where is the position vector; ; is the incoming plane wave with the
wavenumberk along the z axis; is the outgoing spherical wave; θ is the scattering angle (angle between the incident and scattered direction); and is the scattering amplitude. The
dimension of the scattering amplitude is
length. The scattering amplitude is a
probability amplitude; the differential
cross-section as a function of scattering angle is given as its
modulus squared,
The asymptotic form of the wave function in arbitrary external field takes the form[2]
where is the direction of incidient particles and is the direction of scattered particles.
Unitary condition
When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we have[2]
In the centrally symmetric field, the unitary condition becomes
where and are the angles between and and some direction . This condition puts a constraint on the allowed form for , i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if in is known (say, from the measurement of the cross section), then can be determined such that is uniquely determined within the alternative .[2]
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[3]
,
where fℓ is the partial scattering amplitude and Pℓ are the
Legendre polynomials. The partial amplitude can be expressed via the partial wave
S-matrix element Sℓ () and the scattering phase shiftδℓ as