|
|
Line 1: |
Line 1: |
| 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
Back to Central Potential Scattering and Phase Shifts