14.6: 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 \(r<a\), and zero for \(r>a\). It follows that \(\psi({\bf r})\) is zero in the region \(r<a\), which implies that \(u_l =0\) for all \(l\). Thus,

\[\beta_{l-} = \beta_{l+} = \infty, \nonumber \]

for all \(l\). Equation \ref{e17.82} thus gives

\[ \tan \delta_l = \frac{j_l(k\,a)}{y_l(k\,a)}. \label{e17.90} \]

Consider the \(l=0\) partial wave, which is usually referred to as the \(S\)-wave. Equation \ref{e17.90} yields

\[\tan\delta_0 = \frac{\sin (k\,a)/k\,a}{-\cos (k\,a)/ka} = -\tan (k\,a), \nonumber \]

where use has been made of Equations \ref{e17.58a} and \ref{e17.58b}. It follows that

\[ \delta_0 = -k\,a. \label{e17.92} \]

The \(S\)-wave radial wave function is [see Equation \ref{e17.80}]

\[\begin{align} {\cal R}_0(r) &= \exp(-{\rm i}\, k\,a)\, \frac{[\cos (k\,a) \,\sin (k\,r) -\sin (k\,a) \,\cos (k\,r)]}{k\,r}\nonumber\\[4pt] &= \exp(-{\rm i}\, k\,a)\, \frac{ \sin[k\,(r-a)]}{k\,r}.\end{align} \nonumber \]

The corresponding radial wavefunction for the incident wave takes the form [see Equation \ref{e15.49}]

\[\tilde{\cal R}_0(r) = \frac{ \sin (k\,r)}{k\,r}. \nonumber \]

Thus, the actual \(l=0\) radial wavefunction is similar to the incident \(l=0\) wavefunction, except that it is phase-shifted by \(k\,a\).

Let us examine the low- and high-energy asymptotic limits of \(\tan\delta_l\). Low energy implies that \(k\,a\ll 1\). In this regime, the spherical Bessel functions reduce to:

\[\begin{align} j_l(k\,r) &\simeq \frac{(k\,r)^l}{(2\,l+1)!!},\\[4pt] y_l(k\,r) &\simeq -\frac{(2\,l-1)!!}{(k\,r)^{l+1}},\end{align} \nonumber \]

where \(n!! = n\,(n-2)\,(n-4)\cdots 1\) . It follows that

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

It is clear that we can neglect \(\delta_l\), with \(l>0\), with respect to \(\delta_0\). In other words, at low energy, only \(S\)-wave scattering (i.e., spherically symmetric scattering) is important. It follows from Equations \ref{e15.17}, \ref{e17.73}, and \ref{e17.92} that

\[\frac{d\sigma}{d\Omega} = \frac{\sin^2 k\,a}{k^2} \simeq a^2 \nonumber \]

for \(k\,a\ll 1\). Note that the total cross-section

\[\sigma_{\rm total} = \int\frac{d\sigma}{d\Omega}\,d\Omega = 4\pi \,a^2 \nonumber \]

is four times the geometric cross-section \(\pi \,a^2\) (i.e., the cross-section for classical particles bouncing off a hard sphere of radius \(a\)). However, low energy scattering implies relatively long wavelengths, so we would not expect to obtain the classical result in this limit.

Consider the high-energy limit \(k\,a\gg 1\). At high energies, all partial waves up to \(l_{\rm max} = k\,a\) contribute significantly to the scattering cross-section. It follows from Equation \ref{e17.75} that

\[ \sigma_{\rm total} \simeq \frac{4\pi}{k^2} \sum_{l=0,l_{\rm max}} (2\,l+1)\,\sin^2\delta_l. \label{e17.99} \]

With so many \(l\) values contributing, it is legitimate to replace \(\sin^2\delta_l\) by its average value \(1/2\). Thus,

\[\sigma_{\rm total} \simeq \sum_{l=0,k\,a} \frac{2\pi}{k^2} \,(2\,l+1) \simeq 2\pi \,a^2. \nonumber \]

This is twice the classical result, which is somewhat surprising, 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 \(a\) must be deflected. However, in order to produce a “shadow” behind the sphere, there must also be some scattering in the forward direction in order to produce destructive interference with the incident plane-wave. (Recall the optical theorem.) In fact, the interference is not completely destructive, and the shadow has a bright spot (the so-called “Poisson spot” ) in the forward direction. The effective cross-section associated with this bright spot is \(\pi \,a^2\) which, when combined with the cross-section for classical reflection, \(\pi \,a^2\), gives the actual cross-section of \(2\pi \,a^2\) .