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A. BENYOUSSEF and H. Ez-ZAHRAOUY: Magnetic Properties of a Spin-1 king Model 503

phys. stat. sol. (b) 180, 503 (1993)

Subject classification: 75.10

Laborutoire de Magnitisme et Physique des Hautes Energies, Depurtement de Physique, Faculti des Sciences, Rabat ')

Magnetic Properties of a Transverse Spin-1 Ising Model with Longitudinal Crystal Field Interactions

BY A. BENYOUSSEF and H. Ez-ZAHRAOUY ')

The three-dimensional spin-1 king model with crystal field interactions exhibits tricritical behaviour. Using an expansion technique for cluster identities of spin-1 localized spin systems, the influence of transverse magnetic field on this behaviour is studied. Temperature-crystal field phase diagrams are investigated for different values of transverse field. The longitudinal and transversc magnetizations, as well as the quadrupolar moments are calculated. General formulas applicable to structures with arbitrary coordination number N , are given.

1. Introduction

Spin systems are widespread in very different fields of physics, e.g., in the theory of magnetism, superconductivity, nuclear physics, etc. Special methods of theoretical physics are needed to describe the systems since commutation relations for spin components differ in the corresponding relations for both Bose and Fermi systems. The study of phase transitions in the king and Heisenberg models has been the subject of much interest [l to 71. Phase diagrams of such models show various types of multicritical phenomena [3, 4, 71. The Ising model in the presence of a transverse field serves for the study of cooperative phenomena and phase transitions in many physical systems [8 to 101. The diluted three-dimensional spin-1 king model with crystal field interactions is studied by Saber [ l l ] within finite cluster approximation. The spin-1 Ising model with a random crystal field is studied by Benyoussef et al. [12] and by Boccara et al. [13] within a mean field solution. Our aim is to study the influence of a transverse magnetic field on the phase diagram and the magnetic properties of a spin-1 Ising model with crystal field interactions. We use the finite cluster approximation [14, 151 with an expansion technique for cluster identities of spin-1 localized spin systems established by Ez-Zahraouy et al. [16]. The phase diagram for coordination number N = 6 is represented in the T-D space for different values of R, where T , R, and D are temperature, transverse field, and longitudinal crystal field, respectively. Magnetizations and quadrupolar moments are determined for several values of the crystal field at fixed value of the transverse field, for N = 6. In Section 2 we give the method and calculate the state equations. Section 3 is reserved to results and discussion.

I ) B.P. 1014, Rabat, Morocco. 2, International Centre for Theoretical Physics, Strada Costiera, 11 -34700 Trieste, Italy.

504 A. BENYOUSSEF and H. Ez-ZAIIRAOIJY

2. Finite Cluster Approximation

We consider a spin-1 king system in a simple cubic lattice described by a Hamiltonian corresponding to a paramagnet of the longitudinal-axis type in a transverse magnetic field,

where Six and Si, are the x-component and the z-component of a spin-1 operator at site i, respectively, s2 represents the transverse field, and Jij is the exchange interaction between spins at site i and j ; in this paper Jij is constant and equal to J . ( i j ) runs over all nearest-neighbour pairs of spins and D is the crystal field.

Using a single-site cluster approximation in which attention is focused on a cluster comprising just a single selected spin labelled 0, and the neighbouring spins with which it directly interacts, then the Hamiltonian containing 0 is

where

with N

e = c Sj,. j = 1

This single-site Hamiltonian can readily be diagonalized and its eigenvalues and eigenvectors found. The three eigenvectors corresponding to the eigenvalues

( P k = - a r c c o s ( - z ) + - ( k - 1 2 1)n 3 3

and

e = --1/274 3 v 3 + 14p3 + 27q21, 2

where k = 1, 2, 3.

Magnetic Properties of a Transverse Spin-l Ising Model 505

In a representation in which So, is diagonal, the starting point of the single-site cluster approximation is a set of formal identities of the type

where S& is the a-component of the spin operator So raised to the power p, (S&)= denotes the mean value of Sg, for a given configuration c of all other spins, i.e. when all other spins Si (i =+ 0) have fixed values. (. . .) denotes the average over all spin configurations. tr, means the trace performed over So only. p = t/k,T, T is the absolute temperature and k , the Boltzmann constant.

Equations (1 1) are not exact for an Ising system in a transverse field, they have nevertheless been accepted as a reasonable starting point in many studies of that system [17]. Let (SoJc and (,S;Jc denote, respectively, the mean value of So, and S& for a fixed configuration c for all other spins.

To calculate (So,)c and ( S $ J C , one has to effect the inner traces in (11) over the states of the spin at site 0 and this is most easily performed using the eigenstates of (8) as the basic states. In this way, it follows on setting p = 1 and 2 in turn in (11) that

The magnetizations rn, (a = z, x ) and the quadrupolar moments 4, (a = z, x ) are given by

m, = (f,@)> > 4, = (ga(0)) with

506 A. BENYOUSSEF and H. Ez-ZAHRAOUY

where (. . .) denotes the average over all configurations of the spins S j , ( j =+= 0). To calculate ( f , (d)) and (g,(H)) we have used the expansion technique for spin-1 Ising systems as follows [14]:

that contains 3N terms. From

factors of Si", and q factors

Suppose one considers the general product n 1 Sf; i = l N ( 2 p , = O 1

these terms one may collect together all those terms containing of Si,. Such a group is denoted by {Sz , Sz}N,p, q. For example, if N = 4, p = 1, and q = 2, then

{S:, ' 2 ' } 4 , 1 , 2 = S:z(S2zS3z + S 2 z S 4 z + S 3 z S 4 z ) + S:z(S1:S3z + S 1 z S 4 z + S3zS4,) + S ~ z ( S 1 z S 2 z f S 1 z S 4 z f '%zS4z) + S L ( S I J 2 , + S d 3 : + S 2 z S 3 , ) . (18)

Our aim is to expand the functions of (16) and (17) in terms of these {S:, Sz>N,p,q. Thus, if one writes

the problem is to find the coefficients A&)(N) and Bk) (N) . To achieve this it is advantageous to transform the spin-1 system to a spin-4 representation containing the Pauli operators oiz = *l. This may be accomplished by setting Siz = zizoiz with ziz = 0, 1. In this representation, (19) and (20) become

and must hold for arbitrary choices of ziz. Suppose one now chooses the first Y out of the N operators z i z to be unity, and the remainder zero. Then (21) and (22) give

gz ( il giz) = q = n i p = n B ~ ) P , " ) c ; - q { O z > r , q > (24) where ( C J ~ } ~ , ~ is the sum of all possible products of q spin operators, oiz, out of a maximum of r, and the Cy are the binomial coefficients m!/n!(rn - n)!. That is,

Magnetic Properties of a Transverse Spin-1 Ising Model 507

with

The spin-1 problem of (19) and (20) containing N spins has thus been transformed to a spin-1/2 problem containing r spins. The advantage of doing this is that it now enables one to use directly the results already established in [18] for the spin-i system. It may also be noted that whereas the coefficients b f ) ( r ) and d$)(r) for the spin% problem depend on the total number of spins present, the coefficients A f ( N ) and B&)(N) are in fact independent of N , as is clear from (27) and (28). Thus the label N is superfluous and may henceforth be dropped. This could, of course, have been inferred directly from (19) and (20) by setting one of the Si, spins equal to its zero value throughout. Specializing the results of [18] to a single group of r spins, one has for the current problem

where i

P c = O

Fi(Y, q) = C (- 1) cicr-i P 4 - P

3. Results and Discussion

In this section we present results of the Hamiltonian (1) on a simple cubic lattice ( N = 6), especially the dependence of the magnetizations m, and m,, and the quadrupolar moments q, and q, on temperature. Once the coefficients b?)(r) and dF) ( r ) have been calculated, the coefficients AE) and B&) may be found by the following procedure. First, A& and B g are got by setting r = q in (29) and (30). That is,

AE = bF(q), B@ = dF(q). (34)

Then, the other A;) and BE) may be obtained by expressing (29) and (30) as a recurrence relation, namely,

508 A. BENYOUSSEF and H. Ez-ZAHRAOUY

Then the magnetizations m, (a = z , x) and the quadrupolar moments q, (a = z , x) are given for an arbitrary coordination number N , by

Using the simplest approximation of the Zernike decoupling of the type

( S i z S j , ... s k ; ...) = (S,,) (S,,) ... ( S k z ) ... for i + j + k + ... , and seeing that the number of elements of the group {Sq, S z } N , p , q is equal to C:Cf-P, then (37) and (38) become

.. . . y = o p = o

Let us put m = mZ = (S,) and x = q, = (Sf), and replace x in (39) by its expression taken from (40), we obtain an equation for m of the form

m = am + bm3 + ... , (41) where

with xo being the solution of the following equation:

The critical temperature of the second-order transition is determined by a = 1. In the vicinity of a second-order transition the magnetization m, is determined by

1- -a b

m, = -.

At this temperature the transverse magnetization is given by N

m, = 1 A$)C;x{ , p = o

and the quadrupolar moments q, (a = z , x) are given by N

q, = c B$Cfxg. p = o

(44)

(45)

Magnetic Properties of a Transverse Spin-1 king Model 509

The right-hand side of (44) must be positive. If this is not the case the transition is of first order. The point at which a = 1 and b = 0 is the tricritical point. To obtain the expression for b one has to solve (40) for small m. The solution is of the form

x = xo + X l d , (47) where x1 is given by

p = o p = 1

p = o p = 1

The dependence of the magnetizations m, and m,, and the quadrupolar moments q, and q, on temperature for a fixed value of the transverse field SZ are represented for several values of the strength of the longitudinal crystal field D. A first-order transition is characterized by the gap of the longitudinal magnetization m, at the transition temperature T,, shown in Fig. 2 ( D = 2.95,2.85) in agreement with [19]. If the magnetization m, decreases continuously in the vicinity of the transition temperature and vanishes at T = T,, this is the second-order transition. It is seen that the critical crystal field above which a first-order transition appears (Fig. 1) decreases with increasing transverse field. The resulting phase diagram is shown in Fig. 1. From Fig. 1 it can be seen that there exists a tricritical line

2

7

I

Fig. 1. Phase diagram in D- T space for N = 6. TCL is the tricritical line. Dashed line corre- sponds to first-order transition. The numbers accompanying each curve denote the value of ' QIJ

0 2 3 4 D /J -

5 10 A. BENYOUSSFF and H. Ez-ZAHRAOUY

a

0 2 3 0.05

0 2 3 4 JlJ -

Fig. 2. The temperature dependence of a) the longitudinal and b) the transverse magnetization when Q / J = 0.5. The numbers accompanying each curve denote the values of D / J

TCL separating the surfaces of second- and first-order transitions. This line ends on tricritical points in the T-D and D-SZ planes. The temperature dependence of the longitudinal and transverse magnetizations for different values of the longitudinal crystal field is exhibited in Fig. 2 a and b. The transverse magnetization m, increases with the strength of the longitudinal magnetic field at low temperatures and passes through a peak for the first-order

i?!

100

$

0.7:

0 I 2 3 4

2.9:

2 3 4 T1.J -

Fig. 3. The temperature dependence of a) the longitudinal and b) the transverse quadrupolar moment for the cubic lattice when Q / J = 0.5. The numbers accompanying each curve denote the values of D/J

Magnetic Properties of a Transverse Spin-1 Ising Model 511

transition and a cusp for the second-order transition temperature of m, and then falls off rapidly as determined by (45). Fig. 2 a and b also display the extent to which the components of magnetization depend on the lattice coordination number. Finally, in Fig. 3 a and b, the temperature dependence of the longitudinal (4,) and the transverse (q,.) quadrupolar moments is displayed for the simple cubic lattice for a typical value of the transverse field. At low temperature, the longitudinal quadrupolar moment decreases with increasing temperature, while for T > T,, the latter increases with increasing temperature, or decreasing strength of the crystal field, while the transverse quadrupolar moment increases, when increasing the strength of the longitudinal crystal field. In the vicinity of the second-order transition temperature the quadrupolar moments q, (a = z , x) pass through a cusp as determined by (46). We conclude that the tricritical behaviour due to the presence of crystal field interactions disappears for sufficiently large transverse magnetic field s2 in agreement with [7, 191.

Acknowledgements

One of the authors (H. Ez-Zahraouy) wishes to thank Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO, for hospitality at the International Centre for Theoretical Physics, Trieste.

References

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[lo] V. V. ULYANOV and 0. B. ZASLAVSKII, Physics Rep. 216, 179 (1992). [l I] M. SABER, Internal Report Ic/88/261. [12] A. BENYOUSSEF, T. BIAZ, M. SABER, and M. TOUZANI, J. Phys. C 20, 5349 (1987). [13] N. BOCCARA, A. EL-KENZ, and M. SABER, J. Phys.: Condensed Matter 1, 5721 (1989). [14] N. BOCCARA, Phys. Letters A 94, 185 (1983). [IS] A. BENYOUSSEF and N. BOCCARA, J. Phys. C 16, 1143 (1983). [16] H. Ez-ZAHRAOUY, M. SABER, and J. W. TUCKER, J. Magnetism magnetic Mater. 118, 129 (1993). [17] F. C. SA BARRETO, I. P. FITTIPALDI, and B. ZEKS, Ferroelectrics 39, 1103 (1981). [I81 P. TOMCZAK, E. F. SARMENTO, A. F. SIQUEIRA, and A. R. FERCHMIN, phys. stat. sol. (b) 142, 551

[19] A. BENYOUSSEF, H. Ez-ZAHRAOUY, and M. SABER, Physica (Utrecht) 198A, 593 (1993). ( 1987).

(Received June 16, 1993; in revised,form September 28, 1993)