Magnetic properties of the mixed spin transverse Ising model with longitudinal crystal field interactions

phys. stat. sol. (b) 201, 491 (1997) Subject classification: 75.10.Dg; 75.30.Kz

Magnetic Properties

of the Mixed Spin Transverse Ising Model with Longitudinal Crystal Field Interactions

N. Benayad ¹ † and R. Zerhouni

Laboratoire de Physique Th eorique, Facult e des Sciences, Universit e Hassan II, An Chock, B.P. 5366 Ma^ arif, Casablanca, Morocco.

(Received November 6, 1996; in revised form February 11, 1997)

The three-dimensional mixed Ising spin system consisting of spin-1/2 and spin-1 with crystal field interactions exhibits tricritical behaviour. The influence of a transverse field on this behaviour is studied within the finite cluster approximation based on a single-site cluster theory. In order to expand the cluster identity of spin-1, we transformed the spin-1 to spin-1/2 representation contain- ing Pauli operators. We derived the state equations applicable to structures with arbitrary coordi- nation number N. The complete phase diagram in the case of a simple cubic lattice …N ˆ 6†, is investigated. The thermal dependence of the longitudinal and transverse sublattice magnetizations as well as those corresponding to the total magnetization and quadrupolar moments are also stud- ied.

1. Introduction

During several decades there has been considerable interest in the study of pure classical Ising models and their variants because they have been used to describe many physical situations in different fields of physics. Special attention has been focused on the study of two-state and three-state spin systems. In recent years, a particular interest has been devoted to the theoretical study of the above Ising models in a transverse field. We mention that the spin-1/2 transverse Ising model was originally introduced by De Gen- nes [1] as a valuable model for hydrogen-bonded ferroelectrics [2] such as the KH 2 PO 4

type. Since then, it has been applied to physical systems, such as cooperative Jahn-Tel- ler systems [3] like DyVO 4 , ordering in rare earth compounds with a singlet crystal-field ground state [4], and also to some real magnetic materials with strong uniaxial anisotro- py in a transverse field [5]. It has been extensively studied by the use of various techni- ques [6 to 9], including the effective field theory [10, 11] based on a generalized but approximated Callen-Suzuki relation derived by Sa Barreto, Fittipaldi, and Zeks. On the other hand, the spin-1 transverse Ising models have been studied [12 to 16] as well as Ising models with spin higher than one [17 to 22]. The influence of a transverse magnetic field on the magnetic properties of a spin-1 Ising model with longitudinal crystal-field interactions has been investigated [23] using the finite cluster approximation [24, 25].

Recently, attention has been directed to the study of the magnetic properties of a two- sublattice mixed spin Ising system. They are of interest for the following main reasons.

They have less translational symmetry than their single spin counterparts, and are well

1

† Corresponding author.

adapted to study a certain type of ferrimagnetism [26]. It has been shown that the MnNi(EDTA)-6 H 2 O complex is an example of a mixed spin system [27]. The mixed Ising spin system consisting of spin-1/2 and spin-1 had been studied by renormalization group technique [28, 29], by high-temperature series expansions [30], by free-fermion approximation [31] and by finite cluster approximation [32]. The effects of single ion anisotropy on its transition temperature have been investigated by renormalization group method [29], Monte Carlo simulation in the case of square lattice [33], effective field theory with correlations [34] and finite cluster approximation [35]. The two latter methods predict a tricritical behaviour in systems with a coordination number N larger than three. It is important to note here that the exact solution for the transition tem- perature, which is always of second order, can be obtained analytically if the structure of the system is chosen to be a honeycomb lattice …N ˆ 3† [36, 37].

The first purpose of this paper is to study the influence of the transverse field on the phase diagram of the three-dimensional …N ˆ 6† mixed spin-1/2 and spin-1 Ising system with longitudinal crystal field interaction. Such a system may be described by the follow- ing Hamiltonian:

H ˆ ÿ P

hiji J ij s iz S jz ‡ D P

j S _jz ² ÿ W 1 P

i s ix ÿ W 2 P

j S jx ; …1†

where s ia and S ja …a ˆ x; z† are the a-components of spin-1/2 and spin-1 operators at sites i and j, respectively. J ij is the exchange interaction and the first summation is carried out only over nearest-neighbour pairs of spins. D is the single-site crystal field interaction and …W 1 ; W 2 † are transverse fields. The second purpose is to examine the thermal dependences of longitudinal and transverse components of the magnetizations and quadrupolar mo- ments of the system described by the Hamiltonian (1). To this end, we use the finite cluster approximation [24, 25] with an expansion technique for cluster identities of spin-1 localized on one sublattice [38], which correctly accounts for the single-site kinematic relations.

Our presentation is as follows. In Section 2 we describe the theoretical framework and calculate the state equations. In Section 3 we investigate and discuss the phase diagram, in T D W space, and the thermal dependences of the transverse and longitudinal mo- ments of the magnetizations.

2. Finite Cluster Approximation

The theoretical framework to be used in the study of the system described by the Hamil- tonian (1) is the finite cluster approximation (FCA), based on a single-site cluster theo- ry. In this method, attention is focused on a cluster comprising just a single selected spin s 0 …S 0 †, and the nearest-neighbour spins fS 1 ; S 2 ; . . . ; S 6 g …fs 1 ; s 2 ; . . . ; s 6 g†, see Fig. 1, with which it directly interacts. We split the total Hamiltonian (1) into two parts, H ˆ H 0 ‡ H ⁰ , where H 0 includes all parts of H associated with the lattice site 0.

In the present system, H 0 takes the form

H 0s ˆ A 1 s 0z ‡ B 1 s 0x ; …2†

H 0S ˆ A 2 S 0z ‡ B 2 S 0x ‡ DS _0z ² ; …3†

where

A 1 ˆ ÿJ P ^N

j ˆ1 S jz ; B 1 ˆ ÿW 1 ; A 2 ˆ ÿJ P ^N

i ˆ1 s iz ; B 2 ˆ ÿW 2 ; …4†

whether the lattice site 0 belongs to s or S sublattice, respectively. Now, the problem consists in evaluating the sublattice longitudinal and transverse components of the mag- netization and the quadrupolar moments. In order to calculate them, we choose a repre- sentation in which s 0z and S 0z are diagonal and denote by hs 0a i _c …hS _0a ⁿ i _c ; n ˆ 1; 2† the mean value of s 0a …S ⁿ _0a † for a given configuration c of all other spins, i.e. when all other spins s i and S j …i; j 6ˆ 0† have fixed values. Neglecting the fact that H 0 and H ⁰ do not commute, hs 0a i _c and hS ⁿ _0a i _c are given by

hs 0a i _c ˆ Tr s

s 0a exp …ÿbH 0s †

Tr s

exp …ÿbH 0s † …5†

and

hS ⁿ _0a i _c ˆ Tr S

S _0a ⁿ exp …ÿbH 0S †

Tr S

exp …ÿbH 0S † ; …6†

where Tr s

(or Tr S

† means the trace performed over s 0 (or S 0 † only. As usual b ˆ 1=k B T , where T is the absolute temperature and k B the Boltzmann constant. There- fore, the magnetizations m _a ; m a …a ˆ x; z† and the quadrupolar moments q a are given by

m a hhs 0a i _c i ˆ Tr s

s 0a exp …ÿbH 0s † Tr s

exp …ÿbH 0s †

; …7†

m a hhS 0a i _c i ˆ Tr S

S 0a exp …ÿbH 0S † Tr S

exp …ÿbH 0S †

; …8†

and

q a hhS _0a ² i _c i ˆ Tr S

S _0a ² exp …ÿbH 0S † Tr S

exp …ÿbH 0S †

…9†

which can be considered as the starting point of the single-site cluster approximation.

h. . .i denotes the average over all spin configurations. Equations (7) to (9) are not exact.

Nevertheless, they have been accepted as a reasonable starting point [10] for transverse Ising systems and have been successfully applied to a number of interesting transverse Ising systems [10, 11, 13, 14, 16, 19, 22]. We have to emphasize that in the Ising limit …W 1 ˆ W 2 ˆ 0†, the Hamiltonian (1) contains only s iz and S jz . Then, relations (7), (8), and (9) become exact identities.

To calculate hs 0a i _c and hS ⁿ _0a i _c one has first to diagonalize the single-site Hamiltonians

H 0s and H 0S , respectively. H 0s can be written in a diagonal form if we use the following

Fig. 1. a) Nearest neighbours of spin s

. b) Nearest neighbours of spin S

rotation transformation:

s 0z ˆ cos js 0z

ÿ sin js 0x

; …10†

s 0x ˆ sin js 0z

‡ cos js ox

…11†

with

cos j ˆ ÿA 1

‰A ² ₁ ‡ B ² ₁ Š ¹⁼² ; sin j ˆ ÿB 1

‰A ² ₁ ‡ B ² ₁ Š ¹⁼² : …12†

Then, evaluating the inner traces in (5) over the states of the selected spin s 0 , we obtain hs 0z i _c ˆ ÿA 1

2‰A ² ₁ ‡ B ² ₁ Š ¹⁼² tanh b

2 ‰A ² ₁ ‡ B ² ₁ Š ¹⁼²

; …13†

hs 0x i _c ˆ ÿB 1

2‰A ² ₁ ‡ B ² ₁ Š ¹⁼² tanh b

2 ‰A ² ₁ ‡ B ² ₁ Š ¹⁼²

: …14†

H 0S can also readily be diagonalized. Its eigenvalues g k are given by

g _k ˆ ² ₃ …D ‡ p

 h cos …q k † ; k ˆ 1; 2; 3 …15†

with

q k ˆ 1

3 arcos ÿ27 r 2h

‡ 2

3 …k ÿ 1† p ; …16†

h ˆ 3 

p 3

2 ‰27 r ² ‡ j4r ³ ‡ 27 r ² jŠ ¹⁼² ; …17†

and

r ˆ ÿ…A ² ₂ ‡ B ² ₂ † ÿ D ²

3 ; r ˆ ÿ D

3 2A ² ₂ ÿ 2

9 D ² ÿ B ² ₂

: …18†

The corresponding eigenvectors are

jci _k ˆ a k j‡i ‡ b k jÿi ‡ c k j0i …19†

with

a k ˆ  jB 2 …g k ÿ D ‡ A 2 †j p 2

fB ² ₂ ‰…g _k ÿ D† ² ‡ A ² ₂ Š ‡ ‰…g _k ÿ D† ² ÿ A ² ₂ Š ² g ¹⁼² ; …20†

b k ˆ g _k ÿ D ÿ A 2

g _k ÿ D ‡ A 2 a k ; c k ˆ

 2 p

B 2 …g _k ÿ D ÿ A 2 † a k : …21†

Using the above eigenvalues and eigenvectors, to perform the inner traces in (6) over the states of the selected spin S 0 and setting n ˆ 1 and 2, we obtain

hS 0z i _c ˆ P ³

kˆ 1 …a ² _k ÿ b ² _k † exp …ÿbg _k † P ³

k ˆ1 exp …ÿbg _k †

; …22†

hS 0x i _c ˆ 

p 2 P ³

k ˆ 1 …a k ‡ b k † c k exp …ÿbg _k † P ³

kˆ 1 exp …ÿbg _k †

; …23†

hS ² _0z i _c ˆ P ³

k ˆ1 …a ² _k ‡ b ² _k † exp …ÿbg _k † P ³

k ˆ 1 exp …ÿbg _k †

; …24†

hS ² _0x i _c ˆ P ³

k ˆ1

1

2 …a k ‡ b k † ² ‡ c ² _k

exp …ÿbg _k † P ³

k ˆ 1 exp …ÿbg _k †

: …25†

We note that these starting equations for spin-1 are equivalent to those discussed by other authors [39, 40].

From eqs. (13), (14), (22) to (25), we easily observe that the magnetizations m _a ; m a

and quadrupolar moments q a ˆ hS _a ² i …a ˆ x; z† are functions of P

j S jz or P

i s iz . They can be written as

m a ˆ h a P ^N

j ˆ1 S jz

!

* +

; m a ˆ f a P ^N

i ˆ1 s iz

; q a ˆ g a P ^N

i ˆ1 s iz

…26†

with

h a P ^N

j ˆ1 S jz

!

ˆ hs 0a i _c ; …27†

f a P ^N

i ˆ1 s iz

ˆ hS 0a i _c ; …28†

g a P ^N

i ˆ1 s iz

ˆ hS _0a ² i _c : …29†

When calculating the average on the right-hand side of eqs. (26), we use the fact that any function E…s z † and E…S z † of s z …ˆ 1=2† and S z …ˆ 0; 1† can be written as the linear superposition

E…s z † ˆ E 1 ‡ E 2 s z ; …30†

E…S z † ˆ E 1 ‡ E 2 S z ‡ E 3 S _z ² …31†

with appropriate coefficients E 1;2 and E 1;2; 3 . Applying this to all spins s iz and S jz in eqs. (27) to (29), the functions h a ; f a and g a are decomposed as

h a P ^N

jˆ 1 S jz

!

ˆ P ^N

q ˆ0

P

N ÿq

p ˆ0 H _{p; q} ^a fS _z ² ; S z g _{N; p; q} ; …32†

f a P ^N

i ˆ1 s iz

ˆ P ^N

q ˆ0 F _q ^a …N† fs z g _{N; q} ; …33†

g a P ^N

i ˆ1 s iz

ˆ P ^N

q ˆ0 G ^a _q …N† fs z g _{N; q} ; …34†

where fS _z ² ; S z g _{N; p; q} denotes the superposition of all the terms containing p factors of S ² _jz and q factors of S j

z with j 6ˆ j ⁰ . These factors are selected from the set fS 1z ; S 2z ; . . . ; S Nz ; S ² _1z ; S _2z ² ; . . . ; S _Nz ² g. For example, if N ˆ 4; p ˆ 2 and q ˆ 1, then

fS _z ² ; S z g _{4; 2;1} ˆ S 1z …S ² _2z S ² _3z ‡ S _2z ² S _4z ² ‡ S _3z ² S _4z ² † ‡ S 2z …S _1z ² S _3z ² ‡ S ² _1z S ² _4z ‡ S _3z ² S _4z ² †

‡ S 3z …S _1z ² S _2z ² ‡ S ² _1z S ² _4z ‡ S ² _2z S ² _4z †

‡ S 4z …S _1z ² S _2z ² ‡ S ² _1z S ² _3z ‡ S ² _2z S ² _3z † : …35†

fs z g _{N; q} denotes the superposition of all the terms containing q different factors of s iz

selected from the set fs 1z ; s 2z ; . . . ; s Nz g. It may be noted that the coefficients F _q ^a …N†

and G ^a _q …N† for spin-1/2 depend on the nearest-neighbour coordination number, whereas the coefficients H _{p; q} ^a are in fact independent of N.

The problem is to determine the coefficients H _{p; q} ^a ; F _q ^a …N† and G ^a _q …N†. To evaluate H _{p; q} ^a , it is advantageous to transform the spin-1 system to spin-1/2 representation con- taining the Pauli operators s jz ˆ 1 [38], by setting S jz ˆ t jz s jz with t jz ˆ 0; 1. In this representation (32) becomes

h a P ^N

jˆ 1 t jz s jz

!

ˆ P ^N

q ˆ 0

P

N ÿ q

p ˆ 0 H _{p; q} ^a† ft z ; t z s z g _{N; p; q} …36†

which must be satisfied for arbitrary choices of t jz . Now let us choose the first r out of the N operators t jz to be unity, and the remainder zero. Then (36) gives

h a P ^r

jˆ 1 s jz

!

ˆ P ^r

q ˆ 0

P

rÿ q

pˆ 0 H ^a† _{p; q} C _p ^{rÿ q} f s z g _{r; q} ; …37†

where f s z g _{r; q} has the same meaning as fs z g _{N; q} , but s jz and N are replaced by s jz and r, respectively. C _n ^m are the binomial coefficients m!=‰n!…n ÿ m†!Š. That is

h a P ^r

jˆ 1 s jz

!

ˆ P ^r

q ˆ 0 e ^a† _q …r† f s z g _{r; q} …38†

with

e ^a† _q …r† ˆ ^r P ^ÿq

p ˆ0 H _pq ^a† C ^r _p ^ÿq : …39†

As is clear from eq. (39), the coefficients e ^a† _q …r† for the spin-1/2 problem depend on the total number of the present spins. The above transformation of the spin-1 problem of (32) containing N spins to spin-1/2 problem containing r spins, enables us to use di- rectly the results already established in [41] for the spin-1/2 system. Applying these results to the single group of r spins (38), we obtain

e ^a† _q …r† ˆ 1 2 ^r C ^r _q

P ^r

n

ˆ0 C ^r _n

e n

…r; q† h n

a …r ÿ 2n 1 † ; …40†

where

e n

…r; q† ˆ P ⁿ

lˆ 0 …ÿ1† ^l C ⁿ _l

C ^r _q ^ÿn _ÿl

: …41†

Once the coefficients e ^a† _q …r† have been calculated, the coefficients H _{p; q} ^a may be found by the following procedure. First, H _{0; q} ^a† is obtained by setting r ˆ q in (39). That is

A ^a† _0q ˆ e ^a† _q …q† : …42†

By expressing (39) as a recurrence relation, namely H _rÿ ^a† _{q; q} ˆ e ^a† _q …r† ÿ ^rÿ P ^{q ÿ 1}

p ˆ0 H ^a† _pq C _p ^{rÿ q} ; …43†

and using (40) we obtain the coefficients H _{1; q} ^a† ; H _{2; q} ^a† ; . . . ; H ^a† _{N ÿ q; q} when r ˆ q ‡ 1; q ‡ 2; . . . ; N, respectively. Thus, doing this for each value of q…2 f1; 2; . . . ; Ng†, we determine all the coefficients H pq appearing via the right-hand side of (32). On the other hand, to calculate f a P

i s iz

and g a P

i s iz

we use directly the results established in [41] for the spin-1/2 system. Then, the coefficients F _q ^a …N† and G ^a _q …N† are given by

F _q ^a† …N† ˆ 1 2 ^{N ÿq} C ^N _q

P ^N

n

ˆ 0 C _n ^N

e n

…N; q† f n

a 1

2 …N ÿ 2n 2 †

; …44†

G ^a† _q …N† ˆ 1 2 ^{N ÿq} C _q ^N

P ^N

n

ˆ0 C _n ^N

e n

…N; q† g n

a 1

2 …N ÿ 2n 3 †

; …45†

where

e n …N; q† ˆ P ⁿ

l ˆ0 …ÿ1† ^l C ⁿ _l C _q ^N _ÿ ^ÿ _l ⁿ ; n ˆ n 2 or n 3 : …46†

3. Results and Discussion

The sublattice magnetizations m a and m a …a ˆ z; x† and the quadrupolar moments q a …a ˆ z; x† are given by eqs. (26). They are valid for any lattice (arbitrary coordina- tion number N†, and constitute a set of relations, according to which we can study the present system. However, in order to carry out the average over all spin configurations implied in these equations, we have to deal with multispin correlation functions. The problem becomes mathematically untractable if we try to treat them in the spirit of the FCA. In this work, we use the simplest approximation in which the correlations between quantities pertaining to different sites are neglected,

hs iz s kz . . . s 1z i ˆ hs iz i hs kz i . . . hs 1z i ;

hS ^p _jz

S _mz ^p

. . . S _nz ^p4 i ˆ hS _jz ^p

i hS _mz ^p

i . . . hS _nz ^p

i …47†

with i 6ˆ k 6ˆ . . . 6ˆ 1; j 6ˆ m 6ˆ . . . 6ˆ n, and p i ˆ 1 or 2. If this is done, and counting the

number of elements of the sets fS ² _z ; S z g _{N; p; q} and fs z g _{N; q} which are equal, respectively,

to C _p ^N C _q ^{N ÿ p} and C _q ^N , one finds the following coupled equations:

m _a ˆ P ^N

q ˆ 0

P

N ÿ q

p ˆ 0 C ^N _p C _q ^{N ÿ p} H ^a† _pq m ^q _z q _z ^p ; …48†

m a ˆ P ^N

q ˆ0 C ^N _q F _q ^a† …N† m ^q _z ; …49†

q a ˆ P ^N

q ˆ0 C _q ^N G ^a† _q …N† m ^q _z : …50†

In this paper we are first interested in investigating the phase diagram of the system in the case of a uniformly applied transverse field …W 1 ˆ W 2 ˆ W† when the structure of the system is the simple cubic lattice …N ˆ 6†. At high temperature, the longitudinal magnetizations m _z and m z are disordered. Below a transition temperature T c , they order …m z 6ˆ 0 and m z 6ˆ 0†, while the corresponding transverse magnetizations m x and m x are expected to be ordered at all temperatures. To calculate T c , we substitute m z and q z in (48) with their expressions taken from (49) and (50). Then, we obtain an equation for m _z of the form

m _z ˆ am _z ‡ bm ³ _z ‡ . . . ; …51†

where

a ˆ N ² F ₁ ^{z† N} P ^ÿ1

p ˆ0 C ^N _p ^ÿ1 A ^z† _p1 ‰G ^z† ₀ Š ^p …52†

and

b ˆ N ^N P ^ÿ1

p ˆ0 C ^N _p ^ÿ1 H _p1 ^z† ‰NpC ^N ₂ F ₁ ^z† G ^z† ₂ …G ^z† ₀ † ^{p ÿ1} ‡ C ^N ₃ F ₃ ^z† …G ^z† ₀ † ^p Š

‡ N ^{3 N} P ^{ÿ 3}

p ˆ 0 C _p ^N C ₃ ^{N ÿp} H _p3 ^z† ‰F ₁ ^z† Š ³ ‰G ^z† ₀ Š ^p …53†

within this approximation. As usual, the second-order transition surface in T D W space is determined by the condition a ˆ 1. We note here that at this transition, the transverse sublattice magnetizations m x and m x keep in fact finite values which are given by

m _x …T ˆ T c † ˆ P ^N

pˆ 0 C _p ^N H _p0 ^x† q ^p _z ; …54†

m x …T ˆ T c † ˆ F ₀ ^x† …N† : …55†

The magnetization m _z in the vicinity of the second-order transition surface is given by m ² _z ˆ 1 ÿ a

b : …56†

The right-hand side of (56) must be positive. If this is not the case, the transition is of

the first-order and, in the T D W space, the point at which a ˆ 1 and b ˆ 0 charac-

terizes the tricritical point.

In Fig. 2, we represent the phase diagram in the T D W space in the case of a uni- formly applied transverse field W 1 ˆ W 2 ˆ W for a coordination number appropriate to the simple cubic lattice …N ˆ 6†. Let us first indicate that in the T W plane …D ˆ 0†, the critical temperature decreases gradually from its value in the mixed Ising system T c …W ˆ 0†, to vanish at some critical value W c ˆ 3:516 of the transverse field strength.

In the absence of the transverse field …W ˆ 0†, the system reduces to the two-sublattice mixed spin-1/2 and spin-1 Ising model with longitudinal crystal field interaction discussed previously in [34, 35]. In this case, we have to mention that the critical line ends in a tricritical point followed by a first-order transition line. The remaining part …W 6ˆ 0† of the phase diagram shows the influence of the transverse field. In the figure, we plot var- ious transition lines when the strength of the transverse field W takes values less than W c . It is seen that the system keeps a tricritical behaviour only when W is relatively small …0 W=J < 1:165†. The D component of the tricritical point decreases with increasing transverse field and therefore there exists a tricritical line (TCL) separating first- and second-order transition surfaces. When W belongs to the range 1:165 W=J < 3:516, the Fig. 2. Phase diagram in T D ÿ W space for the simple cubic lattice. TCL is the tricritical line.

The dashed line corresponds to a first-order tran- sition. The number accompanying each curve de- notes the value of W=J

Fig. 3. The thermal dependences of the

longitudinal and transverse magnetiza-

tions …m

; m

; M

; a ˆ z; x† for the sim-

ple cubic lattice when the system exhibits

a second-order transition, W=J ˆ 0:5,

D=J ˆ 2

tricritical behaviour disappears and all transitions are always of second order for any value of the crystal field interaction.

Equations (48) to (50) are established for any structure with arbitrary coordination number N. Now, let us use them to investigate the temperature dependence of the longitudinal and transverse components of the sublattice magnetizations m _a and m a …a ˆ z; x† as well as those corresponding to the total magnetization defined by M a ˆ …m _a ‡ m a †=2. In Fig. 3, we plot the longitudinal and transverse magnetizations m a ; m a and M a in the case of a uniformly applied transverse field W ˆ 0:5J for a fixed value of the crystal field D ˆ 2J, which corresponds to a second-order transition. As is shown in the figure, the longitudinal sublattice …m _z ; m z † and the total M z magnetiza- tions decrease continuously in the vicinity of the transition temperature and vanish at T ˆ T c , we note that the role of the transverse field W is to inhibit the ordering of m z ; m z and M z . The behaviours of the corresponding transverse components m x ; m x and M x are also shown in Fig. 3. As is expected, they are ordered at all temperatures. On the other hand, the behaviours of the magnetizations m _a ; m a and M a …a ˆ z; x†, when the system exhibits a first-order transition …W ˆ 0:5J; D ˆ 2:96J†, are depicted in Fig. 4.

The longitudinal components m _z ; m z and M z exhibit a gap at the transition temperature T c . The x-components of the sublattice magnetizations m _x and m x are discontinuous at T c since they depend on the behaviour of m _z and m z , respectively. However, as is clearly shown in the figure, it is important to note that the transverse total magnetization M x

does not exhibit a gap at the first-order transition. This results from the fact that the gaps exhibited by m _x and m x at T c compensate each other completely. As far as we know, such behaviour …M x is continuous at the first-order transition) has not been found in monoatomic and mixed spin transverse Ising models investigated up to now.

Fig. 5 shows how the thermal dependences of the total magnetizations M a depend on the value of the longitudinal crystal field interactions D. The total longitudinal magneti- zation M z , for a fixed value of the transverse field …W ˆ 0:5J† are represented in Fig. 5a for several values of D. As is drawn in this figure a first-order transition is characterized by a gap of M z at the transition temperature (the curve corresponding to D ˆ 2:96J†.

Fig. 4. The thermal dependences of the

longitudinal and transverse magnetiza-

tions …m

; m

; M

; a ˆ z; x† for the sim-

ple cubic lattice, when the system exhi-

bits a first-order transition W=J ˆ 0:5,

D=J ˆ 2:96

We also note that M z decreases with increasing strength of the longitudinal anisotropy.

As is seen from Fig. 5b, and in contrast to M z , the total transverse magnetization M x

increases with the value of D, passes through a cusp at the second-order transition …D ˆ 0; J; 2J; 2:8J† and shows an analytic behaviour when the system exhibits a first- order transition D ˆ 2:96J. We have to point out that below T c , the dependence of M x

on temperature becomes more and more important as the strength of the anisotropy

takes higher values. It is worthy to note here that for a given large value of D; M x is a

decreasing function of the temperature, while in a transverse spin-1 Ising model, the

transverse magnetization increases with temperature [23]. Finally, selected results are

shown in Fig. 6a and b, for the temperature dependences of the longitudinal q z and

Fig. 5. The temperature dependences of

the total a) longitudinal and b) trans-

verse magnetizations for the simple cu-

bic lattice when W=J ˆ 0:5. The num-

ber accompanying each curve denotes

the value of D=J

transverse q x quadrupolar moments in the case of a fixed value of the transverse field …W ˆ 0:5J†. We find that for any strength of D belonging to 0 D=J < 3; q z …q x † de- creases (increases) rapidly from its zero temperature value till the transition tem- perature T c is reached. Here, at T c ; q z and q x pass through a cusp or a peak, the tran- sition is either of second order or first or- der, respectively. Above T c , they vary slightly. As is shown in Fig. 6a and b, the longitudinal quadrupolar moment q z de- creases with increasing values of D, while the transverse component q x increases when the strength of the crystal field increases.

Within the same theoretical framework as used in this paper, we investigate the effect of a random crystal field on the phase dia- gram plotted in Fig. 2. The first results show a variety of interesting phenomena re- sulting from the fluctuation of the crystal field interaction. The work is in progress and will be published elsewhere.

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