Use of Dirac–Coulomb Sturmians of the first order for relativistic calculations of two-photon bound–bound transition amplitudes in hydrogen-like ions

• Published in: Journal of physics. B, Atomic, molecular, and optical physics Vol. 40; no. 5; pp. 807 - 836
• Main Authors: Tetchou Nganso, H M, Kwato Njock, M G
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 14.03.2007; Institute of Physics
• Subjects: Atomic and molecular physics; Corrections to electronic structure; Electronic structure of atoms, molecules and their ions: theory; Exact sciences and technology; Physics; Relativistic and quantum electrodynamic effects in atoms and molecules; Selection rules; Energy-level transitions; H-like ions; Bound bound transition; Dirac equation; Transition amplitudes; Theoretical study; S matrix; Two-photon processes; Matrix elements; Green function; Relativistic corrections
• ISSN: 0953-4075; 1361-6455
• DOI: 10.1088/0953-4075/40/5/001

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 |12S towards the L and M shells (|22S, |32S, |32D3/2 and |32D5/2) 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 l1p = 2 and l2p = 2. The numerical results obtained for the transition from the ground state |12S towards the L and M shells enable us to draw inferences as to the improvements of our method.