A general mathematical formalism for the energy transfer moments and their associated integrals, useful in the study of neutron thermalization, is presented. This formalism has been employed to obtain these quantities for the “general Doppler approximation” case, which represents a large number of approximations that belong to the Doppler class. An exact formula for M2 (the second energy transfer moment weighted by the Maxwellian distribution) is given in terms of binding parameters for the general Doppler case. A new, useful Doppler approximation, which satisfies the Detailed Balance theorem and is based upon the Debye-Waller factor and the specific heat integral, is also formulated. A comparative study has been undertaken of this and three other previously known Doppler cases (the monatomic gas model, the effective temperature, and the Krieger-Nelkin approximations for rotating molecules) in terms of the validity of the Detailed Balance theorem and the asymptotic scattering behavior. Numerical results based upon the Debye frequency distribution of vibrational modes in the Doppler approximation are presented.