short-range parameter $s$, and is averaged over the short-range phase. Langevin approximation depicted by black solid lines, assumes constant reaction probability $P^{\mathrm}{re}$ above the barrier, and no reaction below the barrier.](IntegralC6003.pdf){width="\linewidth"}

![\[partial\](color online) Contribution of few partial waves to reaction rates for universal ($y=1$, top) and nonuniversal case ($y=0.1$, bottom) within the parabolic approximation for the centrifugal barrier averaged over the short range phase. Horizontal black line represents $P^{\mathrm}{re}$](Singlely1C6002 "fig:"){width="0.9\linewidth"} ![\[partial\](color online) Contribution of few partial waves to reaction rates for universal ($y=1$, top) and nonuniversal case ($y=0.1$, bottom) within the parabolic approximation for the centrifugal barrier averaged over the short range phase. Horizontal black line represents $P^{\mathrm}{re}$](Singlely0p1C6002 "fig:"){width="0.9\linewidth"}

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![ \[myfigure3\] Relative corrections of reaction rates given by the quantum analytical model with respect to the standard Langevin approach, calculated for different interaction potentials $1/r^n$ and energies for $y=0.01$.](Korekta3D003.pdf){width="\linewidth"}

The high energy approximations from the previous section do not take into account the presence of shape resonances. If the collision energy is close to the energy of a quasibound state behind the centrifugal barrier, the rates may be significantly modified. At high temperatures the total effect of many shape resonances in different partial waves should result in some average 