Addendum to "Quasiparticle branch mixing rates in superconducting aluminum"

The quasiparticle charge distribution generated in a superconductor by tunnel injection from a normal metal is computed as a function of temperature and injection voltage. The low-voltage limiting form is compared with the distributions obtained from the Tinkham-Clarke and Pethick-Smith chemical potential models. The quasiparticle charge relaxation rate ${\tau }_{Q^{*}}^{-1\mathrm{}}$ due to inelastic scattering is computed for temperatures between $0.3T_c$ and $0.9999T_c$, and for injection voltages between $0.01\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{e}$ and $20\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{e}$. The limit $T\to T_c$ yields the voltage-independent Schmid-Schön and Pethick-Smith result ${\tau }_{Q^{*}}^{-1\mathrm{}}=\frac{\pi {\tau }_{E=0\mathrm{}}^{-1\mathrm{}}\left(T_c\right)\Delta \mathrm{}\left(T\right)\mathrm{}}{4k_{B}T}$, where ${\tau }_{E=0\mathrm{}}^{-1\mathrm{}}\left(T_c\right)$ is the inelastic scattering rate at the Fermi energy and at $T_c$. At temperatures below $T_c$, both the steady-state distribution function and ${\tau }_{Q^{*}}^{-1\mathrm{}}$ depend on the injection voltage.