Abstract |
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A fully relativistic treatment of the S-matrix elements describing two-photon bound - bound transition amplitudes in hydrogen-like ions is undertaken in the present work. Several selected transitions from the ground state vertical bar 1(2)S > towards the L and M shells (vertical bar 2(2)S >, vertical bar 3(2)S >, vertical bar 3(2)D(3/2)> and vertical bar 3(2)D(5/2)>) are described. For that purpose, we use the complete set of relativistic Sturmian functions derived by Szmytkowski ( 1997 J. Phys. B: At. Mol. Opt. Phys. 30 825) from the first-order Sturm-Liouville problems for the Dirac equation. The method followed consists of writing the matrix elements in terms of Green functions expanded over the first-order Dirac - Coulomb Sturmians. Previous approaches used a Sturmian basis associated with the Gell-Mann-Feynman equation. On the other hand, a distinctive feature of our tensor treatment is that the expressions derived are quite general and could be applied to any multipole of the two-photon bound - bound transitions. In the case of dipole transitions, considered also by Szymanowski et al ( 1997 Phys. Rev. A 56 700) in their calculations, the selection rules derived from our method lead to two additional terms related to l(1p) = 2 and l(2p) = 2. The numerical results obtained for the transition from the ground state vertical bar 1(2)S > towards the L and M shells enable us to draw inferences as to the improvements of our method. |