Asymptotic Relations in the Thomas-Fermi-Dirac Atom Model

Asymptotic solutions of the Thomas-Fermi-Dirac equation for the case where the atom radius is vanishingly small (Fermi-Dirac limit) are considered. The limiting situation is examined in which the atomic number vanishes for a constant value of the Thomas-Fermi pressure (without exchange). It is shown that the pressure with exchange from the Thomas-Fermi-Dirac model approaches a finite limit in this case. The results yield limit curves approached for vanishing atomic number, by curves at constant atomic number of the pressure and other thermodynamic quantities with exchange, as a function of the pressure without exchange. The theoretical conclusions are confirmed by numerical data given by others and obtained directly by solution of the Thomas-Fermi-Dirac equation.