9.8: Low Energy Scattering

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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}]

$A_0(r) = \exp(\,{\rm i}\, \delta_0)\,\left[ j_0(k\,r) \cos\delta_... ...elta_0\right] = \frac{ \exp(\,{\rm i} \,\delta_0)\, \sin(k\,r+\delta_0)}{k\,r},$ \ref{1008}

where use has been made of Equations \ref{962}-\ref{963}. The internal wavefunction follows from Equation \ref{991}. We obtain

$ B$ is a constant, and $ E>V_0$. For $ A_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$ \ref{1011}

where

$ A_0(r)$, and its radial derivative at $ \tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan (k'\,a)$ \ref{1013}

for $ \tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh (\kappa\, a)$

\ref{1014}

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).$ \ref{1015}

This yields

$ \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}
Now,
$ k\,a$ ,

$ 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 $ 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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