22.101 Applied Nuclear Physics Solutions to Quiz 3 Prob 1 (a) P(Ω_c) = C 0 ≤ θ_c ≤ π/2 = 0 otherwise ∫ dΩ_c P(Ω_c) = 1 ⇒ 2πC ∫₀^(π/2) sinθ_c dθ_c = 1 ⇒ C = 1/π (b) G(θ_c,θ_b) = ∫ dφ sinθ_c dθ_c P(Ω_c) = sinθ_c dθ_c F(E,E') = ∫ dΩ_c G(θ_c,θ_b) F'(E,E') with F' = E [δ(1+α) + δ(1-α)] cos θ_c ∫ dφ = 2E [δ(1+α) + δ(1-α)] cos θ_c θ_c = π/2, E' = E θ_c = 0, E' = E(1+α) F(E,E') = 2E [δ(1-α) + δ(1+α)] (c) [Diagram: A graph showing the probability distribution F(E,E') as a function of E'. The x-axis represents E', and the y-axis represents F(E,E'). The graph shows two peaks at E' = E and E' = E(1+α).] presents: spherical symmetry (d) Range: present range = E - E(1+α) = E(1-α) When angular range is restricted, expect the energy range to be also restricted. Reduction is a factor of 2 in the present case. Prob 2 [Diagram: A graph showing the intensity of scattered neutrons as a function of their energy (E). The graph has two peaks labeled A and B.] (a) Peak A: elastic scattering of thermal neutrons E' = E dominant process is Bragg diffraction in a crystal Peak B: inelastic (inelastic scattering) by exciting lattice vibrations (phonon emission) Intensity variation with T and θ peak A will vary with θ (Bragg condition, λ=2dsinθ) but will not vary with T peak B will not vary much with θ or α, although the phase absorption (up-scattering by de-exciting lattice vibrations) will be sensitive to T (intensity will increase with increasing T) Prob 3 (a) π⁺ does not undergo β⁺ decay (given). π⁺ must come from pair production by the π⁻ as indicated in the diagram. E > 1.02 MeV (provided) [Note: Another possible internal pair conversion also can occur - one person in the class mentioned this] (b) E(π⁺) = 4.04 - 1.38 = 2.7 GeV E(π⁻) = 1.38 GeV T(π⁺) = 2.7 - 0.14 = 2.56 GeV T(π⁻) = 1.38 - 0.14 = 0.36 GeV (c) Decay modes: π⁺ → μ⁺ + ν_μ allowed, F and G-T π⁺ → e⁺ + ν_e allowed, but highly suppressed π⁺ → π⁰ + e⁺ + ν_e suppressed π⁺ → π⁰ + μ⁺ + ν_μ suppressed π⁺ → π⁰ + π⁰ + e⁺ + ν_e suppressed E_γ is unique

22.101 Applied Nuclear Physics, Solutions to Quiz 3

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