On the relativistic Thomas-Fermi model

The relativistic generalization of the Thomas-Fermi model of the atom is derived. It approaches the usual nonrelativistic equation in the limit Z ? Z_crit, where Z is the total number of electrons of the atom and $\text{Z}{}_{\mathrm{<mtext>crit</mtext>}}\text{=(}\text{3π}\text{4}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{α}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ and α is the fine structure constant. The new equation leads to the breakdown of scaling laws and to the appearance of a critical charge, purely as a consequence of relativistic effects. These results are compared and contrasted with those corresponding to N self-gravitating degenerate relativistic fermions, which for $\text{N ≈ N}{}_{\mathrm{<mtext>crit</mtext>}}\text{=(}\text{3π}\text{4}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(m/m}{}_{\mathrm{<mtext>p</mtext>}}\text{)}{}^3$ give rise to the concept of a critical mass against gravitational collapse. Here m is the mass of the fermion and $\text{m}{}_{\mathrm{<mtext>p</mtext>}}\text{=(?c/G)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is the Planck mass.