16.9: Exercises - Scattering
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An asterisk denotes a harder problem, which you are nevertheless encouraged to try!
1. The UK has two major scattering facilities based at Rutherford Appleton Laboratory in Oxford, the Diamond synchrotron (X-rays) and the Isis spallation source (neutrons).
With reference to the time dependent scattering matrix element
explain.
a) What causes V in each case:
b) Which technique would you use to study (i) the crystal structure of polonium (ii) the crystal structure of deuterated ice (iii) the phonon (vibrational) spectrum of sapphire (Al
2. Particles of mass
Show that, in the first Born approximation, the differential and total cross-sections are given by
where
Hint: the required integral may be obtained by maple or parametric differentiation of the integral
3. Particles are scattered from a box of size
where
Given that this is a model for the interaction of light with a fluctuating dipole in a gas molecule, comment on the presence of the factor of
Considering the Born series (lecture 13), can you guess what the second order term would mean here
4. Evaluate the differential cross-section in the Born approximation for the potential
What happens to the total cross-section for this potential ?
You may assume that:
5. Two beams of unpolarised electrons are scattered from one another. Taking the scattered wavefunction to be
6. * Show that in a classical elastic two-body collision between particles of mass
and hence that the LAB and CM frame differential cross-sections are related by
7. Show that the function
and evaluate E. You may resort to a verbal argument to explain the delta function without evaluating its strength.
Show that for a potential
Under what circumstances is this useful?