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We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R
_{1} and R
_{2 }of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.

The problems of surface magnetism have been investigated for many years. Among them the effects of surface on phase transitions in semi-infinite systems have received increasing interest. Real systems have surfaces, interfaces or boundaries and the translational symmetry is not preserved. This gives a set of surface phase diagrams richer than the infinite bulk one [

Several experimental studies show a critical behaviour of the surface different than the bulk. As example, we mention the work done on mixed compounds NbSe_{2} [

The three-dimensional semi-infinite spin-1 Ising model with a crystal field has been studied [

In the last years, a great attention has been devoted to systems of mixed spins and this is related to their importance in the study of magnetic materials with ferrimagnetic properties [

Recently, this attention has been expanded on systems of mixed spins higher than 1/2, like the case S = 1 and S = 3/2, which has been also studied by several methods, as the mean field approximation (MF) [

Our aim in this present paper is to determine the various types of phase diagram in the semi-infinite system of mixed spins S = 1 and S = 3/2 on the Blume-Capel model [

We consider a two sublattice mixed spin-1 and spin-3/2 Blume-Capel model with different single-ion anisotropies Δ_{A} and Δ_{B} acting on the spin-1 and spin-3/2, respectively. The Hamiltonian of the system is given by

where the sites of sublattice A are occupied by spins

In the Blume-Capel model, the reduced biquadratic interaction K is equal to zero, but we will take it into account due to the renormalization group technique we are using. We also introduce one additional interaction C to obtain self-consistent recursion relations. Thus, the Hamiltonian we will effectively use in the remainder of our work is as follows:

The renormalization does not keep in general the parameters space of the Hamiltonian. The terms added to the original Blume-Capel Hamiltonian model, in addition to their role in the conservation of the parameter space, can be used in the improvement of critical exponents and precision Monte Carlo simulation: scaling correction [

For example, in the Blume-Emery-Griffiths model, the three parameters

To have a more reliable qualitative appreciation of the phase transitions characteristics, we use an approximation of the real space renormalization group, namely the Migdal-Kadanoff one, which combines decimation and bond shifting and is tractable in all space dimensionalities. In order to implement the renormalization machinery, we consider a one-dimensional chain (

The spatial factor rescaling, denoted by b, is chosen as an odd integer to keep the possible sublattice symmetry breaking character of the system. In our present study, we take

After performing the decimation on the two middle spins

with

By using the renormalization group equation, we can make the link between (3) and (4) to obtain

with

Replacing the expressions of the Equations (3) and (4) in Equation (5), and knowing that

Since

For

For

For

For

For

For

The Equations (7) to (12) and the bond-shifting process yield the final renormalized couplings

Numerical analysis of these equations gives the flow in the parameter space of the Hamiltonian. We have already obtained [

Our results in the infinite model can be compared with those obtained by other methods: concerning the existence of tricritical point and its domain according to the values of the ratios of the interactions, we are in agreement with mean field theory result [

Another comparison is possible, with the pure BC model S = 1 or S = 3/2. In this case all those approximations are in agreement with the existence of the tricritical point in the integer spin case, but for the half integer spin, there exist a first order transition between two ordered phases at low temperature and the non existence of disorder in this domain. In reference [

We consider a system consisting of two mixed spin-1 and spin-3/2 sublattices, limited by a surface where the single-ion anisotropies acting on the spin-1 and spin-3/2 are denoted, respectively,

In the bulk, we keep the same notations of Section 2.1. On the surface, the reduced bilinear exchange interaction J, the reduced biquadratic interaction K and the interaction C mentioned in the Hamiltonian (2), are denoted by

Each spin at the surface can interact with a spin located in the bulk and increasingly with its first neighbors at the surface. With this new environment, some new critical properties will appear at the surface which will have recursion equations coupled to the bulk. This latter keeps the same equations as in the infinite model, forming an invariant subspace.

Concerning the surface, we can write the recursion equations in the following compact form [

with

model

We introduce the ratios

phase diagrams are constructed in the plane

In the renormalization procedure, the critical behaviors are derived from fixed points, by evaluating the eingenvalues of the transformation as

In our case, the additive terms of the infinite and semi infinite (bulk terms) model can be considered useful for finding a point of the second order transition line for which the irrelevant field vanishes.

The calculation of the critical temperature of infinite Blume-Capel model with mixed spins S_{A} = 1 and S_{B} = 3_{ }/2 (Section 2.1), in the three-dimensional case d = 3, shows three main types of diagram, labelled I, II and III. These types of diagrams can be classified as follows:

Type I: appears in the diagrams

Type II: appears in the diagrams

Type III: appears in the diagrams

The phase diagrams in the

The second and third types of phase diagrams (

Using renormalization-group calculations in the semi-infinite case (Section 2.2), we have obtained five generic types of phase diagram, illustrated in the

1) For

2) For

3) For

This ordinary phase transition can be first-order, second-order or tricritical.

4) For

5) For

Thereafter, we present the phase diagrams in the

a) For

b) For

c) For

d) For

Let us comment the types of phase diagrams obtained by Migdal-Kadanoff renormalization:

1) The presence of a semi infinite surface gives rise to a variety of new phase diagrams. Are highlighted, ordinary transitions (e.g.

2) Each of these phase transitions is represented by a different fixed point, and that they belongs to a new universality class different from that of the bulk. For example, the surface performs two different types of second order phase transitions. The first has a temperature higher than that of the bulk; this surface transition is described by the fixed point (O_{B}, C_{S}), where O_{B} is the fixed point (disorder) of the bulk [_{S} is that of the surface: C_{S} (J_{S} = 1.55, K_{S} = 0, Δ_{A}(S) = ¥. Δ_{B}_{(S)} = −¥, C_{S} = −0.47). The second possible transition occurs at the same temperature as that of the bulk; it is the special transition, represented by another different fixed point (C_{B}, C_{Sp}). C_{B} is the second order fixed point of the bulk [_{Sp} the surface one with coordinates C_{Sp} (J_{S} = 1.29, K_{S} = 0, Δ_{A}_{(S)} = ¥, Δ_{B}_{(S)} = −¥, C_{S} = −0.074).

3) The topology of the phase diagrams obtained is compared with that already established in the references [

4) In these types of phase diagrams obtained by using the Migdal-Kadanoff approximation, we note the absence of successive (surface/bulk) first order phase transitions and only the successive (surface/bulk) second-order phase transitions can occur. Also, simultaneous phase transitions of different orders are not observed. This was already met in the study of pure semi infinite models, see references [

Lipowski had already mentioned the expected difference between the two approaches (MF and RG), in the study of semi infinite Potts model [_{B} infinite), the surface one is also at 0 K. In particular, the fixed point of the bulk first order phase transition is precisely at 0 K, which causes ordinary first order transitions at surface.

The experimental results confirm the existence of a continuous transition at surface, while the bulk exhibit simultaneously a first order transition, what is called in the literature “surface induced disordering” (SID). This type of transition was highlighted in the Cu3Au alloy, see reference [

During this work, we have studied the pure Blume-Capel model in the semi-infinite case. To achieve our goal, we have determined the global phase diagrams of the mixed spin-1 and spin-3/2 in the semi-infinite system with different single-ion anisotropies acting on the spin-1 and spin-3/2 (on the surface and in the bulk) by using the Migdal-Kadanoff renormalization group technique, which combines decimation (with a space scale ration b = 3) and bond shifting. In the phase diagrams, the critical temperature lines versus single-ion anisotropies are shown. We have classified the various phase diagrams at fixed

A comparison with the types of phase diagrams in the pure semi-infinite model with S = 1 and S = 3/2 obtained by the renormalization and the mean field approaches was performed.

In perspective, we hope that this work could stimulate further theoretical and experimental works on ferrimagnetic systems such as mixed spins with random fields in finite, infinite and semi-infinite systems.

Mohamed El Bouziani,Mohamed Madani,Abou Gaye,Abdelhameed Alrajhi,1 1, (2016) Phase Diagrams of the Semi-Infinite Blume-Capel Model with Mixed Spins (SA = 1 and SB = 3/2) by Migdal Kadanoff Renormalization Group. World Journal of Condensed Matter Physics,06,109-122. doi: 10.4236/wjcmp.2016.62015