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Journal of Physics C Solid State Physics

Rainbow scattering from anisotropic potentials

F Drepper

Published Sep 14, 1977• 2 citations
Publisher

Abstract

The leading term in a semiclassical expansion of the quantum mechanical cross section for potential scattering is obtained for those cases where the classical cross section shows either a fold or an umbilic (hyperbolic or elliptic) type catastrophe. As an intermediate step a narrow wave packet approximation is derived by decomposing the S-matrix into transition amplitudes for wave packets and by expanding the modulus of these amplitudes for large mass.

DOI
10.1088/0022-3719/10/17/026

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The leading term in a semiclassical expansion of the quantum mechanical cross section for potential scattering is obtained for those cases where the classical cross section shows either a fold or an umbilic (hyperbolic or elliptic) type catastrophe. As an intermediate step a narrow wave packet approximation is derived by decomposing the S-matrix into transition amplitudes for wave packets and by expanding the modulus of these amplitudes for large mass.
Investigation of the large angle scattering behaviour of the narrow wave packet approximation is made. First an expression is derived for the scattering amplitude in the narrow wave packet approximation when the potential is arbitrary. Then the same expression is derived for the case when a potential is spherically symmetric. In many aspects the approximation behaves as Jeffrey's approximation. Since the narrow wave packet approximation breaks down in the vicinity of the singularity of a potential, the validity of the approximation for the case when the potential is short range and has a singularity near the origin is investigated.
A semiclassical formula that gives scattering cross sections down to an angular resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$ is derived. The formula describes rainbow peaks, which have widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$, but not other interference effects, which give structure on a scale of order \ensuremath{\Elzxh}. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.
A semiclassical approximation is presented which uses the wave-pocket formulation of collision theory. First an appropriate semiclassical approximation is derived by assuming that the wave packet is and remains narrow during the passage of the interaction region. After this, a process is devised to decompose a macroscopic wave packet into narrow wave packets. The wave function, after the collision, is obtained by summing up the contributions from all the narrow wave packets into which the macroscopic wave packet was decomposed. Finally, using this function, the SSS matrix and the scattering cross section are calculated.