9.7: Hard Sphere Scattering
- Page ID
- 1244
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,
where use has been made of Equations \ref{962}-\ref{963}. It follows that
The corresponding radial wavefunction for the incident wave takes the form
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:
where \( \tan\delta_l = \frac{-(k\,a)^{2\,l+1}}{(2\,l+1) \,[(2\,l-1)!!]^{\,2}}.\)
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
With so many \( \sin^2\delta_l\) by its average value \( \sigma_{\rm total} = \sum_{l=0,k\,a} \frac{2\pi}{k^2} \,(2\,l+1) \simeq 2\pi \,a^2.\)
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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