Theory of finite-temperature magnetism of NiMn. III

Journal of Magnetism and Magnetic Materials 43 (1984) 79-88 North-Holland, Amsterdam

75

T H E O R Y O F F I N I T E - T E M P E R A T U R E M A G N E T I S M OF NiMn. !II

Yoshiro K A K E H A S H I Max Planck lnstitut fur FestkOrperforschung, 7000 Stuttgart 80, Fed. Rep. Germany Received 15 December 1983; in revised form 26 January 1984

The magnetic properties of NiMn alloys at finite temperatures are investigated on the basis of an improved theory of the local environment effect (LEE). Magnetization vs. concentration curves, magnetization-temperature curves, phase diagram, high-field susceptibility and atomic short-range order dependence of the magnetization vs. temperature curves are calculated. The asymmetric behaviour of the high-field susceptibility around the critical concentration of ferromagnetism is explained by the LEE, as well as the itinerant character of the amplitude of the Ni local moment, for the first time. Two self-consistent solutions, corresponding to the high magnetization state and the low magnetization state, are found for a certain intermediate region of atomic order in Ni3Mn. Therefore, the existence of metamagnetism is suggested in that region. The anomalous double-stage magnetization-temperature curves in Ni3Mn alloys are explained by the transition from the high magnetization state to the low magnetization state.

1. Introduction

NiMn alloys show anomalous magnetic behaviour, such as deviation from the Slater-Pauling curve [1-4], large dependence of Atomic Short Range Order (ASRO) on the magnetization and the Curie temperature [1,2,5,6], existence of the spin-glass states and so on [7-10]. The explanation of these experimental facts was first given on the basis of the localized model. Carr and other investigators introduced a ferromagnetic exchange coupling between Ni atoms and between Ni and Mn atoms, and an antiferromagnetic coupling between Mn atoms [3,11-13]. The localized model with such a set of exchange parameters conceptually explains the deviation from the Slater-Pauling curve and the large change of the magnetization with varying ASRO. However, it has been clarified that the simple localized model often leads to unrealistic results in the actual calculations. Particularly, the vanishing of ferromagnetism at 25 at % Mn cannot be described by this simple localized model [14,15]. Recently, the validity of the localized model has been investigated on the basis of the functional integral method [16-20]. The localized model may have a meaning for Mn, but it is

difficult to justify it for Ni. A first investigation based on the itinerant model has been carried out by Hasegawa and Kanamori [21]. They calculated the electronic structure of NiMn alloy in the ground state by the use of the CPA and Hartree-Fock approximations. In spite of the successful application of their theory to other magnetic alloy systems, they could not obtain continuous deviation from the Slater-Pauling curve in NiMn, since they did not take account of the Mn local moments (LM) antip a r a l l e l to the m a g n e t i z a t i o n , in their H a r t r e e - F o c k CPA scheme. Jo took the antiparallel Mn LM into account in a self consistent way and derived a continuous deviation from the Slater-Pauling curve [22]. The simple CPA theory, however, does not explain the strong ASRO dependence of the magnetization, or the local environment effect (LEE), in Ni3Mn alloy. Many cluster-CPA theories as an extension of CPA, have been proposed by Tsukada [23], Miwa [24] and Brouers et al. [25]. More recently, the recursion method of Haydock et al. [26] has been applied to alloys and amorphous systems, in order to take account of the cluster effect [27,28]. Their theories enable us to investi-

0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

80

Y. Kakehashi / Finite-temperature magnetism of NiMn. 111

gate the electronic structure at T = 0, when the local moment configuration is given. However, the most important point in the actual applications to magnetic alloys is how to determine the local moment configuration in a self-consistent way. This is not an easy problem. It becomes serious when there are two Hartree-Fock solutions, which correspond to the parallel- and the antiparallel local moment configuration, as in NiMn alloys. For that reason, nobody was able to investigate until now the Local Environment Effect (LEE) of NiMn alloys at T---0 from the viewpoint of the itinerant electron picture. The extension of CPA to finite temperatures has been carried out by Hasegawa [29], on the basis of the functional integral method. The theory gives an overall description of the finite temperature properties of completely disordered alloys, except for the region of the critical concentration at which ferromagnetism sets in; for example, as in FeNi [30]. Moreover, the theory reduces to the original Hartree-Fock CPA scheme at T--- 0. The present author proposed a finite temperature theory of the LEE (which is referred to as I), on the bases of the Ising model picture [20], extending the CPA theory by Kakehashi [31]. The theory is based on a pair approximation, but the self-consistency of the local moment configuration is taken into account by adopting the method of the distribution function initiated by Matsubara and Katsura et al. for insulator systems [32,33]. The theory of LEE describes the fluctuation of the local moment (LM) with respect to atomic configuration and contains, therefore, the spin-glass state, which is not described by CPA theory or the Single Site Approximation (SSA) [34]. The magnetization vs. concentration curve, strongly variable magnetization with the degree of atomic ordering, concentration dependence of the Curie temperature T~ and the spin-glass temperature Tg and so on, have been investigated by adopting the theory. In the second paper (which is referred to as II), the theory was extended up to the Bethe approximation [35]. The susceptibility behaviour and reductions of Tc and Tg were also examined. Moreover, it has been shown that these quantities strongly depend on the ASRO. The purpose of the present paper is threefold.

First we reexamine the magnetism of NiMn on the basis of the improved theory, which has been proposed in the previous work for FeNi alloys [36]. The itinerant-electron characteristics, such as the LEE on the amplitude, are taken into account in the theory. The second purpose lies in the theoretical investigation of the asymmetric behaviour of the highfield susceptiblity. Okuda et al. have measured the high-field susceptibility of NiMn alloys as functions of the concentration. They found a clear asymmetry around the critical concentration of the disappearance of ferromagnetism [37]. Subsequently, the CPA calculation for the ground state has been performed by Jo [38]. However, this type of asymmetry could not be explained by simple CPA. Therefore it is quite plausible that LEE is important for this phenomenon. We explain the asymmetry by the concentration dependence of the amplitude of Ni LM and LEE. The third purpose is related to the double-stage magnetization vs. temperature curves in the intermediate degree of atomic order in Ni3Mn alloys [5,6]. It is interpreted by experimentalists as the result of the superposition of the macroscopic region with large magnetization and that with the small one. We propose here a different microscopic mechanism for this phenomenon. In the following section, we briefly summarize the calculation scheme and the parameters which are used in numerical calculations. Section 3 is divided into three subsections, according to the threefold purpose of this investigation described above. The results of the calculation of the magnetization vs. concentration curves, magnetization vs. temperature curves, local-moment behaviour, Local Density of States (LDOS) and so on, are given in the first subsection. The second subsection is devoted to the presentation of the result of the calculation for the high-field susceptibility, as a function of the concentration. The mechanism of the asymmetric behaviour of the susceptibility is given there. In the last subsection, the existence of metamagnetism is found for a certain degree of the atomic order in Ni3Mn. The explanation of the double-stage magnetization curves is presented together with the results of the numerical calculation. Both phenomena are closely related to each

Y. Kakehashi / Finite-temperature magnetism of NiMn. I11

other. Finally, we summarize our results in section 4.

~/'~k,(~) --- E.(~) + n~,,e(~) + (z - n ) ~ ( } )

2. Method of calculation

In the tl'~eory of LEE [36], the average LM of the atom a ((me)), the configurational average of the square of the LM ((m~) 2) are self-consistently determined by the following equations for a given effective medium L~- 1:

[--] 2 2 X,,V~,

-- [ ( m , , > 2 '

F(n,z,p~

(1)

).,

(i=1,2)

(2)

n=O

i ~

z--n

E r ( k , n, q e + ) r ( l , z - n, k=0 /=0

x (<,:o>.,,)',

(3)

q~+ = ½(1 +u-~Jo~).

tlbex

-(21-z+

n ~jb e xe~[~)v~, t ~ -x

ee(¢)e

f

Here, E,(~) is the impurity energy functional in the single-site theory [•7,30]. ~@ev(~) ( ~ v~( } ) ) is the atomic (exchange) pair energy functional between atom a with field variable } and the atom -g with magnitude xv of fictitious spin [361:

(~e>,,, = fP~,,t(~)~ d~,

f d 6 exp[ - flq'..,,(6)]

4~:~(¢)--_1 E

(8) v~ey(~, vxr),

(9)

where ~ r ( ~ e , ~r) is the pair energy functional between the neighbouring atoms a and y. On the other hand, one can determine the effective medium L~ ~ as a function of fie and G, by solving the CPA equation:

(5) ,

J

=

Here ( ~ ) (i = 1,2) is defined by

~

fined by f t , - - - ( m , > / x e and G = ~/(----m---a)2//Xa. F(n, z, p ~ ) is the binominal distribution function defined by [ z ! / n ! ( z - n ) ! ] ( p ~ ) " ( 1 - p y " ) ~ - " , where z is the number of nearest neighbours and p~e is the probability of finding an atom a at a neighbouring site of an atom a. It is given by Cowley's ASRO parameter ¢ as p~e = c~ + (1 c,)¢ [39]. c~ is the concentration of the atom a. q~+ shows the probability that the fictitious spin on the atom a with the magnitude G is in the up direction. (~,>,kt is the local moment of an atom of type a at the central site, when k of the fictitious spins among the surrounding n atoms of a point up and when in addition, 1 spins of the remaining z - n atoms of type R point also up:

exp[ - fl'/'~"*'(~)]

(7)

(4)

Here, x~ is the root mean square value of the field variable }~ with respect to the single-site impurity energy functional E~(},), which is defined by x] = fd(; ~2exp(-flE~(~))/fd~exp(-flEe(~)). Normalized quantities fi~ and o~ are respectively de-

Pe.,,(f)

-(2k-

E

(m~f =

<'no>. =

81

(6)

z--n

n~O

k=0 I~0

×F(l,z--n,g~+)fp,.,,t(l~)Ud/i.

(11)

Geo(~0, ~) is the electron Green function at the impurity site occupied by an atom a, with the field variable ~ being in spin state o [40] and Fo is the coherent electron Green function of spin o. DGI2 is the off-diagonal factor in the CPA [40]. In order to solve eqs. (1) and (10), we have adopted the iteration method. First one specifies the model DOS of the pure metal at T = 0 and the ASRO parameter r. Then one can calculate the effective medium L~1 by solving the CPA equation (10), assuming values for ~-~)(= (m,>) and (~2). Next one computes E~(~), x~ and ~-r and therefore P, okt(~), by assuming a value (me>. These functions produce a new set of parameters (m~), (me> ~ and (~2), according to eqs. (1) and

Y. Kakehashi / Finite- temperature magnetism of NiMn. I11

82

(11). This procedure should be iterated until the self-consistency is obtained. In the numerical calculation for NiMn alloy, we adopted the following set of parameters. nNi = 8.967,

WNi = 0.300Ry,

nMn = 6.200,

WMn = 0.415Ry,

~B

T=150K

]

,

UNi = 0.07642Ry, UMn = 0.06077Ry,

where n~, W~ and U~ are, respectively, the d electron number of the pure metal a, the d band width and the exchange energy parameter. These values are the same as in I and II. The fcc model density of states is also the same as the previous one and it is almost reproduced by the Ni local density of states (LDOS) for c = 0.0 in fig. 4a.

0~O~ 3. Results of the calculations

3.1. Concentration- and temperature-dependences of local moments The concentration dependences of the LMs are shown in fig. 1. The magnetization has a maxim u m at a concentration of 10 at% Mn. Then it decreases with increasing concentration and shows a transition from the ferromagnetic state to the spin-glass state, in agreement with experiments [3,7-9]. The overall features are the same as in I. However, the behaviour of each Mn LM specified by the number of the Mn nearest neighbours is considerably different from that calculated previously (see fig. 5a in I), as shown in fig. 2. According to the present calculation, the Mn LM with one or two Mn at the neighbouring site, changes more smoothly with concentration, which seems to be more reasonable. The structure of the LM distribution is rather simple, as seen from fig. 3. In the present case, there exists a broad distribution of LMs due to the atomic configuration, even if the ferromagnetism disappears. This behaviour is quite different to that of the FeNi alloy, where it is absent [36]. In the improved calculation, the LEE on the amplitude of the LM is taken into account. For the present alloys, it is not so important for the Mn LM, as shown in fig. 2c. The LDOS of atom c~ are shown in fig. 4a as a

0.3

0.2

0.1

Ni

Fig. I. Various local moments (LM) as functions of Mn concentrations at T = 150 K. Solids curves are calculated values of ( m r , . ) and (-m~i). Corresponding experimental values at T = 4. 2 K (room temperatures) are denoted by zx and [] (A and II L respectively [3]. Dotted curve is the calculated magnetization. The experimental values are shown by O (4.2 K) and • (room temperatures) [3,4]. The amplitudes of Mn and Ni LM are shown by dashed curves. The root mean square values of the thermal average of Mn and Ni LM are shown by dot-dashed curves.

function of the concentration. A peak which corresponds to the Mn LM being antiparallel to the magnetization appears in the spin-up band above the Fermi level and gradually grows with increase of the Mn concentration. The LDOS of various environments at c = 25 at% Mn are shown in fig. 4b. The LDOS of Mn strongly depend on the LEE. However, their LDOS at the Fermi level are small for every environment, as seen in the figure. Temperature variation of the magnetization and other local quantities are shown in fig. 5. Overall behaviour of the temperature variations of the amplitude of the LM (~/~--~)) and the configurational average of the square of the LM (~/~m~) 2 ) are similar to the previous results in I. Magnetization curves in the high temperature region are also similar. Therefore, the phase diagram proposed

83

Y. Kakehashi / Finite-temperature magnetism of NiMn. 111

ii

I.tB

I

C=0.05

n=O

3

I

I

-2

O

M

11=0

2

0.5

:: ~.~:~:.:;-'.:--:;~ ....

1

c--o.15

I

-2

0

0'.3 ~

N~

oi,

i!

i

i i'N 0

r'l ,

2

C=0.25 ,-J i -~

0

2

t2

-2

c:0.30

T=15

i i i'-q

-3 -2

0

5] 2

1,4 I.tB

Fig. 3. The LM distribution g~(]) for various concentrations. Solid lines show the Mn LM distribution. The dotted lines show the Ni part.

b 0.3

'~/trn:6)n~ I.tB

J-

.I

0.2

0.1

Ni

T = 150 K

n=O

-

!

n=Q

"~i

p r e v i o u s l y h a r d l y changes, as shown in fig. 6. H o w e v e r the m a g n e t i z a t i o n m a x i m u m as a function o f t e m p e r a t u r e does n o t a p p e a r in the p r e s e n t calculation, at least a b o v e 100 K, in d i s a g r e e m e n t with the p r e v i o u s result [31]. This m e a n s that the p r e v i o u s a p p r o x i m a t i o n scheme in I is n o t so g o o d for T_< 200 K. W e leave the p h e n o m e n o n o f the m a g n e t i z a t i o n m a x i m u m in N i 3 M n d i s o r d e r e d all o y discovered b y Kouvel, G r a h a m a n d Becker [41,42] for a future investigation. T h e t e m p e r a t u r e variation o f the L M s in various local e n v i r o n m e n t s is also shown for c = 0.1 in fig. 7. T h e p r e s e n t c a l c u l a t i o n shows a s m o o t h t e m p e r a t u r e variation, even at low temperatures.

o'z ' o'~ ,CNJo

Fig. 2. (a) Concentration dependences of the LM in various environments, ~ at 150 K. The solids (dashed) curves show the Mn (Ni) LM. Integers n in the figure show the coordination number of Mn; (b) concentration dependences of ~/~, the root mean square of the average Mn LM in various environments. Notice that the scale of the vertical line is different from (a) and (c); (c) the amplitude of the LM in various environments.

Y. Kakehashi / Finite-temperature magnetism of NiMn. 111

84

The same holds true for the magnetization. The rapid temperature variation at about 200 K, which appeared in the previous calculation [31], does not exist here. The different temperature variations of

P,

c:o.o

..~

C =0.25

x= -0.33

P,,

C

/\

a

Fig. 4. (a) The c o n c e n t r a t i o n d e p e n d e n c e s of the local density of states ( L D O S ) at T = 150 K. The F e r m i energy c o r r e s p o n d s to ~ = 0. The solid (dotted) curves show the M n (Ni) part; (b) L D O S in various e n v i r o n m e n t s at 150 K for c = 25 at% Mn. Solid curves shows L D O S in the e n v i r o n m e n t with no nearestn e i g h b o u r M n atoms. D a s h e d curves, L D O S with 6 M n nearest neighbors. D o t t e d curves. L D O S with 12 M n nearest neighbours; (c) L D O S with the a t o m i c short range order ( A S R O ) 'r = - 0 . 3 3 3 3 at c = 25 at% M n and T = 150 K. - the M n part. - . . . . . , the N i part.

'.

.... ",i...-..,_ ........ '[ ...............

the LM with different local environments has been recently checked by Yamagata in dilute NiMn alloys, by using the N M R technique [43]. Mn

PMnt

\

b

C=0"25

2

1

-0.t. " " ~ . ' . : , , - ~

20

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

:-.'-....~oo Ry

'i~!/'"':" I V

......... n = 1 2

PNi t

0,5

20 0

20

0

gB Ni

.... -~:~.

...6;':

-"i.:

t~ Ry

Fig. 5. T e m p e r a t u r e v a r i a t i o n s of various LM. The u p p e r part shows the M n LM. The lower part, the N i LM. - - , <--~;

Y. Kakehashi / Finite- temperature magnetism of NiMn. 11I

3.2. Asymmetric behaviour of the high-field susceptibifity

TK NiMn 1500 -

"',,

P

100C

500

0.t,

0.2

85

Ni

Fig. 6. Magnetic phase diagram for completely disordered NiMn alloys. F, G and P, respectively, show the ferromagnetic phase, the glass-like phase and the paramagnetic phase. Solid lines show the present result. The dashed curves show the result in I. The dot-dashed curves show the result of the Bethe approximation in II.

The high-field susceptibility was obtained by numerical differentiation of the magnetization with respect to the magnetic field. The results are shown in fig. 8, together with the experimental data [37] and the result of the CPA [38]. The present result shows two peaks in the ferromagnetic region. They are related to the reversal of the Mn LM with 1 and 2 Mn nearest neighbours (see fig. 2a). If we take account of the further distant LEE, the peaks may be broadened and disappear. The peak around the critical concentration where the ferromagnetism disappears shows a clear asymmetry, in agreement with the experiment. The difference between the calculation and the experiments is not serious. In fact, the induced magnetization seems to be nonlinear in the experiment, even down to the small external field [37]. Thus there is an ambiguity in the interpretation of the measurement in the present case. The origin of the asymmetry is simple, but neither the SSA nor the simple Heisenberg model

× (xl03PB/Ry) PB

C=0.1 I.tB J

1

2

~o 0.5

i

10'0 -1

-2

0 5

-3

Fig. 7. Temperature variation of ( m ~ ) . at c = 10 at% Mn.

0,~;

0.2

Ni

Fig. 8. Concentration dependence of high-field susceptibility per atom at T = 150 K. Solid curve shows the present result of the theory of the LEE. The dotted curves show the result of the simple CPA [22]. The experimental results are shown by the dot-dashed curve and the open circles [37]. The dashed line shows the critical concentration of the vanishing of the ferromagnetism.

Y. Kakehashi / Finite-temperature magnetism of NiMn. HI

86

can describe this phenomenon, because it is caused by both the itinerant characteristic of the amplitude of the Ni LM and the LEE. The behaviour of the Mn LM surrounded by the Ni LM is shown schematically in fig. 9 as an example. The Ni LM have considerably large amplitudes in the ferromagnetic states. Thus a ferromagnetic coupling between the Ni and the Mn atoms is effective (see fig. 9a). However, the amplitude of the Ni LM becomes small in the spin-glass region. Then we have a so-called "dead coupling" between the Mn and Ni atoms. In that case, the Mn LM surrounded by Ni atoms shows a Curie-like behaviour in response to an external magnetic field, which largely contributes to the total susceptibility in the spin-glass regime. The reduction of the amplitudes of Ni LM by such a mechanism results from the induced local moment character, which is due to the singleminimum energy functional ENi(~) [44]. Co LM, for example, has a LM character rather than an induced character (see fig. 2 in ref. [45]). Therefore, we can speculate that CoMn alloys do not show a pronounced asymmetry, although CoMn's magnetization vs. concentration curves are similar to the NiMn alloy.

3.3. Double-stage temperature vs. magnetization curoe

We reinvestigated the ASRO effect in Ni3Mn alloys, by using the improved theory. The present scheme produces the rough structure of the DOS, even for the ASRO parameter corresponding to the C u 3 A u type structure as shown in fig. 4c. This holds, although a simple pair approximation has been made in the calculation [46]. Fig. 10 shows the magnetization and other LMs in NiMn as a function of the ASRO parameter. For a small regime of • (i.e. 0.18 < "r < 0.195), we found two self-consistent solutions (the high magnetization state and the low magnetization state). This means that there is the possibility of the appearance of metamagnetism for a certain intermediate region of the degree of the atomic order. Whether or not this occurs in the experiments is not clear at present [5]. The existence of two solutions provides a mechanism for the double-stage magnetization vs. temperature curvesin Ni3Mn alloys. In other words, we can easily expect a transition to occur with increasing temperatures, from the high magnetization state to the low magnetization state. The

lu8

Ferro

~ 0 o71dnNi> 0

"-~Ni

$.G. QNi~ JM.Ni ~ 0 L

-0.3 Fig. 9. Behaviour of the Mn LM surrounded by Ni LM. The ferromagnetic couplings between M n and Ni LM are effective in the ferromagnetic region (Ferro), while it nearly dies due to the small amplitude of Ni LM in the spin-glass region (SG).

L

I

-0.2

I

1

-0.1

I

I

0

Fig. 10. Various LM as a function of the A S R O (~-) at c = 25 at% Mn and T = 150 K. - - , ~ - ~ - > ; - . - , magnetization; . . . . . . , ~ / ( m ] ) ; . . . . . , ~/~-m~ 2 . The dotted curves in ( r a M , ) m e a n the extrapolated ones.

Y. Kakehashi / Finite-temperature magnetism of NiMn. 111 ~ " (/~B/atom)

Ni3Mn

a

1.0

0.5

I

500

M 1.0

•

1000 TK

N i0.763Mn0.237

b

0 .................O'O"o,..,_x = - 0.31 vO.

"0,0."

°°""°°'o-o 0.5

0 1 ~ I

~.: - 0.25 o.. .o.° i~..

~ : m0.22 Oee,Oi el

"b

O.Oao't =-0.1t. R:b°"(~..,..oo~o.o°"o"h 6 0

°,x~;~oog-o..-o,. o....o.o..n , q ~ 500

I

TK'

Fig, 11. (a) Calculated magnetization vs. temperature curves for various ASRO parameters ~"at c = 25 at%Mn; (b) experimental curves by Marcinkowski and Brown at 23.7 at% Mn [5]. ASRO parameters are estimated from their values of the long range order parameter S by using the approximate relation ,r = - 16c(1 - c)$2/9 [39].

present theory cannot treat the phase stability between them. Therefore, we calculated the magnetization-temperature curves as functions of the ASRO ~, by assuming that the high magnetization states are always stable at low temperatures. The results shown in fig. 11 explain well the doublestage magnetization-temperature curves in the intermediate region of atomic order.

4. Summary Finite temperature magnetism in NiMn alloys was reinvestigated by using an improved theory, in which full integration with respect to the field

87

variable at the central sites is performed. By that, the magnetization vs. concentration curve and the concentration dependence of the various other LMs are reasonably explained. The present calculation shows a normal magnetization vs. temperature curve for all concentrations in disordered NiMn alloys, at least above 100 K, in disagreement with the previous results, in which a magnetization maximum appeared at 150 K in 20 at% Mn alloys. The previous simpler theory does not give good results at low temperatures. Asymmetric behaviour in the high-field susceptibility as a function of the concentration was explained by the present theory of the LEE. It is caused by the LEE and a large reduction of the magnitude of the coupling between Mn and Ni LMs in the spin-glass region, due to a reduction of the amplitude of the Ni LM. We found that the metamagnetism appears in NiaMn alloy for a certain intermediate region of the degree of order as a result of the existence of two possible solutions; that is, a high magnetization state and a low magnetization state. The existence of the metamagnetism in the range of intermediate degree of order has not yet been established in the experiments, although the observed magnetization by Marcinkowski et al. rapidly changes in that region as a function of the ASRO parameter [5]. Detailed experimental investigations in an external magnetic field are highly desirable, for various degrees of atomic order in Ni3Mn. The double-stage magnetization vs. temperature curves for intermediate degree of order in Ni3Mn are closely related to the metamagnetism and were explained by the transition from high magnetization state to low magnetization state with increasing temperature. The transition is caused by the microscopic LEE. The mechanism is quite different from that customarily used for interpretation by experimentalists, where macroscopic regions of large and small magnetizations are assumed [5,6]. Detailed MOssbauer experiments around the transition temperature from the high magnetization state to the low magnetization state may be useful, in order to clarify the validity of our theory. There must be a clear difference in the pattern of the internal field distribution between the microscopic

88

Y. Kakehashi / Finite-temperature magnetism of NiMn. II1

transition and the macroscopic one. Finally, we briefly discuss the spin-glass state in NiMn. The spin-glass state of our scheme is close to the concept of a locally ordered state, where each local moment feels a considerable molecular field, although the spatial distribution of the LMs is random. This state produces a large spin glass transition temperature T~. The frustration effect may reduce Tg, but the local lattice distortion accompanied by the frustration may enhance Tg. The ASRO effect also strongly reduces Tg, as has been shown in II. Thus the actual situation seems to be very complicated. Further theoretical and experimental investigations are highly desired. M6ssbauer measurements especially would be useful for the determination of T~, because the vanishing of the width of the internal-field distribution just corresponds to Tg, as has been pointed out in I and II. The corresponding susceptibility anomaly is not so clear in this system, according to the previous theoretical investigation [35]. Another problem which prevents us from a detailed understanding is the ambiguity in the definitions of the so-called spin-glass state, the mictomagnetism and the superparamagnetism. All the spin-glass theories cannot yet describe the difference between them, because there is only one order parameter ( m i ) 2 u s e d in them. We have to introduce other quantities into the theory, in order to specify the spatial size of the magnetic clusters. This is a problem for the future.

Acknowledgement The author would like to express his sincere thanks to Prof. P. Fulde for refining the manuscript and for valuable discussions.

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