Angular-dependent time delay studies for Na 3s including both dipole and quadrupole in the vicinity of the dipole Cooper minimum

The angular distribution of Eisenbud-Wigner-Smith (EWS) time delay [1-3] for Na 3s photoionization in the region of the 3s E1 Cooper minimum is studied in this work. We obtain the time delays for both pure dipole (E1) transitions and the combined dipole and quadrupole (E1+E2) transitions[4]. To calculate the required wavefunctions, i.e., the initial and ionised state of the atom, we employ the Relativistic Multi-Configuration Dirac Hartree-Fock (RMCDHF) method. The reduced matrix elements are computed using well-established computational tools, namely GRASP [5,6] and RATIP [7]. The helicity fomulation[8,9] is used to determine the transition matrix elements. The interchannel coupling between the E1 and E2 channels is included in separate calculations. The combined effect of E1 and E2 transitions are included following a method similar to that used to study the photoelectron angular distribution using the differential cross section formulae developed by Derevianko et. al.[10].