Third-order isotope-shift constants for alkali-metal atoms and ions

We derive expressions for third-order matrix elements of one- and two-particle operators within the framework of relativistic many-body perturbation theory and use these formulas to evaluate specific-mass isotope shifts and isotope field shifts in atoms and ions with one valence electron. Third-order corrections to isotope-shift constants for $3s,$ $3p,$ and $3d$ levels in sodium and sodiumlike ions with $Z=12\mathrm{}–18$ and $Z=26\mathrm{}$ are given. Applications to $4s,$ $4p,$ $3d,$ and $4d$ levels in K and $4s,$ $4p,$ and $3d$ levels in ${\mathrm{Ca}}^{+}$ are also given. Comparisons are made with previous calculations and with experiment. Including third-order corrections improves the agreement between theoretical and experimental values of the specific-mass isotope-shift constants for the cases considered.