Physics 418, Homework Assignment
Due in Class, Wednesday, February 11, 2004

1. Using a web browser, get access to the ``Table of Nuclides'' from the www link on our website.
   • Make a table of all the isotopes of Carbon, listing
     • the atomic weight, A,
     • the atomic mass (amu)
     • the abundance (%) (only for stable isotopes, blank otherwise)
     • the half life with appropriate units (only for unstable isotopes)
     • the decay mode(s) - show the reaction(s) (only for unstable isotopes)
   • Do the same thing for Iron.
2. Protons with kinetic energy T$_p = 10.8 MeV$ are elastically scattered from Gold nuclei ($\rm ^{197}Au$).
   • Do you need relativistic kinematics for this situation?
   • What is the maximum polar angle for the final state proton?
   • What is the momentum of the proton at this angle?
   • What is the momentum of the recoil nucleus?
   Now change the incident beam particles from protons to electrons. The kinetic energy of the electrons is the same: T$_e = 10.8 MeV$. They are elastically scattered from Gold nuclei.
   • Do you need relativistic kinematics for this situation?
   • What is the maximum polar angle for the final state electron?
   • What is the momentum of the electron at this angle?
   • What is the momentum of the recoil nucleus?
3. Consider the form factor for a spherically symmetric nuclear charge density.
   • Show that it can be written:
     
     $$F(q^2) = \frac{1}{Ze}\frac{4\pi\hbar} {q}\int_{0}^{\infty} \rho(r)\cdot r\cdot\sin\frac{qr}{\hbar} dr$$
     
     

   • What is the requirement on $q$ such that the sine function in the integral can be safely expanded about zero?
   • Do this expansion, and show that
     
     $$F(q^2) = 1 - \frac{1}{6}\frac{q^2}{\hbar^2}<r^2>+...$$
     
     

   Here $r^2$ is the charge radius.
4. In the shell model of the nucleus, an example of the notation for the quantum numbers is $1 s_{1/2}$, where the ``principal'' quantum number n=1, the orbital angular momentum quantum number $\ell = 0$ is an ``$s$''-state, and the total angular momentum is $J = 1/2$. In the same notation, write down the highest-energy state which is just filled with a ``magic number'' = 82 of neutrons. Suppose we add one neutron to this nucleus to form a new isotope. Again in the same notation, what is the state of the newly added neutron? (In this problem, assume the nuclei are in their ground states.)