Mode-to-mode vibrational energy flow in S1 benzene

;chemicalPhysicj 0 N&th_~olland

27. (i978) 127-150 PFblishir;g Cotipany

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-MODE-TO-MODH irI~RA~i0tikU ENERGY FLOW IN s, BENZENE t -. C.S. PARMENTER and K.Y. TANG * Department of Chemistry. Indiana University. Bloomington. Indfana 47401, USA Received 13 April 1977 Resolved fluorescence spectra from low pressures of benzene with nine added gases have been used to follow mode-tomode vibrational relaxation in the S1 state of benzene under “singlecollision” conditions. Cw pumping of the S1 fundamental 6’ (I& = 522 cni’) allows study of collisional vibrational energy flow into each of four channels. Two channels consist of flow into single levels, and the others represent flow into unresolved pairs of levels. The mode-tomode cross sections are much larger than those usually observed in ground electronic states, being near gas kinetic even for partners transferring energy by V-T, R processes alcine. The mode-tomode transfer has highly specific patterns, with roughly seventy percent of the transfer going into the four channels in spite of many other nearby levels. The largest cross sections are always to a level 237 cd above the initial level rather than to a level nearly resonant (AE = 7 cm’) with the initial level. A common pattern of flow occurs for the four gases transferring energy by V-T, R processes alone, and another common pattern is established for the five gases which can also use V-V transfers. With the exception of one channel, V-V resonances with vibrationally complex partners increase cross sections by less than a factor of two over that provided by the V-T, R path. V-V transfers have a similarly small effect on the overall vibrational relaxation rate out of the initial level 6’. Both the flow patterns and the V-V versus V-T, R competitions are accounted for with an extremely simple and general set of propensity rules taken diiectly from SSH calculations made by others for vibrational relaxation in ground electronic states. The rules are based on the degeneracies of the final levels, the number of vibrational quantum changes, and the amount of energy exchanged between viirationaland translational/rotational degrees of freedom. The rules seem general to relaxation in both ground and excited electronic states, whereas large cross sections seem a special property of the excited state. The cross sec:ions for collision partners SFe and perfluorohexane are small relative to those for other partners with similar vibrational complexity and mass.

I. Introduction

Infra-red lasers opened a new class of V-V,T,R energy transfer experiments which have produced abundant data concerning transfer from specific modes in the ground electronic states of small polyatomics as well as from selected levels of diatomics. Numerous reviews show the comprehensive view of ground state vibrational energy flow which has emerged from this work **. Single vibronic level (SVL) fluorescence techniques z

1 Co&ibution No. 2977 from the Chemical Laboratories of Indiana Uniirersity. * Present address: MaxwellLaboratories,&c. 9244 Balboa ** Avenue, San Diego, California 92123, USA. See tif. [l] ; $ese early and recent reviews include refcrences to most of the studied as well as to numerous other summaries. More recent references to the unusually complete studies of transfer in methyl halides are found in ref. [2].

allow replication of these studies in excited electronic states of polyatomic vapors. By pumping a single vibronic level with tuned narrow band cw excitation and observing electronic fluorescence from that and other levels in the excited state, rather fine details of excited state vibrational energy flow can be seen. The short lifetime of a fluorescing excited state is particularly helpful, since pressures can be adjusted so that the electronic decay cuts off the evolution of collisional vibrational energy flow at any desired stage. Even with a weak excitation source such as the tuned xenon arc used in this study, it is quite feasible to see emerging vibrational distributions after on average only one “effective” collision has occurred. Details of efficient mode-to-mode transitions, which are often difficult to observe in the ground state studies, arise clearly from measurements in this pressure range. We here make use of the unusually favorable experimental characteristics of benzene 133 for a study of the

C.S. Patme&.

K. Y. Tmrg/Mode-to-mode

vibtational energy flow in S1 benzene

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and V-T transfer. Thkgriiwth ie&'bands in-the . spectrum is readily apparent, and-it ispos&ie to identify the iibronic levels giving rise to many of those new transitions. The quantitative measurements pf this -- band grdwth in the “single-collision” stage of steady state experiments yields the-cross sections-for the collisional transfer of vibrational energy from the initial level 6l to other specific levels in the S1 state as identified in fig. 1. While studies of this type on diatomjcs reach back into early spectroscopic history, remarkably little has been learned about collisional vibrational flow in poly-

1

atomic excited states. In turn, much of what is known comes from benzene itself. The study:of relax&ion in

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Fig. 1. A schematicshowingevery vibrational ievelup to 1200 cm-’ above the zero point level in Sr benzene. The arrows show the viirational energy flow monitored in these experiments. Thelevel 6l is initially pumped, and collisional transfer into each of the four channels A through D is monitored. Two channels characterize flow to single levels (ignoring degeneraties). The other channels each involve two fmal levels. The energy separation between 6’ and the fmal levels is given in cm -1 . Many levels are intrinsically degenerate; these components have not been depicted. Heavy lines in the level stack indicate Ievel pairs too cIose to show individually_

“one-collision” mode-to-mode vibrational energy flow in its first excited singlet (1B2u) state. Fig. 1 shows every level in the lowest1200 cm-’ of the S, vibronic manifold. The level 6l (L$ = 1) at 522 cm-’ can be selectively pumped to the exclusion of others by tuned excitation isolated from a xenon arc, and the data in the present study concern relaxation from this level. At low pressures (=G0.1 torr) the excited molecules decay electronically prior to significant collisional interactions, and fluorescence displays the distinctive spectrum from molecules in the level 6’. As the experiment is re eated in the presence of an added gas, bands from 6 P diminish and new ones emerge. UItimatety at high added gas pressures, the fluorescence spectrum becomes characteristic of SI molecules in a Boltzmann vibrational distribution. This transformation is shown in fig. 2. The destruc-

tion of the initially pumped level can be monitored by the decline of intensity in any of several strong 6l fluorescence

bands. When combined

with SVL life-

times, these data yield absolute cross sections for total 4l destruction by the combined effects of V-V, V-R,

S, benzene began in 1924 with fluorescence excited by the 2537 A Hg line and has since remained pretty much centered on this fluorescence. Pringsheim and Reimann [4] were the first to distinguish between the low and high pressure spectra, a difference generated by vibrational relaxation exactly analogous to that in fig. 2 except that the initial levels reached by 2537 A

excitation are about 2000 cm-’ above the S, zero point level [5] _Kistiakowsky and Nelles 163 in 1932 studied the transformation more carefully, pointing out that pressures must be reduced to less than about 0.1 torr before the effects of vibrational relaxation become minor. The issue then lay-dormant until Stockburger’s quantitative study in 1962 [7] _This work and later extensions [8--141 pointed at large cross sections for many added gases in vibrational relaxation from the levels pumped by 2537 A excitation. However, their magnitudes could not be set with certainty because reliable lifetimes were not available. The fust lifetime measurements [15-171, showed these cross sections to be commonly near gas kinetic, and somewhat larger than gas kinetic for benzene itself as a collision partner. Several studies describe total vibrational energy transfer out of single vibronic levels other than those pumped with 2537 A Hg radiation [1X-20] _They also fmd high collision efficiences. All results taken together suggest that high efficiencies (near gas kinetic) may be the rule for destruction of S, benzene levels .. by vibrational relaxation. Whik the literature does not contahrdata on the mode-to-mode aspects of vibration+_energy flow ti_ S1 penzene, several &$ies have follG+ed the build.$p of the zeio point level and.of qua&of the low frequencymode vi6 = 237.cm-’ after initial excitation

CL%Pamzenter, K. Y. Tang/Mode-to-mode vibrational energy jhv

in SI benzene

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0.5

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cm-’ Fig. 2. Fluorescence from 0.1 torr of benzene with various pressures of added isopentane. The S1 level 6’ is initially excited by pumping the absorption band 6: (A$ at 38602 cn-’ (vat.) An asterisk marks the excitation position. Added isopentane pressures are indicated to the right. A small band appears at the right edge of the spectra (maximum near 38446 cm-‘) at intermediate pressures. This isemission fmm the level 6’16’ lying 237 cm-’ above the pumped level 6’. Its steady state population first r&s and ihen falls as higher added gas pressures increase the extent of vibrational relaxation during the S1 lifetime.

with the 2537 ii Hg line [7,11,13,14] . They emphasize the stepwise nature of this equilibration, which dissipates about 2M.O Cm-l of vibrational energy. Logan [ll] has quantitative measurements of this equilibration for many collision partners. Growth and decay of levels during vibrational relaxation after 15~16’ excitation has also been reported [18] . Populations of the states 6l and 6l 162,‘each reached by one-quantum -transfers,are observed to pass through maxima during relaxation, showing clearly its stepwise nature.

The many details and subtleties of our mode-tomode measurements of vibrational relaxation after pumping 6’ in S 1 benzene follow. A steady state model for the initial stages of vibrational relaxation is first set forth. We then identify and assign fluorescence growth bands. The quantitative problem ofmonitoring growth band intensities and relating them to relative steady state populations of single levels is next discussed. We then describe the use of the data to derive cross sections for mode-to-mode energy transfers.

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C.S. Pamenter, K. Y. Tong/Mode-to-mode vibrational energy f7ow in Sl benzene

2. The kinetic mode! Narrow-band exciting light tuned to the 6: absorption band pumps the level 6l

B+h&&B**, which may decay by fluorescence or by collision-free nonradiative paths B**&B+hv

0)

f’

** kl B ---_, destruct.

(2)

B** denotes S, benzene in the initiahy pumped Ievel 6*. B is ground state benzene. The quantity (kl + k2)-’ defmes the zero-pressure lifetime of B*“. Vibrational relaxation by collision with either ground state benzene or an added gas M is given by B**+B%B*+B,

(3)

ke B**+M-_B*+M,

(4)

in which B* denotes S, benzene molecules in a distribution of vibrational levels other than 6l. The total “self-relaxation” rate constant is given by k,, and 4 is the total rate constant for vibrational relaxation of B** by M collisions. Collisional destruction of the S, electronic state by S, benzene or by the added gases used in this study is so inefficient that it can be ignored. We describe processes (3) and (4) more explicitly as B** i-BaB*(i)i-B, B** t&&B*(~)

@a) CM,

(4a)

where we identify a specific tinai vibronic IeveI formed in the interaction as B*(i). Thus k, = X&(i) and k4= X,kq(i). No attempt ismade in processes (3) and (4) to identify changes in the internal states of the collision partners B or M. Decay of the level B*(i) is given by processes analogous to those above k; B*(i)-B +hv, cl*) B*(i)- k:

destruct,

(2*)

; B*(i)+BkBB’+B,

(3*)

k4* B*(i)+M-B’+M.

(4*)

B indicates S, benzene molecules in a distribution of vibronic levels excluding the level B*(i) itself. Care is taken in our quantitative experiments to keep the pressures of-B and M low enough so that on average few collisions occur during the lifetime of the excited S, moIecules. Under these conditions, we obse.rve on& the very initial stages of vibrational relaxation, and processes (3a) and (4a) are the dominant routes for the formation of any level B*(i). Collision sequences B**%. B*(j)% B*(i) do not contribute significantly to the population of B*(i). We call such pressure conditions the “one-collision” regime. Processes (3*) and (4*) may comprise an important segment of B*(i) decay in this pressure regime, but they do not contribute effectively to the formation of other levels under observation. In the one-collision regime the steady-state population of B*(i) always remains a small fraction of that of the initially pumped level B** . Thus level formation by processes (3”) or (4*) is expected to be small relative to that by (3) and (4) provided that no single sequence B** % B*o) % B*(i) has an unusually large cross section for the second step. No indications of the latter have occurred. The one-collision regime occurs at pressures below about 0.2 torr for the most efficient collision partners and below about 2 torr for the least efficient partners. The more efficient partners are vibrationally complex molecules which cause not more than about twenty percent of the 6l molectdes to change state at the upper pressure limit. These collision partners appear to reach severaf levels without overwhelming bias to any single level, so that as an upper limit, not more than a quarter of the 6l molecules destroyed by vibrational relaxation reach any single level B*(i). Thus the population ratio [B*(f)] /LB**] does not exceed about 0.05 during quantitative meamrements. The upper limit for the less efficient partners has been set by similar considerations. These limits generally correspond to a little less than one gas kinetic collision during the lifetime of the S, state [at about one torr,Zhi = 80 ns= r(S1)] Other complicating collisional interactions are also absent. The pumping light is of low intensity and its absorption is small (E x 60 f mole” cm-‘) so that excited state concentrations are too Iow for B** - B*” collisions to be important. Wall collisions are precluded by the short S, lifetimes. Thus in the one collision regime, the model describes with reasonable accuracy, the growth and decay of the excited states.

C.S. Pamzenter, K. Y. TmzgfMade-to-modevibrational energy flow

The steady-state approximation gives the standard Stem-Vtilmer relationship for decay of the initially pumped level 6’

‘fo~61Y~#5’) = I+ kJ~, *3 ‘k, PI 11[Ml- (1) I&6’) and $(6l) are the relative fluorescence intensities in a 6l emission band with and without added gas [Ml. The rate constants describing the total decay of 6lwithout specific reference to level-to-level decay channels are derived from this equation. Rate constants describing the population of some specific state B*(i) in vibrational relaxation from 6l are derived from the steady-state equation

in

S1

benzene

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sure 020 torr). A capacitance manometer measured cell pressures continuously. Gases were mixed for 3060 min before use. Fluorescence spectra were taken at added gas increments of about 100-300 mtorr to a maximum pressure of two torr with the less efficient collision partners. Increments were reduced to about 30 mtorr for vibrationally cotiplex partners with a maximum pressure of 0.5 torr. Benzene with stated purity of 99.999% from James Hinton Co. was degassed before use. Research grade He, N2, CO, and CO, were taken from one-liter bulbs without further purification. Other partners were used after vacuum distill&ion.

4. Band intensity measurements The ratio [B*(i)] / [B**] is the actual steady-state population ratio for given [Ml and [B] _I#) and 1&6l) are the relative intensities of a band from B*(i) and from B**. This intensity ratio is converted into the required population ratios by conversion factors@)/ A(61).

3. Experimental Benzene was excited with a bandpass of 40 cm-’ (fwhm) isolated at 38602 cm-’ (vat) from a 500 W Xenon arc lamp with a 0.75 m monochromator. This covers a large part of the rotational contour of the 6: (A$ absorption band so that S, rotational disequilibrium is minimal. Fluorescence was dispersed with a 1.7 m CzernyTurner spectrometer and detected with a photomultiplier whose output was monitored with photon counting circuits. Details of the equipment are given with references elsewhere [2 I] . Quantitative fluorescence band intensities were measured by integrating the area in appropriate regions of fluorescence spectra obtained with a fluorescence band pass of 15-20 cm-l (fwbm). Further details are given later. Exciting lamp intensities were monitored by a photomultiplier viewing the exciting radiation upon its exit from the fluorescence cell. Benzene pressures in added gas experiments were adjusted to 0.1 torr. Foreign gases were leaked into the cell through a needle valve from a high backing pres-

4.1. Decay of emission from the initial level S’(B**) The emission spectra in fig. 2 show a number of prominent 6l bands which could be used to monitor the total destruction of the level initially pumped. The decay of emission in several of these bands as foreign gas is added has been monitored [22] . Each gives essentially the same decay constants when used with eq. (1). The data for this study have been taken from IfoIIf measurements on the band 1:6; whose maximum is at 37618 cm-‘. 4.2. Growth of ernissiou from new levelsB*(i) The kinetics are described by eq. (2) which is based on measurements of If(#I&61). It is thus necessary to isolate and measure the intensity in some band from each level B*(i) studied. Due to spectral conjestion in fluorescence, the intensities are extracted from the spectrum by a process of some complexity_ (a) We fit list B*(i) levels which might be substantial contributors to growth intensity in the one-collision regime, and calculate the positions of the prominent emission bands from each. The compendium reveals that representative transitions from every level can be found within several small regions of the fluorescence spectrum. (b) Each of these regions is examined to learn which levels actually contribute to experimental growth intensity and to learn how to make quantitative measurements of it for various single levels.

...(c).%e ftially l&duce a procedure to identify the separat~:c~ntiibutiolns‘of B**-B collisions and a*~*: M collisions to growth-band intensities_ -With these procedures;we can follow quantitatively the fourstate-to-state vibrational relaxation channels involv%gsixleVels shown in fig_ l..The growth of level 6I 16I lying 237 cm-’ above the initial level (channel D) and the growth of the zero point level (channel A) are each monitored without interference from other levels. The growth of the other four levels is seen only _ by pairs. The emissions from the levels 162 and 4l stand clear of other interference but cannot be separated from each other. Similarly, the growth of the levels 1It and 16’ cannot be separated.

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Fulldetaih of the procedures a&given_ c&where [23] ;-The more important_ a&ecis$re~given below.:. -.,~ -.. j (a) A prosp+us ofgrowthmbands .~ -_.-. It is assumed that the five levels below the 6 ’ level initially pumped can gain significant population withii the one collision regime. All high& le&ls within kTpf 6I are also considered possibiiities. we have further included as contenders those levels-in the next higher 2kT region which can be reached in one qu_antum transitions_ These criteria isolate 17 levels as the most lkely sources of growth intensity: ~~ The dominant emission,bands from e&h Ievel can

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cm+ Pig. 3. Fluorescence spectra in the region 37000-37650 cm-’ after pumping 6’; Upper s&&rurir~i0 cm-’ tlubrescence resolution from a mixture of 0.1 ton benzene and 0.4 torr of isopentane. Lower solid spectrum: 5-c& resolution (and a slower scan rate ) from same mixture. Lower dotted spectrum: 5 cm-r resolution from 0.1 tdrr df benzene without added gas.

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C.B.~&qter,

K: Y. TangjMode-to-madevibrational energy JZowin S1 benzene

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38400

38500

cm-’ Fig. 4. Fluorescence in the region 38200-38500 cm-’ from 0.1 torr of benzene with 0.4 torr of isopentane after pumping the level 6’ _Fluorescence resolution is 5 cm-‘. Tbis region is es&entiaUyempty of fluorescence without the added isopentane.

be predicted with security [24] _Intensity from a level Xa wih occur dominantly in the single progression 1~6~X~. Likewise, the dominant pro ression from levels such as XaYb will be 1~6~X~4 b. From states 61Xa we observe two dominant progressions: 1:6$X: and 1:6tX$‘Other progressions will be less intense by at least an order of magnitude. In the earliest stages of vibrational relaxatfon, only transitions of these types will be observed from any growth level, and their positions comprise the expectations for the first appearance of growth bands. At least one such band from each of the 17 possible growth levels would occur in the spectral regions 37100-37660 cm-’ or 38200-38500 cm-l. These regions are shown in figs. 3 and 4 together with indications of the calculated band positions. (b) 7&e identification and measurement ofgrowth band intensity

The growth of new bands can be clearly seen at 5 cm-’ resolution in the spectra of figs. 3 and 4. The growth maxima are accounted for by transitions from among the 17 levels, but the identity of most is not uniquely defined due to conjestion. The problem is further compounded by the fact that a spectrum with 5 cm-’ resolution requires about 15 h of photon counting so that it is not practical to use this resolution for a large number af quantitative measurements. Quantitative measurements were instead made from spectra with 2O_crr1~~‘resolution,such as the example in fig. 3. While this reduces recording times to 20-30 mm, it

133

also requires that measurements of growth be made of spectrai regions rather than one individual bands. The regions used in this study are labeledA,B, __ in fig. 3. Their boundaries were estimated with knowledge of the approximate rotational contours of the‘ bands comprising a region. The contour of a single band can be seen in the transition 1!6: (region A). The analysis of actual contributions by individual transitions to growth regions is too detailed [23] to be present:d fully here. The results are given below along with the basis fir intensity measurements of ffuorescence from the individual growth levels or Ievel pairs marked in fig. I _ Region A. This is really a ‘decay” region, and 6r _ emission as lt$$, is completely dominant. Region B. Two growth maxima appear in this region. That at higher energy coincides with the 0’ emission 6: _Bands from the levels 6t 16l and 6lll r match approximately the other growth maxima, but a simple argument shows that the 6r 11’ contribution must be ml [23]. The individual contributions of 0’ and 6l16r emissions which comprise region B can be separated in quantitative measurements by monitoring intensity in the 6: 16; band shown in fig. 4. This is the only band of significant intensity in that region at low added gas pressures, so that it gives a clear measure to which 6l16r intensity elsewhere can be scaled. Region C. This is dominated by the 6r band 6;. Other transitions near the maximum are obscured. Regfon D. Transitions from 62, 1 11, 16l, 6’162

and 6’4’ are expected to lie in this region. It is certain that among these only 1 I1 and 16t emissions contribute significantly to intensity. However, no way has been bund to separate 1l1 and 16’ contributions so that the intensities are monitored as a pair. The boundaries of the region are difficult to defiie since the transitions lie on the low energy tail of an intense band from the parent level 6t. The contour of that band (6:) in benzene-foreign gas mixtures was determined by observation of that transition in benzene without added gas under identical instrumental conditions. Region E. Transitions from growth levels 6lll t , 6r 16’, 17l, and 10’ fall in this region. In addition, one of the “weak” bands 6: 16: from the parent level persists with substantial intensity throughout spectra obtained in the one collision regime. Those contributions from the parent level 6l and the growth level 6ll6’ can be separated by reference to emission in other re-

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C.S.Pamenter, K. Y. Tmg/!ode-tkridde vibrotioriirl energyfloiv iriS1 bekene

gions leaving contributions from 6-1 11 1 ,17r, and-lo’. However this residue is too small for quantitative measuremer$, although IO1 emission can be seen clearly at5cm resolution. Region F. While transitions from four states 4l, 16*, 6llO’ and 1 l1 16l lie within this region, only one maximum appears, and we ascribe intensity in the maximum as principally l62 emission. The maximum itself Is coincident with both 162 and 1 1116r emissions, but it can be argued from other fluorescence that . 11’ 16I emission cannot be strong. It is surprising that 4l emission (as 6!4:) does not appear with a distinct maximum. This transition is expected to have intensity about the same as that in the 16’ maximum 6: 165 for equal populations of the levels 4’ and 162. Either the 4r level population remains small during the initial stages ofvibrational relaxation, or an unexpectedly fast nonradiative decay from the levei 41 keeps its emission intensity low. Emission from 6r 10’ in this region can be shown to be nil. fig, 4; Emission in the regiin 38200-38500. In fig. 2, an emission band near 38442 cm-r can be seen fust emerging from the noise and then retreating as added gas pressure increases. This is emission from 61 16i (as 6: 16:) building up in the initial stages of relaxation only to be destroyed at high added gas pressures as multiple collisions ultimately drive the excited state molecules to vrbrational equilibration. The band is shown more clearly iri fig. 4, where it stands clear of other transitions with the exception of the predicted position of the 6’11’ emission band 6:11:. The level 6’ 11 t lies nearly 2.5 kT above the initial level and its population would remain small relative to that of 6l16.l lying only kT above 6’. Thus it is safe to consider the band with maximum at 38442 cm-r as an unambiguous monitor of the relative 6l16l population. The emission in fig. 4 is also valuable for its comment on growth of the levels 6141,61 162, 611 I’, 6r 10’ and 62. All would have prominent bands in this region yet none are observed. Aside from the intrinsic importance of the fig. 4 region as the guide to 6’16r growth, it forms the reference for extraction of 6l16’ emission from regions B and E in fig. 3, so that its accurate measurement is crucial. Unfortunately it is partially reabsorbed since the transition terminates in a low-lying ground state level. We have measured the extent of the reabsorption by

_

studying relative mtensities in fluoresce&e spectra obtained.from various (low) pressures of benzene in the presence ofhigh isopentane pressures. Under these conditions emission occurs from a Boltzmann~distribution of vibrational levels in Sl benzene and the relative vibrational populations are independent of benzene pressure. Thus the relative intensities in such “Boltzmann” spectra must be independent of benzene pressure, save those bands which are attenuated by reabsorption. By comparing band intensities as a function of benzene pressure, correction factors for reabsorption can be obtained. (Absorption changes due to pressure broadening are unimportant.) Fig. 5 shows a comparison of two pairs of transitions. The ratio of 6:/6: 1: intensities is independent of benzene pressure as expected, since the terminating level of each transition is high off the zero point floor of the ground state and neither is reabsorbed. On the other hand, the ratio of 6: 16:/1~6~ intensities shows a marked decline at higher pressures due to reabsorption of the 6: 16: transition_ The reabsorption is exponential with pressure as expected. The intensity ratio extrapolaies to 0.49 at zero pressure which is consistent with the ratio calculated from the known relative Boltzmann populations for the levels 6’ 16l and 6l and relative rate constants for the two radiative transitions.

1; 0

0.1

0.2

Benzene Pressure (torr) Fig. 5. The intensities of fluorescence bands 6: and 6:16: in “Boltzmann” benzene spectraas a fun&on of benzene pressure. The intensities are normalized by comparison with intensity in the 6: 1; band.

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C.S. Parmenter,K. Y. Tang/Mode-to-mode vibrationalenergy jlow in S1 benzene

All if the intensity corrections in regions B and E of fig_3 have used 6: 16: intensities corrected to zero pr&sure by the intensity ratio I(0 iorr)/I(O.l torr) k O-49/0.28 given h-fig. 5. Thi intensity bi6: 1: emission from the S, level l1 lying 922 cm-l above the initial level 6’ is misleading in that it gives an inaccurate impression of the relative populations of the levels 1’ and 6l16l. Experience with Franck-Condon factors and vibronic patterns [24,25] shows that 40-50% of the intensity from I1 will appear in the 6: 1: band whereas only a small fraction of intensity from 6l16l ap ears in the P 6: 16: emission band. Furthermore the 6,16: emission band is partially reabsorbed. The net result is that equal intensities in 6; 1’ .&I 6l16l emissions means a small population of 1P relatik to’that of 6l16l. Furthermore, the relative intensity of 1’ emission in this particular illustration is at a maximum relative to most conditions used for quantitative measurements. The l1 intensity is usually too small to measure, and 1’ populations have not been followed quantitatively.

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similarly are the intensities in a B** fluorescence band with and wit&out the added gas.

The rate constants in brackets of eq. (3) characterize B*(i) decay, but they will not be substantially different

from ‘hose for E** decay. The latter are known and show that ratio will remain small for our pressures. Thus the brackets do not depart far from unity, and it is a close enough approximation to establish the intensity of growth bands resulting from B**-B collisions from C(i>=

@%-9/~(691439.

(4)

Emission in some of the “‘minor” transitions from B** molecules also contributes intensity to some growth regions, and this intensipl can be scaled by eq. (4) as well. In effect, the weak B** contributions and the contributions from B*(i) reached in B**-B collisions constitute a residual base line underlying intensity from B**-M collisions. In every spectrum, this baseline can be scaled directly to intensity in the 0.1 torr benzene spectrum and subtracted to yield the B*(i) intensity coming from B**-M collisions alone.

4.3. Corrections for intensity porn El**-B collisions 5. Conversion of band intensities to level populations All growth band intensities were obtained from mixtures containing 0.1 torr of benzene. This benzene pressure is not quite low enough to preclude observable popdation of some B*(i) levels by B**-B collisions. Corrections have been made so that intensities It.@)can be gotten for only that part of level populations created by B**-M collisions. The partitioning of growth band intensities into the contributions from B**-M and B**-B collisions is simple, since the contributions from the latter can be gauged from intensities seen in pure benzene spectra. Using the previous kinetic model, the ratio of intensity from B*(i) molecules populated by B**-5 collisions in an added gas experiment to the intensity in the pure benzene spectrum becomes

&I

-= pR(i)

[B**] &;

+ k; + k; [Bl )

[B**] n(ky + kz + kz

PI f kz[MI)

1’ -i’

where [B**] M and [B**] o are B** concentrations with and without the added gas ande(6l) and I$?)

The conversions could be made by the relationship [B*(i)] /[B**] = k,(6’y,(i)/k~(j~~6l),

where the radiative rate constants k,($ are those ap- . propriate for the specific transitions being monitored. These constants can be derived from SVL fluorescence spectra, lifetimes and quantum yields, but the data are not available for all levels. An alternate procedure derives populations from the relationship [B*(i)] /[B**] =A(61)I&$4(iY,(61). A(i) are conversion factors determined in experiments which measure band intensities in fluorescence spectra with high added gas pressures to establish an S, Boltzmann vibrational distribution prior to fluorescence. A typical spectrum is shown in fig. 6. The relative band intensities were independent of added isopentane pressure above about 70 torr and of changes in exciting wavelength, which pumped the level 6’ or 1161. It is thought that the spectrum in fig. 6 is truly representative of emission from vibrationally equilibrated S, molecules.

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Fis. 6. A partial Uuorescence spectrum frond b&kne vapor (0.1 torr) in-the presence of isopentane (120 torr) with excitation of the benzene AZ (6:) absorption band. The emission-is from lBzu benzene molecules with a 300 K distribution of vibrational levels.,

The relative intensities

observed in the regions A,

B . . .) are given in table 1. Averages of ineakrements from several spectra with varying added gas pressures

and as such they &e strictly valid for use only when the level pairs have the equilibrium population ratios

.occurring in the BoItzmami +ectruni. Po@Iatioris in

and excitation conditions are reported, and 6: 16: emission intensities have been corrected for self-absorption. The factors&) follow dire&y from the populations and intensities given iti table 1. Two cdnversion factors are based on pairs of levels,

our relaxation

experiments

are clearly none@&brium.

In each case, how&q, the individual-conversion

factors

for tie components of the pair are expected tti be nearly identical-[231 _Thus the factors$ describing these cases should be approximately valid. .-

Table 1 Relative intensities of regions in iheCkH6 Boltzman_?fluorescence spectrum and population convekion factors Region

Dominant emitting level

Observed transition

A(0

6616;

3.8

1.25

6.0

4.5

37.2

26.6b)

23.6

23-4

6’16’ 6l

.tig.3-B

o”

fq.3-D

16’ 11’

$;;t

rig. 3-F

16’ 41

6;16; 4;69

6:

l;&

}

Conversion factor

Observed intensity

6;1:

fig.4 f&.3-A

’

Relative a) Boltzmano population,

18.3

3.0 1.3 .;I.4

.:.

-

.:.!-I-. ; 1-l

: . .

- __

.~..I. __

,-.

., :;~. _..

a) TheIe~els6~16’. 6’. 16l, and 16* havedegeneraciesofi, 2,2. and 3 respedkly. b) Emission kom 6’ 16’ contniutes signiticar;tly to-this region. 6’16’ inten& lias been s&&&d .. .- .~. so that the reported value refers to 0’ emission only.

:.

‘&or&& total iegi& ik&i@~

‘. ;.

-.

~:-

J ::

.y. .._:. -.:.‘_

:-p&

-.

r$~:~&mentt?r, K.Y..T;mglMoae-to~rnod~ vibrational energy frow in S1

Ben&e

137

~: ‘. ‘; .,.

$

_

for destruction. of the S; benzene level 6! by vibrational relaxation plus some calculated parameters

Ob&ved &&&amete~s fo;thk coll&on pair

F&i& gas-

-_

.

b a) k4 b) kl +k2 +k3[B1 (406

to&?)

uOQS

C)

oobs

(A 1

ohs

(torr’) -.

He

0.097

N2

0.10

co

0.13 0.24 .. 0.33

CO2

ocs SF6 iC,jH12 nGF14

nC6H14 n-C7His.

-

1.7. 1.8

1.3 3.2

0.09 0.16

2.3

4.2

0.21

8.6 10.8 9.8

0.4’) 0.49 0.33

4.1 4.7 f)

d)

E e)

p

2

(au)?/2

10.2 912 110 190 335 201

1.95 4.54 4.54 5.30 5.82 7.23 6.12

0.20

3.5

0.55 0.58

9.7 10.2

23

09s 9)

345

32

0.94 h)

413h)

7.96

1.03

18.1

46

1.46

413

6.40

1.12

19.6

51

1.41 i,

282

6.61

a) From plots of eq. (1). lJ) Calculated with (kl + k2T1 = 79 ns 171 and k3 = 5 x 10’ tar? s’ [23]. c) Derive I from k4 = 2.54 X LO7a/p‘11 ton --’ s ’ at T _- 300 K. p is reduced mass in au and D is the cross section in A’. d) ohs =&I, + dM)*, whired is the hard sphere collision diameter. Data from ref. [26] e, Relative magnitudes of the well depth, e, of B**- M collision pairs used. Their derivation is discussed elsewhere [40] _ -1 Cl, f?Corrected for electronic quenching of the ‘Bzu state of benzene, which occurs with a rate constant of about 1.1 X lo6 torr g) Hard spherk dieter from ref. [27] _ h, Value of nCeH14 used for haid -sphere diameter and E. i, Haid’sphere diameter estimated by extrapolation from lower n-alkane values.

6. Results 6.1. Tke rate constantfor vibrationalen&y transfer We seek the following rate constants

:.. Fig. 7.@/If’fagainst [M] according to eq. (1). The benzene : : pressure is 100 mtorr. Triangks “e for added n-heptane and chclesarefor@dedCO.

..,

k4:

B**+M+B*+M.

k4 (i):

B** -I-M + B*(i) + M.

B** is S, benzene in the level 6l, B* is S benzene in a distribution of levels other than 6l, B*i-I) is SI benzene in a specific level (i), and M is a foreign gas. k4: The rate constants k4 are summarized in table 2. They were determined directly from eq. (1) by monitoring the intensity of 6’ emission as foreign gases M were added to 0.1 ?orr of benzene. Fig. 7 gives examples of the data. The constant k4 was derived from a least-squares slope using k + k2)-’ = 79 ns [ 16,171 and k, = 5 X lo7 torr-‘s- 0 233. The slopes of eq. (1) were generally reproducible to

138.

:

*able;

CS. Parmenter, K. Y.-Tang/Mode-to-mode

vibmtional ener& f7ow in S1 benzene

-.

-

~.

Observed rate constants IQ(~) for mode-to-mode vibrational energy transfer in St benzene. Constants are given in units of 113~ -. ~to<‘sl a) Foreign gas

Rate constants .Q(O for specific channels A

Bb)

C =)

D

6’-+0’

6’ + 16’ 6’ + 11’

6’-t4’

6’ +6’16l

6l* 162

k4

Hi?

[<0.041

0.3

[<0.031

1.0

1.7

0.72

[<0.09]

0.4

[<0.07]

1.0

1.8

0.71

co

[
0.3

0.3

1.2

2.3

0.79

co2

[<0.15]

0.8

0.5

2.0

4.1

0.78

OCS

I.1

0.7

05

1.3

4.7

0.75

SF6

0.4

0.9

0.5

0.9

3.5

0.75

GH12

1.6

?.7

1.1

3.6

9.7

0.92

nGF14

0.9

2.1

1.4

3.4

10.2

0.16

nGH14

2.2

2.3

2.6

6.1

18.1

0.73

nC7H16

3.0

3.9

25

6.8

19.6

0.83

b, Zero pressure lifetime of 90 nsused.

within 15% and this represents the possible

error in k4 values relative to each other. The errors in lifetime measurements used to derive (kl + k2) were reported to be not more than five percent [16,17] and although the uncertainty in k3 is large (about 40%), its contribution to the slope in the sum (kl + k2 + k, [B] ) is small, so that the systematic error introduced into the set of k, values by that sum is in the region IO-15%. These

: k4 (a -

N2

a) A eas kinetic value would be about 10 X lo6 tori? s’. Cl ieyo pressure lifetime of 80 ns used.

k,(i):

k4

rate constants

Examples of the data plotted in this form Bre given in fig. 8 for the added gas n-heptane. Slopes of such plots were evaluated via ale&t squares analysis k (i) was deduced from the slopes using kz = 4 X 10 $ torr-Is-’ and (k: f kz) from the observed low pressure lifetime [ 16,171 or estimated if measurements are not available. The lifetime seem to vary little among the lower levels ofthe S, state so that estimates are unlikely to intro-

are given in table 3. The

growth band analysisshowed that fluorescence from some growth levels could not be resolved from that of others. Hence the level-to-levei rate constants k4(i) are reported for “channels” A, B, C, and p, where some channels encompass transfer to pairs of levels as indicated in the table and in fig. 1. The rate constants k4(i) for these channels are derived from B*(i) emission measurements corrected for the contributions from B** + B collisions to B*(i) formation. Elimination of the B**-B collisional route to B*(i) population modifies the kinetic model by the exclusion of process (3a). Eq. (2) becomes

, 5

10

l/bC7H,61 (torr-‘) Fig. 8. A plot of [B**(6’)]/[B*(i)l ag&st l/[M] according to eq. (5). This is a monitor of the growth ofemissitm from the level B*(i) as the level B**(6*) is destroyed:Channels and the specific levels are indicated in the fire:

C.S. pannenter, K.Y. Tang/Mode-to-mode

duce error beyond the scatter of the data. The value of k; was chosen-from the fact that the self-quenching r&e constants L$ in measbrements on several levels all lie in *e range (3-S) X -10’ torr-l s-l. In fact, the total quenching cross section for vibrationally complex collision partners in general seems rather insensitive to the level being destroyed by vibrational relaxation [23] .

vibrational energy jlow in S, benzene

139

described by Ea is appropriate. A weighted sum Eabc combining Ea, Eb, and EC describes the uncertainty in comparing the efficiency of a given gas M for populating the various levels B*(i). The error in the absolute value of Ici is given by a weighted sum of all in the form Eabcd. tainty

Each of the 34 vabes of ki(Q listed in table 2 is an

average of severaldeterminations. A check of the kinetics could in principle be derived by calculating k4(i) values from the intercept kT//c,(i) of eq. (5). However, the intercept is extreme!y sensitive to scatter in the data and the check is quantitative only for k4 (6’ 16l). Values of the latter derived from the inter-

cept tire consistent with values derived from the slope. Four sources of error contribute to the uncertainty in kq(i) values. (Ea) Scatterin repeated measurement of growth band intensities as a function of added gas M determines the relative accuracy of k4(i) for a given level evaluated for a series of M gases. (Eb) Error introduced by improper choice of regional boundaries and uncertainties associated with extraction ofgrowth band intensity for a given transition from other intensity in the region combine to introduce a systematic error common to an entire set of k4(i) values for a given level. (EC) Error by improper conversion factors A(i) introduces a second systematic error common to a set of k4(‘> values for a given level. (Ed) Error in the sum (k; + k; + kf [B]) constitutes a systematic error common to ali k4(i) values reported in table 3. These errors are summarized in table 4. In comparing the relative values of rate constants for population of ti given level B*(i) by a variety of M gases, the uncer-

7. Discussion Two sets of measurements form the principal experimental basis of this study. (a) The rate constants k, (table 2) describe the total rate of collisional transfer from the initial level 6l into the entire field of S L vibrational levels. (b) The rate constants k4(i) (table 3) describe the single channel rates of transfer from the level 6l to some specific level (i) or set of levels in this field. These data are summarized in fig. 9. 7.1. Large absolute cross sections for vibrational energy transfer

The maI;y IR studies of vibrational energy flow in polyatomic ground electronic states provides basis for -

I-

He

N2

co

co2

oiXTrii f~~jt ii fi[ ,‘I”:I[!

Table 4 Estimated errors, E. [in percent of k&i)] contribu!ing to the uncertainties in t(i)_ See text for definition of error SOUICBS

ABCDZ

SF6

Ea

Level

.

Eb

Eta)

Ed

E&c

Eabcd

20

15

15

15

30

33

16’+11’

15

20

15

-15

30

33

16’ +4:

15

15

10

15

23

27

f? 16l

15

10

10

15

21

25

a) Relative to

A(6l).

oo

ABCDX

ABCDI

ABCDZ

ABCDC-

i-C,H,,

n-C,H,4

n-C6F,4

n:ClH,6

Fig. 9. A comparisonof the data with calculations from the propensity rules. The numbers give the total coUision efficiencies of cciUision partners for vibrational energy transfer from the level 6’ into the nearby field of S1 levels. The vertical bars indicate the observed fraction of total vibrational energy transfer which goes into specific channels, with the Z column indicating the fraction into the channels A, B, C and D combined. The crosses indicate the fractions calculated from the propensity rules.

140

C.S. Pannenter, K. Y. Tang/Mode-t&node vibrational energy jlow in S1 benzene

comparison. How does the flow in excited electronic states differ? One cannot securely answer with generalizations since the present study is the only large compendium of mode-to-mode excited state data. But in these data, a very significant difference is seen, and it concerns the general magnitude of the energy flow cross sections. Examine first the cross sections for total relaxation out of the level 6l in table 2. When compared to gas kinetic values, they are unusually large. None falls below about 10% of gas kinetic, and several match or exceed it. One might propose that the large values are reached by summation of many relaxation channels, each with a small cross section. While this may be the case for vibrational relaxation from regions of high level density as encountered in studies of chemically activated complexes $ or in the excited states of larger polyatomics [29], it cannot describe the origin of large rate constants in the present case. To the contrary, the constants for 6’ come principally from just a few channels. This is seen in the mode-to-mode data of table 3, where just four channels alone usually account for 70 to 80% of the 6l relaxation. In the extreme case of He and N2, this fraction is supplied by only two channels. The high efficiency of vibrational relaxation in S, benzene is perhaps best appreciated by looking at some of the specific mode-to-mode cross sections in table 3. For example, the addition of a quantum of vie (channel D) requires a T, R + V exchange of 237 cm-r, yet He adds this quantum in only 18 collisions. The cross sections for N,, CO and CO, are equally large with collision numbers of 11,9, and 5 respectively. These efficiencies exceed by orders of magnitude those which are inferred for comparable transfers in polyatomic ground electronic states. Data for excited electronic states of other polyatomics are sparse. Vibrational relaxation in ‘A,& j glyoxal seems generally rapid [30,31], and a recent measurement shows that removal of a quantum of the CHO-CHO torsional mode v; = 237 cm-’ to reach the zero point level occurs with nearly a hard sphere cross section in collision with Ar [32] . Unfortunately there are no ground state numbers available for explicit comparisons in either benzene or glyoxal. Data are avaaable in a number of diatomics, and one $ See for example,ref. [28].

finds cases’where excited state r&&ion v’ = n -+ U’ =_ n - I (usuahy by rare gas collisions) exceeds gronnd state rates by orders of magnitude. Tire issue has been discussed in a stud? of Lii by Eniren and Ottinger [33] . Comes and Fink’s data for CO offer a particularly cornpelling comparison [34]. In CO; relaxation in the AllI state occurs with collision efficiencies r&ging from 0.01 (He) to 0.6 (Kr). This is nearly a factor of IO6 greater than efficiencies found in ground states [35] _ The difference surely cannot be explamed by the change in vibrational frequencies-alone. On the other hand, fast vibrational relaxation is apparently not a property of all excited states of CO. Marcoux, et al. 1361 report the efficiency of He in relaxing vibrations within the a3n state of CO to be less than 1.5 X 10m7.Vibrational relaxation in the excited and ground states of NO shows also large differences. Relaxation cross sections in the A’Z’ state ofN0 are roughly 0.01 of gas kinetic [37] whereas the ground state cross sections are less than IO4 of gas kinetic. Extensive data describe relaxation in the B3f& state of 5 [38]. As with CO and NO, cross sections for rare gas collisions approach gas kinetic values, but unfortunately ground state data are not available. A detailed study of the BICi state of HD shows also large vibrational relaxation cross sections (u = 3 + u -+ 2) in collision with rare gases [39]. A theoretical account of these qualitative differences between excited and ground state cross sections has not yet been offered. One would suppose that the key may lie in the increased polarizabilities of the excited states which lead to subsmritial increases in intermolecular well depths. Some evidence on this point is given in fig. 10 which shows a plot of ln u for 6’ destruction by vibrational relaxation against (eEnM)l’* where ehlM is a measure of the vapor intermolecular potential weU depth. This correlation holds for many types of energy transfer jnteractions [40] , but in the present case, the relationship is not quite exponential. If such plots are attempted for analogous ground state relaxation data, no correlation whatsoever is found. Thus from this point of view as well as from comparison of efficiencies, the excited and ground state relaxation interactions appear fundamentally different_ An analysis shows that the general correlation [40] is expected for those interactions which occur predominantly on the attractive part of the potential_ This analysis would imply, therefore, that electronic excitation so changes the collision-pair potential that the

C.S. Pannenter.KY. TangfMode-to-mode vibmtionalenergyflow in S1 benzene M-

B**(6’)-vib

relax JW,

100-

1 n-W%6

50-

i,_,,,, n-C& l

/ !

C3%/

c

to-

-CzHzC’z

l0cs

/’ co,-/

/:,

lW,,

%F,

Fig. 10. A plot of the total crosssections for vibrational relaxar12in arbitrary of the level 6r as a function of (EMM) units. The intermolecular M-M welldepth in this form is proportional to the well depth in B**-M collision pairs. tion out

vibrational relaxation interaction tends to switch away from the repulsive wall as one changes from ground to excited state relaxation. 7.2. Propensity rules for collisional mode-to-mode energy flow IA the discussions to come, we use the term V-T jR transfer for processes in which the vibrational change in Sl benzene occurs without vibration changes in the collision partner. The energy difference between the initial and final benzene levels is the vibrational energy AE which must be taken up (or supplied) by translations and rotations of the collision partner. This differs from a common use of V-T, R terminology in ground state studies where it often designates the emptying of vibrational energy (i.e., transitions from excited levels to the zero point level) rather than mode-to-mode flow

141

among excited vrbrationallevels. The term V-V transfer is reserved for processes which change vibrational states in the collision partner as well as in benzene. It is apparent from the energy flow patterns in fig. 9 that strong propensities occur. One sees this from the E column for each gas which indicates that over 70% of the flow is directed into just the four monitored channels, in spite of the large number of nearby final 12vels. In fact the gases using V-T, R transfer alone (He, N2, CO, CO,) bias this 70% or 80% into just two or three levels. Fig. 5)shows also that the propensities establish a rather common flow pattern for collision partners transferring energy by V-T, R exchange_ Channel A, which is loss of the ui fundamental, is not used by the V-T, R gases. Channel B includes transfer to a level nearly resonant with the initially pumped level (AE = 7 cm-‘), yet transfer adding a quantum of vi6 = 237 cm-’ (channel D) is nearly twice as efficient with V-T, R partners. The V-V, T, R gases have 2 number of near rescnances with level separations in S, benzene, and their patterns show this most clearly by the increased participation of channel A. The V-V, T, R flow patterns are remarkably similar from partner to partner. Theory s.swell as experience with ground state vibrational ersrgy transfer suggests that the propensity restrictions are principally associated with the energy AE which n&t be transfered between vibrational and translational or rotational modes and with the required change Av in vibrational quantum numbers. A qualitative but quite consistent replication of the mode-tomode energy flow patterns can be derived from simple rules containing these effects. Our rules are taken from SSH treatment [41] of vi-. brational relaxation as modified by Tanczos [42] and by Stretton [43] . This theory has so far provided the basis for discussion of mode-to-mode relaxation in ground state polyatomics, with variable success. A referee has pointed out that the essential form of the rules can be derived with more generality by noting the consequence of any perturbation F(t)X where F(t) is a force function and X is the oscillator displacement. This gives 2 transition probability in the form cos (A.Ef&if 1’. The matrix elements fi-2K$lJflr) IX. l- give the restrictions on Au and the time integral If maximizes for AE = 0 giving rise to the energy restriction. The specific propensity rules developed from the

142

-._- :-- -..- .

C.S. Parmenter.K. Y. Tang/Mode-zo-modevibnitiotil energy flow in SI b!ruene

-SSH_formulation~describe the relative probabilities for transfer ihto .variOus channels when considering a specific&h.sion partner. (1) The relative probabnity for a given mode-tomode transition in which, say, mode A changes i+ j and mode B (perhaps in the collision partner) changes k-+Iisgivenby

(2) The vibrational level degeneraciesgi and gr eriter in a strai t forward manner. $ and Ui are factors of 10-l for each Av = 1 (3) Ucl change in mode A or B. A Au = 2 change in a mode introduces the factor 10B2 and so forth(4) I(A.E) is a prohibition dependent upon the amount of energy AE switched between V and T,R modes; (a) for processes transfering energy.V + T,R (exothermic, benzene loses vibrational energy) with AE = 0 to 50 cm-l , use the factor 0.6; (b) for such exothermic processes with AE > 50 cm-’ use I(AE) = exp [-1O-2 I AE I] where AE is in cm-l; (c) for processes transfering energy T, R+ V (endothermic), multipIy the factors above by the Boltzmann factor. Rule (1) is the SSH formulation in its simplest form, having set steric factors to unity and ignoring the secondary dependence of the integral I upon temperature, reduced mass of the collision partners, and intermolecular potential parameters. Rule (3) imposes the restrictions due to quantum number changes, with the factor Vi (and similarly @) being the matrix element ](A,1 VlAjj_12 with intermolecular potential V and vibrational agenfunctions Ai and AT Such matrix elements have been calculated for a variety of ground state systems, and we note particularly the results for transfer among modes in CH4 [44] _ Those_calculations show little variability in U2 for different modes, with values alI lying with a factor of 2 to 5 X 10s2 for Au = 1 changes inmodesul,v2,vj, and v4_ Furthermore,muItiple quantum changes introduce nearly multiplicative factors. Similar invariance among modes can be found in other systems, for example CzHa [45]. Thus a tentative basis exists in these calculations to assign a rather general propensity rule associated with Au changes. Rule (4) stems from the fact that the integral I has

its main dependence on the energy gap AE, and as such it is the principal measure of AE restrictions on transition probabilities. We again retreat to ground state calculations, taking the results from analyses of transfer iu CH:iCH4 1463, CHjF*-CH3F 1471, and C%Cl*CH,Cl [48] collisions. The integral is shown in fig. 11 ford&thesesystems. Little variation occurs from system to system and little variation occurs among modes within a given system. Clearly Al? is the dominant factor among the parameters contributing to the integral and we take the results of the CH, F*-CH, F calculations without change to formulate the AE restrictions in our propensity rules. Several a priori considerations would suggest that such SSH theory rules are inappropriate when applied to excited state energy flow in benzene. First, SSH theory is constructed for inelastic collisions operating on repulsive walls, and we are considering processes for which attractive potentials must surely be important

A E (cm-‘) Fig. 11. Calculated values of the integralI forV + T,R transfer of energy aE.

_C.S. Parmenter. K. Y. Tang/%de-to-mode

intbe interaction. Furthermore the SSH calculations used to scale the Au and AE factors are from &stems perhaps better suited to the “breathing sphere” SSH model than is benzene. We do not try to defend our rules in the face of these liabilities, but rather, the principal justification for their use is that they work. 7.3. Application ofpropensity rules How well they work is illustrated in fig. 9, where . the observed patterns of energy flow are compared with patterns calculated directly from the propensity rules without adjustment of any parameters. The flow patterns are reproduced remarkably well, considering the generality of the rules. Not only is the channel-tochannel competition replicated, but also the fraction of total transfer appearing in the four channels combined. It should be emphasized that the rules are derived entirely from consideration of relaxation in ground states. Their successful application to the benzene data therefore suggests that no special effects are encountered in setting the competition among vibrational relaxation channels in the excited electronic state. This is in contrast to separate consideration of the absolute magnitude of the rate constants, where the excited state values seem to differ fundamentally from those generally found in ground states. First consider detailed use of the rules to describe energy flow in collisions with V-T,R partners_ (He, N,, and CO obviously qualify and CO, must be included as well. The CO2 bending mode v; = 667 cm -1 could set up a V-V exchange with the initially pumped benzene level vi = 522 cm-l. The data show, however, that this resonance is ineffective, the CO, rate constant for channel A being too small to observe). To calculate the fractions of transfer into various channels, we have used the rules to calculate the relative probability of transfe’r from 6l to each S, benzene level Iying within 600 cm-’ of the initial level. These relative probabilities are summed to give a measure of the total transfer probability, from which fractions of transfer to various levels can than be calculated. Table 5 shows calculations for all levels up to 6ll6’. Transfer probabilities to higher levels damp out rapidly, with none exceeding seven percent of the probability.for 6l16’. The relative probabmties in table 5 depict a picture of transfer consistent in spirit with the observations.

vibrational energy jlow in 5’1 benzene

143

Transfer is dominated by the level 6’ 16l_ Transfer to the zero point level is by comparison inconsequential. Transfer to the near resonant level 11’ (AI? = -7 cm-‘) is predicted to be only about half as efficient as transfer to the dominant level 6l16l. Finally, the calculations show that channels B, C and D encompass all important transfer processes with the exception of the single channel 6’ + lo1 (AE = 63 cm-‘). This is predicted to be nearly as important as the dominant channel 6’ 16’. Asdescribed earlier, 10’ growth was observed inthe relaxation spectrum, but spectral interferences precluded quantitative measurements. Inspection of table 5 shows how the factorsg, @AU) and I@!?) trade off to establish this pattern of channel competition. As expected, the iarge AE restriction blocks channel A: Dominance of channel D (6l16l; A/?=+237 cm-‘) transfer over the nearly resonant transfer to 11’ (A!?= -7 cm-‘) is achieved by the high degeneracy of the level 6l16l and by the . need to change only a single vibrational quantum. The comparison in tig_ 9 of calculated and observed V-T, R transfers shows more easily the usefulness of the propensity rules in describing mode-to-mode Sow out of the 6’ level. Note the close agreement between the calculated and observed fraction of transfer out of 6l which is accounted for by the sum ofchsnnels A-D. We pressume from the calculations in table 5 that most of the remaining transfer occurs in the single channel 6l + 10’. The fraction of transfer occurring in various channels for collision partners CO and CO, is reproduced with accuracy exceeding expectations con$dering the approximate nature of the propensity rules. We observe good correspondence between calculated and observed transfer in channels D and B for partners He and N,. However, transfer within channel C by these two gases is markedly less efficient than predicted. The reason is unknown. The observed rate constants for channel D and channel B are in nearly constant ratio for the four V-T,52 gases He, N,, CO, and CO,, the ratio being 3,3,4, and 3 respectively. The‘SSH expression describing the constants has two factors U and Iwhich in principle are dependent on the collision partner. The constant channel D/B ratios imply that the energy dependence of1 is not strongly variable among collision partners. This supports the use of AE rules which are insensitive to the identity of collision partners. The constant ratios also imply that the relative coupling strengths of modes

/ :

6’16l

759

237

11’16’

7.52

230

jl

749

42

730

17’

10-l

4 2

1O-3

.0.03 0.03

227

1

1O-2

0.03

208

1

1O-3

-0.04

1

D..

0.005 ~. 0,03

0.063.

-719

197

2

-lo-*

0.05

0.08

_ 711

189

4.

lo*

0.06

j.002

602

80

2

lO-3

d.3

0.0;

10’

585

63

2

IO-’

0.4

11’

515.

-7

1

10-Z

0.6

05

Hi*

414

- 4%

3

10-3

0.6

0.2

B. c

4l

365

-157

1

lo-*

, 0.2

0.2

C

16’

237

-285

2

lo-’

0.06

0.1

0

-522

1

10-l

0.005

0.04

163_ 4l16’

00

us and v16 with the perturbin potential as reflected in the Au factors U2(6) and U9 (16) do not vary strongly among the four collision partners. This again supports the use of rules for V-T;R transfer which are collision partner independent. The gases in the second row of fig. 9 can all use V-V transfer in addition to V-T,R transfer. The propensity rules make an interesting statement about the competition between these transfers. They predict that V-V resonances can boost markedly transfer rates in one channel, but the effect on 6’ relaxation rates is otherwise rather small. The details of this begii with calculations in table 6 which show the explicit V-V effects on transfer to specific levels. We have assumd the most favorable case for V-V transfer, namely that for processes with benzene levels separated by about 200 cm-’ ormore, a near resonant transfer can exchange vibrational energy with a Au = 1 quantum change in the collision partner. This reduces the AE to less than 50 cm-’ _ On thisbasis V-V contributions can supplement the V-T,R transfer for many channels. As an example, consider transfer to 6ll6’. AV-V near resonance-reduces AE to essentially zero, boosting the I(M) factor over that for V-T,R transfer. On the other hand, quantum numbers must change in two modes, and this

.

,o.i

B -A

reduces the U2 factors to 10m2.The net result is that transfer by a V-V process would occur with a probability which is only 0.7 of that for aV-T,R transfer. The total transfer probability to this level is increased by something less than a factor of two by the additional possibility of a V-V retinance. Table 6 shows this to be the general result. With one exception, the V-V transfer, constant does not exceed the V-T,R constant. V-V transfer-does not generally introduce new relaxation channels which were closed in V-T,R transfer. The exception is of.course trarisfer to the zero point level (channel A) where theV-V resonance reduces large Al? (522 &r-l) to nearly zero. Near resonant V-Vtransfer is predicted to bccurwith a rate constant about twelve times greater than that of

V-T,R transfer, allowingchannel A to be in$ortant in collisions with vibrationtiy complex~moleculds. The calculated @&.itioning of~relaxation among channels for gases using V-V;T,R transfer matches welt the experimentafobservations in fig. 9. As expected, -the priricipal difference between t&se gases and V-T; R gases is the increased importance~of cha&el A> The total effect of V1-Vtrairsfei onSrelaxation~out~ of 6’ is derived by cornpa-&g cslculatcd surns~bf $I transfer processes out-of 6’. If we~&ui~dei transfer by V-T,R proct%ses.~one, the~rehitive con&&s ki(i) :

.

C.S_ Pornrenter, K. Y_ To&Mode-to-mode

vibrational energy flow in S1 benzene

145

Calculated knirgy transfer-parameters and relative probabilities for vibrationally complex collision partners which move energy from

6r.b~ V‘-Vand V-T,R processes. The notation (V-T,R) or (V-V) or (V-V,TR) defmes the type of process being considered Final

&

level

(V-T,i)

6’16’ llt16’ 5r 42 17’ 163 4’16’ 10’

AE

237 230 227 208 197 189 80 63

11’ 162 4’ 16’ 00

-7 -48 -157 -285 -522

k4 (0

U2 (Au)

I@EI

k(i) (V-V)

W-V)

w-v)

k4WS-T,R)

k4(,5116’) W-V,T,R)

0 0 0 0 0 0 -

1O-2 lo* lo-3 10” 10-s 105 -

0.19 0.19 0.20 0.22 0.23 0.24 -

0.7 0.7 0.7 0.6 0.5 0.4 0 0

1.0 0.005 0.03 0.003 0.07. 0.002 0.03 0.41

D

0 0

10-s 10-a

-

0 0 0 1.0 12

0.31 0.09 0.10 0.12 0.33

B C C B A

w-v,

““p?$

0.60 0.60

._

c

0.3 -

/ C&B C 5.

0.5 -

,’ ,!

/ 0.1 L

P

Ii

co

.sF,

0.1 -

‘;‘.ocs

0.05 -

/

on5 -

/ CO?’

&2

:eo

2

SF

0.1 -

.ocs SF,

bN,

/ 0.05 - -He /

H.”: /

0.01 ’

/

co,.,’

/

/

Channel

Channel E 4

1

I

6

8

Do,1 , , 2

C~onnefC

4

6

l/2

l/2

p

8

r

I

1

2

/I

Channel D 1

I

I

4

6 A 112

8

p

Fig. 12: Plots of cotlision effciencies_P against reduced mass asp ‘I2 (iIt au’/‘) for ChanIIek B, C and D.P is defmed as (oobservcd/ u as hme&). Note that the scales for efficienci_esare not all identical. Abbreviations have been used for some collision partners: I?-s siCsHrz;Cs isnCsH12iC,isnC;H12;CF isnC~Fr4~Dashedlineshavebeendrawnonl~ to suggest theV-T,R trend on ihe basis of the efficiencies of He, Co, Na CO2 and OCS.

,

(xgu21) sum _to 0.035 (table 5). If in addition we inelude V-V prbcesses,the sum of the rate constants @+V;‘i’,R) bi&n& 0.055 (table 6). Thus; the propensity rules’suggest that V-V-contributions to energy tr&sfer are remarkedly small, boosting total transfer out of 6l by only the factor 0.055/0.035 = 1.6. (The i@usion of levels higher than those in the tables makes -negligible change.) The common flow patterns in fig. 9 for vibrationally complex molecules is also consistent with this picture of minor V-V influences. Although specific \‘-V resonances must be somewhat variable among the collision partners, this cannot be detected very precisely in the data. Ah partners seem much the same. Part of -the reason is probably due to averaging over many resonances, but a major factor must surely be that V-V processes are not generally of dominant importance. 7.4. Competitim between V-V and V-T, R energy transfer In the last paragraphs of the preceding section, arguments were given to indicate that V-V contributions to relaxation out of 6l are no more than comparable to those of V-T,R transfer. With the exception of channel A, vibrational resonances with the collision partner would seem to have only a small effect in vibrationsI relaxation. Here we support these propositions by other arguments. First, in order to gain a more direct view of the added V-V contributions to channels B, C, and D, we have in fig_ 12 plotted the channel efficienciesP of energy transfer against ~_r Ii2 _V-T,R transfer often hasp. with exponential dependence upon p 112 [45,49-541, and fig_ 12 shows that this trend is approximately obeyed for the efficiencies of He, N, CO, CO, and OCS [55] for the channels B and D. We have extrapolated these V-T,R efficiencies to larger p1’2 in order to gauge the expected V-T,R contributions from the larger polyatomics. Only three V-T,R efficiencies guide the extrapolation for channel C, but the plot !ooks entirely similar to that for the other channels. These plots show a uniform pattern of behavior among the three channels. The efficiencies of the three Cs-C7 hydrocarbons lie above those predicted for V-T,R transfer by factors of about two. Thus V-V contributions to each channel appear modest, being about equivalent to V-?‘,R contributions. This is quite-

P

l/P

Fig. 13. A plot of the efficiency P of net vibrational energy transfer out of the St benzene level 6’ against reduced mass as w1j2_The collision partner identity is given at the top of the figure. SF6 and rzCsF14 points are open circles.

in accord with tire expectations from propensity rules. The efficiency of SF, faUs substantially below that predicted for V-T,R. The efficiency of perfluorohexane (n-C6F14) is low relative to hydrocarbon analogs and nearly coincides with that expected for V-T,R transfer alone. These behaviors are special cases to which we return later. Oneobserves a‘similar situation when the efficiencies for total transfer out of 6l are plotted against p1j2. Extrapolation of the V-T,R efficiencies for He, N,, CO and CO, should give a rough measure of the V-T,R efficiencies of the vibrationally compiex partners. The observed total efficiencies (V-V,T,R efficiencies) of those partners are within about a factor of two of the extrapolation, emphasizing once again-the restricted role played by near resonant V-V-processes.It has been noted that efficiencies increase with cr1j2 for-small amounts of-energy tranefkrred V-T,R whereas they de&se for large VTT,R tra&feis [49] . hf. ’ ‘relations,for ground &&epolyatomics_ [49154] &ow the break point at $00 K to ‘be_roughly_k~. Trami-_

CS. Panni?nter,_ K. Y. TangfMode-to-modevibrational energy jlow in SI benzene

tions &thAE(V-T/R) 2 200 cm-r often (but not always!) havean approximately exponential dependence ofP upon y172 with increasing efficiency for greater n_ The comparisons in fig, 13 would by analogy suggest that the channels contributing principally to 6l destruction have small energy gaps, and this is pretty much in accord with the earlier discussions. The correlation of total transfer cross sections with @MM)U2 in fig. 10 also supports the modest role played by V-V transfers. The correlation fails to distinguish . between the V-T,R and the V-V collision partners. No hints of any special resonant effects on 61 rehrxation can be found. 7.5. Special effects T.S.1. Dead resonant V-V transfer with SF6

An especially interesting possibility occurs for resonant V-V transfer in benzene-SF6 collisions: B**(69 + SF6(0) + B*(OO)f SF&;),

AE = 0.

In addition to the energy resonance, the level v: in SFs is triply degenerate. Notwithstanding the encouragement of these parameters, the efficiency of this transfer with SF, remains low in comparison with the hydrocarbons where such remarkably favorable resonances and degeneracies almost certainly do not exist. We can see this reflected also in fig. 9 where relative rates into channels are compared. The fraction of transfer into this process (channel A).is not distinguished when compared to that for the hydrocarbons, nor when compared with the fraction calculated for the process, even with a calculation using g = 1 rather than g = 3. 7.5.2. Inefficient collision partners: SF6 and perfluorohexarre The anomalously low absolute magnitudes of SF, transfer rate constants noted in the plots of figs. 12 and 13 is quite consistent with that observed in ground state SF6 vibrational relaxation [52]. In those cases V-T,R transfer efficiencies in SF: + SF6 collisions fall below~SSH theory predictions, and V-V transfer in SF: + SFe collisions seems also less efficient than expected. The flow patterns in B**-SF, collisions matches the propensity rule predictions and those of other large collision partners (fig. 9) This is consistent w&h the ground state r&&s, in that both V_-V’and

147

V-T,R processes seem equally reduced in magnitude. Note, however, that SF, looks quite normal on the correlation in fig. 10, which stresses interaction potentials among collision partners. The reduced SF, efficiencies seem to follow directly the interaction potentials. Perfluorohexane was used because of an interesting property. As a solvent for condensed phase abSOrptiGn or fluorescence spectra of aromatic hydrocarbons, it tends to preserve the sharp structure seen in vapor spectra [56]. The solvent-aromatic interactions appear to be unusually small. This property now appears also in the vibrational relaxation data, where the perfluorohexane rate constants fail substantially below those for n-hexane in every channel except channel C where they are about equivalent. Similar results are seen in fig. 12 where, in fact, the constants for perfluorohexane never even reach the values expected for V-T,R energy transfer alone. As in the case of SF,, perfluorohexane relaxation of the level 6’ appears rather normal when compared in fig. 13 on a correlation stressing potential energies in the intermolecular interaction_ And as with SFe , the flow patterns with perfluorohexane collisions appear similar to calculations and to those of other collision partners (fig. 9). Thus the reduced transfer efficiencies seem to stem from a reduced interaction with benzene, rather than from more specific characteristics. 7.5.3. A correlation of 6’ relaxation cross sections for hydrocarbons Monson et al. [57] put forward a correlation which was to suggest that vibrational energy removal in‘collisions with hydrocarbons occurs primarily through their methyl groups. In table 7 we show that the cross section for the destruction of the level 6l is nearly a constant function of the ratio of methyl groups to the total number of carbon atoms in the hydrocarbon collision partner. On the other hand, the plots in figs. 12 and 13 argue that there are no special V-V effects due to methyl groups. As a side issue we also show in table 7 the correlation for removing quanta of H-F vibrations , in ground state HF. It works remarkably well. A related correlation of HF relaxation with n for CnH2rr+2 partners has also been noted [61].

148

--. -:

.

C’S_ Pannenter, K. Y. Tahg/b&?e-to-hiode vibrational eneq$ flow in L$ benzene

T&e 7 __ : Corn&ion ofvibratior&relaxation Excited gas

z

Foreign Pk.

. uobs (A*)

wk

HF (v= 1) b,

.

HF(v=l)

HF(u=2)

.__

cross section i with proportion of methyl grotipsin hydr&b& Ratio of m&y1 groups b, for&,, gas to total carbonatoms .

___ chain-~- i_.

~.Noo&&.,j 0 d) - ‘I .- ~. -:

X5.2=)

2:3

10.1

f-G&2

23.3

35..

14.0

nGJJ14

45.5

25

15.2

f%+b6

51.1

2:7

14.6

C3&3

CHLJ

0.2

1:l

03~

Cz H6

0.45

2:2

Q.45

c3 Hs

0.61

2:3

0.40

n-C4Hlo

0.82

2:4

0.41

CH4

0.24

1:l

024

c2 H6

0.23

2:2

023

c3 H8

0.38

2:3

0.25

CHQ

0.87

1:l

0.81

c2 H6

1.15

2:2

1.15

C3H8

1.35

2:3

0.90

_-_-I’

:

.’ .-

-I _

-.

:

_: __ : ___. ‘_ I

--_-

-.

a) Data from ref. [58] _ b, Data from ref. [59]. Cl Data from ref. [60]. d, (LT~~~) times.(Ratio of methyl groups in foreign gas to total carbon atoms.)

7.5.4. Benzene-benzene collisions: E-E contributiohs to vibratiomd relaxation We will report elsewhere on relaxation experiments

with benzene itself as the collision partner. For compa&on wc note some results here. The absolute magnitudes of vibrational relaxation rate constants in benzene-benzene collisions are exceptiohlly large (about four times larger than gas kinetic), and the pattern of

channels contributing to the relaxation is fundamentally different from that of the other collision partners. Channels A-D contain only about 40% of the total cross section as opposed to about 70% for most other partners. A fifth major channel cannot be identified in the growth spectra, so presumably a number of less effective channels combine to yield the large total cross section. The efficiency of vibrational relaxation in benzenebenzene collisions is further emphasized by comparing the rate constants with those for relaxation in the ground

state. Cheng [62] and also Lambert and Rowlinson [63] have followed ground state relaxation using .&und velocity dispersion in high pressure vapors, and find that 100-300

collisions are required to remove a quantum

of v;g (fyj, = 399 cm-‘). In the excited state, a quantum of us (522 cm-l) is removed in one collision, and the addition of a quantum of & (237 cm-‘) requires about three colhsioirs. We have carried out a separate investigation in order to learn about the extent to which the-special mechanism electronic energy transfer can contribute to “vibrational relaxation” in benzene-benienecolliiions [64]. For example if 6l benzene molecules were to transfer electronic but notvibrational energy td a ground state collision partner m another vibrationdstate,-the exchange would appear to the experimentali& as vibrational relaxation; The results are sometihat~iiiconclus&due to uncertainty in carrying,over the results from electronic energy transfer iriCeD: 4 C$-I~ccoll&ion~ tdthe C;Hz + C6 H6 system:Hotiever there is @doubt that electronic

C.S. Parmentet. KY. Tanghode-to-mode vibrational energy flow in S; benzene

energy tranSfer occurs and does so with cross sections near gas kinetic values. Thus it is entirely possible that a major b&ibution to the observed vibrational relaxation is in fact due to electronic energy transfer_ . .i Acknowledgement Dr. G.H. A_tkinson has been most helpful in laying out some of the very initial aspects of this work, and we are greatly indebted to Dr. A.E.W. Knight for some experimental assistance. This work has been supported by a grant from the National Science Foundation.

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150

z.

_‘-

:

;.,

-~C%_ Pa&nter, K-I’. Taqg&ie-to&o&

vibratiod @e&-flo~wminSi-b&k& .^

~_ ji&i%&i&.w.tiynn

1481 F.ti.G~biner’&~G.W. : -..

J Chem

l&n, ,

Phys.58

J. &m_?hys.

(I&)

2781 60 (1974)-

:

:39&: --,- -:-:: -1491 E.IW+tz and G.W. F&m,J. Chem. phys. 58 (1973) 2679. [:Ol S.M. Lee and A.M. Ro~,Chem. Phys. Letters 22 (1973) _

[& :?Kimdtson .’

[j21 [53] [54] [SS]

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pUbliShed.

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_

: . . : ’ y. -.

--

-.

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_