, it is defined as the following low-energy limit: where
is the phase shift of the outgoing spherical wave.
The elastic cross section,
, at low energies is determined solely by the scattering length:
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 outgoing wave, called the s-wave in analogy with the atomic orbital at angular momentum quantum number l=0.
At higher energies one also needs to consider p and d-wave (l=1,2) 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.
The concept of the scattering length can also be extended to potentials that decay slower than
As an example on how to compute the s-wave (i.e. angular momentum
) scattering length for a given potential we look at the infinitely repulsive spherical potential well of radius
The radial Schrödinger equation (
) outside of the well is just the same as for a free particle: where the hard core potential requires that the wave function
The solution is readily found: Here
is the s-wave phase shift (the phase difference between incoming and outgoing wave), which is fixed by the boundary condition
is an arbitrary normalization constant.
, in other words the scattering length for a hard sphere is just the radius.
(Alternatively one could say that an arbitrary potential with s-wave scattering length
has the same low energy scattering properties as a hard sphere of radius
To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section
In scattering theory one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming plane wave along the
According to the probability interpretation of quantum mechanics the differential cross section is given by
(the probability per unit time to scatter into the direction
If we consider only s-wave scattering the differential cross section does not depend on the angle
, and the total scattering cross section is just
The s-wave part of the wavefunction
is projected out by using the standard expansion of a plane wave in terms of spherical waves and Legendre polynomials