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 and Attributions

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

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