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Physical Review A

Comment on ‘‘Lifetime of excited atomic states’’

Viorica Florescu, I. F. Schneider, Ion Nicolae Mihailescu

Published Aug 1, 1988• 16 citations
Publisher

Abstract

We show that the two-photon nonresonant contribution to the lifetime of excited 3s and 3d states of hydrogen, as given by a formula recently derived by Cresser et al. [Phys. Rev. A 33, 1677 (1986)], can be evaluated in an economic way using the analytic expressions of the amplitudes of 3s-1s and 3d-1s two-photon transitions. Our numerical evaluation confirms the numbers given by Cresser et al.

DOI
10.1103/physreva.38.2189

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We show that the two-photon nonresonant contribution to the lifetime of excited 3s and 3d states of hydrogen, as given by a formula recently derived by Cresser et al. [Phys. Rev. A 33, 1677 (1986)], can be evaluated in an economic way using the analytic expressions of the amplitudes of 3s-1s and 3d-1s two-photon transitions. Our numerical evaluation confirms the numbers given by Cresser et al.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).
The relativistic and nonrelativistic approaches for the calculations of the two-photon decay rates of highly excited states in hydrogen are compared. The dependence on the principal quantum number ( n ) of the n s, n d and n p initial states is investigated up to n = 100 for the nonresonant emissions. For the n s states together with the main E1E1 channel the contributions of higher multipoles (M1M1, E2E2 and E1M2) are considered. For the n p states the E1M1 and E1E2 channels are evaluated. Moreover, the simple analytical formula for the E1M1 decay is derived in the nonrelativistic limit.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.
We investigate nonresonant contributions to resonant Rayleigh-scattering cross sections of atoms. The problematic nonresonant contributions set a limit to the accuracy to which atomic spectra determine energy levels. We discuss the off-resonance effects in one-photon transitions. We also show that off-resonance contributions for the 1S–2S two-photon transition in atomic hydrogen are negligible at current and projected levels of experimental accuracy. The possibility of a differential measurement for the detection of off-resonance effects in one-photon transitions in atomic hydrogen is discussed. PACS Nos.: 31.15-p, 12.20Ds