= Finite nuclear size correction
{title2=$\Delta E_{ns}\simeq(2\pi/3)[Ze^2/(4\pi\epsilon_0)]|\psi_{ns}(0)|^2\langle r^2\rangle_{\rm ch}$}
Spreading a point nuclear <electric charge> over a small spherical <nuclear charge distribution> weakens the central attraction and raises electronic <energy levels>. In nonrelativistic first-order <time-independent perturbation theory>, a slowly varying $s$-state contact density gives the displayed shift. For a uniform sphere, $\langle r^2\rangle_{\rm ch}=3R^2/5$, and integration of the difference between spherical and point <Coulomb potentials> gives $\Delta E_{1s}=(4/5)Z^4\mathrm{Ry}(R/a_0)^2$ for a <hydrogen-like ion>. A nonrelativistic $p$ <wavefunction> vanishes at the origin, suppressing its leading shift to order $R^4$. Heavy <atomic nuclei> require relativistic and realistic-charge-distribution corrections. An observed transition <frequency> shifts by the difference of the two level corrections divided by the <Planck constant>.
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