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Quadrupolar ordering and metastable phase diagram in a ferroquadrupolar phase
Physics Letters A 320 (2003) 28–34 www.elsevier.com/locate/pla
Quadrupolar ordering and metastable phase diagram in a ferroquadrupolar phase Ahmet Erdinç, Mustafa Keskin ∗ Department of Physics, Erciyes University, Kayseri 38039, Turkey Received 25 September 2003; received in revised form 4 November 2003; accepted 4 November 2003 Communicated by R. Wu
Abstract The phase transitions of the metastable and unstable branches of order parameters that exist in the ferroquadrupolar phase of the spin-1 Ising system are studied. The metastable phase diagram is presented in the ferroquadrupolar phase. The calculated first- and second-order phase boundaries of the unstable branches of the order parameters are also superimposed on the metastable phase diagram. 2003 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 05.70.Fh; 64.60.Cn; 64.60.My; 75.10.Hk Keywords: Spin-1 Ising system; Pair approximation; Quadrupolar ordering; Ferroquadrupolar phase; Metastable phase diagram
The existence of quadrupolar interactions has been established in several cubic rare-earth intermetallic compounds [1]. The most obvious proof is the possibility of a quadrupolar phase transition, as seen for example in TmCd [2] and TmZn [3]. Moreover, the quadrupolar interactions may act on the nature of the magnetic phase transition in which they may change a second-order phase transition into a first-order one, as observed, e.g., DySb [4], TbP [5] or TmCu [6], or vice versa as in PrMg2 [7]. On the other hand, numerous theoretical works have been worked out concerning the existence of dipolar and quadrupolar phase transitions, especially in the Ising systems, such as spin-1 [8–10], spin-3/2 [11] and spin-2 [12]. Magnetic dipolar and quadrupolar phase transitions in cubic rare-earth intermetallic compounds has been studied, in terms of single-ion susceptibilities and within the Landau theory [13]. Recently, the quadrupolar order in the S = 1 isotropic Heissenberg model with the biquadratic interaction has been studied by the quantum Monte Carlo simulation [14]. Moreover, our recent theoretical work, i.e., equilibrium and nonequilibrium behavior of the spin-1 Ising model [15], shows that the metastable and unstable branches of dipole (magnetic) and quadrupolar moment order parameters in which some of them undergo a firstand second-order phase transitions, exist in the ferroquadrupolar phase. The phase transitions of these metastable
* Corresponding author.
E-mail address: [email protected] (M. Keskin). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.003
A. Erdinç, M. Keskin / Physics Letters A 320 (2003) 28–34
29
and unstable branches of order parameters that occur in the ferroquadrupolar phase were not studied in Ref. [15]. Therefore, the aim of the present Letter is to study the phase transition of these branches of the order parameters and to present the metastable phase diagram in the ferroquadrupolar phase. Moreover, the calculated first- and secondorder phase boundaries of the unstable branches of the order parameters are also superimposed on the metastable phase diagram. The Hamiltonian of the spin-1 Ising model with bilinear and biquadratic exchange interactions is Si Sj − K Qi Qj , βH = −J (1) ij
ij
where Si can take the values +1, 0, −1, Qi can take the values +1, 0, and the symbol ij denotes a nearestneighbor pair of lattice sites. J and K are the bilinear exchange and the biquadratic exchange interactions, respectively. We have introduced interaction parameters J and K that depend on the temperature J → J /kB T and K → K/kB T , where T is the absolute temperature and kB is the Boltzmann factor and kB = 1 is taken. Using the pair approximation of the cluster variation method (PACVM) [16] which is identical to the constantcoupling, Bethe and the two-particle cluster approximations, the order parameters in terms of the bond or pair variables are found [15] S ≡ S = Y11 + Y12 − (Y32 + Y33 ), Q ≡ Q = Y11 + Y33 + 2Y13 − (Y21 + 2Y22 + Y23 ),
(2)
where S is the average magnetization, which is the excess of one orientation over the other orientation, also called the dipole moment, and Q is the quadrupolar moment that is a linear function of the average of the squared magnetization S 2 , defined as 3S 2 − 2. This is different from the definition Q = S 2 . The first definition ensures that the stable values of Q is equal to zero infinite temperature. Applying the PACVM, the free energy Φ per site can be found as [15] 3 3 3 3 γ βF γ = Φ= Eij Yij Yij ln(Yij ) − (γ − 1) Xi ln(Xi ) + βλ 1 − Yij , N 2 2 i,j =1
i,j =1
i=1
(3)
i,j =1
where λ is introduced to maintain the normalization condition. The next to the last term includes the correction of the overcount of the pair term. Moreover Yij bond or pair variables are found within the same method [15] as Y11 = (X1 X1 )γ¯ exp(−E11 )/Z ≡ Y22 = (X2 X2 )γ¯ exp(−E22 )/Z ≡ Y33 = (X3 X3 )γ¯ exp(−E33 )/Z ≡ Y12 = (X1 X2 )γ¯ exp(−E12 )/Z ≡ Y13 = (X1 X3 )γ¯ exp(−E13 )/Z ≡ Y23 = (X2 X3 )γ¯ exp(−E23 )/Z ≡ where γ¯ =
γ −1 γ
e11 , Z e22 , Z e33 , Z e12 ≡ Z e13 ≡ Z e23 ≡ Z
e21 , Z e31 , Z e32 , Z
(γ is the coordination number of lattice, i.e., the number of nearest neighbors), and
(4)
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A. Erdinç, M. Keskin / Physics Letters A 320 (2003) 28–34
Fig. 1. Thermal variations of the order parameters S and Q. Subscript 1 indicates the stable state (solid line), 2 the metastable state (dashed-dotted lines) and 3 the unstable state (dashed lines). Exhibiting a second-order phase transition for the metastable and unstable branches of the magnetization and also a first-order phase transition for the metastable branch of quadrupolar order parameter. TC2 and TC3 are the second-order phase transition or the critical temperatures for S2 and S3 , respectively. TtQ2 indicates first-order phase transition temperatures for Q2 . α = 1.125.
E11 = −(J + K),
E22 = −4K,
E33 = −(J + K),
E12 = E21 = +2K,
E13 = E31 = (J − K),
E23 = E32 = +2K.
These six nonlinear algebraic equations are solved by using the Newton–Raphson method for a fixed T , α = J /K, called the ratio of coupling parameters, and γ = 8 (representing a bcc lattice). After establishing the Yij values, order parameters, namely, S and Q can be obtained easily using Eq. (2). Since the solutions of these equations and their figures, i.e., the thermal variations of S and Q, are discussed in Ref. [15] extensively, we shall only give one interesting figure, i.e., Fig. 1, as an example and discuss it briefly. In Fig. 1, subscript 1 indicates the stable solutions (solid lines), and subscript 2 corresponds to the metastable solutions (dashed-dotted lines), and subscript 3 to the unstable solutions (dashed lines). This classification is done by comparing the free energy values of these solutions and as well as studying the dynamic of the model by the path probability method with pair distribution [17]. This dynamic investigation assures us that we found and defined the metastable and unstable branches of the order parameters completely and correctly. Since these works we are also given in Ref. [15] in detail, we will not give here again. In the figures, TC2 and TC3 are the critical or the second-order phase transition temperatures for the metastable and the unstable branches of the magnetization (S2 and S3 ), respectively. Tt Q2 indicates the first-order phase transition temperatures where the metastable branches of the quadrupolar order parameters jumps to zero discontinuously. The critical temperatures for the metastable and unstable branches of the magnetization in the case of a second-order phase transition are calculated easily and precisely by using the Hessian determinant which is the determination of the second derivative of the free energy with respect to Yij and the determination A can be
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Fig. 2. The values of the Hessian determinant as a function of temperature for J /K = 1.125. The changing sings of the determinant correspond to the critical temperature of the unstable (TC3 ) and metastable (TC2 ) branches of magnetization.
obtained as 1 − 2γ¯ Y11 X1 0 0 A= −γ¯ Y12 X1 −γ¯ Y13 X1 0
0
0
−2γ¯ YX111
−2γ¯ YX111
0
1 − 2γ¯ YX222
0
−2γ¯ YX222
0
−2γ¯ YX222
0
1 − 2γ¯ YX333
0
−2γ¯ YX333
−2γ¯ YX333
−γ¯ YX122
0
1 − γ¯ Y12 X12
−γ¯ YX121
−γ¯ YX122
0
−γ¯ YX131
−γ¯ YX131
1 − γ¯ Y13 X13
−γ¯ YX133
−γ¯ YX232
−γ¯ YX233
−γ¯ YX232
−γ¯ YX233
1 − γ¯ Y23 X23
,
(5)
where X12 = X11 + X12 ; X13 = X11 + X13 and X23 = X12 + X13 . The values of determinant are calculated and plotted as a function of the temperature for only one value of α as an example, α = 1.125, seen in Fig. 2 which corresponds to the second-order phase transition for S2 and S3 . The change in sign of the determinant corresponds to the critical temperature. In the figure, the determinant changes in sing in two different temperatures in which the first one corresponds to the critical temperature for the unstable branch of S, namely TC3 and the second one is the critical temperature for the metastable branch of S, namely TC2 . If one compares Fig. 1 with Fig. 2, one can see that the critical temperatures found using both calculations are exactly the same. However, the critical temperatures are found more easily and precisely with the Hessian determinant calculation. On the other hand, the first-order phase transitions for the metastable and unstable branches of order parameters are the temperatures where order parameters jump to zero discontinuously, seen in Fig. 1. We can now obtain the metastable phase diagram and the phase transitions for the unstable branches of the order parameters in (T /K, J /K) plane. The calculated the first- and second-order phase boundaries of the unstable branches of the order parameters are superimposed on the metastable phase diagram, seen in Fig. 3. In the phase
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Fig. 3. The calculated metastable phase diagram in the (T /K, J /K) plane (dashed-dotted lines). The very thin dashed-dotted lines illustrate a first-order phase transition for the metastable (Q2 ) branch of the quadrupolar order parameter. The thick and very thick dashed-dotted lines illustrate the first- and second-order phase transitions for the metastable (S2 ) branches of the magnetization, respectively. On the other hand, thick and very thick dashed lines are the first- and second-order phase boundaries of the unstable (S3 ) branches of the magnetization, respectively. The dotted lines separate the different subregions. The full and open triangles denote the tricritical points for metastable (S2 ) and unstable (S3 ) branches of dipole moment, respectively. M, M , M
and M
are multicritical points. Region I is the pure quadrupolar phase and regions II, IIa , IIb , . . . , IVc , and IVd are the different subquadrupolar phases.
diagram, the very thin dashed-dotted lines illustrate a first-order phase transition for metastable (Q2 ) branch of quadrupolar order parameter. The thick dashed-dotted and dashed lines illustrate a first-order phase transition for the metastable (S2 ) and unstable (S3 ) branches of the magnetization, respectively. The first-order phase transition lines for S2 and Q2 meet and as well as terminate at the multicritical point, namely M
(α = 3 and T /K = 15.77). Above M
, ferromagnetic and paramagnetic phases exist and only the stable values of order parameters undergo first, a first-order then a second-order phase transitions, hence only equilibrium phase diagram is present [9]. On the other hand, very thick dashed-dotted and dashed lines illustrate a second-order phase transition for the metastable (S2 ) and unstable (S3 ) branches of the magnetization, respectively. The dotted lines separate the different subregions. The full and open triangles denote the tricritical points for metastable (S2 ) and unstable (S3 ) branches of dipole moment, respectively. The system exhibits four multicritical points, namely M, M , M
and M
. The metastable phase diagram contains one pure quadrupolar phase (I) in which is only stable values of Q exists, and many more subquadrupolar phases that are marked with IIa , IIb , . . . , IVc , and IVd . For example, IIIa is the subquadrupolar phase in which the metastable branch of the quadrupolar order parameter that undergoes a firstorder phase transition, exist besides the stable of Q and metastable branch (S2 ) of dipole moment which undergoes a second-order phase transition; or, IVc is the other subquadrupolar phase in which the metastable branches of both order parameters exist and undergo a first-order phase transition, and also unstable branch of magnetization occur and undergoes a first-order phase transition. Moreover, the other subquadrupolar phases are defined similarly.
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In conclusion, we have seen that the metastable phase diagram always exist at the low temperatures that are consistent with the experimental and theoretical works, such as some alloys [18,19], semiconductor [20,21], polymers [22], water [23,24], the ternary system [25], one [10] and two sublattice [26] spin-1 Blume–Emery– Griffiths (BEG) model and also the spin-3/2 model [27] using the lowest approximation of the cluster variation method. Moreover, the first- and second-order phase boundaries for the unstable branches of order parameters IV semiconductor alloys occur at low temperatures, which are also consistent with the works on (AIII B V )1−X C2X [28], one [10] and two [26] sublattice spin-1 BEG model and as well as the spin-3/2 model [27]. We should also mention that the spin-1 Ising BEG model with the Hamiltonian bilinear (J ) and biquadratic (K) and crystalfield interaction (D) has been used to calculate the metastable phase diagram of the Cu–AL–Mn shape-memory alloys [19], semiconductor alloys [20], and as well as the ternary system [25]. Acknowledgements This research was supported by the Research Fond of Erciyes University, Grant numbers 98-51-5 and 00-12-2. References [1] P.M. Levy, P. Morin, D. Schmitt, Phys. Rev. Lett. 42 (1979) 1417; J. Kötzler, G. Raffius, Z. Phys. B 38 (1980) 139. [2] R. Aléonard, P. Morin, Phys. Rev. B 19 (1979) 3868. [3] P. Morin, J. Rouchy, D. Schmitt, Phys. Rev. B 17 (1978) 3684. [4] P.M. Levy, J. Phys. C 6 (1973) 3545. [5] C. Jaussaud, P. Morin, D. Schmitt, J. Magn. Magn. Mater. 22 (1980) 98. [6] J. Kötzler, G. Raffius, A. Loidl, C.M.E. Zeyen, Z. Phys. B 35 (1979) 125. [7] A. Loidl, K. Knorr, M. Müllner, K.H.J. Buschow, J. Appl. Phys. 52 (1981) 1433. [8] H.H. Chen, P.M. Levy, Phys. Rev. B 5 (1973) 4267; M. Keskin, M. Arı, P.H.E. Meijer, Physica A 157 (1989) 1000. [9] M. Keskin, P.H.E. Meijer, J. Chem. Phys. 85 (1986) 7324. [10] M. Keskin, C. Ekiz, J. Chem. Phys. 113 (2000) 5407. [11] J. Sivardière, M. Blume, Phys. Rev. B 5 (1972) 1126. [12] D.K. Ray, J. Sivardière, Phys. Rev. B 18 (1978) 1401; W. Phystasz, Phys. Rev. B 37 (1988) 9813; M. Dudzinski, G. Faith, J. Sznajd, Phys. Rev. B 59 (1999) 13764. [13] P. Morin, D. Schmitt, Phys. Rev. B 27 (1983) 4412. [14] K. Harada, N. Kawashima, Phys. Rev. B 65 (2002) 052403. [15] A. Erdinç, M. Keskin, Physica A 307 (2002) 453. [16] R. Kikuchi, Phys. Rev. 81 (1951) 988; R. Kikuchi, J. Chem. Phys. 60 (1974) 1071. [17] R. Kikuchi, Suppl. Prog. Theor. Phys. 35 (1966) 1. [18] See, e.g., S. Nourbakhsh, P. Chen, Acta Metall. 37 (1989) 2573; H.J. Fecht, Acta Metall. Mater. 39 (1991) 1003; C. Michaelsen, Z.H. Yan, R. Bormann, J. Appl. Phys. 73 (1993) 2249; J. Delamare, D. Lemarchand, P. Vigier, J. Alloys Compd. 216 (1995) 273; J.M. Sanches, M.C. Cadeville, V. Pierron-Bohnes, G. Inden, Phys. Rev. B 54 (1996) 8958; W. Loser, R. Hermann, M. Leonhardt, D. Stephan, R. Bormann, Mater. Sci. Eng. A 224 (1997) 53. [19] E. Obradó, C. Frontera, L. Manosa, A. Planes, Phys. Rev. B 58 (1998) 14245. [20] K.E. Newman, J.D. Dow, Phys. Rev. B 27 (1983) 7495; B.L. Gu, K.E. Newman, P.A. Fedders, Phys. Rev. B 35 (1987) 9135; B.L. Gu, J. Ni, J.L. Zhu, Phys. Rev. B 45 (1992) 4071; J. Ni, S. Iwata, Phys. Rev. B 52 (1995) 3214. [21] See, e.g., J.L. Zilko, J.E. Greene, J. Appl. Phys. 51 (1980) 1560; T. Laoui, M.J. Kaufman, Metall. Trans. A 22 (1991) 2141; J. Ni, B.L. Gu, Solid State Commun. 83 (1992) 757.
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[22] V. Mishra, H. Sperling, Polymer 36 (1995) 3593; V. Mishra, F.E. Du Prez, E. Gosen, E.J. Goethals, L.H. Sperling, J. Appl. Polym. Sci. 58 (1995) 331; V. Mishra, L.H. Sperling, J. Polym. Mater. 13 (1996) 93. [23] See, e.g., P.H. Poole, F. Sciortino, U. Essmann, H.E. Stanley, Nature (London) 360 (1992) 324; P.H. Poole, F. Sciortino, T. Grande, H.E. Stanley, C.A. Angell, Phys. Rev. Lett. 73 (1994) 1632; E.G. Ponyatovsky, V.V. Sinitiyn, T.A. Pozdnyakova, J. Chem. Phys. 109 (1998) 2413. [24] P.G. Debenedetti, Metastable Liquids, Princeton Univ. Press, Princeton, NJ, 1998; D.S. Corti, P.G. Debenedetti, Phys. Rev. E 57 (1998) 4211. [25] J. Ni, B.L. Gu, Phys. Rev. Lett. 79 (1997) 3922. [26] C. Ekiz, M. Keskin, Phys. Rev. B 66 (2002) 54105. [27] O. Özsoy, M. Keskin, Physica A 319 (2003) 404. [28] R. Osório, S. Froyen, A. Zunger, Solid State Commun. 78 (1991) 249; R. Osório, S. Froyen, A. Zunger, Phys. Rev. B 43 (1991) 14055.
Quadrupolar ordering and metastable phase diagram in a ferroquadrupolar phase Ahmet Erdinç, Mustafa Keskin ∗ Department of Physics, Erciyes University, Kayseri 38039, Turkey Received 25 September 2003; received in revised form 4 November 2003; accepted 4 November 2003 Communicated by R. Wu
Abstract The phase transitions of the metastable and unstable branches of order parameters that exist in the ferroquadrupolar phase of the spin-1 Ising system are studied. The metastable phase diagram is presented in the ferroquadrupolar phase. The calculated first- and second-order phase boundaries of the unstable branches of the order parameters are also superimposed on the metastable phase diagram. 2003 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 05.70.Fh; 64.60.Cn; 64.60.My; 75.10.Hk Keywords: Spin-1 Ising system; Pair approximation; Quadrupolar ordering; Ferroquadrupolar phase; Metastable phase diagram
The existence of quadrupolar interactions has been established in several cubic rare-earth intermetallic compounds [1]. The most obvious proof is the possibility of a quadrupolar phase transition, as seen for example in TmCd [2] and TmZn [3]. Moreover, the quadrupolar interactions may act on the nature of the magnetic phase transition in which they may change a second-order phase transition into a first-order one, as observed, e.g., DySb [4], TbP [5] or TmCu [6], or vice versa as in PrMg2 [7]. On the other hand, numerous theoretical works have been worked out concerning the existence of dipolar and quadrupolar phase transitions, especially in the Ising systems, such as spin-1 [8–10], spin-3/2 [11] and spin-2 [12]. Magnetic dipolar and quadrupolar phase transitions in cubic rare-earth intermetallic compounds has been studied, in terms of single-ion susceptibilities and within the Landau theory [13]. Recently, the quadrupolar order in the S = 1 isotropic Heissenberg model with the biquadratic interaction has been studied by the quantum Monte Carlo simulation [14]. Moreover, our recent theoretical work, i.e., equilibrium and nonequilibrium behavior of the spin-1 Ising model [15], shows that the metastable and unstable branches of dipole (magnetic) and quadrupolar moment order parameters in which some of them undergo a firstand second-order phase transitions, exist in the ferroquadrupolar phase. The phase transitions of these metastable
* Corresponding author.
E-mail address: [email protected] (M. Keskin). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.11.003
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29
and unstable branches of order parameters that occur in the ferroquadrupolar phase were not studied in Ref. [15]. Therefore, the aim of the present Letter is to study the phase transition of these branches of the order parameters and to present the metastable phase diagram in the ferroquadrupolar phase. Moreover, the calculated first- and secondorder phase boundaries of the unstable branches of the order parameters are also superimposed on the metastable phase diagram. The Hamiltonian of the spin-1 Ising model with bilinear and biquadratic exchange interactions is Si Sj − K Qi Qj , βH = −J (1) ij
ij
where Si can take the values +1, 0, −1, Qi can take the values +1, 0, and the symbol ij denotes a nearestneighbor pair of lattice sites. J and K are the bilinear exchange and the biquadratic exchange interactions, respectively. We have introduced interaction parameters J and K that depend on the temperature J → J /kB T and K → K/kB T , where T is the absolute temperature and kB is the Boltzmann factor and kB = 1 is taken. Using the pair approximation of the cluster variation method (PACVM) [16] which is identical to the constantcoupling, Bethe and the two-particle cluster approximations, the order parameters in terms of the bond or pair variables are found [15] S ≡ S = Y11 + Y12 − (Y32 + Y33 ), Q ≡ Q = Y11 + Y33 + 2Y13 − (Y21 + 2Y22 + Y23 ),
(2)
where S is the average magnetization, which is the excess of one orientation over the other orientation, also called the dipole moment, and Q is the quadrupolar moment that is a linear function of the average of the squared magnetization S 2 , defined as 3S 2 − 2. This is different from the definition Q = S 2 . The first definition ensures that the stable values of Q is equal to zero infinite temperature. Applying the PACVM, the free energy Φ per site can be found as [15] 3 3 3 3 γ βF γ = Φ= Eij Yij Yij ln(Yij ) − (γ − 1) Xi ln(Xi ) + βλ 1 − Yij , N 2 2 i,j =1
i,j =1
i=1
(3)
i,j =1
where λ is introduced to maintain the normalization condition. The next to the last term includes the correction of the overcount of the pair term. Moreover Yij bond or pair variables are found within the same method [15] as Y11 = (X1 X1 )γ¯ exp(−E11 )/Z ≡ Y22 = (X2 X2 )γ¯ exp(−E22 )/Z ≡ Y33 = (X3 X3 )γ¯ exp(−E33 )/Z ≡ Y12 = (X1 X2 )γ¯ exp(−E12 )/Z ≡ Y13 = (X1 X3 )γ¯ exp(−E13 )/Z ≡ Y23 = (X2 X3 )γ¯ exp(−E23 )/Z ≡ where γ¯ =
γ −1 γ
e11 , Z e22 , Z e33 , Z e12 ≡ Z e13 ≡ Z e23 ≡ Z
e21 , Z e31 , Z e32 , Z
(γ is the coordination number of lattice, i.e., the number of nearest neighbors), and
(4)
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A. Erdinç, M. Keskin / Physics Letters A 320 (2003) 28–34
Fig. 1. Thermal variations of the order parameters S and Q. Subscript 1 indicates the stable state (solid line), 2 the metastable state (dashed-dotted lines) and 3 the unstable state (dashed lines). Exhibiting a second-order phase transition for the metastable and unstable branches of the magnetization and also a first-order phase transition for the metastable branch of quadrupolar order parameter. TC2 and TC3 are the second-order phase transition or the critical temperatures for S2 and S3 , respectively. TtQ2 indicates first-order phase transition temperatures for Q2 . α = 1.125.
E11 = −(J + K),
E22 = −4K,
E33 = −(J + K),
E12 = E21 = +2K,
E13 = E31 = (J − K),
E23 = E32 = +2K.
These six nonlinear algebraic equations are solved by using the Newton–Raphson method for a fixed T , α = J /K, called the ratio of coupling parameters, and γ = 8 (representing a bcc lattice). After establishing the Yij values, order parameters, namely, S and Q can be obtained easily using Eq. (2). Since the solutions of these equations and their figures, i.e., the thermal variations of S and Q, are discussed in Ref. [15] extensively, we shall only give one interesting figure, i.e., Fig. 1, as an example and discuss it briefly. In Fig. 1, subscript 1 indicates the stable solutions (solid lines), and subscript 2 corresponds to the metastable solutions (dashed-dotted lines), and subscript 3 to the unstable solutions (dashed lines). This classification is done by comparing the free energy values of these solutions and as well as studying the dynamic of the model by the path probability method with pair distribution [17]. This dynamic investigation assures us that we found and defined the metastable and unstable branches of the order parameters completely and correctly. Since these works we are also given in Ref. [15] in detail, we will not give here again. In the figures, TC2 and TC3 are the critical or the second-order phase transition temperatures for the metastable and the unstable branches of the magnetization (S2 and S3 ), respectively. Tt Q2 indicates the first-order phase transition temperatures where the metastable branches of the quadrupolar order parameters jumps to zero discontinuously. The critical temperatures for the metastable and unstable branches of the magnetization in the case of a second-order phase transition are calculated easily and precisely by using the Hessian determinant which is the determination of the second derivative of the free energy with respect to Yij and the determination A can be
A. Erdinç, M. Keskin / Physics Letters A 320 (2003) 28–34
31
Fig. 2. The values of the Hessian determinant as a function of temperature for J /K = 1.125. The changing sings of the determinant correspond to the critical temperature of the unstable (TC3 ) and metastable (TC2 ) branches of magnetization.
obtained as 1 − 2γ¯ Y11 X1 0 0 A= −γ¯ Y12 X1 −γ¯ Y13 X1 0
0
0
−2γ¯ YX111
−2γ¯ YX111
0
1 − 2γ¯ YX222
0
−2γ¯ YX222
0
−2γ¯ YX222
0
1 − 2γ¯ YX333
0
−2γ¯ YX333
−2γ¯ YX333
−γ¯ YX122
0
1 − γ¯ Y12 X12
−γ¯ YX121
−γ¯ YX122
0
−γ¯ YX131
−γ¯ YX131
1 − γ¯ Y13 X13
−γ¯ YX133
−γ¯ YX232
−γ¯ YX233
−γ¯ YX232
−γ¯ YX233
1 − γ¯ Y23 X23
,
(5)
where X12 = X11 + X12 ; X13 = X11 + X13 and X23 = X12 + X13 . The values of determinant are calculated and plotted as a function of the temperature for only one value of α as an example, α = 1.125, seen in Fig. 2 which corresponds to the second-order phase transition for S2 and S3 . The change in sign of the determinant corresponds to the critical temperature. In the figure, the determinant changes in sing in two different temperatures in which the first one corresponds to the critical temperature for the unstable branch of S, namely TC3 and the second one is the critical temperature for the metastable branch of S, namely TC2 . If one compares Fig. 1 with Fig. 2, one can see that the critical temperatures found using both calculations are exactly the same. However, the critical temperatures are found more easily and precisely with the Hessian determinant calculation. On the other hand, the first-order phase transitions for the metastable and unstable branches of order parameters are the temperatures where order parameters jump to zero discontinuously, seen in Fig. 1. We can now obtain the metastable phase diagram and the phase transitions for the unstable branches of the order parameters in (T /K, J /K) plane. The calculated the first- and second-order phase boundaries of the unstable branches of the order parameters are superimposed on the metastable phase diagram, seen in Fig. 3. In the phase
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Fig. 3. The calculated metastable phase diagram in the (T /K, J /K) plane (dashed-dotted lines). The very thin dashed-dotted lines illustrate a first-order phase transition for the metastable (Q2 ) branch of the quadrupolar order parameter. The thick and very thick dashed-dotted lines illustrate the first- and second-order phase transitions for the metastable (S2 ) branches of the magnetization, respectively. On the other hand, thick and very thick dashed lines are the first- and second-order phase boundaries of the unstable (S3 ) branches of the magnetization, respectively. The dotted lines separate the different subregions. The full and open triangles denote the tricritical points for metastable (S2 ) and unstable (S3 ) branches of dipole moment, respectively. M, M , M
and M
are multicritical points. Region I is the pure quadrupolar phase and regions II, IIa , IIb , . . . , IVc , and IVd are the different subquadrupolar phases.
diagram, the very thin dashed-dotted lines illustrate a first-order phase transition for metastable (Q2 ) branch of quadrupolar order parameter. The thick dashed-dotted and dashed lines illustrate a first-order phase transition for the metastable (S2 ) and unstable (S3 ) branches of the magnetization, respectively. The first-order phase transition lines for S2 and Q2 meet and as well as terminate at the multicritical point, namely M
(α = 3 and T /K = 15.77). Above M
, ferromagnetic and paramagnetic phases exist and only the stable values of order parameters undergo first, a first-order then a second-order phase transitions, hence only equilibrium phase diagram is present [9]. On the other hand, very thick dashed-dotted and dashed lines illustrate a second-order phase transition for the metastable (S2 ) and unstable (S3 ) branches of the magnetization, respectively. The dotted lines separate the different subregions. The full and open triangles denote the tricritical points for metastable (S2 ) and unstable (S3 ) branches of dipole moment, respectively. The system exhibits four multicritical points, namely M, M , M
and M
. The metastable phase diagram contains one pure quadrupolar phase (I) in which is only stable values of Q exists, and many more subquadrupolar phases that are marked with IIa , IIb , . . . , IVc , and IVd . For example, IIIa is the subquadrupolar phase in which the metastable branch of the quadrupolar order parameter that undergoes a firstorder phase transition, exist besides the stable of Q and metastable branch (S2 ) of dipole moment which undergoes a second-order phase transition; or, IVc is the other subquadrupolar phase in which the metastable branches of both order parameters exist and undergo a first-order phase transition, and also unstable branch of magnetization occur and undergoes a first-order phase transition. Moreover, the other subquadrupolar phases are defined similarly.
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In conclusion, we have seen that the metastable phase diagram always exist at the low temperatures that are consistent with the experimental and theoretical works, such as some alloys [18,19], semiconductor [20,21], polymers [22], water [23,24], the ternary system [25], one [10] and two sublattice [26] spin-1 Blume–Emery– Griffiths (BEG) model and also the spin-3/2 model [27] using the lowest approximation of the cluster variation method. Moreover, the first- and second-order phase boundaries for the unstable branches of order parameters IV semiconductor alloys occur at low temperatures, which are also consistent with the works on (AIII B V )1−X C2X [28], one [10] and two [26] sublattice spin-1 BEG model and as well as the spin-3/2 model [27]. We should also mention that the spin-1 Ising BEG model with the Hamiltonian bilinear (J ) and biquadratic (K) and crystalfield interaction (D) has been used to calculate the metastable phase diagram of the Cu–AL–Mn shape-memory alloys [19], semiconductor alloys [20], and as well as the ternary system [25]. Acknowledgements This research was supported by the Research Fond of Erciyes University, Grant numbers 98-51-5 and 00-12-2. References [1] P.M. Levy, P. Morin, D. Schmitt, Phys. Rev. Lett. 42 (1979) 1417; J. Kötzler, G. Raffius, Z. Phys. B 38 (1980) 139. [2] R. Aléonard, P. Morin, Phys. Rev. B 19 (1979) 3868. [3] P. Morin, J. Rouchy, D. Schmitt, Phys. Rev. B 17 (1978) 3684. [4] P.M. Levy, J. Phys. C 6 (1973) 3545. [5] C. Jaussaud, P. Morin, D. Schmitt, J. Magn. Magn. Mater. 22 (1980) 98. [6] J. Kötzler, G. Raffius, A. Loidl, C.M.E. Zeyen, Z. Phys. B 35 (1979) 125. [7] A. Loidl, K. Knorr, M. Müllner, K.H.J. Buschow, J. Appl. Phys. 52 (1981) 1433. [8] H.H. Chen, P.M. Levy, Phys. Rev. B 5 (1973) 4267; M. Keskin, M. Arı, P.H.E. Meijer, Physica A 157 (1989) 1000. [9] M. Keskin, P.H.E. Meijer, J. Chem. Phys. 85 (1986) 7324. [10] M. Keskin, C. Ekiz, J. Chem. Phys. 113 (2000) 5407. [11] J. Sivardière, M. Blume, Phys. Rev. B 5 (1972) 1126. [12] D.K. Ray, J. Sivardière, Phys. Rev. B 18 (1978) 1401; W. Phystasz, Phys. Rev. B 37 (1988) 9813; M. Dudzinski, G. Faith, J. Sznajd, Phys. Rev. B 59 (1999) 13764. [13] P. Morin, D. Schmitt, Phys. Rev. B 27 (1983) 4412. [14] K. Harada, N. Kawashima, Phys. Rev. B 65 (2002) 052403. [15] A. Erdinç, M. Keskin, Physica A 307 (2002) 453. [16] R. Kikuchi, Phys. Rev. 81 (1951) 988; R. Kikuchi, J. Chem. Phys. 60 (1974) 1071. [17] R. Kikuchi, Suppl. Prog. Theor. Phys. 35 (1966) 1. [18] See, e.g., S. Nourbakhsh, P. Chen, Acta Metall. 37 (1989) 2573; H.J. Fecht, Acta Metall. Mater. 39 (1991) 1003; C. Michaelsen, Z.H. Yan, R. Bormann, J. Appl. Phys. 73 (1993) 2249; J. Delamare, D. Lemarchand, P. Vigier, J. Alloys Compd. 216 (1995) 273; J.M. Sanches, M.C. Cadeville, V. Pierron-Bohnes, G. Inden, Phys. Rev. B 54 (1996) 8958; W. Loser, R. Hermann, M. Leonhardt, D. Stephan, R. Bormann, Mater. Sci. Eng. A 224 (1997) 53. [19] E. Obradó, C. Frontera, L. Manosa, A. Planes, Phys. Rev. B 58 (1998) 14245. [20] K.E. Newman, J.D. Dow, Phys. Rev. B 27 (1983) 7495; B.L. Gu, K.E. Newman, P.A. Fedders, Phys. Rev. B 35 (1987) 9135; B.L. Gu, J. Ni, J.L. Zhu, Phys. Rev. B 45 (1992) 4071; J. Ni, S. Iwata, Phys. Rev. B 52 (1995) 3214. [21] See, e.g., J.L. Zilko, J.E. Greene, J. Appl. Phys. 51 (1980) 1560; T. Laoui, M.J. Kaufman, Metall. Trans. A 22 (1991) 2141; J. Ni, B.L. Gu, Solid State Commun. 83 (1992) 757.
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