Electronic, phonon and magnon specific heats of FePd alloys

Physica B 149 (1988) 209-216 North-Holland, Amsterdam

ELECTRONIC, PHONON AND MAGNON SPECIFIC HEATS OF Fe-Pd ALLOYS

Jian-ping KUANG, Masaaki KONTANI, Masaaki MATSUI a and Kengo ADACHI Department of Physics, Faculty of Science, Nagoya University, Nagoya 464, Japan aDepartment of Iron and Steel Engineering, Faculty of Engineering, Nagoya University, Nagoya 464, Japan

Low temperature specific heat of disordered (fcc and fct) and ordered (CuAu I and Cu3Au types) F e - P d alloys has been measured. The coefficients of electronic, phonon and magnon parts were analysed. The density of states attains a maximum, while the Debye temperature and the spin wave stiffness constant show a minimum at the boundary between the fcc and fct phases (33.4% Pd). For the ordered FePd and FePd3, a lower value of the density of states, and higher values of the Debye temperature and the spin wave stiffness are found. The spin wave stiffness constant for the 32% Pd alloy is also obtained by a thermomagnetic measurement. The ordered FePd gives a value of 93 -+ 10 meV ~2, while FePd 3 110--+ 10 meV ~2 at T = 0. Results are discussed from the standpoints of the itinerant electron ferromagnetism.

1. Introduction

Disordered Fe-Pd alloys form fcc structure m a wide region of the concentration from 25 to 100% Pd at room temperature. All the alloys show ferromagnetism [1-3] and the magnetic properties have been extensively investigated by measurements of magnetic anisotropy [4-6], magnetostriction [7, 8], M6ssbauer effect [9-11] and so on. In fig. 1 the Curie point and the ferromagnetic moments per atom for this system are shown.

In the concentration region from 30 to 35% Pd, the alloys show the invar effect [12] similar to Fe-Ni and Fe-Pt alloys, where the thermal expansion coefficient is extremely small near the Curie temperature. In connection with this effect, a structural transformation from fcc to fct has been reported at lower temperature near the invar region as shown in fig. 2 by X-ray diffraction and electron microscopy methods [13-16]. Since the transformation is thermally reversible (of second order), this is regarded as a thermo-elastic martensite transformation, where the fct phase is located in the region of 28 to

A

E

Fel-x Pdx

A

2.0

400

tO

\ 1000 v

E

~o

200

fit) v

500

"41.0

i

0.0

02

04

X

0 6

0.8

I

1.0

Fig. I. Concentration dependence of the Curie point T c and the saturation moments per atom at 4.2 K in the quenched Fe~ xPdx alloys [3]. In the case of moment, open, closed and half-open circles are data on the fcc, bcc and fct phases respectively.

0.22

I

I

0.26

I

0.30

I

I

1

0.3/,

X Fig. 2. Phase diagram of the quenched Fe~ ~Pdx alloys [15[. T,(x) is the fcc to fct transformation. In the region between the dashed and chained curves, the fct and fcc structures coexist.

0378-4363/88/$03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

210

J.P. Kuang et al. / Specific heat of Fe-Pd alloys

33.4% Pd between bcc and fcc phases at 0 K with a change of the ratio of lattice parameters, c/a, from 0.92 to 1.0. Some investigations on this transformation and on its related p h e n o m e n a have been carried out recently and remarkable features were revealed by means of measurements for magnetostriction [7], magnetic anisotropy [4], elastic constants [7, 17], Mrssbauer effects [10] and neutron scattering [18]. Data on the specific heat [17] were partly reported by us. Thus the mechanism of the transformation was elucidated as a cooperative p h e n o m e n o n accompanied with a lattice softening [17-19]. On the other hand, two kinds of ordered phases, FePd and FePd 3 having the tetragonal CuAu I and cubic Cu3mu type structures, are stabilized in the equilibrium phase diagram. Both show ferromagnetism [2] with slightly different values of the Curie point and the saturation m o m e n t than those of the disordered FePd and FePd 3. The low temperature specific heat only of ordered and disordered FePd 3 has been reported by Bechman et al. [20]. They obtained the coefficients of electronic and lattice heats without the spin wave term. Though many investigations on the magnetic and lattice properties of this alloy system have been done, as mentioned above, there is almost no investigation on the electronic structure in the whole system from both experimental and theoretical standpoints. The phonon and magnon behaviors were partly revealed by neutron inelastic scattering [18] for two kinds of the alloy at higher temperature, but the low temperature properties have not been examined so far. In this paper, in order to make clear the electronic state and the phonon and magnon characteristics systematically for this alloy system, especially, to elucidate the electronic origin on the martensite transformation and the superlattice formation, measurements of the specific heat at low temperature are carried out. Then the density of states for electrons, the Debye temperature and the spin wave stiffness constants will be obtained as a function of the Pd concentration. In addition, the magnon behavior is also examined from the thermomagnetic curve for an alloy with martensite transformation. The

results obtained will be discussed from standpoints of the localized and itinerant electron ferromagnetic theories.

2. Experimental procedure 2.1. Sample preparation

The Fel_xPd x alloys were prepared by melting in an arc furnace with necessary amounts of 99.99% Fe and 99.95% Pd, using a water-cooling cooper crucible in argon atmosphere. After annealing in vacuum at 1100°C for 168 h in a quartz tube, the samples were quenched into iced water. Each specimen was confirmed to have a single fcc structure by X-ray analysis. Thus thirteen kinds of specimens from x = 0.3 to x = 0.80 were prepared. Measurements of the specific heat were made first on these quenched specimens. To prepare the ordered specimens, the sample containing 50% and 75% Pd was heat-treated for 24 h at 1000°C, cooled slowly to 300°C at the rate of 50°C per day, and then furnace-cooled to room temperature. The X-ray diffraction patterns clearly revealed the superlattice reflections corresponding to the CuAu I and Cu3Au type structure for FePd and FePd 3 respectively. 2.2. Specific heat m e a s u r e m e n t

Heat capacity was measured by a calorimeter consisting of a vacuum can immersed in liquid helium in which a disk-shaped addenda made from gold-plated copper is hung. The addenda can be cramped so as to connect or break thermally to the can, by a mechanical heat switch operated from the outside in room temperature. A germanium resistance thermometer is fixed to the addenda and a heater made of manganin wire is wound around it. Resistivity of the heater and the thermometer is 256 and 1242 O at 4.2 K respectively. The specimen in a button form of 2 g in weight was mounted on the addenda face by G E 7031 varnish, the surface of which was polished to contact thermally to the addenda. The measurement unit is composed of three parts: a comput-

211

J.P. Kuang et al. / Specific heat o f F e - P d alloys

ing digital multimeter to observe the voltage drop across the thermometer, a programmable dc current/voltage generator to produce the heat-pulse current and a digital multimeter to monitor the voltage drop across the heater. All the operations were carried out by an on-line computer control to facilitate the speed and precision of the measurement. Thus the specific heat was measured in the temperature range of 1.4 to 20 K.

specific heats are expressed a s / 3 ' T 5 and a ' T 5/2 respectively. These terms will be used as a correction term to fit the experimental data to eq. (1) as shown below. On the other hand, based on the localized spin model, the change of magnetization in the low temperature region is expressed as follows: AM M(O) - M ( T ) M(0) M(0)

= AT3/z

2.3. T h e r m o m a g n e t i c m e a s u r e m e n t

where M ( 0 ) = N g ~ B S and

is a saturation moment

On the other hand, the thermomagnetic curve for a disordered alloy with x = 0.32 was measured by a low frequency (2 Hz) vibration sample magnetometer from 4.2 K to room temperature under a magnetic field of 17 kOe. For this alloy, the magnetization saturates almost completely above 5 kOe at 4.2 K, though this alloy forms fct at this temperature.

3. Results and analyses 3.1. Expressions specific heat and magnetization [21]

The low temperature specific heat of a metallic ferromagnet can be expressed as C = TT

+ / 3 T 3 + o t T 3/2 ,

(1)

where the three terms indicate electronic, phonon and magnon specific heats respectively. The coefficients %/3 and a are given by

2

2

, / = ('rr / 3 ) k B N ( E F ) N A , fl = (12 / 5)~r4kaNa/ O3 ' Ol ~ 0 l l 3 N A ( a 3 / 4 ) k ~ / 2 D -3/2

(2) (3) (4)

where kB, Oil, N ( E v ) , N A , a and D are the Boltzmann constant, the D e b y e temperature, density of states at the Fermi level, EF, total number of atoms, lattice parameter of fcc and spin wave stiffness constant respectively. The next terms to be added to eq. (1) at higher temperature for the phonon and magnon

,

(5)

0.0587a2 (~_~a)3/2 A -

4----if---

(6)

for a fcc magnet and D and S are the spin wave stiffness constant and the spin quantum number respectively. Within this model, A, is taken as a constant and S =/x/2/.%, /x is an averaged moment per atom. The next term at higher temperature appears in B T 5/2 due to the spin wave dispersion curve, say D ' q 4 in htoq = D q - D ' q 4. In the case of an itinerant electron ferromagnet, however, the D value itself undergoes a temperature dependence [22, 23], written as D'(T) = D ( 1 - a'r 2 + b'TS/2) .

Therefore the variation of A M / M ( O ) model is indicated as

(7) in this

A M / M ( O ) = A ' ( T ) T 3/e + C T 2 ,

(8)

where A ' ( T ) contains D ' ( T ) through eq. (6) and the term C T 2 originates from the Stoner excitation. This equation will be applied to the observed thermomagnetic curve. 3.2. Results and analysis o f the specific heat

Following the usual way, we plotted the observed data of Fel_xPdx alloys as C / T v s . T z for the disordered (0.3 <~ x ~< 0.8) and ordered (x = 0.50 and 0.75) specimens. The results are shown in figs. 3a, 3b and 3c. In these curves, the deviations from a straight line can be seen in details. The curves from x = 0.30 to 0.45 turn down,

J.P. Kuang et al. / Specific heat of Fe-Pd alloys

212

while those from 0.50 to 0.80 turn up with increasing T 2 and the m a x i m u m bending occurs at about x = 0.34 and x = 0.75 respectively. On the other hand, most of the curves show m o r e or less a slight change of curvature in the low t e m p e r a t u r e region of T 2 = 30-40. For the or-

60.-

60: C

X=O.8.0~'f"""

..

sJ7 _ __~

.EE 20."

ao;

40.-

"

20.o.65

0'0~

0 Ea0:

100 '

200 ' T 2 (K 2)

30'0

400

-

Fig. 3(a,b,c). The specific heat data of Fel_xPd . alloys plotted as C/T versus T 2. Each solid line is the best-fitted curve of the present analysis.

30.

O.

o

2oo

T 2 ( K 2)

601 b

30o

46o

X ~

4o-

ol

0

100

200 T2(K 2)

300

400

dered specimens, the curve shows an obvious difference from the disordered one. Applying eq. (1), we analysed these data to obtain the y,/3 and a values simultaneously as a function of x. In the data for 0.32 ~< x ~ 0.40, where the downward curvature is seen, the term a T 3/2 contributes considerably, implying a small D value, while in the other curves having a slight curvature at T - 6 K , the contribution is less. The curves for x >/0.5 showing the upward curvature cannot be fitted only by eq. (1) and additions of the higher order terms, such a s / 3 ' T s and a ' T 5/2, a r e necessary. The best-fitted power of T to these data is determined to be T 5 rather than T 5/2. Therefore the upward curvature originates from the higher order phonon term. Fig. 4 shows a concentration dependence of the coefficient of 3' and the D e b y e t e m p e r a t u r e 0o obtained from t h e / 3 value. In this figure, the values [24] for the Fel_xNi x alloys are also exhibited for a comparison, where 0D was calculated from their/3 values. It is r e m a r k a b l e to note that both 3' and 0o values of F e - P d alloys show a prominent p e a k and valley at the boundary between the fcc and fct phases. The increase of 3' with decreasing x in the fcc phase is not so sharp

J.P. Kuang et al. / Specific heat of Fe-Pd alloys

500

213

300

Fel-x Pdx , ordered FePd or FePd 3

+

~1l/

j-

/

~'~"

eo--~ o

4.-. O0 ---

E

r~

---

CD

300

12.'

"~ 200 > o

100

I I

fcti

10.

fcc

A

~ 8.

I

t

200

~6. !

~.4.

I

l"

fctifcc

1 . . . . .

o~ - ~ ~

-

i 2° 1

0.

o

0~3

0:4 0~5 016 0.'7 0'.8 X

Fig. 4. Electronic specific heat coefficient 7 and the Debye temperature OD as a function of x for the disordered Fel_xPdx (solid lines) and Fel_~Ni x alloys [20] (dashed lines). Open circles are data for the ordered FePd and FePd 3. Vertical chained line is the boundary between the fcc and fct phases.

as compared with the case of F e - N i but decreases in the fct phase. The change of 0D is contrary to that of 3, and the curve approaches to 245 K for pure Pd [21]. For the ordered alloys, FePd and FePd3, on the other hand, the lower and the higher values for 3' and 0D respectively are found in both superlattices as compared with the disordered ones and this agrees with ref. [20] for FePd 3. Fig. 5 shows a concentration dependence of the spin wave stiffness constant obtained from the ot values. It is noteworthy also here that the D value takes a minimum at the phase boundary signifying a softening of the stiffness. The D value for x = 0 . 3 2 , for which the martensite transformation takes place at higher temperature, is given to be 93---10 meV A2. The values obtained both from the thermomagnetic curve and from the neutron scattering [18] at room temperature for x = 0.28 and 0.37 are shown in the same figure. For both the superlattices, the

o14

o 0:3

o'.5 X o16

017

o18

Fig. 5. Spin wave stiffness constant D as a function of x in disordered Fe 1 xPd~ (closed circles). Open circles indicate D values for the ordered FePd and FePd 3. The vertical chained line is the boundary between the fcc and fct phases. The closed triangle for x = 0.32 is the value obtained from the thermomagnetic curve in the present experiment (see fig. 7). Closed squares for x =0.28 and 0.36 are the values from neutron scattering [18] at room temperature.

larger value is determined as shown in the figure. The coefficient of the higher order phonon term, /3', for x I> 0.5 including the ordered ones is shown in fig. 6 and it has a maximum at about x = 0.75. The obtained values of 7, /3, a, /3' and the calculated density of states at the Fermi level from T values of the present experiment are summarized in table I.

xlO -s

2.0

Fel-xPdx

A

-61.5

E ~1.0

0.5

0.0

.5'o 'o. o

X

o.- o

o.13o

Fig. 6. Coefficient of the higher order phonon term /3' expressed i n / 3 ' T 5 as a function of x in Fe 1 xPd~.

214

J.P. Kuang et al. / Specific heat o f F e - P d alloys

Table I Values of 3, (mJmol 1K 2), /3 (mJmol 1K 4), /3, (mJ mol 1K 6), a (mJ mo1-1 K -5/z) and N(EF) (states Ry I atom -1) of the disordered Fe 1 xPdxalloys and the ordered FePd and FePd3, where the notations are, respectively, the coefficients for the electronic, phonon, higher order phonon, magnon parts and the density of states. fl' × 10 5

x

y

N(Ev)

fl

ot

0.30 0.32 0.34 0.38 0.40 0.45 0.50 0.55 0.60 0.65 0.68 0.75 0.80

4.45 5.34 5.99 5.34 4.9 s 4.5 z 4.09 3.72 3.39 3.13 3.07 2.74 2.69

25. 7 30. 8 34. 6 30. 8 28. 6 26.1 23. 6 21. 5 19. 6

0.099 0.121 0.133 0.105 0.090 0.077 0.067 0.062 0.059

18. 0

0.049

17. 7 15. 8 15.~

0.060 0.064 0.067

0.138 0.39 o 0.763 0.367 0.138 0.145 0.140 0.169 0.148 0.049 0.156 0.123 0.327

3.26 3.87 7.5 z 5.56 9.34 9.98 6.44

18. 3 10. 2

0.034 0.054

0.081 0.109

4.30 15.08

0.12 Feo.68 Pdo.32 0.10 0.08

AM(r) = A.T312 M{0)

o

A =1"68X10-5 K-3/2

~

o.o,

o

oo4

o

f

J

°

0.02 0.0

0.0

1.'0

210

310

T3/2(K312)

4'.0

5.0

Xl0 3

Fig. 7. Fractional change of the thermomagnetic curve o f Feo.68Pdo.32 plotted as AM(T)/M(O) vs. T 3/2 measured in

17 kOe. T, indicates the transition temperature from fcc to fct structure.

Ordered phase

FePd FePd 3

3.3.

3.18 1.76

T h e r m o m a g n e t i c curve f o r x = 0.32

T h e r m o m a g n e t i c curve of the disordered specimen with x = 0.32 was obtained in the wellsaturated state under a magnetic field of 17 k O e from 4.2 to 300 K. The change of the magnetization, A M ( T ) / M ( O ) , is shown in fig. 7 as a function of T 3/2. Thus, the D value is obtained to be 110_+ 10 meV A2 for a wide t e m p e r a t u r e range up to 180 K ( - T c / 3 ). This value agrees with the value of 93-+ 10 obtained above within errors. The deviation from the straight line, the T 3/2 law, can be fitted as a whole by taking A T 3/2 + B T 5/2 up tb 300 K assuming the localized spin model. In this way, however, D = 133 meV A 2 can be obtained from least square fitting, which is far from 93 meV/~2. Therefore, the reason of deviation should rather be taken as a t e m p e r a t u r e dependence of the D value as given in eqs. (7) and (8) based on the itinerant electron system. On the other hand, according to the spin fluctuation theory in the itinerant electron ferromagnetism developed recently by Moriya [25],

the local m o m e n t , / x = 2S/xB, in metal undergoes a t e m p e r a t u r e dependence. Since this factor affects the deviation from the T 3/2 law in the thermomagnetic curve, we cannot determine the numerical constants separately appearing in the expression for D ( T ) . Our values of D(0) for x = 0 . 3 2 , 93 and l l 0 m e V A 2, determined above, are c o m p a r e d with the results [26] hitherto determined in the other invar alloys, D(0) = 140 meV,£.2 for Fe0.65Ni0.35 and D(0) = 80 meV/~2 for Fe0.75Pt0.25. As a final r e m a r k in this section, it is strange that we could not find any anomaly in the therm o m a g n e t i c curve at the martensite transformation point 7", = 180 K, as seen in fig. 7. Similar effects at T t have also been observed in the thermal expansion [11], electrical resistivity and M6ssbauer effect m e a s u r e m e n t s [27], although this structural transformation was confirmed already by diffraction experiments, magnetic anisotropy and magnetostriction measurements.

4. Discussions In the preceding sections, we have determined the concentration dependence of y, 0D and D for

J.P. Kuang et al. / Specific heat of Fe-Pd alloys 0.30~
4.1. Density of states As shown in fig. 4, the y value in the fcc phase increases with decreasing x but the increase is not so steep compared with the case of F e - N i alloy. This steep increase was explained by the coherent potential approximation theory by Hasegawa and Kanamori [28] and they verified the instability of ferromagnetism at about 35% Ni where the m o m e n t disappears. In the F e - P d alloy, however, the m o m e n t does not disappear but increases continuously to the boundary between the fct and bcc phases in spite of the decrease of y in the fct phase. From such behaviors, it is conceived that the peak and the slope of the majority d spin band, N+(E) in ref. [28], are gentler than that of F e - N i and the instability of the moment cannot break out but the instability can be released by the lattice distortion to the fct structure. Thus with decreasing x, the Fermi level E v can ascend to the slope of N+(E) in the fcc region without the instability of the moments and when x goes across the boundary, the peak of N + ( E ) begins to spread due to the tetragonal distortion, then the y is lowered. On the other hand, the decrease of y in the ordered FePd and FePd 3 can be considered as a change of N+(Ev) a n d / o r N ( E F ) due to the formation of a superlattice Brillouin zone. In either case, it is desirable to calculate the band structure of the F e - P d alloys for both the disordered and ordered states.

4.2. Debye temperature The anomaly of 0o around the phase boundary shown in fig. 4 signifies a lattice softening effect caused by the martensite transformation in the fct phase. This effect extends to the wide region of fcc phase, 0.34 < x < 0.6, as a pre-martensitic transformation, in which local distortions poss-

215

ibly take place. Whereas temperature variations of the elastic constants have been studied in alloys near the transformation point by direct [11, 17] and neutron scattering [18] measurements. Both results confirmed a softening of the constants, especially for a remarkable decrease of Cll-C12 toward the transformation point. Thus the Debye temperature calculated from the observed CH, C12 and C44 is temperature dependent, that is, a decreasing 0D with decreasin~g T. The specific heat curve plotted as C~ T vs. T is affected by this effect with a turn down curvature for the specimen near the boundary. Therefore, the obtained value of D, such as 93 meV ~2 for x = 0.32, is thought to be underestimated. On the other hand, the higher order phonon term expressed as /3'T 5 occurred in the specimens for x I> 0.5 and even for the ordered ones. The appearance of this term is possibly attributed to a deviation from the ordinal p h o n o n spectrum, g(v) ~ V2, in the low frequency 4 region, such as an addition of a v spectrum or some local mode one. The reason is not clear in the present investigation. Examination by neutron inelastic scattering on the phonon dispersion and a calculation of the spectrum are expected in the future.

4.3. Spin wave stiffness The spin wave stiffness showed a similar softening effect as the behavior of 0D near the phase boundary, but the range of the concentration was narrower than 0D, while the y value attained a maximum at the boundary. The stiffness constant D can be calculated from the N~(E) curves and the Fermi levels, E F and E F , for plus and minus spins [29]. The D value thus calculated depends on the details of the curve structure and the location of the levels. Roughly speaking, D is proportional to an area enclosed by an inverse curve of the density of states and a line connecting two points on the curve at E~ and E F. Therefore, when E F goes to a slope point of the peak, the inverse of N(EF) decreases and the area becomes small and when the peak is lowered by the distortion,

216

J.P. Kuang et al. / Specific heat of Fe-Pd alloys

the area increases. Thus the trend of D(x) can be explained qualitatively from the feature of y(x) assumed above. However, to verify such behaviours more in detail, it is necessary to calculate the band structure taking a disordered effect into account for this alloy.

[13] [14] [15] [16]

References [1] M. FaUot, Ann. Phys. 10 (1938) 291. [2] A. Kussmann and K. Jessen, J. Phys. Soc. Japan 17, Suppl. B-1 (1962) 136; Z. Metalkd. 54 (1963) 504 [3] M. Matsui, T. Shimizu, H. Yamada and K. Adachi, J. Magn. Magn. Mat. 15-18 (1980) 1201. [4] M. Matsui, J.P. Kuang, T. Totani and K. Adachi, J. Magn. Magn. Mat. 54-57 (1986) 911. [5] N. Miyata, K. Tomotsune, H. Nakada, M. Hagiwara, H. Kadomatsu and H. Fujiwara, J. Phys. Soc. Japan 55 (1986) 946. [6] N. Miyata, M. Hagiwara, H. Kunitomo, S. Ohishi, Y. Ichiyangi, K. Kuwabara, K. Tsuru, H. Kadomatsu and H. Fujiwara, J. Phys. Soc. Japan 55 (1986) 953. [7] M. Matsui and K. Adachi, J. Magn. Magn. Mat. 31-34 (1983) 115. [8] N. Miyata, T. Kamimori and M. Goto, J. Phys. Soc. Japan 55 (1986) 2037. [9] G. Longwarth, Phys. Rev. 172 (1968) 572. [10] R. Oshima, K. Kosuga, M. Sugiyama and F.E. Fujita, Proc. Intern. Conf. on the Application of the M6ssbauer Effect, 14-18 (1981) 510 (Jaipur, India, December 1981). [11] M. Matsui, T. Shimizu and K. Adachi, Physica l19B (1983) 84. [12] For example, see The Invar Problem, A.J. Freeman and

[17]

[18] [19]

[20] [21] [22] [23] [24] [25]

[26] [27] [28] [29]

M. Shimizu, eds., the special issue of J. Magn. Magn. Mat. 10 (1979) 113-322. T. Sohmura, R. Oshima and F.E. Fujita, Scr. Met. 14 (1980) 855. M. Matsui, H. Yamada and K. Adachi, J. Phys. Soc. Japan 48 (1980) 2161. M. Matsui, K. Adachi and H. Asano, Sci. Rep. RITU A29, Suppl. 1 (1981) 61. R. Oshima, K. Kasuga, M. Sugiyama and F.E. Fujita, Sci. Rep. RITU A29, Suppl. 1 (1981) 67. J.P. Kuang, T. Totani, M. Kontani, M. Matsui and K. Adachi, Proc. Intern. Symp. on Physics of Magnetic Materials, Sendai, Japan (April 8-10, 1987) 520. M. Sato, B.H. Grier, S.M. Shapiro and H. Miyajima, J. Phys. F: Metal Phys. 12 (1982) 2117. K. Adachi and K. Iwahashi, Proc. Intern. Conf. on Martensitic Transformation (The Japan Institute of Metals, 1986), p. 331. C.A. Bechman, W.E. Wallace and R.S. Crage, Phil. Mag. 27 (1973) 1249. For example, see C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966). T. Izuyama and R. Kubo, J. Appl. Phys. 35 (1964) 1074. J. Mathon and E.P. Wohrfarth, Proc. Roy. Soc. A 302 (1968) 409. R. Candron, J.J. Meunier and P. Costa, Solid State Commun. 14 (1974) 975. For example, see T. Moriya, in: Electron Correlation and Magnetism in Narrow Band Systems, T. Moriya, ed. (Springer, Berlin, 1981), pp. 2-50. Y. Ishikawa, S. Onodera and K. Tajima, J. Magn. Magn. Mat. 10 (1979) 183. J.P. Kuang, M. Matsui and K. Adachi, to be published in J. Phys. Soc. Japan. H. Hasegawa and J. Kanamori, J. Phys. Soc. Japan 31 (1971) 382. S. Wakoh, J. Phys. Soc. Japan 30 (1971) 1068.