Photoelectron angular distributions from the subshells of high-Zelements

Numerical predictions are obtained, within a relativistic multipole central-field approximation calculation, for the radial matrix elements and resulting angular distributions of photoelectrons ejected from inner through outer subshells of uranium and for photoelectron energies from near threshold (1 eV) to 100 keV. Some data are also presented for outer subshells of mercury. We confirm that for inner shells higher-multipole effects persist to low energies, while we find that for outer shells such effects become small. For uranium inner $\mathrm{ns}$ subshells the asymmetry parameter ${\beta }_{\mathrm{ns}}$ differs from its nonrelativistic value 2, but not more than 30% (for $n<\mathrm{}4\mathrm{}$), until the energy becomes quite high (the radial matrix elements for $\mathrm{ns}\to \epsilon p_{\frac{1}{2}}$ and $\mathrm{ns}\to \epsilon p_{\frac{3}{2}}$ transitions for these inner shells are similar). For outer $s$ subshells relativistic spin-orbit splittings cause large deviations from the nonrelativistic predictions which can be understood from the magnitudes, zeros, and splittings of the two radial matrix elements. For U $6p_{\frac{1}{2}}$ and $6p_{\frac{3}{2}}$, ${\beta }_{6p}$ has the same qualitative features found in nonrelativistic calculations for outer $p$ shells of other high-$Z$ elements, but the angular distributions from the two states do differ in detail. Such differences in angular distributions of the photoelectrons from the two $j$ substates of given $l$ become smaller for inner $p$ shells. For outer $d$ and $f$ subshells the asymmetry parameter oscillates with photoelectron energy in a manner similar to that found by Manson in nonrelativistic dipole $d$-subshell calculations, and no qualitatively important relativistic effects have been identified; this is due to the decrease of the fine-structure-splitting interval with increasing angular momentum. For high energies the shapes of the angular distributions from a sequence of states of varying principal quantum number $n$, for fixed angular momentum ($\mathrm{JL}$), tend to merge into a common curve because at high energies the matrix elements are determined at small distances where all radial wave functions of given ($\mathrm{JL}$) have the same shape.