Q1: A modern experiment is designed to test Rutherfords cross section. It involves a spherical detector surrounding a thin gold target. The detector has a small hole surround the beam in the region 0 < θ < π/10 and 9π/10 < θ < π. A thin, ideal beam of 10 MeV alpha particles are fired at the target. Corresponding to a total current of 10 pA.
(a) (5 pts) What is the solid angle of this detector?
(b) (5 pts) What is the ratio of alpha particles scattering into a near θ = π/8 and near θ = π/2?
Assume that you can use the Rutherford differential cross section.
(c) (10 pts) What is the rate the detector sees scattered alpha particles? Assume that you can use the Rutherford differential cross section.
(d) (5 pts) If the detector covered a solid angle of 4π, what would the ruther ford cross section be and why?
(e) (5 pts) If a beam of 500 MeV electrons is used, how would the measured cross section change qualitatively and what type of physics could this experiment be sensitive to?
(f) (5 pts) If this high energy beam lost 10 MeV of energy to an excitation of the nucleus, what would the scattering angle θ of the electron be?
Q2: A scattering experiment is performed to measure the form factor of Gold and Aluminum targets.
(a) (5 pts.) Draw what you expect these form factors to look like and describe what their features mean physically.
(b) (5 pts.) Which element’s form factor will have a broader distribution and why?
Q3: Assume a free proton is in its mass eigenstate in a definite momentum state with kinetic energy 20 MeV.
(a) (5 points) What is the protons momentum?
(b) (10 points) In a non-relativistic quantum theory, what is the phase this particle’s wave function gains after 1 second?
Q4: (10 pts) Compute the scattering amplitude in the first born approximations for a hard spherical
shell potential:
V(r) = αδ(x − a).
(5 pts) What does this reduce to in the limit ka << 1?
Q5: (15 pts.) If you are designing a neutrino oscillation experiment. Assume a world in which only
two neutrino flavors exist. Assume that you know the squared mass difference is 1 eV2 and
you know that mixing angle was 33 degrees. Your experiment start a source of 1 GeV electron
neutrinos At what distance would you put a detector to measure the dissapearance of these
electron neutrinos and why?
