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Submitted on 1 Jan 1979
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MAGNETIZATION OF AMORPHOUS MAGNETS
S. Inawashiro, S. Katsura, M. Takahashi
To cite this version:
S. Inawashiro, S. Katsura, M. Takahashi. MAGNETIZATION OF AMORPHOUS MAGNETS.
Journal de Physique Colloques, 1979, 40 (C2), pp.C2-147-C2-148. �10.1051/jphyscol:1979251�. �jpa-
00218649�
MAGNETIZATION OF AMORPHOUS MAGNETS
S. Inawashiro, S. Katsura and M. Takahashi
Department of Applied Physics, Tohoku University, Sendai, Japan
Abstracts.-The temperature dependence of the distribution of the magnetic moment in an amorphous magnet, where the exchange interactions and the magnitude of the magnetic moment at 0 K are random variables, has been investigated. In particular, we considered the two-limiting cases, the one with the fixed spin and the other with the fixed interaction. The mean magnetization is also shown in a figure. The temperature-dependence of the width of the magnetization distribution shows an opposite tendency to each other. It is to be noted that the constant width observed in the experiment can be explained by the simultaneous existence of the both effect.
The distribution of the hyperfine field de- termined from the Mossbauer effect in amorphous fer- romagnets /1-3/ is mainly caused by the distribu- tion of the magnetic moment of an individual atom.
It is considered that the origin of the distribu- tion of the magnetic moment is attributed to the random distribution of the exchange interaction and the distribution of the magnetic moment at 0 K. The problem is analysed in two limiting cases; the case of the random interaction and the fixed value of the spin and the case of the fixed interaction and the random distribution of the magnetic moment at 0 K. An effect of the co-existence of the both dis- tributions is inferred from the results of the two limiting cases.
In the first case, the distribution of the mean field is approximated by 11,kl
R(C) = |6(?- 2SJ .a.S) II P ( J )G(a.)dJ.da. , (l) where J denotes the interaction, a the thermal ave- rage of the normalized spin, P(J) the probability distribution of J, and G(a) the distribution of a.
Using the Fourier-integral representation of delta- function, the equation (1) is transformed to
where z denotes the number of neighbouring spins, and Q(q) the Fourier transform of P(J). The latter equation is also expressed by using R ( Q , i.e.
The equations (2) and (4) constitute the fundamental integral-equations which determine R (£). The equa- tions are essentically the same as that derived by Plefka /5/ in connection with the problem of the spin glasses. It is to be pointed out that the method of the distribution function /4/ is a powerful tool in the treatment of the random spin-system, as the molecular field theory in the pure system.
For a gaussian form of the interaction dis- tribution, i.e.
P(j) = (1/2TT)2 (I/AJ)exp|3(J-J0)2/2(AJ)23, (6) the Fourier transform becomes
Q(q) = exp{iqJ0 - (AJ)2q2/2} . (7) Now, the equation (3) is approximated by
g(k) = exp{i2SkJQp - (Aj)24S2k2q/2} (8) where
p = 0 , and q = O2" " )
Then we obtain a gaussian form of R(£), i.e.
R(?) = (1/2TT)2 0/A£)exp{-(?-£0)2/2(A5)2} (j0) where
C0 = 2zJSQ0 (11)
and
A? = 2S/zq AJ. (12) The mean value of a and the mean square deviation q
are determined self-consistently from
a = f f(?)R(?)dC , (13)
and
q
= J f
2(5)R(5)d5 . (14)
JOURNAL DE PHYSIQUE
Colloque Cl, supplément au n" 3, Tome 40, mars 1979, page C2-147
Résumé.- On a étudié l'influence de la température sur la distribution des moments magnétiques dans un matériau amorphe ou les interactions d'échange et l'amplitude du moment magnétique à OK sont distribués au hasard. Deux cas limites sont analysés particulièrement; l'un où les spins sont fixés l'autre où les interactions sont constantes. L'aimantation moyenne est représentée sur une figure.
La largeur constante de la distribution d'aimantation peut être expliquée par l'existence des deux effets qui sont antagonistes.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979251
c2-148 JOURNAL DE PHYSIQUE
The temperature dependence of
3
is shown in Figure the mean field is obtained in a similar manner to 1 together with one of theexperimental curves 161, the first case. The distribution of the magnetic and the distribution of the thermal-average of nor- moment at a finite temperature is given bymalized spin, G(u), is shown in Figure 2. It is to W(m) =
i
G(~-uS)G(U)~~(S)~U~S,
(16)be emphasized that these results are effectively in where u (S) denotes the distribution of the magne- 0
agreement with the numerical solutions of the fun- tic moment at 0 K. For a gaussian form of u (S) with 0
damental equations. 3 = SO and ~-s$(As)~, the distribution W(m) at se- veral temperatures are shown in Figure 3, together with one of the experimental curves 171.
1.0
Fig. 1 : The temperature-dependence of the averaged 0 0.5 1
,
1.5magnetization o calculated for the fixed spin S =1/2 with z = 12. Dashed line denotes uO, the magnetiza-
tion of the molecular field theory for a uniform Fig. 3 : The temperature dependence of the distribw ferromagnet. A dashed and dotted line represents an tion of the magnetization W(m) for the gaussian experimental curve for Fe7g Bzl 161. distribution of spin with So = 1 and AS = 0.15. A
dashed and dotted line represents an experimental curve for Fe79 B21 171.
Fig. 2 : The distribution of the thermal average of the normalized magnetic moment calculated for the fixed spin S : 112 with z = 12 and AJ/JO = 1/a.
In the second case, the mean field is given by
5
= 2J0$ UjSj.
(15)Assuming that the thermal average of a. is given by J
a Brillouin function of an integral spin independent of the small fluctuation of S the distribution of
j'
Finally, it can be seen from Figures 2 and 3 that each temperature-dependence of the width shows an opposite tendency to each other. It is to be noted that the co-existence of the both effect can explain the approximately constant width as observed in the temperature-dependence of the hyper fine field 181.
References
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32
(1969)K55./2/ Klein, M.W., Phys. Rev. B x (1976) 5008.
/ 3 / Southern, B.W., J. Phys. C : Solid State Phys.
9 (1976) 4011.
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/ 4 / Matsubara, F., Progr. Theor. Phys.
11
(1974) 1 124.151 Plefka,T., J. Phys. F
6
(1976) L327./6/ Takahashi, M., and Koshirmra, M., Japan. J.App1.
Phys. 1711 (1977).
/7/ Takahashi, M., Koshimura, M., and Suzuki, T., ICAME (1978) Kyoto, Japan.
181 Sharon,,T.E. and Tsuei, C.C., Phys. Rev.
(1972) 1047.
