Physics 406 Homework 2
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Physics 406
Homework 2
Due Friday January 27.
Problem 1. Consider elastic collision of a proton of total energy E and mass m off a stationary proton (so called lab frame) .
(a) Determine the maximal possible transverse momentum of the final state proton. Hint: use the c.m. frame to analyze the kinematics.
(15 pts)
(b) Find the emission angles θ 1 , θ2 for the protons in the lab. frame for the scattering with maximal transverse momenta (see Fig.1), as well as their
total energies.
(15 pts)
(c) Analyze how θi depend on E in high energy limit: E 》 m. (10 pts)
θ 2
Figure 1: Sketch of the geometry of pp collisions in the lab frame.
Problem 2. Read the article of Chadwick.
(a) Write equations for energy and momentum conservation for the discussed process.
(10 pts)
(b) Analyze the equations and confirm Chadwick’s statement that if the protons ejected from the hydrogen were due to a Compton-like effect, the incident gamma energy would have to be near 50 MeV and that such a gamma ray would produce recoil nitrogen nuclei with energies up to about
400 KeV.
(20 pts)
(c) What nitrogen recoil energies would be expected for the neutron hy- pothesis?
(10 pts)
The paper is posted on the webpage as chadwick.pdf.
Hints: you can use nonrelativistic approximation for nuclei/ neutron involved in the process; both energy and momentum conservation are important; max- imal energy transfers correspond to collinear kinematics (zero transverse mo- menta for produced particles).
Problem 3. In 2013 the LHC studied collisions of a beam of Pb-208 and a beam of protons. The proton beam had energy E0 =3.5 TeV, while the energy of the Pb beam was E0 . Z/A. (Z is the charge of Pb nucleus, A=208). That is the total energies of the nuclei are E0 . Z . What energy the proton beam should have to reach the same invariant energy for scattering off the stationary lead target?
In the calculation you can neglect corrections of the order mN /E0 (20 pts)
Problem 4. The nuclear density away from the surface is ρ0 = 0.17nucleon/fm3 . Calculate the mass of an orange of diameter 10 cm build of nuclear matter.
(10 pts)
Extra credit problem 1:
Starting from the momentum space expression for the nuclear form factor discussed in class
F (q) = ψ(k)ψ* (k + q)d3 k,
derive the expression in the coordinate space : F (q) = |ψ(r)|2 exp(iq . r)d3r Hint: You have to use the expressions for ψ(k), ψ(k + q) as Fourier trans- forms of the coordinate space waver functions. If you are uneasy with three
dimensional case, consider a one dimensional case.
(20 pts)
Extra credit problem 2:
Calculate the difference of the energy bindings of 3 He and 3 H assum- ing that it originates solely from the Coulomb repulsion between protons in 3 He. (That is assuming thatall other contributions to the NN potential of interaction nucleons in 3 He and 3 H are the same.)
To estimate the Coulomb contribution assuming that the average distance between two nucleons in A = 3 system is 2fm.
(15 pts).
2023-01-28