Theory of neutron scattering from hydrogen in metals involving transitions to excited tunnel-split states

Abstract

A theory is developed for neutron-induced transitions to excited states of a proton trapped at a two-well potential near an interstitial impurity in a metal. The proton wave function is written as a linear combination of the states in each well (tight-binding approximation). Hence, in the main, the theory is limited to the low-lying excitations which lie below the central barrier. Otherwise, the theory is quite general and independent of the details of the potential. General expressions are derived for the differential cross section for transitions from the ground-state doublet to the excited states in terms of $\tilde{F}\left(\overrightarrow{\mathrm{q}}\right)$, the transition form factor. A corresponding quantity, $F\left(\overrightarrow{\mathrm{q}}\right)$, appears in expressions for $\frac{d\sigma }{d\Omega }$ associated with transitions within the ground-state doublet. As in earlier work, concerned only with the ground-state doublet, the effects of asymmetric displacements of the well bottoms due to strains are considered. As is the case for the ground-state doublet, the effect of the asymmetry in the two-well potential is to introduce a general mixing angle for the excited-state doublet describing the admixture of the left-hand and right-hand single-well states. Whereas the effect of nonequal mixing on the inelastic differential cross section for transitions within the ground-state doublet is to diminish it by a factor ${〈{sin}^{2}2\theta 〉}_{\mathrm{av}}$ ($\theta$ is the mixing angle), under suitable experimental conditions, I show that the corresponding cross section for neutron-induced transitions from the ground-state doublet to either component of the excited-state doublet is independent of both $\theta$ and $\phi$ ($\phi$ is the excited-state mixing angle). That is, the latter cross section is independent of $\epsilon$, a measure of the asymmetry, provided $\frac{\epsilon }{\hslash \omega } \ll 1$ ($\hslash \omega$ is the excitation energy). Thus the ratio $\Lambda$ of the higher-energy to the lower-energy inelastic cross section is, on the one hand, enhanced by a factor $\frac{1}{{〈{sin}^{2}2\theta 〉}_{\mathrm{av}}}$ and, on the other, diminished by the quantity ${\left|\tilde{F}\right(\overrightarrow{\mathrm{q}}\left)\right|}^2$. General expressions for $\Lambda$ and for the ratios of various other cross sections are derived in terms of $\tilde{F}\left({\overrightarrow{\mathrm{q}}}^{\prime }\right)$ and $F\left(\overrightarrow{\mathrm{q}}\right)$. These are then estimated for the first excited state by making various simplifying approximations. $\Lambda$ is finite in the limit $\overrightarrow{\mathrm{q}}\to 0$, ${\overrightarrow{\mathrm{q}}}^{\prime }\to 0$, keeping their ratio fixed. If I assume that errors in making the small-$\overrightarrow{\mathrm{q}}$ expansion tend to...