Observation of quantum capture in an ion-molecule reaction

Vogt and Wannier [Phys. Rev. 95, 1190 (1954)] predicted that the capture rate of a polarizable neutral atom or molecule by an ion should increase by a factor of 2 compared to the classical Langevin rate as the collision energy approaches zero. This prediction has not been verified experimentally. The ${\mathrm{H}}_2^{+}$ $+$ ${\mathrm{H}}_2$ reaction is ideally suited to observe this effect because the small reduced mass makes quantum effects related to $s$-wave scattering observable at higher collision energies than in other systems. Moreover, the reaction rate for this barrierless, strongly exothermic reaction follows the classical Langevin capture model down to cold-collision conditions (about $k_{\mathrm{B}}\times 1$ K) and is not affected by short-range interactions. Below this temperature, a strong enhancement of the reaction rate resulting from charge–quadrupole interaction between ${\mathrm{H}}_2^{+}$ and ground-state ortho ${\mathrm{H}}_2$ ($J=1$) was observed. Here we present an experimental study of the reaction of ${\mathrm{H}}_2^{+}$ and para ${\mathrm{H}}_2$ ($J=0$), which has no dipole and no quadrupole moments, at collision energies below $k_{\mathrm{B}}\times 1$ K. We observe an enhancement at the lowest collision energies which is attributed to the quantum enhancement predicted by Vogt and Wannier. Measurements of the reaction of ${\mathrm{HD}}^{+}$ with HD support this conclusion.

Figure 1

(a) Schematic energy-level diagram showing the R(0) and R(1) transitions of the $B$ ${}^1{\mathrm{\Sigma }}_{\text{u}}^{+}(v=3)$–$X$ ${}^1{\mathrm{\Sigma }}_{\text{g}}^{+}(v=0)$ band of ${\mathrm{H}}_2$ used to determine the relative concentration of para and ortho ${\mathrm{H}}_2$ in the enriched para-${\mathrm{H}}_2$ samples by LIF spectroscopy. (b) LIF spectrum of a jet-cooled natural sample of ${\mathrm{H}}_2$. (c), (d) LIF spectra of the para-enriched ${\mathrm{H}}_2$ samples with para:ortho contents of 59%:41% and 80%:20%, respectively, used for the measurements. See text for details.

Figure 2

Collision-energy-dependent relative rate coefficients, normalized to the value of $k_{\mathrm{L}}$ at the highest collision energies, of the ${\mathrm{H}}_2^{+}$ $+$ ${\mathrm{H}}_2$ [(a)–(c)] and ${\mathrm{HD}}^{+}$ $+$ HD [(d)] reactions. The gray and black data points correspond to individual measurements and their averages over the collision-energy ranges separated by the vertical dashed gray lines, respectively. (a)–(c) ${\mathrm{H}}_2^{+}$ $+$ ${\mathrm{H}}_2$ reaction measured with samples of (a) natural ${\mathrm{H}}_2$ [25% para ${\mathrm{H}}_2(J=0)$ and 75% ortho ${\mathrm{H}}_2(J=1)$] and para-enriched ${\mathrm{H}}_2$ samples with compositions of (b) 59(1)% para ${\mathrm{H}}_2(J=0)$ and 41(1)% ortho ${\mathrm{H}}_2(J=1)$, and (c) 80(1)% para ${\mathrm{H}}_2(J=0)$ and 20(1)% ortho ${\mathrm{H}}_2(J=1)$. (d) ${\mathrm{HD}}^{+}$ $+$ HD reaction measured with a sample of HD containing 4.25% (volume %) of each ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$. In (a)–(c), the areas shaded in red and blue correspond to the contributions from ortho ${\mathrm{H}}_2(J=1)$ and para ${\mathrm{H}}_2(J=0)$, respectively, calculated with the theoretical values of the rate coefficients reported in Ref. [12]. In d), the areas shaded in blue, red, green, and orange correspond to the contributions of $\mathrm{HD}(J=0$), $\mathrm{HD}(J=1$), ${\mathrm{H}}_2$, and ${\mathrm{D}}_2$, respectively, also calculated with the rate coefficients reported in Ref. [12]. See text for details.

Figure 3

Collision-energy-dependent rate constants obtained at finite collision-energy resolution from the theoretical reaction rate constants. (a), (c) Theoretical rate constants $f\left(v_{\text{rel}}\right)$ from Ref. [12] (black lines and left-hand-side scale), distributions $g\left(v_{\text{rel}}\right)$ of relative velocities (blue lines and right-hand-side scale), and convolution ($f$ * $g$) of these rate constants with the distribution of relative velocities (red dashed lines, left scale) for the reactions ${\mathrm{H}}_2^{+}+{\text{H}}_2(J=1)$ and ${\mathrm{H}}_2^{+}+{\text{H}}_2(J=0)$, respectively. (b), (d) Theoretical rate constants from Ref. [12] (black lines) and their convolution (red dashed lines) displayed as a function of the collision energy on a logarithmic scale for the reactions ${\mathrm{H}}_2^{+}+{\text{H}}_2(J=1)$ and ${\mathrm{H}}_2^{+}+{\text{H}}_2(J=0)$, respectively.

Figure 4

Rate coefficient of the reaction ${\mathrm{HD}}^{+}$ $+$ $\mathrm{HD}(J=0$) calculated at low collision energies using the charge-dipole and charge-quadrupole capture models described in Refs. [49, 51]. See text for details.

Figure 5

Overview of Raman spectrum of the gas sample extracted from the HD lecture bottle at the end of the measurements (a) and details of the spectrum in the regions of the $v=1\leftarrow v=0$ Stokes Raman band of HD (b), ${\mathrm{H}}_2$ (e), and ${\mathrm{D}}_2$ (f). For comparison, the Raman Stokes bands of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ recorded using pure samples are displayed in (c) and (d). The red and blue lines in (e) and (f) correspond to the pure spectra of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$, scaled by 0.9%, respectively. See text for details.

Figure 6

(a) Comparison of the R(0) transitions in the LIF spectrum of the $B$ ${}^1{\mathrm{\Sigma }}_{\text{u}}^{+}(v=4)$–$X$ ${}^1{\mathrm{\Sigma }}_{\text{g}}^{+}(v=0)$ band of HD and ${\mathrm{D}}_2$ used to determine the relative concentration of the ${\mathrm{D}}_2$ impurity in the supersonic expansion of HD. (b) Comparison of the R(0) and R(1) transitions in the LIF spectrum of the $B$ ${}^1{\mathrm{\Sigma }}_{\text{u}}^{+}(v=4)$–$X$ ${}^1{\mathrm{\Sigma }}_{\text{g}}^{+}(v=0)$ band of HD used to determine the relative population of the $J=0$ and $J=1$ of HD in the supersonic expansion. The dotted lines represent magnifications by a factor of 40. See text for details.