The Quantum Defect of Nonpenetrating Orbits, with Special Application to Al II

The conventional formula for the displacement in energy due to polarization of the atom-core, the major cause of quantum defect in nonpenetrating orbits, is -12e^2-{r^-4}. Modifications of this formula are given which are necessary if the absorption frequencies of the outer electron are not negligible compared with those of the atom-core. These modifications are more important for alkaline earths than alkalis, since in the former the atom-core includes the inner valence electron. The modified formulas require the evaluation of certain mean or "centroid" frequencies, for which approximate methods are given. Our centroid methods are readily adaptable to other problems, such as, for example, derivation of Wigner's formula for the apportionment of the dispersion f-sum between l-1 and l+1. With the aid of the modified polarization formula, the numbers of dispersion electrons for the resonance lines 3s-3p of Al III, Si IV are calculated to be 0. 83, 0. 80, respectively, from the quantum defects of the G terms of Al II, Si III. These values are, we believe, as reliable as those by other, more standard methods. The absolute (but not the relative) term values given in the literature for the spectrum of Si III are shown to be all too low by 9015 cm^-1 due to improper evaluation of the series limit. The calculated values of the quantum defect of the ^3F terms of Al II exclusive of the perturbation by 3p3d, also the values of the interaction matrix elements H (3p3d^3F; 3snf^3F) as computed by wave functions, are found to agree within the limits of error with the values obtained in the preceding paper in connection with the multiplet anomaly. From the behavior of the 3snf^1F terms, it is estimated that the unknown term 3p3d^1F is about 10, 000 cm^-1 beyond the series limit. The centroid modifications of the quadrupole corrections are calculated. The negligible singlet-triplet separation in the G terms of Al II is due to a fortuitous cancellation of penetration and quadrupole effects. The conditions are derived under which Langer's perturbation formula is theoretically valid.