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{\large \bf Physics 499 Winter 2016}\\
Homework Assignment 3\\
Kinematics and Simulation of $\pi - N$ scattering\\
\medskip
Due February 16th
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\bigskip
In this assignment, you will write two programs: One to calculate the center
of mass momentum for a general two-particle interaction and, another to
simulate the outcome of a scattering experiment.\\
\bigskip
\noindent {\bf Problem 1}.\\
Suppose you are interested in a two-body scattering problem where particle 1
collides with particle 2 at rest. After the collision, particle 3 and particle
4 emerge:
\begin{equation}
1 + 2 \rightarrow 3 + 4
\end{equation}
\begin{itemize}
\item {\bf Your program should input} the masses of the 4 particles:
$m_1$, $m_2$, $m_3$, and $m_4$. It should also input the laboratory
kinetic energy of particle $1$. Particle $2$ is at rest in the lab
frame.
\item {\bf Your program should output} the center of mass energy of the system
(i.e. the total invarient energy of the system), and the momentum of particle
$3$ (also $4$) in the center of mass frame.
\end{itemize}
\bigskip
\noindent {\bf Problem 2}.\\
The experiment will be $\pi^+$ particles scattering off of a neutron. The
differential cross section is given by:
\begin{equation}
{{d\sigma} \over {d\Omega}} = |f_0 + f_1 cos(\theta)|^2
\end{equation}
\noindent where $f_0$ and $f_1$ are the "s" and "p" amplitudes for orbital
angular momentum $l=0$ and $l=1$. The angle $\theta$ is the angle of the
scattered pion in the {\it center of momentum (or mass)} frame.
Take for the values of $f_0$ and $f_1$ to be the following:
\begin{eqnarray*}
f_0 & = & 1.1+0.1i \; fm\\
f_1 & = & -1.5 + 0.2i \; fm
\end{eqnarray*}
\noindent You should assume a Gaussian scatter of each data point, with a $\sigma$ of
$x$ times the value of the differential cross section. Your program should let the
user input the value of $x$.
{\bf Your root program should output a plot of the simulated data for angles between
$10^\circ$ and $160^\circ$ in increments of $10$ degrees}. The horizontal axis should be
the scattering angle in units of degrees, and the vertical axis should be the differential
cross section in units of millibarns.
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