Relativistic all-order calculations of Stark-induced $6P_{1/2}-7P{1/2}$ amplitudes in thallium

Stark-induced amplitudes $\alpha$ and $\beta$ of the $6P_{1/2}-7P_{1/2}$ transition in Tl I are calculated in the relativistic single-double (SD) approximation, where single and double excitations of the Pb-like core are summed to all-orders in perturbation theory. Dipole matrix elements and energies for transitions from $6P_{1/2}$ and $7P_{1/2}$ states to $nS_{1/2}$ ($n$=7-10) and $nD_{3/2}$ ($n$=6-9) states are evaluated in the SD approximation and summed to give the dominant contributions to the polarizabilities. Remainders are evaluated using Dirac Fock wave functions. We obtain $\alpha=364\, a_0^3$ and $\beta= -297\, a_0^3$ compared with measured values$^1$ $\alpha=377\pm 8\, a_0^3$ and $|\beta|=313\pm 8\, a_0^3$. A calculation of the Stark shift within the $6P_{1/2}-7S_{1/2}$ transition is also carried out using SD wave functions and differs from precise measurements$^{1,\, 2}$ by about 5\%. \newline \newline $^1$ D. DeMille, D. Budker, and E. D. Commins, Phys.\ Rev.\ A {\bf 50}, 4657 (1994). \newline $^2$ S. C. Doret, P. D. Frieberg, A. J. Speck, D. S. Richardson, and P. K. Majumder, Phys.\ Rev.\ A {\bf 66}, 52504 (2002).