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 $n^2$-1=-(4πρb/$k_0^2$)/[1+ (4πρ$b^2$/${\mathrm{nk}}_0$) ${\mathcal{F}}_0^{\infty }$da e${}_0^{\mathrm{ik}}$asin(${\mathrm{nk}}_0$a)h(a)], where $k_0$ is the incident wave propagation vector, b the nuclear scattering length, ρ the number density of nuclei (ρ≡1/$a_0^3$, 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 $n^2$-1∼(b/$a_0$ ${)}^4$($k_0$ $a_0$ ${)}^{\mathrm{-}2}$ ln[($k_0$ $a_0$ ${)}^{\mathrm{-}1}$]. Thus, the refractive index is exceedingly close to unity unless b is large (a resonance) or $k_0$→0 (ultracold neutrons). The presence of the logarithmic term indicates that randomness in the scattering field apparently reduces the effective dimension.