As a specific example, let us consider scattering by a finite potential well, characterized by for , and for . Here, is a constant. The potential is repulsive for , and attractive for . The external wavefunction is given by [see Equation (10.95)]

(10.118) |

where use has been made of Equations (10.60) and (10.61). The internal wavefunction follows from Equation (10.100). We obtain

where use has been made of the boundary condition (10.101). Here, is a constant, and

(10.120) |

Note that Equation (10.120) only applies when . For , we have

(10.121) |

where

(10.122) |

Matching , and its radial derivative, at yields

for , and

(10.124) |

for .

Consider an attractive potential, for which . Suppose that (i.e., the depth of the potential well is much larger than the energy of the incident particles), so that . As can be seen from Equation (10.124), unless becomes extremely large, the right-hand side of the equation is much less than unity, so replacing the tangent of a small quantity with the quantity itself, we obtain

(10.125) |

This yields

(10.126) |

According to Equation (10.115), the total scattering cross-section is given by

Now,

so for sufficiently small values of ,

(10.129) |

It follows that the total (S-wave) scattering cross-section is independent of the energy of the incident particles (provided that this energy is sufficiently small).

Note that there are values of
(e.g.,
) at which
the scattering cross-section (10.128) vanishes, despite the very strong
attraction of the potential. In reality, the cross-section is not
exactly zero, because of contributions from
partial waves. But,
at low incident energies, these contributions are small. It follows that
there are certain values of
,
, and
that give rise to almost perfect
transmission of the incident wave. This is called the *Ramsauer-Townsend
effect*, and has been observed experimentally [88,4].