9.7: Hard Sphere Scattering

Let us test out this scheme using a particularly simple example. Consider scattering by a hard sphere, for which the potential is infinite for $r>a$ . It follows that $r<a$ , which implies that $l$ . Thus,

$l$. It follows from Equation $\ref{988}$ that $l=0$ partial wave, which is usually referred to as the $\tan\delta_0 = \frac{\sin (k\,a)/k\,a}{-\cos (k\,a)/ka} = -\tan (k\,a),$ $\ref{997}$

where use has been made of Equations $\ref{962}$-$\ref{963}$. It follows that

$s$ -wave radial wave function is $\ref{999}$

The corresponding radial wavefunction for the incident wave takes the form

$l=0$ radial wavefunction is similar to the incident $k\,a$ .

Let us consider the low and high energy asymptotic limits of $k\,a\ll 1$ . In this limit, the spherical Bessel functions and Neumann functions reduce to:

$\simeq \frac{(k\,r)^l}{(2\,l+1)!!},$ $\ref{1001}$ $\simeq -\frac{(2\,l-1)!!}{(k\,r)^{l+1}},$ $\ref{1002}$

where $\tan\delta_l = \frac{-(k\,a)^{2\,l+1}}{(2\,l+1) \,[(2\,l-1)!!]^{\,2}}.$

$\ref{1003}$

It is clear that we can neglect $l>0$ , with respect to $s$ -wave scattering (i.e., spherically symmetric scattering) is important. It follows from Equations $\ref{938}$, $\ref{979}$, and $\ref{998}$ that

$k\,a\ll 1$. Note that the total cross-section $\pi \,a^2$ (i.e., the cross-section for classical particles bouncing off a hard sphere of radius $k\,a\gg 1$ . At high energies, all partial waves up to $\sigma_{\rm total} = \frac{4\pi}{k^2} \sum_{l=0,l_{\rm max}} (2\,l+1)\,\sin^2\delta_l.$ $\ref{1006}$

With so many $\sin^2\delta_l$ by its average value $\sigma_{\rm total} = \sum_{l=0,k\,a} \frac{2\pi}{k^2} \,(2\,l+1) \simeq 2\pi \,a^2.$

$\ref{1007}$

This is twice the classical result, which is somewhat surprizing, because we might expect to obtain the classical result in the short wavelength limit. For hard sphere scattering, incident waves with impact parameters less than $\pi \,a^2$ which, when combined with the cross-section for classical reflection, $2\pi \,a^2$ .

Contributors

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

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