PHY304 |
Particle Physics |
Dr C N Booth |

##
Particle Physics - Homeworks

###
Yukawa potential

**Note. Marks are obtained for a proper explanation of your working, not simply for obtaining the correct numerical or algebraic results!**
Earlier in the course, we used the Born approximation to show
that in the case of scattering with a momentum transfer *q* from a
spherically symmetric potential *V*(*r*), the matrix element
is given by

.
For the case of the Yukawa potential,

(with *R* = *ħ*/*mc*,
and *m* the mass of the exchanged boson mediating the force) show that the
matrix element evaluates to
.

When an electron scatters electromagnetically off a nucleus, the exchanged
boson is the massless photon, and at low energies the nucleus can be considered
to remain effectively at rest. Starting from the definition of __q__,
__q__ = __p___{i} − __p___{f},
derive an expression for *q*^{2} in terms of θ
for elastic scattering, and show that the angular dependence of the scattering
is then given simply by the Rutherford formula

.

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