next up previous
Next: Resonances Up: Scattering Theory Previous: Hard Sphere Scattering

Low Energy Scattering

In general, at low energies (i.e., when $1/k$ is much larger than the range of the potential) partial waves with $l>0$ make a negligible contribution to the scattering cross-section. It follows that, at these energies, with a finite range potential, only $S$-wave scattering is important.

As a specific example, let us consider scattering by a finite potential well, characterized by $V=V_0$ for $r<a$, and $V=0$ for $r\geq a$. Here, $V_0$ is a constant. The potential is repulsive for $V_0>0$, and attractive for $V_0<0$. The outside wavefunction is given by [see Eq. (1310)]

$\displaystyle {\cal R}_0(r)$ $\textstyle =$ $\displaystyle \exp( {\rm i}  \delta_0) \left[
\cos\delta_0 j_0(k r) - \sin\delta_0 y_0(k r) \right]$  
  $\textstyle =$ $\displaystyle \frac{ \exp( {\rm i}  \delta_0)  \sin(k r+\delta_0)}{k r},$ (1332)

where use has been made of Eqs. (1289) and (1290). The inside wavefunction follows from Eq. (1315). We obtain
{\cal R}_0(r) = B  \frac{\sin (k' r)}{r},
\end{displaymath} (1333)

where use has been made of the boundary condition (1316). Here, $B$ is a constant, and
E - V_0 = \frac{\hbar^2  k'^{ 2}}{2 m}.
\end{displaymath} (1334)

Note that Eq. (1333) only applies when $E> V_0$. For $E< V_0$, we have
{\cal R}_0(r) = B  \frac{\sinh(\kappa  r)}{r},
\end{displaymath} (1335)

V_0 - E = \frac{\hbar^2 \kappa^2}{2 m}.
\end{displaymath} (1336)

Matching ${\cal R}_0(r)$, and its radial derivative, at $r=a$ yields
\tan(k a+\delta_0) = \frac{k}{k'}  \tan( k' a)
\end{displaymath} (1337)

for $E> V_0$, and
\tan(k a+ \delta_0) = \frac{k}{\kappa}  \tanh( \kappa  a)
\end{displaymath} (1338)

for $E< V_0$.

Consider an attractive potential, for which $E> V_0$. Suppose that $\vert V_0\vert\gg E$ (i.e., the depth of the potential well is much larger than the energy of the incident particles), so that $k' \gg k$. We can see from Eq. (1337) that, unless $\tan (k' a)$ 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

k a + \delta_0 \simeq \frac{k}{k'} \tan (k' a).
\end{displaymath} (1339)

This yields
\delta_0 \simeq k a \left[ \frac{\tan( k' a)}{k' a} -1\right].
\end{displaymath} (1340)

According to Eq. (1330), the scattering cross-section is given by
\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.
\end{displaymath} (1341)

k' a = \sqrt{ k^2  a^2 + \frac{2  m  \vert V_0\vert  a^2}{\hbar^2}},
\end{displaymath} (1342)

so for sufficiently small values of $k a$,
k'  a \simeq \sqrt{\frac{2  m  \vert V_0\vert  a^2}{\hbar^2}}.
\end{displaymath} (1343)

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 $k' a$ (e.g., $k' a\simeq 4.49$) at which $\delta_0\rightarrow \pi$, and the scattering cross-section (1341) vanishes, despite the very strong attraction of the potential. In reality, the cross-section is not exactly zero, because of contributions from $l>0$ partial waves. But, at low incident energies, these contributions are small. It follows that there are certain values of $V_0$ and $k$ which give rise to almost perfect transmission of the incident wave. This is called the Ramsauer-Townsend effect, and has been observed experimentally.

next up previous
Next: Resonances Up: Scattering Theory Previous: Hard Sphere Scattering
Richard Fitzpatrick 2010-07-20