Frequency-Stabilized Cavity Ring-Down Spectroscopy of O₂ and CO₂ to Support Atmospheric Remote Sensing Citation Long, David Alexander (2012) Frequency-Stabilized Cavity Ring-Down Spectroscopy of O₂ and CO₂ to Support Atmospheric Remote Sensing. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CTFH-GZ09. https://resolver.caltech.edu/CaltechTHESIS:06282011-172416252 Abstract Recent remote-sensing satellite missions have aimed to measure global greenhouse gas concentrations with precisions as demanding as 0.25%. These high-resolution measurements should allow for the quantification of carbon sources and sinks, thus, allowing for a considerable reduction in present carbon cycle uncertainties. To achieve these unprecedented measurement goals will require the most precise body of spectroscopic reference data (i.e., laboratory measurements) ever assembled. In order to aid these missions, we have measured ultraprecise spectroscopic parameters for the (30012) ←(00001) CO 2 band at 1.57 µm and the O 2 A -band at 0.76 µm. These near-infrared transitions are utilized in recent greenhouse gas monitoring missions, with the A -band being employed to derive pressure and temperature profiles. In these investigations we have employed frequency-stabilized cavity ring-down spectroscopy (FS-CRDS), a novel ultrasensitive spectroscopic technique. In the O 2 A -band we have measured magnetic dipole line parameters for 16 O 2 as well as each of the rare isotopologues and have produced calculated, HITRAN-style line lists. Due to the clear presence of collisional narrowing in the spectra, we have utilized the Galatry line profile in these studies and have reported narrowing parameters under self- and air-broadened conditions. We anticipate that the use of these spectral parameters will greatly reduce the uncertainties of atmospheric remote-sensing retrievals. In addition, the spectral fidelity of FS-CRDS allowed us to observe and quantify unresolved hyperfine structure for the 17 O-containing isotopologues. Furthermore, the high sensitivity of FS-CRDS enabled measurements of ultraweak ( S ~10 −30 cm molec. −1 ) electric quadrupole transitions in the A -band, many of which had not previously been observed. Recently we have begun a series of studies of the near-infrared CO 2 transitions. Measurements at low pressures (<40 kPa) have revealed the simultaneous presence of Dicke narrowing and speed dependence of collisional broadening and shifting. In addition, we have demonstrated that the use of the simple Voigt profile (which neglects these effects) in the pressure range will lead to several percent biases in the retrieved Lorentzian width and spectral area. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Oxygen; Carbon Dioxide; Remote Sensing; Cavity Ring-Down Spectroscopy; Collisional Narrowing; Frequency-Stabilized Cavity Ring-Down Spectroscopy; A-band; Near-Infrared Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Okumura, Mitchio Thesis Committee: Blake, Geoffrey A. (chair) Miller, Thomas F. McKoy, Basil Vincent Miller, Charles E. Okumura, Mitchio Defense Date: 8 August 2011 Funders: Funding Agency Grant Number National Science Foundation Graduate Research Fellowship National Defense Science and Engineering Graduate Fellowship UNSPECIFIED NASA Upper Atmospheric Research Program NNG06GD88G NASA Upper Atmospheric Research Program NNX09AE21G NASA Atmospheric Carbon Observations from Space (ACOS) 104127-04.02.02 NIST Greenhouse Gas Measurements and Climate Research Program UNSPECIFIED Record Number: CaltechTHESIS:06282011-172416252 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06282011-172416252 DOI: 10.7907/CTFH-GZ09 Related URLs: URL URL Type Description http://dx.doi.org/10.1016/j.jqsrt.2010.05.011 DOI UNSPECIFIED http://dx.doi.org/10.1016/j.jqsrt.2011.07.002 DOI UNSPECIFIED http://dx.doi.org/10.1103/PhysRevA.81.064502 DOI UNSPECIFIED http://dx.doi.org/10.1103/PhysRevA.80.042513 DOI UNSPECIFIED http://dx.doi.org/10.1063/1.3624527 DOI UNSPECIFIED http://dx.doi.org/10.1007/s00340-011-4518-z DOI UNSPECIFIED Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6531 Collection: CaltechTHESIS Deposited By: David Long Deposited On: 25 Aug 2011 18:48 Last Modified: 03 Oct 2019 23:52 Thesis Files Preview PDF - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

Frequency-Stabilized Cavity Ring-Down Spectroscopy of O2 and CO2 to Support Atmospheric Remote Sensing Thesis by David Alexander Long In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2012 (Defended August 8 2011) ii © 2012 David Alexander Long All Rights Reserved iii Acknowledgements This dissertation is dedicated first and foremost to my family. Without their love and support, this certainly would not have been possible. I would like to especially thank my parents and my brother; you have made the past 26 years truly wonderful. Scientifically I will always be indebted to Mitchio Okumura, Joe Hodges, and Chip Miller; I have learned so much from each of you. Mitchio, thank you for allowing me the freedom to explore the physical chemistry world, but still always being there for discussion and guidance. You taught me how to develop an idea and carry it through. Chip, your enthusiasm is truly infectious and your ability to convey the importance of fundamental science to a broad audience is inspiring. I hope that my collaborations with both of you will span well into the future. Joe, your impact on this work cannot be overestimated. I am so grateful to you for the time you have spent with me. You were never too busy for a conversation or a little help. In four years you transformed me from an undergraduate with no physical chemistry lab experience into the scientist I am today. I am absolutely thrilled to have the opportunity to continue to work with you once I arrive at NIST. Further I would like to note the contributions of my collaborators on this work: Dan Havey, Linda Brown, Herb Pickett, Pin Chen, David Robichaud, Laurence Yeung, Daniel Lisak, Kasia Bielska, Shanshan Yu, Roger van Zee, Jeremie Courtois, Keith Gillis and the entire OCO/OCO-2 science team. The work in this thesis could not have been completed without your great assistance. As well as my committee, Mitchio, Chip, Geoff Blake, Tom Miller, and Vince McKoy, for their insight and advice. And of course, the entire Okumura group, past and present: Kana Takematsu, Matt Sprague, Nathan Eddingsaas, Aaron Noell, David Robichaud, Laurence Yeung, Kathryn Perez, Thinh Bui, Leah Dodson, Kathleen Spencer, Aileen Hui, and Sigrid Barklund. You iv guys made the office a fun place to be and were always around for a non-blueberry coffee, softball, or a little Ernie’s. I will miss Noyes 101, despite the tight confines and mysterious green carpeting. I’d also like to thank my undergraduate collaborators: Mo Hunsen, Jim Anderson, Jim Hedrick, Jane Frommer, Russell Pratt, and Sheryl Hemkin as well as all of my professors at Kenyon College for the excellent preparation. This work would not have been possible without the education and insight you provided. I arrived at Caltech with an very solid foundation, thank you. Liz, this thesis is for you. You were there for unwavering support, friendship, and love. I couldn’t have done this without you. Your wit, sense of humor, intelligence, and compassion are an inspiration. Thank you for putting up with me. I can’t wait to walk across the stage with you come June. See you in DC! v Abstract Recent remote-sensing satellite missions have aimed to measure global greenhouse gas concentrations with precisions as demanding as 0.25%. These high-resolution measurements should allow for the quantification of carbon sources and sinks, thus, allowing for a considerable reduction in present carbon cycle uncertainties. To achieve these unprecedented measurement goals will require the most precise body of spectroscopic reference data (i.e., laboratory measurements) ever assembled. In order to aid these missions, we have measured ultraprecise spectroscopic parameters for the (30012)←(00001) CO2 band at 1.57 μm and the O2 A-band at 0.76 μm. These nearinfrared transitions are utilized in recent greenhouse gas monitoring missions, with the Aband being employed to derive pressure and temperature profiles. In these investigations we have employed frequency-stabilized cavity ring-down spectroscopy (FS-CRDS), a novel ultrasensitive spectroscopic technique. In the O2 A-band we have measured magnetic dipole line parameters for 16O2 as well as each of the rare isotopologues and have produced calculated, HITRAN-style line lists. Due to the clear presence of collisional narrowing in the spectra, we have utilized the Galatry line profile in these studies and have reported narrowing parameters under self- and air-broadened conditions. We anticipate that the use of these spectral parameters will greatly reduce the uncertainties of atmospheric remote-sensing retrievals. In addition, the spectral fidelity of FS-CRDS allowed us to observe and quantify unresolved hyperfine structure for the 17Ocontaining isotopologues. Furthermore, the high sensitivity of FS-CRDS enabled measurements of ultraweak (S~10−30 cm molec.−1) electric quadrupole transitions in the A-band, many of which had not previously been observed. Recently we have begun a series of studies of the near-infrared CO2 transitions. Measurements at low pressures (<40 kPa) have revealed the simultaneous presence of Dicke narrowing and speed dependence of collisional broadening and shifting. In addition, we have demonstrated that the use of the simple Voigt profile (which neglects these effects) in the pressure range will lead to several percent biases in the retrieved Lorentzian width and spectral area. vi Contents Acknowledgements iii Abstract v 1. Introduction 1 Motivation.............................................................................................................. 2 Background ............................................................................................................ 4 1.1. Cavity Ring-Down Spectroscopy .............................................................. 4 1.2. Frequency-Stabilized Cavity Ring-Down Spectroscopy ........................... 5 References.............................................................................................................. 7 2. O2 A-Band Line Parameters to Support Atmospheric Remote Sensing 8 Abstract .................................................................................................................. 9 2.1. Introduction................................................................................................... 10 2.2. Experiment.................................................................................................... 13 2.3. Results and Discussion ................................................................................. 17 2.3.1. Measured Line Intensities ..................................................................... 17 2.3.2. Standard Intensity Model...................................................................... 19 2.3.3. Comparison of Standard Intensity Model to Existing Databases ......... 20 2.3.4. Comparison of Measurements to Standard Intensity Model................. 21 2.3.5. Broadening Coefficients ....................................................................... 26 2.3.5.1. Air-Broadening Coefficients......................................................... 28 2.3.5.2. Self-Broadening Coefficients........................................................ 29 2.3.6. Collisional Narrowing Parameters for the Galatry Profile ................... 30 2.3.7. Pressure-Shifting Coefficients .............................................................. 35 2.4. Conclusions................................................................................................... 35 Acknowledgements.............................................................................................. 36 References............................................................................................................ 37 Appendix............................................................................................................... 41 3. O2 A-band Line Parameters to Support Atmospheric Remote Sensing. Part II: The Rare Isotopologues 46 Abstract ................................................................................................................ 47 3.1. Introduction................................................................................................... 48 3.2. Experiment.................................................................................................... 50 3.3. Results and Discussion ................................................................................. 52 3.3.1. Line Positions........................................................................................ 53 3.3.2. Line Position Calculations .................................................................... 54 3.3.3. Line Intensities...................................................................................... 55 3.3.4. Calculated Line Parameters for the 16O18O and 16O17O Isotopologues 57 3.4. Conclusions................................................................................................... 58 vii Acknowledgements.............................................................................................. 59 References............................................................................................................ 60 Appendix............................................................................................................... 63 4. Cavity Ring-Down Spectroscopy Measurements of Sub-Doppler Hyperfine Structure 76 Abstract ................................................................................................................ 77 4.1. Introduction................................................................................................... 78 4.2. Experiment.................................................................................................... 78 4.3. Results and Discussion ................................................................................. 80 4.4. Conclusions................................................................................................... 85 Acknowledgements.............................................................................................. 85 References............................................................................................................ 87 5. Laboratory Measurements and Theoretical Calculations of O2 A-band Electric Quadrupole Transitions 89 Abstract ................................................................................................................ 90 5.1. Introduction................................................................................................... 91 5.2. Background ................................................................................................... 92 5.2.1. Line Position Calculations .................................................................... 95 5.3. Experiment.................................................................................................... 97 5.3.1. Experimental Apparatus........................................................................ 98 5.3.2. Short-Term Measurement Statistics and Signal Averaging................ 101 5.3.3. Long-Term Averaging of Spectra....................................................... 102 5.3.4. Effect of Long-Term Averaging on Observed Lineshape .................. 105 5.3.5. Analysis of Measured Spectra ............................................................ 106 5.4. Results and Discussion .............................................................................. 108 5.4.1. Electric Quadrupole Line Intensity Measurements............................ 108 5.4.2. Uncertainty Analysis.......................................................................... 113 5.4.3. Electric Quadrupole Line Intensity Calculations............................... 115 5.5. Conclusions................................................................................................. 117 Acknowledgements............................................................................................ 118 References.......................................................................................................... 119 6. The Air-Broadened, Near-Infrared CO2 Line Shape in the Spectrally Isolated Regime: Evidence of Simultaneous Dicke Narrowing and Speed Dependence 122 Abstract .............................................................................................................. 123 6.1. Introduction................................................................................................. 124 6.2. Experiment.................................................................................................. 125 6.3. Results and Discussion ............................................................................... 127 6.4. Conclusions................................................................................................. 134 Acknowledgements............................................................................................ 135 References.......................................................................................................... 137 viii 7. Frequency-Stabilized Cavity Ring-Down Spectroscopy Measurements of Carbon Dioxide Isotopic Ratios 140 Abstract .............................................................................................................. 141 7.1. Introduction................................................................................................. 142 7.2. Experimental Apparatus.............................................................................. 144 7.3. Results and Discussion ............................................................................... 146 7.4. Conclusions................................................................................................. 153 Acknowledgements............................................................................................ 154 References.......................................................................................................... 155 ix List of Tables 2. O2 A-Band Line Parameters to Support Atmospheric Remote Sensing 1. Experimental conditions ............................................................................................. 16 2. Factors used to calculate γself, γair, ηself, ηair, and δ ....................................................... 27 A1. Measured line parameters ......................................................................................... 41 A2. Calculated line list..................................................................................................... 43 3. O2 A-band Line Parameters to Support Atmospheric Remote Sensing. Part II: The Rare Isotopologues 1. Composition of isotopically enriched samples ........................................................... 51 2. Lower state (3Σg-) molecular constants ....................................................................... 55 3. Upper state (1Σg+) molecular constants ....................................................................... 55 A1. Measured 16O17O line positions ................................................................................ 63 A2. Measured 16O18O line positions ................................................................................ 64 A3. Measured 17O2 line positions..................................................................................... 65 A4. Measured 17O18O line positions ................................................................................ 66 A5. Measured 18O2 line positions..................................................................................... 67 A6. Calculated line list for 16O18O................................................................................... 68 A7. Calculated line list for 16O17O................................................................................... 72 5. Laboratory Measurements and Theoretical Calculations of O2 A-band Electric Quadrupole Transitions 1. Spectroscopic constants for the 16O2 3Σg- and 1Σg+ states............................................ 96 2. Calculated transition frequencies for O2 A-band electric quadrupole transitions ....... 97 3. Measured electric quadrupole line intensities........................................................... 113 4. Calculated electric quadrupole line intensities ......................................................... 116 7. Frequency-Stabilized Cavity Ring-Down Spectroscopy Measurements of Carbon Dioxide Isotopic Ratios 1. CO2 isotopologue transitions utilized in this study.................................................... 143 x List of Figures 1. Introduction 1. Frequency-stabilized cavity ring-down spectroscopy.................................................... 5 2. O2 A-Band Line Parameters to Support Atmospheric Remote Sensing 1. Comparison of cavity ring-down spectroscopy techniques ........................................ 14 2. Measured absorption spectrum and Galatry fit for the R19Q20 transition................. 18 3. Comparison of m-dependence of intensities in HITRAN08 to standard model ......... 22 4. Comparison of experimental intensities to calculations ............................................. 23 5. Air-broadening coefficients as a function of J′........................................................... 28 6. Self-broadening coefficients as a function of J′.......................................................... 29 7. Air-narrowing parameter as a function of J′ ............................................................... 33 8. Self-narrowing parameter as a function of J′.............................................................. 34 3. O2 A-band Line Parameters to Support Atmospheric Remote Sensing. Part II: The Rare Isotopologues 1. Measured R-branch spectrum with an isotopically enriched sample.......................... 49 2. Comparison of experimental intensities to calculations ............................................. 57 4. Cavity Ring-Down Spectroscopy Measurements of Sub-Doppler Hyperfine Structure 1. Experimental Gaussian widths and theoretical Doppler widths ................................. 82 2. Measured spectrum and hyperfine structure fit for the 16O17O R3Q4 transition ........ 84 5. Laboratory Measurements and Theoretical Calculations of O2 A-band Electric Quadrupole Transitions 1. Calculated stick spectrum for magnetic dipole and electric quadrupole transitions... 93 2. Energy level diagram for the 16O2 3Σg- and 1Σg+ states................................................ 94 3. Frequency-stabilized cavity ring-down spectroscopy schematic................................ 98 4. Measured Allan deviation ......................................................................................... 102 5. Effect of long-term spectral averaging on baseline noise......................................... 104 6. Doppler-broadened spectrum of the 16O2 PP(21) magnetic dipole transition ........... 105 7. Measured and calculated electric quadrupole self-broadening coefficients ............. 108 8. Effect of long-term averaging on spectral signal-to-noise ratio ............................... 109 9. Measured spectrum of the TS(5) electric quadrupole transition................................ 110 10. Measured spectrum of the NO(21) electric quadrupole transition............................ 111 11. Measured spectral area as a function of O2 number density.................................... 112 xi 6. The Air-Broadened, Near-Infrared CO2 Line Shape in the Spectrally Isolated Regime: Evidence of Simultaneous Dicke Narrowing and Speed Dependence 1. Measured spectrum with various fits of (300012)←(00001) R16 CO2 transition .... 128 2. Pressure dependence of Lorentzian width retrieved with various line profiles ........ 129 3. Pressure dependence of spectral area retrieved with various line profiles ............... 130 4. Comparison of Lorentzian widths retrieved with various line profiles to HITRAN 131 5. Collisional narrowing frequency as a function of pressure ...................................... 133 6. Measured spectrum and multispectrum fit for the R16 CO2 transition..................... 134 7. Frequency-Stabilized Cavity Ring-Down Spectroscopy Measurements of Carbon Dioxide Isotopic Ratios 1. Measured Allan deviation ......................................................................................... 146 2. Survey spectrum of the 6263–6271 cm-1 region ....................................................... 148 3. Measured spectrum and Galatry fit of a 16O12C16O transition .................................. 150 4. Measured spectrum and Galatry fit of a 16O13C18O transition .................................. 151 1 CHAPTER 1 Introduction 2 Motivation Carbon dioxide (CO2) is the dominant anthropogenic greenhouse gas in the Earth’s atmosphere. During the industrial era the CO2 mixing ratio has risen from 280 ppm (parts per million) to greater than 390 ppm, contributing a radiative forcing of 1.66(17) W m-2 [1]. Fortunately, only about fifty percent of the emitted CO2 has contributed to this increase, with the remainder being absorbed by the oceans and biosphere. Due to the great complexity of the global carbon cycle, the precise identity and location of these carbon sinks is not well known. As a result, it is poorly understood how these carbon sinks will respond to rising temperatures and elevated CO2 mixing ratios and thus, the Earth’s susceptibility to further perturbation. Finding answers to these pressing questions will require extremely precise measurements of CO2 mixing ratios (at or below the 1 ppm level, 0.3%) which assesses variations across a wide range of spatial and temporal scales [2]. To meet this challenging measurement target NASA is presently developing two satellite-based remote sensing missions: the Orbiting Carbon Observatory (OCO-2) [3] and Active Sensing of CO2 Emissions over Nights Days and Seasons (ASCENDS). These satellite missions are ideally suited for this challenge, allowing for uninterrupted measurements with global coverage including oceanic and difficult-to-reach sites where ground-based stations are infeasible. In addition, they will have a high measurement precision (<0.3%) and spatial resolution, thus allowing for identification and quantification of present CO2 sources and sinks. Achieving this daunting measurement target will require the most precise spectroscopic reference data (i.e., laboratory measurements) ever assembled [4]. When this measurement goal was set, in many cases the needed, metrology-level spectroscopic precision was in excess of any reported measurement. In order to achieve its precision targets, OCO must employ entire spectral bands, not just the strongest rovibrational transitions as in earlier missions, thus, ultraprecise reference data are required for 3 thousands of transitions across the three utilized bands (the 1.61 and 2.06 μm bands of CO2 and the 0.76 μm band or A-band of O2). 1 In addition, all weak interfering transitions (e.g. hot bands, combination bands, and rare isotopologue transitions) whose intensities are >0.1% of the weakest main band transitions must be measured and quantified. Thus, a laboratory-based spectroscopic technique is required that is not only extremely sensitive but also exhibits very high resolution and spectral resolution. While discussing the OCO measurement goals in 2005, Dr. Charles Miller of NASA’s Jet Propulsion Laboratory (JPL) outlined three fundamental challenges these daunting targets pose to the spectroscopy community [4]: Can absolute (or even relative) intensities and line shape parameters be measured to the required 0.3% accuracy for a polyatomic molecule? Are there new, unanticipated aspects of the typical line position/line intensity/line shape problem that are revealed at this level of scrutiny? Will these investigations reveal new physics or demonstrate a new paradigm for the spectroscopic parameters needed to reproduce atmospheric remote sensing data quantitatively? These challenges motivated this dissertation. Herein we will present our efforts to address these pressing spectroscopic needs through a series of investigations of near-infrared O2 and CO2 transitions. We have employed frequency-stabilized cavity ring-down spectroscopy (FS-CRDS), a novel ultrasensitive technique which is ideally suited for high precision measurements of spectral line shapes. These studies have produced ultraprecise line lists for the O2 A-band (for both the dominant and rare isotopologues) and have quantified subtle line shape effects in both the A-band and the near-infrared CO2 bands including Dicke narrowing and speed dependence. In addition, we have measured hyperfine structure in the 17Ocontaining isotopologues of O2, previously unobserved electric quadrupole transitions, and preformed precision measurements of rare CO2 isotopologues. 1 Detailed discussion of these bands and their role in remote sensing will follow. 4 Background Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) is an ultrasensitive variant of traditional cw-cavity ring-down spectroscopy which was developed in the laboratory of Dr. Joseph Hodges at the National Institute of Standards and Technology (NIST) in Gaithersburg, MD [5,6]. 1.1. Cavity Ring-Down Spectroscopy Cavity ring-down spectroscopy (CRDS) is an established absorbance technique whereby a gas sample is placed within a high finesse optical cavity [7]. Laser radiation is injected into the cavity until the intracavity power reaches a desired level. The light is then turned off (for cw-CRDS this is generally achieved through the use of an acousto-optic modulator, AOM) and the power within the cavity decays in an exponential fashion. Since the transmission is proportional to the intracavity power, the resulting exponential decay can be recorded by a photodiode located after the rear cavity mirror. The time constant of this decay is related to the intracavity losses due to mirror losses and sample absorbance (i.e., a large absorbance leads to a short time constant):  tr 2[(1  R)  L] (1) where R is the mirror reflectivity (assuming a symmetric cavity with two identical mirrors), α is the absorption coefficient, L in the length of the optical cavity, and tr is the round trip time for light in the cavity (i.e., 2L/c). The resulting decay is only a pure, single exponential if a lone cavity mode is excited by the probe laser. Therefore, for high precision, quantitative CRDS it is important that the probe laser be actively modematched to the optical cavity through a servo. Generally, extremely high reflectivity mirrors are utilized (99.9% reflectivity or higher), which results in an enormous increase in effective path length and a complimentary increase in the signal-to-noise ratio. CRDS possesses two fundamental 5 advantages over traditional long path length direct absorbance techniques. Firstly, since we are measuring the time constant of the decay rather than absolute light intensity, laser power fluctuations have a minimal impact. In addition, the use of an optical cavity allows for a very small gas sample volume (~0.2 L in our case), a feature which will be important for isotopic analyses where expensive, isotopically enriched samples are utilized. 1.2. Frequency-Stabilized Cavity Ring-Down Spectroscopy Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) differs from traditional CRDS in two fundamental aspects [5]. Firstly, FS-CRDS is a single-mode technique, so individual cavity modes are selectively excited, thus ensuring quantitative absorption measurements. Secondly, the optical cavity length is actively stabilized with respect to an external frequency reference (see Figure 1a). FIG. 1. Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS). (a) Schematic of the FS-CRDS instrument. (b) Spectral acquisition scheme. During a scan, the probe laser is locked to successive TEM00 modes of the optical cavity. The stabilized comb of resonant frequencies thus serves as an extremely linear, stable, and accurate frequency axis for the measured spectra. Figure reprinted from [8] with permission of the American Physical Society (2009). 6 During a scan the probe laser is actively locked to a given TEM00 cavity mode. After ring-down measurements have been made, the probe laser frequency is then stepped a distance in frequency space given by the free-spectral range (in our case ~200 MHz) to the next TEM00 mode and relocked. This procedure ensures single-mode operation and due to the locked optical cavity length provides an extremely linear, accurate, and stable frequency axis for our spectra. Detailed descriptions of the locking procedures can be found in Refs. [5,6,9]. Note that the entire system has been automated, allowing for spectra to be collected for several days without user intervention [6]. The high sensitivity and spectral fidelity of FS-CRDS make it ideal for ultraprecise line shape studies. Recent FS-CRDS studies have measured line intensities with relative combined uncertainties less than 0.3% [10], Doppler widths to better than 1 part in 6,000 [8], and transition frequencies with uncertainties less than 0.5 MHz (1.5×10−5 cm−1) [11]. Spectral signal-to-noise ratios as high as 28,000:1 have been achieved for relatively weak near-infrared transitions (S~3×10−26 cm molec. −1, or ~8 orders of magnitude weaker than a typical mid-infrared vibrational transition) [12]. In addition, ultraweak transitions [8] (S<4×10−29 cm molec.-1) and unresolved hyperfine structure [13] have been observed and quantitatively measured. 7 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, H. L. Miller (Ed.), Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. (Cambridge University Press, Cambridge, UK and New York, NY, 2007). P. J. Rayner, D. M. O'Brien, Geophys. Res. Lett. 28, 175-178 (2001). D. Crisp, R. M. Atlas, F. M. Breon, L. R. Brown, J. P. Burrows, P. Ciais, B. J. Connor, S. C. Doney, I. Y. Fung, D. J. Jacob, C. E. Miller, D. O'Brien, S. Pawson, J. T. Randerson, P. Rayner, R. J. Salawitch, S. P. Sander, B. Sen, G. L. Stephens, P. P. Tans, G. C. Toon, P. O. Wennberg, S. C. Wofsy, Y. L. Yung, Z. Kuang, B. Chudasama, G. Sprague, B. Weiss, R. Pollock, D. Kenyon, S. Schroll, In: Trace Constituents in the Troposphere and Lower Stratosphere, (Pergamon-Elsevier Science Ltd, Kidlington, 2004). C. E. Miller, L. R. Brown, R. A. Toth, D. C. Benner, V. M. Devi, C. R. Phys. 6, 876-887 (2005). J. T. Hodges, H. P. Layer, W. W. Miller, G. E. Scace, Rev. Sci. Instrum. 75, 849863 (2004). J. T. Hodges, R. Ciurylo, Rev. Sci. Instrum. 76, 023112 (2005). A. O'Keefe, D. A. G. Deacon, Rev. Sci. Instrum. 59, 2544-2551 (1988). D. A. Long, D. K. Havey, M. Okumura, H. M. Pickett, C. E. Miller, J. T. Hodges, Phys. Rev. A 80, 042513 (2009). D. J. Robichaud, Dissertation (Ph. D.), California Institute of Technology, 2008. D. A. Long, D. K. Havey, M. Okumura, C. E. Miller, J. T. Hodges, J. Quant. Spectrosc. Radiat. Transfer 111, 2021-2036 (2010). D. J. Robichaud, J. T. Hodges, P. Maslowski, L. Y. Yeung, M. Okumura, C. E. Miller, L. R. Brown, J. Mol. Spectrosc. 251, 27-37 (2008). D. A. Long, M. Okumura, C. E. Miller, J. T. Hodges, Appl. Phys. B, In press (2011). D. A. Long, D. K. Havey, M. Okumura, C. E. Miller, J. T. Hodges, Phys. Rev. A 81, 064502 (2010). 8 CHAPTER 2 O2 A-Band Line Parameters to Support Atmospheric Remote Sensing This chapter was published as D. A. Long, D. K. Havey, M. Okumura, C. E. Miller, and J. T. Hodges, J. Quant. Spectrosc. Radiat. Transfer, 111, 2021–2036 (2010). Copyright 2010 by Elsevier B.V. 9 Abstract Numerous satellite and ground-based remote sensing measurements rely on the ability to calculate O2 A-band [b1Σg+←X3Σg-(0,0)] spectra from line parameters, with combined relative uncertainties below 0.5% required for the most demanding applications. In this work, we combine new 16O2 A-band R-branch measurements with our previous P-branch observations, both of which are based on frequency-stabilized cavity ring-down spectroscopy. The combined set of data spans angular momentum quantum number, J′ up to 46. For these measurements, we quantify a J-dependent quadratic deviation from a standard model of the rotational distribution of the line intensities. We provide calculated transition wave numbers, and intensities for J′ up to 60. The calculated line intensities are derived from a weighted fit of the generalized model to an ensemble of measured data and agree with our measured values to within 0.1% on average, with a relative standard deviation of ~0.3%. We identify an error in the calculated frequency dependence of the O2 A-band line intensities in existing spectroscopic databases. Other reported line shape parameters include a revised set of lower-state energies, self- and air-pressure-broadening coefficients and self- and air-Dicke-narrowing coefficients. We also report a bandintegrated intensity at 296 K of 2.231(7)×10−22 cm molec.−1 and Einstein-A coefficient of 0.0869(3) s−1. Note that Dr. Joseph Hodges of NIST assisted greatly in the preparation of this publication. His contributions were especially significant in sections 2.3.5–2.3.7. 10 2.1. Introduction For fifty years the O2 A-band [b1Σg+←X3Σg-(0,0)] has been used extensively in atmospheric remote sensing. The A-band has been utilized in the SCIAMACHY [1,2], GOME [3], SAGE III [4], ACE-MAESTRO [5], and GOSAT [6] satellite missions to determine pressure and temperature profiles. In addition, the A-band has been applied to remote measurements of cloud-top heights [7], cloud optical properties [8], surface pressure [9,10], and aerosol optical properties [11]. This band is ideally suited to these remote sensing applications for several reasons: O2 in the atmosphere is uniformly distributed with a well-known mixing ratio, many relatively weak A-band transitions are not saturated for long path lengths, and the band is centered at 762 nm (13,125 cm−1) in a window region largely free of spectral interferences. Due to the importance of O2 spectral parameters to remote sensing applications, there have been numerous laboratory studies of the A-band. Thorough literature summaries can be found in Brown and Plymate [12] and Robichaud et al. [13], and as a result we will only discuss a few relevant studies in detail. In 1987, Ritter and Wilkerson [14] measured 54 transitions with a spectrometer comprising a continuous-wave tunable dye laser (<10−4 cm−1 resolution) in a multipass configuration. They reported line strengths, half-widths, and lineshape parameters which formed the basis for the 1996 edition of the HITRAN database [15]. In 2000, Brown and Plymate [12] used the McMath Fourier-transform infrared spectrometer (FT-IR) at Kitt Peak to measure A-band transitions up to J′ = 22 (see Section 3 for notation) with a wave number resolution of 0.02 cm−1. Line intensities were reported with a relative uncertainty of 2%. Also reported were half-widths and their corresponding temperature dependence. These measurements were utilized in HITRAN 2000 [16] and 2004 [17], with the half-width temperature dependence being preserved in the current edition [18]. Schermaul and Learner [19] used FT-IR with a resolution of 0.0069–0.029 cm−1 to measure intensities and lineshape parameters at room temperature and at 198 K. The resulting spectra were fit using a collisionally narrowed Voigt profile, with line intensities reported having uncertainties of ~0.3%. A comparison of these intensity measurements to calculations showed evidence 11 of a J-dependent slope which was attributed to a rotation-vibration, or Herman-Wallis (HW) effect [20]. Predoi-Cross et al. employed FT-IR and a multispectrum fitting procedure to report intensities and lineshape parameters of N2- [21] and self-broadened [22] transitions. The effects of chosen lineshape profile and line mixing were also examined. In recent years, the A-band has been utilized in measurements with demanding precision requirements. In particular, high-resolution measurements have aimed to determine surface pressure to 0.1% [9,10] and CO2 concentration to 0.3% [23,24,25]. These increasingly precise remote sensing measurements, however, have shown that the present uncertainty in O2 spectroscopic reference data ultimately limits the accuracy of the resulting retrievals [9,26]. To address this problem, Robichaud et al. utilized frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) to measure lineshape parameters [13], positions [27], and pressure-shifting coefficients [28] for many transitions up to J′ = 32 in the P-branch of the A-band. The measured positions were tied to the D1 and D2 atomic transitions of 39K, yielding uncertainties less than 1 MHz (3×10−5 cm−1). Transition line shapes were fit using Galatry line profiles, resulting in line intensity relative uncertainties of ~0.1%-0.5%. It was shown in that study and by Yang et al. [29] that the air- and self-broadening coefficients found in HITRAN 2004, which were based on extrapolations of low-J measurements, were inaccurate at higher-J. In addition, FS-CRDS was utilized to probe b1Σg+←X3Σg-(0,0) transitions of rare O2 isotopologues [30]. This collection of FS-CRDS measurements presently forms the basis for HITRAN 2008 for the band intensities of all listed isotopologues, although the rotational dependences were unchanged from HITRAN 2004. In addition, the air- and selfbroadening coefficients, transition frequencies, and pressure-shifting coefficients found in HITRAN 2008 are based on FS-CRDS measurements of Robichaud et al. [13]. Most recently, we have pushed the absolute detection limit of FS-CRDS to enable quantitative measurements of ultraweak absorption features, including electric quadrupole transitions [31] and high-J [32] magnetic dipole transitions of the 16O2 Aband. In these experiments the ring-down cavity length stabilization enabled long-term 12 averaging of spectra, which reduced the detection limit to ~1.810−11 cm−1. The electric quadrupole transitions were measured and their rotational intensity distribution was modeled. As reported in Long et al., the measured intensities, which had relative uncertainties of <9%, were in agreement with the calculations [31]. The high-J magnetic dipole measurements of Havey et al. extended the range of A-band line parameters to J′ = 50 [32]. The intensities of the ultraweak electric quadrupole and high-J magnetic dipole transitions are more than 5 orders of magnitude weaker than the strongest magnetic dipole transitions discussed herein. The success of these high-sensitivity studies enabled the intensity measurements over the relatively wide range of J reported here. Additionally, in our high-J measurements [32], we noted a J-dependent deviation in the line intensities (from a standard model) consistent with that reported by Schermaul and Learner [19] which was assigned to a HW interaction. The preceding FS-CRDS measurements have reduced the uncertainties associated with present A-band spectroscopic parameters. Unfortunately, in these earlier studies the tuning range of the probe laser limited measurements to the P-branch of the O2 A-band. In this work, we present new FS-CRDS measurements of 34 transitions in the R-branch which are combined with our previous P-branch measurements. Intensity data are fit using a standard model which includes a HW effect for the rotational dependence of the intensities. Thus, we confirm and quantify the HW effect and demonstrate its importance with regard to an accurate description of O2 A-band line intensities. Also, we present data illustrating the J- and branch-dependence of the narrowing parameter. This information is intended to enable the remote sensing community to incorporate more realistic line shape profiles that capture narrowing effects. We provide correlations which are based on fits to our measured line shape parameters for line positions, intensities, collisional narrowing parameters, air- and self-broadening coefficients, and pressure-shifting coefficients. These results are summarized in a new list of calculated line parameters up to J = 60. 13 2.2. Experiment Measurements were made using the frequency-stabilized cavity ring-down spectrometer located at the National Institute of Standards and Technology (NIST) in Gaithersburg, MD. The FS-CRDS technique has been previously described in considerable detail [33,34], and thus we will only briefly summarize the technique here. FS-CRDS is similar to traditional CW-CRDS techniques in that both methods utilize a narrow linewidth continuous wave (CW) laser to excite individual optical resonances (modes) of the ring-down cavity. Both enable the determination of sample absorption coefficient from the measured intensity decay rate and avoid the complexities of multiexponential decays caused by broadband laser excitation [35,36,37]. However, the FS-CRDS and CW-CRDS techniques differ in an important way. In the former, the comb of resonant frequencies and thus cavity length are actively locked against a frequency-stabilized reference laser, whereas in the latter, the cavity modes are unlocked and subject to drift as a result of nanometer-scale changes in the cavity length (see Fig. 1). In CW-CRDS, the cavity length or probe laser frequency is typically modulated by an amount slightly more than the free-spectral range (FSR), thus forcing spectral overlap of the narrowband probe laser with the cavity resonant frequencies [38,39,40], and in other CW-CRDS implementations, the cavity length is modulated with a smaller amplitude and servoed to track frequency tuning of the probe laser [41,42]. For FSCRDS, the probe laser selectively excites the TEMq00 transverse mode (where q is a specific longitudinal mode order), and sub-FSR frequency steps are realized by frequency shifting the reference laser (and hence cavity frequency comb) with an acousto-optic modulator. The result is a high-resolution (<1 MHz) and linear spectrum frequency-axis for FS-CRDS that is immune to frequency drift in the probe laser. For the measurements described herein, the reference laser was a frequencystabilized HeNe laser with a long-term (8 h) frequency uncertainty of less than 1 MHz. The probe laser was an external cavity diode laser (ECDL) with a wavelength range of 759-771 nm and an output power of 6-10 mW. Two separate pairs of custom-made dichroic ring-down mirrors were employed for these measurements. Both pairs of mirrors 14 had reflectivity R = 95% at the reference laser wavelength (633 nm). The higher loss mirrors had a nominal reflectivity for the probe wavelength of 99.98%, which corresponds to an effective path length of 3.5 km, given our 0.74 m cavity (finesse ≈ 15,000). While using these mirrors, the noise equivalent absorption coefficient (NEA) was 6×10-10 cm−1 Hz−1/2. For the measurement of weaker transitions, lower-loss mirrors for the probe beam were utilized having a nominal reflectivity of R = 99.997%. The use of these mirrors reduced the NEA to 2.5×10−10 cm−1 Hz−1/2. We measured the cavity freespectral range to be 201.970(10) MHz for the higher-loss mirrors and 202.495(12) MHz for the lower-loss mirrors using the technique described by Lisak and Hodges [43]. FIG. 1. Comparison of cavity ring-down spectroscopy techniques. (a) CW-CRDS method using an unstabilized cavity and modulation of the laser frequency through one cavity free-spectral range (FSR). Alternatively, the laser frequency is step scanned and the cavity length is modulated to force spectral overlap at each mean frequency step. In the former case, the observed spectrum is instrumentally broadened by the mode spacing. In the latter case, transverse mode competition, laser frequency drift and nonlinearity in the laser frequency tuning can distort the observed spectrum. (b) FS-CRDS method using a stabilized resonator. The probe laser is step-scanned through successive TEMq00 modes. The frequency resolution is given by the stability of the cavity resonances, the step size is equal to the cavity FSR (or smaller using an AOM) and the observed spectrum is subject to minimal instrumental broadening. 15 Spectral scans were made over 10 GHz wide windows centered at an individual transition (note that the Doppler width of 16O2 (full-width half-maximum) in the A-band at 296 K is ~0.85 GHz). For measurements with the higher-loss (2×10−4) mirrors, 50 MHz frequency steps were made, while 200 MHz steps were used with the lower-loss (3×10−5) mirrors. Note that the uncertainty in the frequency axis is limited by the frequency-stabilized HeNe reference laser and is therefore less than 1 MHz. Further details on this frequency stability and the experimental scanning procedure can be found elsewhere [13,33,34]. Temperature measurements were made with a NIST-calibrated 2.4 kΩ thermistor in good thermal contact with the ring-down cavity. The uncertainty in the gas temperature was examined in detail by Havey et al. [32]. It was shown that the combined uncertainty is less than 28 mK (15 mK uncertainty in the measurement and 23 mK uncertainty from the presence of a thermal gradient along the cavity). All measurements were made over the temperature range 299 to 300 K. Pressure was measured by two NIST-calibrated capacitance diaphragm gauges having full-scale responses of 13.3 kPa (100 Torr) and 133 kPa (1,000 Torr). We estimate the relative combined standard uncertainty of this pressure measurement to be less than 0.1%. Measurements were made on static samples of either pure O2, synthetic air (mixture of N2 and O2), or a synthetic air mixture having ~2% O2. The pure O2 had >99.9% purity. The synthetic air was a NIST standard reference material (SRM 2659a, hereafter referred to as “NIST standard air”) having a molar composition of 79.28% N2, 20.720 ± 0.043% O2, 0.0029% Ar, 0.00015% H2O, and 0.001% other compounds. The 2% O2 mixture was also a NIST SRM (2657a, hereafter referred to at “NIST 2% O2”) which had a molar composition of 98.05% N2, 1.9734 ± 0.0053% O2, 0.004% Ar, 0.00011% H2O, and 0.001% other compounds. All samples contained O2 at natural terrestrial isotopic abundance (Ia = 0.995262 for 16O2 [18]). Table 1 summarizes the experimental conditions. For our analysis, we calculate the number density, n, of O2 to be n = (xO2p)/(kBT), where xO2 is the mole fraction of O2, T is temperature, p is total pressure, and kB is the Boltzmann constant. 16 TABLE 1. Experimental Conditions for the O2 A-band R-branch FS-CRDS measurements. J' 2-22 24,26,32 34-42 34, 38-42 samplea NIST 2% NIST Air NIST Air Pure O2 mirror losses (10-6) 200 200 30 30 step size (MHz) 50 50 202 202 finesse 15,000 15,000 105,000 105,000 a All measurements were made over the temperatures range of 299 to 300 K and then corrected to Tref = 296 K. Sample pressure was transition dependent [see Tables A1(a-c)]. The ring-down cavity finesse is based on the measured time constant under empty cavity conditions. We modeled the measured FS-CRDS spectra, denoted by [c ( )]1 as the sum of absorption line shape profiles superimposed on a linear baseline, [c ( )]1   ( )  Lb ( ) 1 . Here, c is the speed of light,  is wave number of the probe laser,  is the measured ring-down time constant,  is the ring-down cavity length, and Lb is the effective base cavity loss comprising all losses other than local line absorption  ( 0 ) in which by the gas sample. We determined the line area l by evaluating  d  is the absorption coefficient found by fitting the line shape to the observed spectrum centered at 0 . Measured values of n and l gave the line intensity as S = Al/(cn). Reported results were obtained by fitting the Galatry line profile [44] to our measured spectra. This lineshape is based on a soft-collision model which includes the effects of collisional (Dicke) narrowing [45]. Line areas, Lorentzian full-width at halfmaximum frequency (  L ), the narrowing frequency ( opt ) [45,46], and a linear baseline were treated as fitted parameters while the Doppler width was constrained to its theoretical value. The narrowing frequency is also referred to as the optical frequency of velocity-changing collisions. Generally, the collisional narrowing parameter   opt / p was calculated from the slope of  opt vs. p. However, it was not possible for some transitions to determine  opt with sufficient accuracy. For some blended transitions  opt 17 was constrained based on a previously established J-dependence for  [13,32] while  opt for a few lower pressure spectra was fixed to a value determined by a linear fit to the higher pressure measurements. 2.3. Results and Discussion The O2 A-band [b1Σg+←X3Σg- (0,0)] is forbidden by quantum mechanical electric dipole selection rules, but is allowed as a magnetic dipole transition [47]. The rotational levels of O2 are fully described by three quantum numbers: N, the rotational angular momentum, S, the spin angular momentum, and J, the total angular momentum (J = N + S). The X3Σg- ground state of O2 is split into three energy levels having J′′ = N′′ 1, N′′ + 1, N′′. The b1Σg+ upper state has only one level whereby J′ = N′. For 16O2, N′ must be even and N′′ must be odd due to spin statistics. The A-band transitions can be described as ∆N N′′ ∆J J′′ and occur in four branches: PP, PQ, RR, and RQ. Both the P- (∆N = 1) and R- (∆N = 1) branches are made up of pairs of transitions which are separated by ~2 cm−1. The P-branch transitions (i.e., PP, PQ) begin at 13,120 cm−1 (762 nm) and decrease in wave number with increasing J′. The spacing between sets of pairs increases with increasing J′, whereas, the distance between sets of pairs decreases in the R-branch, forming a bandhead at 13,165 cm−1. Each transition in the four branches can be specified by an angular momentum index, m, where m = J′′ (PP- and PQ-branches) and, m = J′′+1 (RR- and RQbranches). 2.3.1. Measured Line Intensities Measured line intensities are given at natural terrestrial isotopic abundance and were corrected to Tref = 296 K using S (Tref )  S (T )[Q(T ) / Q(Tref )]e  hcE  / k B (1/ Tref 1/ T ) , where E′′ is the lower state energy expressed as a wave number and Q(T) is the total internal partition function of 16O2 for which Q(Tref) = 215.7739. The resulting intensities and other measured line parameters can be found in Tables A1(a-c). 18 FIG. 2. Top panel: Measured absorption spectrum (symbols) and Galatry fit (line) for the R19Q20 O2 Aband transition. The abscissa represents frequency detuning about line center. The gas mixture was NIST 2% O2, and the total pressure, p = 12.7 kPa. Bottom panel: Fit residuals for Galatry and Voigt profiles. The inset of the top panel shows the Lorentzian width (FWHM) vs. total gas concentration, n. The Voigt fit gives a slope (broadening coefficient) 15.4% lower and spectrum area 1.35% greater in comparison to those values obtained from fitting the Galatry profile. Figure 2 gives a typical measured absorption spectrum and Galatry fit, as well as fit residuals for the Galatry and Voigt profiles. The sample conditions correspond to the NIST 2% O2 sample at a total pressure of 12.7 kPa. The ratio of peak absorption losses to root-mean-square baseline variations is ~1,500:1 for this spectrum, making it possible to clearly distinguish between the two fitted lineshapes. Unlike the Galatry profile, which essentially captures the measured line shape at the baseline noise level, the Voigt profile exhibits relatively large symmetric residuals. We attribute these residuals in the Voigt fit to collisional narrowing effects. This phenomenon also leads to an underestimation of the Lorentzian width shown in the inset of Fig. 2 and which is discussed below. The fitted Voigt profile in Fig. 2 yields a spectrum area which is 1.35% greater than that obtained from fits of the Galatry profile. Similar results were found in a 19 FS-CRDS study of six O2 b1Σg+(1)←X3Σg-(0) transitions (B-band) where Lisak et al. [48] showed that the Voigt profile gave spectrum areas differing from 0.3% to 1% in comparison to those based on the Galatry profile. They showed that the Voigt profile induced a decrease in fitted area for all but one transition. The uncertainty in our measured line intensities is obtained from a quadrature sum of the Type A and Type B uncertainties which arise from random effects and systematic effects, respectively. For measured transitions with J′ ≤ 26, the estimated uncertainties are similar to those found in Robichaud et al. [13]. For each of these transitions the relative combined standard uncertainty was between 0.2% and 0.4%. For these transitions below the bandhead, the dominant uncertainties were fit reproducibility and sample composition. Choice of line profile [48], temperature [32], and pressure uncertainties contributed a small amount (less than 0.1%) to the overall reported uncertainty. For weaker transitions near the bandhead (i.e., those with J′ > 26) the uncertainties are much larger than those found in Robichaud et al. [13] and Havey et al. [32] for similar J′. For these transitions, the measurement accuracy was not limited by the signal-to-noise ratio, but rather by the presence of interfering transitions (including many rare isotopologue transitions). These transitions were often blended with the target transition, thus making quantitative fitting difficult. For the rare isotopologue lines, lineshape parameters found in HITRAN 2008 [18] (which are based upon the measurements of Robichaud et al. [30]) were utilized and were not floated in the fitting procedure. Uncertainties in these rare isotopologue line shape parameters were incorporated into the combined standard uncertainties for the bandhead 16O2 transitions. These fitting challenges ultimately led to relative combined standard uncertainties as large as 4%. In these cases, the fitting uncertainties were dominant and contributions from n, P, T, were negligible. 2.3.2. Standard Intensity Model A physical model for the distribution of line intensities within a given band is a valuable tool for predicting and interpreting molecular spectra. The intensity of absorption for a dipole transition in a band of intensity Sb is given by [49] 20 Sm  Cb Sb gm Lm e  Em hc / k BTref  hc / k BTref (1  e m Q(Tref )b ) . (1) In this expression, g is a degeneracy factor (in this case equal to three given the triplet ground state), Lm is the Hönl-London factor (the transition-dependent factor which captures the non-Boltzmann, J-dependence of the intensities), m is the transition wave number,b is the band center wave number, and Cb is a dimensionless normalization constant close to unity which ensures that Σ Sm = Sb. The temperature-independent Einstein-A coefficients, Am(J′), can also be calculated as [49] Am ( J )  8  c Q(Tref ) m2 Sm (2 J   1)e  Em hc / k BTref  hc / k BTref (1  e m ) . (2) In the present case, the set of Lm are the Hönl-London factors are given by Watson [50] and were calculated using the 16O2 molecular parameters of Long et al. [31]. Henceforth, we refer to the preceding two expressions as the standard intensity model. Below, we compare experimentally measured intensities, corrected to the reference temperature, to the standard intensity model. 2.3.3. Comparison of Standard Intensity Model to Existing Databases In principle, the O2 A-band line intensities in HITRAN (since the 1996 edition) are based upon the standard intensity model [49] and the Einstein A-coefficient of 0.0887 s−1 measured by Ritter and Wilkerson [14]. For HITRAN 2008, this value was reduced by 1% [18] based on the measurements of Robichaud et al. [13]. This recent modification had no effect on the J-dependence of the tabulated intensities. Previously, we identified a discrepancy between J-dependent line intensities predicted by the standard intensity model and those found in the HITRAN 2008 database [32]. Here we confirm that the O2 A-band line intensities in HITRAN 2008 do not follow the standard model given by Eq. 1. Further, our analysis indicates that the frequency dependence of the HITRAN 2008 intensities can be reconciled with Eq. 1 provided one introduces the extra factor, m 3 . Denoting the corresponding database intensities by SHT, 21 we empirically find that (m /b )3 SHT / Sm is essentially a transition-independent constant. See Figure 3. It appears that a mistake was made in calculating the J-dependence of the HITRAN line intensities for the O2 A-band. We conclude that the wave number dependence of the HITRAN 2008 O2 A-band line intensities is unphysical, and consequently with regard to the J-dependence of the line intensities, this error masks physical departures from the standard intensity model. Finally, we emphasize that this discrepancy applies to multiple rovibronic bands of 16O2 and other isotopologues in both the HITRAN and GEISA [51,52] databases. 2.3.4. Comparison of Measurements to Standard Intensity Model For the O2 A-band, J-dependent deviations of the intensities from the standard intensity model have been previously observed by Miller et al. [53], Ritter and Wilkerson [14], and Schermaul and Learner [19]. Schermaul and Learner ascribed this to a Herman-Wallis (HW) rotation-vibration interaction [20], while Balasubramanian and Narayanan [54] have attributed these deviations to a series of rotational perturbations (spin uncoupling, orbit rotation interaction and spin-orbit perturbation). Unfortunately, a HW theory applicable to transitions between electronic states is not presently available. Further, by comparison of their rotational perturbation theory to the measurements of Miller et al., Balasubramanian and Narayanan noted only marginal improvement over the standard model [54]. Therefore, the present theoretical basis for this effect is incomplete. In the following we will denote observed deviations from the standard intensity model as HWlike, while acknowledging that other perturbations are likely present. 22 FIG. 3. Comparison of m-dependence of the intensities given by HITRAN 2008, SHT, to those given by the standard intensity model, Sm. Triangles (Case 1) correspond to the ratio S HT / S m and circles (Case 2) are for the ratio, S HT (m /b ) / S m . The average absolute deviation from unity for Case 2 is 210−4, which 3 we assign to round-off error. In Fig. 4 we present a comparison of the standard intensity model to the present measurements and those of Robichaud et al. [13], Havey et al. [32], Brown and Plymate [12], Ritter and Wilkerson [14], and Schermaul and Learner [19]. In our previous work [32], all data was for m < 0 (P-branch) for which the intensity deviations over this range appear linear. We note that the experimentally determined HW-like slope elucidated by comparison to the HITRAN 2008 intensities [32] had the opposite sign and a smaller magnitude than that obtained in comparison to the standard intensity model. This discrepancy is now attributed to the extra wave number factor discussed above. In the present work, we provide new line intensity measurements for m > 0 (R-branch). Below, all of the NIST FS-CRDS O2 A-band data, which includes the present measurements and those of Robichaud et al. [13] and Havey et al. [32], are treated as a single data set spanning m = 46 to 43. After combining this data, it is apparent that the deviation between the measurements and the standard intensity model varies nonlinearly with m. 23 FIG. 4. Comparison of experimental intensities to calculations performed using Eq. 2 with the Watson [50] Hönl-London factors. Note that the Ritter and Wilkerson measurements (R&W) [14] were corrected to 296 K (from 294 K), the Schermaul and Learner measurements (S&L) [19] were corrected to natural isotopic abundance, while the Brown and Plymate measurements (B&P) correspond directly to those reported in the literature [12]. In addition, the NIST and R&W studies employed Galatry line profiles, while the B&P and S&L studies utilized Voigt line profiles. Top panel: The indicated data sets (symbols), and fitted values (lines) for the global fit of the quadratic HW-like functions fi−1(1 + a1m+a2m2)2, where fi−1 is the fitted scale factor for set i, (i= 1 to 4). The dashed line represents another HW-like fit to previous NIST FS-CRDS data [13,32] combined with the present work yielding, a1 = 2.658010−4 and a2 = 3.362210−6. Bottom panel: The relative difference between the NIST FS-CRDS measurements and the latter fit. The error bars for the NIST FS-CRDS data correspond to ±1 standard deviation and include the quadrature sum of random (Type A) and systematic (Type B) uncertainties. We empirically used the general quadratic formula of Watson [55], FHW  (1  a1m  a2 m 2 ) 2 , (3) which was derived for HW corrections to the intensities of rotation-vibration transitions of linear molecules. We did a global, weighted least-squares fit of the ratio Smeas,i/Sm = fiFHW, using this model and including all four sets of measurements shown in Fig. 4. Here, Smeas,i represents the observed intensities for measurement set i. The coefficients a1 24 and a2 were common fitted parameters, and each measurement set was independently scaled by the fitted parameter, fi, to account for systematic deviations between the experiments. The results are given by the solid curves in the upper panel of Fig. 4. As previously shown [19,32], all of the data exhibit a HW-like effect. Also, the global fit reveals a spread of ~2% in the scaling factors which is indicative of systematic differences between the measured the line intensities. We repeated the fit of the quadratic FHW function to the ratio Smeas/Sm, using only the NIST FS-CRDS data. This result, which is shown in Fig. 4 as the dashed line, differs only slightly from the global fit, yielding a1 = 2.658(88)×10−4 and a2 = 3.36(44)×10−6. Tipping and Bouanich [56] discussed how for dipole transitions, FHW can be expressed as an expansion in powers of m given by  FHW  1   e j m j , (4) j 1 where the jth order coefficient ej is of order  j . Here,  is the ratio of rotational to vibrational spacing equal to (2Be/e). We calculate  = 1.82103, using Be = 1.4377 cm1 and e = 1580.19 cm1 [57] for the X3Σg lower state of 16O2. Expanding Eq. 4 and using our fitted values for a1 and a2 gives the measured expansion coefficients, e1 = 2a1 = 5.310−4, e2 = (a12 + 2a2) = 6.8106, e3 = 2a1a2 = 1.8109 and e4 = a22 = 1.11011, which are of order  ,  ,  , and  , respectively. This order-of-magnitude analysis leads us to conclude that our measured HW-like J-dependent correction factor is consistent with the spectroscopic constants of O2 and the expansion given by Eq. 4. However, when we employ the rotationally perturbed Hönl-London factors of Balasubramanian and Narayanan [54] the observed intensity deviations of the FS-CRDS data are reduced by roughly a factor of 2, suggesting that this mechanism also plays an important role. Therefore, we conclude that a more comprehensive theoretical analysis of the relative contributions of various perturbations is warranted. The precision of the quadratic fit to the FS-CRDS O2 A-band intensity data is illustrated in the bottom panel of Fig. 4, where the standard deviation of the relative difference between the measurements and the theoretical values (over the range m = -38 25 to 38) is 0.34%, with a mean value of 0.1% (m = 46 to 43). In this figure, the three sets of data which comprise the FS-CRDS intensity measurements are delineated and are self-consistent at the 0.3% level. We note that the FS-CRDS intensity measurements reported by Havey et al. [32] comprising the five O2 A-band transitions P47P47 through P51P51, the weakest of which has a measured intensity of 1.10(31)10−30 cm molec. −1 were not included in the above analysis because of their relatively high uncertainties. Nevertheless, these measured intensities agree with the calculated values given in Table A2 to within their measurement uncertainty, thus confirming the extrapolation of the intensity calculation out to at least J  50. The fit described above yielded a band intensity of Sb(296 K) = 2.231(7)×10−22 cm molec.−1, from which we obtained the band-integrated Einstein-A coefficient, Ab =0.0869(3) s−1. This band intensity is lower than those calculated for the previous experimental studies, with Brown and Plymate’s [12] and Schermaul and Learner’s [19] measurements resulting in band intensities ~0.7%-1.5% higher than the present study, and Ritter and Wilkerson’s [14] temperature-corrected value being ~2% higher than our present measurements. See Table 4 of Ref. [22] for a complete collection of earlier measurements of the band-integrated Einstein-A coefficient. Line intensities calculated using the above model, with the HW-like correction described above, can be found below in Table A2 up to J′ = 60 for all four branches. This table also contains line positions and lower state energies calculated using the molecular parameters of Long et al. [31] as well as the Einstein-A coefficients calculated using Eq. 2 and the Hönl-London factors of Watson [50]. Table A2 also includes other line parameters including the broadening coefficients, narrowing parameters, and pressure-shifting coefficients which are discussed below. 26 2.3.5. Broadening Coefficients The air- and self-broadening coefficients,  air and  self , found in HITRAN 2008 [18] are based upon the FS-CRDS measurements of Robichaud et al. [13], which were fit using the empirical correlation of Yang et al. [26]:   A  B 1  c1 J   c2 J 2  c3 J 4 , (5) in which A and B have units of MHz/Pa, c1, c2, and c3 are dimensionless constants, and where the broadening coefficient is defined as    L /(2 p ) . It has been noted in the literature [13,26,32] that the air- and self-broadening correlations found in HITRAN 2004 [17] which were based upon the extrapolation of low-J data, are not accurate at high-J (with deviations of up to 40% at some high-J values). Recently, Havey et al. [32] examined the accuracy of the HITRAN 2008 correlation at high-J (up to J′ = 50) in the Pbranch using FS-CRDS. Excellent agreement was seen between these new measurements and HITRAN 2008 in the overlap region. In addition, new correlations for air- and selfbroadening coefficients were proposed; although these new results are only slightly altered from those of HITRAN 2008. Coefficients for air- [13] and self- [32] broadening correlations given by Eq. 5 are given in Table 2. Tables A1(a-c) give the measured broadening coefficients for the NIST 2% O2, NIST air and pure O2, gas mixtures respectively. Each of these  values was based on fitting the Galatry profile to measured spectra and an average of measurements made at several different pressures, p. We corrected the broadening coefficients to Tref = 296 K according to the temperature dependence found in HITRAN 2008 [18] which is based upon the measurements of Brown and Plymate [12]. The uncertainty in this temperature correction was negligible compared to the fit uncertainties. 27 TABLE 2. Factors used to calculate air- and self-pressure broadening coefficients (γair and γself) and airand self- narrowing parameters (ηair and ηself) given in Table A2. The former were given by   A  B /(1  c1 J   c2 J   c3 J  ) and the latter by      ( J ) . The pressure shifting 2 4 coefficient was calculated from   b0  b1 J   b2 J  . 2 air self B (MHz Pa−1) 1.217E-02 1.021E-02 A (MHz Pa−1) 6.018E-03 7.186E-03 branch sample PP PQ RQ RR PP,PQ,RQ,RR air air air air O2 branch PP,PQ,RQ,RR sample air,O2 c1 c2 c3 7.806E-02 7.180E-02 3.270E-03 2.334E-03 4.184E-06 3.025E-06 b0 (MHz Pa−1) -1.7195E-03   (MHz Pa−1) 0.00992 0.00900 0.00550 0.00601 0.00883 0.43011 0.37063 0.05684 0.08725 0.39532 b1 (MHz Pa−1) -4.5947E-05 b2 (MHz Pa−1) 3.3747E-07 In all cases, a plot of  L as a function of p exhibited good linearity and roughly zero intercept, as would be expected for physical parameters. Linearity was also observed for spectra fit with the Voigt profile. However, the slope and hence broadening coefficient was generally smaller than for the Galatry profile case. This difference in fitted line width between the Galatry and Voigt profiles is a consequence of collisional narrowing effects. In this case, the best-fit Voigt profile compensates for line narrowing by reducing the Lorentzian width and consequently fails to capture the actual line shape. We refer again to Fig. 2 where this effect is illustrated. Here fitting the Voigt profile gave a broadening coefficient 15.4% smaller than that obtained by fitting the Galatry profile. These results are also consistent with the O2 B-band observations of Lisak et al. [48], who showed that use of the Voigt profile can reduce the fitted Lorentzian width by as much as 50% compared to those obtained with the Galatry profile or other line shapes that incorporate line narrowing effects. 28 2.3.5.1. Air-Broadening Coefficients Our O2 A-band R-branch  values for air-broadening, given in Table A1(b), agree in general with the earlier FS-CRDS P-branch measurements to within their combined uncertainties (see Fig. 5). In addition, these present measurements show excellent agreement with HITRAN 2008, with a mean absolute relative deviation of 4% and values typically agreeing to within the experimental uncertainty. Therefore, these present Rbranch measurements serve as an independent validation of the broadening coefficients found in HITRAN 2008. FIG. 5. Air-broadening coefficients as a function of J′ for P- and R-branch transitions in the O2 A-band. All coefficients have been corrected to Tref = 296 K. Lines show correlations from the HITRAN 2004 [17] and 2008 [18] databases. Also, shown are the earlier FS-CRDS P-branch measurements [13,32] and the measurements of Brown and Plymate [12] and Predoi-Cross et al. [21,22]. Note that the present study and the NIST P-branch measurements utilized Galatry line profiles, while the Brown and Plymate and PredoiCross et al. studies employed Voigt line profiles. Differences in air-broadening coefficients for a given J′ were largely due to the presence of interfering lines. Unfortunately, these interferences led to uncertainties that were too large to enable us to differentiate between the correlation found in HITRAN 29 2008 and that of Havey et al. [32]. As a result, this study finds no evidence that the airbroadening coefficients for the O2 A-band R-branch found in HITRAN 2008 need to be revised. Furthermore, these overlapping lines greatly complicated the fitting procedure and limited the number of transitions we were able to quantitatively measure. FIG. 6. Self-broadening coefficients as a function of J′ for P- and R-branch transitions in the O2 A-band. All coefficients have been corrected to Tref = 296 K. The outer two lines show the correlations found in the HITRAN 2004 [17] and 2008 [18] databases. The dotted inner line is the correlation given by Havey et al. [32]. Also, shown are the earlier FS-CRDS P-branch measurements [13,32] and the measurements of Brown and Plymate [12] and Predoi-Cross et al. [21,22]. Note that the present study, the NIST P-branch measurements, and the Predoi-Cross et al. study utilized Galatry line profiles, while the Brown and Plymate study employed Voigt line profiles. 2.3.5.2. Self-Broadening Coefficients Over the range of J′ probed in this study, the self-broadening coefficients given in Table A1(c) were on average 10% larger than the corresponding air-broadening coefficients. As can be seen in Fig. 6, the present R-branch measurements agreed with the earlier FSCRDS P-branch measurements to within their combined uncertainties [13,32]. In addition, for J′ up to ~34 the measured self-broadening coefficients showed excellent agreement with the self-broadening coefficients found in HITRAN 2008, with a mean 30 absolute relative deviation of 4%. The high-J study of Havey et al. [32], however, provide an extended range over which the correlation of Eq. 5 was fit to the data. Compared to the HITRAN 2008 self-broadening values, the resulting correlation begins to differ for J′ > 30, and approaches a value 20% greater for J′ = 50. 2.3.6. Collisional Narrowing Parameters for the Galatry Profile Molecular spectroscopy line lists typically assume Voigt profiles for which the only shape parameters are the Doppler and Lorentzian widths. In recent years it has become apparent that more general line profiles, such as the Galatry profile, are required for demanding atmospheric applications [21,22,26,58]. Although the Galatry profile may not completely account for all physical mechanisms influencing the lineshape, unlike the Voigt profile it provides an additional fit parameter to model collisional narrowing effects, which is important for the O2 A-band (see Figure 2). In particular, the choice of a Galatry over a Voigt profile will affect the fitted line parameters, especially the Lorentzian width. However, when compared to more complex models, (e.g., speeddependent profiles, which require at least two more adjustable parameters), the Galatry profile provides a reasonable compromise between accuracy and computational complexity. Unfortunately, accurately measuring the narrowing parameter, , requires both high spectral resolution with low instrumental broadening and a high signal-to-noise ratio. As a result a typical approach is to assume that  is a constant independent of rotational quantum number and either measure a mean value [14,59] or calculate it as discussed below. As a result, compiled lists of  vs. J are uncommon. Here, we provide data which establish the branch- and J-dependence of the narrowing parameter for the O2 A-band. The Galatry profile [44] was derived in terms of Dicke narrowing effects which are caused by elastic velocity-changing (VC) collisions between the absorber and perturber [45]. Dicke narrowing is a consequence of a reduction in the root-mean-square velocity of the absorber with increasing density [60], which decreases the effective Doppler width. For VC-only collisions, it can be argued from gas kinetic theory that the 31 narrowing parameter =optp is  diff  k BT /(2 ma Dp ) where D and ma are the absorber’s mass diffusion coefficient and mass, respectively. This is the basis for assuming a J-independent narrowing parameter. With regard to mixtures of O2 and N2 considered here, the O2 diffusion coefficient is nearly independent of composition because both constituents have similar molecular weights and diffusion coefficients. We find D(100 kPa, 296K) 0.2 cm2 s−1 [61], and for all three gas mixtures used in this study, we calculate diff = 0.0062 MHz Pa−1. Consequently,  would be constant for all cases considered here (i.e., all branches and gas mixtures) under the VC-only collision assumption. However, inelastic velocity-changing and dephasing (VCD) collisions, which affect both the velocity and the internal state of the absorber, were not considered by Dicke and can result in values for opt that differ from those predicted by kinetic theory. To the extent that there are significant VCD collisions, the collisional narrowing will depend on the internal states of the absorber and perturber. Wehr et al. measured variations in  with J′ with   diff for CO absorption broadened by Ar [62,63]. They argued that only rotationally elastic VC collisions contribute to the Dicke narrowing of an isolated transition, whereas internal-state-changing VCD collisions do not contribute to this effect. Thus, when rotationally inelastic VCD collisions are concurrent with Dicke narrowing, diff should tend to overestimate . Furthermore as discussed by Wehr et al. [62,63], correlation between  and  may occur when such VCD collisions dominate collisional broadening. With this assumption, mutual dependency on J′ of our measured narrowing parameter and broadening coefficients could lead to a correlation of the form,      ( J ), (6) where  is an empirical coefficient [62,63] and  is the narrowing parameter in the limit of high J′ where collisional broadening effects are minimized. Note, that we have neglected the effect of pressure shifting which makes the above relation complex valued. As in references [64,65], we have also relaxed the original constraint,   diff , which assumes Dicke narrowing as the only narrowing effect. As reported by Casa et al. [66] in 32 measurements of self-broadened CO2 lineshapes, other mechanisms such as speeddependent effects may introduce additional narrowing which could cause  to differ from or exceed diff. Our new experimental narrowing parameters for the O2 A-band R-branch are given in Tables A1(a-c). These data are combined with our previous P-branch measurements to form the data set shown in Fig. 7, where the results and fits to the data are grouped by branch and plotted vs. J′ for the air-broadened (also including the NIST 2% O2) mixtures. For each branch, we fit Eq. 6 to the narrowing parameter data by floating  and  and using the air (J′) correlation given by Eq. 5. The fitted values of and  coefficients for each branch are given in Table 2. As can be seen in Fig. 7, although the  values have a relatively high uncertainty and exhibit scatter when plotted against J′, these measurements reveal that  cannot be treated as a constant. These data show a relatively strong N-branch dependence of , with the N = 1 case showing much more variation in  with J′ than that of the N = 1. For the N = 1 branch, the J = 0 case generally has a smaller value of  for a given J′ than the J = 1. However, for the N = 1 branch a slight reduction of the J = 0 measurements of  relative to those of the J = 1 can barely be resolved. We note furthermore, that for the N = 1 measurements, both P- and Q- branches (in J) have  values exceeding diff at high J′, unlike the N = 1 results in which the measured  at high J′ are generally less than diff. For all branches and low J′, the measured  are close to the constant value of 0.00429 MHz Pa−1 reported by Ritter and Wilkerson [14] for the O2 A-band. 33 FIG. 7. Measured air-narrowing parameter, air vs. J′, for O2 A-band. Data include P-branch measurements of Robichaud et al. [13], and Havey et al. [32], as well as the R-branch results obtained here. Gas mixtures correspond to NIST standard air and NIST 2% O2. Temperature was nominally equal to 296 K for all cases. Data for each branch were fit using the correlation function = ∞  air(J′) based on the air-broadening coefficient. The horizontal dashed line in each panel corresponds to diff = 0.0062 MHz Pa1. Fit coefficients, ∞ and , for each case are given in Table 2. The measurements of  for the self-broadened case are shown in Fig. 8. These results include our previous P-branch measurements of Robichaud et al. [13] and Havey et al. [32]. Because of the relatively high scatter and small range in J′, we are unable to distinguish between data from the various branches, although the data reveal a tendency for  to increase with J′. We fit Eq. 6 to all the self-broadened data, yielding  = 0.3953 and  = 0.00883 (MHz Pa-1). Lisak et al. [48] showed that the line narrowing effect in the O2 B-band can be modeled equally well using profiles based on Dicke-narrowing (e.g., Galatry profile) or those that incorporate the speed dependence of collisional broadening. These models 34 give similar line areas and Lorentzian widths. In practice, they found that it is difficult to distinguish between Dicke-narrowing and speed-dependent contributions because of practical limitations in the signal-to-noise ratio of measured spectra and the concomitant uncertainty in the fitted value of opt. They also measured relatively symmetric line shapes and small pressure shifting and therefore concluded that the speed dependence of collisional shifting for spectra in this band was insignificant. Similarly, the present measurements (see Fig. 2) show little evidence of asymmetry in the A-band line shapes. Thus, Galatry profiles that incorporate measured narrowing parameters are expected to provide an accurate representation for O2 A-band spectra. FIG. 8. Measured self-narrowing parameter,  vs. J′, for O2 A-band. Data include P-branch measurements of Robichaud et al. [13], and Havey et al. [32], as well as the R-branch results obtained here. Gas mixtures correspond to pure O2. Temperature was nominally equal to 296 K for all cases. Data for all branches were fit together using the correlation function = ∞  self(J′) based on the self-broadening coefficient. The horizontal dashed line corresponds to diff = 0.0062 MHz Pa1. Fit coefficients are given in Table 2. 35 2.3.7. Pressure-Shifting Coefficients Achieving the necessary precision and/or range of pressure required to resolve changes in line peak position makes the measurement of pressure-shifting coefficient, , challenging. Nevertheless, numerous groups have measured this line parameter in the O2 A-band [12,13,28,67,68] for various mixtures of O2 and N2. The O2 A-band P-branch data from these groups were compared in Robichaud et al. [13] and exhibit relatively wide variation. However, all measurements support a general trend of increasingly negative  as a function of J′. Recently, the high frequency stability of the FS-CRDS method was exploited to measure  with uncertainties up to 5 times lower than previous measurements [13,28]. These P-branch data were fit to a second-order polynomial in J′. This empirical correlation given in Table 2 was used to generate the list of  values for the P- and R- branches given in Table A2. 2.4. Conclusions We used frequency-stabilized cavity ring-down spectroscopy to measure 34 transitions in the R-branch of the O2 A-band and determined line parameters from fits of Galatry profiles to the measured spectra. These measurements complement our previous Pbranch studies and provide a more complete body of data that allows us to formulate a line list up to J′ = 60. We report calculated line positions, intensities, air- and selfbroadening coefficients, narrowing parameters, and pressure-shifting coefficients for the 16 O2 A-band. Low relative standard uncertainties in the line intensity measurements enabled the quantification of a quadratic J-dependent deviation from a standard intensity model. Over the range m = 38 to 38, this calculated line list reproduces the FS-CRDSmeasured line intensities on average to less than 0.1% with a relative standard deviation of ~0.3%. It is anticipated that this new line list will allow for significantly improved remote sensing atmospheric retrievals. 36 Acknowledgements We thank Piotr Masłowski of Nicolaus Copernicus University in Toruń, Poland for useful discussions regarding line shape analysis. David A. Long was supported by the National Defense Science and Engineering Graduate Fellowship. We acknowledge the National Research Council for awarding Daniel Havey a postdoctoral fellowship at the National Institute of Standards and Technology (NIST), Gaithersburg, MD. Part of the research described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Additional support was provided by the Orbiting Carbon Observatory (OCO) project, a NASA Earth System Science Pathfinder (ESSP) mission; the NASA Upper Atmospheric Research Program grant NNG06GD88G and NNX09AE21G; and the NIST Office of Microelectronics Programs. 37 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] S. Corradini, M. Cervino, J. Quant. Spectrosc. Radiat. Transfer 97, 354-380 (2006). B. van Diedenhoven, O. P. Hasekamp, I. Aben, Atmos. Chem. Phys. 5, 21092120 (2005). K. Chance, J. Quant. Spectrosc. Radiat. Transfer 58, 375-378 (1997). L. W. Thomason, G. Taha, Geophys. Res. Lett. 30, 1631 (2003). C. R. Nowlan, C. T. McElroy, J. R. Drummond, J. Quant. Spectrosc. Radiat. Transfer 108, 371-388 (2007). M. Yokomizo, Fujitsu Sci. 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For the line intensities and broadening coefficients, the values in parenthesis are the combined (Type A and Type B) standard uncertainties in the last digit. For the narrowing parameters the values in parenthesis correspond to the fit uncertainty. (a) sample: NIST 2% O2 ~ transition R1R1 R1Q2 R3R3 R3Q4 R5R5 R5Q6 R7R7 R7Q8 R9R9 R9Q10 R11R11 R11Q12 R13R13 R13Q14 R15R15 R15Q16 R17R17 R17Q18 R19R19 R19Q20 R21R21 -1 (cm ) 13126.392003 13128.268803 13131.491440 13133.441040 13136.217103 13138.204803 13140.567420 13142.583320 13144.540760 13146.580460 13148.135226 13150.196626 13151.348659 13153.430459 13154.178835 13156.280235 13156.623169 13158.743569 13158.679005 13160.818105 13160.343426 S (10 cm molec.-1) 1.530(6) 3.57(1) 4.28(1) 6.22(2) 6.30(2) 8.03(3) 7.35(2) 8.80(3) 7.44(2) 8.61(3) 6.79(2) 7.65(3) 5.64(2) 6.25(2) 4.34(1) 4.76(1) 3.09(1) 3.38(1) 2.079(7) 2.249(7) 1.292(5) -24  (MHz Pa-1) 0.0183(1) 0.0174(1) 0.0167(1) 0.0166(1) 0.0158(1) 0.0158(1) 0.0152(1) 0.0148(2) 0.0147(1) 0.0146(1) 0.0144(2) 0.0139(2) 0.0142(2) 0.0135(2) 0.0133(2) 0.0133(1) 0.0122(1) 0.0131(2) 0.0126(2) 0.0123(1) 0.0117(1)  (MHz Pa-1) 0.0047(5) 0.0049(3) 0.0051(5) 0.0050(2) 0.0047(3) 0.0046(2) 0.0050(4) 0.0042(3) 0.0046(4) 0.0043(3) 0.0047(7) 0.0042(6) 0.0053(4) 0.0044(5) 0.0047(5) 0.0045(3) 0.0042(2) 0.0050(8) 0.0056(8) 0.0046(3) 0.0051(1) pressure range (kPa) 4.0-20.0 2.7-8.0 1.3-5.6 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-3.3 1.3-5.6 1.3-5.6 5.6-8.0 2.7-8.0 2.7-8.0 3.3-13.3 4.0-20.0 # scans 8 10 9 5 5 5 5 5 5 5 5 5 6 6 9 9 6 9 9 8 6 # pressures 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 4 4 6 4 42 (b) sample: NIST standard air ~ transition R23R23 R23Q24 R25Q26 R31Q32 R33R33 R33Q34 R35R35 R37R37 R37Q38 R39R39 R39Q40 R41R41 R41Q42 (cm-1) 13161.613145 13163.788745 13164.678607 13164.922553 13161.917130 13164.182130 13160.740806 13159.141488 13161.441788 13157.114405 13159.432305 13154.654303 13156.989803 S (10-26 cm molec-1) 75.9(2) 81.2(2) 44.3(1) 5.04(2) 2.057(8) 2.18(1) 0.83(2) 0.324(2) 0.339(9) 0.118(2) 0.1227(9) 0.0398(5) 0.0420(8)  (MHz Pa-1) 0.0111(3) 0.0115(3) 0.0105(2) 0.0089(2) 0.0074(5) 0.0087(6) 0.0080(9) 0.0073(8) 0.008(1) 0.007(1) 0.008(1) 0.007(1) 0.006(1)  pressure range (kPa) 1.3-3.3 1.3-2.7 1.3-3.3 10.0-12.0 3.3-13.3 3.3-13.3 1.7-6.7 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 # scans 4 4 5 7 3 6 3 4 3 4 3 20 20 # pressures 4 4 4 2 3 3 3 3 3 3 2 3 3  pressure range (kPa) 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 3.3-13.3 # scans 3 3 3 3 3 3 3 3 # pressures 3 3 3 3 3 3 3 3 (MHz Pa-1) 0.005(1) 0.0049(5) 0.005(1) 0.0053(6) 0.0050(5) 0.0056(6) 0.006(2) 0.0055(5) 0.006(2) 0.007(2) Eq. 6 Eq. 6 Eq. 6 (c) sample: pure O2 ~ transition R33R33 R33Q34 R37R37 R37Q38 R39R39 R39Q40 R41R41 R41Q42 -1 (cm ) 13161.917130 13164.182130 13159.141488 13161.441788 13157.114405 13159.432305 13154.654303 13156.989803 S (10 cm molec-1) 20.4(1) 21.7(2) 3.23(2) 3.31(6) 1.17(1) 1.22(1) 0.39(2) 0.41(1) -27  (MHz Pa-1) 0.0090(9) 0.0097(7) 0.0087(9) 0.008(1) 0.0076(8) 0.0079(9) 0.008(1) 0.007(1) (MHz Pa-1) 0.0047(5) 0.0055(5) 0.0054(5) 0.008(3) 0.0039(8) 0.0045(6) 0.008(3) 0.003(1) 43 Table A2. Line parameters for magnetic dipole transitions of the 16O2 A-band. Lower state energies and positions were calculated using the lower and upper state molecular constants found in Long et al. [31] and the definitions found in Rouillé et al. [69]. The intensities, S, which correspond to Tref = 296 K and natural terrestrial isotopic abundance of 16O2, were calculated using Eq. 2 with the Hönl-London factors, L, of Watson [50], a band intensity of 2.231410−22 cm molec.−1 and the positions and lower state energies shown. We estimate ur(S) < 0.4% for m = 38 to 38. Note that the intensities were weighted by the HW-like quadratic term, given by (1+a1m+a2m2)2 with a1 = 2.65810−4 and a2 = 3.362210−6, as described in the text. Einstein-A coefficients were calculated from the corresponding intensities using Eq. 4. The broadening coefficients, γair and γself, the narrowing parameters, ηair and ηself, and the pressure-shifting coefficients, δ, were calculated using the correlations and coefficients given in Table 2. Uncertainties for the broadening and narrowing parameters are comparable to those given in Table A1. Note 1 cm−1 atm−1 = 0.295872 MHz Pa−1. transition L ~ P61Q60 30.3441 (cm-1) 12774.178997 air self  air self S (cm molec.-1) A (s-1) E” (cm-1) (MHz Pa-1) (MHz Pa-1) (MHz Pa-1) (MHz Pa-1) (MHz Pa-1) 4.547E-34 0.02139 5367.87752 6.27E-03 7.47E-03 -3.26E-03 6.68E-03 5.88E-03 5.86E-03 P59P59 30.0000 12790.010354 2.323E-33 0.02192 5029.97839 6.31E-03 7.51E-03 -3.25E-03 7.21E-03 P59Q58 29.3441 12791.508287 2.286E-33 0.02142 5028.48046 6.31E-03 7.51E-03 -3.25E-03 6.66E-03 5.86E-03 P57P57 29.0000 12806.885855 1.108E-32 0.02197 4701.28301 6.35E-03 7.56E-03 -3.23E-03 7.19E-03 5.84E-03 P57Q56 28.3441 12808.400536 1.090E-32 0.02146 4699.76833 6.35E-03 7.56E-03 -3.23E-03 6.65E-03 5.84E-03 P55P55 28.0000 12823.330010 5.013E-32 0.02202 4383.32615 6.41E-03 7.62E-03 -3.22E-03 7.16E-03 5.82E-03 P55Q54 27.3441 12824.861458 4.926E-32 0.02149 4381.79470 6.41E-03 7.62E-03 -3.22E-03 6.63E-03 5.82E-03 P53P53 27.0000 12839.348247 2.149E-31 0.02207 4076.15958 6.47E-03 7.68E-03 -3.20E-03 7.14E-03 5.79E-03 P53Q52 26.3441 12840.896484 2.110E-31 0.02152 4074.61134 6.47E-03 7.68E-03 -3.20E-03 6.60E-03 5.79E-03 P51P51 26.0000 12854.945718 8.726E-31 0.02212 3779.83330 6.55E-03 7.76E-03 -3.17E-03 7.10E-03 5.76E-03 P51Q50 25.3441 12856.510765 8.562E-31 0.02155 3778.26826 6.55E-03 7.76E-03 -3.17E-03 6.57E-03 5.76E-03 P49P49 25.0000 12870.127299 3.356E-30 0.02218 3494.39554 6.64E-03 7.86E-03 -3.15E-03 7.06E-03 5.72E-03 P49Q48 24.3441 12871.709183 3.290E-30 0.02158 3492.81365 6.64E-03 7.86E-03 -3.15E-03 6.54E-03 5.72E-03 P47P47 24.0000 12884.897600 1.222E-29 0.02223 3219.89269 6.76E-03 7.98E-03 -3.12E-03 7.01E-03 5.68E-03 P47Q46 23.3441 12886.496349 1.197E-29 0.02161 3218.29394 6.76E-03 7.98E-03 -3.12E-03 6.50E-03 5.68E-03 5.62E-03 P45P45 23.0000 12899.260968 4.211E-29 0.02229 2956.36938 6.90E-03 8.12E-03 -3.09E-03 6.95E-03 P45Q44 22.3441 12900.876614 4.119E-29 0.02164 2954.75373 6.90E-03 8.12E-03 -3.09E-03 6.44E-03 5.62E-03 P43P43 22.0000 12913.221492 1.373E-28 0.02234 2703.86842 7.07E-03 8.28E-03 -3.05E-03 6.88E-03 5.56E-03 P43Q42 21.3441 12914.854071 1.341E-28 0.02166 2702.23585 7.07E-03 8.28E-03 -3.05E-03 6.38E-03 5.56E-03 P41P41 21.0000 12926.783009 4.233E-28 0.02240 2462.43083 7.29E-03 8.48E-03 -3.02E-03 6.79E-03 5.48E-03 P41Q40 20.3441 12928.432561 4.130E-28 0.02169 2460.78127 7.29E-03 8.48E-03 -3.02E-03 6.30E-03 5.48E-03 P39P39 20.0000 12939.949105 1.234E-27 0.02247 2232.09578 7.55E-03 8.72E-03 -2.98E-03 6.67E-03 5.38E-03 P39Q38 19.3441 12941.615675 1.202E-27 0.02172 2230.42921 7.55E-03 8.72E-03 -2.98E-03 6.20E-03 5.38E-03 P37P37 19.0000 12952.723123 3.397E-27 0.02253 2012.90065 7.86E-03 9.00E-03 -2.94E-03 6.54E-03 5.27E-03 P37Q36 18.3441 12954.406761 3.304E-27 0.02174 2011.21702 7.86E-03 9.00E-03 -2.94E-03 6.09E-03 5.27E-03 P35P35 18.0000 12965.108162 8.838E-27 0.02260 1804.88101 8.24E-03 9.32E-03 -2.89E-03 6.37E-03 5.15E-03 P35Q34 17.3441 12966.808930 8.579E-27 0.02177 1803.18025 8.24E-03 9.32E-03 -2.89E-03 5.95E-03 5.15E-03 P33P33 17.0000 12977.107088 2.170E-26 0.02268 1608.07059 8.70E-03 9.69E-03 -2.84E-03 6.18E-03 5.00E-03 P33Q32 16.3441 12978.825055 2.102E-26 0.02179 1606.35262 8.70E-03 9.69E-03 -2.84E-03 5.78E-03 5.00E-03 P31P31 16.0000 12988.722531 5.029E-26 0.02276 1422.50128 9.23E-03 1.01E-02 -2.79E-03 5.95E-03 4.84E-03 P31Q30 15.3441 12990.457779 4.860E-26 0.02182 1420.76603 9.23E-03 1.01E-02 -2.79E-03 5.58E-03 4.84E-03 44 P29P29 15.0000 12999.956891 1.099E-25 0.02286 1248.20315 9.83E-03 1.05E-02 -2.74E-03 5.69E-03 P29Q28 14.3441 13001.709517 1.059E-25 0.02185 1246.45053 9.83E-03 1.05E-02 -2.74E-03 5.36E-03 4.66E-03 4.66E-03 P27P27 14.0000 13010.812342 2.263E-25 0.02296 1085.20445 1.05E-02 1.10E-02 -2.69E-03 5.41E-03 4.48E-03 4.48E-03 P27Q26 13.3441 13012.582464 2.174E-25 0.02187 1083.43433 1.05E-02 1.10E-02 -2.69E-03 5.12E-03 P25P25 13.0000 13021.290834 4.388E-25 0.02307 933.53157 1.11E-02 1.15E-02 -2.63E-03 5.13E-03 4.30E-03 P25Q24 12.3441 13023.078597 4.200E-25 0.02190 931.74381 1.11E-02 1.15E-02 -2.63E-03 4.87E-03 4.30E-03 P23P23 12.0000 13031.394096 8.005E-25 0.02321 793.20908 1.18E-02 1.19E-02 -2.57E-03 4.86E-03 4.12E-03 P23Q22 11.3441 13033.199679 7.629E-25 0.02193 791.40349 1.18E-02 1.19E-02 -2.57E-03 4.64E-03 4.12E-03 P21P21 11.0000 13041.123637 1.372E-24 0.02336 664.25967 1.23E-02 1.23E-02 -2.50E-03 4.61E-03 3.96E-03 P21Q20 10.3441 13042.947271 1.301E-24 0.02196 662.43604 1.23E-02 1.23E-02 -2.50E-03 4.42E-03 3.96E-03 P19P19 10.0000 13050.480752 2.207E-24 0.02355 546.70424 1.28E-02 1.27E-02 -2.44E-03 4.40E-03 3.81E-03 P19Q18 9.3442 13052.322739 2.079E-24 0.02200 544.86225 1.28E-02 1.27E-02 -2.44E-03 4.24E-03 3.81E-03 P17P17 9.0000 13059.466520 3.324E-24 0.02377 440.56179 1.32E-02 1.30E-02 -2.37E-03 4.23E-03 3.68E-03 P17Q16 8.3442 13061.327268 3.108E-24 0.02204 438.70104 1.32E-02 1.30E-02 -2.37E-03 4.10E-03 3.68E-03 P15P15 8.0000 13068.081811 4.679E-24 0.02406 345.84949 1.36E-02 1.33E-02 -2.30E-03 4.09E-03 3.56E-03 P15Q14 7.3442 13069.961892 4.332E-24 0.02208 343.96941 1.36E-02 1.33E-02 -2.30E-03 3.97E-03 3.56E-03 P13P13 7.0000 13076.327282 6.131E-24 0.02443 262.58268 1.39E-02 1.36E-02 -2.22E-03 3.96E-03 3.45E-03 P13Q12 6.3442 13078.227537 5.605E-24 0.02214 260.68242 1.39E-02 1.36E-02 -2.22E-03 3.87E-03 3.45E-03 P11P11 6.0000 13084.203384 7.445E-24 0.02494 190.77481 1.42E-02 1.39E-02 -2.15E-03 3.83E-03 3.33E-03 P11Q10 5.3442 13086.125129 6.690E-24 0.02221 188.85306 1.42E-02 1.39E-02 -2.15E-03 3.75E-03 3.33E-03 P9P9 5.0000 13091.710358 8.312E-24 0.02569 130.43750 1.45E-02 1.43E-02 -2.07E-03 3.68E-03 3.18E-03 P9Q8 4.3443 13093.655833 7.287E-24 0.02231 128.49202 1.45E-02 1.43E-02 -2.07E-03 3.62E-03 3.18E-03 P7P7 4.0000 13098.848243 8.426E-24 0.02688 81.58050 1.50E-02 1.47E-02 -1.98E-03 3.47E-03 3.00E-03 P7Q6 3.3444 13100.821748 7.110E-24 0.02247 79.60699 1.50E-02 1.47E-02 -1.98E-03 3.44E-03 3.00E-03 2.76E-03 P5P5 3.0000 13105.616870 7.573E-24 0.02913 44.21171 1.57E-02 1.53E-02 -1.90E-03 3.18E-03 P5Q4 2.3446 13107.628463 5.974E-24 0.02276 42.20012 1.57E-02 1.53E-02 -1.90E-03 3.19E-03 2.76E-03 P3P3 2.0000 13112.015868 5.721E-24 0.03497 18.33718 1.67E-02 1.62E-02 -1.81E-03 2.75E-03 2.43E-03 P3Q2 1.3458 13114.100185 3.888E-24 0.02353 16.25286 1.67E-02 1.62E-02 -1.81E-03 2.82E-03 2.43E-03 P1P1 1.0000 13118.044661 3.066E-24 0.08744 3.96108 1.82E-02 1.74E-02 -1.72E-03 2.10E-03 1.95E-03 2.43E-03 R1R1 0.5000 13126.391964 1.531E-24 0.00875 3.96108 1.67E-02 1.62E-02 -1.81E-03 4.55E-03 R1Q2 1.1542 13128.268754 3.566E-24 0.02019 2.08429 1.67E-02 1.62E-02 -1.81E-03 4.55E-03 2.43E-03 R3R3 1.5000 13131.491402 4.282E-24 0.01458 18.33718 1.57E-02 1.53E-02 -1.90E-03 4.64E-03 2.76E-03 R3Q4 2.1554 13133.440971 6.209E-24 0.02095 16.38761 1.57E-02 1.53E-02 -1.90E-03 4.61E-03 2.76E-03 R5R5 2.5000 13136.217029 6.289E-24 0.01683 44.21171 1.50E-02 1.47E-02 -1.98E-03 4.70E-03 3.00E-03 R5Q6 3.1556 13138.204770 8.013E-24 0.02124 42.22397 1.50E-02 1.47E-02 -1.98E-03 4.65E-03 3.00E-03 R7R7 3.5000 13140.567357 7.338E-24 0.01802 81.58050 1.45E-02 1.43E-02 -2.07E-03 4.74E-03 3.18E-03 R7Q8 4.1557 13142.583244 8.797E-24 0.02139 79.56461 1.45E-02 1.43E-02 -2.07E-03 4.68E-03 3.18E-03 R9R9 4.5000 13144.540696 7.437E-24 0.01875 130.43750 1.42E-02 1.39E-02 -2.15E-03 4.77E-03 3.33E-03 R9Q10 5.1558 13146.580459 8.603E-24 0.02149 128.39773 1.42E-02 1.39E-02 -2.15E-03 4.70E-03 3.33E-03 R11R11 5.5000 13148.135152 6.776E-24 0.01925 190.77481 1.39E-02 1.36E-02 -2.22E-03 4.80E-03 3.45E-03 R11Q12 6.1558 13150.196583 7.658E-24 0.02155 188.71338 1.39E-02 1.36E-02 -2.22E-03 4.71E-03 3.45E-03 R13R13 6.5000 13151.348626 5.646E-24 0.01962 262.58268 1.36E-02 1.33E-02 -2.30E-03 4.83E-03 3.56E-03 R13Q14 7.1558 13153.430440 6.277E-24 0.02160 260.50087 1.36E-02 1.33E-02 -2.30E-03 4.73E-03 3.56E-03 R15R15 7.5000 13154.178813 4.344E-24 0.01989 345.84949 1.32E-02 1.30E-02 -2.37E-03 4.85E-03 3.68E-03 R15Q16 8.1558 13156.280200 4.772E-24 0.02163 343.74811 1.32E-02 1.30E-02 -2.37E-03 4.75E-03 3.68E-03 R17R17 8.5000 13156.623203 3.105E-24 0.02010 440.56179 1.28E-02 1.27E-02 -2.44E-03 4.89E-03 3.81E-03 R17Q18 9.1558 13158.743620 3.379E-24 0.02166 438.44137 1.28E-02 1.27E-02 -2.44E-03 4.77E-03 3.81E-03 R19R19 9.5000 13158.679074 2.071E-24 0.02028 546.70424 1.23E-02 1.23E-02 -2.50E-03 4.93E-03 3.96E-03 45 R19Q20 10.1559 13160.818147 2.237E-24 0.02168 544.56517 1.23E-02 1.23E-02 -2.50E-03 4.80E-03 3.96E-03 R21R21 10.5000 13160.343498 1.292E-24 0.02042 664.25967 1.18E-02 1.19E-02 -2.57E-03 4.98E-03 4.12E-03 R21Q22 11.1559 13162.500954 1.387E-24 0.02170 662.10222 1.18E-02 1.19E-02 -2.57E-03 4.83E-03 4.12E-03 R23R23 11.5000 13161.613332 7.559E-25 0.02053 793.20908 1.11E-02 1.15E-02 -2.63E-03 5.04E-03 4.30E-03 R23Q24 12.1559 13163.788973 8.075E-25 0.02171 791.03344 1.11E-02 1.15E-02 -2.63E-03 4.87E-03 4.30E-03 R25R25 12.5000 13162.485220 4.153E-25 0.02063 933.53157 1.05E-02 1.10E-02 -2.69E-03 5.09E-03 4.48E-03 R25Q26 13.1559 13164.678897 4.417E-25 0.02172 931.33790 1.05E-02 1.10E-02 -2.69E-03 4.91E-03 4.48E-03 R27R27 13.5000 13162.955591 2.145E-25 0.02071 1085.20445 9.83E-03 1.05E-02 -2.74E-03 5.15E-03 4.66E-03 R27Q28 14.1559 13165.167191 2.274E-25 0.02172 1082.99285 9.83E-03 1.05E-02 -2.74E-03 4.94E-03 4.66E-03 R29R29 14.5000 13163.020655 1.043E-25 0.02078 1248.20315 9.23E-03 1.01E-02 -2.79E-03 5.20E-03 4.84E-03 R29Q30 15.1559 13165.250091 1.102E-25 0.02173 1245.97372 9.23E-03 1.01E-02 -2.79E-03 4.98E-03 4.84E-03 R31R31 15.5000 13162.676399 4.777E-26 0.02084 1422.50128 8.70E-03 9.69E-03 -2.84E-03 5.25E-03 5.00E-03 R31Q32 16.1559 13164.923607 5.035E-26 0.02173 1420.25407 8.70E-03 9.69E-03 -2.84E-03 5.01E-03 5.00E-03 R33R33 16.5000 13161.918589 2.063E-26 0.02089 1608.07059 8.24E-03 9.32E-03 -2.89E-03 5.29E-03 5.15E-03 R33Q34 17.1559 13164.183518 2.169E-26 0.02173 1605.80566 8.24E-03 9.32E-03 -2.89E-03 5.03E-03 5.15E-03 R35R35 17.5000 13160.742763 8.404E-27 0.02094 1804.88101 7.86E-03 9.00E-03 -2.94E-03 5.32E-03 5.27E-03 R35Q36 18.1559 13163.025375 8.818E-27 0.02173 1802.59840 7.86E-03 9.00E-03 -2.94E-03 5.05E-03 5.27E-03 R37R37 18.5000 13159.144227 3.232E-27 0.02097 2012.90065 7.55E-03 8.72E-03 -2.98E-03 5.35E-03 5.38E-03 R37Q38 19.1559 13161.444496 3.384E-27 0.02173 2010.60039 7.55E-03 8.72E-03 -2.98E-03 5.07E-03 5.38E-03 R39R39 19.5000 13157.118058 1.174E-27 0.02100 2232.09578 7.29E-03 8.48E-03 -3.02E-03 5.37E-03 5.48E-03 R39Q40 20.1559 13159.435962 1.227E-27 0.02172 2229.77787 7.29E-03 8.48E-03 -3.02E-03 5.09E-03 5.48E-03 R41R41 20.5000 13154.659092 4.026E-28 0.02103 2462.43083 7.07E-03 8.28E-03 -3.05E-03 5.39E-03 5.56E-03 R41Q42 21.1559 13156.994619 4.204E-28 0.02172 2460.09530 7.07E-03 8.28E-03 -3.05E-03 5.10E-03 5.56E-03 R43R43 21.5000 13151.761924 1.306E-28 0.02105 2703.86842 6.90E-03 8.12E-03 -3.09E-03 5.41E-03 5.62E-03 R43Q44 22.1559 13154.115067 1.361E-28 0.02171 2701.51528 6.90E-03 8.12E-03 -3.09E-03 5.11E-03 5.62E-03 R45R45 22.5000 13148.420908 4.004E-29 0.02107 2956.36938 6.76E-03 7.98E-03 -3.12E-03 5.42E-03 5.68E-03 R45Q46 23.1559 13150.791663 4.170E-29 0.02170 2953.99863 6.76E-03 7.98E-03 -3.12E-03 5.12E-03 5.68E-03 R47R47 23.5000 13144.630146 1.162E-29 0.02109 3219.89269 6.64E-03 7.86E-03 -3.15E-03 5.43E-03 5.72E-03 R47Q48 24.1559 13147.018515 1.208E-29 0.02169 3217.50432 6.64E-03 7.86E-03 -3.15E-03 5.12E-03 5.72E-03 R49R49 24.5000 13140.383486 3.189E-30 0.02110 3494.39554 6.55E-03 7.76E-03 -3.17E-03 5.44E-03 5.76E-03 R49Q50 25.1559 13142.789473 3.314E-30 0.02168 3491.98955 6.55E-03 7.76E-03 -3.17E-03 5.13E-03 5.76E-03 R51R51 25.5000 13135.674520 8.289E-31 0.02111 3779.83330 6.47E-03 7.68E-03 -3.20E-03 5.44E-03 5.79E-03 R51Q52 26.1559 13138.098134 8.606E-31 0.02167 3777.40969 6.47E-03 7.68E-03 -3.20E-03 5.13E-03 5.79E-03 R53R53 26.5000 13130.496578 2.040E-31 0.02111 4076.15958 6.41E-03 7.62E-03 -3.22E-03 5.45E-03 5.82E-03 R53Q54 27.1559 13132.937827 2.116E-31 0.02165 4073.71833 6.41E-03 7.62E-03 -3.22E-03 5.14E-03 5.82E-03 R55R55 27.5000 13124.842719 4.757E-32 0.02112 4383.32615 6.35E-03 7.56E-03 -3.23E-03 5.45E-03 5.84E-03 R55Q56 28.1559 13127.301617 4.931E-32 0.02164 4380.86725 6.35E-03 7.56E-03 -3.23E-03 5.14E-03 5.84E-03 R57R57 28.5000 13118.705733 1.051E-32 0.02112 4701.28301 6.31E-03 7.51E-03 -3.25E-03 5.46E-03 5.86E-03 R57Q58 29.1559 13121.182295 1.089E-32 0.02162 4698.80645 6.31E-03 7.51E-03 -3.25E-03 5.14E-03 5.86E-03 R59R59 29.5000 13112.078129 2.202E-33 0.02111 5029.97839 6.27E-03 7.47E-03 -3.26E-03 5.46E-03 5.88E-03 R59Q60 30.1559 13114.572371 2.279E-33 0.02160 5027.48415 6.27E-03 7.47E-03 -3.26E-03 5.14E-03 5.88E-03 46 CHAPTER 3 O2 A-band Line Parameters to Support Atmospheric Remote Sensing. Part II: The Rare Isotopologues This chapter was submitted for publication as D. A. Long, D. K. Havey, S. S. Yu, M. Okumura, C. E. Miller, and J. T. Hodges, J. Quant. Spectrosc. Radiat. Transfer, (2011). Copyright 2011 by Elsevier B.V. 47 Abstract Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) was employed to measure over 100 transitions in the R-branch of the b1Σg+←X3Σg-(0,0) band for the rare O2 isotopologues. The use of 17O- and 18O-enriched mixtures allowed for line positions to be measured for the 16O17O, 16O18O, 17O2, 17O18O, and 18O2 isotopologues. Simultaneous fits to the upper and lower states were performed for each isotopologue using the FSCRDS positions supplemented by microwave, millimeter, submillimeter, terahertz, and Raman ground state positions from the literature. Positions, line intensities, pressure broadening parameters, and collisional narrowing parameters are reported for the 16O18O and 16O17O isotopologues which are based upon the present study and our earlier FSCRDS work [Long et al., JQSRT, 111, 2021(2010) and Robichaud et al., JPCA, 113, 13089(2009)]. The calculated line intensities include a term for the observed HermanWallis-like interaction and correct a frequency-dependent error which is present in existing spectroscopic databases. 48 3.1. Introduction The O2 A-band [b1Σg+←X3Σg-(0,0)] is utilized extensively in remote sensing to determine surface pressure [1-2], aerosol and cloud optical properties [3-4], cloud-top heights [5], and optical path length. Recent remote sensing measurements have demonstrated high precision (e.g., surface pressure determinations with a reproducibility better than 0.1% [1-2] and optical path length to better than 0.3% [6-7]), requiring spectroscopic parameters with even higher precision. Additionally, many A-band transitions from the dominant 16 O2 isotopologue are heavily saturated under atmospheric observing conditions. Also, transitions from rare O2 isotopologues have significant absorption strengths, requiring high-precision spectroscopic parameters for these transitions [8]. Despite the importance of O2 rare isotopologues to remote sensing applications, only a few high resolution studies have measured rare isotopologue spectroscopic parameters. Babcock and Herzberg performed the first quantitative analysis of 16O17O and 16 O18O spectroscopic parameters using high-resolution atmospheric spectra [9]. Since these early measurements, a variety of techniques including Fourier-transform spectroscopy (FTS) [10-11], cavity ring-down spectroscopy (CRDS) [12], frequencymodulation spectroscopy (FMS) [13], and noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) [14] have been employed to measure rare isotopologue spectroscopic parameters. A major experimental challenge in these studies has been to obtain sufficient signal-to-noise in the high resolution spectra to retrieve accurate spectroscopic parameters for the rare isotopologues of O2. See Robichaud et al. for a complete literature summary [15]. Recently, Robichaud et al. [15] used frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) to perform the most precise rare isotopologue measurements of O2 to date. These measurements were the first comprehensive study of line intensities and lineshape parameters for the 17 O18O and 18 16 O2 isotopologues. These FS-CRDS 18 measurements presently form the basis for the O O and 16O17O transitions found in the 49 HITRAN 2008 database [16]. 1 The tuning range of the external-cavity diode laser, which was used in the Robichaud et al. FS-CRDS study [15], unfortunately limited our earlier measurements to the P-branch of the b1Σg+←X3Σg-(0,0) band. The present study extends FS-CRDS measurements to the R-branch transitions and uses a 17O-enriched sample to significantly improve the spectroscopic parameters determined for the 17O isotopologues (see Figure 1 for a survey scan). We combine our present results with those of our earlier FS-CRDS studies [15,18] to generate a spectroscopic database for the b1Σg+←X3Σg-(0,0) transitions of 16O18O and 16O17O. 650 550 500 -1 -6 -1 (cτ) (10 cm ) 600 450 400 350 300 250 13125 13130 13135 13140 13145 13150 13155 13160 13165 -1 wave number (cm ) 16O 2 16O17O 16O18O 17O 600 17O18O 550 -1 -6 -1 (cτ) (10 cm ) 2 18O 2 500 450 400 350 300 250 13134 13134.5 13135 13135.5 13136 13136.5 13137 13137.5 13138 13138.5 -1 wave number (cm ) FIG. 1. Upper panel: Recorded absorption spectrum of the R-branch of the O2 A-band at 0.234 kPa (1.75 Torr) 17O-enriched gas (see Table 1 for composition) at 299.5 K. Lower panel: The above scan between 13134.0 and 13138.5 cm1 with isotopologue assignments shown. 1 Contrary to the description found in the HITRAN 2008 publication [16] (which states that the given transition frequencies were taken from Robichaud et al. [15]), the 16O17O transition frequencies found in HITRAN 2008 are identical to those found in HITRAN 2004 [17]. 50 3.2. Experiment The frequency-stabilized cavity ring-down spectrometer utilized for these measurements is located at the National Institute of Standards and Technology (NIST) Gaithersburg, MD and has been described in detail previously [19-20]. Briefly, FS-CRDS differs from traditional cw-CRDS techniques by actively stabilizing the intracavity length to an external frequency reference (in our case a co-resonant frequency-stabilized HeNe laser), which in turn eliminates drift in the cavity’s comb of transmission modes. The probe laser frequency is then stepped from one TEM00 cavity mode to the next, resulting in an extremely linear and stable frequency axis [21]. Each of these modes is separated by the cavity free spectral range (FSR), which is typically determined to better than 1 part in 104. Sub-FSR frequency steps are made by shifting the HeNe laser frequency with an acousto-optic modulator in a double-pass alignment. In the present study, we used this procedure to take ½ FSR frequency steps (~100 MHz) to increase the sampling density of our measured spectra. The probe laser used in the present study was an external cavity diode laser with an output power of 6-10 mW, a wavelength tuning range of 759-771 nm, and a linewidth of 300 kHz (1 s averaging time). Cavity mirrors with reflectivities of 99.98% were employed (cavity finesse ~15,000), leading to a noise-equivalent absorption coefficient of 6×1010 cm1 Hz1/2 for an acquisition rate of 20 Hz and relative uncertainty in the measured decay time of 0.1%. The reference laser was a frequency-stabilized HeNe laser with a long-term frequency uncertainty of 1 MHz (over 8 h). The cavity FSR was determined to be 201.970(10) MHz through the procedure of Lisak and Hodges [22]. Two isotopically enriched O2 samples were employed for these measurements. The first was an 18O-enriched sample (nominally 50% 18O) which had been previously utilized in our measurements of the P-branch O2 isotopologue transitions [15]. The supplier of this sample was unable to measure the isotopologue composition with sufficient accuracy for the present study. Therefore, we performed a mass spectral analysis of this sample using a double-focusing magnetic-sector mass spectrometer with electron impact ionization [15]. A 17O-enriched sample (nominally 50% 17O) was 51 supplied with a mass spectral analysis and isotopologue fractions; nevertheless, we performed an independent mass spectral analysis in order to verify the given composition and assess the relevant uncertainties. For the mass 34 species (i.e. 16O18O and 17O2) the manufacturer atomic composition was utilized to constrain the subsequent fit. The analyses for both samples are given in Table 1. TABLE 1. Composition of isotopically enriched samples Iso. 16 Natural Abundancea 18 O-Enriched Sampleb 17 O-Enriched Samplec Calculated Band Intensityd (cm molec.−1) Measured Band Intensitye (cm molec.−1) O2 0.995262 0.2784 (51) 0.1523(57) 2.231E-22 2.231(7)E-22 O17O 16 0.000742 0.0099 (4) 0.3816(28) 1.663E-25 1.77(7)E-25 18 O O 0.00399141 0.3672 (18) 0.0828(29) 8.947E-25 9.13(5)E-25 17 O2 1.38×10 7 not determined 0.2593(49) 3.093E-29 not determined O18O 1.49×10−6 0.0174 (4) 0.1115(80) 3.340E-28 3.37(8)E-28 4.00178×10−6 0.3271 (32) 0.0125(26) 8.970E-28 8.66(9)E-28 16 17 18 O2 − a Based upon the HITRAN 2004 recommended values [17]. bComposition determined by mass spectrometer. Values in parentheses are standard uncertainties (1σ). Note that 16O18O abundance is actually 16 18 O O + 17O2 (i.e. mass 34). The uncertainty due to this assumption should be less than 0.01%. cSupplier provided composition data. We performed an independent mass spectral analysis in order to assess the uncertainty in the given composition. dBand intensities at Tref = 296 K and natural terrestrial abundance as determined by scaling the FS-CRDS determined 16O2 band intensity by the natural abundance [17]. eBand intensities at Tref = 296 K and natural terrestrial isotopic abundance as determined by a fit to the FS-CRDS rare isotopologue measurements. This fit employed the Gamache et al. intensity model [23] with the HönlLondon factors of Watson [24] and a quadratic Herman-Wallis-like correction [25] as described in the text. Spectral scans of each of the isotopically enriched samples were performed over the R-branch (13125-13165 cm−1) with a nominal 100 MHz (~0.003 cm−1) step size (see Figure 1 for an example survey spectrum).The 18O-enriched sample and 17O-enriched sample were scanned at 0.214 kPa (1.60 Torr) and 0.234 kPa (1.75 Torr), respectively. Pressure was measured with a NIST-calibrated capacitance diaphragm gauge having a full-scale response of 1.33 kPa (10 Torr) and a relative combined standard uncertainty less than 0.1%. The sample cell was placed within an insulated box to mitigate temperature variations. During these scans, temperatures ranged from 299.5 to 300.0 K as measured by a NIST-calibrated 2.4 kΩ thermistor. The temperature uncertainty in these 52 experiments was examined in detail by Havey et al. [26] and shown to be less than 28 mK. The importance of using a Galatry profile [27] (which accounts for collisional narrowing [28]) as opposed to a Voigt profile in quantitative measurements of line intensities and lineshape parameters of the b1Σg+←X3Σg-(0,0) band has been demonstrated by our group and others [18,29-30]. However, for the purposes of determining the line center of isolated transitions, symmetric profiles, such as the Voigt or Galatry, yield identical positions when fit to a given set of data. Therefore, since the primary goal of the present study was to experimentally determine transition frequencies, for computational efficiency all transitions were fit with a Voigt profile. 2 Moreover, for the present case (<0.234 kPa), collisional narrowing should be negligible (<1 MHz) compared to the Doppler width (~850 MHz FWHM), thus ensuring effective equivalence between the Voigt and Galatry profiles. Importantly, our earlier FS-CRDS measurements [15,18,30] retrieved line intensities and lineshape parameters, and therefore, it was necessary to utilize the Galatry profile in those studies. Line intensity, position, and pressure broadening coefficient were fit by leastsquares minimization. For the isotopologues which did not contain 17O, the Gaussian width was constrained to the theoretical Doppler width, whereas for the 17O-containing isotopologues, the Gaussian width was floated to account for unresolved hyperfine splitting. Note that the observed hyperfine structure was the subject of a separate publication [31]. . As was shown in that publication, the unresolved hyperfine structure does not alter the center of mass of the transition and therefore, does not alter the measured transition frequency. 3.3. Results and Discussion The b1Σg+←X3Σg-(0,0) band is triply forbidden as an electric dipole transition but occurs as a magnetic dipole transition [9]. The rotational energy levels of O2 are commonly described by three quantum numbers: N, the orbital angular momentum; S, the spin; and 2 No other spectroscopic line parameters determined in the present study (e.g., intensity, pressure broadening parameter) were reported or utilized in the generation of the line lists (Tables 4-5) given below. 53 J, the total angular momentum (i.e., J = N+S). The X3Σg- ground state of O2 (for N″>0) is split into three levels having J′′ = N′′-1, N′′, N′′+1; while the upper state has only one level with J′ = N′. The 16O2 and 18O2 isotopologues follow Bose-Einstein statistics and only odd levels of N′′ have non-zero spin-statistical weights, whereas the other isotopologues can have any value of N′′. The b1Σg+←X3Σg-(0,0) band magnetic dipole transitions occur as doublets designated as ∆N (N′′) ∆J (J′′), leading to four branches: PP, PQ, RR, and RQ. Our previous FS-CRDS study [15] focused upon the P-branch (i.e., PP- and PQbranches), while this study measured the R-branch. The R-branch of each isotopologue exhibits dense, overlapping rotational structure with a bandhead near 13,165 cm1. Near the bandhead, the isotopologue transitions become heavily blended (especially in isotopically enriched mixtures); thus limiting the range of transitions we can measure quantitatively. 3.3.1. Line Positions Frequencies for the rare isotopologue transitions were measured relative to the 16O2 positions which we recently have measured with FS-CRDS [21]. Those earlier measurements were referenced to the hyperfine components of 39K; thus allowing for combined uncertainties below 1 MHz (~3×105 cm−1). Each rare isotopologue transition was referenced to the frequency of the nearest spectrally isolated 16O2 transition. Rare isotopologue transition frequencies were corrected for pressure shifting using the Jdependent 16O2 pressure shifts [32] since accurate pressure-shifting parameters for the rare isotopologues are not known and could not be accurately determined from the present data. This approximation should introduce negligible uncertainty since the pressure shift was only ~0.5 MHz (1.5×105 cm1) at the low pressures used in the present study. The estimated zero-pressure transition frequencies are given in the Supplementary Information (Tables A1-A5). In some cases the presence of interfering transitions limited the precision with which we could measure transition frequencies. This 54 was included in the experimental uncertainties given in Table A1-A5 and leads to some strong transitions to have larger uncertainties than relatively weak transitions. 3.3.2 Line Position Calculations Molecular constants for each of the rare isotopologues were then determined through a simultaneous fit to the lower (X3Σg-) and upper (b1Σg+) states using SPFIT [33] and the formalism of Rouillé et al. [34]. FS-CRDS measurements (the present study and Robichaud et al. [15]) were used for the b1Σg+←X3Σg-(0,0) band transition frequencies while lower state transition energies were further constrained by including microwave [35-39], millimeter [40-41], submillimeter [41-42], terahertz [43], and Raman [44] measurements. These global fits included a total of 127 [15,35], 203 [15,36-38,42-44], 539 [40] 98 [15,35], and 135 [15,39,41,43-44] transitions for the 16O17O, 16O18O, 17O2, 17 O18O, and 18O2 isotopologues, respectively. 3 The resulting lower and upper state molecular constants and their corresponding uncertainties are given in Tables 2-3. Following HITRAN convention, lower state energies for the 16O2 and 18O2 are set relative to the N″ = 1, J″ = 0 level, whereas for the remaining isotopologues they are relative to the N″ = 0, J″ = 1 level. For comparison molecular constants determined in previous studies have been presented in Tables 8 and 9 of Robichaud et al. [15]. It is interesting to compare these lower state molecular constants to those recently determined by Leshchishina et al. [42-43]. In that study CRDS measurements of the a1Δg←X3Σg-(0,0) were employed in combination with a similar collection of microwave and Raman studies. The FS-CRDS measurements described herein are more precise than those described by Leshchishina et al. [42-43], but a significantly more transitions were measured in that study. Therefore, a comparison allows us to assess the importance of these two separate attributes. In general, the lower state molecular constants agree to within their given uncertainties. The only significant departures are seen for 16O2, whereby the B0 and D0 parameters differ by three and nine times the quadrature sum of the given uncertainties, respectively. In addition, the two sets of parameters generally 3 For the rare isotopologues, the terms H and H0 are not well constrained, resulting in the large uncertainties given. Global fits performed with these parameters fixed to zero led to similar fit residuals. 55 have very similar reported uncertainties. The Leshchishina et al. studies [42-43] report significantly lower uncertainties for a few of the 17O2 parameters, largely due to the very limited number of 17O2 FS-CRDS measurements. TABLE 2. Lower state (X3Σg-) molecular constants determined through the described global fit for each of the O2 isotopologues (Iso.). All values are in cm−1. Lower state energy levels are defined by Rouillé et al. [34]. Values in parentheses are standard uncertainties in the final digit. 16O2 parameters were determined through a global fit as described in Long et al. [47]. Iso. B0 D0 (10-6) H0 (10-12) λ0 λ0 ′ (10-6) λ0′′ (10-12) μ0 (10-3) μ0 ′ (10-9) μ0′′ (10-14) Zero level (cm-1) 1616 1.437676078(29) 4.84178(14) 4.28(19) 1.98475118(5) 1.9470(3) 9.70(3) -8.4253696(58) -8.136(22) -4.04(15) -0.24584 1617 1.3953309(28) 4.541(12) 1.98471099(30) 1.8887(38) -8.176541(78) -7.13(59) -0.46452 1618 1.357852204(61) 4.3141(22) 1.98467434(16) 1.83330(86) -7.956002(11) -7.275(39) -0.47795 1717 1.3529812(23) 3.9(10) 1.98466937(57) 1.82776(88) -7.9278(11) -7.194(90) -0.47975 1718 1.3154968(23) 4.0485(84) 1.98463263(69) 1.7847(96) -7.70718(12) -7.0(1) -0.49403 1818 1.278008487(57) 3.82340(42) 1.984595545(42) 1.72139(23) -7.4866992(46) -6.418(16) -28(22) 3.8(11) 9.57(81) 7.73(24) -3.8(1) -0.07513 TABLE 3. Upper state (b1Σg+) molecular constants determined through the described global fit for each of the O2 isotopologues (Iso.). All values are in cm−1. Upper state energy levels are given by T+BJ(J+1)DJ2(J+1)2+HJ3(J+1)3. Values in parentheses are standard uncertainties in the final digit. 16O2 parameters were determined through a global fit as described in Long et al. [47]. Iso. 1616 1617 1618 1717 1718 1818 T 13122.0057456(89) 13123.803741(92) 13124.793138(23) 13124.92252(26) 13125.92895(11) 13126.516194(33) B 1.39124922(11) 1.3502891(32) 1.31404418(43) 1.309331(24) 1.2730753(29) 1.27800849(41) D (10−6) 5.36909(28) 5.028(31) 4.7858(18) 4.43(88) 4.487(11) 4.2381(13) H (10−12) 0.0165(33) -36(25) 3.3.3 Line Intensities As was previously noted for the 16O2 A-band [18], the calculated line intensities found in the HITRAN 2008 [16] and GEISA [48] databases contain an erroneous frequency dependence (note that the two databases have identical values for all included [b1Σg+←X3Σg-(0,0)] isotopologue transitions). This error masked a quadratic Herman- 56 Wallis (HW)-like rotation-vibration interaction [25,49] which is present in numerous high-resolution studies [18,26,29-30,50-51]. An identical error is present in the 16O18O databases. The 16O17O line intensities in the databases exhibit a different, but still erroneous, frequency dependence. Once the line intensities for the rare isotopologues are correctly calculated using the standard model of Gamache et al. [23] with the Hönl-London factors of Watson [24] and compared to the FS-CRDS measurements [15] there is evidence of a quadratic HWlike deviation. This effect, which is described by the term FHW, is given by Watson [25] as FHW  (1  a1 m  a 2 m 2 ) 2 , (1) where m = -J′′ (P-branch) and m = J′′+1 (R-branch). As seen in Figure 2, the HW-like interaction for the rare isotopologues is nearly identical to that observed for the 16O2 isotopologue. It should be expected that the HW coefficients for the different isotopologues would be very similar. To first order the linear HW coefficient has a μ−1/2dependence (where μ is the reduced mass), while the quadratic HW coefficient has a μ−1dependence [52-53]. As a result, in the most extreme case (i.e., 16O2 vs. 18O2), these coefficients would only differ by ~5% and ~12%, respectively. This subtle effect is below our instrumental sensitivity limit. Therefore, the HW coefficients determined for 16O2 were utilized for all isotopologues when calculating intensities for our line list. Band intensities were determined by a new fit of the measurements of Robichaud et al. [15] using the standard intensity model of Gamache et al. [23] with the quadratic HW like correction given in Eq. 1. These fitted band intensities differ by as much as 7% from those given in Robichaud et al. [15], because those were based on the erroneous Jdependent intensities given in HITRAN 2004 [17]. The present measured band intensities were compared to those calculated based upon the 16O2 measurements, and as can be seen in Table 1 there are no significant differences. As a result, calculated band intensities were utilized in the line list calculations (as described subsequently) instead of those based on the Robichaud et al. measurements [15]. 57 FIG. 2. Comparison of measured FS-CRDS intensities (Si) to intensities calculated using the standard model of Gamache et al. [23] with the Hönl-London factors of Watson [24] (Sm) for the 16O2, 16O18O, and 18 O2 isotopologues. Also shown is a quadratic Herman-Wallis (HW) correlation [25] of the form F=(1+a1m+a2m2)2 where a1 and a2 were determined via a fit to the 16O2 measurements to be 2.6580×104 and 3.3622×106, respectively. As can be seen, this correlation matches the 16O18O and 18O2 measurements to within the experimental uncertainty. The uncertainties shown for the 16-16 isotopologue are relative combined standard uncertainties which include Type A random uncertainties (due to fit precision) and Type B systematic uncertainties (including those uncertainties due to choice of line shape, temperature, pressure, and ring-down cavity free-spectral range). For the rare isotopologues (16-17 and 16-18) the shown uncertainties are relative standard Type A uncertainties which only include uncertainty due to fit precision. 3.3.4. Calculated Line Parameters for the 16O18O and 16O17O Isotopologues We calculated HITRAN-style line parameters for the 16O18O and 16O17O isotopologues using our collected FS-CRDS measurements [15,18,26,30,32]. The results are presented in Tables A6 and A7. Transition frequencies (~ ) and lower state energies (E′′) were calculated using the lower and upper state molecular constants given in Tables 2 and 3. Lower state energies are reported relative to the N″ = 0, J″ = 1 level in accordance with HITRAN convention. Hönl-London factors (L) were also calculated using these molecular constants and the formalism of Watson [24]. Band intensities were then determined by scaling the 16O2 band intensity determined in Long et al. [18] by the natural isotopic abundance [17] as shown in Table 1 (see Calculated Band Intensities). Line intensities were then calculated using these band intensities and the standard intensity model of Gamache et al. [23] with the quadratic 58 HW-like correction as described in Long et al. [18]. As was shown above, the rare isotopologues exhibited a similar HW deviation, thus, the HW coefficients previously determined for 16O2 [18] were used for all of the isotopologues. Additionally, Tables A6 and A7 give calculated intensities for transitions up to J′′ = 41 for the 16O18O and 16O17O isotopologues at Tref = 296 K and natural terrestrial isotopic abundance [17]. Uncertainties for these calculated intensities can be estimated based upon the experimental uncertainties given in Robichaud et al. [15]. Einstein-A coefficients were then calculated through the formalism of Gamache et al. [23] (see Eq. 2 of Long et al. [18]). The earlier FS-CRDS study [15] revealed no discernible difference in pressure broadening for the different isotopologues. Therefore, self- and air-broadening parameters (γ) were calculated based upon the correlations recently determined for the 16 O2 isotopologue [18]. The J-dependent correlations for the collisional narrowing parameters (η) of 16O2 [18] were mass-corrected for the rare isotopologues based on the gas kinetic theory formulation: ηdiff = kBT/(2πmaDp), where ma and D are the absorber’s mass and diffusion coefficient, respectively. Note that this mass correction is in agreement with the experimentally determined mass-dependence of the collisional narrowing parameter determined by Ritter and Wilkerson [29] and Robichaud et al. [15]. The pressure-shifting parameters (δ) for the rare isotopologues were constrained to the 16 O2 correlation [18,32] since we did not measure precise pressure shifts under the limited pressure range of the present experiment. 3.4 Conclusions Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) has been used to measure R-branch transitions in the b1Σg+←X3Σg-(0,0) band for all five rare O2 isotopologues. This study complements our earlier work which focused on the P-branch. The combined FS-CRDS data set was employed to produce a recommended Galatry line list for the 16O18O and 16O17O isotopologues out to J′ = 40 and includes line positions, intensities, pressure broadening parameters, and collisional narrowing parameters. The 59 reported line positions and molecular constants are based upon a simultaneous fit to the upper and lower states which included FS-CRDS measurements and a variety of earlier lower state measurements. The line intensity calculations included a Herman-Wallis-like interaction and correct a frequency-dependent error which is found in present spectroscopic databases. We anticipate that the use of the reported line parameters will significantly reduce remote sensing residuals for these rare isotopologues. Acknowledgements David A. Long was supported by the National Defense Science and Engineering Graduate Fellowship and the National Science Foundation Graduate Fellowship. Daniel K. Havey acknowledges the support of the National Research Council as a postdoctoral fellow at the National Institute of Science and Technology (NIST), Gaithersburg, MD. Part of the research described in this chapter was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Additional support was provided by the Orbiting Carbon Observatory (OCO) project, a NASA Earth System Science Pathfinder (ESSP) mission; the NASA Upper Atmospheric Research Program grant NNG06GD88G and NNX09AE21G; and the NIST Greenhouse Gas Measurements and Climate Research Program. We would also like to acknowledge Dr. Mona Shahgholi for performing the described mass spectral analysis. 60 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] D. M. O'Brien, S. A. English, G. DaCosta, J. Atmos. Ocean. Technol. 14, 105-119 (1997). D. M. O'Brien, R. M. Mitchell, S. A. English, G. A. Da Costa, J. Atmos. Ocean. Technol. 15, 1272-1286 (1998). V. V. Badayev, M. S. Malkevich, Izv. Atmos. Oceanic Phys. 14, 722–727 (1978). A. K. Heidinger, G. L. 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M. Smith, S. A. Tashkun, J. L. Teffo, R. A. Toth, V. G. Tyuterev, J. V. Auwera, P. Varanasi, G. Wagner, J. Quant. Spectrosc. Radiat. Transfer 109, 1043-1059 (2008). R. Herman, R. F. Wallis, J. Chem. Phys. 23, 637-646 (1955). R. Schermaul, R. C. M. Learner, J. Quant. Spectrosc. Radiat. Transfer 61, 781794 (1999). L. R. Brown, C. Plymate, J. Mol. Spectrosc. 199, 166-179 (2000). C. Chackerian, R. H. Tipping, J. Mol. Spectrosc. 99, 431-449 (1983). S. J. Singh, J. Mol. Struct. 127, 203-208 (1985). 63 Appendix TABLE A1. Measured 16O17O (1617) line positions, uncertainties (Unc), and the difference between observed line positions (Obs) and those calculated (Calc) using the parameters found in Tables 2 and 3. All measurements were made on the 17O-enriched sample (see Table 1 for isotopic composition). Assignment ∆N N′′ ∆J J′′ R0Q1 R1R1 R1Q2 R2R2 R3Q4 R4R4 R4Q5 R5R5 R5Q6 R6R6 R6Q7 R7R7 R8R8 R7Q8 R9R9 R8Q9 R10R10 R9Q10 R11R11 R10Q11 R12R12 R11Q12 R13R13 R12Q13 R14R14 R13Q14 R15R15 R14Q15 R16R16 R15Q16 Position (cm-1) 13126.50416 13127.31879 13129.20387 13129.83936 13134.22093 13134.60675 13136.58025 13136.85489 13138.84470 13139.01246 13141.01599 13141.07838 13143.05334 13143.09510 13144.93589 13145.08106 13146.72690 13146.97505 13148.42558 13148.77642 13150.03266 13150.48610 13151.54712 13152.10239 13152.96802 13153.62646 13154.29624 13155.05754 13155.53118 13156.39490 Unc Obs-Calc (10-5 cm-1) (10-5 cm-1) 25 -14 25 -2 25 41 25 54 24 -57 24 -48 24 -6 24 -41 24 1 24 7 48 7 48 9 48 52 48 52 24 14 24 16 24 4 24 11 24 -33 72 -25 24 1 48 15 24 30 24 -25 24 -13 24 -6 48 -12 24 16 24 4 24 -5 Sample 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 64 TABLE A2. Measured 16O18O (1618) line positions, uncertainties (Unc), and the difference between observed line positions (Obs) and those calculated (Calc) using the parameters found in Tables 2 and 3. Measurements were made on either the 17O-enriched (17) or 18O-enriched (18) sample (see Table 1 for isotopic compositions). Assignment ∆N N′′ ∆J J′′ R0Q1 R1R1 R1Q2 R2R2 R2Q3 R3R3 R3Q4 R4R4 R4Q5 R5R5 R5Q6 R6R6 R8R8 R7Q8 R9R9 R8Q9 R10R10 R9Q10 R11R11 R10Q11 Unc Obs-Calc Position (10-5 cm-1) (10-5 cm-1) Sample (cm-1) 13127.42102 24 -19 18 13128.15250 24 -2 18 13130.04493 24 37 18 13130.60487 24 -12 18 13132.53661 24 28 18 13132.96980 24 34 18 13134.92577 24 -26 18 13135.24562 24 -17 18 13137.22106 24 -22 18 13137.43362 24 -19 18 13139.42508 24 24 18 13139.53362 24 28 18 13143.46604 48 -19 17 13143.56077 72 -22 17 13145.29942 47 25 17 13145.49450 48 16 17 13147.04291 72 7 17 13147.33810 72 12 17 13148.69669 72 -29 17 13149.09159 72 -29 17 65 TABLE A3. Measured 17O17O (1717) line positions, uncertainties (Unc), and the difference between observed line positions (Obs) and those calculated (Calc) using the parameters found in Tables 2 and 3. All measurements were made on the 17O-enriched sample (see Table 1 for isotopic composition). Assignment ∆N N′′ ∆J J′′ R1Q2 R2R2 R2Q3 R3R3 R3Q4 R4R4 R4Q5 R5R5 R5Q6 R6R6 R8R8 R7Q8 R9R9 R8Q9 R10R10 R9Q10 R11R11 R10Q11 R11Q12 R12Q13 R14R14 R13Q14 R14Q15 Position (cm-1) 13130.15582 13130.70447 13132.63690 13133.06098 13135.01845 13135.32964 13137.30543 13137.50982 13139.50152 13139.60232 13143.52076 13143.62248 13145.34761 13145.54933 13147.08496 13147.38627 13148.73287 13149.13360 13150.79213 13152.36039 13153.14054 13153.83907 13155.22859 Unc Obs-Calc (10-5 cm-1) (10-5 cm-1) Sample 25 118 17 24 -84 17 24 -40 17 24 -37 17 24 8 17 24 6 17 24 4 17 24 -2 17 72 43 17 24 36 17 48 -26 17 48 -12 17 24 6 17 24 20 17 48 -15 17 72 -4 17 72 -61 17 48 -46 17 72 -13 17 72 -34 17 72 -25 17 72 -20 17 73 94 17 66 TABLE A4. Measured 17O18O (1718) line positions, uncertainties (Unc), and the difference between observed line positions (Obs) and those calculated (Calc) using the parameters found in Tables 2 and 3. All measurements were made on the 17O-enriched sample (see Table 1 for isotopic composition). Assignment ∆N N′′ ∆J J′′ R0Q1 R1R1 R1Q2 R2Q3 R3R3 R3Q4 R4R4 R4Q5 R5R5 R5Q6 R6R6 R7R7 R6Q7 R8R8 R7Q8 R9R9 R8Q9 R9Q10 R11R11 R10Q11 R12R12 R11Q12 R13R13 R12Q13 R14R14 R13Q14 R15R15 R14Q15 R16R16 Unc Obs-Calc Position (10-5 cm-1) (10-5 cm-1) Sample (cm-1) 13128.47552 26 44 17 13129.11188 26 45 17 13131.01134 25 -103 17 13133.42487 101 10 17 13133.77793 24 -52 17 13135.73912 24 -4 17 13135.98394 24 -7 17 13137.96239 24 -5 17 13138.10400 24 -8 17 13140.09715 48 0 17 13140.13850 48 0 17 13142.08746 48 36 17 13142.14478 48 41 17 13143.94964 24 -5 17 13144.10457 24 0 17 13145.72637 24 28 17 13145.97833 24 37 17 13147.76485 24 28 17 13149.01928 24 -20 17 13149.46426 24 -11 17 13150.53621 72 17 17 13151.07750 24 28 17 13151.96521 24 -32 17 13152.60288 24 -12 17 13153.30771 24 0 17 13154.04153 24 2 17 13154.56239 24 6 17 17 13155.39282 24 27 13155.72892 24 -19 17 67 TABLE A5. Measured 18O18O (1818) line positions, uncertainties (Unc), and the difference between observed line positions (Obs) and those calculated (Calc) using the parameters found in Tables 2 and 3. Measurements were made on either the 17O-enriched (17) or 18O-enriched (18) sample (see Table 1 for isotopic compositions). Assignment ∆N N′′ ∆J J′′ R1R1 R1Q2 R3R3 R3Q4 R5R5 R5Q6 R7Q8 R9R9 R11R11 R13R13 R13Q14 R15R15 R17R17 Unc Position (10-5 cm-1) (cm-1) 13129.97558 24 13131.88468 24 13134.50922 24 13136.47434 24 13138.71252 24 13140.70756 24 13144.60107 49 13146.11954 26 13149.31970 26 13152.18353 26 13154.25810 73 13154.70863 28 13156.89267 29 Obs-Calc (10-5 cm-1) Sample 33 18 11 18 -35 18 9 18 7 18 2 18 10 17 49 17 -25 17 -23 17 -2 17 -4 17 -3 17 68 TABLE A6. Line parameters of b1Σg+←X3Σg-(0,0) magnetic dipole transitions for 16O18O (1618). Transition frequencies (~ ) and lower state energies (E′′) were calculated using the molecular parameters determined during the global fit (see Tables 2 and 3) and the formalism of Rouillé et al. [34]. Line intensities (S) at Tref = 296 K and natural terrestrial isotopic abundance [17] were calculated using the standard model of Gamache et al. [23] with the Hönl-London factors (L, also calculated with the molecular parameters of Tables 2 and 3) of Watson [24] and a band intensity Sb = 8.947×1025 cm molec.1. These intensities included a quadratic Herman-Wallis-like correction [25], given by (1+a1m+a2m2)2 where a1= 2.6580×104 and a2= 3.3622×106. Einstein-A coefficients were calculated based upon these intensities as described in the text. Note 1 cm1 atm1 = 0.295872 MHz Pa1. Uncertainties are comparable to those given in Table 4 of Robichaud et al. [15]. Transition L ~ -1 (cm ) S (cm molec.-1) A (s-1) E" (cm-1) γair (MHz Pa-1) γself (MHz Pa-1) δ (MHz Pa-1) ηair (MHz Pa-1) ηself (MHz Pa-1) P41Q40 20.3643 12941.361526 1.487E-30 0.02142 2325.43140 7.29E-03 8.48E-03 -3.02E-03 5.93E-03 5.15E-03 P40P40 20.5000 12945.953401 2.538E-30 0.02213 2216.96355 7.41E-03 8.59E-03 -3.00E-03 6.34E-03 5.11E-03 P40Q39 19.8643 12947.629006 2.478E-30 0.02143 2215.28795 7.41E-03 8.59E-03 -3.00E-03 5.89E-03 5.11E-03 P39P39 20.0000 12952.120128 4.174E-30 0.02216 2109.45667 7.55E-03 8.72E-03 -2.98E-03 6.28E-03 5.07E-03 P39Q38 19.3643 12953.803756 4.071E-30 0.02144 2107.77304 7.55E-03 8.72E-03 -2.98E-03 5.84E-03 5.07E-03 P38P38 19.5000 12958.194499 6.772E-30 0.02219 2004.58270 7.70E-03 8.85E-03 -2.96E-03 6.22E-03 5.02E-03 P38Q37 18.8643 12959.886163 6.599E-30 0.02145 2002.89103 7.70E-03 8.85E-03 -2.96E-03 5.79E-03 5.02E-03 P37P37 19.0000 12964.176885 1.084E-29 0.02222 1902.34587 7.86E-03 9.00E-03 -2.94E-03 6.15E-03 4.96E-03 P37Q36 18.3643 12965.876597 1.055E-29 0.02146 1900.64616 7.86E-03 9.00E-03 -2.94E-03 5.73E-03 4.96E-03 P36P36 18.5000 12970.067641 1.711E-29 0.02225 1802.75031 8.04E-03 9.15E-03 -2.91E-03 6.08E-03 4.91E-03 P36Q35 17.8643 12971.775414 1.665E-29 0.02147 1801.04253 8.04E-03 9.15E-03 -2.91E-03 5.66E-03 4.91E-03 P35P35 18.0000 12975.867106 2.666E-29 0.02228 1705.80000 8.24E-03 9.32E-03 -2.89E-03 6.00E-03 4.84E-03 P35Q34 17.3643 12977.582955 2.591E-29 0.02148 1704.08415 8.24E-03 9.32E-03 -2.89E-03 5.59E-03 4.84E-03 P34P34 17.5000 12981.575607 4.098E-29 0.02232 1611.49884 8.46E-03 9.50E-03 -2.87E-03 5.91E-03 4.78E-03 P34Q33 16.8643 12983.299546 3.979E-29 0.02149 1609.77490 8.46E-03 9.50E-03 -2.87E-03 5.52E-03 4.78E-03 P33P33 17.0000 12987.193454 6.211E-29 0.02236 1519.85059 8.70E-03 9.69E-03 -2.84E-03 5.81E-03 4.71E-03 P33Q32 16.3644 12988.925501 6.024E-29 0.02151 1518.11854 8.70E-03 9.69E-03 -2.84E-03 5.44E-03 4.71E-03 4.63E-03 P32P32 16.5000 12992.720944 9.286E-29 0.02240 1430.85890 8.96E-03 9.89E-03 -2.82E-03 5.71E-03 P32Q31 15.8644 12994.461116 8.997E-29 0.02152 1429.11872 8.96E-03 9.89E-03 -2.82E-03 5.35E-03 4.63E-03 P31P31 16.0000 12998.158360 1.369E-28 0.02243 1344.52730 9.23E-03 1.01E-02 -2.79E-03 5.60E-03 4.55E-03 P31Q30 15.3644 12999.906677 1.325E-28 0.02153 1342.77898 9.23E-03 1.01E-02 -2.79E-03 5.25E-03 4.55E-03 P30P30 15.5000 13003.505971 1.991E-28 0.02248 1260.85922 9.52E-03 1.03E-02 -2.77E-03 5.48E-03 4.47E-03 P30Q29 14.8644 13005.262455 1.924E-28 0.02154 1259.10274 9.52E-03 1.03E-02 -2.77E-03 5.15E-03 4.47E-03 P29P29 15.0000 13008.764033 2.856E-28 0.02252 1179.85797 9.83E-03 1.05E-02 -2.74E-03 5.36E-03 4.39E-03 P29Q28 14.3644 13010.528707 2.756E-28 0.02156 1178.09330 9.83E-03 1.05E-02 -2.74E-03 5.04E-03 4.39E-03 P28P28 14.5000 13013.932787 4.036E-28 0.02257 1101.52674 1.01E-02 1.08E-02 -2.71E-03 5.23E-03 4.30E-03 P28Q27 13.8644 13015.705676 3.890E-28 0.02157 1099.75385 1.01E-02 1.08E-02 -2.71E-03 4.93E-03 4.30E-03 P27P27 14.0000 13019.012462 5.627E-28 0.02262 1025.86861 1.05E-02 1.10E-02 -2.69E-03 5.10E-03 4.21E-03 4.21E-03 P27Q26 13.3644 13020.793595 5.414E-28 0.02158 1024.08748 1.05E-02 1.10E-02 -2.69E-03 4.82E-03 P26P26 13.5000 13024.003273 7.733E-28 0.02268 952.88657 1.08E-02 1.12E-02 -2.66E-03 4.96E-03 4.13E-03 P26Q25 12.8644 13025.792682 7.428E-28 0.02160 951.09716 1.08E-02 1.12E-02 -2.66E-03 4.70E-03 4.13E-03 P25P25 13.0000 13028.905423 1.047E-27 0.02273 882.58346 1.11E-02 1.15E-02 -2.63E-03 4.83E-03 4.04E-03 P25Q24 12.3644 13030.703143 1.004E-27 0.02161 880.78574 1.11E-02 1.15E-02 -2.63E-03 4.59E-03 4.04E-03 P24P24 12.5000 13033.719101 1.398E-27 0.02279 814.96204 1.15E-02 1.17E-02 -2.60E-03 4.70E-03 3.96E-03 69 P24Q23 11.8644 13035.525171 1.338E-27 0.02163 813.15597 1.15E-02 1.17E-02 -2.60E-03 4.47E-03 3.96E-03 P23P23 12.0000 13038.444484 1.840E-27 0.02286 750.02495 1.18E-02 1.19E-02 -2.57E-03 4.57E-03 3.88E-03 P23Q22 11.3644 13040.258949 1.756E-27 0.02164 748.21048 1.18E-02 1.19E-02 -2.57E-03 4.36E-03 3.88E-03 P22P22 11.5000 13043.081737 2.385E-27 0.02293 687.77471 1.21E-02 1.21E-02 -2.54E-03 4.45E-03 3.80E-03 P22Q21 10.8644 13044.904646 2.271E-27 0.02166 685.95181 1.21E-02 1.21E-02 -2.54E-03 4.26E-03 3.80E-03 P21P21 11.0000 13047.631011 3.046E-27 0.02301 628.21376 1.23E-02 1.23E-02 -2.50E-03 4.34E-03 3.72E-03 P21Q20 10.3644 13049.462421 2.894E-27 0.02168 626.38235 1.23E-02 1.23E-02 -2.50E-03 4.16E-03 3.72E-03 P20P20 10.5000 13052.092445 3.832E-27 0.02310 571.34441 1.26E-02 1.25E-02 -2.47E-03 4.23E-03 3.65E-03 P20Q19 9.8644 13053.932422 3.630E-27 0.02169 569.50443 1.26E-02 1.25E-02 -2.47E-03 4.07E-03 3.65E-03 P19P19 10.0000 13056.466167 4.746E-27 0.02319 517.16885 1.28E-02 1.27E-02 -2.44E-03 4.14E-03 3.59E-03 P19Q18 9.3644 13058.314787 4.482E-27 0.02170 515.32023 1.28E-02 1.27E-02 -2.44E-03 3.99E-03 3.59E-03 P18P18 9.5000 13060.752291 5.789E-27 0.02329 465.68920 1.30E-02 1.29E-02 -2.40E-03 4.05E-03 3.52E-03 P18Q17 8.8644 13062.609642 5.447E-27 0.02172 463.83185 1.30E-02 1.29E-02 -2.40E-03 3.92E-03 3.52E-03 P17P17 9.0000 13064.950922 6.949E-27 0.02341 416.90743 1.32E-02 1.30E-02 -2.37E-03 3.98E-03 3.47E-03 P17Q16 8.3644 13066.817107 6.512E-27 0.02175 415.04124 1.32E-02 1.30E-02 -2.37E-03 3.85E-03 3.47E-03 P16P16 8.5000 13069.062149 8.206E-27 0.02354 370.82543 1.34E-02 1.32E-02 -2.33E-03 3.91E-03 3.41E-03 P16Q15 7.8644 13070.937292 7.658E-27 0.02177 368.95028 1.34E-02 1.32E-02 -2.33E-03 3.79E-03 3.41E-03 P15P15 8.0000 13073.086052 9.532E-27 0.02368 327.44497 1.36E-02 1.33E-02 -2.30E-03 3.85E-03 3.35E-03 3.35E-03 P15Q14 7.3644 13074.970302 8.851E-27 0.02180 325.56072 1.36E-02 1.33E-02 -2.30E-03 3.74E-03 P14P14 7.5000 13077.022698 1.089E-26 0.02386 286.76774 1.37E-02 1.35E-02 -2.26E-03 3.79E-03 3.30E-03 P14Q13 6.8644 13078.916235 1.005E-26 0.02183 284.87420 1.37E-02 1.35E-02 -2.26E-03 3.69E-03 3.30E-03 P13P13 7.0000 13080.872144 1.221E-26 0.02405 248.79528 1.39E-02 1.36E-02 -2.22E-03 3.73E-03 3.25E-03 P13Q12 6.3644 13082.775192 1.120E-26 0.02186 246.89223 1.39E-02 1.36E-02 -2.22E-03 3.64E-03 3.25E-03 P12P12 6.5000 13084.634433 1.346E-26 0.02428 213.52906 1.40E-02 1.38E-02 -2.18E-03 3.67E-03 3.19E-03 P12Q11 5.8644 13086.547273 1.225E-26 0.02190 211.61622 1.40E-02 1.38E-02 -2.18E-03 3.59E-03 3.19E-03 P11P11 6.0000 13088.309598 1.455E-26 0.02455 180.97044 1.42E-02 1.39E-02 -2.15E-03 3.61E-03 3.13E-03 P11Q10 5.3644 13090.232590 1.312E-26 0.02194 179.04745 1.42E-02 1.39E-02 -2.15E-03 3.53E-03 3.13E-03 P10P10 5.5000 13091.897660 1.541E-26 0.02488 151.12066 1.43E-02 1.41E-02 -2.11E-03 3.54E-03 3.06E-03 P10Q9 4.8645 13093.831279 1.375E-26 0.02200 149.18704 1.43E-02 1.41E-02 -2.11E-03 3.47E-03 3.06E-03 P9P9 5.0000 13095.398629 1.598E-26 0.02528 123.98087 1.45E-02 1.43E-02 -2.07E-03 3.46E-03 2.99E-03 P9Q8 4.3645 13097.343514 1.408E-26 0.02206 122.03598 1.45E-02 1.43E-02 -2.07E-03 3.41E-03 2.99E-03 P8P8 4.5000 13098.812502 1.619E-26 0.02579 99.55209 1.47E-02 1.45E-02 -2.02E-03 3.37E-03 2.91E-03 P8Q7 3.8645 13100.769551 1.403E-26 0.02215 97.59505 1.47E-02 1.45E-02 -2.02E-03 3.33E-03 2.91E-03 P7P7 4.0000 13102.139268 1.599E-26 0.02645 77.83528 1.50E-02 1.47E-02 -1.98E-03 3.27E-03 2.82E-03 P7Q6 3.3646 13104.109789 1.357E-26 0.02225 75.86476 1.50E-02 1.47E-02 -1.98E-03 3.24E-03 2.82E-03 P6P6 3.5000 13105.378901 1.533E-26 0.02736 58.83125 1.53E-02 1.50E-02 -1.94E-03 3.14E-03 2.72E-03 P6Q5 2.8647 13107.364921 1.266E-26 0.02239 56.84523 1.53E-02 1.50E-02 -1.94E-03 3.13E-03 2.72E-03 P5P5 3.0000 13108.531365 1.422E-26 0.02867 42.54074 1.57E-02 1.53E-02 -1.90E-03 2.99E-03 2.60E-03 P5Q4 2.3649 13110.536270 1.132E-26 0.02260 40.53584 1.57E-02 1.53E-02 -1.90E-03 3.00E-03 2.60E-03 2.46E-03 P4P4 2.5000 13111.596614 1.265E-26 0.03072 28.96436 1.61E-02 1.57E-02 -1.85E-03 2.81E-03 P4Q3 1.8653 13113.626722 9.532E-27 0.02292 26.93426 1.61E-02 1.57E-02 -1.85E-03 2.85E-03 2.46E-03 P3P3 2.0000 13114.574588 1.066E-26 0.03441 18.10264 1.67E-02 1.62E-02 -1.81E-03 2.59E-03 2.29E-03 P3Q2 1.3662 13116.643966 7.359E-27 0.02350 16.03326 1.67E-02 1.62E-02 -1.81E-03 2.66E-03 2.29E-03 P2P2 1.5000 13117.465220 8.323E-27 0.04302 9.95599 1.73E-02 1.67E-02 -1.77E-03 2.32E-03 2.08E-03 P2Q1 0.8697 13119.617605 4.874E-27 0.02494 7.80360 1.73E-02 1.67E-02 -1.77E-03 2.42E-03 2.08E-03 P1P1 1.0000 13120.268426 5.695E-27 0.08603 4.52471 1.82E-02 1.74E-02 -1.72E-03 1.97E-03 1.84E-03 R0Q1 0.6303 13127.421207 3.665E-27 0.01807 0.00000 1.73E-02 1.67E-02 -1.77E-03 4.25E-03 2.08E-03 R1R1 0.5000 13128.152519 2.845E-27 0.00861 4.52471 1.67E-02 1.62E-02 -1.81E-03 4.29E-03 2.29E-03 70 R1Q2 1.1338 13130.044563 6.508E-27 0.01951 2.63267 1.67E-02 1.62E-02 -1.81E-03 4.28E-03 2.29E-03 R2R2 1.0000 13130.604991 5.539E-27 0.01230 9.95599 1.61E-02 1.57E-02 -1.85E-03 4.33E-03 2.46E-03 R2Q3 1.6347 13132.536326 9.137E-27 0.02010 8.02465 1.61E-02 1.57E-02 -1.85E-03 4.31E-03 2.46E-03 R3R3 1.5000 13132.969465 7.984E-27 0.01434 18.10264 1.57E-02 1.53E-02 -1.90E-03 4.37E-03 2.60E-03 R3Q4 2.1351 13134.926032 1.147E-26 0.02042 16.14607 1.57E-02 1.53E-02 -1.90E-03 4.34E-03 2.60E-03 R4R4 2.0000 13135.245790 1.010E-26 0.01565 28.96436 1.53E-02 1.50E-02 -1.94E-03 4.40E-03 2.72E-03 R4Q5 2.6353 13137.221279 1.342E-26 0.02062 26.98888 1.53E-02 1.50E-02 -1.94E-03 4.36E-03 2.72E-03 2.82E-03 R5R5 2.5000 13137.433807 1.181E-26 0.01655 42.54074 1.50E-02 1.47E-02 -1.98E-03 4.43E-03 R5Q6 3.1354 13139.424838 1.495E-26 0.02076 40.54971 1.50E-02 1.47E-02 -1.98E-03 4.37E-03 2.82E-03 R6R6 3.0000 13139.533343 1.309E-26 0.01722 58.83125 1.47E-02 1.45E-02 -2.02E-03 4.45E-03 2.91E-03 R6Q7 3.6355 13141.537898 1.601E-26 0.02086 56.82670 1.47E-02 1.45E-02 -2.02E-03 4.39E-03 2.91E-03 R7R7 3.5000 13141.544215 1.392E-26 0.01772 77.83528 1.45E-02 1.43E-02 -2.07E-03 4.46E-03 2.99E-03 R7Q8 4.1355 13143.560991 1.660E-26 0.02094 75.81850 1.45E-02 1.43E-02 -2.07E-03 4.40E-03 2.99E-03 R8R8 4.0000 13143.466228 1.431E-26 0.01812 99.55209 1.43E-02 1.41E-02 -2.11E-03 4.48E-03 3.06E-03 R8Q9 4.6355 13145.494336 1.675E-26 0.02100 97.52399 1.43E-02 1.41E-02 -2.11E-03 4.41E-03 3.06E-03 R9R9 4.5000 13145.299174 1.429E-26 0.01845 123.98087 1.42E-02 1.39E-02 -2.15E-03 4.49E-03 3.13E-03 R9Q10 5.1356 13147.337981 1.647E-26 0.02105 121.94206 1.42E-02 1.39E-02 -2.15E-03 4.42E-03 3.13E-03 R10R10 5.0000 13147.042835 1.391E-26 0.01872 151.12066 1.40E-02 1.38E-02 -2.18E-03 4.51E-03 3.19E-03 R10Q11 5.6356 13149.091875 1.583E-26 0.02109 149.07162 1.40E-02 1.38E-02 -2.18E-03 4.43E-03 3.19E-03 R11R11 5.5000 13148.696981 1.323E-26 0.01894 180.97044 1.39E-02 1.36E-02 -2.22E-03 4.52E-03 3.25E-03 R11Q12 6.1356 13150.755900 1.491E-26 0.02113 178.91152 1.39E-02 1.36E-02 -2.22E-03 4.44E-03 3.25E-03 R12R12 6.0000 13150.261369 1.232E-26 0.01913 213.52906 1.37E-02 1.35E-02 -2.26E-03 4.53E-03 3.30E-03 R12Q13 6.6356 13152.329893 1.376E-26 0.02116 211.46054 1.37E-02 1.35E-02 -2.26E-03 4.44E-03 3.30E-03 R13R13 6.5000 13151.735743 1.124E-26 0.01929 248.79528 1.36E-02 1.33E-02 -2.30E-03 4.54E-03 3.35E-03 R13Q14 7.1356 13153.813656 1.246E-26 0.02119 246.71737 1.36E-02 1.33E-02 -2.30E-03 4.45E-03 3.35E-03 R14R14 7.0000 13153.119839 1.007E-26 0.01944 286.76774 1.34E-02 1.32E-02 -2.33E-03 4.56E-03 3.41E-03 R14Q15 7.6356 13155.206967 1.109E-26 0.02121 284.68061 1.34E-02 1.32E-02 -2.33E-03 4.46E-03 3.41E-03 R15R15 7.5000 13154.413376 8.847E-27 0.01956 327.44497 1.32E-02 1.30E-02 -2.37E-03 4.57E-03 3.47E-03 R15Q16 8.1356 13156.509579 9.695E-27 0.02122 325.34877 1.32E-02 1.30E-02 -2.37E-03 4.47E-03 3.47E-03 R16R16 8.0000 13155.616062 7.641E-27 0.01968 370.82543 1.30E-02 1.29E-02 -2.40E-03 4.59E-03 3.52E-03 R16Q17 8.6356 13157.721224 8.332E-27 0.02124 368.72026 1.30E-02 1.29E-02 -2.40E-03 4.48E-03 3.52E-03 R17R17 8.5000 13156.727595 6.488E-27 0.01977 416.90743 1.28E-02 1.27E-02 -2.44E-03 4.60E-03 3.59E-03 R17Q18 9.1356 13158.841619 7.045E-27 0.02125 414.79340 1.28E-02 1.27E-02 -2.44E-03 4.49E-03 3.59E-03 R18R18 9.0000 13157.747658 5.418E-27 0.01986 465.68920 1.26E-02 1.25E-02 -2.47E-03 4.62E-03 3.65E-03 R18Q19 9.6356 13159.870463 5.861E-27 0.02127 463.56639 1.26E-02 1.25E-02 -2.47E-03 4.50E-03 3.65E-03 R19R19 9.5000 13158.675921 4.452E-27 0.01994 517.16885 1.23E-02 1.23E-02 -2.50E-03 4.64E-03 3.72E-03 R19Q20 10.1356 13160.807439 4.799E-27 0.02128 515.03734 1.23E-02 1.23E-02 -2.50E-03 4.52E-03 3.72E-03 R20R20 10.0000 13159.512042 3.601E-27 0.02001 571.34441 1.21E-02 1.21E-02 -2.54E-03 4.67E-03 3.80E-03 R20Q21 10.6356 13161.652214 3.869E-27 0.02129 569.20424 1.21E-02 1.21E-02 -2.54E-03 4.53E-03 3.80E-03 R21R21 10.5000 13160.255666 2.867E-27 0.02008 628.21376 1.18E-02 1.19E-02 -2.57E-03 4.69E-03 3.88E-03 R21Q22 11.1356 13162.404443 3.073E-27 0.02130 626.06499 1.18E-02 1.19E-02 -2.57E-03 4.55E-03 3.88E-03 R22R22 11.0000 13160.906424 2.248E-27 0.02014 687.77471 1.15E-02 1.17E-02 -2.60E-03 4.72E-03 3.96E-03 R22Q23 11.6356 13163.063763 2.403E-27 0.02131 685.61737 1.15E-02 1.17E-02 -2.60E-03 4.56E-03 3.96E-03 R23R23 11.5000 13161.463934 1.736E-27 0.02019 750.02495 1.11E-02 1.15E-02 -2.63E-03 4.74E-03 4.04E-03 4.04E-03 R23Q24 12.1356 13163.629799 1.851E-27 0.02131 747.85908 1.11E-02 1.15E-02 -2.63E-03 4.58E-03 R24R24 12.0000 13161.927802 1.321E-27 0.02024 814.96204 1.08E-02 1.12E-02 -2.66E-03 4.77E-03 4.13E-03 R24Q25 12.6356 13164.102161 1.406E-27 0.02132 812.78768 1.08E-02 1.12E-02 -2.66E-03 4.60E-03 4.13E-03 R25R25 12.5000 13162.297618 9.906E-28 0.02028 882.58346 1.05E-02 1.10E-02 -2.69E-03 4.80E-03 4.21E-03 71 R25Q26 13.1356 13164.480442 1.052E-27 0.02132 880.40063 1.05E-02 1.10E-02 -2.69E-03 4.62E-03 4.21E-03 R26R26 13.0000 13162.572959 7.319E-28 0.02033 952.88657 1.01E-02 1.08E-02 -2.71E-03 4.82E-03 4.30E-03 R26Q27 13.6356 13164.764225 7.759E-28 0.02133 950.69530 1.01E-02 1.08E-02 -2.71E-03 4.63E-03 4.30E-03 R27R27 13.5000 13162.753388 5.331E-28 0.02037 1025.86861 9.83E-03 1.05E-02 -2.74E-03 4.85E-03 4.39E-03 R27Q28 14.1356 13164.953075 5.641E-28 0.02133 1023.66893 9.83E-03 1.05E-02 -2.74E-03 4.65E-03 4.39E-03 R28R28 14.0000 13162.838454 3.826E-28 0.02040 1101.52674 9.52E-03 1.03E-02 -2.77E-03 4.87E-03 4.47E-03 R28Q29 14.6356 13165.046543 4.043E-28 0.02134 1099.31865 9.52E-03 1.03E-02 -2.77E-03 4.67E-03 4.47E-03 R29R29 14.5000 13162.827691 2.708E-28 0.02044 1179.85797 9.23E-03 1.01E-02 -2.79E-03 4.90E-03 4.55E-03 R29Q30 15.1356 13165.044166 2.857E-28 0.02134 1177.64149 9.23E-03 1.01E-02 -2.79E-03 4.68E-03 4.55E-03 4.63E-03 R30R30 15.0000 13162.720619 1.889E-28 0.02047 1260.85922 8.96E-03 9.89E-03 -2.82E-03 4.92E-03 R30Q31 15.6356 13164.945467 1.991E-28 0.02134 1258.63437 8.96E-03 9.89E-03 -2.82E-03 4.70E-03 4.63E-03 R31R31 15.5000 13162.516742 1.300E-28 0.02049 1344.52730 8.70E-03 9.69E-03 -2.84E-03 4.94E-03 4.71E-03 R31Q32 16.1356 13164.749950 1.368E-28 0.02134 1342.29409 8.70E-03 9.69E-03 -2.84E-03 4.71E-03 4.71E-03 R32R32 16.0000 13162.215551 8.817E-29 0.02052 1430.85890 8.46E-03 9.50E-03 -2.87E-03 4.96E-03 4.78E-03 R32Q33 16.6357 13164.457109 9.269E-29 0.02135 1428.61734 8.46E-03 9.50E-03 -2.87E-03 4.72E-03 4.78E-03 R33R33 16.5000 13161.816519 5.899E-29 0.02054 1519.85059 8.24E-03 9.32E-03 -2.89E-03 4.98E-03 4.84E-03 R33Q34 17.1357 13164.066417 6.194E-29 0.02135 1517.60069 8.24E-03 9.32E-03 -2.89E-03 4.74E-03 4.84E-03 R34R34 17.0000 13161.319105 3.892E-29 0.02057 1611.49884 8.04E-03 9.15E-03 -2.91E-03 5.00E-03 4.91E-03 R34Q35 17.6357 13163.577337 4.083E-29 0.02135 1609.24061 8.04E-03 9.15E-03 -2.91E-03 4.75E-03 4.91E-03 R35R35 17.5000 13160.722753 2.534E-29 0.02059 1705.80000 7.86E-03 9.00E-03 -2.94E-03 5.01E-03 4.96E-03 R35Q36 18.1357 13162.989311 2.655E-29 0.02135 1703.53344 7.86E-03 9.00E-03 -2.94E-03 4.76E-03 4.96E-03 R36R36 18.0000 13160.026890 1.627E-29 0.02061 1802.75031 7.70E-03 8.85E-03 -2.96E-03 5.02E-03 5.02E-03 R36Q37 18.6357 13162.301768 1.703E-29 0.02135 1800.47543 7.70E-03 8.85E-03 -2.96E-03 4.76E-03 5.02E-03 5.07E-03 R37R37 18.5000 13159.230925 1.030E-29 0.02063 1902.34587 7.55E-03 8.72E-03 -2.98E-03 5.04E-03 R37Q38 19.1357 13161.514119 1.077E-29 0.02135 1900.06268 7.55E-03 8.72E-03 -2.98E-03 4.77E-03 5.07E-03 R38R38 19.0000 13158.334255 6.435E-30 0.02064 2004.58270 7.41E-03 8.59E-03 -3.00E-03 5.05E-03 5.11E-03 R38Q39 19.6357 13160.625760 6.727E-30 0.02134 2002.29119 7.41E-03 8.59E-03 -3.00E-03 4.78E-03 5.11E-03 R39R39 19.5000 13157.336255 3.967E-30 0.02066 2109.45667 7.29E-03 8.48E-03 -3.02E-03 5.06E-03 5.15E-03 R39Q40 20.1357 13159.636069 4.143E-30 0.02134 2107.15685 7.29E-03 8.48E-03 -3.02E-03 4.79E-03 5.15E-03 R40R40 20.0000 13156.236287 2.413E-30 0.02068 2216.96355 7.17E-03 8.38E-03 -3.04E-03 5.07E-03 5.19E-03 R40Q41 20.6357 13158.544406 2.517E-30 0.02134 2214.65543 7.17E-03 8.38E-03 -3.04E-03 4.79E-03 5.19E-03 72 TABLE A7. Line parameters of b1Σg+←X3Σg-(0,0) magnetic dipole transitions for 16O17O (1617). Transition frequencies (~ ) and lower state energies (E′′) were calculated using the molecular parameters determined during the global fit (see Tables 2 and 3) and the formalism of Rouillé et al. [34]. Line intensities (S) at Tref = 296 K and natural terrestrial isotopic abundance [17] were calculated using the standard model of Gamache et al. [23] with the Hönl-London factors (L, also calculated with the molecular parameters of Tables 2 and 3) of Watson [24] and a band intensity Sb = 1.663×1025 cm molec.1. These intensities included a quadratic Herman-Wallis-like correction [25], given by (1+a1m+a2m2)2 where a1= 2.6580×104 and a2= 3.3622×106. Einstein-A coefficients were calculated based upon these intensities as described in the text. Note 1 cm1 atm1 = 0.295872 MHz Pa1. Uncertainties are comparable to those given in Table 6 of Robichaud et al. [15]. Transition L ~ (cm-1) S (cm molec.-1) A (s-1) E" (cm-1) γair (MHz Pa-1) γself δ ηair (MHz Pa-1) (MHz Pa-1) (MHz Pa-1) ηself (MHz Pa-1) P41Q40 20.3546 12935.320248 2.260E-31 0.02323 2389.43349 7.29E-03 8.48E-03 -3.02E-03 6.11E-03 5.31E-03 P40P40 20.5000 12940.090586 3.916E-31 0.02401 2277.92727 7.41E-03 8.59E-03 -3.00E-03 6.53E-03 5.27E-03 P40Q39 19.8546 12941.757977 3.820E-31 0.02324 2276.25988 7.41E-03 8.59E-03 -3.00E-03 6.06E-03 5.27E-03 P39P39 20.0000 12946.425220 6.532E-31 0.02405 2167.46318 7.55E-03 8.72E-03 -2.98E-03 6.47E-03 5.22E-03 P39Q38 19.3546 12948.100865 6.367E-31 0.02326 2165.78753 7.55E-03 8.72E-03 -2.98E-03 6.02E-03 5.22E-03 P38P38 19.5000 12952.665349 1.075E-30 0.02408 2059.70473 7.70E-03 8.85E-03 -2.96E-03 6.41E-03 5.17E-03 P38Q37 18.8546 12954.349261 1.047E-30 0.02327 2058.02082 7.70E-03 8.85E-03 -2.96E-03 5.96E-03 5.17E-03 P37P37 19.0000 12958.811310 1.744E-30 0.02412 1954.65617 7.86E-03 9.00E-03 -2.94E-03 6.34E-03 5.11E-03 P37Q36 18.3546 12960.503501 1.697E-30 0.02328 1952.96397 7.86E-03 9.00E-03 -2.94E-03 5.90E-03 5.11E-03 P36P36 18.5000 12964.863425 2.791E-30 0.02415 1852.32164 8.04E-03 9.15E-03 -2.91E-03 6.26E-03 5.05E-03 P36Q35 17.8546 12966.563910 2.713E-30 0.02330 1850.62115 8.04E-03 9.15E-03 -2.91E-03 5.84E-03 5.05E-03 P35P35 18.0000 12970.822007 4.404E-30 0.02419 1752.70517 8.24E-03 9.32E-03 -2.89E-03 6.18E-03 4.99E-03 P35Q34 17.3546 12972.530801 4.278E-30 0.02331 1750.99638 8.24E-03 9.32E-03 -2.89E-03 5.76E-03 4.99E-03 P34P34 17.5000 12976.687356 6.854E-30 0.02423 1655.81069 8.46E-03 9.50E-03 -2.87E-03 6.09E-03 4.92E-03 P34Q33 16.8546 12978.404475 6.651E-30 0.02332 1654.09357 8.46E-03 9.50E-03 -2.87E-03 5.69E-03 4.92E-03 P33P33 17.0000 12982.459761 1.052E-29 0.02427 1561.64201 8.70E-03 9.69E-03 -2.84E-03 5.99E-03 4.85E-03 P33Q32 16.3546 12984.185223 1.020E-29 0.02334 1559.91655 8.70E-03 9.69E-03 -2.84E-03 5.60E-03 4.85E-03 P32P32 16.5000 12988.139499 1.591E-29 0.02431 1470.20284 8.96E-03 9.89E-03 -2.82E-03 5.88E-03 4.77E-03 P32Q31 15.8546 12989.873323 1.541E-29 0.02335 1468.46902 8.96E-03 9.89E-03 -2.82E-03 5.51E-03 4.77E-03 P31P31 16.0000 12993.726835 2.374E-29 0.02436 1381.49677 9.23E-03 1.01E-02 -2.79E-03 5.77E-03 4.69E-03 P31Q30 15.3546 12995.469042 2.296E-29 0.02336 1379.75457 9.23E-03 1.01E-02 -2.79E-03 5.41E-03 4.69E-03 P30P30 15.5000 12999.222023 3.491E-29 0.02440 1295.52730 9.52E-03 1.03E-02 -2.77E-03 5.65E-03 4.61E-03 P30Q29 14.8546 13000.972635 3.371E-29 0.02338 1293.77668 9.52E-03 1.03E-02 -2.77E-03 5.30E-03 4.61E-03 P29P29 15.0000 13004.625304 5.060E-29 0.02445 1212.29779 9.83E-03 1.05E-02 -2.74E-03 5.52E-03 4.52E-03 P29Q28 14.3546 13006.384347 4.880E-29 0.02339 1210.53874 9.83E-03 1.05E-02 -2.74E-03 5.19E-03 4.52E-03 P28P28 14.5000 13009.936909 7.230E-29 0.02451 1131.81152 1.01E-02 1.08E-02 -2.71E-03 5.39E-03 4.43E-03 P28Q27 13.8546 13011.704410 6.962E-29 0.02340 1130.04402 1.01E-02 1.08E-02 -2.71E-03 5.08E-03 4.43E-03 P27P27 14.0000 13015.157057 1.018E-28 0.02456 1054.07165 1.05E-02 1.10E-02 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13162.006779 2.460E-28 0.02200 837.36040 1.08E-02 1.12E-02 -2.66E-03 4.91E-03 4.25E-03 R24Q25 12.6454 13164.185952 2.620E-28 0.02319 835.18123 1.08E-02 1.12E-02 -2.66E-03 4.74E-03 4.25E-03 R25R25 12.5000 13162.385503 1.828E-28 0.02205 906.84320 1.05E-02 1.10E-02 -2.69E-03 4.94E-03 4.34E-03 R25Q26 13.1454 13164.573392 1.942E-28 0.02319 904.65531 1.05E-02 1.10E-02 -2.69E-03 4.76E-03 4.34E-03 R26R26 13.0000 13162.667199 1.338E-28 0.02209 979.08123 1.01E-02 1.08E-02 -2.71E-03 4.97E-03 4.43E-03 R26Q27 13.6454 13164.863780 1.419E-28 0.02320 976.88465 1.01E-02 1.08E-02 -2.71E-03 4.77E-03 4.43E-03 R27R27 13.5000 13162.851444 9.646E-29 0.02213 1054.07165 9.83E-03 1.05E-02 -2.74E-03 5.00E-03 4.52E-03 R27Q28 14.1454 13165.056692 1.022E-28 0.02320 1051.86640 9.83E-03 1.05E-02 -2.74E-03 4.79E-03 4.52E-03 R28R28 14.0000 13162.937800 6.855E-29 0.02217 1131.81152 9.52E-03 1.03E-02 -2.77E-03 5.02E-03 4.61E-03 R28Q29 14.6454 13165.151696 7.249E-29 0.02320 1129.59762 9.52E-03 1.03E-02 -2.77E-03 4.81E-03 4.61E-03 R29R29 14.5000 13162.925820 4.800E-29 0.02221 1212.29779 9.23E-03 1.01E-02 -2.79E-03 5.05E-03 4.69E-03 R29Q30 15.1454 13165.148346 5.069E-29 0.02321 1210.07526 9.23E-03 1.01E-02 -2.79E-03 4.82E-03 4.69E-03 R30R30 15.0000 13162.815044 3.313E-29 0.02224 1295.52730 8.96E-03 9.89E-03 -2.82E-03 5.07E-03 4.77E-03 R30Q31 15.6454 13165.046185 3.494E-29 0.02321 1293.29615 8.96E-03 9.89E-03 -2.82E-03 4.84E-03 4.77E-03 R31R31 15.5000 13162.605000 2.254E-29 0.02227 1381.49677 8.70E-03 9.69E-03 -2.84E-03 5.09E-03 4.85E-03 R31Q32 16.1454 13164.844743 2.374E-29 0.02321 1379.25703 8.70E-03 9.69E-03 -2.84E-03 4.85E-03 4.85E-03 R32R32 16.0000 13162.295205 1.512E-29 0.02230 1470.20284 8.46E-03 9.50E-03 -2.87E-03 5.11E-03 4.92E-03 R32Q33 16.6454 13164.543538 1.590E-29 0.02321 1467.95451 8.46E-03 9.50E-03 -2.87E-03 4.87E-03 4.92E-03 R33R33 16.5000 13161.885164 9.993E-30 0.02233 1561.64201 8.24E-03 9.32E-03 -2.89E-03 5.13E-03 4.99E-03 R33Q34 17.1454 13164.142076 1.050E-29 0.02321 1559.38510 8.24E-03 9.32E-03 -2.89E-03 4.88E-03 4.99E-03 R34R34 17.0000 13161.374371 6.514E-30 0.02235 1655.81069 8.04E-03 9.15E-03 -2.91E-03 5.15E-03 5.05E-03 R34Q35 17.6454 13163.639853 6.837E-30 0.02321 1653.54521 8.04E-03 9.15E-03 -2.91E-03 4.89E-03 5.05E-03 R35R35 17.5000 13160.762306 4.186E-30 0.02237 1752.70517 7.86E-03 9.00E-03 -2.94E-03 5.16E-03 5.11E-03 R35Q36 18.1454 13163.036352 4.390E-30 0.02321 1750.43112 7.86E-03 9.00E-03 -2.94E-03 4.90E-03 5.11E-03 R36R36 18.0000 13160.048441 2.653E-30 0.02240 1852.32164 7.70E-03 8.85E-03 -2.96E-03 5.18E-03 5.17E-03 R36Q37 18.6454 13162.331043 2.779E-30 0.02321 1850.03903 7.70E-03 8.85E-03 -2.96E-03 4.91E-03 5.17E-03 R37R37 18.5000 13159.232232 1.658E-30 0.02241 1954.65617 7.55E-03 8.72E-03 -2.98E-03 5.19E-03 5.22E-03 R37Q38 19.1454 13161.523386 1.735E-30 0.02321 1952.36501 7.55E-03 8.72E-03 -2.98E-03 4.92E-03 5.22E-03 R38R38 19.0000 13158.313128 1.022E-30 0.02243 2059.70473 7.41E-03 8.59E-03 -3.00E-03 5.20E-03 5.27E-03 R38Q39 19.6454 13160.612828 1.069E-30 0.02321 2057.40503 7.41E-03 8.59E-03 -3.00E-03 4.92E-03 5.27E-03 R39R39 19.5000 13157.290562 6.211E-31 0.02245 2167.46318 7.29E-03 8.48E-03 -3.02E-03 5.21E-03 5.31E-03 R39Q40 20.1454 13159.598804 6.490E-31 0.02320 2165.15494 7.29E-03 8.48E-03 -3.02E-03 4.93E-03 5.31E-03 R40R40 20.0000 13156.163959 3.723E-31 0.02246 2277.92727 7.17E-03 8.38E-03 -3.04E-03 5.22E-03 5.35E-03 R40Q41 20.6454 13158.480739 3.888E-31 0.02320 2275.61049 7.17E-03 8.38E-03 -3.04E-03 4.94E-03 5.35E-03 76 CHAPTER 4 Cavity Ring-Down Spectroscopy Measurements of Sub-Doppler Hyperfine Structure This chapter was published as D. A. Long, D. K. Havey, M. Okumura, C. E. Miller, and J. T. Hodges, Phys. Rev. A, 81, 064502 (2010). Copyright 2010 by the American Physical Society. 77 Abstract Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) was used to measure magnetic dipole transitions in the b1Σg+ ← X 3Σg-(0,0) band of O2. The 17O-containing isotopologues show unresolved hyperfine structure due to magnetic hyperfine splitting in the ground state. The sensitivity and stability of FS-CRDS allow for quantitative subDoppler measurements of the hyperfine constants, even when the hyperfine splittings are much smaller than the Doppler width. Unlike saturation spectroscopy, this linear absorption technique can be applied to weak transitions and employed to quantitatively measure intensities and line shapes. This method may be an attractive approach for measuring unresolved hyperfine structure in excited electronic states. 78 4.1. Introduction Recent advances in near-infrared spectroscopy have enabled measurements with frequency precision previously only possible in the microwave regime [1,2,3]. Unfortunately, large near-infrared Doppler widths often obscure any hyperfine structure. While saturation spectroscopy can resolve individual hyperfine components [2,4], it is by its very nature a nonlinear technique. As a result, saturation spectroscopy leads to distortion of the spectral line shape and cannot be employed to quantitatively measure line-shape parameters [5,6,7]. In addition, saturation spectroscopy is less general than traditional absorption spectroscopy and cannot easily be applied to the study of weak transitions, such as those described herein. Here we describe an alternate approach to the measurement of hyperfine structure in the near-infrared region: the use of a linear and highly sensitive cavity ring-down spectrometer. In particular, the hyperfine splitting of the 17O-containing isotopologues of O2 were measured for magnetic dipole transitions in the b1Σg+ ← X 3Σg-(0,0) band. While individual hyperfine transitions are unresolved, the spectra exhibit a sufficiently large signal-to-noise ratio to enable quantitative measurements of hyperfine coupling constants. We note that despite the large number of recent studies which have measured spectroscopic parameters of the b1Σg+ ← X 3Σg-(0,0) band for 16O2 and the rare isotopologues (see Refs. [1],[8] and the references contained therein), observation of hyperfine splitting has not previously been reported. 4.2. Experiment Measurements were made using the frequency-stabilized cavity ring-down spectrometer (FS-CRDS) located at the National Institute of Standards and Technology (NIST) in Gaithersburg, MD. The FS-CRDS system has been described in detail previously [9,10], and therefore only the most important details are given. FS-CRDS differs from other single-mode cw-CRDS spectroscopies by actively stabilizing the intracavity length to an external frequency reference. This length stabilization in turn stabilizes the cavity’s free spectral range (FSR). The probe laser frequency is then stepped between successive 79 TEM00 cavity modes, resulting in an accurate, stable, and linear frequency axis. FSCRDS has been utilized to measure transition frequencies to better than 0.5 MHz [1] and line intensities to better than 0.3% [11]. Importantly, due to active frequency stabilization, FS-CRDS does not exhibit instrumental broadening and has therefore been able to record O2 Doppler (Gaussian) widths to 143 kHz, which is 1 part in 6,000 of the room-temperature Doppler width (~850 MHz) for this band [12]. For the described measurements, the probe was an external-cavity diode laser with a tuning range of 759–771 nm, an output power of 6–10 mW, and a line width of <1 MHz (<3 ×10−5 cm−1). The frequency reference was a frequency-stabilized HeNe laser with a frequency uncertainty of less than 1 MHz over 8 h. Mirrors with 99.98% reflectivity were used, corresponding to a finesse of ~15,000. The resulting short-term noise equivalent absorption coefficient was 6 ×10−10 cm−1 Hz−1/2, and the cavity FSR was determined to be 201.970(10) MHz. A continuous spectrum was measured over the Rbranch of the b1Σg+ ← X 3Σg-(0,0) band (13,126–13,165 cm−1) with a step size of 100 MHz (~0.003 cm−1). Sub-FSR frequency steps were enabled by frequency shifting the HeNe reference laser through the use of an acousto-optic modulator in a double-pass configuration [9]. A 17O-enriched O2 sample containing 15.23% 16O2, 38.16% 16O17O, 8.28% 16 O18O, 25.93% 17O2, 11.15% 17O18O, and 1.25% 18O2 was used. Measurements were performed at a pressure of 236 Pa (1.77 Torr) and a temperature of 299.5 K. The combined uncertainty (random and systematic) of this temperature measurement is ~28 mK [13]. Spectra were fit using Voigt line-shape profiles, which account for pressure (Lorentzian) and Doppler (Gaussian) broadening. The importance of using the Galatry [14] line profile for high-resolution measurements at higher pressures is well established [15,16]; however, due to the low pressures of this study, Dicke narrowing [17] is less than 1 MHz and was therefore neglected. During the fit, the transition position and intensity were floated while the pressure broadening coefficient was constrained to the 16 O2 known values [11]. We have recently shown that there is no difference between the pressure broadening parameters for the O2 isotopologues to within the experimental 80 uncertainty of 2% [8]; therefore, at the low pressures utilized herein, the use of the 16O2 coefficients for all isotopologues should introduce negligible error. Further experimental details will be given in a separate publication [18]. 4.3. Results and Discussion The b1Σg+ ← X 3Σg-(0,0) band is triply forbidden by quantum mechanical electric dipole selection rules but occurs as a magnetic dipole transition [19]. Due to this magnetic dipole character and the necessary change in multiplicity, the b1Σg+ ← X 3Σg (0,0) band of 16 O2 is very weak with a band intensity of only 2.231(7) × 10−22 cm molec.−1 at 296 K [11]. The rotational levels of O2 are described by three quantum numbers: N is the rotational angular momentum, S is the spin, and J is their sum (i.e., J = N +S). The O2 ground state is a good Hund’s case (b) and is split into three levels (J″ = N″ − 1, N″, N″ + 1), while the upper state has only J″ = N″. For 16O2 and 18O2, only odd levels of N″ are allowed due to spin statistics, while all values of N″ are allowed for the remaining isotopologues. We denote the b1Σg+ ← X 3Σg-(0,0) transitions by ΔN N″, ΔJ J″ with the band divided into four branches (PP, PQ, RR, and RQ). While 16O and 18O have a nuclear spin of I = 0, 17O has I = 5/2, and therefore the 17 O-containing isotopologues exhibit hyperfine structure. In the 3Σg- ground state, this hyperfine splitting is almost entirely magnetic in nature due to the unpaired spins. Miller and Townes demonstrated that any higher order magnetic and nuclear electric quadrupole hyperfine couplings are negligible (i.e., <1 MHz) [20]. This ground-state hyperfine structure has been observed in microwave, millimeter, submillimeter, and electron paramagnetic resonance spectroscopy measurements of the 16O17O [20,21,22], 17O2 [23], and 17O18O [22] isotopologues. The upper state (11Σg+) is a singlet state and therefore does not exhibit a permanent magnetic dipole. Thus, any magnetic hyperfine coupling is the result of a perturbation and will be extremely weak [24]. Additionally, it can be shown that the electronic structure of this state leads to a weak electric quadrupole coupling (on the order of a few megahertz) [24,25,26]. Thus, we assume that all observed hyperfine 81 structure is the result of hyperfine splitting within the 3Σg- ground state. This assumption will introduce at most an error of a few megahertz to the measured hyperfine coupling constants. For 16O17O and 17O18O, the hyperfine energies were calculated by Miller and Townes [24] based upon the formalism of Frosch and Foley [27]: IJ  c  , b  N 1 2N  3  (1) IJ b  c  , N ( N  1) (2) c  IJ   b  , N  2N  1  (3) E J  N 1  EJ N  E J  N 1  where IJ  F ( F  1)  I ( I  1)  J ( J  1) , 2 (4) and F  I  J. (5) The experimentally determined values of the hyperfine coupling constants are b = −101.441(5) MHz, c = 139.73(3) MHz for 16O17O and b = −101.46(1) MHz, c = 139.68(6) MHz for 17O18O [22]. For 16O17O, the hyperfine coupling constants have been theoretically confirmed using both a simple Slater-type orbital formalism [28] and the multi-configuration self-consistent field method [29]. Our spectra do not show resolved hyperfine transitions; however, we do observe as much as 200 MHz (~1/4 the Doppler width) of line broadening in excess of the Doppler width. We ascribe this broadening to the hyperfine structure. We have previously measured the O2 Doppler width to within 143 kHz [12]; as a result, the observed broadening cannot be attributed to instrumental broadening. The R branch of the b1Σg+ ← X 3Σg (0,0) band consists of RR and RQ transitions. The RR-branch transitions begin in ground-state levels which have J″ = N″; thus, their hyperfine splitting is described by Eq. 2. Note that b and c have opposite signs and (b + c) ~ 38 MHz. As a result, minimal hyperfine splitting would be expected for these 82 transitions, especially for large N″. As can be seen in Fig. 1(a), the RR transitions for each isotopologue exhibit minimal broadening, with observed Gaussian widths converging to the Doppler width for N″ > 2. The RQ branch begins in ground-state levels having J″ = N″ + 1. In this case, b dominates the observed hyperfine splitting for N″ > 4. As a result, hyperfine splittings of hundreds of megahertz are expected, with splittings increasing to an asymptotic limit with increasing N″. In Fig. 1(b), it can readily be seen that the expected behavior is observed for each of the isotopologues. FIG. 1. Experimental Gaussian full width at half-maximum (FWHM) and theoretical Doppler widths (dashed horizontal lines) in 17 O-containing megahertz for the isotopologues with fit uncertainties. Each experimental Gaussian width was determined by a Levenberg-Marquardt least-squares fit using a single Voigt line shape with the pressure-broadening parameter constrained to the previously determined 16O2 value [11]. Note that collisional narrowing is negligible (i.e., <1 MHz) at the pressure utilized in this study, although it can be very important at higher pressures [15,16]. Also shown are calculated (solid black lines) broadenings for 16 17 O O and 17O18O based on the model described in the text. (a) RR-branch transitions which exhibit minimal broadening (except for very low-N″ transitions). (b) RQ-branch transitions exhibit significant broadening due to the large hyperfine splitting (note the different y-axis scales between the two figures). The hyperfine splitting of the 17O2 transitions exhibit a dependence on the parity of N″ due to symmetry constraints. For both branches, the model performs well for transitions with N″ > 1. For 17O2, the parity of N″ has a large impact upon the observed splitting. 17O2 has two I = 5/2 nuclei and therefore has a total nuclear spin of Itotal = 5, 4, 3, 2, 1, 0. The symmetry of 17O2 dictates that odd values of N″ can only have even values of Itotal and even values of N″ can only have odd values of Itotal [23]. Thus, even values of N″ have 83 Itotal = 5, 3, 1, and odd values of N″ have Itotal =4, 2, 0. Since larger values of Itotal give rise to larger splitting, we would expect even values of N″ to have a larger splitting [as can be seen in Fig. 1(b)]. The observed broadening can be analyzed with a simple intensity model to determine b and c. In the high-J case, only hyperfine transitions with ΔJ = ΔF are significant, and the intensities of these transitions are roughly proportional to (2F + 1) [24]. This high-J limit was utilized to calculate the relative intensities of the hyperfine structure components for each of the measured transitions. The hyperfine structure positions were calculated as shown previously. The hyperfine spectrum could then be readily calculated. It can readily be shown algebraically that for all J = N ±1,0,  (2 F  1) E ( F )  0, J (6) F where each sum is over all allowed F quantum numbers (i.e., F = I + J, I + J − 1, . . . ,|I − J |) and EJ is defined as in Eqs. 1-3. Thus, the calculated hyperfine structure does not shift the spectral center of mass. For 16O17O and 17O18O, these calculated frequency splittings and intensities were then utilized to simulate the broadened line shape as a sum of six separate hyperfine profiles whose Gaussian widths were fixed to the theoretical Doppler width for the given isotopologue. As can be seen in Fig. 1, this model reproduces the observed broadening very well for N″ > 3. For N″ ≤ 3, the assumptions made regarding the relative intensities of the hyperfine structure break down and more complicated expressions are required. Figure 2 shows a measured spectrum which has been fit by a set of six Voigt line profiles (one for each hyperfine component). Each of these hyperfine profiles had its Gaussian width set equal to the theoretical Doppler width and the pressure-broadening parameter fixed to the value determined for 16O2 [11]. Thus, the only two fitted parameters were the position of the spectral center of mass and the total integrated area (i.e., all relative hyperfine coupling constants were held fixed). As can be seen in Fig. 2, this model captures the observed behavior to within the baseline noise level. 84 FIG. 2. Upper panel: Measured absorption given by (cτ)−1 where c is the speed of light and τ is the measured ring-down decay time (symbols) and fitted spectrum (solid black line) for the R3Q4 16O17O transition, at 236 Pa (1.77 Torr) and 299.5 K. The sum of six Voigt profiles corresponding to the hyperfine structure were fit to the measurements (solid black line) using a Levenberg-Marquardt least-squares fitting algorithm. Each of these Voigt profiles [shown in solid colored (gray) lines with larger values of F″ corresponding to higher intensity hyperfine components] had its relative position and intensity fixed to values calculated using the model described in the text. Each Voigt profile’s Gaussian width was constrained to the theoretical Doppler width, and the pressure-broadening parameter was set equal to the measured 16O2 value [11]. Thus, only the position of the spectral center of mass and the total integrated area of the composite spectrum were floated. Also shown is a single Voigt profile whose Doppler width was constrained to the theoretical 16O17O value (dashed gray line). Lower panel: Fit residual for the sum of hyperfine structure. Based on the peak height and the root-mean-square fit residual, the signal-to-noise ratio of this spectrum is ~825:1. We were able to determine the b and c hyperfine coupling constants by leastsquares fits of the model to our measured spectra. For a fixed value of N″, it is not possible to determine both b and c in this fashion because these two quantities appear as differences in Eqs. 1 and 2, and as a consequence they are completely correlated for a given transition. However, b and c can be determined by a global multispectrum fit of spectra spanning a range of N″. Thus, the hyperfine model presented here was simultaneously fit to five measured spectra with b and c treated as global parameters. 85 These data correspond to the R3Q4, R4Q5, R5Q6, R8Q9, and R13Q14 transitions of 16 O17O. Assuming the individual hyperfine transitions were Voigt profiles having the theoretical Doppler width and J-dependent broadening coefficients given in [11], we obtained b = −100.93(30) MHz and c = 129.8(40), which differ from the more accurate microwave measurements of Cazzoli et al. [22] by approximately 0.5 and 10 MHz, respectively. With c constrained to the value determined by Cazzoli et al. (139.73 MHz), the multispectrum least-squares fit yielded b = −101.64(10) MHz. This value of b is in good agreement (difference < 200 kHz) with that reported by Cazzoli et al. [22]. We attribute this small difference to the finite signal-to-noise ratio of the measured absorption and to variations in the frequency of our reference laser to which the ring-down cavity was locked. 4.4. Conclusions We emphasize that the hyperfine coupling constants derived from the measured splittings are directly related to important (and often difficult to calculate [29]) electronic structure properties such as the electronic density at the nucleus with nonzero nuclear spin, |ψ(0)|2, and r 3 [30]. In the b1Σg+ ← X 3Σg (0,0) band, the hyperfine structure occurs due to the ground state; however, FS-CRDS can similarly measure hyperfine structure which is dominated by the upper state. As a result, the described technique may provide a new approach to the measurement of hyperfine coupling constants of electronic excited states, measurements which can be difficult by traditional pure rotational spectroscopy. Acknowledgements D. A. Long was supported by the National Science Foundation and Department of Defense. D. K. Havey acknowledges the support of the National Research Council as a postdoctoral fellow at NIST, Gaithersburg, MD. Part of the research described in this report was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Additional support was provided by the Orbiting Carbon Observatory (OCO) 86 project, a NASA Earth System Science Pathfinder (ESSP) mission, and the NASA Upper Atmospheric Research Program Grants NNG06GD88G and NNX09AE21G. 87 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] D. J. Robichaud, J. T. Hodges, P. Maslowski, L. Y. Yeung, M. Okumura, C. E. Miller, L. R. Brown, J. Mol. Spectrosc. 251, 27-37 (2008). L. S. Chen, J. Ye, Chem. Phys. Lett. 381, 777-783 (2003). S. Falke, E. Tiemann, C. Lisdat, H. Schnatz, G. Grosche, Phys. Rev. A 74, 032503 (2006). G. Giusfredi, S. Bartalini, S. Borri, P. Cancio, I. Galli, D. Mazzotti, P. De Natale, Phys. Rev. Lett. 104, 110801 (2010). D. Lisak, J. T. Hodges, Appl. Phys. B 88, 317-325 (2007). J. Y. Lee, J. W. Hahn, Appl. Phys. B 79, 653-662 (2004). S. S. Brown, H. Stark, A. R. Ravishankara, Appl. Phys. B 75, 173-182 (2002). D. J. Robichaud, L. Y. Yeung, D. A. Long, M. Okumura, D. K. Havey, J. T. Hodges, C. E. Miller, L. R. Brown, J. Phys. Chem. A 113, 13089-13099 (2009). J. T. Hodges, H. P. Layer, W. W. Miller, G. E. Scace, Rev. Sci. Instrum. 75, 849863 (2004). J. T. Hodges, R. Ciurylo, Rev. Sci. Instrum. 76, 023112 (2005). D. A. Long, D. K. Havey, M. Okumura, C. E. Miller, J. T. Hodges, J. Quant. Spectrosc. Radiat. Transfer 111, 2021-2036 (2010). D. A. Long, D. K. Havey, M. Okumura, H. M. Pickett, C. E. Miller, J. T. Hodges, Phys. Rev. A 80, 042513 (2009). D. K. Havey, D. A. Long, M. Okumura, C. E. Miller, J. T. Hodges, Chem. Phys. Lett. 483, 49-54 (2009). L. Galatry, Phys. Rev. 122, 1218-1223 (1961). K. J. Ritter, T. D. Wilkerson, J. Mol. Spectrosc. 121, 1-19 (1987). L. R. Brown, C. Plymate, J. Mol. Spectrosc. 199, 166-179 (2000). R. H. Dicke, Phys. Rev. 89, 472-473 (1953). D. A. Long, D. K. Havey, S. S. Yu, M. Okumura, C. E. Miller, J. T. Hodges, J. Quant. Spectrosc. Radiat. Transfer, Submitted. (2011). H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167-190 (1948). S. L. Miller, C. H. Townes, Phys. Rev. 90, 537-541 (1953). P. Gerber, Helv. Phys. Acta 45, 655-682 (1972). G. Cazzoli, C. D. Esposti, P. G. Favero, G. Severi, Nouvo Cimento B 62, 243-254 (1981). G. Cazzoli, C. D. Esposti, B. M. Landsberg, Nuovo Cimento D 3, 341-360 (1984). C. H. Townes, A. K. Schawlow, Microwave Spectroscopy (Dover Publications, New York, 1975). C. H. Townes, Phys. Rev. 71, 909-910 (1947). C. H. Townes, B. P. Dailey, J. Chem. Phys. 17, 782-796 (1949). R. A. Frosch, H. M. Foley, Phys. Rev. 88, 1337-1349 (1952). M. Kotani, Y. Mizuno, K. Kayama, E. Ishiguro, J. Phys. Soc. Jpn. 12, 707-736 (1957). B. F. Minaev, Spectroc. Acta A 60, 1027-1041 (2004). 88 [30] S. L. Miller, C. H. Townes, M. Kotani, Phys. Rev. 90, 542-543 (1953). 89 CHAPTER 5 Laboratory Measurements and Theoretical Calculations of O2 A-band Electric Quadrupole Transitions This chapter was published as D. A. Long, D. K. Havey, M. Okumura, H. M. Pickett, C. E. Miller, and J. T. Hodges, Phys. Rev. A, 80, 042513 (2009). Copyright 2009 by the American Physical Society. 90 Abstract Frequency-stabilized cavity ring-down spectroscopy was utilized to measure electric quadrupole transitions within the 16O2 A-band, b 1Σg+ ← X 3Σg- (0,0). We report quantitative measurements (relative uncertainties in intensity measurements from 4.4% to 11%) of nine ultraweak transitions in the NO, PO, RS, and TS branches with line intensities ranging from 3×10−30 to 2×10−29 cm molec.−1. A thorough discussion of relevant noise sources and uncertainties in this experiment and other cw-cavity ring-down spectrometers is given. For short-term averaging (t < 100 s), we estimate a noise-equivalent absorption of 2.5×10−10 cm−1 Hz−1/2. The detection limit was reduced further by co-adding up to 100 spectra to yield a minimum detectable absorption coefficient equal to 1.8×10−11 cm−1, corresponding to a line intensity of ~2.5×10−31 cm molec.−1. We discuss calculations of electric quadrupole line positions based on a simultaneous fit of the ground and upper electronic state energies which have uncertainties <3 MHz, and we present calculations of electric quadrupole matrix elements and line intensities. The electric quadrupole line intensity calculations and measurements agreed on average to 5%, which is comparable to our average experimental uncertainty. The calculated electric quadrupole band intensity was 1.8(1)×10−27 cm molec.−1 which is equal to only ~8×10−6 of the magnetic dipole band intensity. Note that Dr. Joseph Hodges of NIST assisted greatly in the preparation of this chapter. His contributions were especially significant in sections 3.3. Also Dr. Herb Pickett performed the line intensity calculations and provided some textual description. 91 5.1. Introduction In 1961, Noxon observed the first electric quadrupole transitions of molecular oxygen in the (subsequently named) Noxon band a 1Δg←b 1Σg+ [1]. In 1981, electric quadrupole transitions within the electronic ground state of O2 were identified by Goldman et al. [2] in the solar spectra of Niple et al. [3] and observed in the laboratory spectra of Reid et al. [4]. Quadrupole transitions within the O2 A-band, b 1Σg+ ← X 3Σg- (0,0), were observed in the solar spectra of Brault [5]. Brault was able to produce line positions and intensities for eight transitions (N″ = 5 to 19) within the TS branch of the 16O2 A-band. See Background for notation. The only laboratory observation of an electric quadrupole line within the O2 A-band was performed by Naus et al. who measured the TS(9) line with a signal-to-noise ratio (SNR) of ~2:1 using pulsed cavity ring-down spectroscopy (CRDS) [6]. The O2 A-band is of special significance due to its extensive use over the past 50 years in remote sensing of the Earth’s atmosphere. Measurements of this band have the potential to yield surface pressures to 0.1% [7] and enable accurate CO2 remote sensing (~0.3%) by constraining surface pressure and photon path length in these retrievals. Use of the O2 A-band has been expanded in recent years to ground-based observations of solar spectra such as those of the TCCON network [8]. Also, solar absorption by this band is used in satellite sensing of atmospheric gases by the GOME [9], SCIAMACHY [10,11], GOSAT [12] missions and was to be used in the OCO [13,14,15] mission. Although the electric quadrupole transitions of the A-band are 105 times weaker than the corresponding magnetic dipole transitions, they can be observed in atmospheric spectra [5,16] reaching 1% absorbance at high solar zenith angles. Failure to include electric quadrupole absorptions in O2 A-band remote sensing retrievals will lead to systematic errors in the assessment of photon path lengths, cloud or aerosol optical depths, and atmospheric pressure at the scattering height [7]. The situation is compounded since many of the electric quadrupole transitions occur underneath the much stronger magnetic dipole transitions. For these same reasons, the failure to include electric quadrupole transitions in fits of the line shape and line mixing parameterizations 92 of the magnetic dipole transitions may result in systematic errors in the representation of the magnetic dipole features. We present quantitative laboratory-based CRDS measurements which constitute observations of ultraweak (line intensities ranging from 3×10−30 to 2×10−29 cm molec.−1) electric quadrupole transitions in the NO, PO, RS, and TS branches of the O2 A-band. We note that Brault [5] measured isolated TS lines which lie above the R-branch bandhead. Figure 1 shows the calculated positions and intensities of the four electric quadrupole (and magnetic dipole) branches, where the electric quadrupole intensities were calculated as described in Section 4.3. The TS lines are relatively easy to observe in comparison with detection of NO, PO, and RS lines that are nearly degenerate with the magnetic dipole transitions—as are reported in the present study. We also discuss pertinent short- and long-time noise sources in this, and other, cw-CRDS experiments. Line intensities are assigned and combined (absolute) uncertainties in line intensity are estimated by evaluating random and systematic measurement uncertainties. Finally, we present theoretical calculations of line intensities and positions and compare these predictions to our experimental results. 5.2. Background The O2 A-band, with a band origin at wave number ~ = 13,122 cm−1 (λ = 762 nm), is the strongest near-infrared absorption feature in the Earth’s atmosphere. Although the transition is triply forbidden by quantum mechanical electric dipole selection rules (gerade-gerade, singlet-triplet, and Σ+-Σ−), magnetic dipole and electric quadrupole transitions are allowed [17,18]. The magnetic dipole transitions are 107 times weaker than a typical electric dipole transition and the electric quadrupole transitions are a further 105 times weaker (see Section 4.3) than the magnetic dipole transitions. 93 FIG. 1. Stick spectra of magnetic dipole (upper panel) and electric quadrupole (lower panel) 16O2 A band line intensities at T=296 K. Positions for the electric quadrupole lines are given in Table 2. Transitions from the present study and those of Brault [5] as indicated. The symbol * indicates the transition observed by Naus et al. [6]. Note the difference in scales of the vertical axes. The rotational energy levels of molecular oxygen are described by three quantum numbers: N, the rotational angular momentum, S, the spin angular momentum, and J, the total angular momentum (J = N+S). We therefore, describe the A-band transitions by ΔN ΔJ(N″), where the double prime indicates the lower state. An energy level diagram for the O2 A-band is shown in Figure 2, illustrating that there are three J levels for every N in the triplet ground state. For even values of N (in the case of 16O2 and 18O2) the spinstatistical weight is 0 and hence only odd N are allowed. The PP, PQ, RR, and RQ branches, corresponding to |ΔJ| = 0,1 are both magnetic dipole and electric quadrupole allowed. Given the weakness of the electric quadrupole transition moments, we do not expect them to contribute significantly to line intensities in the |ΔJ| = 0,1 branches. The |ΔJ| = 2 electric quadrupole transitions correspond to four branches: NO, PO, RS, and TS. The TS and NO branches have |ΔN| = 3, whereas the PO and RS branches have |ΔN| = 1. As 94 can be seen in Figure 1, these two |ΔN| = 1 branches are heavily masked by the P- and Rbranches of the magnetic dipole transitions, respectively, with line positions differing, on average, by only 0.2 to 0.3 cm−1. Only the N″ = 1 transition for each of these two branches is isolated from its sister magnetic dipole transition and can therefore be feasibly measured in this laboratory experiment. FIG. 2. Energy level diagram of molecular oxygen (only for 16O2 and 18O2) for the X 3Σg− ground state and the b 1Σg+ excited state. The |ΔJ|=0,1 branches (PP, PQ, RR, RQ) are shown on the left for ground state rotational angular momentum, N (odd), and total angular momentum, J. These branches are both magnetic dipole (MD) and electric quadrupole (EQ) symmetry allowed. Similarly, on the right four symmetryallowed electric quadrupole transition branches measured in this work are shown (PO, NO, RS, TS). 95 5.2.1. Line Positions Calculations The positions of the electric quadrupole transitions were calculated based on a simultaneous fit of the ground X 3Σg- and upper b 1Σg+ states using SPFIT [19]. Microwave [20,21,22,23], far-infrared [24], and Raman [25,26,27,28] positions for the v = 0 and v = 1 levels of the ground state were taken from the literature and used with their reported uncertainties. The A-band positions reported by Robichaud et al. [29] yielded parameters for the upper state and also improved the distortion constants for the ground state compared to fits using only the ground state transitions. The final fit included 327 transitions and yielded a reduced root-mean-square (rms) uncertainty of 0.843, indicating that the relative weighting of the experimental data was slightly conservative. Note that we switched assignments for the (N′ ,J′←N″ ,J″) = (11,11←13,13) and (11,10←13,12) transitions at ~ = 1,482.4971 cm−1 and 1,482.4739 cm−1, respectively, compared to the assignments reported by Brodersen and Bendtsen [27]. This resulted in a 62% reduction in the reduced rms of the total ground-state-only fit. Our fitted parameters for both lower and upper states as well as those of Rouillé et al. [25] (lower state) and Robichaud et al. [29] (upper state) are given in Table 1, and our predicted electric quadrupole positions are given in Table 2. Uncertainties in our predicted positions of the electric quadrupole transitions are predicted to be less than 3 MHz (10−4 cm−1) since these transitions are based on the same energy levels as the magnetic dipole transitions (Figure 2) [29]. However, when we compare our calculated positions to the calculations of Naus et al. [6], we find that their positions are systematically ~300 MHz (0.01 cm−1) lower than ours for both the magnetic dipole and electric quadrupole transitions. We believe that pressure shifting is the root of this discrepancy. Our line positions were calculated based upon laboratory measurements which have been corrected for pressure shifting, while no such correction was applied to the solar observations of Brault [5] and Babcock et al. [17] which formed the basis for the Naus et al. calculations. Brown and co-workers [30] have previously noted a similar discrepancy between the magnetic dipole line positions of several high-resolution laboratory studies [30,31,32] and those of Babcock et al. The direction and magnitude of the observed shift is consistent with the reported pressure 96 shifting of Robichaud et al. [33] (approximately −0.002 MHz Pa−1 or −0.007 cm−1 atm−1) for the P-branch of the A-band magnetic dipole transitions. Thus, we believe that the calculations of Naus et al. [6] would agree with ours if the former were corrected for atmospheric pressure shifting. Additionally, the value reported by Naus et al. for the R S(1) line position is approximately 1 cm−1 low compared to both our reported value and our recalculation of this position using their specified spectroscopic constants. TABLE 1. Spectroscopic constants (cm−1) of the X 3Σg− and b1Σg+ states of 16O2. Lower state energy levels are defined in Ref. [25] and those of the upper state are given by v0+B0J(J+1)−D0J2(J+1)2+H0J3(J+1)3. Numbers in parentheses represent 1σ uncertainty in units of the least significant digit. X3Σgstate This work Rouillé et al. B0 1.437676078(29) 1.437676476(77) D0 4.84178(14)×10-6 4.84256(63)×10-6 H0 4.28(19)×10-12 2.80(16)×10-12 λ0 1.98475118(5) 1.984751322(72) λ0 ′ 1.9470(3)×10-6 1.94521(50)×10-6 λ0 ″ 9.70(3)×10-12 1.103(41)×10-11 μ0 –8.4253696(58)×10-3 –8.425390(13)×10-3 μ0′ –8.136(22)×10-9 –8.106(32)×10-9 μ0″ –4.04(15)×10-14 –4.70(19)×10-14 b1Σg+ state This work Rouillé et al. ν0 13122.0057456(89) 13122.00575(1) B0 1.39124922(11) 1.3912496(1) D0 5.36909(28)×10-6 5.3699(3)×10-6 H0 1.65(33)×10-14 –1.80(2)×10-12 97 As described below, having accurate knowledge of the line positions (i.e., those given in Table 2) facilitated our efforts to unambiguously locate the electric quadrupole lines. This information, combined with accurate measurements of probe laser frequency, made it sufficient to interrogate the absorption spectrum over narrow spectral ranges centered on each expected line position. TABLE 2. Calculated vacuum wave numbers (cm−1) for the electric quadrupole transitions within the NO, P O, RS, and TS branches of the 16O2 A band using our lower and upper state rotational parameters given in Table 1. Transitions probed in this study are underlined. We use the designation ΔNΔJ(N″), such that NO(13) indicates the [(N′ = 10)←(N″ = 13) , (J′ = 12)←(J″ =14)] transition. N″ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 N O 13105.75288 13088.15293 13070.22159 13051.93672 13033.29479 13014.29577 12994.94055 12975.23027 12955.16606 12934.74895 12913.97982 12892.85936 12871.38808 12849.56627 P O 13119.92145 13113.96544 13107.60461 13100.86413 13093.75012 13086.26481 13078.40910 13070.18320 13061.58694 13052.61982 13043.28109 13033.56974 13023.48451 13013.02394 13002.18633 R S 13130.35305 13133.57572 13138.22862 13142.54086 13146.48617 13150.05690 13153.24888 13156.05889 13158.48395 13160.52106 13162.16713 13163.41892 13164.27298 13164.72571 13164.77328 T S 13147.74429 13164.04113 13179.92388 13195.41358 13210.51223 13225.21793 13239.52744 13253.43688 13266.94194 13280.03801 13292.72019 13304.98336 13316.82215 13328.23096 13339.20396 5.3. Experiment All measurements were made at the National Institute of Standards and Technology (NIST), in Gaithersburg, Maryland using the frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) technique. This technique, which enables precise and accurate measurements of both the absorption coefficient and spectrum frequency axis, has been described previously [34,35]. See Figure 3(a) for a schematic of the setup. The 98 fundamental distinction between FS-CRDS and traditional cw-CRDS is that the cavity length is actively locked through the use of a co-resonant reference laser beam. In addition, the probe beam frequency is selectively locked to longitudinal modes of the TEM00 cavity mode. FIG. 3. Experimental configuration. (a) Principal components of the frequency-stabilized cavity ring-down spectrometer, including probe and reference lasers, ring-down cavity, and length-stabilization servo. (b) Spectral acquisition scheme illustrating how the probe laser is scanned over successive longitudinal modes of the resonant cavity. The equally spaced modes are separated by ~200 MHz and form a frequency axis whose long-term precision is limited by the stability of the reference laser. With this approach, the absorption spectrum is discretely sampled at fixed intervals as indicated by the solid circles. 5.3.1. Experimental Apparatus The reference laser was a frequency-stabilized HeNe laser (λ = 633 nm) with a long-term (8 h time interval) frequency variation less than 2 MHz and output power of 1 mW. The probe beam was from a continuous-wave external-cavity diode laser (ECDL) with a Litmann-Metcalf resonator configuration. A Faraday isolator (−40 dB of isolation) was mounted directly on the probe laser housing. The ECDL output was single-mode and had a typical output power of 6–10 mW over the wavelength range λ = 759–771 nm. We used 99 custom-made, dichroic ring-down cavity mirrors with nominal transmission losses of 3×10−5 and 0.05 at λ = 765 nm and 633 nm, respectively. Losses at the probe wavelength corresponded to a cavity finesse, F~105,000, and the cavity length was 74 cm, yielding an effective optical path length of ~25 km. The mirror losses were ~7 times smaller compared to those of our previous FS-CRDS O2 A-band studies [29,36]. However, the short-term detection limit (discussed below) was only ~4 times lower than previously demonstrated because of reduced signal intensity of the decay signals. The internal volume of the ring-down cavity was ~0.2 L, and all internal surfaces (except for Cu gaskets used in vacuum-compatible knife-edge seals) were made of electro-polished stainless steel. We performed all experiments at room temperature and made no effort to control the temperature of the ring-down cell. The sample temperature, T, was measured using a NIST-calibrated 2.4 kΩ thermistor mounted in good thermal contact with an external wall of the ring-down cavity. Uncertainty in the measured sample temperature was limited by temperature gradients within the cavity and was estimated to be less than 0.2 K. Cell wall temperature varied by at most 0.2 K during the course of a day, and the average temperature for all spectra was 295.25±0.3 K. All measurements were made on O2 with >99.9% purity and natural isotopic abundance. We measured gas pressure, p, using two NIST-calibrated capacitance diaphragm gauges with full-scale responses of 13.3 and 133 kPa, respectively. We estimated the relative combined standard uncertainty in the pressure measurement to be less than 0.2% of the reading over the entire pressure range. Spectra were acquired by locking the probe laser frequency to successive longitudinal modes of the frequency-stabilized ring-down cavity as shown in Figure 3(b). As was previously demonstrated [29,33], this approach provides an exceptionally linear spectrum frequency axis with a step size given by νFSR, where FSR is the cavity freespectral range. The spectral resolution was limited by the stability of the HeNe reference laser. To measure absolute frequencies and the cavity FSR, we used a commercial Michelson-interferometer wave meter having a precision of 30 MHz (Δ~ = 0.001 cm−1). We determined the mean cavity FSR to be 202.569(15) MHz by measuring the frequency 100 steps of nearly 1000 spectra, each spanning a range of 10 GHz. The 15 kHz uncertainty in FSR is due to variations in the effective path length of the ring-down cavity caused by concomitant changes in the sample index of refraction as the pressure was varied between 2.5 and 30 kPa. The wave meter was calibrated against the measured position of the P P(11) 16O18O A-band magnetic dipole transition. This line is within 0.15 cm−1 of the N O(5) electric quadrupole line, and its position was recently determined with an uncertainty of 4×10−5 cm−1 through the FS-CRDS method [37] by using hyperfine components of the D1 and D2 lines of 39K as an absolute frequency reference [29]. We used a variable-frequency (200 MHz ± 10 MHz) acousto-optic modulator (AOM) to provide a first-order diffracted beam that was mode-matched into the ringdown cavity. A Si-PIN photoreceiver (bandwidth of 700 kHz and responsivity of 0.3 V W−1) located at the cavity output measured probe beam transmission. Ring-down decay events were initiated by transmission bursts having amplitudes exceeding an adjustable threshold. These bursts triggered TTL pulses from a digital delay generator, switching off the AOM power (decay time <50 ns) and shifting the AOM frequency by 10 MHz. The resulting ring-down signals were digitized by a 12-bit analog-to-digital board at a rate of 25 M samples/s and had maximum decay times of 80 μs. The decays were acquired at a nominal acquisition rate, facq = 10 Hz. To obtain the decay time constant, τ, we fitted an exponential decay plus a DC baseline to each signal using the algorithm of Halmer et al. [38]. Spectra could be continuously measured over a 24 h interval without operator intervention. We employed two modes of laser tuning (coarse and fine). Coarse tuning was performed through the use of a stepper-motor-actuated grating (9 GHz minimum movement), while fine tuning was done via a piezoelectric transducer (PZT)-actuated mirror (full range of 60 GHz). The stepper motor, PZT, and wave meter were automatically used in combination to move the laser frequency to the desired starting point for an individual scan. Once this location was found, the PZT was employed to step the laser frequency over successive cavity longitudinal modes, qm. A typical scan spanned a total of ~10 GHz, and took roughly ten to fifteen minutes. Fifty individual 101 ring-down time constants were averaged at each frequency step. The spectrum frequency axis was given by ΔqmνFSR where Δqm was the number of mode steps through which the laser was tuned relative to the beginning of the scan. At least ten complete spectra were then co-added to produce a final averaged spectrum. The effects of these two separate averaging steps will be explored in detail in the next two sections 5.3.2. Short-Term Measurement Statistics and Signal Averaging It is instructive to consider both short- and long-term averaging statistics to evaluate system performance and quantify detection limits and measurement precision. With regard to short times, we use the noise-equivalent absorption coefficient εNEA given by 1/ 2   / c 2 f acq  . Here στ represents the standard deviation of the observed time constant about its mean value,  , and c is the speed of light. For a given averaging time, Δtav, (corresponding to a set of ring-down signals acquired at fixed laser frequency) the uncertainty in the mean value of the base loss,  , is u ( )   NEA /(t av )1 / 2      / c 2 n1d/ 2 , where nd is the total number of measured decay signals. We found that   /  was typically less than 0.2%, leading to a εNEA = 2.5×10−10 cm−1 Hz−1/2. For the averaging time employed, Δtav = 5 s, this corresponded to an uncertainty in the absorption coefficient of αmin,acq = 1.1×10−10 cm−1. For sufficiently long averaging timescales, the noise-equivalent absorption coefficient cannot be used to predict detection limits because of slow drift in system variables. We observed that our system exhibited a nominally linear drift in  over a time span of several hours given by  1 d / dt ~ 2×10−5 s−1. The Allan variance, which was originally formulated to quantify the performance of ultrastable clocks [39], provides a useful measure of long-term system stability and has been applied to analyze detection limits in laser absorption spectroscopy [40]. We measured the Allan variance under empty-cavity conditions. The results for our experiment are shown in Figure 4 and reveal an optimal averaging time of 150 s. For averaging timescales smaller than 150 s, the Allan variance varied inversely with Δtav consistent with the definition of εNEA, whereas 102 for longer averaging timescales, the Allan variance reached a minimum due to system drifts. When evaluated at the optimum averaging time, the Allan variance analysis yields a minimum uncertainty in the absorption coefficient of αmin = 4×10−11 cm−1, which corresponds to the peak loss from a self-broadened A-band line with an intensity S~6×10−31 cm−1 molec.−1 at p = 13.3 kPa (100 Torr). Numerical simulations indicate that the optimal averaging time in our Allan variance is consistent with our observed drift rate. FIG. 4. Measured Allan deviation (square root of the Allan variance) of the cavity losses as a function of averaging time, Δtav. The quantity nd is the number of measured decay signals, and the ring-down signal acquisition rate is facq. The dashed line tangent to the Allan deviation corresponds to εNEA=2.5×10−10 cm−1 Hz−1/2 and for short averaging times gives an uncertainty in  , u(  ), which is proportional to nd−1/2. 5.3.3. Long-Term Averaging of Spectra In addition to short-time averaging of successive ring-down time constants, ensembles of complete spectra were also averaged. It is important to note that this is a simple average; no baseline correction or other preanalysis was performed. It was found that, over time, 103 the broadband baseline absorbance drifted (as is to be expected due to thermo-mechanical drift); however, the integrated area of the relevant absorbance feature was independent of this drift. This is an important feature of FS-CRDS which allows spectra to be co-added, thereby reducing the amplitude of temporally uncorrelated residual features which vary on a timescale shorter than our single-spectrum acquisition time (10–15 min). The base losses of ring-down spectrometers typically have a weak wavelength dependence that drifts slowly with time. The baseline spectrum of the empty cavity arises from λ-dependence in the mirror reflectivity and from secondary resonant cavities (etalons). Spectral and temporal variation in the base losses is a manifestation of coupling between cavities of low finesse and the main high-finesse ring-down cavity [41], and consequently the measured decay time τ is influenced by all resonant cavities along the optical axis of the probe beam. The most important secondary cavities are formed by pairs of surfaces where the first surface corresponds to one of the high-reflectivity ringdown cavity mirrors and the other corresponds to a normal-incidence surface in the probe beam path. Examples of the latter include: mode-matching lenses, view ports, detector faceplates, ring-down mirror input surfaces, etc. Note that a pair of coupled cavities will be mutually resonant at a given frequency, when the round-trip path difference is an integral multiple of λ/2. This effect gives rise to periodicity in the base losses as a function of optical frequency. Fourier analysis of ring-down spectra reveals the amplitudes and characteristic spatial frequencies (and hence path lengths) associated with these secondary resonators. Given that these etalons can involve optical elements external to the ring-down cavity, they are sensitive to slow thermo-mechanical variations in optical path length, therefore contributing to drift in the observed ring-down losses. We evaluated the baseline noise of a composite spectrum by calculating the residuals to a quadratic fit in a region largely free of absorption features. The resulting rms baseline (in units of cm−1) as a function of number of spectra averaged, ns, can be found in Figure 5. The overall trend is a steady reduction in the rms baseline noise to a minimum value of 1.8×10−11 cm−1. This trend persists for an averaging time of approximately 10 h and shows no evidence of leveling off, indicating that additional 104 gains could be achieved by further averaging. As shown in Figure 5, a power-law fit to the data gives a slightly smaller decay exponent of μ = 0.46 than would be expected from an ergodic ns-1/2 dependence. Note the rms baseline for ns = 1 is consistent with the singlespectrum αmin,acq value shown in Figure 4. Kassi et al. [42] similarly observed that coadding cw-CRDS spectra reduced rms baseline noise in measurements made with a fibered distributed feedback (DFB) diode laser at 1.58 μm. However, they found a slightly faster decay of noise in their setup (μ = 0.61), which they assigned to the presence of slowly varying “interference fringes” whose movement was driven by thermo-mechanical drift. FIG. 5. Decrease in rms baseline noise (symbols) with increasing number of co-added spectra, ns. The solid red line corresponds to a power-law fit yielding the decay exponent, μ=0.46 and the constant Afit~αmin,acq=1.1_10−10 cm−1. The dashed blue line represents an ergodic decay proportional to ns−1/2. The minimum value of the rms baseline corresponds to the peak loss of a self-broadened A band line with S~2.5×10−31 cm−1 molec.−1 at p=13.3 kPa (100 Torr). 105 5.3.4. Effect of Long-Term Averaging on Observed Lineshape One possible deleterious result of spectra averaging is that the averaged spectrum may exhibit a nonphysical line shape due to instrumental broadening. To examine instrumental broadening, we measured the PP(21) transition of 16O at ~ = 2 13,041.1236 cm−1 both with and without the frequency stabilization at p = 67 Pa. At this pressure the line shape was essentially Doppler broadened. The frequency-stabilized case is shown in Figure 6. A Galatry profile [43] was fit to the measured spectra to account for residual pressure broadening as well as collisional narrowing effects that were observable at the high SNR of these measurements. We used the recent line shape measurements of Robichaud et al. [36] for the broadening and narrowing coefficients of this line, which were 0.0118 and 0.0039 MHz Pa−1, respectively. FIG. 6. Frequency-stabilized, Doppler-broadened spectrum of the 16O2 PP(21) magnetic dipole transition at ~ =13 041.1236 cm−1 for p=67 Pa, T=294.987(40) K, and an effective path length of 25 km. Upper panel shows the measured (symbols) and Galatry line shape fit (solid line) to a single spectrum. Lower panel shows the fit residuals for the single spectrum, and those corresponding to the fit of an averaged spectrum comprising 28 spectra (blue line with open symbols). SNR for the averaged spectrum was 18,000:1 with rms noise in the baseline absorption coefficient of 3.9×10−11 cm−1. 106 We observed significant differences between the stabilized and unstabilized spectra with respect to the measured Doppler width and center frequency determinations. In the stabilized case, the mean difference between the measured Doppler width and the calculated value (~0.85 GHz) was 143 kHz, with a standard deviation of 329 kHz. In the unstabilized case, the difference was three times larger (379 kHz) and the measured standard deviation was six times larger (1.96 MHz). Apparent drifts in the transition center frequency as large as ~2 MHz min−1 were observed which resulted from continuous movement of the cavity modes over the 10 min duration of each spectrum acquisition. These effects of cavity drift were mitigated in the stabilized case, for which we measured a standard deviation in fit-derived center frequency of only 200 kHz. The residual scatter in center frequency for the stabilized case was driven in part by variations (occurring on the timescale of spectrum acquisition) in the HeNe laser reference frequency. These results illustrate that with frequency stabilization of the ring-down cavity, the measured Doppler width and center frequency are stable to within 0.5 MHz. Therefore, co-adding multiple spectra obtained over time spans of several hours should introduce negligible instrumental broadening in the present experiment. We conclude that our measurements of the weak electric quadrupole lines, obtained by co-adding multiple spectra, can be readily modeled using standard line shapes [44]. 5.3.5. Analysis of Measured Spectra Higher-order line shape phenomena such as collisional narrowing [43] and speeddependent effects [45] were not expected to be detectable for measurements of the ultraweak electric quadrupole lines. Therefore, we modeled all these lines as Voigt profiles. The observed spectra, given by the measured quantity n0(cτ)-1, comprised absorption by O2 and all other system losses. Here n0 is the broadband (nonresonant) index of refraction of the cavity medium, which we assumed to be unity. The electric quadrupole lines were superimposed on a baseline (linear plus sinusoidal terms) to capture variations in the local loss arising from wings of other lines and etaloning effects. 107 The spectrum frequency axis was ν(qm) = ν0 + qmνFSR, where ν0 was the starting frequency given by the wave meter. The electric quadrupole line intensities are given by S = A/(nc), where A is the fitted area of the electric quadrupole peak, and n is the measured sample number density equal to p(kBT)−1, in which kB is the Boltzmann constant. The Gaussian width, wG, was calculated in terms of the measured temperature, transition frequency, and 16 O2 molecular mass. For the lines considered here, wG, expressed as a full width at half- maximum (FWHM), was nominally 0.85 GHz. Given the relatively low SNR of the electric quadrupole spectra, it was not possible to retrieve the Lorentzian width, wL, from the fitting procedure with sufficient accuracy. Instead, we constrained wL to be 2γp, where γ is the pressure broadening coefficient. This constraint also ensured self-consistency in the fit-derived areas for a given line as the sample pressure was varied. We used the empirical J-dependent correlation of Yang et al. [46] with the parameters found in the self-broadening measurements of O2 A-band magnetic dipole P-branch spectra of Robichaud et al. [36] to calculate γ as   A B , 1  c1 J   c 2 J  2  c3 J  4 (1) where J′ is the upper state angular momentum, A = 0.004481 MHz Pa−1, B = 0.01269 MHz Pa−1, and c1, c2, and c3 equal 0.03723, −7.86×10−4, and 1.05×10−6, respectively. Note that these parameters are presently utilized to calculate the broadening coefficients found in HITRAN 2008 [47]. 108 FIG. 7. Measured (filled symbols) and calculated (line) self-broadening coefficient, γ, vs. J′. To test the applicability of Eq. 1, we relaxed the above constraint on wL and subsequently retrieved γ from fits to the spectra. Figure 7 shows the results of these measured γ values and the predicted J′-dependent widths given by Eq. 1. Although the spread in the measurements is relatively large compared to the predicted variation with J′, the mean γ values derived from the fits agrees well with the above correlation. The effect of uncertainty in γ on the uncertainty of the measured line intensities will be addressed in the Section 4.2. 5.4. Results and Discussion 5.4.1. Electric Quadrupole Line Intensity Measurements We used FS-CRDS to measure six electric quadrupole transitions in the NO branch and one transition each in the PO, RS, and TS branches of the O2 A-band. Except for the TS(5) transition, none of these transitions has previously been detected. We found good agreement between our observed and calculated line positions, with a mean deviation within the standard uncertainty of our calibrated frequency reference (60 MHz or 0.002 cm−1). 109 FIG. 8. Increase in SNR of average absorption spectrum with increasing number of averaged spectra. Central peak corresponds to the NO(19) electric quadrupole transition of 16O2 (~ = 12,955.1661 cm−1, S = 6.36×10−30 cm molec.−1) at p = 12.7 kPa. The middle and top spectra are offset in the vertical direction by 0.0005 and 0.0010 units, respectively. As discussed in the Experiment section, 50 ring-down acquisitions were averaged per frequency step within an individual spectrum. At least ten complete spectra were then co-added to produce a final averaged spectrum. The effect of the reduction in baseline noise level achieved by co-adding electric quadrupole spectra can be seen in Figure 8. It is apparent that a feature, which is only arguably observable in a single scan given the high noise level, is clearly visible with a SNR 10:1 after averaging twenty scans. Figure 9 shows the relatively isolated TS(5) transition, which was previously measured by Brault [5] and which we readily measured with a SNR of 50:1. 110 FIG. 9. TS(5) electric quadrupole line of 16O2 (~ = 13,179.9239 cm−1) at p = 12.7 kPa, T = 299.4 K. Symbols represent measured values and the line is a fit based on the Voigt line shape. The measurements correspond to the average of ten scans, giving a SNR~50:1. The reduction in noise with signal averaging allows the quantitative observation of ultraweak absorption features within the wings of much stronger lines. The benefits of signal averaging are illustrated in Figure 10, which shows an average of 23 individual spectra, for the NO(21) electric quadrupole transition. This is the weakest line probed in the present study. Nevertheless, it can clearly be seen in the wings of a 16O2 hot band line. The absorption peak of the electric quadrupole line is 3.4×10−10 cm−1 and the rms baseline is 2.1×10−11 cm−1, giving a peak SNR of 16:1. 111 FIG. 10. Upper panel gives the NO(21) electric quadrupole line (~ = 12,934.7490 cm−1 , S = 3.70×10−30 cm molec.−1) in the wings of the PQ(11) hot band, magnetic dipole transition of 16O2 (~ = 12,934.5516 cm−1, S = 2.87×10−27 cm molec.−1). Symbols represent measured values and the lines are fitted Voigt line shapes. Bottom panel shows the fit residuals. The spectrum comprises an average of 23 files, taken at p = 12.7 kPa. Averaged spectra were acquired at several pressures between 3.33 and 26.7 kPa, with each spectrum yielding a measurement of line intensity. The mean line intensity for each transition (see Table 3) was obtained by a weighted average over all measurements taken at various pressures. We used a normalized weighting factor proportional to (nj + εj−2)1/2, where nj is the number of co-added spectra comprising the jth measurement, and εj(pj) is the fractional uncertainty in the jth measurement arising from systematic error in γ (as discussed below). We observed linearity between peak area, Ai, and O2 number density, n, for all measured transitions. This result is illustrated in Figure 11, where the solid lines intersecting the origin correspond to the calculated peak areas based on the mean line intensity for each case (i.e., Ai ,calc  n S i c ). 112 FIG. 11. Fit-derived peak areas (symbols), Ai, vs O2 number density, n, for the NO branch electric quadrupole lines given in Table 2. The solid lines correspond to the expected areas based on the weighted mean line intensity for each case. The open symbols with abscissae near n = 3.1×1018 molec. cm−3, are repeat measurements of the respective peak areas. The repeat data were obtained several weeks after the original data using another probe laser nominally identical to the original one. For completeness, the measured intensities were also corrected to the standard reference temperature Tr = 296 K using the relation: S (Tr )  S (T ) Q(T ) e  hcE  /( k BTr ) , Q(Tr ) e  hcE  /( k BT ) (2) where E″ is the lower state energy, and Q(T) is the total partition function for 16O2. Lower state energies for the measured transitions are given in Table 3. Since most measurements were made within 1 K of 296 K, the corrections required to adjust the measured intensities to Tr were all less than 0.7%, far below our experimental measurement uncertainty. 113 TABLE 3. Measured electric quadrupole line intensities at 296 K (S in units of 10×−30 cm molec.−1), lower state energies (E″ in cm−1) and standard relative uncertainties, [uc,r(S) in %] for the 16O2 A band. The Type B uncertainties assume a 10% relative uncertainty in the self-broadening coefficient, γ. The quantities, nT and <ns> represent the number of measurements of S and the corresponding average number of co-added spectra, respectively. Transition E″ S Type A Type B O(5) 42.2001 21.2 8.9 7.4 4.9 15 21 O(13) 260.6824 16.8 5.0 3.0 3.9 6 13 N O(15) N 343.9694 11.6 5.2 4.0 3.3 9 27 N O(17) 438.7010 8.53 4.2 2.3 3.5 4 19 N O(19) 544.8623 6.36 5.6 3.9 4.0 4 17 O(21) 662.4361 3.65 6.3 5.1 3.7 5 33 P O(1) 2.0843 35.8 4.1 3.0 2.8 1 12 R S(1) 0 3.83 11 9.9 4.7 1 45 S(5) 42.2240 20.5 4.4 3.0 3.3 1 10 N N T uc,r(S) nT <ns> 5.4.2. Uncertainty Analysis There are a number of observables in our experiment that do not make a significant contribution to the combined uncertainty in line intensity. These include measurements of the FSR, the spectrum frequency axis, and the sample temperature, pressure and number density. We estimate the relative combined uncertainty in S driven by uncertainties in these quantities to be less than 0.5%. There remain three dominant sources of uncertainty: (1) line shape effects influenced by collisional narrowing and self-broadening, (2) the base losses of the ring-down cavity, and (3) the total (base plus absorption) loss of the system. The first is a systematic (Type B) uncertainty, and the latter two can be treated as statistical uncertainties (Type A) that can be reduced by least-squares fits and signal averaging [48]. Our previous analysis of high SNR O2 A-band spectra [36] revealed collisional (Dicke) narrowing line shape effects which we modeled using the Galatry profile [43]. In the present study, we estimated that our choice of the Voigt profile contributes a relative uncertainty of ~2% to S, with a larger uncertainty arising from our choice of broadening 114 parameter. We found the systematic uncertainty in S arising from uncertainty in γ by numerical evaluation of the sensitivity parameter, d ln S/d ln γ. This parameter represents the fractional change in line intensity per fractional change in broadening parameter and can be readily found by repeating the fits to the quadrupole transitions over a range of γ values. From this analysis we obtain d ln S/d ln γ ~ ηp, where η is line dependent and ranges from 1.7 to 2.1 Pa−1. Assuming a 10% relative uncertainty in γ, we find the relative uncertainty in S arising from error in γ to be less than 5% for all cases. Variations in the base loss occurring on timescales that are comparable to the scan time can give rise to spurious structure in individual spectra. This type of effect may not be captured by a model that incorporates a simple linear or sinusoidal baseline. However, when taken over an ensemble of spectra, residual variations in base losses tend to diminish inversely with the square root of the number of spectra, consistent with a Type A uncertainty. Likewise, noise in the ring-down signals occurring in the photoreceiver and A/D systems, limits the precision with which the ring-down time constant can be measured. This effect is manifest as an intrinsic noisiness in the spectra that is apparent near the detection limit. It too tends to be uncorrelated from one scan to the next, and hence both effects can be reduced by signal averaging as illustrated in Figure 5. The rms deviation in S about the ensemble mean S is a measure of the combined Type A uncertainty. For NO-branch lines (which are those for which several repeat measurements were made), Table 3 shows that this quantity ranges from 2.3% to 7.4%. We also repeated the peak area measurements as a further check on our measurement precision and reproducibility. These experiments were repeated for each line several weeks after the original data were taken (open symbols in Figure 11), and data were acquired using a different probe laser that was nominally identical to the first. We found an average relative difference between the new S and those based on previously determined S of 1.3% with a standard deviation of 4.1%. These differences are consistent with the rms deviations in S that were found in the original data set. Combining, the Type A and Type B uncertainties in quadrature, we estimate the relative combined standard uncertainty for most of the NO-branch electric quadrupole line 115 intensities to be ~6%. By comparison, this relative uncertainty is only about 20 times greater than those of the O2 A-band magnetic dipole lines [36]; despite the fact that the latter transitions are five orders of magnitude stronger than the former. These data also indicate that it would be possible to detect much weaker lines using the present technique. Our weakest measured line had an intensity of 3.7×10−30 cm molec.−1 with a Type A uncertainty of 5%. If we assume a best-case measurement precision of 2.5% (improved over the present case by a fourfold increase in signal averaging time or by a twofold reduction in mirror losses), we estimate a SNR of ~2:1 for a line having an intensity of 2×10−31 cm molec.−1. This limit is approximately 100 times lower than the detection limit reported by Naus et al. [6] in their CRDS measurement of the TS(9) O2 electric quadrupole line. 5.4.3. Electric Quadrupole Line Intensity Calculations The SPCAT program [19] was modified to calculate the electric quadrupole matrix elements. Intensities for the |ΔN| = 1,3 electric quadrupole transitions were calculated for N″ = 1 to 29 (with estimated values for M1, Q1, and Q3). The values of Q1 and Q3 were determined in a least-squares fit of the predicted electric quadrupole intensities to those we measured. A similar procedure was used to determine M1 from the previously measured magnetic dipole transition intensities [36]. The fitted transition moments were M1 = 6.369(8)×10−25 J T−1 [0.06868(8) Bohr-magnetons], Q1 = 4.15(13)×10−42 C m2 [0.0124(4) D Å], Q3 = 2.61(8)×10−42 C m2 [0.00783(23) D Å], where the value in parentheses denotes the standard uncertainty in the last digit. We recalculated the predicted intensities for all four electric quadrupole branches using the fitted values of M1, Q1, and Q3. The average absolute difference between the present electric quadrupole measurements and the calculations is ~5%, which is comparable to our reported average experimental uncertainty. 116 TABLE 4. Calculated intensities (at 296 K, in units of 10−30 cm molec.−1) for the electric quadrupole transitions within the NO, PO, RS, and TS branches of the 16O2 A band. Fit uncertainties (mean absolute relative deviation between our measurements and calculated intensities) of these intensities are ~5%. Transitions observed in this study are underlined. N″ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 N O 11.4 18.7 21.8 21.9 19.7 16.3 12.5 8.83 5.86 3.62 2.11 1.15 0.60 0.29 P O 35.9 76.5 102.3 114.0 112.3 100.4 82.4 62.7 44.4 29.4 18.2 10.6 5.78 2.97 1.44 R S 4.09 51.2 80.9 96.6 98.9 90.9 76.2 58.8 42.1 28.2 17.6 10.3 5.68 2.93 1.42 T S 6.09 14.0 19.8 22.7 22.8 20.7 17.2 13.2 9.46 6.32 3.95 2.31 1.27 0.661 0.318 The results of these intensity calculations for all four electric quadrupole branches can be found in Table 4. Figure 1 shows a stick spectrum based on the present calculations with all observed transitions indicated. Using the calculated transition moments, we calculated the transition band intensities: SMD = 2.25(2)×10−22 cm molec.−1 and SEQ = 1.8(1)×10−27 cm molec.−1. It is important to note that the |ΔN| = 1 transitions (i.e., PO and RS branches) are four times stronger than the |ΔN| = 3 transitions (i.e., NO and TS branches) as illustrated in Figure 1, thus confirming the 1934 prediction of van Vleck [49]. As a result, while the ratio of intensities between a |ΔN| = 3 branch and a given magnetic dipole branch is 3×10−6 (as was reported by Brault [5]), the ratio of intensities for a |ΔN| = 1 electric quadrupole branch is 12×10−6. After summing intensities over all branches, we find that the ratio of electric quadrupole and magnetic dipole band intensities is 8×10−6. 117 On average, the measurements of Brault [5] are systematically low by 15% compared to the present calculations for the TS electric quadrupole transition intensities, displaying increasing discrepancy with large J. A deviation of ~12% was observed between Brault’s TS(5) measurement and our own. These disparities are consistent with the quadrature sum of the 10% uncertainty in effective path length reported by Brault and the uncertainties in our calculations and measurements. 5.5. Conclusions We have reported laboratory observations, quantitative intensity measurements, and calculations of electric quadrupole transitions in the O2 A-band. These measurements may serve as a benchmark for ultrasensitive spectroscopic techniques. The experiment exploited the high sensitivity, spectral fidelity, and long-term stability of frequencystabilized cavity ring-down spectroscopy; a technique that enabled intensity measurements of ultraweak lines (S~3×10−30 to 2×10−29 cm molec.−1) with unprecedented accuracy (4.4% to 11% relative combined uncertainty). Our calculations included the electric quadrupole line positions (uncertainties less than 3 MHz) and and line intensities. The line intensity calculations and measurements agreed to within our average experimental uncertainty of 6%. Our calculated positions and intensities are sufficiently accurate to model the electric quadrupole contribution to the O2 A-band which may be observed in long-path atmospheric spectra. In addition, we have demonstrated how such high-sensitivity and low uncertainty measurements can add to our understanding of diatomic spectroscopy and provide a basis for comparison with present day computational models. Finally, the measurements of other ultraweak transitions in atmospherically relevant species may have possible future implications for increasingly sensitive remote sensing measurements. 118 Acknowledgements David A. Long acknowledges the support of the National Defense Science and Engineering Graduate Fellowship. Daniel K. Havey received support from the National Research Council as a postdoctoral fellow at the National Institute of Standards and Technology (NIST). The research at the Jet Propulsion laboratory (JPL), California Institute of Technology, was performed under contract with National Aeronautics and Space Administration. 119 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] J. F. Noxon, Can. J. 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Copyright 2011 by the American Physical Society. 123 Abstract Frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) was employed to measure air-broadened CO2 line shape parameters for transitions near 1.6 μm over a pressure range of 6.733 kPa. The high sensitivity of FS-CRDS allowed for the first measurements in this wavelength range of air-broadened line shape parameters on samples with CO2 mixing ratios near those of the atmosphere. The measured airbroadening parameters show several percent deviations from values found in the HITRAN 2008 database. Spectra were fit with a variety of models including the Voigt, Galatry, Nelkin-Ghatak, and speed-dependent Nelkin-Ghatak line profiles. Clear evidence of collisional narrowing was observed, which if unaccounted for can lead to several percent biases. Furthermore, it was observed that only the speed-dependent Nelkin-Ghatak line profile was able to model the spectra to within the instrumental noise level because of the concurrent effects of collisional narrowing and speed dependence of collisional broadening and shifting. 124 6.1. Introduction Recent carbon dioxide remote sensing missions have set unprecedented precision targets as demanding as 0.3% [1-2] (change in atmospheric CO2 mixing ratio of ~1 μmol mol−1). These applications will require a detailed and thorough understanding of the near-infrared spectral line shapes of CO2 and O2 over a wide range of atmospherically relevant temperatures and pressures [3]. Of particular importance are higher-order effects such as collisional narrowing, speed-dependent effects and line mixing which are yet to be incorporated into the vast majority of atmospheric retrievals. These effects will play an even greater role in next generation, active sensing missions (such as NASA’s Active Sensing of CO2 Emissions over Nights Days and Seasons, ASCENDS) where individual transitions in the near-infrared spectral region will be employed, rather than entire bands. While near-infrared CO2 transitions have been extensively studied (for example see Refs. [4-20] and the references found therein), only recently have non-Voigt line profiles begun to be employed [9-14,19-20]. In a recent Fourier-transform spectroscopy (FTS) study, clear evidence of line narrowing was observed, and further it was found that the use of the Voigt profile (which does not include this effect) limited the accuracy of the measured spectroscopic parameters to ~1% [8]. These findings led to a reanalysis of the FTS measurements using a speed-dependent Voigt profile with line mixing which resulted in a ~2% increase in the measured broadening parameters [9-10]. More recently Casa et al. used a laser absorption spectrometer to measure lowpressure (<5 kPa), self-broadened CO2 line shapes near 5,000 cm−1 [19]. Unlike previous FTS studies of CO2 line shapes, they were able to clearly distinguish various line profiles because of the relatively high signal-to-noise ratio (~10,000:1) and spectral resolution (~1 MHz) of the measurements. However, they were unable to completely describe the observed line shapes with either the Voigt, Galatry, Nelkin-Ghatak, speed-dependent Voigt, correlated Galatry or correlated Nelkin-Ghatak profiles. Notably, none of these theoretical profiles incorporated both collisional narrowing and speed-dependent effects. This lack of consistency between experiment and theory motivated the present work in which we apply frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) to the 125 measurement of near-infrared CO2 line shapes. As previously demonstrated, the sensitivity and spectral fidelity of FS-CRDS make it an ideal technique for detailed study of subtle line shape effects [21-26]. Furthermore, the use of multispectrum fitting [24,2728], which simultaneously fits spectra across a wide pressure range, can enable the collisional narrowing and speed-dependent effects to be separated and quantified. Below we present FS-CRDS pressure-dependent line shape measurements of an air-broadened near-infrared CO2 transition near 6,360 cm−1, and we compare our measured Lorentzian broadening parameters to literature values. We also show that both collisional narrowing and speed-dependent effects need to be taken into account to best describe the observed pressure dependence of the line shape. 6.2. Experiment Measurements were made using the frequency-stabilized cavity ring-down spectrometer located at the National Institute of Standards and Technology in Gaithersburg, Maryland [29-30]. This particular FS-CRDS instrument has been described earlier [31], therefore, we will discuss only the most salient details herein. FS-CRDS differs from traditional cw-CRDS in two fundamental ways. Firstly, the intracavity distance is actively stabilized with respect to a reference laser. This length stabilization in turn stabilizes the entire comb of ring-down cavity transmission modes, thus, providing an exceptionally linear, stable, and accurate frequency axis for our spectra. In addition, FS-CRDS is a single-mode technique which allows for quantitative ring-down measurements. FS-CRDS has been employed to measure ultraweak transitions [32-33], hyperfine structure [25], rare isotopes [22,31,34], and to produce ultraprecise spectroscopic reference data [21-22,26,34-35]. For the measurements described herein the probe laser was an external cavity diode laser with an output power of 1518 mW and the reference laser was a frequencystabilized HeNe laser with a long-term stability of 1 MHz (8 h). The ring-down mirrors had reflectivities of ~99.997% at the probe wavelength, corresponding to a finesse of ~105,000. Three hundred ring-down time constants were collected and averaged at each 126 frequency step leading to a minimum detectable absorption coefficient of 3.5×10−11 cm−1 (see Figure 1 of Chapter 7 for an Allan variance plot with the given system parameters). For the wavelength region investigated here, the sensitivity of FS-CRDS enables high signal-to-noise-ratio CO2 absorption spectra to be acquired on air samples with CO2 mixing ratios close to the current atmospheric background value (~390 μmol mol−1). This capability of FS-CRDS is in contrast to similar FTS measurements which require CO2 mixing ratios at the several percent level. Therefore, air-broadening parameters can be directly measured with FS-CRDS. In the present study, measurements were made using NIST SRM 1676, a blend of carbon dioxide in air with a CO2 mixing ratio of 367.81(37) μmol mol−1. This dry gas sample contained nominally 20.8% oxygen and 0.98% argon. At pressures above 40.0 kPa (300 Torr) line mixing (line coupling) became apparent. Therefore, at pressures near or exceeding 40.0 kPa it is no longer possible to treat the transitions as being spectrally isolated. The present study aimed to examine CO2 line shape behavior in the spectrally isolated pressure regime, therefore, measurements were made over the pressure range 6.733 kPa (50250 Torr). No evidence of line mixing was observed in these spectra. A future study is forthcoming which will utilize these low pressure results in combination with line mixing to model the CO2 line shape at and above 101.325 kPa. The sample pressure was measured using a NIST-calibrated diaphragm capacitance gauge having a full-scale response of 133.3 kPa (1,000 Torr) and a relative standard uncertainty below 0.1%. All measurements were made at room temperature with no attempt made to control the cell temperature. The cell temperature was monitored using a NIST-calibrated 2.4 kΩ platinum resistance thermistor. The combined standard uncertainty in the cell temperature has recently been shown to be ~30 mK [33]. The measured spectra were fit with a variety of line profiles including the Voigt profile (VP), the Galatry profile (GP) [36], the Nelkin-Ghatak profile (NGP) [37-38], the speed-dependent Voigt profile (SDVP) [39], and the speed-dependent Nelkin-Ghatak 127 profile (SDNGP) [38,40]. The physical basis and mathematical form of these line profiles is described in Refs. [23,41], therefore, this will be only briefly discussed herein. The VP is a convolution of a Gaussian and Lorentzian line profiles and therefore accounts for Doppler and pressure broadening by treating these effects as statistically independent. The GP and NGP include the contribution of collisional (Dicke) narrowing [42], which makes the observed Gaussian width narrower than the classical Doppler width. In the case of a relatively light perturber, the GP is more applicable than the NGP because the former uses a soft-collision model, whereas the latter uses a hard-collision model of velocity-changing collisions. The SDVP does not include collisional narrowing, but generalizes the VP by taking into account the dependence of collisional width and shift upon absorber speed. These effects usually cause line narrowing and line asymmetry, respectively. Of the line profiles considered here, the SDNGP is the most complex, because it includes both collisional narrowing and speed dependence. 6.3. Results and Discussion In the present study, we measured transitions within the (30012)←(00001) CO2 band which is centered at wavelength  = 1.576 nm. We focused upon the R16 transition which is located at wave number  = 6,359.9673 cm−1 and which has been proposed for use in NASA’s ASCENDS active sensing mission. The top panel of Figure 1 shows the measured absorption spectrum (symbols) and SDNGP fit (line) for the R16 transition, and the residuals obtained by individually fitting the VP, GP, SDVP, NGP, and SDNGP to the same measured spectrum are given in the bottom panels of this figure. Inspection of the VP residuals (note the unique y-axis scale for this residual panel) indicates that this profile underestimates the measured peak intensity and fails to capture the measured line shape. We attribute this lack of agreement to line narrowing. In contrast, the GP, NGP, and SDVP each model a mechanism of line narrowing (either collisional narrowing or speed dependence), and therefore fitting any one of these profiles to the measured spectrum leads to vastly reduced residuals when compared to the VP. However, only the SDNGP, which includes both collisional narrowing and speed-dependent narrowing, is 128 able to model the spectrum to within the instrumental noise level, resulting in a spectrum signal-to-noise ratio (ratio of peak absorption to standard deviation of fit residuals) of ~5,000:1. FIG. 1. Upper panel gives the measured FSCRDS spectrum of the (30012)←(00001) R16 transition of CO2 and the single-spectrum SDNGP fit to these data. The sample gas was air with a CO2 molar fraction of 367.81(37) μmol mol−1 at a total pressure of 13.3 kPa. Bottom panels are the corresponding fit residuals for the indicated line profiles. Fitting the VP to the measured spectrum leads to large, systematic residuals which are attributed to collisional narrowing. Note the tenfold difference in the vertical axis scale between the VP residual plot and those of the other cases. Use of the GP, NGP, and SDVP reduces the spectrum residuals approximately tenfold, however, only the SDNGP is able to model the spectrum to within the instrumental noise level. The spectrum signal-to-noise ratio (ratio of peak absorption to standard deviation of the fit residuals) for the SDNGP fit is ~5,000:1. The choice of line profile utilized to model the measured spectra can have a considerable impact on the fitted spectroscopic parameters obtained at a given pressure. As can be seen in Figures 2 and 3, fitting the VP to the measured spectra leads to significant deviations with respect to the measured Lorentzian width and spectrum area (proportional to line intensity). These deviations are especially large at low pressure (e.g., 6.7 kPa) where the Lorentzian width and spectrum area obtained through the fitting the Voigt profile deviate by as much as 6% and 2%, respectively from those obtained with the SDNGP fits. In addition, we note that the Lorentzian width and line area parameters obtained from VP fits do not converge with those obtained using the other line profiles 129 considered here, even at higher pressures. Furthermore, in the case of the VP, the fitted parameters have considerably larger uncertainties than those obtained with more advanced line profiles that are fit to the same data. Fitted parameters obtained with the GP, NGP, and SDVP are rather similar to those obtained with the SDNGP. Excluding the VP fits, we found that the maximum deviation in fitted Lorentzian width and spectrum area were at most 1.5% and 0.5%, respectively. FIG. 2. Pressure dependence of the ratio of the air-broadened Lorentzian width, Γ, obtained by fitting various line profiles (VP, GP, NGP, SDVP) to that obtained with the SDNGP (denoted by ΓSDNGP). These data correspond to the (30012)←(00001) R16 transition of CO2. Fitting spectra with the VP systematically underestimates the Lorentzian width and at a pressure of 7 kPa leads to a deviation of as much as 6%. Although the ratio ΓVP/ΓSDNGP increases with pressure, it does not converge to unity even at p = 40 kPa. In contrast, ΓNGP/ΓSDNGP is insensitive to pressure and has a mean value of 1.0014(9). Interestingly, for all the line profiles evaluated here the fitted  varies linearly with pressure, p, and as shown in Figure 4 the slope, d/dp, is similar for all cases. Note that d/dp corresponds to the air-broadening parameter, γ. However, with regard to the set of (p) obtained from VP fits, we find that there is an unphysical, negative-valued, yintercept, which implies the occurrence of a nonlinear relationship for (p) at low pressure. We attribute this result to pressure-dependent line narrowing effects that are not modeled by the VP. This mode of analysis may have a profound impact upon 130 laboratory and atmospheric determinations of γ. Specifically, when γ is evaluated as the ratio, /p, where  is found by VP fits to a spectrum at a single pressure, then this measurement will tend to underestimate the true broadening parameter. In the present case, the magnitude of this effect is nominally 2%. Theoretical calculations using the GP to simulate line narrowing were performed and verify these experimental findings regarding the VP. For the air-broadened CO2 transition considered here, it is expected that the value of d/dp observed at pressures below 6.7 kPa would diverge from the higher pressure value. FIG. 3. Pressure dependence of the ratio of the spectrum area, A, obtained by fitting various line profiles (VP, GP, NGP, SDVP) to that obtained with the SDNGP (denoted by ASDNGP). These data correspond to the air-broadened (30012)←(00001) R16 transition of CO2. Fitting spectra with the VP systematically underestimates the spectrum area, and at a pressure of 7 kPa use of the VP leads to a deviation of as much as 2% from the SDNGP value. The SDVP-fitted areas are on average about 1.5% greater than ASDNGP, whereas the GP and NGP fitted areas are within 0.02% and 0.04%, respectively, of ASDNGP. Regardless of the line profile employed, the air-broadening parameters obtained from the present measurements show large, several-percent deviations from those found in the HITRAN 2008 database [43] (see Figure 4). The HITRAN 2008 air-broadening parameters in this band are based upon FTS measurements of CO2-air mixtures with several percent CO2 [10]. As the self-broadening parameters of CO2 are generally ~40% 131 larger than the air-broadening parameters [43], these results were extrapolated to natural abundance of CO2. We speculate that either this extrapolation or other line shape effects (e.g., line mixing) are the source of this deviation. Note that the SDNGP-retrieved, airbroadening parameter for the R16 transition is 21.75(9) kHz Pa−1 (0.0735(3) cm−1 atm−1) at T = 296 K. We observed similar deviations from the HITRAN 2008 air-broadening parameters for the R14, R18, P16, P18 transitions of CO2; each of which was low from the HITRAN 2008 parameters by 0.9%2.7%. 22.25 22.00 -1 (kHz Pa ) HITRAN SDVP NGP VP GP SDNGP (SS) SDNGP (MS) 21.75 21.50 FIG. 4. Measured air-broadening parameter, γ, at T = 296 K for the (30012)←(00001) CO2 R16 transition. The labels indicate the line profiles that were used, and SDNGP(SS) and SDNGP(MS) correspond to the single-spectrum fit and multi-spectrum fit cases, respectively for the SDNGP. Except for the multispectrum fit case, these results were obtained by fitting the profiles to FS-CRDS spectra over the pressure range 6.734 kPa and subsequently evaluating  = dΓ/dp from each set of fitted Γ values. The airbroadening parameter for the SDNGP(MS) case is 21.76(1) kHz Pa−1. Note that each of the fitted values is ~2% lower than the HITRAN 2008 value, which is equal to 22.16 kHz Pa−1 [43]. The factor 1 cm-1 atm-1  295.872 kHz Pa−1 can be used to convert these values into the conventional units for γ (cm−1 atm−1). The pressure dependence of the narrowing frequency obtained by fitting the GP to our measured spectra can be found in Figure 5. If collisional narrowing were the only source of line narrowing, we would expect to see a linear dependence on pressure. However, in the present case we observe a quadratic dependence on pressure (similar behavior is observed when the NGP is employed). This non-linearity is indicative of the simultaneous occurrence of collisional (Dicke) narrowing and speed-dependent narrowing (e.g., Refs. [23,44-47]). 132 The SDNGP accounts for both of these effects and is therefore the appropriate line profile for these measurements. However, since both collisional narrowing and speed dependence narrow the total line width, their fitted values can become correlated. Therefore, it is necessary to utilize multi-spectrum fitting [24] to simultaneously fit spectra across a given range of pressures, whereby the two sources of narrowing can be separated based on their different pressure dependencies. Specifically, the collisional narrowing frequency is given by  nar   p in which  is the collisional narrowing parameter, whereas the quadratic approximation speed-dependent parameters aw and as are independent of pressure [48]. This multispectrum fitting procedure produces a single value for each of the narrowing and asymmetry parameters (i.e., η, aw , and as ). With this approach, the SDNGP profiles were simultaneously fit to four spectra spanning the pressure range 1434 kPa, to yield fitted values for these three parameters. The measured spectra, the fitted profiles, and the respective sets of fit residuals are shown Fig. 6. From this multi-spectrum fit of the SDNGP we find η = 6.18(21) kHz Pa−1, aw = 0.042(8) and as = 0.08(2). Note that the spectrum signal-to-noise ratio ranges from 2,150 to 3,600 and is not as high as that obtained with the SDNGP single-spectrum fit shown in Figure 1. We assign this reduction in fit quality to pressure-dependent variations in the spectrum baseline that cannot be fully taken into account with a fixed baseline model. 133 FIG. 5. Collisional narrowing frequency, nar, as a function of pressure for the air-broadened (30012)←(00001) R16 CO2 transition. For single-spectrum fits of the GP (square symbols) the nar values have an unphysical, quadratic pressure dependence caused by the concurrence of collisional (Dicke) narrowing and speed-dependent narrowing. The indicated narrowing frequencies (red circles) are obtained when the same spectra are simultaneously fit over the entire pressure range with the SDNGP. This multispectrum analysis uses the SDNGP subject to the constraint that nar = ηSDNGPp. Importantly, the fitted collisional narrowing parameter (the slope of the line formed by the red circles, shown with its standard uncertainty in dashed red lines), agrees well with the diffusion-based value, ηdiff [49]. We find that ηSDNGP = 6.18(21) kHz Pa−1 and ηdiff = 5.44(27) kHz Pa−1. When all velocity changing collisions contribute to the line narrowing, the collisional narrowing parameter can be calculated based on the mass diffusion coefficient of the absorber, D, and pressure and temperature, as ηdiff = kBT/(2πmaDp). For dilute mixtures of CO2 in air we estimate that ηdiff = 5.44(27) kHz Pa−1 (0.0184(9) cm−1 atm−1) [49], and as can be seen in Figure 5, the collisional narrowing parameter, ηSDNGP which was obtained from the multi-spectrum fit agrees well (relative difference of <15%) with ηdiff. This agreement supports our use of the SDNGP for describing the collisional narrowing effect [45] in the present case and is consistent with the underlying assumption (required also for the GP and NGP) that all velocity changing collisions fully contribute to the line narrowing. Therefore, it appears unlikely that models accounting for correlations between velocity- and phase-changing collisions such as the CNGP [38] or the CSDNGP [50] are required to describe our measured CO2 line shapes in this pressure range. 134 FIG. 6. Measured and modeled absorption spectra of airbroadened (30012)←(00001) R16 CO2 transition. The top panel shows the measured spectra (symbols) and multispectrum SDNGP fit (lines). For clarity, the three highest-pressure spectra are successively offset in the vertical direction in increments of 2.510−7 cm−1. The bottom four panels (increasing pressure from top to bottom) give the residuals of the SDNGP global fit (measured spectrum – fitted profile) for each pressure. Prior to implementing the multi-spectrum fit, each individual spectrum is corrected for a baseline etalon. Casa et al. recently observed that fitting the GP to self-broadened CO2 spectra (near  = 2.0 μm) over the range p = 0.74.0 kPa led to collisional narrowing parameters that exceeded ηdiff by roughly a factor of two [19]. This discrepancy between measured and calculated narrowing parameters can be explained by the simultaneous occurrence of collisional narrowing and speed-dependent effects, as found in the present study. We anticipate that a multi-spectrum fit of the SDNGP to their measured spectra, equivalent to that presented here, would yield a physically consistent collisional narrowing parameter that is either equal to or less than ηdiff. 6.4. Conclusions Air-broadened near-infrared CO2 transitions were measured with frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) at CO2/air mixing ratios close to those in the atmosphere. Special emphasis was placed upon the R16 transition within the (30012)←(00001) band (located near 6360 cm−1) which has been proposed for use in NASA’s ASCENDS active sensing mission. For the measured transitions, these were the 135 first measurements of the air-broadening parameters at near-atmospheric mixing ratios. We find that our measured broadening parameters differed by up to several percent relative to those in the HITRAN 2008 database. The measured spectra showed unambiguous evidence of collisional narrowing. Failure to account for this narrowing (i.e., through the application of the commonly used Voigt line profile) led to biases of several percent in the measured broadening parameters and intensities and to relatively large uncertainties by comparison to those obtained with more complex line profiles. In addition, it was found that both collisional narrowing and speed dependence of collisional broadening and shifting played a significant role in this pressure range (6.733 kPa). Of the profiles considered here, the SDNGP (which includes both of these effects) was the only one that modeled the spectra to within the instrumental noise level. Finally, excellent agreement was observed between the fitted and mass diffusion-based narrowing parameters when SDNGPs were simultaneously fit to spectra acquired over a range of pressure. This result indicated that the use of correlation line profile models such as the correlated-NGP or correlated-SDNGP is likely unnecessary in the present case. For the pressure range considered, we found no evidence of line mixing, although in general, we expect that more comprehensive models which include line mixing effects will be required to accurately predict the evolution of CO2 absorption spectra for pressures exceeding ~40 kPa. Acknowledgements David A. Long was supported by the National Science Foundation and National Defense Science and Engineering Graduate Fellowships. Daniel K. Havey was supported by a National Research Council postdoctoral fellowship at the National Institute of Technology (NIST), Gaithersburg, MD. Part of the research described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Additional support was provided by the Orbiting Carbon Observatory (OCO) project, a NASA Earth System Science Pathfinder (ESSP) mission; the NASA Upper Atmospheric Research 136 Program grants NNG06GD88G and NNX09AE21G; the NASA Atmospheric Carbon Observations from Space (ACOS) grant 104127-04.02.02; and the NIST Greenhouse Gas Measurements and Climate Research Program. Research was also supported by the Polish MNISW Project No. N N202 1255 35. 137 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] D. Crisp, R. M. Atlas, F. M. Breon, L. R. Brown, J. P. Burrows, P. Ciais, B. J. Connor, S. C. Doney, I. Y. Fung, D. J. Jacob, C. E. Miller, D. O'Brien, S. Pawson, J. T. Randerson, P. Rayner, R. J. Salawitch, S. P. Sander, B. Sen, G. L. Stephens, P. P. Tans, G. C. Toon, P. O. Wennberg, S. C. Wofsy, Y. L. Yung, Z. Kuang, B. Chudasama, G. Sprague, B. Weiss, R. Pollock, D. Kenyon, S. 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Robert, J. M. Thuet, J. Bonamy, S. Temkin, Phys. Rev. A 47, R771-R773 (1993). 140 CHAPTER 7 Frequency-Stabilized Cavity Ring-Down Spectroscopy Measurements of Carbon Dioxide Isotopic Ratios This chapter was published as D. A. Long, M. Okumura, C. E. Miller, and J. T. Hodges, Appl. Phys. B, In press: doi: 10.1007/s00340-011-4518-z. Copyright 2011 by Springer. 141 Abstract Carbon dioxide (CO2) isotopic ratios on samples of pure CO2 were measured in the 1.6 μm wavelength region using the frequency-stabilized cavity ring-down spectroscopy (FSCRDS) technique. We present CO2 absorption spectra with peak signal-to-noise ratios as high as 28,000:1. Measured single-spectrum signal-to-noise ratios were as high as 8,900:1, 10,000:1, and 1,700:1 for 13C/12C, 18O/16O, and 17O/16O, respectively. In addition, we demonstrate the importance of utilizing the Galatry line profile in the spectrum analysis. The use of the Voigt line profile, which neglects the observed collisional narrowing, leads to large systematic errors which are transition dependent and vary with temperature and pressure. While the relatively low intensities of CO2 transitions near  = 1.6 μm make this spectral region non-optimal, the sensitivity and stability of FS-CRDS enabled measurement precision comparable to those of other optical techniques which operate at far more propitious wavelengths. These results indicate that a FS-CRDS spectrometer designed to probe CO2 bands near wavelengths of 2.0 or 4.3 μm could achieve significantly improved precision over the present instrument and likely be competitive with mass spectrometric methods. 142 7.1. Introduction For over half of a century, stable isotope measurements have been utilized to quantify sources and sinks of CO2 [1]. Recent simultaneous measurements of 18O/16O and 17O/16O isotopic ratios in CO2 have elicited considerable interest due to their use as a tracer of stratospheric dynamics through the observed mass-independent isotope effect [2]. In addition, isotopic fractionation measurements are important in partitioning CO2 emissions between natural and anthropogenic sources and deducing exchange between the biosphere, ocean, and atmosphere [3]. Isotope-ratio mass spectrometry (IR-MS) is the standard for isotope fractionation measurements. Commercial, dual-inlet IR-MS instruments routinely achieve measurement precisions better than 0.1‰ for measurements of 13C/12C and 18O/16O isotopic ratios (e.g., [4]). Unfortunately, IR-MS has a number of technical limitations: it is expensive, bulky, time intensive, sample destructive, and requires careful sample preconcentration to remove isobaric interferences. In addition, IR-MS analysis is incompatible with molecules such as H2O and SO2 which have a tendency to stick to surfaces. Thus, isotopic analysis of these species requires an initial chemical conversion which increases the complexity and overall uncertainty of the measurement. Finally, measurements of 17O/16O ratios in CO2 by IR-MS are greatly complicated by the presence of the 16O13C16O isobar. This necessitates an initial conversion of CO2 to O2 which introduces an uncertainty of approximately ±0.1‰ and precludes a simultaneous measurement of the 13C/12C ratio [5]. These difficulties with IR-MS have motivated the development of isotope ratio measurements based on laser absorption spectroscopy [6]. These new laser-based spectroscopic techniques have significant potential for inexpensive, in situ measurements. In addition, since individual isotopologues exhibit distinct spectra, unlike IR-MS methods they are not susceptible to isobaric interference. As these techniques have recently been reviewed [6,7], we will only discuss a few recent developments in laser-based isotope ratio measurements of CO2. For measurements of 13 C/12C ratios, laser-based 143 spectroscopic techniques have begun to rival IR-MS, with commercially available instruments offering precisions better than 0.3‰. For 18O/16O, precisions as high as 0.05‰ have been achieved through the use of quantum cascade lasers which can probe the CO2 fundamental band at  ~ 4.3 μm [8]. The low isotopic abundance of 16O12C17O presents a significant challenge to spectroscopic measurements (see Table 1 for natural isotopic abundances). As a result, only three previous studies have measured the 17O/16O isotopic ratio of CO2 [9,10,11]. The highest precision was achieved by Castrillo et al. [11] who measured samples of 2% CO2 in exhaled breath. They utilized wavelength- modulation spectroscopy with a liquid-nitrogen-cooled quantum cascade laser emitting at  ~ 4.3 μm. Short term precisions of 0.5‰ and 0.6‰ were achieved for the 18O/16O and 17 O/16O isotopic ratios, respectively. Unfortunately, the measurements exhibited large, systematic deviations of 11.5‰ and 13.0‰ for the two isotopic ratio measurements, respectively. These were attributed to either non-linearity in the second-harmonic detection (as they were operating outside of the linear Beer’s law regime) or non-linearity in their HgCdTe detector. TABLE 1. CO2 isotopologue (Iso.) transitions utilized in this study. Positions, intensities (at near-natural isotopic abundance as given in Ref. [12], na, and T = 296 K), and lower state energies (E′′) are from the HITRAN 2008 database [12]. The shown Type A (random) uncertainties capture the Galatry fit uncertainty as calculated based upon the spectral signal-to-noise ratio (SNR). The uncertainty in the corresponding ratio measurement due to temperature, uδ, is a Type B (systematic) uncertainty which was calculated as shown in Eq. 1. Note that the uncertainties in p and the ring-down cavity free spectral range are not present as these will cancel in the isotopic ratio measurements. The quantity ADiff = 1000(AGP/AVP-1), where AGP and AVP are the spectral areas derived using the GP and VP, respectively. (30013)←(00001) Position (cm−1) 6263.369778 E′′ (cm−1) 1722.9412 Intensity (cm/molec.) 3.156E-26 Type A unc. (‰) 0.03 uδ (‰) - (30012)←(00001) 6270.249101 994.2387 8.48E-27 0.11 0.33 8900:1 20 R22 (30012)←(00001) 6269.931712 186.271 1.938E-26 0.09 0.69 10000:1 20 P37 (30012)←(00001) 6265.793803 532.0822 1.403E-27 0.59 0.54 1700:1 40 Iso. na Trans. Band 626 0.98420 R66 636 0.01106 R50 628 0.0039471 627 0.000734 SNR 28000:1 We have recently constructed a frequency-stabilized cavity ring-down spectrometer in order to perform ultrasensitive, high-resolution lineshape studies of CO2 combination bands near  = 1.6 μm [13]. These laboratory measurements will assist ADiff (‰) 10 144 remote sensing measurements of atmospheric CO2 concentrations [14,15]. We have taken this opportunity to evaluate the potential of frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) for measurements of CO2 isotopic ratios. While this wavelength range is non-optimal due to the relative weakness of these transitions (two orders of magnitude weaker than the transitions at  ~ 2.0 μm and five orders of magnitude weaker than those of the fundamental band at  ~ 4.3 μm), this study allows us to compare FS-CRDS to other optical methods in the determination of isotopic ratios. In addition, the results of this study enable us to estimate the precision and accuracy of a FS-CRDS measurements made at more advantageous wavelengths. 7.2. Experimental Apparatus Measurements were made using the frequency-stabilized cavity ring-down spectrometer located at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland. The FS-CRDS spectrometer utilized herein is similar to that which was previously employed to measure the O2 A-band at  ~ 0.762 μm [16,17,18,19,20,21,22,23]. In the present experiment we used a different external-cavity diode laser with an output centered near  = 1.6 μm, thus enabling us to probe nearinfrared CO2 transitions. Measurements in this spectral region also required a different set of highly reflective cavity ring-down mirrors and an InGaAs in place of a Si-PIN detector. All other aspects of the present FS-CRDS experiment including the cavity and probe laser frequency locking and data collection methods were identical to those employed for our previous O2 A-band studies. Because of the similarities of the two sets of experiments, in this chapter we will only present the most pertinent details. FS-CRDS differs from traditional cw-cavity ring-down spectroscopy in two fundamental ways [24,25]. Firstly, the optical cavity length is actively stabilized to an external frequency reference, in our case a frequency-stabilized HeNe with a long-term stability of ~1 MHz (3×10−5 cm−1) over 8 h. Secondly, FS-CRDS is a single-mode cavity ring-down technique. Individual resonances of the fundamental transverse mode (TEM00) of the ring-down cavity are selectively excited, allowing for accurate, quantitative 145 measurements of the total cavity losses. During a spectral scan, the probe laser frequency is locked to successive longitudinal modes with ring-down time constants collected at each frequency step. This leads to an extremely stable, accurate, and linear frequency axis. We have recently demonstrated the capability to measure line positions with uncertainties less than 0.5 MHz (1.5×10−5 cm−1) [16], Doppler (Gaussian) widths to better than 1 part in 6,000 [21], and line intensities with relative uncertainties less than 0.3% [18,23]. Importantly, the FS-CRDS spectrometer is completely automated such that spectra can be taken over a broad spectral region without operator intervention [25]. The probe source was an external-cavity diode laser with a wavelength tuning range of 1.571.63 μm and an output power of 1518 mW. The ring-down cavity mirrors had nominal reflectivities of 99.997%, corresponding to a finesse of ~105,000 and an effective path length of ~25 km. The relative standard deviation of the measured ringdown time constant was ~0.05%, which for these mirrors and cavity length leads to a relative standard deviation in the absorption coefficient of 2.1×10−10 cm1. The cavity’s free-spectral range was determined to be 203.097(22) MHz via the procedure of Lisak and Hodges [26]. The FS-CRDS system stability was assessed through a measurement of the Allan variance [27], a measure which was originally derived as a performance metric for atomic clock standards. Figure 1 shows that as successive ring-down time constants are averaged the Allan deviation (square-root of the Allan variance) decreases roughly in proportion to nd−1/3, where nd is the number of decays in an averaging bin. This result differs from the frequently observed nd−1/2 dependence of Allan deviation on bin size. For a 1 s averaging time (corresponding to nd = 30 obtained at an acquisition rate of 30 Hz), the minimum detectable absorption (MDA) was 8×10−11 cm−1. After averaging ~3,000 ring-down time constants the Allan variance reaches a minimum, indicating the presence of system drifts on the ~100 s timescale. Averaging over 3,000 successive decay events leads to an MDA of 1.4×1011 cm1, which is a fifteen-fold decrease relative to the single-shot case. For the measurements discussed herein 300 averages were employed, corresponding to an MDA of 3.5×1011 cm1. 146 FIG. 1. Allan deviation  (square-root of the Allan variance) of the empty cavity losses as a function of the number of measured ring-down decays, nd, in an averaging bin. For nd < 3,000 the Allan deviation of these data is proportional to ~nd−1/3. Averaging for 1 s, which at an acquisition rate of 30 Hz corresponds to an average over 30 ring-down time constants, reduces the standard uncertainty in the absorption coefficient from 2.1×1010 cm1 to 8×1011 cm1. The minimum in the Allan deviation occurs at nd ~ 3,000 (100 s averaging time) and yields an MDA of 1.4×1011 cm1. Note that this Allan deviation corresponds to an ensemble of ring-down decay time measurements obtained at a single data-point in the absorption spectrum. Measurements were made on a sample of pure CO2 at natural isotopic abundance and a nominal pressure of 6.67 kPa (50 Torr). We measured the sample pressure with a NIST-calibrated capacitance diaphragm gauge having a relative standard uncertainty of less than 0.1% and a full-scale response of 133 kPa (1,000 Torr). The ring-down cavity cell wall temperature was measured with a NIST-calibrated 2.4 kΩ thermistor. Spectra were collected at room temperature (298.60299.22 K) with no attempt made to actively control the cell temperature. 7.3. Results and Discussion Measurements were made in the 6,2636,271 cm-1 wave number region (see Fig. 2). We selected this spectral region because it provides transitions from multiple isotopologues with comparable absorption strengths. This approach allows us to make the most precise measurements of the isotopic ratios [28,29,30], although one disadvantage is that 147 isotopologue transitions of comparable absorption typically have very different lower state energies and, therefore, different temperature dependencies. The uncertainty in the measured isotopic ratio, uδ, caused by the temperature uncertainty, uT, can be calculated as [29]: u  E  uT kT T (1) where ∆E′′ is the difference in lower state energies of the pair of transitions probed, T is the temperature, and k is the Boltzmann constant. It has been shown that the temperature uncertainty in these FS-CRDS measurements is ~28 mK [20]. For the 16O13C16O, 16 O12C18O, and 16O12C17O isotopologue transitions utilized in the present study (see Table 1), this temperature uncertainty leads to Type B (systematic) uncertainties in the isotopic ratio measurements of ~0.47‰, ~0.83‰ and ~0.70‰, respectively. Figure 2 shows a spectral survey of the 6,2636,271 cm-1 wave number region and a spectrum which was calculated using the Voigt line profile parameters found in the HITRAN 2008 database [12]. Approximately 170 transitions were observed in this survey representing the 16 O12C16O, 16 O13C16O, 16 O12C18O, 16 O12C17O, 16 O13C18O isotopologues with intensities ranging from 7.8×1026 to 3.6×1029 cm molec1. All observed transitions are in the HITRAN 2008 database with the exception of the 16 O13C16O (14411)←(00001) R38 transition. This transition is listed in the database of Perevalov et al. [31] with an intensity of S ~ 3.04×1028 cm molec.1. 148 FIG. 2. Survey of the 6,2636,271 cm1 wave number region. a) FS-CRDS measurement (symbols) and calculated HITRAN 2008 spectrum [12] (lines) using the Voigt line profile for a pure CO2 sample at natural isotopic abundance, a pressure of 6.67 kPa (50 Torr), and a temperature of 298.733 K. b) Corresponding stick spectrum of the four dominant isotopologues calculated through the use of HITRAN 2008. This spectral region was chosen to minimize the absorption difference between isotopologue transitions. Transitions utilized in the present study to determine isotopic ratios are marked by arrows. The 16O13C18O transition is unmarked but it is located at 6270.3543 cm−1 [12]. All measured transitions (including interfering transitions) were fit with a Galatry profile (GP) [32] which accounts for Dicke (collisional) narrowing [33]. The Doppler width was constrained to the theoretical value based upon the measured temperature and molecular mass, while the other parameters (e.g., transition area, Lorentzian width, 149 narrowing frequency) were floated during the resulting fit. Because the measured spectra exhibit collisional narrowing, we find that fitting the Voigt profile (VP) to our data incurs large, systematic residuals (see Fig. 3). Thus, use of the GP (as opposed to the commonly used VP) is critical for the present application. More specifically, the use of the VP also results in a measured area (and therefore CO2 number density) which is on average ~20‰ lower than with the GP, an error which is far greater than our experimental imprecision. However, the difference in transition areas measured using the two profiles is not identical for the isotopologue transitions considered here but rather varies between 10‰ and 40‰, thus, removing the possibility of a constant correction factor. This is not surprising as collisional narrowing parameters are known to exhibit J-dependence (e.g., [23,34,35]). In addition, this deviation is expected to be both pressure and temperature dependent. An example spectrum of the 16O12C16O R66 (30013)←(00001) transition (at natural isotopic abundance in a pure CO2 sample at 6.67 kPa and 298.929 K) and the corresponding Galatry fit is shown in Fig. 3. This transition is located at a wave number of 6,263.369778 cm1 with an intensity of 3.156×1026 cm molec.1 [12]. The signal-tonoise ratio (SNR, defined as the ratio of peak absorption to root-mean-square fit residuals) is 28,000:1, leading to a precision in the area measurement of 0.03‰. The highest SNR previously reported was ~10,000:1 by Casa et al. [34] who utilized an intensity-stabilized diode laser absorption spectrometer to probe transitions at  ~ 2.0 μm. Note that the transition shown in Fig. 3 is ~40,000 times weaker than the transitions probed in the Casa et al. study. 150 FIG. 3. a) Measurement and Galatry line profile (GP) fit of the 16O12O16O (30013)←(00001) R66 transition. Sample conditions are at T = 298.929 K, p = 6.67 kPa (50 Torr) for pure CO2 at natural isotopic abundance. This transition is located at wave number 6,263.369778 cm1 with an intensity of 3.156×1026 cm molec.1 [12]. b) Voigt line profile (VP) fit residual. All line profile parameters were floated except for the Doppler width which was fixed. The presence of collisional narrowing leads to a large systematic residual which is ~100 times greater than the instrumental noise level. c) GP fit residual. All line profile parameters were floated except for the Doppler width which was fixed. Note the difference in scale between the GP and VP fit residuals. The GP accounts for collisional narrowing leading to a signal-to-noise ratio of 28,000:1 with a precision in the area measurement of 0.03‰. The root-mean-square of the Galatry residual is 1×1010 cm1. The fitted areas determined using the two line profiles differ by 9‰, which is far greater than the measurement precision. Figure 4 shows an example spectrum of one of the doubly-substituted CO2 isotopologues, 16O13C18O, with a SNR of 20:1. The interfering transitions have been fitted and subtracted from the shown spectrum. This transition is located at 6,270.3543 cm1 with an intensity of only 3.588×10-29 cm molec.-1 at natural isotopic abundance [12]. The Type A (random) uncertainty of the measured area resulting from the Galatry fit is ~50‰, demonstrating the capability of FS-CRDS to make quantitative measurements of ultraweak absorption features. This transition is eleven orders of magnitude weaker than 16 O12C16O fundamental transitions in the mid-infrared and is one of the weakest 151 transitions to be quantitatively measured in the laboratory. Unfortunately, this precision pales in comparison to modern IR-MS measurements, which can achieve external precisions as high as 0.01‰ for this isotopologue through the use of large, preconcentrated samples and long averaging times (~24 h) [36]. FIG. 4. Measurement and Galatry fit of the 16O13O18O (30011)←(00001) P12 transition at T = 298.814 K and p = 6.67 kPa (50 Torr) of CO2 at natural isotopic abundance. This doubly-substituted isotopologue transition is located at a wave number of 6,270.3543 cm1 with an intensity of 3.588×1029 cm molec.1 [12]. In this background-subtracted spectrum the signal-to-noise ratio is 20:1. We found single-spectrum fit precisions of 0.11‰, 0.09‰ and 0.59‰ for the 13 12 C/ C, 18O/16O and 17O/16O ratios, respectively. These single-spectrum precisions are given by the inverse of the SNR of the fitted spectra and therefore do not include systematic uncertainties. They represent the highest precision which could be achieved with the present instrument, for transitions of the given line intensities, in the absence of confounding effects such as temperature uncertainties and interfering transitions. The corresponding uncertainties due to temperature can be calculated, as shown in Eq. 1. 152 Once added in quadrature with the shown single-spectrum fit precisions, the corresponding uncertainties are 0.35‰, 0.70‰ and 0.80‰ for the 13C/12C, 18O/16O and 17 O/16O ratios, respectively. The presence of interfering transitions, which leads to additional unconstrained fit parameters and a more weakly constrained area determination, can degrade the reproducibility of the isotope ratio measurement. In order to quantify the magnitude of this effect, several repeated measurements of the chosen isotopologue transitions were performed to measure the distribution of fitted areas. During these fits, the pressure broadening and collisional narrowing parameters were fixed to our previously measured values. For the 16O13C16O, 16O12C18O and 16O12C17O transitions the relative standard deviations of the ensemble of measured areas were 4‰, 0.5‰, and 10‰, respectively. These values were comparable to the estimated relative uncertainties in the fitted areas, yet much larger than the single-spectrum fit precisions. This indicates that uncertainties from interfering transitions, temperature effects, etalons etc. dominate the uncertainty in the spectrum area determination. For the 16O12C18O transition this uncertainty is lower than the previously estimated precision, indicating that neighboring transitions play a negligible role in this measurement. This was to be expected as all neighboring transitions are orders of magnitude weaker than the 16O12C18O transition in this spectral region. However, for the 16O13C16O and 16O12C17O transitions, it is clear that neighboring transitions significantly impact the reproducibility of the isotopic ratio. Thus, the dominant uncertainty in the 18O/16O measurement is due to temperature while for the 13 C/12C and 17O/16O measurements it is due to the presence of interfering transitions. This suggests two areas in which the precision of a future FS-CRDS-based isotope ratio measurement could be improved. Firstly, the introduction of active temperature control to minimize the temperature gradient is needed if the 18O/16O measurement is to be improved. For example, by reducing the temperature uncertainty to the few millikelvin-level, the uncertainty in the isotopic ratio measurement due to temperature, uδ, could be reduced below the single-spectrum fit precision. Secondly, by moving to a more optimal wavelength range, we could access transitions which are orders 153 of magnitude stronger (two orders of magnitude for  ~ 2.0 μm or five orders of magnitude for  ~ 4.3 μm). Distributed-feedback diode lasers are readily available at  ~ 2.0 μm and external-cavity quantum-cascade lasers (which operate without cryogenic cooling) are now available at  ~ 4.3 μm. By moving to  ~ 2.0 μm, the stronger transitions would allow us to utilize a lower pressure in our measurements, where the relevant isotopologue transitions would be significantly more isolated. This would eliminate the dominant uncertainty in the 13C/12C and 17O/16O measurements. In addition, the spectral SNRs for the 16O12C17O and 16O13C18O transitions would be significantly higher due to the increased intensities in this range. It is worth noting that this would not be the case for the 16O12C16O and 16O12C18O transitions where we are already utilizing nearly the complete dynamic range of the instrument. The use of the fundamental band at ~4.3 μm would lead to an additional three orders of magnitude increase in spectral intensity and allow for CO2 isotope ratios to be measured at natural abundance (i.e., ~385 ppm) with precisions similar to those predicted for pure CO2 at  ~ 2.0 μm. 7.4. Conclusions We employed frequency-stabilized cavity ring-down spectroscopy (FS-CRDS) on pure samples of CO2 to optically measure CO2 isotopic ratios at  ~ 1.6 μm. Although this spectral region is non-optimal, the sensitivity and stability of FS-CRDS led to singlespectrum fit precisions (based on spectrum SNR) of 8,900:1, 10,000:1 and 1,700:1 for the 13 C/12C, 18O/16O and 17O/16O ratios, respectively. 16O12C16O transitions were measured with SNRs as high as 28,000:1. We found that fitting Voigt profiles to the measured spectra significantly compromises the accuracy of the measured isotopic ratios. This is a result of collisional narrowing of the observed lineshapes; an effect which can be modeled using the Galatry profile. The fitted areas derived using Voigt and Galatry profiles differ by 10‰40‰ for the measured transitions, which is far greater than the experimental precision. We will focus future developments upon the  ~ 2.0 μm band. These transitions are about two orders of magnitude stronger than those discussed herein and are readily accessible through the use of distributed feedback diode lasers. With the 154 temperature stability demonstrated here, this should enable simultaneous measurements of 13C/12C, 18O/16O, and 17O/16O on pure CO2 samples with precision at the sub0.5‰ level. Acknowledgements David A. Long was supported by the National Science Foundation and National Defense Science and Engineering Graduate Fellowships. 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