Isotope shifts and transition frequencies for the S and P states of lithium: Bethe logarithms and second-order relativistic recoil

Abstract

Isotope shifts and total transition frequencies are calculated for the $2\phantom{\rule{0.16em}{0ex}}^{2}S\ensuremath{-}3\phantom{\rule{0.16em}{0ex}}^{2}S$ transition of the lithium isotopes $^{6}\mathrm{Li}, ^{7}\mathrm{Li}, ^{8}\mathrm{Li}, ^{9}\mathrm{Li}$, and the halo nucleus $^{11}\mathrm{Li}$. The accuracy is improved for previously calculated relativistic and quantum electrodynamic corrections, and in particular a disagreement for the Bethe logarithm is resolved for the ground $^{2}S$ state. Our previous result is confirmed for the $2\phantom{\rule{0.16em}{0ex}}^{2}P$ state. We use the pseudostate expansion method to perform the sum over virtual intermediate states. Results for the second-order relativistic recoil term of order ${\ensuremath{\alpha}}^{2}{(\ensuremath{\mu}/M)}^{2}$ Ry are shown to make a significant contribution relative to the theoretical uncertainty, but because of accidental cancellations the final result for the isotope shift is nearly unchanged. However, the spin-orbit term makes an unexpectedly large contribution to the splitting isotope shift (SIS) for the $2\phantom{\rule{0.16em}{0ex}}^{2}P_{1/2}\ensuremath{-}2\phantom{\rule{0.16em}{0ex}}^{2}P_{3/2}$ fine structure, increasing the theoretical value for the $^{6}\mathrm{Li}\ensuremath{-}^{7}\mathrm{Li}$ isotopes to $0.556\phantom{\rule{0.16em}{0ex}}31(7)\ifmmode\pm\else\textpm\fi{}0.001$ MHz. A comparison is made with high-precision measurements and other calculations for the SIS and for the total $2\phantom{\rule{0.16em}{0ex}}^{2}S\ensuremath{-}3\phantom{\rule{0.16em}{0ex}}^{2}S$ transition frequency.