Energy and Angular-Momentum Redistribution in Hydrogen Migdal Ionization

Abstract

MARVEL's first direct observation of Migdal ionization in neutron scattering marks an experimental milestone and opens a new avenue for probing electronic response to nuclear recoil. Hydrogen, both the simplest atom and a constituent of its molecular target, provides a controlled benchmark. Within the nonrelativistic sudden approximation, we compute its energy-differential and integrated ionization probabilities with the full recoil phase. At the recoil parameter $β\equiv v_N/(αc)=1.04$, the full-to-dipole spectral ratio rises from $0.43$ when the emitted-electron energy is $1\%$ of the hydrogen binding energy to $5.0$ when it is $2.3$ times that energy. Near $β=10$, partial waves with orbital angular momentum $\ell\ge3$ carry more than $90\%$ of the calculated continuum probability. An independent bound-state-closure evaluation verifies the absolute normalization: at eight recoil values, its ionization probabilities agree with the continuum-integrated results to relative discrepancies below $4.4\times10^{-7}$. These results extend the hydrogen dipole response to the fast-neutron regime and establish energy and angular-momentum redistribution as linked consequences of resolving the recoil phase across an atom.