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There is a significant exception to the energy independence of the scattering crosssection at low
incident energies described in the previous section. Suppose that the quantity
is slightly less than
. As the incident energy increases,
,
which is given by Equation (10.129), can reach the value
. In this case,
becomes infinite, so we can no longer assume that the righthand side of Equation (10.124) is small. In fact,
at the value of the incident energy
at which
, it follows from Equation (10.124) that
,
or
(because we are assuming that
).
This implies that

(10.130) 
Note that the total scattering crosssection now depends on the energy. Furthermore, the
magnitude of the crosssection is much larger than that given in Equation (10.128)
for
(because
, whereas
).
The origin of the rather strange behavior described in the previous paragraph is easily explained. The condition

(10.131) 
is equivalent to the condition that a spherical well of depth
possesses a bound state at zero energy. Thus, for a potential
well that satisfies the preceding equation, the energy of the scattering system
is essentially the same as the energy of the bound state. In this situation,
an incident particle would like to form a bound state in the potential
well. However, the bound state is not stable, because the system has a small
positive energy. Nevertheless, this sort of resonant scattering
is best understood as the capture of an incident particle to form
a metastable bound state, followed by the decay of the bound state
and release of the particle. The crosssection for resonant scattering
is generally far higher than that for nonresonant scattering.
We have seen that there is a resonant effect when the phaseshift of
the Swave takes the value
. There is nothing special about
the
partial wave, so it is reasonable to assume that there
is a similar resonance when the phaseshift of the
th partial
wave is
. Suppose that
attains the value
at the incident energy
, so that

(10.132) 
Let us expand
in the vicinity of the resonant energy:
Defining

(10.134) 
we obtain

(10.135) 
Recall, from Equation (10.93), that the contribution of the
th partial wave
to the total scattering crosssection is

(10.136) 
Thus,

(10.137) 
which is known as the BreitWigner formula [16]. The variation of
the partial crosssection,
, with the incident energy has
the form of a classical resonance curve. The quantity
is
the width of the resonance (in energy). We can interpret the
BreitWigner formula as describing the absorption of an incident particle
to form a metastable state, of energy
, and lifetime
.
Next: Elastic and Inelastic Scattering
Up: Scattering Theory
Previous: LowEnergy Scattering
Richard Fitzpatrick
20160122