基于Thomas-Fermi-Kirzhnits模型的物态方程研究

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

本文针对丝阵Z箍缩等高能量密度物理实验的数值模拟研究,建立了一种适用温度、密度范围宽的三项式半经验物态方程.三项式半经验物态方程包括零温自由能项,电子热贡献项和离子热贡献项.零温自由能项采用多项式拟合的方法确定.多项式系数通过多项式计算的结果与高压缩比区域和压缩比为1时零温Thomas-Fermi-Kirzhnits模型计算的结果对应相等得到.离子对物态方程的热贡献采用一种准谐振模型,此谐振模型可以描述离子在固态相中的行为,并且在高温度、低密度区域趋近于理想气体物态方程.电子对物态方程的热贡献采用含温Thomas-Fermi-Kirzhnits模型计算.利用所建立的三项式半经验物态方程计算了铝的等温压缩曲线,并与实验数据做了对比.给出了很宽温度、密度范围内铝的压强,其数据与相应的SESAME数据库数据做了对比.

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.