  Theories and experiments on dirty superconductors are sophisticated but
important for both fundamentals and applications. It becomes more challenging
when magnetic fields are present, because the field distribution, the electron
density of states, and the superconducting pairing potentials are nonuniform.
Here we present tunneling microspectroscopic experiments on NbC single crystals
and show that NbC is a homogeneous dirty superconductor. When applying magnetic
fields to the sample, we observe that the zero-energy local density of states
and the pairing energy gap follow an explicit scale relation proposed by de
Gennes for homogeneous dirty superconductors in high magnetic fields.
Surprisingly, our experimental findings suggest that the validity of the scale
relation extends to magnetic field strengths far below the upper critical field
and call for new nonperturbative understanding of this fundamental property in
dirty superconductors. On the practical side, we use the observed scale
relation to drive a simple and straightforward experimental scheme for
extracting the superconducting coherence length of a dirty superconductor in
magnetic fields.
