PHYS 620 Classical Mechanics II - University of Delaware. Fall 2018 Homework 9 Solutions.

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PHYS 620 Classical Mechanics II - University of Delaware. Fall 2018 Homework 9 Solutions.PHYS620 Fall 2018 Homework 9 Solutions. 1. The cross section for scattering a certain nuclear particle by a co... pper nucleus is 2.0 barns (if needed, look up the definition of a barn). If 109 of these particles are fired through a copper foil of thickness 10 μm, how many particles are scattered? (Copper’s density is 8.9 g/cm3 and its atomic mass is 63.5). 2. The differential scattering cross section for 6.5 MeV alpha particles at 120° off a silver nucleus is about 0.5 barns/sr. If a total of 1010 alphas impinge on a silver foil of thickness 1 μm and if we detect the scattered particles using a counter of area 0.1 mm2 at 120° and 1 cm from the target, about how many scattered alphas should we expect to count? (Silver has density 10.5 g/cm3 and atomic mass of 108.) 3. One can set up a two dimensional scattering theory, which could be applied to puck projectiles sliding on an ice rink and colliding with various target obstacles. The cross section σ would be the effective width of a target, and the differential cross section dσ/dθ would be the number of projectiles scattered in the angle dθ. (a) Show that the two dimensional analog of sin θ θ d d b db d σ = Ω is . d db d d σ θ θ = (b) Now consider the scattering of a small projectile off a “hard” sphere of radius R pinned down to the ice. Find the differential scattering cross section. (c) By integrating the differential cross section, show that the total cross section is 2R as expected. 14. One of the first observations that suggested his nuclear model of the atom to Rutherford was that several alpha particles got scattered by metal foils into the backward hemisphere, π/2 ≤ θ ≤ π – an observation that was impossible to explain on the basis of α b 𝜃𝜃 rival atomic models, but emerged naturally from the nuclear model. In an early experiment, Geiger and Marsden measured the fraction of incident alphas scattered into the backward hemisphere off a platinum foil. By integrating the Rutherford cross section over the backward hemisphere show that the cross section for scattering with θ ≥ 90° should be 4πσ0(E). Using the following numbers, predict the ratio Nsc(θ ≥ 90°)/Ninc: thickness of platinum foil ≈ 3 μm, density 21.4 g cm-3, atomic weight = 195, atomic number = 78, energy of incident alphas = 7.8 MeV. Compare your answer with their estimate that “of the incident α particles about 1 in 8000 was reflected”. 5. Consider the scattering of a particle of energy E by a fixed repulsive 1/r3 force field, with potential energy U = γ/r2. Find θ in terms of b and show that the differential cross section is 6. Consider the elastic scattering of a projectile that is heavier than the target, that is λ = m1/m2 > 1. (a) Draw a diagram showing the relationship between the initial and final momenta in the CM and LAB frames. Show that there are two values of the CM scattering angle for each value of the lab scattering angle. (b) What are the two CM angles that correspond to θLAB = 0? In terms of this example, explain why there is this twofold ambiguity in θCM when λ > 0. (c) Plot θLAB as a function of θCM for the case that λ = 2. (d) Use the picture from part (a) to find an expression for the maximum possible of θLAB for a given value of λ. Check that your answer is correct for the case that λ = 1. [Show More]

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