Internal energy distributions from nitrogen dioxide fluorescence. 1

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9890

J. Phys. Chem. 1993, 97, 9890-9903

Internal Energy Distributions from Nitrogen Dioxide Fluorescence. 1. Cumulative Sum Method H. S. Johnston,' C. E. Miller, B. Y. Oh, K. 0. Patten, Jr., and W. N. Sisk Department of Chemistry, University of California at Berkeley, and Chemical Sciences Division, Lowrence Berkeley Laboratory. Berkeley, California 94720 Received: November 22. 1991; In Final Form: July 13, 1993'

This article describes a method of obtaining information about the internal energy (E)distribution of a fluorescing population of nitrogen dioxide, NO2*, from its dispersed spectrum between 400 and 840 nm. We show that two fluorescing populations of the same average energy but different energy spread give statistically significant differences in their observed cumulative sum spectra, although the differences are small. Broadly spread distributions of NO2* internal energy are produced by photolysis of RNO2 moleculesand by collisionaldeactivation of monoenergetically excited NO2. The cumulative sum fluorescence spectrum from a broadly distributed internal energy population is represented as a weighted combination of monoenergetically excited cumulative sum fluorescence spectra. A cumulative sum spectrum utilizes all of the data, is positive and single valued, and smoothly, monotonically increases with decreasing observation energy. By differentiation of the cumulative sum spectrum, the original spectrum is recovered undistorted. Unlike a structured monoenergetic fluorescence spectrum, the cumulative sum is well approximated by a simple algebraic expression, Z(E,X), where E is the internal energy of NO2* and X are the photon energies of the observed spectrum. LIF spectra of NO,, which are essentially collision-free by virtue of low pressure and short observation times, are observed at 22 excitation wavelengths between 400 and 673 nm, and these provide the two parameters of I(E,X). A matrix equation relates the observed cumulative sum spectrum of such data to the internal energy distribution, D(Ei), of the associated fluorescing NO2* ensemble. This equation is solved indirectly, representing D(Ei) by trial functions whose parameters are iteratively optimized to the data via a nonlinear least-squares algorithm. From the shape of the derived distribution, we evaluate the average internal energy, the internal energy spread, and the skewness of the population of molecules that fluoresce. We find the method is 20 times less sensitive to spread than to average energy. We test the method against synthetic spectra including realistic amounts of noise, and at the 95% confidence level, the average energy is recovered better than 1%, the spread better than 2092, and the skewness within about a factor of 2.

Introduction In a series of articles, Troe and co-workers used absorption and emission spectroscopy to study energy transfer by collisions involving highly excited polyatomic molecules (see references in article 2 of this series, the following paper in this issue). The following quotation' represents an important aspect of one of their methods: "Ultraviolet absorption spectroscopy of excited polyatomicmolecules provides an accessto the averagevibrational energy (E)of highly excited species. Spectra of microcanonical, canonical, and intermediate ensembles have been shown experimentally, and rationalized theoretically, to coincide nearly completely when their average energies (E) agree. This fact, on the one hand, prevents conclusions on the details of the energy distributions; on the other hand, a method is available which services to measure average vibrational energy nearly independently of the distribution". We propose that our analysis of the cumulative sums of NO2 fluorescencespectra revealsthe small differencesbetween spectra arising from populations with equal average internal energy (E) but with different spread and that these small differences are statistically significant. In essential agreement with the above quotation, we observe that the fluorescence spectra have a strong dependence on the mean of the internal energy distribution and a weak dependence on the spread and on the skewness. By deliberately introducing unrealistically large amounts of noise in testing our method, we confirm that the averagevibrational energy is "nearly independent" of features that can cause the measured spread to be lost in noise. The complexities of the absorption and emission spectra of nitrogen dioxide in the visible region are well documented.2-5 The Abstract published in Aduance ACS Abstracts, September 1, 1993.

purpose of these four articles is to use simple concepts, low resolution, and considerations of only energy in order to derive, test, and use a new method of obtaining information about the internal energy population of highly excited nitrogen dioxide molecules. There are a number of chemically interestingsystems, among them RNO2 photodissociation and the collisional deactivation of monoenergetically prepared NO2 ensembles, which produce broad, a priori undefined internal energy distributions and visible fluorescence continua. Usually, such emission is analyzed by spectroscopically identifying the emitting molecular eigenstates and then using the appropriate Franck-Condon and Honl-London factors to obtain the population of each emitting state. Such an analysis is currently not feasible for NO2 due to the extensive spectral perturbations mentioned above. Furthermore, the ergodic nearest-neighbor spacing statistics exhibited by the NO2 fluorescence excitation spectrum indicate that "for our data sample no good vibrational and electronic quantum numbers exist apart from energy, which could serve to label the vibronic states." 4 In 1986, we reported preliminary explorations of a method to deconvolute multicomponent N02* fluorescence spectra into a linear combination of monoenergetic contributions,6 and the present article supersedes that article in all respects and provides major improvementsof experimental procedures: (i) At constant resolution, the present method has about 400 times better throughput of light, and observed fluorescing molecules have undergone about 0.003-0.01 'hard-sphere" collisions, whereas the older method involved 1-3 such collisions. (ii) Here, the sensitivity of the entire system is calibrated against an absolute emission source, but not before, which explains the difference in empirical fitting functions. (iii) The earlier study employed an uncertain extrapolationof the NO2 fluorescence lifetimefunction

0022-3654/93/2097-9890$04.00/0 @ 1993 American Chemical Society

The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9891

Internal Energy from NO2 Fluorescence Oo80

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Figure 3. An experimental apparatus used for the study is depicted in block diagram.

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Figure 2. The calculated NO2 potential energy level diagram shown as a function of 0-N-O bending angle, adopted from Gillispie et a1.8 The dissociation energy (NO2 NO + 0)is indicated as a line at 25 132 cm-1.

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from 600 to 800 nm. We now have measured NO2 fluorescence lifetimes for excitation wavelengths out to 746 33111.’ Figure 1shows a low-resolution absorption spectrum of nitrogen dioxide in the visible region.3 Figure 2 presents the calculated NO2 potential energy diagrams as a function of 0-N-O bending angle.* Of the electronic states depicted, 2B2 and 2B1 contribute to the NO2* visible emission. Douglas2 suggested that strong nonadiabatic coupling between vibrational levels of appropriate symmetries, but different zero order electronic states, causes the wave function of the fluorescing molecules to have a large componentof the highly vibrationallyexcited 2A1electronic state.

Experimental Section Figure 3 shows a block diagram of the experimental apparatus used for studyingdispersedfluoresame (LIFand PIF) of nitrogen dioxide. The NO2* fluorescence is collected with anf/2 quartz lens and focused onto the input slit of a 1-m monochromator (Interactive Technology Model CT-103), equipped with a large, 1200 lines/” grating blazed at 500 nm and quartz slit lenses. The entrance slit lens focusesthe external collecting lens onto the entrance mirror of the monochromator, and the exit slit lens focuses theoutputmirror of the monochromator onto theexternal lens that focuses the beam on the photomultiplier tube (PMT). The slit lenses permit full use of the 50-mm-high slits of the monochromator and greatly increase the throughput of light, At 2 mm input and output slit widths, the monochromator resolution is about 1.6 nm in the visible. The dispersedfluorescenceis focused onto a PMT (RCA C-31034A), which is cooled to -30 OC to

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Figure4. One exampleof instrumentresponseprofileobtained by placing a hot tungsten strip before the collection optics, dispersing through the monochromator,and detectingwith PMT. By measuringthe temperature of the tungsten lamp and knowing the tungsten emissivity at that temperature! we found the instrument response at each wavelength of observation. Response profiles are used to correct the experimentally observed fluorescence, so that equal signal means an equal number of photons s-l at all wavelengths of a spectrum.

reducedarkcurrent. A homeassembled amplifier (AvantekGPD 461,462, and 463) sends the signal from the PMT to a gated integrator/boxcar averager (SRS 250) for time-resolved signal detection. An IBM PC-AT computer, equipped with an A/D board (Data Translation 2801A), provides data analysis and plotting (ASYST software). The computer system monitors sample pressure and laser power throughout the scan for normalizing each spectrum. It, typically, averages 100 shots at 10-Hz repetition rate at each wavelength of observation and advances the monochromator by 2 nm to the next observation wavelength. A special tungsten lamp,operated at constant controlled current and voltage, is used to obtain the spectral response of the system (Le., net effect of PMT, monochromator grating and spectral sensitivity, transmittance of optics, etc.). An optical pyrometer measures the surface temperature of the large tungsten strip in the lamp, and standard tables give the tungsten emissivity? The system is calibrated in terms of photon s-1. Figure 4 displays one example of instrument response function; each experimental system requires its own calibration. During the LIF studies, the scattered laser spectrum from an empty cell is subtracted from the observed NO2 fluorescence. Matheson NO2 of 99.5% purity is purified by repeated t r a p to-trap distillation with retention of the middle third, which is either stored in a darkened bulb or kept in a trap at liquid nitrogen

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The Journal of Physical Chemistry, Vol. 97, No. 39, 1993

TABLE I: Experimental Conditioap in the Laser-Induced Fluorescence of NO2 and Wavelength-Dependent V d m of tbe Parameter 8 As Found with the Hot-Band Energy H Fixued at lo00 cm-1 no. of X/nm cases PlmTorr laser E l d u/1t20 cm-* a ~~

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68.0 56.7 67.4 52.0 63.4 54.7 64.3 41.1 42.0

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0.270 i 0.022 0.305 i 0.086 0.336 i 0.045 0.373 i 0.101 0.708 i 0.258 0.289 i 0.065 0.382 i 0.052 0.320 i 0.023 0.215 i 0.039 0.258 f 0.050 0.153 i 0.015 0.275 i 0.020 0.138 f 0.007 0.115 i 0.022 0.258 i 0.025 0.256 f 0.025 0.374 f 0.050 0 . 1 5 0 i 0.024 0.151 i 0.030 0.126 i 0.024 0.170 i 0.089 0.179 0.058

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ns. temperature. An excimer pumped dye laser (Lumonics Model EPD-330) provides excitation wavelengths from 399.8 to 672.6 nm. The spectral line width of the dye laser, estimated from the observed line width of the nitric oxide two-photon LIF spectrum, is approximately 0.8 cm-l. The beam, limited with an iris before the fluorescence cell, is 3 mm in diameter. A Scientech 360001 meter measures the laser power. A photodiode (EGCG GPD 100) and a boxcar averager (SRS 250) monitor shot-to-shot variations of the laser pulse energy scattered off the input window. A capacitance manometer (Baratron model 3 10AHS-10) measures the sample pressure. Method of Data Analysis

Observed NO2 Laser-Induced Fluorescence (LIF). Typically, the systemcollectsdispersedfluorescenceintensityinto230equally spaced wavelength bins from 380 to 840 nm. The NOz fluorescence quantum yield changes abruptly from 4 = 1.O to 4 < 10-6 at the NO 0 dissociationthreshold (25 130 cm-'/397.9 nm); this defines the low-wavelength limit xdi,(high-energy limit Xdiu) for spectral analysis. The red limit of the photomultiplier tube (about 840 nm) or twice the wavelength of the threshold of observed fluorescence (to avoid overlapping orders of the grating monochromator), whichever is lower, sets the upper wavelength limit A d (lower photon energy limit Xrd):

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Figure 5. (A) An example of raw data, the nitrogen dioxide fluorescence spectrum,excited at 466.3 nm and observed up to 820 nm. The spectrum is proportional to photons s-I nm-I. The wavelength scale is given at the bottom of the figure, and the photon energies are given at the top of the figure. (B) Cumulative sum of this fluorescence spectrum.

from wavelength X/nm to photon energy, X/cm-1. In terms of both wavelength and wavenumber, Figure 5A gives an example of the unprocessed data, the dispersed fluorescencefrom nitrogen dioxide (photons s-l/nm) that was laser excited at 466.3 nm (21 445 cm-l). This figure is typical of collision-free NO2 fluorescence spectra, showing peaks that are only partly interpretable and an underlying congested region that is primarily uninterpretable in terms of quantum numbers. The threshold for observed fluorescence, Or, is not the laser photon energy Y, (which coincides with the strongest peak on the figure) but rather the laser photon energy plus the ro-vibrational energy H in the room-temperature molecules. In subsequent derivations, we use the index i to indicate the threshold energy of a laser-excited fluorescenceand the bin indexes j or k for fluorescence spectra, each on the same absolute scale, 1-200:

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Table I presents the experimental conditions of the LIF experiments; NO2 LIF spectra were generated at 22 excitation wavelengthsfrom 399.8 to 672.6 nm. Wecorrect for experimental variables, such as laser fluence, sample pressure, and the instrument response. The total fluorescence intensity observed for a given spectrum is used to normalize the intensities of individual cells in that spectrum. The result is that prior to data analysis the total fluorescenceintensity of each spectrum is unity. The relations, X = 107/Aand AX = -107AA/Xz, convert the data

1 , 2 , 3,..., i fluorescence spectrum, s,i 1 , 2 , 3 , ...,j

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As determined by the laser energy, a fluorescencespectrum starts at any value of the index i and runs with the index j to Xd. From this construction, it follows that j 2 i throughout the course of one spectrum. Energy decreases as a bin index increases. The fluorescence intensity element for the LIF excited to 8, and observed at emission energy Xj is represented by sfj (in units of photons s-1 collected in data bin j). The experimental LIF spectrum is then given by

Internal Energy from NO2 Fluorescence

The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9893

observed LIF spectrum = S~,, where i is constant, j = i ... red, j 2 i

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spectrum as intensive variable = su/aX,, photons s-l/(cm-l)cncrgy (3) where AX,is the variable-wavenumber bin width. The cumulative sum of a fluorescence spectrum, excited to energy bin i and observed over the bins j = i to j = k, is

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0 where the number of elements is S,k and in si/ is the same when k = red. The cumulative sum of the complicated spectrum of Figure SA is shown in Figure 5B. The sum of all the LIF data from the threshold of its appearance to the red limit of observations is Sm, the last member of the set Sik. Sm is the quantity that would be measured with a non-dispersed fluorescence signal and a photon counter equally sensitive to photons at all wavelengths. Observed Photolysis-InducedFluorescence (PIF). Ultraviolet photolysis of several RNOz molecules produces electronically excited mol%ecules, N02*, which emit across the visible spectrum. We camed out such studiesunder almost collision-free conditions (low pressures and short times in room temperature flow-tube experiments). The monochromator disperses the PIF spectrum into data cells at photon energies X,. We correct for variations in laser fluence and other experimental features in the same way as for the LIF spectra. Figure 6A shows an NO2* emission spectrum from N205 photolysis at 248.5 nm. For this experiment, the pressure of N205 is 10 mTorr, the delay time is 10 ns, and the observation interval is 30 ns. These conditions correspond to a collision probability of 0.008 before the termination of the observations (0.8-nm mutual NOz*-N205 collision diameter). The laser photon energy is 40240 f 100cm-l, the OzNO-NO2 bond energy is 7460 cm-l, and the available energy M above that needed to break the bond is

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Energy/lOOO cm-' Figme6. Observed dispersed NO2 fluorescence following N2Osphotolysis at 248 nm, 10mTorr, 10-nsdelay time, 30-11sobservation time, and 0.008 hard-sphere collision probability. (A) Directly observed fluorescence (photons s-l nm-I), structured line. (B) Cumulative sum of the

fluorescence (proportionalto photons-s-I), continuous line. The smooth

line on (A) and the circles on (B) are the calculated functions based on parameters n and U, found according to (Ne) and (28-32).

M = EavailablC = hv - RNO, bond energy = 32780 f 100 cm-' ( 5 ) which is greater than the 0-NO bond energy (25 130 cm-l). There is some fluorescence at the dissociation energy of NO*, and the PIF spectrum spreads across the visible region. The apparent structure in the red region is noise. By analogy with (3), the normalized PIF data are represented as observed PIF data = p,, j = 1,2, ... red

(6)

spectrum as intensive variable = pj/AXj, photons s-l/(cm-l)cncrgy (7) and the cumulative sum of the PIF spectrum is irk

The cumulative sum of the PIF data of Figure 6A is shown by the line in Figure 6B. Model-Free Consideration of Errors and Signal-to-Noise

Ratios Statistical Errors of the Cumulative Sum Spectra. We carried out three to five LIF replicate experiments at most of the wavelengths of excitation (Table I). At a given laser excitation wavelength, we normalize the area under each of three to five spectra (compare Figure 5A) from the threshold of observation

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Wavelength / nm Figure 7. Empirical standard deviation per bin of cumulative sum fluorescence spectra,where in replicateexperiments NO2 is photoexcited at 405.3 nm under almost collision-freeconditions (5-6 mTorr, 30-ns delay time, 30-11s observation time). The numerical values are relative to 1.O00, the value of the cumulative sum at 800 nm.

to 800 nm, form the cumulative sum of each spectrum, and find the small sample, standard deviation (U{k) among the three to five points in each wavelength bin of the cumulative sums. Figure 7 is a typical example, where the laser excitation is at 405 nm and the numerical value of a standard deviation is relative to = 1.000. The signal approaches zero at the blue limit at 400 nm, and normalization makes all the spectra equal at the red limit at 800 nm. The standard deviation per bin has a broad slowly varying region above the value of 0.01 from about 500 to 720 nm, and its maximum value is 0.016. The average standard deviation per bin over the 200 bins is 0.0105. Baseline Error. The baseline of a LIF or PIF spectrum is taken to be the average of the 2&30 data bins obtained to the blue of the threshold of the fluorescence. Typically, the empirical standard deviation per bin, ubi,,, is 0.012 f 0.006 times the maximum value of an LIF or PIF spectrum. Based on 18 bins,

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Figure 8. Signal-to-noise ratio (S/N) of successive differences in the cumulativesum spectra of collisionallydeactivated Nq*,which is excited at 435 nm. (A, top) The differencebetween CDA-1 and CDA-3 (Figure 2 of article 2 of this series) divided by the standard deviation (Figure 7). (B, bottom) The same procedure applied to CDA-1 and CDA-4.

the baseline error is the standard error of estimate of this mean, U w i n e = ~ b n / d 1 6= 0.003 f 0.0015, relative to the maximum signal. Signal-to-NoiseRatio between Successive Collisionally Deactivated Cumulative Sum Spectra. Figures 1 and 2 of article 2 of this series show four fluorescence spectra and four cumulative sum spectra, normalized at 800 nm, which correspond to NO2 excitation at 435 nm (22 990 cm-1) and subject to collisional deactivation (CDA) by 2,4,8, and 12 Lennard-Jones collisions, labeled respectively CDA-1, CDA-2, CDA-3, and CDA-4. The problem of interest here is how the cumulative sum spectra change with molecular collisions. The four cumulative sum spectra (Figure 2 of article 2) are distinctly separated from each other. At each wavelength bin between 400 and 800 nm of Figure 2 of article 2, we divide the difference (CDA- 1 minus CDA-3) by the local standard deviation of cumulative sum spectra (Figure 7) to give anempirical signal-tenoiseratio (S/N), Figure 8A. Between 400 and 435 nm, the data are at wavelengths below the excitation laser at 435 nm, and both the signal and the error are very small. At wavelengths above that of the excitation laser, S/N rapidly climbs to values above 2, is greater than 4.5 over more than half the range, reaches a maximum of about 6, and plummets toward zero between 760 and 800 nm. Similarly, we find S/N for the difference between CDA-1 and curve CDA-4 of Figure 2 of article 2 (Figure 8B). It rises from about zero at 430 nm to about 10 above 600 nm, is greater than 8 over about half the range, and falls towards zero between 770 and 800 nm. These comparisons show that the differences between the observed cumulative sum spectra (Figure 2, article 2) are statistically significant, with substantial regions having SIN in the 2-10 range. Spectroscopic Evidence for Spread of the Internal Energy Populrtion Significant Difference between Fluorescence Spectra Having Same Average Energy and Merent Intend Energy Spread.From the physics of collisional deactivation of highly excited molecules, we presume that the spectra considered in the paragraph above and displayed in Figures 1and 2 of article 2 arise from populations of fluorescing molecules of successively lower average vibrational

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Figure 9. (A, top) Cumulative sums for two fluorescence spectra, where

the populations of fluorescing molecules have the same average internal energy (about 18 300 cm-1) but are prepared in different ways. The smooth curve is the CDA-4 of Figure 2 of article 2 (photoexcited at 435 nm and hasundergone 12Lennard-Joncscollisions).Themorestructured curve is a nascent LIF spectrum excited at 553 nm (18 090 cm-l). (B, bottom) The S/Nof differencebctwcen these two cumulative sum spectra as found by (9). energy and increasing spread. It is generally agreed that spectroscopic absorption and emission methods are capable of obtaining the average energy of a population of highly excited NO2 molecules,l and in this section we use average energies based on our method, which is described in subsequent sections of this paper. Figure 9A shows two cumulative sum spectra. The relatively smooth curve is CDA-4, which was excited at 435 nm (22 990 cm-I), has undergone 12 Lennard-Jones collisions, and, according to our analysis of the fluorescence spectrum, has average internal energy of about 18 300 cm-l (546 nm). The more structured curve is a nascent LIF spectrum excited at 553 nm (18 090 cm-I); these two cumulative sum spectra have internal energy populations with about the same average internal energy ( E ) . At each wavelength bin, we take thedifference between the two curves in Figure 9A and divide by the corresponding standard deviation (see Figure 7), which gives the signal-to-noise ratio between the curves as shown in Figure 9B.

S/N = Slk(obs., collisionally deactivated, (E))Slk(ObS., LIF with same ( E ) ) / u,(based on obs. LIF, Fig. 6) (9) Figure 9B (12 collisions) shows a broad, positive, significant S/N (up to 7) at wavelengths below 630 nm and a broad negative signal with significant S/N (between 5 and 10) above 650 nm. We have two other CDA experiments (also shown in Figures 1 and 2 of article 2) that have average energies that correspond closely to nascent LIF experiments. CDA-1 has undergone two Lennard-Jones collisionsand has average internal energy of 22 120 cm-1; a nascent LIF experiment at 449.9 nm prepares a population with internal energy of 22 230 cm-1. CDA-3 has undergoneeight Lennard-Jonescollisions and has average internalenergy of 19 960 cm-l; a nascent LIF experiment at 501.6 nm prepares a population with internal energy of 19 935 cm-1. Using (9) at each of 200 wavelength bins, we find the SIN difference between the CDA cumulative sum spectrum and the LIF cumulative sum spectrum,

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Figure 10. (A, top) S/Ndifferencebetween cumulative sums of CDA-1 (photoexcited at 435 nm, has undergone two Lennard-Jonescollisions, (E) = 22 12O~m-~)andnascentLIFat449.9or22 230cm-I. (B,bottom) S/N difference between cumulative sums of CDA-3 (photoexcited at 435 nm, has undergone eight Lennard-Jones collisions, (E) = 19 960 cm-1) and nascent LIF at 501.6 or 19 935 cm-’ (compare Figure 9B).

each with a NO2* population of the same average energy: Figure 10A (two collisions) shows a positive peak with S/N of 1.4 and a large region with negative values of S/N between 1.5 and 2.5, which is marginally statistically significant. Figure 10B (eight collisions) shows a narrow, positive, significant S/N of about 3 at wavelengths below 530 nm and a broad, negative, significant S/N between 2 and 5 at wavelengths above 530 nm. Including Figure 9B, we have large regions of statistically significant observable differences between pairs of spectra that have the same average energy. Figure 9A provides a qualitative interpretation of Figures 9B and 10A, 9B in terms of the internal energy distributions of the fluorescing molecules. For the LIF spectrum in Figure 9A, the internal energy E of the monoenergetic population of fluorescing N02* is given by the laser wavelength (553 nm, 18 090 cm-*); the monoenergetic population emits over a wide range of photon energies below 18 090 cm-I; and the cumulative sum builds up toward longer wavelengths in a fixed manner, given by the structured line. The threshold of the CDA-4 spectrum is about 450nm (22 OOOcm-l),anduptothethresholdoftheLIFspectrum the cumulative sum of CDA-4 has statistically significantpositive values, directly demonstrating some spread in the internal energy population on the high-energy side of the average. Beginning at the LIF threshold, the LIF cumulative sum builds up faster than that of the CDA-4 spectrum, which directly demonstrates that the concentration of N02* at ( E )is less for the CDA-4 population than for the LIF population. Above 670 nm (about 15 000 cm-l) and moving toward the red, the CDA-4 cumulative sum slowly catches up with the LIF cumulative sum, which indicates the opening up of new channels of CDA-4 fluorescence by N02* at internal energies substantially lower than (E). The end point of the cumulative sum is the integral of all the photons that are emitted between the threshold of fluorescence and 800 nm, and the LIF and CDA-4 spectra are scaled to have the same end point at 800 nm and thus the same total number of emitted photons. The two spectra in Figure 9A are conspicuously different, Figure 9B shows that these differences are statistically significant, and this analysis shows that these differences are caused by different

internal energy spread of the two fluorescing populations. This spread, which is confidently expected on physical grounds, has a statistically significantspectroscopicsignature. The conclusion of this paragraph is that differences in population spread give rise to statistically significant differences in observed cumulative sum spectra (compare the first paragraph of the Introduction). The changes of our cumulative sum spectra ascribable to spread of the internal energy population have a signal-to-noise ratio of about 2-5 in the individual data bins, which is painfully low precision. When about 150 data, each with S/N of 2-5, are used to find the average and the spread (our method, see below), the final results may have greater S/N by a factor up to d148. In view of the paucity of experimental measurements of second moments of collisionally deactivated populations (see article 2 of this series), even a low precision value may be interesting. The rest of this article is devoted to deriving and judging a method of estimating the internal energy distribution of an ensemble of fluorescing NO2 molecules, from which we evaluate and use the average energy, the spread, and the skewness. Interpretationof the Cumulative Sum LIF Data Approximationof LIF Cumulative Sum Spectrum by a Simple Empirical Function. When we try to fit the fluorescencespectrum of Figure 5A, for example, by a two- or three-parameter empirical function, the least-squares procedure treats the spectral structure as noise, and the desired energy profile is distorted by leastsquares minimization. We found that procedures to smooth the spectrum of Figure 5A, such as moving averages or Fourier transform filtering,distort the information of the original spectrum depending on the degree to which the smoothing is carried out. The cumulative sum of the spectrum, Figure 5B, is a relatively smooth, positive, single-valued monotonically increasing function that contains undistorted all the information of the spectrum, and it appears to be well suited for approximation by a simple empirical function. In the limit of small steps, the cumulative sum is the running integral as a function of its upper bound. For our functions it is well behaved, and upon differentiation it returns the original spectrum. Over our range of observations, the simple analytical function I(~X , ~-,) or Ilk approximates the cumulative sum of the nascent NO2 LIF spectrum, (4) or Figure 5B: Zi& = (2 - [2

+ 2Z( Y&) + z(y,J&)21exP[-Z(Y,J&)ll (10)

where Z(yi&) = ( yi + H - Xk)/(ai(fi + H)) if X , C (Y,+ H), otherwise, 2 = 0; yi = photon energy of excitation laser in units of cm-l, an experimental quantity; Ei = internal energy of NO**, a conceptual quantity, later related to yi; H = thermal internal energy of ground-state molecule; Xj or xk = photon energy of observed fluorescence (two subscripts permit double sums); ai = dimensionless parameter fitted to the data; and 19, = yi H = threshold of LIF and cumulative sum spectrum. The empirical function,Zik, is the exact integral of an increasing power law times a decreasing exponential

+

[2 - (2 + 2 2 ,

+ Z:)

exp[-Z,]] = r Z 2 exp(-2) d Z

from which we have an indirect approximation to the LIF spectrum: ( s i j / a X j > Uj= L(Y,,x,) Uj= C Z ~ exp(-z)

d~ (1 l a )

where Ci = C(Yi)is a factor to put the functional representation

9896

Johnston et al.

The Journal of Physical Chemistry, Vol. 97, No. 39, 1993

of the spectrum and theobserved spectrum on the same numerical scale, dZ = -dX/Ba, and Z = 0 when X = 6 or when X 2 25 130 cm-1. The observed cumulative sums (4) are then represented by the function: -0.6 O

s/k

= c/eP/z/k

A

(12)

and the sum of the spectral data from threshold to the red limit of observations is

SIR = C,ePh

In a

(13)

Dividing (12) by (13), we get the relation used to fit the parameters, H and a,, to the data: (14) We use a nonlinear least-squares program that carries out a FORTRAN search over two parameters to find the optimum values of a and H by minimizing the error of the sum for a spectrum produced by laser excitation energy, yi:

L

-2.4 I

I

16

18

I

I

20 22 Energyll000 cm"

I

24

1 26

Figure 11. The parameter u as a function of laser photon energy E at constant hot-band energy H of lo00 cm-l. At each wavelength of observation, the average value and twice the standard deviation are indicated.

where n is the number of data cells between the threshold energy 8 and the red wavelength limit. There is a physical basis for the parameter H. When excited by a laser, an isolated NO2 molecule near absolute zero would radiate only to the red of its excitation energy. A molecule at room temperature, as in these experiments, shows fluorescence to the blue of the excitation energy due to thermal rotationvibrational energy. The observed threshold fluorescence is at higher energy than the excitation energy. The superposition of the various rotational states is well represented by the classical mechanical, normalized Boltzmann distributionlo BOltZ(EvR) dEvR = (EvR/kT)e-EYR'kTd(EvR/kT) (16) This function goes asymptotically to zero at high energies, but the apparent threshold is where the fluorescence intensity just exceedsthe noise level of the observed fluorescence. Considering the signal-to-noise ratio in these experiments (compare Figure SA), the expected threshold is at an energy about 1000 cm-1 higher than the laser energy. We used the fitting program to optimize II for fixed values of the hot-band energy at each of 800, 900,1000,1100,and 1200cm-I. Theminimumsumofthesquares of the error (1 5 ) over all excitation energies is for H = 1000cm-1, which is used for the rest of this analysis. Table I gives the a values averaged at each excitation wavelength. Out of several empirical functions connecting the parameter a and laser excitation energy yi, we selected the one plotted on Figure 11:

H = 1000 cm-I a, = exp[(7.29 f 2.69) X lO-'Y,/cm-'

(2.986 f 0.623)] (17)

Figure 11 shows large scatter of points, which we ascribe to the narrow laser line picking out some of the high-resolution details that we are trying to average out and avoid. It would be better if our laser had about lOO-cm-' line width, in which case we think the points in Figure 11 would show less scatter. Figure 12 gives examples of various nascent LIF data fitted asdescribed above. For eachofthree wavelengths (421.0,511.6,

Energyll 000 cm" Energyll 000 cm-' Figure 12. Observed laser-induced fluorescence (photons s-1 nm-1) from nitrogen dioxide under near-collision-free conditions by virtue of low pressure (5 mTorr) and fast detection (30-nsdelay and 30-ns observation time) and the cumulative sum of the data including all peaks and continua: (panels a, b) 421 .O nm, (panels c, d) 5 11.6 nm, (panels e, f) 591.1 nm. Excitation wavelength indicated by arrow in a, c, e. Solid lines: (a, c, e) directly observed fluorcacence spectra, (b, d, f ) cumulative sum of data. Calculated functions: (b, d, f) solid circles at regular intervals, (IO); (a, c) dashed lints, first derivative of (IO).

591.1 nm), Figure 12A,C,E give the observed LIF spectra,

SR,(3), as solid lines. In each of Figure 12B,D,F, the cumulative sum of the observed data, St,/Sm, (4). is a solid line, and the fitted running integral of the empirical function,I,,/Im, is presented as filled circles. The first derivativeof the fitted running integrals gives the dashed line in Figure 12A,C. The agreement of the cumulative sum of the data with the running integral of the empirical function is good at all wavelengths. Relation of LIF Cumulative Sum to Physical Properties. The cumulativesum S,k is related to the zero-pressureradiativelifetime

Internal Energy from NO2 Fluorescence of N02*. When NO2 is excited by a narrow band laser at an energy Y,, the collision-free fluorescence decays with a lifetime or rate constant l/rl characteristic of energy Ei:

The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9897 (CDA) molecules that were originallyprepared in a narrow energy band, the photons collected in a wavenumber bin, pl, arise from a range of molecular internal energies, El, all undergoing transitions in the same wavenumber band:

qsij, i-1

pi = The value of 1/Ti is independent of the observation energy XI or range of observation energies. The value of 1 / q is formally expressed as the sum of the Einstein coefficient of spontaneous emission11 over the full fluorescence range:

For many molecules the Einstein A coefficientis a straightforward expression containing factors of u3, k12l2,and I(uIu")I~,and in these cases accurate estimates of l / r i are possible using (19). However, the breakdown of the Born-Oppenheimer approximation and the severe vibronic perturbations induced in highly excited NO2 molecules make calculations of 1/71 using the All formalism neither practical nor accurate. Instead, we determine 1/71 as a function of laser energy, Y,, which is closely related to internal energy, Et, using the recently measured experimental values of the zero pressure limit fluorescence rate constant'

+

l/Ti 0.504(yi-9710) 7960 (20) where Y, is expressed in cm-' and 1/71 has the units s-l. The nascent fluorescence spectrum excited to a narrow (- lOO-cm-') band of energies Ei is detected at energies Xp The observed number of photons s-l, si,,is proportional to the Einstein coefficientll for spontaneous emission, All, and to the number density of excited molecules in energy band i: su = aAifl1 (21) where a is an unknown geometric and units factor. A , here is not a state-testate Einstein coefficient; it is an average over populated internal energy states within the width of bin i and an average over the fluorescence spectrum having photon energy within the width of bin j . The Franck-Condon overlaps in one such average All surely come from a wide range of N02* configuration space. The sum of the elements of the fluorescence spectrum (21) from the threshold near theexcitation energy over the full fluorescence range 0' = all) is expressed by

where the second equation follows from (19). This summation over the entire spectrum is expressed in terms of our observable &k, our integrated function (1 1) with xk = 0, and relation (12):

Substituting (22) into (23) and solving for Sfk,we find

At constant fluorescence observation bin k in a multienergy system, (24) relates &k to the various internal energies Et, each of which fluoresces at photon energy xk.

Approxlmrtion of tbe PIF Data by Sums of the Empirical Function Derivation of System of Linear Eq~rtionsThat in Principle Wvea for the Internal Energy Population. With photolysisinduced fluorescence (PIF) or with collisionally deactivated

E, 2 X,,i I j

(25)

i-

The cumulative sum of the PIF spectrum (8), Pk, at wavenumber bin k is defined as the sum of pI from 1 to k,and the application of (25) to (7) gives a double summation:

By virtue of the condition j 1 i (24), si, is a triangular matrix, and the second double summation in (26) is equal to the first double summation. The order of summation in the second double summation (26) may be interchanged, which gives the double summation below. This second summation is identified by (4) as s i k :

The physical interpretation of (25) is that a PIF spectrum may be expressed as a linear combination of LIF spectra, and the physical interpretation of (27) is that the cumulative sum of a PIF spectrum may be expressed as a linear combination of cumulative sums of LIF spectra. Replacing SIk in (27) by its functional representation on the right-hand side of (24), we get a system of simultaneous linear equations:

for which the matrix expression is

The triangular matrix M is a known function, the array P is known data, and the array y is proportional to N,/rl containing the unknown relative population of excited NO2 molecules as a function of internal energy El. Using synthetic data built out of the empirical functions, (10) or (1 la), we construct complicated, multipeaked distributions for the array aNI/Ti= yi and carry out the indicated summations (28) to generate an array of synthetic data yPkn. We put these synthetic YPknas if data into (28) and solve for the relative populationyi. The arbitrary 200-bin input populationsare entered to three significant figures, and the 200-bin recovered distributions are identical with this input. This test is not a circular argument: a large amount of input information is lost in the summation of 200 X 200/2 weighted terms (10) to form 200 synthetic spectral terms (28), and the recovery of this information is a test of the ultimate possibilities of the method and demonstrates that our matrix M (29) is not singular nor numerically almost singular. Using experimental PIF or CDA data, we find that experimental error in the observed cumulative sums, Pk, causes the solutions of this set of linear equationsto oscillate between positive and negative values. We are unable to solve this set of linear equations using experimental data, and so we turn to an indirect method using trial functions with a small number of adjustable parameters.

9898 The Journal of Physical Chemistry, Vol. 97, No. 39, 19'93 Indirect Method for Solving for the Relative Internal Energy Population We assume that the product of terms, aNi/ri in (28), is distributed according to prescribed distribution functions, F(E), which depend on two or three parameters:

aNi/ri = FParamctm(EJ 'E,

(30)

We use three different functions for F(E): the Gaussian, the kernel of the gamma function, and the biexponential; and we vary the parameters to minimize the square of errors between the calculated and observed cumulative sums of PIF data: error2 = error' =

-

(Pk(Ca1C) - Pk(0bS))'

(31)

1 X n-2

0:

RaErXd (JR=,

NO + 0 -NO2* +NO2 N02*+M-N02+M

s25000,1 2500

F(Ei - R) Boltz(R) r(Ei) dR) X

d(E,) (33) to obtain the relative probability a ( N i / N ) for internal energy between Ei and Et + dEi. A FORTRAN program was written to carry out the operations above, This program minimizes the sumof squares error between calculated and observed cumulative sum of PIF data (3 1) as it varies the parameters of the distribution function. The program includes special features to assure that it locates the global rather than a local minimum in parameter space, and it has methods to avoid underflow and overflow numerical errors in dealing with exponential functions. This method uses all the data with no smoothing or interpolation. The information content of our method goes to zero as the internal energy nears the lower end of the observed fluorescence range. The optical system spectral response decreases rapidly below 14 OOO cm-1 (Figure 4), the fluorescence rate constant 1/ r i has been measured only down to 13 500 cm-l, the monoenergetic LIF calibrations only go down to 15 000 cm-l, and the integral goes to zero at 12 500 cm-l. Nominal populations for internal energies below 14 000 cm-l are especially to be suspected. In one instance, we can use observed spectral data, Sm/StA, to check the ratio of our calculated integrated function over our range of observations, Zm,relative to our calculated function over the full range, ZiA. Bradburn and Lilenfeld'z in their Table I give the chemiluminescence spectrum of NO1 over the range, 600026000 cm-1, produced by the reactions

+ hv

or N O + O + M

(34)

Their spectrum appears to be slightly red-shifted by collisional deactivation, but, even so, it is useful for this comparison. We translate this spectrum from (photons s-l)/nm to (photons s-l)/(cm-l)-w. Stair and Kennealy13 measured this spectrum from 8000 to 3000 cm-1, at which point they judged the vibronic spectrum to be essentially at its end. We scale Figure 5 of Stair and Kennealy from 6000 to 3000 cm-I, connect it to Bradbum and Lilenfeld's lowest energy point at 6000 cm-1, and draw a smooth curve through the combined data from 3000 to 25 000 cm-1. By integrating under this curve, we find Sm,the observed number of photons s-1 from 25 000 to 12 500 cm-1, and Si*, the observed number of photons s-1 from 25 000 to 3000 cm-l, which we compare with our functional form over the same ranges (1 1):

(

where AEi is the energy bin width. This procedure successfully gives an array of numbers, yi, which we interpret as aNi/ri. Multiplying through by the known values of q,we get aNi. Because of the unknown constant a, we can only obtain a relative population, aNi. When this method is used to find the internal energy population of a nascent laser-induced fluorescence, it returns essentially a delta function; that is, all the population is found in one energy bin. The excitedenergy state has additionalspread,the Boltzmann distribution of ground-state ro-vibrational thermal energy, EVR (16). In all cases, we broaden the internal energy populations derived from (3 1) to include the thermal spread. The probability of EVR,given by (16), is multiplied by the probability of E = ELnt - EVRand then integrated with a local variable R (not to be confused with the red limit bin index) over all values of EVR: D(Ei) d(Ei)

Johnston et al.

s25000,3000

( ) lkWd z25000,12500

= 0.446,

z25000,3000

= 0.433

function

(35) When we connect our calculated fluorescence spectrum to this combined observed spectrum at 12 500 cm-', the two curves agree from 12 500 cm-1 to the maximum at about 9000 cm-l, but the two curves do not agree on the low-energy side of the maximum. This testshows thattheratiosofareasunder thecurves,asspecified by (35), are in reasonably good agreement, which gives a partial verification of (24). Differences between ( S z s o o ~ ~and )~ (I~SOO~,.~,) are a source of systematic error. If these experiments are repeated, it is advisableto measure the fluorescence spectrum down to 3000 cm-l and formulate a better empirical function than (10) or to measure absolute fluorescence quantum yields from 25 000 to 12 500 cm-l. Distribution Functions. ME). and Their Parameters. Three shape functions are used for'F(h) for all cases discussed in this and the following three articles: (i) the Gaussian function, (ii) a function related to the kernel of the gamma function, and (iii) the biexponential function. Beyond the standard two parameters of each of these functions, we use a third parameter related to Mof ( 5 ) as the origin of the gamma and biexponential functions. The Gaussian function is exp[-(H - ~ ) ~ / 2 2 1

F(E) =

JLdexp[-(fi

- E)2/202] d E

if E I M a n d = 0 if E > M (36) where the adjustable parameters are p , the mean; u, the spread; and M, the blue cutoff of the function to yield a truncated Gaussian function spanning the range of observations. The full Gaussian function is symmetric; the truncated Gaussian is not. Another distribution function used here is related to the kernel of the gamma function, Pe-' dt. The natural variable U in the desired weighting function F(E) is the difference between the total available energy M ( 5 ) and the internal energy E: U Z M - E i f E I M and U = O i f E > M (37) The coefficient weighting function is based on F(E) d E = -(M and its several variants

= U'e-'I'

- E)"eiM-a/"

dE

(384

dU, where U = M- E by definition (38b)

= s(*')( U/s)"e-""

d( V/s)

(38c)

Defining t as U/s,one sees the relation to the kernel of the gamma function.

= s("+l)r"e-' dr = dH1)times kernel of r(n+l) (38d) Since U,,, =U ,, = ns, one obtains an especially useful form

Internal Energy from NO2 Fluorescence

The Journal of Physical Chemistry, Vo1. 97. No. 39, 1993 9899

= fi$')[ (U/ U,)e4u/um)] " d( U/U,)

0.0151

(38e)

which has the following desirable properties: (a) Unlike the Gaussian function, it has a finite upper energy bound at E = M , as does the excess energy after photolysis of RNOz. In cases of collisional deactivation, M is varied as an adjustable parameter to locate the threshold of the fluorescence. (b) It can closely approximate a Gaussian function for certain ranges of its parameters. (c) Unlike the Gaussian function, it can be highly asymmetric on either the high-energy or low-energy side of its maximumvalue. Thenormalizeddistribution function with (38e) uses U, and n as parameters, and it is

,

-

I

,

,

,

,

,

,

,

,

,

,

I

... ........

1

0.010

~ 0 . 0 0 5

[(U/U,)e4v/vm)]"A(U/U,) 12

(39)

Where (37) defines U,the biexponential function is e-V/'ld( 1 - e-V//3 F(E) = E - X d

T

E=Z 132

InscwPi0n Test of Our Method against Synthetic Fluorescence Spectra in Terms of the First Wee Moments. Equation (25) regards a PIF spectrum to be a sum of LIF spectra arising from an internal energy distribution determined by the nature of the system and to be found by further considerations. Synthetic spectra are constructed from a modified form of (25), in which we specify an internal energy population: 1'1

i = 172,

200 (41)

I= 1

'

18

20

22

24

EnUgy/rooo an-'

(40)

e-V/'ld( 1 - e-"')

= C(Snoisy),jCvi,p,J,,

16

M p e 13. A "box" population function, ymPt of (41), used to generate a synthetic fluorescencespectrum from replicates of one real nascent LIF spectrum observed at 449.9 nm. This spectrum is processed through our program, and the population, ymmt of (41), is derived with a Gaussian function, a gamma kernel function, and a biexponential function. The best tit in each case is shown.

where r is the constant for the rising exponential, d is the constant for the decaying exponential, and M is the origin of the biexponential function. The biexponential function is always asymmetric.

@synhaic)j

14

where the array snobyis either actual nascent LIF spectra (3) or our functional form of LIF spectra (1 la and 13) to which we add various types of random noise, and the array yhputis the assigned, artificial internal-energy distribution. Our procedure starts with a PIF or with a CDA spectrum. The array psynth& is put into our usualcomputer program,which uses Gaussian, gamma kemel, and biexponential functions, that is, Fpnmctcn(E)AE of (30 to 40), to findan arrayyouWt. For bothyhputand yoUtpIt,wecalculate the first moment M Ior mean or average energy (E), the second central moment M2,((E- (E ) ) z ) ,from which we find the spread, and the third central moment from which weevaluate the skewness M3/(M2)3/z. In this study, we test the accuracy with which our method recovers the first three moments of Yinput in its found

trial function Input r kernel Gaussian biexpo.

lo3 error I: mean/cm-l (error) spread/cm-l (error) 19049 1775 0.937 19301 (+1.3%) 2195 (+24%) 0.964 19328 (+1.5%) 2191 (+23%) 2.04 19108 (+0.3%) 2282 (+29%)

The box distribution is symmetric, the Gaussian is symmetric, the biexponential function is asymmetric, and the gamma kernel can be either symmetric or asymmetric. The Gaussian and gamma functions give almost identical curves, but they are offset from the center of the box, perhaps because of the inappropriateness of the single LIF function (449.9 nm) used to generate the synthetic box spectrum, which should involve input LIF spectra that change with increasing wavelength. Being of different symmetry from the box, the biexponential function has a large error sum. Figure 14 shows a triangular yinput,spanning the same range as Figure 13. hobyis built from 23 observed LIF spectra excited by laser lines at 10 different wavelengths between 449.9 and 591.1 nm (Table I), which is a much more appropriate hY than that used in Figure 13. Each observed LIF spectrum was used three or four times, with different assigned threshold energies. The spectra are weighted according to the triangular function and according to the factor Z,~/Z,A (28), where i is the threshold energy assigned to each of the 87 spectra that are superimpmed to give psyntbctio.The triangular function is asymmetric, and only youtputfrom the asymmetric biexponential function is shown on this figure. The least-squares fit has the parameter M, origin of the biexponential function at 22 773 cm-', which is above the origin of the triangle and appears to the eye to be a poor choice. The fit is repeated with Mfixed at 22 000 and 22 500 cm-I; these two have slightly larger error sums than the least-squares M and give comparable moments:

Ywtput.

In Figure 13, wUy is constructed entirely from one observed nascent LIF spectrum excited at 449.9 nm, which includes any baseline offset, baseline noise, signal noise, and the systematic structure of the spectrum (compare Figure 5A). ymPtis a box function spanning the range 15974-22124 cm-l, which uses 87 copies of the LIF spectrum each offset 2 nm from the other. Processingp ~ ~ t h through & our usual method, we find three curves as ywtpt from the Gaussian, gamma kernel, or biexponential functions. It is impossible for the trial functions to fit the shape of the box, but all three functions find the correct average energy within +1.5% and find the spread within +29%:

~ l c m - 3103 error I:

mean/cm-l (error) 20074 input

spread/cm-l (error) 1450 input

22 OOO 22 500 22 773

20073 (-0.005%) 20124 (+0.25%) 20095 (+0.10%)

1750 (+21%) -1.36 (-140%) 1627 (+12%) -1.15 (-103%) 1803 (+24%) -1.06 (-87%)

8.14 7.78 7.60

skewness (error) -0.566 input

All threevalues of Mgive good agreement with the average energy, the spread is strongly influenced by the low-energy tail below 16 000 and is in error by 12-24%. The skewness has the correct sign, but its absolute value is high by about a factor of 2. In the language of our functions (10) and (1 la), the re-use of one real spectrum in Figure 13 corresponds to using a constant

9900 The Journal of Physical Chemistry, Vol. 97, No. 39, 199’3

It -

Ma

__-__-W M - 2 2 7 7 3

0.03

12

i

M-22000

14

16

18 20 EmoyHOOo m”

22

24

Figure 14. Atriangularpopulationfunction,y~of(41),usedtogcneratc a synthetic fluorescence spectrum from 23 real observed nascent LIF spectra between 450 and 591 nm. This synthetic spectrum is processed through our program, and the population, yaw, is derived with a biexponential function. The best fit is found with the origin, M,of the biexponentialfunction at 22 773 cm-1; the populations for two fixedvalues

of M are included. RELATIVE POPULATION (ASSIGNED)

Johnston et al. multiplicative random noise where each of 50 noise-free synthetic LIF (1 la) is multiplied by a different array of random numbers equally distributed above and below 1.OOO; (c) additive random noise where an array of random numbers equally distributedabove and below zero is added to the noise-free synthetic LIF spectra, including the baseline; and (d) the baseline of a PIF spectrum is increased or decreased by an assigned amount. One of the authors of this paper constructs the synthetic spectra (using the program Mathematica, Wolfram,Research) and another carries out the FORTRAN calculations without knowledge of the input population. The results of these studies are summarized in Table 11. In this paragraph, we consider cases where %* of (41) is built from noise-free functions. When yeplt is Gaussian and F(E) of (30) is Gaussian, the output mean and spread differ from input mean and spread by less than 1 cm-1 (Table 11). When ybpt is a box and F(E)is a Gaussian, the output mean is 0.8% too large and the spread is 3.8% too large. When yhpt is a triangle and F(E)is any of the three, the maximum error of the output average is +0.6% and +1.5% for the spread; but the skewness error is much larger: -5.8% for the truncated Gaussian, -1 1% for the gamma function, and -1 49% for the biexponential. Figure 16 shows an application of multiplicative noise to a synthetic spectrum based on the box distribution. The smooth line in the top panel is one monoenergetic “LIF” function (out of 50 in yinput),and the noisy curve is this spectrum multiplied by an array of random numbers between 0.4 and 1.6, that is, a 60% amplitude multiplicative error. This noise is intended to represent the effect of real spectral structure (compare Figure SA). Theother 49 input spectra have successivelydifferent input thresholds and a different array of random numbers. The synthetic PIF spectrum including this noise is in the middle panel and should be compared with Figure 6A. The cumulative sum, given by the bottom panel of Figure 16, is to be compared with Figure 6B. The population recovered from this synthetic noisy spectrum has average energy that differs from the input value of 0.9% and has a spread that differs from the input value by -5.2%. A similar 60% multiplicative error applied to the triangle distribution yields a population that differs from the input: mean, 0.5%; spread, 6.1%; and skewness, -54%. The baseline of a LIF or PIF spectrum is taken to be the average of the 20-30 data bins obtained to the blue of the threshold of the fluorescence, and such a value is subject to the random fluctuations of the zero-signal noise. However, a baseline error acts as a systematic error throughout the course of one run, and it changes the slope of a cumulative sum. Random baseline offsets imposed on the 50 LIF synthetic spectra have small effect on the calculated PIF spectrum and on the population. Baseline offsets applied to a synthetic PIF spectrum produce significantdistortions of the inferred population (Table 11). When yinpltis Gaussian and F(E) of (30) is Gaussian, a shift of the baseline of twice the standard deviation of the baseline error changes the output, as follows: baseline shift up, mean error = 0.2%, spread error = +22% baseline shift down, mean error = -0.24, spread error = -27%. Where Yinplt is the triangle and F(E)of (30) is the gamma kernel, a shift of the baseline of one standard deviation of the baseline error changes the output, as follows: baseline shift up, mean error = 0.7%, spread error = +7.2%, skewness error = +61%; baseline shift down, mean error = 0.5%. spread error = -4.O%, skewness error = -40%. In this case, a 2 u m shift changes the spread by 11% and the skewness by 101%. In a number of cases, we add extremely large amounts of noise in constructing synthetic spectra to see what it takes to force our method to fail. Figure 17 shows an application of additive noise to a synthetic spectrum based on the triangle distribution. The noisy curve in the top panel is one monoenergetic “ L I P spectrum (out of 50) to which an array of random numbers between -5Ouh and +Soubin has been added to the baseline and to the spectrum, where 5oCrbin is 50 times the standard deviation of the observed

0’0

WAVENUMBER I CM-1

Figure 15. Three shapes of artificial input distributions, ybpltrto make synthetic spectra. (i) The Gaussian input function corresponds to one

of our fitting functions and provides the least severe test for these three input functions. (ii) The box is symmetric; its third moment is zero. It testa how well ow method gives the average and spread for a shape strongly unlike any of the fitting functions. (iii) The right triangle is asymmetric, and it tests how accurately our method reavers average, spread, and skewness for a shape strongly unlike any of the fitting functions. See Table I1 for mean, spread, and skewness obtained from fitting noise-free and noisy input functions. value of the parameter a, and the use of multiple real spectra in Figure 14correspondsto using variable parameter a. The recovery of information is better in the latter case, but there is fairly good recovery of information even in the first case. This comparison shows that our method is not sensitive to the parameter a, and the great spread in its values in Figure 11 is probably not a serious matter. Our method recovers major features of the underlying pop ulation of synthetic data constructed from real LIF data (Figures 13 and 14). Even for nonphysical box and triangle input populations, it gives good estimates of the average energy, fair estimates of the spread, and factor-of-2 estimates of the skewness. These simulated spectra include the spectral structure of real NO1 fluorescence, the random noise, and the systematic errors of our experimental system. To identify the sensitivity of our method to various types of error, we carry out a series of studies in which we add errors one-by-one to noise-free synthetic spectra. We use three input distribution functions, ybput,spanning 50 bins in the range 18000-22000 cm-I: (i) a Gaussian, (ii) a box function, and (iii) a right triangle as shown in Figure 15. These synthetic spectra include one of four types of “noise”: (a) no noise; (b)

Internal Energy from NO2 Fluorescence

The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9901

TABLE II: Test of Information Recovery from Synthetic Fluorescence Spectrac Yhpt

Gauss

(41) noise code0

NoN

X2096b X40@ UP~~buOliw Dn2-

NoN

box

fitting function Gauss

Gauss

X20W X40%b X60%b triangle

NoN NoN NoN x20w X40%b X60%b UP~~buOliw Dnlup15u~e DnlSukli,

C C

*SO% +50%

C

C

gamma Gaud biexp biexp biexp gamma gamma gamma gamma gamma gamma biexp

mean (E) 20191 20191 20191 20191 20233 20148 201 12 2027 1 20263 20280 20285 20766 20893 20893 20844 20872 20872 20863 209 14 20873 21315 20599 20796 20796

(A%) (input (0)

(0) (0) (0.2) (-0.2) (input) (+0.8) (+0.8) (+0.8)

(+0.9) (input) (+0.6) (+0.6) (+0.4) (+OS) (+OS) (+OS)

(+0.7) (+OS)

(+2.6) (-0.8)

(+0.1) (+OS)

spread 642 642 649 671 78 1 469 1133 1176 1188 1122 1074 927 941 937 923 997 855

984 994 890 1619 57 796 792

(A% (input)

skewness

(A%)

-0.566 -0.533

(input)

-0.504

(+I 1) (-149) (-149) (-149) (-54) (+61)

(0) (+1.2) (+4.3) (+22) (-27) (input) (+3.8) (+4.8) (-1.0) (-5.2) (input) (+1.5) (+l.l) (-0.4) (+7.6) (-7.8) (+6.1) (+7.2) (-4.0) (+75) (-94) (-14) (-14)

-1.41 -1.41 -1.41 -0.870 -0.219 -0.790 -0.252 -0.209 -1.98 -1.99

(-5.8)

(-40) (+55)

(+63 (-250) (-250)

* Random errors are introduced into the input functions according to the code: NoN,no noise; Xee%, multiplicative random error applied to input LIF functions were ee is error amplitude relative to maximum of LIF function (Figure 16); +SOut,i,,, random errors, in the range between -50 to +50 times the observed standard deviation of bins along the baseline, added to baseline and to calculated PIF function (Figure 17); Upe-, the upward offset of the baseline, e is multiple of the standard error of estimate of the baseline; Dnea, the downward offset of the baseline. This error function imitates the structure in LIF spectra (compare Figures SA and 16). Unrealistically large errors assumed to find conditions where our method breaks down. d This has lowest error sum of the three fitting functions for this test. e The synthetic fluorescing populations are distributed with the shapes of a Gaussian, a box, and a triangle (Figures 13, 14, 15). Each spectrum is processed by the program for finding the population from PIF spectra, and the results are compared in terms of the first two or three moments of the derived distribution. A% is percentage deviation from input. (All energies are cm-1.) error of the baseline per bin. The synthetic PIFspectrum including this noise is shown in the middle panel, and the cumulative sum is given by the bottom panel, where most of the random noise has canceled out. For this case, the errors are mean, +0.1%; spread, -14%; skewness, -250%. Even this large application of additive noise has virtually no effect on the mean, a nonserious effect on the spread, but the skewness error is large. Also, we consider baseline offsets, up and down, 15 times the standard deviation of the baseline error: baseline shift up, mean error = 2.6%, spread error = +75%, skewness error = +55%; baseline shift down, mean error = 4.896, spread error = -94%, skewness error = +63%. We find the mean of the population to be insensitive even to such baseline offsets, but the large baseline offsets (f15ui-1i,,~) overwhelm the ability of our method to find the spread. We conclude that the largest source of error in our population spreads is offset of the baseline in the observed PIF or CDA spectra. Considering the study of synthetic spectra as a whole, we find that the recovered averageinternal energy is insensitive to imposed random errors and to baseline offsets, even ridiculously large ones. The spread is highly sensitive to offsets of the baseline and only moderately sensitive to multiplicative noise. The skewness is highly sensitive to all errors. When synthetic spectra are constructed from observed nascent LIF spectra according to the triangle input function (Figure 14), the percentage errors between input and output moments of the internal energy population are comparable to the differencesproduced by purely syntheticspectra with a 2u baseline offset and with spectral structure imitated by 60% multiplicative random errors: source of synthetic spectra mean error spread error skewness error real nascent LIF