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14.7: Low-Energy Scattering

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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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    This page titled 14.7: Low-Energy Scattering is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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