The energy-dependent thermal diffusion equation is considered in a region free of external sources. Two cases of experimental interest are calculated. The first of these is the steady-state condition where an eigenvalue problem for the thermal diffusion length is obtained. The associated eigenfunction is the neutron spectrum. The second case, which is mathematically identical to the first, is the exponential decay in time of the thermal flux in a pulsed source experiment. The neutron leakage is assumed to be describable by a single eigenvalue for the buckling. In this case the eigenvalue is the decay constant of the flux.When the ratio of absorption cross section to transport mean free path decreases with energy in the thermal region, the first case will give a“diffusion hardening,”and the second case a“diffusion cooling”of the neutron spectrum compared to a Maxwellian distribution at the moderator temperature. These effects are investigated quantitatively for the model of a heavy gaseous moderator.