Neutron refractive index: A Fermi-Huygens theory

A multiple-scattering calculation of the neutron refractive index is performed by an extension of the Fermi-Huygens technique. The extension involves projecting the problem into a one-dimensional walk by integrating out the transverse coordinate in a semi-infinite medium and then partially summing parts of the walk to infinite order. The square of the refractive index is given by n²-1=-(4{π}{ρ}b/k₀²)/[1+ (4{π}{ρ}b²/nk₀) F₀^∞da e₀ⁱᵏasin(nk₀a)h(a)], where k₀ is the incident wave propagation vector, b the nuclear scattering length, {ρ} the number density of nuclei ({ρ}{≡}1/a₀³, say), and h(a)=g(a)-1, where g(a) is the pair distribution function. The results parallel those obtained by constitutive equation methods, and offer a physical picture of local-field effects. When the mean scattering length vanishes (total incoherence), correlated multiple scattering yields n²-1{~}(b/a₀{)}⁴$(${k}₀a₀ )^-2 ln[(k₀{a}₀)^-1]. Thus, the refractive index is exceedingly close to unity unless b is large (a resonance) or k₀{→}0 (ultracold neutrons). The presence of the logarithmic term indicates that randomness in the scattering field apparently reduces the effective dimension.