A beam of alpha particles ( ) scatters elastically off free carbon-12 nuclei ( ) held…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Momentum, energy and conservation laws · Gravitation and central-force motion

Action and Symmetry, Ch Scattering in a central field

Problem

A beam of alpha particles (m1=4 um_1 = 4\,\mathrm{u}) scatters elastically off free carbon-12 nuclei (m2=12 um_2 = 12\,\mathrm{u}) held in a thin target. Model each carbon nucleus as a hard sphere of radius a=3.0 fma = 3.0\,\mathrm{fm} so that the centre-of-mass deflection function is χ=π−2arcsin⁡(b/a)\chi = \pi - 2\arcsin(b/a). ((a)) Find the impact parameter bb that gives a centre-of-mass deflection χ=60∘\chi = 60^\circ. ((b)) The beam delivers 2.0×1052.0\times10^5 alphas per second onto a foil with 1.5×10231.5\times10^{23} carbon nuclei per square metre. A detector is set to catch particles with χ=60∘\chi = 60^\circ; the cone it accepts covers a centre-of-mass solid angle of 5.0×10−4 sr5.0\times10^{-4}\,\mathrm{sr}. Using the hard-sphere result dσ/dΩ=a2/4\mathrm{d}\sigma/\mathrm{d}\Omega = a^2/4 (the same at every angle), find the count rate. ((c)) Find the laboratory-frame deflection angle χlab\chi_{\rm lab} corresponding to χ=60∘\chi = 60^\circ.

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