Nondipole effects in time delay of photoelectrons from atoms, negative ions, and endohedrals

We investigate the nondipole effects in time delay of photoelectrons emitted by many-electron atoms, negative ions, and respective endohedrals. We present the necessary general formulas in the frame of the random phase approximation with exchange (RPAE) applied to atoms, negative ions, and properly adjusted to endohedrals. We concentrate on low photon energy region, where nondipole effects are very small in the cross sections, but become observable in angular distributions. We not only derive the formulas for nondipole effects in time delay but also perform corresponding numeric calculations. We demonstrate how the nondipole corrections can be isolated in experiment. Concrete calculations are performed for noble gas atoms Ar and Xe, isoelectronic to them negative ions Cl$^-$ and I$^-$ and endohedrals Ar(Cl$^-$)@C$_{60}$ and Xe(I$^-$)@C$_{60}$. We found that the forward–backward photoelectron time delay differences give direct information on nondipole effects. They proved to be quite measurable and prominently affected by the presence of the fullerenes shell.