SUBMILLIMETER WAVE/THZ TECHNOLOGY AND ROTATIONAL SPECTROSCOPY OF SEVERAL MOLECULES OF ASTROPHYSICAL INTEREST
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By Ivan R. Medvedev, M.S. ***** The Ohio State University 2005
Approved by Dissertation Committee:
Professor Frank C. De Lucia, Adviser
Professor Eric Herbst
Professor Thomas Humanic ______Professor Linn Van Woerkom Adviser Graduate Program in Physics ABSTRACT
A compact, high resolution spectrometer operating in the millimeter and submillimeter wave range has been developed based on concepts of signal detection and understanding of the molecular systems whose study is desired in numerous applications.
The novel FASSSTER (FAst Scan Submillimeter Spectroscopic Technique with
Electronic Reference) technique employs electronic frequency markers for frequency
calibration. The new frequency calibration scheme replaces the optical calibration used
by the existing FASSST spectrometer. Moreover, the small size, low power consumption,
and the potential of very low cost make this approach attractive for a number of important applications.
A number of improvements and modifications in the realm of software and hardware have been introduced to the existing FASSST spectrometer. The accuracy of the frequency calibration is now comparable to the accuracy of the spectrometers using more
traditional principle of PLL (Phase Lock Loop) frequency stabilization and source
modulation
The approach presented for the assignment of rovibrational spectra,
CAAARS(Computer Aided Assignment of Asymmetric Rotor Spectra), provides an
integrated tool to simplify and speed up spectral analysis. CAAARS combines visual,
interactive, mouse-assisted line assignment with real-time fitting of the assigned
transitions to the spectroscopic constants of an appropriate rovibrational Hamiltonian. ii The ease of operation and flexibility in the choice of an appropriate theoretical model make it a powerful tool in the hands of a spectroscopist. Its advanced user interface, capable of displaying multiple traces of the experimental spectrum for unambiguous line identification, is a logical extension of the Loomis-Wood approach to line assignment.
CAAARS implements user-specified sorting to select subsets of the predicted transitions for transparent assignment and manipulation. The current version of CAAARS is available for downloading at http://www.physics.ohio-state.edu/~medvedev/caaars.htm.
The room temperature millimeter and submillimeter wave spectra of diethyl ether
(C2H5OC2H5), oxiranecarbonitrile (H2COC(H)CN), and ethyl formate (HCOOC2H5) were
measured using FASSST (FAst Scan Submillimeter Spectroscopy Technique)
spectrometer. The vibrational ground states of the oxiranecarbonitrile, trans and gauche
conformers of ethyl formate, and trans-trans and trans-gauche conformers of diethyl
ether were analyzed. The results of this investigation can now be used for identification of diethyl ether, oxiranecarbonitrile, and ethyl formate in the interstellar medium.
iii
To my parents
iv ACKNOWLEDGMENTS
I wish to thank my adviser, Dr. Frank De Lucia, for providing me with a unique
opportunity to work in his research laboratory, for intellectual stimulation, support, and
research ideas, which this thesis is based on. His expertise and patience helped me to get through all the ups and downs of the graduate school. His tact, never-ending optimism and sincere desire to help people are exemplary. I am really proud and thankful for being a member of the Microwave group at the Ohio State University
I would also like to express my deepest gratitude Dr. Brenda Winnewisser and Dr.
Manfred Winnewisser for their support, guidance, and inspiration. The knowledge and plethora of scientific ideas they introduced to our group when they joined us in 1999 are simply invaluable. Their ideas and help were essential to a large fraction of my research.
Among many other things, they taught me how to be persistence in pursuing scientific goals and still have fun.
I thank Dr. Eric Herbst for theoretical discussions and all the help and support he provided over my years in the graduate school.
I would like to thank Dr. Markus Behnke for all the work we did together. His experimental skills and expertise in chemistry were priceless to our research. I am grateful to Markus and his wife Susanne for their friendship and help.
v I greatly enjoyed working with Dr. Paul Helminger during his summer visits. Our endless endeavors to improve the FASSST spectrometer led to an unprecedented quality of the nitric acid spectra.
I thank my colleagues and friends Dr. Zbigniew Kisiel, Dr. Doug Petkie, Dr. John
Hoftiezer, and Dr. Atsuko Maeda for their help and advice, and for being wonderful people to work with.
I am grateful to Andrey Meshkov, TJ Ronningen, Laszlo Sarkozy, David Graff and all other members of our group for being wonderful friends and colleagues.
The support stuff at OSU is excellent. Everyone in the front office, electronics shop, machine shop shop go out of there way to help graduate students.
And last but not the least I thank my beloved family - my dear wife, Anna Fisher, for her love and support; my parents Tatyana and Roman, and my uncle Nikolai Medvedev for their love, care, and faith in me.
vi VITA
December 5, 1975….. Born - Moscow, Russia 1995 – 1996………… Research Associate, Institute for Physical Problems, Russian Academy of Sciences, Moscow, Russia 1996………………… BA in General Physics and Mathematics 1996 – 1998………… Research Associate, Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, Russia 1998………………… MA in General Physics and Mathematics 1998 – 2000………… Graduate Teaching Associate, The Ohio State University 2000 – present………. Graduate Research Associate, The Ohio State University
PUBLICATIONS
Research Publication
1. Winnewisser, M., Medvedev, I. R., Sastry, K.V.L.N., Koput, J., Butler, R. A.H., De Lucia, F.C. & Herbst, E. (2005, in print). “The Millimeter- and Submillimeter-Wave Spectrum of Cyanoformamide.” The Astrophysical Journal Supplement Series.
2. Medvedev, I. R., Behnke, M. & De Lucia, F. C. (2005). “Fast analysis of gases in the submillimeter/terahertz with ‘absolute’ specificity.” Applied Physics Letters, 86, 154105.
3. Medvedev, I. R., Winnewisser, M., Winnewisser, B.P., De Lucia, F. C. & Herbst, E. (2005). “The Use Of CAAARS (Computer Aided Assignment Of Asymmetric Rotor Spectra) In The Analysis Of Rotational Spectra.” Journal of Molecular Structure, 742, 229-236.
4. Medvedev, I., Winnewisser, M., De Lucia, F. C., Herbst, E., Bialkowska- Jaworska, E., Pszczolkowski, L., & Kisiel, Z. (2004). “The millimeter- and
vii submillimeter-wave spectrum of the trans–gauche conformer of diethyl ether.” Journal of Molecular Spectroscopy, 228, 314-328.
5. Behnke, M., Medvedev, I., Winnewisser, M., De Lucia, F. C., & Herbst, E., (2004). “The Millimeter- and Submillimeter-Wave Spectrum of Oxiranecarbonitrile.” The Astrophysical Journal Supplement Series, 152, 97 – 101.
6. Medvedev, I., Winnewisser, M., De Lucia, F. C., Herbst, E., Yi, E., Leong , L., Bettens, R., Bialkowska-Jaworska, E., Desyatnyk, O., Pszczólkowski, L., & Kisiel, Z. (2003). “The Millimeter- and Submillimeter-Wave Spectrum of the trans-trans Conformer of Diethyl Ether (C2H5OC2H5).” The Astrophysical Journal Supplement Series, 148, 593 – 597.
FIELDS OF STUDY
Major Field: Chemical Physics
viii TABLE OF CONTENTS
ABSTRACT ………………………………………………………………………... ii ACKNOWLEDGMENTS ……………………………………………………….... v VITA ………………………………………………………………………………... vii LIST OF TABLES ………………………………………………………………… xiii LIST OF FIGURES ……………………………………………………………….. xv INTRODUCTION …………………………………………………………………. 1 CHAPTER 1. THE FASSST SPECTROMETER …………………………………. 6 1.1 BACKWARD WAVE OSCILLATORS …………………………… 7 1.2 RING CAVITY AND INTERFERENCE FRINGES ………………. 10 1.3 RING CAVITY ALIGNMENT PROCEDURE ……………………. 13 1.4 ABSORPTION CELL ………………………………………………. 16 1.5 DETECTORS AND PREAMPLIFIERS …………………………… 16 1.6 DATA ACQUISITION SYSTEM ………………………………….. 17 1.7 CALIBRATION PROCEDURE ……………………………………. 18 1.8 ANALYSIS OF THE EXPERIMENTAL SPECTRAL LINE SHAPE OF THE FASSST ………………………………………….. 20 1.9 FREQUENCY MEASUREMENT ACCURACY OF THE FASSST SPECTROMETER ………………………………………………….. 23 1.10 MODIFICATIONS AND ENHANCMENTS OF THE CALIBRATION SOFTWARE ……………………………………... 33 CHAPTER 2. THE USE OF CAAARS (COMPUTER AIDED ASSIGNMENT OF ASYMMETRIC ROTOR SPECTRA) IN THE ANALYSIS OF ROTATIONAL SPECTRA …………………………………………. 38 2.1 INTRODUCTION …………………………………………………...38 2.2 CURRENTLY USED ASSIGNMENT STRATEGIES …………….. 40 2.3 THE CAAARS CONCEPT …………………………………………. 42 2.4 CAAARS FEATURES AND STRUCTURE ………………………. 45
ix 2.5 ASSIGNMENT OF SPECTRA WITH CAAARS ………………….. 51 2.6 DISCUSSION ………………………………………………………. 56 CHAPTER 3. THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS-TRANS CONFORMER OF DIETHYL ETHER (C2H5OC2H5) …………………………………... 58 3.1 INTRODUCTION …………………………………………………...58 3.2 EXPERIMENTAL CONSIDERATIONS …………………………...60 3.3 SPECTRAL ANALYSIS …………………………………………… 62 3.4 DISCUSSION ………………………………………………………. 66 CHAPTER 4. THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS-GAUCHE CONFORMER OF DIETHYL ETHER ………………………………………………….. 68 4.1 INTRODUCTION …………………………………………………...68 4.2 ROTATIONAL SPECTRUM AND ANALYSIS …………………... 71 4.3 DISCUSSION ………………………………………………………. 77 CHAPTER 5. THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF OXIRANECARBONITRILE ……………………. 80 5.1 INTRODUCTION …………………………………………………...80 5.2 EXPERIMENTAL CONSIDERATIONS …………………………...83 5.3 SPECTRAL ANALYSIS …………………………………………… 84 5.4 DISCUSSION ………………………………………………………. 88 CHAPTER 6. THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS- AND GAUCHE- CONFORMERS OF ETHYL FORMATE …………………………………………….. 89 6.1 INTRODUCTION …………………………………………………...89 6.2 EXPERIMENTAL CONSIDERATIONS …………………………...93 6.3 SPECTRAL ANALYSIS …………………………………………… 94 6.4 DISCUSSION ………………………………………………………. 98 CHAPTER 7. THE SOLID STATE FASSSTER (FAst Scan Submillimeter Spectroscopic Technique with Electronic Reference) SPECTROMETER ………………………………………………….. 106 7.1 INTRODUCTION …………………………………………………...106 7.2 Voltage Tunable Oscillator (VTO) …………………………………. 109 7.3 VTO TUNING VOLTAGE SWEEPER ……………………………. 114
x 7.4 FREQUENCY REFERENCE ………………………………………. 117 7.5 DIRECTIONAL COUPLER AND MIXERS ………………………. 118 7.6 HARMONIC COMB GENERATOR ………………………………. 119 7.7 DETECTORS ……………………………………………………….. 120 7.8 FREQUENCY MULTIPLIER CHAIN …………………………….. 122 7.9 ELECTRONIC FREQUENCY MARKERS ………………………... 123 7.10 SHAPE ANALYSIS OF THE FREQUENCY MARKERS GENERATED BY THE FASSSTER SPECTROMETER …………. 126 7.11 DATA ACQUISITION SOFTWARE ……………………………… 129 7.12 CALIBRATION SOFTWARE ……………………………………... 130 7.13 FREQUENCY MEASUREMENT ACCURACY OF THE FASSSTER SPECTROMETER ...………………………………….. 133 7.14 SENSITIVITY OF THE FASSSTER SPECTROMETER …………. 136 7.15 DISCUSSION ………………………………………………………. 137 CHAPTER 8. THE FUNDAMENTAL AND REAL LIMITS TO SPECTROMETER SENSITIVITY IN THE SUBMILLIMETER SPECTRAL REGION ………………………………………………. 138 8.1 SOURCES OF NOISE ……………………………………………… 138 8.1.1 INTRODUCTION …………………………………………... 138 8.1.2 FLUCTUATIONS OF THE BLACKBODY RADIATION FIELD ……………………………………………………….. 141 8.1.3 SIGNAL FLUCTUATION LIMIT (PHOTON SHOT NOISE) ……………………………………………………… 144 8.1.4 THERMAL NOISE …………………………………………. 145 8.1.5 CURRENT SHOT NOISE ………………………………….. 145 8.1.6 1/f NOISE …………………………………………………… 146 8.1.7 AMPLIFIER NOISE ………………………………………... 146 8.1.8 OTHER SOURCES OF NOISE …………………………….. 149 8.2 DETECTION OF RADIATION ……………………………………. 149 8.2.1 SQUARE LAW DETECTORS ……………………………... 149 8.2.2 FIGURES OF MERIT FOR DETECTORS ………………… 151 8.2.2.1 SPECTRAL RESPONSE …………………………… 151 8.2.2.2 RESPONSIVITY …………………………………… 151
xi 8.2.2.3 DETECTIVITY AND NOISE EQUIVALENT POWER ……………………………………………... 152 8.2.2.4 FREQUENCY RESPONSE ………………………… 153 8.2.3 DETECTION SENSITIVITY LIMITED BY BACKGROUND FLUCTUATIONS ………………………. 154 8.2.4 DETECTION SENSITIVITY LIMITED BY SIGNAL PROCESSING ……………………………………………… 157 8.2.5 DETECTION SENSITIVITY OF A REAL SPECTROMETER ………………………………………….. 160 8.3 ABSORPTION CAUSED BY MOLECULAR ROTATION ………. 161 8.3.1 LINESHAPE CONSIDERATIONS ………………………... 162 8.3.2 POWER SATURATION …………………………………… 166 8.3.3 SUMMARY ……...…………………………………………. 169 APPENDIX A. FORMAT OF THE FILE SPECIFYING THE SORTING SCHEME USED BY CAAARS ……………………………………. 171 APPENDIX B. BLACKBODY RADIATION …………………………………….. 174 APPENDIX C. RADIATION FROM AN EXTENDED LAMBERTIAN SOURCE 177 APPENDIX D. RANDOM SIGNAL THEORY …………………………………… 180 APPENDIX E. TRANSITION PROBABILITIES AND ABSORPTION COEFFICIENT FOR A TWO LEVEL SYSTEM ………………….. 184 APPENDIX F. CALCULATION OF THE TRANSITION PROBABILITY USING TIME-DEPENDENT PERTURBATION THEORY ……… 187 APPENDIX G. CALCULATION OF THE DIPOLE MOMENT MATRIX ELEMENT FOR A LINEAR RIGID ROTOR MOLECULE………………………………….……………………... 189 BIBLIOGRAPHY …………………………………………………………………. 192
xii LIST OF TABLES
Table Page 1 Specifications of commercially available backward wave oscillators ………... 9 2 Assigned and fitted transition frequencies of tt-diethyl ether in the vibrational ground state …………………………………………………………………… 63 3 Spectroscopic parameters for tt-diethyl ether in the vibrational ground state ... 64 4 Predicted transition frequencies of tt-diethyl ether in the vibrational ground state …………………………………………………………………………… 67 5 Assigned and fitted transition frequencies of trans-gauche diethyl ether in its vibrational ground state ……………………………………………………….. 78 6 The experimental and the calculated spectroscopic constants for the two conformers of diethyl ether …………………………………………………… 79 7 Assigned and fitted transition frequencies of oxiranecarbonitrile in its vibrational ground state ……………………………………………………….. 85 8 Spectroscopic parameters for oxiranecarbonitrile in its vibrational ground state …………………………………………………………………………… 86 9 Predicted transition frequencies of oxiranecarbonitrile in its vibrational ground state …………………………………………………………………… 87 10 Assigned and fitted transition frequencies of the trans conformer of ethyl formate in its vibrational ground state ………………………………………... 100 11 Assigned and fitted transition frequencies of the gauche conformer of ethyl formate in its vibrational ground state ………………………………………... 101 12 Spectroscopic parameters for the trans conformer of ethyl formate in its vibrational ground state ……………………………………………………….. 102 13 Spectroscopic parameters for the gauche conformer of ethyl formate in its vibrational ground state ……………………………………………………….. 103 14 Predicted transition frequencies of the trans conformer of ethyl formate in its vibrational ground state ……………………………………………………….. 104 15 Predicted transition frequencies of the gauche conformer of ethyl formate in its vibrational ground state ……………………………………………………. 105
xiii 16 Specifications of the selected commercially available voltage tunable oscillators ……………………………………………………………………... 113
17 Accuracy of the transition frequencies of sulfur dioxide (SO2) measured with the FASSSTER spectrometer corresponding to different orders of the polynomial interpolation between frequency markers. The first column contains catalog frequencies of SO2 transitions taken from the JPL catalog [1]. Columns 2 to 6 contain differences between catalog and measured frequencies. Standard deviations of the residuals are at the bottom of the table 135
18 Minimum detectable number densities of CH3F, H2O, H2S, HNO3, OCS, and SO2 calculated for the following experimental conditions: temperature T = 290 K, electrical bandwidth BW = 10 kHz, optical bandwidth 1 THz, detector responsivity 1000 V/W, detector resistance10 kΩ , amplifier noise as shown in Figure 32, absorption cell length 1m. Only transitions below 1 THz were considered …………………………………………………………………….. 165 19 Possible branch name components that correspond to various combinations of sorting and naming instructions ………………………………………………. 173 20 Definitions of radiometric terms ……………………………………………… 174
xiv LIST OF FIGURES
Figure Page 1 Diagram of the FASSST spectrometer. The interference fringes are produced in the present set-up by a ring cavity arrangement ……………………………. 8 2 Alignment of the ring cavity of the FASSST spectrometer. The new alignment significantly reduces the feed-through of the cavity fringes into the signal channel. The spacing of the ring cavity markers is ~9.2 MHz ………………... 15 3 Simulated experimental line shape profiles of the FASSST spectrometer corresponding to sweep rates of the radiation source ranging from 5GHz/s (green trace) to 40Hz/s (blue trace). The frequency roll-offs used for this calculation were fLF = 10kHz and fHF = 30kHz. The FWHM of the original line shape (black trace) equals 500kHz. The FWHM of the first derivative line profile used for convolution was equal to 500KHz and corresponded to the sweep rate of 10GHz/s ………………………………………………………… 22 4 Comparison between FASSST and traditional PLL mm-wave spectra of trans- gauche diethyl ether, covering the expanded bandhead of the Ka = 14←13 Q- branch. The PLL recording has a somewhat higher resolution, although the FASSST lineshape is, on average, found to be only about 50% broader ……... 24 5 The accuracy of calibration of the FASSST spectrum as determined from (a) comparison of measured line frequencies from FASSST and PLL spectra of diethyl ether, and (b) comparison of FASSST frequencies with frequencies calculated from the best spectroscopic constants for the ground (open circles) and excited vibrational states (triangles) of trans-trans diethyl ether, the ground state of trans-gauche diethyl ether (squares), and SO2 (full circles). The shaded line in (b) denotes the average of the dependence obtained in (a) ... 26 6 Small-scale structure of the sweep rate for sweeps upwards and downwards in frequency, as displayed in traces (a) and (b), respectively. Average of up- down sweep peak positions minus down sweep peak positions (c). Average of up-down sweep peak positions minus up sweep peak positions (d). FASSST peak positions using sweep up in frequency only minus PLL peak frequencies (e). PLL peak positions minus FASSST peak positions using average of up- down sweeps (f) ……………………………………………………………….. 28
xv 7 Accuracy of the FASSST spectrometer frequency measurement for 2 different calibration strategies. Observed minus calculated frequencies of SO2 spectral lines for up-down averaged peak frequencies are presented in trace a). Only transitions of SO2 below 365 GHz were used to calibrate spectrum used in trace a). Systematic trends caused by atmospheric water line dispersion are clearly visible. Trace b) shows results of a new calibration, which supplements up-down averaging with fitting for the atmospheric relative humidity and correcting experimental frequency assignment. Markers c) indicate positions of the strongest spectral lines of water with their relative intensities …………. 30 8 Change of the index of refraction δn inside calibration ring cavity associated with presence of atmospheric water vapor was calculated for STP and 50% relative humidity using Liebe’s model ………………………………………… 32 9 CAAARS menu options that become available at Igor Pro start up …………... 45 10 Example of the file containing the sorting and branch naming instructions …... 46 11 “Select Branch” dialog window provides information about selected transitions and access to predicted and assigned branches. Figure 11-I: Activation of the selection of the predicted and assigned branches is achieved with the buttons “Select Predicted” and “Select Assigned”, respectively. Figure 11-II: Tab selection specifies branch which is used to center consecutive traces of the CAAARS assignment display. Figure 11-III: CAAARS can assist user in plotting Fortrat diagrams as well as intensity, error and energy plots. Figure 11–IV: Information about individual transitions of the selected branch is displayed in the bottom part of the CAAARS dialog .. 48 12 In this example the CAAARS assignment window displays eight different spectral traces of the millimeter wave spectrum of cyanoformamide, NCCONH2. The frequency axis refers to the bottom trace that has zero frequency shift shown on the right hand side of the display window. All commands and program options of Igor Pro are available for all traces. Figure 12-I – Navigation buttons help to search for a particular transition as well as modify the appearance of the assignment display. Figure 12-II – Cursors provided by Igor Pro are used for the assignment of experimental peaks to selected predicted transitions. Figure 12-III – Information about cursor positions assists user in locating the cursors ………………………………………….. 52 13 Molecular structure of tt-diethyl ether, and its orientation in the inertial axis system ………………………………………………………………………….. 62 14 Comparison of the geometries of the trans-trans and the trans-gauche conformers of diethyl ether. Both structures are given in the principal axis systems of inertia together with the orientation of the respective components of the permanent electric dipole moments …………………………………….. 69
xvi 15 Part of the millimeter-wave spectrum of diethyl ether. The top four traces show the theoretical frequency and intensity predictions for both trans-trans and trans-gauche conformer transitions, using the spectroscopic constants in Table 6 and the respective dipole moment components discussed in the text. The bottom trace represents the experimental spectrum ………………………. 70
16 Log10 (Intensity) versus frequency plot for some Q and R branches of trans- gauche diethyl ether b-type transitions. The second subscript number of the R ' branches represents (J″-Ka″-Kc″). The sensitivity of the FASSST spectrometer was calculated from the signal-to-noise ratio of SO2 reference lines …………. 73 17 An accidental near-coincidence between a Q branch of the trans-gauche conformer and a Q branch of the trans-trans conformer of diethyl ether. In the trans-trans Q branch each line consists of an unresolved asymmetry doublet, whereas in the trans-gauche Q branch each line represents an unresolved quartets arising from overlapped b- and c-type doublets. The two Q branches are characterized by considerably differing values of the Ka quantum number owing to rather different values of the A rotational constant in the two conformers. The upper plot shows the calculated line positions and intensities, while the lower trace displays part of the experimental spectrum …………….. 75 18 b- and c-type Q-branch transitions of the trans-gauche conformer of diethyl ether in the submillimeter-wave region. The upper plot represents the calculated line positions and intensities, while the bottom trace shows the experimental spectrum ………………………………………………………… 76 19 Computed molecular structure of oxirancarbonitrile, and its orientation in the inertial axis system …………………………………………………………….. 81 20 The geometries of the trans and the gauche conformers of ethyl formate. Both structures are given in the principal axis systems of inertia. The structures are based on the calculated geometries of Peng et al. and realized via the PMIFST program of Kisiel (http://info.ifpan.edu.pl/~kisiel/prospe.htm) ………………. 90 21 Diagram of the FASSSTER spectrometer. The frequency markers are produced by down-conversion of the VTO signal and mixing with the signal of the harmonic comb generator ………………………………………………. 108 22 Circuit design of the voltage sweeper used by the FASSSTER spectrometer … 114 23 A full cycle of the signals synthesized and digitized by the data acquisition computer: a) digitized up-down voltage ramp signal generated by the voltage sweeper; b) synthesized and digitized integrator input signal; c) synthesized integrator reset signal; d) digitized frequency marker signal ………………….. 116 24 Noise spectrum of the WR-3.4ZBD detector at 428 µW of detected power and 10 kHz – 1MHz of electrical bandwidth ………………………………………. 121 25 Detector signal as a function of frequency with radiation chopped at 500 Hz ... 123
xvii 26 Frequency markers produced by mixing of the 10.585 GHz frequency reference with the VTO’s signal followed by mixing with the harmonic comb spaced by 10 MHz. The frequency scale is multiplied by 24. The frequency marker corresponding to the reference frequency (10.585 GHz) is missing. The frequencies of every marker are calculated with respect to the missing marker, the frequency of which is known. The intensity is in arbitrary units .... 124 27 Frequency marker at the 24th harmonic of 10.355 GHz. The shape of the marker is determined by the low pass filter with high frequency roll-off set at 100 kHz. Intensity is in arbitrary units ………………………………………… 125 28 Simulated experimental shapes of two frequency markers produced by the FASSSTER spectrometer corresponding to two values of the phase shift ϕ0 between the VTO’s down-converted signal and the signal of the comb generator. The frequency roll-offs used for this calculation were νLF = 100 Hz and νHF = 100 kHz. The sweep rate of the VTO was set at 400 MHz/s ..……… 128 29 Peaks of the frequency marker signal array above the threshold equal to 250 (blue dots) are used to determine the positions of the marker centers ………… 132 30 Single scan of methanol spectrum recorded with FASSST and FASSSTER spectrometer. Traces a) and c) demonstrate the noise levels of the FASSST and FASSSTER spectrometer respectively. Trace b) and d) show the intensities of the strongest spectral lines ………………………………………. 136 31 Thermal detector exposed to a thermal background and to a shield radiation…. 143 32 Noise figure contours of Stanford Research Systems SR560 low noise preamplifier ……………………………………………………………………. 147 33 Equivalent circuit of an amplifier and a signal source ………………………… 147 34 Bolometer electrical circuit ……………………………………………………. 150 35 Power transmitted through the absorption cell ………………………………... 154 36 Dependence of αmin and SNmax on the source power …………………………... 159 37 Dependence of αmin and SNmax on detector the responsivity ……………….. 159 38 Dependence of αmin and SNmax on the detector resistance R …………………... 159 39 Dependence of the saturation power on the pressure inside the absorption cell (left figure), and on the absorption cell radius (right figure) for the rotational J= 40→ 41 transition of OCS in its vibrational ground state …………………. 169 40 Radiation source and detector relative placement ……………………………... 177 41 Radiation from a lambertian disk ……………………………………………… 177 42 Detector elements and focusing optics ………………………………………… 178
xviii 43 Two level system. Bul and Blu are the induced emission and absorption Einstein’s coefficients. Alu is the spontaneous emission Einstein’s coefficient. nu and nl are number densities of particles in the upper and lower states.……... 184
xix INTRODUCTION
The unique rovibrational spectral fingerprint inherent in every rotating polar molecule
and the strong maxima of their interaction strengths with electromagnetic radiation in the
millimeter and submillimeter / THz wave region [2] make this spectral region particularly
advantageous for the study of these molecules. Spectroscopic analysis of high resolution
submillimeter wave molecular spectra is used to obtain precise quantum mechanical
descriptions of molecular systems, while line shape analysis provides information about
intermolecular interactions. Additionally, the availability of such high resolution spectra
makes an unambiguous identification of molecular species possible. Millimeter and
submillimeter wave spectroscopy have long been used to identify and observe molecules present in the atmosphere and interstellar medium and the number of applications in which THz spectroscopy is used is still expanding.
Since the discovery of the first interstellar molecule (hydroxyl radical OH) in 1963 more than 100 molecules have been detected in the interstellar medium [3]. These molecules are most commonly found in ‘dense’ molecular interstellar clouds where they are shielded from optical and ultraviolet radiation. The hot cores of the interstellar molecular clouds which are associated with high-mass star-formation regions exhibit temperatures that rise to 100-300 K as stellar formation occurs. The rise in temperature is thought to contribute to the production of some complex organic molecules. The rotational spectrum of a molecule can be used to estimate its mass column density as well 1 as the absolute rotational excitation temperature. Both of these parameters are critical for
understanding of the chemical processes occurring in the interstellar medium. To account
for the effect of the thermal distribution within a molecular cloud upon the spectral line
intensity, the exact rovibrational energy level structure of a molecule must be known.
Thus, experimental measurements of the rovibrational spectrum of a molecule in the
laboratory, commonly carried out at a single sample temperature, are followed by a spectroscopic analysis intended to provide the quantum mechanical description of the
system allowing predictions of the spectral line intensity at an arbitrary temperature.
The spectrometer based on the FAst Scan Submillimeter Spectroscopy Technique
(FASSST) developed in recent years in this laboratory [4, 5] was used in the course of
this work to measure millimeter and submillimeter wave spectra of three organic
molecules which are thought to be present in detectable abundances in the interstellar
medium. The FASSST spectrometer provides the user with a very broad spectral
coverage, high sensitivity and high spectral resolution. The speed of measurements is
superior to any other measurement technique used in this frequency range. Chapter 1 provides an updated description of the FASSST spectrometer operation as well as an
account of the improvements and modifications introduced to the spectrometer in the
course of this work.
The FASSST spectrometer has proven to be very effective in conducting
spectroscopic measurements in a variety of spectroscopic studies [6-11]. A wide acceptance of the FAst Scan Submillimeter Spectroscopy Technique as a spectroscopic and analytical tool is precluded by the large physical dimensions of the original system
and by the high voltage and magnetic field requirements of the backward wave oscillator
2 radiation source employed by the FASSST spectrometer. The growth of technology in
this spectral region to support wireless communication, collision avoidance radar, and
other applications made possible the development of a compact all solid state system,
described in Chapter 7. This system uses an active multiplier chain to multiply the
frequency of the radiation generated by a voltage tunable oscillator (VTO) operating around 10 GHz. The electronic reference markers generated via mixing of the VTO’s signal with the signal of a harmonic comb generator are used for frequency calibration, hence leading to the acronym for the new technique - FASSSTER (FAst Scan
Submillimeter Spectroscopic Technique with Electronic Reference). The system developed in the course of this work has the potential to become the first low cost compact analytical tool in the millimeter and submillimeter / THz wave region.
The amount of spectral information contained in the millimeter and submillimeter
wave spectrum of complex polar molecules such as those mentioned here can be
overwhelming even for an expert spectroscopist. The high density of spectral lines, in
part caused by the presence of the transitions corresponding to excited vibrational states,
makes spectral analysis challenging and time consuming. The need for an automated tool
intended to assist a spectroscopist in assigning spectral lines to the transitions of an
appropriate quantum mechanical model was recognized early in the course of this work.
Such a tool, named CAAARS [12] (computer aided assignment of asymmetric rotor spectra), was first introduced during work on the rotational spectrum of the trans-trans
conformer of diethyl ether. It has now been enhanced with a Loomis-Wood type plotting
capability and was tested and improved while working on a variety of asymmetric rotor
3 spectra [9-11, 13]. Chapter 2 introduces the reader to CAAARS as well as the concepts
one has to deal with while assigning the rotational spectrum of an asymmetric rotor.
During the work on the new spectrometer design several molecules were studied, of the type for which FASSSTER is planned, using the improved FASSST spectrometer and
CAAARS.
Chapters 3 and 4 discuss the study of the vibrational ground state of the trans-trans
and gauche-trans conformers of diethyl ether (CH3CH2OCH2CH3) [9, 10] respectively.
Diethyl ether is thought to be present in the interstellar medium [14] because its simpler
sibling, dimethyl ether is found in high abundance in a number of interstellar sources.
Similarly, the simpler sibling to oxiranecarbonitrile (H2COC(H)CN), the
unsubstituted oxirane ring, is found in a variety of hot cores. Chapter 5 discusses the
spectroscopic study of its vibrational ground state [11]. Unlike diethyl ether and ethyl
formate, also studied in the course of this work, oxiranecarbonitrile is not commercially
available and had to be synthesized.
The fact that methyl formate, often labeled as “interstellar weed”, is found in a
number of hot cores justifies our interest in the rotational spectrum of the vibrational
ground state of the trans and gauche conformers of ethyl formate (HCOOC2H5) discussed
in Chapter 6.
The description of the novel spectrometer design presented in Chapter 7 would not be
complete without a discussion of the fundamental and real limits to spectrometer
sensitivity in the submillimeter wave region. Chapter 8 concludes the wide range of
4 topics covered by this work and discusses noise properties behind the design considerations for a spectrometer operating in the millimeter and submillimeter wave range.
5 CHAPTER 1
THE FASSST SPECTROMETER
The early developments of the FASSST (Fast Scan Submillimeter Spectroscopy
Technique) approach was reported in 1997 by Petkie et al. [4] and later by Albert et al. in
2001 [5]. Substantial advances in the FASSST spectrometer system have been made in
course of this work both in the realm of spectrometer and computer hardware and in the
calibration and spectroscopic software to facilitate data reduction and the assignment of
complex molecular spectra.
The FASSST technology makes use of backward-wave oscillators (BWOs) as
radiation sources for the millimeter- and submillimeter wave region. The intrinsic
property of a BWO is that its output frequency can be tuned via the electron acceleration
voltage over a wide range (10 to 100 GHz) in a very short time of ~1 s with a remarkably
high spectral purity, thus freezing out most frequency instabilities. The BWOs used in
our experiment were produced by ISTOK Research and Development Company of
Fryazino, Moscow Region, Russia.
A block diagram of the current FASSST spectrometer is shown in Figure 1. The essential elements are: a) a radiation source (BWO) placed in the field of a strong permanent magnet or electromagnet; b) power supplies for BWO and magnet; c) an aluminum
6 absorption cell 6 m in length and 15 cm in diameter; d) a reference-gas cell with a length
of 60 cm; e) two beam splitters and a cavity that produces the interference fringes which will be converted into frequency markers by the calibration procedure; f) two InSb hot- electron detectors, one for the signal channel and one for the reference channel; g) a
sweeper unit, which produces the electron acceleration voltage applied to the BWO slow-
wave structure; h) an amplifier with adjustable frequency roll-off; and i) a data acquisition system with computer. We will discuss the characteristics and properties of some of these spectrometer constituents.
1.1 BACKWARD WAVE OSCILLATORS
A set of three BWOs operating in the frequency ranges from 108 to 190, 189 to 252,
260 to 295 GHz and 297 to 366 GHz is routinely used to acquire millimeter and
submillimeter wave spectra with the FASSST spectrometer. A complete list of
commercially available BWO tubes is presented in Table 1. BWOs operating in the
frequency range below 190 GHz are air cooled and come pre-assembled with a
permanent magnet. The higher frequency tubes are mounted in an electromagnet
adjustable from 0.5 T to 1.1 T, depending on the requirements of each tube. The output
waveguide has a horn antenna attached to it for collimation of the radiation. A set of
lenses made of Teflon and/or polyethylene is used to improve the collimation of the
microwave beam.
7 Magnet Mylar beam splitter 1 Glass rings used to suppress reflections Lens Reference BWO InSb detector 1 gas cell Lens Aluminum cell: length 6 m; Length ~60 cm diameter 15 cm Path of microwave radiation Mylar beam
splitter 2 Reference channel preamplifier InSb
Preamplifier roll-off Frequency detector 2
Interference fringes Spectrum Signal channel
8 Stepper motor Ring cavity: L~15 m Data acquisition Stainless steel rails system Computer Slow wave structure sweeper Trigger channel /Triangular waveform channel Filament voltage power supply High voltage power supply
Figure 1: Diagram of the FASSST spectrometer. The interference fringes are produced in the present set-up by a ring cavity arrangement.
Frequency Power Power Wave- BWO MHz Magnetic range min. typ. Voltage/V guide Cooling model /Volt field/T /GHz /mW /mW type
OB-69 33-55 12 350-1200 25 WR-22 Air OB-70 52-79 12 400-1200 33 WR-15 Air OB-71 78-119 6 500-1500 41 WR-10 Air OB-76 128-142 20 1100-1500 35 WR-6 Air OB-861 118-178 6 500-1500 60 WR-6 Air OB-241 177-263 1-10 20-50 1000-4000 28 WR-10 Water 0.7 1 9 OB-30 258-375 1-10 10-20 1000-4000 39 WR-10 Water 0.8 OB-321 370-535 1-5 4-15 1000-5000 41 WR-10 Water 1 OB-801 530-714 1-5 4-15 1500-6000 40 WR-15 Water 1 OB-81 690-850 1-5 4-15 1500-6000 35 WR-15 Water 1 OB-82 790-970 1-3 3-10 1500-6000 40 WR-15 Water 1 OB-83 900-1100 1-3 3-10 1500-6000 44 WR-15 Water 1.1
1This tube is mounted and used as radiation source of the FASSST spectrometer.
Table 1: Specifications of commercially available backward wave oscillators.
The choice of the operating parameters of the FASSST spectrometer, such as sweep rate and the bandwidth of the signal processing channel, is determined by the response times of the system components. The main sources of noise are fluctuations in the BWO output frequency that are caused in part by the 60 Hz modulation introduced by the AC filament current, and in part by mechanical vibrations of the cavity or by any perturbation with a time constant smaller than the time of the sweep across a single spectral feature.
The frequency source is operated in a free-running mode. By applying a triangular voltage wave form to the electron acceleration electrodes of the BWO, the output frequency can be swept rapidly. The short term frequency stability is ~10 kHz. Thus, spectral absorption features with a width of 500 kHz (Doppler width) are not significantly broadened. The intrinsic spectral purity of BWOs and their high voltage sensitivity require an electronic filtering of the electron acceleration voltage to obtain a residual noise of less than 2 mV. This filtering is carried out in the sweep generator. A sweep rate of 10 to 20 GHz/s freezes the inherent low frequency fluctuations in the BWO’s output frequency on the time scale of the sweep across a single absorption line. Furthermore, a high sweep rate results in a signal recovery frequency that largely eliminates 1/f noise and the need for a superimposed high modulation frequency and lock-in detection.
1.2 RING CAVITY AND INTERFERENCE FRINGES
In order to achieve an accurate frequency calibration of the individual absorption lines in the FASSST spectra, we use the well-predicted line positions of a reference gas, which is usually sulfur dioxide contained in the second absorption cell as shown in Figure
10 1. The known frequencies of the reference gas lines used in the calibration procedure
enable us to determine the frequency spacing between adjacent cavity interference fringes
(cavity markers) and their absolute frequency positions. The frequency of each individual interference fringe is calculated from a count of cavity markers, and interpolation between the positions of adjacent fringes relative to the position of an unknown absorption line yields the calculated line position.
In order to enhance the experimental flexibility, a ‘compact’ Fabry-Perot cavity was designed and assembled (see Figure 1). The new cavity design consists of two vertical aluminum mirror support plates spaced approximately 3 m apart. Each mirror support carries 6 copper mirrors and is mounted on a movable platform which can slide along two solid stainless steel rails. The diameter of each copper mirror is 15 cm and its radius of curvature 3 m. This arrangement of the new mirror system provides flexibility in the cavity configuration. The initial configuration of 11 traversals resulted in a cavity length of 33 m and a spacing of ~5 MHz between adjacent interference fringes. The problem with this setup is, however, that part of the radiation inside the cavity could couple back into the spectrometer system at the beam splitter, thus, traveling all the way back to the radiation source, reflecting and entering the spectral channel.
The problem of the power reflections was resolved by averaging consecutive scans and incrementally changing the total length of the cavity between sweeps. A stepper motor was attached to one of the aluminum mirror support plates to change the spacing between the plates by ~1 µm, which changes the frequency positions of the fringes by
~100 kHz at 300 GHz. The co-addition of scans significantly reduces the effective fringe feed-through.
11 In order to further reduce the feed-through of the cavity fringes into the signal
channel, the alignment mode of the mirrors was changed to form a ring cavity (see Figure
2). In this configuration the wave front of the radiation coupled into the cavity can only
travel in one direction through the beam splitter. If the ring cavity is well aligned, the
system does not exhibit any feed-through of the cavity fringes into the signal channel.
The cavity marker spacing increased to ~9.2 MHz. No significant change in the
calibration accuracy was observed. The length of the ring cavity is still routinely
incrementally changed between consecutive scans to prevent cavity fringe signal from
propagating into the spectrum signal.
Absorption of radiation by atmospheric water strongly affects the sharpness of the
ring cavity markers. The effect is so strong that measurements are almost impossible
around the water line at 556.94 GHz where attenuation of the microwave radiation is
~10000 db/km, which leads to a very low quality factor of the cavity. To reduce this
effect and make measurements above 500 GHz possible, a temporary enclosure, made out
of Mylar®, was build around the ring cavity. This enclosure is now routinely purged with
dry nitrogen to reduce the humidity inside the ring cavity.
A more reliable and air-tight version of the calibration cavity enclosure is currently
being built. In the new version only one mirror, attached to a movable support, will be
stepped using a stepper motor. This arrangement will reduce possible oscillations caused by stepping the entire six mirror support implemented in the current version. The new placement of the calibration cavity with respect to the rest of the FASSST spectrometer
will enhance the efficiency of coupling of the electromagnetic radiation into the cavity.
To ensure air-tightness each outer surface plane of the enclosure containing the new ring
12 cavity enclosure will be fitted with a Mylar® sheet stretched across an aluminum frame,
which will have rubber seals along its perimeter.
1.3 RING CAVITY ALIGNMENT PROCEDURE
Ring cavity alignment procedure involves three major steps.
First a small flat optical mirror (~1/2 inch is size) is attached to the center of every
cavity mirror to accomplish a rough alignment of the cavity using a laser. A laser pointer is held by hand at the center of mirror #12 and aimed at the center of mirror #1 (see
Figure 2). Alignment of mirror #1 is adjusted to aim the reflected laser beam at mirror #2.
The laser pointer is then moved to mirror #1 and aimed at mirror #2. These steps are
consecutively repeated for the remaining mirrors until the desired alignment was
achieved. After that, this approach is used for the rough alignment of the beam splitters
by placing the laser pointer at the output of the BWO and two optical mirrors at the
centers of the beam splitters.
The rough optical alignment if followed by the initial microwave alignment done
using the radiation produced by OB24 backward wave oscillator tube (see Table 1). The
oscillation frequency of the BWO was set in the operating range of the WR-3.4ZBD
Schottky diode detector (220-325 GHz) produced by Virginia Diodes [15]. The diode detector block has a horn antenna for collecting and coupling of the radiation into the input waveguide leading to the detector element. The horn antenna is similar to the one used for collimation of radiation at the output of the BWO tubes. It consists of a conical horn gradually tapered into a rectangular waveguide which matches the size of the input waveguide of the detector block. The entrance diameter of the horn antenna is ~3/4 inch, 13 which allows us to probe the wave front profile produced by the microwave radiation
source rather accurately. The radiation of the BWO is chopped using an optical chopper.
This allows us to monitor the relative magnitude of the detected signal using an
oscilloscope.
The diode detector is placed consecutively at the center of every optical element in
the microwave radiation path starting at the BWO tube. The preceding optical element
was aligned to achieve the maximum strength of the detected signal. By doing this we
were able to observe relatively broad (~5 MHz) ring cavity markers by sweeping the
BWO tube.
The third precision stage of alignment of the cavity mirrors is carried out by
monitoring the strength of the cavity markers (interference fringes). The frequency of the
BWO is swept over a narrow frequency range. The signal of the InSb bolometer placed at
the detector coupling hole (see Figure 2) is monitored using an oscilloscope. As many as
eight iterations of the alignment of each mirror are necessary to achieve the highest
quality and highest signal to noise ratio of the markers. The resulting FWHM of a cavity
marker at 240 GHz was ~300 kHz. Thus, the quality factor Q equals ~8× 105 .
In order to suppress higher order spatial modes inside the calibration cavity, a ring made of microwave-absorbing material is routinely attached to one of the mirrors. The inner diameter of the ring matches the size of the lowest-order spatial mode supported by the ring cavity. Since mode size gets smaller at higher frequencies, different rings are
used with different BWO tubes.
14 6 7
2 11
4 5
8 3
12 1 Detector
15 coupling hole 10 9
Path of microwave Mylar beam Radiation splitters Magnet
BWO
Lens
Figure 2: Alignment of the ring cavity of the FASSST spectrometer. The new alignment significantly reduces the feed-through of the cavity fringes into the signal channel. The spacing of the ring cavity markers is ~9.2 MHz. 1.4 ABSORPTION CELL
In order to facilitate heating and cooling of the gas cell, the previously used glass absorption cell was replaced with a 6 m long, 15 cm diameter aluminum pipe with Teflon windows. The insertion of glass rings (see Figure 1) into the aluminum cell substantially reduces reflections on the cell wall and thus smoothes the baseline signal. The present version of the absorption cell is heated with 8 heating tapes wound around the aluminum tube, which allows us to reach maximum temperature of ~200C. The uniformity of the temperature is currently controlled with appropriate settings of 4 variable transformers (2 tapes per transformer). A new version of temperature control hardware, which is currently being developed, will consist of 10 semi-cylindrical heating elements, placed along the absorption cell. Each heating element is capable of providing 1000 Watts of heating power. The cell temperature will be maintained by means of continuous heating and cooling with liquid nitrogen and controlled by digital temperature controllers.
1.5 DETECTORS AND PREAMPLIFIERS
Two hot electron InSb bolometers cooled to liquid 4He temperature were used as detectors. With a bandwidth of 1 MHz such a detector is capable of reproducing ~5x105
Doppler limited spectral features per second. The Doppler line width is ~500 kHz at 300
GHz for diethyl ether. This translates into 250 GHz/s maximum scan rate based on the detector specifications. However, the typical scan rate implemented is 10 to 20 GHz/s.
This results in a detector signal peaking at ~20 kHz.
16 Each InSb bolometer is connected to a QMC low noise preamplifier. For the spectrum channel this preamplifier is followed by a SRS SR560 amplifier with adjustable band pass. The low frequency roll-off is typically set to decrease the baseline variations and eliminate most of the 1/f noise. This selection transforms the Doppler limited (Gaussian) line shape function of the absorption signal into an approximate first derivative. The high frequency roll-off, typically set at 30 kHz, decreases the high frequency noise. For the reference channel the QMC preamplifier is followed by a SRS SR560 amplifier with a band pass of DC to 1 MHz.
1.6 DATA ACQUISITION SYSTEM
A dual processor workstation (Athlon 1.8 GHz, 2 Gbytes RAM) is currently used to control the experiment. The dual CPU configuration allows us to acquire and process large quantities of data simultaneously. For example, 400 scans over a 60 GHz spectral interval at a resolution of 10 data points/MHz produce ~2 Gbytes of data files. We employ a DVD writer to routinely backup the recorded spectra. A digitizer (NI PCI-6110) provides, in addition, independent variable gain control for four available channels. The digital output of the board is used to control the stepping motor for changing the length of the ring cavity between consecutive scans.
Data acquisition is triggered by a signal from the high voltage BWO sweeper unit.
The data acquisition system digitizes the input from four channels: 1) signal (spectrum) voltage, 2) reference voltage (cavity fringes), 3) voltage ramp signal and 4) voltage trigger signal. The spectrometer data acquisition software is written in Labview [16] and
17 has several operational modes. For example, if the software is operating in the averaging mode it is capable of enabling the stepper motor to change the cavity length. All acquired data for each individual scan or sweep are saved as individual files for later processing by the calibration software. The frequency-voltage function depends critically on the thermal environment of the BWO. Thus, it was found mandatory to calibrate each sweep separately before the individual sweeps could be co-added. Additionally, the latest version of the data acquisition system allows us to record spectra both up and down in frequency, providing a marked improvement in the accuracy of the measured line frequencies. It should be noted that any scan rate variations of the BWO tubes alter the shapes and widths of all detected spectral lines and shifts their positions. This effect is discussed in more detail in what follows.
1.7 CALIBRATION PROCEDURE
The calibration software, like an earlier version which was discussed in [4], is based on the Igor Pro© package [17], which provides a convenient graphical and programming interface. In the averaging mode, the software automatically calibrates each individual scan and co-adds all calibrated scans.
The following steps are undertaken to calibrate individual data files. Trigger channel information is used to identify the margins of the sweep. The initial estimate of the frequency range is obtained from the data in the voltage ramp channel and the stored
BWO frequency–voltage characteristics. The spectral channel data are then convolved with a first derivative Gaussian line shape to reduce the baseline signal and to transform
18 the signal of the spectral line shape into an approximate second derivative line profile.
The calibration lines, usually SO2 frequencies and intensities taken from the JPL catalog
[18], are identified in the spectrum by a cross correlation technique. This information is then used together with the fringe peak positions in a least-squares fit to calculate just two parameters, the starting frequency of the scan and the cavity marker spacing. In the last step, the calibration software calculates the frequency position for each data point in the spectrum data array.
The data averaging mode has become the standard for recording spectra with the
FASSST spectrometer. The feed-through of the interference fringe signal into the spectrum channel is greatly and effectively reduced by averaging scans with different settings of the cavity length. The double benefit of averaging is an improved signal-to- noise ratio of the final spectrum and reduced fringe feed-through in comparison with one single scan recorded with the earlier version of the FASSST system.
The latest modification of the calibration software was introduced to overcome the effect of atmospheric water vapor inside the ring cavity (see Figure 1). The index of refraction (see [19, 20]) inside the calibration cavity is strongly affected around water absorption lines present in the operational frequency range of the FASSST spectrometer.
The fringe intervals are therefore not constant in frequency in these regions. To account for apparent frequency shifts caused by this phenomenon the FASSST spectrometer calibration software was modified to perform fitting for water vapor number density together with the starting frequency of the scan and cavity marker spacing. This will be described in detail in Section 1.8.
19 1.8 ANALYSIS OF THE EXPERIMENTAL SPECTRAL LINE SHAPE OF THE FASSST
The FASSST spectrometer makes use of a free-running BWO tube swept by a triangular voltage wave form to record spectra with later optical calibration which involves matching of experimental peaks with well-predicted line positions of a reference gas. Any nonlinearity in the frequency-voltage characteristic of the radiation source [21] will result in a varying time of sweep through individual spectral features, thus, affecting the Fourier content of the electrical signal reaching the frequency roll-off preamplifier in the signal channel (see Figure 1). Since the frequency roll-off limits of the preamplifier are fixed for the duration of a scan, the signal at the output of the spectrometer will strongly depend upon matching between preamplifier bandwidth and the instantaneous sweep rate of the radiation source. For the purpose of this calculation we will assume that the spectral line shape has a Gaussian profile, which holds for the Doppler limited case.
Additionally, since the detectors used by the FASSST spectrometer have ~ 1 MHz of electrical bandwidth one can safely assume that if the time of sweep through a spectral feature is longer than 1 µs, the detector will not distort the experimental line shape.
To within a normalizing factor the electrical signal at the output of the detector S(t) is given by
2 SR ln 2 2 −−()tt0 ()δν /2 2 St()= e , (1.1) where t0 is the time when the frequency of the radiation source is equal to the center frequency of the spectral line, SR is the instantaneous sweep rate of the radiation source,
δν is the FWHM (full width at half maximum) of the spectral line, and t is time. To 20 derive equation 1.1 we assumed that the frequency of the radiation source ν is a linear function of time,
ν = SR() t−+ t00ν , (1.2) where ν0 is the frequency corresponding to the center of the spectral line.
The filter response function representing the roll-off amplifier can be expressed as a product of a high pass response function and a low pass response function
f f 1+ i LF 1− i f f Hf()= HF , (1.3) c 2 ⎛⎞f ⎛⎞fLF 1+ ⎜⎟1+ ⎜⎟ ⎝⎠f ⎝⎠fHF where f is the frequency of the electrical signal at the input of the roll-off preamplifier, and fLF and fHF are electrical frequencies corresponding to the low and high frequency limits of the roll-off preamplifier respectively. The Fourier content of the electrical signal at the output of the detector is thus modified according to equation 1.3 and is recorded by the ADC of the data acquisition board.
Further line shape processing happens at the calibration step via convolution of the digitized signal with the first derivative of the Gaussian line profile. This, as was described earlier, reduces the baseline signal and transforms the signal of the spectral line shape function into an approximate second derivative line profile. According to the convolution theorem [22], the Fourier transform of a convolution is the product of the
Fourier transforms of the individual functions. In order to simulate the experimental line shape of the FASSST spectrometer, the following steps can thus be undertaken:
1) The Fourier transform of the electrical signal at the output of the detector (1.1)
must be multiplied by the filter response function (1.3). 21 0.5
0.4 Second derivative of the original line
0.3 Lineshape corresponding to the sweep rate of 5 GHz/s
0.2
Lineshape corresponding 0.1 to the sweep rate of 40 GHz/s
0.0
-0.1
-0.2
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Frequency relative to the center of the line / MHz
Figure 3: Simulated experimental line shape profiles of the FASSST spectrometer corresponding to sweep rates of the radiation source ranging from 5GHz/s (green trace) to 40Hz/s (blue trace). The frequency roll-offs used for this calculation were fLF = 10kHz and fHF = 30kHz. The FWHM of the original line shape (black trace) equals 500kHz. The FWHM of the first derivative line profile used for convolution was equal to 500KHz and corresponded to the sweep rate of 10GHz/s.
2) The resulting expression must be multiplied by the Fourier transform of the first
derivative of the Gaussian profile, corresponding to a fixed (average) line
width and scan rate.
3) The inverse Fourier transform of the resulting expression must be calculated. The
real part of the complex expression obtained corresponds to the simulated
experimental line shape.
22 The supporting calculations were carried out using Mathematica® software package
[23]. The results of these calculations are presented in Figure 3.
The analysis of the results presented in Figure 3 reveals that broadening as well as frequency shift of the peak position of the experimental line profile increases as the sweep rate changes towards higher values. Both of these effects are strongly influenced by the roll-off amplifier settings. Thus, if the sweep rate of the radiation source is not maintained constant one can expect a degrading effect of the sweep rate variation upon the accuracy of frequency measurements of the FASST spectrometer.
1.9 FREQUENCY MEASUREMENT ACCURACY OF THE FASSST SPECTROMETER
Acquisition of extensive rotational spectra of diethyl ether, discussed in more detail
Chapters 3 and 4, with two different spectrometers employing the same radiation sources allowed us to conduct a precision instrumental comparison. In particular, it was possible to benchmark the performance of the FASSST spectrometer against a spectrometer using the more traditional principle of PLL (Phase Lock Loop) frequency stabilization and source modulation. The latter instrument is located at the Institute of Physics, Polish
Academy of Science, Warszawa, Poland [9, 24-26].
23
Figure 4: Comparison between FASSST and traditional PLL mm-wave spectra of trans- gauche diethyl ether, covering the expanded bandhead of the Ka = 14←13 Q-branch. The PLL recording has a somewhat higher resolution, although the FASSST lineshape is, on average, found to be only about 50% broader.
Some useful information can be gleaned from Figure 4, which presents a direct comparison of the same spectral segment recorded with the two spectrometers. It is apparent that the PLL spectrometer has somewhat higher resolution which is, in principle, only limited by the Doppler profile of the absorption lines. The FASSST line shape is visibly broader, primarily due to the filtering and signal convolution made in the detection channel. In spite of these additional factors, it is found that the FASSST line shape is, on average, only about 50% broader than that of the PLL. In the densest part of the spectrum in Figure 4 the broader FASSST line shape results in more pronounced
24 blending, while in the case of more isolated lines it should not affect the precision of frequency measurement, especially since the FASSST spectrum is recorded at a very good signal-to-noise level.
The second issue is the accuracy of transition frequencies derived from the FASSST spectrum. This is addressed in Figure 5, where external calibration of the FASSST frequencies has been made in two different ways. Figure 5 (a) is a plot of simple frequency differences between frequency measurements of the same lines in FASSST and PLL spectra. This plot results from correlation of raw peak finder outputs and rejection of evident mismatches. Some of the apparent noise and the remaining outliers are attributable to differences in blending of lines in spectra from the two spectrometers.
Nevertheless a general clear trend of systematic frequency deviations is visible. In order to confirm that there is no effect coming from the PLL frequencies, a second calibration is shown in Figure 5 (b).
This calibration is based on the assumption that frequencies of identified lines in the range of Figure 5 are expected to be calculable to better or much better than 0.10 MHz, since precise spectroscopic constants have already been determined for several of the involved spectroscopic species. Thus, a difference plot between the FASSST and the predicted frequencies would be expected to parallel the plot in Figure 5 (a), as is indeed found to be the case. The plots in Figure 5 can therefore be regarded as an absolute frequency calibration of the FASSST spectrum of diethyl ether over a frequency span of almost 6 GHz, and allow for much useful insight. The root mean square deviation in
Figure 5 (a), with inclusion of the outliers, is calculated to be 0.125 MHz, about twice that found for the entire data set, substantiating the assumption of 0.10 MHz normally
25 made for the accuracy of FASSST line frequencies. Interestingly the plots in Figure 5 are not centered on zero but have an average positive deviation of 0.06 MHz. The error waveform is also seen to be a rather long range one, with a period in the region of 0.5
GHz and, in the worst but relatively rare cases, the deviation from the true frequency may exceed 0.3 MHz. On the other hand, while locally the frequency deviations have a systematic nature, these are expected to become statistical for the large data sets normally extracted from FASSST spectra. It is clear that a factor of two improvement in the
FASSST calibration procedure would take the frequency accuracy of the FASSST spectra to a level hitherto thought to be possible only with PLL techniques.
Figure 5: The accuracy of calibration of the FASSST spectrum as determined from (a) comparison of measured line frequencies from FASSST and PLL spectra of diethyl ether, and (b) comparison of FASSST frequencies with frequencies calculated from the best spectroscopic constants for the ground (open circles) and excited vibrational states (triangles) of trans-trans diethyl ether, the ground state of trans-gauche diethyl ether (squares), and SO2 (full circles). The shaded line in (b) denotes the average of the dependence obtained in (a).
26 The systematic trends shown in Figure 5 suggest that it is possible to improve the accuracy of the frequency measurement of the FASSST system. As a first step, we employed a statistical analysis and individually calculated the error distributions and the rms deviations of the PLL data and the FASSST data in the fit of the trans-gauche diethyl ether (see Chapter 4). We found for both data sets an error distribution close to Gaussian with 0.052 MHz and 0.079 MHz rms deviations, respectively, roughly consistent with our
2:1 accuracy estimate. This indicates that the spectral region displayed in Figure 5 probably exhibits some of the largest deviations in the measurements of the present
FASSST data set. It is thus a good region for tests of the efficacy of measures to eliminate the type of errors discussed above.
The frequency dependence of the systematic trends displayed in Figure 6c, 6d, and 6e is similar to the reproducible small-scale [4] variation in the frequency-voltage characteristics of the BWO which are reproduced in Figure 6a and b. It suggests that the sweep rate variation needs to be considered more carefully. Because of time delays and phase shifts in the signal processing, the peak positions of the experimental line shapes which approximate second derivatives, are shifted in time. If all lines, including the calibration lines, have the same line width, and if the sweep rate is constant during the sweep, the delay occurring in both the calibration lines and the spectroscopic lines to be measured is equal. In order to test this hypothesis, we determined the actual sweep rates in up and down frequency sweeps, shown in Figure 6a and 6b. There exists an expected similarity between the up and down sweep rates as a function of frequency. The small difference between the sweep rate excursions in Figure 6a and 6b can be explained by the difference in the thermal history of the BWO in the up and down sweep
27
Figure 6: Small-scale structure of the sweep rate for sweeps upwards and downwards in frequency, as displayed in traces (a) and (b), respectively. Average of up-down sweep peak positions minus down sweep peak positions (c). Average of up-down sweep peak positions minus up sweep peak positions (d). FASSST peak positions using sweep up in frequency only minus PLL peak frequencies (e). PLL peak positions minus FASSST peak positions using average of up-down sweeps (f).
modes. Our test (diethyl ether) data-set consisted of the co-addition of 100 scans up in frequency and 100 scans down in frequency. With the traditional solution of averaging the peak frequencies obtained from the up and down sweeps, we obtained the deviations that are plotted in Figure 6c and 6d. When this is done, Figure 6f results, showing that the systematic shifts displayed in Figure 5 and Figure 6e have been eliminated (rms deviation
~34 kHz) when the up and down averaged line positions of the FAAAST data are subtracted from the PLL data.
28 As a result of the critical analysis of the frequency measurement accuracy of the
FASSST spectrometer, we now routinely record and co-add scans both up and down in frequency, and then average the peak positions from the two co-added spectra.
This significant improvement in calibration accuracy allowed us to observe another systematic trend in frequency measurement accuracy previously obscured by the scatter caused by the effect of the variation in the radiation source sweep rate (see Figure 7a).
When up-down averaged frequencies of SO2 spectral peaks were compared with their catalog [1] frequencies (see Figure 7a) it became clear that in order to further improve calibration accuracy, the atmospheric water dispersion effect must be taken into account when calibrating spectra taken with the FASSST spectrometer.
The index of refraction inside the calibration ring cavity is strongly affected around water absorption lines present in the operational frequency range of the FASSST spectrometer. To account for frequency shifts caused by this phenomenon we used the empirical model derived by Hans J. Liebe [27]. We only consider dispersion effects caused by the presence of water vapor since the dry air contribution is smaller by several orders of magnitude. In his model Liebe takes into account contributions of the 30 strongest spectral lines of water in the frequency range below 1THz. Figure 8 shows the change of the refraction index caused by atmospheric water vapor calculated for STP
(standard temperature and pressure) using this model. There is clearly a strong correlation between Figure 7 a) and Figure 8.
29 a) b) 0.4 c)
0.2 /MHz ∆ν
0.0
-0.2
310 320 330 340 350 360 370 380 Frequency/GHz
Figure 7: Accuracy of the FASSST spectrometer frequency measurement for 2 different 1 calibration strategies. Observed minus calculated frequencies of SO2 spectral lines for up-down averaged peak frequencies are presented in trace a). Only transitions of SO2 below 365 GHz were used to calibrate spectrum used in trace a). Systematic trends caused by atmospheric water line dispersion are clearly visible. Trace b) shows results of a new calibration, which supplements up-down averaging with fitting for the atmospheric relative humidity and correcting experimental frequency assignment. Markers c) indicate positions of the strongest spectral lines of water with their relative intensities.
The change of the refraction index is strongest in the vicinity of the water line frequencies. Due to the width of water lines at atmospheric pressure (FWHM ~ 5 GHz) and the slow drop-off of the pressure-broadened Lorentzian line profile, the refraction index is significantly affected even in regions distant from water line frequencies. The contribution of the water line at 556.936 GHz, which is almost 50 times stronger than the380.197 GHz line, increases the refraction index over the entire frequency range
1 Frequencies of the calculated transitions are available at http://spec.jpl.nasa.gov/ 30 presented in Figure 8. Thus, inclusion of the 30 strongest spectral lines of water in the frequency range below 1 THz into the model improves the overall frequency accuracy significantly.
The FASSST spectrometer calibration software matches spectral lines in the spectrum channel to those of the reference gas spectrum and assigns a fractional cavity fringe index to each matched line by interpolating between indices of the closest adjacent cavity fringes. The new calibration procedure then fits pairs of frequencies and assigned fractional marker indices to the function given by Equation 1.4, which allows us to
th calculate frequency of the N (N can be fractional) calibration cavity marker fN as follows
ffN0 +δ ⋅ fN = , (1.4) 1(,,,)+δ nfn hTP
where f0 is the starting frequency of a scan, δf is the cavity marker spacing, and the change of the index of refraction δn is a function of frequency fn, relative humidity h, temperature T and pressure P. This fit now has therelative humidity h, together with the starting frequency of the scan f0, and cavity marker spacing δf as free parameters. The frequencies of every data point in the spectrum are calculated using Equation 1.4 and fractional fringe index N assigned to every data point. The results of the improved calibration procedure are presented in Figure 7b. For the frequency region shown in
Figure 7 the rms deviation of frequency changed from 137 kHz for trace a) to 43 kHz for trace b). As one can see, frequency deviations associated with atmospheric water vapor dispersion are now virtually eliminated.
31
5
-6 4 n=n-1)/10 δ
( 3
2
310 320 330 340 350 360 370 380 Frequency/GHz
Figure 8: Change of the index of refraction δn inside calibration ring cavity associated with presence of atmospheric water vapor was calculated for STP and 50% relative humidity using Liebe’s model [27].
In an attempt to further improve the accuracy of frequency measurement we investigated the effect of changing the interpolation scheme between frequency markers.
Second and third order interpolations did not lead to any significant improvement which indicates that the remaining measurement uncertainty is dominated by the uncertainty of the reference gas frequencies used for calibration (~30-50 kHz).
It will not be an overstatement to say that the accuracy of the FASSST spectrometer frequency measurement is now comparable to, if not better than, the accuracy of the PLL technique and is mainly limited by the accuracy of the reference gas line frequencies.
32 1.10 MODIFICATIONS AND ENHANCMENTS OF THE CALIBRATION SOFTWARE
FASSST spectrometer performance depends strongly upon the algorithms used for calibrating experimental spectra. Most of the hardware improvements described earlier in the text led to modifications and enhancements of the calibration procedure. Since the calibration package evolved greatly since it was first introduced [4, 21] an updated description of its operation will provide the basis for understanding the underlying logic and help in extending its capabilities. The algorithm used for calibration of spectra recorded with the FASSST spectrometer consists of five main parts.
1) The first part is responsible for loading experimental spectra from files created
with data acquisition software. An additional flag is now added to the header
of the recorded data file by the data acquisition software to distinguish up and
down in frequency files. Once the data file is loaded by calibration software
this flag is analyzed and if necessary the loaded data arrays are inverted. After
that the trigger channel is used to identify the margins of the spectrum to be
processed. Also, rough frequency limits of the loaded spectrum are calculated
using voltage channel information and the updated frequency/voltage curve of
the BWO tube used to record the selected spectrum.
2) The main purpose of the second part of the calibration package is to identify
and extract usable pieces of the loaded spectrum and cavity fringe arrays. Also,
the convolution the spectrum with the first derivative of the Gaussian line
profile takes place at this stage of the calibration process.
33 3) At the third step of the calibration, peaks in the spectrum and cavity fringes
arrays are identified numerically. Also, the user can now specify an intensity
cut-off for spectral peaks. Every peak with intensity above cut-off will be
erased from the peak list. This feature is necessary to eliminate saturated lines
from being used by the calibration software, since line centers for such lines
can not be identified with sufficient accuracy. Additionally, every point of the
spectrum array is assigned a fractional index by interpolating between indices
of the closest cavity markers. This index array is then used at later stages of
calibration. Calibration software now allows a user to select a first, second or
third order interpolation scheme for creating this index array.
4) The fourth part of the calibration calculates the starting frequency of the scan f0
by cross correlating the experimental spectrum with the spectrum created from
the catalog line positions and intensities of the reference gas. The success of
this operation strongly depends upon the initial guess of the cavity marker
spacing, which is held constant at this stage of calibration. Manual calculation
of the marker spacing might be necessary after any realignment of the
calibration cavity. It is done by recording a spectrum of reference gas, visually
identifying 2 spectral lines spaced by frequency ∆f, and using the following
equation
BS δ f = CM ∆f , (1.5) BSSL ⋅ N
where δf is the cavity marker spacing, BSCM is the bin spacing between peaks
of two selected cavity markers, BSSL is the bin spacing between the peaks of
34 the selected spectral lines and N is the number of cavity marker intervals
separating the selected cavity markers.
5) The last part of the calibration procedure performs assignment of the reference
gas spectrum to the peaks of the experimental lines followed by the least
squares fitting for relative humidity h together with the starting frequency of
the scan f0 and cavity marker spacing δf as described in the previous section.
Room temperature and pressure are kept constant, while fitting, and can be set
to actual experimental values by the user. If necessary, the relative humidity h
can also be kept constant.
The assignment is performed automatically by choosing the strongest peak
within the user-specified frequency “window” surrounding the selected
reference line position. This straightforward assignment can be supplemented
and checked by the intensity analysis step, which involves calculation of the
ratio of catalog and experimental intensities. Assigned transitions that do not
fall into the region surrounding the peak of the intensity ratio distribution are
eliminated from the assigned line list. By iteratively limiting the size of the
frequency “window”, performing line assignment, and fitting, successively
more accurate values of h, f0 and δf are calculated. The fact that averaging of
the up and down in frequency scans can only be performed after each of the
scans is calibrated justifies the selection of the final frequency “window” size
used for line final assignment. It was demonstrated earlier that the rms
deviation of the frequency differences before up-down averaging is equal to
125 kHz (see Figure 5). The final frequency “window” size used for calibrating 35 spectra was chosen to be 150 kHz. This guaranties that lines are not rejected
due to the frequency uncertainty caused by variation in the sweep rate of the
radiation source.
The array containing frequencies corresponding to every data point of the
spectrum intensity array is created at the very last step of the calibration
sequence according to Equation 1.4.
The software stores the parameters used for calibrating the last loaded spectrum.
Thus, in order to perform co-addition the user must calibrate interactively the first scan of a data set. To initiate the co-addition sequence, the user can specify several parameters:
(1) the minimum total number of assigned lines, (2) the minimum percentage of the total frequency range containing all of the assigned lines, and (3) the maximum value of the second derivative of the array containing bin indices of the calibration cavity marker peaks. These parameters are used to monitor the quality of the calibration. If the actual value of either of the first two checks parameter is lower than the user-specified value, the spectrum is not used for co-addition. The third parameter serves as a measure of the uniformity of the bin spacing between fringes of the calibration cavity. Any discontinuity in the sequence of cavity markers registers on the graph of the second derivative of the array containing indices of the calibration cavity marker peaks. Thus, if the actual value of the third parameter exceeds the user-specified value this spectrum is also eliminated.
On average the new calibration algorithm accepts about 95 to 100 percent of the files recorded for co-addition.
It has become standard to record multiple (~200) scans up and down in frequency, co- add them separately and then to perform up-down averaging of the experimental peak
36 positions. The latter step is done by means of a standalone procedure which was written using Igor Pro programming language. At first, the peaks in up and down scans that are closest in frequency are paired. The algorithm described below is used to verify the correctness of the matching between the up and down peaks.
1) The up and down spectra are linearized in frequency such that frequency
spacing between adjacent data points is uniform.
2) Pieces of the experimental spectrum (usually ~30MHz) surrounding the paired
peaks are extracted and cross-correlated using functions provided by Igor Pro.
Upon completion of the cross-correlation an array of length (2N – 1) is created,
where N is the number of elements in each extracted piece. This array contains
the results of the cross-correlation.
3) The positions of the maxima of the correlated array are calculated. If the peak
closest to the center of the correlated array is also the strongest peak of this
array, then the matching between the up and down peaks is successful.
Otherwise the pair is eliminated from the peak list.
4) The up-down averaged frequency is then calculated from the up and down
peak lists and placed into a separate array.
This algorithm has proven to be quite effective. The majority of the rejected peaks correspond to various baseline features or blended and very weak spectral lines.
37 CHAPTER 2
THE USE OF CAAARS (COMPUTER AIDED ASSIGNMENT OF ASYMMETRIC ROTOR SPECTRA) IN THE ANALYSIS OF ROTATIONAL SPECTRA
2.1 INTRODUCTION
A modern day spectroscopist has access to a broad range of experimental techniques
[2, 4, 19, 22, 28-30] to record high-resolution spectra in the millimeter, submillimeter, infrared and visible wavelength ranges, all of which are capable of producing large amounts of information in a relatively short period of time. For example, the room temperature rotational spectrum of cyanoformamide NCCONH2 [13], which will be used as an example in this chapter, acquired with a FASSST spectrometer [4], [9], contains tens of thousands of spectral lines distributed over several hundred gigahertz.
The assignment of such high resolution spectra to rovibrational transitions involves an iterative, interactive search for matching sets of spectral features which can be represented by some mathematical model (i.e., effective Hamiltonian) of the dynamics of the molecular system studied. The search involves handling large data arrays, which consist of the full set of recorded resolution elements, the list of identified absorption/emission features with their positions and intensities, and sets of quantum numbers uniquely identifying a molecular transition between two discrete energy levels.
38 This process requires experience, but remains time-consuming even for the expert.
Various interactive assignment programs or program packages have been developed over the last twenty years. However, many spectroscopists have remained skeptical of the usefulness of such methods. One justified reason for skepticism about their usefulness is that the first generation of such programs displayed line positions, but could not usefully display the actual spectrum simultaneously with the assignment schemes. This suppression of the full information in the original spectrum required frequent interruption of the assignment process to refer to the spectrum. Furthermore, the most widely used assignment programs employ very simple theoretical models for the identified series of spectroscopic transitions, generally based on a polynomial expansion in J(J + 1) of the energy levels, where J is the overall rotational quantum number, or even more simply, the
Ritz combination principle. Therefore, currently a second stage of data processing must be carried out separately to achieve the association of the full set of quantum numbers with each frequency and the reduction of the assigned transitions to the parameters of a rovibronic Hamiltonian, generally using customized stand-alone programs or software packages. Iteration of the assignment, fitting and prediction steps must be carried out until a desired level of assignment and fitting accuracy is achieved.
Modern tabletop personal computers can complete fitting and prediction of transition frequencies and intensities very fast. Thus, the assignment step, which requires informed human input, clearly is the rate-determining part of the process. Until the assignment of spectra becomes fully automated (see, for example, [31]) the spectroscopist will have to look to interactive programs to simplify and partially automate the assignment process as much as possible.
39 We believe that the assignment technique developed in the course of this work should succeed in satisfying spectroscopists, by exploiting, even enhancing, the full information content of a spectrum and accelerating dramatically the processes of pattern recognition, association of an assigned transition with its full set of quantum numbers, least-squares- fitting to a rigorous model, and the important step of error evaluation.
2.2 CURRENTLY USED ASSIGNMENT STRATEGIES
Any interactive assignment scheme is a way of facilitating pattern recognition. Line positions, intervals between lines, and line intensities are the elements of the patterns sought. The ability to distinguish these patterns strongly depends on the scheme for graphically depicting the spectral information in the assignment procedure. If one draws the rovibrational spectrum with a resolution of 1 MHz/mm (roughly one line width per mm, in the submillimeter wave range), 100 gigahertz (a large portion of a rotational or rovibrational band) of the spectrum will stretch 100 meters in length. Often, one is trying to assign spectral series spread over such a range. The adjacent elements can not be conveniently displayed at the same time. A powerful approach to circumvent this problem is to plot several traces showing the positions of candidate features, one above the other, offset in frequency by an amount equal to the spacing of adjacent elements in the expected spectral series. This approach, combined with fitting of spectral series under consideration to a polynomial function, first introduced by Loomis and Wood in 1928
[32], was used sporadically with hand-drawn plots up through 1968, when computer programs using plotters or just line printers began to be exploited to make the diagrams,
40 including some indication of the line intensities [33-35]. The advent of personal computers led inevitably to the development of interactive Loomis-Wood assignment programs at the Giessen laboratory [36] and elsewhere [37-41]. These programs have proven to be highly effective tools for the assignment of rovibrational spectra [42-45].
The programs of that generation use line positions and intensities to identify any spectral series of transition frequencies which can be represented in the form of a power series in
(J + 1), where J is the overall rotational quantum number. Though the model is strictly valid only for a linear or symmetric top molecule, for asymmetric top molecules it works well for J-sequences of transitions with all but the lowest values of Ka (or Kc). However, all these programs lack the ability to identify all of the quantum numbers or to convert the output of the “Loomis-Wood” program to the input file format of a general asymmetric rotor fitting program such as SPFIT [18], ASFIT [46] or AWAT/SWAT [47]. The approach recently taken by Urban et al. [48] extends the pattern recognition capability of the Loomis-Wood algorithm, but still relies on a simple polynomial fit. Additionally some of these existing programs still use a DOS-based interface which limits resolution, making the display of the full experimental information in a multiple-trace display virtually impossible.
The most complete assignment package currently available, to our knowledge, is that of Moruzzi [49, 50]. This package, RITZ, defines series with symmetric top quantum numbers plus a distinction of asymmetry split components, and is oriented to finding series of energy levels, which it fits with polynomial expansions, like the Loomis-Wood programs, or with an asymmetric rotor Hamiltonian (Watsonian), in its newest version. It exploits the Ritz combination principle and parabolic extrapolation to find new levels and
41 thus transitions, has several options for plotting candidate lines and even a Loomis–Wood option. It has limited generality, however, and is not easy to use for ground state pure rotational spectra, nor can it parameterize resonances.
Streamlined procedures based on the traditional assignment methods have been implemented in various laboratories [46], [51], [18].
A modernization of the Giessen Loomis-Wood program was recently implemented using Igor Pro [52]. This is worthy of mention because it pointed out the useful strategy of using the graphics tools and data management methods offered by multipurpose software packages such as Igor Pro to consider a new approach to the assignment task.
2.3 THE CAAARS CONCEPT
In a step towards matching the experimental capabilities of the FASSST spectrometer
[4], [9], in its realm of recording spectra, the CAAARS (computer aided assignment of asymmetric rotor spectra) package has been developed for the data reduction of rotationally resolved molecular spectra. The CAAARS package combines display of the original spectrum on the computer screen, in addition to peak positions, at any desired resolution, either in a single trace or in multiple traces (Loomis-Wood display), with real- time calculations –fitting of line positions and prediction including relative intensities – using the full Hamiltonian appropriate for the particular molecular system, for as many states as needed.
The flexibility of CAAARS makes it relevant for pure rotational, rovibrational or rovibronic band spectra. The predictions are rigorously applicable for linear, symmetric
42 or asymmetric top molecules, diatomics of any desired Hund’s case, with and without resolved hyperfine structure, in excited rovibronic states, and with or without accidental resonances, with or without a quantitative model of the dynamics of the molecule. This flexibility is largely due to the choice of the SPFIT and SPCAT programs for the analysis of the data [18]. Any state or set of states that can be described by SPFIT can be assigned interactively with CAAARS. However, CAAARS can be coupled with any other analysis program for molecular systems not adequately represented by SPFIT/SPCAT. The package has been optimized so far for asymmetric rotors (thus the acronym), but extending it comfortably to other systems is relatively trivial, as will be seen below. The package is implemented on the basis of the Windows version of Igor Pro® [17] and currently uses SPFIT and SPCAT [18] without modification.
The main features of CAAARS are listed below.
• The experimental spectrum and list of peak positions are read, stored, and
automatically displayed upon opening every subsequent assignment session.
• The output of the prediction routine SPCAT [18], including line positions and
intensities with all relevant quantum numbers, calculated with values of
parameters for the fully rigorous relevant model Hamiltonian, can be loaded by
CAAARS with a mouse-click.
• Predicted and assigned transitions are sorted according to a user-defined sorting
scheme which provides the organizing principle for the manipulation of the
information used and generated by the program. Each subset or series of
transitions, referred to as a branch in what follows, corresponds to the same
43 values of the sorting parameters and is given a unique name based on the sorting
rules loaded at the startup of an assignment process.
• The complete experimental spectrum or an excerpt at any desired resolution,
markers at the peak positions determined from that spectrum, and markers for the
predicted and assigned transitions with their intensities are superimposed in a
common trace. Multiple traces, as many as desired, with centers corresponding to
the calculated frequencies of consecutive elements of a selected branch, can be
stacked in a single display providing a Loomis-Wood diagram [36]. The
maximum number of traces displayed is equal to the number of elements in the
selected branch.
• The assignment is carried out interactively using a cursor operated by a mouse,
with various options such as have proved efficient in a Loomis-Wood program,
providing functions emulating the steps followed in the assignment procedures of
master spectroscopists. The capabilities of Igor Pro can be fully exploited to
provide more flexibility than in previous assignment programs.
• All transitions, in all assigned branches, are fitted simultaneously at a mouse click
to the parameters of the rovibrational Hamiltonian using SPFIT [18] in the
background. A new set of predictions is generated automatically following each
fit and displayed on the screen.
• Access to each of the arrays active in the “experiment” (the entire assignment and
fitting process for a given spectrum) is available, and individual branches, or
errors, may be plotted without exiting the assignment program.
44 2.4 CAAARS FEATURES AND STRUCTURE
CAAARS consists of two interacting program parts. The part dealing with data manipulation and assignment as well as the CAAARS dialog operation was written in
C++ and compiled with Microsoft Visual Studio under Windows XP. The second part, encompassing procedures to initialize and display the data used by CAAARS, was written in the Igor Pro programming language. After installation both parts are automatically loaded and become available when Igor Pro is started.
Figure 9: CAAARS menu options that become available at Igor Pro start up.
Listed below are the most significant workflow steps in its operation.
2.4.1. Once Igor Pro is launched the CAAARS menu shown in Figure 9 becomes
available. The first menu item “Load Sort File” (see Figure 9) is also the first option
that must be selected. It loads the sorting scheme, a text file which contains sorting
and branch-naming instructions used for managing predicted and assigned transitions.
45 For an asymmetric rotor, this means sorting according to transition type (a, b and c) and up to six user-specified sum/difference combinations of quantum numbers according to any desired sorting hierarchy. The contents of a sorting scheme file generally used for the case of the rotational spectrum of a near-prolate asymmetric top such as cyanoformamide are shown in Figure 10. The short text gives the code specifying the sorting hierarchy, and the branch naming conventions. It is convenient to have several files containing sorting instructions set up for one or more given types of molecules, so that one can easily switch from one to the other. The simplicity of the sorting strategy is one component that enhances the flexibility of the entire
CAAARS package, and the branch structure in the organization of both the predicted and assigned lists of lines is essential for moving through the data in the display and for inspection of the data lists that are always instantly accessible.
Figure 10: Example of the file containing the sorting and branch naming instructions.
A full description of how to write a sorting scheme file is given in the Appendix
A.
46 2.4.2. By selecting the second menu item “Load Predictions” (see Figure 9) the user
loads the predictions produced by SPCAT. After the predicted transitions are loaded
they are sorted into branches according to the sorting scheme specified in the file
loaded in the previous step. CAAARS then sorts the predicted branches in descending
order of the maximum intensity within the branch and places the result in a separate
table to assist the user in selecting the most intense and therefore promising branch
for assignment. This feature is particularly helpful in the beginning of the assignment
process, in progressing through a reasonable sequence of branches in terms of ease of
assignment.
2.4.3. Upon choosing the option “Select Branch” (see Figure 9) the user activates the
dialog window shown in Figure 11, which is the work-horse of CAAARS. It allows
the selection of the branches to be marked as predicted and assigned. It controls the
specifications for updating the assignment display and provides information about
currently selected subsets of transitions. It also provides buttons for the most common
operations.
47
Figure 11: “Select Branch” dialog window provides information about selected transitions and access to predicted and assigned branches. Figure 11-I: Activation of the selection of the predicted and assigned branches is achieved with the buttons “Select Predicted” and “Select Assigned”, respectively. Figure 11-II: Tab selection specifies branch which is used to center consecutive traces of the CAAARS assignment display. Figure 11-III: CAAARS can assist user in plotting Fortrat diagrams as well as intensity, error and energy plots. Figure 11–IV: Information about individual transitions of the selected branch is displayed in the bottom part of the CAAARS dialog.
2.4.4. In the upper part of the dialog window (see Figure 11 – I) the user can select one
assigned branch and one or two predicted branches to work with simultaneously.
Selection of “all” assigned or predicted transitions is also possible (see Figure 11 – I).
Activation of the selection of the predicted and assigned branches is achieved with
the buttons “Select Predicted” and “Select Assigned”, respectively. Additionally, if
the “Create Branch Waves” checkbox is marked (see Figure 11 – III) the intensity,
frequency and selected quantum number arrays will be created for the branch
specified by tab selection (see Figure 11 – II). This feature is intended to simplify
plotting of Fortrat and intensity diagrams which have proven to be extremely helpful
in the early stages of the assignment process [9].
48 Clicking on one of the three branch tabs (Figure 11 – II) implements a second
level of selection. At this point information about individual transitions in a selected
branch, defined by tab selection (see Figure 11 – II), becomes available in the lower
part of the dialog (see Figure 11 – IV). Branch name, frequency of the selected
transition within the branch, its quantum numbers and the frequency interval to the
next and previous transitions are displayed. The use of Igor Pro cursors allows any
other transition to be detailed with the “Get Cursor” button.
2.4.5. Once the “Update Display” button is pressed (see Figure 11), CAAARS displays
the single or multi-trace graph of intensity versus frequency shown in Figure 12,
analogous to a Loomis-Wood diagram. Each trace contains the experimental
spectrum and conveniently color-coded markers for all the experimental peaks,
assigned experimental transitions, and calculated primary and secondary branches.
The centers of the ascending traces correspond to the frequencies of the subsequent
transitions at higher point index in the selected branch (see Figure 11 - IV). This
approach makes possible an easy identification of the series of experimental
transitions corresponding to the predicted branch. Features of this display used in the
assignment process will be discussed in the next section.
2.4.6. Menu option “Sort Assigned” performs sorting of the assigned transitions
according to the selected sorting algorithm. This feature may be occasionally
necessary, when the assigned line list has been changed by hand, since under normal
operating conditions sorting is done automatically as soon as each new line is
assigned.
49 2.4.7. Upon completion of a set of assignments, or when new predictions are needed, the
remaining options in the CAAARS menu are relevant. The “Save Line File” feature
saves the assigned line list in a text (ASCII) file, sorted by branches, in the format
used by SPFIT.
2.4.8. To achieve proper weighting of blended lines in SPFIT, the assigned line list must
be sorted by frequency. The feature “Save Line File (Sorted in Frequency)” saves the
assigned line list, sorted according to increasing frequencies, in the format used by
SPFIT.
2.4.9. The menu option “Run Fit” saves the assigned list and executes a batch file which
runs both the fitting and prediction programs, SPFIT and SPCAT.
2.4.10. For convenience CAAARS stores all file paths once they have been selected.
“Reset File Names” clears stored path information so that selection of a different set
of files becomes possible (another vibrational state, another conformer, an
impurity…).
2.4.11. For safety reasons the “Save Assigned” option offers saving of the assigned
transitions in a text file. It is prudent to use this at regular intervals, especially in the
learning stage.
2.4.12. “Load Assigned” loads assigned transitions in the format of those saved at the
previous step.
2.4.13. The option “Set Default Experimental Error” is used to define the default
accuracy of the experimental measurements. The experimental accuracy of any
individual transition can be modified manually.
50 2.4.14. When a second vibrational state or species is to be assigned in the same spectrum,
the lines assigned to the first species can be flagged with markers color coded for
“Previously Assigned”.
2.4.15. To close a session, the entire “experiment” can be saved in Igor Pro file format.
Reopening of the experiment brings up the display and active arrays, including the list
of assigned lines, at the state they were in when it was closed.
2.5 ASSIGNMENT OF SPECTRA WITH CAAARS
The actual assignment process is initiated by displaying the spectrum with choices as specified in Section 2.4.4. After assigned, primary and secondary branches are selected, by clicking on the “Update display” button the user instructs CAAARS to draw a display as illustrated in Figure 12, which corresponds to the choices shown in Figure 11. This display corresponds to the “Number of traces” parameter specified in the “Select Branch” dialog. Each trace contains the experimental spectrum, and superimposed on it color- coded stick spectra corresponding to the positions and intensities of the experimental peaks, as well as the assigned, primary and secondary branch transitions (see figure legend added at the lower left of Figure 12). In Figure 11 - I, under “Assigned Branch”,
“all” has been checked, so all assigned lines are marked. The amplitude scale is automatically adjusted to the number of traces requested, maintaining the visibility of whatever amplitude range had been previously set. The frequency axis at the bottom of
51 52
Figure 12: In this example the CAAARS assignment window displays eight different spectral traces of the millimeter wave spectrum of cyanoformamide, NCCONH2. The frequency axis refers to the bottom trace that has zero frequency shift shown on the right hand side of the display window. All commands and program options of Igor Pro are available for all traces. Figure 12-I – Navigation buttons help to search for a particular transition as well as modify the appearance of the assignment display. Figure 12-II – Cursors provided by Igor Pro are used for the assignment of experimental peaks to selected predicted transitions. Figure 12-III – Information about cursor positions assists user in locating the cursors. the display corresponds to the frequencies of the bottom trace, while its center corresponds to the frequency of the transition specified in the “Select Branch” dialog (see
Figure 11 - IV). For the case demonstrated in Figure 12, the center frequency is that of the 26th element of the ‘b,p,R,7,-1’ branch, which is the primary branch in this case. The element of the primary branch with point index equal to 26 in CAAARS notation corresponds to the rotational transition 406,34 ← 397,33 of cyanoformamide in its ground vibrational state. The centers of the successively higher traces correspond to the predicted frequencies of correspondingly higher-index elements of this same branch, since
“Primary Branch” tab is selected (see Figure 11 - III). For the reader’s convenience the working image is supplemented in Figure 12 by the frequency shifts of each successive trace. The secondary branch is seen to curve off the screen to the lower right-hand side. If the “Secondary Branch” tab were selected, then the secondary branch ‘b,r,R,6,0’ would be centered on the screen and the primary branch ‘b,p,R,7,-1’ would curve off the screen to the lower left. The same display features are possible for the “Assigned Branch”. This feature allows us to follow splittings or perturbations involving two or three branches on the screen simultaneously.
At any time, navigation buttons created by CAAARS (see Figure 12 – I) can be used to modify the appearance of the display. One can zoom in and out in intensity or frequency and shift the centers of either axis. In addition, one can set the centers to specific values, or shift the axis centers to the position of a cursor, as indicated in Figure
12 – II. This feature is extremely useful in finding any particular transition in the actual spectrum. Setting the center of the display to the desired transition frequency places this transition in the middle of the bottom trace.
53 Igor Pro provides two standard cursors, A and B, shown in Figure 12 – II.
Information about the placement of these cursors, including the names of the data arrays
(called ‘waves’ in Igor Pro) on which the cursors are placed, point indices and coordinates, is provided in the lower left corner of the display window (see Figure 12 –
III).
These cursors are exploited by CAAARS for making assignments. CAAARS provides two distinct ways to assign a transition.
1. One cursor is placed with the mouse on an experimental peak in any of the traces
(see Figure 12). The second cursor is placed on the marker for the chosen
predicted transition, in either the primary or secondary predicted branch, and the
“Add Peak” button is pressed (see Figure 11).
2. If the experimental peak to be assigned is the closest one to the marker for the
prediction, then either cursor can be placed on the predicted transition (either
Primary or Secondary) and the “Add Closest” button is pressed (see Figure 11).
In both cases, “Add” means that 1) the experimental frequency and intensity together with all transition quantum numbers as given in the prediction file are placed in the assigned line list, 2) the assigned line list is resorted so that the lines for each branch are in sequence and the branches are in sequence, and 3) the screen is updated to display the assignment: a marker corresponding to “Assigned Branch” is placed on the experimental peak (depending on the choices selected, see Figure 11).
If a transition must be erased from the assigned line list, either cursor can be placed on the “Assigned” marker of the line that is to be deleted and the “Delete Peak” button is
54 pressed (see Figure 11). The display will again be automatically updated. There are several built-in safeguards that prevent the most common accidental wrong assignments.
After a certain number of transitions have been assigned, the user can fit them to the parameters of the rovibrational Hamiltonian as described in Sections 2.4.7 to 2.4.9. The first time “Run Fit” and “Load Predictions” are used, CAAARS will check the file selections with a dialogue box. Subsequent use of these options in the same session will use the same files, so that refitting the entire set of assigned data – 20 lines or 6000 lines
– and updating the display with the new predictions involve exactly two mouse-clicks.
At any time, the “Select Branch” dialogue box can be used to shift to an entirely different set of branches. The assigned transitions can be monitored by displaying them in the form of a table, and if necessary, the assigned line list can be modified using standard features of Igor Pro. As the assignment process progresses, new parameters may have to be freed or defined in the parameter file for SPFIT. This is done by calling up the file with your favorite text editor, making the necessary modifications, and returning to
CAAARS to run the fit.
The point at which the assignment process can be considered complete can be very effectively monitored using the CAAARS display. Outliers, odd line shapes, branches dwindling into the noise, lines too strong due to blends, can all be seen at a glance in the stacked display of the original spectral data.
55 2.6 DISCUSSION
The approach to the assignment of the rovibrational spectra realized in CAAARS has been developed and tested while working on a variety of asymmetric rotor spectra [9-11,
13] and one quasi-linear molecule. Assignment rates – including fitting and error diagnosis – of several hundred lines a day is typical. Not only does it make spectral assignment fast, but it also simplifies the search for complex spectral patterns. The user can take full advantage of current technology for a visual approach to line assignment using the actual spectrum. The flexible sorting principle, while developed with user convenience in mind, also makes CAAARS applicable to a wide range of spectroscopic problems, as noted in Section 2.3. It can be used without any modifications for handling data recorded with any broad banded high resolution instrument in any frequency or wavenumber region.
The organization of the predicted and assigned lines into branches allows the user to plot easily Fortrat diagrams (overall rotational quantum number J vs. frequency), intensity plots (intensity vs. frequency), R- or P-branch clusters, (Ka or Kc vs. frequency for a given J), errors, reduced energy plots, perturbed branch behavior, and whatever else is found useful for a given spectrum.
Although the current version of CAAARS uses SPFIT/SPCAT, relatively simple changes to the source code can make it compatible with the use of any other package used for fitting and prediction of spectroscopic high resolution data. The ease of writing routines in Igor Pro, or integrating routines written in C++ into CAAARS, means that the user can write additional routines for extending the options. For example, one user in our
56 laboratory has already written a routine to use an alternative way of marking previously assigned lines, by suppressing the line position markers on those lines. Routines for using combination differences, a second difference method of filtering the spectrum [53], the
Ritz combination principle, or other assignment techniques, can all be added as the need arises. A classic Loomis-Wood display can be achieved by simply suppressing the experimental spectrum, if desired.
The current version of CAAARS is available for downloading at http://www.physics.ohio-state.edu/~medvedev/caaars.htm. Its operation has not been fully tested with versions of Igor Pro older than version 5, and modifications to the code might be necessary for proper operation of all functions.
57 CHAPTER 3
THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS-TRANS CONFORMER OF DIETHYL ETHER (C2H5OC2H5)
3.1 INTRODUCTION
Diethyl ether (CH3CH2OCH2CH3) is a well-known anesthetic in the terrestrial operating room, as well as a popular solvent in organic chemistry. It is also a likely interstellar molecule because its simpler sibling – dimethyl ether (CH3OCH3) is found in high abundance in hot-core type sources within dense interstellar clouds [54]. Hot cores, which are associated with high-mass star-formation regions, exhibit temperatures that rise to 100-300 K as stellar formation occurs. The increase in temperature causes the evaporation of icy mantles from interstellar dust grains [55]. The mantles contain large amounts of the simplest alcohol, methanol (CH3OH), and possibly its more complex analog ethanol (C2H5OH), presumably produced via reactions starting from CO-ice [14].
Following evaporation, high abundances of methanol and ethanol in the gas phase can lead to the efficient production of ethers after protonated ions of these alcohols are formed [56]. For example, the well-studied reactions [57]
+ + CH3OH + C2H5OH2 → CH3O(H)C2H5 + H2O (3.1) 58 + + C2H5OH + C2H5OH2 → C2H5O(H)C2H5 + H2O (3.2)
produce protonated ethyl methyl and diethyl ether, respectively, after which dissociative recombination reactions form the neutral ethers. The results of simple models show that fractional abundances of approximately 10-8 - 10-10 can be achieved in hot cores for these more complex ethers [14, 56]. Such abundances are detectable today in the millimeter wave range.
The detection of interstellar diethyl ether requires its previous spectral analysis in the laboratory, but studies of the rotational spectrum of diethyl ether have been extremely limited. Since the spectrum in the 15 - 30 GHz frequency range was studied in 1971 by
Hayashi et al. [58] and in 1974 by Hayashi & Kuwada [59], no higher frequency transitions have been reported. In addition to the assignment of transition frequencies for the normal isotopomer, Hayashi & Kuwada [59] were able to assign analogous transitions in five other isotopomers containing deuterium and 13C nuclei, while Hayashi & Adachi
[60] reported the rotational spectrum of the 18O isotopomer. From these data, the structure of the molecule was determined, and it was found to be in the so-called trans- trans (tt) conformer, where the carbon atoms lie in a zigzag structure in which they are most strongly separated. A picture of this conformer is shown in Figure 13. All of the heavy atoms and two hydrogens lie in a plane while the remaining pairs of hydrogen atoms are symmetrically disposed about this plane. The most recent geometry has been deduced by Kuze et al. [61] from the combined results of rotational spectroscopy and electron diffraction. Hayashi & Kuwada [59] determined the dipole moment of tt-diethyl ether to be 1.061(18) Debye using Stark effect measurements. The dipole moment lies
59 along the b principal axis, so that rotational transitions obey b-type selection rules [2].
Moreover, unlike the case of ethyl methyl ether [62] diethyl ether shows no signs of splitting due to internal rotation of the methyl groups. Thus, the rotational spectrum can be analyzed by standard semi-rigid rotor techniques involving centrifugal distortion [2].
Although determining the frequencies of millimeter-wave lines by extrapolation of low frequency data is somewhat risky, Charnley et al. [56] have claimed the apparent detection of diethyl ether in W51 e1/e2, Orion-KL, and Sgr B2(N) based on such an extrapolation (Kuan, private communication). However, an unequivocal detection of as complex an interstellar molecule as diethyl ether requires laboratory data closer to the frequency range of use by radioastronomers. We have measured and assigned over 1000 new rotational lines for the molecule ranging in frequency up to 350 GHz and in rotational quantum number J up to 90. A data set consisting of 1105 old and new lines was fitted with a total of 13 parameters to experimental accuracy. The resulting spectroscopic constants were used to predict accurately more than 4000 unmeasured lines through 400 GHz in frequency.
3.2 EXPERIMENTAL CONSIDERATIONS
Three different groups contributed to this work, and a total of four different spectrometers were used. Diethyl ether is readily available, given its commercial importance. Measurements in the 75-110 GHz range were undertaken with the spectrometer at the National University of Singapore. This instrument contains an Agilent
2-20 GHz synthesized sweeper as the primary source of radiation. The radiation is multiplied into the 75-110 GHz range with an Agilent millimeter-wave source module,
60 and enters a Stark absorption cell, before being detected with a Schottky diode. Phase- sensitive detection and frequency modulation were utilized in the study of diethyl ether.
The experimental accuracy of lines measured in Singapore is estimated to be 0.050 MHz.
Two spectrometers in the Institute of Physics, Polish Academy of Sciences, Warsaw were also used in our study of diethyl ether. New measurements at low frequencies, specifically in the 7.0-17.3 GHz range, were carried out at sub-Doppler resolution with a supersonic expansion, cavity Fourier transform instrument [63]. The nominal measurement uncertainty of this spectrometer is 0.002 MHz. This spectrometer was also used to measure the electric dipole moment of diethyl ether by employing the recently developed Stark electrode configuration for generating a more uniform electric field within the microwave absorption cavity [64]. The new value of the dipole moment is
1.0976(9) Debye. This value has been given a conservative estimate of uncertainty, as discussed by Kisiel et al. [65]. The experimental value can be compared with an ab initio value of 1.125 Debye calculated1 at the MP2/6-31G** level with the program PC-
GAMESS2, a version of the GAMESS (US) package of quantum chemical programs [66].
Millimeter-wave measurements in Warsaw were made in the range 265-342 GHz with a spectrometer that employs ISTOK backward-wave oscillators (BWOs) used in a standard source-modulated configuration with phased-lock stabilization for the source
[24]. The nominal frequency accuracy of these measurements is 0.050 MHz.
Millimeter- and submillimeter-wave transitions through 366 GHz were studied at
Ohio State using the “Fast Scan Submillimeter Spectroscopy Technique” (FASSST) [4,
1 Calculation were carried out by Zbigniew Kisiel at the Institute of Physics, Polish Academy of Sciences, Warsaw. 2 PC-GAMESS (Granovsky 1997) is available at http://classic.chem.mus.su/gran/gamess/ 61 67] (see Chapter 1). The present spectral data of diethyl ether spectrum were recorded with the co-addition of 400 scans. It should be noted that only scans upwards in frequency were recorded and are used in this work.
Figure 13: Molecular structure of tt-diethyl ether, and its orientation in the inertial axis system.
3.3 SPECTRAL ANALYSIS
Altogether 1099 lines were assigned as arising from rotational transitions in the ground vibrational state of the lowest energy, tt conformer of diethyl ether, virtually all of which had not been studied previously. In some instances, these lines include blends between two or more specific rotational transitions. The strategy for either including or removing blends from the data set is as follows. When two components are extremely close together, in other words much closer than the experimental resolution, only one is included. When the splitting is comparable with spectroscopic resolution, one fits a
62 J’ Ka´ Kc´ J" Ka" Kc" Frequency/MHz Obs.–Calc./MHz Uncert./MHz Reference 1 1 1 2 0 2 7020.7865 -0.0017 0.0020 5 6 0 6 5 1 5 11300.1859 -0.0003 0.0020 5 1 1 0 1 0 1 15854.2840 0.0007 0.0040 5 2 1 1 2 0 2 15997.6897 -0.0011 0.0040 5 7 0 7 6 1 6 16027.0545 0.0028 0.0020 5 3 1 2 3 0 3 16214.6024 -0.0029 0.0040 5 4 1 3 4 0 4 16507.1861 0.0035 0.0040 5 5 1 4 5 0 5 16878.2882 0.0058 0.0040 5 6 1 5 6 0 6 17331.4583 0.0094 0.0040 5 7 1 6 7 0 7 17870.7000 -0.1813 0.1000 1 8 1 7 8 0 8 18501.1800 -0.2086 0.1000 1 9 1 8 9 0 9 19228.1500 -0.1742 0.1000 1
63 8 0 8 7 1 7 20802.8200 -0.2808 0.1000 1 2 1 2 1 0 1 24261.3700 -0.0982 0.1000 1 9 0 9 8 1 8 25622.2700 -0.1649 0.1000 1 3 1 3 2 0 2 28394.3100 0.0997 0.1000 1 10 0 10 9 1 9 30478.5400 -0.0288 0.1000 1 4 1 4 3 0 3 32457.1800 -0.1261 0.1000 1 43 2 41 43 1 42 75533.3750 -0.0142 0.0500 2 53 3 50 53 2 51 76824.6740 0.0143 0.0500 2 44 2 42 44 1 43 78903.8230 -0.0402 0.0500 2
NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. A portion is shown here for guidance regarding its form and content. The references are defined in the full table.
Table 2: Assigned and fitted transition frequencies of tt-diethyl ether in the vibrational ground state. mean of the two frequencies calculated for the blended components. For lines split by an amount greater than the spectroscopic resolution, both lines are included with the standard uncertainty in frequency.
Parameter Experimental Valuea A (MHz) 17596.15648(46) B (MHz) 2244.225553(76) C (MHz) 2101.793429(85) ∆J (kHz) 0.273924(24) ∆JK (kHz) –3.64882(28) ∆K (kHz) 85.9166(28) δJ (kHz) 0.0329800(69) δK (kHz) 0.51277(66) ΦJ (Hz) 0.0000749(23) ΦJK (Hz) −0.002632(34) ΦKJ (Hz) −0.01776(80) ΦΚ (Hz) 0.1381(71) φJ (Hz) 0.00002671(76) σb 0.926 c Nfit 1105
aUncertainties in parentheses (in units of the last digit of the fitted value). bWeighted standard deviation (dimensionless). c The number of distinct frequency lines in the fit.
Table 3: Spectroscopic parameters for tt-diethyl ether in the vibrational ground state.
Since the dipole moment lies along the b principal axis, the rotational transitions obey b-type selection rules; namely, ∆J = 0, ±1; ∆Ka = ±1, 3...; ∆Kc = ±1, 3... [2]. Here, J is the quantum number that represents the overall rotation of the molecule, while Ka and Kc are pseudo-quantum numbers signifying the projection of angular momentum along top symmetry axes in the prolate and oblate symmetric-top limits. The description of a
64 rotational level of an asymmetric top via the labels J, Ka, and Kc is unique. Transitions in which the overall rotational quantum number J does not change comprise Q-branches, while transitions in which J does change by unity are designated as R- and P-branch transitions, depending upon whether the higher state in energy has the higher or lower J value, respectively. Because the structure of the tt conformer of diethyl ether is that of a near-prolate asymmetric rotor, the spectrum is of a fairly standard variety, with well- known methods used to assign the quantum numbers. The main difficulties here were the high density of the spectral lines, the existence of at least one other conformer, and rotational transitions in excited vibrational states.
The assigned lines were fitted to a set of spectroscopic parameters (“constants”) that appear in various terms of the Watson type A-reduced asymmetric-top Hamiltonian, which includes both the angular momentum of a rigid body and centrifugal distortion corrections [2, 68]. The fitting was accomplished via a least-squares, iterative technique.
The program used, known as ASFIT, was developed by the Warsaw group, and is available at the website (http://info.ifpan.edu.pl/~kisiel/prospe.htm).
The result of our fit to the experimental data is shown in Table 2, where quantum numbers, measured frequencies, residuals (observed – calculated frequencies), estimated uncertainties, and references to the spectrometers used for each line are included.
Blended transitions are designated with an upper case B. (Only the first 21 lines of Table
2 appear in the text; for the complete version, see the attached CD which contains the supplementary materials.) The lines are fit on average to the experimental uncertainties.
The spectroscopic constants determined by the fit are shown in Table 3. Given the high values of the angular momentum included in the fitted transitions, the small number of
65 rotational parameters testifies to the rigidity of diethyl ether in its lowest energy conformer. Since there is no evidence of splitting due to internal rotation of the methyl hydrogen atoms, internal rotation was not considered in the analysis.
3.4 DISCUSSION
With the spectroscopic constants in Table 3, the frequencies of many transitions not analyzed in the laboratory both within and outside the measured frequency range were predicted. Predictions were made with the program SPCAT (Pickett 1991).1 They are listed in Table 4 (found in its entirety on the attached CD which contains the supplementary materials) through 400 GHz, along with the rotational quantum numbers, the estimated uncertainties, the intensities, a lower case “b” designating b-type transitions, and the upper state energies Eupper/k in K. The intensities (in units of Debye squared) are expressed as the product of the line strength S [19] and the square of the b-
type dipole moment measured here (µb = 1.0976 Debye). Only transitions with J ≤ 80,
2 2 Eupper/k < 1000 K, and intensities µb S over 1 Debye are included in Table 4. The assigned and fitted lines listed in Table 2 are also included in Table 4 (with their predicted rather than actual frequencies) as long as they meet the selection criteria, so that the intensities and upper state energies of these lines are available. The claimed detection of diethyl ether in W51 e1/e2, Orion-KL, and Sgr B2(N) by Charnley et al. [56] could not be confirmed because the transition frequencies used for the detection were not reported by the authors.
1 The program is available online at http://spec.jpl.nasa.gov/ 66 2 1 2 J´ Ka´ Kc´ J" Ka" Kc" Frequency/MHz Uncertainty/MHz µb S /D Type (Eupper/k) /K 25 3 22 24 4 21 98.3684 0.0055 5.346 b 74.740 47 6 42 46 7 39 915.7571 0.0217 10.079 b 262.772 47 6 41 46 7 40 950.6953 0.0216 10.079 b 262.774 60 9 52 61 8 53 1021.6595 0.0443 13.067 b 443.230 60 9 51 61 8 54 1023.5878 0.0443 13.067 b 443.231 24 4 20 25 3 23 1412.3287 0.0067 5.313 b 74.737 31 5 27 32 4 28 1424.4018 0.0088 6.832 b 122.460 76 10 67 75 11 64 1425.5200 0.0841 16.313 b 686.333 76 10 66 75 11 65 1425.6465 0.0841 16.313 b 686.333 11 2 9 12 1 12 1556.2481 0.0095 2.126 b 16.823 13 1 13 12 2 10 1557.3549 0.0112 2.236 b 19.412 40 5 36 39 6 33 1830.1800 0.0146 8.583 b 190.157 67 79 6 74 80 3 77 1832.7271 0.1808 1.632 b 687.950 31 5 26 32 4 29 1850.3189 0.0093 6.826 b 122.461 40 5 35 39 6 34 1971.5389 0.0143 8.585 b 190.163 17 3 14 18 2 17 2006.5857 0.0065 3.743 b 38.755 19 2 18 18 3 15 2007.8084 0.0073 3.965 b 42.613 4 0 4 3 1 3 2014.3309 0.0007 1.868 b 2.085 67 10 58 68 9 59 2021.5985 0.0607 14.560 b 551.065 67 10 57 68 9 60 2022.0359 0.0607 14.560 b 551.065 25 3 22 24 4 21 98.3684 0.0055 5.346 b 74.740
1Line strengths for b-type transitions multiplied by the square of the dipole moment in Debye. NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. A portion is shown here for guidance regarding its form and content.
Table 4: Predicted transition frequencies of tt-diethyl ether in the vibrational ground state. CHAPTER 4
THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS-GAUCHE CONFORMER OF DIETHYL ETHER
4.1 INTRODUCTION
Our interest in the trans-gauche conformer followed from our earlier study of the millimeter-wave and submillimeter-wave spectrum of the trans-trans conformer discussed in the preceding Chapter. After the assignment in the laboratory of over 1000 lines arising from rotational transitions in the ground vibrational state of the trans-trans conformer, it became clear that at least one more conformer of diethyl ether must exist together with rotational transitions arising from molecules in excited vibrational states of perhaps both conformers. Calculated structures of the trans-gauche and trans-trans conformers are depicted in Figure 14.
A detailed analysis based on data from two spectrometers has yielded over 1000 lines belonging to the ground vibrational state of the trans-gauche conformer. To the best of our knowledge, the rotational spectrum of this conformer had not been studied previously. The co-existence of spectral lines from both conformers is clearly illustrated in Figure 15, which shows a part of the millimeter-wave spectrum we have measured around 208 GHz. The rotational spectrum of the trans-gauche form of diethyl ether
68 significantly differs from that of its trans-trans counterpart, a fact that has made the analysis of the many spectral lines recorded easier than it might have been. Unlike the trans-trans form, which possesses a dipole component along only one axis, the trans- gauche conformer has three non-zero components. Moreover, rotational constants of the two conformers are sufficiently different so that strong overlapping of lines is often avoided.
Figure 14: Comparison of the geometries of the trans-trans and the trans-gauche conformers of diethyl ether. Both structures are given in the principal axis systems of inertia together with the orientation of the respective components of the permanent electric dipole moments.
69 Prior to our experimental work, the trans-gauche conformer had been studied by electron diffraction [61] and Raman spectroscopy in the gas phase, and by liquid-phase vibrational spectra [69, 70]. All of those studies indicate that at room temperature the percentage of ether in this conformer is approximately 25-35 %, and that it lies approximately 5-6 kJ mol−1 or 418-502 cm−1 above the ground state conformer, in agreement with ab initio determinations [61]. In astronomical units, the conformer lies approximately 600-700 K above the ground conformer, making its detection in hot molecular cores a distinct possibility if the tentative detection of the trans-trans form is correct.
Figure 15: Part of the millimeter-wave spectrum of diethyl ether. The top four traces show the theoretical frequency and intensity predictions for both trans-trans and trans- gauche conformer transitions, using the spectroscopic constants in Table 6 and the respective dipole moment components discussed in the text. The bottom trace represents the experimental spectrum.
70 The final composite FASSST spectrum of diethyl ether ranges from 108 GHz to 366
GHz with small frequency gaps from 252 GHz to 260 GHz and 295 to 297 GHz. The present spectral data of diethyl ether spectrum were recorded with the co-addition of 400 scans. It should be noted that only scans upwards in frequency were recorded and used in this work, before software for up-down averaging was developed. In view of the availability of the FASSST spectrum, PLL spectra (see Section 3.2) were only recorded with the OB-30 source in the spectral region from 252 GHz to 342 GHz.
4.2 ROTATIONAL SPECTRUM AND ANALYSIS
Results from previous gas phase electron diffraction work [61] and vibrational spectroscopy of diethyl ether [69, 70] suggest that the room temperature population of the trans-gauche conformer is near 30%. Supporting ab initio calculations1 were carried out at the MP2/6-31G** level of theory with PC-GAMESS [66, 71]. The trans-gauche conformer was calculated to be less stable by ∆E = 5.4 kJ mol−1 and to have a somewhat larger total dipole moment, 1.25 D, than the value of 1.09 D found for the trans-trans conformer. However, in contrast to the trans-trans species, where the total dipole moment is along the twofold principal inertial b axis, as can be seen in Figure 14, the dipole moment of the trans-gauche species is oriented in the inertial axis system in such a way that there are three non-zero, but relatively small dipole moment components. With the calculated values of the three components, µa = 0.409 D, µ b= 0.792 D, and µc = 0.872
1 Calculation were carried out by Zbigniew Kisiel at the Institute of Physics, Polish Academy of Sciences, Warsaw 71 D, b-type and c-type transitions would be expected to be dominant in the trans-gauche spectrum, as is indeed indicated in Figure 15. Nonetheless, the spectrum of the trans- gauche conformer was expected to be rather complex and at an intensity level appreciably lower than that of the trans-trans conformer.
The rotational spectrum of the trans-gauche conformer was identified and assigned in two steps. In the first step, a search was made for b- and c-type Q branches at a spacing close to the value 2A−B−C = 22.6 GHz predicted from calculated rotational constants.
The Q branches are indicated in Figure 16 for b-type transitions, where, instead of a classic FORTRAT diagram, we plotted the logarithm of b-type transition intensities for R and Q branches versus frequency. Figure 16 also displays the sensitivity limits of the
FASSST spectrometer for a single scan and for 100 co-added scans. The spacing of 22.6
GHz was sufficiently different from the analogous spacing of 30.8 GHz found in the trans-trans conformer so that an unambiguous assignment was possible. An example of
r the superimposed trans-gauche b-and c-type Q13 branches is shown in Figure 17 with the experimental spectrum in the bottom trace and calculated frequency and intensity predictions in the top trace. This particular Q branch is accidentally close in frequency to
r the b-type Q9 branch of the trans-trans species so that initially it was considered to be a vibrational satellite of the latter. As it turned out, and is indicated in Figure 17, the two nearby Q branches are characterized by considerably different Ka quantum numbers.
r Figure 18 shows clearly the lack of obvious structure of the combined b- and c-type Q15 - branch pattern at the higher value of Ka = 15.
72
Figure 16: Log10 (Intensity) versus frequency plot for some Q and R branches of trans- gauche diethyl ether b-type transitions. The second subscript number of the R branches ' represents (J″-Ka″-Kc″). The sensitivity of the FASSST spectrometer was calculated from the signal-to-noise ratio of SO2 reference lines.
Once such a sequence of Q branches with spacing close to 22 GHz was identified, which confirmed that the trans-gauche conformer is indeed observable, an attempt was made to assign transitions in the strongest b- and c-type R branches. Since the trans- gauche conformer is a near prolate rotor with an asymmetry parameter κ = −0.972, such transitions were expected to be those with the smallest values of the Ka quantum number.
Some difficulty was encountered in assigning such transitions since they do not fall into obvious patterns, but are in the form of intermeshed combs, each comb with spacing close to B + C = 4.8 GHz, as shown in Figure 16. Assignment of several such sequences was finally achieved and, allowed the much weaker a-type transitions to be identified. A 73 further factor to consider in the analysis was that the strongest transitions often consisted of unresolved doublets or quartets involving both b- and c-type transitions. The transitions belonging to the R branches of the trans-gauche conformer, although readily measurable, give rise to patterns that are only discernible with some difficulty, since the spectrum is dominated by stronger lines of the trans-trans conformer, as can be seen from Figures 15 and 17.
The Q branches and several b-type R-branches were in fact first identified in the PLL spectra and an initial set of spectroscopic constants was determined. Those constants were then used as input data for the computer aided assignment of asymmetric rotor spectra (CAAARS) program (see Chapter 2), whereupon a much larger number of lines measured with the FASSST system could rapidly be assigned.
The final data set acquired for trans-gauche diethyl ether is reported in Table 5 as electronically available supplementary material and consists of 1090 measured lines, many of which are multiplets. Unresolved multiplets with splittings >20 kHz were not included in the fit. Thus, 2130 individual transitions are entered in Table 5. The PLL spectrometer provided 188 lines or 550 transitions, while the remaining 902 lines or 1580 transitions were obtained from the FASSST spectrum. A portion of Table 5 is shown here for guidance regarding its form and content.
Transition frequencies range from 108 to 366 GHz, with values of quantum number
J" in the range 4 − 77 for R-branch transitions and 11 − 79 for Q-branch transitions. The measured transition frequencies were fitted with constants in Watson’s asymmetric rotor
Hamiltonian [68], in the A-reduction and the Ir-representation. The fitting was accomplished using SPFIT program [18] available at http://spec.jpl.nasa.gov. 74
Figure 17: An accidental near-coincidence between a Q branch of the trans-gauche conformer and a Q branch of the trans-trans conformer of diethyl ether. In the trans-trans Q branch each line consists of an unresolved asymmetry doublet, whereas in the trans- gauche Q branch each line represents an unresolved quartets arising from overlapped b- and c-type doublets. The two Q branches are characterized by considerably differing values of the Ka quantum number owing to rather different values of the A rotational constant in the two conformers. The upper plot shows the calculated line positions and intensities, while the lower trace displays part of the experimental spectrum.
The fitted constants are listed in Table 6, in which they are compared with values resulting from our ab initio calculation, as well as with corresponding values for the trans-trans conformer. The FASSST lines were weighted in the fit using an assumed frequency measurement error of 0.1 MHz, and an error of 0.05 MHz was used for the
PLL lines. The overall deviation of fit 0.063 MHz, is seen to be acceptable although, in contrast to the trans-trans conformer, it was necessary to resort to constants up to the octic level of centrifugal distortion. In fact the constants in Table 6 represent the most
75
Figure 18: b- and c-type Q-branch transitions of the trans-gauche conformer of diethyl ether in the submillimeter-wave region. The upper plot represents the calculated line positions and intensities, while the bottom trace shows the experimental spectrum.
economical set of constants, since a further small improvement in the standard deviation of the fit is still possible by adding more constants. The need for octic constants is not surprising since all quartic constants for the trans-gauche conformer are greater than those for the trans-trans conformer, and the important sextic constants HJK, HKJ, and HK are all an order of magnitude greater. The appreciable increase in centrifugal distortion in the trans-gauche form is understandable since this conformer is confined within a considerably shallower potential well. The barrier to the trans-trans form is calculated to be only 386 cm−1, whereas in the reverse direction this is 837 cm−1. The quartic
76 centrifugal distortion constants are all seen to be in fair agreement with values calculated from the harmonic force field. The poorest correspondence, for ∆JK and δK , is more likely to be the result of anharmonicity than of some spectroscopic interaction.
4.3 DISCUSSION
The trans-gauche conformer has been unambiguously identified in the rotational spectrum of diethyl ether, as confirmed by the correspondence between the observed and calculated rotational and quartic centrifugal distortion constants listed in Table 6.
Although the trans-gauche conformer constitutes an important fraction of the room temperature population of diethyl ether, it will be less populous with diminishing temperature, as in interstellar medium [9].
77 J´ Ka´ Kc´ J" Ka" Kc" Type Frequency/MHz Blend O-C/MHz Uncert./MHz 34 13 22 35 12 23 b 108819.6620 B -0.0179 0.10 34 13 21 35 12 24 b 108819.6620 B -0.0179 0.10 34 13 22 35 12 24 c 108819.6620 B -0.0179 0.10 34 13 21 35 12 23 c 108819.6620 B -0.0179 0.10 23 1 23 22 1 22 a 108836.5600 0.0663 0.10 31 2 29 30 3 27 c 108968.1530 0.1035 0.10 54 6 48 54 5 49 b 109116.5010 0.0132 0.10 53 6 48 53 5 48 c 109305.3630 0.0852 0.10 58 5 54 58 4 55 b 109632.4360 0.0264 0.10 56 4 53 56 3 54 b 110130.7650 0.0663 0.10 53 6 47 53 5 48 b 110427.4170 0.0916 0.10
78 23 1 23 22 0 22 b 110587.9006 -0.0142 0.10 52 6 47 52 5 47 c 110734.9040 0.0292 0.10 17 2 15 16 1 15 c 111115.3040 0.0420 0.10 23 7 16 22 7 15 a 111274.6630 0.0700 0.10 23 6 18 22 6 17 a 111296.2190 B 0.0148 0.10 23 6 17 22 6 16 a 111296.2190 B -0.0119 0.10 52 6 46 52 5 47 b 111642.8370 0.0018 0.10 24 1 24 23 1 23 a 113530.8150 0.0926 0.10 42 15 28 43 14 29 b 114402.8670 B -0.0692 0.10 42 15 27 43 14 30 b 114402.8670 B -0.0692 0.10
NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. A portion is shown here for guidance regarding its form and content.
Table 5: Assigned and fitted transition frequencies of trans-gauche diethyl ether in its vibrational ground state.
trans-trans conformer [10] trans-gauche conformer obs. calc.a obs. calc.a A / MHz 17596.15648(48) 13572.0904(11) 13710.0 B / MHz 2244.225553(76) 2495.41982(17) 2492.0 C / MHz 2101.793429(85) 2340.38842(15) 2334.4 ∆J / kHz 0.273924(24) 0.295 0.769546(56) 0.844 ∆JK / kHz -3.64882(28) -3.996 -6.32356(94) -4.375 ∆K / kHz 85.9166(28) 89.93 106.416(10) 104.9 δJ / kHz 0.0329800(69) 0.0367 0.134543(24) 0.167 δK / kHz 0.51277(66) 0.595 2.8683(39) 8.797 HJ / Hz 0.0000749(23) 0.0000634(76) HJK / Hz -0.002632(34) 0.1589(25) HKJ / Hz -0.01776(80) -1.5100(81) HK / Hz 0.1381(71) -1.038(45) hJ / Hz 0.00002671(76) -0.0003068(39) hJK / Hz 0.0 -0.12143(95) hK / Hz 0.0 17.92(23) LJJK / mHz 0.0 -0.00633(45) LJK / mHz 0.0 -0.0067(11) LKKJ / mHz 0.0 -0.0795(80) LK / mHz 0.0 -1.0700(76) lKJ / mHz 0.0 -0.622(44) b Nfit 1010 1090 σ fit / kHz 84 63 c σw 0.926 0.736
a Calculated at the MP2/6-31G** level of theory. b The number of distinct frequency lines in the fit. c Deviation of fit per unit weight.
Table 6: The experimental and the calculated spectroscopic constants for the two conformers of diethyl ether.
79 CHAPTER 5
THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF OXIRANECARBONITRILE
5.1 INTRODUCTION
Oxiranecarbonitrile (H2COC(H)CN), known more precisely as 2,3-epoxy- propionitrile, is a cyclic molecule consisting of an ethylene oxide (CCO) ring, with one carbon atom connected to two hydrogen atoms and the other carbon atom connected to one hydrogen atom and a cyanide (CN) group. The structure of this molecule is shown in
Figure 19. It has been suggested that oxiranecarbonitrile is an important prebiotic precursor for the formation of biological molecules. In particular, the reaction of oxiranecarbonitrile and inorganic phosphate in aqueous solution leads to the formation of glycolaldehyde phosphate in high yield [72]. This compound turns out to be an effective precursor for the synthesis of sugar phosphates and, as Eschenmoser & Loewenthal [73] have shown, its reaction with formaldehyde in alkaline, aqueous solution forms ribose
2,4-diphosphate, the backbone component of p-RNA.
A simpler sibling to oxiranecarbonitrile, the oxirane ring (C2H4O) itself, commonly known as ethylene oxide, has been found in SgrB2(N) [74] and subsequently in the hot cores NGC6334F, G327.3-0.6, G31.41+0.31, and G34.3+0.2 [75]. Its fractional abundance relative to molecular hydrogen is on the order 6 x 10-11 in SgrB2(N) and 80 between 2 x 10-10 and 6 x 10-10 in the other sources. An oblate asymmetric rotor, oxirane is a planar molecule with a dipole moment of 1.88 Debye directed along the b principal axis so that its rotational transitions obey b-type selection rules [2].
Figure 19: Computed molecular structure of oxirancarbonitrile, and its orientation in the inertial axis system.
The substitution of a hydrogen atom attached to the ring by a CN group destroys the two-fold symmetry of the species and results in two possible forms of the molecule, known as enantiomers, which are non-superimposable mirror images of each other, both with the same geometry and spectral frequencies. A mixture of the “right-handed” and
“left-handed” enantiomers, used in this study, is known as racemic. In terms of the rotational spectrum the structure of oxiranecarbonitrile is a prolate asymmetric rotor with
81 components of the electric dipole moment along all three principal axes. The dipole moment along the a principal axis possesses a value of 2.985 Debye, which is about twice as large as the dipole moments along the b and c axis - 1.662 and 1.522 Debye, respectively [76]. As with the spectrum described in Chapter 4 the expected spectrum for this molecule is more complex than a typical rigid asymmetric top since the rotational transitions now obey a-, b- and c-type selection rules. Because there is no internal rotation in this molecule and the oxirane ring is fairly rigid the rotational spectrum can be analyzed by standard semi-rigid rotor techniques involving centrifugal distortion [2].
The synthesis of oxiranecarbonitrile in interstellar clouds is uncertain. In the realm of ion-molecule processes, one possibility is a ring-closure reaction between protonated
+ ethyl cyanide (CH3CH2CNH ) and atomic oxygen to form a hydrogen-rich ion, followed by dissociative electron recombination [74]. Other possibilities, based on suggestions from Pitch et al. [72], include reactions between either the radical CH2CN or HCCN and formaldehyde (H2CO):
CH2CN + H2CO ö H2COC(H)CN + H, (5.1)
HCCN + H2CO ö H2COC(H)CN + hν . (5.2)
To the best of our knowledge, none of these processes has been studied in the laboratory.
Crude estimates indicate that detectable amounts of oxiranecarbonitrile can be produced in hot core-type sources with a fractional abundance of perhaps 10−10 .
The detection of interstellar oxiranecarbonitrile requires its previous spectral analysis in the laboratory. Up to now, the spectrum has only been studied in the 8-40 GHz 82 frequency range [76]. In a first search for interstellar oxiranecarbonitrile based on these data from the laboratory the species was not detected [74].
A successful detection of interstellar oxiranecarbonitrile will probably require laboratory data at higher frequencies than 40 GHz. In the course of this work such higher- frequency laboratory data for the vibrational ground state of oxiranecarbonitrile has been provided. We measured and assigned over 1300 rotational lines for the molecule ranging in frequency up to 360 GHz and in rotational quantum number J up to 95. A data set of
1460 old and new lines has been fitted to experimental accuracy with a total of 14 parameters. The resulting spectroscopic constants have been used to predict accurately more than 21000 unmeasured lines through 400 GHz in frequency. This work has been done in collaboration with Dr. Markus Behnke.
5.2 EXPERIMENTAL CONSIDERATIONS
Oxiranecarbonitrile is not commercially available, and its preparation was accomplished in our laboratory. The more traditional synthetic pathway is the alkali- catalyzed epoxidation and oxidation using a nitrile as co-reactant [77]. This preparation turns out to be time consuming with a yield of less than 60%. Since only small amounts of sample are needed for our spectroscopic studies, we have chosen a more convenient, fast, one-step process [78]. Here, the epoxidation of acrylonitrile in aqueous hypochlorite solution at moderately low temperature (<15°C) followed by intense “vacuum” distillation (a boiling point of 43°C at a pressure of 10 torr) to give oxiranecarbonitrile of high purity with a 30% yield. 83 Millimeter- and submillimeter-wave measurements were made in the range 110-360
GHz using the “Fast Scan Submillimeter Spectroscopic Technique” (FASSST) [4, 67].
The obtained spectra are calibrated optically with a Fabry-Perot cavity containing a reference gas. The nominal measurement accuracy of the FASSST spectrometer is 0.100
MHz. For our investigations on oxiranecarbonitrile, we averaged 400 scans in the entire frequency range. Each scan was calibrated with sulfur dioxide as calibration gas.
5.3 SPECTRAL ANALYSIS
In total, 1340 new lines belonging to the ground vibrational state of oxiranecarbonitrile were assigned using the CAAARS software package (see Chapter 2).
In some instances, lines are in reality blends between two or more specific rotational transitions. When blended transitions are predicted to lie closer together than the experimental resolution, at most one frequency was included. When the splitting is comparable with the spectroscopic resolution, the blended transitions were not included in the fit. For fully resolved lines (split by an amount greater than the spectroscopic resolution of 500 kHz), both lines were included with the standard uncertainty in frequency.
With a Ray’s asymmetry parameter [2] κ value of −0.9792, oxiranecarbonitrile shows the spectrum of a near-prolate asymmetric rotor. The main difficulties were the high density of the spectral lines, due to the presence of a-, b- and c-type transitions, and rotational transitions in excited vibrational states.
84
J’ Ka´ Kc´ J" Ka" Kc" Frequency/MHz Obs. – Calc./MHz Blend 16 1 16 15 1 15 108823.437 0.028 17 0 17 16 1 16 109175.864 0.087 25 2 23 24 3 22 109385.480 0.051 16 0 16 15 0 15 109495.732 0.045 16 2 15 15 2 14 110127.074 -0.027 16 5 12 15 5 11 110301.182 0.013 B 16 5 11 15 5 10 110301.182 -0.001 B 16 3 14 15 3 13 110345.685 0.000 16 3 13 15 3 12 110396.608 0.004 15 1 15 14 0 14 110477.136 0.002 16 2 14 15 2 13 110891.661 0.049 16 1 15 15 1 14 111284.334 0.018 54 5 49 54 4 50 111338.394 0.049 52 5 47 52 4 48 114849.825 -0.001 17 1 17 16 1 16 115600.381 0.000 26 2 24 25 3 22 115685.292 0.031 50 5 46 50 4 46 115851.893 0.009 17 0 17 16 0 16 116250.961 0.056 11 2 10 10 1 9 116546.259 -0.021 16 1 16 15 0 15 116570.759 -0.055
NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. A portion is shown here for guidance regarding its form and content.
Table 7: Assigned and fitted transition frequencies of oxiranecarbonitrile in its vibrational ground state.
The assigned transitions were fitted to the set of spectroscopic parameters that appear in various terms of the Watson asymmetric-top Hamiltonian, in the A-reduction and the
Ir-representation [2, 68]. In addition to the newly assigned lines, we included 120 microwave transitions from previous work [76]. The fitting was accomplished via a least- squares technique using the SPFIT program [18] available for downloading at http://spec.jpl.nasa.gov. The experimental uncertainty for the microwave frequency lines
(< 40 GHz) was set at 30 kHz while uncertainty for the FASSST data was set at 100 kHz.
85 Parameter Experimental Value1 Experimental Value2 A / MHz 18456.28276(117) 18456.2955(29) B / MHz 3524.704148(223) 3524.70497(46) C / MHz 3367.811043(216) 3367.81108(42) ∆J / kHz 1.149250(163) 1.14702(60) ∆JK / kHz –1.74870(73) -1.7094(90) ∆K / kHz 76.0380(197) 75.858(23) δJ / kHz –0.0502282(253) -0.04956(7) δK / kHz 1.3987(177) 1.392(29) ΦJ / mHz 2.113(36) ΦJK / mHz −15.964(232) ΦKJ / Hz 0.1640(52) ΦΚ / Hz 0.447(91) φJ / mHz −0.3073(71) φJK / mHz −4.7(36) rms /kHz 48.9 23 3 Nfit 1460 120
1This work. Uncertainties in parentheses (in units of the last digit of the fitted value). 2Previous study by Müller & Bauder [76]. 3 The number of distinct frequency lines in the fit.
Table 8: Spectroscopic parameters for oxiranecarbonitrile in its vibrational ground state.
The results of our fit in comparison with the experimental data are shown in Table 7, where quantum numbers, measured frequencies, and residuals (observed – calculated frequencies) for each line are included. Blended transitions assigned to the same frequency are designated with an upper case B and have been weighted in the fit according to the line intensities. Only the first 20 lines of Table 7 are shown in the text; for the complete version see the attached CD which contains the supplementary materials. The spectroscopic constants determined by the fit are shown in Table 8.
86 Frequency Uncertainty (E /hc) J´ K ´ K ´ J" K " K " µ 2S1/D2 Type2 upper a c a c /MHz /MHz a,b,c /cm-1 30 1 30 30 0 30 1224.413 0.001 115.40 c 105.885 29 1 29 29 0 29 1418.674 0.001 110.62 c 99.100 49 2 48 49 1 48 1632.228 0.003 154.55 c 281.928 28 1 28 28 0 28 1639.780 0.001 105.84 c 92.540 48 2 47 48 1 47 1881.900 0.003 149.66 c 270.786 27 1 27 27 0 27 1890.391 0.001 101.05 c 86.205 47 2 46 47 1 46 2165.788 0.003 144.78 c 259.867 26 1 26 26 0 26 2173.136 0.001 96.28 c 80.095 46 2 45 46 1 45 2487.651 0.004 139.89 c 249.172 25 1 25 25 0 25 2490.518 0.002 91.48 c 74.209 24 1 24 24 0 24 2844.795 0.002 86.72 c 68.549
87 45 2 44 45 1 44 2851.438 0.004 134.99 c 238.702 23 1 23 23 0 23 3237.833 0.002 81.96 c 63.114 44 2 43 44 1 43 3261.235 0.004 130.05 c 228.456 22 1 22 22 0 22 3670.951 0.002 77.21 c 57.904 43 2 42 43 1 42 3721.191 0.004 125.12 c 218.434 21 1 21 21 0 21 4144.745 0.002 72.52 c 52.920 31 8 24 32 7 25 4230.367 0.009 12.81 b 146.064 31 8 23 32 7 25 4230.367 0.009 10.94 c 146.064 31 8 24 32 7 26 4230.383 0.009 10.94 c 146.064
1Line strengths for a-, b- or c-type transitions multiplied by the square of the dipole moment component in Debye along the a, b or c principal axis. 2Type of transition – a, b, or, c. NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. A portion is shown here for guidance regarding its form and content.
Table 9: Predicted transition frequencies of oxiranecarbonitrile in its vibrational ground state. The root-mean-square deviation (rms) of 49 kHz for the resulting fit is a factor of two smaller than our estimated experimental uncertainty of the FASSST spectrometer. This can be explained by the implemented data averaging scheme as well as the exclusion of some blended lines. This high precision has been confirmed by other recent measurements.
5.4 DISCUSSION
The spectroscopic constants in Table 8 are in excellent accord with the earlier values of Müller & Bauder [76]. This allowed us to predict the frequencies of many transitions not analyzed in the laboratory both within and outside the measured frequency range.
Predictions, made with the program SPCAT [18], are listed in Table 9 (found in its entirety on the attached CD which contains the supplementary materials) through 400
GHz, along with the rotational quantum numbers, the estimated uncertainties, the intensities, a lower case a, b or c designating a-, b- or c-type transitions, and the upper
-1 state energies Eupper/(hc) in cm . The intensities (in units of Debye squared) are expressed as the product of the line strength S [19] and the square of the a-, b- or c-type dipole- moment components taken from the literature [76], depending on the type of transition.
-1 Only transitions with J ≤ 95, Eupper/(hc) § 400 cm (575 K), uncertainty § 300 kHz, and
2 2 intensity µa,b,c S ¥ 8 Debye are included in Table 9. The assigned and fitted lines listed in Table 7 are also included in Table 9 with their predicted rather than actual frequencies as long as they meet the selection criteria, so that the intensities and upper state energies of these lines are available.
88 CHAPTER 6
THE MILLIMETER- AND SUBMILLIMETER-WAVE SPECTRUM OF THE TRANS- AND GAUCHE- CONFORMERS OF ETHYL FORMATE
6.1 INTRODUCTION
Ethyl formate (HCOOC2H5) is a likely candidate for interstellar detection because its simpler sibling – methyl formate (HCOOCH3), often labeled as an “interstellar weed” – is found in a number of hot cores of giant molecular clouds [79-81]. In fact, methyl formate has just recently been found in the smaller “corinos” of two low-mass protostars [82, 83].
Although the chemical synthesis of ethyl formate in the interstellar medium is uncertain, it is thought to be produced via the synthesis of its protonated ion through the gas-phase reaction between protonated ethyl alcohol and formaldehyde
+ + [C2H5OH2] + H2CO → [HC(OH)OC2H5] + H2 (6.1) followed by dissociative recombination of the ion:
+ [HC(OH)OC2H5] + e → HCOOC2H5 + H . (6.2)
Calculations based upon this reaction path [14] reveal that detectable fractional abundances of ethyl formate (~10-10) can be accumulated within 104- 105 years in hot
89
Figure 20: The geometries of the trans and the gauche conformers of ethyl formate. Both structures are given in the principal axis systems of inertia. The structures are based on the calculated geometries of Peng et al. [84] and realized via the PMIFST program of Kisiel (http://info.ifpan.edu.pl/~kisiel/prospe.htm).
90 cores. It should be noted, however, that an experimental and theoretical study a reaction of analogous to that described above between protonated methanol and formaldehyde to form protonated methyl formate [85] revealed that this reaction does not occur at a significant rate. The reaction had previously been thought to lead to the production of methyl formate [86].The synthesis of methyl formate in hot cores and corinos is thus no longer understood, so it is unlikely that we can be sure how ethyl formate is synthesized.
An unambiguous interstellar identification of ethyl formate depends upon laboratory spectral analysis. The rotational spectrum of ethyl formate has been studied previously in both the microwave [87-89] and millimeter-wave [90] ranges of the electromagnetic spectrum. While several conformers, or rotational isomers, of ethyl formate are possible
[84, 91, 92], only two have been observed in the experimental studies of the rotational spectrum. The trans conformer, which corresponds to the global minimum of energy, possesses a skeletal planar form, while the gauche conformer is produced mainly by an out-of-plane rotation of the ethyl group around the O-C bond, as can be seen in Figure 20, which contains structures for the two conformers. Both of these conformers have the ethyl (C2H5) group cis to the terminal oxygen atom, where the word cis indicates that the atoms or groups of atoms are on the same side of the molecule. The proper names for the observed conformers in the IUPAC system are [sp,ap] and [sp,sc] [84] but we will retain the simpler terms trans and gauche. It should be noted that there are two equivalent gauche forms corresponding to rotation in two different directions. In other molecules, these two gauche forms split into symmetric and antisymmetric combinations (see, e.g.
[93]) but this splitting has not been detected here because of a high barrier to interconversion.
91 Riveros & Wilson [87] successfully assigned rotational transitions belonging to the vibrational ground state of both the trans and gauche conformers of ethyl formate as well as transitions belonging to several excited vibrational states in the frequency range up to
35 GHz. Since ethyl formate, unlike methyl formate [94], shows no signs of splitting due to the internal rotation of the methyl group, the rotational spectrum of the conformers can be analyzed by standard semi-rigid techniques involving centrifugal distortion in the absence of strong perturbations [2]. In addition to their analysis of the spectrum, Riveros
& Wilson [87] used Stark effect measurements to determine the dipole moment components µi for the two conformers, where the subscript i refers to the principal axes a, b, and c. For trans ethyl formate, they found that µa = 1.85 Debye (D) and µb = 0.69 D, whereas for the gauche form they found all three components of the dipole moment to be non-zero; viz., µa = 1.44 D, µb = 1.05 D, and µc = 0.25 D. Finally, based on relative intensities and other data, Riveros & Wilson [87] determined the gauche conformer to lie
0.186(60) kcal mol-1 (1 kcal mol-1 = 503 K) above the trans conformer with an energy barrier of 1.10(3) kcal mol-1 relative to the trans form. Following this work, some details of the torsional potential around the O-C bond were studied by Meyer & Wilson [88] while Kaushik [89] analyzed the spectrum to obtain quartic centrifugal distortion constants.
After the pioneering microwave studies, Demaison et al. [90] extended the study of the vibrational ground state transitions of trans and gauche ethyl formate through 242
GHz, well into the millimeter-wave region of the spectrum (30 GHz – 300 GHz). The most recent geometries were deduced by Peng et al. [84] based on a joint analysis of electron diffraction, rotational and vibrational spectroscopic information, and quantum 92 chemical calculations. It is these structures for the trans and gauche conformers of ethyl formate that are shown in Figure 20, together with the directions of the principal axes.
Based on their observation that at room temperature, ethyl formate consists of 61% trans and 39% gauche, Peng et al. [84] estimated the excitation energy of the gauche conformer to be 0.3(2) kcal mol-1, in reasonable agreement both with the result of their quantum chemical result – 0.5 kcal mol-1 – and the microwave value. Peng et al. [84] also calculated the barrier to interconversion between the two types of conformers to be 1.3 kcal mol-1, in excellent agreement with the earlier value.
Relying on values for the frequencies of submillimeter-wave lines (300 GHz – 1
THz) obtained by extrapolation from lower frequencies can be risky. So, the purpose of the work reported here is to extend the known frequency range of the rotational spectrum of ethyl formate. We have succeeded in assigning 565 new spectral lines belonging to the vibrational ground state of the trans conformer of ethyl formate and 997 such transitions belonging to the gauche conformer at frequencies through 377 GHz.
6.2 EXPERIMENTAL CONSIDERATIONS
A commercial 98% pure sample of ethyl formate was used without any additional purification. The spectrum was taken at room temperature with 11 mtorr of sample pressure in the absorption cell, so as to achieve maximum spectral resolution and sensitivity.
Millimeter- and submillimeter-wave transitions in the frequency range from 106 GHz to 377 GHz, with small gaps from 252 GHz to 260 GHz and from 292 GHz to 303 GHz,
93 were studied using the Fast Scan Submillimeter Spectroscopic Technique (FASSST) [4,
9, 67]. The present data of the ethyl formate spectrum were recorded with the co-addition of 100 scans. It should be noted that only scans upwards in frequency were recorded, since the spectrum was recorded early in this work. To improve the accuracy of the frequency measurement, each scan was later calibrated with a correction for the dispersion caused by the atmospheric water vapor present in the calibration cavity of the
FASSST spectrometer.
6.3 SPECTRAL ANALYSIS
In both of the conformers studied here, ethyl formate is an asymmetric top, which is a semi-rigid molecule with three different values for its moments of inertia directed along the three principal axes, with the order Ia [2]. Transitions in which J does not change are known as Q branches, while those in which J increases or decreases by 1 are known as R and P branches, respectively. So- called a-type transitions (∆Ka = 0,±2,…, ∆Kc = ±1, ±3,…) are associated with a non-zero dipole component along the a axis, b-type transitions (∆Ka = ±1,±3,…, ∆Kc = ±1, ±3,…) with a non-zero component along the b axis, and c-type transitions (∆Ka = ±1,±3,…, ∆Kc 94 = 0, ±2,…) with a non-zero component along the c axis. Although there is no general analytic formula for the energy levels of even a perfectly rigid asymmetric top, the energy levels and spectrum of a semi-rigid top can be determined numerically by matrix methods using an effective for a given vibrational state Hamiltonian written in terms of the so-called rotational and centrifugal distortion parameters multiplied by suitable angular momentum operators [2]. The particular treatments of centrifugal distortion generally utilized are due to Watson [68]. With Ray’s asymmetry parameter [2] κ values of -0.9571 for the trans conformer and -0.8149 for the gauche conformer, ethyl formate shows the spectrum of a near-prolate asymmetric rotor for both conformers. The main difficulties in the assignment were associated with the high density of the spectral lines, due to the presence of both a - and b-type transitions for the trans conformer and a-, b- and c-type transitions for the gauche form, as well as rotational transitions belonging to various excited vibrational states of both conformers. The rotational constants of the trans and gauche conformers of ethyl formate are sufficiently different so that overlapping of lines was generally avoided. Assignments were initially based on predictions for lower frequency data [87, 90] and following a bootstrap procedure with the CAAARS software package (see Chapter 2) [12]. Partially blended lines, in which the splitting was comparable with the spectroscopic resolution, were not included in the fit. For fully resolved lines (split by an amount greater than the spectroscopic line width of ~1 MHz), both lines were included with the estimated uncertainty in frequency of the FASSST spectrometer - 150 kHz (see Chapter 1 for the discussion of the frequency uncertainty before the up-down averaging) [9]. Along 95 with the new lines measured here, many of the millimeter-wave lines of Demaison et al. [90] were re-measured. We were able to analyze transitions with rotational quantum numbers J up to 71 and 72 for the gauche and trans conformers respectively. The majority of the assigned transitions for the trans form correspond to a-type R branches, and the strongest b-type P and R branches could also be identified at high J. We were not able to identify any transitions of the b-type Q branches. For the gauche form, on the other hand, such Q branches were assigned. For this conformer, only a- and b-type transitions were assigned because the c-component of the dipole moment, while non-zero, is more than four times smaller than the a- or b- components. Additionally, many transitions obeying c-type selection rules are blended with b-type transitions. In addition to newly assigned lines, we included the microwave transitions of Riveros & Wilson [87] and selected millimeter- wave transitions of Demaison et al. [90] not assigned here in global data sets for the trans and gauche conformers. The data sets were fitted to the Watson asymmetric-top Hamiltonian, in the A-reduction and the Ir-representation [2, 68]. To estimate the experimental uncertainty in frequency of the microwave transitions (not reported by Riveros &Wilson [87]), we took 100 kHz as a conservative measure based on fits to the microwave data. The global fitting was accomplished via a least-squares technique using the SPFIT program [18], available at http://spec.jpl.nasa.gov. The results of our fits in comparison with the experimental data are shown in Table 10 for the trans conformer and Table 11 for the gauche conformer. In these tables, measured frequencies and their uncertainties are listed along with assigned quantum numbers, and residuals (observed – calculated frequencies). The source of the measured 96 lines is also listed by a number under the column titled “Comment.” Blended transitions assigned to the same frequency are designated with an upper case B and have been weighted in the fit according to the predicted line intensities; these appear only for the gauche form. Only the first 21 lines of Tables 10 and 11 are shown in the text; the complete versions are available in electronic form. The twenty one lines shown in the abbreviated Table 11 are chosen to exhibit blending, in this case an unresolved asymmetry doublet. The spectroscopic constants and root-mean-square (rms) deviations determined by the fits are shown in Tables 12 and 13 for the trans and gauche forms, respectively. The numbers in parenthesis represent 1σ uncertainties in the parameters. The root-mean- square deviations of the fits are 72 kHz and 74 kHz for the trans and gauche forms. In addition to our spectroscopic constants, those determined by Demaison et al. [90] are included for comparison. For the trans conformer, our constants are in excellent agreement with those of Demaison et al. [90], the one exception being the quartic distortion constant ∆K. Note that the large uncertainty in the A rotational constant and the distortion constant ∆K comes from the lack of Q-branches assigned. In total, we utilized only two more fourth-order constants than the older treatment. For the gauche conformer, on the other hand, we were forced to use many more centrifugal distortion constants, and the Hamiltonian contains terms through 10th order. The necessity of including higher- order centrifugal distortion terms can, most likely, be explained by a relatively weak global perturbation involving a low-lying vibrational state or states, but this line of investigation has not been pursued at this time. 97 For the trans conformer, we observed two perturbations in the rotational spectrum of the vibrational ground state. One is a sharp local resonance, which only affects several consecutive transitions belonging to the a-type R branch with both Ka′ and Ka′′ equal to 16. We chose not to include the affected transitions, 4816,32 - 4716,31 through 5316,36 - 5216,3, in the assigned line list. The relative energy of the lower state for the affected -1 -1 transitions ranges from 332 cm (478 K) up to 380 cm (547 K). Another more extended perturbation affects transitions belonging to all a-type R branches with values of Ka′ = Ka′′ above 17. The effect increases with Ka and J, with the relative energy of the lower state well above 300 cm-1 (432 K). We were not able to fit the perturbed frequencies by including higher order centrifugal distortion constants, and so the lines are also not in our fit. 6.4 DISCUSSION The spectroscopic constants determined in this study have been used to predict accurately more than 3000 unmeasured transitions of the trans conformer and more than 10000 transitions of the gauche conformer. Tables 14 and 15 (found in their entirety on the attached CD which contains supplementary materials) list these predicted frequencies through 378 and 380 GHz for the trans and gauche conformers, respectively, along with their rotational quantum numbers, estimated uncertainties, intensities, a lower case a, b or c designating a-, b- or c-type transitions, and upper state energies Eupper/(k) in K. The intensities (in units of Debye squared) are expressed as the product of the line strength S [19] and the square of the a-, b- or c-type dipole-moment components taken from the 98 literature [87], depending on the type of transition. Only transitions with Eupper/(k) § 1000 2 2 K, uncertainty § 500 kHz, and intensity µa,b,c S ¥ 1 Debye are included in Tables 14 and 15. The assigned and fitted lines listed in Tables 10 and 11 are also included in Tables 14 and 15 with their predicted rather than actual frequencies as long as they meet the selection criteria, so that the intensities and upper state energies of these lines are available. The rotational partition function for each conformer of ethyl formate in its vibrational ground state is approximated well by the standard rigid asymmetric top formula (/)kT h 3/2 q = π 1/2 , (6.3) ABC where the rotational constants A, B, and C are in units of frequency. With the values in Tables 12 and Table 13, we obtain values for the (dimensionless) partition function q is 14.621 T1.5 for the trans conformer and 15.189 T1.5 for the gauche conformer. There are several complications to the use of these partition functions, however. First, it is likely that both conformers are present in hot cores and second, radiative excitation can populate excited vibrational states for each conformer. Since it is assumed by laboratory investigators that collisional processes can interconvert the trans and gauche forms, one must consider a global partition function q such that q = qtrans + 2exp(−∆E /T )qgauche (6.4) where ∆E is the excitation energy of the two degenerate gauche conformers. With the latest estimated value of approximately 150 K for this energy, one can see that even at 99 100 K, both conformers will be appreciably populated. Also, since there are low-lying excited torsional states for both conformers [84, 87, 92], these must also be taken into account in the individual rovibrational partition functions for the two conformers if radiative excitation is present. Relevant rotational constants for the excited states are found in [87]. 100 Frequency Obs. – Calc. Uncert. J´ K ´ K ´J"K"K" Comment a c a c /MHz /MHz /MHz 2 1 2 1 1 1 10642.2600 0.0926 0.1 1 2 0 2 1 0 1 10962.4400 -0.0005 0.1 1 2 1 1 1 1 0 11293.2900 -0.0486 0.1 1 2 1 1 2 0 2 15498.2600 -0.0990 0.1 1 3 1 3 2 1 2 15959.9500 0.0154 0.1 1 3 1 2 3 0 3 16004.6500 0.0530 0.1 1 3 0 3 2 0 2 16430.4000 0.0129 0.1 1 3 2 2 2 2 1 16451.6700 0.0145 0.1 1 3 2 1 2 2 0 16472.8200 -0.0192 0.1 1 4 1 3 4 0 4 16697.9300 0.0172 0.1 1 3 1 2 2 1 1 16936.6200 -0.0052 0.1 1 101 5 1 4 5 0 5 17593.7100 0.0572 0.1 1 4 1 4 3 1 3 21273.8000 0.0028 0.1 1 8 1 7 8 0 8 21692.0400 0.0563 0.1 1 4 0 4 3 0 3 21882.4800 0.0188 0.1 1 4 2 3 3 2 2 21931.4100 0.0614 0.1 1 4 2 2 3 2 1 21984.0500 -0.1948 0.1 1 4 1 3 3 1 2 22575.7400 -0.0371 0.1 1 9 1 8 9 0 9 23601.5800 -0.0164 0.1 1 5 1 5 4 1 4 26582.5900 0.0119 0.1 1 5 0 5 4 0 4 27313.5600 0.0107 0.1 1 NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. The comments are defined in the full table. Table 10:Assigned and fitted transition frequencies of the trans conformer of ethyl formate in its vibrational ground state. Frequency Obs. – Calc. Uncert. J´ K ´ K ´ J" K "K" Blend Comment a c a c /MHz /MHz /MHz 32 6 27 31 6 26 227298.7803 -0.0270 0.15 2 26 6 21 25 5 20 229821.2185 -0.1258 0.15 2 33 5 29 32 5 28 230328.4069 -0.0315 0.15 2,3 32 5 27 31 5 26 230779.8434 -0.0137 0.15 2 27 6 22 26 5 21 231849.4916 -0.1142 0.15 2 33 5 29 32 4 28 232027.1954 0.0269 0.15 2 34 4 31 33 4 30 232545.3359 -0.0810 0.15 2,3 34 3 31 33 3 30 232578.6857 -0.0772 0.15 2,3 36 1 36 35 0 35 232996.0594 B 0.1209 0.15 2,3 36 0 36 35 0 35 232996.0594 B 0.1213 0.15 2,3 33 19 15 32 19 14 233604.2278 0.0018 0.15 2,3 102 33 18 16 32 18 15 233611.8800 0.0255 0.15 3 33 20 14 32 20 13 233612.2600 -0.0572 0.15 3 33 21 13 32 21 12 233633.6872 -0.0068 0.15 2,3 33 17 17 32 17 16 233638.3873 0.0118 0.15 2,3 33 22 12 32 22 11 233666.4739 0.0185 0.15 2,3 33 16 18 32 16 17 233688.0347 0.0244 0.15 2,3 33 23 11 32 23 10 233709.1107 0.0191 0.15 2,3 33 24 10 32 24 9 233760.4328 0.0471 0.15 2,3 33 15 19 32 15 18 233766.6515 0.1443 0.15 2,3 33 14 20 32 14 19 233882.0284 0.1154 0.15 2,3 NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. The comments are defined in the full table. Table 11: Assigned and fitted transition frequencies of the gauche conformer of ethyl formate in its vibrational ground state. Parameter Current Value1 Previous Value2 A / MHz 17746.6525(278) 17746.733(28) B / MHz 2904.73234(69) 2904.72853(72) C / MHz 2579.14710(56) 2579.14641(56) ∆J / kHz 0.624672(146) 0.62326(19) ∆JK / kHz -3.52970(266) -3.5435(33) ∆K / kHz 52.97(105) 81.80(48) δJ / kHz 0.101856(129) 0.10127(11) δK / kHz -0.8983(109) -0.881(17) ΦJ / mHz 0.0971(164) - ΦKJ / Hz -1.0656(89) -1.1343(74) φJ / mHz 0.0288(164) - rms /kHz 72 87 3 Nfit 654 96 1This work. Uncertainties in parentheses (in units of the last digit of the fitted value). The parameter “rms” stands for the root-mean-square deviation of the fit. 2Demaison et al. [90]. 3 The number of distinct frequency lines in the fit. Table 12: Spectroscopic parameters for the trans conformer of ethyl formate in its vibrational ground state. 103 Parameter Current Value1 Previous Value2 A / MHz 9985.58186(299) 9985.6518(99) B / MHz 3839.60989(99) 3839.6071(35) C / MHz 3212.86762(87) 3212.8738(32) ∆J / kHz 5.93910(91) 5.9493(44) ∆JK / kHz - 32.3373(38) -32.3791(75) ∆K / kHz 78.7548(167) 85.81(29) δJ / kHz 2.00186(36) 2.0004(15) δK / kHz 7.1776(187) 7.431(28) ΦJ / Hz -0.05934(40) -0.055(19) ΦJK / Hz 0.9169(39) 0.769(40) ΦKJ / Hz -4.1566(119) -3.664(126) ΦΚ / Hz 7.649(43) - φJ / Hz -0.025606(175) -0.02616(87) φJK / Hz -0.0735(112) - φK / Hz 1.785(42) LJ / µHz -2.898(58) LJJK / mHz 0.02458(77) LJK / mHz -0.4155(123) LKKJ / mHz 1.2523(313) LK / mHz -1.467(55) lJ / µHz -1.3596(258) lJK / mHz -0.02004(183) lK / mHz -1.860(99) PKJ / µHz 0.04159(117) PKKJ / µHz -0.11757(180) PK / µHz 0.1494(204) pJK / µHz -0.02624(149) rms /kHz 74 102 3 Nfit 1067 88 1This work. Uncertainties in parentheses (in units of the last digit of the fitted value). The parameter “rms” stands for the root-mean-square deviation of the fit. 2Demaison et al. [90]. 3 The number of distinct frequency lines in the fit. Table 13: Spectroscopic parameters for the gauche conformer of ethyl formate in its vibrational ground state. 104 2 1 Frequency Uncertainty µa,b S 2 Eupper J´ Ka´ Kc´J"Ka"Kc" 2 Type /K /MHz /MHz /D k 22 3 19 22 3 20 8085.6607 0.0562 2.400 a 73.608 31 4 27 31 4 28 8516.6806 0.0604 2.966 a 142.986 26 4 23 25 5 20 9816.8968 0.3317 1.981 b 104.200 20 3 17 19 4 16 10013.3262 0.1881 1.538 b 62.109 23 3 20 23 3 21 10175.6029 0.0675 2.246 a 79.775 21 3 19 20 4 16 10188.9903 0.1862 1.582 b 67.417 41 5 36 41 5 37 10237.9434 0.0623 3.386 a 246.188 32 4 28 32 4 29 10583.5430 0.0711 2.810 a 151.554 2 1 2 1 1 1 10642.1674 0.0016 5.125 a 1.486 15 2 13 15 2 14 10851.0796 0.0557 1.560 a 34.888 2 0 2 1 0 1 10962.4405 0.0014 6.831 a 0.789 105 16 4 12 17 3 15 11274.4987 0.1865 1.222 b 47.372 2 1 1 1 1 0 11293.3386 0.0019 5.125 a 1.533 51 6 45 51 6 46 11538.4800 0.0832 3.813 a 377.304 17 2 16 16 3 13 12099.6454 0.1225 1.164 b 42.991 21 5 17 22 4 18 12194.8313 0.3327 1.599 b 78.895 26 4 22 25 5 21 12306.8190 0.3305 1.995 b 104.315 42 5 37 42 5 38 12525.9915 0.0722 3.230 a 257.435 24 3 21 24 3 22 12613.9284 0.0797 2.100 a 86.219 1Line strengths for a-, b-type transitions multiplied by the square of the dipole moment component in Debye along the a or b principal axis. 2Type of transition – a, or b. NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. Table 14: Predicted transition frequencies of the trans conformer of ethyl formate in its vibrational ground state. 2 1 Frequency Uncertainty µa,b,c S 2 Eupper J´ Ka´ Kc´ J" Ka"Kc" 2 Type /K /MHz /MHz /D k 16 4 12 164 13 6853.1677 0.0229 3.163 a 51.654 25 6 19 25 6 20 6970.5105 0.0249 4.441 a 122.510 2 1 1 2 0 2 7444.9710 0.0039 2.615 b 1.370 12 3 9 12 3 10 7559.8909 0.0226 2.375 a 29.663 30 7 23 30 7 24 7937.8104 0.0321 4.892 a 174.468 3 1 2 3 0 3 8535.9162 0.0054 3.380 b 2.430 21 5 16 215 17 8555.9890 0.0272 3.598 a 87.031 35 8 27 35 8 28 8670.4614 0.0400 5.352 a 235.555 40 9 31 40 9 32 9133.4247 0.0433 5.827 a 305.739 45 10 35 45 10 36 9299.2540 0.0397 6.320 a 384.981 17 4 13 174 14 9968.0784 0.0298 2.806 a 57.605 106 26 6 20 26 6 21 10031.2634 0.0325 4.035 a 131.579 4 1 3 4 0 4 10132.7653 0.0081 3.887 b 3.839 9 2 7 9 2 8 10861.2356 0.0241 1.410 a 16.893 13 3 10 133 11 10879.8252 0.0294 2.077 a 34.228 31 7 24 31 7 25 11236.8356 0.0416 4.475 a 185.277 17 8 10 16 9 7 11369.0767 0.0300 1.210 b 71.786 17 8 9 16 9 8 11369.2256 0.0300 1.209 b 71.786 56 12 44 56 12 45 11867.7873 0.0853 6.939 a 589.805 1Line strengths for a-, b-type transitions multiplied by the square of the dipole moment component in Debye along the a, b or c principal axis. 2Type of transition – a, b or c. NOTE: This table is available in its entirety on the attached CD which contains the supplementary materials. Table 15: Predicted transition frequencies of the gauche conformer of ethyl formate in its vibrational ground state. CHAPTER 7 THE SOLID STATE FASSSTER (FAst Scan Submillimeter Spectroscopic Technique with Electronic Reference) SPECTROMETER. 7.1 INTRODUCTION The unique spectral fingerprint of every polar molecule and the strong maxima of their interaction strengths with the electromagnetic radiaion in the millimeter and submillimeter wave region [2] make this spectral region particularly advantageous for the study of these molecules. Laboratory investigations have included the spectroscopy of small (few atoms) fundamental species, ions, and reactive radicals; studies of high temperature systems, excited states, and plasmas; and investigation of collisional energy transfer, pressure broadening and low temperature systems [95-98]. Additionally, a number of more applied applications have been proposed including analytical chemistry, industrial safety, and chemical sensing [30]. The potential analytical use of the molecular rotational fingerprint in microwave region was recognized and discussed 50 years ago by Townes and Schawlow [19]. About 25 years later Hewlett Packard introduced a commercial millimeter wave instrument, but during its limited production it gained considerably more favor with spectroscopists interested in molecular structure than with the analytical community. Due to its size (it 107 filled a small room), cost (several hundred thousand dollars), and limited capability (it operated below 50 GHz) it was not widely suited for analytical use. The FAst Scan Submillimeter Spectroscopic Technique (FASSST) [4] provides a powerful, fast and simple way of conducting spectroscopic measurements in the millimeter and submillimeter wave region. A new, compact, all solid state system employing the FAst Scan Submillimeter Spectroscopic Technique has been developed in the course of this work. A key element of the development of the new, compact system was the replacement of the large (~20 m) Fabry-Perot or ring cavity previously used for frequency calibration in the original Backward Wave Oscillator (BWO) tube implementation (see Chapter 1). This was accomplished with a compact, scanning electronic system which is described in this chapter. The resulting extension of the FAst Scan Submillimeter Spectroscopic Technique, called FASSSTER (FAst Scan Submillimeter Spectroscopic Technique with Electronic Reference), uses electronically generated reference markers for frequency calibration. The new system architecture replaces the complexity and slow speed of the phase locked multiplier chains typical of microwave spectrometers with the fast scan of a solid state voltage tunable oscillator (VTO) and fast, parallel digitization of the signal and reference channels. Additionally, the choice of a low voltage VTO eliminates bulky magnets and high-voltage power supplies employed by the BWO-based FASSST spectrometer. The desired frequency of the radiation field is realized by a solid state multiplier chain. As in the case of the BWO- based FASSST spectrometer, the simplicity of the hardware is complemented by sophisticated calibration software. 108 VTO sweeper VTO Frequency Frequency RF Mixer 1 LO reference 10.285 – 10.885 GHz IF standard 10.585 GHz Frequency Directional synthesizer coupler RF Harmonic ×8 multiplier Mixer 2 LO Amplifier 1 10 MHz comb IF W-band generator ×24 Amplifier 2 W-band amplifier low pass filter 75-110 GHz 100 kHz ×3 multiplier Computer W-band Frequency roll-off Gas cell Detector preamplifier Figure 21: Diagram of the FASSSTER spectrometer. The frequency markers are produced by down-conversion of the VTO signal and mixing with the signal of the harmonic comb generator. The block diagram of the FASSSTER spectrometer is shown in Figure 21. The main components of the system are: a) a VTO radiation source tunable between 10.285 GHz and 10.885 GHz; b) a sweeper which produces a voltage ramp applied to the VTO; c) a directional coupler; d) a frequency multiplier chain; e) a copper absorption gas cell 4 feet in length and 0.75 inches in diameter; f) a detector unit; g) an external frequency standard; h) a tunable frequency synthesizer operating near the center of the VTO’s range (10.585 GHz), which is used for down-conversion of the VTO signal; i) a harmonic comb generator, which provides a comb of reference frequencies; j) frequency mixers; k) an 109 amplifier with adjustable roll-off; and l) a data acquisition system with computer. We will describe the characteristics and properties of some of these spectrometer constituents. 7.2 Voltage Tunable Oscillator (VTO) Backward wave oscillators (BWOs) set high standards for a radiation source in the millimeter and submillimeter wave range. The high spectral purity (<20 kHz), high power output (see Table 1) and broad spectral coverage of a BWO make it the radiation source of choice for spectroscopic applications. A wide adoption of these tubes is encumbered by high voltage requirements, high power dissipation which requires water cooling, and by a requirement of an external magnetic field. The choice of a low-frequency VTO as a radiation source has advantages as well as drawbacks. The biggest advantages of VTOs are low voltage (~10 V), low power consumption (~10W), low heat dissipation and commercial availability. Additionally, the fact that frequency multiplication starts at frequencies close to 10 GHz allowed us to develop a novel calibration scheme based on the down-conversion of the VTO radiation signal followed by mixing with the signal of a harmonic comb generator. While the fundamental phase noise, which is commonly used as a measure of spectral purity for VTOs, is low, the actual frequency stability of the radiation source depends on the noise characteristics of the frequency-tuning voltage signal. If we assume that the frequency voltage characteristic of a VTO is linear, then the short term frequency stability δf is related to the short term voltage stability δV as δ f δV = , (7.1) f V 110 where f is the radiation frequency and V is the tuning voltage. Thus, if we assume that f = 300 GHz, δf = 100 kHz, and V = 10 V this requires δV equal to ~3 µV. To achieve this level of the desired frequency noise, the voltage ramp is generated by means of an integrator with a time constant of about 1 s. This suppresses very effectively voltage fluctuations on the time scale of a sweep through individual spectral feature (~50 µs) and allows us to achieve the desired short term voltage stability. The fact that the VTO signal is multiplied by the multiplier chain leads to an increase of the phase noise [99]. Noise associated with the multiplier chain leads mostly to AM modulation of the resulting signal, while noise associated with the tuning voltage leads mostly to FM modulation. Additionally, while the output power level of the currently available VTOs is rather flat across their operational range, multiplier chains introduce power fluctuations, resulting in a varying power level across the final frequency range. Despite all these factors the achievable level of spectral purity (~50 kHz) attained with commercial VTOs followed by multipliers is fully satisfactory for spectroscopic applications, if we take into consideration that the Doppler limited line width for most molecules is about 500 kHz at 300 GHz at room temperature. Furthermore, power fluctuations at the output of the multiplier chain have a periodicity of a few gigahertz, which allows us to faithfully reproduce local intensity pattern for most molecular species. In the process of the development of the FASSSTER spectrometer we tested various types of VTOs as candidates to be used as radiation sources: 1) YIG (Yttrium-Iron Garnet) based oscillators; 2) coaxial resonator VTOs; and 3) transistor based VTOs with integrated resonator. 111 YIG oscillators [100] can be tuned over wide frequency ranges (a typical YIG based VTO can be swept form 8 GHz to 18 GHz) with an external DC magnetic field. YIG resonators also have a high Q factor resulting in a low phase noise (-123 dBc at 100 kHz offset) when used in an oscillator. Finally, the tuning characteristic of a YIG oscillator is linear, since the frequency is directly proportional to the applied DC magnetic field. Thus, to generate a linear frequency sweep one has to linearly change the current through the tuning coils of a YIG VTO. Building a stable current supply capable of sweeping at a rate of 50 mA/s, to achieve 20 GHz/s sweep rate when multiplied up by 24 (see Figure 21), (20 MHz/mA tuning rate is typical of YIG oscillators) over a ~0.5 A range proved to be challenging. Thus, we turned our attention to the resonator-based VTOs. The output frequency of a resonator-based VTO is tuned by adjusting a varactor [101] voltage. This, in part, precludes this type of VTO’s from having as broad a tuning range as YIG-based VTOs. Unlike YIGs, the frequency tuning input of this type of VTOs has a high impedance. Coaxial resonator VTOs use a piece of microstrip/coaxial waveguide as resonator. VTOs of this type, manufactured by MITEQ (http://www.miteq.com), employ a free- running coaxial resonator VTO with its output multiplied to achieve the final output frequency. The maximum tuning bandwidth that can be achieved with this device is about 5% of the operational frequency. The phase noise typical for this type of VTOs is ~ -110 dBc/Hz at 100 kHz offset. Two interchangeable VTOs purchased from MITEQ are used by the FASSSTER spectrometer. A unit tunable between 9.312 GHz and 9.812 GHz (MITEQ model number CRM-9562-15P-SP), is followed by an integrated microwave amplifier, which produces 112 22 dBm of power. When multiplied by 24 (see Figure 21), it produces radiation from 223.5 GHz to 235.5 GHz. Another VTO tunable from 10.285 GHz to 10.885 GHz (MITEQ model number CRM-10585-15P) covers the range from 246.8 GHz to 261.2 GHz. Its output power varies between 14 dBm and 16 dBm. Thus, when attached to the multiplier chain, these VTOs require different sets of attenuators to match the optimum input power of the x8 multiplier (see Figure 21). Both VTOs require a 15 V (0.5 A) DC power supply. They are tuned by a voltage varying between 0 V and 15 V. Transistor-based VTOs with an integrated resonator can be purchased from a number of manufacturers. They come as unmounted chips as well as packaged devices. Table 16 contains parameters of various VTOs produced by MITEQ (http://www.miteq.com), MICRONETICS (http://www.micronetics.com), and HITTITE (http://www.hittite.com). As can be seen from Table 16, VTOs produced by MITEQ exhibit lower phase noise values, which can be explained in part by the higher quality factor of the coaxial resonator relative to the quality factor of the resonators integrated onto transistor based VTOs produced by HITTITE and MICRONETICS. Development of a source based upon transistor based VTOs which would meet input frequency and power requirements of the multiplier chain (see Figure 21), would involve design of a circuit containing multipliers, amplifiers and filters. Due to the complexity of such a design we resorted to using VTOs produced by MITEQ. 113 Phase VTO Tuning Output Coverage Power noise at frequency voltage frequency of VTO avg. 100 kHz Multiplier range range range H-band2 /dBm offset1 / GHz / V / GHz / % /(dBc/Hz) MITEQ 9.312 - 223.5 - 0 - 15 22 - 82 24 10 CRM-9562-15P-SP 9.812 235.5 MITEQ 10.285 - 246.8 - 0 - 15 15 - 82 24 12 CRM-10585-15P 10.885 261.2 MICTONETICS 5.0 - 240 - 0 - 20 10 - 74 48 62 MW500-1262 6.5 312 MICTONETICS 2.25 - 216 - 114 0 - 17 7 -63 96 100 MW500-1114 3.55 340 HITTITE 5.8 - 278 - 0 -10 10 -76 48 41 HMC358MS8G 6.8 326 1The value of the phase noise corresponds to the output frequency of the multiplier chain 2H-band extends from 217 GHz to 333 GHz Table 16: Specifications of the selected commercially available voltage tunable oscillators. 7.3 VTO TUNING VOLTAGE SWEEPER In order to generate a tuning voltage ramp which would satisfy the spectroscopic short term voltage stability requirements (see Section 7.2), the circuit shown in Figure 22 was developed. This circuit is powered by a +/-20 V stable power supply attached to connector J1 (see Figure 22). The design of this stable power supply was borrowed from the circuit design of the voltage sweeper used by the BWO-based FASSST spectrometer. Figure 22: Circuit design of the voltage sweeper used by the FASSSTER spectrometer. The integrating circuit responsible for generation of a triangular voltage wave form, which is applied to the VTO, is implemented on the basis of a IVC-102 precision 115 integrating amplifier chip labeled U1 in Figure 22. The integrating time constant is determined by the values of resistor R1 = 20 kΩ and capacitor C3 = 47 µF and is equal to 0.94 seconds. The integrator requires two input signals, as shown in Figure 23: 1) The signal that undergoes integration (see Figure 23-b) is applied to connector J3. 2) The reset signal applied to the reset pin of the integrator chip to drain any leftover charge from the integrating capacitor (see Figure 23-c) is applied to connector J2. Both of these signals (see Figure 23-b,c) are synthesized by a program written using the Labview development system [16] and applied to connectors J2 and J3 via analog outputs of the multifunction data acquisition board NI PCI-6115 (http://www.ni.com) installed in the data acquisition computer. The amplitude of the rectangular signal which undergoes integration (Figure 23-b) determines the sweep rate, while its duration defines maximum amplitude of the resulting voltage sweep. A full cycle of the tuning voltage wave form is shown in Figure 23-a together with synthesized wave forms used for integration and digitized frequency markers, which are discussed in what follows. Summation of the integrator output and the variable voltage offset defined by the setting of the variable resistor R3 takes place at the operational amplifier U3. The resulting voltage wave form, shown in Figure 23-a, corresponds to the offset voltage equal to 14 V. If a constant tuning voltage is required, the integrating circuit can be bypassed altogether by the setting of switch S1. The resulting voltage wave form is applied to connector J5. Connector J4 provides the same signal buffered off by the operational amplifier U5 for digitization and monitoring. 116 Figure 23: A full cycle of the signals synthesized and digitized by the data acquisition computer: a) digitized up-down voltage ramp signal generated by the voltage sweeper; b) synthesized and digitized integrator input signal; c) synthesized integrator reset signal; d) digitized frequency marker signal. The sweep voltage parameters are adjusted for every VTO employed by the FASSSTER spectrometer to achieve a frequency scan rate at the output of the multiplier chain of ~ 20 GHz/s. For the Doppler limited line shape this translates into ~ 50 µs of integration time per Doppler limited spectral feature. 117 7.4 FREQUENCY REFERENCE The external frequency standard (see Figure 21) used by the FASSSTER spectrometer is the model Xli GPS Time & Frequency Generator produced by Symmetricom, Inc. (http://www.symmetricom.com), which combines a rubidium oscillator with GPS synchronization for the highest possible short and long term stability. The 10 MHz signal generated by the Xli GPS Time & Frequency Generator unit is used as an external clock signal for the reference frequency (see Figure 21) synthesizer unit MLSL-0912 produced by Micro Lambda Wireless, Inc. (http://microlambdawireless.com). The MLSL-0912 synthesizer puts out 10 (+/-2) dBm of power and can be tuned in 500 kHz steps from 9 to 12 GHz. Its frequency is controlled by a standalone program, written in C++, communicating with the unit through a serial port of the data acquisition computer. The MLSL-0912 operates near the center of the frequency range of the VTO employed by the FASSSTER spectrometer and provides the signal necessary for down-conversion of the VTO’s frequency at Mixer 1 (see Figure 21). The 10 MHz reference signal produced by the Xli GPS Time & Frequency Generator unit is also used as an external clock signal for the DS345 function generator produced by Stanford Research Systems, Inc. (http://www.srsys.com), which is used as a driver for the harmonic comb generator (see Figure 21). The DS345 provides an opportunity to adjust the harmonic comb spacing and the driving power applied to the comb generator and commonly puts out a 10 MHz sine wave signal with 17 dBm of power. The combination of the MLSL-0912 synthesizer and the DS345 function generator give us a lot of flexibility in choosing experimental conditions. The MLSL-0912 unit 118 could easily be replaced with a phase locked source while the DS345 unit could be replaced with an amplifier, which would make the overall system a lot cheaper and simpler. 7.5 DIRECTIONAL COUPLER AND MIXERS The FASSSTER spectrometer uses a C2068-10 wideband directional coupler produced by MAC Technology, Inc. (http://www.mactechnology.com) for coupling a fraction of the VTO signal into the part of the spectrometer responsible for frequency calibration. The C2068-10 operates in the frequency range form 7 to 18 GHz. The ratio of the power measured at the output port relative to that measured at the coupled port (coupling parameter) equals 10 dB for this unit. The C2068-10 has an insertion loss of 0.5 dB. For the CRM-10585-15P VTO produced by MITEQ (see Table 16), which will be used as an example for the following discussion, the average coupled power equals 5 dBm and the average transmitted power reaching the input of the multiplier chain equals 9 dBm. The mixer (M14A double-balanced mixer, produced by M/A-COM http://www.macom.com) responsible for mixing the VTO’s signal with the reference signal is labeled ‘Mixer 1’ in Figure 21. The 10.585 GHz reference signal generated by the MLSL-0912 synthesizer is attenuated by a 10dB attenuator, for better matching with the RF input power, and is applied to the M14A mixer. The output of the M14A mixer is applied to the input of ‘Mixer 2’ (see Figure 21). 119 ‘Mixer 2’ (see Figure 21) a model ZFM-1H produced by Mini-Circuits (http://www.minicircuits.com) and responsible for mixing the down-converted VTO’s signal from the output of ‘Mixer 1’ with the amplified (by ‘Amplifier 1’) output signal of the harmonic comb generator. 7.6 HARMONIC COMB GENERATOR The FASSSTER spectrometer employs the harmonic comb generator model 8406A produced by Hewlett Packard (no longer in production). It uses a step recovery diode for harmonic generation. In the forward-biased mode of operation, step recovery diodes behave as noraml diodes. However, when back-biased, they continue to conduct due to stored charge carriers. When the stored charge carriers are depleted the diode shuts off abruptly, thus, forming a short pulse. When driven with an alternating signal a step recovery diode produces a sequence of short pulses at the frequency of the driving signal, which in the frequency domain corresponds to a harmonic frequency comb. The 8406A unit generates a harmonic comb spaced by 10 MHz usable up to 2 GHz (the intensity drop off is less than 10 db). The output of the harmonic comb generator is amplified by the ZFL-2000 amplifier (Mini-Circuits, http://www.minicircuits.com), which provides 20 dB of gain in the frequency range from 10 MHz up to 2GHz. The resulting signal is applied to the input of the ‘Mixer 2’ (see Figure 21). 120 7.7 DETECTORS Both a solid state detector and hot electron InSb bolometers cooled to liquid 4He temperature (see Section 1.5) have been used to record data with the FASSSTER spectrometer. The best performance among solid state detectors tested by us so far is demonstrated by the Schottky diode detectors produced by Virginia Diodes, Inc. (http://www.virginiadiodes.com). Their detector model WR-3.4ZBD operates in the H- band (220 to 330 GHz). Its typical NEP (noise equivalent power) (see Section 8.2.2.3) of 10-10W/Hz-1/2 is inferior to the NEP typical for InSb detectors (10-12W/Hz-1/2). The noise spectrum of these solid state detectors is dominated by 1/f noise (see Section 8.1.6) for electrical frequencies below 100 kHz (Figure 24), which is typical of semiconductor components. Despite the higher noise values, WR-3.4ZBD exhibits excellent responsivity (2000 V/W average), high saturation power (several milliwatts), and higher (at high enough input power levels) output bandwidth than InSb detectors. The detector is followed by an SRS SR560 amplifier with adjustable band-pass. As in the case of the BWO-based FASSST spectrometer, the low frequency roll-off is typically set at 10 kHz to decrease the baseline variations and eliminate most of the 1/f noise. This selection transforms the Doppler (Gaussian) line shape function of the absorption signal into an approximate first derivative. The high frequency roll-off, typically set at 30 kHz, decreases the high frequency noise. The WR-3.4ZBD detector block has a horn antenna for collecting and coupling of the radiation into the input waveguide of the detector (see Section 1.3). 121 Figure 24: Noise spectrum of the WR-3.4ZBD detector at 428 µW of detected power and 10 kHz – 1MHz of electrical bandwidth. In order to match the noise performance of the diode detector to that of an InSb detector one can shift the signal retrieval to higher electrical frequencies, to overcome 1/f noise, which will require higher frequency amplifiers and faster digitizers for data acquisition. Furthermore, one would have to integrate the signal longer than in the case of the InSb detectors. However, the fact that solid state detectors operate at room temperature and do not require any cryogenic equipment to operate makes them an attractive alternative for certain spectroscopic applications. Additionally, they saturate at higher power, thereby compensating for their lower NEP. 122 7.8 FREQUENCY MULTIPLIER CHAIN The frequency multiplier chain used by the FASSSTER spectrometer consists of 3 parts: 1) ×8 W-band (output frequency) multiplier model AW-8X (Spacek Labs, Inc., http://www.spaceklabs.com); 2) W-band amplifier model SPW-15-10 (Spacek Labs, Inc., http://www.spaceklabs.com ); and 3) ×3 W-band (input frequency) multiplier model WR3.4x3 (Virginia Diodes, Inc., http://www.virginiadiodes.com). The WR3.4x3 multiplier block has a horn antenna at its output for collimation of radiation. These components are matched in input/output power levels such that the system puts out ~500 µW of power averaged over the H-band (220 to 330 GHz). Information about frequency dependence of the power output of the source can be gleaned from Figure 25, which shows detector signal as a function of frequency with radiation chopped at 500 Hz. For the case shown in Figure 25, the power level of the signal intercepted by the detector is below saturation and, thus, the detector operates in the square law regime (see Section 8.2), which means that signal strength at the output of the detector is proportional to the power of the detected radiation. Several factors, such as standing waves along the path of the radiation, as well as horn antenna effects, influence coupling of the source radiation onto the detector element. The results shown in Figure 25 indicate that the power of the radiation reaching the detector element varies by more than a factor of two over the frequency range of ~10 GHz, while on the scale of ~1 GHz this power fluctuation is much smaller. 123 Figure 25: Detector signal as a function of frequency with radiation chopped at 500 Hz. 7.9 ELECTRONIC FREQUENCY MARKERS This section details the way in which frequency markers used to calibrate the frequency are generated and recorded with the FASSSTER spectrometer. The output signal of the VTO, which is being swept over a certain frequency range, is mixed at ‘Mixer 1’ (see Figure 21) with the signal of the frequency reference synthesizer, which is generating a signal at a frequency close to the center of the VTO’s sweeping range (10.585 GHz). This down-converts the signal of the VTO, so that the resulting signal has zero frequency at the moment when the VTO’s frequency equals the frequency of the reference signal (10.585 GHz). The down-converted signal is then mixed (see Figure 21, 124 ‘Mixer 2’) with the output of the harmonic comb generator with frequency modes spaced by 10 MHz. Again, the resulting signal has zero frequency at the moments when the frequency of the signal equals one of the frequencies of the individual comb modes (Integer Number x 10 MHz). The resulting signal is then passed through the SRS SR560 amplifier (Stanford Research Systems, http://www.srsys.com ) with the low-frequency roll-off set at DC level and the high-frequency roll-off set at 100 kHz. In this case the SR560 unit is used as a low-pass filter with variable gain. 24 × 10.585 GHz 24 × 10 MHz Figure 26: Frequency markers produced by mixing of the 10.585 GHz frequency reference with the VTO’s signal followed by mixing with the harmonic comb spaced by 10 MHz. The frequency scale is multiplied by 24. The frequency marker corresponding to the reference frequency (10.585 GHz) is missing. The frequencies of every marker are calculated with respect to the missing marker, the frequency of which is known. The intensity is in arbitrary units. 125 Figure 26 shows the frequency markers spaced by 10 MHz at the fundamental frequency with a marker missing at the VTO’s frequency of 10.585 GHz (reference frequency). The fact this marker is not present is explained by the fact that the fundamental frequency of the harmonic comb is at 10 MHz. The frequency scale is multiplied by 24 to correspond to the output frequency of the multiplier chain. Figure 27 shows the frequency range surrounding the frequency marker located at 10.335 GHz at the fundamental frequency. The phase information is preserved, so that by numerically determining the positions of the central feature of each digitized frequency marker one can interpolate between adjacent markers to determine frequencies of every point in the spectrum. 24 ×100 kHz Figure 27: Frequency marker at the 24th harmonic of 10.355 GHz. The shape of the marker is determined by the low pass filter with high frequency roll-off set at 100 kHz. Intensity is in arbitrary units. 126 7.10 SHAPE ANALYSIS OF THE FREQUENCY MARKERS GENERATED BY THE FASSSTER SPECTROMETER The computational simulation of the experimental shape of the frequency markers generated by the FASSSTER spectrometer is very similar to the analysis of the experimental spectral line shape of the FASSST spectrometer presented in Section 1.8. The results presented in Section 1.8 are totally applicable to the experimental spectral line shape of the FASSSTER spectrometer and will not be repeated in this chapter. For the purpose of this calculation we will assume that frequency of the signal at the output of ‘Mixer 2’ (see Figure 21) is a linear function of time ν ()tSRtt= (− 0 ), (7.2) where ν is the radiation frequency and SR is the instantaneous sweep rate of the radiation source. The signal at the output of ‘Mixer 2’ can be expressed as the real part of the following complex function itt(2πνϕ ( )+ 0 ) St()= Se0 , (7.3) where S0 is the amplitude of the signal and ϕ0 is an arbitrary phase between the down- converted VTO’s signal and the signal of the comb generator. These two signals are not coherent, thus ϕ0 changes from one frequency marker to another. The filter response function representing the roll-off amplifier can be expressed as a product of the high pass and low pass response functions 127 f f 1+ i LF 1− i f f Hf()= HF , (7.4) c 2 ⎛⎞f ⎛⎞fLF 1+ ⎜⎟1+ ⎜⎟ ⎝⎠f ⎝⎠fHF where f is the frequency of the electrical signal at the input of the filter, and fLF and fHF are electrical frequencies corresponding to the low and high frequency limits respectively of the roll-off preamplifier. The Fourier content of the electrical signal at the output of the detector is modified according to equation 7.4 and is later recorded by the data acquisition board. In order to simulate the experimental line shape of the frequency markers produced by the FASSSTER spectrometer, the following steps must be undertaken: 1) The Fourier transform of the electrical signal at the output of the ‘Mixer 2’ (Equation 7.3) must be multiplied by the filter response function (Equation 7.4). 2) The inverse Fourier transform of the resulting expression must be calculated. The real part of the complex expression obtained corresponds to the simulated experimental line shape. The supporting calculations were carried out using the Mathematica® software package [23]. The results of these calculations are presented in Figure 28. 128 ϕ0 = 70 ϕ0 =−30 Figure 28: Simulated experimental shapes of two frequency markers produced by the FASSSTER spectrometer corresponding to two values of the phase shift ϕ0 between the VTO’s down-converted signal and the signal of the comb generator. The frequency roll-offs used for this calculation were νLF = 100 Hz and νHF = 100 kHz. The sweep rate of the VTO was set at 400 MHz/s. The results presented in Figure 28 reveal a small frequency shift of the marker’s centers as well as slight asymmetry, both of which are caused by the presence of the low frequency roll-off. The best match between experimental and calculated profiles corresponds to the low frequency roll-off equal to 100 Hz. As in the case of the BWO-based FASSST spectrometer (see Section 1.9) we implemented averaging of up and down in frequency scans to achieve the maximum accuracy of the frequency measurement. 129 7.11 DATA ACQUISITION SOFTWARE A personal computer (Intel 1 GHz, 1.5 Gbytes of RAM) is currently used to control the experiment. A digitizer NI PCI-6115 (http://www.ni.com) provides independent variable gain control for four channels. The analog outputs of the board are used to apply the synthesized voltage waveforms to the VTO sweeper circuit described in Section 7.3. Data acquisition is triggered by the signal applied to the integrating input of the VTO sweeper. The data acquisition system digitizes the input from four channels: 1) the spectrum signal, 2) the frequency marker signal, 3) the voltage ramp signal and 4) the trigger voltage signal taken directly from the analog output of the data acquisition board. The spectrometer data acquisition software is written in Labview [16] and has several operational modes. For example, if the software is operating in the averaging mode it is capable of recording multiple scans which are later co-added by the calibration software. All acquired data for each individual scans are initially saved as individual files for later off-line processing by the calibration software. Each data file contains scans up and down in frequency. Each sweep is calibrated separately before the individual scans are co- added. 130 7.12 CALIBRATION SOFTWARE The calibration software of the FASSSTER spectrometer is based on the IgorPro© package [17], which provides a convenient graphical and programming interface. In the averaging mode as for the FASSST spectrometer, the software automatically calibrates each individual scan and co-adds all calibrated scans. The algorithm used for calibration of the spectra recorded with the FASSSTER spectrometer consists of four main parts. 1) The first part is responsible for loading of the experimental data from the files created with the data acquisition software. 2) The main purpose of the second part of the calibration package is to identify and extract the needed portions of the loaded spectrum and frequency marker arrays. The trigger channel is used to identify the margins of the segments to be extracted. Up and down in frequency scan are separated into separate sets of arrays at this stage. Also, an optional convolution of the spectrum with the first derivative of the Gaussian line profile takes place at this stage of the calibration process. 3) In the third step of calibration, the centers of the frequency markers are identified. This is done by means of a robust yet simple algorithm described at the end of this section. 4) In the final step of the calibration, the frequencies of all data point in the spectrum array for both up and down in frequency scans are calculated. First the software calculates and compares differences between bin-indices of the adjacent 131 frequency marker centers identified in the previous step. The largest gap corresponds to the markers adjacent to the missing marker (see Figure 26), which corresponds to the instant when the frequency of the VTO was equal to the reference frequency (see Figure 21). Using the value of the reference frequency provided as an input parameter, center frequencies of every frequency marker are calculated. Using the user specified order of the interpolation polynomial, the software calculates frequencies of every data point by means of polynomial interpolation between frequencies of the adjacent markers. For the nth order polynomial interpolation there exist n ways to select n frequency markers surrounding each data point. Each subset of markers defines a polynomial functions which can be used to calculate the frequency of a particular point. The calibration software allows a user to perform an interpolation for any chosen subset of n frequency markers as well as to calculate the average of interpolation results for all n polynomial functions corresponding to a given interpolation order n. The resulting frequency scale is then multiplied by the multiplication factor of the multiplier chain provided by a user. The software stores parameters used for the calibration of the last loaded spectrum. Thus, in order to perform averaging, the user must calibrate the first file of a data set interactively. In the averaging mode the software then calibrates each subsequent scan, adjusts the frequency scales and adds points of the spectral arrays corresponding to the same frequency. The numerical determination of the frequency marker centers mentioned in 3), above, is done in the following three steps: 132 a) First, every peak of a frequency marker signal above a certain threshold is located and its intensity stored. Figure 29 shows the result of this step for one of the markers and with the value of the threshold set to 250. Every recorded frequency marker must have peak intensity above the threshold, since the same value of the threshold is used to identify centers of all markers. Figure 29: Peaks of the frequency marker signal array above the threshold equal to 250 (blue dots) are used to determine the positions of the marker centers. b) The separation between adjacent peaks (blue dots in Figure 29) is then analyzed in search of the largest differences. The positions of the peaks between those largest differences allow us to roughly identify the margins of the frequency markers. 133 c) The section of the frequency marker array between left-most and right-most peaks belonging to the same marker is extracted and duplicated into a separate array, which is reversed such that its last element becomes its first. The reversed and the original arrays are then cross-correlated. The position of the central maximum of the cross-correlated waveform contains information about the position of the marker center, which is then calculated. 7.13 FREQUENCY MEASUREMENT ACCURACY OF THE FASSSTER SPECTROMETER To estimate the accuracy of frequency measurements with the FASSSTER spectrometer we measured the spectrum of sulfur dioxide in the frequency range from 247 to 257 GHz. The peak frequencies of spectral lines were obtained using averaged (see Section 7.12) polynomial interpolations of the 1st, 2nd, 3rd, 4th, and 5th order between markers. The comparison of the experimental results to the catalog frequencies of SO2 taken from JPL database [1] is presented in Table 17. Transition from the linear to the 2nd order interpolation leads to a reduction in the standard deviation of the residuals by almost an order of magnitude. This is explained by the fact that the frequency voltage dependence of the VTO employed by the FASSSTER spectrometer is best fitted not by a linear function, but by a higher order polynomial. Table 17 indicates that the best result is obtained by using the 4th order polynomial interpolation. The accuracy of the frequency measurement demonstrated by the FASSSTER spectrometer is fully comparable to the accuracy of the BWO-based FASSST 134 spectrometer (see Section 1.9) despite the fact that the frequency is obtained by interpolation between frequency markers spaced by 240 MHz. An interpolation this long in the case of the BWO-based FASSST spectrometer would lead to a much higher uncertainty in the frequency measurement (see Figure 6). This significant difference in the behavior of the BWOs and solid state VTO can be explained by the relative physical dimensions of the two types of radiation sources. The small scale structure of the frequency-voltage dependence of BWOs exhibits a rough periodicity of 1 to 2 GHz (see Figure 6). This can be explained by the complex standing wave pattern present within the BWO tube with spacing between maxima of 2 GHz, which corresponds to the physical size of a resonator equal to ~ 7.5 cm. This size is defined by the distance from the back of the BWO tube to the output of the waveguide transition. The standing waves within the tube influence the interaction of the electrons with the slow wave structure, thus, modifying the output frequency. The much smaller physical size of resonators employed by solid state VTOs (for some VTOs it is less than a millimeter) significantly increases the frequency spacing of the standing wave modes supported, which leads to an apparent smoothness of the frequency-voltage dependence over the implemented frequency range. 135 Catalog 1st order 2nd order 3rd order 4th order 5th order transition interpolation interpolation interpolation interpolation interpolation frequency/MHz residual/Hz residual/Hz residual/Hz residual/Hz residual/Hz 247169.767 -33070.258 -4763.903 11669.743 4893.336 6991.646 247275.015 -57332.563 -39442.052 -14486.174 -21477.848 -19433.615 247485.480 -147572.352 -12138.909 6484.024 -600.511 -524.018 248057.401 -83945.199 10129.060 9660.523 9740.813 9984.620 248830.824 -355857.364 -3881.620 -5383.997 -4963.166 -4518.354 248995.115 -37209.827 -3767.633 -3366.190 -3352.027 -3283.361 250816.786 -470581.407 16604.592 11132.431 832.797 -6770.072 251199.675 -286737.625 -28313.188 -19704.126 -11067.115 -2736.694 251210.586 -345457.112 -33803.075 -23537.224 -13447.164 -3813.482 251428.543 -136321.785 30622.767 21258.138 15302.367 11362.395 251450.183 -206802.739 56220.468 41131.477 31673.648 25677.471 252563.891 -186921.904 5576.742 4616.725 3822.109 5557.559 252731.065 -343441.143 14296.553 17607.208 18696.933 19280.923 253935.885 -827173.776 -80556.880 -63559.602 -60360.383 -65927.174 253956.567 -815823.916 -7105.090 10493.388 16522.533 20273.408 254194.874 -521681.510 11547.604 7276.444 8741.389 9159.462 254280.536 -1473.692 200.730 188.133 183.841 241.518 254283.319 -15212.159 -4651.201 -4724.701 -4747.122 -4732.745 255553.303 -73930.130 19325.817 24542.115 29809.669 35826.959 255595.335 -127974.087 -18422.634 -13266.070 -7684.964 -2071.260 255818.414 -53327.768 27420.798 21762.724 16752.917 11547.434 255958.045 -7939.794 -5165.677 -5506.240 -5602.253 -5773.002 256246.946 -44462.066 -162.611 475.644 3480.698 5413.591 257099.966 -66037.905 9096.728 17844.522 25892.570 20843.742 257318.869 -54059.713 22508.568 11003.087 20714.044 21585.400 257420.280 -10405.414 24196.599 12539.987 7487.851 4924.082 Standard deviation / kHz 234.1 26.7 19.9 18.6 18.7 Table 17: Accuracy of the transition frequencies of sulfur dioxide (SO2) measured with the FASSSTER spectrometer corresponding to different orders of the polynomial interpolation between frequency markers. The first column contains catalog frequencies of SO2 transitions taken from the JPL catalog [1]. Columns 2 to 6 contain differences between catalog and measured frequencies. Standard deviations of the residuals are at the bottom of the table. 136 7.14 SENSITIVITY OF THE FASSSTER SPECTROMETER Figure 30: Single scan of methanol spectrum recorded with FASSST and FASSSTER spectrometer. Traces a) and c) demonstrate the noise levels of the FASSST and FASSSTER spectrometer respectively. Trace b) and d) show the intensities of the strongest spectral lines. To assess the detection capabilities of the solid state spectrometer, two single scans of the methanol spectrum were recorded, one with the BWO based FASSST spectrometer and one with the FASSSTER spectrometer employing a solid state detector. The results of this comparison are presented in Figure 30. The signal to noise ratio for the strongest lines of methanol measured with the FASSSTER spectrometer in the frequency range shown in Figure 30 equals ~500. The signal to noise ration for the strongest lines recorded with the FASSST spectrometer is at 137 least ten times higher. Lines that are hidden in noise in the FASSSTER spectrum are recorded with a good signal to noise ratio by the FASSST spectrometer. Several factors contribute to this difference in sensitivity. The FASSST spectrometer uses hot electron InSb bolometers cooled to liquid 4He temperature to record spectra, which is less noisy than the solid state detector employed by the FASSSTER spectrometer. The absorption cell of the FASSST spectrometer is 5 times longer that the absorption cell of the FASSSTER spectrometer. The power level at the output of the multiplier chain used by the FASSSTER spectrometer does not exceed 500 µW. This power level is below the saturation limit of the solid state detector. Thus, increase in the radiation power would lead to an increase in signal-to-noise ratio assuming that no molecular saturation occurs. Finally, noise in the solid state multiplier chain contributes to the overall noise level of the FASSSTER spectrometer. 7.15 DISCUSSION A small, low power consumption, all solid state system has been developed which is capable of recording high resolution spectra in the millimeter and submillimeter wavelength range. This system allows a user to take advantage of the unique fingerprints possessed by every polar molecule for an unambiguous identification of these molecules. Moreover, the system architecture is fundamentally simple. With the growth of technology in this spectral region to support wireless communication, collision avoidance radar, and other applications, a system modeled on that described here has the potential to become very low cost in the near future. 138 CHAPTER 8 THE FUNDAMENTAL AND REAL LIMITS TO SPECTROMETER SENSITIVITY IN THE SUBMILLIMETER SPECTRAL REGION 8.1 SOURCES OF NOISE 8.1.1 INTRODUCTION The ultimate sensitivity of any detection system is limited by the presence of noise. The magnitude of the weakest signal the system can detect is determined by the amount of noise and its power spectrum. To have a good understanding of the system performance one must look into the physical processes responsible for the generation of signal fluctuations in every part of the experimental set up. This section describes factors determining sensitivity of a spectrometer operating in the millimeter and submillimeter wave range such as FASSSTER, allowing an estimate of minimum detectable number density of some typical molecules. A typical submillimeter wave range spectrometer consists of four distinct parts: (1) a radiation source, (2) an absorption cell, (3) a detector, and (4) signal processing electronics. Consequently, noise sources can be subdivided into the following categories: (1) radiation and background noise, (2) detector noise, and (3) amplifier and data 139 acquisition noise. Fluctuations of the experimental conditions inside the absorption cell can affect the sensitivity of the spectrometer, but in the case when time constant associated with these phenomena is much larger than the observation time these fluctuations can be neglected. In an absorption spectroscopy experiment one has to start with a reliable source of radiation. The power supply and frequency control circuits have certain amount of noise at their outputs, which affects the stability of the output frequency and power. Dealing with power supply noise can be a challenging task, but the phenomenon itself does not impose any fundamental limit on the sensitivity of the spectrometer and will not be considered. Another origin of noise associated with the monochromatic radiation sources arises from the fact that the emission of photons is a random near instantaneous process. The ultimate limit of the radiation source stability is set by the fluctuation of the photon arrival rate to the detector commonly referred to as the signal fluctuation limit or the photon shot noise. The presence of the radiant background in the detector field of view gives rise to the background noise. The treatments of the background noise and of the photon shot noise are very similar to each other. The difference arises from the fact that photon emission probability follows Poisson distribution, while the fluctuation of the blackbody mode population obeys Bose-Einstein statistics. Photon shot noise and background noise will be discussed in more detail in Sections 8.1.3 and 8.1.2 respectively. Most detectors used in the millimeter and submillimeter wave range utilize semiconductors to convert the high frequency signal into an electrical signal. Indium 140 Antimonide (InSb) [102] [103] is a widely used material for producing bolometer chips in this frequency range. An InSb detector element is cooled to the liquid helium temperature in order to increase its responsivity to the incident radiation. The incident radiation does not raise electrons from the valence band to the conduction band, but causes intra-band transitions, which change significantly the distribution of electrons within the band, hence changing the average mobility of the electrons (extrinsic photoconductivity [104]). Cooling of InSb detectors also reduces the amount of thermal noise associated with the detector chip. Another class of detectors used in the millimeter and submillimeter wave range utilizes the photovoltaic effect [104]. The detectors that fall into this category are photodiodes and Schottky barrier photodiodes. The current-voltage characteristics of p-n junctions in photodiodes exhibit rectification which allows us to use them as detection elements. The following sources of noise are commonly associated with detector elements, (1) 1/f noise, (2) current shot noise, and (3) thermal noise. A more detailed discussion of the noise properties of semiconductor detectors will be presented in Sections 8.1.4 - 8.1.6. The last category of noise sources, which can define noise properties of the detection system, is amplifier and data acquisition noise. Amplifiers and all modern day electronics make heavy use of semiconductor components which contribute to the overall noise level of the amplification and digitization equipment. Current noise, commonly referred to as 1/f noise, current shot noise, and thermal (Johnson [105] / Nyquist [106]) noise, the major contributors to the noise properties of the electronics components, will be discussed in Sections 8.1.4 through 8.1.8. The dynamic range of the digitizer used in the experiment 141 can be the determining factor for the system sensitivity, but proper choice of the system operational parameters can help to avoid this situation. Supporting derivations are presented in Appendices B-G and will be referred to as B- G in what follows. 8.1.2 FLUCTUATIONS OF THE BLACKBODY RADIATION FIELD As follows from equations (D.10) and (D.18) (presented in Appendix D) the 1 dispersion of the radiant flux fluctuations Pd reaching the detector can be expressed as ∞ [()]δ Pt−==δνν P2 R (0)2() S d . (8.1) dd ∫ 0 The power spectral density2 S()ν can be expressed in terms of the radiant flux fluctuations emitted by the blackbody (D.21) as +∞ 2 SPtedt()νδ= ()− jt2πν . (8.2) ∫ d −∞ If we assume that photon emission is an instantaneous process, radiant energy fluctuation equals 2 δνPtddd()= h ( N−− N ) δ ( t t0 ), (8.3) where ν is the radiation frequency, and the fluctuation of the total number of photons Nd received by the detector element at a given frequency ν during the observation period ∆t equals 1 For definition of radiant flux see Table 20. 2 For definition of power spectral density see Appendix D. 142 2 11dNm c 2 ()NNdd−=∑∑ nn (1)(1)sin +=+ nnπ Admaxθ . (8.4) ∆∆ttddirrections modes νπ4 where n is the average number of photons per mode. See Appendices B and C for derivations leading to equation 8.4. Fluctuation of the number of photons emitted by a blackbody into a single spatial mode is direction independent. Thus, the resulting fluctuation of the total number of photons reaching the detector (equation 8.4) is a product of the fluctuation of the number dN of photons in a single mode (B.9) and three factors. The factor m is the spatial modes dν c density (B.4). The factor represents the fact that photons are moving isotropically at 4π the speed of light into a 4π solid angle (B.2). The fact that the detector collects photons 2 from an extended source justifies the inclusion of the third factor π Admaxsin θ (see Appendix C). If we define the electrical bandwidth of the system as the inverse of the observation time 1 BW = , (8.5) ∆t substitution of (8.3) and (8.4) into (8.2) yields the following expression dN c Sh()νν=+ ( )22m A sin θ nnBW ( 1) . (8.6) dν 4 dmax Substitution of (B.4) and (B.9) into equation 8.6 yields the following expression for the power spectral density: 2πνhe24hkTν / SA()νθ= sin2 BW. (8.7) ce2/2dmax(1)hkTν − 143 In the long wavelength limit ( hkTν ) for the case, when only one mode of the blackbody radiation is accessible to the detector, equation (8.6) becomes SkTBW()ν ≈ ( )2 . (8.8) Integration of equation 8.1 over the entire frequency range yields ∞∞4π he2/hkTν [()]2()δ Pt−=δννθ P224 S d = A sin BW ν d ν dd∫∫ dmaxce2/2(1)hkTν − 00 5 (8.9) 4π ⎛⎞kT ⎜⎟ 15 ⎝⎠h 25 = 8sin Admaxθσ BWk T . For the case represented in Figure 31 the dispersion of the power fluctuations at the detector element equals ∞ [()]2()8δ Pt−=δννσθ P252552 S d = ABWkT (sin ++− T T (1sin)) θ. (8.10) dd∫ d b dsh 0 Cold shield, Tsh θ Detector, Td Background, Tb Figure 31: Thermal detector exposed to a thermal background and to a shield radiation Equation (8.8) becomes 22 [()]()δ Ptdd−δν P=∆ kTBW , (8.11) 144 where BW is the electrical bandwidth and ∆ν is the optical bandwidth accessible to the system. For further information on the subject of this section see [20, 107, 108]. 8.1.3 SIGNAL FLUCTUATION LIMIT (PHOTON SHOT NOISE) Fluctuations in the emission of a monochromatic radiation source can be analyzed using the same approach as the fluctuations of the blackbody radiation. The emission of the photons is a random process probability of which is described by the Poisson distribution. It can be shown that the dispersion in the number of emitted photons equals ()NN− 2 = N , (8.12) where the average emission rate N can be expressed in terms of the source power Ps and photon energy hν as P N = s . (8.13) hν If we assume that photon emission is an instantaneous process, the radiant energy fluctuation equals 2 δ Ptd ()=− hνδδνν ( N N ) BWt ( −− t00 ) ( ) , (8.14) where BW according to equation (8.5) is the electrical bandwidth of the system. After we combine (8.14), (8.2) and (8.1) we get 2 [()]2δδPtdd−= P hPBW ν s. (8.15) For further information on the subject of this section see [109]. 145 8.1.4 THERMAL NOISE Random thermal motion of the charge carriers in any resistive material produces fluctuations in the electrical signal commonly referred to as thermal or Johnson noise. As was first shown by Nyquist [106] the rms of the voltage fluctuations due to thermal noise is given by the expression 2 VN = 4 kT R BW, (8.16) where R is the electrical resistance of the conductor, T is the absolute temperature, and BW is the electrical bandwidth, which is inversely proportional to the observation time. For further information on the subject of this section see [20, 105, 106]. 8.1.5 CURRENT SHOT NOISE Statistical fluctuations in the number of charge carriers in any material give rise to the fluctuations in the electrical signal called current shot noise. As in the case of the photon shot noise described in Section 8.1.3 these fluctuations obey Poisson statistics. The dispersion of the current fluctuations in is given by the expression 2 ieiN = 2 BW, (8.17) where e is the charge of electron, i is the average value of the electrical current in the circuit, and BW is the electrical bandwidth. For further information on the subject of this section see [20, 110, 111]. 146 8.1.6 1/f NOISE Unlike the previously discussed shot noise, thermal noise, and blackbody background noise, 1/f noise is not a white noise, which means that its power spectrum varies with frequency. 1/f noise, also known as current noise in semiconductors and flicker noise in electron tubes, is dominant at low electrical frequencies. The origin of this noise source is not well understood, but an empirical expression for these fluctuations has been established Kiα i2 = BW , (8.18) N f β where K is a proportionality factor, α ≈ 2 , and β ≈ 1. For further information on the subject of this section see [20, 110, 111]. 8.1.7 AMPLIFIER NOISE Modern amplifiers consist of a large number of active and passive components, most of which make use of semiconductors. In order to characterize the noise properties of an amplifier, the manufacturer often provides noise figure contours with the amplifier specification. Noise figure contour plots such as that presented in Figure 32 allow us to calculate the noise figure, defined below, for a given combination of the electrical frequency and the source resistance at a given temperature (290 K). For the source resistance Rs =Ω10 k and frequency equal to 10 kHz, Figure 32 yields NF ≈ 0.5 db . 147 Figure 321: Noise figure contours of Stanford Research Systems SR560 low noise preamplifier. As follows from Figure 33, the source can be presented as an ideal resistor Rs followed by a thermal noise voltage source Vn . The amplifier can be considered as an ideal amplifier preceded by an equivalent source noise Va . Figure 33: Equivalent circuit of an amplifier and a signal source 1 Figure is available in the online manual for SR560 low noise preamplifier at http://www.srsys.com 148 The noise figure NF can than be defined by 22 2 ⎛⎞VVn+ a ⎛(/) VG out ⎞ ⎛⎞ VG out / NF ==10 log10⎜⎟22 10 log 10 ⎜ ⎟ = 20 log 10 ⎜⎟, (8.19) ⎝⎠VVVnnn ⎝ ⎠ ⎝⎠ where Vout is the noise signal at the output of the amplifier and G is the amplifier gain. The noise figure is a measure of the amount of noise added by the amplifier to the input signal. As follows from the definition (8.19), NF = 0 db when the amplifier is noiseless. The noise figure contours presented in Figure 32 can be used to obtain the noise figure parameter in the case when the source of the thermal noise is at 290 K. If the 4 source is at a different temperature Ts , which is the case for the He cooled InSb hot electron bolometer, the noise figure can be calculated as ⎛⎞NF /10 290 TKs=290 NF =+10log10 ⎜⎟ 1 10 − 1 , (8.20) Ts ( ) ⎝⎠Ts where NF is the NF value obtain from Figure 32. Thus, for the conditions Ts = TKs =290 4.2 K, Rs =Ω 10 k and frequency equal to 10 kHz, we obtain NF ≈ 9.3 db . Even though the value of the noise figure is significantly different, the rms of the voltage fluctuation produced by the amplifier is the same for any Ts 2/102NF VVan=−(10 1) . (8.21) For further information on the subject of this section see [112, 113]. 149 8.1.8 OTHER SOURCES OF NOISE Generation-Recombination noise [20] in semiconductor materials, which is the dominant source of fluctuations at intermediate electrical frequencies, is characterized by a power spectrum that decreases rapidly beyond a characteristic frequency proportional to the inverse time of the charge carrier lifetime. By its nature generation-recombination noise is very close to shot noise since it arises from statistical fluctuations of the charge carrier number density in semiconductor materials. At low frequencies this noise is exceeded by 1/f noise, which is the dominant source of noise at low electrical frequencies. External noise sources are introduced into the experimental system via a number of mechanisms, the most common among which are capacitive coupling, inductive coupling, resistive coupling (also known as ground loops) and microphonics. Good experimental design can help to minimize, and possibly to avoid, the effect of stray signals coupling into the detection channel. 8.2 DETECTION OF RADIATION 8.2.1 SQUARE LAW DETECTORS The two most commonly used types of square law detectors in the millimeter and submillimeter wavelength range are bolometers[114] and diode detectors[104]. Even though these kinds of detectors rely on different physical phenomena to convert high frequency signals into electrical signals, both bolometers and diode detectors can be 150 classified as square law detectors in certain cases. The output signal voltage level of the square law detector is proportional to the amount of the incident power reaching the detection element. Consider Figure 34. A bolometer is biased such that a small change in the resistance of the detector caused by the radiation will produce a noticeable signal change at the input of the amplifier. The resistivity of the bolometer element changes in response to the heating caused by the incident radiation. In the regime when the resistance of the bolometer chip is proportional to the change in its temperature, the output signal strength will be proportional to the amount of the absorbed power (Figure 34). Figure 34: Bolometer electrical circuit Diode detectors make use of the nonlinearity of the current-voltage dependence to rectify the signal. The magnitude of the output signal is proportional to the incident power due to the quadratic term in the current-voltage dependence (low power approximation), which puts the diode detector as well as the bolometer into the square law detector category. 151 For further information on the subject of this section see [104]. 8.2.2 FIGURES OF MERIT FOR DETECTORS 8.2.2.1 SPECTRAL RESPONSE Spectral response describes the way the signal at the output of a detector changes as a function of frequency of the incident radiation. For background limited detection this parameter sets the limit on the optical bandwidth accessible to the detector (see equation 8.11). Spectral response of photon detectors usually grows rapidly for frequencies above the semiconductor absorption limit, peaks and drops off linearly at higher frequencies. Thermal detectors, which respond to incident power, ideally exhibit a constant spectral response, which is wavelength independent. 8.2.2.2 RESPONSIVITY The purpose of a detector is to generate an electrical signal in response to incident radiation. Responsivity is defined as the ratio of the detector output signal Vs and the radiant power P intercepted by the detector element, V = s . (8.22) P Responsivity of an ideal square law detector will be constant for any value of the incident power. 152 8.2.2.3 DETECTIVITY AND NOISE EQUIVALENT POWER As follows from equation (8.9), the rms of the power fluctuation in the case of the background limited detection is proportional to the detector area Ad, electrical bandwidth BW and field of viewsin2 θ , where θ is the maximum aperture angle. 22 Pnd∼ sin θ ABW. (8.23) The ultimate sensitivity of the detector is limited by the presence of noise. In order to quantify the least discernable power one can define the noise equivalent power (NEP) as the amount of power in watts of the signal in 1 Hz of electrical bandwidth that produces signal-to-noise ratio of one at the detector output. The units of NEP are thus [NEP] = watt . (8.24) Some manufacturers and authors define NEP as the minimum detectable power per square root bandwidth. When defined this way, NEP has the dimension of watts per (hertz)1/2. For the case of the background-limited detection one can see that NEP will be a function of detector geometry. Detectivity D* is defined as an area-independent figure of merit ()ABW 1/2 D* = d . (8.25) NEP cm Hz1/2 [D *]=≡ Jones . (8.26) watt Unless otherwise specified, the detector field of view is hemispherical ( 2π ster). 153 Jones [115] has introduced D** as a figure of merit shich eliminates the necessity to define the field of view in the case of background-limited detection. This is defined as D* 2π D**= . (8.27) sin θ cm Hz1/2 ster 1/2 [**]D = . (8.28) watt 8.2.2.4 FREQUENCY RESPONSE The ability of the detector to respond to a modulated signal is characterized by the frequency response. Frequency response is defined as the dependence of the detector output signal on the modulation frequency of the incident radiation. The response time τ can be defined as 1 τ = , (8.29) 2π f3 db where f3 db is the frequency at which the signal power is 3 db below the value at zero modulation frequency. For further information on the subject of section 8.2.2 see [104, 115]. 154 8.2.3 DETECTION SENSITIVITY LIMITED BY BACKGROUND FLUCTUATIONS Ps ∆PPP=−s d Power −αL PPeds= Time (frequency) Figure 35: Power transmitted through the absorption cell In an absorption spectroscopy experiment, power emitted by the radiation source Ps transverses a region which contains the sample (absorption cell). Let us assume that the absorption cell is 1m in length (L = 1m) and the cell itself does not attenuate the high frequency signal. As the output frequency of the radiation source is tuned, the transmitted power level will be reduced at frequencies corresponding to the resonant frequencies of the molecular absorption. At these frequencies the total transmitted power reaching the detector element is equal to −α L ∆=PPPsd − = P s(1 − e ) , (8.30) where α is the molecular absorption coefficient. In the case of a small absorption (8.30) becomes ∆PPL≈ sα . (8.31) 155 The presence of background noise (Section 8.1.2) and photon shot noise (Section 8.1.3) will degrade the detector sensitivity. In order to correctly estimate the magnitude of the weakest detectable signal one has to take into account that the power reaching the detector element is determined by the field strength which is the sum of the field strength of the source radiation and the field strength of the background radiation. So for the detected power one can write 2 PPPPdssb≈±±( δδ) , (8.32) where Pd is the power reaching the detector, Ps is the power of the radiation source, δ Ps is the fluctuation caused by the source photon shot noise, and δ Pb is the power fluctuation caused by the background radiation. If one does the multiplication, (8.32) becomes PPds≈±δ P s ±2 PPPPP sbsbbδδδδ ± ± . (8.33) To identify the leading contributors to the signal fluctuation, we will estimate every term in this expression. For the parameters T = 300 K, optical bandwidth ∆=ν 1 THz = 1012 Hz , and electrical bandwidth BW = 10 kHz, equation (8.11) yields −13 δνPkTBWb =∆≈×410 W. (8.34) −3 12 For the photon shot noise for Ps ≈10 W , and ν == 1 THz 10 Hz , equation (8.15) yields −10 δνPhPBWss=≈210 W. (8.35) Term δ PPsδ b can be neglected since it is much smaller than PPsδ b . To estimate the 2 PPsδ b term correctly we have to take into account that due to the mixing of the 156 monochromatic source radiation with the thermal background radiation the accessible optical bandwidth in this case will be the same as the electrical bandwidth BW. −10 2PPsbδ ==× 2 PkTBW s 410 W. (8.36) Thus, to a good level of accuracy, fluctuation of the power at the detector element δ Pd is the sum of the mixing term 2 PPsδ b and the photon shot noise δ Ps δ PPds≈+δδ2 PP sb. (8.37) The maximum achievable signal to noise ratio for the case specified is 6 Ps 210× SNmax =≈ . (8.38) δ Pd 1 The magnitude of the smallest detectable signal caused by the molecular absorption is determined by the following expression ∆PPLPmin≈≈sα min δ d , (8.39) where Ps = 1mW is the power of the radiation source and αmin is the smallest detectable value of the absorption coefficient. Rearranging terms in (8.39) yields δ Pd αmin ≈ . (8.40) PLs For the condition chosen above −9-1 αmin ≈×510 cm. (8.41) The estimate of αmin that can be detected with the FASSST spectrometer yields αmin ~ 10-8 cm-1. The decrease in sensitivity can be explained by the saturation of the real He4 cooled InSb bolometer at the chosen source power Ps = 1mW. For further information on the subject of this section see [19, 108]. 157 8.2.4 DETECTION SENSITIVITY LIMITED BY SIGNAL PROCESSING To estimate the sensitivity of a spectrometer limited by signal processing we will have to consider contributions from the following sources of noise: (1) fluctuations in the background radiation, (2) Johnson noise in the detector element, (3) current shot noise, and (4) amplifier noise. We will assume that the detector is a He4 cooled InSb bolometer with resistance R =Ω10 k , the responsivity of the detector equals 1000 V/W , the electrical bandwidth BW = 10 kHz, and the radiation power Ps = 1 mW. The rms of the voltage fluctuations in the signal channel caused by the background fluctuations is −7 VPPPnB≈+ (δδ s2510V s b ) ≈×. (8.42) According to equation (8.16) the rms of the voltage fluctuations caused by thermal noise in the detector element equals −7 VkTRBWnJ =≈×4 1.6 10 V . (8.43) P If we calculate current through the detector element as i =≈s 10−4 A , then R according to (8.17) the rms of the voltage fluctuations caused by current shot noise will be −6 VeiBWRePBWRnS==2 2 s ≈×5.610 V. (8.44) From Figure 32 we can estimate that noise figure NF=0.5db for the input resistance R =Ω10k and the frequency bandwidth 10 kHz. Substitution of these numbers into 158 equation 8.21 with the rms of the thermal noise of a 10 kΩ resistor at 290 K gives the rms of the amplifier noise −7 VnA ≈×4.4 10 V . (8.45) The rms of the overall fluctuation at the output of the amplifier will be 2222− 6 VVVVVnTotal=+++≈× nB nJ nS nA 5.6 10 V . (8.46) The above equation indicates that under the chosen experimental conditions the current shot noise limits the sensitivity of the He4 cooled detector followed by the low noise preamplifier, the noise figure of which is defined by the noise figure contours presented in Figure 32. The maximum achievable signal to noise ratio in this case equals Ps 175000 SNmax =≈, (8.47) VnTotal /1 and value of the absorption coefficient corresponding to the minimum detectable number density is VnTotal / −8-1 αmin ≈=×5.6 10 cm . (8.48) PLs To analyze the effect of changing the source power P0, detector resistance R or responsivity on the values of the maximum signal to noise ratio SNmax and minimum absorption coefficient αmin consider Figures 36-38. As one can see, a decrease in radiation power or detector responsivity leads to a monotonic decrease in the sensitivity of the spectrometer. The dependence of the sensitivity on the detector resistance can be explained by the minimum value of the amplifier noise for input resistance of 500Ω. 159 175000 3 µ10-6 max 150000 -6 SN 2.5 µ10 = 1000 V/W = 1000 V/W 125000 2 µ10-6 R =Ω10 k R =Ω10 k -1 100000 1.5 µ10-6 /cm 75000 min α 1 µ10-6 50000 5 µ10-7 25000 0 0 0.001 0.005 0.01 0.05 0.1 0.5 1 ratio noise to signal Maximum 0.001 0.005 0.01 0.05 0.1 0.5 1 Source power/mW Source power/mW Figure 36: Dependence of αmin and SNmax on the source power 6µ10-7 175000 max -7 5µ10 SN 150000 4µ10-7 125000 Psource = 1 mW -1 100000 -7 R =Ω10 k /cm 3µ10 min Psource = 1 mW 75000 α 2µ10-7 R =Ω10 k 50000 1µ10-7 25000 Maximum signalMaximum to noise ratio 200 400 600 800 1000 200 400 600 800 1000 Responsivity/[V/W] Responsivity/[V/W] Figure 37: Dependence of αmin and SNmax on detector the responsivity 5.5µ10-8 500000 max 5µ10-8 SN 450000 -8 4.5µ10 Psource = 1 mW 400000 -1 Psource = 1 mW -8 = 1000 V/W /cm 4µ10 350000 = 1000 V/W min α 3.5µ10-8 300000 3µ10-8 250000 2.5µ10-8 200000 -8 2µ10 signal to noise ratioMaximum 10 50 100 500 1000 5000 10000 10 50 100 500 1000 5000 10000 Detector resistance/Ω Detector resistance/Ω Figure 38: Dependence of αmin and SNmax on the detector resistance R 160 8.2.5 DETECTION SENSITIVITY OF A REAL SPECTROMETER Even though all of the above results are certainly applicable to a real spectrometer, more factors have to be taken into account when dealing with a real-life detection system. Attenuation of the signal caused by the absorption cell will reduce the sensitivity of the spectrometer. Detector power saturation, detector spectral response, and 1/f noise, while hard to access analytically, do affect the performance of the experiment. For instance, 1/f noise can be the limiting factor for the performance of a real diode detector. Responsivity of a real thermal detector has frequency dependence and usually goes down at higher frequencies. Additionally, increase in power can saturate the detector element, thus reducing its sensitivity. At higher power levels deviation from square law behavior will become increasingly noticeable, thus limiting the applicability of the equations derived above. However, despite all of these factors, calculation of the fundamental limits gives a reliable estimate of spectrometer performance. The results and examples given in this chapter are relevant to the FASSSTER system. 161 8.3 ABSORPTION CAUSED BY MOLECULAR ROTATION In order to estimate the minimum detectable number densities of real gases we need calculate the absorption coefficients of the transitions which will be used for detection. We will calculate the rotational absorption coefficient of carbonyl sulfide (OCS) in its vibrational ground state under the assumption that the molecule is a rigid linear rotor (see Appendices E, F, and G), which holds well for OCS. For the purposes of this discussion we will assume that all the molecules are in the vibrational ground state, which is a good approximation at room temperature. Calculations of the absorption coefficient of a linear rigid rotor are presented in Appendices E, F, and G. In what follows we will use the results of these calculations. Carbonyl sulfide is a linear molecule with a permanent electric dipole moment of 0.715 Debye and rotational constant B (see equation G.4) equal to 6081.4921 MHz. As follows from equation G.10, at room temperature T = 300 K peak absorption corresponds to Jmax ≈ 40 . In order to achieve maximum detection sensitivity for OCS a measurement should thus be conducted around ν = 2BJ (+= 1) 498.682 GHz . Calculation of the absorption coefficient (G.9) for the transition J = 40→ 41gives −−11 1 α40→ 41 ≈×n 6.57 10 cm Hz , (8.49) where n is the number density of OCS in units of cm-3. 162 8.3.1 LINESHAPE CONSIDERATIONS As follows from equation (8.31), a change in power due to the molecular absorption in the limit of a small absorption equals ∆PPL≈ sα . (8.50) Thus, when the detector operates in the square law regime the experimental signal will be proportional to the value of the absorption coefficient at a given frequency. The line shape function α(,νν0 )defines the frequency dependence of the absorption coefficient. As dimensions of the absorption coefficient obtained in the previous section (equation 8.49) indicate, this value is equal to the integral of the lineshape function over frequency, because it was derived under the assumption that all the molecules absorb at the same frequency. The signal to noise ratio calculation requires knowledge of the maximum value of the experimental signal caused by the absorption. To obtain the maximum value of the absorption coefficient, corresponding to the peak of the experimental line shape, we have to integrate this line shape over frequency and equate it to (8.49). Doppler broadening determines the line shape at low pressures (usually below 10 mtorr) when the binary collision rate is low. The thermal velocity distribution of molecules in the laboratory frame of reference causes a Doppler shift in the absorption frequencies, thus broadening the line. We can assume that our experimental line shape is Doppler limited, since we are trying to estimate the minimum detectable number density. The Doppler limited line shape is defined as 163 2 ⎛⎞νν− −ln 2⎜⎟0 ⎝⎠δν d αννd (,0max )= α e , (8.51) where the Doppler line width equals 1/2 1/2 2ν 0 ⎛⎞2ln2NkTA -7 ⎛⎞ T δννd =≈×⎜⎟3.58 10 ⎜⎟ 0 . (8.52) cM⎝⎠ ⎝⎠ M The symbol M in the last part of equation (8.52) represents the molecular mass in atomic mass units, NA is Avogadro’s number, and ν 0 is the resonant frequency in the molecular frame of reference. ∞ If we integrate α (,νν )d ν and equate this to (G.9), we obtain an expression for ∫ d 0 −∞ the absorption coefficient corresponding to the maximum of the experimental line shape ln 2 1 ααmax= JJ→+ 1 . (8.53) π δν d For the J= 40→ 41 transition of OCS −17 -1 αmax =×n 7.98 10 cm , (8.54) where n is the number density of OCS in units of cm-3. As was shown in Section 8.2.3 the minimum detectable number density, for the case limited by background fluctuations, corresponds to the value of the absorption coefficient −9-1 αmin ≈×510cm. (8.55) Thus, the minimum detectable number density in this case equals 510× -9 n ≈=×cm-3 6.3 10 7 cm -3 , (8.56) min 7.98× 10-17 which corresponds to the pressure in the absorption cell of 2.5× 10−12 atm . 164 For the case limited by the signal processing we get −8-1 αmin ≈×5.6 10 cm , (8.57) and the minimum detectable number density is about 10 times larger, 5.6× 10-8 n ≈=×cm-3 7 10 8 cm -3 . (8.58) min 7.98× 10-17 Table 18 shows the minimum detectable number densities calculated in this manner for CH3F [116, 117], H2O [118], H2S [119], HNO3 [7], OCS [120] and SO2 [121]. Values of the absorption coefficients and transition frequencies for CH3F were calculated using SPFIT/SPCAT [18] package available at http://spec.jpl.nasa.gov [93]. For H2O, H2S, HNO3, and SO2 values of the absorption coefficients were taken from the JPL online catalog1. For OCS the calculated values of the absorption coefficients were used. 1 http://spec.jpl.nasa.gov 165 Background fluctuation limited case Signal processing limited case Detection Minimum Minimum Molecule frequency Pressure in detectable Pressure in detectable /GHz absorption number density absorption cell/atm number density cell/atm /cm-3 /cm-3 6 -13 7 -12 CH3F 816 4.9×10 1.9×10 5.4×10 2.1×10 1 6 -13 7 -12 H2O 984 8.8×10 3.5×10 9.8×10 3.9×10 e 7 -13 8 -12 H2S 993 1.6×10 6.4×10 1.8×10 7.18×10 e 8 -12 9 -11 HNO3 394 1.5×10 5.9×10 1.7×10 6.5×10 OCSe 498 6.2×107 2.5×10-12 7.01×108 2.7×10-11 e 7 -12 9 -11 SO2 484 9.1×10 3.6×10 1×10 4×10 1Values of the absorption coefficients are available online at http://spec.jpl.nasa.gov 166 Table 18: Minimum detectable number densities of CH3F, H2O, H2S, HNO3, OCS, and SO2 calculated for the following experimental conditions: temperature T = 290 K, electrical bandwidth BW = 10 kHz, optical bandwidth 1 THz, detector responsivity 1000 V/W, detector resistance10 kΩ , amplifier noise as shown in Figure 32, absorption cell length 1m. Only transitions below 1 THz were considered. 8.3.2 POWER SATURATION To understand the effect of the increasing source power level on the sensitivity of the spectrometer one has to consider molecular power saturation. Excited molecules have to relax to the lower state either through spontaneous emission, which is rather weak in the millimeter and submillimeter wavelength range (see Appendix E), or through collisional relaxation processes. Since the number of molecules is limited, at a certain power level complete saturation will occur, when the amount of absorbed power is completely determined by the relaxation rate, because the number of the molecules in the upper state is equal to the number of molecules in the lower state. As follows from equation E.6 the net change of power in the radiation field equals dP= Pul−= P lu hvB lu u() n u − n l dV . (8.59) The kinetic equation for the population of the upper state nu of a two level system (see Appendix E) with the inclusion of the collisional relaxation term becomes dn n uu= −+−Bun Bun , (8.60) dt ulτ where τ is the mean time between collisions, B is the induced emission/absorption Einstein’s coefficient, u is the radiant energy density and nu and nl are the number densities of particles in the upper and lower states. For a steady state the number densities of the particles in the upper and lower states are related as Bu nnul= . (8.61) Bu + 1 τ 167 Substitution of (8.61) into (8.59) yields Bu dP=−=− hvBu() n n dV hν n dV . (8.62) ul l1+ Buτ For the case when Buτ 1 (saturation limit) the net power change becomes independent of the radiant power density 1 dP≈− hν n dV . (8.63) l τ To estimate a lower saturation limit we will define the start of saturation when the induced transition rate becomes equal to the relaxation rate Buτ = 1. (8.64) As follows from equation (F.7) 3 18π 2 B = µ , (8.65) 2 ∑ i lu δν 3h ixyz= ,, where δν is the experimental linewidth. The respective contributions of the Doppler broadening δν D and collision/pressure broadening δν PB lead to the following approximate expression for δν : 22 δνδνδν≈+D PB . (8.66) As was demonstrated by Srivastava et al. [122], pressure self-broadening for the vibrational ground state rotational transitions of OCS is described by δν PB ≈ 5.8 p [MHz] , (8.67) where p is the pressure inside the absorption cell in mtorrs. 168 The mean time between collisions τ can be estimated from the uncertainty principle to be τ ≈ 1 . (8.68) δν PB As follows from (E.8) the spectral radiant energy density equals P u = , (8.69) cS where Ps is the power emitted by the radiation source, c is the speed of light, and S is the cross-section of the absorption cell. Thus, saturation power can be expressed from (8.64) as −1 ⎛⎞3 18π 2 PcS= ⎜⎟µ δν . (8.70) saturation⎜⎟223h2 ∑ i lu PB ⎝⎠δνDPB+ δν ixyz= ,, For the rotational J= 40→ 41 transition of OCS in its vibrational ground state, saturation power as a function of the pressure and the radius of the absorption cell is presented in Figure 39 169 50 Absorption cell diameter 15 cm Pressure in the cell 10 mtorr 10 20 10 1 5 0.1 2 Saturation power/mW Saturation power/mW Saturation 1 0.01 0.2 0.5 1 2 5 10 1 2 5 10 20 50 Pressure/mtorr Absorption cell radius/mm Figure 39: Dependence of the saturation power on the pressure inside the absorption cell (left figure), and on the absorption cell radius (right figure) for the rotational J= 40→ 41 transition of OCS in its vibrational ground state. Increasing the source power beyond saturation level will not lead to an increase in signal, but will further amplify the contribution of the background thermal fluctuations, as follows from (8.39). Thus, a decrease in signal to noise ratio will occur. Higher power levels can also have a detrimental effect on the performance of the detector element. Saturation, low responsivity and low resistance of the detector at high power levels (detector saturation)can reduce the sensitivity of the spectrometer. 8.3.3 SUMMARY It was demonstrated that the sensitivity of a spectrometer operating in the submillimeter wave range is limited by the noise at the signal processing stage (detector and amplifier). 170 At low power levels amplifier noise becomes the major contributor to the system noise level, while an increase in power can lead to molecular and detector saturation which lower the signal to noise ratio. For a real detection system, the 1/f noise, the attenuation of the absorption cell, as well as collisional pressure broadening and wall effects should not be neglected. 171 APPENDIX A FORMAT OF THE FILE SPECIFYING THE SORTING SCHEME USED BY CAAARS The sorting routine for CAAARS is written in C++ and optimized in order for it to be efficient for large lists of frequencies. The contents of the sorting scheme file selected by the “Load Sort Scheme” option are used as input to this routine. The file for the case of a near-prolate asymmetric top is shown in Figure 10 as an example. The first line instructs CAAARS that in the present case three quantum numbers per state will be used, J, Ka and Kc (SPFIT and SPCAT use up to 12 quantum numbers). Any number in the text lines that follow refers to the index of a quantum number in the output of the prediction program SPCAT, numbered 0 to (2×quantum numbers per state) -1. For the case presented in Figure 10 the indices 0 through 5 indicate quantum numbers J', Ka', Kc' , J'', Ka'', and Kc''. Each subsequent line defines a sorting step, in the order of the desired sorting hierarchy. Each line contains four fields delimited by separator signs ‘|’: (1) sorting instruction field with parameters in parentheses, (2) branch naming instruction field, (3) branch naming inclusion field, and (4) comment field. (1) The sorting instruction can have the following values: (i) ‘t’, (ii) ‘s’ or (iii) the index of a quantum number. 172 (i) ‘t’ stands for transition type and has as parameters the indices of the four quantum numbers Ka', Kc', Ka'', and Kc'' which are used, following the electric dipole transition selection rules, to determine the transition type (a, b, or c), which is then the actual sorting parameter. (ii) ‘s’ stands for a sum or difference of quantum numbers specifying the sorting parameter. It can have any number of indices of quantum numbers as parameters. A minus sign in front of an index indicates subtraction. (iii) An index means that the remaining instructions in that line apply to the value of the indicated quantum number, which will itself be the sorting parameter. (2) The branch naming instruction field can have the following values: (i) ‘l’ – lower case letter, (ii) ‘L’ – upper case letter and (iii) ‘n’ - number. Table 19 contains typical letters that go into branch names corresponding to legitimate sorting instructions for an asymmetric rotor molecule. (3) The branch naming inclusion field, which instructs CAAARS whether or not to include the sorting parameter into the branch name, has two possible values: (i) ‘y’ – yes, and (ii) ‘n’ – no. A branch name consists of a series of letters and numbers, separated by commas, corresponding to the sorting parameters which have been specified to contribute to the branch name. Thus, the sorting scheme presented in Figure 10 corresponds to the following naming algorithm: transition type contributes a letter to the branch name, ∆Ka 173 contributes a letter, ∆J contributes a letter, Ka'' contributes a number, and the sum J'' - Ka'' - Kc'', which distinguishes the two components of transitions split by asymmetry, contributes a number; J' is not included in the branch name. The last instruction sets J' as the running index of a given branch. A possible branch name in this case would be ‘b,p,R,7,-1’. It corresponds to an b-type transition with ∆Ka = -1, ∆J = 1, Ka'' = 7, and J'' - Ka'' - Kc'' = -1. The user can easily write further naming tables and sorting scheme files for cases involving, for example, hyperfine splitting. Naming instruction Sorting instruction ‘l’ ‘L’ ‘n’ ‘t’ a, b, c 1 A, B, C 1 - Value of ‘s’ …, p, q, r, …2 …, P, Q, R, …2 sorting parameter 1Standard spectroscopic notation for transition type [19]. 2Standard spectroscopic notation [19]: ‘p’ corresponds to ‘s’=-1, ‘q’ corresponds to ‘s’=0, ‘r’ corresponds to ‘s’=+1 and so on. Table 19: Possible branch name components that correspond to various combinations of sorting and naming instructions. 174 APPENDIX B BLACKBODY RADIATION Any body at a non zero temperature emits electromagnetic radiation. A blackbody can be defined as a body that absorbs all the radiation it intercepts. In order to stay in thermal equilibrium with its surrounding it has to emit energy in the form of electromagnetic radiation. Quantity Symbol Definition Unit Radiant energy Q ∫F dt J Radiant energy density u dQ/dV J/m3 Radiant flux (power) F, P dQ/dt W Radiant exitance M dF/dA W/m2 Irrandiance E dF/dA W/m2 2 2 Radiance L d F/dAproj dW W/m sr Radiant Intensity I dF/dW W/sr Table 20: Definitions of radiometric terms Any blackbody is a lambertian source, which means that its radiance does not depend on the viewing angle. For a lambertian source integration over a solid angle allows us to relate radiant exitance (M), radiance (L) and radiant energy density (u) as cu M = , (B.1) 4 175 cu L = , (B.2), 4π where c is the speed of light in vacuum. The radiation field obeys Bose-Einstein statistics. Average number of photons per spatial mode equals 1 n = . (B.3) ehkTν / −1 The density of spatial modes in the frequency interval ν to ν+dν is given by the following expression 8πν 2 dN= dν . (B.4) m c3 Thus, the average radiation energy density in the frequency interval ν to ν+dν can be expressed as du= hv n dNm . (B.5) If we rewrite this in terms of average energy density per unit frequency, we obtain du8πν h 3 u == . (B.6) ν dν ce3/()hkTν −1 Integration of uν over frequency leads to the Stefan-Boltzmann law ∞ 2π 5k 4 u = u dν = T 4 = σT 4 , where σ = 5.67 10-8 W/m2K4. (B.7) ∫ ν 3 2 0 15h c Equation (B.6) substituted into equation (B.2) leads to the Planck radiation law dL2 hν 3 L == . (B.8) ν dν ce2/()hkTν −1 For every spatial mode the rms of the number of photons per special mode equals 176 hkTν / 2 2 e ⎧(/),kT hνν h kT ()(1)nn−= nn +=hkTν /2 ≈⎨ (B.9). (1)e − ⎩n, hν kT Since fluctuations for different spatial modes are independent, fluctuation of the total density of photons can be expressed as dN ()N− N2 =+=+∑ nn (1)(1)m nn (B.10), modes dν where Nm is the total density of spatial modes (B.4). For further information on the subject of this section see [20, 108, 110, 111, 123]. 177 APPENDIX C RADIATION FROM AN EXTENDED LAMBERTIAN1 SOURCE R dAs dAd θs θd Figure 40: Radiation source and detector relative placement. R r z θ θmax dAs Figure 41: Radiation from a lambertian disk. 1 For definition of lambertian source see Appendix B 178 From the definition of radiance L (Table 20) we get cosθ ∂A ∂=∂2Φ LAcosθθ ∂=∂Ω LA cos dd. (C.1) dssdssR2 Since the radiation source is parallel to the detector plane the following expressions are valid θsd==θθ,2,tan,/cos dA s = π rdr r = z θ R = z θ ddθ θ (C.2) dr== d( z tanθπθ ) z , dA = 2 z tan z . cos22θ s cos θ After we substitute (C.2) into (C.1) we get θmax Φ ==2cossinsinπ LAθθθπ d LA 2 θ. (C.3) dd∫ dmax 0 Lens Detector D F θmax Ad Figure 42: Detector elements and focusing optics If we assume that the detector area is equal to the area of a diffraction limited spot, 2 ⎛⎞F Ad ∼ ⎜⎟λ , (C.4) ⎝⎠D 179 D2 /4 sin2 θ = , (C.5) max FD22+ /4 which is a good assumption in millimeter and submillimeter wavelength range, then for the power reaching the detector we get 2 2 ⎛⎞F 1 Φd ∼ πλL ⎜⎟ 2 . (C.6) ⎝⎠D ⎛⎞F 41⎜⎟+ ⎝⎠D In the case when F>>D, which also is true in most practical cases, (C.6) becomes 2 Φd ∼ Lλ . (C.7) The result can be interpreted as follows: In the case when the detector area is equal to the area of a diffraction limited spot, increase of the collecting optics size will not lead to an increase in the sensitivity of the spectrometer, but will result in a higher angular resolution of the instrument. For further information on the subject of this section see [123]. 180 APPENDIX D RANDOM SIGNAL THEORY Calculation of the power fluctuation associated with random noise can be done using a correlation approach to random signal analysis. In order to proceed with the definition of the correlation function, the nth central moment of probability distribution of a random variable x has to be defined as +∞ µ =−=−F((x mxmpxdx )nn ) ( ) ( ) , (D.1) n ∫ −∞ where p(x) is the probability distribution and m is the statistical average of x. +∞ mx==F()∫ xpxdx () . (D.2) −∞ A closer look reveals that the dispersion or variance of x is equal to the 2nd order central moment of the probability distribution 22 σµx ==2 F((xm − ) ) . (D.3) The correlation function, which is a measure of the statistical similarity between two random processes, is defined as +∞ R(,xx )=−F (( xmxm )( −= )) ( xmxmpxxdxdx − )( − )(,) . (D.4) 12 1122∫∫ 1122 1212 −∞ 181 The autocorrelation function for a random signal is given by the following expression +∞ R(,t t )=−F ((() xt mt ())(() xt −=− mt ( ))) (() xt mt ())(( xt ) − mt ( ))(,, pxt t ) dx. 121122∫ 1122 12 −∞ (D.5) A random process is called stationary if pxt(,12 , t )= pxt (, 1++τ , t 2 τ ), (D.6) and is called stationary in a wide sense if R(,tt12 )= R (),ττ =t 1− t 2. (D.7) A stationary random process is called ergodic if its time average is equal to its statistical average. 1 T mxt==() lim xtdt () . (D.8) T →∞ ∫ T 0 If a random process is ergodic, then a long enough piece of its representation is a typical statistical representative of the process. The study of this piece can reveal information characterizing the process. The above statement is definitely true for the background fluctuations and the photon shot noise, which are discussed in Section 8.1. Thus, for the purposes of this discussion we will imply that the random processes under consideration are ergodic by nature. It can be shown that R()ττ= xtxt ()(+− ) m2 , (D.9) and that 22 2 R(0)= σ x =−xt ( ) m. (D.10) 182 Term m2 in the above equations represents constant power component of the random process signal and under experimental conditions is usually filtered out using high pass filter at the input of the amplifier. Thus, we will assume that m=0. If process x(t) is periodic it can be expressed in term of its Fourier transform coefficients as +∞ jn2/π t T xt()= ∑ an e , (D.11) n=−∞ where 1 T axtedt= () − jn2/π t T . (D.12) n ∫ T 0 It can be shown that +∞ 2 jn2/π t T Rae()τ = ∑ n . (D.13) n=−∞ If we introduce power spectral density as +∞ 2 1 San()νδννν= ∑ n0 (−=0 ), , (D.14) n=−∞ T the following equation is obtained by calculating the Fourier transform of the correlation function +∞ +∞ R()τ ed− j2πντ τδννν=−= a2 ( n ) S () . (D.15) ∫ ∑ n 0 −∞ n=−∞ Thus, for a periodic function x(t) its power spectral density and its correlation function are a Fourier-transform pair. +∞ SRed()ν = ∫ ()ττ− j2πντ , (D.16) −∞ 183 +∞ R()τ = ∫ Se ()ννj2πντ d. (D.17) −∞ The following result is of a great importance, since it allows us to calculate the dispersion of the random process from the knowledge of its spectral density function +∞ ∞ R(0)==∫∫Sd (ν )ννν 2 Sd ( ) . (D.18) −∞ 0 For a non periodic process time domain representation and its Fourier transform are related as +∞ X ()ν = ∫ xte ()− jt2πν dt, (D.19) −∞ +∞ x()tXed= ∫ (ν ) jt2πν ν , (D.20) −∞ Thus, the spectral density function is related to the Fourier transform of the signal as +∞ 2 SX()νν== ()2 ∫ xtedt ()− jt2πν . (D.21) −∞ For further information on the subject of this section see [109, 124]. 184 APPENDIX E TRANSITION PROBABILITIES AND ABSORPTION COEFFICIENT FOR A TWO LEVEL SYSTEM u nu Blu Bul Alu ν l nl Figure 43: Two level system. Bul and Blu are the induced emission and absorption Einstein’s coefficients. Alu is the spontaneous emission Einstein’s coefficient. nu and nl are number densities of particles in the upper and lower states. To calculate the absorption coefficient we will consider the interaction of radiation with an idealized two level system represented in Figure 43. The kinetic equation describing the population of the lower state that interacts with radiation can be expressed as dn l =−B un + Bun + An, (E.1) dt luνν l ul u lu u 185 where uν is the radiant energy density per unit frequency. By comparing the expression for uν , obtained from (E.1) for a steady state when the system is in thermal equilibrium, and the expression for the blackbody radiant energy density (B.6) we get BBul= lu , 8πνh 3 (E.2) AB= . luc3 ul hν For the millimeter and submillimeter wavelength range, when 1, the kT contribution of the spontaneous emission is rather weak and can be neglected. Alu 1 hν . =≈hkTν / 1 (E.3) Buul ν e−1 kT Radiant power absorbed in volume dV equals PdVnuBhlu= l lu ν . (E.4) Similarly, the emitted power can be expressed as PdVnuBhul= u ul ν . (E.5) The net change of power in the radiation field is dP= Pul−= P lu hvB lu u() n u − n l dV . (E.6) As follows from equation (E.6), in the case when the population of the lower level is smaller than the population of the upper level, there will be a net amplification of the radiant field. The absorption coefficient is defined as 1 dP α =− . (E.7) Pdx If we take into account that the radiant energy density equals 186 P u = , (E.8) cS where S is the cross-sectional area of the volume element dV dV=Sdx, (E.9) then the expression for the absorption coefficient becomes αlu= −−hBν lu()/ n u n l c. (E.10) For further information on the subject of this section see [2, 125]. 187 APPENDIX F CALCULATION OF THE TRANSITION PROBABILITY USING TIME-DEPENDENT PERTURBATION THEORY Calculation of the probability of a transition caused by the interaction of a molecule with an electromagnetic field requires that this interaction be treated as a time-dependent perturbation. The time-dependent Schrödinger equation for the unperturbed system has form ∂Ψ Htˆ Ψ=−() 0n , (F.1) 00n it∂ iEn t / where Ψ=00nn()teψ form a full basic set. If the interaction of the molecule’s permanent electric dipole moment µ with the electric field E is defined as a correction to the unperturbed Hamiltonian ˆ HE1 =− iµ , (F.2) then the Schrödinger equation becomes ∂Ψ ()()HHˆˆ+Ψ=− t , (F.3) 01 it∂ where Ψ(t ) can be expressed as a linear combination of Ψ0n ()t Ψ=()tatt∑ nn () Ψ0 () , (F.4) If all the population of the u-l system is initially in the lower state, 188 a (0)= 1, l (F.5) au (0)= 0, and the distribution of the radiation is isotropic, then the time dependence of the probability of the l-u transition can be expressed as 3 228π at() = µ u, (F.6) ui2 ∑ lu 3h ixyz= ,, 2 where u is the radiant energy density, and ∑ µi lu is the transition dipole moment ixyz= ,, matrix element. This expression leads to the formula for the Blu Einstein’s coefficient 3 8π 2 B = µ . (F.7) lu2 ∑ i lu 3h ixyz= ,, Thus, the absorption coefficient (E.10) equals 3 −hkTν / 8π 2 ανlu=− n f l (1 e ) ∑ µ i lu , (F.8) 3ch ixyz= ,, where n is the total number density of the particles and fl is the fraction of particles in the lower state. For further information on the subject of this section see [125]. 189 APPENDIX G CALCULATION OF THE DIPOLE MOMENT MATRIX ELEMENT FOR A LINEAR RIGID ROTOR MOLECULE For the case of a linear rigid rotor molecule the radial part of the wave function can be neglected, leading to the following form of the Schrödinger equation in spherical coordinates 22⎡⎤11∂∂ΨΨ⎛⎞ ⎢⎥⎜⎟sinθ + 22+Ψ=E 0, (G.1) 2sinI ⎣⎦θθ∂∂⎝⎠ θ sin θϕ ∂ where I is the non zero (end-over–end) moment of inertia. Separation of the angular components of the wave function yields 1/2 1 iMϕ ⎡⎤(2JJM+− 1)( )! M Ψ=JM(,θ ϕθ )eP⎢⎥ J (cos), (G.2) 2π ⎣⎦2(JM+ )! where J is the total angular momentum quantum number, M is the z-component of the M angular momentum quantum number ( M < J ), and PJ (cosθ ) is an associated Legendre function. The energy levels are quantized as EBJJJ = (1)+ , (G.3) where 190 2 B = . (G.4) 2I Each energy level in this case has gJJ = 2+ 1 degenerate M sublevels. The fraction of the molecules in the J-th energy level can be calculated as −EkTJ / geJ hB −+hBJ(1)/ J kT fJeJ =≈+∞ (2 1) . (G.5) −EkTn / kT ∑ gen n=0 2 To calculate the transition dipole moment matrix element ∑ µi lu we need to ixyz= ,, calculate the following integral for all the spatial components of the dipole moment ππ2 Μ=ΨΨ(,JMJ , ',') M* (,)θ ϕµ (,)sin θϕ θ d ϕθ d iJMiJM∫∫ 00 , (G.6) ixyz= ,, where µi is a component of the permanent dipole moment in the space-fixed coordinate system, µx = µθsin cos ϕ , µ y = µθϕsin sin , (G.7) µµz = cos θ . 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