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Submitted on 1 Jan 1988
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ISING FERROMAGNET WITH THREE-SITE
FOUR-SPIN INTERACTION
N. Uryû, T. Iwashita
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, dbcembre 1988
ISING FERROMAGNET WITH THREE-SITE FOUR-SPIN INTERACTION
N. UryCi (l) and T. Iwashita (')
(l) Department of Applied Science, Faculty of Engineering, Kyshu University, Fukuoka 812, Japan (') College of General Education, Kyushu Tokai University, Kumamoto 862, Japan
Abstract. - For the Ising ferromagnet with S = 3/2 and S = 2, the effects of the threesite four-spin interaction on the magnetization and the Curie temperature have been investigated with the use of three-spin model approximation. It has
been shown that the effects are large for the Ising ferromagnet with large spin.
Recently, the effects of the higher-order spin in- teractions such as the biquadratic exchange interac- tion (s:.s~)'
,
three-site four-spin interaction ( % . S j ) ( S j - S k ) and four-site four-spin interaction ( & . S j ) (SksSj) on the magnetic properties of both ferro- andantiferromagnets have been discussed extensively by many investigators [l-41. In the previous studies we
discussed the appearance of the higher-order spin in- teractions of the type (Si.Sj)' and ( S i . S j ) ( S j . S k ) and
estimated the order of magnitude of these terms t o be one tenth or one thousandth of the Heisenberg ex- change interaction [5].
The present authors have studied the effects of the three-site four-spin interaction on the magnetization and the Curie temperature of the Ising ferromagnet with S = 1 [6]. Since this interaction is expressed as the fourth power of a spin operator S , its effects are ex-
pected to be large for the spin system with large spin. The number of paths of the interaction ( % . S j ) ( S j . S k )
is z ( z - 1 ) and that of the interactions ( & . S j ) and ( ~ i . s j ) ' is z, where z is the number of nearest neigh- bours. Therefore, the effects of the three-site four-spin interaction may be large for the ferromagnet with large value of z. In the present paper, we extend the cal- culation to the Ising ferromagnets with S = 3 / 2 and S = 2 on bcc lattice ( z = 8) and investigate the effects of the three-site four-spininteraction on the magne- tization and the Curie temperature. The calculations are made by employing the three-spin model approxi- mation 171, and the results for S = 3 / 2 and S = 2 are compared with each other.
Let us consider the Ising ferromagnet in which both the bilinear exchange and the three-site four-spin in- teractions are assumed. Then, the spin Hamiltonian for the present system is written as follows:
The second term in this Hamiltonian comes from the relation
J! ( S i z s j x ) ( s j z s k z )
+
J' ( S j x s k t ) ( S i t s j z ) =Here we have also assumed that the coefficient J' of
the three-site four-spin interaction for spins S ; , S j and S k on a straight line is equal to that for three spins not
on a straight line.
The Hamiltonian ( 1 ) may be approximated by mak-
ing use of the three-spin model as
H3 = -J (SizSjz
+
SjzSkz) 2 ~ ' ~ i z ~ ; ~ ~ k ~ --U (Six
+
Ska)-
bSjx - C (S:,+
S:,)-
dS&-e -k
sjX
S,&,) -f
(six
't S: skZ) (3)where
a = ~ ( z - ~ ) ( s , ) + ~ J ' ( z - 1 ) ' ( ~ : ) ,
b = J ( z - 2 ) ( S Z ) + 2 ~ ' ( z - 1 ) ( e - 2 ) ( ~ ~ ) ,
In the above expression, (S,) and (S:) denote the
thermal average of S;, and Si",sjz, respectively. The
above decomposition in equations ( 3 ) and ( 4 ) can only
be made for loose-packed lattices such as sq, sc and bcc lattices.
For the Ising spin system of S = 3 / 2 and S = 2, 64
and 125 eigenfunctions and the corresponding eigenval-
ues of H3 are obtained, respectively. By making use of
these eigenvalues, the thermal averages (S,) and (S:)
can be calculated as a function of temperature from the following equations:
1
Cm
3
(S=,+
Sjz+
Sbz) exp (-OX,)(S,) = 9 (5)
C ,
exp (-OX,)where @ = l / l c ~ T and k g is the Boltzmann constant.
C8
-
1546 JOURNAL DE PHYSIQUEIn the case of J'
#
0, the magnetization (S,) and the Curie temperature Tc are obtained by solving thesimultaneous equations (5) and (6). These values are calculated by numerical method for both positive and negative values of J'.
Firstly, the temperature dependence of (S,) for the Ising spin system of S = 312 has been calculated and the results are shown in figure 1. As can be seen, the temperature dependence of (S,) shows an abnormal behaviour, especially in the low temperature region for
Fig. 1. - The temperature depencence of (S,) for the Ising ferromagnet of S = 312 on bcc lattice with various values of three-site four-spin interaction J' in the range of -0.0255
5
J' 5 0.01 J.This abnormal behaviour suggests that the ferromag- netic ground state becomes unstable at J' N -J/ ( 2 (2
-
1)s2)
and below this value the complicated ferrimagnetic state may appear as the ground state. It is also concluded that the transition becomes to be of the first-order for J'>
0.07 J.Secondly, the temperature dependence of (S,) for the Ising spin system of S = 2 has been calculated and the results are shown in figure 2. The temperature
0 1 0 2 0
-
k s T 3 0 JFig. 2. - The temperature dependence o
f\
(S ) for the Isingz .
ferromagnet of S = 2 on bcc lattice with various values of three-site four-spin interaction J' in the range of -0.02 J 5 J'
5
0.03J.dependence of (S,) also shows an abnormal behaviour for
J' 21 - J / (2 (2 - 1)
s2)
= .-0.018 J.It also turns out that the transition is of the first order for J' 2 0.03 J.
On the other hand, J' has been shown theoretically to be of the order of magnitude of one tenth or one thousandth of the Heisenberg exchange interaction [5].
Therefore, the instability of the ferromagnetic ground state and the first-order transition may be realizable for the Ising ferromagnet with S
>
912.Furthermore, the J'-dependence of T,/T: for the
Ising ferromagnets with S = 312 and S = 2 on bcc lattice is shown in figure 3. Here, T: denotes the Curie
temperature of the Ising ferromagnet only with a bilin- ear exchange interaction J. It turns out that the three- site four-spin interaction 5' affects largely the Curie temperature T,, and that the effect of J' on Tc is large for the Ising ferromagnet with large spin.
Fig. 3.
-
The J'-dependence of the reduced Curie temper- ature T,/T; for the Ising ferromagnets with S = 312 andS = 2 on bcc lattice. The value of T: denotes the Curie temperature only with a bilinear exchange interaction.
[l] Brown, H. A., Phys.
Rev.
B 4 (1971) 115. [2] Adler, J. and Oitmaa, J., J. Phys. C 1 2 (1979)575.
[3] Aksamit, J. and Westwanski, B., J. Phys. C 14 (1981) 255.
[4] Roger, M., Hetherington, J. H. and Delrieu, J. M.,
Rev.
Mod. Phys. 55 (1983) 1.(51 Iwashita, T. and U r y ~ , N., J. Phys. G 17 (1984) 855.
[6] Iwashita, T. and Uryfi, N., Phys. Lett. A 125 (1987) 5.
