The electric quadrupole channel of the 7.8 eV $~^{229}\mathrm{Th}$ transition

The unique isomeric transition at 7.8 eV in $~^{229}\mathrm{Th}$ has a magnetic dipole ($M1$) and an electric quadrupole ($E2$) multipole mixing. So far, the $E2$ component has been widely disregarded. Here, we investigate the nuclear physics nature and the impact of the $E2$ decay channel for the nuclear coupling to the atomic shell based on the newest theoretical predictions for the corresponding reduced nuclear transition probabilities. Our results show that the contribution of the $E2$ channel is dominant or at least of the same order of magnitude for internal conversion or electronic bridge transitions involving the atomic orbitals $7p$, $6d$ and $5f$. Notable exceptions are the internal conversion of the $7s$ electron and the electronic bridge between the electronic states $7s$ and $7p$, for which the $M1$ component dominates by two to three orders of magnitude. Caution is therefore advised when considering isomeric excitation or decay via nuclear coupling to the atomic shell, as the involved orbitals determine which multipole transition component dominates.