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The transverse crystal-field effects of the mixed spin Ising bilayer system
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 269 (2004) 245–258
The transverse crystal-field effects of the mixed spin Ising bilayer system K. Htoutou, A. Ainane*, M. Saber D!epartement de Physique, Facult!e des Sciences, Universit!e Moulay Ismail, B.P. 4010, Mekn"es 4010, Morocco Received 31 May 2003
Abstract Using the effective field theory with a probability distribution technique that accounts for the self-spin-correlation functions, the critical temperature and the layer longitudinal magnetizations of a ferromagnetic or ferrimagnetic mixed Ising bilayer system with both spin-12 and spin-1 (or spin-32) in a transverse crystal field and their averages are examined. These quantities as functions of the strength of the transverse crystal field and the exchange interactions are calculated numerically for the cubic lattice and some interesting results are obtained. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Dg; 75.10.Hk; 75.40.Mg Keywords: Transverse crystal field; Mixed-spin Ising model; Bilayer system; Ferrimagnet
1. Introduction During the last few years much effort has been directed towards the study of critical phenomena in mixed-spin Ising systems consisting of spin-12 and spin-S of magnitude S ðS > 12Þ: They are of great interest, because new and possibly useful properties are expected. The systems have less translational symmetry than their single-spin counterparts since they consist of two interpenetrating inequivalent sublattices, and are well adopted to study a certain type of ferrimagnetism, namely molecular-based magnetic materials which are of current interest [1–3]. The mixed spin system has also been discussed for the regular ferrimagnetic case [4–6], and in amorphous binary solids [7] in order to explain the experimental results in the ferrimagnetic amorphous oxides in which Fe3þ ions are included [8,9]. The model for different values of S ðS > 12Þ with longitudinal crystal field has been studied extensively by a variety of techniques, namely the exact and the approximate methods [10–16] as well as the hightemperature series expansion method [17], mean field approximation (MFA) [18,19], effective field theory with correlations [20–24], real-space renormalization group approximation (RSRGA) [25] and Monte-Carlo simulation [26–28]. *Corresponding author. Fax: +212-5-536808. E-mail address: [email protected] (A. Ainane). 0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.07.002
ARTICLE IN PRESS K. Htoutou et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 245–258
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On the other hand, a tricritical behavior in the mixed Ising model which takes into account the influence of longitudinal crystal field or transverse field has been examined by many authors [29–32]. Recently, the mixed Ising model with spin-12 and spin-1 has been studied in transverse random field (TRMIM) using the pair approximation with discretized path-integral representation [4–6] and a tricritical behavior has been found. This result is in contradiction with the results reported in Refs. [29,30]. Laaboudi et al. [31] have investigated the study of the TRMIM with both spin-12 and spin-1, within the effective field theory and no tricritical behavior has been observed. The results are in agreement with those obtained in Refs. [29,30]. By using the effective-field theory, in the one- and two-spin cluster approximation, Bobak and Jascur [24] have discussed the existence of a tricritical point (TCP) for the mixed spin-12 and spin-1 Ising system with longitudinal crystal field anisotropy. They claim to have found tricritical behavior only for the square lattice case contrary to the results obtained in Refs. [15–18,32], that predict a second-order phase transition for all the values of the longitudinal crystal-field parameter, for both the honeycomb and square lattices. In this work, we investigate the effect of the transverse crystal field mixed Ising ferromagnetic or ferrimagnetic bilayer system composed of two monolayers ðL ¼ 1Þ with both spin-12 and spin-1 (or spin-32), within the framework of the effective field theory with correlations with a probability distribution technique that accounts for the single-site kinematic relations. We investigate the effect of the transverse crystal field on the magnetic properties and the phase diagrams of the system. We will see that there is no tricritical behavior. Our results are in contradiction with the results reported in Ref. [33], where a tricritical behavior has been found. This may be due to the fact that the authors neglect the quadrupolar moment for the spin-1 and 32 operator in their calculations. The outline of this paper is as follows. In Section 2, we briefly describe the model and the formulation of the system. The numerical results of the phase diagrams are presented in Section 3. Section 4 is devoted to a brief conclusion.
2. Model and formulation: mixed Ising model We consider a bilayer system composed of two different magnetic monolayers A and B with LA ¼ 1 and LB ¼ 1: Each monolayer is defined on the x–y plane and with pseudo-spin sites on a cubic lattice. The ferromagnetic mixed Ising model with transverse crystal field is described by a Hamiltonian of the form X X X X 2 H ¼ JA siz sjz JB Smz Snz JAB siz Smz Dx Smx ; ð1Þ ði;jÞ
ðm;nÞ
ði;mÞ
ðmÞ
s i of magnitude s ¼ 12 at site i; Smz and Smx denote, where siz denotes the z component of a quantum spin ~ ~m of magnitude S ¼ 1 (or 3) at site m and Dx is the respectively, the z and x components of a quantum spin S 2 transverse crystal field acting on Smx : The first three sums are carried out only over nearest-neighbor pairs of spins. The coupling strength between nearest-neighboring spins in AðBÞ is denoted by JA (JB ), while JAB stands for the exchange coupling between the nearest-neighbor spins across the interface. In order to relate our system to some experimental data for RE/TM multilayer systems, we take JA > JB > 0: That is to say, A layers consist of TM atoms and B layers are made up of RE atoms. Then, the exchange interaction JA between A atom pairs results from the direct interaction and JB between B atom pairs is considered to be due to the indirect interaction and JAB (or 3d–4f indirect interaction) is taken to be negative [33]. The method to be used is the effective field theory in which attention is focussed on a cluster comprising just a single selected spin, labeled 0; and the neighboring spins with which it directly interacts. To this end, the Hamiltonian is split into two parts, H ¼ H0 þ H 0 ; where H0 is that part of the Hamiltonian containing the spin 0; namely 2 H0 ¼ As0z þ BS0z þ CS0x
ð2Þ
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P PN 0 PN PN 0 with A ¼ JA N j¼1 sjz þ JAB n¼1 Snz JAB m¼1 Smz ; B ¼ JB i¼1 siz and C ¼ Dx ; where N and N0 are the numbers of nearest neighbors in the plane and between adjacent planes, respectively. 2.1. Mixed Ising model with spin-12 and spin-1 The longitudinal magnetizations and the longitudinal quadrupolar moment of the system are approximately given by [33,34] 1 1 tanh bA ¼ hFz ðAÞi; mAz ¼ hs0z ic ¼ ð3Þ 2 2 mBz ¼ hS0z ic *
+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Bsinh ðb=2Þ 4B2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4B2 þ D2x ½2coshððb=2Þ 4B2 þ D2x Þ þ expððb=2ÞDx Þ
¼ hGz ðB; Dx Þi;
qz ¼
D
ðS0z Þ2
*
ð4Þ
E c
+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cosh ðb=2Þ 4B2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2coshððb=2Þ 4B2 þ D2x Þ þ expððb=2ÞDx Þ ¼ hHz ðB; Dx Þi;
ð5Þ
where mAz ; mBz and qz are, respectively, the longitudinal magnetizations and quadrupolar order parameter of the layers A and B; b ¼ 1=kB T where kB is the Boltzmann constant (we take kB ¼ 1 for the sake of simplicity) and T is the absolute temperature, and hyi indicates the usual canonical ensemble thermal average for a given configuration. In the mean field approximation one would simply replace these spin operators by their thermal values. However, it is at this point that a substantial improvement to the theory is made by noting that the spin opertors have a finite set of base states, with the result that the averages over the functions Fz ; Gz ; Hz can be expressed as an average over a finite polynomial of spin operators belonging to the neighboring spins. This procedure can be effected by the combinatorial method and correctly accounts for the single site kinematic relations. Up to this point, the right-hand sides of Eqs. (3)–(5) will contain multiple spin-correlation functions. To perform thermal averaging on the right-hand sides of Eqs. (3)–(5), one now follows the general approach described in Ref. [36]. Thus, with the use of the integral representation method of the d Dirac’s distribution, Eqs. (3)–(5) can be written in the form Z Z 1 dt expðiotÞ P P exp itJA sjz hexpðitJAB Smz Þi; mAz ¼ doFz ðoÞ ð6Þ j m 2p
mBz ¼
qz ¼
Z
Z
1 doGz ðo; Dx Þ 2p
doHz ðo; Dx Þ
1 2p
Z
Z
dt expðiotÞ P PhexpðitJB Snz ÞihexpðitJAB siz Þ
ð7Þ
dt expðiotÞ P PhexpðitJB Snz ÞihexpðitJAB siz Þi;
ð8Þ
n
n
i
i
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where
1 1 Fz ðoÞ ¼ tanh bo ; 2 2
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Bsinh ðb=2Þ 4o2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gz ðo; Dx Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4o2 þ D2x ½2coshððb=2Þ 4o2 þ D2x Þ þ expððb=2ÞDx Þ
ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cosh ðb=2Þ 4o2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hz ðo; Dx Þ ¼ : 2coshððb=2Þ 4o2 þ D2x Þ þ expððb=2ÞDx Þ
ð11Þ
In the derivation of Eqs. (6)–(8), the commonly used approximation has been made according to which the multispin correlation functions are decoupled into products of the spin averages (the simplest approximation of neglecting the correlations between different sites has been made). That is D E D E 2 sj1 z Sj2 z ysj3 z D sj1 z Sj22 z y sj3 z ð12Þ for j1 aj2 aj3 : On the basis of Eqs. (6)–(8) and with the use of the probability distribution of the spin variables siz and Sz (for details, see [35,36]): 1 1 1 Pðsiz Þ ¼ ð1 2miz Þd siz þ þ ð1 þ 2miz Þd siz ð13Þ 2 2 2 and PðSiz Þ ¼ 12½ðqiz miz ÞdðSiz þ 1Þ þ 2ð1 qiz ÞdðSiz Þ þ ðqiz þ miz ÞdðSiz 1Þ
ð14Þ 2 with mz ¼ hSiz i and qz ¼ Siz ; we obtain the following set of equations for the layer longitudinal magnetizations: mAz ¼ 2NN0
N0 NX N X 0 m1 X
CmN CmN10 CmN20 m1 2m2 ð1 2mAz Þm
m¼0 m1 ¼0 m2 ¼0
ð1 þ 2mAz ÞNm ðqz mBz Þm1 ð1 qz Þm2 ðqz þ mBz ÞN0 m1 m2 JA JAB ðN 2mÞ Fz ðN0 2m1 m2 Þ ; 2 JA mBz ¼ 2NN0
N0 X N Nm X X1
ð15Þ
m 1 2m2 ð1 2m CmN0 CmN1 CmNm Az Þ 2
m¼0 m1 ¼0 m2 ¼0
ð1 þ 2mAz ÞN0 m ðqz mBz Þm1 ð1 qz Þm2 ðqz þ mBz ÞNm1 m2 JB JAB Gz JA ðN 2m1 m2 Þ þ ðN0 2mÞ ; Dx ; JA 2JA
ð16Þ
where N ¼ 4 and N0 ¼ 2 in the case of a cubic lattice which is considered here and Ckl are the binomial coefficients, Ckl ¼ l!=k!ðl kÞ!: The equation of the longitudinal quadrupolar moment is obtained by susbstituting the function Gz by Hz in the expression of the layer B longitudinal magnetization. This yields qz ¼ mBz ½Gz ðy; Dx Þ-Hz ðy; Dx Þ
ð17Þ
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Eqs. (15)–(17) are of the form mAz ¼ a10 mA;z þ a01 mB;z þ a30 m3A;z þ a21 m2A;z mBz þ a12 mA;z m2Bz þ a41 m4A;z mB;z þ a32 m3A;z m2Bz ;
ð18Þ
mBz ¼ b10 mB;z þ b01 mA;z þ b30 m3B;z þ b21 m2B;z mAz þ b12 mB;z m2Az þ b41 m4B;z mA;z þ b32 m3B;z m2Az ;
ð19Þ
qz ¼ q0 þ q1 m2B;z þ q2 mA;z mB;z þ q3 m2A;z þ q4 m4B;z þ q5 mA;z m3Bz þ q6 m2B;z m2A;z þ q7 m4B;z m2Az ;
ð20Þ
where the coefficients aij ; bij and qj are given in the appendix. By combining Eqs. (18) and (19), we have in the vicinity of the second-order transition mBz ¼ amBz þ bm3Bz þ ?
ð21Þ
qz Eq0 þ q1 m2Bz ;
ð22Þ
and where a ¼ b10 þ
a01 b01 ð1 a10 Þ
ð23Þ
b ¼ b30 þ
a01 b12 a01 b01 a01 a21 a2 a30 b21 þ þ a12 þ þ 01 : ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ2
ð24Þ
and
The second-order phase transition line is then determined by the conditions a ¼ 1 and b/0:
ð25Þ
The magnetization mBz is given by 1a : ð26Þ m2Bz ¼ b The right-hand side of Eq. (26) must be positive. If it is not the case, the transition is of first order, and hence the point at which a ¼ 1 and b ¼ 0 is the tricritical point. 2.2. Mixed Ising model with spin-12 and spin-32 The Hamiltonian of the system is given by Eq. (2). In this case the application of the effective field theory to the mixed spin-12 and spin-32 case is similar to the spin-12Dand except that we have to deal with three p spin-1, E ordering parameters in B monolayer, defined by mpz ¼ S0z ; when p ¼ 1; 2; 3 for the B monolayer. c r The starting point of the single-site cluster theory is a set of formal identities of the type p p trace0 expðbHB ÞS0z S0z ¼ ; ð27Þ trace0 ½expðbHB Þ
when 2 ð28Þ HB ¼ BS0z þ CS0x PN PN 0 with B ¼ JB n¼1 Snz JAB i¼1 siz and C ¼ Dx ; where the single angular brackets denote the thermal average for a fixed spatial configuration of the occupied sites. Evaluation of the inner traces over the
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selected spin in Eq. (27) yields the following equation: p 1 S0z ¼ p Fp S1z ; S2z ; y; SNz ; s1z ; y; sN0 z ¼ Fp ðxÞ 2 P PN 0 with x ¼ JB N n¼1 Snz JAB i¼1 siz and P4 expðbEk ÞhCk jS0z jCk i ; F1 ðxÞ ¼ k¼1 P4 k¼1 expðbEk Þ P4 2 k¼1 expðbEk Þ Ck ðS0z Þ Ck F2 ðxÞ ¼ ; P4 k¼1 expðbEk Þ P4 3 k¼1 expðbEk Þ Ck ðS0z Þ Ck F3 ðxÞ ¼ ; P4 k¼1 expðbEk Þ
ð29Þ
ð30Þ
ð31Þ
ð32Þ
where Ek and jCk i are the eigenvalues and eigenvectors of the Hamiltonian HB : We begin with a discussion of a function F ðSi Þ with Si ¼ 32; y; 32 involving the variables of a single spin. Because the spin has a finite set of bases states, one can expand the function as follows: F ðSiz Þ ¼ a0 þ a1 Siz þ a2 Siz2 þ a3 Siz3 :
ð33Þ
By considering the eigenstates of Siz in turn, the following equations are then generated F ð3=2Þ ¼ a0 þ 32a1 þ 94a2 þ 27 8 a3 ;
ð34Þ
F ð1=2Þ ¼ a0 þ 12a1 þ 14a2 þ 18a3 ;
ð35Þ
F ð1=2Þ ¼ a0 12a1 þ 14a2 18a3 ;
ð36Þ
F ð3=2Þ ¼ a0 32a1 þ 94a2 27 8 a3 :
ð37Þ
On substituting the quantities ai ; as extracted from these equations, into Eq. (32), the following result is obtained: 1 F ðSiz Þ ¼ 48 ½ð3 2Siz þ 12Siz2 þ 8Siz3 ÞF ð3=2Þ
þ 3ð9 þ 18Siz 4Siz2 8Siz3 ÞF ð1=2Þ þ 3ð9 18Siz 4Siz2 þ 8Siz3 ÞF ð1=2Þ ð3 þ 2Siz þ 12Siz2 8Siz3 ÞF ð3=2Þ : It may be noted that in the special case when F ðSiz Þ is just eaSiz ; this equation reduces to a 1 a 1 3a 3a sinh eaSiz ¼ cosh 9cosh 27sinh Siz 8 2 2 12 2 2 a a 1 3a 1 3a þ cosh cosh Siz2 þ sinh 3sinh Siz3 2 2 2 3 2 2
ð38Þ
ð39Þ
which is the Van der Waerden type identity for spin-32 used by Kaneyoshi et al. [37]. If one effects the thermal and configurational averaging on both sides of Eq. (32) the following result is obtained: 3
hhF ðSiz Þii ¼
2 X 3 Siz ¼2
PðSiz ÞF ðSiz Þ;
ð40Þ
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where the distribution function PðSiz Þ is given by 1 PðSiz Þ ¼ 48 ð3 2m þ 12q þ 8rÞd
3 Siz ;2
þ 3ð9 18m 4q þ 8rÞd
þ 3ð9 þ 18m 4q 8rÞd
1 Siz ;2
1 Siz ;2
þ ð3 þ 2m þ 12q 8rÞd
3 Siz ;2
ð41Þ
and m ¼ hhSiz iir ; q ¼ Siz2 r and r ¼ Siz3 r : It is important to stress that Eq. (37) is exact and thus it correctly accounts for the kinematic relations of the spin operators. We now generalize this result to the situation where the function F contains spin operators pertaining to several sites. In particular, in effective field theories for both spin-12 and spin-32 bilayer system one is interested in functions of the type ! N0 N X X F ðxÞ ¼ F JB Snz JAB siz : ð42Þ n¼1
i¼1
Using the integral representation method of the d Dirac’s distribution, the layer longitudinal moments of the system are given by mAz ¼ 2N 48N0
N0 NX N X 0 m1 N0 m 1 m2 X X m¼0 m1 ¼0 m2 ¼0
CmN CmN10 CmN20 m1 CmN30 m1 m2 ð1 2mAz Þm
m3 ¼0
ð1 þ 2mAz ÞNm ð3 þ 2mBz þ 12qz 8rÞm1 ð27 54mBz 12qz þ 24rÞm2 ð27 þ 54mBz 12qz 24rÞm3 ð3 2mBz þ 12qz þ 8rÞN0 m1 m2 m3 JA JAB ðN 2mÞ þ Fz ð3N0 6m1 4m2 2m3 Þ ; 2 JA mpz ¼ 48N 2N0
N0 X N Nm 1 m2 X X1 Nm X m¼0 m1 ¼0 m2 ¼0
ð43Þ
1 1 m2 CmN0 CmN1 CmNm CmNm ð1 2mAz Þm 2 3
m3 ¼0
ð1 þ 2mAz ÞN0 m ð3 þ 2mBz þ 12qz 8rÞm1 ð27 54mBz 12qz þ 24rÞm2 ð27 þ 54mBz 12qz 24rÞm3 ð3 2mBz þ 12qz þ 8rÞNm1 m2 m3 JA JB JAB Fp ðN0 2mÞ þ ð3N 6m1 4m2 2m3 Þ ; ð44Þ 2 JA JA where mBz ¼ hhSiz iir ; q ¼ Siz2 r and r ¼ Siz3 r : One has thus arrived at four coupled equations for the ordering parameters mAz ; mBz ; q and r that can be solved directly by numerical iteration without any further algebraic manipulations. This is the advantage of introducing the probability distribution technique. The same equations hold for a general lattice coordination number, N; so results for different structures may be obtained without carrying out the detailed algebra encountered when employing other techniques.
3. Results and discussion In this paper, we take JA as the unit of energy and we introduce the reduced exchange interactions R1 ¼ JB =JA and R2 ¼ JAB =JA : In generality, we assume that the critical temperature of the A monolayer is higher than that of the B monolayer, that is, JB oJA ðR1 o1Þ: We are now able to study the phase diagrams and magnetizations of the ferromagnetic and ferrimagnetic mixed spin Ising model in a transverse crystal field on a cubic lattice.
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3.1. Phase diagrams 3.1.1. Mixed spin-12 with spin-1 case In this section some results are presented by solving Eq. (25) numerically. In Fig. 1, the transition tempareture Tc =JA of the bilayer system is plotted as a function of the transverse crystal field, R1 and R2 (R2 o0). First, let us examine the variation of the transition temperature Tc =JA versus the strength of the transverse crystal field Dx =JA : The results are shown in Fig. 1a when R1 ¼ 0:5 for different values of R2 ¼ 1; 0:5 and 0:05: It is seen that Tc =JA increases when we increase Dx =JA for negative values, passes through a maximum to saturate for an infinitely large value of Dx =JA at lower temperatures. The maximum of Tc =JA is reached approximately at Dx =JA ¼ 1:6; 1:2; 0:9 for R2 ¼ 1; 0:5; 0:05 respectively. We can also remark that, in contrast with the system in a transverse field and longitudinal crystal field (Blume–Capel model) the phase diagrams in this case are not symmetric with Dx : It is found, contrary to the result obtained in Ref. [33], that the system studied here does not exhibit a tricritical behavior. We have studied the effect of the exchange interactions on the critical temperature of the bilayer system in the presence of the transverse crystal field, in particular for Dx =JA ¼ 5: Our results for the transition temperature as a function of R1 and R2 are shown in Fig. 1b and c, respectively. As usual, we see that the system exihibits different types of phases transitions. In Fig. 1b, for the case R2 ¼ 0:05; the critical temperature Tc =JA increases with increasing R1 : The curves a, b and c for R2 ¼ 1:5; 1; 0:5; respectively, are qualitatively the same as for R2 ¼ 0:05 (curves d). We have also found contrary to the results reported in Ref. [33], that the critical points do not appear for different values of R1 : The typical results for different values of R2 are presented in Fig. 1c.
3.5
3.0
1.8 a
3.0 2.5
DX /JA =5
1.6
Dx /JA =5
R1 =0.5
a
b
1.2
kBTC /JA
kBTC /JA
kBTc /JA
2.5 1.4
a 2.0 b
2.0 b 1.5
c
1.5
c
c
1.0 1.0
1.0
-8
(a)
-6
-4
-2
0
Dx/JA
2
4
6
0.5
8
0.0
(b)
d
d
0.8
0.5 0.5
1.0
R1
1.5
2.0
0.0
(c)
0.5
1.0
1.5
2.0
|R 2 |
Fig. 1. (a) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðDx =JA ; Tc =JA Þ plane for typical values of R2 when the value of R1 is fixed at R1 ¼ 0:5; R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðR1 ; TC =JA Þ plane for typical values of R2 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R2 ¼ 1:5 (curve a), R2 ¼ 1 (curve b), R2 ¼ 0:5 (curve c), R2 ¼ 0:05 (curve d). (c) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðjR2 j; TC =JA Þ plane for typical values of R1 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R1 ¼ 1:5 (curve a), R1 ¼ 1 (curve b), R1 ¼ 0:5 (curve c), R1 ¼ 0:05 (curve d).
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3.1.2. Mixed spin-12 with spin-32 case In this section we are interested in studying the phase diagrams of the system with spin-12 and spin-32: In order to determine the transition temperature Tc ; we can linearize and easily solve Eqs. (43)–(44), since mAz -0; mBz -0 and r-0 when the temperature approaches the Curie temperature Tc : This leads to a matrix equation 0 1 0 10 1 A010 A001 mAz A100 1 mAz B C B CB C B010 1 B001 A@ mBz A ¼ 0; M @ mBz A ¼ @ B100 ð45Þ r r100 r010 r001 1 r where the coefficients Aijk ; Bjjk and rijk are given in the appendix. The second-order transition temperature Tc can be determined from det M ¼ 0:
ð46Þ
As seen in Figs. 2a–c, the results are qualitatively the same as for the spin-12 and spin-1 case. In particular no tricritical phenomena have been observed. We can also see from Fig. 2a that the transition temperature does not fall to zero at a certain value of Dx for R2 ¼ 1 and R2 ¼ 0:5: Our results are in contradiction with the results reported in Ref. [33], where a tricritical behavior has been found and the transition temperature falls to zero at a certain value of the transverse crystal field. This may be due to the fact that the authors neglect the quadrupolar moment q and the higher order moment r for the spin-32 operator in their calculations. Notice that the phase diagrams obtained do not change when the sign of R2 is changed from the ferromagnetic interaction (R2 > 0) to the ferrimagnetic one (R2 o0). In other words, the phase diagrams can be studied in the same manner when studying the magnetization of the bilayer system consisting of the two monolayers coupled ferromagnetically or ferrimagnetically. 3.5
2.5
3.0
a b
3.0
R1=0.5
2.5
2.0
DX /JA=5.0
c
Dx /JA=5
a
2.5
2.0
kBTC /JA
2.0
KBTC /JA
b
1.5
kBTC /JA
a
c
1.0
1.5
b c
d
1.5
d
1.0
0.5 1.0
0.5 -8
(a)
0.5
-6
-4
-2
0
Dx/JA
2
4
6
0.0 0.0
8
(b)
0.6
1.2
R1
0.0 0.0
1.8
(c)
0.6
1.2
1.8
|R2|
Fig. 2. (a) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðDx =JA ; TC =JA Þ plane for typical values of R2 when the value of R1 is fixed at R1 ¼ 0:5: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðR1 ; TC =JA Þ plane for typical values of R2 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R2 ¼ 1:5 (curve a), R2 ¼ 1 (curve b), R2 ¼ 0:5 (curve c), R2 ¼ 0:05 (curve d). (c) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðjR2 j; TC =JA Þ plane for typical values of R1 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R1 ¼ 1:5 (curve a), R1 ¼ 1 (curve b), R1 ¼ 0:5 (curve c), R1 ¼ 0:05 (curve d).
ARTICLE IN PRESS K. Htoutou et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 245–258
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0.5
0.25
b
a
c
mAz
c
0.20
0.0 0.15
a
|Mz|
b
Dx /JA=0 R1=0.05
c
mBz
0.10
-0.5
c
b
a
Dx /JA=0 R1=0.05
0.00 0.0
-1.0 0.5
(a)
1.0
KBT/JA
b
0.05
1.5
(b)
0.5
1.0
1.5
kBT/JA
Fig. 3. (a) The thermal dependences of the monolayer magnetizations (mAz and mBzz Þ in the ferrimagnetic bilayer system with sA ¼ 12 and SB ¼ 1 when Dx =JA ¼ 0 and R1 ¼ 0:05: The value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) The jMz j versus T curve of (a).
3.2. Longitudinal magnetizations Let us study here the temperature dependence of the longitudinal magnetizations mAz ; mBz and the total longitudinal magnetization in the system defined by Mz ¼
mAz þ mBz : 2
ð47Þ
Figs. 3a and b Figs. 4a and b show the thermal variations of mAz ; mBz and jMz j of the system with both spin-12 and spin-1 with Dx =JA ¼ 0 and 0.5. It is seen that the presence of a transverse crystal field (Dx =JA ¼ 0:5Þ; of course, causes a reduction in the longitudinal magnetization of the monolayer B and average magnetization at T ¼ 0 for different values of R2 (R2 ¼ 1; 0:5; 0:05). Then the role of the transverse crystal field is to destroy one saturated magnetization. On the other hand, we remark from Fig. 3b that the compensation behavior due to the competition between the crystal field and exchange interaction constants R2 (R2 /0) can be obtained in the jMz j curves labeled b and c. These results clearly show that when jR2 j decreases, the compensation temperature becomes smaller. In Fig. 4b we show some typical results for the variation of the jMz j with the transverse crystal field (Dx =JA ¼ 0:5) for different values of R2 : It is seen that the compensation points disappear with the increase in the transverse crystal field in the jMz j curves labeled c; it is reasonable since the transverse crystal field affects the compensation curve. In Fig. 5a and b, by fixing the temperature at Tc =JA ¼ 0:5 and exchange interaction R1 ¼ 0:05; the transverse crystal field dependence of the absolute average longitudinal magnetization is clearly observed for the ferromagnetic and ferrimagnetic bilayer systems.
ARTICLE IN PRESS K. Htoutou et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 245–258 0.5
0.25
0.4
mAz
255
a
0.3
b
c
a
0.2
c
0.20
0.1 0.0 -0.1
c
b
0.15
a
-0.2
-0.4
mBz
b
|Mz|
-0.3
Dx/JA=0.5 R1=0.05
0.10
Dx/JA=0.5 R1=0.05
-0.5 -0.6 -0.7
0.05
-0.8
b
-0.9 -1.0 0.0
0.5
1.0
kBT/JA
(a)
0.00 0.0
1.5
0.5
1.0
1.5
kBT/JA
(b)
Fig. 4. (a) The thermal dependences of the monolayer magnetizations (mAz and mBz Þ in the ferrimagnetic bilayer system with sA ¼ 12 and SB ¼ 1 when Dx =JA ¼ 0:5 and R1 ¼ 0:05: The value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c) (b) The jMz j versus T curve of (a).
0.30
0.7
a
k BT/JA =0.5 R1=0.05 0.24
0.6
c
a 0.5
b
0.18
|Mz|
|Mz|
b 0.12
0.4
b a
c
0.3
0.06
0.2
0.1
0.00 -8
(a)
-4
0
Dx /JA
4
-8
8
(b)
-4
0
4
8
Dx /JA
Fig. 5. The total magnetization jMz j versus Dx =JA of the bilayer system with sA ¼ 12 and SB ¼ 1; when kb TC =JA ¼ 0:5 and R1 ¼ 0:05: (a) In the ferrimagnetic system when the value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) In the ferromagnetic system when the value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c).
ARTICLE IN PRESS K. Htoutou et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 245–258
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4. Conclusion In this work, we have investigated the phase diagrams of the mixed spin-12 and spin-1 (or spin-32) Ising ferromagnetic or ferrimagnetic bilayer system in a transverse crystal field consisting of the two different magnetic monolayers A (s ¼ 12) and B (S ¼ 1 or 32) coupled by the interlayer exchange interaction JAB by using the effective field theory with correlations with the probability distribution technique. We have shown in Figs. 1 and 2 that the transition tamperature Tc decreases when we increase the transverse crystal field Dx : No tricritical behavior has been observed. Our results are in contradiction with those obtained in Ref. [33]. which show a tricritical behavior. Further, we have shown in Figs. 3–5 that this mixed-spin ferrimagnetic system may exhibit compensation behavior, depending on the value of JAB and Dx :
Acknowledgements This work has been carried out with the support of PROTARS III: D1208.
Appendix spin-12 and spin-1 case: The coeffcients aij ; bij and qij are given by aij ¼ 2
NN0
m1 N0 m m Nm N0 NX N X 0 m1 X 1 m2 X XX X m¼0 m1 ¼0 m2 ¼0 j1 ¼0 j2 ¼0 j3 ¼0
m N m m CjNm Cj31 Cj4 0 1 2 ð1Þj1 þj3 ð1 2
bij ¼ 2NN0
0 m2 j3 j4 qz Þm2 qN di;j1 þj2 dj3 þj4 ;j Fz ðyA Þ; z
m¼0 m1 ¼0 m2 ¼0 j1 ¼0 j2 ¼0 j3 ¼0
j4 ¼0
m1 Nm m N N0 X N Nm 0 m X 1 m2 X X1 X X X
m Nm m Cj31 Cj4 1 2 ð1Þj1 þj3 ð1
2m2 þj1 þj2 CmN CmN10 CmN20 m1 Cjm1 ðA1Þ
1 C m C N0 m 2m2 þj1 þj2 CmN0 CmN1 CmNm j1 j2 2
j4 ¼0
2 j3 j4 qz Þm2 qNm dj;j1 þj2 dj3 þj4 ;i Gz ðyB ; Dx Þ; z
ðA2Þ
and qij ¼ bij ðGz -Hz Þ and where 1 ðN 2mÞ R2 ðN0 2m1 m2 Þ ; 2 R2 yB ¼ JA R1 ðN 2m1 m2 Þ þ ðN0 2mÞ : 2
yA ¼ JA
ðA3Þ
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spin-12 and spin-32 case: The coeffcients Aijk ; Bijk ; Qijk and rijk are given by Aijk ¼ 2N 48N0
m3 N0 mX m1 X m2 X m Nm N0 NX N X 0 m1 N0 m 1 m2 m3 1 m2 X X X XX m¼0 m1 ¼0 m2 ¼0
k1 ¼0 k2 ¼0 k3 ¼0
m3 ¼0
m2 m3 j6 mX 3 j5 N0 m1 X 1 j3 mX 2 j4 mX
j1 ¼0 j2 ¼0 j3 ¼0 j4 ¼0 j5 ¼0
j6 ¼0 m
m
CmN CmN10 CmN20 m1 CmN30 m1 m2 Cjm1 CjNm Cj31 Cj42 2
k4 ¼0
m N m m m m j N m m m j m j m j Cj53 Cj6 0 1 2 3 Ck11 3 Ck22 4 Ck33 5 Ck40 1 2 3 6 ð1Þj1 þj4 þj6 þk1 þk3 2j1 þj2 þj3 þj6 8k1 þk4 24k2 þk3 54j4 þj5 ð3 þ 12qz ÞN0 m2 m3 j3 j6 k1 k4
ð27 12qz Þm2 þm3 j4 j5 k2 k3 di;j1 þj2 dj3 þj4 þj5 þj6 ;j dk1 þk2 þk3 þk4 ;k Fz ðyA Þ; Bijk ¼ 2N0 48N
m3 NmX m1 X m2 X m N N0 X N Nm 0 m X 1 m2 X 1 m2 m3 X X1 Nm X X m¼0 m1 ¼0 m2 ¼0
m3 ¼0
mX 3 j5 Nm1 m 2 m3 j6 1 j3 mX 2 j4 mX X k1 ¼0 k2 ¼0 k3 ¼0
j1 ¼0 j2 ¼0 j3 ¼0 j4 ¼0 j5 ¼0
j6 ¼0
1 1 m2 CmN0 CmN1 CmNm CmNm Cjm1 CjN2 0 m Cj31 2 3
m
k4 ¼0
m Nm m m m j Nm m m j m m j m j Cj42 Cj53 Cj6 1 2 3 Ck11 3 Ck22 4 Ck33 5 Ck4 1 2 3 6 ð1Þj1 þj4 þj6 þk1 þk3 2j1 þj2 þj3 þj6 8k1 þk4 24k2 þk3 54j4 þj5 ð3 þ 12qz ÞNm2 m3 j3 j6 k1 k4
ð27 12qz Þm2 þm3 j4 j5 k2 k3 di;j1 þj2 dj3 þj4 þj5 þj6 ;j dk1 þk2 þk3 þk4 ;k F1z ðyB ; Dx Þ;
ðA5Þ
Qijk -Bijk ðF1 -F2 Þ;
ðA6Þ
rijk -Bijk ðF1 -F3 Þ
ðA7Þ
with JA ðN 2mÞ þ R2 ð3N0 6m1 4m2 2m3 Þ ; 2 JA R1 ð3N 6m1 4m2 2m3 Þ þ R2 ðN0 2mÞ : yB ¼ 2
yA ¼
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
ðA4Þ
O. Kahn, Molecular Magnetism, VCH, New York, 1993. T. Kaneyoshi, Y. Nakamura, J. Phys. Condens. Matter 10 (1998) 3003. T. Kaneyoshi, Y. Nakamura, S. Shin, J. Phys. Condens. Matter 10 (1998) 7025. X. M. Weng, Z.Y. Li, Phys. Stat. Sol. (b) 197 (1996) 487. T. Kaneyoshi, E.F. Sarmento, I.P. Fittipaldi, Jpn. J. Appl. Phys. 27 (1988) L690. T. Kaneyoshi, J. Appl. Phys. 64 (1988) 2545. T. Kaneyoshi, Solid State Commun. 69 (1989) 91. M. Sugimoto, N. Hiratsuka, Jpn. J. Appl. Phys. 21 (1982) 197. G. Srinivasan, B. Uma Maheshwar Rao, J. Zhao, M.S. Sechra, Appl. Phys. Lett. 59 (1991) 372. C. Domb, Adv. Phys. 9 (1960) 1489. L.L. Goncalves, Phys. Scr. 32 (1985) 248. T. Kaneyoshi, J. Phys. Soc. Japan 56 (1987) 2675.
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Journal of Magnetism and Magnetic Materials 269 (2004) 245–258
The transverse crystal-field effects of the mixed spin Ising bilayer system K. Htoutou, A. Ainane*, M. Saber D!epartement de Physique, Facult!e des Sciences, Universit!e Moulay Ismail, B.P. 4010, Mekn"es 4010, Morocco Received 31 May 2003
Abstract Using the effective field theory with a probability distribution technique that accounts for the self-spin-correlation functions, the critical temperature and the layer longitudinal magnetizations of a ferromagnetic or ferrimagnetic mixed Ising bilayer system with both spin-12 and spin-1 (or spin-32) in a transverse crystal field and their averages are examined. These quantities as functions of the strength of the transverse crystal field and the exchange interactions are calculated numerically for the cubic lattice and some interesting results are obtained. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10.Dg; 75.10.Hk; 75.40.Mg Keywords: Transverse crystal field; Mixed-spin Ising model; Bilayer system; Ferrimagnet
1. Introduction During the last few years much effort has been directed towards the study of critical phenomena in mixed-spin Ising systems consisting of spin-12 and spin-S of magnitude S ðS > 12Þ: They are of great interest, because new and possibly useful properties are expected. The systems have less translational symmetry than their single-spin counterparts since they consist of two interpenetrating inequivalent sublattices, and are well adopted to study a certain type of ferrimagnetism, namely molecular-based magnetic materials which are of current interest [1–3]. The mixed spin system has also been discussed for the regular ferrimagnetic case [4–6], and in amorphous binary solids [7] in order to explain the experimental results in the ferrimagnetic amorphous oxides in which Fe3þ ions are included [8,9]. The model for different values of S ðS > 12Þ with longitudinal crystal field has been studied extensively by a variety of techniques, namely the exact and the approximate methods [10–16] as well as the hightemperature series expansion method [17], mean field approximation (MFA) [18,19], effective field theory with correlations [20–24], real-space renormalization group approximation (RSRGA) [25] and Monte-Carlo simulation [26–28]. *Corresponding author. Fax: +212-5-536808. E-mail address: [email protected] (A. Ainane). 0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.07.002
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On the other hand, a tricritical behavior in the mixed Ising model which takes into account the influence of longitudinal crystal field or transverse field has been examined by many authors [29–32]. Recently, the mixed Ising model with spin-12 and spin-1 has been studied in transverse random field (TRMIM) using the pair approximation with discretized path-integral representation [4–6] and a tricritical behavior has been found. This result is in contradiction with the results reported in Refs. [29,30]. Laaboudi et al. [31] have investigated the study of the TRMIM with both spin-12 and spin-1, within the effective field theory and no tricritical behavior has been observed. The results are in agreement with those obtained in Refs. [29,30]. By using the effective-field theory, in the one- and two-spin cluster approximation, Bobak and Jascur [24] have discussed the existence of a tricritical point (TCP) for the mixed spin-12 and spin-1 Ising system with longitudinal crystal field anisotropy. They claim to have found tricritical behavior only for the square lattice case contrary to the results obtained in Refs. [15–18,32], that predict a second-order phase transition for all the values of the longitudinal crystal-field parameter, for both the honeycomb and square lattices. In this work, we investigate the effect of the transverse crystal field mixed Ising ferromagnetic or ferrimagnetic bilayer system composed of two monolayers ðL ¼ 1Þ with both spin-12 and spin-1 (or spin-32), within the framework of the effective field theory with correlations with a probability distribution technique that accounts for the single-site kinematic relations. We investigate the effect of the transverse crystal field on the magnetic properties and the phase diagrams of the system. We will see that there is no tricritical behavior. Our results are in contradiction with the results reported in Ref. [33], where a tricritical behavior has been found. This may be due to the fact that the authors neglect the quadrupolar moment for the spin-1 and 32 operator in their calculations. The outline of this paper is as follows. In Section 2, we briefly describe the model and the formulation of the system. The numerical results of the phase diagrams are presented in Section 3. Section 4 is devoted to a brief conclusion.
2. Model and formulation: mixed Ising model We consider a bilayer system composed of two different magnetic monolayers A and B with LA ¼ 1 and LB ¼ 1: Each monolayer is defined on the x–y plane and with pseudo-spin sites on a cubic lattice. The ferromagnetic mixed Ising model with transverse crystal field is described by a Hamiltonian of the form X X X X 2 H ¼ JA siz sjz JB Smz Snz JAB siz Smz Dx Smx ; ð1Þ ði;jÞ
ðm;nÞ
ði;mÞ
ðmÞ
s i of magnitude s ¼ 12 at site i; Smz and Smx denote, where siz denotes the z component of a quantum spin ~ ~m of magnitude S ¼ 1 (or 3) at site m and Dx is the respectively, the z and x components of a quantum spin S 2 transverse crystal field acting on Smx : The first three sums are carried out only over nearest-neighbor pairs of spins. The coupling strength between nearest-neighboring spins in AðBÞ is denoted by JA (JB ), while JAB stands for the exchange coupling between the nearest-neighbor spins across the interface. In order to relate our system to some experimental data for RE/TM multilayer systems, we take JA > JB > 0: That is to say, A layers consist of TM atoms and B layers are made up of RE atoms. Then, the exchange interaction JA between A atom pairs results from the direct interaction and JB between B atom pairs is considered to be due to the indirect interaction and JAB (or 3d–4f indirect interaction) is taken to be negative [33]. The method to be used is the effective field theory in which attention is focussed on a cluster comprising just a single selected spin, labeled 0; and the neighboring spins with which it directly interacts. To this end, the Hamiltonian is split into two parts, H ¼ H0 þ H 0 ; where H0 is that part of the Hamiltonian containing the spin 0; namely 2 H0 ¼ As0z þ BS0z þ CS0x
ð2Þ
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P PN 0 PN PN 0 with A ¼ JA N j¼1 sjz þ JAB n¼1 Snz JAB m¼1 Smz ; B ¼ JB i¼1 siz and C ¼ Dx ; where N and N0 are the numbers of nearest neighbors in the plane and between adjacent planes, respectively. 2.1. Mixed Ising model with spin-12 and spin-1 The longitudinal magnetizations and the longitudinal quadrupolar moment of the system are approximately given by [33,34] 1 1 tanh bA ¼ hFz ðAÞi; mAz ¼ hs0z ic ¼ ð3Þ 2 2 mBz ¼ hS0z ic *
+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Bsinh ðb=2Þ 4B2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4B2 þ D2x ½2coshððb=2Þ 4B2 þ D2x Þ þ expððb=2ÞDx Þ
¼ hGz ðB; Dx Þi;
qz ¼
D
ðS0z Þ2
*
ð4Þ
E c
+ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cosh ðb=2Þ 4B2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2coshððb=2Þ 4B2 þ D2x Þ þ expððb=2ÞDx Þ ¼ hHz ðB; Dx Þi;
ð5Þ
where mAz ; mBz and qz are, respectively, the longitudinal magnetizations and quadrupolar order parameter of the layers A and B; b ¼ 1=kB T where kB is the Boltzmann constant (we take kB ¼ 1 for the sake of simplicity) and T is the absolute temperature, and hyi indicates the usual canonical ensemble thermal average for a given configuration. In the mean field approximation one would simply replace these spin operators by their thermal values. However, it is at this point that a substantial improvement to the theory is made by noting that the spin opertors have a finite set of base states, with the result that the averages over the functions Fz ; Gz ; Hz can be expressed as an average over a finite polynomial of spin operators belonging to the neighboring spins. This procedure can be effected by the combinatorial method and correctly accounts for the single site kinematic relations. Up to this point, the right-hand sides of Eqs. (3)–(5) will contain multiple spin-correlation functions. To perform thermal averaging on the right-hand sides of Eqs. (3)–(5), one now follows the general approach described in Ref. [36]. Thus, with the use of the integral representation method of the d Dirac’s distribution, Eqs. (3)–(5) can be written in the form Z Z 1 dt expðiotÞ P P exp itJA sjz hexpðitJAB Smz Þi; mAz ¼ doFz ðoÞ ð6Þ j m 2p
mBz ¼
qz ¼
Z
Z
1 doGz ðo; Dx Þ 2p
doHz ðo; Dx Þ
1 2p
Z
Z
dt expðiotÞ P PhexpðitJB Snz ÞihexpðitJAB siz Þ
ð7Þ
dt expðiotÞ P PhexpðitJB Snz ÞihexpðitJAB siz Þi;
ð8Þ
n
n
i
i
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where
1 1 Fz ðoÞ ¼ tanh bo ; 2 2
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Bsinh ðb=2Þ 4o2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gz ðo; Dx Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4o2 þ D2x ½2coshððb=2Þ 4o2 þ D2x Þ þ expððb=2ÞDx Þ
ð10Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2cosh ðb=2Þ 4o2 þ D2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hz ðo; Dx Þ ¼ : 2coshððb=2Þ 4o2 þ D2x Þ þ expððb=2ÞDx Þ
ð11Þ
In the derivation of Eqs. (6)–(8), the commonly used approximation has been made according to which the multispin correlation functions are decoupled into products of the spin averages (the simplest approximation of neglecting the correlations between different sites has been made). That is D E D E 2 sj1 z Sj2 z ysj3 z D sj1 z Sj22 z y sj3 z ð12Þ for j1 aj2 aj3 : On the basis of Eqs. (6)–(8) and with the use of the probability distribution of the spin variables siz and Sz (for details, see [35,36]): 1 1 1 Pðsiz Þ ¼ ð1 2miz Þd siz þ þ ð1 þ 2miz Þd siz ð13Þ 2 2 2 and PðSiz Þ ¼ 12½ðqiz miz ÞdðSiz þ 1Þ þ 2ð1 qiz ÞdðSiz Þ þ ðqiz þ miz ÞdðSiz 1Þ
ð14Þ 2 with mz ¼ hSiz i and qz ¼ Siz ; we obtain the following set of equations for the layer longitudinal magnetizations: mAz ¼ 2NN0
N0 NX N X 0 m1 X
CmN CmN10 CmN20 m1 2m2 ð1 2mAz Þm
m¼0 m1 ¼0 m2 ¼0
ð1 þ 2mAz ÞNm ðqz mBz Þm1 ð1 qz Þm2 ðqz þ mBz ÞN0 m1 m2 JA JAB ðN 2mÞ Fz ðN0 2m1 m2 Þ ; 2 JA mBz ¼ 2NN0
N0 X N Nm X X1
ð15Þ
m 1 2m2 ð1 2m CmN0 CmN1 CmNm Az Þ 2
m¼0 m1 ¼0 m2 ¼0
ð1 þ 2mAz ÞN0 m ðqz mBz Þm1 ð1 qz Þm2 ðqz þ mBz ÞNm1 m2 JB JAB Gz JA ðN 2m1 m2 Þ þ ðN0 2mÞ ; Dx ; JA 2JA
ð16Þ
where N ¼ 4 and N0 ¼ 2 in the case of a cubic lattice which is considered here and Ckl are the binomial coefficients, Ckl ¼ l!=k!ðl kÞ!: The equation of the longitudinal quadrupolar moment is obtained by susbstituting the function Gz by Hz in the expression of the layer B longitudinal magnetization. This yields qz ¼ mBz ½Gz ðy; Dx Þ-Hz ðy; Dx Þ
ð17Þ
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Eqs. (15)–(17) are of the form mAz ¼ a10 mA;z þ a01 mB;z þ a30 m3A;z þ a21 m2A;z mBz þ a12 mA;z m2Bz þ a41 m4A;z mB;z þ a32 m3A;z m2Bz ;
ð18Þ
mBz ¼ b10 mB;z þ b01 mA;z þ b30 m3B;z þ b21 m2B;z mAz þ b12 mB;z m2Az þ b41 m4B;z mA;z þ b32 m3B;z m2Az ;
ð19Þ
qz ¼ q0 þ q1 m2B;z þ q2 mA;z mB;z þ q3 m2A;z þ q4 m4B;z þ q5 mA;z m3Bz þ q6 m2B;z m2A;z þ q7 m4B;z m2Az ;
ð20Þ
where the coefficients aij ; bij and qj are given in the appendix. By combining Eqs. (18) and (19), we have in the vicinity of the second-order transition mBz ¼ amBz þ bm3Bz þ ?
ð21Þ
qz Eq0 þ q1 m2Bz ;
ð22Þ
and where a ¼ b10 þ
a01 b01 ð1 a10 Þ
ð23Þ
b ¼ b30 þ
a01 b12 a01 b01 a01 a21 a2 a30 b21 þ þ a12 þ þ 01 : ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ ð1 a10 Þ2
ð24Þ
and
The second-order phase transition line is then determined by the conditions a ¼ 1 and b/0:
ð25Þ
The magnetization mBz is given by 1a : ð26Þ m2Bz ¼ b The right-hand side of Eq. (26) must be positive. If it is not the case, the transition is of first order, and hence the point at which a ¼ 1 and b ¼ 0 is the tricritical point. 2.2. Mixed Ising model with spin-12 and spin-32 The Hamiltonian of the system is given by Eq. (2). In this case the application of the effective field theory to the mixed spin-12 and spin-32 case is similar to the spin-12Dand except that we have to deal with three p spin-1, E ordering parameters in B monolayer, defined by mpz ¼ S0z ; when p ¼ 1; 2; 3 for the B monolayer. c r The starting point of the single-site cluster theory is a set of formal identities of the type p p trace0 expðbHB ÞS0z S0z ¼ ; ð27Þ trace0 ½expðbHB Þ
when 2 ð28Þ HB ¼ BS0z þ CS0x PN PN 0 with B ¼ JB n¼1 Snz JAB i¼1 siz and C ¼ Dx ; where the single angular brackets denote the thermal average for a fixed spatial configuration of the occupied sites. Evaluation of the inner traces over the
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selected spin in Eq. (27) yields the following equation: p 1 S0z ¼ p Fp S1z ; S2z ; y; SNz ; s1z ; y; sN0 z ¼ Fp ðxÞ 2 P PN 0 with x ¼ JB N n¼1 Snz JAB i¼1 siz and P4 expðbEk ÞhCk jS0z jCk i ; F1 ðxÞ ¼ k¼1 P4 k¼1 expðbEk Þ P4 2 k¼1 expðbEk Þ Ck ðS0z Þ Ck F2 ðxÞ ¼ ; P4 k¼1 expðbEk Þ P4 3 k¼1 expðbEk Þ Ck ðS0z Þ Ck F3 ðxÞ ¼ ; P4 k¼1 expðbEk Þ
ð29Þ
ð30Þ
ð31Þ
ð32Þ
where Ek and jCk i are the eigenvalues and eigenvectors of the Hamiltonian HB : We begin with a discussion of a function F ðSi Þ with Si ¼ 32; y; 32 involving the variables of a single spin. Because the spin has a finite set of bases states, one can expand the function as follows: F ðSiz Þ ¼ a0 þ a1 Siz þ a2 Siz2 þ a3 Siz3 :
ð33Þ
By considering the eigenstates of Siz in turn, the following equations are then generated F ð3=2Þ ¼ a0 þ 32a1 þ 94a2 þ 27 8 a3 ;
ð34Þ
F ð1=2Þ ¼ a0 þ 12a1 þ 14a2 þ 18a3 ;
ð35Þ
F ð1=2Þ ¼ a0 12a1 þ 14a2 18a3 ;
ð36Þ
F ð3=2Þ ¼ a0 32a1 þ 94a2 27 8 a3 :
ð37Þ
On substituting the quantities ai ; as extracted from these equations, into Eq. (32), the following result is obtained: 1 F ðSiz Þ ¼ 48 ½ð3 2Siz þ 12Siz2 þ 8Siz3 ÞF ð3=2Þ
þ 3ð9 þ 18Siz 4Siz2 8Siz3 ÞF ð1=2Þ þ 3ð9 18Siz 4Siz2 þ 8Siz3 ÞF ð1=2Þ ð3 þ 2Siz þ 12Siz2 8Siz3 ÞF ð3=2Þ : It may be noted that in the special case when F ðSiz Þ is just eaSiz ; this equation reduces to a 1 a 1 3a 3a sinh eaSiz ¼ cosh 9cosh 27sinh Siz 8 2 2 12 2 2 a a 1 3a 1 3a þ cosh cosh Siz2 þ sinh 3sinh Siz3 2 2 2 3 2 2
ð38Þ
ð39Þ
which is the Van der Waerden type identity for spin-32 used by Kaneyoshi et al. [37]. If one effects the thermal and configurational averaging on both sides of Eq. (32) the following result is obtained: 3
hhF ðSiz Þii ¼
2 X 3 Siz ¼2
PðSiz ÞF ðSiz Þ;
ð40Þ
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where the distribution function PðSiz Þ is given by 1 PðSiz Þ ¼ 48 ð3 2m þ 12q þ 8rÞd
3 Siz ;2
þ 3ð9 18m 4q þ 8rÞd
þ 3ð9 þ 18m 4q 8rÞd
1 Siz ;2
1 Siz ;2
þ ð3 þ 2m þ 12q 8rÞd
3 Siz ;2
ð41Þ
and m ¼ hhSiz iir ; q ¼ Siz2 r and r ¼ Siz3 r : It is important to stress that Eq. (37) is exact and thus it correctly accounts for the kinematic relations of the spin operators. We now generalize this result to the situation where the function F contains spin operators pertaining to several sites. In particular, in effective field theories for both spin-12 and spin-32 bilayer system one is interested in functions of the type ! N0 N X X F ðxÞ ¼ F JB Snz JAB siz : ð42Þ n¼1
i¼1
Using the integral representation method of the d Dirac’s distribution, the layer longitudinal moments of the system are given by mAz ¼ 2N 48N0
N0 NX N X 0 m1 N0 m 1 m2 X X m¼0 m1 ¼0 m2 ¼0
CmN CmN10 CmN20 m1 CmN30 m1 m2 ð1 2mAz Þm
m3 ¼0
ð1 þ 2mAz ÞNm ð3 þ 2mBz þ 12qz 8rÞm1 ð27 54mBz 12qz þ 24rÞm2 ð27 þ 54mBz 12qz 24rÞm3 ð3 2mBz þ 12qz þ 8rÞN0 m1 m2 m3 JA JAB ðN 2mÞ þ Fz ð3N0 6m1 4m2 2m3 Þ ; 2 JA mpz ¼ 48N 2N0
N0 X N Nm 1 m2 X X1 Nm X m¼0 m1 ¼0 m2 ¼0
ð43Þ
1 1 m2 CmN0 CmN1 CmNm CmNm ð1 2mAz Þm 2 3
m3 ¼0
ð1 þ 2mAz ÞN0 m ð3 þ 2mBz þ 12qz 8rÞm1 ð27 54mBz 12qz þ 24rÞm2 ð27 þ 54mBz 12qz 24rÞm3 ð3 2mBz þ 12qz þ 8rÞNm1 m2 m3 JA JB JAB Fp ðN0 2mÞ þ ð3N 6m1 4m2 2m3 Þ ; ð44Þ 2 JA JA where mBz ¼ hhSiz iir ; q ¼ Siz2 r and r ¼ Siz3 r : One has thus arrived at four coupled equations for the ordering parameters mAz ; mBz ; q and r that can be solved directly by numerical iteration without any further algebraic manipulations. This is the advantage of introducing the probability distribution technique. The same equations hold for a general lattice coordination number, N; so results for different structures may be obtained without carrying out the detailed algebra encountered when employing other techniques.
3. Results and discussion In this paper, we take JA as the unit of energy and we introduce the reduced exchange interactions R1 ¼ JB =JA and R2 ¼ JAB =JA : In generality, we assume that the critical temperature of the A monolayer is higher than that of the B monolayer, that is, JB oJA ðR1 o1Þ: We are now able to study the phase diagrams and magnetizations of the ferromagnetic and ferrimagnetic mixed spin Ising model in a transverse crystal field on a cubic lattice.
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3.1. Phase diagrams 3.1.1. Mixed spin-12 with spin-1 case In this section some results are presented by solving Eq. (25) numerically. In Fig. 1, the transition tempareture Tc =JA of the bilayer system is plotted as a function of the transverse crystal field, R1 and R2 (R2 o0). First, let us examine the variation of the transition temperature Tc =JA versus the strength of the transverse crystal field Dx =JA : The results are shown in Fig. 1a when R1 ¼ 0:5 for different values of R2 ¼ 1; 0:5 and 0:05: It is seen that Tc =JA increases when we increase Dx =JA for negative values, passes through a maximum to saturate for an infinitely large value of Dx =JA at lower temperatures. The maximum of Tc =JA is reached approximately at Dx =JA ¼ 1:6; 1:2; 0:9 for R2 ¼ 1; 0:5; 0:05 respectively. We can also remark that, in contrast with the system in a transverse field and longitudinal crystal field (Blume–Capel model) the phase diagrams in this case are not symmetric with Dx : It is found, contrary to the result obtained in Ref. [33], that the system studied here does not exhibit a tricritical behavior. We have studied the effect of the exchange interactions on the critical temperature of the bilayer system in the presence of the transverse crystal field, in particular for Dx =JA ¼ 5: Our results for the transition temperature as a function of R1 and R2 are shown in Fig. 1b and c, respectively. As usual, we see that the system exihibits different types of phases transitions. In Fig. 1b, for the case R2 ¼ 0:05; the critical temperature Tc =JA increases with increasing R1 : The curves a, b and c for R2 ¼ 1:5; 1; 0:5; respectively, are qualitatively the same as for R2 ¼ 0:05 (curves d). We have also found contrary to the results reported in Ref. [33], that the critical points do not appear for different values of R1 : The typical results for different values of R2 are presented in Fig. 1c.
3.5
3.0
1.8 a
3.0 2.5
DX /JA =5
1.6
Dx /JA =5
R1 =0.5
a
b
1.2
kBTC /JA
kBTC /JA
kBTc /JA
2.5 1.4
a 2.0 b
2.0 b 1.5
c
1.5
c
c
1.0 1.0
1.0
-8
(a)
-6
-4
-2
0
Dx/JA
2
4
6
0.5
8
0.0
(b)
d
d
0.8
0.5 0.5
1.0
R1
1.5
2.0
0.0
(c)
0.5
1.0
1.5
2.0
|R 2 |
Fig. 1. (a) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðDx =JA ; Tc =JA Þ plane for typical values of R2 when the value of R1 is fixed at R1 ¼ 0:5; R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðR1 ; TC =JA Þ plane for typical values of R2 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R2 ¼ 1:5 (curve a), R2 ¼ 1 (curve b), R2 ¼ 0:5 (curve c), R2 ¼ 0:05 (curve d). (c) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 1 in the ðjR2 j; TC =JA Þ plane for typical values of R1 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R1 ¼ 1:5 (curve a), R1 ¼ 1 (curve b), R1 ¼ 0:5 (curve c), R1 ¼ 0:05 (curve d).
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3.1.2. Mixed spin-12 with spin-32 case In this section we are interested in studying the phase diagrams of the system with spin-12 and spin-32: In order to determine the transition temperature Tc ; we can linearize and easily solve Eqs. (43)–(44), since mAz -0; mBz -0 and r-0 when the temperature approaches the Curie temperature Tc : This leads to a matrix equation 0 1 0 10 1 A010 A001 mAz A100 1 mAz B C B CB C B010 1 B001 A@ mBz A ¼ 0; M @ mBz A ¼ @ B100 ð45Þ r r100 r010 r001 1 r where the coefficients Aijk ; Bjjk and rijk are given in the appendix. The second-order transition temperature Tc can be determined from det M ¼ 0:
ð46Þ
As seen in Figs. 2a–c, the results are qualitatively the same as for the spin-12 and spin-1 case. In particular no tricritical phenomena have been observed. We can also see from Fig. 2a that the transition temperature does not fall to zero at a certain value of Dx for R2 ¼ 1 and R2 ¼ 0:5: Our results are in contradiction with the results reported in Ref. [33], where a tricritical behavior has been found and the transition temperature falls to zero at a certain value of the transverse crystal field. This may be due to the fact that the authors neglect the quadrupolar moment q and the higher order moment r for the spin-32 operator in their calculations. Notice that the phase diagrams obtained do not change when the sign of R2 is changed from the ferromagnetic interaction (R2 > 0) to the ferrimagnetic one (R2 o0). In other words, the phase diagrams can be studied in the same manner when studying the magnetization of the bilayer system consisting of the two monolayers coupled ferromagnetically or ferrimagnetically. 3.5
2.5
3.0
a b
3.0
R1=0.5
2.5
2.0
DX /JA=5.0
c
Dx /JA=5
a
2.5
2.0
kBTC /JA
2.0
KBTC /JA
b
1.5
kBTC /JA
a
c
1.0
1.5
b c
d
1.5
d
1.0
0.5 1.0
0.5 -8
(a)
0.5
-6
-4
-2
0
Dx/JA
2
4
6
0.0 0.0
8
(b)
0.6
1.2
R1
0.0 0.0
1.8
(c)
0.6
1.2
1.8
|R2|
Fig. 2. (a) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðDx =JA ; TC =JA Þ plane for typical values of R2 when the value of R1 is fixed at R1 ¼ 0:5: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðR1 ; TC =JA Þ plane for typical values of R2 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R2 ¼ 1:5 (curve a), R2 ¼ 1 (curve b), R2 ¼ 0:5 (curve c), R2 ¼ 0:05 (curve d). (c) Phase diagrams of the bilayer system with sA ¼ 12 and SB ¼ 32 in the ðjR2 j; TC =JA Þ plane for typical values of R1 when the value of Dx =JA is fixed at Dx =JA ¼ 5: R1 ¼ 1:5 (curve a), R1 ¼ 1 (curve b), R1 ¼ 0:5 (curve c), R1 ¼ 0:05 (curve d).
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0.5
0.25
b
a
c
mAz
c
0.20
0.0 0.15
a
|Mz|
b
Dx /JA=0 R1=0.05
c
mBz
0.10
-0.5
c
b
a
Dx /JA=0 R1=0.05
0.00 0.0
-1.0 0.5
(a)
1.0
KBT/JA
b
0.05
1.5
(b)
0.5
1.0
1.5
kBT/JA
Fig. 3. (a) The thermal dependences of the monolayer magnetizations (mAz and mBzz Þ in the ferrimagnetic bilayer system with sA ¼ 12 and SB ¼ 1 when Dx =JA ¼ 0 and R1 ¼ 0:05: The value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) The jMz j versus T curve of (a).
3.2. Longitudinal magnetizations Let us study here the temperature dependence of the longitudinal magnetizations mAz ; mBz and the total longitudinal magnetization in the system defined by Mz ¼
mAz þ mBz : 2
ð47Þ
Figs. 3a and b Figs. 4a and b show the thermal variations of mAz ; mBz and jMz j of the system with both spin-12 and spin-1 with Dx =JA ¼ 0 and 0.5. It is seen that the presence of a transverse crystal field (Dx =JA ¼ 0:5Þ; of course, causes a reduction in the longitudinal magnetization of the monolayer B and average magnetization at T ¼ 0 for different values of R2 (R2 ¼ 1; 0:5; 0:05). Then the role of the transverse crystal field is to destroy one saturated magnetization. On the other hand, we remark from Fig. 3b that the compensation behavior due to the competition between the crystal field and exchange interaction constants R2 (R2 /0) can be obtained in the jMz j curves labeled b and c. These results clearly show that when jR2 j decreases, the compensation temperature becomes smaller. In Fig. 4b we show some typical results for the variation of the jMz j with the transverse crystal field (Dx =JA ¼ 0:5) for different values of R2 : It is seen that the compensation points disappear with the increase in the transverse crystal field in the jMz j curves labeled c; it is reasonable since the transverse crystal field affects the compensation curve. In Fig. 5a and b, by fixing the temperature at Tc =JA ¼ 0:5 and exchange interaction R1 ¼ 0:05; the transverse crystal field dependence of the absolute average longitudinal magnetization is clearly observed for the ferromagnetic and ferrimagnetic bilayer systems.
ARTICLE IN PRESS K. Htoutou et al. / Journal of Magnetism and Magnetic Materials 269 (2004) 245–258 0.5
0.25
0.4
mAz
255
a
0.3
b
c
a
0.2
c
0.20
0.1 0.0 -0.1
c
b
0.15
a
-0.2
-0.4
mBz
b
|Mz|
-0.3
Dx/JA=0.5 R1=0.05
0.10
Dx/JA=0.5 R1=0.05
-0.5 -0.6 -0.7
0.05
-0.8
b
-0.9 -1.0 0.0
0.5
1.0
kBT/JA
(a)
0.00 0.0
1.5
0.5
1.0
1.5
kBT/JA
(b)
Fig. 4. (a) The thermal dependences of the monolayer magnetizations (mAz and mBz Þ in the ferrimagnetic bilayer system with sA ¼ 12 and SB ¼ 1 when Dx =JA ¼ 0:5 and R1 ¼ 0:05: The value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c) (b) The jMz j versus T curve of (a).
0.30
0.7
a
k BT/JA =0.5 R1=0.05 0.24
0.6
c
a 0.5
b
0.18
|Mz|
|Mz|
b 0.12
0.4
b a
c
0.3
0.06
0.2
0.1
0.00 -8
(a)
-4
0
Dx /JA
4
-8
8
(b)
-4
0
4
8
Dx /JA
Fig. 5. The total magnetization jMz j versus Dx =JA of the bilayer system with sA ¼ 12 and SB ¼ 1; when kb TC =JA ¼ 0:5 and R1 ¼ 0:05: (a) In the ferrimagnetic system when the value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c). (b) In the ferromagnetic system when the value of R2 is changed: R2 ¼ 1 (curve a), R2 ¼ 0:5 (curve b), R2 ¼ 0:05 (curve c).
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4. Conclusion In this work, we have investigated the phase diagrams of the mixed spin-12 and spin-1 (or spin-32) Ising ferromagnetic or ferrimagnetic bilayer system in a transverse crystal field consisting of the two different magnetic monolayers A (s ¼ 12) and B (S ¼ 1 or 32) coupled by the interlayer exchange interaction JAB by using the effective field theory with correlations with the probability distribution technique. We have shown in Figs. 1 and 2 that the transition tamperature Tc decreases when we increase the transverse crystal field Dx : No tricritical behavior has been observed. Our results are in contradiction with those obtained in Ref. [33]. which show a tricritical behavior. Further, we have shown in Figs. 3–5 that this mixed-spin ferrimagnetic system may exhibit compensation behavior, depending on the value of JAB and Dx :
Acknowledgements This work has been carried out with the support of PROTARS III: D1208.
Appendix spin-12 and spin-1 case: The coeffcients aij ; bij and qij are given by aij ¼ 2
NN0
m1 N0 m m Nm N0 NX N X 0 m1 X 1 m2 X XX X m¼0 m1 ¼0 m2 ¼0 j1 ¼0 j2 ¼0 j3 ¼0
m N m m CjNm Cj31 Cj4 0 1 2 ð1Þj1 þj3 ð1 2
bij ¼ 2NN0
0 m2 j3 j4 qz Þm2 qN di;j1 þj2 dj3 þj4 ;j Fz ðyA Þ; z
m¼0 m1 ¼0 m2 ¼0 j1 ¼0 j2 ¼0 j3 ¼0
j4 ¼0
m1 Nm m N N0 X N Nm 0 m X 1 m2 X X1 X X X
m Nm m Cj31 Cj4 1 2 ð1Þj1 þj3 ð1
2m2 þj1 þj2 CmN CmN10 CmN20 m1 Cjm1 ðA1Þ
1 C m C N0 m 2m2 þj1 þj2 CmN0 CmN1 CmNm j1 j2 2
j4 ¼0
2 j3 j4 qz Þm2 qNm dj;j1 þj2 dj3 þj4 ;i Gz ðyB ; Dx Þ; z
ðA2Þ
and qij ¼ bij ðGz -Hz Þ and where 1 ðN 2mÞ R2 ðN0 2m1 m2 Þ ; 2 R2 yB ¼ JA R1 ðN 2m1 m2 Þ þ ðN0 2mÞ : 2
yA ¼ JA
ðA3Þ
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spin-12 and spin-32 case: The coeffcients Aijk ; Bijk ; Qijk and rijk are given by Aijk ¼ 2N 48N0
m3 N0 mX m1 X m2 X m Nm N0 NX N X 0 m1 N0 m 1 m2 m3 1 m2 X X X XX m¼0 m1 ¼0 m2 ¼0
k1 ¼0 k2 ¼0 k3 ¼0
m3 ¼0
m2 m3 j6 mX 3 j5 N0 m1 X 1 j3 mX 2 j4 mX
j1 ¼0 j2 ¼0 j3 ¼0 j4 ¼0 j5 ¼0
j6 ¼0 m
m
CmN CmN10 CmN20 m1 CmN30 m1 m2 Cjm1 CjNm Cj31 Cj42 2
k4 ¼0
m N m m m m j N m m m j m j m j Cj53 Cj6 0 1 2 3 Ck11 3 Ck22 4 Ck33 5 Ck40 1 2 3 6 ð1Þj1 þj4 þj6 þk1 þk3 2j1 þj2 þj3 þj6 8k1 þk4 24k2 þk3 54j4 þj5 ð3 þ 12qz ÞN0 m2 m3 j3 j6 k1 k4
ð27 12qz Þm2 þm3 j4 j5 k2 k3 di;j1 þj2 dj3 þj4 þj5 þj6 ;j dk1 þk2 þk3 þk4 ;k Fz ðyA Þ; Bijk ¼ 2N0 48N
m3 NmX m1 X m2 X m N N0 X N Nm 0 m X 1 m2 X 1 m2 m3 X X1 Nm X X m¼0 m1 ¼0 m2 ¼0
m3 ¼0
mX 3 j5 Nm1 m 2 m3 j6 1 j3 mX 2 j4 mX X k1 ¼0 k2 ¼0 k3 ¼0
j1 ¼0 j2 ¼0 j3 ¼0 j4 ¼0 j5 ¼0
j6 ¼0
1 1 m2 CmN0 CmN1 CmNm CmNm Cjm1 CjN2 0 m Cj31 2 3
m
k4 ¼0
m Nm m m m j Nm m m j m m j m j Cj42 Cj53 Cj6 1 2 3 Ck11 3 Ck22 4 Ck33 5 Ck4 1 2 3 6 ð1Þj1 þj4 þj6 þk1 þk3 2j1 þj2 þj3 þj6 8k1 þk4 24k2 þk3 54j4 þj5 ð3 þ 12qz ÞNm2 m3 j3 j6 k1 k4
ð27 12qz Þm2 þm3 j4 j5 k2 k3 di;j1 þj2 dj3 þj4 þj5 þj6 ;j dk1 þk2 þk3 þk4 ;k F1z ðyB ; Dx Þ;
ðA5Þ
Qijk -Bijk ðF1 -F2 Þ;
ðA6Þ
rijk -Bijk ðF1 -F3 Þ
ðA7Þ
with JA ðN 2mÞ þ R2 ð3N0 6m1 4m2 2m3 Þ ; 2 JA R1 ð3N 6m1 4m2 2m3 Þ þ R2 ðN0 2mÞ : yB ¼ 2
yA ¼
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
ðA4Þ
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