Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces

We consider scattering from a corrugated hard surface Z with random moving perturbations (a ‘rippling mirror’). Kirchhoff’s approximation enables the classical limit, diffraction effects and incoherence to be treated within the same framework. The classical rainbow is a curve V in the two-dimensional space of deflections G; we study the topology of V and show that it has cusps whose positions are sensitive to the form of Z. Classically the scattering is singular on Z but diffraction softens the singularities; we give the diffraction functions to be used near and on smooth parts and cusps of W, and derive criteria for the observability of rainbow structure (taking account of surface periodicity which quantizes G). Random thermal perturbations of I: blur the diffracted beams; we introduce a simple approximation for the blurring function, and this suggests a simple method for inverting experimental data to obtain the ‘surface phonon spectrum’, even in cases where ‘multiphonon processes’ dominate.