Scattering by a central potential

Figure 4: The Lennard-Jones 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 (see Fig. 5). 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.

Figure 5: Quantities involved in the scattering of a particle by a central potential.

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

$$\frac{d\sigma}{d\Omega}=\frac{b}{\sin{\Theta}}\vert\frac{db}{d\Theta}\vert.$$ (35)

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.

In the following we are going to consider a Lennard-Jones potential:

$$V(r)=4V_0[(\frac{a}{r})^{12}-(\frac{a}{r})^6],$$ (36)

The potential is attractive for long distances, and strongly repulsive approaching the core (see Fig. 4), with a minimum occurring at $r_{min}=2^{(1/6)}a$ with a depth $V_0$.