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The differential scattering crosssection
is simply
the modulus squared of the scattering amplitude
. The
total crosssection is given by
where
. It follows that

(981) 
where use has been made of Equation (967). A comparison of this result with
Equation (979) yields

(982) 
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

(983) 
where

(984) 
is the
th partial crosssection: i.e., the contribution to the
total crosssection from the
th partial wave. Note that the maximum
value for the
th partial crosssection occurs when the phaseshift
takes the value
.
Next: Determination of PhaseShifts
Up: Scattering Theory
Previous: Partial Waves
Richard Fitzpatrick
20130408