The differential scattering cross-section is simply the modulus squared of the scattering amplitude . The total cross-section is given by
where . It follows that
where use has been made of Equation (967). A comparison of this result with Equation (979) yields
since . This result is known as the optical theorem. It is a reflection of the fact that the very existence of scattering requires scattering in the forward ( ) direction in order to interfere with the incident wave, and thereby reduce the probability current in this direction.
It is usual to write
is the th partial cross-section: i.e., the contribution to the total cross-section from the th partial wave. Note that the maximum value for the th partial cross-section occurs when the phase-shift takes the value .
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)