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2 Ising system on the Bethe lattice
Physics Letters A 353 (2006) 121–129 www.elsevier.com/locate/pla
Mixed spin-3/2 and spin-5/2 Ising system on the Bethe lattice Erhan Albayrak ∗ , Ali Yigit Erciyes University, Department of Physics, 38039 Kayseri, Turkey Received 2 November 2005; received in revised form 24 December 2005; accepted 26 December 2005 Available online 4 January 2006 Communicated by R. Wu
Abstract In order to study the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system, we have used the exact recursion relations on the Bethe lattice. The system was studied for the coordination numbers with q = 3, 4, 5 and 6, and the obtained phase diagrams are illustrated on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J |, the reduced crystal field of the sublattice with spin-5/2, and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J |, the reduced crystal field of the sublattice with spin-3/2, for q = 3 only, since the cases corresponding to q = 4, 5 and 6 reproduce results similar to the case for q = 3. In addition we have also presented the phase diagram with equal strengths of the crystal fields for q = 3, 4, 5 and 6. Besides the second- and first-order phase transitions, the system also exhibits compensation temperatures for appropriate values of the crystal fields. In this mixed spin system while the second-order phase transition lines never cut the reduced crystal field axes as in the single spin type spin-3/2 and spin-5/2 Ising models separately, the first-order phase transition lines never connect to the second-order phase transition lines and they end at the critical points, therefore the system does not give any tricritical points. In addition to this, this mixed-spin model exhibits one or two compensation temperatures depending on the values of the crystal fields, as a result the compensation temperature lines show reentrant behavior. © 2006 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 05.70.Fh; 64.60.Cn; 75.10.Hk Keywords: Mixed spin; Bethe lattice; Reentrant behavior; Compensation temperature
1. Introduction Intense experimental studies have accumulated in the recent years in the area of molecular-based magnetic materials, and the magnetic properties, i.e. the molecular magnetism, have become an active area of scientific interest. These materials include bimetallic molecular-based magnetic materials in which two kinds of magnetic atoms alternate regularly have exhibited ferrimagnetic properties and therefore they are well interpreted by the use of mixed-spin systems [1]. That is why, there has been a great deal of interest in studying two sublattice mixed spin-σ and spin-S (σ = S) Ising ferrimagnetic (J < 0) or Ising ferromagnetic systems (J > 0). These ferrimagnetic materials, as mentioned above the molecular based magnetic materials, may exhibit compensation temperatures, i.e. the temperature at which the net magnetization goes to zero, in contrast to Ising models containing only one type of spins and could be handled theoretically by the use of mixed spins. Many combinations of the mixed-spin Ising ferrimagnetic systems are possible, but the less studied ferrimagnetic materials includes only the half-integer mixed spin systems. Maybe the simplest of this includes spin-1/2 and spin-3/2 system which was studied within the framework of the effective-field theory with correlations on the honeycomb lattice [2], on a square lattice by using the effective-field theory [3,4], again on a square lattice with a Monte Carlo algorithm [5] and on the Bethe lattice [6] with the coordination numbers q = 3, 4, 5 and 6, but no interesting behaviors was observed since this mixed spin system only exhibits second-order phase transitions. The next * Corresponding author.
E-mail address: [email protected] (E. Albayrak). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.077
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possible mixing of the half integer spins is the mixing of spin-1/2 and spin-5/2, unfortunately this system was not studied as far as in our knowledge but in any case even if we do not present the phase diagrams belonging to this mixed system in here we have studied it in search of any interesting critical behavior but we only find that the system gives second-order phase transition temperatures as in the case of mixed spin-1/2 and spin-3/2. Therefore, the next possibility includes the mixing of spin-3/2 and spin-5/2, which was studied using the effective field theory with correlations with interlayer coupling [7]. It should also be mentioned that this mixed-spin system maybe fictitious for at least up to this date, since as far as our search concerns we were not able to find a real system containing spin-3/2 and spin-5/2, but at least it does deserve to be studied for theoretical reasons. Besides, Co/Cu (001) system seem to behave Ising like system, therefore, it is justified to use the Ising type model to study the statistical behaviors of this mixed-spin system [8]. As a result the purpose of the present work is to obtain all the thermodynamic functions of interest in terms of the recursion relations and then is to study the thermal variations of the sublattice magnetizations and the quadrupole moments for the sublattices with spin-3/2 and with spin-5/2 to obtain the phase diagrams on the Bethe lattice for q = 3, 4, 5 and 6 and it was found that the system exhibits a very rich phase diagram as would be seen in the upcoming sections. The remaining part of the Letter is constructed as follows. In Section 2, the formulation of the problem is given and the exact expressions for the order-parameters, dipolar and the quadrupolar moments for the sublattices with spin-3/2 and spin-5/2, are obtained in terms of the recursion relations. In Section 3, the definitions of the critical temperatures and their formulations in terms of the recursion relations, the exact expressions for the Curie or the second-order phase transition temperature, the free energy, which is used to find the first-order phase transition temperatures, and the definition of the compensation temperature, are given. Finally, in the last section besides the phase diagram on the (kTc /|J |, D/|J |) plane, i.e. DA = DB = D, for q = 3, 4, 5 and 6, we have also presented the phase diagrams on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J | for q = 3 only, and we have concluded this section with a brief summary and a discussion. 2. The formulation in terms of the exact recursion relations We consider the mixed-spin Blume–Capel Ising ferrimagnetic system, J < 0, whose Hamiltonian includes the bilinear interaction of the sublattices with spin-3/2 and spin-5/2 and the crystal fields DA and DB acting on that sites, respectively, and it is given as σi sj − DA σi2 − DB sj2 , H = −J (1) i,j
i
j
where each σi located at site i is a spin-3/2 with four discrete spin values, i.e. ±3/2 and ±1/2 and each sj located at site j is a spin-5/2 and can take six discrete spin values ±5/2, ±3/2, and ±1/2. For this mixed spin-3/2 and spin-5/2 system, the Bethe lattice is arranged in such a way that the central spin is spin-3/2, σ0 , the next generation is spin-5/2, s1 , and the next generation is again spin-3/2, σ2 , and so on to infinity. Since the Bethe lattice is an infinite Cayley tree, a connected graph without circuits, one can avoid introducing the boundary conditions by considering the system in the thermodynamic limit, when n → ∞, n is the number of the shells from root, i.e. the central site with σ0 , to the boundary. The Bethe lattice gets its name from the fact that its partition function is exactly that of an Ising model in the Bethe approximation [9]. We should also point out that the cluster variation method in the pair approximation studies on regular lattices yield results that are exact for the same model on the Bethe lattice [10,11]. In Eq. (1), the first sum runs over all the nearest-neighbor pairs, i.e. between the sites with spin-3/2 and spin-5/2, the second sum runs over all the spin-3/2 sites and the last summation runs over the sites with spin-5/2. In order to obtain the formulation of this mixed-spin system, it is the common idea to start with the partition function, therefore, the partition function is given as P (Spc) = exp β J σi sj + DA σi2 + DB sj2 , Z= e−βH = (2) {σ,s}
Spc
i,j
i
j
where P (Spc) is considered as an unnormalized probability distribution over the spin configurations, Spc ≡ {σ, s}, etc. and σi and sj refer to the spin values at site i and site j , respectively. If the Bethe lattice is cut in some central point deep inside with a spin σ0 , then it splits up into q identical branches whose number depends on the number of nearest-neighbors or the coordination numbers. Each of these branches is a rooted tree at the central-spin σ0 . This implies that P ({σ0 }), Spc = {σ0 }, is the spin-configuration with the spin value σ0 at the central site, may be written as
(k) P {σ0 } = exp βDA σ02 × , Qn σ0 s1 q
(3)
k=1
since the central-spin is spin-3/2, when Bethe lattice is cut at that site the central spin σ0 can interact only with the crystal field (k) (k) DA and where {s1 } denotes the spin-configurations of the kth branch of the Bethe lattice starting with s1 other than the central spin σ0 , the suffix n expresses the fact that the subtree has n-shells, i.e. the number of steps from the root to the boundary sites. In
E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129
Eq. (3) the function Qn in the product is given as (k) (k) 2 Qn σ0 s1 = exp β J σ0 s1 + J σi sj + DA σ i + DB sj2 , i,j
i
123
(4)
j
where σi and sj refer to the spins of the site i and j of the subtree (other than the central spin σ0 , which is a spin-3/2). Site 1 with spin-5/2 is the site next to the central cite σ0 . The first summation in Eq. (4) is over all edges of the subtree other than the edge (0, 1) and the summation over i is over all sites with spin-3/2 other than the central site. Now if the subtree, e.g. the upper subtree, is cut at the site 1 next to 0, then it also decomposes into q pieces: one being “trunk” (0, 1) and the rest are the identical branches. Each of these branches is a subtree like the original, but with n − 1 shells and q − 1 neighbors. Thus q−1 (k) (k) 2 (k) (l) (k) Qn−1 s1 τ2 , = exp β J σ0 s1 + DB s1 × Qn σ0 s1
(5)
l=1 (l)
(k)
where {τ2 } denotes the spin-configurations (other than s1 ) on the lth branch of the subtree. The prime on the function Qn−1 in the product is used to distinguish the sublattices with spin-3/2 and spin-5/2 since they have common spin values ±1/2. If now the upper subtree is cut at site 2 again with spin-3/2 next to site 1, then it again decomposes into q pieces: one again being the next trunk (1, 2) and the rest are the identical branches, again with n − 1 shells and q − 1 neighbors. Therefore Q n−1
(k) (l) (l) s1 τ2 = exp β J s1 σ2
(l) 2
× + DA σ 2
q−1
(l) (m) , Qn−2 τ2 r3
(6)
m=1
where {r3(m) } indicates all the spins (other than τ2(l) ) on the next mth branch of the upper sub-subtree. As a result, one takes n steps from root to the boundary on the Bethe lattice and in the thermodynamic limit n → ∞ should be taken. Adding the shells in this manner, the Bethe lattice is set up in such a way that the central-spin is spin-3/2 with q-neighbors of spin-5/2, the next generation with spin-5/2 has q − 1 neighbors with spin-3/2, and the next generation with spin-3/2 has q − 1 neighbors of spin-5/2, and so on to infinity. It should be mentioned that the choice of the central spin has no effect on the system, i.e. the behaviors of the system parameters do not change whether one chooses the spin-3/2 or spin-5/2 as the central spins Ref. [11]. Now we define, Qn σ0 {s1 } , g n σ0 = (7) {s1 }
where the summation over s1 , chosen to be spin-5/2 with spin values ±5/2, ±3/2 and ±1/2, means that there is a certain probability for the spin σ0 to be in a certain spin value, it is a spin-3/2 with spin values ±3/2 and ±1/2, as a result of its interaction with the external parameters and with its q nearest-neighbors of s1 . Now we are ready to obtain the exact recursion relations, in order to do so let us start with Eq. (6) and sum over all the discrete spin values σ2 , which is a spin-3/2, thus we obtain
q−1 (s1 ) = exp β J σ2 s1 + DA σ22 gn−2 (σ2 ) . gn−1 (8) σ2
As we know σ2 can take four discrete values ±3/2 and ±1/2, since it is a spin-3/2, therefore summing over these spin values and then inserting the spin values for spin-5/2, with six discrete spin values ±5/2, ±3/2 and ±1/2, one can obtain six different gn−1 (s1 ) for each of the spin values s1 . Therefore, when s1 = ±5/2
q−1 (±5/2) = exp β ±5J σ2 /2 + DA σ22 gn−2 (σ2 ) gn−1 σ2
q−1
q−1 = eβ(±15J /4+9DA /4) gn−2 (+3/2) + eβ(∓15J /4+9DA /4) gn−2 (−3/2)
q−1
q−1 + eβ(±5J /4+DA /4) gn−2 (+1/2) + eβ(∓5J /4+DA /4) gn−2 (−1/2) ,
(9)
and when s1 = ±3/2 gn−1 (±3/2) =
q−1 exp β ±3J σ2 /2 + DA σ22 gn−2 (σ2 )
σ2
q−1
q−1
q−1 = eβ(±9J /4+9DA /4) gn−2 (+3/2) + eβ(∓9J /4+9DA /4) gn−2 (−3/2) + eβ(±3J /4+DA /4) gn−2 (+1/2)
q−1 + eβ(∓3J /4+DA /4) gn−2 (−1/2) , (10) and for the last two values of s1 = ±1/2, we get
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E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129 gn−1 (±1/2) =
q−1 exp β ±J σ2 /2 + DA σ22 gn−2 (σ2 )
σ2
q−1
q−1
q−1 = eβ(±3J /4+9DA /4) gn−2 (+3/2) + eβ(∓3J /4+9DA /4) gn−2 (−3/2) + eβ(±J /4+DA /4) gn−2 (+1/2)
q−1 + eβ(∓J /4+DA /4) gn−2 (−1/2) . (11) Now in order to calculate the gn (σ0 ), we take Eq. (5) together with Eq. (7), summing over all s1 , which is a spin-5/2, one can obtain
q−1 exp β J σ0 s1 + DB s12 gn−1 (s1 ) . gn (σ0 ) = (12) s1
Since σ0 can take the spin values ±3/2 and ±1/2, and s1 can take the values ±5/2, ±3/2, and ±1/2, therefore, one can obtain four different gn (σ0 ) for four possible values of σ0 , then for σ0 = ±3/2
q−1 exp β ±3J s1 /2 + DB s12 gn−1 (s1 ) gn (±3/2) = s1
q−1
q−1 = eβ(±15J /4+25DB /4) gn−1 (+5/2) + eβ(∓15J /4+25DB /4) gn−1 (−5/2)
q−1
q−1
q−1 + eβ(±9J /4+9DB /4) gn−1 (+3/2) + eβ(∓9J /4+9DB /4) gn−1 (−3/2) + eβ(±3J /4+DB /4) gn−1 (+1/2)
q−1 + eβ(∓3J /4+DB /4) gn−1 (−1/2) , (13) and finally for σ0 = ±1/2, we obtain
q−1 exp β ±J s1 /2 + DB s12 gn−1 (s1 ) gn (±1/2) = s1
q−1
q−1
q−1 = eβ(±5J /4+25DB /4) gn−1 (+5/2) + eβ(∓5J /4+25DB /4) gn−1 (−5/2) + eβ(±3J /4+9DB /4) gn−1 (+3/2)
q−1
q−1
q−1 (−3/2) + eβ(±J /4+DB /4) gn−1 (+1/2) + eβ(∓J /4+DB /4) gn−1 (−1/2) . + eβ(∓3J /4+9DB /4) gn−1 (14) Now we are ready to introduce the recursion relations as the ratios of the gn functions for spin-3/2 as Xn =
gn (+3/2) , gn (−1/2)
Yn =
gn (−3/2) , gn (−1/2)
Zn =
gn (+1/2) gn (−1/2)
(15)
and for spin-5/2 as the ratios of gn−1 functions
An−1 = Dn−1 =
gn−1 (+5/2) gn−1 (−1/2) gn−1 (−3/2) gn−1 (−1/2)
,
Bn−1 =
,
En−1 =
gn−1 (−5/2) gn−1 (−1/2) gn−1 (+1/2) gn−1 (−1/2)
,
Cn−1 =
gn−1 (+3/2) gn−1 (−1/2)
,
.
(16)
One can easily obtain the explicit expressions of the recursion relations by using Eqs. (9)–(11) and (13)–(16), unfortunately, their expressions are too long to be given in here. These recursion relations may not have any physical meaning, but they do reflect the critical behaviors of the system. In order to obtain the order-parameters in terms of these recursion relations, i.e. M3/2 and Q3/2 for the sublattice with spin-3/2 and, M5/2 and Q5/2 for the sublattice with spin-5/2, the magnetizations and the quadrupolar order-parameters, respectively, we use the usual definitions given for the central-spin as σ0 P {σ0 } , Q = Z −1 σ02 P {σ0 } , M = Z −1 (17) {σ0 }
{σ0 }
where Z and P ({σ0 }) are given by Eqs. (2) and (3), respectively, and σ0 is the central-spin. Since the central-spin is chosen to be spin-3/2, the magnetization or the dipole moment for the central-spin can be obtained from Eqs. (2), (3) and (17) as
q σ0 exp βDA σ02 gn (σ0 ) , M3/2 = Z −1 (18) σ0
which is easily expressed in terms of the recursion relations by using Eqs. (2), (13)–(15) and (17) as
M3/2 =
q
q
q
3/2e9β dA /4 (Xn − Yn ) + 1/2eβ dA /4 (Zn − 1) , q q q e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1)
(19)
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and similarly the quadrupolar order parameter for the central-spin, i.e. spin-3/2, is obtained from the second of Eq. (17) or replacing σ0 with σ02 in Eq. (18) in terms of the recursion relations as
Q3/2 =
q
q
q
9/4e9β dA /4 (Xn + Yn ) + 1/4eβ dA /4 (Zn + 1) . q q q e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1)
(20)
In order to obtain the order-parameters for the sublattice with spin-5/2 starting from the first shell of the Bethe lattice, one either carries out the whole calculation by choosing the spin-5/2 as the central-spin or simply by using the definitions given in Eq. (17) and noting that deep inside the Bethe lattice all the sites with the same kinds of spins are equivalent, therefore, one may easily adapt the equations to obtain the formulations of the order-parameters for the spin-5/2 directly, so the magnetization is calculated as
M5/2 =
q
q
q
q
q
5/2e25β dB /4 (Al − Bl ) + 3/2e9β dB /4 (Cl − Dl ) + 1/2eβ dB /4 (El − 1)
q
q
q
q
q
e25β dB /4 (Al + Bl ) + e9β dB /4 (Cl + Dl ) + eβ dB /4 (El + 1)
(21)
,
and the quadrupolar order parameter for spin-5/2 is given as
Q5/2 =
q
q
q
q
q
25/4e25β dB /4 (Al + Bl ) + 9/4e9β dB /4 (Cl + Dl ) + 1/4eβ dB /4 (El + 1)
q
q
q
q
q
e25β dB /4 (Al + Bl ) + e9β dB /4 (Cl + Dl ) + eβ dB /4 (El + 1)
,
(22)
where l is equal to n − 1. The thermal variations of the order-parameters now could be studied in terms of the recursion relations. In order to do so, first one needs to obtain the recursion relations in terms of the system parameters, i.e. the reduced crystal fields DA /|J | and DB /|J | for the sublattices with spin-3/2 and spin-5/2, respectively, and the coordination number q, by varying the temperature and then inserting the obtained results in Eqs. (19)–(22) the thermal variations of the order-parameters could be studied in detail. It should be mentioned that the numerical calculations are carried out by using an iteration procedure for given initial values of the recursion relations and for each given values of the reduced crystal fields and the coordination numbers. The phase diagrams of the system is then obtained by studying the thermal variations of the order-parameters and are going to be illustrated for q = 3 (for two cases; i.e. DA = DB and DA = DB ), 4, 5 and 6 (for only DA = DB ) in the last section. 3. The definitions and the formulations of the critical temperatures The thermal behaviors of the order-parameters play a crucial role in obtaining the phase diagrams of the system: when the magnetization curves go to zero continuously separating the ferrimagnetic phase from the paramagnetic phase is the second-order phase transition or the Curie temperature, i.e. the temperature at which magnetizations become zero, and when there is a magnetization jump to either zero or to another value is the first-order phase transition temperature, i.e. the temperature at which the magnetization jump occurs. Besides these two critical temperatures there is another temperature called as the compensation temperature and is defined as the temperature when the net magnetization becomes zero or the sublattice magnetization curves cut each other before the critical temperatures. Therefore, in order to find the second- and first-order phase transition temperatures to obtain the phase diagrams and also the lines of the compensation temperatures one has to study the thermal behaviors of the order-parameters. Thus we have studied the thermal variations of the order-parameters for the mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system for q = 3, 4, 5 and 6 in order to obtain the phase diagrams on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J |, besides the phase diagrams with equal strengths of the crystal fields. The second-order phase transition temperature, Tc , is the temperature at which either of the sublattice magnetizations become zero continuously, separating the ferrimagnetic phase from the paramagnetic phase. Therefore, by using the expressions for the magnetizations given in Eqs. (19) and (21) one can obtain the exact formulation of the second-order phase transition temperatures by setting M3/2 or M5/2 separately to zero: q q q M3/2 = 3/2e9β dA /4 Xn − Yn + 1/2eβ dA /4 Zn − 1 = 0 (23) and q q q q q M5/2 = 5/2e25β dB /4 An−1 − Bn−1 + 3/2e9β dB /4 Cn−1 − Dn−1 + 1/2eβ dB /4 En−1 − 1 = 0.
(24)
It is clear that Eq. (23) for the sublattice with spin-3/2 has a simple solution, i.e. Xn = Yn and Zn = 1, which means that at the second-order phase transition temperature the probability of spins being up and spins being down has to be equal to each other. This implies that at Tc the condition gn (+3/2) = gn (−3/2) and gn (+1/2) = gn (−1/2) must be satisfied. Similarly, the last equation for the sublattice with spin-5/2 also has a simple solution and it is given as An−1 = Bn−1 , Cn−1 = Dn−1 and En−1 = 1 which also (+5/2) = gn−1 (−5/2), gn−1 (+3/2) = gn−1 (−3/2) and gn−1 (+1/2) = gn−1 (−1/2) must also be satisfied. implies that at Tc , gn−1 It should be mentioned that the latter condition is readily obtained from the first condition as maybe easily seen from the recursion relations given in Eqs. (15) and (16).
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Now it should also be clear that far from the boundary sites, i.e. deep inside the Bethe lattice, all sites with spin-3/2 and also the sites with spin-5/2 are equivalent in their kinds. Therefore, one can set the sites to be equal as n = n − 2 = n − 4 = · · · and n − 1 = n − 3 = n − 5 = · · ·, or simply drop the subscripts form the recursion relations, to obtain three equations to determine the Curie temperatures explicitly using Eqs. (15) and (16) as
q−1 q−1 Xn = Yn = e25βc dB /4 cosh(15βc /4)An−1 + e9βc dB /4 cosh(9βc /4)Cn−1 + eβc dB /4 cosh(3βc /4)
−1 q−1 q−1 × e25βc dB /4 cosh(15βc /4)An−1 + e9βc dB /4 cosh(9βc /4)Cn−1 + eβc dB /4 cosh(3βc /4) , (25)
An−1 = Bn−1 =
q−1
e9βc dA /4 cosh(3βc /4)Xn−2 + eβc dA /4 cosh(βc /4)
Cn−1 = Dn−1 =
q−1
e9βc dA /4 cosh(15βc /4)Xn−2 + eβc dA /4 cosh(5βc /4)
q−1
(26)
e9βc dA /4 cosh(9βc /4)Xn−2 + eβc dA /4 cosh(3βc /4) q−1
,
e9βc dA /4 cosh(3βc /4)Xn−2 + eβc dA /4 cosh(βc /4)
.
(27)
One may numerically carry out the calculations to obtain the second-order phase transition temperatures or Curie temperatures by using these last three equations. We instead first obtained the values of the recursion relations iteratively from Eqs. (15) and (16), and then used the results in Eqs. (19)–(22) to obtain the sublattice magnetizations and quadrupole moments, respectively. Then, the Curie temperature is the temperature when the sublattice magnetizations go to zero, i.e. M3/2 = 0 and M5/2 = 0. In order to calculate the first-order phase transition temperatures, we need the free energy expression, so using the definition of the free energy F = −kT ln Z and Eqs. (2), (9)–(16) in thermodynamic limit as n → ∞, since then the sites with spin-3/2 and the sites with spin-5/2 are equivalent in their species, one can obtain the free energy expression in terms of the recursion relations as,
1 q − 1 β (−3/4+9dA /4) q−1 q−1 q−1 Xn + eβ (3/4+9dA /4) Yn + eβ (−1/4+dA /4) Zn + eβ (1/4+dA /4) ln e F /J = − β 2−q
1 q q q q−1 q−1 ln eβ (−5/4+25dB /4) An−1 + eβ (5/4+25dB /4) Bn−1 + ln e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1) + 2−q
β (−3/4+9dB /4) q−1 β (3/4+9dB /4) q−1 β (−1/4+dB /4) q−1 β (1/4+dB /4) . +e Cn−1 + e Dn−1 + e En−1 + e (28) The first-order phase transition temperatures are determined from a free energy analysis. It should be mentioned that in solving the recursion relations by iteration, i.e. Eqs. (15)–(16), one has to assign an initial value for each of the recursion relations. Therefore, varying the initial values may result in different solutions for all the thermodynamic functions including the free energy. As a result, the temperature at which the free energy values are equal to each other is the first-order phase transition temperature. The compensation temperature, Tcomp , can be located by finding the crossing points between the absolute values of the sublattice magnetizations, M3/2 (Tcomp ) = M5/2 (Tcomp ) (29) or when the net magnetization goes to zero MNET = M3/2 (Tcomp ) − M5/2 (Tcomp ) = 0 with the conditions
sign M3/2 (Tcomp ) = − sign M5/2 (Tcomp ) ,
(30)
Tcomp < Tc .
(31)
These conditions ensure that at Tcomp the two sublattice magnetizations cancel each other, whereas at the critical temperature, Tc , both sublattice magnetizations also the net magnetization go to zero. We are now ready to study the phase diagrams of this mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system in depth on the (kT /|J |, DA /|J |) plane for constant values of DB /|J |, on the (kT /|J |, DB /|J |) plane for constant values of DA /|J | and on the (kT /|J |, D/|J |) plane for the equal strengths of the reduced crystal fields, DA = DB , including the lines of compensation temperatures. 4. The phase diagrams and the conclusions In this last section we present the results of the previous sections and illustrate the phase diagrams on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kT /|J |, DB /|J |) plane for constant values of DA /|J | for q = 3 only, and with equal strengths of the reduced crystal fields, DA = DB , that is on the (kT /|J |, D/|J |) plane including also q = 4, 5 and 6 for
E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129
127
Fig. 1. The phase diagram for equal values of the sublattice crystal fields, DA = DB = D, in the (kT /|J |, D/|J |) plane. The solid, dashed and grey dotted-dashed lines refer to the second- and first-order phase transition lines and the lines of the compensation temperatures, respectively. The lines are labeled with the q values.
the mixed spin-3/2 and spin-5/2 Ising system. In the phase diagrams, while the solid- and dashed-lines are used for the secondand first-order phase transitions occurring at T = Tc and T = Tt , respectively, the gray dashed-dotted lines refer to compensation temperatures, T = Tcomp . In Fig. 1 we have depicted the phase diagram for the case with equal strengths of crystal fields for the sublattices. As seen in the figure, all the phase transition lines are quite similar to each other for q equals to 3, 4, 5 and 6, except as the coordination number increases the critical behaviors are seen to occur at higher temperatures which is expected. As the values of the reduced crystal field D/|J | → ∞(−∞) the second-order phase transition temperatures reach the limiting values at about 6.882 (0.462), 10.9 (0.73), 14.77 (0.989) and 18.6 (1.244) for q = 3, 4, 5 and 6, respectively. These lines separate the ferrimagnetic region (below the lines) from the paramagnetic region (above them). The first-order phase transition lines start from about D/|J | ∼ = −3, −2.5, −2 and −1.5, respectively, and as the values of D/|J | are lowered for each q then the Tt values gradually decreases. These lines do not connect to the second-order lines instead they end at critical points and separate the ferri(±1/2) region from the ferri(±3/2, ±5/2) regions. Besides these critical temperatures, the system also shows one or two compensations depending on the values of D/|J |. One compensation is seen to occur at lower values of D/|J | and kT /|J | where they connect to the first-order lines and as the D/|J | is increased the system starts giving two compensations one of which follows the previous line and the other starts from the second-order phase transition lines and as D/|J | is increased further these two lines combine to each other for all q. Therefore the compensation temperature lines exhibit reentrant phenomenon. For the final illustrations, we have shown the phase diagrams for different values of the reduced crystal field strengths; that is Fig. 2(a) and (b) are obtained on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and Fig. 2(c) and (d) on the (kT /|J |, DB /|J |) plane for constant values of DA /|J |, respectively, for q = 3. It should be mentioned that, even if we have studied the phase diagrams when q = 4, 5 and 6 for different crystal field values, DA /|J | = DB /|J |, their phase diagrams are not illustrated in here since they are qualitatively similar to the case with q = 3 and only the numerical values are different. In Fig. 2(a), as DA /|J | → ∞(−∞) for DB /|J | → ∞ the second-order temperatures reach a constant value at 6.8924 (2.306) and for DB /|J | → −∞ the constant value of kTc /|J | is at 1.39 (0.462). Similarly, in Fig. 2(c), as DB /|J | → ∞(−∞) for DA /|J | → ∞ the second-order temperatures reach a constant value at 6.891 (1.382) and for DA /|J | → −∞ the constant value of kTc /|J | is at 2.3 (0.462). In both of these figures as the values of DB /|J |, in Fig. 2(a), and DA /|J |, in Fig. 2(c), becomes more and more negative the second-order phase transition temperatures are seen at lower values. Fig. 2(b) shows the phase diagram in the (kT /|J |, DA /|J |) plane for constant values of DB /|J | which are used to label the lines. The second-order lines from top to bottom and the first-order lines from left to right are obtained for the constant values of the DB /|J | = −1, −1.075, −1.15, −1.225, −1.3, −1.4, −1.5 and −2. As seen, the first-order lines start from higher temperatures for higher negative values of DA /|J | and lower negative values of DB /|J | and as the values of DA /|J | is increased the first-order phase transition temperatures decrease and eventually they disappear. It should also be mentioned that these lines do not connect to its conjugate second-order lines instead they start from their critical points. The system gives two compensations at higher temperatures and one compensation at lower temperatures for lower negative values of DB /|J | and for positive values of DA /|J | and as DA /|J | is decreased one of the Tcomp decreases from above and the other one increases from below and they combine with each other at their lowest possible value of DA /|J |, as a result the compensation lines show reentrant behavior. This behavior continues until the value of DB /|J | equals to −1.311 and then the system starting from this value only gives one compensation temperature and finally the compensation temperatures disappear for about DB /|J | ∼ = −1.5. In the last figure for q = 3, i.e. Fig. 2(d), the roles of the sublattice crystal fields were interchanged. The behaviors of the first-order phase transition lines, which are labeled with the values of DA /|J |, are similar to those of in Fig. 2(b). The second-order lines and the lines of the compensation temperatures are obtained from top to bottom for DA /|J | = 4, 3, 2, 1, 0,
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E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129
(a)
(b)
(c)
(d)
Fig. 2. While the phase diagrams on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | are given in (a) and (b), the roles of the crystal fields were switched in (c) and (d). (a) and (c) only show the second-order lines which are labeled with DB /|J | and DA /|J | values, respectively. The second-order lines from top to bottom and the first-order lines from left to right are obtained for the constant values of the DB /|J | = −1, −1.075, −1.15, −1.225, −1.3, −1.4, −1.5 and −2 in (b). Similarly, the second-order lines and the lines of the compensation temperatures are obtained from top to bottom for DA /|J | = 4, 3, 2, 1, 0, −1, −1.3, −1.5, −1.65, −1.9 and −2 in (d).
−1, −1.3, −1.5, −1.65, −1.9 and −2. In this case the system gives two compensations one of them starts from higher negative values of DA /|J | at high temperatures and as DA /|J | becomes more positive Tcomp decreases and the other one start from below and increases as DA /|J | is increased then it combines with the previous line, and outside this region that is towards left for lower negative values of DB /|J | the system gives only one compensation and the lines of which combine at about DB /|J | ∼ = −1.5 and they terminate there. It should be obvious from our previous discussion for the case with equal crystal fields, where we have observed that as the coordination number increases the critical behaviors occur at higher temperatures otherwise the qualitative behaviors of the phase lines are very similar for all q, we should also expect that the phase lines in the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and in the (kT /|J |, DB /|J |) plane for constant values of DA /|J | must be also similar for all q. Indeed this is the case, since we have also studied the phase diagrams in these planes for different crystal fields for the sublattices, but we did not illustrate them in here because of the qualitative similarities with the q = 3 case. Unfortunately as we mentioned in the introduction of this Letter there is only one work available [7] for comparison reasons, even if their phase diagrams are not studied in a great detail as we have done, however, their results show similar critical behaviors. The compensation temperature lines as we have obtained in Fig. 2 are similar with the Fig. 1(a) of [7] and all the second-order lines are qualitatively similar, but they have not obtained the first-order lines, therefore, the comparison is not possible. In conclusion, we have studied the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system by the use of exact recursion relations on the Bethe lattice. As a final word, we should say that this mixed spin system compared the spin-1/2 and spin-3/2 or spin-5/2 mixed-spin system, which present only second-order phase transition lines, give much richer phase diagrams including the first-order phase transition lines and the lines of the compensation temperatures.
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References [1] O. Khan, Molecular Magnetism, VCH, New York, 1993; T. Mallah, S. Thiébaut, M. Vardegauer, P. Veillet, Science 262 (1993) 1554; H. Okawa, N. Matsumoto, H. Tamaki, M. Ohba, Mol. Cryst. Liq. Cryst. 233 (1993) 257; C. Mathoniére, C.J. Nutall, S.G. Carling, P. Day, Inorg. Chem. 35 (1996) 1201; M. Drillon, E. Coronado, D. Beltran, R. Georges, J. Chem. Phys. 79 (1983) 449. [2] W. Jiang, G.Z. Wei, Z.H. Xin, Physica A 293 (2001) 455. [3] N. Benayad, et al., Ann. Phys. 5 (1996) 387. [4] A. Bobák, M. Jurˇcišin, J. Phys. IV France 7 (1997) C1-179. [5] G.M. Buendia, R. Cardano, Phys. Rev. B 59 (1999) 6784. [6] E. Albayrak, A. Alçi, Physica A 345 (2005) 48. [7] Q. Zhang, G. Wei, Y. Gu, Phys. Status Solidi B 242 (2005) 924. [8] Q. Jiang, H.N. Yang, G.C. Wang, J. Appl. Phys. 79 (8) (1996) 5122. [9] M. Kurata, R. Kikuchi, T. Watari, J. Chem. Phys. 21 (1953) 434; C. Domb, Adv. Phys. 9 (1960) 283. [10] J.W. Tucker, J. Magn. Magn. Mater. 195 (1999) 733. [11] E. Albayrak, M. Keskin, J. Magn. Magn. Mater. 261 (2003) 196; E. Albayrak, A. Yigit, Physica A 349 (2005) 471; E. Albayrak, A. Yigit, Phys. Status Solidi B 242 (2005) 1510.
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Mixed spin-3/2 and spin-5/2 Ising system on the Bethe lattice Erhan Albayrak ∗ , Ali Yigit Erciyes University, Department of Physics, 38039 Kayseri, Turkey Received 2 November 2005; received in revised form 24 December 2005; accepted 26 December 2005 Available online 4 January 2006 Communicated by R. Wu
Abstract In order to study the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system, we have used the exact recursion relations on the Bethe lattice. The system was studied for the coordination numbers with q = 3, 4, 5 and 6, and the obtained phase diagrams are illustrated on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J |, the reduced crystal field of the sublattice with spin-5/2, and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J |, the reduced crystal field of the sublattice with spin-3/2, for q = 3 only, since the cases corresponding to q = 4, 5 and 6 reproduce results similar to the case for q = 3. In addition we have also presented the phase diagram with equal strengths of the crystal fields for q = 3, 4, 5 and 6. Besides the second- and first-order phase transitions, the system also exhibits compensation temperatures for appropriate values of the crystal fields. In this mixed spin system while the second-order phase transition lines never cut the reduced crystal field axes as in the single spin type spin-3/2 and spin-5/2 Ising models separately, the first-order phase transition lines never connect to the second-order phase transition lines and they end at the critical points, therefore the system does not give any tricritical points. In addition to this, this mixed-spin model exhibits one or two compensation temperatures depending on the values of the crystal fields, as a result the compensation temperature lines show reentrant behavior. © 2006 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 05.70.Fh; 64.60.Cn; 75.10.Hk Keywords: Mixed spin; Bethe lattice; Reentrant behavior; Compensation temperature
1. Introduction Intense experimental studies have accumulated in the recent years in the area of molecular-based magnetic materials, and the magnetic properties, i.e. the molecular magnetism, have become an active area of scientific interest. These materials include bimetallic molecular-based magnetic materials in which two kinds of magnetic atoms alternate regularly have exhibited ferrimagnetic properties and therefore they are well interpreted by the use of mixed-spin systems [1]. That is why, there has been a great deal of interest in studying two sublattice mixed spin-σ and spin-S (σ = S) Ising ferrimagnetic (J < 0) or Ising ferromagnetic systems (J > 0). These ferrimagnetic materials, as mentioned above the molecular based magnetic materials, may exhibit compensation temperatures, i.e. the temperature at which the net magnetization goes to zero, in contrast to Ising models containing only one type of spins and could be handled theoretically by the use of mixed spins. Many combinations of the mixed-spin Ising ferrimagnetic systems are possible, but the less studied ferrimagnetic materials includes only the half-integer mixed spin systems. Maybe the simplest of this includes spin-1/2 and spin-3/2 system which was studied within the framework of the effective-field theory with correlations on the honeycomb lattice [2], on a square lattice by using the effective-field theory [3,4], again on a square lattice with a Monte Carlo algorithm [5] and on the Bethe lattice [6] with the coordination numbers q = 3, 4, 5 and 6, but no interesting behaviors was observed since this mixed spin system only exhibits second-order phase transitions. The next * Corresponding author.
E-mail address: [email protected] (E. Albayrak). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.077
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possible mixing of the half integer spins is the mixing of spin-1/2 and spin-5/2, unfortunately this system was not studied as far as in our knowledge but in any case even if we do not present the phase diagrams belonging to this mixed system in here we have studied it in search of any interesting critical behavior but we only find that the system gives second-order phase transition temperatures as in the case of mixed spin-1/2 and spin-3/2. Therefore, the next possibility includes the mixing of spin-3/2 and spin-5/2, which was studied using the effective field theory with correlations with interlayer coupling [7]. It should also be mentioned that this mixed-spin system maybe fictitious for at least up to this date, since as far as our search concerns we were not able to find a real system containing spin-3/2 and spin-5/2, but at least it does deserve to be studied for theoretical reasons. Besides, Co/Cu (001) system seem to behave Ising like system, therefore, it is justified to use the Ising type model to study the statistical behaviors of this mixed-spin system [8]. As a result the purpose of the present work is to obtain all the thermodynamic functions of interest in terms of the recursion relations and then is to study the thermal variations of the sublattice magnetizations and the quadrupole moments for the sublattices with spin-3/2 and with spin-5/2 to obtain the phase diagrams on the Bethe lattice for q = 3, 4, 5 and 6 and it was found that the system exhibits a very rich phase diagram as would be seen in the upcoming sections. The remaining part of the Letter is constructed as follows. In Section 2, the formulation of the problem is given and the exact expressions for the order-parameters, dipolar and the quadrupolar moments for the sublattices with spin-3/2 and spin-5/2, are obtained in terms of the recursion relations. In Section 3, the definitions of the critical temperatures and their formulations in terms of the recursion relations, the exact expressions for the Curie or the second-order phase transition temperature, the free energy, which is used to find the first-order phase transition temperatures, and the definition of the compensation temperature, are given. Finally, in the last section besides the phase diagram on the (kTc /|J |, D/|J |) plane, i.e. DA = DB = D, for q = 3, 4, 5 and 6, we have also presented the phase diagrams on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J | for q = 3 only, and we have concluded this section with a brief summary and a discussion. 2. The formulation in terms of the exact recursion relations We consider the mixed-spin Blume–Capel Ising ferrimagnetic system, J < 0, whose Hamiltonian includes the bilinear interaction of the sublattices with spin-3/2 and spin-5/2 and the crystal fields DA and DB acting on that sites, respectively, and it is given as σi sj − DA σi2 − DB sj2 , H = −J (1) i,j
i
j
where each σi located at site i is a spin-3/2 with four discrete spin values, i.e. ±3/2 and ±1/2 and each sj located at site j is a spin-5/2 and can take six discrete spin values ±5/2, ±3/2, and ±1/2. For this mixed spin-3/2 and spin-5/2 system, the Bethe lattice is arranged in such a way that the central spin is spin-3/2, σ0 , the next generation is spin-5/2, s1 , and the next generation is again spin-3/2, σ2 , and so on to infinity. Since the Bethe lattice is an infinite Cayley tree, a connected graph without circuits, one can avoid introducing the boundary conditions by considering the system in the thermodynamic limit, when n → ∞, n is the number of the shells from root, i.e. the central site with σ0 , to the boundary. The Bethe lattice gets its name from the fact that its partition function is exactly that of an Ising model in the Bethe approximation [9]. We should also point out that the cluster variation method in the pair approximation studies on regular lattices yield results that are exact for the same model on the Bethe lattice [10,11]. In Eq. (1), the first sum runs over all the nearest-neighbor pairs, i.e. between the sites with spin-3/2 and spin-5/2, the second sum runs over all the spin-3/2 sites and the last summation runs over the sites with spin-5/2. In order to obtain the formulation of this mixed-spin system, it is the common idea to start with the partition function, therefore, the partition function is given as P (Spc) = exp β J σi sj + DA σi2 + DB sj2 , Z= e−βH = (2) {σ,s}
Spc
i,j
i
j
where P (Spc) is considered as an unnormalized probability distribution over the spin configurations, Spc ≡ {σ, s}, etc. and σi and sj refer to the spin values at site i and site j , respectively. If the Bethe lattice is cut in some central point deep inside with a spin σ0 , then it splits up into q identical branches whose number depends on the number of nearest-neighbors or the coordination numbers. Each of these branches is a rooted tree at the central-spin σ0 . This implies that P ({σ0 }), Spc = {σ0 }, is the spin-configuration with the spin value σ0 at the central site, may be written as
(k) P {σ0 } = exp βDA σ02 × , Qn σ0 s1 q
(3)
k=1
since the central-spin is spin-3/2, when Bethe lattice is cut at that site the central spin σ0 can interact only with the crystal field (k) (k) DA and where {s1 } denotes the spin-configurations of the kth branch of the Bethe lattice starting with s1 other than the central spin σ0 , the suffix n expresses the fact that the subtree has n-shells, i.e. the number of steps from the root to the boundary sites. In
E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129
Eq. (3) the function Qn in the product is given as (k) (k) 2 Qn σ0 s1 = exp β J σ0 s1 + J σi sj + DA σ i + DB sj2 , i,j
i
123
(4)
j
where σi and sj refer to the spins of the site i and j of the subtree (other than the central spin σ0 , which is a spin-3/2). Site 1 with spin-5/2 is the site next to the central cite σ0 . The first summation in Eq. (4) is over all edges of the subtree other than the edge (0, 1) and the summation over i is over all sites with spin-3/2 other than the central site. Now if the subtree, e.g. the upper subtree, is cut at the site 1 next to 0, then it also decomposes into q pieces: one being “trunk” (0, 1) and the rest are the identical branches. Each of these branches is a subtree like the original, but with n − 1 shells and q − 1 neighbors. Thus q−1 (k) (k) 2 (k) (l) (k) Qn−1 s1 τ2 , = exp β J σ0 s1 + DB s1 × Qn σ0 s1
(5)
l=1 (l)
(k)
where {τ2 } denotes the spin-configurations (other than s1 ) on the lth branch of the subtree. The prime on the function Qn−1 in the product is used to distinguish the sublattices with spin-3/2 and spin-5/2 since they have common spin values ±1/2. If now the upper subtree is cut at site 2 again with spin-3/2 next to site 1, then it again decomposes into q pieces: one again being the next trunk (1, 2) and the rest are the identical branches, again with n − 1 shells and q − 1 neighbors. Therefore Q n−1
(k) (l) (l) s1 τ2 = exp β J s1 σ2
(l) 2
× + DA σ 2
q−1
(l) (m) , Qn−2 τ2 r3
(6)
m=1
where {r3(m) } indicates all the spins (other than τ2(l) ) on the next mth branch of the upper sub-subtree. As a result, one takes n steps from root to the boundary on the Bethe lattice and in the thermodynamic limit n → ∞ should be taken. Adding the shells in this manner, the Bethe lattice is set up in such a way that the central-spin is spin-3/2 with q-neighbors of spin-5/2, the next generation with spin-5/2 has q − 1 neighbors with spin-3/2, and the next generation with spin-3/2 has q − 1 neighbors of spin-5/2, and so on to infinity. It should be mentioned that the choice of the central spin has no effect on the system, i.e. the behaviors of the system parameters do not change whether one chooses the spin-3/2 or spin-5/2 as the central spins Ref. [11]. Now we define, Qn σ0 {s1 } , g n σ0 = (7) {s1 }
where the summation over s1 , chosen to be spin-5/2 with spin values ±5/2, ±3/2 and ±1/2, means that there is a certain probability for the spin σ0 to be in a certain spin value, it is a spin-3/2 with spin values ±3/2 and ±1/2, as a result of its interaction with the external parameters and with its q nearest-neighbors of s1 . Now we are ready to obtain the exact recursion relations, in order to do so let us start with Eq. (6) and sum over all the discrete spin values σ2 , which is a spin-3/2, thus we obtain
q−1 (s1 ) = exp β J σ2 s1 + DA σ22 gn−2 (σ2 ) . gn−1 (8) σ2
As we know σ2 can take four discrete values ±3/2 and ±1/2, since it is a spin-3/2, therefore summing over these spin values and then inserting the spin values for spin-5/2, with six discrete spin values ±5/2, ±3/2 and ±1/2, one can obtain six different gn−1 (s1 ) for each of the spin values s1 . Therefore, when s1 = ±5/2
q−1 (±5/2) = exp β ±5J σ2 /2 + DA σ22 gn−2 (σ2 ) gn−1 σ2
q−1
q−1 = eβ(±15J /4+9DA /4) gn−2 (+3/2) + eβ(∓15J /4+9DA /4) gn−2 (−3/2)
q−1
q−1 + eβ(±5J /4+DA /4) gn−2 (+1/2) + eβ(∓5J /4+DA /4) gn−2 (−1/2) ,
(9)
and when s1 = ±3/2 gn−1 (±3/2) =
q−1 exp β ±3J σ2 /2 + DA σ22 gn−2 (σ2 )
σ2
q−1
q−1
q−1 = eβ(±9J /4+9DA /4) gn−2 (+3/2) + eβ(∓9J /4+9DA /4) gn−2 (−3/2) + eβ(±3J /4+DA /4) gn−2 (+1/2)
q−1 + eβ(∓3J /4+DA /4) gn−2 (−1/2) , (10) and for the last two values of s1 = ±1/2, we get
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E. Albayrak, A. Yigit / Physics Letters A 353 (2006) 121–129 gn−1 (±1/2) =
q−1 exp β ±J σ2 /2 + DA σ22 gn−2 (σ2 )
σ2
q−1
q−1
q−1 = eβ(±3J /4+9DA /4) gn−2 (+3/2) + eβ(∓3J /4+9DA /4) gn−2 (−3/2) + eβ(±J /4+DA /4) gn−2 (+1/2)
q−1 + eβ(∓J /4+DA /4) gn−2 (−1/2) . (11) Now in order to calculate the gn (σ0 ), we take Eq. (5) together with Eq. (7), summing over all s1 , which is a spin-5/2, one can obtain
q−1 exp β J σ0 s1 + DB s12 gn−1 (s1 ) . gn (σ0 ) = (12) s1
Since σ0 can take the spin values ±3/2 and ±1/2, and s1 can take the values ±5/2, ±3/2, and ±1/2, therefore, one can obtain four different gn (σ0 ) for four possible values of σ0 , then for σ0 = ±3/2
q−1 exp β ±3J s1 /2 + DB s12 gn−1 (s1 ) gn (±3/2) = s1
q−1
q−1 = eβ(±15J /4+25DB /4) gn−1 (+5/2) + eβ(∓15J /4+25DB /4) gn−1 (−5/2)
q−1
q−1
q−1 + eβ(±9J /4+9DB /4) gn−1 (+3/2) + eβ(∓9J /4+9DB /4) gn−1 (−3/2) + eβ(±3J /4+DB /4) gn−1 (+1/2)
q−1 + eβ(∓3J /4+DB /4) gn−1 (−1/2) , (13) and finally for σ0 = ±1/2, we obtain
q−1 exp β ±J s1 /2 + DB s12 gn−1 (s1 ) gn (±1/2) = s1
q−1
q−1
q−1 = eβ(±5J /4+25DB /4) gn−1 (+5/2) + eβ(∓5J /4+25DB /4) gn−1 (−5/2) + eβ(±3J /4+9DB /4) gn−1 (+3/2)
q−1
q−1
q−1 (−3/2) + eβ(±J /4+DB /4) gn−1 (+1/2) + eβ(∓J /4+DB /4) gn−1 (−1/2) . + eβ(∓3J /4+9DB /4) gn−1 (14) Now we are ready to introduce the recursion relations as the ratios of the gn functions for spin-3/2 as Xn =
gn (+3/2) , gn (−1/2)
Yn =
gn (−3/2) , gn (−1/2)
Zn =
gn (+1/2) gn (−1/2)
(15)
and for spin-5/2 as the ratios of gn−1 functions
An−1 = Dn−1 =
gn−1 (+5/2) gn−1 (−1/2) gn−1 (−3/2) gn−1 (−1/2)
,
Bn−1 =
,
En−1 =
gn−1 (−5/2) gn−1 (−1/2) gn−1 (+1/2) gn−1 (−1/2)
,
Cn−1 =
gn−1 (+3/2) gn−1 (−1/2)
,
.
(16)
One can easily obtain the explicit expressions of the recursion relations by using Eqs. (9)–(11) and (13)–(16), unfortunately, their expressions are too long to be given in here. These recursion relations may not have any physical meaning, but they do reflect the critical behaviors of the system. In order to obtain the order-parameters in terms of these recursion relations, i.e. M3/2 and Q3/2 for the sublattice with spin-3/2 and, M5/2 and Q5/2 for the sublattice with spin-5/2, the magnetizations and the quadrupolar order-parameters, respectively, we use the usual definitions given for the central-spin as σ0 P {σ0 } , Q = Z −1 σ02 P {σ0 } , M = Z −1 (17) {σ0 }
{σ0 }
where Z and P ({σ0 }) are given by Eqs. (2) and (3), respectively, and σ0 is the central-spin. Since the central-spin is chosen to be spin-3/2, the magnetization or the dipole moment for the central-spin can be obtained from Eqs. (2), (3) and (17) as
q σ0 exp βDA σ02 gn (σ0 ) , M3/2 = Z −1 (18) σ0
which is easily expressed in terms of the recursion relations by using Eqs. (2), (13)–(15) and (17) as
M3/2 =
q
q
q
3/2e9β dA /4 (Xn − Yn ) + 1/2eβ dA /4 (Zn − 1) , q q q e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1)
(19)
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and similarly the quadrupolar order parameter for the central-spin, i.e. spin-3/2, is obtained from the second of Eq. (17) or replacing σ0 with σ02 in Eq. (18) in terms of the recursion relations as
Q3/2 =
q
q
q
9/4e9β dA /4 (Xn + Yn ) + 1/4eβ dA /4 (Zn + 1) . q q q e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1)
(20)
In order to obtain the order-parameters for the sublattice with spin-5/2 starting from the first shell of the Bethe lattice, one either carries out the whole calculation by choosing the spin-5/2 as the central-spin or simply by using the definitions given in Eq. (17) and noting that deep inside the Bethe lattice all the sites with the same kinds of spins are equivalent, therefore, one may easily adapt the equations to obtain the formulations of the order-parameters for the spin-5/2 directly, so the magnetization is calculated as
M5/2 =
q
q
q
q
q
5/2e25β dB /4 (Al − Bl ) + 3/2e9β dB /4 (Cl − Dl ) + 1/2eβ dB /4 (El − 1)
q
q
q
q
q
e25β dB /4 (Al + Bl ) + e9β dB /4 (Cl + Dl ) + eβ dB /4 (El + 1)
(21)
,
and the quadrupolar order parameter for spin-5/2 is given as
Q5/2 =
q
q
q
q
q
25/4e25β dB /4 (Al + Bl ) + 9/4e9β dB /4 (Cl + Dl ) + 1/4eβ dB /4 (El + 1)
q
q
q
q
q
e25β dB /4 (Al + Bl ) + e9β dB /4 (Cl + Dl ) + eβ dB /4 (El + 1)
,
(22)
where l is equal to n − 1. The thermal variations of the order-parameters now could be studied in terms of the recursion relations. In order to do so, first one needs to obtain the recursion relations in terms of the system parameters, i.e. the reduced crystal fields DA /|J | and DB /|J | for the sublattices with spin-3/2 and spin-5/2, respectively, and the coordination number q, by varying the temperature and then inserting the obtained results in Eqs. (19)–(22) the thermal variations of the order-parameters could be studied in detail. It should be mentioned that the numerical calculations are carried out by using an iteration procedure for given initial values of the recursion relations and for each given values of the reduced crystal fields and the coordination numbers. The phase diagrams of the system is then obtained by studying the thermal variations of the order-parameters and are going to be illustrated for q = 3 (for two cases; i.e. DA = DB and DA = DB ), 4, 5 and 6 (for only DA = DB ) in the last section. 3. The definitions and the formulations of the critical temperatures The thermal behaviors of the order-parameters play a crucial role in obtaining the phase diagrams of the system: when the magnetization curves go to zero continuously separating the ferrimagnetic phase from the paramagnetic phase is the second-order phase transition or the Curie temperature, i.e. the temperature at which magnetizations become zero, and when there is a magnetization jump to either zero or to another value is the first-order phase transition temperature, i.e. the temperature at which the magnetization jump occurs. Besides these two critical temperatures there is another temperature called as the compensation temperature and is defined as the temperature when the net magnetization becomes zero or the sublattice magnetization curves cut each other before the critical temperatures. Therefore, in order to find the second- and first-order phase transition temperatures to obtain the phase diagrams and also the lines of the compensation temperatures one has to study the thermal behaviors of the order-parameters. Thus we have studied the thermal variations of the order-parameters for the mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system for q = 3, 4, 5 and 6 in order to obtain the phase diagrams on the (kTc /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kTc /|J |, DB /|J |) plane for constant values of DA /|J |, besides the phase diagrams with equal strengths of the crystal fields. The second-order phase transition temperature, Tc , is the temperature at which either of the sublattice magnetizations become zero continuously, separating the ferrimagnetic phase from the paramagnetic phase. Therefore, by using the expressions for the magnetizations given in Eqs. (19) and (21) one can obtain the exact formulation of the second-order phase transition temperatures by setting M3/2 or M5/2 separately to zero: q q q M3/2 = 3/2e9β dA /4 Xn − Yn + 1/2eβ dA /4 Zn − 1 = 0 (23) and q q q q q M5/2 = 5/2e25β dB /4 An−1 − Bn−1 + 3/2e9β dB /4 Cn−1 − Dn−1 + 1/2eβ dB /4 En−1 − 1 = 0.
(24)
It is clear that Eq. (23) for the sublattice with spin-3/2 has a simple solution, i.e. Xn = Yn and Zn = 1, which means that at the second-order phase transition temperature the probability of spins being up and spins being down has to be equal to each other. This implies that at Tc the condition gn (+3/2) = gn (−3/2) and gn (+1/2) = gn (−1/2) must be satisfied. Similarly, the last equation for the sublattice with spin-5/2 also has a simple solution and it is given as An−1 = Bn−1 , Cn−1 = Dn−1 and En−1 = 1 which also (+5/2) = gn−1 (−5/2), gn−1 (+3/2) = gn−1 (−3/2) and gn−1 (+1/2) = gn−1 (−1/2) must also be satisfied. implies that at Tc , gn−1 It should be mentioned that the latter condition is readily obtained from the first condition as maybe easily seen from the recursion relations given in Eqs. (15) and (16).
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Now it should also be clear that far from the boundary sites, i.e. deep inside the Bethe lattice, all sites with spin-3/2 and also the sites with spin-5/2 are equivalent in their kinds. Therefore, one can set the sites to be equal as n = n − 2 = n − 4 = · · · and n − 1 = n − 3 = n − 5 = · · ·, or simply drop the subscripts form the recursion relations, to obtain three equations to determine the Curie temperatures explicitly using Eqs. (15) and (16) as
q−1 q−1 Xn = Yn = e25βc dB /4 cosh(15βc /4)An−1 + e9βc dB /4 cosh(9βc /4)Cn−1 + eβc dB /4 cosh(3βc /4)
−1 q−1 q−1 × e25βc dB /4 cosh(15βc /4)An−1 + e9βc dB /4 cosh(9βc /4)Cn−1 + eβc dB /4 cosh(3βc /4) , (25)
An−1 = Bn−1 =
q−1
e9βc dA /4 cosh(3βc /4)Xn−2 + eβc dA /4 cosh(βc /4)
Cn−1 = Dn−1 =
q−1
e9βc dA /4 cosh(15βc /4)Xn−2 + eβc dA /4 cosh(5βc /4)
q−1
(26)
e9βc dA /4 cosh(9βc /4)Xn−2 + eβc dA /4 cosh(3βc /4) q−1
,
e9βc dA /4 cosh(3βc /4)Xn−2 + eβc dA /4 cosh(βc /4)
.
(27)
One may numerically carry out the calculations to obtain the second-order phase transition temperatures or Curie temperatures by using these last three equations. We instead first obtained the values of the recursion relations iteratively from Eqs. (15) and (16), and then used the results in Eqs. (19)–(22) to obtain the sublattice magnetizations and quadrupole moments, respectively. Then, the Curie temperature is the temperature when the sublattice magnetizations go to zero, i.e. M3/2 = 0 and M5/2 = 0. In order to calculate the first-order phase transition temperatures, we need the free energy expression, so using the definition of the free energy F = −kT ln Z and Eqs. (2), (9)–(16) in thermodynamic limit as n → ∞, since then the sites with spin-3/2 and the sites with spin-5/2 are equivalent in their species, one can obtain the free energy expression in terms of the recursion relations as,
1 q − 1 β (−3/4+9dA /4) q−1 q−1 q−1 Xn + eβ (3/4+9dA /4) Yn + eβ (−1/4+dA /4) Zn + eβ (1/4+dA /4) ln e F /J = − β 2−q
1 q q q q−1 q−1 ln eβ (−5/4+25dB /4) An−1 + eβ (5/4+25dB /4) Bn−1 + ln e9β dA /4 (Xn + Yn ) + eβ dA /4 (Zn + 1) + 2−q
β (−3/4+9dB /4) q−1 β (3/4+9dB /4) q−1 β (−1/4+dB /4) q−1 β (1/4+dB /4) . +e Cn−1 + e Dn−1 + e En−1 + e (28) The first-order phase transition temperatures are determined from a free energy analysis. It should be mentioned that in solving the recursion relations by iteration, i.e. Eqs. (15)–(16), one has to assign an initial value for each of the recursion relations. Therefore, varying the initial values may result in different solutions for all the thermodynamic functions including the free energy. As a result, the temperature at which the free energy values are equal to each other is the first-order phase transition temperature. The compensation temperature, Tcomp , can be located by finding the crossing points between the absolute values of the sublattice magnetizations, M3/2 (Tcomp ) = M5/2 (Tcomp ) (29) or when the net magnetization goes to zero MNET = M3/2 (Tcomp ) − M5/2 (Tcomp ) = 0 with the conditions
sign M3/2 (Tcomp ) = − sign M5/2 (Tcomp ) ,
(30)
Tcomp < Tc .
(31)
These conditions ensure that at Tcomp the two sublattice magnetizations cancel each other, whereas at the critical temperature, Tc , both sublattice magnetizations also the net magnetization go to zero. We are now ready to study the phase diagrams of this mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system in depth on the (kT /|J |, DA /|J |) plane for constant values of DB /|J |, on the (kT /|J |, DB /|J |) plane for constant values of DA /|J | and on the (kT /|J |, D/|J |) plane for the equal strengths of the reduced crystal fields, DA = DB , including the lines of compensation temperatures. 4. The phase diagrams and the conclusions In this last section we present the results of the previous sections and illustrate the phase diagrams on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and on the (kT /|J |, DB /|J |) plane for constant values of DA /|J | for q = 3 only, and with equal strengths of the reduced crystal fields, DA = DB , that is on the (kT /|J |, D/|J |) plane including also q = 4, 5 and 6 for
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Fig. 1. The phase diagram for equal values of the sublattice crystal fields, DA = DB = D, in the (kT /|J |, D/|J |) plane. The solid, dashed and grey dotted-dashed lines refer to the second- and first-order phase transition lines and the lines of the compensation temperatures, respectively. The lines are labeled with the q values.
the mixed spin-3/2 and spin-5/2 Ising system. In the phase diagrams, while the solid- and dashed-lines are used for the secondand first-order phase transitions occurring at T = Tc and T = Tt , respectively, the gray dashed-dotted lines refer to compensation temperatures, T = Tcomp . In Fig. 1 we have depicted the phase diagram for the case with equal strengths of crystal fields for the sublattices. As seen in the figure, all the phase transition lines are quite similar to each other for q equals to 3, 4, 5 and 6, except as the coordination number increases the critical behaviors are seen to occur at higher temperatures which is expected. As the values of the reduced crystal field D/|J | → ∞(−∞) the second-order phase transition temperatures reach the limiting values at about 6.882 (0.462), 10.9 (0.73), 14.77 (0.989) and 18.6 (1.244) for q = 3, 4, 5 and 6, respectively. These lines separate the ferrimagnetic region (below the lines) from the paramagnetic region (above them). The first-order phase transition lines start from about D/|J | ∼ = −3, −2.5, −2 and −1.5, respectively, and as the values of D/|J | are lowered for each q then the Tt values gradually decreases. These lines do not connect to the second-order lines instead they end at critical points and separate the ferri(±1/2) region from the ferri(±3/2, ±5/2) regions. Besides these critical temperatures, the system also shows one or two compensations depending on the values of D/|J |. One compensation is seen to occur at lower values of D/|J | and kT /|J | where they connect to the first-order lines and as the D/|J | is increased the system starts giving two compensations one of which follows the previous line and the other starts from the second-order phase transition lines and as D/|J | is increased further these two lines combine to each other for all q. Therefore the compensation temperature lines exhibit reentrant phenomenon. For the final illustrations, we have shown the phase diagrams for different values of the reduced crystal field strengths; that is Fig. 2(a) and (b) are obtained on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and Fig. 2(c) and (d) on the (kT /|J |, DB /|J |) plane for constant values of DA /|J |, respectively, for q = 3. It should be mentioned that, even if we have studied the phase diagrams when q = 4, 5 and 6 for different crystal field values, DA /|J | = DB /|J |, their phase diagrams are not illustrated in here since they are qualitatively similar to the case with q = 3 and only the numerical values are different. In Fig. 2(a), as DA /|J | → ∞(−∞) for DB /|J | → ∞ the second-order temperatures reach a constant value at 6.8924 (2.306) and for DB /|J | → −∞ the constant value of kTc /|J | is at 1.39 (0.462). Similarly, in Fig. 2(c), as DB /|J | → ∞(−∞) for DA /|J | → ∞ the second-order temperatures reach a constant value at 6.891 (1.382) and for DA /|J | → −∞ the constant value of kTc /|J | is at 2.3 (0.462). In both of these figures as the values of DB /|J |, in Fig. 2(a), and DA /|J |, in Fig. 2(c), becomes more and more negative the second-order phase transition temperatures are seen at lower values. Fig. 2(b) shows the phase diagram in the (kT /|J |, DA /|J |) plane for constant values of DB /|J | which are used to label the lines. The second-order lines from top to bottom and the first-order lines from left to right are obtained for the constant values of the DB /|J | = −1, −1.075, −1.15, −1.225, −1.3, −1.4, −1.5 and −2. As seen, the first-order lines start from higher temperatures for higher negative values of DA /|J | and lower negative values of DB /|J | and as the values of DA /|J | is increased the first-order phase transition temperatures decrease and eventually they disappear. It should also be mentioned that these lines do not connect to its conjugate second-order lines instead they start from their critical points. The system gives two compensations at higher temperatures and one compensation at lower temperatures for lower negative values of DB /|J | and for positive values of DA /|J | and as DA /|J | is decreased one of the Tcomp decreases from above and the other one increases from below and they combine with each other at their lowest possible value of DA /|J |, as a result the compensation lines show reentrant behavior. This behavior continues until the value of DB /|J | equals to −1.311 and then the system starting from this value only gives one compensation temperature and finally the compensation temperatures disappear for about DB /|J | ∼ = −1.5. In the last figure for q = 3, i.e. Fig. 2(d), the roles of the sublattice crystal fields were interchanged. The behaviors of the first-order phase transition lines, which are labeled with the values of DA /|J |, are similar to those of in Fig. 2(b). The second-order lines and the lines of the compensation temperatures are obtained from top to bottom for DA /|J | = 4, 3, 2, 1, 0,
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(a)
(b)
(c)
(d)
Fig. 2. While the phase diagrams on the (kT /|J |, DA /|J |) plane for constant values of DB /|J | are given in (a) and (b), the roles of the crystal fields were switched in (c) and (d). (a) and (c) only show the second-order lines which are labeled with DB /|J | and DA /|J | values, respectively. The second-order lines from top to bottom and the first-order lines from left to right are obtained for the constant values of the DB /|J | = −1, −1.075, −1.15, −1.225, −1.3, −1.4, −1.5 and −2 in (b). Similarly, the second-order lines and the lines of the compensation temperatures are obtained from top to bottom for DA /|J | = 4, 3, 2, 1, 0, −1, −1.3, −1.5, −1.65, −1.9 and −2 in (d).
−1, −1.3, −1.5, −1.65, −1.9 and −2. In this case the system gives two compensations one of them starts from higher negative values of DA /|J | at high temperatures and as DA /|J | becomes more positive Tcomp decreases and the other one start from below and increases as DA /|J | is increased then it combines with the previous line, and outside this region that is towards left for lower negative values of DB /|J | the system gives only one compensation and the lines of which combine at about DB /|J | ∼ = −1.5 and they terminate there. It should be obvious from our previous discussion for the case with equal crystal fields, where we have observed that as the coordination number increases the critical behaviors occur at higher temperatures otherwise the qualitative behaviors of the phase lines are very similar for all q, we should also expect that the phase lines in the (kT /|J |, DA /|J |) plane for constant values of DB /|J | and in the (kT /|J |, DB /|J |) plane for constant values of DA /|J | must be also similar for all q. Indeed this is the case, since we have also studied the phase diagrams in these planes for different crystal fields for the sublattices, but we did not illustrate them in here because of the qualitative similarities with the q = 3 case. Unfortunately as we mentioned in the introduction of this Letter there is only one work available [7] for comparison reasons, even if their phase diagrams are not studied in a great detail as we have done, however, their results show similar critical behaviors. The compensation temperature lines as we have obtained in Fig. 2 are similar with the Fig. 1(a) of [7] and all the second-order lines are qualitatively similar, but they have not obtained the first-order lines, therefore, the comparison is not possible. In conclusion, we have studied the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system by the use of exact recursion relations on the Bethe lattice. As a final word, we should say that this mixed spin system compared the spin-1/2 and spin-3/2 or spin-5/2 mixed-spin system, which present only second-order phase transition lines, give much richer phase diagrams including the first-order phase transition lines and the lines of the compensation temperatures.
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