Quantum-defect model of a reactive collision at finite temperature

We consider a general problem of inelastic collision of particles interacting with power-law potentials. Using quantum-defect theory we derive an analytical formula for the energy-dependent complex scattering length, valid for arbitrary collision energy, and use it to analyze the elastic and reactive collision rates. Our theory is applicable for both universal and nonuniversal collisions. The former corresponds to the unit reaction probability at short range, while in the latter case the reaction probability is smaller than one. In the high-energy limit we present a method that allows us to incorporate quantum corrections to the classical reaction rate due to the shape resonances and the quantum tunneling.

Figure 1

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 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.

Figure 2

Contribution of few partial waves to reaction rates for the 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 $P^{\mathrm{re}}$.

Figure 3

Quantum corrections to the classical reaction rates due to the contribution of the shape resonances. Presented results are averaged over the short-range phase, as its value becomes unimportant at large energies.

Figure 4

Relative corrections of reaction rates given by the quantum analytical model with respect to the classical approach, calculated for different interaction potentials $1/r^n$, energies, and amplitudes of reactions $y$.

Figure 5

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 $y=0.01$.

Figure 6

Reactive rate vs collision energy for distinguishable particles with van der Waals interaction at different reaction amplitudes $y$. The $s$ values are chosen to be close to $p$-wave and $d$-wave shape resonances. The dashed lines show classical approximation (33).

Figure 7

Same as in Fig. 6, but averaged over thermal distribution. The shape resonances can still be seen, especially for low reactivity. The dot-dashed lines show the results obtained using parabolic approximation (48).

Figure 8

Thermally averaged elastic rates for the same parameters as in Figs. 6 and 7. The dashed lines show the approximate high energy result given by Eq. (43). The dotted lines show the low-energy $s$-wave limit. Small discrepancy is due to thermal averaging and $p$-wave contribution.