Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field
The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic field h via the recursion relations method. A ground-state phase diagram is constructed which may be needful to explore important regions of the temperature phase diagrams of a model. The...
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irk-123456789-1561992019-06-19T01:31:11Z Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field Karimou, M. Yessoufou, R.A. Oke, T.D. Kpadonou, A. Hontinfinde, F. The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic field h via the recursion relations method. A ground-state phase diagram is constructed which may be needful to explore important regions of the temperature phase diagrams of a model. The order-parameters, the corresponding response functions and internal energy are thoroughly investigated in order to typify the nature of the phase transition and to get the corresponding temperatures. So, in the absence of the magnetic field, the temperature phase diagrams are displayed in the case of an equal crystal-field on the (kBT /|J|, D/|J|) plane when q = 3, 4, 5 and 6. The model only exhibits the second-order phase transition for appropriate values of physical parameters of a model. Дослiджується змiшана спiн-1/2 i спiн-5/2 модель на гратцi Бете в присутностi магнiтного поля h, використовуючи метод рекурсивних спiввiдношень. Побудована фазова дiаграма основного стану, яка може бути використана для дослiдження цiкавих областей температурної фазової дiаграми моделi. Для того, щоб прокласифiкувати природу фазового переходу та отримати вiдповiднi температури, детально дослiджено параметри порядку, вiдповiднi функцiї вiдгуку i внутрiшню енергiю. Так, при вiдсутностi магнiтного поля, температурнi фазовi дiаграми є представленi для випадку однакового кристалiчного поля на площинi (kBT /|J|,D/|J|), коли q = 3, 4, 5 i 6. Модель демонструє тiльки фазовий перехiд другого роду для вiдповiдних значень фiзичних параметрiв моделi. 2016 Article Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field / M. Karimou, R.A. Yessoufou, T.D. Oke, A. Kpadonou, F. Hontinfinde // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33003: 1–15. — Бібліогр.: 39 назв. — англ. 1607-324X DOI:10.5488/CMP.19.33002 arXiv:1609.04687 PACS: 05.50.+q, 05.70.Ce, 64.60.Cn, 75.10.Hk, 75.30.Gw http://dspace.nbuv.gov.ua/handle/123456789/156199 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic
field h via the recursion relations method. A ground-state phase diagram is constructed which may be needful
to explore important regions of the temperature phase diagrams of a model. The order-parameters, the corresponding response functions and internal energy are thoroughly investigated in order to typify the nature of
the phase transition and to get the corresponding temperatures. So, in the absence of the magnetic field, the
temperature phase diagrams are displayed in the case of an equal crystal-field on the (kBT /|J|, D/|J|) plane
when q = 3, 4, 5 and 6. The model only exhibits the second-order phase transition for appropriate values of
physical parameters of a model. |
| format |
Article |
| author |
Karimou, M. Yessoufou, R.A. Oke, T.D. Kpadonou, A. Hontinfinde, F. |
| spellingShingle |
Karimou, M. Yessoufou, R.A. Oke, T.D. Kpadonou, A. Hontinfinde, F. Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field Condensed Matter Physics |
| author_facet |
Karimou, M. Yessoufou, R.A. Oke, T.D. Kpadonou, A. Hontinfinde, F. |
| author_sort |
Karimou, M. |
| title |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field |
| title_short |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field |
| title_full |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field |
| title_fullStr |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field |
| title_full_unstemmed |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field |
| title_sort |
bethe approach study of the mixed spin-1/2 and spin-5/2 ising system in the presence of an applied magnetic field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156199 |
| citation_txt |
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field / M. Karimou, R.A. Yessoufou, T.D. Oke, A. Kpadonou, F. Hontinfinde // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33003: 1–15. — Бібліогр.: 39 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
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| first_indexed |
2025-07-14T08:34:59Z |
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2025-07-14T08:34:59Z |
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1837610673337532416 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 3, 33003: 1–15
DOI: 10.5488/CMP.19.33003
http://www.icmp.lviv.ua/journal
Bethe approach study of the mixed spin-1/2 and
spin-5/2 Ising system in the presence of an applied
magnetic field
M. Karimou1, R.A. Yessoufou1,2∗, T.D. Oke1,2, A. Kpadonou1,3, F. Hontinfinde1,2
1 Institute of Mathematics and Physical Sciences (IMSP), Republic of Benin
2 Department of Physics, University of Abomey-Calavi, Republic of Benin
3 University of Natitingou, École Normale Supérieure, Republic of Benin
Received January 18, 2016, in final form May 13, 2016
The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic
field h via the recursion relations method. A ground-state phase diagram is constructed which may be needful
to explore important regions of the temperature phase diagrams of a model. The order-parameters, the cor-
responding response functions and internal energy are thoroughly investigated in order to typify the nature of
the phase transition and to get the corresponding temperatures. So, in the absence of the magnetic field, the
temperature phase diagrams are displayed in the case of an equal crystal-field on the (kBT /|J |, D/|J |) plane
when q = 3,4, 5 and 6. The model only exhibits the second-order phase transition for appropriate values of
physical parameters of a model.
Key words:magnetic systems, thermal variations, phase diagrams, magnetic field, second-order transition
PACS: 05.50.+q, 05.70.Ce, 64.60.Cn, 75.10.Hk, 75.30.Gw
1. Introduction
Over the last five decades, the Ising model has been one of the most largely used models to describe
critical behaviors of several systems in nature. Lately, numerous extensions have been made in the
spin-
1
2 Ising model to describe a wide variety of systems. For example, the models consisting of mixed
spins with different magnitudes are interesting extensions, forming the so-called mixed-spin Ising class
[1, 2]. Beyond that, magnetic materials have several important technological applications: they find wide
use in information storage devices, microwaves communication systems, electric power transformers
and dynamo, and high-fidelity speakers [3–6]. Thus, in response to an increasing demand placed on the
performance of magnetic solids, there has been a surge of interest inmolecular-basedmagnetic materials
[7–10]. Indeed, the discovery of these materials [11] has been one of the advances in modern magnetism.
Manymagnetic materials have two types ofmagnetic atoms regularly alternating which exhibit ferrimag-
netism. In this context, a good description of their physical properties is given by means of mixed-spin
configurations. The interest in studying magnetic properties of these materials is due to their reduced
translational symmetry rather than to their single-spin counterparts, since they consist of two interpene-
trating sublattices. Thus, ferrimagnetic materials are of great interest due to their possible technological
applications and from a fundamental point of view. These materials are modelled using mixed-spin Ising
models that can be built up by infinite combinations of different spins.
In literature there exist many studies on mixed-spin Ising systems which intend to clarify the mag-
netic properties of magnetic systems. In this regard, there has been great interest in the study of mag-
netic properties of the systems formed by two sublattices with different spins and crystal-field interac-
tions [12]. Theoretically, such systems have been widely analysed using several numerical approaches,
∗
E-mail: yesradca@yahoo.fr, olorire2012@gmail.com
©M. Karimou, R.A. Yessoufou, T.D. Oke, A. Kpadonou, F. Hontinfinde, 2016 33003-1
http://dx.doi.org/10.5488/CMP.19.33003
http://www.icmp.lviv.ua/journal
M. Karimou et al.
e.g., effective-field theory [13–18], mean-field approximation [19–23], renormalization-group technique
[24, 25], numerical simulations based on Monte-Carlo [26–30] and exact recursion equations [31–38]. A
somewhat newer interest is to extend such investigations into a more general mixed-spin Ising model
with one constituent spin-
1
2 and the other constituent spin-
5
2 . To this end, Deriven et al. [18] used an
effective-field theory with correlation to study the same model and got interesting results. Recently, Guo
et al. [17] studied the thermal entanglement of the same model by means of the concept of negativity and
also got interesting results concerning the effects of the magnetic field on the entanglement.
In the present paper, we use the exact recursion equations technique to examine themagnetic proper-
ties of themixed spin-
1
2 and spin-
5
2 Isingmodel with equal crystal-field on the Bethe lattice in the presence
of a longitudinal magnetic field. The aim of this work is to investigate the effect of the crystal-field and
the magnetic field on the physical magnetic properties of the model.
The remainder of this paper is arranged as follows. In section 2, the description of the model on the
Bethe lattice is clarified. Furthermore, the order-parameters, the corresponding response functions and
the internal energy are expressed in terms of recursion relations. In the next section, some definitions
of the critical temperature of the model are explained. In section 4, we present some illustrations and
discuss in detail the numerical results. We finally conclude in the last section .
2. Description of the model on the Bethe lattice
The mixed spins system on the Bethe lattice is shown in figure 1. We consider the mixed spin-
1
2 and
spin-
5
2 system consisting of two sublattices A and B. The sites of sublattice A are occupied by atoms of spins
Si , where Si =± 1
2 . Those of the sublattice B are occupied by atoms of spins σ j , where σ j =± 5
2 ,± 3
2 ,± 1
2 . In
our case, the Bethe lattice is arranged in such a way that the central spin is spin-
1
2 and the next generation
spin is spin-
5
2 and so on to infinity. Thus, the Ising Hamiltonian of such a model on the Bethe lattice may
be written as:
H =−J
∑
〈i , j 〉
σ j Si −D
∑
j
σ2
j −h
(∑
i
Si +
∑
j
σ j
)
; (2.1)
J < 0 is the bilinear exchange coupling interaction strength. D and h are, respectively, the single-ion
anisotropy or the crystal-field and the longitudinal magnetic field acting on the spins of the model.
In order to formulate the problem on the Bethe lattice, the partition function is the main ingredient
Figure 1. The mixed spin Ising model consisting of two different magnetic atoms with spins values si = 1
2
and σ j = 5
2 , respectively, defined on the Bethe lattice with coordination number q = 3.
33003-2
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
which is given as:
Z =∑
exp
{
β
[
J
∑
〈i , j 〉
σ j Si +D
∑
j
σ2
j +h
(∑
i
Si +
∑
j
σ j
)]}
. (2.2)
If the Bethe lattice is cut at the central spin S0 , it splits into q disconnected pieces. Thus, the partition
function on the Bethe lattice can be written as:
Z =∑
S0
exp
[
β(hS0)
]
g q
n (S0), (2.3)
where S0 is the central spin value of the lattice, gn(S0) is the partition function of an individual branch
and the suffix n represents the fact that the sub-tree has n shells, i.e., n steps from the root to the bound-
ary sites. If we continue to cut the Bethe lattice on the sites σ1 and S2 which are respectively the near-
est and the next-nearest of the central spin S0 , we can obtain the recurrence relations for gn(S0) and
gn−1(σ1) as:
gn(S0) = ∑
{σ1}
exp
[
β
(
JS0σ1 +Dσ2
1 +hσ1
)][
gn−1(σ1)
]q−1 , (2.4)
gn−1(σ1) = ∑
{S2}
exp
[
β (JS2σ1 +hS2)
][
gn−2(S2)
]q−1 . (2.5)
Now, we explicitly calculate some gn(S0) and gn−1(σ1) as follows:
gn
(
±1
2
)
= ∑
{σ1}
exp
[
β
(
± J
2
σ1 +Dσ2
1 +hσ1
)][
gn−1(σ1)
]q−1
= exp
[
β
(
±5J
4
+ 25
4
D + 5
2
h
)][
gn−1
(
5
2
)]q−1
+exp
[
β
(
∓5J
4
+ 25
4
D − 5
2
h
)][
gn−1
(
−5
2
)]q−1
+exp
[
β
(
±3J
4
+ 9
4
D + 3
2
h
)][
gn−1
(
3
2
)]q−1
+exp
[
β
(
∓3J
4
+ 9
4
D − 3
2
h
)][
gn−1
(
−3
2
)]q−1
+exp
[
β
(
± J
4
+ 1
4
D + 1
2
h
)][
gn−1
(
1
2
)]q−1
+exp
[
β
(
∓ J
4
+ 1
4
D − 1
2
h
)][
gn−1
(
−1
2
)]q−1
, (2.6)
gn−1
(
±5
2
)
= ∑
{S2}
exp
[
β
(
±5J
2
S2 +hS2
)][
gn−2(S2)
]q−1
= exp
[
β
(
±5J
4
+ h
2
)][
gn−2
(
1
2
)]q−1
+exp
[
β
(
∓5J
4
− h
2
)][
gn−2
(
−1
2
)]q−1
, (2.7)
gn−1
(
±3
2
)
= ∑
{S2}
exp
[
β
(
±3J
2
S2 +hS2
)][
gn−2(S2)
]q−1
= exp
[
β
(
±3J
4
+ h
2
)][
gn−2
(
1
2
)]q−1
+exp
[
β
(
∓3J
4
− h
2
)][
gn−2
(
−1
2
)]q−1
, (2.8)
gn−1
(
±1
2
)
= ∑
{S2}
exp
[
β
(
± J
2
S2 +hS2
)][
gn−2(S2)
]q−1
= exp
[
β
(
± J
4
+ h
2
)][
gn−2
(
1
2
)]q−1
+exp
[
β
(
∓ J
4
− h
2
)][
gn−2
(
−1
2
)]q−1
. (2.9)
After calculating all the gn(S0) and gn−1(σ1), we can define the recursion relations for the spin- 12 as:
Zn = gn
( 1
2
)
gn
(− 1
2
) ,
33003-3
M. Karimou et al.
and for the spin-
5
2 as:
An−1 =
gn−1
( 5
2
)
gn−1
(− 1
2
) , Bn−1 =
gn−1
(− 5
2
)
gn−1
(− 1
2
) , Cn−1 =
gn−1
( 3
2
)
gn−1
(− 1
2
) ,
Dn−1 =
gn−1
(− 3
2
)
gn−1
(− 1
2
) , En−1 =
gn−1
( 1
2
)
gn−1
(− 1
2
) . (2.10)
To investigate our model, we define two order-parameters, the magnetization M and the corresponding
quadrupolar momentQ. For the sublattice A, the sublattice magnetizationMA is defined by:
MA = Z−1
A
∑
{S0}
S0exp(βhS0)g q
n (S0). (2.11)
After some mathematical manipulations, the sublattice magnetizationMA is explicitly given by:
MA =
exp
(
βh
2
)
Z q
n −exp
(
−βh
2
)
2
[
exp
(
βh
2
)
Z q
n +exp
(
−βh
2
)] . (2.12)
In the same way, we also calculate the two order-parameters for the sublattice B as follows:
MB =
M ′
B
M 0
B
, QB =
Q ′
B
Q0
B
,
where:
M ′
B
= 5exp
(
25
4
βD
)[
exp
(
5
2
βh
)
Aq
n−1 −exp
(
−5
2
βh
)
B q
n−1
]
+3exp
(
9
4
βD
)[
exp
(
3
2
βh
)
C q
n−1
−exp
(
−3
2
βh
)
Dq
n−1
]
+exp
(
1
4
βD
)[
exp
(
1
2
βh
)
E q
n−1 −exp
(
−1
2
βh
)]
, (2.13)
M 0
B
= 2exp
(
25
4
βD
)[
exp
(
5
2
βh
)
Aq
n−1 +exp
(
−5
2
βh
)
B q
n−1
]
+2exp
(
9
4
βD
)[
exp
(
3
2
βh
)
C q
n−1
+exp
(
−3
2
βh
)
Dq
n−1
]
+2exp
(
1
4
βD
)[
exp
(
1
2
βh
)
E q
n−1 +exp
(
−1
2
βh
)]
, (2.14)
Q ′
B
= 25exp
(
25
4
βD
)[
exp
(
5
2
βh
)
Aq
n−1 +exp
(
−5
2
βh
)
B q
n−1
]
+9exp
(
9
4
βD
)[
exp
(
3
2
βh
)
C q
n−1
+exp
(
−3
2
βh
)
Dq
n−1
]
+exp
(
1
4
βD
)[
exp
(
1
2
βh
)
E q
n−1 +exp
(
−1
2
βh
)]
, (2.15)
Q0
B
= 4exp
(
25
4
βD
)[
exp
(
5
2
βh
)
Aq
n−1 +exp
(
−5
2
βh
)
B q
n−1
]
+4exp
(
9
4
βD
)[
exp
(
3
2
βh
)
C q
n−1
+exp
(
−3
2
βh
)
Dq
n−1
]
+4exp
(
1
4
βD
)[
exp
(
1
2
βh
)
E q
n−1 +exp
(
−1
2
βh
)]
. (2.16)
In order to determine the compensation temperature, one has to define the global magnetizationMT
of the model which is given by:
MT = MA+MB
2
. (2.17)
To really study the model in detail and single out the effect of the crystal-field and of the applied mag-
netic field on the magnetic properties of the model, we have also examined the thermal variations of the
33003-4
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
response functions, i.e., the susceptibilities, the specific heat and the internal energy defined respectively
by:
χTotal =χA+χB =
(
∂MA
∂h
)
h=0
+
(
∂MA
∂h
)
h=0
, (2.18)
C =−β2 ∂
2(−βF ′)
∂β2 , (2.19)
U
N |J | = −kBT 2 ∂
∂T
(
F ′
kBT
)
, (2.20)
where F ′
is the free energy of the model.
3. Definition of the critical temperature
The Curie temperature or the second-order transition temperature Tc is the temperature at which
both sublattice magnetizations and the global magnetization continuously go to zero. Tc separates the or-
dered ferrimagnetic phase (F) from the disordered paramagnetic phase (P). At Tc , one can obtain explicit
expressions of the recursion relations as follows:
for the spin-
1
2 ,
Zn = 1, (3.1)
and for the spin-
5
2 ,
An−1 = Bn−1 =
cosh
(
5βJ
4
)
cosh
(
βJ
4
) , Cn−1 = Dn−1 =
cosh
(
3βJ
4
)
cosh
(
βJ
4
) , En−1 = 1. (3.2)
In addition to the thermal variations of the order-parameters and the global magnetization of the model,
we also calculate and analyze the free energy F ′
of the model in order to identify the first-order transition
temperature Tt. Thus, using the definition of the free energy F ′ =−kBT ln(Z ) in the thermodynamic limit
(n →∞) and in order to introduce the recursion relations, we can rewrite the free energy as:
F ′/J =− 1
β′
[
q −1
2−q
lnF1 + 1
2−q
lnF2 + lnF3
]
, (3.3)
where β′ =βJ , F1 = gn−1(−1/2)/g q−1
n (−1/2), F2 = gn(−1/2)/g q−1
n−1 (−1/2) and F3 = Z /g q
n (−1/2).
After somemathematical manipulations, the free energy expression in terms of recursion relations is
explicitly given by:
F ′/J =− 1
β′
(
q −1
2−q
ln
{
exp
[
β
(
− J
4
+ h
2
)]
Z q−1
n +exp
[
β
(
J
4
− h
2
)]})
− 1
β′
{
ln
[
exp
(
β
h
2
)
Z q
n +exp
(
−βh
2
)]}
− 1
β′
(
1
2−q
ln
{
exp
[
β
(
−5J
4
+ 25D
4
+ 5h
2
)]
Aq−1
n−1 +exp
[
β
(
5J
4
+ 25D
4
− 5h
2
)]
B q−1
n−1
+exp
[
β
(
−3J
4
+ 9D
4
+ 3h
2
)]
C q−1
n−1 +exp
[
β
(
3J
4
+ 9D
4
− 3h
2
)]
Dq−1
n−1
+exp
[
β
(
− J
4
+ D
4
+ h
2
)]
E q−1
n−1 +exp
[
β
(
J
4
+ D
4
− h
2
)]})
. (3.4)
33003-5
M. Karimou et al.
We also investigate the compensation temperature Tcomp at which the global magnetization vanishes
while both sublattice magnetizations cancel each other. Tcomp is found by locating the crossing point
between the absolute values of sublattice magnetizations, i.e.,
|MA(Tcomp)| = |MB(Tcomp)|. (3.5)
Considering different definitions of the critical temperature, we can now investigate the thermal vari-
ations of the calculated thermodynamical quantities of interest and display the temperature phase dia-
grams of the model for q = 3,4,5 and 6.
4. Numerical results and discussions
In this section, we present and discuss the results we obtained for the thermal variations of the
order-parameters, the response functions, the internal energy and the temperature phase diagrams of the
model. We begin discussions with the ground-state phase diagram which is necessary for understanding
the obtained temperature phase diagrams.
4.1. Ground-state phase diagram
Before presenting the numerical results for the temperature dependence of magnetic properties of
the model, we first investigate the ground-state phase diagram. The ground-state structure of the model
can be represented by comparing the values of the energy H0 for different spin configurations which can
be expressed as:
H0 = Sσ− 1
q |J |
[
Dσ2 +h(S +σ)
]
. (4.1)
We only get eleven possible pairs of spins. Computational calculations of the corresponding energies
in the (h/q |J |, D/q |J |) plane yields the ground-state phase diagram displayed in figure 2. This diagram
shows some interesting features of the model, in particular, the existence of eight multicritical points
Figure 2. Ground-state phase diagram of the mixed spin-
1
2 and spin-
5
2 Ising ferrimagnetic model with the
same crystal-fieldD for the two sublattices in the (h/q|J |, D/q|J |) plane. There exist eleven stable phases.
Along theD/q|J |-axis and for all values of q , two hybrid phases may appear at the multicritical points B4
and B5.
33003-6
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
(B1,B2, · · · ,B8) and coexistence lines where the spin pair energy of some phases is the same. In the ab-
sence of the magnetic field, for all values of q and D/q |J |, MB shows five saturation values whereas
for MA, ± 1
2 is the only saturation value. Thus, we get the ferrimagnetic phases F(± 1
2 ,∓ 5
2 ), F(± 1
2 ,∓ 3
2 ),
F(± 1
2 ,∓ 1
2 ) and at the borders of these phases, two hybrid phases: F(± 1
2 ,∓1), F(± 1
2 ,∓2) at the multicriti-
cal points B4 and B5, respectively. These hybrid phases should correspond to cases where the sublattice
B is half-half covered by spins of the two neighboring phases. It is important to indicate that the ground-
state phase diagram is very useful because it helps to check the reliability of the theoretical results and
to classify different phase domains of the model for the temperature dependence phase diagrams.
4.2. Thermal variations of the order-parameters, the response functions and the in-
ternal energy
As it is explained above, thermal variations of the order-parameters, the response functions and the
internal energy for the present model were calculated in terms of recursion relations. Thermal varia-
tions of the order-parameters are crucial in obtaining the temperature dependence phase diagrams of the
model. Thus, when the magnetization curves go to zero continuously separating the ferrimagnetic phase
from the paramagnetic phase, one gets the second-order phase transition or Curie temperature, i.e., the
temperature at which magnetizations become zero. In the case of a jump in the magnetization curves
Figure 3. Sublattice magnetizations of the model as functions of the reduced temperature kBT /|J | when
q = 3,4 and 6 for various values of the crystal-field interactions D/|J |. Panel (a): Curves are displayed for
q = 3 and selected values of D/|J | indicated on the curves. Panel (b): curves are displayed for q = 4 and
selected values of D/|J | indicated on the curves. Panel (c): curves are displayed for q = 6 and selected
values of D/|J | indicated on the curves. For all values of q , the sublattice magnetization curves are all
continuous.
33003-7
M. Karimou et al.
followed by a discontinuity of the first derivative of the free-energy F ′
, one gets a first-order transition
temperature. Besides these two temperatures, there is another temperature referred to as compensation
temperature defined as the temperature where the global magnetization becomes zero prior to the criti-
cal temperature. Therefore, in order to identify transitions and compensation lines, one should study the
thermal behaviors of the considered thermodynamical quantities of the model. Now, we present some
results on the thermal behaviors of the order-parameters, the response functions and the internal energy
in the the absence of the magnetic field h when q = 3,4 and 6.
Figure 3 displays some thermal variations of the sublattice magnetizations M1/2 and M5/2 when
q = 3,4 and 6 for selected values of the crystal-field D/|J |. From panels (a) to (c), we have depicted
the thermal behaviors of sublattice magnetizations M1/2 and M5/2 as functions of the temperature for
selected values of D/|J | when q = 3,4 and 6. The results are in perfect agreement with the ground-state
phase diagram concerning the saturation values. Indeed,M1/2 increases from its unique saturation value
± 1
2 with increasing temperature whereasM5/2 shows five saturation values. The behaviors of the sublat-
tice magnetizationsM1/2 andM5/2 are quite similar. We notice that all the curves are continuous and the
Curie temperature Tc at which both magnetizations curves go to zero increases with the strength of the
crystal-field D/|J | and the coordination number q . Moreover, by comparing figure 3 to figure 4 of [18],
the sublattice magnetizations show similar thermal variations.
To explain in detail the results obtained in figure 3, we have investigated the thermal variations of
order-parameters, the corresponding response functions and the internal energy.
Figure 4. Thermal variations of sublattice magnetizations and corresponding susceptibilities are calcu-
lated for q = 3,4,6 and the reduced crystal-field D/|J | = 1 as shown from panel (a) to panel (c). Values
of the physical parameters considered for the system are indicated in different panels. Tc indicates the
second-order temperature.
33003-8
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
Thus, on the one hand, as shown in figure 4, we have displayed the thermal behaviors of the sub-
lattice magnetizations and corresponding susceptibilities when q = 3,4,6 and D/|J | = 1. In this figure,
one can notice that the model only exhibits the second-order phase transition, and the transition tem-
perature Tc at which the transition occurs increases with an increasing coordination number q . Here,
Tc separates the ferrimagnetic phase F(± 1
2 ,∓ 5
2 ) from the paramagnetic phase (P) and Tc/|J | = 2.1724 (re-
spectively Tc/|J | = 3.2575 and 5.1398) for q = 3 (respectively for q = 4 and 6). Also, one remarks that for
T → Tc , χ1/2 →−∞ whereas χ5/2 →+∞. For T > Tc , the susceptibility χ1/2 rapidly increases whereas
the susceptibility χ5/2 rapidly decreases when the temperature increases and is very far from the Curie
temperature Tc , χ1/2 → 0 and χ5/2 → 0.
On the other hand, to really confirm that the model only exhibits the second-order transition for all
values of q , we have plotted in figure 5 the temperature dependence variations of the specific heat and
the internal energy for various values of the crystal-field as indicated in the figure. Both the specific
heat and the internal energy rapidly increase with an increasing temperature and make a peak without
jump discontinuities at the same Tc . By increasing the strength of the crystal-field and the coordination
number, the Tc at which the transition occurs, increases and this is easily observed by comparing the
results from different panels of figure 5. The results obtained in this figure also confirm that the model
only presents second-order transition for all values of the coordination number q .
Figure 5. Thermal variations of the specific heat and internal energy are calculated for q = 3,4 and
selected values of the crystal-field D/|J | as shown in the figures from panel (a) to panel (d). Values of
the physical parameters considered for the system are indicated in different panels. Analysis of different
panels of this figure shows that the model only exhibits a second-order transition.
33003-9
M. Karimou et al.
Let us now discuss the thermal variations of sublattice magnetizations, the corresponding response
functions and the internal energy of the system in the presence of a longitudinal magnetic field h.
Figure 6 expresses the effects of an applied magnetic field h on the magnetic properties of the model
when q = 3 and D/|J | = 1.0 for selected values of h/|J |. From panel (a) to panel (b), the sublattice magne-
tizations and the global magnetization continuously fall from their saturation values to non-zero values
when the temperature increases. The remaining values of magnetizations are more important when the
value of the appliedmagnetic field h/|J | increases. Thus, one can observe that the system does not present
any transition when h/|J | , 0. It is important to indicate that in the case of h/|J | = 0, the model exhibits
the second-order transition at a Curie temperature Tc/|J | = 2.1724, where the two sublattice magnetiza-
tions and the global magnetization continuously go to zero after falling from their saturation values at
T = 0. In panels (c)–(e), we have displayed the temperature dependence of the total susceptibility χT , the
specific heatC and the internal energyU . One can see from these panels that the response functions and
the internal energy also indicate a second-order transition which occurs at the same Tc/|J | as in the case
Figure 6. Thermal behaviors of the sublattice magnetizations, the global magnetization, the correspond-
ing response functions and the internal energy of the model when D/|J | = 1.0 and q = 3 for selected
values of the magnetic field h/|J | as shown in different panels. From the analysis of different panels, one
can conclude that the model only shows temperature phase transition when h/|J | = 0. For h/|J | , 0, the
remaining magnetizations are more important when the value of the magnetic field h/|J | increases.
33003-10
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
Figure 7. Temperature variations of the response functions of themodel for selected values of the crystal-
field D/|J | as indicated on different curves illustrated when h/|J | = 0.5 and q = 3,4.
of h/|J | = 0. For h/|J | , 0 and T > Tc , the response functions exhibit a maximum and the height of the
maximum decreases when the value of the applied magnetic field increases.
To show the effect of D/|J | on the system properties for h/|J | , 0, we have illustrated in figure 7
the thermal variations of the response functions for some values of the system parameters: h/|J | = 0.5;
q = 3,4 and varying D/|J |. Considering the different panels of figure 7, one observes that the response
functions show interesting behaviors. Indeed, the two studied response functions globally show a max-
imum at a certain value of the temperature. This temperature increases with the coordination number
and the strength of the crystal-field. It is important to mention that the height of the maximum of the two
response functions also increases by increasing the strength of the crystal-field D/|J | but the opposite
holds when the coordination number q increases.
In order to investigate the low-temperature magnetic properties of the model, we plotted the sublat-
tice magnetizations and the global magnetization at kBT /|J | = 0.05 for selected values of the crystal-field
as functions of the field h as shown in figure 8. In figure 8 (a) where D/|J | = −1 and q = 3, M1/2 shows
two step-like magnetization plateaus (M1/2 =− 1
2 , 1
2 ) whereas M5/2 and MT respectively show three and
four step-like magnetization plateaus (M5/2 = 1
2 , 3
2 , 5
2 ) and (MT = 0, 1
2 ,1, 3
2 ). Also, from figure 8 (b) where
D/|J | = 0 and q = 3, only M1/2 and MT present two step-like magnetization plateaus (M1/2 =− 1
2 , 1
2 ) and
(MT = 1, 3
2 ). The obtained results are consistent with the ground-state phase diagram displayed in fig-
ure 2. We also investigated the global magnetization as a function of the temperature and obtained some
33003-11
M. Karimou et al.
Figure 8. M1/2, M5/2 and MT plotted as functions of the magnetic field h for selected values of the
crystal-field when q = 3 as indicated in different panels.
compensation types of Ising model. Figure 9 shows temperature dependencies of the global magnetiza-
tion MT for selected values of the crystal-field when q = 3. As seen in figure 9, the model exhibits five
types of compensation behaviors, namely Q-, R-, S-, L- and P-type compensation behaviors as classified in
the extended Néel nomenclature [39].
4.3. Finite-temperature phase diagrams
Considering all the above calculations, we can illustrate the temperature phase diagrams of themodel.
So in figure 10, we have constructed the phase diagrams of the system in the (D/|J |, kBT /|J |) plane in the
absence of a magnetic field h when q = 3,4,5 and 6. In different phase diagrams, the solid line indicates
the second-order transition line. Two filled triangles indicate two multicritical points B4 and B5 found in
the ground-state phase diagram displayed in figure 2.
From this figure, some interesting properties of the system are singled out. Indeed, for all values of
the coordination number q , from panel (a) to panel (d) where q = 3,4,5 and 6, respectively, the transition
lines are only of the second-order separating the ferrimagnetic phase (F) which is a mixture of five differ-
ent ferrimagnetic phases from the paramagnetic phase (P) and become constant for D/|J | < −q/4. One
can observe that: (1) When D/|J | > −q/8, the second-order phase transition turns from ferrimagnetic
phase F(± 1
2 ,∓ 5
2 ) to a disordered paramagnetic phase P. (2) For −q/4 < D/|J | < −q/8, the second-order
phase transition is from the ferrimagnetic F(± 1
2 ,∓ 3
2 ) to the paramagnetic phase P. (3)WhenD/|J | < −q/4,
the second-order phase transition is from the ferrimagnetic phase F(± 1
2 ,∓ 1
2 ) to the paramagnetic phase
33003-12
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
Figure 9. MT as a function of the temperature for selected values of the crystal-field when q = 3 as
indicated in different panels. The model shows the Q-, R-, S-, L- and P-types of compensation behaviors as
classified in the extended Néel nomenclature.
P. (4) For D/|J | = −q/8 respectively D/|J | = −q/4, the second-order transition phase is from the hybrid
phase F(± 1
2 ,∓2) respectively the hybrid phase F(± 1
2 ,∓1) to the paramagnetic phase P.
It is important to mention that figure 10 presents some resemblances concerning the second-order
transition lines with figure 3 of [18]. Moreover, by increasing the value of the coordination number q , the
ferrimagnetic domain F becomes important.
5. Conclusion
In this work, the study of the mixed spin-
1
2 and spin-
5
2 Ising ferrimagnetic model on the Bethe lattice
in the presence of a longitudinal magnetic field is undertaken via exact recursion equations method. All
the thermodynamical quantities of interest are calculated as functions of recursion relations.
The ground-state phase diagram of the model is displayed as shown in figure 2. From this phase
diagram, we have found eleven existing and stable phases and along the D/q |J |-axis, two particular hy-
brid phases appear at the two multicritical points B4 and B5. The ground-state phase diagram is con-
sidered and used as a guide in obtaining different temperature phase diagrams. In the presence and in
the absence of a longitudinal magnetic field h, we investigated thermal variations of sublattice magneti-
zations, global magnetization, the corresponding response functions and the internal energy as seen in
figures 3–9. From these figures, the order-parameters in most cases showed a usual decay with thermal
fluctuations. By using thermal behaviors of the considered order-parameters, and by analysing the cor-
responding response functions and the internal energy, the nature of different phase transitions encoun-
tered is identified. This enables us to construct and to discuss in detail different temperature dependence
phase diagrams in the case of equal crystal-field interactions as shown in figure 10. The model shows
rich physical properties, namely the second-order transition and multicritical points for all values of the
crystal-field interactions and for all values of the coordination number q .
33003-13
M. Karimou et al.
Figure 10. Temperature phase diagrams of themodel in the (D/|J |, kBT /|J |) plane. The solid line indicates
the second-order transition line. Panel (a): q = 3; panel (b): q = 4; panel (c): q = 5 and panel (d): q = 6.
Here, the model only presents second-order transition for all values of q . The multicritical points B4 and
B5 which respectively indicate the positions of the hybrid phases F(± 1
2 ,∓1) and F(± 1
2 ,∓2), respectively
separate the ferrimagnetic phases F(± 1
2 ,∓ 1
2 ), F(± 1
2 ,∓ 3
2 ) and F(± 1
2 ,∓ 5
2 ).
Finally, it is useful to mention that different results found here are compared to those reported in
some references, and some topological similarities are shown, especially with those found in [18] where
the same model is investigated by means of the effective-field theory with correlation. However, during
our investigation we have not found any compensation temperature.
References
1. Thompson C.J., Mathematical Statistical Mechanics, Princeton University Press, New Jersey, 1992.
2. Strečka J., Jaščur M., Acta Phys. Slovaca, 2015, 65, 235, and references therein.
3. White R.M., Science, 1985, 229, 11; doi:10.1126/science.229.4708.11.
4. Wood R., Understanding Magnetism, Tab Books Inc, Blue Ridge Summit, PA, 1988.
5. Köster E., J. Magn. Magn. Mater., 1993, 120, 1; doi:10.1016/0304-8853(93)91274-B.
6. Lueck L.B., Gilson R.G., J. Magn. Magn. Mater., 1990, 88, 227; doi:10.1016/S0304-8853(97)90032-9.
7. Molecular Magnetism: New Magnetic Materials, Itoh K., Kinoshita M. (Eds.), Kodansha, Tokyo, 2000.
8. Molecular Magnets. Recent Highlights, Linert W., Verdaguer M. (Eds.), Springer-Verlag, Wien, 2003.
9. Gatteschi D., Adv. Mater., 1994, 6, 635; doi:10.1002/adma.19940060903.
10. Miller J.S., Epstein A.J., Chem. Eng. News, 1995, 73, 30; doi:10.1021/cen-v073n040.p030.
11. Kahn O., Molecular Magnetism, VCH, New York, 1993.
12. Bobák A., Physica A, 1998, 258, 140; doi:10.1016/S0378-4371(98)00233-7.
13. Benyoussef A., El Kenz A., Kaneyoshi T., J. Magn. Magn. Mater., 1994, 131, 179; doi:10.1016/0304-8853(94)90026-4.
14. Bobák A., Jurc̆išin M., Physica A, 1997, 240, 647; doi:10.1016/S0378-4371(97)00044-7.
15. De Oliveira D.C., Silva A.A.P., de Albuquerque D.F., de Arruda A.S., Physica A, 2007, 386, 205;
doi:10.1016/j.physa.2007.07.073.
33003-14
http://dx.doi.org/10.1126/science.229.4708.11
http://dx.doi.org/10.1016/0304-8853(93)91274-B
http://dx.doi.org/10.1016/S0304-8853(97)90032-9
http://dx.doi.org/10.1002/adma.19940060903
http://dx.doi.org/10.1021/cen-v073n040.p030
http://dx.doi.org/10.1016/S0378-4371(98)00233-7
http://dx.doi.org/10.1016/0304-8853(94)90026-4
http://dx.doi.org/10.1016/S0378-4371(97)00044-7
http://dx.doi.org/10.1016/j.physa.2007.07.073
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system
16. Kaneyoshi T., Physica A, 1994, 205, 677; doi:10.1016/0378-4371(94)90229-1.
17. Guo K.T., Xiang S.H., Xu H.Y., Li X.L., Quantum Inf. Process., 2014, 13, 1511; doi:10.1007/s11128-014-0745-7.
18. Deviren B., Keskin M., Canko M.O., Physica A, 2009, 388, 1835; doi:10.1016/j.physa.2009.01.032.
19. Da Cruz Filho J.S., Godoy M., de Arruda A.S., Physica A, 2013, 392, 6247; doi:10.1016/j.physa.2013.08.007.
20. Miao H., Wei G., Geng J., J. Magn. Magn. Mater., 2009, 321, 4139; doi:10.1016/j.jmmm.2009.08.018.
21. Mohamad H.K., Domashevskaya E.P., Klinskikh A.F., Physica A, 2009, 388, 4713; doi:10.1016/j.physa.2009.08.014.
22. Mohamad H.K., Int. J. Adv. Res., 2014, 2, No. 9, 442.
23. Kaneyoshi T., Chen J.C., J. Magn. Magn. Mater., 1991, 98, 201; doi:10.1016/0304-8853(91)90444-F.
24. Quadros S.G.A., Salinas S.R., Physica A, 1994, 206, 479; doi:10.1016/0378-4371(94)90319-0.
25. El Bouziani M., Gaye A., Jellal A., Physica A, 2013, 392, 689; doi:10.1016/j.physa.2012.10.007.
26. Buendia G.M., Liendo J.A., J. Phys.: Condens. Matter, 1997, 9, 5439; doi:10.1088/0953-8984/9/25/011.
27. Godoy M., Figueiredo W., Phys. Rev. E, 2002, 66, 036131; doi:10.1103/PhysRevE.66.036131.
28. Cambui D.S., Arruda A.S., Godoy M., Int. J. Mod. Phys. C, 2012, 23, 1240015; doi:10.1142/S0129183112400153.
29. Feraoun A., Zaim A., Kerouad M., Physica B, 2014, 445, 74; doi:10.1016/j.physb.2014.03.071.
30. Z̆ukovic̆ M., Bobák A., J. Magn. Magn. Mater., 2010, 322, 2868; doi:10.1016/j.jmmm.2010.04.043.
31. Yessoufou R.A., Bekhechi S., Hontinfinde F., Eur. Phys. J. B, 2011, 81, 137; doi:10.1140/epjb/e2011-10825-7.
32. Kple J., Yessoufou R.A., Hontinfinde F., Afr. Rev. Phys., 2012, 7, 319.
33. Yigit A., Albayrak E., Chinese Phys. B, 2012, 21, 020511; doi:10.1088/1674-1056/21/2/020511.
34. Ekiz C., Phys. Lett. A, 2007, 367, 483; doi:10.1016/j.physleta.2007.03.038.
35. Albayrak E., Yigit A., Phys. Lett. A, 2006, 353, 121; doi:10.1016/j.physleta.2005.12.077.
36. KarimouM., Yessoufou R., Hontinfinde F., Int. J. Mod. Phys. B, 2015, 29, 1550194; doi:10.1142/S0217979215501945.
37. Ekiz C., J. Magn. Magn. Mater., 2006, 307, 139; doi:10.1016/j.jmmm.2006.03.059.
38. Ekiz C., Commun. Theor. Phys., 2009, 52, 539; doi:10.1088/0253-6102/52/3/30.
39. Ekiz C., Strečka J., Jašc̆ur M., Cent. Eur. J. Phys., 2009, 7, 509; doi:10.2478/s11534-009-0043-7, and references
therein.
Вивчення змiшаної спiн-
1
2 i спiн-
5
2 моделi Iзiнга на гратцi
Бете пiд дiєю магнiтного поля
М. Карiму1, Р.А. Єссуфу1,2, Т.Д. Оке1,2, A. Кпандону1,3, Ф. Онтiнфiнде1,2
1 Iнститут математики i фiзичних наук (IMSP), Республiка Бенiн
2 Фiзичний факультет, Унiверситет м. Абомей-Калавi, Республiка Бенiн
3 Унiверситет м. Наттiнгу, Еколь нормаль сюпер’єр, Республiка Бенiн
Дослiджується змiшана спiн- 1
2 i спiн- 5
2 модель на гратцi Бете в присутностi магнiтного поля h, використо-
вуючи метод рекурсивних спiввiдношень. Побудована фазова дiаграма основного стану, яка може бути
використана для дослiдження цiкавих областей температурної фазової дiаграми моделi. Для того, щоб
прокласифiкувати природу фазового переходу та отримати вiдповiднi температури, детально дослiдже-
но параметри порядку, вiдповiднi функцiї вiдгуку i внутрiшню енергiю. Так, при вiдсутностi магнiтного
поля, температурнi фазовi дiаграми є представленi для випадку однакового кристалiчного поля на пло-
щинi (kBT /|J |,D/|J |), коли q = 3,4,5 i 6. Модель демонструє тiльки фазовий перехiд другого роду для
вiдповiдних значень фiзичних параметрiв моделi.
Ключовi слова: магнiтнi системи, температурнi змiни, фазовi дiаграми, магнiтне поле, перехiд другого
роду
33003-15
http://dx.doi.org/10.1016/0378-4371(94)90229-1
http://dx.doi.org/10.1007/s11128-014-0745-7
http://dx.doi.org/10.1016/j.physa.2009.01.032
http://dx.doi.org/10.1016/j.physa.2013.08.007
http://dx.doi.org/10.1016/j.jmmm.2009.08.018
http://dx.doi.org/10.1016/j.physa.2009.08.014
http://dx.doi.org/10.1016/0304-8853(91)90444-F
http://dx.doi.org/10.1016/0378-4371(94)90319-0
http://dx.doi.org/10.1016/j.physa.2012.10.007
http://dx.doi.org/10.1088/0953-8984/9/25/011
http://dx.doi.org/10.1103/PhysRevE.66.036131
http://dx.doi.org/10.1142/S0129183112400153
http://dx.doi.org/10.1016/j.physb.2014.03.071
http://dx.doi.org/10.1016/j.jmmm.2010.04.043
http://dx.doi.org/10.1140/epjb/e2011-10825-7
http://dx.doi.org/10.1088/1674-1056/21/2/020511
http://dx.doi.org/10.1016/j.physleta.2007.03.038
http://dx.doi.org/10.1016/j.physleta.2005.12.077
http://dx.doi.org/10.1142/S0217979215501945
http://dx.doi.org/10.1016/j.jmmm.2006.03.059
http://dx.doi.org/10.1088/0253-6102/52/3/30
http://dx.doi.org/10.2478/s11534-009-0043-7
Introduction
Description of the model on the Bethe lattice
Definition of the critical temperature
Numerical results and discussions
Ground-state phase diagram
Thermal variations of the order-parameters, the response functions and the internal energy
Finite-temperature phase diagrams
Conclusion
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