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Correlated effective-field theory of the site-diluted Ising model
1 ELSEVIER
Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
Jourill oI ~laPitism
A t materials
Correlated effective-field theory of the site-diluted Ising model A. Bobfik *, M. Jag~ur Department of Theoretical Physics and Geophysics, Faculty of Natural Sciences, P.J. Safarik University, Moyzesova 16, 041 54 Ko~ice, Slovak Republic
Received 21 December 1993
Abstract
The quenched site-diluted Ising model is considered within the framework of a correlated effective-field theory. The dependence with temperature and magnetic concentration of the correlation parameter and thermodynamic quantities of interest are presented and discussed for a square lattice. The present formalism yields satisfactory results, in particular the effects, on the susceptibility and specific heat, of the coexisting finite and infinite clusters are exhibited. It is shown that the correlation parameter exhibits peculiar behaviour, away from the critical region, as a function of temperature for selected values of concentration of magnetic atoms.
1. Introduction
For many years now there have been extensive studies of the disordered magnetic materials. One kind of disorder appears in site-diluted ferromagnets, where the perfect crystallographic lattice has lattice sites occupied at random by magnetic atoms. Major points of interests in the theoretical and experimental studies of such disordered magnets are the concentration dependence of the critical temperature and value of the critical concentration of magnetic atoms below which the system will not exhibit a transition to magnetic order. The problem has been studied using a variety of approximations and mathematical techniques (see, e.g. Ref. [1]). Recently there has been renewed interest in the study of site-diluted Ising models by the use of effective field theory [2-13] in which all the single-site kinematic relations are correctly accounted for, but multispin correlation functions between various spins are entirely decoupled. This is achieved most frequently by the use of single- [14] or two-site [15] exact formal identities and by employing the differential (integral) operator technique or the combinatorial method. On the other hand, a correlated effective field theory [16], developed for the s p i n - l / 2 Ising model on a regular lattice, partially takes into account the effects of multispin correlations. The theory yields values for the critical temperature and other thermodynamical quantities that are identical to those of the Bethe-Peierls approximation, although the approach is completely different in its formulation f r o m
* Corresponding author. 0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00296-4
106
A. Bobdk, M. Ja~ur / Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
that method. Apart from studies of the pure system, extensions of this theory have also been applied to the site-diluted ferromagnet [17], anisotropic systems [18,19] and semi-infinite systems [20]. In this paper, the correlated effective-field theory is applied to the quenched site-diluted s p i n - l / 2 Ising model on a square lattice. The relevant thermodynamical quantities, namely the correlation parameter, spontaneous magnetization, critical temperature, susceptibility, internal energy and specific heat are all investigated within this framework. Although all these thermodynamical quantities, apart from of the internal energy and the specific heat, were studied in Ref. [17], our results differ from those presented there, for two reasons. First, in the derivation of the coupled equations for the averaged magnetization and the correlation parameter in Ref. [17] the difference between a conditional configurational average and a non-conditional configurational average was ignored. Second, the results of Ref. [17] were based on the use of a configurational averaging procedure that neglected the correlation between the site occupancy number for a particular site and the configuration-dependent thermal average of the spin operator at that site. A general discussion of the importance of including these correlations in the study of site-diluted Ising systems has recently been given in Ref. [9]. In the present paper, we present results based on the adoption of the proper configurational averaging procedure for a site-diluted Ising model, proposed in Ref. [6]. The plan of this paper is as follows. In Section 2 we introduce the formalism of the correlated effective-field approximation for the site-diluted Ising model. In Section 3 the theory is applied to the diluted Ising ferromagnet on a square lattice and the relations for the most relevant thermodynamical quantities are obtained. In Section 4 numerical results of such quantities are presented and discussed. Our concluding remarks are presented in Section 5.
2. Formulation
We consider the site-diluted Ising model in an applied external magnetic field H. The Hamiltonian of the system is
,~= i EJij~i~jsisj _ nE~isi ' i,j
(1)
i
where Jij is the exchange interaction, ~i is the random variable that takes the value unity or zero, depending on whether the site i is occupied by a magnetic atom or not, and s i = + 1. The starting point of the theory is the set of exact formal identities, of the type discussed in Ref. [14] for a pure system, that exist for the thermal average ({f;}~isi) of a single spin:
({f~},~si)=(~i{fi } tanh[/3,i(~j J~j~jsy+H)]),
(2)
where /3 = 1/(kBT) and {fi} represents any function of the Ising variables except si. To recast this equation in a more convient form, the differential operator technique [21] may be employed, to give
({ fi} ~iSi) = (~i{ fi} H [ ~j cosh(Otij ) + ~jsj sinh(Otij ) + 1 - ~j] ) tanh( x + h) I x =0,
J
where D = O/Ox is the differential operator, have used the relation exp(a~j) = ~j exp(a) + 1 - ~j,
tij =/3Jij, and
(3)
h =/3H. In deriving Eq. (3) from Eq. (2), we (4)
together with the Van der Waerden identity for the s p i n - l / 2 Ising model. It should be noted that Eq. (3) is for a fixed configuration of occupied sites, so the thermal averages are configurationally dependent.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
107
Therefore, the next step is to carry out the configurational averaging, to be denoted by (...)~. Then Eq. (3) for a lattice having only nearest-neighbour interactions, J, can be written as follows: z
<({fi}~iSi))c = (<~i{fi} H [~j cosh(Dt) + ~jsi sinh(Dt) + 1 - ~j]))c tanh(x + h) I x=0,
(5)
j=l
where z is the coordination number and t =/3J. Since we are interested in the thermodynamical quantities, let us expand the right-hand side of Eq. (5) with respect to h and retain only its first-order terms:
((~i{fi}R))c + ((~,{f,}L))~h,
(6)
R = I-I [~j cosh(Dt) + ~j.sj sinh(Dt) + 1 - ~ j ] tanh(x) I ~=0, j=l
(7)
(({fi}~iSi))c =
where z
and z
L = I ~ [~i cosh(Dt) + ~jsj sinh(Dt) + 1 - e j ] sech2(x) I x=0.
(8)
j=l
Eq. (6) can generate many kinds of identities, which give relations among spin correlation functions, on substituting some Ising variable functions for {fi}. Among them, upon setting {fi} "- 1, Eq. (6) reduces to m = ((~iR))c + ((~iL))ch,
(9)
and by setting {fi} = ~:ysj, we obtain ,1~= ( ( ~ i ~ j s j R ) )c d¢- ( (~il~jsjL ) )~h,
(10)
where m = (<~isi))~ and e = ((l~isigysj)) ~ are the averaged magnetization and short-range order parameter, respectively, expressed in terms of multisite correlation functions, which are yet undetermined. In order to evaluate Eqs. (9) and (10), many authors have introduced the effective-field approximation which decouples next-neighbour spin correlations:
((XjXk"" Xt))c = ((Xj))c((Xk))c''' ((Xt))c,
(11)
where j m k # ... # l are the nearest-neighbours of a site i and xj = gj.sj.. Instead of using the decoupling approximation (11), let us extend the concept of a correlated effective field [16,22] to the site-diluted Ising model as follows: ~jSj -~ ( (~jSj) )c "1- X( ~iS i -- ( (~iSi) )c) ,
(12)
where j denotes the nearest-neighbour site of a certain central site i, and A is a temperature-, concentration- and external field-dependent correlation parameter. It should be noted here that we have adopted a form of correlated effective-field different from that originally proposed for the site-diluted Ising model in Ref. [17]. Substituting the relation (12) into Eqs. (9) and (10) yields equations for the averaged magnetization rn and the short-range order parameter e as functions of the correlation parameter, reduced temperature, external magnetic field and concentration of magnetic atoms. In order to determine A uniquely, another equation is needed. In particular, we can use the three-site identity, obtained on putting {fi} = ~jSj~kSk in Eq. (6),
( <~isi~jSj~kSk >>¢= ( (~i~jSj~kSkR> )~ + ( (~i~jSjgkSkL ) )~h.
(13)
A. Bobdk, M. Ja~ur /Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
108
Thus, the magnetization, short-range order parameter and the correlation parameter can be determined by solving the coupled Eqs. (9), (10) and (13) for any coordination number z defining the crystallographic structure. However, for simplicity, we discuss below the square lattice with z = 4, although with an appropriate z-value any other lattice could be discussed as well.
3. Application to the square lattice
In this section we study the thermodynamical properties of the diluted Ising ferromagnet on a square lattice. From Eq. (9), the averaged magnetization, for the system with z = 4, reduces to m = 4 p [ p a R 1+ 3 p 2 ( 1 - p ) g 2 + 3 p ( 1 - p ) 2 R 3 + ( 1 - p ) a g 4 ] m
+4[PRs+(l_p)R6]f.+(p[p4Ll+4pa(l_p)L2+6p2(1 + 4 p ( 1 - p ) 3 L 4 + (1 _p)4] + 6p[ p2L 5 + 2 p ( 1 - p ) L
- p ) 2L 3
(14)
6 + ( 1 - p)2L7] + toL8)h ,
where p = (~:i)c is the concentration of magnetic atoms, p = ((¢i¢/S/~kSk)) ~ and z = ((¢i(;jSj(~kSkCtSt))¢, j ~ k 4: l = 1-~ 4, are the averaged two- and three-site correlation functions, respectively, and to = ((¢i¢1S1¢2S2¢3S31~4S4))c is the averaged four-site correlation function. The coefficients R~ (v = 1 ~ 6) and L~ (v = 1-8) can be easily calculated by applying the mathematical relation e x p ( y D ) f ( x ) = f ( x + y), and are given in the Appendix. Applying the correlated approximation (12) to the multisite spin correlation functions appearing in Eq. (14), we obtain
( A - Blm2)m + Clh = 0,
(15)
with
A 1 = 4 p [ p a R I + 3 p 2 ( 1 - p ) R 2+ 3p(1 - p ) 2R 3 + (1 - p ) a R 4 ] + 4[pg 5 + (1-p)g6]a
1 - 1,
B 1 = - 4 [ p R 5 + (1 - p ) R 6 ] b l ,
Cl=p[p4Ll +4pa(l_p)LE+6p2(1
(16) - p ) 2 L 3 + 4p(1 - p ) 3 L 4 + (1 -19) 4]
+ 6(c 0 + clm2)[ p2L 5 + 2p(1 - p ) L 6 + (1 - p ) 2 L 7 ] + (d o + dxm 2 + d2m4)Ls, where a 1 = (1 - 3p)A 3 + 3pA 2, b I = (3 - p ) A 3 - 3(2 - p ) A 2 + 3(1 - p ) ; t +p,
CO -~-p~t2,
C1 = -- (2 --p)A 2 + 2(1 - p ) A + p ,
d0=PA 4,
d~= - 2 ( 2 - 3 p ) A 4 + 4 ( 1 - 3 p ) A 3 + 6 p A
(17) 2,
d E= - ( 4 - p ) A 4 + 4 ( 3 - p ) A 3 - 6 ( 2 - p ) A 2 + 4 ( 1 - p ) A + p . On the other hand, from Eq. (13), in the same way as that used in derivation of Eq. (15), we can obtain for the square lattice the following equation:
( A 2 - B2m2)m + C2h = O,
(18)
A. Bobdk, M. Jcd~ur/Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
109
with A2 = 2p2[ p2(R1 + R5 ) + p ( 1 - p ) ( 2 R
2 + R6) + ( 1 - p ) 2 R 3 ]
+ 2 [ p ( R , + R s ) + ( 1 - p ) R 2 ] a 1 - ( 1 - 2p)A 2 - 2pA, B 2= - 2 [ p ( R 1+R5) + ( 1 - p ) R 2 ] b , + (1 - A ) 2, C2 = p3[ p2L5 + 2p(1 - p ) L
(19)
(1 _p)2L7]
6 "at-
+(Co + c , m 2 ) [ p 2 ( L t + 4L 5 +L8 ) + 2 p ( 1 - p ) ( L 2 + 2L6) + ( 1 - p ) 2 L 3 ] + (do + d i m 2 + d2m4)L5.
Using Eqs. (15) and (18), we can evaluate the magnetization, critical temperature, critical concentration, correlation parameter and susceptibility associated with the diluted Ising ferromagnet on a square lattice. 3.1. Magnetization, correlation parameter and critical temperature
From Eqs. (15) or (18), for h = 0, the averaged magnetization m is given by m 2 =A1/B 1
(20)
m 2 = A 2 / B 2,
(21)
or from which we obtain the following equation for the correlation parameter A as a function of p and t: Am
A2
B1
B2 •
(22)
It should be noted that the parameter A evaluated from (22) is only valid below a critical temperature Tc(p), since in order to determine A we used the averaged magnetization. For temperatures above To(p), another equation will be: obtained later (see Eq. (30)). At the critical temperature T = To(p) , the averaged magnetization reduces to zero. Consequently, from (20) and (21), we obtain A~=0
for v = 1, 2,
(23)
from which the critical temperature as a function of concentration and correlation parameter Ac at T = T~(p) can be calculated. Here, it is worth noting that for p = 1 (pure system), we find 1 2 (24) A e4 t _ 1 , t e l = In 2 ' Ac = 71, where t c =J/(kBT~). The results (24) are equivalent to those derived in Ref. [19] for the special case J1 --J2. 3.2. Magnetic susceptibility
The initial magnetic susceptibility is defined by X = lira 0m H--,0 OH
t am h=O" J ah
(25)
110
A. Bobdk, M. Ja~ur ~Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
Differentiating both sides of Eqs. (15) and (18) with h, we obtain ( A v - 3 B ~ m 2 ) - ~~m h=0 + m ( X v - m
2Y ~ ) -~A ~ h=O+Q=O,
(26)
where OA~ OA '
X~=
OB~ Y~= OA
forv=l,2.
(27)
Therefore, the inverse initial susceptibility below a critical temperature is given by (Jx)-
1 A 1-A2F t C 1 --
for T < r c,
(28)
C2F
with X 1 - m2Y1
F
X2
(29)
__ m E Y 2 .
On the other hand, the inverse paramagnetic susceptibility, through Eq. (26), is given by (jX)-I
A1 A2 . . . . . . tC 1 tC 2
for T _> Tc.
(30)
By solving Eq. (30), the correlation parameter A can be evaluated for the region of T >/Tc. Finally, by the use of A, the inverse paramagnetic susceptibility (30) can be calculated. 3.3. Internal energy and specific heat The internal energy per site is defined by U = -2Je.
(31)
where e is the short-range order parameter (10), which can be evaluated by applying our basic assumption, Eq. (12), to Eq. (10), for the case of z -- 4 and h = 0. Then, the magnetic contribution to the specific heat per site can be obtained from relation C = oU/OT.
(32)
On the basis of the equations evaluated in this section, the numerical calculations of thermodynamical quantities for the diluted square lattice have been made and the results are presented and discussed in Section 4.
4. Numerical results and discussion
We plot the critical temperature Tc(P) and the correlation parameter At(p) against the concentration of magnetic atoms in Figs. 1 and 2, respectively, by solving Eqs. (23). We have found that at the critical concentration Pc at which Tc reduces to zero is given by Pc -- 0.4740. This value for Pc represents an improvement on the effective-field theory of Pc -- 0.4284 [6,23] and is somewhat closer to the best value of Pc = 0.590 obtained by the series expansion method [24]. The value of Ae(p) decreases from the value Ac = ~1 at p = 1 to the critical value Ac = 0.1873 at p =Pc (see Fig. 2). It should be noted here that these results differ from those obtained within the former version correlated effective-field theory [17], where a rapid increase from the value Ac -- ~1 at p = 1 to the critical value Ac = 0.734 at Pc = 0.5642 was found.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105"117
.
0
-
•
,
-
.
-
•
.
111
-
2.5 2.0
~.~ 1.5 1.0 0.5 0.0 0.4
0.5
0.6
0.7 P
0.8
0.9
.0
Fig. 1. Critical temperature as a function of the concentration of magnetic atoms.
The reason for this is the difference between the configurational averaging adopted here and that employed in Ref. [17] (for a review, see Ref. [9]). The thermal behaviour of the correlation parameter is shown in Fig. 3, through Eq. (22) below To(p), and Eq. (30) above T¢(p), for selected values of p. We observe that the curve for p = 1 increases from the value of zero at T -- 0, passes through a maximum value at T¢(p), and decreases monotonically with increasing temperature. But this behaviour is changed for p < 1. Namely, the parameter A has a finite value, even at T - - 0, which decreases on decreasing the concentration of magnetic atoms. On the other hand, very near the critical concentration, the value of A at T = 0 weakly increases with decrease of p.
Xc 0.3
0.25
0.2 I I II
0.15
i p¢ = 0.474 I i
0.4
.
0.5
.
.
.
0.6
.
.
.
0.7 P
.
.
.
0.8
0.9
1.0
Fig. 2. Critical correlation parameter as a function of the concentration of magnetic atoms.
112
A. Bobdk, M. Ja~ur / Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
0.35 X
0.8
0.,3
0.7
0.25 o.2
0.15 0.1
[
o.o 0.0
0.0
.
0.5
.
.
1.0
.
.
.
1.5
2.0
r/rc Fig. 3. Temperaturedependenceof the correlationparameterfor selectedconcentrationsp.
This latter peculiar behaviour of the correlation parameter is also observed above the critical temperature. This anomalous behaviour away from the vicinity of the critical temperature may be related to the fact that spatial fluctuations of the local magnetization, on the basis of Eq. (12), were totally neglected. Indeed, for the diluted ferromaguet, the magnitude of the thermal averages (gisi) should be different at each site, because of the random configuration of nearest neighbours. However, near to the critical temperature T¢ the thermal spin correlation length diverges, and then the spin system as a whole reacts cooperatively and does not feel anything from the random environment of nearest neighbours (see also Ref. [25], where similar arguments are presented). Therefore, at To, the spatial fluctuations of the local magnetization may be neglected and our basic assumption (Eq. (12)) should be appropriate in the vicinity of To. On the other hand, away from the critical region, the spatial fluctuations of the local magnetization may appear because of the finite thermal spin correlation length. The neglect of these fluctuations in Eq. (12), therefore, may be the cause of the unusual behaviour of the correlation parameter far away from T¢. In Fig. 4, the magnetization as a function of temperature as obtained by solving the coupled Eqs. (20) (or (21)) and (22), is shown for several values of p. We can see that the effect of decreasing concentration of magnetic atoms is an increase in the depression of magnetization over the entire temperature range for T < T¢, as observed in diluted and amorphous ferromagnets [26]. This result is in contradiction with the conclusion of Ref. [17] that the magnetization near the critical concentration may increase with decrease in the concentration of magnetic atoms. The reason for this difference is that in Ref. [17] the authors only considered Eq. (9) for an occupied central site (i.e. ~i = 1). However, there is evidently an inconsistency in this, a s ((si))c ((]£i) in the notation of Ref. [17]) and ((sj))~ ((/zi+ , ) on the right-hand side of Eq. (7) of Ref. [17]) are then not the same quantity. The former is conditional on the spin being present; the latter is not. Thus both quantities cannot be replaced by m. The same defect is also present in another equation used in Ref. [17] which can be obtained from (13) by putting ~:~=gj = ~k = 1. Moreover, the decoupling approximation ((~jsj))c - (~j)~((sj))~, which ignores the fact that the (sj) are configurationally dependent [9] was used in Ref. [17]. Therefore, we believe that the approach used in Ref. [17] is not appropriate for analyzing the site-diluted Ising model since the results obtained are probably very dependent on the approximate procedures mentioned above.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
1.0
~,,~=
113
1.0
0.8
o
0.4
5
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
:r/r~ Fig. 4. Temperature dependence of the spontaneous magnetization for selected concentrations p.
The temperature dependence of the inverse magnetic susceptibility is shown in Fig. 5 for selected values of p. We observed that for the concentration Pc < P < 1, the susceptibility exhibits two divergences: one at T - - 0 due to finite clusters, and another at Tc corresponding to an infinite duster. In contrast, in the pure system ( p = 1) X ~ 0 as T ~ 0, and only one has the Curie-Weiss peak. We notice that these effects have also been observed for both bond-diluted [27] and site-bond-correlated [28] Ising models.
2s IIo9 2.o[I
~.~
I
^.
0°1o
/I
0.9
//
o:~\///
0.5
o o ~ 0.0
0.5
1.0
1.5
2.0
r/ro Fig. 5. Temperature dependence of the inverse magnetic susceptibility for typical values of concentration p.
114
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
2.0 0.9
1.5
0.8 '1 1"0"
0.7
0.5
0.0 0.0 0.5
1.0
1.5
2.0
2.5
3.0
3.5
k, T/J Fig. 6. Temperature dependence of the internal energy per site for several concentrations p.
The internal energy (or short-range order) is illustrated in Fig. 6. The survival of short-range order at
T > To(p) is manifested in these curves by the observed discontinuity in their derivatives at To(p), which become less pronounced as p decreases towards Pc" In Fig. 7 the specific heat is shown for selected values of p. As is usual in effective-field theories, the well known logarithmic (at least for p = 1) singularity is not recovered. Despite the fact that the singularity which appears is of an incorrect type, an essential phenomenon is dearly exhibited. We refer to the fact that, for Pc < P -< 0.8038, tWO different contributions to the specific heat are present (the singular one coming from the unique infinite cluster, and the regular one coming from the isolated finite
2.0 -------~
~
-
-
0.9
1.5 0.8 ,re
1.0
0.7 0.6
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
,3.0 ,3.5
k,B T / J Fig. 7. Temperature dependence of the specific heat per site for several concentrations p.
A. Bobdk, M. Ja~Eur/Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
115
0.25
0.2 ~=0.5 *~ 0.15
0.5 0.05
0.0
o.o
0.3
'
'
1.o
1.5
i
2.0
2.5
sr/J Fig. 8. The same as Fig. 7, but for valuesof concentrationaboveand belowthe criticalconcentrationPc = 0.4740.
clusters), whereas for 0.8038 < p < 1, the finite duster contribution becomes negligible and the specific heat only has a peak at To(p). On the other hand, for 0 < p
5. Conclusions
We have discussed the quenched site-diluted spin-l/2 Ising ferromagnet on a square lattice. Within a correlated effective-field theory which differs from that introduced in [17], we calculate the correlation parameter and the most relevant thermodynamical quantities, namely the phase diagram in the concentration-temperature space, spontaneous magnetization, magnetic susceptibility, internal energy and specific heat. Contrary to Ref. [17], our approach takes into account the correlation between the site occupancy number for a particular site and the configuration-dependent thermal average of the spin at that site. Some interesting effects (see Figs. 5, 7 and 8) arise in the thermal behaviours of both susceptibility and specific heat due to the eventual coexistence in the system, of an infinite cluster with finite ones (whose respective weights depend on the site concentration). It is important to remark here that although the correlation parameter exhibits some anomalous behaviour away from the critical region, the thermodynamical quantities obtained behave normally and are quite satisfactory. As discussed in Section 4, the anomalous behaviour of the correlation parameter probably arises because the present treatment based on approximation (12) neglects the spatial fluctuations of the local magnetization. These effects will be considered in a forthcoming paper. Finally, we have studied for simplicity the physical properties of the site-diluted Ising ferromaguet on a square lattice. However, qualitative features due to site dilution are expected to remain even for higher spin dimensionality and other lattices.
A. Bobdk, M. Jag~ur/Joumal of Magnetism and Magnetic Materials 136 (1994) 105-117
116
Appendix T h e coefficients R~ (v -- 1 - 6) a n d L , (v = 1 - 8) in Eqs. (14) a n d (18) are as follows: R 1 - sinh(Dt)
c o s h 3 ( D t ) t a n h ( x ) I x=O = ~ [ t a n h ( 4 / )
+ 2 tanh(2t)],
R 2 - sinh(Dt)
c o s h 2 ( D t ) t a n h ( x ) I x=O = ¼ [ t a n h ( 3 t ) + t a n h ( t ) ] ,
R 3 = sinh(Dt)
cosh(Ot)
t a n h ( x ) I x=O = ½ t a n h ( 2 t ) ,
R 4 - s i n h ( D t ) t a n h ( x ) I x=O = t a n h ( t ) , R 5 - sinh(Dt) 3 cosh(Dt)
(A1)
t a n h ( x ) I ~=o = -~[ t a n h ( a t ) - 2 t a n h ( 2 t ) ] ,
R 6 = s i n h 3 ( D t ) t a n h ( x ) I x=O = ¼ [ t a n h ( 3 t ) - 3 t a n h ( t ) ] ; and L1 = c o s h 4 ( D t ) s e c h 2 ( x ) I x=o = gl [ s e c h Z ( 4 t ) + 4 s e c h 2 ( 2 t ) + 3] L 2 - c o s h 3 ( D t ) s e c h 2 ( x ) I ~=o = ¼ [ s e c h 2 ( 3 t ) + 3 s e c h E ( t ) ] , L 3 - c o s h 2 ( D t ) s e c h Z ( x ) I ~=o = X [ s e c h 2 ( 2 t ) + 1], L 4 ---- c o s h ( D t )
s e c h 2 ( x ) I x=o = s e c h 2 ( t ) ,
L 5 --- s i n h 2 ( D t ) c o s h 2 ( D / ) s e c h 2 ( x ) I ~=0 = ~ [ s e c h 2 ( a t ) - 1], L 6 ~
sinh2(Dt) cosh(Dt)
L 7 --~
s i n h 2 ( D t ) s e c h 2 ( x ) I x_o = ½ [ s e c h 2 ( 2 t ) - 1],
(A2)
s e c h 2 ( x ) I x=o = ¼ [ s e c h 2 ( 3 t ) - s e c h 2 ( t ) ] ,
L s -= sinh 4 ( D t ) sech 2 ( x ) [ x = o = 1 [sech 2 (4 t) - 4 sech 2 (2 t ) + 3].
Acknowledgement This w o r k was s u p p o r t e d by g r a n t no. 1 / 2 4 8 / 9 3 .
References [1] R.B. Stinchcombe, in: Phase Transitions and Critical Phenomena, vol. 7, eds. C. Domb and J.L. Lebowitz (Academic Press, London, 1983). [2] N. Boccara, Phys. Lett. 94A (1983) 185. [3] O.F. De Alcantara Bonfim and I.P. Fittipaldi, Phys. Lett. 98A (1983) 199. [4] T. Kaneyoshi, R. Honmura, I. Tamura and E.F. Sarmento, Phys. Rev. B 29 (1984) 5121. [5] A. Benyoussef, N. Boccara and M. Saber, J. Phys. C: Solid State Phys. 18 (1985) 4275. [6] T. Balcerzak, A. Bob~ik, J. Mielnicki and V.H. Truong, Phys. Stat. Solidi (b) 130 (1985) 183. [7] A. Bob~ik and J. Karaba, Phys. Stat. Solidi (b) 142 (1987) 575. [8] G. Wiatrowski, T. Balcerzak and J. Mieinicki, J. Magn. Magn. Mater. 71 (1988) 197. [9] J.W. Tucker, J. Magn. Magn. Mater 102 (1991) 144. [10] S. Mockov~iak, M. Ja~ur and A. Bobfik, Phys. Stat. Solidi (b) 166 (1991) K25. [11] M. Saber and J.W. Tucker, J. Magn. Magn. Mater. 102 (1991) 287. [12] M. Saber and J.W. Tucker, J. Magn. Magn. Mater. 114 (1992) 11. [13] A. Bob~ik, S. Mockov~iak and J. Sivulka, Phys. Stat. Solidi (b) 176 (1993) 477. [14] H.B. Callen, Phys. Lett. 4 (1963) 161. [15] A. Bobfik and M. Ja~ur, Phys. Stat. Solidi (b) 135 (1986) K9.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117 [16] T. Kaneyoshi, I.P. Fittipaldi, R. Honmura and T. Manabe, Phys. Rev. B 24 (1981) 481. [17] T. Kaneyoshi, I. Tamura and R. Honmura, Phys. Rev. B 29 (1984) 2769. [18] T. Kaneyoshi and I. Tamura, Phys. Rev. B 25 (1982) 4679. [19] R. Honmura, Phys. Rev. B 30 (1984) 348. [20] I. Tamura, E.F. Sarmento, I.P. Fittipaldi and T. Kaneyoshi, Phys. Star. Solidi (b) 118 (1983) 409. [21] R. Honmura and T. Kaneyoshi, J. Phys. C: Solid State Phys. 12 (1979) 3979. [22] M.E. Lines, Phys. Rev. B 9 (1974) 3927. [23] T. Kaneyoshi, R. Honmura and I. Tamura, Phys. Rev. B 29 (1984) 5121. [24] M.F. Sykes and J.W. Essam, Phys. Rev. 133 (1964) A310. [25] M. F~ihnle, G. Herzer, H. Kronmiiller, R. Meyer, M. Saile and T. Egami, J. Magn. Magn. Mater. 38 (1983) 240. [261 T. Kaneyoshi, Amorphous Magnetism (CRC Press, Boca Raton, 1984). [27] R. Honmura, E.F. Sarmento and C. Tsallis, Z. Phys. B 51 (1983) 355. [28] A.A.P. Da Silva and F.G. Brady Moreira, Phys. Rev. B 40 (1989) 10986.
117
Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
Jourill oI ~laPitism
A t materials
Correlated effective-field theory of the site-diluted Ising model A. Bobfik *, M. Jag~ur Department of Theoretical Physics and Geophysics, Faculty of Natural Sciences, P.J. Safarik University, Moyzesova 16, 041 54 Ko~ice, Slovak Republic
Received 21 December 1993
Abstract
The quenched site-diluted Ising model is considered within the framework of a correlated effective-field theory. The dependence with temperature and magnetic concentration of the correlation parameter and thermodynamic quantities of interest are presented and discussed for a square lattice. The present formalism yields satisfactory results, in particular the effects, on the susceptibility and specific heat, of the coexisting finite and infinite clusters are exhibited. It is shown that the correlation parameter exhibits peculiar behaviour, away from the critical region, as a function of temperature for selected values of concentration of magnetic atoms.
1. Introduction
For many years now there have been extensive studies of the disordered magnetic materials. One kind of disorder appears in site-diluted ferromagnets, where the perfect crystallographic lattice has lattice sites occupied at random by magnetic atoms. Major points of interests in the theoretical and experimental studies of such disordered magnets are the concentration dependence of the critical temperature and value of the critical concentration of magnetic atoms below which the system will not exhibit a transition to magnetic order. The problem has been studied using a variety of approximations and mathematical techniques (see, e.g. Ref. [1]). Recently there has been renewed interest in the study of site-diluted Ising models by the use of effective field theory [2-13] in which all the single-site kinematic relations are correctly accounted for, but multispin correlation functions between various spins are entirely decoupled. This is achieved most frequently by the use of single- [14] or two-site [15] exact formal identities and by employing the differential (integral) operator technique or the combinatorial method. On the other hand, a correlated effective field theory [16], developed for the s p i n - l / 2 Ising model on a regular lattice, partially takes into account the effects of multispin correlations. The theory yields values for the critical temperature and other thermodynamical quantities that are identical to those of the Bethe-Peierls approximation, although the approach is completely different in its formulation f r o m
* Corresponding author. 0304-8853/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00296-4
106
A. Bobdk, M. Ja~ur / Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
that method. Apart from studies of the pure system, extensions of this theory have also been applied to the site-diluted ferromagnet [17], anisotropic systems [18,19] and semi-infinite systems [20]. In this paper, the correlated effective-field theory is applied to the quenched site-diluted s p i n - l / 2 Ising model on a square lattice. The relevant thermodynamical quantities, namely the correlation parameter, spontaneous magnetization, critical temperature, susceptibility, internal energy and specific heat are all investigated within this framework. Although all these thermodynamical quantities, apart from of the internal energy and the specific heat, were studied in Ref. [17], our results differ from those presented there, for two reasons. First, in the derivation of the coupled equations for the averaged magnetization and the correlation parameter in Ref. [17] the difference between a conditional configurational average and a non-conditional configurational average was ignored. Second, the results of Ref. [17] were based on the use of a configurational averaging procedure that neglected the correlation between the site occupancy number for a particular site and the configuration-dependent thermal average of the spin operator at that site. A general discussion of the importance of including these correlations in the study of site-diluted Ising systems has recently been given in Ref. [9]. In the present paper, we present results based on the adoption of the proper configurational averaging procedure for a site-diluted Ising model, proposed in Ref. [6]. The plan of this paper is as follows. In Section 2 we introduce the formalism of the correlated effective-field approximation for the site-diluted Ising model. In Section 3 the theory is applied to the diluted Ising ferromagnet on a square lattice and the relations for the most relevant thermodynamical quantities are obtained. In Section 4 numerical results of such quantities are presented and discussed. Our concluding remarks are presented in Section 5.
2. Formulation
We consider the site-diluted Ising model in an applied external magnetic field H. The Hamiltonian of the system is
,~= i EJij~i~jsisj _ nE~isi ' i,j
(1)
i
where Jij is the exchange interaction, ~i is the random variable that takes the value unity or zero, depending on whether the site i is occupied by a magnetic atom or not, and s i = + 1. The starting point of the theory is the set of exact formal identities, of the type discussed in Ref. [14] for a pure system, that exist for the thermal average ({f;}~isi) of a single spin:
({f~},~si)=(~i{fi } tanh[/3,i(~j J~j~jsy+H)]),
(2)
where /3 = 1/(kBT) and {fi} represents any function of the Ising variables except si. To recast this equation in a more convient form, the differential operator technique [21] may be employed, to give
({ fi} ~iSi) = (~i{ fi} H [ ~j cosh(Otij ) + ~jsj sinh(Otij ) + 1 - ~j] ) tanh( x + h) I x =0,
J
where D = O/Ox is the differential operator, have used the relation exp(a~j) = ~j exp(a) + 1 - ~j,
tij =/3Jij, and
(3)
h =/3H. In deriving Eq. (3) from Eq. (2), we (4)
together with the Van der Waerden identity for the s p i n - l / 2 Ising model. It should be noted that Eq. (3) is for a fixed configuration of occupied sites, so the thermal averages are configurationally dependent.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
107
Therefore, the next step is to carry out the configurational averaging, to be denoted by (...)~. Then Eq. (3) for a lattice having only nearest-neighbour interactions, J, can be written as follows: z
<({fi}~iSi))c = (<~i{fi} H [~j cosh(Dt) + ~jsi sinh(Dt) + 1 - ~j]))c tanh(x + h) I x=0,
(5)
j=l
where z is the coordination number and t =/3J. Since we are interested in the thermodynamical quantities, let us expand the right-hand side of Eq. (5) with respect to h and retain only its first-order terms:
((~i{fi}R))c + ((~,{f,}L))~h,
(6)
R = I-I [~j cosh(Dt) + ~j.sj sinh(Dt) + 1 - ~ j ] tanh(x) I ~=0, j=l
(7)
(({fi}~iSi))c =
where z
and z
L = I ~ [~i cosh(Dt) + ~jsj sinh(Dt) + 1 - e j ] sech2(x) I x=0.
(8)
j=l
Eq. (6) can generate many kinds of identities, which give relations among spin correlation functions, on substituting some Ising variable functions for {fi}. Among them, upon setting {fi} "- 1, Eq. (6) reduces to m = ((~iR))c + ((~iL))ch,
(9)
and by setting {fi} = ~:ysj, we obtain ,1~= ( ( ~ i ~ j s j R ) )c d¢- ( (~il~jsjL ) )~h,
(10)
where m = (<~isi))~ and e = ((l~isigysj)) ~ are the averaged magnetization and short-range order parameter, respectively, expressed in terms of multisite correlation functions, which are yet undetermined. In order to evaluate Eqs. (9) and (10), many authors have introduced the effective-field approximation which decouples next-neighbour spin correlations:
((XjXk"" Xt))c = ((Xj))c((Xk))c''' ((Xt))c,
(11)
where j m k # ... # l are the nearest-neighbours of a site i and xj = gj.sj.. Instead of using the decoupling approximation (11), let us extend the concept of a correlated effective field [16,22] to the site-diluted Ising model as follows: ~jSj -~ ( (~jSj) )c "1- X( ~iS i -- ( (~iSi) )c) ,
(12)
where j denotes the nearest-neighbour site of a certain central site i, and A is a temperature-, concentration- and external field-dependent correlation parameter. It should be noted here that we have adopted a form of correlated effective-field different from that originally proposed for the site-diluted Ising model in Ref. [17]. Substituting the relation (12) into Eqs. (9) and (10) yields equations for the averaged magnetization rn and the short-range order parameter e as functions of the correlation parameter, reduced temperature, external magnetic field and concentration of magnetic atoms. In order to determine A uniquely, another equation is needed. In particular, we can use the three-site identity, obtained on putting {fi} = ~jSj~kSk in Eq. (6),
( <~isi~jSj~kSk >>¢= ( (~i~jSj~kSkR> )~ + ( (~i~jSjgkSkL ) )~h.
(13)
A. Bobdk, M. Ja~ur /Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
108
Thus, the magnetization, short-range order parameter and the correlation parameter can be determined by solving the coupled Eqs. (9), (10) and (13) for any coordination number z defining the crystallographic structure. However, for simplicity, we discuss below the square lattice with z = 4, although with an appropriate z-value any other lattice could be discussed as well.
3. Application to the square lattice
In this section we study the thermodynamical properties of the diluted Ising ferromagnet on a square lattice. From Eq. (9), the averaged magnetization, for the system with z = 4, reduces to m = 4 p [ p a R 1+ 3 p 2 ( 1 - p ) g 2 + 3 p ( 1 - p ) 2 R 3 + ( 1 - p ) a g 4 ] m
+4[PRs+(l_p)R6]f.+(p[p4Ll+4pa(l_p)L2+6p2(1 + 4 p ( 1 - p ) 3 L 4 + (1 _p)4] + 6p[ p2L 5 + 2 p ( 1 - p ) L
- p ) 2L 3
(14)
6 + ( 1 - p)2L7] + toL8)h ,
where p = (~:i)c is the concentration of magnetic atoms, p = ((¢i¢/S/~kSk)) ~ and z = ((¢i(;jSj(~kSkCtSt))¢, j ~ k 4: l = 1-~ 4, are the averaged two- and three-site correlation functions, respectively, and to = ((¢i¢1S1¢2S2¢3S31~4S4))c is the averaged four-site correlation function. The coefficients R~ (v = 1 ~ 6) and L~ (v = 1-8) can be easily calculated by applying the mathematical relation e x p ( y D ) f ( x ) = f ( x + y), and are given in the Appendix. Applying the correlated approximation (12) to the multisite spin correlation functions appearing in Eq. (14), we obtain
( A - Blm2)m + Clh = 0,
(15)
with
A 1 = 4 p [ p a R I + 3 p 2 ( 1 - p ) R 2+ 3p(1 - p ) 2R 3 + (1 - p ) a R 4 ] + 4[pg 5 + (1-p)g6]a
1 - 1,
B 1 = - 4 [ p R 5 + (1 - p ) R 6 ] b l ,
Cl=p[p4Ll +4pa(l_p)LE+6p2(1
(16) - p ) 2 L 3 + 4p(1 - p ) 3 L 4 + (1 -19) 4]
+ 6(c 0 + clm2)[ p2L 5 + 2p(1 - p ) L 6 + (1 - p ) 2 L 7 ] + (d o + dxm 2 + d2m4)Ls, where a 1 = (1 - 3p)A 3 + 3pA 2, b I = (3 - p ) A 3 - 3(2 - p ) A 2 + 3(1 - p ) ; t +p,
CO -~-p~t2,
C1 = -- (2 --p)A 2 + 2(1 - p ) A + p ,
d0=PA 4,
d~= - 2 ( 2 - 3 p ) A 4 + 4 ( 1 - 3 p ) A 3 + 6 p A
(17) 2,
d E= - ( 4 - p ) A 4 + 4 ( 3 - p ) A 3 - 6 ( 2 - p ) A 2 + 4 ( 1 - p ) A + p . On the other hand, from Eq. (13), in the same way as that used in derivation of Eq. (15), we can obtain for the square lattice the following equation:
( A 2 - B2m2)m + C2h = O,
(18)
A. Bobdk, M. Jcd~ur/Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
109
with A2 = 2p2[ p2(R1 + R5 ) + p ( 1 - p ) ( 2 R
2 + R6) + ( 1 - p ) 2 R 3 ]
+ 2 [ p ( R , + R s ) + ( 1 - p ) R 2 ] a 1 - ( 1 - 2p)A 2 - 2pA, B 2= - 2 [ p ( R 1+R5) + ( 1 - p ) R 2 ] b , + (1 - A ) 2, C2 = p3[ p2L5 + 2p(1 - p ) L
(19)
(1 _p)2L7]
6 "at-
+(Co + c , m 2 ) [ p 2 ( L t + 4L 5 +L8 ) + 2 p ( 1 - p ) ( L 2 + 2L6) + ( 1 - p ) 2 L 3 ] + (do + d i m 2 + d2m4)L5.
Using Eqs. (15) and (18), we can evaluate the magnetization, critical temperature, critical concentration, correlation parameter and susceptibility associated with the diluted Ising ferromagnet on a square lattice. 3.1. Magnetization, correlation parameter and critical temperature
From Eqs. (15) or (18), for h = 0, the averaged magnetization m is given by m 2 =A1/B 1
(20)
m 2 = A 2 / B 2,
(21)
or from which we obtain the following equation for the correlation parameter A as a function of p and t: Am
A2
B1
B2 •
(22)
It should be noted that the parameter A evaluated from (22) is only valid below a critical temperature Tc(p), since in order to determine A we used the averaged magnetization. For temperatures above To(p), another equation will be: obtained later (see Eq. (30)). At the critical temperature T = To(p) , the averaged magnetization reduces to zero. Consequently, from (20) and (21), we obtain A~=0
for v = 1, 2,
(23)
from which the critical temperature as a function of concentration and correlation parameter Ac at T = T~(p) can be calculated. Here, it is worth noting that for p = 1 (pure system), we find 1 2 (24) A e4 t _ 1 , t e l = In 2 ' Ac = 71, where t c =J/(kBT~). The results (24) are equivalent to those derived in Ref. [19] for the special case J1 --J2. 3.2. Magnetic susceptibility
The initial magnetic susceptibility is defined by X = lira 0m H--,0 OH
t am h=O" J ah
(25)
110
A. Bobdk, M. Ja~ur ~Journal of Magnetism and MagneticMaterials 136 (1994) 105-117
Differentiating both sides of Eqs. (15) and (18) with h, we obtain ( A v - 3 B ~ m 2 ) - ~~m h=0 + m ( X v - m
2Y ~ ) -~A ~ h=O+Q=O,
(26)
where OA~ OA '
X~=
OB~ Y~= OA
forv=l,2.
(27)
Therefore, the inverse initial susceptibility below a critical temperature is given by (Jx)-
1 A 1-A2F t C 1 --
for T < r c,
(28)
C2F
with X 1 - m2Y1
F
X2
(29)
__ m E Y 2 .
On the other hand, the inverse paramagnetic susceptibility, through Eq. (26), is given by (jX)-I
A1 A2 . . . . . . tC 1 tC 2
for T _> Tc.
(30)
By solving Eq. (30), the correlation parameter A can be evaluated for the region of T >/Tc. Finally, by the use of A, the inverse paramagnetic susceptibility (30) can be calculated. 3.3. Internal energy and specific heat The internal energy per site is defined by U = -2Je.
(31)
where e is the short-range order parameter (10), which can be evaluated by applying our basic assumption, Eq. (12), to Eq. (10), for the case of z -- 4 and h = 0. Then, the magnetic contribution to the specific heat per site can be obtained from relation C = oU/OT.
(32)
On the basis of the equations evaluated in this section, the numerical calculations of thermodynamical quantities for the diluted square lattice have been made and the results are presented and discussed in Section 4.
4. Numerical results and discussion
We plot the critical temperature Tc(P) and the correlation parameter At(p) against the concentration of magnetic atoms in Figs. 1 and 2, respectively, by solving Eqs. (23). We have found that at the critical concentration Pc at which Tc reduces to zero is given by Pc -- 0.4740. This value for Pc represents an improvement on the effective-field theory of Pc -- 0.4284 [6,23] and is somewhat closer to the best value of Pc = 0.590 obtained by the series expansion method [24]. The value of Ae(p) decreases from the value Ac = ~1 at p = 1 to the critical value Ac = 0.1873 at p =Pc (see Fig. 2). It should be noted here that these results differ from those obtained within the former version correlated effective-field theory [17], where a rapid increase from the value Ac -- ~1 at p = 1 to the critical value Ac = 0.734 at Pc = 0.5642 was found.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105"117
.
0
-
•
,
-
.
-
•
.
111
-
2.5 2.0
~.~ 1.5 1.0 0.5 0.0 0.4
0.5
0.6
0.7 P
0.8
0.9
.0
Fig. 1. Critical temperature as a function of the concentration of magnetic atoms.
The reason for this is the difference between the configurational averaging adopted here and that employed in Ref. [17] (for a review, see Ref. [9]). The thermal behaviour of the correlation parameter is shown in Fig. 3, through Eq. (22) below To(p), and Eq. (30) above T¢(p), for selected values of p. We observe that the curve for p = 1 increases from the value of zero at T -- 0, passes through a maximum value at T¢(p), and decreases monotonically with increasing temperature. But this behaviour is changed for p < 1. Namely, the parameter A has a finite value, even at T - - 0, which decreases on decreasing the concentration of magnetic atoms. On the other hand, very near the critical concentration, the value of A at T = 0 weakly increases with decrease of p.
Xc 0.3
0.25
0.2 I I II
0.15
i p¢ = 0.474 I i
0.4
.
0.5
.
.
.
0.6
.
.
.
0.7 P
.
.
.
0.8
0.9
1.0
Fig. 2. Critical correlation parameter as a function of the concentration of magnetic atoms.
112
A. Bobdk, M. Ja~ur / Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
0.35 X
0.8
0.,3
0.7
0.25 o.2
0.15 0.1
[
o.o 0.0
0.0
.
0.5
.
.
1.0
.
.
.
1.5
2.0
r/rc Fig. 3. Temperaturedependenceof the correlationparameterfor selectedconcentrationsp.
This latter peculiar behaviour of the correlation parameter is also observed above the critical temperature. This anomalous behaviour away from the vicinity of the critical temperature may be related to the fact that spatial fluctuations of the local magnetization, on the basis of Eq. (12), were totally neglected. Indeed, for the diluted ferromaguet, the magnitude of the thermal averages (gisi) should be different at each site, because of the random configuration of nearest neighbours. However, near to the critical temperature T¢ the thermal spin correlation length diverges, and then the spin system as a whole reacts cooperatively and does not feel anything from the random environment of nearest neighbours (see also Ref. [25], where similar arguments are presented). Therefore, at To, the spatial fluctuations of the local magnetization may be neglected and our basic assumption (Eq. (12)) should be appropriate in the vicinity of To. On the other hand, away from the critical region, the spatial fluctuations of the local magnetization may appear because of the finite thermal spin correlation length. The neglect of these fluctuations in Eq. (12), therefore, may be the cause of the unusual behaviour of the correlation parameter far away from T¢. In Fig. 4, the magnetization as a function of temperature as obtained by solving the coupled Eqs. (20) (or (21)) and (22), is shown for several values of p. We can see that the effect of decreasing concentration of magnetic atoms is an increase in the depression of magnetization over the entire temperature range for T < T¢, as observed in diluted and amorphous ferromagnets [26]. This result is in contradiction with the conclusion of Ref. [17] that the magnetization near the critical concentration may increase with decrease in the concentration of magnetic atoms. The reason for this difference is that in Ref. [17] the authors only considered Eq. (9) for an occupied central site (i.e. ~i = 1). However, there is evidently an inconsistency in this, a s ((si))c ((]£i) in the notation of Ref. [17]) and ((sj))~ ((/zi+ , ) on the right-hand side of Eq. (7) of Ref. [17]) are then not the same quantity. The former is conditional on the spin being present; the latter is not. Thus both quantities cannot be replaced by m. The same defect is also present in another equation used in Ref. [17] which can be obtained from (13) by putting ~:~=gj = ~k = 1. Moreover, the decoupling approximation ((~jsj))c - (~j)~((sj))~, which ignores the fact that the (sj) are configurationally dependent [9] was used in Ref. [17]. Therefore, we believe that the approach used in Ref. [17] is not appropriate for analyzing the site-diluted Ising model since the results obtained are probably very dependent on the approximate procedures mentioned above.
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
1.0
~,,~=
113
1.0
0.8
o
0.4
5
0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
:r/r~ Fig. 4. Temperature dependence of the spontaneous magnetization for selected concentrations p.
The temperature dependence of the inverse magnetic susceptibility is shown in Fig. 5 for selected values of p. We observed that for the concentration Pc < P < 1, the susceptibility exhibits two divergences: one at T - - 0 due to finite clusters, and another at Tc corresponding to an infinite duster. In contrast, in the pure system ( p = 1) X ~ 0 as T ~ 0, and only one has the Curie-Weiss peak. We notice that these effects have also been observed for both bond-diluted [27] and site-bond-correlated [28] Ising models.
2s IIo9 2.o[I
~.~
I
^.
0°1o
/I
0.9
//
o:~\///
0.5
o o ~ 0.0
0.5
1.0
1.5
2.0
r/ro Fig. 5. Temperature dependence of the inverse magnetic susceptibility for typical values of concentration p.
114
A. Bobdk, M. Ja~ur /Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
2.0 0.9
1.5
0.8 '1 1"0"
0.7
0.5
0.0 0.0 0.5
1.0
1.5
2.0
2.5
3.0
3.5
k, T/J Fig. 6. Temperature dependence of the internal energy per site for several concentrations p.
The internal energy (or short-range order) is illustrated in Fig. 6. The survival of short-range order at
T > To(p) is manifested in these curves by the observed discontinuity in their derivatives at To(p), which become less pronounced as p decreases towards Pc" In Fig. 7 the specific heat is shown for selected values of p. As is usual in effective-field theories, the well known logarithmic (at least for p = 1) singularity is not recovered. Despite the fact that the singularity which appears is of an incorrect type, an essential phenomenon is dearly exhibited. We refer to the fact that, for Pc < P -< 0.8038, tWO different contributions to the specific heat are present (the singular one coming from the unique infinite cluster, and the regular one coming from the isolated finite
2.0 -------~
~
-
-
0.9
1.5 0.8 ,re
1.0
0.7 0.6
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
,3.0 ,3.5
k,B T / J Fig. 7. Temperature dependence of the specific heat per site for several concentrations p.
A. Bobdk, M. Ja~Eur/Journal of Magnetism and Magnetic Materials 136 (1994) 105-117
115
0.25
0.2 ~=0.5 *~ 0.15
0.5 0.05
0.0
o.o
0.3
'
'
1.o
1.5
i
2.0
2.5
sr/J Fig. 8. The same as Fig. 7, but for valuesof concentrationaboveand belowthe criticalconcentrationPc = 0.4740.
clusters), whereas for 0.8038 < p < 1, the finite duster contribution becomes negligible and the specific heat only has a peak at To(p). On the other hand, for 0 < p
5. Conclusions
We have discussed the quenched site-diluted spin-l/2 Ising ferromagnet on a square lattice. Within a correlated effective-field theory which differs from that introduced in [17], we calculate the correlation parameter and the most relevant thermodynamical quantities, namely the phase diagram in the concentration-temperature space, spontaneous magnetization, magnetic susceptibility, internal energy and specific heat. Contrary to Ref. [17], our approach takes into account the correlation between the site occupancy number for a particular site and the configuration-dependent thermal average of the spin at that site. Some interesting effects (see Figs. 5, 7 and 8) arise in the thermal behaviours of both susceptibility and specific heat due to the eventual coexistence in the system, of an infinite cluster with finite ones (whose respective weights depend on the site concentration). It is important to remark here that although the correlation parameter exhibits some anomalous behaviour away from the critical region, the thermodynamical quantities obtained behave normally and are quite satisfactory. As discussed in Section 4, the anomalous behaviour of the correlation parameter probably arises because the present treatment based on approximation (12) neglects the spatial fluctuations of the local magnetization. These effects will be considered in a forthcoming paper. Finally, we have studied for simplicity the physical properties of the site-diluted Ising ferromaguet on a square lattice. However, qualitative features due to site dilution are expected to remain even for higher spin dimensionality and other lattices.
A. Bobdk, M. Jag~ur/Joumal of Magnetism and Magnetic Materials 136 (1994) 105-117
116
Appendix T h e coefficients R~ (v -- 1 - 6) a n d L , (v = 1 - 8) in Eqs. (14) a n d (18) are as follows: R 1 - sinh(Dt)
c o s h 3 ( D t ) t a n h ( x ) I x=O = ~ [ t a n h ( 4 / )
+ 2 tanh(2t)],
R 2 - sinh(Dt)
c o s h 2 ( D t ) t a n h ( x ) I x=O = ¼ [ t a n h ( 3 t ) + t a n h ( t ) ] ,
R 3 = sinh(Dt)
cosh(Ot)
t a n h ( x ) I x=O = ½ t a n h ( 2 t ) ,
R 4 - s i n h ( D t ) t a n h ( x ) I x=O = t a n h ( t ) , R 5 - sinh(Dt) 3 cosh(Dt)
(A1)
t a n h ( x ) I ~=o = -~[ t a n h ( a t ) - 2 t a n h ( 2 t ) ] ,
R 6 = s i n h 3 ( D t ) t a n h ( x ) I x=O = ¼ [ t a n h ( 3 t ) - 3 t a n h ( t ) ] ; and L1 = c o s h 4 ( D t ) s e c h 2 ( x ) I x=o = gl [ s e c h Z ( 4 t ) + 4 s e c h 2 ( 2 t ) + 3] L 2 - c o s h 3 ( D t ) s e c h 2 ( x ) I ~=o = ¼ [ s e c h 2 ( 3 t ) + 3 s e c h E ( t ) ] , L 3 - c o s h 2 ( D t ) s e c h Z ( x ) I ~=o = X [ s e c h 2 ( 2 t ) + 1], L 4 ---- c o s h ( D t )
s e c h 2 ( x ) I x=o = s e c h 2 ( t ) ,
L 5 --- s i n h 2 ( D t ) c o s h 2 ( D / ) s e c h 2 ( x ) I ~=0 = ~ [ s e c h 2 ( a t ) - 1], L 6 ~
sinh2(Dt) cosh(Dt)
L 7 --~
s i n h 2 ( D t ) s e c h 2 ( x ) I x_o = ½ [ s e c h 2 ( 2 t ) - 1],
(A2)
s e c h 2 ( x ) I x=o = ¼ [ s e c h 2 ( 3 t ) - s e c h 2 ( t ) ] ,
L s -= sinh 4 ( D t ) sech 2 ( x ) [ x = o = 1 [sech 2 (4 t) - 4 sech 2 (2 t ) + 3].
Acknowledgement This w o r k was s u p p o r t e d by g r a n t no. 1 / 2 4 8 / 9 3 .
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