TFmix: A high-precision implementation of the finite-temperature Thomas–Fermi model for a mixture of atoms

Abstract

In this work we present a TFmix code intended for numerical calculation of the thermal part of electronic thermodynamic properties of a mixture of elements by the finite-temperature Thomas–Fermi model. The code is based on analytical models for both first and second derivatives of Helmholtz thermodynamic potential. All numerical calculations are made within a controlled high accuracy: tests for thermodynamic consistency give at least 11 coinciding decimal digits. The code calculates thermodynamic functions on a regular grid of isotherms and isochores; at each grid point some extensive parameters and the number of free electrons are output both for the whole mixture and for each component. Other extensive or intensive thermodynamic properties, including pressure, entropy, isochoric and isobaric heat capacities, isothermal and adiabatic sound velocities can be easily calculated from the information available at each grid point. Several unit systems are provided for convenience. A cross-platform graphical user interface is developed to simplify the use of the code. Program summaryProgram Title: TFmix, version 1.0Program Files doi:http://dx.doi.org/10.17632/mc3vj77jfn.1Licensing provisions: GPLv3Programming language: C, PythonNature of problem: Any substance consists of elements so its equation of state contains a contribution of electronic gas. Thermodynamics of the electronic gas in a mixture of ions and electrons has been studied in many approaches. Thermodynamic properties of a uniform ideal electron gas can be calculated using the well-known analytical model of Fermi-gas. On the other hand, models of electron gas which take into account interaction effects are quite complicated and require sophisticated computational techniques. Even a simplified semiclassical Thomas–Fermi model is based upon the numerical solution of a non-linear boundary problem. Two main issues of the Thomas–Fermi model restrict its usage: uncontrolled accuracy of calculated thermodynamic functions (especially second derivatives of a thermodynamic potential), and unphysical behavior of the model at relatively low temperatures.Solution method: Each atom in the mixture is surrounded by a spherical cell. The radii of the cells are fitted to equalize the chemical potentials of all atoms. A guaranteed accuracy of first derivatives of the thermodynamic potential is provided by a transformation of integrals over the Thomas–Fermi potential to a system of differential equations. One of equations in the system is the Thomas–Fermi equation. Second derivatives of the thermodynamic potential are calculated similarly with the only difference that a corresponding derivative of the Thomas–Fermi equation is used in the system of differential equation. To avoid the unphysical behavior of the Thomas–Fermi model at low temperatures we extract a thermal contribution to thermodynamic properties which vanishes at zero temperature. To eliminate the error which appears from the subtraction of the cold part at low temperatures we use asymptotic expressions for thermodynamic functions and the Thomas–Fermi equation. The code calculates regular tables of thermodynamic functions on a grid of isotherms and isochores including second derivatives of a thermodynamic potential. This information is necessary for astrophysical applications, for continuum mechanics simulation of processes in plasma and for the creation of wide-range equations of state. A graphical user interface is provided with the code and allows to specify input parameters, to perform calculations and to plot the results.Additional comments including restrictions and unusual features: GSL library version 1.16 or 2.x is required for compilation; matplotlib Python library is required to run the graphical user interface.