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Magnetization of a random binary alloy with short-range order of atomic arrangement
Physica I14B (1982) 217-222 North-Holland Publishing Company
217
MAGNETIZATION OF A RANDOM BINARY ALLOY WITH SHORT-RANGE ORDER OF ATOMIC ARRANGEMENT
S. I N A W A S H I R O * Department of Mathematics and School of Physics, University of Melbourne, Parkville, Victoria, 3052, Australia and J. H A R A D A t School of Physics, University of Melbourne, Parkville, Victoria, 3052, Australia Received 22 February 1982 A model for a random magnetic alloy with short-range order for atomic arrangement is presented. In the model, magnetic interactions are distributed according to certain probabilities derived from the short-range order. Theoretical treatment is given in an approximation where the probability distribution of magnetic interaction is replaced by a Gaussian form. The magnetization of a particular random AuXin alloy is calculated using short-range order obtained recently by X-ray diffuse scattering and by assuming a set of plausible values for the magnetic interaction. The result shows that the main features of the magnetization of the random alloy are well reproduced.
1. Introduction In the theories of a r a n d o m m a g n e t i c system, a m o d e l is usually a s s u m e d in such a way that magnetic interactions are distributed r a n d o m l y with a certain probability distribution [1, 2]. L a c k of k n o w l e d g e of the probability distribution in a real system has b e e n o n e of the main difficulties in the application of the t h e o r y to any r a n d o m m a g n e t i c system. In disordered binary alloys, it is well k n o w n that the W a r r e n - C o w l e y s h o r t - r a n g e o r d e r ( S R O ) p a r a m e t e r with respect to an atomic a r r a n g e m e n t can be d e t e r m i n e d by a F o u r i e r analysis of s h o r t - r a n g e o r d e r diffuse scatterings
* o n leave from the Department of Applied Physics, Tohoku University Aza Aoba, Sendai 980, Japan. t on leave from the Department of Applied Physics, Nagoya University, Nagoya 464, Japan. 0378-4363/0000-0000/$02.75 0
1982 N o r t h - H o l l a n d
o b t a i n e d by X - r a y or n e u t r o n diffraction experim e n t s [3]. Since the S R O p a r a m e t e r s are closely related to the o c c u p a t i o n probability of an a t o m s e p a r a t e d by a certain lattice vector f r o m another atom, it is possible to obtain a probability distribution of magnetic interaction for the diso r d e r e d state of the alloy. A l t h o u g h this state m a y not be thermally stable, we will refer to it simply as a r a n d o m alloy in this paper. Besides, we m a y h a v e a k n o w l e d g e of the o r d e r e d magnetic alloy with the s a m e atomic composition, if chemical o r d e r - d i s o r d e r phase transition is r e c o g n i z e d in the binary system. F r o m the point of view of statistical mechanics of r a n d o m systems, we stress the i m p o r t a n c e of studying binary alloys which exhibit b o t h o r d e r e d and d i s o r d e r e d a r r a n g e m e n t s of atoms, because such a r a n d o m spin system can be easily o b t a i n e d by quenching the specimen f r o m its stable state to low temperature. It is very interesting to apply the
S. lnawashiro and J. Harada / Magnetization of a random binary alloy
218
theoretical approach so far developed for the r a n d o m magnetic system to such alloys and to c o m p a r e the theoretical results with experimental ones. Recently, one of the authors (J.H.) together with other collaborators has succeeded in obtaining interesting short-range order a r r a n g e m e n t for Mn atoms in a disordered Au41Vln alloy [4]. This has motivated us to the present theoretical study. The purpose of this p a p e r is to provide a model for a r a n d o m magnetic alloy on the basis of S R O p a r a m e t e r s which could be determined experimentally f r o m the analysis of X-ray and neutron diffuse scatterings and, thereafter, to show that the t e m p e r a t u r e d e p e n d e n c e of the magnetization of a r a n d o m Au4Mn alloy can be explained by the model. In section 2, a model for the r a n d o m alloy is presented and an approximation in terms of socalled "spin-glass m o d e l " is described. In section 3, a r a n d o m alloy Au4Mn is examined based upon the treatment given in section 2. Section 4 is devoted to discussions.
2. Model and approximate solutions Atomic arrangement in a binary alloy system of A and B atoms can be described by introducing the Warren--Cowley short-range order (SRO) p a r a m e t e r 0txYtm,which are given by the Fourier transform of the diffuse scattering. According to Cowley the S R O p a r a m e t e r [5] is defined as xY_
Ol l m n - -
1 - - -p - xY ,
(X, Y = A, B ) ,
(2.1)
Xy
where Pl.,n X Y is the probability of finding a Y atom at a distance Rtmn = lal + ma2+
ha3
(2.2)
from an X a t o m located at the origin, and xy denotes the composition of the Y atomic species. It is to be noted that
AB
BA
XAP t,,, = XBP t,,, .
(2.3)
For a completely disordered system without any S R O arrangement, which is called the Laue monotonic disordered state, it is easily seen that
(2.4)
x Y =- X y , Ptmn
and therefore all the S R O parameters are zero except a~0~ = 1 ( X ~ Y). These probabilities for a system with S R O should be, therefore, significantly different from (2.4). Let us denote the magnetic interaction between these X and Y atoms as XY Jr,,,,.
(2.5)
F r o m the definition we see that J xlmn Y vanishes when X or Y, or both, are nonmagnetic. For the case of the alloy in which only the B atomic species has a spin while the A atomic species does not, we can disregard the distribution of the A atomic species. AuStin r a n d o m alloy, as will be discussed later, belongs to this category. If we neglect m a n y - b o d y correlations for the atomic arrangement, we can construct a model of the r a n d o m alloy in such a way that the magnetic interaction J t xY m n takes place on one of the z bonds emerging from a B atom with the probability
Plmn -- Zlmn Z
BB Ptm.,
where z t , , , denotes the n u m b e r of the neighbours and
z = ~ zo,,,, •
(2.6)
(lmn)th
(2.7)
T h e summation in (2.7) should be taken over a finite range of the interactions. This model is still too complicated to be subjected to a simple theoretical treatment. Let us characterize this r a n d o m system by introducing
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
two parameters, the average magnetic interaction and the root mean square deviation of the interaction, respectively, as following: (2.8)
Y = ~'~ P o , . J , . , . . F
_ -11/2
A J = [ Z p ImnJ2mn - j 2 j
(2.9)
.
The system may be approximated by a random magnetic system with a Gaussian distribution of interaction:
p(j) =
1 exp/-I" ( J - y)2] X/2--~ AJ t_ 2(A J) 2 J"
(2.10)
This is the well-known model for the spin glass presented by Edward and Anderson [1]. In the limit of an infinite coordination number with Y = 1z J°-->O
and
A?
AJ = ~ - f -->0, (as z -->oo),
(2.11)
keeping Jo and AJ as constants, solutions for the S = ½system are given by [6, 7] m =~
21g
3. Ordered and disordered state of Au4Mn The ordered Au4Mn alloy is stable below the chemical order-disorder phase transition poinl (120 K) and has an NijVlo type structure [8, 9] in which Mn atoms are arranged in linear chain., occupying the second neighbours along the [001] direction, as shown in fig. 1. In this structure there is no Mn atom which is linked with the firsl neighbour, though an Mn atom is connected with two second neighbour Mn atoms located along the chain, and with eight third neighbour Mn atoms located at 4-surrounding chain. Although magnetic interactions in the alloy would be of longrange, due to the R K K Y interaction, we confine ourselves, for simplicity, to the range of interaction up to third neighbours. On the other hand, the random Au4Mn alloy is of a simple FCC [10] structure and is obtained by quenching it after appropriate annealing at a temperature above the order-disorder transition point. The specimen, however, shows peculiar diffuse scatterings, on electron [9], X-ray [4, 11] and neutron diffraction [12] patterns, indicating the existence of short-range ordering of the Mn atom. From the accurate analysis of the X-ray diffuse scattering, Mn-atom arrangement has been studied and the following SRO parameters
-~ tanh(m]o + A J ~ / q x ) e -x2n dx, (2.12)
q = ~ ' ~1
O0
-® [tanh(mJo + AJX/qx)] 2 e -x2n dx (2.13)
A.
OOM.
where m and q are the random average and the random mean square of the magnetic moment per magnetic atom, respectively. Here ]0 corresponds to z.~ and A] to ~v/zzAJ in our finite z system, i.e., (2.14)
Jo = z ~ , p t m , J ~ m , , _ 11/2
Aj = V'z E
P ~ , J t,,n -
j2 j
.
(2.15)
Fig. 1. A t o m i c a r r a n g e m e n t p r o j e c t e d on the (001) plane for o r d e r e d Au,tMn alloy. Thin circles r e p r e s e n t the a r r a n g e m e n t of a t o m s o n t h e z = 12plane. T h e tetraogonal unit cell for ordered Au4Mn is a = V ~ 2 , avcc = 6.448 A, c = 4.028 A; F C C unit cell: aFcc = 4.068 A.
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
220
have been reported [4]: a nBA o= -0.168, a~o~ = 0.222, BA = Or211
(3.1)
0.042
F r o m (2.1), we have BB Pn0 = 0.0656, BB P20o = 0.378,
(3.2)
BB--
P2n - 0.234. In contrast to the case of ordered state, there is a nonvanishing probability of finding the M n Mn coupling connected with first neighbours, and also the probabilities of finding the higher order coupling are significantly different f r o m the value of a completely r a n d o m case. F r o m (2.6) and (3.2), it follows that Pl = Pn0 = 0.0187, /92 = p200 = 0.0539,
(3.3)
P3 = P2n = 0.1335, where we have used ZI10 =
12,
z200 = 6,
Z211 =
24
as that of the ordered alloy [13]. T h e items (b) and (c) can be interpreted that a partial cancellation of the magnetization takes place as a result of antiparallel alignment of Mn spins in an appropriate n u m b e r in the r a n d o m alloy. Although its definite mechanism remains still unknown, we believe that antiferromagnetic interactions between Mn atoms play an important role in the mechanism. Further (a) can be understood as a result of the competition between the ferromagnetic and antiferromagnetic interactions. Since the ordered alloy is ferromagnetic, the second and the third neighbour interactions are supposed to be ferromagnetic. It seems plausible to assume a little weaker interaction for the third neighbours than for the second neighbours. T h e absence of the first neighbour coupling in the ordered alloy and the presence of the same in the r a n d o m alloy, as shown in (3.3), lead us to assume that a strong antiferromagnetic interaction exists between the first neighbours Mn atoms in the r a n d o m alloy. If the magnetic interactions between Mn atoms can be assumed to depend only on the distance between the spins irrespective of the ordered and disordered states of the alloy, the magnetic interactions can be described using two p a r a m e t e r s a and y as following:
(3.4) J1 = Jllo = - a J2,
and 2" =
J2 = J200, Z l l 0 " J - 2200 + Z211 =
42.
(3.5)
T h e magnetic p r o p e r t i e s of the quenched Au4Mn r a n d o m alloy are characterized as follows: (a) the Curie t e m p e r a t u r e is a b o u t half that of the ordered alloy [13]; (b) the magnetization at low t e m p e r a t u r e is also about half that of the ordered alloy [13], and (c) the effective n u m b e r of Bohr magnetons of the r a n d o m alloy (determined by m e a s u r e m e n t of paramagnetic susceptibilities) is almost the same
(3.6)
J3 = .12,, = y J 2 .
The mean field value of the Curie t e m p e r a t u r e of the ordered alloy is then given by k T c = (2 + 8 y ) J 2
(3.7)
and that of the r a n d o m alloy by k T R = Jo = Z ( - p , a
+ p2+ PaY)J2.
(3.8)
Now it is quite interesting to see whether it is
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
possible or not to fit the magnetization vs. temperature curve by adjusting the p a r a m e t e r s a and y. For a few sets of ot and y, we have evaluated kTc/J2, kT~/J2, A J/J2, TR/Tc and J0/AJ, as shown in table I. As an example, a calculated magnetization for a = 2.65 and y = 0.7 is plotted against t e m p e r a t u r e in fig. 2, together with the result of the measurement. T h e magnetization for the ordered alloy is also shown in the same figure. T h e ratio of the calculated Curie t e m p e r a t u r e of the disordered alloy to that of the ordered alloy is about one half, and the calculated value of the m a x i m u m magnetization of the disordered alloy is about a half of that of the ordered alloy, in agreement with the main features of the experimental results. However, Table I T h e Curie temperatures and parameters of the model for a few sets of a and y
ct, y
kTc/J2 kT~/J2(=Jo/J2)
A~/2
T~/Tc
~/0/A.]r
0.8, 2.90 0.7, 2.65 0.6, 2.40
8.40 7.60 6.80
3.46 3.18 2.91
0.532 0.540 0.550
1.29 1.29 1.29
221
calculated magnetization for the r a n d o m alloy decreases at low t e m p e r a t u r e when t e m p e r a t u r e decreases further. This discrepancy will be discussed in section 4. It should also be mentioned that similar fitting of the magnetization curve is obtained for other sets of p a r a m e t e r s a and y, for instance, a = 2.90 and y = 0.8, and a = 2.40 and y -- 0.6.
4. Discussions
W e have presented a model of a r a n d o m magnetic alloy with short-range orders f o r atomic arrangement. For a r a n d o m binary alloy, the probability distribution of magnetic interaction is given by a discrete distribution, i.e.,
P ( J ) = ~ plm,,8(J -- Jtmn) + p o S ( J ) ,
(4.1)
where
1.0
4.47 4.11 3.74
M(T) /M(O)
• ~DERED M(T)/M(O)
~.
0.5
0.0
o
0.5
1.0
T/Tc
Fig. 2. Magnetization of the regular and the r a n d o m alloys Au4Mn. Results of m e a s u r e m e n t by Ido are d e n o t e d by black circles for the r a n d o m alloy [13]. Calculated values are represented by full lines.
Po = 1 - ~ p,m,.,
(4.2)
which denotes the probability of vanishing interaction. In section 2, w e h a v e replaced the probability distribution (4.1) by the Gaussian distribution with the same average magnetic interaction and the same variance as calculated f r o m (4.1). It should be emphasized that in spite of our drastic a p p r o x i m a t i o n - a d o p t i n g the Gaussian distribution for the i n t e r a c t i o n - t h e main features of the magnetic behaviour of the r a n d o m Au4Mn is well reproduced in this calculation. Therefore, we believe that our approach as given in this p a p e r is in the right direction as a whole. As for the decrease of calculated magnetization at low t e m p e r a t u r e mentioned in section 3, it may be a serious problem in the approximation made in this p a p e r or m o r e fundamentally in our construction of the model. First of all the effect of neglect of long-ranged and oscillatory R K K Y
222
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
interactions may be considered as one possible reason, since we have taken the interactions only up to the third neighbours. But such termination for an infinite series of interactions would not cause serious effects, because it is expected that the inclusion of these long-range interactions could be covered to some extent by adjusting the parameters a and 3' in the present calculation, without affecting the magnetization essentially. The decrease of the calculated magnetization at low temperature is then considered to arise from the reduction of the magnetization, as a whole, due to the increase in number of antiparallel alignments of local magnetic moments at lattice sites. This may be attributed to our drastic approximation in which the continuous Gaussian distribution was adopted instead of the discrete probability distribution for the magnetic interaction, or more fundamentally to the neglect of many-body correlation of atomic arrangement in our model. These problems are left to future investigations. According to the measurements of paramagnetic susceptibilities of the ordered and the random Au4Mn, the effective numbers of Bohr magneton are pen = 4.8 and 4.5, respectively. Therefore, the use of the Brillouin function with S = 2 would be more appropriate than the Brillouin function with S = ½which have been used in the present simplified treatment. However, it is not considered that the magnitude of spin would essentially influence the results obtained. Anyway, refinement of theory along these lines would be desirable and will be left to the next stage of approximation. It seems appropriate here to comment that for a simple discrete probability distribution with two interactions - J and a few coordination numbers (z = 3 and 4), solutions in a pair approximation at absolute zero temperature were discussed in detail [14]. Extension to finite temperatures and to a more general discrete probability distribution with a larger coordination number is required. The discrete probability distribution could be
determined experimentally by an analysis of Xray and neutron diffuse scatterings, as mentioned in this paper. Therefore, we finally stress the importance of experimental and theoretical studies of random alloy systems with short-range orders, in order to understand the physical properties of random magnetic systems. We hope that problems along these lines will be further studied in the future.
Acknowledgments The authors wish to thank the School of Physics and the Department of Mathematics, University of Melbourne, for offering them an opportunity to collaborate in this work and for the hospitality during their stay. One of the authors (S.I.) wishes to thank the A R G C for their support. J.H. is grateful for the Sir Thomas Lyle Fellowship which enabled him to stay at the University of Melbourne.
References [1] S.F. Edwards and P.W. Anderson, J. Phys. 1::5 (1975) 965. [2] S. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. [3] B.E. Warren, X-ray Diffraction (Addison-Wesley, 1969). [4] H. Suzuki, J. Harada, T. Nakashima and K. Adachi, submitted to Acta Cryst. A. [5] J.M. Cowley, J. Appl. Physics21 (1950) 24. [6] D. Sherrington and S. Kirkpatrick; Phys. Rev. Lett. 35 (1975) 1972; Phys. Rev. B17 (1978) 4384. [7] C.J. Thompson, J. Stat. Phys., to be published. [8] A. Kussmann and E. Raub, Z. Metallkde. 47 (1956) 9. [9] D. Watanabe, J. Phys. Soc. Japan 15 (1960) 1251. [10] W. Koster and R.E. Hummel, Z. Metallkde. 55 (1964) 175. [11] P. Fiirnrohr, J.E. Epperson and V. Gerold, Z. Metallkde. 71 (1980) 403. [12] T. Nakashima, S. Mizuno, T. Ido, K. Sato, S. Mitani and K. Adachi, J. Phys. Soc. Japan 43 (1977) 1870. [13] T. Ido, Thesis (1973), Nagoya University, Chikusa-ku, Nagoya, Japan. [14] S. Inawashiro and S. Katsura; Physica 100A (1980) 24.
217
MAGNETIZATION OF A RANDOM BINARY ALLOY WITH SHORT-RANGE ORDER OF ATOMIC ARRANGEMENT
S. I N A W A S H I R O * Department of Mathematics and School of Physics, University of Melbourne, Parkville, Victoria, 3052, Australia and J. H A R A D A t School of Physics, University of Melbourne, Parkville, Victoria, 3052, Australia Received 22 February 1982 A model for a random magnetic alloy with short-range order for atomic arrangement is presented. In the model, magnetic interactions are distributed according to certain probabilities derived from the short-range order. Theoretical treatment is given in an approximation where the probability distribution of magnetic interaction is replaced by a Gaussian form. The magnetization of a particular random AuXin alloy is calculated using short-range order obtained recently by X-ray diffuse scattering and by assuming a set of plausible values for the magnetic interaction. The result shows that the main features of the magnetization of the random alloy are well reproduced.
1. Introduction In the theories of a r a n d o m m a g n e t i c system, a m o d e l is usually a s s u m e d in such a way that magnetic interactions are distributed r a n d o m l y with a certain probability distribution [1, 2]. L a c k of k n o w l e d g e of the probability distribution in a real system has b e e n o n e of the main difficulties in the application of the t h e o r y to any r a n d o m m a g n e t i c system. In disordered binary alloys, it is well k n o w n that the W a r r e n - C o w l e y s h o r t - r a n g e o r d e r ( S R O ) p a r a m e t e r with respect to an atomic a r r a n g e m e n t can be d e t e r m i n e d by a F o u r i e r analysis of s h o r t - r a n g e o r d e r diffuse scatterings
* o n leave from the Department of Applied Physics, Tohoku University Aza Aoba, Sendai 980, Japan. t on leave from the Department of Applied Physics, Nagoya University, Nagoya 464, Japan. 0378-4363/0000-0000/$02.75 0
1982 N o r t h - H o l l a n d
o b t a i n e d by X - r a y or n e u t r o n diffraction experim e n t s [3]. Since the S R O p a r a m e t e r s are closely related to the o c c u p a t i o n probability of an a t o m s e p a r a t e d by a certain lattice vector f r o m another atom, it is possible to obtain a probability distribution of magnetic interaction for the diso r d e r e d state of the alloy. A l t h o u g h this state m a y not be thermally stable, we will refer to it simply as a r a n d o m alloy in this paper. Besides, we m a y h a v e a k n o w l e d g e of the o r d e r e d magnetic alloy with the s a m e atomic composition, if chemical o r d e r - d i s o r d e r phase transition is r e c o g n i z e d in the binary system. F r o m the point of view of statistical mechanics of r a n d o m systems, we stress the i m p o r t a n c e of studying binary alloys which exhibit b o t h o r d e r e d and d i s o r d e r e d a r r a n g e m e n t s of atoms, because such a r a n d o m spin system can be easily o b t a i n e d by quenching the specimen f r o m its stable state to low temperature. It is very interesting to apply the
S. lnawashiro and J. Harada / Magnetization of a random binary alloy
218
theoretical approach so far developed for the r a n d o m magnetic system to such alloys and to c o m p a r e the theoretical results with experimental ones. Recently, one of the authors (J.H.) together with other collaborators has succeeded in obtaining interesting short-range order a r r a n g e m e n t for Mn atoms in a disordered Au41Vln alloy [4]. This has motivated us to the present theoretical study. The purpose of this p a p e r is to provide a model for a r a n d o m magnetic alloy on the basis of S R O p a r a m e t e r s which could be determined experimentally f r o m the analysis of X-ray and neutron diffuse scatterings and, thereafter, to show that the t e m p e r a t u r e d e p e n d e n c e of the magnetization of a r a n d o m Au4Mn alloy can be explained by the model. In section 2, a model for the r a n d o m alloy is presented and an approximation in terms of socalled "spin-glass m o d e l " is described. In section 3, a r a n d o m alloy Au4Mn is examined based upon the treatment given in section 2. Section 4 is devoted to discussions.
2. Model and approximate solutions Atomic arrangement in a binary alloy system of A and B atoms can be described by introducing the Warren--Cowley short-range order (SRO) p a r a m e t e r 0txYtm,which are given by the Fourier transform of the diffuse scattering. According to Cowley the S R O p a r a m e t e r [5] is defined as xY_
Ol l m n - -
1 - - -p - xY ,
(X, Y = A, B ) ,
(2.1)
Xy
where Pl.,n X Y is the probability of finding a Y atom at a distance Rtmn = lal + ma2+
ha3
(2.2)
from an X a t o m located at the origin, and xy denotes the composition of the Y atomic species. It is to be noted that
AB
BA
XAP t,,, = XBP t,,, .
(2.3)
For a completely disordered system without any S R O arrangement, which is called the Laue monotonic disordered state, it is easily seen that
(2.4)
x Y =- X y , Ptmn
and therefore all the S R O parameters are zero except a~0~ = 1 ( X ~ Y). These probabilities for a system with S R O should be, therefore, significantly different from (2.4). Let us denote the magnetic interaction between these X and Y atoms as XY Jr,,,,.
(2.5)
F r o m the definition we see that J xlmn Y vanishes when X or Y, or both, are nonmagnetic. For the case of the alloy in which only the B atomic species has a spin while the A atomic species does not, we can disregard the distribution of the A atomic species. AuStin r a n d o m alloy, as will be discussed later, belongs to this category. If we neglect m a n y - b o d y correlations for the atomic arrangement, we can construct a model of the r a n d o m alloy in such a way that the magnetic interaction J t xY m n takes place on one of the z bonds emerging from a B atom with the probability
Plmn -- Zlmn Z
BB Ptm.,
where z t , , , denotes the n u m b e r of the neighbours and
z = ~ zo,,,, •
(2.6)
(lmn)th
(2.7)
T h e summation in (2.7) should be taken over a finite range of the interactions. This model is still too complicated to be subjected to a simple theoretical treatment. Let us characterize this r a n d o m system by introducing
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
two parameters, the average magnetic interaction and the root mean square deviation of the interaction, respectively, as following: (2.8)
Y = ~'~ P o , . J , . , . . F
_ -11/2
A J = [ Z p ImnJ2mn - j 2 j
(2.9)
.
The system may be approximated by a random magnetic system with a Gaussian distribution of interaction:
p(j) =
1 exp/-I" ( J - y)2] X/2--~ AJ t_ 2(A J) 2 J"
(2.10)
This is the well-known model for the spin glass presented by Edward and Anderson [1]. In the limit of an infinite coordination number with Y = 1z J°-->O
and
A?
AJ = ~ - f -->0, (as z -->oo),
(2.11)
keeping Jo and AJ as constants, solutions for the S = ½system are given by [6, 7] m =~
21g
3. Ordered and disordered state of Au4Mn The ordered Au4Mn alloy is stable below the chemical order-disorder phase transition poinl (120 K) and has an NijVlo type structure [8, 9] in which Mn atoms are arranged in linear chain., occupying the second neighbours along the [001] direction, as shown in fig. 1. In this structure there is no Mn atom which is linked with the firsl neighbour, though an Mn atom is connected with two second neighbour Mn atoms located along the chain, and with eight third neighbour Mn atoms located at 4-surrounding chain. Although magnetic interactions in the alloy would be of longrange, due to the R K K Y interaction, we confine ourselves, for simplicity, to the range of interaction up to third neighbours. On the other hand, the random Au4Mn alloy is of a simple FCC [10] structure and is obtained by quenching it after appropriate annealing at a temperature above the order-disorder transition point. The specimen, however, shows peculiar diffuse scatterings, on electron [9], X-ray [4, 11] and neutron diffraction [12] patterns, indicating the existence of short-range ordering of the Mn atom. From the accurate analysis of the X-ray diffuse scattering, Mn-atom arrangement has been studied and the following SRO parameters
-~ tanh(m]o + A J ~ / q x ) e -x2n dx, (2.12)
q = ~ ' ~1
O0
-® [tanh(mJo + AJX/qx)] 2 e -x2n dx (2.13)
A.
OOM.
where m and q are the random average and the random mean square of the magnetic moment per magnetic atom, respectively. Here ]0 corresponds to z.~ and A] to ~v/zzAJ in our finite z system, i.e., (2.14)
Jo = z ~ , p t m , J ~ m , , _ 11/2
Aj = V'z E
P ~ , J t,,n -
j2 j
.
(2.15)
Fig. 1. A t o m i c a r r a n g e m e n t p r o j e c t e d on the (001) plane for o r d e r e d Au,tMn alloy. Thin circles r e p r e s e n t the a r r a n g e m e n t of a t o m s o n t h e z = 12plane. T h e tetraogonal unit cell for ordered Au4Mn is a = V ~ 2 , avcc = 6.448 A, c = 4.028 A; F C C unit cell: aFcc = 4.068 A.
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
220
have been reported [4]: a nBA o= -0.168, a~o~ = 0.222, BA = Or211
(3.1)
0.042
F r o m (2.1), we have BB Pn0 = 0.0656, BB P20o = 0.378,
(3.2)
BB--
P2n - 0.234. In contrast to the case of ordered state, there is a nonvanishing probability of finding the M n Mn coupling connected with first neighbours, and also the probabilities of finding the higher order coupling are significantly different f r o m the value of a completely r a n d o m case. F r o m (2.6) and (3.2), it follows that Pl = Pn0 = 0.0187, /92 = p200 = 0.0539,
(3.3)
P3 = P2n = 0.1335, where we have used ZI10 =
12,
z200 = 6,
Z211 =
24
as that of the ordered alloy [13]. T h e items (b) and (c) can be interpreted that a partial cancellation of the magnetization takes place as a result of antiparallel alignment of Mn spins in an appropriate n u m b e r in the r a n d o m alloy. Although its definite mechanism remains still unknown, we believe that antiferromagnetic interactions between Mn atoms play an important role in the mechanism. Further (a) can be understood as a result of the competition between the ferromagnetic and antiferromagnetic interactions. Since the ordered alloy is ferromagnetic, the second and the third neighbour interactions are supposed to be ferromagnetic. It seems plausible to assume a little weaker interaction for the third neighbours than for the second neighbours. T h e absence of the first neighbour coupling in the ordered alloy and the presence of the same in the r a n d o m alloy, as shown in (3.3), lead us to assume that a strong antiferromagnetic interaction exists between the first neighbours Mn atoms in the r a n d o m alloy. If the magnetic interactions between Mn atoms can be assumed to depend only on the distance between the spins irrespective of the ordered and disordered states of the alloy, the magnetic interactions can be described using two p a r a m e t e r s a and y as following:
(3.4) J1 = Jllo = - a J2,
and 2" =
J2 = J200, Z l l 0 " J - 2200 + Z211 =
42.
(3.5)
T h e magnetic p r o p e r t i e s of the quenched Au4Mn r a n d o m alloy are characterized as follows: (a) the Curie t e m p e r a t u r e is a b o u t half that of the ordered alloy [13]; (b) the magnetization at low t e m p e r a t u r e is also about half that of the ordered alloy [13], and (c) the effective n u m b e r of Bohr magnetons of the r a n d o m alloy (determined by m e a s u r e m e n t of paramagnetic susceptibilities) is almost the same
(3.6)
J3 = .12,, = y J 2 .
The mean field value of the Curie t e m p e r a t u r e of the ordered alloy is then given by k T c = (2 + 8 y ) J 2
(3.7)
and that of the r a n d o m alloy by k T R = Jo = Z ( - p , a
+ p2+ PaY)J2.
(3.8)
Now it is quite interesting to see whether it is
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
possible or not to fit the magnetization vs. temperature curve by adjusting the p a r a m e t e r s a and y. For a few sets of ot and y, we have evaluated kTc/J2, kT~/J2, A J/J2, TR/Tc and J0/AJ, as shown in table I. As an example, a calculated magnetization for a = 2.65 and y = 0.7 is plotted against t e m p e r a t u r e in fig. 2, together with the result of the measurement. T h e magnetization for the ordered alloy is also shown in the same figure. T h e ratio of the calculated Curie t e m p e r a t u r e of the disordered alloy to that of the ordered alloy is about one half, and the calculated value of the m a x i m u m magnetization of the disordered alloy is about a half of that of the ordered alloy, in agreement with the main features of the experimental results. However, Table I T h e Curie temperatures and parameters of the model for a few sets of a and y
ct, y
kTc/J2 kT~/J2(=Jo/J2)
A~/2
T~/Tc
~/0/A.]r
0.8, 2.90 0.7, 2.65 0.6, 2.40
8.40 7.60 6.80
3.46 3.18 2.91
0.532 0.540 0.550
1.29 1.29 1.29
221
calculated magnetization for the r a n d o m alloy decreases at low t e m p e r a t u r e when t e m p e r a t u r e decreases further. This discrepancy will be discussed in section 4. It should also be mentioned that similar fitting of the magnetization curve is obtained for other sets of p a r a m e t e r s a and y, for instance, a = 2.90 and y = 0.8, and a = 2.40 and y -- 0.6.
4. Discussions
W e have presented a model of a r a n d o m magnetic alloy with short-range orders f o r atomic arrangement. For a r a n d o m binary alloy, the probability distribution of magnetic interaction is given by a discrete distribution, i.e.,
P ( J ) = ~ plm,,8(J -- Jtmn) + p o S ( J ) ,
(4.1)
where
1.0
4.47 4.11 3.74
M(T) /M(O)
• ~DERED M(T)/M(O)
~.
0.5
0.0
o
0.5
1.0
T/Tc
Fig. 2. Magnetization of the regular and the r a n d o m alloys Au4Mn. Results of m e a s u r e m e n t by Ido are d e n o t e d by black circles for the r a n d o m alloy [13]. Calculated values are represented by full lines.
Po = 1 - ~ p,m,.,
(4.2)
which denotes the probability of vanishing interaction. In section 2, w e h a v e replaced the probability distribution (4.1) by the Gaussian distribution with the same average magnetic interaction and the same variance as calculated f r o m (4.1). It should be emphasized that in spite of our drastic a p p r o x i m a t i o n - a d o p t i n g the Gaussian distribution for the i n t e r a c t i o n - t h e main features of the magnetic behaviour of the r a n d o m Au4Mn is well reproduced in this calculation. Therefore, we believe that our approach as given in this p a p e r is in the right direction as a whole. As for the decrease of calculated magnetization at low t e m p e r a t u r e mentioned in section 3, it may be a serious problem in the approximation made in this p a p e r or m o r e fundamentally in our construction of the model. First of all the effect of neglect of long-ranged and oscillatory R K K Y
222
S. Inawashiro and J. Harada / Magnetization of a random binary alloy
interactions may be considered as one possible reason, since we have taken the interactions only up to the third neighbours. But such termination for an infinite series of interactions would not cause serious effects, because it is expected that the inclusion of these long-range interactions could be covered to some extent by adjusting the parameters a and 3' in the present calculation, without affecting the magnetization essentially. The decrease of the calculated magnetization at low temperature is then considered to arise from the reduction of the magnetization, as a whole, due to the increase in number of antiparallel alignments of local magnetic moments at lattice sites. This may be attributed to our drastic approximation in which the continuous Gaussian distribution was adopted instead of the discrete probability distribution for the magnetic interaction, or more fundamentally to the neglect of many-body correlation of atomic arrangement in our model. These problems are left to future investigations. According to the measurements of paramagnetic susceptibilities of the ordered and the random Au4Mn, the effective numbers of Bohr magneton are pen = 4.8 and 4.5, respectively. Therefore, the use of the Brillouin function with S = 2 would be more appropriate than the Brillouin function with S = ½which have been used in the present simplified treatment. However, it is not considered that the magnitude of spin would essentially influence the results obtained. Anyway, refinement of theory along these lines would be desirable and will be left to the next stage of approximation. It seems appropriate here to comment that for a simple discrete probability distribution with two interactions - J and a few coordination numbers (z = 3 and 4), solutions in a pair approximation at absolute zero temperature were discussed in detail [14]. Extension to finite temperatures and to a more general discrete probability distribution with a larger coordination number is required. The discrete probability distribution could be
determined experimentally by an analysis of Xray and neutron diffuse scatterings, as mentioned in this paper. Therefore, we finally stress the importance of experimental and theoretical studies of random alloy systems with short-range orders, in order to understand the physical properties of random magnetic systems. We hope that problems along these lines will be further studied in the future.
Acknowledgments The authors wish to thank the School of Physics and the Department of Mathematics, University of Melbourne, for offering them an opportunity to collaborate in this work and for the hospitality during their stay. One of the authors (S.I.) wishes to thank the A R G C for their support. J.H. is grateful for the Sir Thomas Lyle Fellowship which enabled him to stay at the University of Melbourne.
References [1] S.F. Edwards and P.W. Anderson, J. Phys. 1::5 (1975) 965. [2] S. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. [3] B.E. Warren, X-ray Diffraction (Addison-Wesley, 1969). [4] H. Suzuki, J. Harada, T. Nakashima and K. Adachi, submitted to Acta Cryst. A. [5] J.M. Cowley, J. Appl. Physics21 (1950) 24. [6] D. Sherrington and S. Kirkpatrick; Phys. Rev. Lett. 35 (1975) 1972; Phys. Rev. B17 (1978) 4384. [7] C.J. Thompson, J. Stat. Phys., to be published. [8] A. Kussmann and E. Raub, Z. Metallkde. 47 (1956) 9. [9] D. Watanabe, J. Phys. Soc. Japan 15 (1960) 1251. [10] W. Koster and R.E. Hummel, Z. Metallkde. 55 (1964) 175. [11] P. Fiirnrohr, J.E. Epperson and V. Gerold, Z. Metallkde. 71 (1980) 403. [12] T. Nakashima, S. Mizuno, T. Ido, K. Sato, S. Mitani and K. Adachi, J. Phys. Soc. Japan 43 (1977) 1870. [13] T. Ido, Thesis (1973), Nagoya University, Chikusa-ku, Nagoya, Japan. [14] S. Inawashiro and S. Katsura; Physica 100A (1980) 24.