Rotational state changes in collisions of diatomic molecular ions with atomic ions

TL;DR

The study addresses how diatomic molecular ions acquire rotational excitations in single Coulomb collisions with atomic ions, leveraging a large separation of translational and rotational energy scales to treat translation classically and rotation quantum mechanically. The authors derive per-collision excitation formulas for apolar and polar molecules, using perturbation theory for apolar species and an adiabatic framework for polar species, supplemented by numerical TDSE solutions. Key findings show quadrupole interactions dominate apolar rotational excitation in relevant regimes, while polar excitation is governed by adiabatic dynamics that depend on the dipole moment and field strength, with high-field behavior resembling libration around the field line. These results enable spectroscopic inference of molecular parameters from collision-induced rotational transitions and provide a foundation for predicting cumulative rotational excitation during sympathetic cooling cycles in hybrid ion systems, as elaborated in the companion paper.

Abstract

We investigate rotational state changes in a single collision of diatomic molecular ions, polar or apolar, with an atomic ion. Rotational state changes may occur since the angular degree of freedom of the molecular ions interacts with the electric field due to the atomic ion. Thanks to the very different time and energy scales of translational and rotational motion, we may treat the collision classically and describe only the rotations quantum mechanically. We first investigate a number of example systems numerically and then derive closed-form approximations for the rotational excitation per collision, depending on the scattering energy and the molecular parameters. These findings provide the basis for estimating the accumulated rotational excitation in sympathetic cooling of molecular ions by laser-cooled atomic ions [arXiv:2410.22458 ] which involves many single collisions.

Figures (14)

• Figure 1: (Color online) Scattering geometry of a collision between a molecular ion and a laser-cooled atomic ion. Initially, the scattering pair are very far apart such that the vector $\vec{r}$ coincides with the $\hat{z}$-axis which thus also defines the quantization axis ($\beta = 0$). $b$ is the impact parameter, $r_0$ the closest distance between the scattering pair and $\theta_{sc}$ the scattering angle. $\theta_a$ is the angle between the molecular axis, $\hat{\rho}$, and $\vec{r}$. Since $\vec{r}$ changes its direction during scattering, the electric field of the atomic ion, indicated in red will acquire an $\hat{x}$-component for all but head-on collisions ($b=0$).
• Figure 2: (Color online) The electric field (in atomic units) due to the atomic ion felt by the molecular ion for head-on ($b=0$) scattering at $E=2.5$ eV, with the Coulomb field transformed to a temporal field by Eq. \ref{['eq:t_r1']} (evaluated numerically), and compared to the Lorentz form, Eq. \ref{['eq:Lorentzfield']}.
• Figure 3: (Color online) Rotational excitations are due to the Coulomb field of the atomic ion coupling to the (a) dipole moment for polar molecular ions and (b) to the polarizability and quadrupole moment for apolar molecular ions. These different interactions give rise to the different selection rules depicted in the two panels. For head-on collisions only $\Delta m=0$ (blue arrows) are allowed.
• Figure 4: (Color online) Population excitation for apolar molecular ions as a function of time of a head-on scattering event with energy $E$ indicated in the panels. (a) For low scattering energies the motion approaches the adiabatic limit, as is clearly seen for H$_2^+$ / Be$^+$-scattering. Here the left scale is for N$_2^+$ / Ca$^+$ and the right scale is for H$_2^+$ / Be$^+$. (b) For higher scattering energies adiabaticity is lost, and the excitation is to a large degree determined by the product $\chi_Q\kappa$. The dynamics is qualitatively the same for other apolar molecular species.
• Figure 5: (Color online) Comparison between final population excitation as obtained from numerical integration of the Schrödinger equation generated by the Hamiltonian Eq. \ref{['eq:ham2']} and as obtained from PT taking only the quadrupole interaction into account (absolute square of Eq. \ref{['eq:excQuad']}). The shaded region shows the maximum deviation due to the polarizability interaction, where the excitation takes values in $|c_Q|^2 + |c_{\Delta\alpha}|^2 \pm 2|c_Q||c_{\Delta\alpha}|$, with $|c_{\Delta\alpha}|$ the absolute square of Eq. \ref{['eq:c_np02']}. The solid lines show the full excitation given by numerical simulations, and the dashed and dotted lines show the final population on individual $m$-states for $j=2$.