Theory of excitation bands of hydrogen in bcc metals and of their observation by neutron scattering

I consider the possibility that the excited-state oscillator wave functions of dilute hydrogen in bcc metals overlap sufficiently with nearest-neighbor occupancy sites so as to produce hydrogenic energy bands, analogous to electronic energy bands in narrow-band semiconductors. The theory is motivated by the experiments of Magerl et al. as well as the earlier observation of ground-state tunnel splitting by Wipf et al., demonstrating quantum coherence in the motion of the hydrogen, despite the necessity of correlated motion by the surrounding metal atoms. Because of the latter complication, the relevant overlap integrals are not calculated from first principles. The band structures are given for the first (nondegenerate) and second (doubly degenerate) excitations ${\ensuremath{\omega}}_{\mathrm{I}}$ and ${\ensuremath{\omega}}_{\mathrm{II}}$ of the local oscillators, modulo a few irreducible overlap integrals, which are then determined by comparison with experiment. The fact that the experimental bandwidths for inelastic neutron scattering from dilute hydrogen in V, Nb, and Ta satisfy $\ensuremath{\Gamma}(\mathrm{V})g\ensuremath{\Gamma}(\mathrm{Nb})g\ensuremath{\Gamma}(\mathrm{Ta})$ at room temperature (Rush, Magerl, and Rowe) finds a natural explanation in the theory. It is shown that the ${\ensuremath{\omega}}_{\mathrm{I}}$ and ${\ensuremath{\omega}}_{\mathrm{II}}$ bandwidths satisfy $\frac{\ensuremath{\Delta}{E}_{\mathrm{II}}}{\ensuremath{\Delta}{E}_{\mathrm{I}}}=(\frac{{H}_{\mathrm{II}}}{{H}_{\mathrm{I}}})\ensuremath{\Upsilon}$, where ${H}_{\mathrm{I}}$ and ${H}_{\mathrm{II}}$ are irreducible overlap integrals and $\ensuremath{\Upsilon}$ is an (almost) universal constant for H in bcc metals, determined (essentially) by the geometry of the tetrahedrally coordinated hydrogen occupancy sites. On the basis of the band structure that I obtain, I estimate that $\ensuremath{\Upsilon}\ensuremath{\simeq}\frac{3}{4}$. Based upon physical reasoning, the relation $(\frac{{H}_{\mathrm{II}}}{{H}_{\mathrm{I}}})={(\frac{{\ensuremath{\omega}}_{\mathrm{II}}}{{\ensuremath{\omega}}_{\mathrm{I}}})}^{2}$ is proposed. Given the (model-consistent) empirical result, $\frac{{\ensuremath{\omega}}_{\mathrm{II}}}{{\ensuremath{\omega}}_{\mathrm{I}}}\ensuremath{\simeq}{2}^{\frac{1}{2}}$, this leads to the prediction $\frac{\ensuremath{\Delta}{E}_{\mathrm{II}}}{\ensuremath{\Delta}{E}_{\mathrm{I}}}\ensuremath{\simeq}\frac{3}{2}$, to be compared with the neutron-measured ratios $\frac{{\ensuremath{\Gamma}}_{\mathrm{II}}}{{\ensuremath{\Gamma}}_{\mathrm{I}}}=1.3 \mathrm{and} 2.0$ for dilute hydrogen trapped at O and N impurities in Nb metal at $T=4 \mathrm{and} 10$ K, respectively. The variation in $\frac{{\ensuremath{\Gamma}}_{\mathrm{II}}}{{\ensuremath{\Gamma}}_{\mathrm{I}}}$ is attributed to perturbations of the intrinsic hydrogen bands by the trapping impurities, which are necessary for low-temperature observation, if one is to prevent coagulation of the hydrogen atoms into the $\ensuremath{\epsilon}$ phase of NbH. The differential cross section for inelastic neutron scattering from hydrogen in band states is related theoretically to that for H in local oscillator states. With appropriate rescaling, the band structure that I obtain for hydrogen can also be applied to the case of trapped positive muons in bcc metals.