Theory of polarization-averaged core-level molecular-frame photoelectron angular distributions: III. New formula for p- and s-wave interference analogous to Young's double-slit for core-level photoemission from hetero-diatomic molecules

We present a new variation of Young's double-slit formula for polarization-averaged molecular-frame photoelectron angular distributions (PA-MFPADs) of hetero-diatomic molecules, which may be used to extract the bond length. So far, empirical analysis of the PA-MFPADs has often been carried out employing Young's formula in which each of the two atomic centers emits a $s$-photoelectron wave. The PA-MFPADs, on the other hand, can consist of an interference between the $p$-wave from the X-ray absorbing atom emitted along the molecular axis and the $s$-wave scattered by neighboring atom, within the framework of Multiple Scattering theory. The difference of this $p$-$s$ wave interference from the commonly used $s$-$s$ wave interference causes a dramatic change in the interference pattern, especially near the angles perpendicular to the molecular axis. This change involves an additional fringe, urging us to caution when using the conventional Young's formula for retrieving the bond length. We have derived a new formula analogous to Young's formula but for the $p$-$s$ wave interference. The bond lengths retrieved from the PA-MFPADs via the new formula reproduce the original C-O bond lengths used in the reference $ab$-$initio$ PA-MFPADs within the relative error of 5 %. In the high energy regime, this new formula for $p$-$s$ wave interference converges to the ordinary Young's formula for the $s$-$s$ wave interference. We expect it to be used to retrieve the bond length for time-resolved PA-MFPADs instead of the conventional Young's formula.