Asymptotic Relations between the Thomas—Fermi—Dirac and Thomas—Fermi Atom Models. I. Case of Low Atomic Number

Abstract

An asymptotic relation between the pressure with exchange on the Thomas—Fermi—Dirac atom model and the corresponding pressure without exchange on the Thomas—Fermi model, obtained by Gilvarry for the limit of vanishing atomic number Z, is derived by a direct physical argument. As compared to the original derivation, this procedure obviates the need for solutions of the differential equations of the models and for discussion of termination of the series so obtained. The virial theorem is used to delimit the domain of validity of the result, and thus to indicate the circumstances under which correction terms are necessary from the solution of March and the general asymptotic solution of Rijnierse for the Thomas—Fermi—Dirac equation at small atom radius. The limiting form for the radius of an isolated Thomas—Fermi—Dirac atom as a function of the atomic number in the case where Z approaches zero is determined on the basis of the Jensen boundary condition. This expression yields an immediate interpretation of the parameter entering the connection between the pressures with and without exchange. The results are discussed in terms of homology transformations of the Thomas—Fermi—Dirac and Thomas—Fermi equations. Corresponding asymptotic forms for other thermodynamic functions (parameters associated with the energy and the equation of state, as well as the compressibility) are considered also for the case of vanishing atomic number.