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 wave-function is given by [see Eq. (1289)]

$${\cal R}_0(r) = \exp(\,{\rm i}\, \delta_0)\,\left[ \cos\delta_0\,j_0(k\,r) - \sin\delta_0\,y_0(k\,r) \right]$$ (1311)

$$= \frac{ \exp(\,{\rm i} \,\delta_0)\, \sin(k\,r+\delta_0)}{k\,r},$$ (1312)

where use has been made of Eqs. (1268) and (1269). The inside wave-function follows from Eq. (1294). We obtain

$${\cal R}_0(r) = B \,\frac{\sin (k'\,r)}{r},$$ (1313)

where use has been made of the boundary condition (1295). Here, $B$ is a constant, and

$$E - V_0 = \frac{\hbar^2 \,k'^{\,2}}{2\,m}.$$ (1314)

Note that Eq. (1313) only applies when $E> V_0$. For $E< V_0$, we have

$${\cal R}_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$$ (1315)

where

$$V_0 - E = \frac{\hbar^2 \kappa^2}{2\,m}.$$ (1316)

Matching ${\cal R}_0(r)$, and its radial derivative, at $r=a$ yields

$$\tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan( k'\,a)$$ (1317)

for $E> V_0$, and

$$\tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh( \kappa\, a)$$ (1318)

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. (1317) 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

$$k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).$$ (1319)

This yields

$$\delta_0 \simeq k\,a \left[ \frac{\tan( k'\,a)}{k'\,a} -1\right].$$ (1320)

According to Eq. (1309), the scattering cross-section is given by

$$\sigma_{\rm total} \simeq \frac{4\pi}{k^2}\, \sin^2\delta_0 =4\pi \,a^2\left[\frac{\tan (k\,'a)}{k'\,a} -1\right]^2.$$ (1321)

Now

$$k'\,a = \sqrt{ k^2 \,a^2 + \frac{2 \,m \,\vert V_0\vert\, a^2}{\hbar^2}},$$ (1322)

so for sufficiently small values of $k\,a$,

$$k' \,a \simeq \sqrt{\frac{2\, m \,\vert V_0\vert\, a^2}{\hbar^2}}.$$ (1323)

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 (1321) 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.