Study structure transition temperature of some rare-Earth metals

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0038 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 106-111 This paper is available online at STUDY STRUCTURE TRANSITION TEMPERATURE OF SOME RARE-EARTH METALS Dang Thanh Hai and Vu Van Hung Vietnam Education Publishing House Abstract. Using the statistical moment method (SMM), we calculated the Helmholtz free energy of hcp, bcc and fcc phases at zero pressure of some rare-earth metals at a wide range of temperatures. The stable phase of Gd, Tb, Sc, Y and Yb metals can be determined by examining the Helmholtz free energy at a given temperature, i.e. the phase that gives the lowest free energy will be stable. These findings on the structural phase transition temperature are in agreement with previous experiments. Keywords: structural transition temperature, rare-earth metals, statistical moment method. 1. Introduction The structural transformation temperature of materials is of great interest to those working with solid state sciences and technologies. The structural and electronic properties [1] show that the 17 rare-earth metals exist in one of five crystal structures. At room temperature, nine exist in a hexagonal close-packed (HCP) structure, four in a double c-axis hcp (DHCP) structure, two in a face-centered cube (FCC), and one in both a body-centered cube (BCC) and rhombic (Sm-type) structures. This distribution changes according temperature and pressure as many of the elements go through a number of structural phase transitions. Many scientists have been interested in the study of thermodynamic properties and the elasticity of rare earth metals. In this study [2], the phonon spectrum and isothermal elastic constants of the rare earth metals La, Yb, Ce, and Th were calculated. Born M and J.R Oppenheimer [3] used the method of density-functional theory (DFT) and the equation of state at various pressures to calculate the lattice constant, molar volume, c/a ratio, the total energy and the temperature of the structural phase transition of Ce, Th, Pu, Am and their alloys. Rao R. S, Godwal B. K, and Sikka S. K investigated the electronic structure of thorium, and its FCC-BCC phase transition at high pressure [4]. Baria and Jani studied phonon dispersion curves, phonon density, Debye temperature, the Gru¨neisen constant and elastic constants, and compared the atomic volume depending on pressure, and the ion-ion interactions of La, Yb, Ce and Th [5]. In the research [6], it was shown that the heat capacity depends on the temperature of Ce, and the atomic volume Received August 15, 2016. Accepted September 14, 2016. Contact Dang Thanh Hai, e-mail address: dthai@nxbgd.vn 106 Study structure transition temperature of some rare-earth metals depends on pressure. Landa A and Soderlind P created a phase diagram for Ce, Th and Pu using the density functional theory (DFT) [7]. However, few scientists have looked at the structure transit temperature of rare earth metals. The authors of this study have looked at the structural transformation of rare-earth metals using the moment method in quantum statistical mechanics, hereafter referred to as the statistical moment method (SMM) [8-13]. This work focuses on the transition structure temperature of Gd, Tb, Sc, Y and Yb metals using the statistical moment method. 2. Content 2.1. Theory As we know, the structural phase transition between the a-phase and the b-phase when a chemical potential is µ balances µa = µb ⇔ ψa + PV a = ψb + PV b. (2.1) In formula (2.1), ψa and ψb are the free energy of the metal structure in the a-phase and b-phase. Free energy ψa, ψb are determined using statistical moment methods. At zero pressure, the formula (2.1) is rewritten as follows: ψa = ψb. (2.2) In expression (2.2), free energy ψa, ψb corresponding to structure a-phase and b-phase is a function of temperature T . Thus, we can determine the structural phase transition temperature by plotting the dependence of the free energy of the a-phase and b-phase on temperature with the same coordinate axes. The intersecting point of two line graphs allow us to determine the transition temperature Tc of the two phases. On the other hand, the free energy is calculated using the formula: ψ = U − TS, (2.3) where S is the entropy. At the transition temperature T = Tc structures, the free energy ψ of the a-phase and b-phase are equal, and using eq.(2.3), we have Ua − U b = T (Sa − Sb)⇔ ∆U = T∆S. (2.4) Thus, another way that we can determine the structural phase transition temperature between the a-phase and b-phase is plotting∆U, T∆S the dependence of temperature on the same coordinate axes. The two-line intersect allows us to identify structural phase transition temperature. Free energy ψ, interaction U and entropy S of the crystal are determined using the statistical moment method. According to thermodynamic theory, the phase with the lowest free energy will be favorable at each temperature. With the aid of the free energy formula ψ, at a wide range of temperatures T 107 Dang Thanh Hai and Vu Van Hung for the a-phase and b-phase for some rare-earth metals, we evaluate the components of ψ which consist of the internal energy U and entropy S: ∆ψ = ψa − ψb = (Ua − U b)− T (Sa − Sb). (2.5) At the temperature where ∆ψ > 0, the a-phase structure of a rare-earth metal is stable. At the temperature where ∆ψ < 0, the b-phase structure of a rare-earth metals is also stable. The temperature when ∆ψ = 0 or in other words when ∆U = T∆S, is called the transition temperature and both phases coexist. Therefore, if we plot the difference of these components, for example ψb = ψb for the rare-earth metal, as the functions of temperature, their interceptive point will denote the transition temperature. We determine the free energy, entropy and potential interactions of metal using statistical moment methods. From the results of referents [10, 11], we deduce the free energy of the metal atoms N using the following expression: ψ = U0 + ψ0 + 3N { θ2 k2 [ γ2x 2coth2x− 2γ1 3 ( 1 + xcothx 2 )] + + 2θ2 k4 [ 4 3 γ22xcothx ( 1 + xcothx 2 ) − 2(γ21 + 2γ1γ2) ( 1 + xcothx 2 ) + (1 + xcothx) ]} , (2.6) where ψ0 = 3Nθ [ x+ ln ( 1− e−2x)] (2.7) U0 = N 2 ∑ i ϕi0(|ri|) (2.8) k = 1 2 ∑ i ( ∂2ϕi0 ∂u2iβ ) . (2.9) γ1 = 1 48 ∑ i ( ∂4ϕi0 ∂u4iβ ) , γ2 = 6 48 ∑ i ( ∂4ϕi0 ∂u2iα∂u 2 iβ ) , (2.10) and the entropy S of the metal of the form: S = S0 + 3NkBθ k2 [ γ1 3 ( 4 + xcothx+ x2 sinh2x ) − 2γ2x 3cothx sinh2x ] , (2.11) where S0 = 3NkB [ xcothx− ln(2sinhx)]. (2.12) 108 Study structure transition temperature of some rare-earth metals 2.2. Results and discussions To calculate the transition structrure temperature, we use the Lennard-Jones (LI) type of pair potentials: φ(r) = D0 n−m { m ( r0 r )n − n ( r0 r )m} . (2.13) Table 1. Parameters n,m,D and r0 determined using the experimental data [14] Metal n m D/kB ,K r0(A) Gd 8.76 3.02 8059.11 3.5800 Tb 8.64 2.91 7867.74 3.5200 Sc 7.72 4.48 7659.47 3.2500 Y 9.61 3.55 8546.35 3.5846 Y b 8.78 3.98 3112.36 3.8800 Using the Maple program with the values of parameters D and r0 determined by experimental data [14], we calculate the Helmholtz free energy ψ of the HCP and BCC phases for Sc, Y, Gd and Tb metals (or FCC and BCC for Yb metal). 1300 1400 1500 1600 1700 1800 -9.6 -9.4 -9.2 -9.0 -8.8 -8.6 F r e e e n e r g y o f G d ( 1 0 - 1 9 J ) Temperature (K) BCC HCP Gd T c = 1599K T exp = 1508K 1400 1500 1600 1700 1800 1900 -9.6 -9.4 -9.2 -9.0 -8.8 -8.6 F r e e e n e r g y o f T b ( 1 0 - 1 9 J ) Temperature (K) BCC HCP Tb T c = 1587K T exp = 1562K 1400 1500 1600 1700 1800 1900 2000 -8.6 -8.4 -8.2 -8.0 -7.8 F r e e e n e r g y o f S c ( 1 0 - 1 9 J ) Temperature (K) BCC HCP Sc T c = 1692K T exp = 1610K 1400 1500 1600 1700 1800 1900 2000 2100 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 -0.88 F r e e e n e r g y o f Y ( 1 0 - 1 9 J ) Temperature (K) BCC HCP Y T c = 1834K T exp = 1751K Figure 1. Temperature dependence of the Helmholtz free energy for Gd, Tb, Sc and Y metals 109 Dang Thanh Hai and Vu Van Hung Table 2. Transition structure temperature of Gd, Tb, Sc, Y and Yb metals Metal Tc(K)[SMM] Texp(K)[15] Metal Tc(K)[SMM] Texp(K)[15] Gd 1599 1508 Y 1834 1751 Tb 1587 1562 Y b 1031 1068 Sc 1692 1610 Figure 1 shows the temperature dependence of Helmholtz energies of Gd, Tb, Sc and Y metals for BCC and HCP phases with the use of dashed and solid curves respectively. In Figure 1, it is clearly shown that the Helmholtz energy of the HCP phase is lower than the Helmholtz energy of the BCC phase in the temperature region lower than Tc and the BCC phase becomes more stable than the HCP phase in the temperature region higher than Tc. The transition temperature Tc of the HCP→ BCC transformation of Gd, Tb, Sc and Y metals in the presence of SMM are Tc(Gd) = 1599K , Tc(Tb) = 1587K , Tc(Sc) = 1692K and Tc(Y) = 1834K . The present value Tc of Gd, Tb, Sc and Y metals is still higher than the experimental value Texp(Gd) = 1508K , Texp(Tb) = 1562K , Texp(Sc) = 1610K and Texp(Y) = 1751K [15]. 800 900 1000 1100 1200 1300 1400 -5.7 -5.6 -5.5 -5.4 -5.3 -5.2 -5.1 -5.0 -4.9 -4.8 -4.7 F r e e e n e r g y o f Y b ( 1 0 - 1 9 J ) Temperature (K) BCC FCC Yb T c = 1031K T exp = 1068K Figure 2. Temperature dependence of the Helmholtz free energy for Yb metal Figure 2 shows the temperature dependence of Helmholtz energies of Ybmetal for BCC and FCC phases with the use of dashed and solid curves respectively. In Figure 2, it is clearly shown that the Helmholtz energy of the FCC phase is lower than the Helmholtz energy of the BCC phase in the temperature region lower than Tc(Yb) ≈ 1031K, and BCC phase becomes more stable than FCC phase with temperature region higher than Tc(Yb) ≈ 1031K. The transition temperature Tc of the FCC→ BCC transformation of Yb metal in the presence of SMM is 1031K. 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