Effects of Chemical Binding on Nuclear Recoil

The recoil of a chemically bound nucleus is considered for slow neutron scattering and for the resonant absorption of neutrons or gamma rays. The Doppler-broadened resonance line shape is derived in terms of the time-dependent self-correlation function describing the motion of a nucleus due to the interatomic forces. This explicitly relates the resonance line shape to the differential scattering cross section for slow neutrons in the Fermi pseudopotential approximation. Within this formulation an expansion for large nuclear recoil is naturally suggested. For the case of a crystal, this expansion can be directly related to the expansion associated with the central limit theorem of probability theory and can therefore be proved to be asymptotic in nature. The expansion parameter is ${\left(\frac{K_{\mathrm{av}}}{R}\right)}^{\frac{1}{2}}$, where $K_{\mathrm{av}}$ is the average kinetic energy of a nucleus and $R$ is the recoil energy for a free nucleus at rest. The leading term of the expansion is the weak binding limit originally obtained by Lamb. In this limit the Doppler-broadened line shape is the same as would obtain for an ideal monatomic gas of the same mass with an effective temperature $T^{\prime }=\left(\frac{2}{3}\right)K_{\mathrm{av}}$. For noncrystalline systems, a similar expansion with the same leading term can be obtained by a rearrangement of the terms in an expansion used by Wick to study the slow neutron total cross section. The relation of the present expansion to Wick's expansion is discussed.