Journal of the Korean Physical Society, Vol. 56, No. 4, April 2010, pp. 1046∼1050 Determination of the Critical Behavior of the Antiferromagnetic Interaction Spin-1 System in an External Magnetic Field by Using the Constant-coupling Approximation Ahmet Erdinc¸∗ and Osman Canko Department of Physics, Erciyes University, Kayseri, 38039, Turkey The critical behavior of the antiferromagnetic interaction spin-1 system model in an external magnetic field is studied by using the constant-coupling approximation for four different nearest- neighbors coupling constant. First, we have investigated the thermal variations of the sublattice magnetizations for different values of the interaction parameters in an external magnetic field. The model exhibits distinct critical region, including the first-order, the second-order tricritical point, and the zero critical point region. Second, we have calculated the phase diagrams. Finally, a discussion and a comparison of the phase diagrams are given. PACS numbers: 05.50.+q, 05.70.Fh, 64.60.Cn, 75.10.Hk Keywords: Constant-coupling approximation, Phase diagram, BEG model, Antiferromagnetism DOI: 10.3938/jkps.56.1046 I. INTRODUCTION rochlore lattice with nearest- and next-nearest- neighbors intereactions and well site dilution by using the general- ized constant coupling (GCC) method. They calculated The constant-coupling approximation (CCA), intro- the magnetic properties and found them to be in excel- duced by Kasteleijn and van Kranendonk [1–3] and by lent agreement with the Monte Carlo (MC) calculations Radcliffe [4], provides a method for calculating equi- [10]. In a recent work, Rausch and Nolting calculated librium properties of spin systems. The method at- the thickness-dependent Curie temperature of thin fer- tempts to deal with correlations that are usually ne- romagnetic films by using the Oguchi cluster method [11] gleted by molecular field approximations, and some and the CCA [12]. results of the constant-coupling approximation, e.g., critical-temperature values, represent an improvement over molecular field results. In addition to the origi- II. FORMULATION IN TERMS OF THE nal work of Kateleijn and van Kranendonk, the general CONSTANT COUPLING cluster expansion for Heisenberg and Ising systems has APPROXIMATION been shown to provide a common framework for arriving at both the molecular field and the constant-coupling approximation. Then, Glauber proposed the time- The effective Hamiltonian in an external magnetic field dependent statistics of the one-dimensional Ising model for a single-spin on the k-th site can be written in the with spin-S = 1/2 [5]. Obokata extended Glauber’s the- form [6] ory to the case of S = 1, developed the time-dependent CCA, and studied the equilibrium spin system with S = H = −zλS − zµS2 − DS2 − hS , (1) 1 by using the static CCA [6]. Falk obtained the for- k k k k k mulation of the CCA for S = 1/2 [7]. Chen and Lee where Sk located at site k is spin-1 with three discrete studied the ferromagnetic exchange-interaction model for spin values, i.e. ±1 and 0, and z is the number of the general spins by using the CCA and found that a first- nearest neighbors, i.e., the coordination number. Two order transition occurred for all spin S ≥ 1 [8]. Fechner parameters λ and µ represent the exchange interaction and Pikula examined some thermodynamic properties with the nearest neighboring spins, D is a single-ion of an antiferromagnetic spin-1/2 system with two dis- anisotropy constant or crystal-field interaction, where h tinct anisotropic exchange interactions in each case for = gµBH and H is the applied magnetic field along the two different nearest-neighbor coupling constant by using z-axis, g Land´efactor, and µB is the Bohr magneton. the CCA [9]. Garcia-Adeva and Huber studied the py- Next, let us assume that the effective Hamiltonian for two spins or a pair Sk and Sl a nearest neighbors can be ∗E-mail: [email protected] written in form -1046- Determination of the Critical Behavior of the Antiferromagnetic Interaction ··· – Ahmet Erdinc¸ and Osman Canko -1047- The normalized density matrices belonging to each sublattice, because each spin only interacts with its Hkl = −2JSkSl − (z − 1)λ(Sk + Sl) nearest-neighbors, can be written, and the averages of 2 2 2 2 −(z − 1)µ(Sk + Sl ) the Sk,Sl,Sk , and Sl are, respectively, given as 2 2 −D(Sk + Sl ) − h(Sk + Sl), (2) A B where J represents the bilinear or coupling constant be- hSki = T r(Skρk ), hSli = T r(Slρl ), tween the two magnetic atoms and is assumed to be neg- hS2i = T r(S2ρA), hS2i = T r(S2ρB). (6) ative (antiferromagnetic). In the ferromagnetic case, the k k k l l l magnetic interactions between magnetic atoms is posi- tive and favors a paralel alignment of the atomic mag- Now, let us consider the effective Hamiltonian for two netic moments. However, in the antiferromagnetic case, spins Sk and Sl that are nearest neighbors to other as the interactions cause a antiparallel arrangement of mag- two sublattices. Then, the effective Hamiltonian can be netic moments. Therefore, the lattice of magnetic atoms written in the form should be divided into two equivalent interpenetrating sublattice, i.e., A and B, such that A atoms have only B AB atoms as nearest-neighbor and vice versa. Hkl = −2JSkSl − (z − 1)(λASk + λBSl) The CCA, higher mean-field theory, is applied to ob- 2 2 −(z − 1)(µASk + µBSl ) tain a quantitative improvement of the result of the −D(S2 + S2) − h(S + S ). (7) mean-field theory. Furthermore, the mean-field theory k l k l is the best effective field that can be devised for a single- atom model and may not be suitable for interpenetrat- ing sublattice models [11]. Now, let us consider a two- The normalized density matrix for the two sublattices sublattice, i.e. A and B, spin-1 system and apply the is constant coupling approximation method. For a two- sublattice system, the effective Hamiltonian in an exter- AB AB nal magnetic field for a single spin on the k-th and l-th AB −βHkl −βHkl ρkl = e /T re . (8) sites can be defined as follows: 2 2 The averages of the Sk,Sl,Sk, and Sl are then given A 2 2 by Hk = −zλASk − zµASk − DSk − hSk, (3) B 2 2 Hl = −zλBSl − zµBSl − DSl − hSl, (4) hS i = T r(S ρAB), hS i = T r(S ρAB), where λA, λB, µA, and µB are exchange interactions for k k kl l l kl 2 2 AB 2 2 AB the A and the B sublattices, respectively. The normal- hSki = T r(Skρkl ), hSl i = T r(Sl ρkl ). (9) ized density matrices for the k-th and the l-th spins as two sublattices are given by Both the single-particle Hamiltonian and the pair Hamiltonian predict the same order parameters [8]. A −βHA −βHA Hence, from the self-consistence conditions that the four ρ = e k /T re k , k expressions for S ,S ,S2 , and S2 in Eq. (6) must be B B k l k l B −βHl βHl ρl = e /T re . (5) equal to Eq. (9), respectively, we have β(zµA+D) e sinh β(zλA + h) β(zµ +D) 2e A cosh β(zλA + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} sinh βt1 + eβ{z µA+D} sinh βta2 + e2β{−J+z µ +D} sinh βta3 = 0 0 0 0 0 eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βta3 + eβz (−λA+λB )]} + 2eβ{z µA+D} cosh βta4 + 2eβ{z µB +D} cosh βta5 + 1 β(zµB +D) e sinh β(zλB + h) β(zµ +D) , 2e B cosh β(zλB + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} sinh βt1 + eβ{z µB +D} sinh βtb2 + e2β{−J+z µ +D} sinh βta3 = 0 0 0 0 0 , eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βtb3 + eβz (λA−λB )]} + 2eβ{z µB +D} cosh βtb4 + 2eβ{z µB +D} cosh βtb5 + 1 -1048- Journal of the Korean Physical Society, Vol. 56, No. 4, April 2010 β(zµA+D) e cosh β(zλA + h) β(zµ +D) 2e A cosh β(zλA + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} cosh βt1 + eβ{z µA+D} cosh βta2 + e2β{−J+z µ +D} cosh βta3 = 0 0 0 0 0 eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βta3 + eβz (−λA+λB )]} + 2eβ{z µA+D} cosh βta4 + 2eβ{z µB +D} cosh βta5 + 1 β(zµB +D) e cosh β(zλB + h) β(zµ +D) 2e B cosh β(zλB + h) + 1 0 0 0 0 0 eβ{2J+z µ +2D} cosh βt1 + eβ{z µB +D} cosh βtb2 + e2β{−J+z µ +D} cosh βtb3 = 0 0 0 0 0 , eβ{z µ +2D}{2e2βJ cosh βt1 + e−2βJ [2 cosh βtb3 + eβz (λA−λB )]} + 2eβ{z µB +D} cosh βtb4 + 2eβ{z µB +D} cosh βtb5 + 1 (10) with λA = JMA, λB = JMB, µA = KQA, µB = KQB, -1.0 and h/|J| = 1.5. As seen in the figure, sublattice 0 0 0 z = z − 1, µ = µA + µB, t1 = z (λA + λB) + 2h, ta2 = magnetizations undergo only a second-order phase tran- 0 0 0 2J + 2z λA + h, ta3 = z (λA − λB), ta4 = z (−λA + λB), sition at the N´eel temperature (TN ), at which the an- 0 0 0 ta5 = z λB +h, tb2 = 2J+2z λB +h, tb3 = z (−λA+λB), tiferromagnetic ordering disappears for values of z = 3, 0 0 tb4 = z (λA − λB), and tb5 = z λA + h, K is the bi- 4, 6, and 8.