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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#10, Rev. B, due Tuesday, 2019-12-10 }
\end{center}

\vspace{.3 in}

{\bf 1. Cross sections}

{\bf \quad a)} 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 cross section?  Calculate the total cross section $\sigma$ 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 \quad b)} Calculate the Rutherford cross section of $\alpha$ particles $(Z_{2}=2)$ scattering from the nucleus of gold atoms $(Z_{1}=79)$ in a thin foil due to the electric force $F=Z_{1}Z_{2}e^{2}/4\pi\epsilon_{0}r^{2}$.  This inverse square law has the same form as gravity, and the trajectories have the same paths as Kepler motion (H09 \#3) with positive total energy $E>0$.  The repulsive potential corresponds to the opposite branch of the hyperbola than the attractive potential.

{\bf \quad c)} A rainbow is formed by rays of sunlight refracting into raindrops, internally reflecting, and refracting back out toward the viewer.  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 $\theta_{0}$ in the usual manner as the {\it caustic} where $d\theta/ds|_{\theta_{0}}=0$.  [bonus: Calculate the differential cross section, and show it peaks near $\theta_{0}$.]

[bonus: The mechanical analog of this wavelike phenomenon is a popular model in nuclear scattering called the {\it optical potential} $V(r)=-V_{0}\,\theta(a-r)$: a constant potential $V=-V_{0}$ inside the sphere $r<a$ and a free particle $V=0$ outside.  The step function $\theta(a-r)$ across the surface of the sphere causes a finite impulse which decreases the velocity of the particle as it enters the sphere.  What is the relation between the index of refraction $n$ of a light ray, and the ratio $E/V_{0}$ of total energy to potential well depth for a particle of mass $m$, to have the same bend angle into the sphere as the corresponding refraction?]

\newpage
{\bf 2. Nuclear Magnetic Resonance} (NMR) is a high-precision spectroscopy technique developed in 1946 by Edward Purcell and Felix Bloch.  This technology is used in Magnetic Resonance Imaging (MRI) machines.  In NMR, the spin $\bs s$ of an atomic nucleus precesses about a constant magnetic field $B_{0}$ similar the precession of a spinning top.  This characteristic resonance called the {\it Larmor frequency} $\omega_{L}=\gamma B$ is proportional to $B_{0}$ and to the gyromagnetic ratio $\gamma$ of the magnetic moment $\bs\mu$ to the angular momentum $\bs s$ of the nucleus, ie. $\bs\mu=\gamma\bs s$.

{\bf \quad a)} Show that $\gamma=e/2m$ for a charged point particle in a circular orbit, independent of radius $r$.  This does not hold for a composite particle; we define its {\it g-factor} by $\gamma=g\cdot e/2m$, ie. how much larger than for a point particle of the same mass and charge.  A pointlike quantum mechanical particle with spin $s=\hbar/2$ has $g=2$.  For example, the electron has $g_{e}=2.002319$, which is slightly larger than 2 due to vacuum polarization.  The neutron also has spin $s=\hbar/2$, but $g_n = -3.826$ due to its internal quark structure.  Thus its magnetic moment is $\mu_{n}=g(e/2m)(\hbar/2) = (g/2)\mu_{N} = -1.91~\mu_{N}$. The unit of magnetic moment used in nuclear physics is the {\it nuclear magneton} $\mu_{N}=e\hbar/2m_{p}$, which equals the magnetic moment of a pointlike proton with orbital angular momentum $\ell=\hbar$ (p-orbital).
              
{\bf \quad b)} Solve the classical equation of motion (the Bloch equation)
\begin{equation} \label{bloch}
\frac{d\bs s}{dt} = \bs \tau = \bs \mu_n\times \bs B = \gamma_n \bs s \times \bs B
\end{equation}
for the {\it Larmor precession} of a neutron in a constant magnetic field $\bh z B_0$, with initial spin $\bs s_0$ at $t=0$.  Compare this with the precession of a spinning top.

{\bf \quad c)} The mechanical equations of motion are simplified in a rotating reference frame, where \scalebox{.8}{$\pmatrix {\bh x' & \bh y'}=\pmatrix {\bh x & \bh y} \pmatrix{\cos\omega t & - \sin\omega t \cr \sin\omega t & \cos\omega t}$}.  \href{http://bit.ly/2TNaBQ8}{[Rabi, Ramsey, Schwinger, Rev.  Mod.  Phy.  {\bf 26}, 167, 1954]}.  Since the operator $\bs\omega dt \times$ generates this rotation, the time derivative becomes $\frac{d\bs s}{dt} = \frac{d\bs s'}{dt} + \bs\omega \times \bs s$, where $\frac{d\bs s'}{dt}$ is with respect to components in the rotating frame.  Substitute $\frac{d\bs s}{dt}$ into Eq.~\ref{bloch} and show that the form remains the same except for the replacement of $\bs B$ with the effective field $\bs B'=\bs B+\bs \omega/\gamma_n$.  Note this field is zero if the frame is rotating at the Larmor frequency $\bs\omega_L = -\gamma_n \bs B$, and thus the spin remains constant $\bs s'=\bs s_0$.  Reconcile this picture with the solution in the static frame.

{\bf \quad d)} The $z$-component of spin $s_{z}$ does not change in a constant magnetic field $\bh z B_0$ (called a {\it holding field} because it preserves the spin state).  To transition the spin from up to down, we must use an oscillatory (RF) field $B_1(\bh x\cos\omega t + \bh y\sin\omega t)$.  In the rotating frame with angular velocity $\bh z\omega$, show that the total field is $\bs B'=\bh z' (B_0+\omega/\gamma_n)+\bh x' B_1$, which is constant.  Let $\theta$ be the angle between $\bs B'$ and $\bh z$, and let the initial spin be $\bs s_0=\bh z$.  Show that the $z$-component of the spin varies as $s_z(t) = s_0(\cos^2\theta + \sin^2\theta \cos \gamma_nB't) = s_0(1-2\sin^2\theta\sin^2(\gamma_n B' t/2))$, which oscillates at the {\it Rabi flopping frequency} $\omega_{R}=\gamma_nB'$.  Plot the amplitude of oscillation as a function of the RF frequency $\omega$ and note the resonance at $\omega=\omega_L$.  [bonus: Plot the 3-d trajectory of $\bs s$ in the lab frame over half a Rabi cycle.]

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