High-pressure EXAFS Debye-Waller Factors of Metals

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VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 76-80 High-pressure EXAFS Debye-Waller Factors of Metals Nguyen Viet Tuyen1, Tran Thi Hai2, Nguyen Thi Hong2, Phan Thi Thanh Hong3, Ho Khac Hieu4,* 1 VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam 2 Hong Duc University, Thanh Hoa, Vietnam 3 Hanoi Pedagogical University No 2, Vinh Phuc, Vietnam 4 Duy Tan University, Da Nang, Vietnam Received 05 December 2017 Revised 16 February 2017; Accepted 20 March 2017 Abstract: The anharmonic correlated Debye model has been developed to investigate the pressure effects on the extended X-ray absorption fine structure (EXAFS) Debye-Waller factors of metals. The recent well-established Grüneisen parameter expressions have been applied to formulate the pressure-dependent analytical expressions of the effective spring constant, correlated Debye frequency and temperature. Combing with the anharmonic correlated Debye model, the expression of EXAFS Debye–Waller factor under pressure can be derived. Numerical calculations, performed for Fe and Cu metals show reasonable agreement with experiments. Keywords:EXAFS, Debye-Waller factors, Debye model, Anharmonicity, Pressure 1. Introduction One of the most effective methods for investigation the structure and thermodynamic properties of crystals is extended X-ray absorption fine structure (EXAFS) [1]. The anharmonic EXAFS provides information on structural and thermodynamic properties of substances. The EXAFS oscillation has been 2 analyzed by means of cumulant expansion approach containing the second cumulant σ2 σ which is an important factor in EXAFS analysis since the thermal lattice vibrations affect sensitively on the XAFS amplitudes through the factor exp 2σ 2k 2 . The second cumulant corresponds to the parallel mean square relative displacement or Debye-Waller factor (DWF). The EXAFS is sensitive to temperature and pressure [2] which can make changes of cumulants including DWF, which in turn lead to uncertainties in physical information taken from EXAFS. In the recent years, the remarkable developments of EXAFS techniques permit the experiments with unprecedented accuracy under extreme conditions of high pressure and temperature. In 2011, Hung et al. have developed the anharmonic correlated Einstein model to determine the DWF of crystals at high pressure [3]. However, this model is still limited. Recently the anharmonic correlated Debye model _______  Corresponding author. Tel.: 84-983036087 Email: hieuhk@gmail.com 76 N.V. Tuyen et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 76-80 77 (ACDM) has been used to calculate the temperature dependence of EXAFS cumulants of crystals at zero pressure [4]. The purpose of this work is to develop the ACDM for calculation and analysis of pressure effects on EXAFS DWF of metals at a given temperature. 2. Theory On the theoretical determination of EXAFS DWF, many great ideas has been put forward such as correlated Einstein model [5], statistical moment method [6],.... In line with the Debye model, Hung et al. developed the ACDM and successfully investigated the temperature-dependent EXAFS cumulants, including DWF [4]. In this model, the second cumulant has been derived as a 2πkeff σ2 πa ω q 0 1 Z q 1 Z q dq , (1) where q is the phonon wave number, a is the lattice constant, M is the mass of composite atoms, and k 0eff is the effective local force constant, Z q frequency and has the form as ω q ω0D and temperature θ 0D are ω0D 2 k0eff M sin exp β ω q with ω q qa 2 π . The correlated Debye frequency a q , 2 k0eff M ; θ0D is the phonon vibration ω0D kB , respectively. In this work, the ACDM will be developed to investigate the pressure (and volume) dependence of EXAFS DWF through the local force constant keff M ωD2 4 by considering the definition of the Grüneisen parameter in Debye model as ln ωD γG lnV , (2) where V is the volume of crystal, ωD is the Debye frequency depending on V (and also pressure P ). At low pressure, the Grüneisen parameter of material can be seen as constant. However, previous works [7, 8] showed that the Grüneisen parameter reduced gradually when pressure increased. Recently, through the consideration of low- and ultra-high-pressure limits in the Thomas-Fermi approximation, Burakovsky et al. proposed an analytic model of the Grüneisen parameter of solid at all densities as[9, 10] γG 12 here V0 and η 13 γ1η V V0 γ 2 ηq with γ1, γ 2 , q const, q 1 , (3) are the crystal volume at zero pressure and volume compression, respectively. By making the combination of the Eq. (3) with Eq. (2) and taking the integration, we derived the volume-dependent expressions of the Debye frequency ωD and temperature θD , respectively, as ωD η ω0 D η 1/ 2 exp 3γ1 η1/3 1 γ2 q ηq 1 , (4) N.V. Tuyen et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 76-80 78 θD η θ 0D η 1/ 2 exp 3γ1 η1/3 γ2 1 ηq q 1 , (5) ω0D kB of material at ambient pressure can where the Debye frequency ω0D and temperature θ0D be gathered from experiments or determined from the correlated Debye model [4]. In pursuance of investigation of the pressure effects on the thermodynamic quantities, we need to know the equation-of-state (EOS) of crystal. There are many EOSs that have been used on studying thermodynamic properties at high pressure of materials such as Birch-Murnaghan [11], Holzapfel [12]... In the work of Cohen et al. [13], the Vinet equation is found to be the most accurate one at high compression. The well-established Vinet EOS has the form as [14] P 3K 0 η 2/3 η1/3 exp 1 3 ' K 2 0 1 1 η1/3 , (6) where K 0 and K 0 are correspondingly the isothermal bulk modulus and its first-pressure derivative. 3. Results and discussion In this section, the expressions derived in the previous section will be used to numerically calculate thermodynamic quantities including the Debye frequency and temperature, and DWF of copper and iron metals. For the sake of simplicity, in the present work, the interatomic potential between two intermediate atoms is assumed to be the Morse potential V r De 2α r r0 2e α r r0 , where α describes the width of the potential, D is dissociation energy, and r0 is the equilibrium distance of the two atoms. It is obviously that the indispensable input parameters required to determine the thermodynamic quantities as functions of compression η (and pressure P ) are the isothermal bulk modulus K 0 , the first-pressure derivative K 0 and γ1, γ 2 ,q of Grüneisen parameter. The bulk modulus and its first-pressure derivative can be gathered from experiments while the values of γ1, γ 2 ,q could be obtained by fitting Eq. (3) with experimental data of copper [15] and iron [16]. The Morse potential parameters, K 0 and K 0 , and fitting parameters γ1, γ 2 ,q of copper and iron are shown in the Table 1. Table 1. Morse potential parameters D, α [17]; isothermal bulk modulus K 0 and its first-pressure derivative K 0 ; and fitting parameters γ1, γ 2 ,q of copper and iron. Metals γ1 γ2 q α (Å-1) D (eV) K 0 (GPa) K 0' Cu -2.6667 4.1935 1.1941 1.3588 0.3429 133.41 5.37 Fe -0.1603 1.4092 1.0003 1.3885 0.4174 148.4 6.126 N.V. Tuyen et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 76-80 79 In Fig. 1, we display the experimental Grüneisen parameters of copper [15] and iron [16] and our fitting curves. As it can be seen in this figure, the Grüneisen parameters of metals can be well described by the Eq. (3) up to high compression. Using the derived parameters γ1, γ 2 ,q , we continue calculating the pressuredependent Debye frequency and temperature, and then the EXAFS DWF of copper and iron. Fig1. Experimental Grüneisen parameters of copper and iron and our fitting curves. The pressure dependence of EXAFS second cumulants of copper and iron at room temperature are presented in Fig. 2 & 3. From the Fig. 2, we can see that the DWF curves of metals are almost similar and diminish gradually when pressure increases. These results will affect the EXAFS amplitude. Here, we should be noted that in the pressure below 12 GPa, iron is in the α phase with body-centered cubic structure while copper is in face-centered cubic structure. At pressure P 0 , the DWF of Cu and Fe are, 8.6 10 3 Å2 and 9.2 10 3 Å2, respectively. Up to pressure 12 GPa, the DWF σ 2 are reduced and have the values 6.6 10 3 Å2 and 7.5 10 3 Å2, correspondingly. This phenomenon can be explained that when pressure increases the vibration of atoms will be limited and it results in the reduction of atomic mean-square relative displacement (or DWF). In Fig. 3, because of the lack of DWF experimental data of these two metals, we present the change in DWF σ2 P σ2 0 along with those of calculations for metallic copper by Freund et al. [18]. It can be seen in this figure, results of our developed ACDM are in very good agreement with those of previous studies up to 10 GPa but with lower values. Fig. 2. EXAFS DWF of copper and iron as functions of pressure. Fig. 3. Change in DWF σ2 P σ 2 0 of copper as a function of pressure. 80 N.V. Tuyen et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 76-80 4. Conclusions In this work, the ACDM has been developed to investigate the EXAFS Debye-Waller factors of metals at pressure. The pressure-dependent analytical expressions of the Debye frequency and temperature, EXAFS Debye–Waller factor have been derived. We have performed numerical calculations for copper and iron metals. Theoretical calculations are in very good agreement with those of previous data verifying our developed theory. Our calculations show that the Debye-Waller factor of metals diminish gradually, and then reduce the EXAFS amplitude when pressure increases. This approach could be used to verify as well as analyze the future high-pressure experiments. It also could be applied to study other thermodynamic parameters in EXAFS theory in the near future. Acknowledgments This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2014.12. References [1] G. Bunker, Application of the ratio method of EXAFS analysis to disordered systems, Nucl. Instruments Methods Phys. Res. 207 (1983) 437–444. [2] R. Ingalls, G.A. Garcia, E.A. 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