Slater sum for central field problems characterized by its s-wave component alone

Abstract

For the hydrogenlike atom, with central potential −Z/r, partial differential equations exist for the Slater sum Z(r,β) [β=(kBT)−1] and for its s-wave (l=0) component Z0(r,β). It is shown that Z can be eliminated, to lead to a result in which Z(r,β) is solely characterized by Z0(r,β). A similar situation is exhibited for the three-dimensional isotropic harmonic oscillator, for which closed forms of both Z(r,β,ω) and Z0(r,β,ω) can be obtained explicitly. Finally, a third central field problem is considered in which independent electrons are confined within a sphere of radius R, but are otherwise free. We are able to derive explicitly for this model the s-wave component Z0(r,β,R). The full Slater sum Z(r,β,R) then is also analyzed in some detail.