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{\large \bf Physics 499}\\
Homework Assignment 2\\
Nuclear Shell Model\\
\medskip
Due Friday April 26th
\end{center}
\begin{tabbing}
Problem : \= Nuclear Shell Model \\
Reference: \> Am. J. Phys. 68, 848 (Sept. 2000).
\end{tabbing}
\medskip
For this assignment you will determine (numerically) the allowed bound state
energies for a neutron and a proton confined within a nucleus. To determine
the allowed energies, solve the radial part of the discrete Schroedinger equation
as derived in lecture:
\begin{equation}
u(i+1) = 2u(i) - u(i-1) + \Delta^2 {{l(l+1)} \over {r^2}} u(i) +
{{2m \Delta^2} \over {\hbar^2}} (V(i) - E) u(i)
\end{equation}
\noindent for the energies $E$ of all bound states.
For the strong potential that a neutron and a proton will experience,
we will use a spherical square well potential, as we did for the
$\Lambda$ particle:
\begin{eqnarray*}
V(r) & = & -V_0 \hspace{2cm} r \le R \\
& = & 0 \hspace{2cm} r > R
\end{eqnarray*}
\noindent The proton will have in addition to the strong potential, an
electrostatic potential. We will take this potential to be that due to
a uniformly charged sphere of radius $R$ and total charge $Ze$:
\begin{eqnarray*}
V_{Coulomb}(r) & = & Ze^2 {{3R^2-r^2} \over {2R^3}} \hspace{1cm} if \; r \le R \\
& = & {{Ze^2} \over r} \hspace{1cm} if \; r>R
\end{eqnarray*}
\noindent In your calculation, use the following values:
Take the nuclear radius to be $R = 1.28 A^{1/3}$ fm,
$m_{neutron} \approx m_{proton} \approx 940 \; MeV/c^2$; $\hbar c = 197.33 \; MeV-fm$,
and $V_0 = 50 \; MeV$.
\newpage
Using the method discussed in lecture, find all the allowed energy levels for neutrons
and protons for the following values of $A$ and $l$:\\
\centerline{\bf Neutrons}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$A$ & $l=0$ & $l=1$ & $l=2$ & $l=3$ \\
\hline
12 & & & & \\
16 & & & & \\
28 & & & & \\
34 & & & & \\
40 & & & & \\
\end{tabular}
\end{center}
\centerline{\bf Protons}
\begin{center}
\begin{tabular}{c|c|c|c|c}
$A$ & $l=0$ & $l=1$ & $l=2$ & $l=3$ \\
\hline
12 & & & & \\
16 & & & & \\
28 & & & & \\
34 & & & & \\
40 & & & & \\
\end{tabular}
\end{center}
\noindent Note: There may be more than one energy for a particular value of $l$, and
for the smaller nuclei there may be no bound states for larger $l$.
\bigskip
\noindent Your computer code should ask the user to input $A$, $Z$, $l$, $V_0$, and
the starting value for the energy. Your code should output the bound state
energy that is just above the starting energy. For the largest
nucleus ($A=40$), what is the ordering of the energy levels?\\
You should turn in (e-mail) two files: your computer code that will run in
either gcc or ROOT, and a file discussing your results. For the discussion
file, you can use straight text (*.txt) or latex. No *.doc files.
Be sure your name is somewhere in each file you e-mail to me.
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