Elsevier

Chemical Physics Letters

Volume 430, Issues 4–6, 30 October 2006, Pages 271-276
Chemical Physics Letters

Improved relations for vibrational relaxation in gases

https://doi.org/10.1016/j.cplett.2006.08.142Get rights and content

Abstract

Explicit expressions are presented for calculating vibration-to-translation (VT) energy conversion probabilities, essential in molecular laser isotope separation. VT conversions in molecular collisions occur by two mechanisms: (1) high-energy impact transfers prevailing at higher temperatures, and (2) Van der Waals-bonding encounters followed by (pre-)dissociations at lower temperatures. While mechanism (1) has been studied for over fifty years culminating in the Schwartz–Slawsky–Herzberg relation, a useful analytic expression for (2) has so far been lacking. An improved dimer formation theory developed by the author together with molecular pre-dissociation physics now provides a VT conversion relation for mechanism (2), which correctly predicts observations.

Graphical abstract

Explicit expressions for the probability of two mechanisms of radiationless vibrational deexcitation of gaseous molecules are presented. The well-studied relaxation catalyzed by impact collisions prevails at higher temperatures, while at lower temperatures the less explored process catalyzed by dimerization and predissociation dominates.

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Introduction

This Letter presents an improved model for calculating relaxation rates of vibrational energy in gases. Over a decade ago, the author experimented with laser-irradiated gas mixtures of UF6 and XR to affect isotope enrichments by promoting a chemical reaction of laser-excited 235UF6 with RX = HBr, HCl, NO, etc. [1], [2]. Of fundamental importance in these so-called CHEMLIS studies is the relaxation rate of vibrationally excited molecules. Vibrational energy is mostly lost by vibration-to-translation (VT) or vibration-to-vibration (VV) energy transfers in collisions with other molecules. Vibrational relaxation by photon emission is usually much slower, while VV transfers in CHEMLIS are minimized with high RX/UF6 ratios and avoidance of vibrational resonances of RX with UF6.

VT energy transfer processes have been analyzed for more than half a century by Landau–Teller, Jackson–Mott, Rapp–Sharp, Schwartz–Slawsky–Herzfeld (SSH), and many others [App A, Ref. [3]]. The SSH formula is most useful for practical applications, but requires a correction for low-energy vibrations and has a low temperature limit discussed below.

Fig. 1 shows a plot of the vibrational deexcitation probability PSSH versus temperature T for a mixture of excited UF6(ω3) molecules diluted in different gases, as calculated by the (corrected) SSH formula. According to the SSH curves in Fig. 1, at low temperatures the lifetime of excited iUF6(ω3) molecules is sufficiently long (VT probability low) for a significant number of iUF6:XR complexes to form, before vibrational energy is lost. In isotope separation, low temperatures are desirable to minimize overlap of isotopic absorption bands, but large-scale condensation of gaseous iUF6 must be avoided. This condition can be attained briefly in supersonic jets of iUF6/XR gas mixtures [2]. It was hoped that for some iUF6:XR complexes, an atomic re-arrangement such as iUF5R + XF would result if the pumped vibrational energy exceeded the activation barrier. Experiments based on this concept were disappointing however and indicated that vibrational lifetimes of UF6 were much shorter than predicted by SSH theory. From this result it was surmised that the laser-pumped vibrational energy was not effectively used to promote reaction, and instead was communicated to the gas mixture at a much more rapid rate than SSH theory predicted.

Dimer formation rates and lifetimes were examined next, resulting in an improved theory of dimer formation [3]. Conventional wisdom held that dimers could only form in 3-body collisions to allow simultaneous conservation of energy and momentum. A third body was needed to carry off excess energy to allow Van der Waals-bonding [4]. However with a 3-bodies-only theory, one finds that calculated dimer formation rates are too low to explain observations. The author showed that more frequent 2-body collisions by a sub-population of low-energy molecules in the Boltzmann distribution can form dimers at a much higher rate [3]. In this case excess energy is carried off by dimer rotation after a bonding transition in the top of the intermolecular potential well.

Researchers at the University of Nijmegen in the Netherlands (and others before them) proved that dissociation of SF6:SF6 and SF6:Ar dimers occurs within microseconds after vibrational excitation by resonant laser photons [5], [6]. They showed that excited homo-dimers and hetero-dimers break apart to the original monomers in sub-microseconds due to the ‘vibrational predissociation’ process [7], [8]. Lee [9] had proposed, and Van den Bergh [10] proved for SF6, that isotope-selective vibrational excitation of dimers in super-cooled jets can induce isotope separations. Later analysis indicated that the lifetime of a QF6:XY dimer formed by a pre-excited monomer QF6 (Q = S, U, W, etc.) and XY should be about the same as that of a (QF6:XY) dimer excited after dimer formation, since the resonant vibrational excitation energy is almost the same. Van den Bergh proved experimentally that this is indeed the case [10], discarding earlier beliefs one had to excite dimers rather than monomers for isotope separation (monomers have higher photon absorption cross-sections). QF6:XY dimers can form just as readily as QF6:XY dimers. During Van der Waals attraction in QF6:XY dimer formation (DF), the fast internal vibration in QF6 does not couple to the much slower vibrations of the intermolecular potential. But after bonding, this energy is rapidly distributed over the dimer in the same way as it would for (QF6:XY). After some pre-dissociation (PD) Lissajous motions in the dimer, the original vibrational energy of QF6 is converted into translational energy of the two dimer partners which recoil off each other [5], [6], [7], [8], [9], [10], [11]. One implication of this scenario is that chemical reactions in laser pre-excited QF6:XR complexes are doomed to failure because the excited hetero-dimer flies apart in sub-microseconds before (slower moving) atoms in the complex have time to undergo an atomic re-arrangement.

Because PD rates are much faster than DF rates, the dimer-induced VT conversion rate kDF−PD is effectively determined by the dimer formation rate kDF, that is kDF−PD  kDF. With implementation of a 2-body dimerization theory, there is better experimental agreement at low temperatures, where DF rates are higher than SSH rates [11]. In the past, experimenters had suspected increased dimer formation at lower temperatures, but lacked an adequate analytic model. In conclusion, in analyzing vibrational energy relaxation,one must consider both DF and SSH processes plotted in Fig. 1, Fig. 2.

Section snippets

VT relaxation formulas

The VT relaxation rate kVT (s−1) of excited molecules is given by the product of the collision rate kc (s−1) and the total VT probability PVT = PSSH + PDF, that iskVT=kcPVT=kc(PSSH+PDF)Assuming that excitable QF6 is diluted in carrier or co-reactant gas G, the contact collision rate kc is given bykc=nGσcQ/Gu¯Q/G,s-1,where nG is the density (cm−3) of molecules G, σcQ/G = 1/4π(soQ + soG)2 is the contact collision cross-section (cm2) for QF6 + G collisions, and the relative molecular velocity u¯Q/G is given

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