Elastic Scattering at Resonance from Bound Nuclei

Abstract

The usual Debye-Waller factor, $〈\exp [-i({\mathrm{k}}_f-{\mathrm{k}}_0\left)·\mathrm{r}\right]〉$, which when multiplying the fixed-scatterer amplitude gives the nonresonant elastic-scattering amplitude from a bound scatterer, is generally applicable only for fast collisions, i.e., if the "collision time," $\hslash \left(\frac{d}{\mathrm{dE}}\right)$ (scattering phase shift), is much less than a characteristic vibration time, ${{\omega }_m}^{-1\mathrm{}}$, of the bound scatterer about its mean position. In the opposite extreme case, where the "collision time" is very long compared to ${{\omega }_m}^{-1\mathrm{}}$ (slow collisions), there is negligible correlation between the positions of absorption and subsequence re-emission (for an atom bound in a crystal), and the "Debye-Waller" factor becomes $〈\exp (-i{\mathrm{k}}_f·\mathrm{r})〉〈\exp \left(i{\mathrm{k}}_0·\mathrm{r}\right)〉$. If the scatterers' surroundings exhibit cubic symmetry, the extremes as well as all intermediate cases give the same factor for 90° scattering angle. In the case of medium collisions, collision time ≈ vibration time, the elastic scattering amplitude becomes sensitive to the detailed vibrational spectrum of the bound scatterer. For nonresonant scattering the collision times are of the order of transit times (x ray across the atom for Thomson scattering, neutron across the nucleus for neutron potential scattering) and are thus fast collisions.