F: Comparison of X‐rays with neutrons

Abstract

The relationship given in Eq. (3.1) between the refractive index, n, and the scattering length density ρr 0 can be derived in a different way when considering neutrons entering a material. Here the material is viewed as a continuum of nuclei, each having the scattering length b, and with a density of ρ. An incident particle that experiences a potential V(r) from the material will be scattered. The scattering length b of a particle is related to the scattering potential V(r) in first-order perturbation theory by V(Q) = 4π 2 2m n b Here V(Q) is the Fourier transform of the scattering potential: V(Q) = V(r) e i Q·r dr and as usual Q = k − k is the wavevector transfer in the scattering process. This relation appears plausible when it is recalled that (a) there is a phase difference of Q · r between the scattering from volume elements around the origin and around r; (b) the scattering can be thought of as a weighted superposition of the scattering from such volume elements, the weight being V(r) times the phase factor; (c) the dimensionality in the equation relating b and V(Q) is correct, i.e. the term (2 /2m n) occurs naturally. Of course, the factor of 4π must rely on a more rigorous treatment. Fermi suggested that one could define a pseudo-potential between thermal neutrons and nuclei in such a way that this general first-order perturbation result would reproduce the correct scattering length. The nuclear scattering of neutrons is due to a short range potential between the neutron and the nucleus. The range of the potential is extremely short (of order 10 −15 m) in comparison with the wavelength of Elements of Modern X-ray Physics, Second Edition. Jens Als-Nielsen and Des McMorrow