Scattering by a central potential

In this section, we will investigate the classical scattering of a particle of mass $m$ by a central potential. In a scattering event, the particle, with initial kinetic energy $E$ and impact parameter $b$ approaches the potential from a large distance. It is deflected during its passage near the force center and eventually emerges with the same energy, but moving at an angle $\Theta$ with respect to the original direction. This problem is very similar in many aspects to the orbital motion, but in this case the potential is repulsive, and it is not necessarily a function of the inverse square of the distance. The energy and momentum are conserved, and the trajectory lies in the plane.

Our basic interest is on the deflection function $\Theta (b)$, giving the final scattering angle $\Theta$ as a function of the impact parameter. This function also depends upon the incident energy. The differential cross section for scattering at an angle $\Theta$, $d\sigma / d\Omega$ is an experimental observable that is related to the deflection function by

Thus, if $d\Theta /db$ can be computed, the cross section is known.

Expressions for the deflection function can be found analytically only for a few potentials, so that numerical methods usually must be employed. There are some simplification that can me made using the fact that the angular momentum is conserved, which connects the angular and the radial motion, making the problem one-dimensional. However, in this section we are going to use the tools learned in the previous sections, and solve the four first-order differential equations for the two coordinates and their velocities in the $xy$ plane.

Quantities involved in the scattering of a particle by a central potential.

In the following we are going to consider a Lennard-Jones potential: $$V(r)=4V_0[(\frac{a}{r})^{12}-(\frac{a}{r})^6],$$ The potential is attractive for long distances, and strongly repulsive approaching the core (see Fig. [lennard-jones]), with a minimum occurring at $r_{min}=2^{(1/6)}a$ with a depth $V_0$.

The Lennard-Jones potential

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. ([cross]) 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 at 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.