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Phase diagrams and tricritical behavior of a diluted mixed spin Ising model in a random field
,d~ Jeurnalof magnetic ELSEVIER
Journal of Magnetismand Magnetic Materials 151 (1995) 45-53
,i~
materials
Phase diagrams and tricritical behavior of a diluted mixed spin Ising model in a random field T. Kaneyoshi Department of Natural Science lnformatics, School of Informatics and Sciences, Nagoya University, 464-01 Nagoya, Japan
Received 12 August 1994; in final revised form 17 March 1995
Abstract A theoretical study of a diluted mixed spin-l/2 and spin-S (S > 1) Ising model in a random field is presented on the basis of the effective-field theory. The general formulations for determining the phase diagram (transition temperature and tricritical point) of the system with a crystal-field interaction are discussed. In particular, the phase diagrams of the system with S = 3 / 2 are examined in detail on the honeycomb lattice and some interesting phenomena are found, such as the appearance of a tricritical point on the honeycomb lattice.
1. Introduction The problem of the random-field Ising model (RFIM) has been investigated in last years both theoretically and experimentally because it helps to simulate many interesting but complicated problems [1-5]. A fundamental question whether the random field destroys long-range order or not was the lower critical dimensionality d t in the system. Now it is well established that d I = 2 [6]. Also, it is well known that for the symmetric distribution of random field with a minimum at zero field a tricritical point may appear in the phase diagram, although the appearance seems to depend on the coordination number z [7-9]. On the other hand, the mixed spin Ising model consisting of s p i n - l / 2 and spin-S (S > 1 / 2 ) has been introduced as a simple model showing a certain type of ferrimagnetism because of the complexity of the structures in real ferrimagnets. Theoretically, an important point in studying this model is that the
exact solution of transition temperature can be obtained analytically when the structure of the system is chosen to be a honeycomb lattice ( z = 3) [10]. Furthermore, the system with S an integer (S = 1, 2 . . . . ) is of particular interest, since the existence of tricritical behavior is predicted in the system with z > 3 [11] while the tricritical behavior does not exist in the system with S a half-integer (S = 3 / 2 , 5 / 2 . . . . ) [12]. Also, one should notice that the magnetic properties of a ferrimagnetic mixed spin1 / 2 and spin-3/2 Ising system may express many characteristic behaviors being different from those of the corresponding mixed s p i n - l / 2 and spin-1 Ising system [13]. Recently, the magnetic properties of a diluted mixed s p i n - l / 2 and spin-1 Ising system with a crystal-field interaction [14] have been examined by the use of the effective-field theory [15]. In particular, reentrant phenomena have been observed for a certain range of crystal-field interaction. The tricritical behavior did not exist in the system with z = 3
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T. Kaneyoshi/Journal of Magnetisra and Magnetic Materials 151 (1995) 45-53
46
while it might appear for z > 3. As far as we know, however, a diluted mixed spin-l/2 and spin-S (S > 1) Ising system has not been studied. The aim of this work is to give the general expression for evaluating the phase diagram of a diluted mixed spin-l/2 and spin-S ( S > 1) Ising model in a random field within the new formulation of the effective-field theory [16]. In Section 3, we present it when S = 3/2. Also, it is clarified how the formulation can be extended to the system with S > 3/2. In the Appendix, the formulation for the case of S = 2 is derived as an example. In Section 4, the phase diagrams of the diluted mixed spin-l/2 and spin-3/2 Ising system in a random field on honeycomb lattice are obtained numerically.
2. Formulation
We at first consider a diluted mixed spin Ising system with a crystal-field interaction D consisting of spin-l/2 and spin-3/2. The Hamiltonian of the system is given by
,7~ = - J E ~, ~iS~1~z - D E ~i( SZ)2 (q)
i
- E n , e , s : - Enje i
;,
(1)
j
where S z takes the values + 3 / 2 and + 1/2, ~f can be + 1 / 2 or - 1 / 2 , and the first summation is carried out only over nearest-neighbor pairs of spins. J is the exchange interaction and ~i is a random variable which takes the value of unity or zero, depending on whether the site i is occupied by a magnetic atom (with a probability p) or not (with a probability 1 - p ) . The H,~ ( a = i or j) is the random field applied at the site a, which is assumed to be randomly distributed according to the bimodal independent probability distribution function P(H~,) as
P(H,)=½[6(H,,-H)+6(H,,+H)].
diluted mixed Ising spin system with a coordination number z: m ~ < ~i>r
= [p{cosh(a/2) + 20, sinh(a/2)} + 1 _ p ] Z
x (x) lx=0,
(3)
r
= [p{ A( a) + B( a)m + C( a)q + D( a)r} + 1 - p ] zj~(x) ] x=o,
(4)
where ( . . . ) , denotes the random configurational average and ( ~ ) r = p ( a = i or j). a=JV and V= O/Ox is the differential operator. The functions F(x) and f(x) are defined by
F( x) = f P( Hi)F( x +Hi) dH,,
(5a)
]( x) -- f e(Hj)F(x + Hi) dHy,
(55)
with f(x) = ltanh(fix~2),
(6)
1 3 sinh(3flx/2) + exp( - 2Dfl ) sinh(/3x/2) F ( x ) = 2 cosh(3flx/2)+exp(-2Dfl)cosh(flx/2) '
(7)
where fl = 1/ktsT. Here, the parameters q and r in Eq. (4) are defined by z 2 <~i<(S i ) >>r q--
<~i<(SZ)3>>r ,
< ~i>r
r=
, < ~i)r
which are given, within the context of the EFT, by
q = [p{cosh(a/2) + 2o" sinh(a/2)} + 1 _ p ] Z
xG(x) Ix=0,
(8)
r = [p{cosh(a/2) + 20" sinh(a/2)} + 1 - p ] Z
xK(x) lx=o,
(9)
with (2)
Within the new formulation of the effective-field theory (EFT) [12,13,16], we obtain the following expressions for the averaged magnetizations of the
G,( x) = f e ( Hi)G( x +Hi) dH i,
(lOa)
R( x) = f P( Hi)R( x +Hi) dHi.
(lOb)
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
where the functions G(x) and R ( x ) are 1 9cosh(3flx/2)+exp(-2Dfl)cosh(flx/2) c o s h ( 3 f l x / 2 ) + exp( - 2Dfl)cosh(/3x/2)
G(x) = ~
3. P h a s e
diagram
(11) '
sinh(3flx/2) + exp( - 2D/3 ) s i n h ( / 3 x / 2 ) 8 cosh(3flx/2)+exp(-2Dfl) c o s h ( ~ x / 2 ) ' 1 27
R(x) = -
(12) The total magnetization M of the system is then M = -P~(m + o'),
47
We are here interested in finding the condition determining the second-order phase transition line in the diluted mixed spin-l/2 and spin-3/2 system. Expanding the right hand side of Eqs. (3), (8) and (9), we have m = 2pZKlO-+ 8p 3
z! K2tr 3 +
3!(z-3)!
• • • ,
(13)
(16a) z!
where N is the number of magnetic atoms. Now, the parameters A(a), B(a), C(a) and D(a) in Eq. (4) result from the usage of the Van der Waerden identity of S = 3 / 2 [16]:
r = 2pzLlo'+ 8p 3
exp(aST) = A ( a ) + B( a)S7 + C( a ) ( SZ) 2
where the coefficients Ka, are given by
+ D ( a ) (SZ) 3
(14)
L 2 O'3 +
3!(z-3)!
z! q=qo+4p22!(z_2)!qa~r2+ K2,
•
••,
...
(16b)
(16c)
Li, L2, qo and ql
K 1 = sinh(a/Z)[p cosh(a/2) + 1 -p]~-I
with
A ( a ) = I[9 c o s h ( a / 2 ) - c o s h ( a a / 2 ) ] ,
(15a)
B(a) = ~[27 sinh(a/2) - sinh(aa/2)],
(15b)
C ( a ) = ~1[cosh(aa/2) - cosh(a/2)] ,
(15c)
D ( a ) = ½[sinh(aa/2) - 3 s i n h ( a / 2 ) ] .
(150)
× F ( x ) I x=0,
(17a)
K 2 = sinh3(a/2) [ p c o s h ( a / 2 ) + 1 - p] z-3 × i f ( x ) I x=o,
(lVb)
L 1 = sinh(a/Z) [ p cosh(a/Z) + 1 - p] ~-' In general, for the diluted mixed spin system with S > 3 / 2 the exponential operator (or exp(aS~)) can be represented as the sum of 2S + 1 independent variables. Even within the EFT, therefore, one must introduce the 2S independent spin variables z
<~i
( ~i)r
, ....
v =
(17c)
L 2 = sinh3(a/2)[p cosh(a/2) + 1 -p]Z 3 x R ( x ) Ix=o, qo = [P c o s h ( a / 2 ) + 1 - p ] Z G ( x ) lx=o,
(17d) (17e)
ql = sinh2(a/2)[ P c o s h ( a / 2 ) + 1 - p ] ~-2
z 2S
<~i<(S i )
>>r
x R ( x ) I x=O,
>>r
X G ( x ) I x=0.
( ~i)r
in the equation of o- as well as the 2S + 1 parameters, like Eq. (4) for S = 3/2, while the equations of m, q . . . . and v have the same forms as those of Eqs. (4), (8) and (9) for S = 3 / 2 except that the functions F(x), G ( x ) and R ( x ) in them have different forms depending on the value of S. From this reason, it is not so difficult to generalize the expression of evaluating the phase diagram in the diluted mixed spin-l/2 and spin-3/2 Ising system with a random field to the system with S > 3 / 2 , when it is derived for the system with S = 3/2.
(17f)
Substituting the relations (16) into (4), we have O-= ~o-+bo
-3 +
• . .
or
o'2 = (1 - ~ ) / b ,
(18)
where the coefficients a and b are given by
-~ = 2( pz)Z[ K , B ( a) + L i D ( a ) ]
× [p{A(a) + qoC(a)} + 1 -p]:-~]'(X) lx=O (19)
48
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
and
with z!
= 8p4z3!(z-- 3)! [K2B(a) +L2D(a)]
A ( a ) = 118 sinh(a) - sinh(2a)],
(24a)
B(a) = ~2[16 cosh(a) - cosh(2a) - 15],
(24b)
× [p{A(a) +qoC(a)} + 1 -p]Z-'f(X)[x=O z! + 8pSzZ(z - 1) 2 ! ( z - 2)! qlC(a)
C ( a ) = 1 [sinh(2a) - 2 sinh( a ) ] ,
(24c)
D ( a ) = l [ c o s h ( 2 a ) - 4 cosh(a) + 3].
(24d)
× [K1B(a ) +L1D(a)]
Within the EFT, the averaged magnetization o- is given by
× [p{A(a) +qoC(a)} + 1 -p]Z-2f(X) lx=O z! + 8 p 6 z 3 3 ! ( z - 3)[ [glB(a) +LID(a)]3
or= [p{1
+ A( a)m + B( a)q + C( a)r + D( a)v}
+ 1 - p] Z?(x) I x:0,
(25)
where the order parameter v is
X [p{a(a) + qoC(a)} + 1 - p ] Z - 3 f ( x ) lx=o. (20) The second-order phase transition line is then determined by the condition [11]: ~=1
and
/9<0,
and
2=0
(22)
is the tricritical point. At this place, it is worth commenting on the following facts. The transition temperature Tc determined from Eq. (21) is independent of the sign of J. The relation is valid for both ferromagnetic ( J > 0) and ferrimagnetic ( J < 0) cases. Within the EFT, the two new order parameters q and r naturally appear in the equation of o- because of the usage of Eq. (14) for S = 3 / 2 . This is not the case of the standard mean-field theory. It is one of the reasons why the present framework provides better results than the standard mean-field theory. From these discussions, one may easily understand how the present formulation for the system with S = 3 / 2 can be generalized to the diluted mixed s p i n - l / 2 and spin-S Ising system in a random field with a certain value of S larger than S = 3 / 2 . For instance, the Van der Waerden identity for S = 2 [11] is given by exp( aSz)
= 1 + a( a)S z + B(a)(SZ) 2 + C(a)(S{) 3 + D ( a ) '
(23)
(Ci)r
= [p{cosh(a/2)
(21)
In the vicinity of the second-order phase transition line, the right-hand side of (18) must be positive. If it is not the case, the transition is of the first order, and hence the point at which ~=1
(~/((SZ)455r v--
+ 2or sinh(a/2)} + 1 -p]Z
× U ( x ) l x=0
(26)
with
U( x) = f n( Hi)U( x +Hi) dH i.
(27)
Then, the equations of m, q and r have the same forms as Eqs. (3), (8) and (9), although the functions F(x), (x) and R(x) in them must be replaced by the new functions for S = 2. These functions as well as U(x) are given in the Appendix. In this way, by following the same procedure as that of S = 3 / 2 we can obtain the same equation (or (18)) as that of S = 3 / 2 . The transition temperature and tricritical point in a diluted mixed s p i n - l / 2 and spin-2 Ising system can be also determined from the conditions (21) and (22). The coefficients a and b for S = 2 are given in the Appendix. Finally, the coefficients a and b of a diluted mixed s p i n - l / 2 and spin-5/2 Ising system in a random field can be easily derived in the same way as the above procedures by the use of the exact Van der Waerden identity [17] for S = 5 / 2 although they are not given here.
4. Numerical results Let us investigate the phase diagram of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on honeycomb lattice (z = 3) by solv-
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
ing the general expressions given in Section 3 numerically. As far as we know, it has not been clarified whether the tricritical behavior may exist or not in the system although the tricritical behavior has not been obtained in some random-field Ising systems with z = 3 (or honeycomb lattice) [7,8]. The results (Tc versus D/J) of the system with H = 0.0 (zero random field) are at first shown in Fig. 1 by changing the value of p, in order to clarify the statistical accuracy of the present formulation (or the EFT). The curve labeled p = 1.0 is equivalent to the previous result (Fig. 1 in Ref. [12]) for the pure ( H = 0.0, p = 1.0) mixed s p i n - l / 2 and spin-3/2 Ising system with z = 3 in which it is compared with the exact solution of the honeycomb lattice [10]. Thus, it indicates that the present formulation gives reasonable results. Furthermore, the curves in Fig. 1 change continuously from a constant value for a large negative D to a constant value for a large positive D. It comes from the following fact: For a large negative value of D (or D/J = - 3 . 0 ) the spin state of S 7 is in the S~ = + 1 / 2 at T = 0 K. But, it is in the S~ = + 3 / 2 state for a large positive D (or D/J = 3.0). For H = 0.0, the parameter b in Eq. (18) is always negative for any value of D. The diluted mixed spin system in zero random field does not exhibit the tricritical behavior. This is also equivalent to the previous one [12,13] for the pure mixed spin system. 20 H =0.0
I~.,T.
P:1.0
0.8
0.6
- 3.o
_0 J
3.0
Fig. 1. The transition temperature of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system with a crystal-field constant D on the honeycomb lattice (z = 3) is plotted as a function of D / J , when H / J (zero random field) and the concentration p is changed.
49
1.0 0.5
P=0.8
//
0.6 0.5
f
J -3.0
0 J
310
Fig. 2. The transition temperatures of the diluted mixed spin- 1/2 and spin-3/2 Ising system with a fixed concentration p ( p = 0.8) on the honeycomb lattice (z = 3) are plotted as a function of D / J , when three values of H / J are selected.
On the other hand, when a finite random field is applied, the phase diagram (T¢ versus D/J) of the system may be changed dramatically. A typical phase diagram of the diluted mixed spin system is given in Fig. 2 by fixing the value of p at p = 0 . 8 and selecting the three values of H. The result for H/J = 0.5 has a form similar to those in Fig. 1. When the value of H/J increases from H/J = 0.5, however, the Tc curve starts to exhibit some characteristic features, as depicted in the figure. In particular, they indicate that for a certain value of H larger than H/J = 0.5 the reentrant phenomenon can be obtained in the system with appropriate values of p and D/J, just as the cases of the diluted mixed s p i n - l / 2 and spin-1 Ising systems, and the amorphization of mixed s p i n - l / 2 and spin-1 Ising systems [8,18]. Now, from Figs. 1 and 2 one may imagine that the tricritical behavior does not exist in the present system with z = 3, like the previous results of mixed s p i n - l / 2 and spin-1 Ising system with z = 3 as well as s p i n - l / 2 (or spin-l) RFIM with z = 3 [7,8,12-14]. In fact, the situation is completely different in the present system and we can find the tricritical point at which the phase transition changes from second order to first order, when some appropriate conditions are satisfied. The typical phase diagrams (Tc versus H/J) are depicted in Fig. 3 by selecting the three values of D/J and changing the value of p. In these
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
50 Tc
~
P.I.0
O =3.0
0,9
\ /
",
0.5
1.0
1.5
H J
(b) 0.0 1.5
1.0
o
0.5
0
0
0.5
~bl
1.0
1.5
(c) -3.0
0.5
_H .I
figures, the solid and dashed lines represent respectively the second-order and first-order transition lines and hence the black points denote the tricritical points. In this way, the results indicate that the tricritical point may appear in the diluted mixed s p i n - l / 2 and spin-3/2 Ising system with a random field even on the honeycomb lattice (z = 3), when the system satisfies the conditions D/J>_. 0 and p > 0.8. In particular, Fig. 3(c) does not exhibit the tricritical behavior, since the spin state of S~ in the system with D / J = - 3.0 is given by S~ = + 1 / 2 at T = 0 K; the system behaves just like the diluted s p i n - l / 2 RFIM. As has been discussed in Ref. [7], it is therefore reasonable that the system with D / J = - 3 . 0 does not exhibit the tricritical behavior on the honeycomb lattice with z = 3. In other words, the appearance of the tricritical point in the present system with z = 3 obtained in Fig. 3(a,b) is a new finding. Thus, the tricritical behavior of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the honeycomb lattice depends on the value of D / J (or whether the spin state of S~ in the system is in the S z = + 3 / 2 ) as well as the concentration p ( p > 0.8). In Fig. 4, the critical temperature is plotted as a function of p, for fixed values of D / J ( D / J = 0.0 for Fig. 4(a), - 0 . 7 5 for Fig. 4(b) and - 3 . 0 for Fig. 4(c)) and selected values of H/J. In the figures, solid and dashed lines denote the second-order and first-order transition lines, and black circles in Fig. 4(a) express the tricritical points. In each figure, when H becomes a little larger than H / J = 0.5, the reentrant phenomenon may be observed, which result is consistent to those of Fig. 3. In particular, the Tc curve labeled a ( H / J = 0.0) in Fig. 4(c) reduces to zero at the critical concentration Pc = 0.5575. The value Pc is equal to that obtained for the diluted s p i n - l / 2 Ising ferromagnet with z = 3 [19] on the basis of the EFT [15] within the Zemike approxima-
O.B
0.5
Fig. 3. The transition temperature of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system on the honeycomb lattice (z = 3) is plotted as a function of H / J , when the value of p is changed and the value of D / J is fixed at D / J = 3.0 for Fig. 3(a), D / J = 0.0 for Fig. 3(b) or D / J = - 3 . 0 for Fig. 3(c). In Figs. 3(a,b), the dashed lines denote the first-order transition lines and hence the black points represent the tricritical points.
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
(a)
1"¢
-7
51 pi, •
0.6
•~ =0.0 1.0
\
0.5575 -=
.
\
\
0.5
H
"S" = 0 . 0 dr---= 0 . 0 2
t
0.45
I
1.0
,o[
0.9
0.8
o17
o.5
t o.5 ~=0.5277
(b)
-
0
~
05
J
Fig. 5. The variation of the critical value p* with the value of net on the honeycomb lattice which is obtained by fixing. T at a very low temperature ( T / J = 0.02) in Eq. (21). The actual critical concentration Pc of the system is expected to be represented by the thin vertical lines and bold horizontal lines.
J
0.5
o.5 t
, - 0.5
D / J in the diluted mixed spin-l/2 and spin-3/2 Ising ferromag-
0.7 5
_r,
0 1.0
i - 1.0
0.9
0.8
0.7
p
0.6
(c)
0.4
~=0.0
tion [20]. The fact also indicates that the spin state of S z in the present system with D/J = - 3 . 0 is in the S/~= + 1 / 2 at T = 0 K. On the other hand, the critical concentration Pc of the curve a (H/J = 0.0) in Fig. 4(a) for D/J = 0.0 is given by Pc = 0.5277. The above results indicate that the critical concentration Pc of the diluted mixed s p i n - l / 2 and spin3 / 2 Ising system in the zero random field may be changed with the variation of D/J. In Fig. 5, therefore, the critical concentration p* of the diluted mixed spin system with H/J = 0.0 and z = 3 at which Tc may go to zero is obtained by fixing T at a very low temperature (T/J = 0.02), in order to avoid the numerical errors when solving Eq. (21) at T = 0 K. As is seen from the figure, the critical concentration p * is given by the same value (pc = 0.5575) in the whole region of D except the special region ( - 0.8 < D/J < 0.05). The result expresses that the
0.2
O0
0.9
0.8
0.7
P
0.6
1" 0.5 P,=0.5575
Fig. 4. The transition temperature of the diluted mixed spin-l/2 and spin-3/2 Ising system on the honeycomb lattice is plotted as a function of p, when the value of D / J is fixed at D / J = 0.0 for Fig. 4(a), D / J = - 0 . 7 5 for Fig. 4(b) or D / J = - 3 . 0 for Fig. 4(c). In Fig. 4(a), the dashed lines denote the first-order transition lines and hence the black points represent the tricritical points.
52
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
value of p* obtained in Fig. 5 is nothing but the critical concentration Pc, namely p * = Pc, especially for the horizontal parts of the figure. In other words, the figure indicates that the real Pc obtained by putting T/J = 0.0 into Eq. (21) must be varied in a stepwise way, as depicted by thin lines in the figure. Also, one should notice that the characteristic feature of Pc similar to that of Fig. 5 has been obtained recently in a diluted spin-3/2 Blume-Capel model [21,22]. Here, an interesting point is that the Pc value of the diluted mixed spin system for D/J > 0 or D/J < - 0.75 is independent of whether the spin state of S[ is in the S iz _-- _+3/2 or in the S[ = + 1 / 2 at T = 0 K, although the Tc value is clearly dependent on it.
5. Conclusions In this work, we have studied a diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the basis of the effective-field theory (EFT). In Section 2, the general formulation of the system has been presented. In Section 3, we have discussed how the phase diagram of a diluted mixed s p i n - l / 2 and spin-S (S > 1) Ising system can be derived within the framework of the EFT. In particular, by solving the formulation of the diluted mixed spin-1/2 and spin-3/2 system on the honeycomb lattice numerically we have obtained a variety of interesting phenomena in the phase diagram, as shown in Section 4. We have shown that the phase diagrams of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the honeycomb lattice (z = 3) include the tricritical point in the T-H plane as well as in the T-P plane, depending on the values of D/J (D/J>f 0) and p ( p > 0.8), in contrast to mixed s p i n - l / 2 and spin-1 Ising systems with z = 3 as well as s p i n - l / 2 (or spin-l) RFIM with z = 3. Furthermore, the reentrant phenomenon has appeared for a certain range of H in the T-p plane. In particular, an anomalous change of Pc has been obtained in the diluted mixed spin system with zero random field for a special region of D/J, namely - 0.75 < D/J < 0.0. Except for the special region of D, the critical concentration of the mixed spin system with z = 3 is equivalent to that of the diluted s p i n - l / 2 Ising system with z = 3.
Finally, experimentally there has been growing interest in preparation and characterization of molecular/organic polymer-based materials exhibiting ferro- and ferrimagnetic behaviors in recent years [23-25]. These new materials are considered to be well described by a pure (or diluted) mixed s p i n - l / 2 and spin-S Ising (or Heisenberg) model. In particular, amorphous V(TCNE) x • y(CH2CI 2) [23] seems to belong a class of a diluted mixed spin system investigated in this work. Furthermore, just like diluted antiferromagnets [3,4], a random field may exist in the system and play an essential role for the magnetic properties. On the other hand, in real diluted materials, such as amorphous V(TCNE) x • y(CH 2CI 2) [23], it may be reasonable to include the concentration of the two types and also the two random fields on the different types of spins. The inclusion of different concentrations and random fields in the present work can be done straightforwardly, which will be investigated in a future work.
Appendix The coefficients ~ and b in Eq. (18) for a diluted mixed s p i n - l / 2 and spin-2 Ising system in a random field are given by
~ = 2( zp)Z[ K1A( a) + L1C( a)] ×[p{1 +
qoB(a) + V,D(a)} + 1 _ p ] Z - 1
x f ( x ) l =o
(A.1)
and z! b = 8p4z.,z J!tz-3)! ×[p{1
[K2A(a ) +LzC(a)]
+ qoB( a) + V,D( a)} + 1 _p]Z-1
x / ( x ) I x=0 + 8 p S z 2 ( z - 1)
z! 2 ! ( z - 2)!
× [qlB(a) + V2D(a)] [KIA(a) +LIC(a)]
X[p{1 + qoB( a) + V,D( a)} + 1 _ p]Z- 2
x/(x) l,=o z! + 8p6z3 3!( z - 3)
! [ K1A(a) + t,C(a)] 3
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
53
X[ p{1 +qoB(a) + V,D(a)} + l - p ] z-3
References
xf(x) lx=o,
[1] Y. Imry, J. Stat. Phys. 34 (1984) 849. [2] A. Aharony, J. Magn. Magn. Mater. 54-57 (1986) 27; Phys. Rev. B 18 (1978) 3318 [3] R.J. Birgeneau, Y. Shapira, G. Shirane, R.A. Cowley and H. Yoshizawa, Physica B 137 (1986) 83. [4] D.P. Belanger, S.M. Regende, A.R. King and V. Jaccarino, J. Appl. Phys. 57 (1985) 3294. [5] D. Andelman, Phys. Rev. B 27 (1983) 3079. [6] J.Z. Imbrie, Physica A 140 (1986) 291. [7] H.E. Borge and P.R. Silva, Physica A 114 (1987) 561; Phys. Stat. Sol. (b) 121 (1984) K25. [8] E.F. Sarmento and T. Kaneyoshi, Phys. Rev. B 48 (1993) 3232. [9] O. Entin-Wohlman and C. Hartztein, J. Phys. A 18 (1985) 315; M. Kaufman, P. Klunizinger and A. Khurana, Phys. Rev. B 34 (1986) 4766. [10] L.L. Gon~alves, Phys. Scripta 32 (1985) 291. [11] T. Kaneyoshi, J, Phys. Soc. Jpn. 56 (1987) 2675; T. Kaneyoshi, Physica A 205 (1994) 677. [12] T. Kaneyoshi, M. Ja~ur and P. Tomczak, J. Phys. Condens. Matter 4 (1993) L653. [13] T. Kaneyoshi, M. Ja~ur and P. Tomczak, J. Phys. Condens. Matter 5 (1993) 5331. [14] A. Benyoussef, A. E1 Kenz and T. Kaneyoshi, J. Magn. Magn. Mater. 131 (1994) 173; 179. [15] R. Honmura and T. Kaneyoshi, J. Phys. C. 12 (1979) 3979; T. Kaneyoshi, Acta Phys. Polon. 83 (1993) 703. [16] T. Kaneyoshi, J.W. Tucker and M. Ja.~(3ur, Physica A 186 (1992) 495. [17] T. Kaneyoshi and M. Ja~ur, Phys. Stat. Sol. (b) 175 (1993) 225. [18] T. Kaneyoshi, Z. Phys. B 71 (1988) 109. [19] Z.Y. Li and C.Z. Yang, Solid State Commun. 56 (1985) 445. [20] F. Zernike, Physica 7 (1940) 565. [21] T. Kaneyoshi and M. Ja~ur, Phys. Stat. Sol. (b) 173 (1992) K37; J. Magn. Magn. Mater. 130 (1994) 29. [22] M. Kerouad, M. Saber and J.W. Tucker, Phys. Stat. Sol. (b) 180 (1993) K23. [23] J.M. Manriquez, G.T. Yee, R.S. McLean, A.J. Epstein and J.S. Miller, Science 252 (1991) 1415. [24] P. Zhou, G.B. Morin, A.J. Epstein R.S. McLean and J.S. Miller, J. Appl. Phys. 73 (1993) 6569. [25] T. Mallah, S. Thiebaut, M. Verdaguer and P. Veillet, Science 262 (1993) 1554.
(A.2)
where the parameters A(a), B(a), C(a) and D(a) are given by Eq. (24). The coefficients q0, ql, Ki (i = 1, 2) and L i (i = 1, 2) have the same forms as Eq. (17) except that the functions F(x), R(x) and
G(x) are replaced by
F(x) 4 sinh(2/3x) + 2 exp( - 3D/3 ) sinh(/3x) 2 cosh(2/3 x ) + 2 exp( - 3 D/3 ) cosh(/3 x) + exp( - 4D/3 ) ' (A.3a)
R(x) 16 sinh(2/3x) + 2 exp( - 3D/3 ) sinh(fix) 2 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) + exp( - 4D/3 ) '
(A.3b) G(x) 8 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) 2 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) + exp( - 4D/3 ) ' (A.3c) O n the o t h e r h a n d , n e w p a r a m e t e r s V l a n d Va h a v e a p p e a r e d f r o m t h e e x p a n s i o n o f v ( o r 26), n a m e l y z!
u=VI+4p42t(z_2)!V2o'2-F
""
,
(A.4)
which are given by V,=[pcosh(a/2)+l-p]ZU(x) V2 = s i n h 2 ( a / 2 ) [ p
x=0,
cosh(a/2) + 1 _ p ] z-2
xU(x) l =o. (A.S) The function U ( x ) is then defined by U(x) 32 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) 2 cosh(2/3 x) + 2 exp( - 3D/3 ) cosh(/3 x) + exp( - 4D/3 )
(A.6)
Journal of Magnetismand Magnetic Materials 151 (1995) 45-53
,i~
materials
Phase diagrams and tricritical behavior of a diluted mixed spin Ising model in a random field T. Kaneyoshi Department of Natural Science lnformatics, School of Informatics and Sciences, Nagoya University, 464-01 Nagoya, Japan
Received 12 August 1994; in final revised form 17 March 1995
Abstract A theoretical study of a diluted mixed spin-l/2 and spin-S (S > 1) Ising model in a random field is presented on the basis of the effective-field theory. The general formulations for determining the phase diagram (transition temperature and tricritical point) of the system with a crystal-field interaction are discussed. In particular, the phase diagrams of the system with S = 3 / 2 are examined in detail on the honeycomb lattice and some interesting phenomena are found, such as the appearance of a tricritical point on the honeycomb lattice.
1. Introduction The problem of the random-field Ising model (RFIM) has been investigated in last years both theoretically and experimentally because it helps to simulate many interesting but complicated problems [1-5]. A fundamental question whether the random field destroys long-range order or not was the lower critical dimensionality d t in the system. Now it is well established that d I = 2 [6]. Also, it is well known that for the symmetric distribution of random field with a minimum at zero field a tricritical point may appear in the phase diagram, although the appearance seems to depend on the coordination number z [7-9]. On the other hand, the mixed spin Ising model consisting of s p i n - l / 2 and spin-S (S > 1 / 2 ) has been introduced as a simple model showing a certain type of ferrimagnetism because of the complexity of the structures in real ferrimagnets. Theoretically, an important point in studying this model is that the
exact solution of transition temperature can be obtained analytically when the structure of the system is chosen to be a honeycomb lattice ( z = 3) [10]. Furthermore, the system with S an integer (S = 1, 2 . . . . ) is of particular interest, since the existence of tricritical behavior is predicted in the system with z > 3 [11] while the tricritical behavior does not exist in the system with S a half-integer (S = 3 / 2 , 5 / 2 . . . . ) [12]. Also, one should notice that the magnetic properties of a ferrimagnetic mixed spin1 / 2 and spin-3/2 Ising system may express many characteristic behaviors being different from those of the corresponding mixed s p i n - l / 2 and spin-1 Ising system [13]. Recently, the magnetic properties of a diluted mixed s p i n - l / 2 and spin-1 Ising system with a crystal-field interaction [14] have been examined by the use of the effective-field theory [15]. In particular, reentrant phenomena have been observed for a certain range of crystal-field interaction. The tricritical behavior did not exist in the system with z = 3
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0304-8853(95)00403-3
T. Kaneyoshi/Journal of Magnetisra and Magnetic Materials 151 (1995) 45-53
46
while it might appear for z > 3. As far as we know, however, a diluted mixed spin-l/2 and spin-S (S > 1) Ising system has not been studied. The aim of this work is to give the general expression for evaluating the phase diagram of a diluted mixed spin-l/2 and spin-S ( S > 1) Ising model in a random field within the new formulation of the effective-field theory [16]. In Section 3, we present it when S = 3/2. Also, it is clarified how the formulation can be extended to the system with S > 3/2. In the Appendix, the formulation for the case of S = 2 is derived as an example. In Section 4, the phase diagrams of the diluted mixed spin-l/2 and spin-3/2 Ising system in a random field on honeycomb lattice are obtained numerically.
2. Formulation
We at first consider a diluted mixed spin Ising system with a crystal-field interaction D consisting of spin-l/2 and spin-3/2. The Hamiltonian of the system is given by
,7~ = - J E ~, ~iS~1~z - D E ~i( SZ)2 (q)
i
- E n , e , s : - Enje i
;,
(1)
j
where S z takes the values + 3 / 2 and + 1/2, ~f can be + 1 / 2 or - 1 / 2 , and the first summation is carried out only over nearest-neighbor pairs of spins. J is the exchange interaction and ~i is a random variable which takes the value of unity or zero, depending on whether the site i is occupied by a magnetic atom (with a probability p) or not (with a probability 1 - p ) . The H,~ ( a = i or j) is the random field applied at the site a, which is assumed to be randomly distributed according to the bimodal independent probability distribution function P(H~,) as
P(H,)=½[6(H,,-H)+6(H,,+H)].
diluted mixed Ising spin system with a coordination number z: m ~ < ~i>r
= [p{cosh(a/2) + 20, sinh(a/2)} + 1 _ p ] Z
x (x) lx=0,
(3)
= [p{ A( a) + B( a)m + C( a)q + D( a)r} + 1 - p ] zj~(x) ] x=o,
(4)
where ( . . . ) , denotes the random configurational average and ( ~ ) r = p ( a = i or j). a=JV and V= O/Ox is the differential operator. The functions F(x) and f(x) are defined by
F( x) = f P( Hi)F( x +Hi) dH,,
(5a)
]( x) -- f e(Hj)F(x + Hi) dHy,
(55)
with f(x) = ltanh(fix~2),
(6)
1 3 sinh(3flx/2) + exp( - 2Dfl ) sinh(/3x/2) F ( x ) = 2 cosh(3flx/2)+exp(-2Dfl)cosh(flx/2) '
(7)
where fl = 1/ktsT. Here, the parameters q and r in Eq. (4) are defined by z 2 <~i<(S i ) >>r q--
<~i<(SZ)3>>r ,
< ~i>r
r=
, < ~i)r
which are given, within the context of the EFT, by
q = [p{cosh(a/2) + 2o" sinh(a/2)} + 1 _ p ] Z
xG(x) Ix=0,
(8)
r = [p{cosh(a/2) + 20" sinh(a/2)} + 1 - p ] Z
xK(x) lx=o,
(9)
with (2)
Within the new formulation of the effective-field theory (EFT) [12,13,16], we obtain the following expressions for the averaged magnetizations of the
G,( x) = f e ( Hi)G( x +Hi) dH i,
(lOa)
R( x) = f P( Hi)R( x +Hi) dHi.
(lOb)
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
where the functions G(x) and R ( x ) are 1 9cosh(3flx/2)+exp(-2Dfl)cosh(flx/2) c o s h ( 3 f l x / 2 ) + exp( - 2Dfl)cosh(/3x/2)
G(x) = ~
3. P h a s e
diagram
(11) '
sinh(3flx/2) + exp( - 2D/3 ) s i n h ( / 3 x / 2 ) 8 cosh(3flx/2)+exp(-2Dfl) c o s h ( ~ x / 2 ) ' 1 27
R(x) = -
(12) The total magnetization M of the system is then M = -P~(m + o'),
47
We are here interested in finding the condition determining the second-order phase transition line in the diluted mixed spin-l/2 and spin-3/2 system. Expanding the right hand side of Eqs. (3), (8) and (9), we have m = 2pZKlO-+ 8p 3
z! K2tr 3 +
3!(z-3)!
• • • ,
(13)
(16a) z!
where N is the number of magnetic atoms. Now, the parameters A(a), B(a), C(a) and D(a) in Eq. (4) result from the usage of the Van der Waerden identity of S = 3 / 2 [16]:
r = 2pzLlo'+ 8p 3
exp(aST) = A ( a ) + B( a)S7 + C( a ) ( SZ) 2
where the coefficients Ka, are given by
+ D ( a ) (SZ) 3
(14)
L 2 O'3 +
3!(z-3)!
z! q=qo+4p22!(z_2)!qa~r2+ K2,
•
••,
...
(16b)
(16c)
Li, L2, qo and ql
K 1 = sinh(a/Z)[p cosh(a/2) + 1 -p]~-I
with
A ( a ) = I[9 c o s h ( a / 2 ) - c o s h ( a a / 2 ) ] ,
(15a)
B(a) = ~[27 sinh(a/2) - sinh(aa/2)],
(15b)
C ( a ) = ~1[cosh(aa/2) - cosh(a/2)] ,
(15c)
D ( a ) = ½[sinh(aa/2) - 3 s i n h ( a / 2 ) ] .
(150)
× F ( x ) I x=0,
(17a)
K 2 = sinh3(a/2) [ p c o s h ( a / 2 ) + 1 - p] z-3 × i f ( x ) I x=o,
(lVb)
L 1 = sinh(a/Z) [ p cosh(a/Z) + 1 - p] ~-' In general, for the diluted mixed spin system with S > 3 / 2 the exponential operator (or exp(aS~)) can be represented as the sum of 2S + 1 independent variables. Even within the EFT, therefore, one must introduce the 2S independent spin variables z
<~i
( ~i)r
, ....
v =
(17c)
L 2 = sinh3(a/2)[p cosh(a/2) + 1 -p]Z 3 x R ( x ) Ix=o, qo = [P c o s h ( a / 2 ) + 1 - p ] Z G ( x ) lx=o,
(17d) (17e)
ql = sinh2(a/2)[ P c o s h ( a / 2 ) + 1 - p ] ~-2
z 2S
<~i<(S i )
>>r
x R ( x ) I x=O,
>>r
X G ( x ) I x=0.
( ~i)r
in the equation of o- as well as the 2S + 1 parameters, like Eq. (4) for S = 3/2, while the equations of m, q . . . . and v have the same forms as those of Eqs. (4), (8) and (9) for S = 3 / 2 except that the functions F(x), G ( x ) and R ( x ) in them have different forms depending on the value of S. From this reason, it is not so difficult to generalize the expression of evaluating the phase diagram in the diluted mixed spin-l/2 and spin-3/2 Ising system with a random field to the system with S > 3 / 2 , when it is derived for the system with S = 3/2.
(17f)
Substituting the relations (16) into (4), we have O-= ~o-+bo
-3 +
• . .
or
o'2 = (1 - ~ ) / b ,
(18)
where the coefficients a and b are given by
-~ = 2( pz)Z[ K , B ( a) + L i D ( a ) ]
× [p{A(a) + qoC(a)} + 1 -p]:-~]'(X) lx=O (19)
48
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
and
with z!
= 8p4z3!(z-- 3)! [K2B(a) +L2D(a)]
A ( a ) = 118 sinh(a) - sinh(2a)],
(24a)
B(a) = ~2[16 cosh(a) - cosh(2a) - 15],
(24b)
× [p{A(a) +qoC(a)} + 1 -p]Z-'f(X)[x=O z! + 8pSzZ(z - 1) 2 ! ( z - 2)! qlC(a)
C ( a ) = 1 [sinh(2a) - 2 sinh( a ) ] ,
(24c)
D ( a ) = l [ c o s h ( 2 a ) - 4 cosh(a) + 3].
(24d)
× [K1B(a ) +L1D(a)]
Within the EFT, the averaged magnetization o- is given by
× [p{A(a) +qoC(a)} + 1 -p]Z-2f(X) lx=O z! + 8 p 6 z 3 3 ! ( z - 3)[ [glB(a) +LID(a)]3
or= [p{1
+ A( a)m + B( a)q + C( a)r + D( a)v}
+ 1 - p] Z?(x) I x:0,
(25)
where the order parameter v is
X [p{a(a) + qoC(a)} + 1 - p ] Z - 3 f ( x ) lx=o. (20) The second-order phase transition line is then determined by the condition [11]: ~=1
and
/9<0,
and
2=0
(22)
is the tricritical point. At this place, it is worth commenting on the following facts. The transition temperature Tc determined from Eq. (21) is independent of the sign of J. The relation is valid for both ferromagnetic ( J > 0) and ferrimagnetic ( J < 0) cases. Within the EFT, the two new order parameters q and r naturally appear in the equation of o- because of the usage of Eq. (14) for S = 3 / 2 . This is not the case of the standard mean-field theory. It is one of the reasons why the present framework provides better results than the standard mean-field theory. From these discussions, one may easily understand how the present formulation for the system with S = 3 / 2 can be generalized to the diluted mixed s p i n - l / 2 and spin-S Ising system in a random field with a certain value of S larger than S = 3 / 2 . For instance, the Van der Waerden identity for S = 2 [11] is given by exp( aSz)
= 1 + a( a)S z + B(a)(SZ) 2 + C(a)(S{) 3 + D ( a ) '
(23)
(Ci)r
= [p{cosh(a/2)
(21)
In the vicinity of the second-order phase transition line, the right-hand side of (18) must be positive. If it is not the case, the transition is of the first order, and hence the point at which ~=1
(~/((SZ)455r v--
+ 2or sinh(a/2)} + 1 -p]Z
× U ( x ) l x=0
(26)
with
U( x) = f n( Hi)U( x +Hi) dH i.
(27)
Then, the equations of m, q and r have the same forms as Eqs. (3), (8) and (9), although the functions F(x), (x) and R(x) in them must be replaced by the new functions for S = 2. These functions as well as U(x) are given in the Appendix. In this way, by following the same procedure as that of S = 3 / 2 we can obtain the same equation (or (18)) as that of S = 3 / 2 . The transition temperature and tricritical point in a diluted mixed s p i n - l / 2 and spin-2 Ising system can be also determined from the conditions (21) and (22). The coefficients a and b for S = 2 are given in the Appendix. Finally, the coefficients a and b of a diluted mixed s p i n - l / 2 and spin-5/2 Ising system in a random field can be easily derived in the same way as the above procedures by the use of the exact Van der Waerden identity [17] for S = 5 / 2 although they are not given here.
4. Numerical results Let us investigate the phase diagram of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on honeycomb lattice (z = 3) by solv-
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
ing the general expressions given in Section 3 numerically. As far as we know, it has not been clarified whether the tricritical behavior may exist or not in the system although the tricritical behavior has not been obtained in some random-field Ising systems with z = 3 (or honeycomb lattice) [7,8]. The results (Tc versus D/J) of the system with H = 0.0 (zero random field) are at first shown in Fig. 1 by changing the value of p, in order to clarify the statistical accuracy of the present formulation (or the EFT). The curve labeled p = 1.0 is equivalent to the previous result (Fig. 1 in Ref. [12]) for the pure ( H = 0.0, p = 1.0) mixed s p i n - l / 2 and spin-3/2 Ising system with z = 3 in which it is compared with the exact solution of the honeycomb lattice [10]. Thus, it indicates that the present formulation gives reasonable results. Furthermore, the curves in Fig. 1 change continuously from a constant value for a large negative D to a constant value for a large positive D. It comes from the following fact: For a large negative value of D (or D/J = - 3 . 0 ) the spin state of S 7 is in the S~ = + 1 / 2 at T = 0 K. But, it is in the S~ = + 3 / 2 state for a large positive D (or D/J = 3.0). For H = 0.0, the parameter b in Eq. (18) is always negative for any value of D. The diluted mixed spin system in zero random field does not exhibit the tricritical behavior. This is also equivalent to the previous one [12,13] for the pure mixed spin system. 20 H =0.0
I~.,T.
P:1.0
0.8
0.6
- 3.o
_0 J
3.0
Fig. 1. The transition temperature of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system with a crystal-field constant D on the honeycomb lattice (z = 3) is plotted as a function of D / J , when H / J (zero random field) and the concentration p is changed.
49
1.0 0.5
P=0.8
//
0.6 0.5
f
J -3.0
0 J
310
Fig. 2. The transition temperatures of the diluted mixed spin- 1/2 and spin-3/2 Ising system with a fixed concentration p ( p = 0.8) on the honeycomb lattice (z = 3) are plotted as a function of D / J , when three values of H / J are selected.
On the other hand, when a finite random field is applied, the phase diagram (T¢ versus D/J) of the system may be changed dramatically. A typical phase diagram of the diluted mixed spin system is given in Fig. 2 by fixing the value of p at p = 0 . 8 and selecting the three values of H. The result for H/J = 0.5 has a form similar to those in Fig. 1. When the value of H/J increases from H/J = 0.5, however, the Tc curve starts to exhibit some characteristic features, as depicted in the figure. In particular, they indicate that for a certain value of H larger than H/J = 0.5 the reentrant phenomenon can be obtained in the system with appropriate values of p and D/J, just as the cases of the diluted mixed s p i n - l / 2 and spin-1 Ising systems, and the amorphization of mixed s p i n - l / 2 and spin-1 Ising systems [8,18]. Now, from Figs. 1 and 2 one may imagine that the tricritical behavior does not exist in the present system with z = 3, like the previous results of mixed s p i n - l / 2 and spin-1 Ising system with z = 3 as well as s p i n - l / 2 (or spin-l) RFIM with z = 3 [7,8,12-14]. In fact, the situation is completely different in the present system and we can find the tricritical point at which the phase transition changes from second order to first order, when some appropriate conditions are satisfied. The typical phase diagrams (Tc versus H/J) are depicted in Fig. 3 by selecting the three values of D/J and changing the value of p. In these
T. Kaneyoshi /Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
50 Tc
~
P.I.0
O =3.0
0,9
\ /
",
0.5
1.0
1.5
H J
(b) 0.0 1.5
1.0
o
0.5
0
0
0.5
~bl
1.0
1.5
(c) -3.0
0.5
_H .I
figures, the solid and dashed lines represent respectively the second-order and first-order transition lines and hence the black points denote the tricritical points. In this way, the results indicate that the tricritical point may appear in the diluted mixed s p i n - l / 2 and spin-3/2 Ising system with a random field even on the honeycomb lattice (z = 3), when the system satisfies the conditions D/J>_. 0 and p > 0.8. In particular, Fig. 3(c) does not exhibit the tricritical behavior, since the spin state of S~ in the system with D / J = - 3.0 is given by S~ = + 1 / 2 at T = 0 K; the system behaves just like the diluted s p i n - l / 2 RFIM. As has been discussed in Ref. [7], it is therefore reasonable that the system with D / J = - 3 . 0 does not exhibit the tricritical behavior on the honeycomb lattice with z = 3. In other words, the appearance of the tricritical point in the present system with z = 3 obtained in Fig. 3(a,b) is a new finding. Thus, the tricritical behavior of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the honeycomb lattice depends on the value of D / J (or whether the spin state of S~ in the system is in the S z = + 3 / 2 ) as well as the concentration p ( p > 0.8). In Fig. 4, the critical temperature is plotted as a function of p, for fixed values of D / J ( D / J = 0.0 for Fig. 4(a), - 0 . 7 5 for Fig. 4(b) and - 3 . 0 for Fig. 4(c)) and selected values of H/J. In the figures, solid and dashed lines denote the second-order and first-order transition lines, and black circles in Fig. 4(a) express the tricritical points. In each figure, when H becomes a little larger than H / J = 0.5, the reentrant phenomenon may be observed, which result is consistent to those of Fig. 3. In particular, the Tc curve labeled a ( H / J = 0.0) in Fig. 4(c) reduces to zero at the critical concentration Pc = 0.5575. The value Pc is equal to that obtained for the diluted s p i n - l / 2 Ising ferromagnet with z = 3 [19] on the basis of the EFT [15] within the Zemike approxima-
O.B
0.5
Fig. 3. The transition temperature of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system on the honeycomb lattice (z = 3) is plotted as a function of H / J , when the value of p is changed and the value of D / J is fixed at D / J = 3.0 for Fig. 3(a), D / J = 0.0 for Fig. 3(b) or D / J = - 3 . 0 for Fig. 3(c). In Figs. 3(a,b), the dashed lines denote the first-order transition lines and hence the black points represent the tricritical points.
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
(a)
1"¢
-7
51 pi, •
0.6
•~ =0.0 1.0
\
0.5575 -=
.
\
\
0.5
H
"S" = 0 . 0 dr---= 0 . 0 2
t
0.45
I
1.0
,o[
0.9
0.8
o17
o.5
t o.5 ~=0.5277
(b)
-
0
~
05
J
Fig. 5. The variation of the critical value p* with the value of net on the honeycomb lattice which is obtained by fixing. T at a very low temperature ( T / J = 0.02) in Eq. (21). The actual critical concentration Pc of the system is expected to be represented by the thin vertical lines and bold horizontal lines.
J
0.5
o.5 t
, - 0.5
D / J in the diluted mixed spin-l/2 and spin-3/2 Ising ferromag-
0.7 5
_r,
0 1.0
i - 1.0
0.9
0.8
0.7
p
0.6
(c)
0.4
~=0.0
tion [20]. The fact also indicates that the spin state of S z in the present system with D/J = - 3 . 0 is in the S/~= + 1 / 2 at T = 0 K. On the other hand, the critical concentration Pc of the curve a (H/J = 0.0) in Fig. 4(a) for D/J = 0.0 is given by Pc = 0.5277. The above results indicate that the critical concentration Pc of the diluted mixed s p i n - l / 2 and spin3 / 2 Ising system in the zero random field may be changed with the variation of D/J. In Fig. 5, therefore, the critical concentration p* of the diluted mixed spin system with H/J = 0.0 and z = 3 at which Tc may go to zero is obtained by fixing T at a very low temperature (T/J = 0.02), in order to avoid the numerical errors when solving Eq. (21) at T = 0 K. As is seen from the figure, the critical concentration p * is given by the same value (pc = 0.5575) in the whole region of D except the special region ( - 0.8 < D/J < 0.05). The result expresses that the
0.2
O0
0.9
0.8
0.7
P
0.6
1" 0.5 P,=0.5575
Fig. 4. The transition temperature of the diluted mixed spin-l/2 and spin-3/2 Ising system on the honeycomb lattice is plotted as a function of p, when the value of D / J is fixed at D / J = 0.0 for Fig. 4(a), D / J = - 0 . 7 5 for Fig. 4(b) or D / J = - 3 . 0 for Fig. 4(c). In Fig. 4(a), the dashed lines denote the first-order transition lines and hence the black points represent the tricritical points.
52
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
value of p* obtained in Fig. 5 is nothing but the critical concentration Pc, namely p * = Pc, especially for the horizontal parts of the figure. In other words, the figure indicates that the real Pc obtained by putting T/J = 0.0 into Eq. (21) must be varied in a stepwise way, as depicted by thin lines in the figure. Also, one should notice that the characteristic feature of Pc similar to that of Fig. 5 has been obtained recently in a diluted spin-3/2 Blume-Capel model [21,22]. Here, an interesting point is that the Pc value of the diluted mixed spin system for D/J > 0 or D/J < - 0.75 is independent of whether the spin state of S[ is in the S iz _-- _+3/2 or in the S[ = + 1 / 2 at T = 0 K, although the Tc value is clearly dependent on it.
5. Conclusions In this work, we have studied a diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the basis of the effective-field theory (EFT). In Section 2, the general formulation of the system has been presented. In Section 3, we have discussed how the phase diagram of a diluted mixed s p i n - l / 2 and spin-S (S > 1) Ising system can be derived within the framework of the EFT. In particular, by solving the formulation of the diluted mixed spin-1/2 and spin-3/2 system on the honeycomb lattice numerically we have obtained a variety of interesting phenomena in the phase diagram, as shown in Section 4. We have shown that the phase diagrams of the diluted mixed s p i n - l / 2 and spin-3/2 Ising system in a random field on the honeycomb lattice (z = 3) include the tricritical point in the T-H plane as well as in the T-P plane, depending on the values of D/J (D/J>f 0) and p ( p > 0.8), in contrast to mixed s p i n - l / 2 and spin-1 Ising systems with z = 3 as well as s p i n - l / 2 (or spin-l) RFIM with z = 3. Furthermore, the reentrant phenomenon has appeared for a certain range of H in the T-p plane. In particular, an anomalous change of Pc has been obtained in the diluted mixed spin system with zero random field for a special region of D/J, namely - 0.75 < D/J < 0.0. Except for the special region of D, the critical concentration of the mixed spin system with z = 3 is equivalent to that of the diluted s p i n - l / 2 Ising system with z = 3.
Finally, experimentally there has been growing interest in preparation and characterization of molecular/organic polymer-based materials exhibiting ferro- and ferrimagnetic behaviors in recent years [23-25]. These new materials are considered to be well described by a pure (or diluted) mixed s p i n - l / 2 and spin-S Ising (or Heisenberg) model. In particular, amorphous V(TCNE) x • y(CH2CI 2) [23] seems to belong a class of a diluted mixed spin system investigated in this work. Furthermore, just like diluted antiferromagnets [3,4], a random field may exist in the system and play an essential role for the magnetic properties. On the other hand, in real diluted materials, such as amorphous V(TCNE) x • y(CH 2CI 2) [23], it may be reasonable to include the concentration of the two types and also the two random fields on the different types of spins. The inclusion of different concentrations and random fields in the present work can be done straightforwardly, which will be investigated in a future work.
Appendix The coefficients ~ and b in Eq. (18) for a diluted mixed s p i n - l / 2 and spin-2 Ising system in a random field are given by
~ = 2( zp)Z[ K1A( a) + L1C( a)] ×[p{1 +
qoB(a) + V,D(a)} + 1 _ p ] Z - 1
x f ( x ) l =o
(A.1)
and z! b = 8p4z.,z J!tz-3)! ×[p{1
[K2A(a ) +LzC(a)]
+ qoB( a) + V,D( a)} + 1 _p]Z-1
x / ( x ) I x=0 + 8 p S z 2 ( z - 1)
z! 2 ! ( z - 2)!
× [qlB(a) + V2D(a)] [KIA(a) +LIC(a)]
X[p{1 + qoB( a) + V,D( a)} + 1 _ p]Z- 2
x/(x) l,=o z! + 8p6z3 3!( z - 3)
! [ K1A(a) + t,C(a)] 3
T. Kaneyoshi/Journal of Magnetism and Magnetic Materials 151 (1995) 45-53
53
X[ p{1 +qoB(a) + V,D(a)} + l - p ] z-3
References
xf(x) lx=o,
[1] Y. Imry, J. Stat. Phys. 34 (1984) 849. [2] A. Aharony, J. Magn. Magn. Mater. 54-57 (1986) 27; Phys. Rev. B 18 (1978) 3318 [3] R.J. Birgeneau, Y. Shapira, G. Shirane, R.A. Cowley and H. Yoshizawa, Physica B 137 (1986) 83. [4] D.P. Belanger, S.M. Regende, A.R. King and V. Jaccarino, J. Appl. Phys. 57 (1985) 3294. [5] D. Andelman, Phys. Rev. B 27 (1983) 3079. [6] J.Z. Imbrie, Physica A 140 (1986) 291. [7] H.E. Borge and P.R. Silva, Physica A 114 (1987) 561; Phys. Stat. Sol. (b) 121 (1984) K25. [8] E.F. Sarmento and T. Kaneyoshi, Phys. Rev. B 48 (1993) 3232. [9] O. Entin-Wohlman and C. Hartztein, J. Phys. A 18 (1985) 315; M. Kaufman, P. Klunizinger and A. Khurana, Phys. Rev. B 34 (1986) 4766. [10] L.L. Gon~alves, Phys. Scripta 32 (1985) 291. [11] T. Kaneyoshi, J, Phys. Soc. Jpn. 56 (1987) 2675; T. Kaneyoshi, Physica A 205 (1994) 677. [12] T. Kaneyoshi, M. Ja~ur and P. Tomczak, J. Phys. Condens. Matter 4 (1993) L653. [13] T. Kaneyoshi, M. Ja~ur and P. Tomczak, J. Phys. Condens. Matter 5 (1993) 5331. [14] A. Benyoussef, A. E1 Kenz and T. Kaneyoshi, J. Magn. Magn. Mater. 131 (1994) 173; 179. [15] R. Honmura and T. Kaneyoshi, J. Phys. C. 12 (1979) 3979; T. Kaneyoshi, Acta Phys. Polon. 83 (1993) 703. [16] T. Kaneyoshi, J.W. Tucker and M. Ja.~(3ur, Physica A 186 (1992) 495. [17] T. Kaneyoshi and M. Ja~ur, Phys. Stat. Sol. (b) 175 (1993) 225. [18] T. Kaneyoshi, Z. Phys. B 71 (1988) 109. [19] Z.Y. Li and C.Z. Yang, Solid State Commun. 56 (1985) 445. [20] F. Zernike, Physica 7 (1940) 565. [21] T. Kaneyoshi and M. Ja~ur, Phys. Stat. Sol. (b) 173 (1992) K37; J. Magn. Magn. Mater. 130 (1994) 29. [22] M. Kerouad, M. Saber and J.W. Tucker, Phys. Stat. Sol. (b) 180 (1993) K23. [23] J.M. Manriquez, G.T. Yee, R.S. McLean, A.J. Epstein and J.S. Miller, Science 252 (1991) 1415. [24] P. Zhou, G.B. Morin, A.J. Epstein R.S. McLean and J.S. Miller, J. Appl. Phys. 73 (1993) 6569. [25] T. Mallah, S. Thiebaut, M. Verdaguer and P. Veillet, Science 262 (1993) 1554.
(A.2)
where the parameters A(a), B(a), C(a) and D(a) are given by Eq. (24). The coefficients q0, ql, Ki (i = 1, 2) and L i (i = 1, 2) have the same forms as Eq. (17) except that the functions F(x), R(x) and
G(x) are replaced by
F(x) 4 sinh(2/3x) + 2 exp( - 3D/3 ) sinh(/3x) 2 cosh(2/3 x ) + 2 exp( - 3 D/3 ) cosh(/3 x) + exp( - 4D/3 ) ' (A.3a)
R(x) 16 sinh(2/3x) + 2 exp( - 3D/3 ) sinh(fix) 2 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) + exp( - 4D/3 ) '
(A.3b) G(x) 8 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) 2 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) + exp( - 4D/3 ) ' (A.3c) O n the o t h e r h a n d , n e w p a r a m e t e r s V l a n d Va h a v e a p p e a r e d f r o m t h e e x p a n s i o n o f v ( o r 26), n a m e l y z!
u=VI+4p42t(z_2)!V2o'2-F
""
,
(A.4)
which are given by V,=[pcosh(a/2)+l-p]ZU(x) V2 = s i n h 2 ( a / 2 ) [ p
x=0,
cosh(a/2) + 1 _ p ] z-2
xU(x) l =o. (A.S) The function U ( x ) is then defined by U(x) 32 cosh(2/3x) + 2 exp( - 3D/3 ) cosh(/3x) 2 cosh(2/3 x) + 2 exp( - 3D/3 ) cosh(/3 x) + exp( - 4D/3 )
(A.6)