Analytic properties of the relativistic Thomas-Fermi equation and the total energy of atomic ions

The analytic properties of solutions of the relativistic Thomas-Fermi equation which tend to zero at infinity are first examined, the neutral-atom solution being a member of this class. A new length is shown to enter the theory, proportional to the square root of the fine-structure constant. This information is used to develop a perturbation expansion around the neutral-atom solution, corresponding to positive atomic ions with finite but large radii. The limiting law relating ionic radius to the degree of ionization is thereby displayed in functional form, and solved explicitly to lowest order in the fine-structure constant. To embrace this knowledge of heavy positive ions, as well as results from the one-electron Dirac equation, a proposal is then advanced as to the analytic form of the relativistic total energy E(Z,N) of an atomic ion with nuclear charge Ze and total number, N, of electrons. The fact that, for N>1, the nucleus is known only to bind Z+n electrons, where n is 1 or 2, indicates nonanalyticity in the complex Z plane, represented by a circle of radius Z∼N. Such nonanalyticity is also a property of the nonrelativistic energy derived from the many-electron Schrödinger equation. The relativistic theory, however, must also embody a second type of nonanalyticity associated with the known property for N=1 that the Dirac equation predicts electron-positron pair production when the electronic binding energy becomes equal to twice the electron rest mass energy. This corresponds to a second circle of nonanalyticity in E(Z,N), and hence to a Taylor-Laurent expansion of this quantity in the atomic number Z. The relation of this expansion to the Layzer-Bahcall series is finally discussed.