Answer Happy

Need parts E and F.
In the lecture notes, we did an example of "hard-sphere" scattering. Although we did this for effectively billiard ball type particles, the same result matches the law of reflection from geometric optics. In this problem, we'll consider "soft-sphere" scattering which models the laws of refraction from geometric optics. The algebra and geometry of soft- sphere scattering is relatively straightforward, to set-up, and although the algebra gets a little hairy at the end, it's mostly just trig identities to get things in the final useful forms. This problem also comes up in quantum mechanics, so if you have, or plan to, take(n) quantum mechanics, you can say you saw it here first! To define a soft-sphere, consider the following potential: {v=0 V = 0 r>a = - V₁ r ≤a where a is the radius of the sphere, Vo is the potential at the boundary, so effectively this a spherical step function. Suppose a particle is in- coming on the soft sphere from the "left" with velocity v₁ and impact parameter s. We will parameterize the scattering of the particle by two angles in the lab frame 0₁, the angle between the incident particle and the vector to where the particle strikes the soft sphere, and 02, the angle between −ô and 72, the velocity vector of the particle after it's scattered by the surface of the soft sphere. In this problem, I'll use the convention that v; = = |vi|. a) Sketch and then write down, in polar coordinates relative to the center of the soft-sphere, 7₁. (Ans: vi =−01 cos 0ır + vi sin 010). b) Do the same for V₂. (Ans: -v₂ cos 0₂r + v₂ sin 0₂Ô) c) The potential is radially symmetric, so any force on the particle must act in the direction which implies that linear momentum is conserved in the direction. Use conservation of momentum and the above results to derive a formula for v2 in terms of v₁, 01, 02. d) Use conservation of energy before and after the collision with the soft-sphere's surface and your results from above to derive a for- mula for the ratio sin 0₁/ sin 02 in terms of E, V₁.(Reminder: The particle always has kinetic energy (T = 1/2mv²) and at the be- ginning of the problem we gave the formula for V(r)). e) A result from geometric optics that you have likely encountered, is Snell's law of refraction which states V2 n1 sin 02 sin 0₁ V1 N2 where the n₁, n₂ are the index of refraction of the two materials, V₁, V2 are the speed of light in the medium, and 0₁,02 are the angles measured from the normal of the boundary. Assuming for free space n = 1, what is the effective index of refraction of the soft sphere? Evidently, refraction is a classical soft-sphere scattering process. = f) Using the analogy to Snell's law, and geometry, what are the im- pact parameters and the scattering angle in terms of a, 0₁, 02? (Ans: s= = a sin 0₁ and = 2 (0₁ — 0₂)). o(O): = The usual formula for the differential scattering cross-section is (from Goldstein and our class notes) = S ds sin Ꮎ | dᎾ