Low energy scattering

At low energies (i.e., when $1/k$ is much larger than the range of the potential) partial waves with $l>0$, in general, make a negligible contribution to the scattering cross-section. It follows that, at these energies, with a finite range potential, only $s$-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 wave-function is given by [see Eq. (972)]

$$A_0(r) = \exp( {\rm i} \delta_0) \left[ j_0(k r) \cos\delta_0 - \eta_0(k r) \sin\delta_0\right]$$ (994)

$$= \frac{ \exp( {\rm i} \delta_0) \sin(k r+\delta_0)}{k r},$$ (995)

where use has been made of Eqs. (948)-(949). The inside wave-function follows from Eq. (977). We obtain

$$A_0(r) = B \frac{\sin k'r}{r},$$ (996)

where use has been made of the boundary condition (978). Here, $B$ is a constant, and

$$E - V_0 = \frac{\hbar^2 k'^2}{2 m}.$$ (997)

Note that Eq. (996) only applies when $E>V_0$. For $E<V_0$, we have

$$A_0(r) = B \frac{\sinh\kappa r}{r},$$ (998)

where

$$V_0 - E = \frac{\hbar^2 \kappa^2}{2 m}.$$ (999)

Matching $A_0(r)$, and its radial derivative at $r=a$, yields

$$\tan(k a+\delta_0) = \frac{k}{k'} \tan k'a$$ (1000)

for $E>V_0$, and

$$\tan(k a+ \delta_0) = \frac{k}{\kappa} \tanh \kappa a$$ (1001)

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 $k' \gg k$. It follows from Eq. (1000) 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

$$k a + \delta_0 \simeq \frac{k}{k'} \tan k'a.$$ (1002)

This yields

$$\delta_0 \simeq k a \left( \frac{\tan k'a}{k'a} -1\right).$$ (1003)

According to Eq. (992), the scattering cross-section is given by

$$\sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sin^2\delta_0 =4\pi a^2\left(\frac{\tan k'a}{k'a} -1\right)^2.$$ (1004)

Now

$$k'a = \sqrt{ k^2 a^2 + \frac{2 m \vert V_0\vert a^2}{\hbar^2}},$$ (1005)

so for sufficiently small values of $k a$,

$$k' a \simeq \sqrt{\frac{2 m \vert V_0\vert a^2}{\hbar^2}}.$$ (1006)

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 (1004) 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.