Electron-optical properties of atomic fields

We present a unified discussion and illustrations of the electron-optical aspects of electron penetration into, or escape from, the inner region of atoms. Both processes may focus or defocus the amplitudes of wavefunctions and shift their phases, as manifested in countless phenomena ranging from level shifts to $\beta$-decay rates. A background survey begins by discussing the Fermi-Segrè formula for hyperfine splittings and emphasizes the interplay of hydrogenic and WKB approximations. The Phase-Amplitude Method, which determines amplitude ratios and phase shifts directly, proves useful for interpreting the systematics of these parameters along the Periodic System. We present results of survey calculations, carried through the Periodic System using Hartree-Slater potential fields, of: (a) $\frac{{\alpha }_l\mathrm{}\left(0\right)\mathrm{}}{{\alpha }_l\left(\infty \right)}$, the ratio of the wavefunction's amplitude at $r=0\mathrm{}$ to that outside the atom; (b) ${\delta }_l\mathrm{}(E=0\mathrm{})\mathrm{}$ and ${\frac{d{\delta }_l}{\mathrm{dE}}|}_{E=0\mathrm{}}$, the phase shift and its energy derivative at $E=0\mathrm{}$, and (c) the changes in ${\delta }_l\mathrm{}(E=0\mathrm{})\mathrm{}$ and $\frac{{\alpha }_l\mathrm{}\left(0\right)\mathrm{}}{{\alpha }_l\left(\infty \right)}$ induced by either a unit perturbation localized near $r=0\mathrm{}$ or a relativistic correction. Thus we provide a broad mapping of certain fundamental parameters based on rather crude but realistic calculations. These results are meant to serve as a bench mark in surveying problems and in checking new results, while standard methods are preferable for working out specific applications accurately.