Accurate and efficient methods for the evaluation of vacuum-polarization potentials of orderZαandZα2

Abstract

Rational approximations are presented for the second-order (Uehling) and fourth-order (Kallen-Sabry) vacuum-polarization potentials in configuration space. These approximations may be applied to point-charge or finite-inducing-charge distributions. For the second-order potential, two approximations are given, with nominal accuracies of nine and four figures for the range 0 & r & oo. For the fourth-order potential, one approximation of about three-figure accuracy is given for the range 0 & r &4, .Although the solution for the lowest-order electron-positron vacuum-polarization potential (second-order correction to the photon propagator) around a point charge was reduced to quadrature' in 1935, no entirely satisfactory means of evaluating this result in routine problems has yet been given. This potential is needed for several applications, including the calculation of exotic (and electronic) atom energy levels, and the calculation of some charged-particle scattering cross sections. The best methods of evaluation available to date are two expansions about r= 0 (r is the distance from the point charge). One is a series due to McKinley. ' The other is a different expansion' based on the so-called Glauber technique, ' a technique which is apparently due to Rarita and Sommerfield' and which involves exponential integrals.Both of these methods fail at large r, and they suffer from unnecessary computational effort for the level of accuracy that they yield. In the present note we report two rational approximations to this function. These approximations are valid over the entire range 0( r&~, and they represent considerably improved levels of accuracy and/or efficiency. In addition, we report a somewhat less accurate polynomial fit to the fourth-order vacuum-polarization function.The second-order vacuum-polarization potential around a static point charge Za, acting on a particle of charge -e, is 2 g~g2 2r i', (r) = --K, 3 pr 2 where fd'r'p(r') =Z. If p(r) is spherically symmetric, the angular integrations may be performed to obtain U, (r) =--' dr'r'p(r') K,r r'~-2

QED radiative corrections and many-body effects in atoms: vacuum polarization and binding energy shifts in alkali metals

“…For example, the code of Hnizdo [24] or the expansion of Fullerton and Rinker [25]. We will instead consider the case of the step-function density (homogeneous charge distribution) and reduce the double integration, Eq.…”

A fully relativistic approach for calculating atomic data for highly charged ions

“…(2. 22). The values correspond to including generalized Breit interaction plus other QED corrections and use nuclear size (all of which are discussed in the next section), while G are obtained without additions or corrections.…”

“…\.~).20) and (~3. 22) as the initial and final states the reduced matrix element (f3tlt I I p(l) II f3~1:). However, as noted previously, the tional filled subshells inactive electrons make no contribution.…”

“…QED corrections were implemented in same manner as in the Grant using the work of F\tllerton and Rinker [22] and Mohr [23,24]. The nu merical procedure for including corrections our program is that the diagonal matrix elements of these interactions are added after diagonalization of the Dirac-Coulomb Hamiltonian, eq.…”

“…22) where l(Ry) is the ionization energy in Rydbergs and '/1, c/l is the incident electron kinetic energy in threshold units. In order to test the utility of such a reduced cross section for RDW calculations, we first considered ionization of electrons residing in sllbshells having principal quantum numbers n = 3, 4 and 5.…”