Final state electron distributions of $^6$Li$^+$ following $\beta$-decay of $^6$He

When the isotope $^6$He undergoes $\beta$--decay into $^6$Li$^+$ ($^6{\rm He} \rightarrow {^6{\rm Li}^+} + e^- + \bar{\nu}_e$), there is a sudden change in the Coulomb potential binding the atomic electrons. In response, a given initial electronic state of $^6$He relaxes into all possible final electronic states of $^6$Li$^+$. In this work we calculate the redistribution function. The method of calculation involves expanding the initial state of $^6$He in terms of a complete set of final states of $^6$Li$^+$. A Hylleraas-type basis set was used to create a pseudospectrum (up to 797 states) which spans both the bound and continuum states of Li$^+$. Starting from both the $1\;^1S_0$ and $2\;^3S_1$ initial states $^6$He, the excitation probabilities to the final $^6$Li$^+$ states, as well as the total ionization probabilities, were calculated in the sudden approximation with neglect of nuclear recoil. The transition probabilities from the $1\;^1S_0$ initial state are compared with those of Wauters and Vaeck [Phys.\ Rev.\ C {\bf 53}, 497 (1996)] and Frolov and Ruiz [Phys.\ Rev.\ A {\bf 82}, 042511 (2010)]. Finally, the transition probabilities show resonant behavior near the energies corresponding to doubly-excited states.