Relation between the energy-weighted sum rules for nuclear photoabsorption and nuclear muon capture

A relation between the energy-weighted sum rules for nuclear photoabsorption and nuclear muon capture is derived and is used to show that, at least in the case of nuclei with $Z=A-Z$, the mean energy of the neutrinos emitted in muon capture does not vary appreciably as $A$ and $Z$ vary. The Foldy-Walecka expression for the total muon-capture rate is rederived on the basis of the energy-weighted sum rule and is used to show that, at least in the case of nuclei with $Z=A-Z$, and apart from the difference in ground-state energy between the nuclei [$Z-1\mathrm{}$, $A=2Z$] and [$Z$, $A=2Z$], the mean energy of excitation of [$Z-1\mathrm{}$, $A=2Z$] due to muon capture is essentially equal to the energy of the giant dipole resonance in [$Z$, $A=2Z$].