Z. Phys. B - C o n d e n s e d Matter 77, 339-341 (1989) Condensed
Zeitschrift M a t t e r ffJr Physik B
9 Springer-Verlag 1989
Two-dimensional mixed spin Ising models
with bond dilution and random _+ J interactions*
N. Benayad 1, **, A. Kliimper 1, j. Zittartz 1, and A. Benyonssef 2 1 Institut ffir Theoretische Physik, Universit/it zu K61n,
Federal Republic of G e r m a n y
2 Facult6 des Sciences, Laboratoire de Magn6tisme, Rabat, M o r o c c o Received July 26, 1989
Using the finite cluster approximation we study a mixed spin model (spins a = 1/2 and S = 1) on a square lattice with nearest-neighbour and crystal field interactions. The near- est-neighbour couplings K u are assumed to be independent r a n d o m variables with distri- bution, P (Ku) = p 6 (K u - K) + (1 - p) 6 (K u - eK), where K > 0. We investigate the cases
= 0 and e = - 1 corresponding to bond dilution and to r a n d o m +_ J interactions, respec- tively. In certain ranges of p the phase diagrams exhibit tricritical behaviour and re- entrance.
I. Introduction
In recent years disordered Ising models were studied with particular interest in the influence of disorder on critical behaviour. F o r instance the two-dimen- sional spin-one Ising system containing crystal field interaction or biquadratic exchange was investigated [1, 2]. The purpose of this paper is to present an ex- tension of this study to a mixed spin Ising system ( a = 1/2 and S = 1), with r a n d o m nearest-neighbour interactions as well as crystal field interactions, by using the finite cluster approximation (F.C.A) [3].
The corresponding pure system was studied [4] by using an effective-field theory with correlations. It was shown that a tricritical point exists if the coordination number z is larger than 3.
In this work we separately treat two kinds of dis- order in a mixed spin Ising system on a square lattice and investigate their influence on the critical behav- iour. The reduced Hamiltonian of our model takes the form
-fill= Z K u a i S j - flA Z $2" (i)
<it> i
* Supported by the agreement of cooperation between the D F G and the C N R - M a r o c
** O n leave from Facult6 des Sciences I, Univ6rsit6 H a s s a n II, Ca- sablanca, Morocco; and Laboratoire de Magn6tisme, Universit6 de Rabat, Morocco
The underlying lattice is composed of two interpene- trating sublattices, one occupied by spins with spin m o m e n t a = _+ 1/2 and the other one is occupied by spins with spin m o m e n t S = 0 , + 1. The summation in (1) extends over all pairs of nearest-neighbour sites in the lattice. The reduced nearest-neighbour interac- tions Kij are assumed to be independent r a n d o m vari- ables with distribution
P (Ku) = p 6 (Kij-- K) + (1 -- p) 6 (K u - a K), (2) where K > 0.
We consider in detail the bond diluted model (a--0) and the model with r a n d o m _+ J interactions (~ = - 1). The second term on the right hand side of (1) represents the interaction with the crystal field.
In Sect. II phase diagrams in the A - K - 1 plane (crys- tal field and temperature) are calculated for different values of p. One should note that the Hamiltonian (1) is equivalent to a mixed spin Ising system with biquadratic exchange
H = - Z J j iSj+ A Z 2.
(i j ) (i j )
II. Finite cluster approximation and results
We consider a particular spin a o (So) and denote the mean value of ao(So) by (ao)c((So)c), while all other
K 1 r ~ I I I 1.4
1.2
i
0 0 . 5 1. 1.5 2o
L~/a Fig. 1. The phase diagrams of the bond-diluted mixed spin Ising model in the A - - K - 1 plane. Second order transition lines are shown for different values of the concentration p. They end in tricritical points, depicted as dots
spins 0.~ and Si(i=~O) and all interactions Kij are kept
fixed. We obtain, similarly to the procedure in [5], (0.o)c= 89189 S 1 --}- K 0 2 S 2 + K 0 3 S 3 q- K 0 4 $4)]
(So)~ =
sinh (K o 10.t + K o 2 0.2 + K o 3 rr3 + K o , 0.4)
(3)
14)
.6
89 exp (flA) + cosh (K o 1 o'1 + Ko2 o"2 -I- Ko30- 3 + Ko4 0.,)"
To derive the magnetizations per site m, and m s for a- and S-sublattices, respectively, one must average the right hand sides of (3) and (4) over $1 . . . $4 and 0.1, ..., o'4, respectively. This is done by using an ap- propriate distribution [2, 5] for the spins preserving self-correlations like (0.~) = 1, but still neglecting cor- relations between different spins in spirit of the F.C.A.
Finally the disorder in the system is taken into ac- count by averaging over all couplings Kii by using the distribution (2). A detailed description of this pro- cedure can be found in [5], from which work most formulas can be taken over, e.g. (17, 18) in [-5] are identical to (3, 4) after replacing a term 89 by 89 exp(flA).
In particular we obtain an equation for rn~ of the form
rnr + Em3 +... (5)
.8
340
K-1
.2
p
Fig. 2. The variation of the tricritical temperature with dilution pa- rameter p
K ~1 I I I I
1 . 4
1.2
0 0 . 5 1. 1.5 2,
~/a
Fig. 3. The phase diagrams of the random _+J interaction model in the A - K-1 plane. Second order transition lines are shown for different values of the concentration p. They end in tricritical points, depicted as dots
where
gt(K,p, c~)=8A~(A2+3B2A3+3 C 2 . ~ +/~2 X3), b-(K, p, c 0 = 144 A1 (B2 B3 + 2 C2 B3 A3 + D2 B3 A32)
+ 8 B~(A: + 3 B2A 3 + 3 C2A~ +/)2 x3)
+ 128 2~ (E 2 + F: A3). (6)
. 8
I i K 9
. 4
.3
. 2
I I I 0
p .9061
Fig. 4. The variation of the tricritical temperature with the concen- tration p of + J bonds
Expressions for A1, B1, ... can be taken from Appen- dix B in [-5] where only the definition of the function
exp(/?A)]_ 1 f ( x ) must be changed to f ( x ) = 1 -~ 2 cosh(x)] "
F r o m (5) the line of second order para-ferromag- netic phase transitions is determined by
~ i = 1, E~O.
(7)
The magnetization m~ in the ferromagnetic vicinity of this line is given by m~ z = ( 1 - d)/b-. A line of second order transition can end in a tricritical point which is characterized by
a = 1, b - ~ 0.
(8)
Using this scheme we have obtained phase diagrams for bond dilution and r a n d o m _+ J interactions.
a) Bond dilution (~ = O)
The corresponding phase diagrams for different values of p are represented in Fig. 1. F o r zero crystal
341 field interaction [5] the bond percolation threshold p* is 0.4132 which may be compared with the exact value 0.5 [-6]. F o r 0 . 6 8 2 9 < p < 1 there is a tricritical point. Its variation in the p - K - 1 plane is represented in Fig. 2. F o r p* < p < 0 . 6 8 2 9 the transition is always of second order. In this case the second order transi- tion temperature goes to zero at two values of the crystal field A1 = 89 and A 2-= 1 if the concentration p varies in the ranges p * < p < 0 . 4 8 2 1 9 and 0.48219
< p < 0 . 6 8 2 9 , respectively. We note that for some values of p the second order transition line exhibits re-entrance. The dependence on p-ranges is certainly related to the approximations.
b) Random +_J interactions ( a = - 1)
The corresponding phase diagrams for different values of p are represented in Fig. 3. F o r zero crystal field the long range ferromagnetic order is destroyed if the concentration of the • J bonds is lower than p*=0.8160. F o r 0 . 9 0 6 1 < p < 1 there is a tricritical point. Its variation in the p - - K -1 plane is shown in Fig. 4. F o r p * < p < 0 . 9 0 6 1 the transition is always of second order. In this case the second order transi- tion temperature goes to zero at two values of the crystal field A 1 = 1 and A 2 = 2 if the concentration of the _ J bonds belongs to the ranges p* < p < 0.8870 and 0.8870<p < 0.9061, respectively. We note that in the first range of p the transition lines exhibit re-en- trance.
References
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2. Benayad, N., Benyoussef, A., Boccara, N.: J. Phys. C: Solid State Phys. 18, 1899 (1985)
3. Boccara, N.: Phys. Lett. 94A, 185 (1983) 4. Kaneyoshi, T.: J. Phys. Soc. Jpn. 56, 2675 (1987)
5. Benayad, N., Klfimper, A., Zittartz, J., Benyoussef, A.: Z. Phys.
B - Condensed Matter 77
6. Sykes, M.F., Essam, J.W.: Phys. Rev. 133A, 310 (1964) N. Benayad, A. Kliimper, J. Zittartz
Institut fiir Theoretische Physik Universit/it zu K61n
Ziilpicher Strasse 77 D-5000 K61n 41
Federal Republic of Germany A. Benyoussef
Facult6 des Sciences Laboratoire de Magn6tisme B.P. 1014
Rabat Marocco
