Neutron refractive index: A Fermi-Huygens theory.

Abstract

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=-(4\ensuremath{\pi}\ensuremath{\rho}b/${k}_{0}^{2}$)/[1+ (4\ensuremath{\pi}\ensuremath{\rho}${b}^{2}$/${\mathrm{nk}}_{0}$) ${\mathcal{F}}_{0}^{\ensuremath{\infty}}$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, \ensuremath{\rho} the number density of nuclei (\ensuremath{\rho}\ensuremath{\equiv}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\ensuremath{\sim}(b/${a}_{0}$${)}^{4}$(${k}_{0}$${a}_{0}$ ${)}^{\mathrm{\ensuremath{-}}2}$ ln[(${k}_{0}$${a}_{0}$${)}^{\mathrm{\ensuremath{-}}1}$]. Thus, the refractive index is exceedingly close to unity unless b is large (a resonance) or ${k}_{0}$\ensuremath{\rightarrow}0 (ultracold neutrons). The presence of the logarithmic term indicates that randomness in the scattering field apparently reduces the effective dimension.