Thermodynamics of the Thomas-Fermi Atom at Low Temperature

Abstract

It is shown that an expression for the Helmholtz function on the Thomas-Fermi atom model permits one, in the limit of low temperature $T$, to expand the total energy as an asymptotic series in ${T}^{2}$. The leading term of the expansion is a known analytic expression for the energy at $T=0$, from which one can calculate the pertinent thermodynamic functions for this case in their dependence on the atom radius and on the boundary and initial parameters associated with a solution of the zero-temperature Thomas-Fermi equation. Published values of these parameters are fitted by functions of the atom radius having the correct asymptotic behavior in the two limiting cases of an infinitesimal and an infinite atom. This procedure makes thermodynamic functions in this case directly available. Thermodynamic functions for the case of a first-order temperature perturbation are derived from the corresponding term in the asymptotic expansion of the energy. They depend (in addition to dependence on parameters of the unperturbed atom) on two parameters derived from solution of the differential perturbation equation. These parameters differ according as the differential perturbation equation is solved under a condition of fixed atomic volume or of zero initial slope (which is computationally more convenient and corresponds to published solutions). The asymptotic forms of the parameters in the first case are determined for the two limits of an infinitesimal and an infinite atom. The boundary and initial parameters corresponding to fixed atomic volume are expressed in terms of the parameters corresponding to a solution under zero initial slope, and are evaluated in three cases from published solutions. These values are fitted by functions of the atom radius which have the proper limiting behavior for an infinitesimal and an infinite atom. Thus, approximate values of the first-order temperature perturbations of thermodynamic functions become directly available.