Beyond universality: Parametrizing ultracold complex-mediated reactions using statistical assumptions

We have calculated accurate quantum reactive and elastic cross sections for the prototypical barrierless reaction ${\mathrm{D}}^{+}+{\mathrm{H}}_2(v=0,j=0)$ using a modified hyperspherical scattering method. The considered kinetic energy ranges from the ultracold to the Langevin regimes. A reaction rate coefficient practically constant in no less than eight orders of magnitude is obtained. The availability of accurate results for this system allows one to test the quantum theory by Jachymski et al. [K. Jachymski, M. Krych, P. S. Julienne, and Z. Idziaszek, Phys. Rev. Lett. 110, 213202 (2013)] in a nonuniversal case. The short-range reaction probability is rationalized using statistical model assumptions and related to a statistical factor. This provides a means to estimate one of the parameters that characterizes ultracold processes from first principles.

Figure 1

Sketch of the intrinsic reaction path and the rovibrational states involved in the reaction at the studied energies. $A\left(E\right)$ and $B\left(E\right)$ are the number of incoming and outgoing channels, respectively (1 and 3 for $J=0)$.

Figure 2

Calculated reaction and elastic total cross section for the collision ${\mathrm{D}}^{+}+{\mathrm{H}}_2(v=0,j=0)$ compared with the Langevin prediction. The inset shows the comparison of the reaction cross sections with the experimental data [28] and the result of the statistically corrected Langevin model.

Figure 3

Reaction and elastic partial cross sections at the indicated $J(=l)$ values for the ${\mathrm{D}}^{+}+{\mathrm{H}}_2$ ($v=0,j=0)$ collision in the cold and ultracold regimes.

Figure 4

Real and imaginary parts of the scattering length, ${\tilde{a}}_l\left(k\right)={\alpha }_l\left(k\right)-i{\beta }_l\left(k\right)$, obtained in the calculations for the four lowest partial waves. The absolute value of $\alpha$, mostly negative, is plotted. The values for $J=1$, 2, and 3 (in continuous lines) are compared with the predictions from the model in Ref. [9] (in dashed lines), calculated using Eqs. (3, 4, 5), assuming for $s$ and $y$ the same values that have been obtained for $J=0$.