[Computer] Consider a point projectile moving in a fixed, spherical force
whose potential energy is
$$U(r)=\left\\{\begin{array}{ll}
-U_{0} & (0 \leq r \leq R) \\
0 & (R < r)
\end{array}\right.\quad\quad\quad\quad\quad\quad(14.60)$$
where \(U_{\mathrm{o}}\) is a positive constant. This so-called spherical well
represents a projectile which moves freely in either of the regions \(r < R\)
and \(R < r_{1}\) but, when it crosses the boundary \(r=R_{1}\) receives a
radially inward impulse that changes its kinetic energy by \(\pm
U_{\mathrm{o}}\left(+U_{\mathrm{o}}\right.\) going inward, \(-U_{\circ}\) going
outward). (a) Sketch the orbit of a projectile that approaches the well with
momentum \(p_{\mathrm{o}}\) and impact
parameter \(b < R .\) (b) Use conservation of energy to find the momentum \(p\) of
the projectile inside the well \((r < R)\). Let \(\zeta\) denote the momentum
ratio \(\zeta=p_{\mathrm{o}} / p\) and let \(d\) denote the projectile's distance
of closest approach to the origin. Use conservation of angular momentum to
show that \(d=\zeta b\). (c) Use your sketch to prove that the scattering angle
\(\theta\) is
$$\theta=2\left(\arcsin \frac{b}{R}-\arcsin \frac{\zeta
b}{R}\right).\quad\quad\quad\quad\quad\quad(14.60)$$
This gives \(\theta\) as a function of \(b\), which is what you need to get the
cross section. The relation depends on the momentum ratio \(\zeta,\) which in
turn depends on the incoming momentum \(p_{\mathrm{o}}\) and the well depth
\(U_{\mathrm{o}} .\) Plot \(\theta\) as a function of \(b\) for the case that
\(\zeta=0.5 .\) ( \(\mathbf{d}\) ) By differentiating \(\theta\) with respect to
\(b\), find an expression for the differential cross section as a function of
\(b\), and make a plot of \(d \sigma / d \Omega\) against \(\theta\) for the case
that \(\zeta=0.5 .\) Comment. [Hint: To plot as a function of \(\theta\) you don't
need to solve for \(b\) in terms of \(\theta ;\) instead, you can make a
parametric plot of the point \((\theta, d \sigma / d \Omega)\) as a function of
the parameter b running from 0 to \(R\).] (e) By integrating \(d \sigma / d
\Omega\) over all directions, find the total cross section.
