Scattering: Cross-Sections and the Born Approximation

A cross-section is an effective interaction area, not necessarily the target's geometric silhouette. For a uniform beam and a thin target of N independent scatterers, a small angular bin gives dσ/dΩ approximately C/(jNtΔΩ). Here C is the event count corrected for background and efficiency, j is incident particles per area per time, t is exposure time and ΔΩ is the bin's solid angle. Solid angle is the area a direction cone cuts out on a unit sphere. The units are area per steradian.

For nonrelativistic elastic scattering from local V(r), first Born replaces the full wave in the interaction integral by the incident plane wave. With q=k_f-k_i, f_B(q)=-m/(2pi hbar²) integral exp(-iq·r)V(r)d³r. The spatial Fourier transform sums V(r) weighted by exp(-iq·r). Here m is particle mass, or reduced mass for two bodies. The k vectors are wave vectors; momentum transfer is hbar q. In this convention, dσ/dΩ=|f_B|².

Intensity gives |f|² and loses f's complex phase. Limited angles sample only part of the transform, so the potential is not generally reconstructed uniquely. Here patterns are absolute differential cross-sections obtained by dividing corrected counts by jNtΔΩ, not curves scaled to unit total. Different absolute patterns need not imply different spatial shapes: changing the strength of a same-shaped potential can change intensity. Control the experimental conditions.

For a central potential V(r), expand the wave in angular-momentum channels and solve each radial equation. The resulting partial-wave phase shifts describe scattering without assuming a weak potential. Include enough contributing channels and impose the radial solutions' scattering boundary conditions. Partial waves are not an automatic replacement for every noncentral or multichannel scattering problem.

First Born theory is useful when replacing the interaction-region wave by the incident plane wave changes the result only a little. Strength, range and incident energy all matter; resonances can make a perturbative estimate poor. Comparing a suitable dimensionless parameter, higher Born terms or an exact calculation can test the approximation. A strong-potential result needs a regime-specific check, not more decimal places.