  The neutrino mass experiment KATRIN uses conversion electrons from the 32.2
keV transition of the nuclear isomer $^{\mathrm{83m}}$Kr for calibration.
Comparing the measured energies to the appropriate literature values allows for
an independent evaluation of the energy scale, but the uncertainties in some of
the literature values obtained by gamma spectroscopy are a limiting factor.
Building upon the already excellent linearity of KATRIN's energy scale, this
paper proposes a novel method for determining the $^{\mathrm{83m}}$Kr
transition energies via high-precision electron spectroscopy. Notably, the
method makes use of conversion electrons from the 41.6-keV direct transition of
$^{\mathrm{83m}}$Kr to its ground state in addition to conversion electrons
from the much more frequent cascade of a 32.2-keV and a 9.4-keV transition. By
implementing this method, KATRIN may be able to deliver order-of-magnitude
improvements in precision over current $^{\mathrm{83m}}$Kr transition energy
literature values.
