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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#9, Rev. A, due Tuesday, 2021-11-23 }
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\vspace{.3 in}

{\bf 1. Chalkboard scattering} Perform a scattering experiment to measure the cross-sectional area $\sigma$ of each of $T$ identical targets drawn randomly on a chalkboard.  Throw as many racquetballs as you can and record the number $H$ of hits versus $M$ of misses.  What other measurements do you need to extract the total cross section $\sigma$ of the target?  Calculate $\sigma$ and its statistical uncertainty from your data and compare to the actual area of the target.  Note that the cross section includes information about both the target and the scatterer---what area are we actually measuring experimentally?  How could you generalize this experiment to measure a differential cross section?


{\bf 2. Rutherford scattering} of $\alpha$ particles $(Z=2)$ from the atoms $(Z=79)$ in a thin foil of gold due is due to the Coulomb repulsion $F=Z_{1}Z_{2}e^{2}/4\pi\epsilon_{0}r^{2}$ of the nuclei in the target.  This inverse square law has the same form as gravity, with the same trajectories as the unbound Kepler problem (H08 \#3, with positive total energy $E>0$).  The repulsive potential corresponds to the opposite branch of the hyperbola than the attractive potential.

{\bf \quad a)} Calculate the Rutherford cross section $d\sigma/d\Omega$ as a function of scattering angle $\theta$ and energy $E$ of an $\alpha$-particle scattering from a gold nucleus.  You may assume that the $\alpha$ particle is energetic enough that electron screening is negligible.

{\bf \quad b)} What is the probability that an $\alpha$ particle would reflect within $1^{\circ}$ of straight backwards from a $1~\mu$m thick gold foil?


{\bf 3.} A {\bf rainbow} is formed by rays of sunlight refracting through raindrops, internally reflecting, and refracting back out toward the viewer.

{\bf \quad a)} Calculate the scattering angle $\theta$ as a function of the impact parameter $s$ for a given index of refraction $n$ of the raindrop.  Calculate the rainbow angle $\theta_{0}$ as the {\it caustic} where $d\theta/ds|_{\theta_{0}}=0$.  Much of the incoming light scatters in this direction because of the zero derivative.

{\bf \quad b)} Calculate the cross section $d\sigma/d\Omega$ as a function of $s$ (not $theta$!).  Show that the peak in the cross section at $s=s_{0}$ occurs at the same angle $\theta_{0}=\theta(s_{0})$ as part a).  Explain why the inside of a rainbow is brigher than the outside.

{\bf \quad c)} [bonus: Calculate the scattering angle $\theta$ for $m$ internal reflections (eg. double rainbow).]


{\bf 4. Optical potential}

The {\it optical potential} $V(r)=V_{0}\,\theta(a-r)$ is the mechanical analog of rainbow scattering.  It describes particles that slow down [or speed up] as they enter a spherical target and return to their original speed as they exit the other side.  The constant potential $V_{0}$ accounts for the loss [gain] of kinetic energy inside the sphere of radius $a$.  The gradient $\delta(a-r)$ of the step function $\theta(a-r)$ represents a finite impulse at the surface of the sphere due short range repulsion/attraction from material inside the target.  This causes a deflection from Snell's on the particle's wave function just the same as the optical effect.

{\bf \quad a)} What is the 'index of refraction' $n=v/v_{0}$ of a particle in terms ratio $E/V_{0}$ of total energy to potential well height/depth for a particle of mass $m$?

{\bf \quad b)} Calculate the scattering cross section $d\sigma/d\Omega$ as a function of $s$ assuming no internal reflections, using the method of problem \#3.

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