Systematics of nondipole effects on photoelectron angular distributions for 3s and 4s photoionization in the region of quadrupole Cooper minima

Dipole effects are dominant in low energy photoeffect integrated cross sections, but quadrupole effects on differential cross sections (DCS) can be quite substantial, due to the presence of dipole-quadrupole interferences terms in the angular distribution parameters $\gamma $ and $\delta $. At low photon energies, quadrupole effects are seen in the parameter $\gamma $, dependent on dipole and quadrupole matrix elements and the cosines of phase shift differences. Calculations have been performed over a broad range of Z and a number of Cooper minima in $3s\rightarrow \epsilon d$ and $4s\rightarrow \epsilon d$ quadrupole transitions have been found for photon energies below 100 eV. The locus of these quadrupole matrix zeros (QMZ) have been mapped out; they appear above the ionization threshold for Z's of 11-30 for 3s and 19-50 for 4s. The nondipole photoelectron angular distribution parameter $\gamma $ has been calculated for these Z's, and significant variations, as a function of photon energy, have been found. In particular, $\gamma = 0$ at the location of the QMZ, and also is zero where the cosine of the p-d phase shift difference vanishes. On the other hand, $\gamma $ has a maximum in the vicinity of dipole Cooper minima. The combined effects of these three types of zeros create the pattern of variations of $\gamma $ as a function of energy and Z.