with one of them negative (see the statement). So we see that the presence of an exchange interaction of the opposite sign does not necessarily change the character of the magnetic ground state as suggested by some authors on the basis of less detailed models (see [4] and [lo]).

For Ax = 2 the magnetic ground state is of a mixed kind and is depicted in Pig. 4a, together with the effective exchange interactions projected on the vectors y,. The distribution of effective T':' is given in Fig. 4b. The temperature dependence of 3 = 2 of /N is depicted in Fig. 5 and of f o ( T ) in Fig. 6. From it

we see that the assumption a, = CT is valid in the case of Ax = 0.5 only for T < 3 and in the case of Ax = 2 is not valid a t all. Nearly for every Ax =+ 0 is wrong for !Z' z T,.

- - ~ _ _ _ _ ~ ~~ ~ ~~ -_ 1 -

1 -8.78 ~ 4.12 1 23.48 1 , 39.6

f

-0.09

204 + + 1%

a b C

Fig. 4. The case of Ax = 2. a) The orientation of yfvectors and the distribution of J i l ( y i y g ) for the magnetic ground state. b) The distribution of TY for the ground state. c) Number-

ing of spins for the purpose of a detailed study of their behaviour

602 J. SLECHTA

Fig. 5. The temperature dependence of 5 for Ax = 0, 0.5, and 2

Common features:

a) In all cases is fo(TM) > fTM.

1 ) ) For Ax + 0 both 1';; and Tll are higher then those for Ax = 0. This is caused by the fact that the changes of J ( R ) for equal fluctuations on either side of R = 4 are different in magnitude (a different slope) (see Fig. 1).

c) If the solution of (6) is projected on the ferromagnetic ground state M ( T ) cc 3(T) and so its temperature dependence has the same character as Z. However, in the case of the antiferroniagnetic ground state the temperature dependence of M is given in Fig. 7 . The uncompensated spins a t T = 0 are due to the finite size of the studied clusters. Taking the value M ( 0 ) as a reference level (it is nearly equal to zero for an infinite cluster) we see that AM = M ( T ) - - M ( 0 ) is not equal to zero for T < TI, (so behaving as a weak ferriniagnet). The deviation from zero is a measure of the disorder. If the criterion M ( T ) = 0 for all temperatures is taken as the definition of an amorphous antiferroniagnets then a very few amorphous materials exactly fulfil this condition. However, their weaker definition M ( 0 ) = 0 should be fulfilled quite often. With the increased disorder we get systems with the ground state neither ferromagnetic nor antiferroniagnetic (see [lo]) and M(0) + 0 even for an infinite cluster. They also behave as weak ferrimagnets. They have been named mixed ferromagnets or antiferromagnets (see [13]).

d) In all cases Ti\ < T,. To understand the reason for it we give here in Fig. 8 the detailed temperature behaviour of all o, for the rase of Ax = 2. The spins are numbered as in Fig. 4c. From it we may recognize several interesting features :

- For all spins, except number 6 and 8, or(!!') =/= 0 for T > TF. For example for T = 19.8 = TF still holds 0 4 ( T ) = 0.5. The reason for it is that its neigh-

Fig. 6. The temperature dependence of f, for Ax = 0.5 and 2

TfK) -

Study of Structurally Disordered Single-Element Nagnetic Materials 603

TiKi - e-

l b-

05

0 10 20 30 40 TiK) ---

Fig. 7. The temperature dependence of ill( T) for Ax = 0.5 and 2

Fig. 8. The case Ax = 2. The temperature dependence of E( T ) (thick curve) and of( T )

(thin numbered curves)

bours have a very high T:' and so impose on spin 9 a strong local effective field. It means that spins with high T:' support spins with low T:'. This property propagates throughout the system as we may see comparing Fig. 8 and 4h. This is the main reason for the fact that Ti\ < T,. - A very interesting behaviour can be noticed in the case of spin number 2, for which Ti1 is niuch lower than any T!' of the rest of the spins in the system. For temperature T > 20 is o2 N 0 and this spin behaves as if it was nearly magnetically free. The possibility of the existence of such free spins in magnetically ordered systems is very important for an understanding of the anomalous Kondo effect in amorphous single element niagnetical materials (sec [ 161) and this. faet will be diwuseed in more detail elsewhere [ 171. - If we order Ti: by their magnitudes we find that o, do not retain the same ordering when the temperature is changed. From it follows that it is not possible to assume of - 1';' (see e.g. [lo]) and the fluctuations in of are not connected with the disorder in the exchange interaction in a simple way.

3.2 Random distribution of exchange interactions

In the previous section we notice hhat in all three examples T& < This means that if one calculates of conrktently from the system (6), one gets in the case of the ferromagnetic ground state a higher Curie temperature T, than that obtained for the same structure from a simpler theory assuming that o = uf for all f (e.g. [2 , 41). This suggests that if we take into account the fluctuation of the exchange integrals only, the Curie temperature increases with increased fluctuations. To show this more obviously we modelled the exchange interaction directly by a random number generator, rather than indirectly as in the

T a b l e 2

AJ

0 2.5

10 22 35

Ti'" ~-

26.6 26.6 26.6 26.6 26.6

28.8 28.64 30.84 37.94 41.00

f Tna

0.25 0.267 0.359 0.574 0.591

604 J. SLECHTA

~~~~~

0 2.08 6.84

12.5

+,I +[I 4 +I 14-78

- , i 3 +[I +I 2.332

+ 4611 + z m + 2.53 + 2847 + 14.18 - 785

+ 4325 + 4733 + 24.06 + 1857 + 79149 + r4.36

AJ=22 n=35

Fig. 9. The ground state orientation of yf vectors and distribution of Tj' for different AJ

~ ~ ~

32 34.5 ' 0.1732 32 ~ 37.8 I 0.2488 32 43 I 0.357 32 I 52 1 0.543

previous Section 3.1. The mean value of the exchange interaction was kept constant. The amplitude of the fluctuations of J ( r ) is characterised by the half width A J of the interval from which the random numbers were generated. As a reference system was taken that with AJ = 0. We shall present here results for two-dimensional clusters of nine and twenty five spins and three-dimensional clusters of twenty seven spins:

Two-dimensional cluster - 9 spins: The distribution of TT' and direction of y, are given in Pig. 9 and the values of the characteristic parameters are in Table 2. For A J = 35 we obtain again a mixed ferromagnet. In Fig. 10 is depicted the total magnetisation for different A J norrnalised such that in all cases = 100. We see that with growing fluctuations it is a nearly straight line.

Two-dimensional cluster - 25 spins: The values of the characteristic parameters are given in the Table 3.


r m - Fig. 10. The temperature dependence of a(T) for different AJ (in reduced temperature

scale (Tar == 100 for all A J ) )

Three-dimensional cluster - 27 spins: The values of the characteristic parameters are given in Table 4.

T a b l e 4

0 2.01 6.18

10.6

40 ’ 42.5 ’ ‘0.2079 40 42 0.223 40 ~ t i ~ 0.314 40 0.450

We see from the results included in Tables 1 to 4 that the best characteristic of the disorder in amorphous magnetic materials is f T x . From this follows that the detailed behaviour of their magnetic properties depends on the higher correlations. Previous theories of amorphous systems have tried to correlate their measured magnetic properties with the structural information contained in the pair-pair correlation function. However we can see from the results above that what really determines the magnetic behaviour of the amorphous materials is the distribution of TfM which is related to the higher spatial correlations of spins.

To compare results for clusters of different size we shall calculate the ratio of T , for two chosen f T M of comparable magnitude from each Table 2 to 4. We obtain :

T&I(fTi% = o.*j74, - - 37.9 = T & f ( f p M = 0 25) 28.5 two-dim.: 9 spins: -

y’ i i ( fTM = 0.5431) 52 = 1.37, - 25 spins: - - ~

T,( fp , = 0.2488) 37.8

T,(fT, = 0.450) 53 T,(f rpM = 0.2079) 42.5

three-dim. : 27spins: ~ - _ _ - - ~ = 1.24.

To complete the comparison we give here a similar ratio from the self-consistent calculation of two-dimensional clusters of nine spins imbedded in an effective

medium [15]. It is T,(fTM = 0 . 5 0 6 ) / T s r ( f ~ ~ , = 0.204) = - = 1.33. 60 45

606 J. SLECHTA

From the numbers we conclude that the relative results concerning Tnf are valid also for a bulk material with homogeneous fTY similar to those for which the ratio was calculated.

The rest of the results presented in this paper represent a detailed behaviour of spins in a characteristic cluster (near a void) in such a material. Those results illustrate one general feature of nonlinear problems in amorphous materials which was formulated byKrumhaus1 (see [HI) . From i t we quote: “. . . in nonlinear random systems it is likely that we must be content with solution of spe- cific examples for guidance when we wish to go beyond discussing average fields in such systems”.

4. Conclusion

The main surprising result of our calculation is that T , increases due to the effect of pure fluctuations of the exchange integrals. This result is in contradiction with findings of the other authors [l to 3 , 5 , 191 and is direct consequence of relaxing the condition a, = a for all f . A similar criticism is valid for [4, 6, 211. For example in [4] there is a study of a similar system of equations to (6) for the case of ferromagnetic ground state. However in due course of the analysis the assumption <St) = (8) is made. Therefore regardless how exactly the spatial fluctuations of J are treated some results there have to be taken with caution.

A further inadequacy of previous theories is that they do not provide for the possibility of a detailed restructuring of the ground state with growing fluctuations of the exchange interaction. This leads to the incorrect result that for a J > J,, (see [2j and [19]) the magnetic order disappears. It is neither true that a small amount of an exchange interaction of the opposite sign needs to change the character of the magnetic state [4, 16, 20, 71.

Our results are important for interpretation of experimental data. If the experimental results show a decrease of T , it is then due to two possi-

bilities: a) a reduction of the number of nearest neighboiirs Z and b) a decrease in average T!’ due to the change in spin-spin distances. The first effect may be estimated froin diffraction data. Tf the change in Z cannot explain the observed decrease of T, then we have to make conclusion about the change in J ( R ) and so possibly obtain information about its R-dependence. We see that the same experimental data can yield a different information if analysed from the point of view of different theoretical models.

For example, the temperature dependence of the total magnctisation flattens with an increase of fluctuations in agreement with other models. However, we showed that what matters are not fluctuations in the exchange interaction but in TTf. From that point of view it is possible to have a better understanding of the flat niagnetisation curve of Ki85P15 after annealing (see [9]). If we assume a strong r-dependence of J ( r ) it is possible that the highly disordered boundaries of the microcrystal grains of the size 50 A could produce a very large fluctuation in Tf” and so the flattening. Moreover, a small change in crystalline positions, which may cause the observed diffraction line broadening can also produce large fluctuations in TfM.

The closest results to those given in this paper in relation to the dependence of T , on the fluctuations of the exchange integral is in [ 191, however the authors


there did not study the important question of the restructuring of the magnetic ground state and did not realized the importance of the higher order structural correlations for the magnetic hehaviour of the amorphous systems.

Acknoroledgemen ts

The author would like to thank the Science Research Council for the support of this work and Drs. R. V. Aldridge and J. C. Wright for valuable discussion in the direction of the connection of this calculation with experiment.

The nuiiierical calculation was done on ICL 1900E computer of UEA.

Ref cremes [l] K. HANDRICH, phys. stat. sol. 32, KO5 (1969). [2] K. HANDRICH and S. KOBE, Acta Phys. Polon. A38, 819 (1970). [3] R. HASEGAWA, phys. stat. sol. (b) 44, 613 (1971). [4] T. KANEYOSHI, J. Phys. C 6, 3180 (1973). 151 C. G. MONTGOMERY, J. I. KRUGEL, and R. 11. STUBBS, Phys. Rev. Letters 25, 669

[6] T. KANEYOSHI and R. HONMURA, J. Phys. C 3, L65 (1972). [7] A. W. SIMPSOK, phys. stat. sol. 40, 207 (1970). [8] 8. V. TYABLIKOV, Methods of Quantum Theory of Magnetism, Plenum Press, Xew

[9J A. W. SIMPSON and D. R. BRAMBLEY, phys. stat. sol. (b) 49, 685 (1972).

(1970).

York 1967.

[lo] S. KOBE and K. HANDRICH, phys. stat. sol. (b) 54, 663 (1972). t l l] J. M. LUTTINGER, Phys. Rev. 81, 1015 (1951). 1123 J. @ART, Effective Field Theories of Magnetism, Sanders, Philadelphia, London 1966. 1131 J. ~ L E C K T A and R. V. ALDRIDGE, J. Phys. F 2, 1,132 (1972). [14] J. SLECHTA, J. Phys. F 4, 1148 (1972). [I51 J. SLECHTA, to be published. [I61 P. K. LEUNG, ?J. SLECHTA, and J. G. WRIGHT, J. Phys. F 4, L21 (1974). [I71 J. SLECHTA, to be published. 1181 J. K. KRUMHAUSL, The Proceedings of the International Symposium on Aniorphous

[I91 S. KOBE and K. HANDRICH, Soviet l’hys. - Solid State 13, 734 (1971). Lao] T. K4NEYOsHr, J. Phys. C 6, L19 (1973). [21] R. HARRIS, &I. PLISHXE, and M. J. ZUCKERIVIANN, Phys. Rev. Letters 31, 160 (1973).

Magnetism, Detroit, Michigan 1972 I p. 24).

(Received Noveinber 8, 1974)

a study of structurally disordered single-element magnetic materials beyond the homogeneous...

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