Phy5645/Cross Section Relation: Difference between revisions

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Consider the scattering of a particle from a real spherically symmetric potential. If <math>\frac{\mathrm{d} \sigma (\theta) }{\mathrm{d} \Omega }</math> is the differential cross section and <math>\sigma</math> is the total cross section, show that
<math>\sigma \leq \frac{4\pi}{k}\sqrt{\frac{\mathrm{d} \sigma (0) }{\mathrm{d} \Omega }}</math>
for a general central potential using the partial-wave expansion of the scattering amplitude and the cross section.
Solution:
The differential cross section is related to the scattering amplitude through
The differential cross section is related to the scattering amplitude through



Revision as of 23:39, 2 September 2013

The differential cross section is related to the scattering amplitude through

Since

therefore,

On the other hand, from the optical theorem we have

For a central potential the scattering amplitude is

and, in terms of this, the differential cross section is

The total cross section is

Since we can write

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