from the region of energies close to the top of the barrier. This yields $$\label{Kel2}
K^{el,(2)} = \frac{2 h}{\mu k} \sum_{\ell < \ell_{{\mathrm}{t}}}(2 \ell +1) \frac{
\tan^2 \xi(E,\ell) + y^2}{\left(1+\tan^2 \xi(E,\ell)\right) \left( 1+ y\right)^2}.$$ In the considered range of angular momenta the phase shifts $\xi(E,\ell)$ vary strongly with the angular momentum. Taking this into account, we can treat $\xi(E,\ell)$ as a random variable in this regime, and we can perform an average assuming uniformly distributed phase shifts $$\frac{1}{\pi} \int_0^\pi d \xi \frac{
\tan^2 \xi + y^2}{\left(1+\tan^2 \xi\right) \left( 1+ y\right)^2} = \frac{1+y^2}{2(1+y)^2}$$ Substituting this into we get $$K^{el,(2)}=\frac{2h}{\mu k} \frac{1+y^2}{2(1+y)^2} \left(\ell_{\mathrm}{t} +\frac12\right)^{2}.$$

In order to calculate $\ell_{\mathrm}{t}$ we can use the semiclassical expression again, substituting it into , which gives $$2\ell_{\mathrm}{t}+1=\left( \frac{3\Gamma(n-1)}{\Gamma^2(n/2)} \right)^{1/(n-1)}\left(\frac{E}{E_n}\right)^{(n-2)/(2n-2)}.$$ This finally yields the result for the total elastic rate constant $$\begin{split}
\label{Kelhigh}
K^{el} \approx &\frac{h R_n}{\mu} \left( \frac{\pi^2}{32(n-2)} +\frac{1+y^2}{4(1+y)^2} \right)\\
&\times\left( \frac{3 \Gamma(n-1)}{\Gamma^2(n/2)} \right)^{\frac{2}{n-1}}\left( \frac{E}{E_n} \right)^{\frac{n-3}{2n-2}}.
\end{split}$$

Approximate treatment including shape resonances {#apprwkb}
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![ \[Ferm\] (color online) Reaction probability calculated for a parabolic potential fitted to the actual centrifugal barrier of the van der Waals potential versus the angular momentum squared $\ell(\ell+1)$ (red solid dashed and green dashed lines). The result is independent of the 