Hard-Sphere Scattering

(10.103) |

for all . Equation (10.97) yields

In fact, this result is most easily obtained from the obvious requirement that . [See Equation (10.95).]

Consider the
partial wave, which is usually referred to as the *S-wave*.
Equation (10.105) gives

(10.105) |

where use has been made of Equations (10.60) and (10.61). It follows that

The S-wave radial wave function is

(10.107) |

[See Equation (10.95).] The corresponding radial wavefunction for the incident wave takes the form

(10.108) |

[See Equations (10.79), (10.80), (10.94), and (10.107).] It is clear that the actual radial wavefunction is similar to the incident wavefunction, except that it is phase-shifted by .

Let us consider the low- and high-energy asymptotic limits of . Low energy corresponds to . In this limit, the spherical Bessel functions and Neumann functions reduce to

(10.109) | ||

(10.110) |

where [1]. It follows that

(10.111) |

It is clear that we can neglect , with , with respect to . In other words, at low energy, only S-wave scattering (i.e., spherically symmetric scattering) is important. It follows from Equations (10.28), (10.81), and (10.107) that

(10.112) |

for . Note that the total cross-section,

(10.113) |

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 de Broglie wavelengths, so we would not expect to obtain the classical result in this limit.

Consider the high-energy limit, . At high energies, by analogy with classical scattering, the scattered particles with the largest angular momenta about the origin have angular momenta (i.e., the product of their incident momenta, , and their maximum possible impact parameters, ). Given that particles in the th partial wave have angular momenta , we deduce that all partial waves up to contribute significantly to the scattering cross-section. It follows from Equation (10.90) that

Making use of Equation (10.105), as well as the asymptotic expansions (10.62) and (10.63), we find that

(10.115) |

In particular,

(10.116) |

Hence, it is a good approximation to write

(10.117) |

[59]. This is twice the classical result, which is somewhat surprising, because we might expect to obtain the classical result in the short-wavelength limit. In fact, for hard-sphere scattering, all incident particles with impact parameters less than are 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. The effective cross-section associated with this forward scattering is , which, when combined with the cross-section for classical reflection, , gives the actual cross-section of [95].