General Concept
When a slow particle scatters off a short ranged scatterer (e.g. an impurity in a solid or a heavy particle) it cannot resolve the structure of the object since its de Broglie wavelength is very long. The idea is that then it should not be important what precise potential one scatters off, but only how the potential looks at long length scales. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodynamics), where one expands in the angular momentum components of the outgoing wave. At very low energy the incoming particle does not see any structure, therefore to lowest order one has only a spherical symmetric outgoing wave, the so called s-wave scattering (angular momentum ). At higher energies one also needs to consider p and d-wave scattering and so on. The idea of describing low energy properties in terms of a few parameters and symmetries is very powerful, and is also behind the concept of renormalization.
Read more about this topic: Scattering Length
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