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, with a finite-range potential, only S-wave (i.e., spherically symmetric) scattering is important at such energies.

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 (10.95)]

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

$$= \frac{ \exp(\,{\rm i} \,\delta_0)\, \sin(k\,r+\delta_0)}{k\,r},$$ (10.118)

where use has been made of Equations (10.60) and (10.61). The internal wavefunction follows from Equation (10.100). We obtain

$$A_0(r) = B \,\frac{\sin (k'\,r)}{r},$$ (10.119)

where use has been made of the boundary condition (10.101). Here, $B$ is a constant, and

$$E - V_0 = \frac{\hbar^{\,2} \,k'^{\,2}}{2\,m}.$$ (10.120)

Note that Equation (10.120) only applies when $E>V_0$ . For $E<V_0$ , we have

$$A_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$$ (10.121)

where

$$V_0 - E = \frac{\hbar^{\,2} \kappa^{\,2}}{2\,m}.$$ (10.122)

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

$$\tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan (k'\,a)$$ (10.123)

for $E>V_0$ , and

$$\tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh (\kappa\, a)$$ (10.124)

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 $k' \gg k$ . As can be seen from Equation (10.124), unless $\tan (k'\,a)$ becomes extremely large, the right-hand side of the equation is much less than unity, so replacing the tangent of a small quantity with the quantity itself, we obtain

$$k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).$$ (10.125)

This yields

$$\delta_0 \simeq k\,a \left[ \frac{\tan( k'\,a)}{k'\,a} -1\right].$$ (10.126)

According to Equation (10.115), the total scattering cross-section is given by

$$\sigma_{\rm total} \simeq \frac{4\pi}{k^{\,2}} \sin^2\delta_0 =4\pi \,a^{\,2}\left[\frac{\tan (k'\,a)}{k'\,a} -1\right]^{\,2}.$$ (10.127)

Now,

$$k'\,a = \sqrt{ k^{\,2} \,a^{\,2} + \frac{2 \,m \,\vert V_0\vert\, a^{\,2}}{\hbar^{\,2}}},$$ (10.128)

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

$$k' \,a \simeq \sqrt{\frac{2\, m \,\vert V_0\vert\, a^{\,2}}{\hbar^{\,2}}}.$$ (10.129)

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., $k'\,a\simeq 4.493$ ) at which the scattering cross-section (10.128) 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 $\vert V_0\vert$ , $a$ , 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 [88,4].