At low energies (i.e., when is much larger than the range of the potential) partial waves with , in general, make a negligible contribution to the scattering cross-section. It follows that, at these energies, with a finite range potential, only -wave scattering is important.
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 (986)]
where use has been made of Equations (962)-(963). The internal wavefunction follows from Equation (991). We obtain
where use has been made of the boundary condition (992). Here, is a constant, and
Note that Equation (1009) only applies when . For , we have
Matching , and its radial derivative at , yields
for , and
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 . It follows from Equation (1013) that, unless becomes extremely large, the right-hand side is much less that unity, so replacing the tangent of a small quantity with the quantity itself, we obtain
According to Equation (1006), the scattering cross-section is given by
so for sufficiently small values of ,
It follows that the total ( -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 , and the scattering cross-section (1017) 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.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)