Solution of the Temperature-Perturbed Thomas-Fermi Equation

Abstract

An analytic solution of the temperature-perturbed Thomas-Fermi equation of $n\mathrm{th}$ order is given in terms of quadratures on the unperturbed solution corresponding to zero temperature. The boundary and initial parameters corresponding to a solution, necessary for the thermodynamics, are given as explicit integrals in terms of the unperturbed solution, on the basis of a boundary condition for fixed atomic volume. Thus, every thermodynamic function at low temperature can be written as an asymptotic series, either in the even or the odd powers of the temperature, with coefficients which can be expressed in terms of quadratures on the zero-temperature solution. In particular, the coefficients of the asymptotic series corresponding to the basic thermodynamic functions (pressure, energy, entropy, enthalpy, Helmholts and Gibbs functions) can be expressed algebraically in terms of such quadratures. These results extend and generalize a similar result obtained by M. G. Mayer, restricted to the first-order temperature-perturbed case. The theory is applied in detail to the first-order case. From six neutral-atom zero-temperature solutions due to Feynman, Metropolis, and Teller, accurate values of the corresponding boundary and initial parameters of the first-order temperature-perturbed equation are obtained for the case of fixed atomic volume under the temperature perturbation.