Time-resolved Fourier transform infrared spectroscopy of ... · Time-resolved Fourier transform infrared spectroscopy of optically pumped carbon monoxide Elke Pl onjes 1, Peter Palm

Time-resolved Fourier transform infrared spectroscopyof optically pumped carbon monoxide

Elke Pl�onjes 1, Peter Palm 1, Andrey P. Chernukho 2, Igor V. Adamovich *,J. William Rich

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43220-1107, USA

Received 26 July 1999

Abstract

The paper discusses measurements of vibration-to-vibration (V±V) energy transfer rates for CO±CO using time-

resolved step-scan Fourier transform infrared spectroscopy of optically pumped carbon monoxide. In the experiments,

time evolution of all vibrational states of carbon monoxide excited by a CO laser and populated by V±V processes (up

to v � 40) is monitored simultaneously. The V±V rates are inferred from these data using a kinetic model that in-

corporates spatial power distribution of the focused laser beam, transport processes, and multi-quantum V±V pro-

cesses. Although the model predictions agree well with the time-dependent step-scan relaxation data, there is variance

between the model predictions and the up-pumping data, however. Comparison of calculations using two di�erent sets

of V±V rates with experimental spectra showed that the use of the semi-empirical V±V rates of DeLeon and Rich

provides better agreement with experiment. It is also shown that the multi-quantum V±V rates among high vibrational

quantum numbers, calculated by Cacciatore and Billing, are substantially overpredicted. The results provide some new

insight into nonequilibrium vibrational kinetics, and also demonstrate the capabilities of the step-scan Fourier trans-

form spectroscopy for time-resolved studies of molecular energy transfer processes and validation of theoretical rate

models. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Nonequilibrium vibrational kinetics of diatomicand small polyatomic molecules has been the focusof attention for many years in gas discharge plas-mas, molecular lasers, upper atmosphere chemis-try, and gas dynamic ¯ows [1,2]. The rate of energytransfer between the vibrational molecular modes

and the ``external'' modes of rotation and trans-lation is, in particular, a determining process inmany high enthalpy ¯uid environments. The de-tails of the actual distribution of energy among thevibrational quantum states is important in a morerestricted range of problems, but is certainly key innon-thermal plasma chemical reactor design, inpredicting radiation from supersonic nozzle ex-pansions, and in the design of a variety of molec-ular gas lasers [3]. This energy distribution isprimarily controlled by vibration-to-vibration(V±V) energy exchange processes [4,5],

AB�v� �AB�wÿ k� ! AB�vÿ k� �AB�w�; �1�

Chemical Physics 256 (2000) 315±331

www.elsevier.nl/locate/chemphys

* Corresponding author.1 Visiting Scholar, Insitut f�ur Angewandte Physik, Univer-

sity of Bonn, Germany.2 Visiting Scholar, Lykov Heat and Mass Transfer Institute,

Minsk, Belarus.

0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 0 9 6 - 3

which for a broad range of parameters are knownto be much faster than vibration-to-translation(V±T) relaxation,

AB�v� �M! AB�vÿ k� �M: �2�In Eqs. (1) and (2), AB and M stand for diatomicmolecule and atom, respectively, v and w are vib-rational quantum numbers, k is the number ofquanta transmitted in a collision. The rates of thenear-resonance V±V exchange processes of Eq. (1)among the high vibrational quantum levels(v � w; v;w� 1) are of particular importance,since molecules on the high vibrational levels canalso participate in nonequilibrium chemical reac-tions as well as produce electronic excitationand ionization. In particular, the vibration-to-electronic (V±E) processes, in which energy istransferred from highly vibrationally excited levelsof the ground electronic state to a low-lying ex-cited electronic state, with subsequent ultravioletand visible band radiation, have been observed inoptical pumping experiments in both CO [6±8] andNO [9,10]. Ionization in collisions of two vibra-tionally excited CO molecules pooling their ener-gies together has been studied in Refs. [11,12].Predictive analysis of kinetics of these energytransfer processes, coupled to the high vibrationallevel populations, require precise knowledge of theV±V rates.

The present paper addresses recent measure-ments of the V±V rates in strongly vibrationallyexcited carbon monoxide (CO). The V±V rates forCO±CO have previously been measured using avariety of spectroscopic techniques (see Ref. [6]and references therein), including experiments byBrechignac and co-workers [13±15] and DeLeonand Rich [6] who inferred the near-resonance V±Vrates up to v 6 29 and v 6 35, respectively. Themain advantage of the present study over theprevious experiments is the use of time-resolvedstep-scan Fourier transform infrared (FTIR)emission spectroscopy. Step-scan Fourier trans-form (FT) spectrometers, such as the system usedin the present study, have become available onlyrelatively recently. These instruments provide theability to record emission spectra in a broadwavelength range with high time resolution. Thetime resolution is limited by the response time of

the detector system and data acquisition capabili-ties. In the present study, a fast response InSbdetector with a high bandwidth DC preampli®erallows time resolution of up to a few nanoseconds.In other words, instead of measuring the time-dependent emission signal in a narrow wavelengthrange, the population of each radiating vibra-tional±rotational state can be monitored simulta-neously. The step-scan technique requires arepetitively pulsed experiment, which is arrangedin the present study; note that the technique couldnot be used for single-pulse experiments, such as inshock tunnels. An emission experiment is advan-tageous with this instrument, since absorptionspectroscopy introduces noise complications fromthe incident absorption source. In addition, FTspectrometers provide throughput and multiplexadvantages compared to conventional narrow-slitmonochromators, improving the signal-to-noiseratio at high spectral resolution. Step-scan FTspectroscopy has been recently used to study en-ergy transfer in vibrationally excited NO2 [16,17].

Thus, in the present study, the time evolution ofall vibrational states populated by the V±V pro-cesses up to extreme vibrational disequilibrium canbe monitored simultaneously.

2. Experimental setup

Fig. 1 shows a schematic of the experimentalsetup for the study of vibrational energy transferin CO. A CO laser is used to irradiate a gas mix-ture of CO and Ar, which slowly ¯ows through theshown pyrex glass optical absorption cell. Theresidence time of the gas mixture in the cell isabout 1 s. The liquid nitrogen cooled CO laser wasdesigned in collaboration with the University ofBonn and fabricated at Ohio State. It produces asubstantial fraction of its power output on thev � 1! 0 fundamental band component in theinfrared. The laser can operate at more than 100 Wcontinuous wave (CW) power. However, in thepresent experiment, the laser is typically operatedat 10 W CW broadband power on the lowest tenfundamental bands, with up to �0.3 W on thev � 1! 0 component (Table 1). The output onthe lowest bands (1! 0 and 2! 1) is necessary to

316 E. Pl�onjes et al. / Chemical Physics 256 (2000) 315±331

begin the absorption process in cold CO (initiallyat 300 K) in the cell.

The present use of CO laser pumped absorptioncells to study the V±V process is a further devel-opment of a technique with a considerable amountof literature [6±11,18].

In the time-resolved experiments, intended toobtain a more precise measurement of the mech-anism and rates of the V±V exchange processes,the laser beam is interrupted by a mechanicalchopper, giving a nearly square wave input of laserpower to the cell (Fig. 2). After the laser is turnedon, the lower states, v 6 10, are populated by di-rect resonance absorption of the pump radiation incombination with the much more rapid redistri-bution of population by the V±V exchange pro-cesses. The time-dependent transmitted laser pulseshapes are also shown in Fig. 2 for di�erent CO

partial pressures in the cell. The V±V processesthen continue to populate the higher vibrationallevels above v � 10, which are not directly coupledto the laser radiation. When the laser is turned o�,the vibrational levels are depopulated, againmainly by the V±V energy transfer, and thevibrational energy distribution approaches theBoltzmann distribution at the translational tem-perature. The chopper is operated at a low fre-quency of 13 Hz with a low duty cycle of �1/11, sothat the laser pulse duration is 6.8 ms, and the timebetween the pulses is about 75 ms. The low dutycycle is deliberately chosen to allow completevibrational relaxation between the laser pulses.

In the steady-state measurements, the laser re-mains on all the time, so that a complete steadystate is reached. Note that the vibrational energystored in the molecule is constantly converted into

Fig. 1. Schematic of the experimental setup.

E. Pl�onjes et al. / Chemical Physics 256 (2000) 315±331 317

heat both in V±V and V±T processes. However,the large heat capacity of the Ar diluent, as well asconductive and convective cooling of the gas ¯ow,allow us to control the translational/rotationalmode temperature in the cell. Even in steady-stateconditions, when the average vibrational modeenergy of the CO would correspond to a fewthousand degrees Kelvin, the temperature neverrises above a few hundred degrees. Thus, a strongdisequipartition of energy can be maintained in thecell, characterized by very high vibrational modeenergy and a low translational/rotational modetemperature. Similar nonequilibrium conditionsexist in a variety of rapid supersonic expansions, inglow plasma discharges, and in a number of otherthermodynamic environments. The present setupallows us to study the energy transfer and kineticprocesses in a closely controlled environment,without the complications of numerous electronimpact processes which occur in electric dischargesor the experimental di�culties of creating andcontrolling a supersonic ¯ow.

As shown in Fig. 1, the population of the vib-rational states of the CO in the cell is monitored byinfrared emission spectroscopy. For this purpose,a Bruker step-scan FT IFS 66 spectrometer withstep-scan and rapid-scan capabilities is used torecord the spontaneous emission from the COfundamental, ®rst and second overtone bandsthrough a window on the side of the cell. The FTspectrometer can be used for recording of bothsteady-state and time-resolved spectra. Steady-state spectra are typically recorded in rapid-scanmode at a spectral resolution of 0.25 cmÿ1. Intime-resolved measurements, a few hundred ``timeslices'' are recorded in step-scan mode with a lowerspectral resolution of 8.0 cmÿ1, and a time reso-lution ranging from 5 to 50 ls, thereby spanning atime period of a few ms to a few tens of ms.

The Gaussian laser beam, which has a diameterof �0.5 cm, does not have to be focused to providesubstantial vibrational mode energy in the cellgases. However, in the present experiments, it isfocused to increase the power loading per COmolecule and accelerate the V±V up-pumping,providing an excitation region in the cell of �1 mmdiameter. The measurements reported here aremade at the same Ar partial pressure of 100 Torr,

Table 1

CO laser spectrum

Laser line Power (W)

1! 0 P(15) 0.29

2! 1 P(15) 0.62

2! 1 P(16) 0.57

3! 2 P(14) 0.79

3! 2 P(15) 1.52

4! 3 P(13) 0.31

4! 3 P(14) 1.14

4! 3 P(15) 0.80

5! 4 P(13) 0.24

5! 4 P(14) 0.95

5! 4 P(15) 0.35

6! 5 P(12) 0.42

6! 5 P(13) 0.20

6! 5 P(14) 0.56

7! 6 P(12) 0.32

7! 6 P(13) 0.44

8! 7 P(12) 0.45

9! 8 P(11) 0.13

9! 8 P(12) 0.10

10! 9 P(10) 0.05

10! 9 P(12) 0.09

Fig. 2. Transmitted CO laser pulse shape at di�erent CO partial

pressures in the cell.


and at seven CO partial pressures varying from 0.5to 3.5 Torr.

3. Kinetic model and the V±V rate parametrization

To interpret the results of both steady-state andtime-resolved measurements, and to infer the V±Vrates for CO±CO, we use a state-speci®c kineticmodel of excitation and relaxation of opticallypumped anharmonic oscillators in inhomogeneousmedia. It is based on the master equation modeldescribed in detail in Ref. [18]; two signi®cantupgrades are (i) incorporation of laser power dis-tribution and transport processes (di�usion andheat conduction) across the Gaussian laser beam,and (ii) incorporation of the multi-quantum V±Vprocesses (jkj > 1 in Eq. (1)). The model evaluatesthe time-dependent vibrational level populationsin CO±Ar mixture excited by a laser beam:

onv�r; t�ot

� 1

roor

rDonv�r; t�

or

� �� VVv � VTv

� SRDv � VEv � PLv;

onv�r; t�or

��r�0

� 0; nv�r; t�jr�1 � nv�r; t�jt�0 � nv�T0�;

v � 0; vmax; �3�

qcp

oT �r; t�ot

� 1

roor

rkoT �r; t�

or

� ��HVR;

oT �r; t�or

��r�0

� 0; T �r; t�jr�1 � T �r; t�jt�0 � T0:

�4�In Eqs. (3) and (4), nv�r; t� is the population of thevth vibrational level of CO; nv�T0� is the initialequilibrium population at T0� 300 K; nCO is theCO concentration; r is the distance from the beamaxis; D and k are the di�usion and heat transfercoe�cients, respectively; q and cp are the densityand speci®c heat at constant pressure, respectively.The rest of the notation is the same as in Ref. [18]:VV, vibration±vibration term; VT, vibration±translation term; SRD, spontaneous radiative de-cay (infrared); VE, vibration±electronic coupling;PL, laser pumping; HVR, gas heating by vibra-

tional relaxation. The explicit expressions for theseterms are given in Ref. [18] and will not be re-peated here, with the exception of the V±V term,which in the presence of the multi-quantum pro-cesses is modi®ed as follows:

VVv �Xw;k

Q�v;wÿ k ! vÿ k;w�nvnwÿk

ÿXw;k

Q�vÿ k;w! v;wÿ k�nvÿknw;

jDkj 6 Dvmax: �5�

In Eq. (5), Q�v;wÿ k ! vÿ k;w� is the rate of V±V energy exchange. Since the ¯ow velocity is quiteslow, a few cm sÿ1, convective cooling in the en-ergy equation (4) is neglected.

The objective of upgrading the model is tostudy the e�ects of nonuniform intensity distribu-tion across the laser beam and vibrational energytransport by di�usion [19], as well the multi-quantum energy transfer, on the excitation andrelaxation processes.

The Gaussian intensity distribution across thelaser beam results in a faster V±V pumping nearthe beam axis. Pump power is highest on the axis,and it is here where the vibrational level popula-tions most rapidly reach steady state. There isslower excitation far from the axis, where pumppower is less. The same qualitative scenario (i.e.faster relaxation near the axis) can be observedduring the relaxation after the laser is turned o�.This occurs because V±V exchange between twoexcited molecules is a nonlinear kinetic processwith a rate proportional to the product of the twolevel populations. For example, �dn2=dt�VV �QVV�1; 1! 2; 0�n1n1, and the ®rst-order ``timeconstant'' sVV � �QVV�1; 1! 2; 0�n1�ÿ1 6� const isinversely proportional to n1, which tends to in-crease with the pump laser power density. Notethat emission spectroscopy is essentially a line-of-sight measurement. The line-of-sight integration ofthe time-dependent signal by the spectrometer willcreate an impression that the high level popula-tions rapidly rise and decay, because of the con-tribution of the central portion of the beam wherethe power density is high and kinetics is fast. Thedescribed e�ect does not appear in linearor quasilinear kinetic processes, such as V±T


relaxation or V±V exchange with the ground vib-rational state, v � 0. For these two cases, the ®rst-order time constants, sVT � �PVT�1! 0�N �ÿ1

andsVV � �QVV�1; 0! 0; 1�n0�ÿ1

, are indeed constant(n0 changes fairly weakly during the excitation)and independent of the laser power.

Focusing of the laser beam is likely to reducethis e�ect due to the more rapid di�usion of vib-rationally excited molecules across the narrowbeam. However, a greater e�ect of di�usion whenthe beam diameter is reduced will further compli-cate the analysis. For example, previous time-resolved experiments [6,18] show that the typicalrise and decay times for the infrared radiation fromthe high CO vibrational levels excited by the fo-cused laser beam are of the order of a few milli-seconds. On the other hand, simple estimates showthat the characteristic time for di�usion of vibra-tional energy out of the excited volume is of thesame order of magnitude: sdiff � R2=D � 10 ms.Here, R � 1 mm is the focused laser beam diame-ter, and D � 1 cm2 sÿ1 is the di�usion coe�cient ofCO at P � 100 Torr. This results in a competitionbetween the V±V energy transfer and energytransport by di�usion.

Finally, three-dimensional semiclassical trajec-tory calculations of the V±V rates for CO±CO byCacciatore and Billing [20] predict the rates of thenear-resonance multi-quantum processes collisionsinvolving exchange of two or more quanta (jkjP 2in Eq. (1)) to be comparable with those of thesingle-quantum processes for high vibrationallevels (v;w > 10 in Eq. (1)). This prediction isconsistent with the results of the previous experi-ments [6], which inferred the near-resonance sin-gle-quantum V±V rates to be an order ofmagnitude greater than the gas kinetic collisionfrequency. This result might be attributed to thelatent contribution of the multi-quantum pro-cesses, not accounted for in the kinetic model used.

All this suggests that the spatial power distri-bution, di�usion, and multi-quantum relaxationmight well be signi®cant factors in the presentexperiments.

In the present model, the rates of the energytransfer processes, except for CO±CO, V±V rates,are the same as described in Ref. [18]. The V±Vrates are the remaining parameters in the model

calculations that are only known with some un-certainty. Therefore, they are varied to providebetter agreement with the experiment. In the cal-culations, two di�erent sets of rates are used. Inthe ®rst baseline series of modeling calculations,we used a V±V rate model based on the analyticparametrization suggested by Je�ers and Kelley[21], developed from the perturbation theory forthe cross-sections, and correlated with the body ofexperimental data obtained in several previousmeasurements (see Ref. [6] and references therein).These rates constitute Set I (Fig. 3). This ratemodel is applicable only for single-quantum V±Vexchange processes. The explicit V±V rate expres-sions used can be found in Appendix A. In thesecond series of calculations, we used an analyticnonperturbative theory of vibrational energytransfer [22±24] that provides a convenient para-metrization of the results of the trajectory calcu-lations by Cacciatore and Billing [20], consideredto be the most reliable theoretical data available.This ``forced harmonic oscillator'' (FHO) theorytakes into account the coupling of many vibra-tional states during a collision and is thereforeapplicable for multi-quantum processes. Since the

Fig. 3. Set I and Set II single-quantum rates at T � 300 K.


published set of the CO±CO rates [20] is somewhatincomplete, we have used the computer codesDIDIAVDIDIAV and DIDIEXDIDIEX developed by Billing [25,26]to extend the calculations up to jkj 6 5, for thesame CO±CO potential as used in Ref. [20]. Thecomparison of these rate data with the FHOparametrization, given in Appendix A, is shown inFigs. 3 and 4. These V±V rates constitute Set II.

In addition, the code DIDIEXDIDIEX was also used tocalculate the rates of the asymmetric one-by-twoquanta near-resonance V±V exchange process(Fig. 5):

CO�0� � CO�w� ! CO�1� � CO�wÿ 2�;w � 35±50: �6�

This process, incorporated in all subsequent cal-culations (using both sets of V±V rates), was sug-gested by Napartovich and co-workers to be veryimportant in kinetics of high vibrational levels ofCO [27]. Note that calculations of the rates ofprocess (6) require the use of accurate CO spec-troscopic constants [28]; otherwise, the quantumnumber at which the rate (6) reaches maximumcan be shifted by a few levels.

We emphasize the necessity of the use of ana-lytic parametrization for rate inference. Even ifonly transitions between adjacent quantum levelsare incorporated in the time-resolved experimentanalysis, there are potentially 40� 40 speci®c ratesinvolved for a pump up to v � 40. As is wellknown, deconvolution of all the speci®c rates fromthe time-dependent population measurementsalone would require far greater accuracy thanachieved in the present data. We therefore requiretheoretical models of the quantum number de-pendence of these rates, as discussed above, togreatly reduce the number of parameters involved.

The system of Eqs. (3) and (4) for 50 vibrationallevels of CO is solved using a standard solver forsti� partial di�erential equations, PDECOLPDECOL [29]. Inthe calculations, a 31-point nonuniform grid, withmost points located near the beam axis, is typicallyused. The laser line intensity distributions aregiven by the equation Ii�r� � 0:5�I0i � I0t� �2=pR2 exp�ÿ2r2=R2��, where I0i and I0t are the inci-dent and the transmitted line intensities in W, andparameter R� 0.28 mm in the Gaussian intensitydistribution across the focused laser beam iscalculated by the code STRAHLSTRAHL developed at

Fig. 4. Set II: three-quantum rates at T � 300 K. Fig. 5. Asymmetric near-resonance V±V rates.


University of Bonn [30]. 3 A synthetic spectrumcode is then used to generate the model spectra.Rotational level populations are assumed to be inequilibrium with the translational temperatureT(r). Both the vibration±rotation level populationsand the emission intensity are integrated along theoptical path of the spectrometer (i.e. across thelaser beam). The code uses the spectroscopic datafor CO molecule [28] and accurate Einstein coef-®cients for spontaneous emission and absorptioncoe�cients for the CO infrared bands [31] as in-puts. As usual, the synthetic spectra are correctedfor the blackbody-calibrated instrument responsefunction.

4. Results and discussion

4.1. Steady-state measurements

Under the conditions described in Section 2,with the laser left on, a highly nonequilibriumdistribution of vibrational energy is created in theCO in the cell. The second overtone, ®rst overtone,and part of the fundamental emission from COexcited by the focused laser beam is shown in Fig. 6,as recorded by the FT spectrometer at a resolutionof 0.25 cmÿ1. The second overtone bands can beseen at the highest frequencies on the left(m > 4300 cmÿ1), the ®rst overtone bands domi-nate at the lower frequencies (2250 cmÿ1 < m <4300 cmÿ1), and the high frequency tail of the R-branch of the v � 1! 0 fundamental is the tallpeak on the right. A long wavelength cut-o� ®lteris used to prevent any more of the very intensefundamental band emission at frequenciesm < 1950 cmÿ1 from being recorded by the detec-tor, and swamping the overall signal. While thisresolution is approximately 1=10 the rotationalspacing of the CO molecule, there is a dense arrayof individual vibrational±rotational lines, due tothe overlapping of the various band components.The vibrational population distribution functions(VDF) inferred from such high-resolution spec-

trum using the standard technique [32] are shownin Figs. 7 and 8 for di�erent CO partial pressures.Typically, the vibrational level populations forv < 30 are inferred with the accuracy of a few percent, while for v P 30 the accuracy becomessomewhat worse, up to 20±30%. The translational/

Fig. 6. High-resolution steady-state CO FT spectrum: PCO �0:5 Torr.

Fig. 7. Experimental and theoretical steady-state CO distribu-

tion functions at PCO � 0:5 Torr.

3 U. Sterr, W. Ertmer, University of Bonn, Germany, private

communication.


rotational temperatures at these conditions, in-ferred from the rotationally resolved R-branch ofthe 1! 0 fundamental band with the accuracy of�10 K, are also shown in Fig. 8. The distributionsshown in these ®gures are the well-known ``V±Vpumped'' distributions [4,5], maintained by therapid redistribution of vibrational energy by theV±V processes, which pump energy into the highervibrational levels. They are obviously extremelynon-Boltzmann, and characterized by high popu-lation of the upper vibrational levels. Althoughthere are more than 80 bound vibrational levels inthe CO ground electronic state, only the lowest 40levels are populated by the V±V process. Note thatV±V pumping of CO above v � 40 has not beenachieved in the previous experiments with vibra-tionally excited CO [6,8,11,18,33±35] in a broadrange of translational temperatures T� 90±700 K,regardless of available pumping power. This in-cludes measurements in electric discharges and inlaser absorption cells (both in gas and in liquidphase).

The reason for this persistent termination of theup-pumping has been discussed in the literature

for some time [6,8,27,33±36]. The three most likelykinetic processes responsible for this e�ect are (i)asymmetric one-by-two quanta near-resonanceV±V exchange (6) [27], (ii) near-resonance V±Eenergy transfer from v � 40 of the ground elec-tronic state, X 1R, to the low levels of the excitedelectronic state A 1P,

CO�X 1R; v � 40� � CO! CO�A 1P; v � 0� � CO

CO�A 1P� ! CO�X 1R� � hm

CO�A 1P� �M! CO�X 1R� �M �7�

suggested in Ref. [6] based on the measurements oftime-resolved radiation from CO fourth positiveelectronic bands A 1P! X 1R, coupled with thepopulation of v � 35, and (iii) rapid V±T relax-ation on the products of vibrationally inducedchemical reactions initiated by the process,

CO�v� � CO�w� ! CO2 � C; �8�

such as C, C2, C2O, etc. (vibrationally inducedchemistry was ®rst observed in optically pumpedCO in Ref. [37]). Finally, both theoretical calcu-lations of the V±T rates for CO±CO [20] andmeasurements of the V±T rates CO±Ar at highvibrational levels [38] show them to be far too slowto a�ect the vibrational level populations atv � 40, especially at the low temperatures ofT� 100±300 K. Note that dramatic increase of thestate-speci®c vibrational relaxation rates above athreshold vibrational quantum level, induced bychemical reactions or intermolecular electroniccoupling, has also been previously observed in NO�v > 14� [39], O2 �v > 26� [40], and NO2 (vibra-tional energy greater than 10,000±12,000 cmÿ1)[41].

To analyze the e�ect of these processes on theobserved up-pumping termination, we calculatedthe steady-state distributions when turning pro-cesses (6) and (7) on and o� separately, for bothsets of V±V rates discussed in Section 3. The re-sults are summarized in Figs. 7 and 8. First, cal-culations using the V±V rates of Set I show thatthe asymmetric V±V exchange (6) truncates theVDF at higher quantum numbers than observed in

Fig. 8. Experimental and theoretical steady-state vibrational

distribution functions at di�erent CO partial pressures.


the experiment, i. e. at v � 45, where the asym-metric V±V rate reaches maximum (Fig. 5). Re-placing process (6) by V±E energy transfer (7)predicts VDF truncation in satisfactory agreementwith the low temperature data (Fig. 7). However,as the CO partial pressure and the translationaltemperature both increase, the experimental VDFcrashes at the lower quantum numbers than pre-dicted using the near-resonance V±E transition (7)(Fig. 8).

These results are consistent with the studies ofthe V±E energy transfer in CO by Wallaart andco-workers [8]. First, Wallaart et al. concludedthat the measured steady-state populations ofCO(A1P; v � 1±11) best correlate not with theground state level v � 40 but with v � 33, andtherefore suggested a more complicated multi-stepmechanism of CO(A 1P) population via media-tion of the triplet electronic states with excitationenergies between 7 and 8 eV. In addition, theirmeasurements of the CO(A 1P) concentrations,nCO�A� � 105±106 cmÿ3, lead us to conclude thatthe maximum power removed by both radiationand collisional quenching of the A state, isnCO�A�EA/s � 10ÿ5±10ÿ4 W cmÿ3, which is negligi-ble compared with the absorbed laser power of2 W. Here, EA� 8 eV is the excitation energy of theA state, and s � 10ÿ8 s is its radiative lifetime,which at P� 100 Torr is comparable with thecollisional lifetime. On the other hand, in ourprevious paper [19], we have shown that an en-ergy sink at the level va � 40 that cuts o� the V±Vpumped VDF with the Treanor minimum atv0 � 10 must remove a much larger fraction of thepower added to the vibrational mode of the os-cillator, �1ÿ xe�va� v0��=�1ÿ 2xev0� � 80%, theother 20% being removed by the V±V exchange.These estimates demonstrate that regardless of thedetails of kinetics of the V±E transitionX 1R! A 1P, it is unlikely to be responsible forthe up-pumping truncation at the conditions ofexperiments [8] (T� 700 K), since it apparentlydoes not remove enough power from the vibra-tional mode of CO. We note, however, that theCO(A 1P; v) populations measured in Ref. [8]might conceivably be underestimated due to thestrong self-absorption of the CO fourth positivebands.

Quite obviously, chemical reactions such asEq. (8) can truncate the V±V pumped distributiononly if the reaction rate coe�cient, kr, is compa-rable with the near-resonance V±V rate that sustainsthe VDF at v � 40, kr � QVV � 10ÿ10 cm3 sÿ1. Thisgives an unrealistic reaction rate krnCO�v�nCO�w� �10ÿ10 � 1014 � 1014 � 1018 l cmÿ3 sÿ1, or �100 TorrCO per second. We therefore tentatively suggestthat the VDF cuto� at v < 40 can be at leastpartially attributed to the fast V±T relaxation ofCO on the products of chemical reaction (8). Thismechanism may be most prevalent at the highercell temperatures. At the same time, previous ex-perimental data (in particular, the VDF crash atv � 40 observed at cryogenic temperaturesT � 90±100 K [33±35], when the V±T relaxationbecomes extremely slow) still suggest that the V±Etransition (8) might also play a role in the VDFtruncation at low temperatures.

Returning to the discussion of Fig. 7, we notethat calculations using the V±V rates of Set IIdemonstrate a similar although somewhat weakere�ect of processes (6) and (7) on the VDF atv � 40±45. In addition, the slope of the VDF cal-culated using Set II V±V rates becomes muchsteeper than in the experiment (Fig. 7). VaryingDvmax in the calculations showed that this occursdue to a rapid increase of the multi-quantum V±Vrates at high vibrational quantum numbers, eventhough the rates of the single-quantum processesof Set II are somewhat slower than those of Set I(Figs. 3 and 4). It therefore appears that Set IIsubstantially overpredicts the multi-quantum V±Vrates at the high v's.

Before we proceed with the discussion of thetime-resolved measurements, we tentatively con-clude that the use of the rates of Set I together withthe V±E process (7) instead of the asymmetric V±Vexchange (6) provide better agreement with thelow-temperature steady-state data. We would liketo emphasize that, apparently, the details of theVDF truncation kinetics, which are still not wellunderstood, have little e�ect on the vibrationalpopulations at v < 40 (Fig. 7). This fact allowsinference of the V±V rates from the time-resolvedmeasurements. Finally, Fig. 9, which shows theVDFs and the values of the ®rst level vibrationaltemperature,


Tv � xe�1ÿ 2xe�ln fo=f1� � ; �9�

at di�erent distances from the beam axis, demon-strates the strong spatial nonuniformity of themodeled V±V pumped region.

4.2. Time-resolved measurements

Fig. 10 shows a typical time-resolved step-scanspectrum obtained in the experiment. In this three-dimensional projection, emission intensity in ar-bitrary units is on the vertical axis, frequency incmÿ1 is on the horizontal axis, and the diagonalaxis is the time in microseconds. The overtonebands (Section 4.1) are used as the primary diag-nostic to infer the vibrational state populations, asthe gas is optically thin on all these transitions.The resolution here is 8 cmÿ1, so that the rota-tional line structure is not resolved, although theindividual vibrational band component structure isclearly seen. The time resolution is 50 ls here, andevery spectrum obtained at these intervals is beingdisplayed. The broadband structure visible at t � 0is the blackbody background, subtracted in thesubsequent analysis. The laser is switched on att � 0, reaches its steady state intensity in about200 ls, and is switched o� at t � 6:6 ms (Fig. 2). Inaddition to the low time resolution spectrumshown in Fig. 10, a 5 ls time resolution spectrum istaken for each CO partial pressure to better re-solve the initial 2 ms stage of the up-pumping afterthe laser is turned on. For each step-scanCO emission spectrum, a time-dependent laser

Fig. 9. Steady-state vibrational distribution functions at dif-

ferent distances from the beam axis: PCO � 0:5 Torr.

Fig. 10. Set of time-resolved CO emission spectra; resolution 8 cmÿ1, time resolution 50 ls.


spectrum was also measured; the time-dependenttransmitted laser power obtained from these laserspectra is shown in Fig. 2.

Figs. 11 and 12 display slice-by-slice compari-sons between the experimental and the syntheticspectra using the model described in Section 3 andthe ®rst set of the V±V rates. Although the spectraare shifted along the y-axis for illustrative purpose,they are shown in the same scale. In this experi-ment, the CO partial pressure is 3.5 Torr. One cansee that although the model is in very goodagreement with the time-resolved relaxation data(Fig. 12), it fails to reproduce the time-dependentspectra during the excitation (Fig. 11, only twosynthetic spectra at t� 0.45 ms and t� 4.45 ms areshown). Basically, the model predicts much fasterlaser radiation absorption and the CO up-pump-ing after the laser is turned on, so that the V±Vpumped distribution reaches the steady statewithin a few hundreds of microseconds instead ofa few milliseconds (Fig. 11). The same kind ofdisagreement between theoretical and experimen-

tal excitation spectra has been also observed atother CO partial pressures.

Fig. 13 displays the slice-by-slice comparisonwith the relaxation data at P � 0:5 Torr. The scalein this ®gure is di�erent from the scale used inFigs. 11 and 12 (the emission intensity is roughlyproportional the CO partial pressure). However,all time slices within each ®gure are still shown inthe same scale.

It is disappointing that the model is at variancewith the time-resolved excitation data. Note thatvarying the adjustable parameters in the Set Iparametrization or the use of Set II had not re-sulted in substantial improvement of the agree-ment. This fact, as well as good agreement of themodel with the relaxation data in a wide range ofCO partial pressures (Figs. 12 and 13), suggeststhat the observed slow up-pumping is not relatedto the V±V rates. On the other hand, the timedependence of the absorbed laser power (e.g.shown in Fig. 2), and therefore the overall rate ofCO up-pumping were found to be very sensitive tothe laser spectrum. We therefore believe that theobserved slow excitation is due to the following

Fig. 11. Time-resolved CO emission spectra during excitation:

PCO � 3:5 Torr. (Ð) experiment; (- - -) calculations using Set I

of the V±V rates (only two synthetic spectra, at t � 0:55 ms and

at t � 4:45 ms are shown). Time slices shown in the legend are

bottom to top.

Fig. 12. Time-resolved CO emission spectra during relaxation.


of the V±V rates. Time slices shown in the legend are top to

bottom.


two e�ects. First is the laser ``line hopping'' (i.e.laser output power change on individual lines)detected in the time-resolved laser spectra, which issigni®cant even though the total laser power re-mains fairly stable (Fig. 2). Second is the fast V±Trelaxation on the chemical reaction products,which is the ``bottleneck'' at the ®rst stage of ex-citation when the rate of the V±V process thattriggers the up-pumping, CO�1� � CO�1� !CO�0� � CO�2�, is comparable with that of theV±T relaxation, CO�1� �M! CO�0� �M. Sincethe V±T relaxation rates for CO±CO and CO±Arare extremely slow, relaxation on reaction prod-ucts may change the e�ective rate of the V±T re-laxation by several orders of magnitude, therebydelaying the onset of the up-pumping. We plan toverify these assumptions in our on-going time-re-solved experiments on optical pumping of NO [42].In these experiments, a stable single-line CO laseris used, while the rapid self-relaxation NO±NO isthe dominant V±T process with well-known rates[39], so there should be much less uncertainty in theup-pumping delay. Finally, even though achievingbetter agreement with the experimental excitation

data is certainly desirable, we believe that the re-laxation data allow more straightforward infer-ence of the V±V rates. Indeed, in this case, themodel does not need to incorporate the laser-related processes (i.e. absorption and inducedradiation), and the model predictions are less de-pendent on the laser spectrum and intensity pro-®le, which reduces the uncertainty in the inferredV±V rates.

Fig. 14 shows the time-resolved relaxationspectra using the second set of the V±V rates forthe conditions shown in Fig. 13 (at PCO� 0.5 Torr).Comparing these two ®gures, one can see thatthe use of the rates of Set II results in a somewhatworse agreement with the experiment, just as hasbeen observed in the steady-state calcula-tions (Section 4.1). In particular, the radiationintensity from v > 20 �m < 3400 cmÿ1� calculatedusing the Set II rates is noticeably weaker thanboth in the experiment and in the Set I calcula-tions (Figs. 13 and 14). This is consistent with thelower steady-state vibrational populations calcu-lated using the rates of Set II, compared with theSet I results (Fig. 7). We therefore con®rm the

Fig. 13. Time-resolved CO emission spectra during relaxation:



bottom.

Fig. 14. Time-resolved CO emission spectra during relaxation:

PCO � 0:5 Torr. (Ð) experiment; (- - -) calculations using Set II


bottom.


conclusion of Section 4.1 that calculations byCacciatore and Billing substantially overpredictthe multi-quantum V±V rates among the highvibrational quantum numbers. Note that recentsemiclassical calculations by Coletti and Billing[43], where a more accurate potential energy sur-face has been used, also predict somewhat lowervalues of the V±V rates.

Fig. 15 displays the time-dependent VDFs forthe conditions of Fig. 12, i.e. relaxation atPCO� 3.5 Torr (solid lines). It shows that therelaxation occurs through a succession of distri-butions with long plateaus, known to be domi-nated by near-resonance V±V exchange [4,5].Another well-pronounced feature is the develop-ment of a total population inversion at a latestage of the relaxation process, for v > 10, cre-ated by the di�usion of the vibrationally excitedmolecules out of the beam region (compare withthe steady-state VDFs at large distances from thebeam center shown in Fig. 9). Fig. 15 also showsthat the calculated vibrational level populationsdrop by a factor of 20±100 during the relaxation,

i.e. that the experimental step-scan data cover asubstantial portion of the entire relaxation pro-cess. Dashed lines in Fig. 15 show the vibrationaldistributions calculated assuming a spatially uni-form excited region and neglecting di�usion. Thelatter calculation starts from a VDF that is closeto the line-of-sight integrated VDF predicted bythe model incorporating nonuniformity and dif-fusion, and the V±V rates are the same in bothcases. Qualitative di�erence between these twocalculations can be easily seen. Relaxation in anonuniform region occurs much faster because ofthe higher vibrational populations near the beamaxis (see discussion in Section 3 and Fig. 9). Thevibrational level populations calculated withoutnonuniformity and di�usion drop by only a fac-tor of 4 and 5, which is inconsistent with thestep-scan relaxation spectra (Fig. 12). Also, inthis calculation the total population inversiondoes not form. This clearly shows the importanceof taking into account spatial nonuniformity ofthe excited region as well as the transport pro-cesses.

Parametric calculations using the rates of Set I,where we replaced the V±E transfer (7) by theasymmetric V±V exchange (6) showed that boththe VDF for v < 40 and the synthetic spectrashown in Figs. 12 and 13 are primarily in¯uencedby the V±V rates, and only weakly sensitive to therates of other energy transfer processes. Finally,varying the adjustable parameters for Set I shownin Appendix A showed that their values chosen inRef. [6] provide the most consistent agreementwith the step-scan relaxation spectra forPCO� 0.5±3.5 Torr.

The time-resolved measurements discussed inthis section are made at translational temperaturesfairly close to room temperature. Indeed, fromFig. 16, which shows the temperature change atthe beam axis during the excitation and relaxation,one can see that the maximum temperature rise atPCO� 3.5 Torr is about 100 K. However, goodagreement between the steady-state data and themodeling calculations for v < 30 in the tempera-ture range T� 400±730 K (Fig. 8) shows that theSet I rate parametrization is also applicable atthese temperatures. As has been discussed in Sec-tion 4.1, the di�erence between the calculated and

Fig. 15. Vibrational distribution functions during the relax-

ation at PCO � 3:5 Torr. Time slices shown in the legend are

from top to bottom.


the experimental VDFÕs at v > 30 in Fig. 8 is dueto the e�ect of the processes unrelated to the V±Vexchange.

5. Summary

In the present paper, time-resolved step-scanFTIR of optically pumped CO is used to study thekinetics of vibrational energy transfer processes(primarily V±V exchange) in strongly vibrationallynonequilibrium CO. Time evolution of all vibra-tional states populated by the V±V processes up toextreme vibrational disequilibrium is monitoredsimultaneously. The V±V rates are inferred fromthese data using a kinetic model that incorporatesthe pump laser spatial power pro®le and transportprocesses, and allows incorporation of multi-quantum V±V processes. The model predictionsagree well with the step-scan relaxation spectra.Comparison of calculations using two di�erentsets of V±V rates with experimental spectrashowed that the use of the V±V rates of Set Iprovides better agreement with experiment. It is

also shown that the multi-quantum V±V rates ofSet II (i.e. analytic parametrization of semiclassicaltrajectory calculations by Cacciatore and Billing)are overpredicted at high vibrational quantumnumbers.

These results provide new insight intononequilibrium vibrational kinetics, and alsodemonstrate the capabilities of the time-resolvedstep-scan FT spectroscopy for studies of molecularenergy transfer processes and validation of theo-retical rate models.

Acknowledgements

This work was supported by the AFOSR SpacePropulsion and Power Program, Grant No.F49620-96-1-0184. Partial support from the OhioBoard of Regents Investment Fund is grate-fully acknowledged. The Department of Com-merce SABIT program supported the visit ofMr. A. Chernukho to Ohio State. The consulta-tion and assistance of Dr. V. Borodin, Prof. Y.Ionikh, Prof. W. Urban, Prof. V. Subramaniam,and Prof. S. Zhdanok, are gratefully acknowl-edged.

Appendix A

A.1. DeLeon±Rich single-quantum V±V rate para-metrization (Set I) [6,18]

Q�v;wÿ 1! vÿ 1;w� � Z�Sv;vÿ1wÿ1;w � Lv;vÿ1

wÿ1;w�eÿDE=2T ;

DE � 2xexe�vÿ w�;

Sv;vÿ1wÿ1;w � S�T � v

1ÿ xevw

1ÿ xewF �kv;vÿ1

wÿ1;w�;

F �k� � 1

2�3ÿ eÿ2k=3�eÿ2k=3;

kv;vÿ1wÿ1;w � 2ÿ3=2

��cT

rDEj j;

Lv;vÿ1wÿ1;w � L�T � gv;vÿ1

g1;0

� �2 gwÿ1;w

g1;0

� �2

exp

�ÿ DE2

bT

�;

Fig. 16. Translational temperature at the beam axis.


gv;vÿ1

g1;0

� �2

� a� 1

a� 3ÿ 2v

� �2

� v�a� 2ÿ 2v��a� 4ÿ 2v�a�a� 3ÿ v� ; a � 1=xe:

Gas-kinetic collision frequency, Z � 3� 10ÿ10

�T =300�1=2cm3 sÿ1.

Parameters: short-range interaction ± S�T � �4:93� 10ÿ4 �T =300� cm3 sÿ1, c � 0:456 Kÿ1; andlong-range interaction ± L�T � � 5:37� 10ÿ3

�300=T � cm3 sÿ1, b � 40:36 K.

A.2. FHO parametrization of Cacciatore±BillingV±V rates (Set II) [20,24]

Q�v;wÿ k ! vÿ k;w�

�Sv;vÿk

wÿk;w

� �k

k! 1� 32

Sv;vÿkwÿk;w

k�1

� �k�1

�Lv;vÿk

wÿk;w

� �k

k! 1� 32

Lv;vÿkwÿk;w

k�1

� �k�1;

Sv;vÿkwÿk;w � S�T � v!

�vÿ k�!w!

�wÿ k�!� �1=k

F �kv;vÿkwÿk;w�;

F �k� � 12�3ÿ eÿ2k=3�eÿ2k=3;

kv;vÿkwÿk;w � 2ÿ3=2

��cT

rDEj jk

;

Lv;vÿkwÿk;w � L�T � v!

�vÿ k�!w!

�wÿ k�!� �1=k

� exp

ÿ �DE=k�2

kb��Tp

!;

DE � 2kxexe�vÿ w�:

Parameters: short-range interaction ± S�T � �8:73� 10ÿ4 �T=300� cm3 sÿ1, c � 0:475 Kÿ1; andlong-range interaction ± L�T � � 3:36� 10ÿ3

�300=T �1=2cm3 sÿ1, b � 1386 K3=2.

A.3. Asymmetric V±V rate parametrization (usedwith both V±V rate sets) [24]

Q�0;w! 1;wÿ 2� � Z�S1;0wÿ2;w � L1;0

wÿ2;w�xew�wÿ 1�

2;

S1;0wÿ2;w � S�T �F �k1;0

wÿ2;w�;

L1;0wÿ2;w � L�T �exp

�ÿ DE2

b��Tp

�:

S�T �; L�T �; c; b, and F �k� are the same as inAppendix A.2.

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