Intensity Measurements and Collision-Broadening Coefficients for the Oxygen A Band Measured by Intracavity Laser Absorption Spectroscopy

Journal of Molecular Spectroscopy 201, 188 –197 (2000) doi:10.1006/jmsp.2000.8096, available online at http://www.idealibrary.com on

Intensity Measurements and Collision-Broadening Coefficients for the Oxygen A Band Measured by Intracavity Laser Absorption Spectroscopy Shengfu Yang,* Manjula R. Canagaratna,* Scott K. Witonsky,* Stephen L. Coy,* Jeffrey I. Steinfeld,* R. W. Field,* and Alexandre A. Kachanov† *Department of Chemistry and G. R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and †Laboratoire de Spectrome´trie Physique, CNRS UMR 5588, Universite´ J.-Fourier, Grenoble I, B.P. 87, 38402 St. Martin d’He´res Cedex, France Received July 15, 1999; in revised form February 25, 2000

High-sensitivity, high-resolution intracavity laser absorption spectroscopy (ICLAS) has been used to measure line intensities, nitrogen-broadening coefficients, and self-broadening coefficients in the A band (b 1 ⌺ g⫹ 4 X 3 ⌺ g⫺ ) of oxygen. Both linear cavity and ring cavity ICLAS configurations were used for these measurements, and the results were intercompared. The results were compared to values measured using long-path multiple-reflection cells by K. D. Ritter and T. D. Wilkinson [J. Mol. Spectrosc. 121, 1–19 (1987)] and L. Brown and C. Plymate, [J. Mol. Spectrosc. 199, 166 –179 (2000)]. New results are included for weakly absorbing transitions, not observed in the earlier measurements, such as high rotational states (up to J ⫽ 39), hot-band transitions (v⬘ ⫽ 1 4 v⬙ ⫽ 1), and isotopically substituted species ( 18O 2 and 16O 18O). Isotopic variants ( 16O 2, 18 O 2, and 16O 18O) have similar broadening coefficients for corresponding rotational levels, but the self-broadening coefficients are larger in the hot band (v⬘ ⫽ v⬙ ⫽ 1) as compared with v⬘ ⫽ v⬙ ⫽ 0 transitions. An ECS-EP scaling analysis of the v⬘ ⫽ v⬙ ⫽ 0 self-broadening data accurately represents the available data, with the exception of the N ⫽ 0 and N ⫽ 1 levels. © 2000 Academic Press

Key Words: oxygen A band; intensities; pressure broadening; scaling laws; intracavity laser absorption spectroscopy. I. INTRODUCTION

Measurements on the atmospheric oxygen A band (b 1 ⌺ g⫹ (v⬘ ⫽ 0) 4 X 3 ⌺ g⫺ (v⬙ ⫽ 0)) are utilized in campaigns by NASA and other international space agencies to determine temperature, pressure, and density profiles in the atmosphere (1, 2). In order for such profiles to be retrievable from the measurements, highly accurate linestrength and lineshape parameters must be available for oxygen A-band transitions. The oxygen A band is an extremely weak magnetic-dipole transition ( f ⬵ 2.5 ⫻ 10 ⫺10 ), so sensitivity-enhancement techniques must be employed to obtain quantitative line parameters for these transitions. This typically involves use of a multiple-reflection “White” cell in conjunction with a continuum light source and a high-resolution spectrograph (3), a tunable narrowband dye laser (4), or a Fourier transform spectrometer (5–7). A path length on the order of tens to hundreds of meters can be obtained in such experiments, but this is still insufficient to permit measurements to be made on weak features of this band. The intracavity laser absorption spectroscopy (ICLAS) technique (8 –13) allows us to overcome this limitation. ICLAS exploits the sensitivity of a laser with a broad gain bandwidth to narrowband intracavity losses such as narrow absorption lines of a vapor sample that either fills the cavity or is contained in an intracavity absorption cell. With this technique, effective pathlengths of tens to hundreds of kilometers through the sample can be achieved, overcoming many of the sensitiv-

ity limitations noted above. By careful control of the ICLAS generation time t g 1 and data acquisition procedure (12), it is possible to carry out quantitative lineshape measurements, as has been demonstrated for water vapor (14, 15), methane (16), and the (v⬘ ⫽ 2) 4 (v⬙ ⫽ 0) band of the O 2 b 4 X transition (␥ band) (17, 18). In this communication, we report ICLAS measurements on selected transitions in the oxygen A band. Our results are compared with benchmark measurements on intensities and linewidths for this band (7), and pressure-broadening parameters are reported for high J lines, hot bands (b(v⬘ ⫽ 1) 4 X(v⬙ ⫽ 1)), and isotopically substituted oxygen ( 18O 2 and 16O 18O). These results are compared with current models for pressurebroadening of molecular absorption lines. II. EXPERIMENTAL DETAILS AND METHOD OF ANALYSIS

(a) Experimental Procedure Two different cavity configurations were employed in these measurements using the intracavity laser absorption spectromThe generation time t g is the time elapsed from the beginning of the spectral evolution of the laser pulse to the moment at which the spectrum is observed. t g can be determined precisely from the delay between two acoustooptic modulator (AOM) pulses, the first of which directs the pump laser beam into the gain medium (the Ti:sapphire crystal, in our case) and the second which directs a 2-␮s “slice” of the output beam into the spectrograph.

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ICLAS MEASUREMENTS IN THE OXYGEN A BAND

FIG. 1. Intracavity laser absorption spectrum of N ⫽ 13 and N ⫽ 15 doublets in 0.104 Torr of oxygen (t g ⫽ 80 ␮s). This is an uncorrected experimental spectrum that includes contributions from atmospheric oxygen in the beam path (see text). In this figure (and all others) the branch notation is ⌬N ⌬J N⬙ ( J⬙).

eter furnished by Science Solutions Inc. Measurements made between October of 1998 and February of 1999 were performed with a standing-wave, linear-cavity (LC) ICLAS system similar to the system described in Ref. (19). A 15-W argon-ion laser (Coherent INNOVA Sabre DBW15) is used to pump a standing-wave Ti:sapphire laser. The total length of the laser cavity is 221 cm; the length of the sample region is 99 cm, giving an occupation ratio of 0.45. Measurements made between October of 1999 and January of 2000 were performed with a traveling-wave, ring-cavity (RC) configuration achieved by reconfiguration of the existing linear cavity, a procedure similar to that described in Ref. (13). The total round-trip length of the cavity was 487 cm in this case, and the total round-trip length through the sampling region was 260 cm. In both configurations, the laser is dispersed with a 2.5-m grating spectrograph used in double-pass mode, giving a resolution of 0.02 cm ⫺1 and detected with a silicon– diode array. The generation time t g is controlled by two acousto-optic modulators. The first gates the pump laser and the second gates the output. In all of the measurements, 100 spectra were averaged to increase the S/N ratio. Using the ring configuration (RC), intensities of the N ⫽ 13 and N ⫽ 15 doublets 2 shown in Fig. 1 were measured at a pressure of 0.104 Torr, varying t g between 80 and 160 ␮s. Intensities of the N ⫽ 23 and N ⫽ 25 doublets were measured at a pressure of 0.58 Torr, and t g was varied from 50 to 500 ␮s. 2 Several different sets of spectroscopic notation have been used in the literature to describe the oxygen A band. We adopt the standard convention, viz., N ⫽ effective rotational angular momentum and J ⫽ total angular momentum including spin. Thus, P Q 15 (14) denotes the {(N⬘ ⫽ 14) 4 (N⬙ ⫽ 15), ( J⬘ ⫽ 14) 4 ( J⬙ ⫽ 14)} transition.

189

In addition, pressure was varied from 0.10 to 4.37 Torr at a fixed t g of 60 ␮s to verify intensity measurements for the N ⫽ 23 and N ⫽ 25 doublets. Intensity measurements of the N ⫽ 37 and N ⫽ 39 doublets and the hot band were carried out using a pressure of 292 Torr and a range of generation times between 50 and 240 ␮s. Several sets of measurements were carried out on pressurebroadening coefficients for selected transitions in the oxygen A band. Nitrogen-broadening coefficients of the 16O 2 N ⫽ 9 and N ⫽ 11 doublets and self-broadening coefficients of high N and hot-band transitions were determined using both the LC and RC configurations. Measurements on the N ⫽ 9 and N ⫽ 11 doublets were carried out with a generation time of 80 ␮s, keeping the oxygen pressure constant at 0.8 Torr and varying the nitrogen pressure from 50 to 770 Torr. Self-broadening measurements on the much weaker high N and hot-band transitions were carried out by keeping the generation time constant at 65 ␮s (LC) or 80 ␮s (RC) and varying the oxygen pressure between a few up to ca. 720 Torr. Self-broadening measurements on the oxygen isotopomers with N ⫽ 9 to 13 were carried out using the linear ICLAS configuration with t g ⫽ 65 ␮s, an oxygen pressure of 0.8 Torr, and nitrogen pressures between 50 and 770 Torr. In previous quantitative line intensity measurements using ICLAS (14, 15, 17, 18), the apparatus function was determined either by recording a spectrum of a single-mode HeNe laser or by operating the Ti:sapphire or dye-laser cw and utilizing a series of intracavity e´talons to record the single frequency spectrum of the laser. To determine the apparatus function during our measurements, a 1-cm thick e´talon (3% reflectivity) was placed inside the laser cavity and the e´talon spectrum, consisting of a progression of peaks, was recorded at a generation time of 300 ␮s. This e´talon spectrum was then fitted to reproduce the apparatus function pixel by pixel (see Section II(b)), thus best representing the apparatus function’s dependence on frequency. This also accounts for nonuniformity in the diode-array detector. Residual background absorption due to atmospheric oxygen was also corrected for in the analysis. An aluminum purge box flushed with argon was used to remove oxygen from the laser cavity in which the sample was contained; this procedure worked very well, reducing background absorption to around 0.1% of its normal atmospheric value. However, it was not feasible to purge the beam path between the ICLAS output coupler and the diode-array detector in the spectrograph. To correct for the small residual background absorption from the unpurged beam path, a background spectrum was first recorded and then subtracted from the spectrum of each sample. Oxygen was taken from a cylinder furnished by BOC Gases with a stated purity of 99.994%. For the measurements on isotopically substituted oxygen, an enriched sample was purchased from Cambridge Isotope Laboratories. The sample had a chemical purity of ⬎98% with 46.7 atom % 16O, 2.6 atom % 17 O, and 50.7 atom % 18O. The desired 16O 18O species consti-

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(b) Global Fitting Procedure To obtain accurate intensity and linewidth data from measured ICLAS records, a number of contributing factors must simultaneously be taken into account. First is the lineshape function for the transition itself. Considering only Doppler and collision broadening, one obtains the well-known (20) Voigt line profile,

V共 x, y兲 ⫽

FIG. 2. Intracavity laser absorption spectrum of 18O-enriched molecular oxygen showing 16O 18O and 16O 2 absorption lines in the 13 082–13 096 cm ⫺1 region. The very weak unassigned features are due to 16O 17O and 17O 18O. The total oxygen pressure is 90.4 Torr and t g ⫽ 65 ␮s.

tuted about 50% of the sample, with the balance consisting of 25% each of 16O 2 and 18O 2. The sample also contained very small amounts (⬍1%) of 16O 17O and 17O 18O; the small unidentified features in Fig. 2 correspond to these isotopic species. A region of the A-band spectrum was selected in which absorption features from all three isotopic varieties could be readily distinguished (see Fig. 2). Nitrogen was also BOC research grade, purity 99.998%. Gas pressures were measured using one of three gauges (Baratron type 121A or 122A absolute pressure transducer) depending on the desired pressure regime. A series of measurements was performed to assure the relative calibration of the gauges. A 0 –1000 Torr head was used in all pressurebroadening measurements. The line-intensity measurements on the N ⫽ 37 and N ⫽ 39 doublets and the hot band were performed with a 0 –1000 Torr head. The intensity measurements on N ⫽ 23 and N ⫽ 25, using the ring cavity, were done using a 0 –100 Torr head. A 0 –1 Torr head was utilized for the ring-cavity intensity measurements of the N ⫽ 13 and N ⫽ 15 doublets. All measurements with the linear cavity were carried out at ambient temperature, which was typically 22.5°C (295.6 K) as determined by a laboratory thermometer. The temperature did not vary by more than ⫾2°C during the course of the measurements and does not represent a significant source of error in the analysis. Ringcavity measurements on pressure broadening and intensity measurements in the hot band and N ⫽ 37 and N ⫽ 39 doublets were done at 24.5°C, intensities of the N ⫽ 23 and N ⫽ 25 doublets at 24.0°C, and intensities of the N ⫽ 13 and N ⫽ 15 doublets at 20.3°C. In each case, the appropriate temperature was used to relate pressure to the total number density of the absorber.

y ␲

冕

⫹⬁

⫺⬁

exp共⫺t 2 兲 dt, y 2 ⫹ 共 x ⫺ t兲 2

[1]

where the frequency displacement from line center x ⫽ [( ␯ ⫺ ␯ 0 )/ ␥ D] (ln 2) 1/2, y ⫽ ( ␥ L/␥ D) (ln 2) 1/2, ␥ D is the Doppler half-width (HWHM), and ␥ L is the Lorentzian half-width of the line. In the infrared and optical regions, collisions may sometimes lead to a narrowing of the linewidth instead of a broadening (Dicke narrowing (21)). If the lifetime of the upper molecular level is long compared to the mean time between successive collisions, the velocity of the oscillator is often altered by elastic collisions and the mean velocity component is smaller than in the absence of such collisions, resulting in a smaller Doppler shift. This line-narrowing effect is frequently represented by the Galatry line profile (22). The Galatry profile describes broadening by thermal motion and state-perturbing collisions in the soft-collision model for velocity-changing collisions. It has the form G共 x, y, z兲 ⫽

1

冑␲

冉冕

⫹⬁

Re

0

再

dt exp ⫺ixt ⫺ yt

⫹

⫽

1

冑␲

冋

Re

1 关1 ⫺ zt ⫺ exp共⫺zt兲兴 2z 2

冎冊

1 共1/ 2z兲 ⫹ y ⫺ ix

冉

⫻ M 1; 1 ⫹

1 y ⫺ ix 1 ; 2 2 ⫹ 2z z 2z

冊册

[2]

,

where M is the confluent hypergeometric function and z is the narrowing parameter in the soft-collision model. A global fitting program to reduce raw data to corrected lineshapes was developed in the MATLAB programming language. The fitting procedure involves the following steps: (i) The relative sensitivity of the detector elements in the array detector varies from pixel to pixel. To normalize the relative sensitivity, the spectrum of an incandescent source was recorded. The output of this source is nearly constant over the range of each recording, so that pixel sensitivities derived from this measurement can be used to normalize the sensitivity of each pixel.

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ICLAS MEASUREMENTS IN THE OXYGEN A BAND

191

TABLE 1a Intensities (ⴛ10 ⴚ5 cm ⴚ2/atm) for Low N Transitions in Oxygen A Band

Note. Last digit uncertainties (2␴) are shown in parentheses.

FIG. 3. Simulation of the oxygen spectrum in Fig. 2, using the global fitting program described in Section II(b). The spectrum was fitted as several groups of adjacent lines, so that the flat regions in the “Residual” panel correspond to gaps between these groups.

(ii) A correction must be made for residual oxygen in the beam path between the output coupler and the detector. For the strongest O 2 lines measured, residual oxygen contributes a broad absorption, independent of the pressure in the sample cell. Line parameters for residual absorption were determined by fits to a Lorentzian lineshape. (iii) A smooth variation in the baseline is evident in Figs. 1, 2, and 3. This results from cavity buildup effects in the ICLAS laser (23) and self-narrowing at longer generation times. The baseline variation, which approximates the output of the laser free of sample absorption, was represented as a low-order polynomial fit. Although pixel sensitivity correction and a slowly varying polynomial should accurately represent the baseline variation, some additional uncorrected baseline fine structure was frequently present, which did not appear to vary with either sample pressure or generation time. This may reflect nonlinearity in the array response and is the major contributor to the residual noise shown in Fig. 3. (iv) The ICLAS apparatus function, which describes the limiting lineshape of a very narrow spectral feature, is primarily a property of the monochromator and, to some degree, of the optical alignment into the monochromator. The apparatus function is recorded by placing a solid e´talon at normal incidence into the laser cavity. The resulting spectrum consists of a series of sharp peaks across the pixel array. These peaks are fit to fourth-order Hermite–Gaussian polynomials. 3 The param-

eters usually vary slightly across the array and are interpolated to find values for each pixel as the spectrum is being fitted. Finally, groups of individual lines were fitted to determine broadening parameters. Ritter and Wilkinson (4) noted that using a Galatry lineshape, which includes the Dicke-narrowing correction, is necessary for accurate results. We found the Dicke-narrowing parameter z to be small, varying from 0.005 cm ⫺1 at p ⫽ 0.1 Torr to less than 0.001 cm ⫺1 at p ⬎ 1 Torr. Therefore, all low-pressure intensity measurements were fitted with the Galatry profile, but at the higher pressures (up to 1 atm) used for the pressure-broadening measurements, the Voigt lineshape (which is more efficiently computed in our fitting algorithm) sufficiently represented the data. The quality of the fit is shown in Fig. 3, wherein it may be seen that the mean residuals (experimental ⫺ simulated spectrum) are less than 1%. The results of intensity measurements are presented in Tables 1a–1c, and the collision-broadening coefficients can be found in Tables 1d–1f. All results are discussed in the following section. The last-digit uncertainties shown in the tables are 2␴ limits derived from the standard deviation of the fits. The calculations and error estimates were spot-checked and verified using several different statistical analysis programs.

TABLE 1b Intensities (ⴛ10 ⴚ5 cm ⴚ2/atm) for Intermediate N Transitions in Oxygen A Band

3

Since the shape of the frequency distribution can vary significantly as a function of peak position with respect to individual array pixels, an analytical representation of the apparatus function is necessary for highest accuracy. In an aberration-free spectrograph, the apparatus function closely approximates a Gaussian. Thus a very natural choice for the analytical representation is a set of Hermite–Gaussian polynomials, as are commonly used to describe transverse laser cavity modes. These are a complete set of functions well localized around the center position, so we expect that such a series will converge rapidly.

Note. Last digit uncertainties (2␴) are shown in parentheses.

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TABLE 1c Intensities (ⴛ10 ⴚ8 cm ⴚ2/atm) for High N and Hot-Band Transitions in Oxygen A Band

TABLE 1e N 2-Broadening Parameters (cm ⴚ1/atm) for Oxygen Isotopomers

Note. Last digit uncertainties (2␴) are shown in parentheses.

III. RESULTS

(a) Intensity Measurements on 16O 2 The principal advantage of ICLAS in line parameter measurements is its ability to obtain quantitative data for very weakly absorbing transitions, as has been amply demonstrated in earlier measurements (14 –18). To validate the accuracy of our apparatus and data analysis methods, however, it is necessary to compare our results with benchmark measurements using other techniques. In the case of the oxygen A band, several sets of measurements are available using long-path absorption cells (4, 5, 7). Because the minimum usable effective path length of ICLAS exceeds the path length range of most passive absorption studies, the weakest lines reported in other studies are often too strongly absorbed to be measured accurately with our instrument. For example, intensity parameters derived for the N ⫽ 13 and N ⫽ 15 doublets shown in Fig. 1 are given in Table 1a and compared with previous TABLE 1d Comparison of Measured N 2-Broadening Parameters (cm ⴚ1/atm) for Oxygen A-Band Lines

Note. Last digit uncertainties (2␴) are shown in parentheses.

Note. Last digit uncertainties (2␴) are shown in parentheses following the reported value. The wavenumber of each line (in cm ⫺1) is given in parentheses below the line-broadening value.

measurements on the same transitions by Brown (LB, 7), Schermaul and Learner (SL, 5), and Ritter and Wilkinson (RW, 4). The latter measurements are in good mutual agreement to within their quoted standard errors. Our measurements using the ring cavity are generally 5– 8% smaller than the average of the other measurements. Several factors that could contribute to this discrepancy were investigated. Since the laser typically requires ⬃5 ␮s to reach threshold once the first AOM is switched open, this quantity was subtracted from t g in the analysis to give accurate effective path lengths in the absorption cell. Scattered light or overlapping orders from the echelle grating in the spectrograph may possibly cause excess light to fall on the detector, producing an artificially high values of I/I 0 . More consistent results are obtained with higher N (and therefore less strongly absorbing transitions). Using the ring configuration, we carried out intensity measurements on N ⫽ 23 and N ⫽ 25 doublets in the oxygen A band. Figure 4 shows intensity versus t g for these lines. For the weaker transitions with N ⫽ 25, the intensity remains linear up to t g ⫽ 500 ␮s; for the stronger (N ⫽ 23) transitions, the intensity deviates significantly when t g ⬎ 280 ␮s, which indicates that the intensity measurement of the RC-ICLAS is linear when the

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ICLAS MEASUREMENTS IN THE OXYGEN A BAND

193

TABLE 1f Self-Broadening Parameters (cm ⴚ1/atm) for High N and Hot-Band Transitions in Oxygen A Band

FIG. 5. Linear fit of intensity versus pressure of N ⫽ 23 and N ⫽ 25 doublets. Derived intensity parameters are listed in Table 1b.

Note. Last digit uncertainties (2␴) are shown in parentheses.

peak absorption is below 70%. A similar saturation effect was noted in Ref. (13). Note that only the solid data points in the figure are used in the linear fit to derive intensity parameters. Figure 5 shows intensity measurements at fixed generation time while varying pressure in the sample cell, which is linear over the pressure range investigated. Table 1b lists the results for the N ⫽ 23 and N ⫽ 25 doublets and compares them with the work of Ritter and Wilkinson (RW, 4), which is the only other reported intensity measurement for these lines. Our ICLAS measurements bracket the RW values with those measured by varying t g about 2–5% smaller and those measured by

FIG. 4. Linear fit of intensity vs. t g for N ⫽ 23 and N ⫽ 25 doublets. Solid data points are included in the linear fit to derive the intensity parameters listed in Table 1b. Data marked by open triangles (‚ and ƒ) are excluded from the fit because of saturation effects.

varying pressure about 2–5% larger. In general, the three data sets agree within the 2␴ standard errors. In the case of high N and hot-band transitions, no other measurements are available for comparison, but calculated intensity values for these lines are reported in the HITRAN database (24). Table 1c reports the intensities of three high N transitions and four hot-band transitions near 12 950 cm ⫺1. While the results for the high N transitions agree with the HITRAN values to within 3%, the intensities for the hot-band transitions are significantly larger (8 –10%) than the values given in HITRAN. This suggests that the value of the transition moment used in HITRAN for the hot-band (v⬘ ⫽ v⬙ ⫽ 1) transitions may be too low. (b) Nitrogen-Broadening Parameters for 16O 2, 18O 2, and 16 18 O O Pressure-broadened linewidths for the N ⫽ 9 and N ⫽ 11 A-band doublets were measured using both the LC and RC configurations. The pressure-broadening coefficients for the 16 O 2 lines are given in Table 1d and compared with Brown’s measurements for the same set of lines (7). The mutual agreement between our two data sets and the LB measurements is excellent, thereby further validating the ICLAS technique and our fitting procedure for linewidth determination. Nitrogen-broadening coefficients were determined for all the isotopic lines of oxygen shown in Fig. 2 using the LC configuration. The results are given in Table 1e. For the most part, there are no significant differences among the broadening coefficients (for a given N) for 16O 2, 18O 2, and 16O 18O. In Schermaul’s study (6) of “self” broadening of natural abundance 16O 18O, i.e., broadening by 16O 2, he found that the broadening coefficients were some 25% smaller than for selfbroadening of the 16O 2 lines. This is not easily explained in view of the fact that the mixed isotope has all rotational levels occupied (as can readily be seen in Fig. 2), while in the 16 –16

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FIG. 6. Self-broadened linewidths for high N and hot band transitions measured in the RC configuration. See Table 1f for broadening coefficients derived from these data. The prefix “1” indicates the A-band fundamental (0 4 0), the prefix “2” indicates the hot band (1 4 1).

and 18 –18 isotopes, only the odd rotational levels are allowed by symmetry. A simple “energy-gap” scaling argument might suggest that the higher “density of states” and correspondingly smaller energy spacings between adjacent rotational levels in the mixed isotope of oxygen would facilitate rotationally inelastic collisions, which are primarily responsible for collision broadening. While this could be partially mitigated by the lack of “resonance” between 16O 18O and 16O 2, such “resonance defects” are small and should not make a large difference for inelastic collision probabilities. The fact that all three isotopic varieties of oxygen have similar self-broadening coefficients suggests a simple interpretation, which is that collision-propensity rules dictate that ⌬N ⫽ 2 inelastic transitions are dominant in the rotational relaxation matrix for all three isotopic species. Rotational levels in the mixed isotope are therefore not strongly coupled to N levels of opposite parity in the manifold.

FIG. 7. Self-broadening coefficients for the O 2 A band, including measurements by Ritter and Wilkinson (open symbols, 4) and high N transitions measured in this work (filled symbols). R-branch transitions (‚) have 具N典 ⫽ odd integer ⫹ 0.5 and P-branch transitions (ƒ and ) have 具N典 ⫽ odd integer ⫺ 0.5. Separate polynomial fits are shown for the P-branch lines (solid curve) and R-branch lines (dotted curve), indicating that P-branch linewidths are marginally larger than R-branch linewidths.

angular-momentum-based scaling laws, as discussed in the following section. Self-broadening coefficients for the hot-band (v⬘ ⫽ 1) 4 (v⬙ ⫽ 1) transitions, shown in Table 1f, are somewhat larger than the values for corresponding rotational lines of the (v⬘ ⫽ 0) 4 (v⬙ ⫽ 0) band, also given in Table 1f. A couple of simple and semiquantitative explanations may be offered to explain this effect. The first derives from the fact that collision broadening in oxygen results largely from quadrupole interactions, and the quadrupole moment increases with the average size of the molecule. A simple calculation of the outer turning point in the v ⫽ 1 level, using the well-known oxygen molecular potential, gives a 4% increase for ␦ 具r max典 in going from v ⫽ 0 to v ⫽ 1, which is comparable to the increase seen in the line-broadening

(c) Self-Broadening of High N and “Hot-Band” Transitions The extremely high sensitivity of ICLAS, as compared with other measurement techniques, makes it possible to obtain data on extremely weak transitions, such as those originating from levels with low-thermal populations, viz., high-rotational states and vibrationally excited levels. Linewidth measurements on such transitions are displayed in Fig. 6, and the results are given in Table 1f. Again, measurements using the LC and RC configurations are in very good agreement with each other. As shown in Figs. 7 and 8, the high N results extrapolate smoothly from previous measurements (4, 7), which did not access levels above N ⫽ 33. This monotonic decrease with N is typical of

FIG. 8. Best fit of ECS–EP scaling law to collision-broadening data shown in Fig. 7 with the parameters given in the text.

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ICLAS MEASUREMENTS IN THE OXYGEN A BAND

coefficient. In addition, oxygen in vibrationally excited levels may undergo exchange of vibrational energy in collisions with ground state molecules, which could contribute to the hot-band linewidths. Vibrational deactivation by this “energy swap” mechanism occurs at a rate of 0.6 –2 ⫻ 10 6 s ⫺1 Torr ⫺1, which corresponds to a broadening contribution of 0.0024 – 0.008 cm ⫺1/atm, also comparable with the difference between v ⫽ 0 and v ⫽ 1 broadening coefficients.

tributions from the basis rates. The ECS scaling relationship was first proposed by DePristo et al. (25b) in the following form for downward-going population-transfer rates for a diatomic molecule, k共 j 4 j⬘兩T兲 ⫽ 关 j⬘兴

冘 k共0 4 L兩T兲关L兴冉 L

j j⬘ L 0 0 0

冊 A共 jj⬘兩L兲, 2

[3]

IV. DISCUSSION AND CONCLUSIONS

Pressure broadening of molecular absorption lines includes contributions from all collision-induced processes affecting the upper and lower levels of the transition. The number of such processes can be very large, even for a simple molecule such as oxygen. The density matrix of a system describes its current state and is completely measurable in principle. The number of elements in the density matrix is equal to the square of the number of accessible levels and sublevels. Since collisions can transform the value of one density matrix element to a number of possible destination states, the size of the rate matrix can increase even faster than the square of the number of accessible levels, even after all symmetry-based simplifications have been applied. Measurable collision properties include single-level properties, such as depopulation and elastic reorientation rates, and two-level properties such as linewidths, lineshifts, stateto-state inelastic population transfer rates, coherence transfer rates, and rovibrational dephasing rates. These rates are necessary to understand energy flow in molecules, to infer densities and partial pressures in reactive systems and in planetary atmospheres, and even to explain population inversions seen in interstellar molecules. They also afford a stringent test of ab initio and empirical intermolecular potential functions, methods for scattering calculations, and our understanding of angular momentum symmetries. Even if experimental measurements could be easily reproduced using the largest possible ab initio-based scattering calculation, doing so would not be very enlightening, would not generalize very well to new systems, and in fact is not feasible for any but the simplest room-temperature systems. Several approaches have been developed to reduce the large number of measurable parameters to a few parameters in a sensible physical model (25). These vary from empirical energy-gap models to more complete theory-based angular momentum scaling laws. A tractable starting point for angular momentum scaling is the infinite-order sudden (IOS) approximation. The IOS approximation, which assumes a very short collision duration, expresses the rates for all possible inelastic processes in terms of a small number of population transfer rates, called basis rates [k(0 4 j兩T)], between each angular momentum state and the rotationless state. Since this simple model has proven inadequate to represent many of the available measurements, the IOS has been elaborated to an energycorrected sudden (ECS) approximation that corrects the con-

where j⬘ ⬍ j, (:::) is a 3-J symbol, and [ j] ⫽ 2j ⫹ 1. Detailed balance is applied to obtain upward rates: 共2j ⫹ 1兲e ⫺E j/k BT k共 j⬘ 4 j兩T兲

[4]

⫽ 共2j⬘ ⫹ 1兲e ⫺E j⬘/k BT k共 j 4 j⬘兩T兲. Without the factor A( jj⬘兩L), Eq. [3] is the exact result in the IOS limit. A( jj⬘兩L) is an adiabaticity factor or energy correction that takes into account the duration of the collision. DePristo et al. (24) derived an approximation for this factor as A 共DP兲 共 jj⬘兩L兲 ⫽

冋 册 ⍀ DP j ⍀ LDP

2

共 j⬘ ⬍ j兲,

[5]

where

冋

⍀ DP ⫽ 1⫹ j

1 6

冉

␻ j, j⫺⌬j ␶ c 2

冊册 2

⫺1

[6]

␶ c ⫽ l c /V៮ R

[7]

V៮ R ⫽ 共8k B T/ ␲␮ 兲 1/ 2 ,

[8]

and ⌬j is the step to the closest collisionally accessible level. The collision duration ␶ c is often re-expressed in terms of an effective impact parameter l c , which has been found to be reasonably independent of temperature (26a, b). The effective impact parameter is the only empirically adjusted parameter in A( jj⬘兩L). The effective energy gap ␻ j, j⫺⌬j is expressed in units of radians/s and is given by

␻ j, j⫺⌬j ⫽ 共E j ⫺ E j⫺⌬j 兲/ប.

[9]

Rigid rotor approximations to the energy levels are sufficient for this type of scaling analysis. The adiabaticity factor plays a crucial role in accurately modeling systems dominated by short-range collisions. This is clearly the case for collisions with oxygen when neither collision partner has a dipole moment. Both Bonamy et al. (27) and DePristo et al. (24) use approximations to derive adiabaticity factors, but the behavior of the Bonamy factor is more reason-

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196

YANG ET AL.

able for high J lines. The Bonamy factor has recently been applied to the CO 2–Ar system by Thibault et al. (28). They computed ab initio basis rates from a good potential and compared scaling laws to these computations and to experimental data. In this case, the Bonamy factor was found to represent the data more accurately. This factor is given by

A 共B兲 共 jj⬘兩L兲 ⫽

⍀ Bj ⍀ LB

共 j⬘ ⬍ j兲,

[10]

where

冋

⍀ Bj ⫽ 1 ⫹

1 3

冉

␻ j, j⫺⌬j ␶ c 2

冊册 2

⫺1

.

[11]

When interactions are dominated by long-range dipole– dipole interactions, or for transfer rates between low j levels, the amount of angular momentum transferred in a collision is small, and it is possible to determine the basis rates from experimental data (29 –31). In short-range collisions, the amount of angular momentum transferred may be large. In this case an empirical scaling law is often used to determine the basis rates from a still smaller number of parameters (26). The ECS-EP basis rate scaling is

k共0 4 L兩T兲 ⫽ A 0

冉 冊 T0 T

N

e ⫺ ␤ E L/k BT . 关L共L ⫹ 1兲兴 ␥

1 2

冘 k共 j 4 j 兩T兲 ⫹ 21 冘 k共 j 4 j 兩T兲 1

j⫽j 1

1 ⫽ 关k j1 共T兲 ⫹ k j2 共T兲兴, 2

k ⬙N ⫹ k⬘N ⫺1 ⬎ k ⬙N ⫺1 ⫹ k⬘N

or

k ⬙N ⫺ k ⬙N ⫺1 ⬎ k⬘N ⫺ k⬘N ⫺1 .

[12]

All basis rates are determined by four parameters: an overall scale factor A 0 , a temperature scaling exponent N, the L scaling exponent ␥, and the L cutoff factor ␤. The ECS-P form takes ␤ ⫽ 0. Linewidths, which are coherence properties, have been frequently approximated by purely inelastic contributions, neglecting purely M-changing collision events and rovibrational dephasing. Using this approximation, the linewidth can be expressed as

␥ 共 j 1 7 j 2兲 ⫽

(32) with B⬙ ⫽ 1.42629 cm ⫺1. In the b 1 ⌺ g⫹ (v⬘ ⫽ 0) upper state, J⬘ ⫽ N⬘ and B⬘ ⫽ 1.400 cm ⫺1 (33). To the extent that spin–rotation coupling is smaller than the rotational level spacing, we may expect that the electron spin in the triplet ground state fine-structure component is not changed by collisions. This assumption was made in a detailed analysis of the M dependence of EPR linewidths (30, 31). If that is the case, the number of collisionally accessible levels is the same in the ground state as it is in the b 1 ⌺ g⫹ (v⬘ ⫽ 0) state. This assumption is tested and supported in Fig. 7, which combines our 16O 2 high N self-broadening data with that of Ritter and Wilkinson (4). If collisional behavior is independent of fine structure, and the collision dynamics of the upper state are very similar to those of the ground state, the observed linewidths will depend only on the average N quantum number of the transition. Figure 7 shows the principal dependence to be on the average value of N, except for transitions at very low average N. The lowest N values in the ground state are those with the strongest spin–rotation coupling. Because of this decoupling, in applying ECS scaling we may use N for the angular momentum value rather than J. The smoothed fit to the linewidths, shown in Fig. 7, is slightly larger for P lines than for R lines, especially at low N. This can be interpreted, using Eq. [13], as

2

j⫽j 2

[13]

where k j (T) is the total depopulation rate out of level j. In the X 3 ⌺ g⫺ (v⬙ ⫽ 0) state, oxygen levels are labeled by the rotational angular momentum N and by the total angular momentum J ⫽ N ⫹ S. For use in the scaling laws, we have approximated the ground state energies up to N ⫽ 60 by BN(N ⫹ 1). Energies were taken from the JPL line catalog

If the relaxation rates in the ground and excited states are qualitatively similar, but the rates in one level are uniformly greater or smaller than those in the other level, then the above relationship implies that the ground state rates are the faster ones. We see that the deviation between P- and R-branch rates is greater at low N. The increased mixing of fine-structure components at low N in the ground state may be responsible for the faster relaxation by opening up a spin-changing collisional pathway not available in the excited state. The M dependence of the linewidths in the EPR study (33) contradicts the assumption implicit in Eq. [13] that all contributions to oxygen pressure broadening arise solely from inelastic processes. Any M dependence of the linewidths is proof that pure M-changing collisions (⌬N ⫽ ⌬J ⫽ 0) are occurring, because only such collisions may contain an M dependence (29 –31). Analysis of the EPR linewidths indicates that pure M-changing collisions constitute about 8 –10% of the collision events. Nevertheless, these processes may be expected to occur at approximately the same rates in the upper and lower states and can be accounted for by small adjustments in the ECS parameters resulting from the fit. The best ECS-EP fits to the combined data with the Bonamy form for the correction function are shown in Fig. 8. The parameters that provide the best fit, with their associated last-digit uncertainties (in parentheses), are

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ICLAS MEASUREMENTS IN THE OXYGEN A BAND

l c ⫽ 1.56共1兲 Å

␥ ⫽ 0.728共2兲 ␤ ⫽ 0.055共1兲 A 0 ⫽ 0.162共1兲 cm ⫺1/atm. The basis rates from this model could be used to predict other experimental results, such as state-to-state population transfer rates. Wodtke and co-workers (34) have used stimulated-emission pumping (a.k.a. “Pump-Dump-Probe”) techniques to study relaxation from high-vibrational levels of oxygen, but without rotational state resolution. Measurements of state-to-state rovibrational energy transfer rates and propensity rules in oxygen would provide a benchmark measurement for these rates, and thereby improve our understanding of pressure broadening in oxygen’s atmospheric absorption bands, as has been shown to be the case for ozone (35). ACKNOWLEDGMENT The ICLAS instrument was funded through a Defense University Research Instrumentation Grant DAAG55-97-1-0040. The research using this instrument is supported by Grant NAG5-3977 from NASA’s Upper Atmosphere Research program, Mission to Planet Earth, and by the Alliance for Global Sustainability project, “Spectroscopic Approach to Local and Global Management of the Earth’s Atmosphere.” Dr. Canagaratna was supported by a Dreyfus Foundation Postdoctoral Fellowship in Environmental Chemistry. Dr. Yang has received partial support from NSF Grant CHE97-30852 (to R.W.F.).

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