Using the Born Approximation, calculate the differential cross-section for a beam of particles of mass m scattered by a potential: . You may use:
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The Born approximation gives the scattering amplitude as: [ f(\theta) = -\frac{2m}{\hbar^2} \frac{1}{4\pi} \int e^{i \mathbf{q} \cdot \mathbf{r}} V(r) , d^3r ] where (\mathbf{q} = \mathbf{k}' - \mathbf{k}) is the momentum transfer, with magnitude (q = 2k \sin(\theta/2)). Given the potential: [ V(r) = V_0 \frac{a}{r} e^{-r/a} ] The scattering amplitude in Born approximation simplifies (spherically symmetric potential) to: [ __________ ________ _____ _______ ____.
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