Behavior of positive ions in extremely strong magnetic fields

The Thomas-Fermi statistical theory yields a total energy for positive ions with $N$ electrons and atomic number $Z$ in a very strong magnetic field $B$ of the form $E_{\mathrm{TF}}\mathrm{}(Z,N,B)=-Z^{\frac{9}{5}}{B}^{\frac{2}{5}}f\left(\frac{N}{Z}\right)$. It is first shown that for small $\frac{N}{Z}$, $f\left(\frac{N}{Z}\right)$ is proportional to ${\left(\frac{N}{Z}\right)}^{\frac{3}{5}}$. The Coulomb field energy is also calculated in the statistical limit as $E_{\mathrm{TF}}^{\mathrm{Coulomb}}\mathrm{}(Z,N,B)=\mathrm{const}\times Z^{\frac{9}{5}}{B}^{\frac{2}{5}}{\left(\frac{N}{Z}\right)}^{\frac{3}{5}}$, and the constant is obtained. Finally, the relation to the $\frac{1}{Z}$ expansion for ions in very strong magnetic fields is established.