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Equation of State for Superconductors Based on Finite Strain Theory Under High Pressure H 6299 A.K. Pandey/ Elixir Crystal Research Tech. 42 (2012) 6299-6305 Available online at www.elixirpublishers.com (Elixir International Journal) Crystal Research Technology Elixir Crystal Research Tech. 42 (2012) 6299-6305 Equation of state for superconductors based on finite strain theory under high pressure H. K. Rai 1 and A.K. Pandey 2 1Solid state chemistry Lab, Sri Jai Narain (P.G.) College, Lucknow 2Department of Physics, Dr. Virendra Swarup Memorial Troust group of institutions, Unnao. ARTICLE INFO ABSTRACT Article history: We have reformulated equations of state (EOS) for superconductors based on Lagrangian Received: 29 October 2011; and Eulerian strains following the method developed by Stacey. The expressions thus Received in revised form: obtained are used conveniently to assess the validity of various EOS for different types of 2 January 2012; superconductors. The logarithmic EOS based on the Hencky measure of finite-strain is also Accepted: 13 January 2012; modified by including the higher terms arising from the fourth-order contribution in the Taylor series expansion of the free energy. In the present work, pressure of five different Keywords superconductors viz. YBa2Cu3O6.6, YBa2Cu3O6.93, YBa2Cu3O7, Bi2Sr2CaCu2O8 and Finite strain, Nd1.32Sr0.41Ce0.27CuO3.96 has been computed at different compression ranges using five Superconductors, different EOS based on finite strain theory viz. Third order Eulerian EOS, Third order EOS, Lagrangian EOS, Second order Born-Mie EOS, Third order Born-Mie EOS and Bardeen Eulerian strain, EOS. the obtained results are compared with available experimental data to test the validity Hencky strain. and success of these equation of states. © 2012 Elixir All rights reserved. Introduction The equation of state (EOS) is fundamentally important in The application of single-crystal X-ray diffraction studying the high pressure properties of solids, which causes techniques to several specimens that are being compressed structural transition. The importance of high pressure within a single, hydrostatic pressure chamber offers a precise investigations for high TC superconductors was recognized after method for determining the relative compressibility’s of the discovery of superconductivity at 30K in La-Ba-Cu-O different crystals [1-4]. In this way, it is possible to resolve the system by Bednorz and Muller [12]. The superconducting effects on the equation of state of varying composition, transition temperature of this system increases by 10K under the nonstoichiometry, site disordering, or other subtle factors. pressure of 15Kbar, which was not observed in any known Hence, the finite-strain theory of equations of state has been superconductors [13]. In fact this led to the successful idea of extended to the analysis of such data [5-7] . replacing La by smaller Y, resulting to YBa2Cu3O7 The purpose of this paper is to illustrate the finite-strain superconductors [14] with Tc at 90K. analysis of relative compressibility by applying it to the recently Since the discovery of the high-temperature published data for the high-pressure superconducting phase. In superconductors, there have been an enormous number of geophysics and high-pressure research, experimental or studies designed to increase the critical temperature and to theoretical data consisting of pressure, temperature, and volume provide an understanding of the mechanical responsible for triplets PVT are parameterized to a functional form for ease of superconductivity. It has been conformed that, in the mineral interpolation and extrapolation. These equations of state are then YBa2Cu3O7, a single phase YBa 2Cu 3O9-x (x ≈2) is responsible used to compute phase diagrams, or are used in geodynamic or for the high transition temperature. Borges et a [15] observed shock compression modeling, etc. [8], comprehensively reviews that pressure has a positive effect on superconducting transition equation-of-state formulations, and comprehensive reviews and temperature (Tc) and is enhanced at a rate of about 0.09 K/kbar comparisons of equations of state are given by Hama and for YBa 2Cu 3O6 and 0.19 K/kbar for YbBa 2Cu 3Ox. Suito[9], Stacey [10], and Duffy and Wang [11]. The primary In order to test equation-of-state formulations, it is purpose of this paper is to discuss the reasons for the accuracy of important to study large compressions, because most common the most-used formulation, based on finite strain theory and equations of state will work reasonably over small compression compare with the available experimental data for ranges. Thus we choose to study the very compressible superconductors. Here we show that at high compressions, and superconducting materials YBa 2Cu 3O6.6 ,YBa 2Cu 3O6.93 , for highly compressible materials the third order Birch YBa 2Cu 3O7, Bi 2Sr 2CaCu 2O8 and Nd 1.32 Sr 0.41 Ce 0.27 CuO 3.96 .The Murnaghan equation of state is considerably more accurate. The results turn out to be significant, not only as an illustrative above equations of state are appropriate for isothermal data. To example but also because they address empirical conclusions include thermal expansivity there are main approaches to fit that have been emerging from research in Geophysics and isotherms, and then tabulate or fit the parameters V0, K0, and condensed matter physics. K0´as functions of temperature. Since experimental data are not Theoretical always collected along isotherms, that method is most amenable Finite-strain theory has been applied extensively to to analysis of theoretical results. problems in geophysics. The resulting equations are called semi Tele: E-mail addresses: [email protected] © 2012 Elixir All rights reserved 6300 A.K. Pandey/ Elixir Crystal Research Tech. 42 (2012) 6299-6305 empirical because they contain parameters that have to be will depend critically upon choice of definition of strain. determined from experiment. The theory relates strain, or Different choices lead to non equivalent equations of state. compression, to pressure. Eulerian and Hencky measures of strain It is well known result [16,17] of continuum mechanics that The commonly used Eulerian measure of finite strain is the finite strain theory may be written equivalently in terms of based on the consideration of the distance element ds 2 and on the any of an infinite number of different definitions of the ‘strain’. analysis of its change during deformation, the final deformed For the case of an elastic solid (i.e. one possessing a free energy state being the reference state. The result is usually expressed in function which is uniquely defined in terms of deformation and terms of displacement field u i as [22,23] temperature), this result implies, specifically, that the stress- 1 ∂U ∂U ∂U ∂U strain relation may be written equivalently in an infinite number ()e = i + j − k k (9) of ways. E ij 2 ∂x ∂x ∂x ∂x R-R0≡ eR 0 (1) j i j i Then the stress-strain relation is Hydrostatic pressure produces an isotropic (negative) ∂F l compressional Eulerian strain εE [24], which can be written as σ = 1( + )e 1( + )e (2) 3/2 ∂η V V T 1 0 (10) Here R is a vector to a point in the deformed lattice; R is ε = 1 − 0 E 2 V the vector to the same point in the undeformed reference state, denoted in this section by a subscript zero. The components of both are referred to the same set of cartesian axes. e is the finite The Lagrangian strain, εL referred to the initial sate is strain tensor defined by equation (1). σ is the homogeneous 3/2 1 V stress at the point R. 1 is the unit 3×3 tensor. (1) ij =δij . F is the ε = −1 (11) Helmholtz free energy, or ‘elastic energy’ function. η is the L 2 V symmetric finite strain tensor defined in relation to e by 0 ηij = (1/2)(e ij +e ji +e mj emi ) (3) It may be noticed that if V goes to zero (i.e., for very large Equation (2) is exact, i.e. independent of the size of e. It is compressions), the langrangian strain tends to find the value derived by Leibfried and Ludwig [18] and Murnaghan [19]. -1/2, which makes its use in appropriate at high pressure. Alternatively, the strain f can be defined implicitly by Structural geologists and materials scientists use a different R - R 0= f R (4) definition of strain [25, 26]. The reference state is neither the And then the stress-strain relation is initial nor the final configuration, but the instantaneous ∂F 1 configuration of the body being deformed. In uniaxial σ = ()1 − 2ε (5) ε∂ V deformation, as the instantaneous length l of the body is T increased by an infinitesimally small increment dl, the ratio dl/l Where the symmetric tensor Є defined in relation to f by is considered as an increment of the current state of strain Єij = (1/2)(f ij +f ji -fmj fmi ) (6) dl Equation (5) is also exact, though its generality is restricted dε = (12) to special situations, including the important cases of isotropic l bodies and of pure strains unaccompanied by rotations [20]. The When the body goes from length l 0 to length l the total finite relations between strain and volume change is: strain or natural strain, also called the Hencky measure of strains 2 (V/V 0) = det (1+2 η)| (7) [27-29], is obtained by integration, l l dl l (13) 2 ε = ∫dε = ∫ =ln (V 0/V) = det(1-Є)| (8) H l l l l V is specific volume and V 0 is volume at zero pressure. 0 0 0 The difference between these two definitions of strain given The Hencky strain is not generally used because the integration by equation (1) and equation (4) is clearly in the selection of the is considered possible only when the increment principal axes do characteristic lengths used as coefficients of the strain.
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