A beam of particles of energy scatters off a fixed repulsive centre with potential

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch Scattering in a central field

Problem

A beam of particles of energy EE scatters off a fixed repulsive centre with potential V(r)=γ/r2V(r)=\gamma/r^2, where γ>0\gamma>0. ((a)) Use the orbit integral for the deflection function to show that χ(b)=π(1−bb2+γ/E).\chi(b)=\pi\left(1-\frac{b}{\sqrt{b^2+\gamma/E}}\right). ((b)) Find dσ/dΩ\mathrm{d}\sigma/\mathrm{d}\Omega as a function of χ\chi, and evaluate it at χ=90∘\chi=90^\circ for γ/E=100 fm2\gamma/E=100\ \mathrm{fm^2}. ((c)) Find the cross-section σ(χ>χmin⁡)\sigma(\chi>\chi_{\min}) for scattering beyond χmin⁡\chi_{\min}, evaluate it for χmin⁡=10∘\chi_{\min}=10^\circ with the same γ/E\gamma/E, and say what happens to it as χmin⁡→0\chi_{\min}\to0 and why.

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