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 external wavefunction is given by [see Equation (986)]

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

where use has been made of Equations (962)-(963). The internal wavefunction follows from Equation (991). We obtain

$$A_0(r) = B \,\frac{\sin (k'\,r)}{r},$$ (1009)

where use has been made of the boundary condition (992). Here, $B$ is a constant, and

$$E - V_0 = \frac{\hbar^2 \,k'^{\,2}}{2\,m}.$$ (1010)

Note that Equation (1009) only applies when $E>V_0$ . For $E<V_0$ , we have

$$A_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$$ (1011)

where

$$V_0 - E = \frac{\hbar^2 \kappa^2}{2\,m}.$$ (1012)

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

$$\tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan (k'\,a)$$ (1013)

for $E>V_0$ , and

$$\tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh (\kappa\, a)$$ (1014)

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 Equation (1013) 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

$$k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).$$ (1015)

This yields

$$\delta_0 \simeq k\,a \left[ \frac{\tan( k'\,a)}{k'\,a} -1\right].$$ (1016)

According to Equation (1006), the scattering cross-section is given by

$$\sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sin^2\delta_0 =4\pi \,a^2\left[\frac{\tan (k'\,a)}{(k'\,a)} -1\right]^{\,2}.$$ (1017)

Now,

$$k'a = \sqrt{ k^2 \,a^2 + \frac{2 \,m \,\vert V_0\vert\, a^2}{\hbar^2}},$$ (1018)

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

$$k' a \simeq \sqrt{\frac{2\, m \,\vert V_0\vert\, a^2}{\hbar^2}}.$$ (1019)

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 (1017) 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 $k$ that give rise to almost perfect transmission of the incident wave. This is called the Ramsauer-Townsend effect, and has been observed experimentally.