Wigner threshold law in photodetachment and photoionization

The Wigner Threshold law describes the yield or cross-section of an ionization or detachment process where the incoming particles are just above the energy required for the reaction to occur (the threshold). It seems that Wigner described the case where there are only two outgoing particles (such as the ones you mention), and others extended his theory to three or more outgoing particles. Certainly papers seem to be out there describing threshold laws for a variety of processes.

All of these reactions have of course been studied in great detail using more powerful methods - the threshold laws allow some convenient simplifications.

As for your centrifugal potential - I can't derive it but I can't see why it would be different for ionization compared to detachment.

When dealing with dipole transition one is usually interested in the electronic dipole matrix element $langle Psi_{f} |mu|Psi_{i} rangle$ where $i$ and $f$ denote the initial and final state, respectively, and $mu$ is the dipole operator.

The photodetachment and photoionization process differs from bound to bound electronic transition because the final state $Psi_{f}$ corresponds to a electronic continuum. In this case, $Psi_{f}$ can be expressed as a product between the wave function describing the cationic (for photoionization) or neutral (for photodetachment) core, and the outgoing electron. Thus $Psi_{f}=Psi_{core}Psi_{ele}(KE)$ where $KE$ is the kinetic energy of the ejected electron. The rate or the cross section for such process is proportional to the square of the matrix element times the density of states of the outgoing electron: $$sigma propto langle Psi_{core}Psi_{ele}(KE) |mu|Psi_{i} rangle rho(KE)$$ $KE$ depends on the photon energy $hbaromega$ and the detachment energy (related to either electroaffinity or ionization energy for photodetachement and ionization, respectively) through $KE=hbaromega-DE$. It has been shown by Wigner that, in the case where $KE$ is small (hence $hbaromega$ small), the cross section of such process mostly relies on the probability that the outgoing electron tunnels through a centrifugal barrier arising from the effective potential $V_{eff}$ "felt" by the electron (same expression you gave). For ionization process $V(r)$ will have a Coulomb form that varies with $r^{-1}$ whereas for photodetachment process $V(r)$ will fall with higher power of $r$. In particular, for the last case, the electronic integral squared varies as $KE^{L}$ when assuming only (not 100 % here) charge dipole-induced interaction between the neutral core and the outgoing electron (which is the case for atomic anion for example). Since $rho(KE)$ varies as $KE^{1/2}$, the photodetachment cross section near threshold becomes $$sigma_{PD} propto KE^{1/2}KE^{L}=KE^{1/2+L} $$ The cross section increases with increasing $L$ since the centrifugal barrier depends on $L(L+1)$. Consequently a "$s$-like" ejected electron leads to a larger cross section. Some corrections are needed when dealing with molecules because $V(r)$ will also depend on other interaction terms if, for example, the molecule possesses a dipole or quadrupole moment. This expression of the cross section is not valid when considering ionization processes since $V(r)$ falls with $r^{-1}$. If I remember correctly, the later becomes constant.

For further reading see the original paper of Wigner : https://journals.aps.org/pr/abstract/10.1103/PhysRev.73.1002

And/or the following article: https://www.osapublishing.org/ao/abstract.cfm?uri=ao-19-23-4080