Next: Low energy scattering Up: Scattering theory Previous: Determination of phase-shifts

## Hard sphere scattering

Let us test out this scheme using a particularly simple example. Consider scattering by a hard sphere, for which the potential is infinite for , and zero for . It follows that is zero in the region , which implies that for all . Thus,
 (981)

for all . It follows from Eq. (974) that
 (982)

Consider the partial wave, which is usually referred to as the -wave. Equation (982) yields

 (983)

where use has been made of Eqs. (948)-(949). It follows that
 (984)

The -wave radial wave function is
 (985)

The corresponding radial wave-function for the incident wave takes the form
 (986)

It is clear that the actual radial wave-function is similar to the incident wave-function, except that it is phase-shifted by .

Let us consider the low and high energy asymptotic limits of . Low energy means . In this regime, the spherical Bessel functions and Neumann functions reduce to:

 (987) (988)

where . It follows that
 (989)

It is clear that we can neglect , with , with respect to . In other words, at low energy only -wave scattering (i.e., spherically symmetric scattering) is important. It follows from Eqs. (923), (965), and (984) that
 (990)

for . Note that the total cross-section
 (991)

is four times the geometric cross-section (i.e., the cross-section for classical particles bouncing off a hard sphere of radius ). However, low energy scattering implies relatively long wave-lengths, so we do not expect to obtain the classical result in this limit.

Consider the high energy limit . At high energies, all partial waves up to contribute significantly to the scattering cross-section. It follows from Eq. (967) that

 (992)

With so many values contributing, it is legitimate to replace by its average value . Thus,
 (993)

This is twice the classical result, which is somewhat surprizing, since we might expect to obtain the classical result in the short wave-length limit. For hard sphere scattering, incident waves with impact parameters less than must be deflected. However, in order to produce a shadow'' behind the sphere, there must be scattering in the forward direction (recall the optical theorem) to produce destructive interference with the incident plane-wave. In fact, the interference is not completely destructive, and the shadow has a bright spot in the forward direction. The effective cross-section associated with this bright spot is which, when combined with the cross-section for classical reflection, , gives the actual cross-section of .

Next: Low energy scattering Up: Scattering theory Previous: Determination of phase-shifts
Richard Fitzpatrick 2006-02-16