Some Exact Radial Integrals for Dirac-Coulomb Functions

Abstract

The zero energy loss Dirac-Coulomb integrals are evaluated using the technique of contour integration. The expressions obtained have a closed analytic form, showing that these integrals are formally similar to the corresponding classical and nonrelativistic quantum mechanical, zero energy loss integrals which also have exact elementary solutions.Application of the zero energy loss Dirac-Coulomb integrals occurs in inelastic electron scattering and similar problems. The investigation of the finite energy loss Dirac-Coulomb integrals requires a study of the zero energy loss integrals as a preliminary. 2 L. C. Biedenharn and P. J. Brussaard, Coulomb Excitation (to be published by Oxford University Press). .' K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Wmther, Rev. Mod. = 2 L -2 k L [r(L)]21 r(l + 1 + il1) I. (7) r(2L) r(l + L + 1 + il1) 9 G. Breit and M. H. Hull in Handbuch der Physik edited by S.

Transformation of bound states of relativistic hydrogenlike atoms into a two-component form

“…Refs. [33][34][35][36]. For the analytical results for the diagonal matrix elements, we refer the reader to Appendix B.…”

DWBA theory for elastic scattering of polarized electrons from heavy unpolarized nuclei

“…When recoil is neglected (i.e. k f = k i ), the second integral in (4.1) can be calculated analytically according to the formulae given by Reynolds et al [48]. With the normalization (2.15) we have…”

“…where G κ (Rey) and F κ (Rey) are the Coulomb-Dirac partial waves as defined in [48]. In order to speed up the calculations for small angular momentum numbers (|κ| < κ max ) it was suggested [32] that the infinite integral in (2.13), starting from R m , can be calculated by means of an (asymptotic) series expansion, the only ingredient being the knowledge of the boundary values g κ (R m ) and f κ (R m ).…”

Elastic electron scattering and the nuclear magnetization distribution

“…(7) and (8) The calculational procedure is to integrate numerically the radial equations [Eq. (4)] from the origin to the nuclear cutoff radius R and compare the values of f"(r) and g"(r) to the point Coulomb wave functions and extract the phase shifts as outlined above.…”

“…The phases are inserted into Eqs. (7) and (8) and the series reduction method of Yennie, Ravenhall, and Wilson' is used to speed convergence of the partial-wave series for the amplitudes. This procedure works quite well and enables us to calculate charge scattering all the way to 180'.…”