The effect of surface roughness on the phase of neutrons specularly reflected at grazing incidence

The effect of surface roughness on the specular intensity of a reflected neutron beam can be calculated using the distorted wave Born approximation. For Gaussian roughness the result obtained is the familiar expression of Nevot and Croce. When the distribution of surface heights, z, is proportional to cosh/2(z/2d), the calculated reflectance is sinh(pi d k sub 0 -k sub 1)/ sinh(pi d k sub 0 +k sub 1). Here k sub 0 and k sub 1 are the components of neutron wavevectors perpendicular to the interface, evaluated outside and inside the reflecting medium, and d is a measure of the amplitude of the roughness. This result for the reflectance differs by a phase factor from that obtained in an exact calculation for an interface with the corresponding graded refractive index. The phase difference between the two expressions is small, scaling as approx. d sup 3, and usually has little effect on measured reflectivities of multilayered structures. Ways in which the phase could be determined are discussed.