The steady-state space-energy distribution of thermal neutrons in homogeneous media is considered in the diffusion approximation. From the general form of the solution it is shown that the asymptotic (in space) distribution depends upon the source distribution and under different circumstances can be qualitatively different. The relation of the asymptotic flux to the usually found infinite medium spectrum is exhibited. For a heavy gas moderator the asymptotic flux from a source of limited extent is shown to be much softer than an infinite medium spectrum in the same material.