Study on equation of state based on Thomas-Fermi-Kirzhnits model

Abstract

A wide-range semi-empirical equation of state is constructed for numerical simulation of high-energy density experiments, such as, wire-array Z-pinch etc. The equation of state consists of zero-temperature free energy term, and thermal contributions of electron and ion. Thomas-Fermi model, which was firstly put forward by Thomas and Fermi, is initially developed to study the electron distribution of multi-electron atoms. Since its advent, this model has been widely used in solid-state physics, atomic physics, astrophysics and equation of state computations. It is a particularly important model to describe the behavior of matter under extreme conditions of high temperature and high density. This model provides reasonably accurate results that are validated experimentally for some thermodynamic quantities, such as the pressure. However, the Thomas-Fermi model yields a pressure of a few GPa under normal density even at very low temperature, and the pressure is always positive, indicating an obvious limitation of this model. Kirzhnits has evaluated the influence of quantum effect and exchange effect on temperature-dependent Thomas-Fermi model and their contributions to the Thomas-Fermi equation of state. Basically, the Thomas-Fermi model with its quantum and exchange corrections which is called Thomas-Fermi-Kirzhnits model, can be applied to calculate the thermal contribution of electrons to the thermodynamic functions, which can lower the pressure given from the Thomas-Fermi model. The zero-temperature free energy term in the semi-empirical equation of state is described by a polynomial expression. The coefficients of the polynomial expression is calculated by using zero-temperature Thomas-Fermi-Kirzhnits model and the relation between thermodynamic quantities. A quasi-harmonic model is adopted to describe the behavior of ions. It is originally applied to calculate the contribution of ions in the condensed state. However, the quasi-harmonic model is close to an ideal equation of state in the high-temperature and low-density region. This model makes the description of the behavior of ions in the phase transition from the solid state to plasma state be approximated. Thomas-Fermi-Kirzhnits model is adopted to calculate the thermal contribution of electrons. The semi-empirical equation of state has the advantages of less calculation and clear physical concepts. Experimental data of isothermal compression at 300 K is fruitful and accurate. They can be used to verify the results of the semi-empirical equation of state. An isothermal compression curve is calculated by the present work and compared with experimental data. The pressures over a wide-range of temperature and density are derived and compared with corresponding data of SESAME database. The trajectory of the electrical explosion of aluminum is demonstrated from solid state to ideal plasma state.