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There is a significant exception to the independence of the crosssection on energy mentioned above. Suppose that the quantity
is slightly less than . As the incident energy increases, ,
which is given by Eq. (1342), can reach the value . In this case,
becomes infinite, so we can no longer assume that the righthand side of Eq. (1337) is small. In fact, it follows from Eq. (1337) that
at the value of the incident energy
when then we also have
,
or
(since we are assuming that ).
This implies that

(1344) 
Note that the crosssection now depends on the energy. Furthermore, the
magnitude of the crosssection is much larger than that given in Eq. (1341)
for
(since ).
The origin of this rather strange behaviour is quite simple. The condition

(1345) 
is equivalent to the condition that a spherical well of depth
possesses a bound state at zero energy. Thus, for a potential
well which satisfies the above 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, since the system has a small
positive energy. Nevertheless, this sort of resonance scattering
is best understood as the capture of an incident particle to form
a metastable bound state, and the subsequent decay of the bound state
and release of the particle. The crosssection for resonance scattering
is generally much larger than that for nonresonance scattering.
We have seen that there is a resonant effect when the phaseshift of
the wave 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

(1346) 
Let us expand in the vicinity of the resonant energy:
Defining

(1348) 
we obtain

(1349) 
Recall, from Eq. (1308), that the contribution of the th partial wave
to the scattering crosssection is

(1350) 
Thus,

(1351) 
This is the famous BreitWigner formula. 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: About this document ...
Up: Scattering Theory
Previous: Low Energy Scattering
Richard Fitzpatrick
20100720