- Email: [email protected]
Phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field
Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
Phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field L.L. Denga, S.L. Yana,b,c,* b
a Department of Physics, Suzhou University, 215006 Suzhou, China Provincial Laboratory of Thin Film Materials, Suzhou University, Suzhou 215006, China c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 19 March 2002; received in revised form 3 June 2002
Abstract The bond dilution spin-1 transverse ferromagnetic Ising model with random crystal field is investigated in the framework of effective field theory (EFT). The general expressions of magnetization are derived for lattice with any coordination number z: In particular, we calculate the phase diagrams of a square lattice. The second-order phase transition lines and the tricritical points are presented in different conditions for the square lattice. Special emphasis is placed on the influence of bond dilution, random crystal field and the transverse field on the phase diagrams. We discover some unusual phenomena. New results obtained in this paper are discussed in detail. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.Dg; 75.10.Jm; 75.40.Cx Keywords: Phase transition features; Bond dilution; Transverse field; Ferromagnetic Ising spin system; Random crystal field
1. Introduction In the past few decades there has been an increasing number of studies dealing with the critical properties of the magnetic spin system. Investigations of the magnetic spin system make use of two main models in local conditions, that is, Ising model or Heisenberg model. It is well known that the critical and magnetic properties of the Ising spin system have been studied for a long time and still attracts significant attention [1–5]. This is because the Ising model is a simple but fruitful model. On the other hand, the crystal field, quantum tunnel effect and some disorder distributions have been considered in the Ising model. A lot of investigations have proved that the Ising model is very useful for various systems, such as one-, two- and three-component fluids, ternary alloys [6], 3He–4He mixtures [7–9] and so on. Many different phase transition features are presented.
*Corresponding author. Department of Physics, Suzhou University, 215006 Suzhou, China. Fax: +86-512-5231918. E-mail address: [email protected] (S.L. Yan). 0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 4 9 0 - 0
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
139
A strategy in exploring the phase transition features has been the mapping of the ferromagnetic spin system associated with different interactions (including transverse field and crystal field interaction) and different disorder distributions (including dilution and randomness). In general, two different kinds of the Ising model are frequently adopted. The first model is often called the transverse Ising model (TIM). The TIM was originally introduced by Gennes [10] as a pseudo-spin model of hydrogen-bonded ferroelectrics. Then it was applied to several other systems, such as cooperative Jahn–Teller systems [11] and ferromagnets with strong uniaxial anisotropy in a transverse magnetic field [12]. In this model the quantum tunnel effect is taken into consideration, which is described as a transverse field O in the Hamiltonian. So the TIM is also a simple quantum spin model. People have learned that the transverse field works against the ordering. Some studies indicate that the transition temperature of the system can be depressed to zero by increasing the transverse field to a critical value Oc [13–15]. The second model is so-called the Blume–Capel model (BCM). The model considers the magnetic spin system with crystal field and suits for spin-S (S > 12). The previous works show that the first-order phase transition line and a tricritical point (TCP) can be discovered in the regions of a large negative crystal field, when the phase transition temperature is plotted as a function of the crystal field [16]. Its rich variations have been investigated by many kinds of techniques. On the other hand, when various random distributions are introduced, critical features of the abovementioned two models have plenty of changes. People have learned that the randomness of the crystal field can change phase diagrams or depress the TCP [27,28]. Moreover, the existence of bond disorder can also produce a powerful role for phase diagrams of the system [17,18]. Further, the different form of the random crystal field distribution plays an important role in the determination of the order of phase transition [19–22]. Recently, an ideal and typical procedure that deals with magnetic Ising system combines the TIM and BCM [23]. The investigations involving the disorder distribution have made much progress in the theoretical aspect. Jiang has discussed the bond-diluted transverse spin-1ferromagnetic system in the presence of crystal field [24]. Wang et al. has studied phase diagram of the random TIM with crystal field [25]. We have also examined the critical properties of the transverse Ising system with dilution crystal field [26]. The above studies had displayed very unusual behaviours mainly because the random distribution in the magnetic system plays an important role. In order to get further messages, it is necessary to consider various disorder distributions simultaneously. More recently, we notice that Pupa and Diep have studied the BCM with both the bond dilution and the random crystal field by means of Monte Carlo technique and given some meaningful messages [27]. However, they did not give the whole phase diagrams due to a large calculation and deal with an important quantum tunnel effect. The present authors deal with the BCM of two disorder distributions as well and obtain some outstanding results [28]. In this paper, we pay much attention to the bond-diluted transverse ferromagnetic Ising spin-1 system with random crystal field interaction and study the phase transition features including the second-order transition line, the reentrant phenomenon and the TCP in the framework of the effective field theory (EFT). Both, bond dilution and random crystal field distribution, may lead to some new and rich results in the presence of quantum tunnel effect. Influence of two different random distributions on the phase transition features will show distinct characteristic. To our knowledge, the above subject has not been investigated in the previous works. Therefore, we need a more profound consideration of competition between the different disorder conditions, the transverse field and crystal field. Our aim is to obtain some new and complete phase transition features and provides some help for resultant experimental and theoretical researches. In this work the bond dilution and the symmetrical distribution of the random crystal field is introduced. The paper is divided into four sections. In Section 2, we briefly present the basic theory of the EFT. In Section 3, the detailed numerical results and discussions in the different conditions are given. And we give a simple conclusion in the last section.
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
140
2. Theory For a spin-1 bond-dilution transverse ferromagnetic Ising system with random crystal field, its Hamiltonian is described by X X X H¼ Jij Siz Sjz O Six Di ðSiz Þ2 ; ð1Þ i;j
i
where Six and Siz are components of spin-1 operators at site i; which can take the value 71 and 0. Jij is the exchange interaction between the nearest neighbour sites, assumed to be positive. Thus, the first summation runs over all pairs of nearest neighbour sites. O and Di represent the transverse field and the crystal field, respectively. And the latter two summations in the Hamiltonian involve all sites. We assume that Jij and Di satisfy independent dilution and random distribution, respectively, PðJij Þ ¼ pdðJij JÞ þ ð1 pÞdðJij Þ;
ð2Þ
PðDi Þ ¼ tdðDi DÞ þ ð1 tÞdðDi þ DÞ;
ð3Þ
where 0ppp1:0 and 0ptp1:0: Let p denotes the concentration of bond dilution and t indicates the concentration of random crystal field. For the present system, the Hamiltonian can be separated into Hi which includes all parts of H associated with the ith site and H 0 which does not connect with the ith site. Thus, the eigenvalues of Hi can be calculated. The standard procedure then leads to the following expectations of averaged magnetization for an above-mention system in the framework of EFT: ** ++ z Y m ¼ //Siz SSr ¼ ½ðSjz Þ2 coshðJij rÞ þ Sjz sinhðJij rÞ þ 1 ðSjz Þ2 F ðxÞjx¼0 : ð4Þ j
r
The parameter q (quadrupolar moment) is given by ** ++ z Y 2 2 2 q ¼ //ðSiz Þ SSr ¼ ½ðSiz Þ coshðJrÞ þ Siz sinhðJrÞ þ 1 ðSiz Þ GðxÞjx¼0 ; j
ð5Þ
r
where r ¼ q=qx is a differential operator. The inner /?S indicates the canonical thermal average and the out /?Sr denotes the bond dilution average for Eq. (2). Functions F ðxÞ and GðxÞ are written as Z F ðxÞ ¼ PðDi Þf ðx; Di Þ dDi ; ð6Þ GðxÞ ¼
Z
PðDi Þgðx; Di Þ dDi ;
where PðDi Þ is a random crystal field distribution for Eq. (3), f ðx; Di Þ and gðx; Di Þ are as follows: 3 X 2bDi f ðx; Di Þ ¼ exp 2bCE1 ðnÞ 3 n¼1 3 2x 2 Di x Di x3 þ ð7=2ÞDi O2 x E1 ðnÞ E2 ðnÞ 3C 27 BC " #1 3 X 2bDi exp 2bCE1 ðnÞ ; 3 n¼1
ð7Þ
ð8Þ
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
gðx; Di Þ ¼
3 X n¼1
141
2bDi exp 2bCE1 ðnÞ 3
2Di 2 ð1=2ÞO2 x2 D2i x2 þ x4 ð1=2ÞO4 2 E1 ðnÞ þ E2 ðnÞ 27 3 9C BC " #1 3 X 2bDi exp 2bCE1 ðnÞ ; 3 n¼1
ð9Þ
where E1 ðnÞ ¼ cos
y þ 2pðn 1Þ ; 3
ð10Þ
E2 ðnÞ ¼ sin
y þ 2pðn 1Þ ; 3
ð11Þ
1 3 A ¼ 27 Di 13Di x2 þ 16Di O2 ;
ð12Þ
B ¼ 19½3ðD2i x þ x3 Þ2 þ 34D2i O4 þ 15D2i O2 x2 þ 9O2 x4 þ 9O4 x2 þ 3O6 1=2 ;
ð13Þ
C ¼ ð19D2i þ 13O2 þ 13x2 Þ1=2 ;
ð14Þ
y ¼ arccosðA=C 3 Þ;
ð15Þ
b ¼ 1=kB T:
ð16Þ
In order to evaluate the Eqs. (4) and (5), we need to adopt the decoupling approximation, //Siz ðSjz Þ2 ?Slz SSr E//Siz SSr //ðSjz Þ2 SSr ?//Slz SSr ; for iaja?al: This is because the problem of multispin correlation is mathematically intractable. With such a procedure, expanding the right-hand sides of Eqs. (4) and (5), the averaged magnetization m and quadrupolar moment q can be obtained: m ¼ ½q/coshðJij rÞSr þ m/sinhðJij rÞSr þ 1 q z F ðxÞjx¼0
ð17Þ
q ¼ ½q/coshðJij rÞSr þ m/sinhðJij rÞSr þ 1 q z GðxÞjx¼0 :
ð18Þ
and Here, we are interested in studying the phase transition features of the present system. If we combine Eqs. (17) and (18), we can get the following self-consistent equation: m ¼ am þ bm3 þ cm5 þ ?:
ð19Þ
In the vicinity of the second-order phase transition line, we retain only the linear term of the self-consistent equation because m is small enough. Thus, the second-order phase transition line is determined by a¼1
and
bo0:
ð20Þ
The averaged magnetization m can be given by 1a : ð21Þ m2 ¼ b Here, the right-hand side of (21) must be positive, thus the parameter bo0: If it is not the case, the phase transition line is first order. So the point at which a ¼ 1 and b ¼ 0 is TCP. a and b are given by the following equations: a ¼ z/sinhðJij rÞSr ½q0 /coshðJij rÞSr q0 þ 1 z1 F ðxÞjx¼0
ð22Þ
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
142
and b ¼ zðz 1Þq1 /sinhðJij rÞSr ½/coshðJij rÞSr 1 ½1 þ q0 /coshðJij rÞSr q0 z2 F ðxÞjx¼0 zðz 1Þðz 2Þ /sinh3 ðJij rÞSr ½1 þ q0 /coshðJij rÞSr q0 z3 F ðxÞjx¼0 ; þ 3! where q0 and q1 satisfy the solutions of q0 ¼ ½q0 /coshðJij rÞSr q0 þ 1 z GðxÞjx¼0
ð23Þ
ð24Þ
and f ; 1e in which e and f are defined by q1 ¼
ð25Þ
e ¼ z½/coshðJij rÞSr 1 ½1 þ q0 /coshðJij rÞSr q0 z1 GðxÞjx¼0
ð26Þ
and zðz 1Þ /sinh2 ðJij rÞSr ½q0 /coshðJij rÞSr þ 1 q0 z1 GðxÞjx¼0 : ð27Þ 2! The Jij in the above expressions is given by independent dilution distribution in Eq. (2). z is the coordination number of a lattice. Eqs. (20)–(27) are the general types of the second-order phase transition and the longitudinal magnetization for the bond-dilution transverse ferromagnetic spin-1 Ising system with random crystal field interaction by considering the nearest neighbour exchange interaction. In this paper, for simplicity we will only study the square lattice with z ¼ 4; although any lattices with an arbitrary coordination number z can be also discussed in the same way. Of course, we know that, with increasing the coordination number z such as simple cubic (z ¼ 6) and body centre cubic (z ¼ 8) lattices, it is obvious that the coefficients expanded in Eqs. (22)–(24), (26) and (27) will add and become more complex. Hence, the study by selecting a square lattice is not only simple, but can also enlighten us to learn the physical status of a higher coordination number lattice. In the next section, we will focus on the phase transition features of the present system for a square lattice, because the existence of two disorder distributions may result in new and rich phenomena. Some phenomena are not discovered in the previous works. We hope that it is significant to clarify the phase transition features of ferromagnetic spin system in complex conditions and to promote further experimental and theoretical research. f ¼
3. Numerical results and discussion By solving the relations (20) numerically, we can obtain a series of phase diagrams of the system and discuss the phase transition features. Firstly, we consider the situation that the transverse field O equals to zero. The Curie temperatures versus negative crystal field for the bond concentrations p ¼ 1:0; 0.8 and 0.6 are depicted in Figs. 1(a)–(c), when the values of random crystal field concentration t are changed. For t ¼ 1:0; it is a pure BCM. The system shows a TCP [20]. With the decreasing of the random concentration t; the tricritical temperature is depressed and reaches to zero monotonously at a critical concentration tc : From our exact calculations, the TCP exists in the ranges of 1:0XtX0:674; while the second-order phase transition temperature goes to zero at two values of the crystal field D=J ¼ 3:0 and 4:0; if random concentration t varies in the ranges 0:673XtX0:585 and 0:584XtX0:572; respectively. When the value of t becomes smaller than a critical value t ; the second-order phase transition lines extend to D-N: The t equals to 0.572 in the present situation. That is to say, for tp0:571; the present system is always in ferromagnetic state at low
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
143
Fig. 1. Curie temperature dependencies of the crystal field for bond concentration p values of (a) 1.0, (b) 0.8, and (c) 0.6 in the absence of transverse field O=J: The numbers on the curves are the values of random crystal field concentration t:
temperature, even though an infinite negative crystal field does not destroy the long-range order. The spin states of all sites are in the status S ¼ 71; while S ¼ 0 state is absent. In this case, the random BCM reduces to the spin-12 Ising model. The reentrant phenomenon is another interesting problem. In Fig. 1(a) we can see that, with decreasing of the concentration of random crystal field, the reentrant phenomena can be observed gradually and become more and more strong, even if there exist the TCPs. On the other hand, a new zigzag transition line occurs at t ¼ 0:673: Then the zigzag transition line shall be depressed but the reentrant phenomena still exist until t ¼ 0:585: When t ¼ 0:584; another zigzag transition line can be observed again. As shown in Fig. 1(a), the system displays very complex and interesting reentrant phenomena at low temperature. The results may be attributed to the randomness of the crystal field. In Fig. 1(b), we can see that the phase diagram has some new variety because of the introduction of the bond dilution. Firstly, the TCP exists in ranges of 1:0XtX0:684: The TCP is affected by the bond-dilution concentration. We can also see that the scope of the random crystal field concentration has a small decrease to compare with Fig. 1(a). Secondly, the phase transition temperature goes to zero at negative crystal field D=J ¼ 2:0 and 3:0; if random concentration varies in ranges of 0:683XtX0:560 and 0:559XtX0:5; respectively. In addition, it is worthwhile paying
144
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
Fig. 2. Variation of random crystal field concentration t with bond concentration p about the existence of TCP.
attention to the fact that all of the second-order phase transition lines for D-N at pure bond status extend to finite negative values of crystal field at bond dilution condition (see the curve labeled 0.5). A simple physical reason is mainly because the crystal field is increased from D-N to finite value in the pressure of bond dilution. Thus, some sites have S ¼ 0 state. The ordered phase will be depressed. When bond-dilution concentration p continues decreasing, we find that the scope of the random crystal field concentration that affects the existence of TCP becomes larger, as depicted in Fig. 1(c). From our calculation, the TCP exists in the range of 1:0XtX0:638 at p ¼ 0:6: This deals with the fact that the crystal field correlation between sites is subjected to a large influence when bond concentration becomes large, so that we expect a depression of the TCP given by a larger random crystal field concentration. Now, it is clear that the TCP depends on the random concentration t and disappears at a certain value of tc : Of course, tc will change due to the introduction of the bond dilution. Fig. 2 shows the behaviour of the critical concentration tc as a function of bond-dilution concentration p: We see that the random crystal field concentration tc has a fluctuation. The curve increases slightly in the beginning and then decreases monotonically and has a minimum value at p ¼ 0:598 and increases rapidly in pp0:597: In this case, a reasonable interpretation is that the connectivity of the magnetic interactions is weakened and decreased the crystal field correlation between all sites in the presence of bond dilution. This means that competition between the bond dilution and the random crystal field is very meaningful. Of course, the form of the symmetrical distribution of the random crystal field is also an important factor. Finally, a large bond concentration can generate a strong depression for the transition lines so that the value of the random crystal field t increases rapidly and the TCP does not appear at all. Figs. 3(a)–(c) express the behaviours of the Curie temperature dependencies of the crystal field D with bond concentration p ¼ 1:0; 0.8 and 0.6 at O=J ¼ 0:8 for various values of the random concentration t: From the comparison of Figs. 3(a) and 1(a), we find that the tricritical behaviour is depressed in Fig. 3(a). The range in which the TCP occurs is 1:0XtX0:810; which is much smaller than that of Fig. 1(a). At the same time, we can observe that the reentrant phenomena are depressed quickly and become very weak. Two fixed values of the negative crystal field disappear at T ¼ 0 K. But the transverse field cannot effectively influence the second-order phase transition lines that extend to D-N; while the bond dilution can depress it strongly (see the lines labeled 0.5 in Figs. 1(a), (b) and 3(a)). When the two disorder factors are introduced into the present system at one time in Figs. 3(b) and (c), some complex phenomena disappear at low temperature and the phase transition lines become very simple and rule. Hence, we infer that the transverse field can effectively destroy the reentrant phenomena and depress TCP, while it is not very sensitive to depress the second-order phase transition lines when there exists a larger random crystal field
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
145
Fig. 3. Curie temperature dependencies of the crystal field for bond concentration p values of (a) 1.0, (b) 0.8, and (c) 0.6 with the transverse field O=J ¼ 0:8: The numbers on the curves are the values of random crystal field concentration t:
concentration. However, the influence of the bond-dilution concentration on second-order phase transition lines is very huge, while the influence on the reentrant phenomena and the TCP is not too large. Certainly, the varieties of the phase diagrams are very apparent in the presence of both bond dilution and random crystal field. In order to further discuss the phase transition features of the present system, the phase diagrams in (T; DÞ plane with a fixed t ¼ 0:65 for the values of the transverse field O=J ¼ 0:0 and 0.8 are presented in Figs. 4(a) and (b), when the values of the bond-dilution concentration p are changed. In Fig. 4(a), we see that the transition lines are of the second order and the TCP does not exist when pX0:7: When the bonddilution concentration becomes large, the system shows the tricritical behaviour (see the curve labeled p ¼ 0:6). With further dilution of bond concentration, the TCP is depressed again. The bond percolation threshold is pc ¼ 0:3976; that is to say the order phase will not appear in pppc : Here, the role of bond dilution is very particular and important. The reason may come from the fact that the TCP appear mainly because the existence of bond concentration is able to decrease the crystal field correlation between all sites,
146
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
Fig. 4. Curie temperature dependencies of crystal field with (a) O=J ¼ 0:0 and (b) O=J ¼ 0:8 for a fixed random crystal field concentration t ¼ 0:65: The numbers on the curves are the values of bond concentration p:
while a large bond concentration can depress strongly the transition lines again. Fig. 4(b) shows that the order phase range is smaller than that of Fig. 4(a). And the tricritical behaviour cannot be observed in Fig. 4(b). The reason is due to the introduction of the transverse field. Eventually, we need to say that symmetrical form of the whole phase diagrams mention above can be obtained if the random concentration t is in ranges of 0:5XtX0: The uniform variety of phase diagrams will turn negative crystal field parameter (D) into (þD).
4. Conclusion In this paper, we have investigated the phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field. Compared with the previous works, because we consider two kinds of disorder factors at one time, the new and rich phase transition features have been discovered and some significant results have been obtained: (1) With the decreasing of the random
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
147
concentration t; the tricritical behaviour is depressed and several kinds of reentrant phenomena can be observed. When tpt ; the system remains the order phase at low temperature. (2) Although the bond concentration p can affect the scope of the random concentration t that the TCP exists and can strongly depress the order phase, the effect of the reentrant phenomena is weak. The effect of both the bond dilution and the random crystal field on the tricritical points has expressed a new phenomenon. (3) The existence of the transverse field depresses the TCP and destroys the reentrant phenomena strongly. However, it is not very sensitive to depress the second-order phase transition lines under a larger random crystal field concentration condition. (4) The phase diagrams are from complex to simple due to the common role of both the transverse field and the bond dilution. We believe that a real ferromagnetic spin system may exist in these disorder factors, so our numerical results and discussions are meaningful and necessary. Finally, we hope that the results here may be helpful for further research.
Acknowledgements This research was supported in part by the Education Bureau Natural Science Foundation of Jiangsu Province (Grant No. 00SJB140003) and in part by the Thin Film Materials Key Laboratory Open Foundation of Jiangsu Province (Grant No. K2022).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
A. Aharony, Phys. Rev. B 18 (1978) 3318. C.Z. Yang, Z.Y. Li, Phys. Status Solidi B 146 (1988) 223. S.L. Yan, C.Z. Yang, J. Magn. Magn. Mater. 205 (1999) 357. G.M. Buendia, E. Machado, Phys. Rev. B 61 (2000) 14686. M. Barati, A. Ramazani, Phys. Rev. B 62 (2000) 12130. R.A. Tahir-Kheli, Phys. Rev. 169 (1968) 517. M. Blume, V.J. Emery, R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. A. Maritan, M. Cieplak, M.R. Swift, F. Toigo, J.R. Banavar, Phys. Rev. Lett. 69 (1992) 221. C. Buzano, A. Maritan, A. Pelizzola, J. Phys.: Condens. Matter 6 (1994) 327. P.G. de Gennes, Solid State Commun. 1 (1963) 132. R.J. Euiott, G.A. Gehring, M.P. Matoemoff, et al., J. Phys. C 4 (1971) l179. R.B. Stinchcombe, J. Phys. C 6 (1073) 2459. E.F. Sarmento, R. Bechara Muniz, S.B. Cavalcanti, Phys. Rev. B 36 (1987) 529. E.F. Sarmento, T. Kaneyoshi, Phys. Rev. B 39 (1989) 9555. J.L. Zhong, et al., Phys. Status Solidi B 160 (1990) 329. A.F. Siqueira, I.P. Fittipaldi, Physica A 138 (1986) 592. K. Hui, A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507. G.M. Zhang, C.Z. Yang, Acta Phys. Sin. 42 (1993) 128. T. Kaneyoshi, J. Mielnicki, J. Phys.: Condens. Matter 2 (1990) 8773. T. Kaneyoshi, Phys. Status Solidi B 170 (1992) 313. N. Benayard, A. Benyoussef, N. Boccara, J. Phys. 18 (1995) 1899. N.S. Branco, B.M. Boechat, Phys. Rev. B 56 (1997) 11673. X.F. Jiang, J.L. Li, J.L. Zhong, C.Z. Yang, Phys. Rev. B 47 (1993) 827. X.F. Jiang, J. Magn. Magn. Mater 134 (1994) 167. Y.Q. Wang, Z.Y. Li, Acta Phys. Sin. 44 (1995) 811. L.L. Deng, S.L. Yan, Chin. Phys. 11 (2002) 383. I. Pupa, H.T. Diep, J. Magn. Magn. Mater. 224 (2001) 85. S.L. Yan, L.L. Deng, Physica A 308 (2002) 301.
Phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field L.L. Denga, S.L. Yana,b,c,* b
a Department of Physics, Suzhou University, 215006 Suzhou, China Provincial Laboratory of Thin Film Materials, Suzhou University, Suzhou 215006, China c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 19 March 2002; received in revised form 3 June 2002
Abstract The bond dilution spin-1 transverse ferromagnetic Ising model with random crystal field is investigated in the framework of effective field theory (EFT). The general expressions of magnetization are derived for lattice with any coordination number z: In particular, we calculate the phase diagrams of a square lattice. The second-order phase transition lines and the tricritical points are presented in different conditions for the square lattice. Special emphasis is placed on the influence of bond dilution, random crystal field and the transverse field on the phase diagrams. We discover some unusual phenomena. New results obtained in this paper are discussed in detail. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.Dg; 75.10.Jm; 75.40.Cx Keywords: Phase transition features; Bond dilution; Transverse field; Ferromagnetic Ising spin system; Random crystal field
1. Introduction In the past few decades there has been an increasing number of studies dealing with the critical properties of the magnetic spin system. Investigations of the magnetic spin system make use of two main models in local conditions, that is, Ising model or Heisenberg model. It is well known that the critical and magnetic properties of the Ising spin system have been studied for a long time and still attracts significant attention [1–5]. This is because the Ising model is a simple but fruitful model. On the other hand, the crystal field, quantum tunnel effect and some disorder distributions have been considered in the Ising model. A lot of investigations have proved that the Ising model is very useful for various systems, such as one-, two- and three-component fluids, ternary alloys [6], 3He–4He mixtures [7–9] and so on. Many different phase transition features are presented.
*Corresponding author. Department of Physics, Suzhou University, 215006 Suzhou, China. Fax: +86-512-5231918. E-mail address: [email protected] (S.L. Yan). 0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 4 9 0 - 0
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
139
A strategy in exploring the phase transition features has been the mapping of the ferromagnetic spin system associated with different interactions (including transverse field and crystal field interaction) and different disorder distributions (including dilution and randomness). In general, two different kinds of the Ising model are frequently adopted. The first model is often called the transverse Ising model (TIM). The TIM was originally introduced by Gennes [10] as a pseudo-spin model of hydrogen-bonded ferroelectrics. Then it was applied to several other systems, such as cooperative Jahn–Teller systems [11] and ferromagnets with strong uniaxial anisotropy in a transverse magnetic field [12]. In this model the quantum tunnel effect is taken into consideration, which is described as a transverse field O in the Hamiltonian. So the TIM is also a simple quantum spin model. People have learned that the transverse field works against the ordering. Some studies indicate that the transition temperature of the system can be depressed to zero by increasing the transverse field to a critical value Oc [13–15]. The second model is so-called the Blume–Capel model (BCM). The model considers the magnetic spin system with crystal field and suits for spin-S (S > 12). The previous works show that the first-order phase transition line and a tricritical point (TCP) can be discovered in the regions of a large negative crystal field, when the phase transition temperature is plotted as a function of the crystal field [16]. Its rich variations have been investigated by many kinds of techniques. On the other hand, when various random distributions are introduced, critical features of the abovementioned two models have plenty of changes. People have learned that the randomness of the crystal field can change phase diagrams or depress the TCP [27,28]. Moreover, the existence of bond disorder can also produce a powerful role for phase diagrams of the system [17,18]. Further, the different form of the random crystal field distribution plays an important role in the determination of the order of phase transition [19–22]. Recently, an ideal and typical procedure that deals with magnetic Ising system combines the TIM and BCM [23]. The investigations involving the disorder distribution have made much progress in the theoretical aspect. Jiang has discussed the bond-diluted transverse spin-1ferromagnetic system in the presence of crystal field [24]. Wang et al. has studied phase diagram of the random TIM with crystal field [25]. We have also examined the critical properties of the transverse Ising system with dilution crystal field [26]. The above studies had displayed very unusual behaviours mainly because the random distribution in the magnetic system plays an important role. In order to get further messages, it is necessary to consider various disorder distributions simultaneously. More recently, we notice that Pupa and Diep have studied the BCM with both the bond dilution and the random crystal field by means of Monte Carlo technique and given some meaningful messages [27]. However, they did not give the whole phase diagrams due to a large calculation and deal with an important quantum tunnel effect. The present authors deal with the BCM of two disorder distributions as well and obtain some outstanding results [28]. In this paper, we pay much attention to the bond-diluted transverse ferromagnetic Ising spin-1 system with random crystal field interaction and study the phase transition features including the second-order transition line, the reentrant phenomenon and the TCP in the framework of the effective field theory (EFT). Both, bond dilution and random crystal field distribution, may lead to some new and rich results in the presence of quantum tunnel effect. Influence of two different random distributions on the phase transition features will show distinct characteristic. To our knowledge, the above subject has not been investigated in the previous works. Therefore, we need a more profound consideration of competition between the different disorder conditions, the transverse field and crystal field. Our aim is to obtain some new and complete phase transition features and provides some help for resultant experimental and theoretical researches. In this work the bond dilution and the symmetrical distribution of the random crystal field is introduced. The paper is divided into four sections. In Section 2, we briefly present the basic theory of the EFT. In Section 3, the detailed numerical results and discussions in the different conditions are given. And we give a simple conclusion in the last section.
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
140
2. Theory For a spin-1 bond-dilution transverse ferromagnetic Ising system with random crystal field, its Hamiltonian is described by X X X H¼ Jij Siz Sjz O Six Di ðSiz Þ2 ; ð1Þ i;j
i
where Six and Siz are components of spin-1 operators at site i; which can take the value 71 and 0. Jij is the exchange interaction between the nearest neighbour sites, assumed to be positive. Thus, the first summation runs over all pairs of nearest neighbour sites. O and Di represent the transverse field and the crystal field, respectively. And the latter two summations in the Hamiltonian involve all sites. We assume that Jij and Di satisfy independent dilution and random distribution, respectively, PðJij Þ ¼ pdðJij JÞ þ ð1 pÞdðJij Þ;
ð2Þ
PðDi Þ ¼ tdðDi DÞ þ ð1 tÞdðDi þ DÞ;
ð3Þ
where 0ppp1:0 and 0ptp1:0: Let p denotes the concentration of bond dilution and t indicates the concentration of random crystal field. For the present system, the Hamiltonian can be separated into Hi which includes all parts of H associated with the ith site and H 0 which does not connect with the ith site. Thus, the eigenvalues of Hi can be calculated. The standard procedure then leads to the following expectations of averaged magnetization for an above-mention system in the framework of EFT: ** ++ z Y m ¼ //Siz SSr ¼ ½ðSjz Þ2 coshðJij rÞ þ Sjz sinhðJij rÞ þ 1 ðSjz Þ2 F ðxÞjx¼0 : ð4Þ j
r
The parameter q (quadrupolar moment) is given by ** ++ z Y 2 2 2 q ¼ //ðSiz Þ SSr ¼ ½ðSiz Þ coshðJrÞ þ Siz sinhðJrÞ þ 1 ðSiz Þ GðxÞjx¼0 ; j
ð5Þ
r
where r ¼ q=qx is a differential operator. The inner /?S indicates the canonical thermal average and the out /?Sr denotes the bond dilution average for Eq. (2). Functions F ðxÞ and GðxÞ are written as Z F ðxÞ ¼ PðDi Þf ðx; Di Þ dDi ; ð6Þ GðxÞ ¼
Z
PðDi Þgðx; Di Þ dDi ;
where PðDi Þ is a random crystal field distribution for Eq. (3), f ðx; Di Þ and gðx; Di Þ are as follows: 3 X 2bDi f ðx; Di Þ ¼ exp 2bCE1 ðnÞ 3 n¼1 3 2x 2 Di x Di x3 þ ð7=2ÞDi O2 x E1 ðnÞ E2 ðnÞ 3C 27 BC " #1 3 X 2bDi exp 2bCE1 ðnÞ ; 3 n¼1
ð7Þ
ð8Þ
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
gðx; Di Þ ¼
3 X n¼1
141
2bDi exp 2bCE1 ðnÞ 3
2Di 2 ð1=2ÞO2 x2 D2i x2 þ x4 ð1=2ÞO4 2 E1 ðnÞ þ E2 ðnÞ 27 3 9C BC " #1 3 X 2bDi exp 2bCE1 ðnÞ ; 3 n¼1
ð9Þ
where E1 ðnÞ ¼ cos
y þ 2pðn 1Þ ; 3
ð10Þ
E2 ðnÞ ¼ sin
y þ 2pðn 1Þ ; 3
ð11Þ
1 3 A ¼ 27 Di 13Di x2 þ 16Di O2 ;
ð12Þ
B ¼ 19½3ðD2i x þ x3 Þ2 þ 34D2i O4 þ 15D2i O2 x2 þ 9O2 x4 þ 9O4 x2 þ 3O6 1=2 ;
ð13Þ
C ¼ ð19D2i þ 13O2 þ 13x2 Þ1=2 ;
ð14Þ
y ¼ arccosðA=C 3 Þ;
ð15Þ
b ¼ 1=kB T:
ð16Þ
In order to evaluate the Eqs. (4) and (5), we need to adopt the decoupling approximation, //Siz ðSjz Þ2 ?Slz SSr E//Siz SSr //ðSjz Þ2 SSr ?//Slz SSr ; for iaja?al: This is because the problem of multispin correlation is mathematically intractable. With such a procedure, expanding the right-hand sides of Eqs. (4) and (5), the averaged magnetization m and quadrupolar moment q can be obtained: m ¼ ½q/coshðJij rÞSr þ m/sinhðJij rÞSr þ 1 q z F ðxÞjx¼0
ð17Þ
q ¼ ½q/coshðJij rÞSr þ m/sinhðJij rÞSr þ 1 q z GðxÞjx¼0 :
ð18Þ
and Here, we are interested in studying the phase transition features of the present system. If we combine Eqs. (17) and (18), we can get the following self-consistent equation: m ¼ am þ bm3 þ cm5 þ ?:
ð19Þ
In the vicinity of the second-order phase transition line, we retain only the linear term of the self-consistent equation because m is small enough. Thus, the second-order phase transition line is determined by a¼1
and
bo0:
ð20Þ
The averaged magnetization m can be given by 1a : ð21Þ m2 ¼ b Here, the right-hand side of (21) must be positive, thus the parameter bo0: If it is not the case, the phase transition line is first order. So the point at which a ¼ 1 and b ¼ 0 is TCP. a and b are given by the following equations: a ¼ z/sinhðJij rÞSr ½q0 /coshðJij rÞSr q0 þ 1 z1 F ðxÞjx¼0
ð22Þ
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
142
and b ¼ zðz 1Þq1 /sinhðJij rÞSr ½/coshðJij rÞSr 1 ½1 þ q0 /coshðJij rÞSr q0 z2 F ðxÞjx¼0 zðz 1Þðz 2Þ /sinh3 ðJij rÞSr ½1 þ q0 /coshðJij rÞSr q0 z3 F ðxÞjx¼0 ; þ 3! where q0 and q1 satisfy the solutions of q0 ¼ ½q0 /coshðJij rÞSr q0 þ 1 z GðxÞjx¼0
ð23Þ
ð24Þ
and f ; 1e in which e and f are defined by q1 ¼
ð25Þ
e ¼ z½/coshðJij rÞSr 1 ½1 þ q0 /coshðJij rÞSr q0 z1 GðxÞjx¼0
ð26Þ
and zðz 1Þ /sinh2 ðJij rÞSr ½q0 /coshðJij rÞSr þ 1 q0 z1 GðxÞjx¼0 : ð27Þ 2! The Jij in the above expressions is given by independent dilution distribution in Eq. (2). z is the coordination number of a lattice. Eqs. (20)–(27) are the general types of the second-order phase transition and the longitudinal magnetization for the bond-dilution transverse ferromagnetic spin-1 Ising system with random crystal field interaction by considering the nearest neighbour exchange interaction. In this paper, for simplicity we will only study the square lattice with z ¼ 4; although any lattices with an arbitrary coordination number z can be also discussed in the same way. Of course, we know that, with increasing the coordination number z such as simple cubic (z ¼ 6) and body centre cubic (z ¼ 8) lattices, it is obvious that the coefficients expanded in Eqs. (22)–(24), (26) and (27) will add and become more complex. Hence, the study by selecting a square lattice is not only simple, but can also enlighten us to learn the physical status of a higher coordination number lattice. In the next section, we will focus on the phase transition features of the present system for a square lattice, because the existence of two disorder distributions may result in new and rich phenomena. Some phenomena are not discovered in the previous works. We hope that it is significant to clarify the phase transition features of ferromagnetic spin system in complex conditions and to promote further experimental and theoretical research. f ¼
3. Numerical results and discussion By solving the relations (20) numerically, we can obtain a series of phase diagrams of the system and discuss the phase transition features. Firstly, we consider the situation that the transverse field O equals to zero. The Curie temperatures versus negative crystal field for the bond concentrations p ¼ 1:0; 0.8 and 0.6 are depicted in Figs. 1(a)–(c), when the values of random crystal field concentration t are changed. For t ¼ 1:0; it is a pure BCM. The system shows a TCP [20]. With the decreasing of the random concentration t; the tricritical temperature is depressed and reaches to zero monotonously at a critical concentration tc : From our exact calculations, the TCP exists in the ranges of 1:0XtX0:674; while the second-order phase transition temperature goes to zero at two values of the crystal field D=J ¼ 3:0 and 4:0; if random concentration t varies in the ranges 0:673XtX0:585 and 0:584XtX0:572; respectively. When the value of t becomes smaller than a critical value t ; the second-order phase transition lines extend to D-N: The t equals to 0.572 in the present situation. That is to say, for tp0:571; the present system is always in ferromagnetic state at low
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
143
Fig. 1. Curie temperature dependencies of the crystal field for bond concentration p values of (a) 1.0, (b) 0.8, and (c) 0.6 in the absence of transverse field O=J: The numbers on the curves are the values of random crystal field concentration t:
temperature, even though an infinite negative crystal field does not destroy the long-range order. The spin states of all sites are in the status S ¼ 71; while S ¼ 0 state is absent. In this case, the random BCM reduces to the spin-12 Ising model. The reentrant phenomenon is another interesting problem. In Fig. 1(a) we can see that, with decreasing of the concentration of random crystal field, the reentrant phenomena can be observed gradually and become more and more strong, even if there exist the TCPs. On the other hand, a new zigzag transition line occurs at t ¼ 0:673: Then the zigzag transition line shall be depressed but the reentrant phenomena still exist until t ¼ 0:585: When t ¼ 0:584; another zigzag transition line can be observed again. As shown in Fig. 1(a), the system displays very complex and interesting reentrant phenomena at low temperature. The results may be attributed to the randomness of the crystal field. In Fig. 1(b), we can see that the phase diagram has some new variety because of the introduction of the bond dilution. Firstly, the TCP exists in ranges of 1:0XtX0:684: The TCP is affected by the bond-dilution concentration. We can also see that the scope of the random crystal field concentration has a small decrease to compare with Fig. 1(a). Secondly, the phase transition temperature goes to zero at negative crystal field D=J ¼ 2:0 and 3:0; if random concentration varies in ranges of 0:683XtX0:560 and 0:559XtX0:5; respectively. In addition, it is worthwhile paying
144
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
Fig. 2. Variation of random crystal field concentration t with bond concentration p about the existence of TCP.
attention to the fact that all of the second-order phase transition lines for D-N at pure bond status extend to finite negative values of crystal field at bond dilution condition (see the curve labeled 0.5). A simple physical reason is mainly because the crystal field is increased from D-N to finite value in the pressure of bond dilution. Thus, some sites have S ¼ 0 state. The ordered phase will be depressed. When bond-dilution concentration p continues decreasing, we find that the scope of the random crystal field concentration that affects the existence of TCP becomes larger, as depicted in Fig. 1(c). From our calculation, the TCP exists in the range of 1:0XtX0:638 at p ¼ 0:6: This deals with the fact that the crystal field correlation between sites is subjected to a large influence when bond concentration becomes large, so that we expect a depression of the TCP given by a larger random crystal field concentration. Now, it is clear that the TCP depends on the random concentration t and disappears at a certain value of tc : Of course, tc will change due to the introduction of the bond dilution. Fig. 2 shows the behaviour of the critical concentration tc as a function of bond-dilution concentration p: We see that the random crystal field concentration tc has a fluctuation. The curve increases slightly in the beginning and then decreases monotonically and has a minimum value at p ¼ 0:598 and increases rapidly in pp0:597: In this case, a reasonable interpretation is that the connectivity of the magnetic interactions is weakened and decreased the crystal field correlation between all sites in the presence of bond dilution. This means that competition between the bond dilution and the random crystal field is very meaningful. Of course, the form of the symmetrical distribution of the random crystal field is also an important factor. Finally, a large bond concentration can generate a strong depression for the transition lines so that the value of the random crystal field t increases rapidly and the TCP does not appear at all. Figs. 3(a)–(c) express the behaviours of the Curie temperature dependencies of the crystal field D with bond concentration p ¼ 1:0; 0.8 and 0.6 at O=J ¼ 0:8 for various values of the random concentration t: From the comparison of Figs. 3(a) and 1(a), we find that the tricritical behaviour is depressed in Fig. 3(a). The range in which the TCP occurs is 1:0XtX0:810; which is much smaller than that of Fig. 1(a). At the same time, we can observe that the reentrant phenomena are depressed quickly and become very weak. Two fixed values of the negative crystal field disappear at T ¼ 0 K. But the transverse field cannot effectively influence the second-order phase transition lines that extend to D-N; while the bond dilution can depress it strongly (see the lines labeled 0.5 in Figs. 1(a), (b) and 3(a)). When the two disorder factors are introduced into the present system at one time in Figs. 3(b) and (c), some complex phenomena disappear at low temperature and the phase transition lines become very simple and rule. Hence, we infer that the transverse field can effectively destroy the reentrant phenomena and depress TCP, while it is not very sensitive to depress the second-order phase transition lines when there exists a larger random crystal field
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
145
Fig. 3. Curie temperature dependencies of the crystal field for bond concentration p values of (a) 1.0, (b) 0.8, and (c) 0.6 with the transverse field O=J ¼ 0:8: The numbers on the curves are the values of random crystal field concentration t:
concentration. However, the influence of the bond-dilution concentration on second-order phase transition lines is very huge, while the influence on the reentrant phenomena and the TCP is not too large. Certainly, the varieties of the phase diagrams are very apparent in the presence of both bond dilution and random crystal field. In order to further discuss the phase transition features of the present system, the phase diagrams in (T; DÞ plane with a fixed t ¼ 0:65 for the values of the transverse field O=J ¼ 0:0 and 0.8 are presented in Figs. 4(a) and (b), when the values of the bond-dilution concentration p are changed. In Fig. 4(a), we see that the transition lines are of the second order and the TCP does not exist when pX0:7: When the bonddilution concentration becomes large, the system shows the tricritical behaviour (see the curve labeled p ¼ 0:6). With further dilution of bond concentration, the TCP is depressed again. The bond percolation threshold is pc ¼ 0:3976; that is to say the order phase will not appear in pppc : Here, the role of bond dilution is very particular and important. The reason may come from the fact that the TCP appear mainly because the existence of bond concentration is able to decrease the crystal field correlation between all sites,
146
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
Fig. 4. Curie temperature dependencies of crystal field with (a) O=J ¼ 0:0 and (b) O=J ¼ 0:8 for a fixed random crystal field concentration t ¼ 0:65: The numbers on the curves are the values of bond concentration p:
while a large bond concentration can depress strongly the transition lines again. Fig. 4(b) shows that the order phase range is smaller than that of Fig. 4(a). And the tricritical behaviour cannot be observed in Fig. 4(b). The reason is due to the introduction of the transverse field. Eventually, we need to say that symmetrical form of the whole phase diagrams mention above can be obtained if the random concentration t is in ranges of 0:5XtX0: The uniform variety of phase diagrams will turn negative crystal field parameter (D) into (þD).
4. Conclusion In this paper, we have investigated the phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field. Compared with the previous works, because we consider two kinds of disorder factors at one time, the new and rich phase transition features have been discovered and some significant results have been obtained: (1) With the decreasing of the random
L.L. Deng, S.L. Yan / Journal of Magnetism and Magnetic Materials 251 (2002) 138–147
147
concentration t; the tricritical behaviour is depressed and several kinds of reentrant phenomena can be observed. When tpt ; the system remains the order phase at low temperature. (2) Although the bond concentration p can affect the scope of the random concentration t that the TCP exists and can strongly depress the order phase, the effect of the reentrant phenomena is weak. The effect of both the bond dilution and the random crystal field on the tricritical points has expressed a new phenomenon. (3) The existence of the transverse field depresses the TCP and destroys the reentrant phenomena strongly. However, it is not very sensitive to depress the second-order phase transition lines under a larger random crystal field concentration condition. (4) The phase diagrams are from complex to simple due to the common role of both the transverse field and the bond dilution. We believe that a real ferromagnetic spin system may exist in these disorder factors, so our numerical results and discussions are meaningful and necessary. Finally, we hope that the results here may be helpful for further research.
Acknowledgements This research was supported in part by the Education Bureau Natural Science Foundation of Jiangsu Province (Grant No. 00SJB140003) and in part by the Thin Film Materials Key Laboratory Open Foundation of Jiangsu Province (Grant No. K2022).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
A. Aharony, Phys. Rev. B 18 (1978) 3318. C.Z. Yang, Z.Y. Li, Phys. Status Solidi B 146 (1988) 223. S.L. Yan, C.Z. Yang, J. Magn. Magn. Mater. 205 (1999) 357. G.M. Buendia, E. Machado, Phys. Rev. B 61 (2000) 14686. M. Barati, A. Ramazani, Phys. Rev. B 62 (2000) 12130. R.A. Tahir-Kheli, Phys. Rev. 169 (1968) 517. M. Blume, V.J. Emery, R.B. Griffiths, Phys. Rev. A 4 (1971) 1071. A. Maritan, M. Cieplak, M.R. Swift, F. Toigo, J.R. Banavar, Phys. Rev. Lett. 69 (1992) 221. C. Buzano, A. Maritan, A. Pelizzola, J. Phys.: Condens. Matter 6 (1994) 327. P.G. de Gennes, Solid State Commun. 1 (1963) 132. R.J. Euiott, G.A. Gehring, M.P. Matoemoff, et al., J. Phys. C 4 (1971) l179. R.B. Stinchcombe, J. Phys. C 6 (1073) 2459. E.F. Sarmento, R. Bechara Muniz, S.B. Cavalcanti, Phys. Rev. B 36 (1987) 529. E.F. Sarmento, T. Kaneyoshi, Phys. Rev. B 39 (1989) 9555. J.L. Zhong, et al., Phys. Status Solidi B 160 (1990) 329. A.F. Siqueira, I.P. Fittipaldi, Physica A 138 (1986) 592. K. Hui, A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507. G.M. Zhang, C.Z. Yang, Acta Phys. Sin. 42 (1993) 128. T. Kaneyoshi, J. Mielnicki, J. Phys.: Condens. Matter 2 (1990) 8773. T. Kaneyoshi, Phys. Status Solidi B 170 (1992) 313. N. Benayard, A. Benyoussef, N. Boccara, J. Phys. 18 (1995) 1899. N.S. Branco, B.M. Boechat, Phys. Rev. B 56 (1997) 11673. X.F. Jiang, J.L. Li, J.L. Zhong, C.Z. Yang, Phys. Rev. B 47 (1993) 827. X.F. Jiang, J. Magn. Magn. Mater 134 (1994) 167. Y.Q. Wang, Z.Y. Li, Acta Phys. Sin. 44 (1995) 811. L.L. Deng, S.L. Yan, Chin. Phys. 11 (2002) 383. I. Pupa, H.T. Diep, J. Magn. Magn. Mater. 224 (2001) 85. S.L. Yan, L.L. Deng, Physica A 308 (2002) 301.