Role of phonon anharmonicity and electronic free energy on thermodynamic properties of fcc ytterbium

Abstract

Although many thermophysical properties of lanthanide family members are similar, ytterbium is an exception. The coefficient of thermal expansion for Yb is almost three times as large as the coefficients for other rare-earth metals, a clear manifestation of anharmonicity. In the present study, therefore, the influence of the phonon anharmonicity and the electronic free energy on thermal expansion and other thermodynamic properties of ytterbium has been investigated from absolute zero to the melting temperature (T). We used first principles density functional perturbation theory combined with thermal perturbative treatment for including intrinsic anharmonicity. Modeling Yb as an anharmonic oscillator as suggested by Oganov and Dorogokupets [J. Phys. Condens. Matter 16, 1351 (2004)], an anharmonic vibrational contribution is included in a parametric way. It has allowed us to unveil the effect of anharmonicity connecting the low-temperature quantum correction up to the high-temperature classical value. Furthermore, due to the complex behavior of 4f-shell electrons, the electronic excitation was computed through the fixed electronic density-of-state approximation. It remains two orders of magnitude higher than the anharmonic lattice term. Combining these contributions, we can evaluate several T-dependent but zero-pressure thermodynamic properties of Yb in its fcc phase. Since, at lower pressure, anharmonicity increases with temperature, one of the objectives of this study is to examine the importance of anharmonicity in determining these properties. For instance, an excellent agreement is found for the linear thermal expansion for the entire temperature range, whereas other properties such as entropy, bulk moduli, thermal Grüneisen parameter, and the phonon frequency shifts are also in agreement with the reported findings; notably, the discrepancy observed in the enthalpy and specific heats at high-T is discussed. A detailed analysis has suggested that an additional contribution from point defects is needed, like vacancy formation, for an accurate calculation of specific heat, while higher-order terms in temperature-dependent perturbative series are mandatory for enthalpy. Electronic contribution remains positive for caloric properties. The theory of anharmonic phonon–phonon interaction and computed thermal expansion of the crystal was finally used to analyze renormalized phonon frequency. The significant objective of the study is to elucidate the role of electronic agitations and intrinsic phonon thermal stress as a physical mechanism over and above the dominating volume expansion effect, which is responsible for restricting an overwhelming quasi-harmonic thermal expansion and a rapid decrease in bulk moduli close to melting. We propose that a quantitative agreement for entropy and enthalpy requires a delicate balance between T2-dependent and higher-order terms in an anharmonic perturbation series.

Structural and mass transport properties of liquid ytterbium in the temperature range 1123 K–1473 K

“…According to the Lindemann law, melting occurs when the root-mean-square vibration amplitude exceeds a threshold value (ranging between 0.05 and 0.2) of the fraction of interatomic distance a 0 , i.e., u 2 = C L a 0 , C L being the Lindemann constant. In our previous paper, we reported the thermophysical properties of Yb in the fcc phase [38] up to normal melting temperature (T M = 1100 K) by combining selfconsistent DFT calculations with the anharmonic phonon contribution. Having obtained the total Helmholtz free energy F (V, T M ), including the phonon anharmonic part, [38] we can estimate C L as follows.…”

“…In our previous paper, we reported the thermophysical properties of Yb in the fcc phase [38] up to normal melting temperature (T M = 1100 K) by combining selfconsistent DFT calculations with the anharmonic phonon contribution. Having obtained the total Helmholtz free energy F (V, T M ), including the phonon anharmonic part, [38] we can estimate C L as follows. Following Joshi et al, [48,49] we have computed the effective vibrational force constant at T M and thereby determined θ e on replacing E e by F (V, T M ).…”

“…( 4). The only pseudopotential parameter r c and the Born-Mayer parameters (α and γ) are tuned to optimize the PDC at RT equilibrium volume with experimental [38] and DFT-based results. [39] These parameters are given in Table 1, and they are used for the entire range of temperature that we explore in the study.…”

“…[32][33][34][35][36][37] An important feature in treating the liquid metals within such empirical local pseudopotential formalism is the determination of the potential parameters. In our recent paper, [38] we have deduced the effect of phonon anharmonicity and electron free energy on the thermodynamic properties of fcc Yb within the density functional theory (DFT). It appears for Yb that Lindemann's criterion is applicable within the prescribed accuracy, and as elaborated below, it is sensible to deduce the potential parameters from the vibrational response of the materials.…”

“…It appears for Yb that Lindemann's criterion is applicable within the prescribed accuracy, and as elaborated below, it is sensible to deduce the potential parameters from the vibrational response of the materials. In the present study, the pseudopotential parameters are determined by fitting the experimental [39] and DFT-based phonon dispersion relation [38] at room temperature (RT). Once the pseudopotential parameters were obtained, they were kept constant for further calculations.…”