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Optical Theorem
The differential scattering crosssection,
, is simply
the modulus squared of the scattering amplitude,
. [See Equation (10.28).] The
total scattering crosssection is defined as
where
. It follows that

(10.89) 
where use has been made of Equation (10.65). A comparison of the preceding expression with
Equation (10.81) reveals that

(10.90) 
because
[1]. This result is known as the optical theorem [107,73],
and is a consequence 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 that direction.
It is conventional to write

(10.91) 
where

(10.92) 
is termed the
th partial scattering crosssection: that is, the contribution to the
total scattering crosssection from the
th partial wave. Note that (at fixed
) the maximum
value for the
th partial scattering crosssection occurs when the associated phaseshift,
, takes the value
.
Next: Determination of PhaseShifts
Up: Scattering Theory
Previous: Partial Waves
Richard Fitzpatrick
20160122