Thomas-Fermi equation revisited: A variation on a theme by Majorana

Abstract

Majorana found a way to exploit the scaling properties of the Thomas-Fermi equation for converting this second-order differential equation into one of first order. We explore his method for the familiar neutral-atom solution and extend it to the solution that is relevant for weakly ionized atoms. Various integrals and other quantities with importance for atomic physics are recalculated and their values compared with the ones obtained in the 1980s by more tedious numerical procedures.

Figures (4)

• Figure 1: The TF functions of the two kinds. Curve a is the graph of ${x\mapsto F(x)}$ for case (i) and curve b is the graph of ${\Phi(x)/\Lambda^2\mapsfrom x}$ for case (ii).
• Figure 2: Graphs of the functions ${t\mapsto u(t)}$ and ${s\mapsto v(s)}$.
• Figure 3: The sums in (\ref{['eq:SA2']}) and (\ref{['eq:SB2a']}) as functions of $t$ or $s$. Curve a is the graph of ${t\mapsto2\gamma\bigl(U(t)+\overline{U}\bigr)+\log(1-t)}$; curve b is the graph of ${s\mapsto(2\sigma/3)\bigl(V(s)+\overline{V}\bigr)+\log(1-s)}$.
• Figure 4: Lin-log plot of the coefficients in (\ref{['eq:SA1']}), (\ref{['eq:SA2']}), (\ref{['eq:SB1']}), and (\ref{['eq:SB2a']}) for ${1\leq n\leq200}$ (or $5\leq n\leq204$ for $b_n$). The large-$n$ slopes are those of geometric sequences with convergence for ${\mathopen{\boldsymbol{|}}1-t\mathclose{\boldsymbol{|}}\lesssim1.20}$ and ${\mathopen{\boldsymbol{|}}1-s\mathclose{\boldsymbol{|}}\lesssim1.84}$, respectively, for the corresponding power series.