A beam of particles scatters classically from a fixed attractive potential well of range
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion
Action and Symmetry, Ch Scattering in a central field
Problem
A beam of particles scatters classically from a fixed attractive potential well of range a. Solving the orbit integral numerically gives the deflection function χ(b) tabulated below, with χ measured so that a negative value means the particle is thrown to the opposite side of the incident direction from a repulsive centre; the physically observed scattering angle is always ∣χ∣, between 0 and 180∘. b/a: 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 2.0 χ (degrees): 180, 150, 100, 40, −10, −35, −32, −22, −14, −6
((a)) Locate the glory: by linear interpolation between the two bracketing table entries, give the impact parameter (to two significant figures, in units of a) at which χ(b) passes through zero, and the observed angle there.
((b)) Locate the rainbow: from the table, give the impact parameter and observed angle at the stationary point of χ(b).
((c)) Explain in one or two sentences why dσ/dΩ=(b/sinχ)∣db/dχ∣ diverges at each of these two impact parameters, and why a real detector of finite solid angle nonetheless records a finite count there.