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Canadian Journal of Physics

The two-photon approximation for the four-photon decay of the 4d excited state in hydrogenThis paper was presented at the International Conference on Precision Physics of Simple Atomic Systems, held at École de Physique, les Houches, France, 30 May – 4 June, 2010.

Dmitry A. Solovyev, Leonti N Labzowsky

Published Jan 1, 2011• 3 citations
Publisher

Abstract

The method of two-photon approximation for multiphoton decay is applied to the decay of the 4d state in the hydrogen atom. In the process of four-photon decay the two-photon contribution that leads to radiation escape, from the interaction with matter, is considered. This may be helpful for a strict description of the recombination process in the hydrogen atom and, in principle, for the history of hydrogen recombination in the early universe.

DOI
10.1139/p10-073

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The method of two-photon approximation for multiphoton decay is applied to the decay of the 4d state in the hydrogen atom. In the process of four-photon decay the two-photon contribution that leads to radiation escape, from the interaction with matter, is considered. This may be helpful for a strict description of the recombination process in the hydrogen atom and, in principle, for the history of hydrogen recombination in the early universe.
The 4 s -1 s two-quantum decay in the hydrogen atom is calculated in the framework of quantum electrodynamics with allowance for cascades. It is shown that the pure two-photon contribution cannot be separated from the contribution of cascade processes. Interference between the transitions of these two types is comparable in magnitude with the pure two-photon contribution. The separation of pure two-photon contributions is discussed recently in connection with the role of two-photon decays in the separation of radiation from matter in the early Universe.
The problem of the evaluation of the two-photon decay width of excited states in hydrogen is considered. Two different approaches to the evaluation of the width including cascades channels are employed: the summation of the transition probabilities for various decay channels and the evaluation of the imaginary part of the Lamb shift. As application, the two-photon decay channels for the 3s3s3s level of the hydrogen atom are evaluated, including the cascade transition probability $3s\text{\ensuremath{-}}2p\text{\ensuremath{-}}1s$. An important role is assigned to the two-photon decays in astrophysics context, since processes of this kind provide a possibility for the decoupling of radiation and matter in the early universe. We demonstrate the ambiguity of separation of the ``pure'' two-photon contribution and criticize the existing methods for such a separation.
The hyperfine levels for the ground state of hydrogen are normally connected by magnetic dipole transitions. However, for astrophysical applications, it may be that two-photon electric dipole processes are also important over cosmological distances. This thesis sets up a general mathematical formalism for the calculation of two-photon processes within the hyperfine structure of an atom with spin 1/21/21/2, and applies it to the ground state of hydrogen. Rates are calculated for both spontaneous emission and absorption of radiation. Due to an extraordinary degree of cancellation in the matrix elements, the rates turn out to be too small to have a significant impact on astrophysical processes involving the absorption and emission of radiation. The results are nevertheless important in reducing the uncertainty due to possible contributions from these processes. Other two-photon processes involving Raman scattering are also considered and the relevant cross sections calculated. The calculations of the decay rate, absorption coefficient and Raman scattering cross section are extended to heavier hydrogenic ions as a function of the nuclear charge ZZZ, using an approximate interpolation formula for the hyperfine splitting in the ground state. Results are presented for the hydrogenlike ions from He+^++ to Pb81+^{81+}81+. Modifications for the case of isotopes with a nuclear spin greater than 1/21/21/2 are briefly considered
The two-photon decay mode of hydrogenic atoms from an arbitrary state (n1{n}_{1}n1​, l1{l}_{1}l1​, m1{m}_{1}m1​) to an arbitrary state (n2{n}_{2}n2​, l2{l}_{2}l2​, m2{m}_{2}m2​) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the 2s2s2s metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.
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.