Coherent optical transient study of molecular collisions: Theory and observations

Abstract

It has been shown recently that molecular gas samples excited with coherent light can display a variety of transient phenomena, similar to those found in nuclear magnetic resonance. This article elucidates how these coherence effects can be used to isolate or unfold molecular collision mechanisms that normally remain hidden within the optical line shape. Elastic collisions, for example, are easily detected here in two-pulse photon echo experiments for a $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F vibration-rotation transition. The echo-decay function which provides a signature for the velocity-changing collision diffusion mechanism, is not just a simple exponential in time but exhibits an $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution for short times and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\Gamma}t)$ decay for long times. This behavior, which is unknown heretofore, contrasts with spin echoes in molecular liquids where Brownian motion leads only to the cubic decay law. An $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ behavior can be understood in terms of a solution of the Fokker-Planck equation which describes the effect of Brownian motion on echo decay. Such a treatment is valid for small phase excursions; in the case of a gas, this implies the Doppler phase factor $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\ll}1$, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ is the propagation vector of light, $\ensuremath{\Delta}u$ is a characteristic velocity jump for a binary collision, and $\ensuremath{\tau}$ is the echo-pulse delay time. When $k\ensuremath{\Delta}u\ensuremath{\tau}\ensuremath{\gg}1$ as in the long-time regime, the Fokker-Planck solution fails. We, therefore, present a new solution to the Boltzmann transport equation using a weak collision model and find agreement with the entire echo time dependence observed. The echo measurements indicate very small changes in longitudinal velocity per $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F-$^{13}\mathrm{C}$${\mathrm{H}}_{3}$F collision, i.e., $\ensuremath{\Delta}u=85$ cm/sec, thereby justifying the weak-collision model. The total elastic collision cross section is 430 ${\mathrm{\AA{}}}^{2}$. It follows that elastic collisions lead to velocity thermalization in a time of \ensuremath{\sim}5 sec when the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F pressure is 1 mTorr. A comparison is also made of the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F dephasing time ${\ensuremath{\tau}}_{2}$ in a coherent Raman beat decay, which is independent of velocity-changing collisions, with the longitudinal decay time ${T}_{1}$. Here, ${T}_{1}$ represents the molecule-optical interaction time, due largely to jumps in molecular rotation ($J$) and orientation ($M$) state, and is obtained from a delayed nutation measurement. The fact that the pressure dependent part of ${\ensuremath{\tau}}_{2}={T}_{1}$ shows that ${T}_{1}$ is also independent of velocity diffusion. Futhermore, when a hole is burned in the Doppler distribution, population recovery must be due to inelastic rather than elastic collisions. Optical Carr-Purcell echoes, multiple pulse echoes, provide a direct measure of the $^{13}\mathrm{C}$${\mathrm{H}}_{3}$F transverse dephasing time ${T}_{2}$ without the effect of elastic collisions while being sensitive to "phase interrupting collisions." Again, we find that ${T}_{2}={T}_{1}$ so that phase interruptions are negligible. Had such a process dominated the two-pulse echo, an $\mathrm{exp}(\ensuremath{-}\frac{t}{{T}_{2}})$ damping would have been noticed with no $\mathrm{exp}(\ensuremath{-}K{t}^{3})$ contribution. Thus, the present study covers several new aspects of molecular collisions. It represents the first detailed examination of velocity-changing collisions by coherence methods and without the complication of Doppler broadening.