Extension of the Thomas-Fermi-Dirac Statistical Theory of the Atom to Finite Temperatures

Abstract

The Thomas-Fermi theory of the atom is generalized to include the effects of temperature as well as exchange. This leads to a nonlinear integral equation for the Fermi electron-momentum distribution function, and the usual Poisson equation for the electron-density distribution. Analytical solutions of the integral equation are given for the limiting cases of near-degeneracy and complete nondegeneracy, and a numerical method of calculating solutions in the intermediate case is described. A complete discussion of the thermodynamics of the atom is given; in particular, it is shown that the Gibbs free energy is the product of the number of electrons and the electronic chemical potential (Fermi energy), despite statements which have been made to the contrary. Numerical results have verified the virial theorem for all $Z$, $T$, and atomic volumes. The ratio of the calculated energy for $T=\mathcal{P}=0\mathrm{}$ to the experimental total ionization energy varies from 2.07 for H down to 1.33 for Al, and is presumably still closer to unity for higher-$Z$ elements. Some numerical results are given for iron over the density range 0.1 to 10 times normal and for values of $\mathrm{kT}$ from 0 to 1000 ev. Pressures, energies, and entropies are lower than the corresponding values calculated without exchange by as much as 40% at $kT=10\mathrm{}$ ev, by up to 10% at $kT=100\mathrm{}$ ev, and by only negligible amounts at $kT=1000\mathrm{}$ ev.