9.8: Low Energy Scattering
- Page ID
- 1245
At low energies (i.e., when \( l>0\) , in general, make a negligible contribution to the scattering cross-section. It follows that, at these energies, with a finite range potential, only \( V=V_0\) for \( V=0\) for \( V_0\) is a constant. The potential is repulsive for \( V_0<0\) . The external wavefunction is given by [see Equation \ref{986}]
where use has been made of Equations \ref{962}-\ref{963}. The internal wavefunction follows from Equation \ref{991}. We obtain
where
for \( \tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh (\kappa\, a)\)
for \( E>V_0\) . Suppose that \( k' \gg k\) . It follows from Equation \ref{1013} that, unless \( k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).\) This yields Now,\ref{1015} \( \sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sin^2\delta_0 =4\pi \,a^2\left[\frac{\tan (k'\,a)}{(k'\,a)} -1\right]^{\,2}.\) \ref{1017} ,\( k\,a\)
Note that there are values of \( k'a\simeq 4.49\) ) at which \( l>0\) partial waves. But, at low incident energies, these contributions are small. It follows that there are certain values of \( k\) that give rise to almost perfect transmission of the incident wave. This is called the Ramsauer-Townsend effect, and has been observed experimentally.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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