Exercise 2.3

1. Before beginning any numerical computation, it is important to have some idea of what the results should look like. Sketch what you think the deflection function should look like at relatively low energies, $E \leq V_0$, where the the peripheral collisions at large $b\leq r_{max}$ will take place in a predominantly attractive potential and the more central collisions will ``bounce'' against the repulsive core. What happens at much higher energies $E\gg V_0$, where the attractive pocket in $V$ can be neglected? Note that the values of $b$ where the deflection function has a maximum or a minimum, Eq. (35) shows that the cross section should be infinite,as occurs in the rainbow formed when light scatters from water drops.

2. Write a program that calculates, for a given kinetic energy $E$, the deflection function solving the equations of motion a a number of equally spaced $b$ values between 0 and $r_{max}$.

3. Use your program to calculate the deflection function for scattering from a Lennard-Jones potential at selected values of $E$ ranging from $0.1V_0$ to $100V_0$. Reconcile your answers in step 1) with the results obtained. Calculate the differential cross sections a function of $\Theta$ at these energies.

4. If your program is working correctly you should observe for energies $E \leq V_0$ a singularity in the deflection function where $\Theta$ appear to approach $-\infty$ at some critical value of $b$, $b_{crit}$, that depends on $E$. This singularity, which disappears when $E$ becomes larger that about $V_0$ is characteristic of "orbiting", and the scattering angle becomes logarithmically infinite. What happens is that the particle spends a very long time spiralling around the center. Calculate some trajectories around this point and convince yourself that this is precisely what's happening. Determine the maximum energy for which the Lennard-Jones potential exhibits orbiting by solving the correct set of equations involving $V$ and its derivatives.