Applying Majorana’s scaling transformation to Thomas–Fermi solutions with fixed ionization boundary conditions

Determine how to employ Majorana’s scaling (homology-invariant) transformation to analyze the family of solutions of the Thomas–Fermi differential equation f''(x)=x^{-1/2}f(x)^{3/2} (with f(x)≥0) that satisfy the boundary conditions f(0)=1, f(x0)=0, and −x0 f'(x0)=q for 0<q<1, where the positive radius x0 depends on the ionization parameter q.

Background

The paper develops Majorana’s scaling approach for the Thomas–Fermi (TF) equation, converting the second-order TF equation into first-order formulations for two classical cases: (i) the neutral-atom solution F(x) with F(0)=1 and asymptotics F(x)~144/x^3 as x→∞, and (ii) the weakly ionized-atom solution Φ(x) defined on 0<x≤1 with Φ(1)=0. For both, the authors derive convergent power series and compute physical integrals and energy coefficients.

Beyond these two cases, there is a broader family of TF solutions characterized by a finite radius x0 at which the potential vanishes and a boundary slope fixed by −x0 f'(x0)=q with 0<q<1 (interpreted as the degree of ionization). The authors explicitly note that it remains unknown how Majorana’s transformation can be used to study this family, indicating an open direction for extending the first-order reduction and associated parameterizations.