Wigner threshold law in photodetachment and photoionization I am writing this question here because I have a problem in understanding the Wigner Threshold law in Photodetachment and Photoionization.
The Wigner Threshold Law is given by:
$$\sigma=E^{L+1/2},$$
where $\sigma$ is the photodetachment cross section, E is the kinetic energy of the detached electron from the anion, and L is the detached electron angular momentum. 
I have the following questions:
What is the threshold in this law? For this threshold law to be valid, should the energy be above or below the threshold? Where the literature says "near threshold", does this mean above or below the threshold - and how far?
I found the  threshold law for photodetachment but I couldn't find a threshold law for photoionization. Is there any threshold law for photoionization? It is known that the ejected electron and the neutral core (in the case of photodetachment) have an effective potential which is a sum of the interaction potential and a centrifugal potential:
$$V_{\mathrm {eff}}=V(r) + h/2mr^2 [l(l+1)].$$
Is this centrifugal potential (second term) still valid in the case of photoionization? 
PS: If anyone can suggest a textbook or any other reference, having the derivation and details about the Wigner law on a fundamental level (a graduate level so that I can understand the basics very well), that would be good.
 A: 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.
A: 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 $\hbar\omega$ and the detachment energy (related to either electroaffinity or ionization energy for photodetachement and ionization, respectively) through $KE=\hbar\omega-DE$. It has been shown by Wigner that, in the case where $KE$ is small (hence $\hbar\omega$ 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 
