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{\large \bf Physics 499}\\
Homework Assignment 3\\
Meson-Nucleus elastic scattering\\
\medskip
Due Friday May 10th
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\begin{tabbing}
Problem : \= $K^+-^{12}C$ elastic scattering \\
Reference: \> Phys. Rev. C30, 1256-1266 (1984).
\end{tabbing}
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For this assignment you will determine (numerically) the elastic scattering
cross section for $K^+$ scattering off of nuclei. You will use the
Born approximation, which involves solving the following integral numerically:
\begin{equation}
\int_0^\infty V(r) r sin(qr) dr
\end{equation}
\noindent where $V(r) = V_0/(1+e^{(r-R)/a})$, and $q=2pc/(\hbar c)sin(\theta /2)$.
In this expression, $\theta$ is the scattering angle, $p$ is the $K^+$ momentum,
and $R=1.28A^{1/3} \; fm$. We will take $a=0.6 \; fm$. The scattring amplitude,
$f(\theta )$ from the Born approximation is:
\begin{equation}
f(\theta ) = -{1 \over {\hbar c}} {{mc^2} \over {pc}} {1 \over {sin(\theta /2)}}
\int_0^\infty V(r) r sin(qr) dr
\end{equation}
\noindent and the differential cross section is
\begin{equation}
{{d \sigma} \over {d \Omega}} = |f(\theta )|^2
\end{equation}
I have attached the data for $K^+-^{12}C$ elastic scattering for a kaon lab momentum
of $800 \; MeV/c$. Note: the kaon momentum in the center of mass frame is
$740 \; MeV/c$. The rest mass of the $K^+$ is $493 \; MeV/c^2$.\\
\noindent Choose one of the following two assignments:\\
\noindent {\bf Assignment 3a}:(the gcc or ROOT compiler)\\
Write a computer program that calculates the differential cross section,
$|f(\theta )|^2$ for $K^+-^{12}C$ elastic scattering at $K^+$ momentum
$800 \; MeV/c$. Your program should ask the user to input the
potential strength $V_0$. Your program should then output to the screen the
cross section for angles $10^\circ \rightarrow 38^\circ$ at two degree
increments.\\
For what value of $V_0$ (if any) do you get a reasonable fit to the data in this angular range?\\
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\noindent {\bf Assignment 3b}:(the ROOT compiler only)\\
Write a program in ROOT that will make a publication quality graph of
the $K^+-^{12}C$ elastic scattering data plus error bars from the
data of Marlow et. al., that is on the next page. In addition to
your *.C program, you should e-mail me a *.eps file of your final
graph.
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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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\includegraphics[width=14cm]{marlow.png}
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