Vibrational and Rotational Relaxation of Iodine
The Journal of Physical Chemistty, Vol. 83, No. 8, 1979
947
Vibrational and Rotational Relaxation of Iodine in Seeded Supersonic Beams G. ,M, McClelland,t K. L. Saenger, J. J. Valentinl,$ and D. R. Herschbach” Department of Chemistry, Harvard University, Cambridge, Massachusetts 02 138 (Received August 22, 1978) Publication costs assisted by the National Science Foundation
Laser-induced fluorescence has been used to measure the vibration-rotation state distributions of molecular iodine in seeded beams expanded from a supersonic nozzle. Four monatomic, four diatomic,and eight polyatomic gases were used as diluents. For an 0.017-cm nozzle operated at 300 K and total pressures from 30 to 1000 torr, the observed vibration-rotation distributions (for u” = 0-3 and J” 0-50) were always found to have a Boltzmann form, characterized by temperatures of TvlbN 300-45 K and Trot 150-3 K. The functional dependence of Troton the total source pressure is governed by the specific heat ratio for the diluent gas and nearly coincides with a simple model for the terminal translational temperature of the expansion. The observed Tdb data are shown to require cross sections for vibrational relaxation which remain constant or increase with decreasing temperature. This cannot be reconciled with the usual impulsive Landau-Teller model; at low temperatures the discrepancy is several orders of magnitude. Relative values for the vibrational relaxation cross sections for different diluent gases were determined by using a series of binary gas mixtures composed of approximately 700 torr of helium and 0-50 torr of another diluent gas. The results show the relaxation cross sections vary by more than a factor of 300 and increase markedly in the order atoms < diatomic molecules < polyatomic molecules. The very efficient vibrational relaxation may be a consequence of the low relative translational energy (typically, T,,,, = 30 K) achieved in the expansion. This fosters formation of metastable or van der Waals complexes, allowing multiple 12-M collisions which can strongly enhance the vibrational relaxat ion.
Introduction In the closing sentence of his book,l Don Bunker wrote: “The field of recombination reactions and allied phecontinue, l by its unpredictability, to lend color nomena ...wil to the study of elementary reaction rates”. This statement, typically both whimsical and accurate, epitomized his Ph.D. thesis research on the classic problem of iodine atom recombinationa2Among several themes he developed from this work was a close study of termolecular collision mechanics and intramolecular energy transfer3 which soon led him to undertake his trajectory calculations for unimolecular and bimolecular reaction^.^^^ The experiments reported in this paper were also originally motivated b:y the iodine atom recombination problem. Our plan was to dissociate molecular iodine a t high pressure and temperature (-800 torr and 1700 K) and expand the vapor through a supersonic nozzle which would foster recombination of the iodine atoms. According to theoretical estimates, the number of collisions experienced during the supersonic expansion should be far too small to allow full vibration-rotation relaxation of the newly formed molecules.6 By measuring the vibrationrotation distribution of the recombinant diatoms, we hoped also to elucidate unresolved questions pertaining to the inverse process of molecular diss~ciation.~ However, as in similar experiments with supersonic alkali beams,8z9 in fact we found no high vibrational excitation. This led us to study instead the “allied phenomena” of collisional relaxation of molecular iodine in seeded beams. The experimental method is laser-induced fluorescence,1° which is used to measure the vibration-rotation state distribution of I2 in its ground X’Zg+electronic state by inducing and observing emission from the B3110+u excited state. Two kinds of relaxation studies are described
-
‘Department of Chemistry, Stanford University, Stanford, CA 94305.
University of California, Los Alamos Scientific Laboratory, Los Alamos, NM 87545. 0022-365417912083-0947$0 1.oo/o
here: (1) In experiments with I2 seeded in a diluent gas and the nozzle operated a t 300 K and total pressures between 30 and 1000 torr, we find the relaxation produces well-defined vibrational and rotational temperatures that vary markedly with the source pressure and identity of the diluent gas. These data permit Tvb and Trotto be selected within a wide range by suitable choice of experimental conditions. (2) In experiments using binary mixtures as the diluent gas, relative values for the vibrational relaxation cross sections for different diluent species were determined. Thirteen gases were examined, ranging from helium to butane. The relaxation cross sections increase markedly in the order atoms C diatomic molecules < polyatomic molecules; the range of variation in fJ\.ib is more than a factor of 300. There are three recent comparable studies of collisional relaxation in seeded beams. Fitch et al.I1 also used laser-induced fluorescence to determine Tvlband Trotfor I2 seeded in He, H2, or D2 a t extremely high source pressures of -100 atm. Mariella et a1.12and Bennewitz and Buess13 used electric resonance spectroscopy to measure Tvlbfor selected rotational states of alkali halides seeded in several gases a t modest source pressures of 5800 torr. Our more extensive data are consistent with these studies. Comparison with model calculations pertaining to the number of “effective hard sphere collisions” in the supersonic expansion shows that in the region of low Tdbthe experimental data require vibrational relaxation more efficient by several orders of magnitude than the usual impulsive Landau-Teller mechanism. It is suggested that this strongly enhanced relaxation may be associated with the low relative translational energy (typically, T,, 30 K) achieved in the expansion. In this regime of “cold translation”, formation of transient collision complexes or van der Waals molecules can play an important role. In particular, we consider a “quasi-resonant complex” m0de1.l~ In this, the acceleration due to the weak attractive van der Waals force can foster redistribution of the initial rotational and translational energy. The “cold” collision
-
0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
948
('::\.,
FLUORESCENCE COLLECTIQN LASER
ILLBEAM -4
1
DYE LASER
-
"
AVERffiERS 1%-I
8)
' q Q F R U h ' GATED INTEGRATOR
REGION
M
JyLR
A
LASER MONITOR
I
!]
I
'G
FLUORESCENCE DETAIL
L
McClelland et al.
Tk Ls,
?PRESSURE
' G + E N T I,
RESERVOIR
RESERVOIR
FHOTOMULTIKIER INTEGRATOR DIVIDER
O - ~ ~ E L E R
Figure 1. Plan view (not to scale) of molecular beam apparatus, including components for laser-induced fluorescence and gas handling system. Needle values labeled A, B, and C are used in procedures for introducing diluent gas as described in text.
complex often will not have enough energy in the separation coordinate to permit it to dissociate. The complex must then undergo further "stuttering" encounters before enough energy is reshuffled into a separation coordinate to permit decomposition. Calculations based on a statistical model show that this mechanism can substantially enhance the relaxation probability. Likewise, formation of van der Waals molecules provides another efficient relaxation mechanism.15
Apparatus and Procedure Figure 1 gives a schematic view of the apparatus. The main components include a gas handling system, a supersonic nozzle to produce a free jet expansion into a vacuum chamber, a tunable dye laser to excite I2 molecules, and a photomultiplier and gated electronics to detect fluorescence. Molecular B e a m Production. The gas line which conveys mixtures of I, and diluent gases to the nozzle is made of monel and stainless steel and operated at room temperature. For experiments using a single diluent gas, the diluent pressure is maintained at about 1100 torr behind a needle valve (labeled A in Figure 1). This valve is adjusted to provide the desired nozzle pressure in the range 10-800 torr, as measured by two mechanical gauges (one for pressures below 400 torr, the other above). On its way to the nozzle, the diluent gas is sent through a column of crystalline iodine, 1.75 cm in diameter and 20 cm long. The diluent thus emerges nearly saturated with I2 at its vapor pressure of 0.3 torr at 300 K. Experiments using binary mixtures as the diluent gas require a more elaborate procedure. A stream of He is sent through the iodine column and the other component of the diluent is maintained at 1800 torr behind another needle valve (C in Figure 1). To produce the gas mixture, the flow of He + I2 is first shut off (by closing valve B) and the flow of the other diluent component is adjusted (by setting valve C) to give a suitable pressure in the nozzle. Then valve B is opened to add He + Iz to the flow until the total nozzle pressure reaches 760 torr (with valve A left wide open). Since the pressures on either side of valve C always differ by considerably more than a factor of 2, the valve operates in a regime where the mass flow rate of the other diluent component is unchanged by opening valve B. The partial pressure does change, however; the somewhat elaborate calibration procedure required to evaluate this is described in Appendix A.
The supersonic nozzle consists of an 0.017 f 0.002 cm diameter hole drilled through a 0.020 cm thick nickel disk which is welded on the end of a 0.47 cm diameter nickel tube. A stainless steel collar is soldered on the tube to hold tantalum heating wires in ceramic inserts. The nozzle temperature is measured with a chromel-alumel thermocouple spot-welded to the end of the tube. The vacuum chamber containing the nozzle is evacuated by a baffled 6-in. diffusion pump and a 10-in. pump in a contiguous chamber; the total pumping speed at the nozzle is -2500 L s-'. With the nozzle flow off, the background pressure in the vacuum chamber is -5 X lo* torr whereas it rises torr when the nozzle is operated at -800 torr, to -3 X the highest source pressure used. Laser Excitation and Fluorescence Detection. The dye laser employed to excite the I2 fluorescence is of the Hansch type.16 The bandwidth is narrowed to 51 cm-* by use of a telescope (but no etalon). The wavelength is scanned by rotating the grating by means of a stepping motor. A 60-kW N2laser17pumps the dye laser to produce 10-ns pulses at 10 Hz. Coumarin 495 dye is used for the range 17 800-18 160 cm-l at a bandwidth of 0.4 cm-l and rhodoamine B is used for 16 260-16 480 cm-l at a 0.9-cm-l bandwidth. The laser intensity is monitored by diverting about 5% of the beam via a beam splitter to a photodiode. The main beam passes through a lens and enters and exits the vacuum chamber through windows mounted on long sidearms which are baffled to reduce the scattered light reaching the detector.ls The lens collimates the laser beam to a diameter of 0.2 cm at its intersection with the molecular beam, 1.4 cm in front of the nozzle. An EM1 9658 photomultiplier views this intersection region at right angles to both the laser beam and the molecular beam. Fluorescence from the intersection region is focused onto the photomultiplier by a lens 6.3 cm in diameter and located 6.7 cm above the plane of the crossed beams; this lens collects light from a zone in the beam plane which is 1.4 cm in diameter. The electronics for the detection system consists of analog modules timed by TTL integrated circuits. The photomultiplier output is integrated from 0.35 to 4.0 ps after the laser pulse by an Optical Electronics 9081 gated integrator triggered by scattered light from the N2 laser which is monitored by a photodiode. The delay of 0.35 ps avoids detection of scattered light but does not appreciably affect detection of the fluorescence from 1, molecules (with a radiative lifetime of 0.6-1.1 ps).19 Both the integrated fluorescence intensity and the integrated signal from the photodiode monitoring the laser intensity are averaged by low pass filters with 1.0-stime constants. The fluorescence intensity is then normalized to the laser intensity by an analog module, to compensate for fluctuations in laser power, and this intensity ratio is displayed on a strip-chart recorder. The primary source of noise is wavelength jitter of the dye laser, although photon shot noise becomes important at the lowest signal levels. The fluorescence signals ranged from a maximum of lo5 photoelectrons per laser pulse (for the 23-0 transition of I, seeded in He a t 800 torr) to 10 photoelectrons per laser pulse (for high u" levels at low vibrational temperatures). Recombination Experiments. For the unsuccessful recombination experiments a quite different beam source was used, similar to a halogen atom source described elsewhere.20 The nozzle was a resistively heated graphite tube with a 0.025 cm diameter X 0.031 cm long orifice. The graphite tube was heated to 150C~2000K to dissociate pure 12, at 660-1140 torr pressure, or a mixture of 1 2 (18-100 torr) and argon (160-230 torr). For these experiments the
She Journal of Physical Chemistry, Voi. 83, No. 8, 1979
Vibrational and Rotational Relaxation of Iodine
TABLE I: Vibronic Transitions Used in LIF Analysisa
100 I
CO
P-760 Torr
___
~-
(VI-v”
75 I
sample T, determlnatlon
He P-900 Torr
23-0 20-0 25-1 22-1 8-1 7-1 10-2 9-2 12-3 11-3
N
h
3 0I
178565cm-’
181554cm
’
949
163597cm-’
164693crn-’
Flgure 2. Excitation spectrum of the B31To+,-X’Z~ system of 12. The ordinate gives the fluorescence intensity and the abscissa the laser frequency. The lower two spectra are fast scans taken with He as the diluent, using as laser dyes coumarin 495 (left) and rhodamine B (right). The vibronic transitions are labeled as V’-V’’. The inset shows a slow scan of the 23-0 transition, with CO as the diluent. The rotational temperature is derived as described in the text from A€:’ and the excitation energies at 30 and 75% of peak intensity.
iodine source was contained in the auxiliary chamber attached to the main chamber of the fluorescence apparatus (see Figure 1). This auxiliary chamber was isolated from the main chamber, in which the laser-induced fluorescence is Observed, by a removable bulkhead. A stainless steel or graphite skimmer (0.081 cm diameter orifice) was installed in the bulkhead to define the iodine beam. The auxiliary chamber was pumped by a 4300 L/s diffusion pump and a !13 cm diameter by 58 cm long liquid nitrogen cryogenic shroud. The nozzle was placed approximately 13 cm from the fluorescence collection region. Elaborate baffling in the main vacuum chamber protected the photomultiplier from light radiating through the skimmer. In addition to the coumarin 495 and rhodamine B dyes used in the relaxation experiments, rhodamine 6G + cresyl violet perchlorate, rhodamine B + nile blue A perchlorate, and IR 125 (Eastman) were used to extend the spectral region. This allowed us to scan from approximately 10 700 to 10900 cm-l and from 14 100 to 19400 cm-l in searching for transitions involving high vibrational excitation.
Results and Analysis Figure 2 shows typical laser-induced fluorescence spectra obtained in the relaxation experiments. The lower two spectra show a series of vibronic bands obtained with a single “fast” scan (100 s), while the inset shows a “slow” scan (also 100 8 ) acrow a single band, the 23-0 transition. The laser bandwidth is about 0.4 cm-l, allowing resolution of the high J , but not low J , transitions within the vibronic band. In all cases a single scan through each spectrum gave adequate S/N ratio for data analysis. However, the spectra were often repeated to check for temporal drift in the fluorescence and to ensure reproducibility of the spectra. Reproducibility was found to be within statistical uncertainty, and no temporal drift (over times long enough to record several spectra) was ever observed. The vibronic transitions used in analysis of the rotational and vibrational temperatures are labeled (u ’u“, with u the vibrational level of the electronically excited B state, u” that of the ground X state). The relative intensities are given in Table I. Rotational Temperature Determination. Iodine rotational temperatures for these I,-diluent gas expansions were determined by analysis of the intensity profile of individual vibronic bands, as illustrated in the inset of
)
I(
VI-vi’
)
(1.000) 0.828 0.313 0.463 0.0129 0.00432 0.0565 0.0342 0.0760 0.0670
dye C C C C
R R R R
R R
a Normalized fluorescence intensities I( v ’ - v ’ ’ ) are listed relative t o the 23-0 transition; all pertain t o the reference temperature of 268 K. The laser dye employed is indicated b y C (coumarin 4 9 5 ) or R (rhodamine B).
Figure 2. The method employs the energy shifts AEr from the band head at the 75 and 30% intensity points on the smoothed intensity profile. According to a classical approximationz1 for the rotational profile
I(AEr)= explEB”/(B”- B?l(AEr/h~Trot))
(1)
This is valid for small, dissimilar values of the rotational constant for the ground (B’? and excited (B? states. The 75 and 30% points thus directly yield the rotational temperature via Trot
=
(AE,3O” - AEr75W)(B”/(B”B?J/(hBIn (0.75/0.30)) (2) The accuracy of this procedure was checked by applying it to simulated iodine laser-induced fluorescence spectra calculated by convoluting the laser bandwidth profile and electronics time constant over all P and R branch lines weighted by the proper Boltzmann factor. A J-dependent predissociation correctionlg was also included. For low rotational temperatures (a few degrees Kelvin), the temperature extracted from the simulated spectra using eq 2 was within 10% of the temperature used to compute the spectrum, while for higher rotational temperatures the agreement was better, within 6%. Although eq 2 will yield a value of Troteven for a non-Boltzmann rotational state distribution, visual inspection of the rotational distributions obtained in these experiments never revealed any obvious non-Boltzmann behavior. In some cases as many as 12 vibronic band profiles were analyzed to determine Trotfor a specific diluent gas at a given temperature and source pressure, while in other cases only a single vibronic transition was examined. No systematic deviations were found in the Tmtvalues extracted from different bands and the agreement was typically f20%. The reproducibility of Trotdetermined from replicate measurements of a given vibronic band was usually better than 10%. The final data were obtained chiefly from the 23-0 and 20-0 bands, since these offered the most intensity. Figure 3 summarizes the rotational temperature behavior as a function of nozzle pressure, Po. The lines shown represent linear least-squares regression fits to the data for each diluent gas. Actual data points are not shown to avoid congestion a t the many places at which the linear plots cross. However, the linear fits to the data are quite good. For all diluent gases except C4H10and CzF6 the coefficient of determination for the regression analysis was greater than 0.95. C4H10and CzF, had coefficients of determination of 0.71 and 0.32, respectively. The lines are characterized by two parameters which are listed in Table 11. These are the slope, S* = d(log TrOt)/d(logE‘&), and the value of Tmt*,the rotational temperature corresponding
950
McClelland et al.
The Journal of Physical Chemistry, Vol. 83,No. 8, 7979
TABLE 11: Parameters for Rotational Relaxationa diluent S*(measd) S*(calcd) He Ne Ar
-1.17 t 0.04 - 0 . 8 7 t 0.01 - 0 . 7 9 t 0.02
H2
co CH, C*H,
C,H,
Ttra,(calcd)
*
- 0.80 - 0.80
0.25
-0.80
26.8 2.4 5.7 t 0.1 3.07 t 0.07
-1.14 0.02 - 0 . 7 7 9 t 0.001 - 0 . 5 0 t 0.06 - 0 . 5 0 + 0.04
-0.58 - 0.57 - 0.57 -0.57
24.5 t 0.8 11.39 i 0.03 16.2 t 2.3 16.7 + 1.1
0.21 2.1
-0.46 t -0.36 t -0.34t -0.27 i -0.42 t -0.04 t
- 0.47
23.4 34.4 50.9 66.1 24.1 58.2
*
D2 N2
T*,,t(measd)
0.07 0.02 0.04 0.12 0.03 0.04
-0.32 -0.23 - 0.17 -0.27 -0.16
t 1.6
t 0.8 t 3.5
1.3 1.6
10 10 13 23 24 39 14
n
rz
21 25 37
5 5 6
0.980 0.999 0.997
28 27 43 40
3 3 4 7
1.000 1.000 0.968 0.974
47
4 6 5 4 4 4
0.958 0.991 0.963 0.707 0.990 0.317
U,
8'
61 80
n-C4H10 + 20 79 k 1.0 69 CF, C'F, + 4.8 a Lines fitted to experimental data (Figure 3) are characterized by a slope, S*(measd) = ( d log Tr,t/d log P O D )and a temperature parameter, T*,t(meas), t h e value of Trot a t P O D= 1 8 6 torr (corresponding t o log P O D= 0.5). The uncertainties listed indicate i one standard deviation. Also given are values of S*(calcd) calculated from a theoretical model for t h e supersonic expansion and literature values o f the specific heat ratios a t 300 K.51The quantity T*tra,(calcd) is the theoretical prediction for t h e terminal translational temperature (also evaluated a t log P O D= 0.5) based o n an E value of unity and y values derived from S*(measd). All temperatures are in Kelvin. The quantity u denotes t h e collision cross section used in computing T*trans(calcd); it is simply n d 2 ,with d the van der Waals radius for t h e diluent molecule as estimated f r o m t h e Lennard-Jones potential.'" The number o f (Trot, logP,D) points for each gas is denoted by n, and t h e quality of t h e linear fit to t h e data is indicated by r 2 , the "coefficient of determination". Pressure, Torr
TABLE 111: Functions Involved in Mach Number Formulasa
257
12,
513 715 1.30 917 1.20 1.10 1.05
2.0-
1.5-
2
a
i 0.5
1
O 0 -10
-0.5
00 0 5 1.0 log,,(PD, crn-torr)
1.5
Flgure 3. Experimental results for rotational relaxation of I, seeded in various diluent gases. Lines show observed variation of the rotational temperature of I, with product of nozzle diameter and source pressure (as a log-log plot). Data points are omitted for clarity; a typical error bar (A one standard deviation) is shown at the lower left. Derived quantities are given in Table 11.
to log P a = 0.5 (or Po = 186 torr, since D = 0.017 cm). Table I1 also gives values of the slope parameter S* derived from an approximate theoretical treatment of the supersonic expansion.22 We expect that the rotational relaxation is so efficient that Trotwill not differ much from the terminal translational temperature T,,,,, attained in the expansion. This terminal temperature is related to the nozzle temperature To,specific heat ratio y = C,/C,, and terminal Mach number by The approximate treatment predicts
Ma, = F(y)(Kno/e)-(T-')/Y
(4)
where Kno is the Knudsen number (ratio of mean free path
3.26 3.65 3.90 3.96 4.29 5.25 6.44
2.03 2.48 2.80 2.85 3.38 4.75 7.04
See eq 5, 19, and 2 1 of the text.
to nozzle diameter) at the nozzle throat. The quantity denotes the maximum fractional change in the mean random velocity per collision. The function F(y) appears not to be available in the literature but can be worked out from formulas given by Anderson and Fenn.22 Thus F(y) = 2'Y-')/Yy([~y(y- 1)](7-')/7[72(y- 1)A2(y)]'/T]-'/' (5) where the function A(y) was first defined by Ashkenas and Sherman.23 Numerical values are given in Table 111. Although F(y) is often replaced by a constant in treatments of supersonic expansions, for our application this factor has an important role; it varies by more than a factor of 3 over the range of interest. Thus, for large Ma,, we obtain log Ttrans= log [2To/(y - 1)1[2(y - l ) / ~log l [ P f l ~ ~ / k ~ T-o2I log F(T) (6) where To is the nozzle temperature, Po is the source pressure, and D is the nozzle diameter. The quantity u is the hard sphere collision cross section for the diluent-diluent interaction. If Trotand Ttransare assumed to differ by no more than a multiplicative constant throughout the expansion, the slope parameter according to this model is given simply by S" = -2(y - l ) / y (7) This predicts a slope of -0.80 for y = 513 (i.e., a monatomic diluent, three translational degrees of freedom), and -0.56 for y = 7/5 (diatomic rotor with three translational and two rotational degrees of freedom). As seen in Table 11, the agreement between predicted and observed behavior is quite good for Ne, Ar, N2, and CO, while significant
Vibrational and Rotational Relaxation of Iodine
deviations occur for He, Hz,and D2 Agreement between predicted and observed behavior is also good for the polyatomic alkane molecules, although for the heavier alkanes and CF4 the observed slopes suggest that the predicted slopes, S*(calcd), based on room temperature y values, might be better calculated from y’s measured at lower temperatures. The one major deviation occurs for C,F,; the observed slope of -0.04 is too small to be reconciled with the model, which predicts S* = -0.156. However, the overall agreement for the S* parameter must be regarded as rather good for such an extremely simple model for rotational relaxation. Table I1 also includes values of the terminal transrational temperature T*trans(calcd)corresponding to log P& = 0.5, as calculated from eq 6 using collision cross sections derived from the Lennard-Jones potential parameters for the diluent gases,%y values derived from the experimental S*, and an E value of unity. A value of E can be obtained if one knows the terminal translational temperature: E = [T*tIan,(calcdwith E = 1)/T*,,,,,(measd)](’i2)yi(y-’) Anderson and F e n d 2 have found that t is about 0.25 for an argon expansion. If TI, N Tkm,E values obtained from the above expression with T*,Jmeasd) replacing T*,Ians(measd) might be expected to be around 0.25. Indeed this is the case. With the exception of the light gases Hzand He, which had E of the order of 0.02, our E values were all in the range 0.1-0.5. Thus for these heavier gases, our data are consistent with facile rotationaltranslational transfer. Similar results have been found in electric deflection studies of rotational relaxation of alkali halides seeded in rare gases.25 It should perhaps be noted also that as yet the approximations involved in eq 6 have not been thoroughly tested. The agreement between the rotational temperature and the calculated terminal translational temperature hence offers favorable evidence for the theoretical treatment22of the transition between continuum and molecular flow in seeded supersonic expansions. Vibrational Temperature Determination. The evaluation of Tvlbinvolves, comparison of intensities for several u’-u’’ vibronic bands. The method requires some means to eliminate the dependence on the vibrational level u’of the excited electronic state in order to extract a reduced signal proportional to the population N(u’? of level uf’ of the ground electronic state. The total laser-induced fluorescence intensity for a vibronic band can be written as8 I(u’,u ’9 = N(u’?IL(XL)IA(u ’,u’?IF(u (8) where IL(XL) is the laser intensity at the absorption wavelength, IA(u’u’? is the transition strength, and IF(u9 is the fraction of the absorbed light detected as fluorescence. This fraction is a function of the Franck-Condon factor for emission, the wavelength dependent efficiency of the photomultiplier, and the excited state lifetime relative to the gated light collection time. The laser intensity I L ( A L ) is eliminated from the observed fluorescence signals by the divider in the detection electronics. The IA(u’,u’? and IF(u? factors can be calculated but it is simpler and more accurate to eliminate them by comparison with a reference spectrum recorded at some known vibrational tem perature, TIef.Thus, the analysis employs a reduced intensity R(u”,Tref) I(u’,u ‘?/Iref(U’,u’? = N(u’?/Nref(u’? (9) If the vibrational populations have a Boltzmann distribution, a plot of log R(u’? or log N(u’? vs. the vibrational energy, E(u’?, should be linear. The slope, m =
The Journal of Physical Chemistty, Vol. 83, No. 8, 1979
951
Vlbrational quantum number, v ” I 2 3 I
-2
1
0
Argon
I
11 -I
P = 270 Torr
400 600 Vibrational Energy, cm-’
200
4
000
Figure 4. Determination of vibrational temperature. Ordinate scale (log) gives ratio of observed fluorescence intensity I( v’, v”) to that for reference system (Ar diluent at T = 268 K); as discussed in text, this is equivalent to the corresponding ratio of populations, N( v”), in the ground electronic state. The lower abscissa scale gives the vibrational energy (above zero-point level), /+,Eo; the upper scale gives the corresponding v”quantum number. Each data point is labeled by the vibrational quantum number v‘of the excited electronic state. The slope of the solid line gives the best estimate of the vibrational temperature, according to eq 10; slopes of the dashed lines provide values used as limits for the error bars shown in Figures 5 and 6.
d log R(u’?/dE(u’? is related to the vibrational temperature Tvlbcharacterizing the N(u’? vibrational state distribution by Tvib = (1/TIef- 2.303hBm)-l (10) where hB is the Boltzmann constant. As written, eq 8 gives the laser-induced fluorescence intensity integrated over the rotational profile of the u’-v’’ band. However, since we found that for a specific diluent gas at a given pressure different vibronic bands have the same rotational temperature within experimental uncertainty, there appears to be no discernible coupling of the rotational and vibrational populations. The peak intensities at the rotationally unresolved band origins thus can be used for I(u’,u’? in eq 8 and 9. Using these peak intensities rather than integrating over the rotational profiles of the bands is a more accurate method of determining the relative vibrational populations whenever the rotational temperature is high enough to produce considerable overlapping of the bands. The relative populations of vibrational states u” = 0 to u” = 3 were determined for each diluent gas at several pressures by analyzing the laser excitation spectra of the B-X vibronic transitions listed in Table I. Plots of log R(u’? vs. E(u’? for three experimental runs are shown in Figure 4. Vibrational temperatures calculated from the slopes of these lines via eq 10 are given to the right of each plot in boldface characters. The values shown in lightface above and below represent maximum and minimum vibrational temperatures compatible with the data. These maximum and minimum values were determined from the slopes of lines (shown dashed) passing through log R = 0 at u” = 0 and the data points which have respectively the largest positive and negative deviations from the best-fit line (shown solid). In determining the “best-fit’’ line through the log R data some points were weighted more than others, in accord with our relative confidence in the data. At relatively high
952
McClelland et al.
The Journal of Physical Chemistty, Vol. 83, No. 8, 1979
Pressure, Torr 1000
100
350
It 1
600
I
300Y
p 250e
550-
2
500-
I-
Y
I
P
-
@ 450-
s
E ij400-c
200:
150-
-
+
5 300 ‘2501
Ii0l
200
too
:
“
0
5 -10
-05
00
05
IO
15
tog. (PD, cm-torr)
Figure 5. Experimental results for vibrational relaxation of 1, seeded in Ar, N, and n-butane. Ordinate gives 1, vibrational temperature (best estimates derived as illustrated in Figure 4), abscissa gives the log of product of nozzle diameter and pressure. All data pertain to nozzle temperature of 300 K except for indicated points for Ar and N2 which pertain to 600 K. Typical error bars are shown whenever these bars exceeded the size of the symbol used to plot the point.
rotational temperatures the (u’-1) and (u’-2) bands were considerably overlapped, decreasing the accuracy with which I(u’,l) and I(u’,2) could be determined, so I(u’,O) and I(u’,3) were weighted more in determining the slopes. At very low vibrational temperatures, the observed intensities I(u’,2) and I(u’,3) were due almost entirely to ambient room temperature background in the vacuum chamber, so in this regime the vibrational temperature was determined solely from the I(u’,O) and I(u’,l) data. As indicated in Figure 4, the reference spectrum Iref(u’,u’? pertains to Tref= 268 K. This reference spectrum was obtained from a nozzle expansion of an Ar/12 mixture at 38 torr. For this system the rotational cooling was sufficient to allow precise measurement of Iref(u’,u’? for all the transitions given in Table I, yet the vibrational cooling was slight and hence the vibrational populations remain near their room temperature values. The value of Trefwas determined by comparing the ratio I(23,0)/1(11,3) for a series of expansions with decreasing argon pressure, corresponding to T. approaching 298 K. The 1(23,0) and 1(11,3) band intensities were used here because they remained relatively free of overlap from other bands as the rotational temperature approached 298 K (when the beam source pressure Po 0). The reference was determined in this manner to be Tref= 268 f 8 K. A less accurate estimate was derived from a comparison of the 1(23,0)/ 1(11,3) ratio for the expansion with Ar at 38 torr, with the same ratio for a “bulb” spectrum at 298 K. This gave Tref = 277 K, only 4% higher. The low pressure expansion spectrum was a superior choice for Iref(v’,u’’), since it could be determined accurately for all the transitions listed in Table I, whereas for the bulb spectrum this was clearly not
-
1
1
-
T 1.5
0 log ,c (PD, cm-torr)
Figure 7. Experimental results for vibrational relaxation of I, seeded in CH,, CzH6, C3H6, C(CH,), CF,, and C,Hs. Data pertain to 300 K. Format as in Figure 5; error bars (not shown) are similar in magnitude.
possible due to severe band overlap in many places. The fractional error in the vibrational temperature derived from eq 10 is given by ATvib/Tvib
(Tvib/Tref)(A.Tref/T,ef)
(I1)
The fractional error in Tvibthus decreases as Tvibdecreases, and auy error in determining Trefonly becomes troublesome for Tvib > TreFThe reference distribution, Nre&u’?, and all other measured N(u”)distributions appear to be Boltzmann, since log R(u”)vs. E(u’? plots such as shown in Figure 4 are linear over orders of magnitude variations in R(u’? for all the diluent gases studied. Figures 5-7 show the variation of vibrational temperature with source pressure for the various diluent gases. The curves are merely hand drawn; no analytic functional behavior is assumed. The only “boundary condition” imposed is that the curves smoothly converge to Toas PO 0 (or log P$ -a). The data for nitrogen, argon, and butane diluents serve to indicate the typical behavior. At both nozzle temperatures of 300 and 600 K and all nozzle pressures N2 yields lower vibrational temperatures than Ar. For butane the situation is much less simple; at some nozzle pressures butane appears to yield a lower Tvib than
-
-
The Journal of Physical Chemistty, Vol. 83, No. 8, 1979 953
Vibrational and Rotational Relaxation of Iodine
Nz, a t other pressures a higher Tvib. However, the differences are not very pronounced, being within or nearly within experimental uncertainty a t most points. The pressure dependence of Tvib varies considerably for the other diluent gases studied. Although the vibrational relaxation is clearly riot simple, a few general aspects can be noted. It is useful to evaluate T*vlb, the vibrational temperature at log POD = 0.5 (Po= 186 torr). We find that T*vib ranges from 216 to 240 K for monatomic diluents; from 105 to 165 K for diatomic diluents; is 65 K for the only triatomic diluent CO,; and ranges from 75 to 230 for the polyatomic diluents, with the fluorocarbons giving the highest temperatures. A clear trend for T*,b of monatomic > diatomic > triatomic is evident. However, the polyatomic diluents give a range of T*vlb as wide as the combined range for the monatomic, diatomic, and triatomic species. Relative Vibrational Relaxation Cross Sections. The collisional relaxation of an iodine molecule in a supersonic expansion depends strongly on the heat capacity ratio y of the diluent, which governs the temperature and density profile in the expansion. It is thus not feasible to obtain relative cross sections for vibrational relaxation by merely comparing Tvihat a fixed nozzle pressure for diluents with different y values. However, since many gases are much better relaxers than He, mixing a small amount of such a gas with He can greatly enhance relaxation while leaving y nearly unchanged.ll We have used this technique to obtain relative vibrational relaxation cross sections. In these experiments, Tvib was determined from the relative peak heights of the 23-0 and 25-1 vibronic bands, as a function of the partial pressure of the sample gas mixed with He at 760 torr total pressure. The procedure used to evaluate relative cross sections from this data is derived in Appendix 13. The ratio of vibrational relaxation cross sections of the sample gas and He is given by avib(S)/
gvib(He) = ksNs/pPr
(12)
Here Ns is the number of S + I2hard sphere collisions and ks is the vibrational relaxation effectiveness. The product k a s is therefore the number of “successful” vibrationally relaxing S + I2 collisions. The denominator contains a factor defined by P = a lrl
[ ~ , ( ~ o ) / ~ v ( ~ l ) I / ~(13) Po
This partial derivative is evaluated for a pure He expansion near 760 torr and contains the ratio of the initial (stagnation) and terminal (measured) vibrational energies of Iz, pertaining to the source temperature To and the observed T I = Tvib,respectively. We measured the value p = 6.9 X torre1. A reduced pressure Pr is defined by Pr
= (m/mH,)1’2(pHe+I,/Ys+12)1’2pO(s)
(14)
where Po(S)is the partial pressure of S in the nozzle, m denotes the average molecular mass of the mixed diluent gas, and p denotes the reduced mass. The first mass factor normalizes to the flow rate for pure He diluent; a large value of f i slows the flow and allows more collisions. Likewise the second factor involving the reduced mass normalizes to the collision rate for He at a given temperature and density. The number of “successful” vibrationally relaxing S I2 collisions is given by KsNs = In [Ev(Ti)/Ev~Tii)] P[(m/mHe)1’2Po(He) - Ptotl (15) where Po(He) is the partial pressure of He in the nozzle. The vibrational energies appearing in eq 13 and 15 are the
+
TABLE IV: Relaxation
diluent
Relative Cross Sections f o r Vibrational
T= 150 K T = Tmh, K
T * ~
He Ne Ar Kr
226 225 219 24 2
(1)
1.1
