A general formalism for determining the lower time eigenvalues associated with a decaying pulse of neutrons in a finite multiplying as well as nonmultiplying medium has been developed. This formalism is based upon the expansion of each energy eigenfunction by a complete sum of the associated Laguerre polynomials of first order. The eigenvalues are expressed in terms of the energy transfer moments of the scattering kernel of the medium, weighted by the Maxwellian distribution. The importance of the first eigenvalue in the establishment of the final asymptotic energy distribution is discussed. In the case of a nonabsorbing infinite medium, the reciprocal of the first eigenvalue is shown to be equal to the thermalization time constant, with which the Maxwellian velocity distribution of neutrons is attained. The thermalization time constant was estimated for various moderators. For the heavy-gas case, the thermalization time constant was was found to be equal to (1.274 ° ζ∑s0υ0)−1. It is also established in this study that only two polynomials are required to obtain the relation between the thermalization time constant and the diffusion cooling coefficient derived previously from the Rayleigh-Ritz variational principle. The formalism presented in this paper is general and avoids the concept of neutron temperature in defining the thermalization time constant. The decay of a neutron pulse in a nonmultiplying medium is discussed in detail. For the case of multiplying medium, an analysis of an experiment is presented to indicate the importance of the time-dependent nonleakage probability in the expression of the zeroth eigenvalue.