$$\require{cancel}$$

# 14.7: 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 Equation ([e17.80])] \begin{aligned} {\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]\nonumber\\[0.5ex] &= \frac{ \exp(\,{\rm i} \,\delta_0)\, \sin(k\,r+\delta_0)}{k\,r},\end{aligned} where use has been made of Equations ([e17.58a]) and ([e17.58b]). The inside wavefunction follows from Equation ([e17.85]). We obtain $\label{e17.103} {\cal R}_0(r) = B \,\frac{\sin (k'\,r)}{r},$ where use has been made of the boundary condition ([e17.86]). Here, $$B$$ is a constant, and $E - V_0 = \frac{\hbar^{\,2} \,k'^{\,2}}{2\,m}.$ Note that Equation ([e17.103]) only applies when $$E>V_0$$. For $$E<V_0$$, we have ${\cal R}_0(r) = B \,\frac{\sinh(\kappa\, r)}{r},$ where $V_0 - E = \frac{\hbar^{\,2}\, \kappa^{\,2}}{2\,m}.$ Matching $${\cal R}_0(r)$$, and its radial derivative, at $$r=a$$ yields $\label{e17.107} \tan(k\,a+\delta_0) = \frac{k}{k'} \,\tan( k'\,a)$ for $$E>V_0$$, and $\tan(k\,a+ \delta_0) = \frac{k}{\kappa} \,\tanh( \kappa\, a)$ for $$E<V_0$$.

Consider an attractive potential, for which $$E>V_0$$. Suppose that $$|V_0|\gg E$$ (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 Equation ([e17.107]) 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 $k\,a + \delta_0 \simeq \frac{k}{k'}\,\tan (k'\,a).$ This yields $\delta_0 \simeq k\,a \left[ \frac{\tan( k'\,a)}{k'\,a} -1\right].$ According to Equation ([e17.99]), the scattering cross-section is given by $\label{e17.111} \sigma_{\rm total} \simeq \frac{4\pi}{k^{\,2}}\, \sin^2\delta_0 =4\pi \,a^{\,2}\left[\frac{\tan (k\,'a)}{k'\,a} -1\right]^{\,2}.$ Now, $\label{e17.112} k'\,a = \sqrt{ k^{\,2} \,a^{\,2} + \frac{2 \,m \,|V_0|\, a^{\,2}}{\hbar^{\,2}}},$ so for sufficiently small values of $$k\,a$$, $k' \,a \simeq \sqrt{\frac{2\, m \,|V_0|\, a^{\,2}}{\hbar^{\,2}}}.$ 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 ([e17.111]) 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 .

# Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
