Thermal damping of quantum oscillations ($R_T=X/\sinh X,X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$)

A finite-temperature measurement averages the oscillatory response over the derivative of the Fermi-Dirac distribution. For the fundamental harmonic and a locally constant cyclotron effective mass, this gives $R_T=X/\sinh X$ with $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$. The oscillations become small when thermal smearing substantially exceeds their Landau-level spacing.