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Radiative Corrections to the Ground-State Energy of the Helium Atom

1957; American Institute of Physics; Volume: 108; Issue: 5 Linguagem: Inglês

10.1103/physrev.108.1256

1536-6065

P. K. Kabir, E. E. Salpeter,

Atmospheric Ozone and Climate

The radiative corrections of relative order ${Z}^{2}{\ensuremath{\alpha}}^{3}$ (absolute order ${Z}^{4}{\ensuremath{\alpha}}^{3}$ ry), corresponding to the Lamb shift terms arising from the nuclear Coulomb potential, and some of the $Z{\ensuremath{\alpha}}^{3}$ corrections, arising from radiative interactions between the electrons, are calculated for the ground-state energy of the helium atom. The ${Z}^{2}{\ensuremath{\alpha}}^{3}$ corrections are all calculated, but of the $Z{\ensuremath{\alpha}}^{3}$ corrections, which are expected to be much smaller, we retain only those containing $\mathrm{ln}\ensuremath{\alpha}$ as a factor---these may be estimated rather easily---and neglect the rest, regarding unity as being small compared to $\mathrm{ln}\ensuremath{\alpha}$.The ${Z}^{2}{\ensuremath{\alpha}}^{3}$ corrections require the calculation of an "average excitation energy" similar to the one defined by Bethe for the hydrogen atom. It is found that virtual transitions to states $(1s)\ifmmode\times\else\texttimes\fi{}(\mathrm{continuous} p)^{1}P$ are the most important, and these are calculated using the momentum matrix-elements obtained by Huang, who used a six-parameter Hylleraas wave function for the ground state, and a product wave function with $Z=2$ for the $s$-electron and $Z=1$ for the $p$-electron (full screening), for the excited states. Transitions to states other than $(1s)(\mathrm{continuous} p)^{1}P$ are also considered, and the value of the "average excitation energy" for helium is found to be 80.5\ifmmode\pm\else\textpm\fi{}10 ry, where the limits represent an estimate of the probable error of the result. The radiative correction to the ionization potential, which will be the difference of the corrections for the two-electron atom and the ion, is found to be -1.26\ifmmode\pm\else\textpm\fi{}0.2 ${\mathrm{cm}}^{\ensuremath{-}1}$, where the error includes an estimation of the $Z{\ensuremath{\alpha}}^{3}$ terms which are not calculated.The corresponding radiative corrections are calculated, less accurately, also for ${\mathrm{Li}}^{+}$, and the results generalized for helium-like ions of higher $Z$ by an extrapolation formula.