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The far-infrared absorption spectrum of low temperature hydrogen gas Wishnow, Edward H. 1993

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THE UNIVERSITY OF BRITISH COLUMBIAFebruary 1993© Edward Hyman Wishnow, 1993THE FAR-INFRARED ABSORPTION SPECTRUM OF LOWTEMPERATURE HYDROGEN GASByEdward Hyman WishnowB.A., Reed College, 1980M.Sc., University of British Columbia, 1985A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardIn presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.PhysicsThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:/14-4), 4/3AbstractThe far-infrared absorption spectrum of normal hydrogen gas has been measured from 20-320 cm' (A=500-31 pm), over the temperature range 21-38 K, and the pressure range0.6-3 atmospheres. The spectra cover the very weak and broad collision-induced trans-lational absorption band of 112 which at these low temperatures is observed well isolatedfrom the 112 rotational lines. Translational absorption occurs when two molecules collideand absorb a photon via a transient induced dipole moment. The molecules emerge fromthe collision with altered translational energies, and the rotational, vibrational, and elec-tronic energy states remain unaffected. The present spectra are the lowest temperature,lowest pressure, and highest resolution studies of the 112 translational spectrum.In order to observe the weak translational absorption band, a long pathlength multi-reflection absorption cell (White cell'), cooled by the continuous flow of helium vapour,has been designed and constructed. The cell has an f/10 optical beam that allows longwavelength radiation to be transmitted, with low diffraction losses, over an optical path ofup to 60 m. The cell is coupled to a Fourier transform interferometer and 112 spectra areobtained at a spectral resolution of 0.24 cm-1,10 times higher than previous experiments.Low temperature absorption spectra are due to not only transitions between molecu-lar translational energy states, but also rotational transitions between the bound statesof the van der Waals complex formed by two hydrogen molecules. The integrated absorp-tion of the measured 112 translational spectrum is consistent with the binary absorptioncoefficient calculated using the Poll and Van Kranendonk theory of collision-induced ab-sorption. The calculation is based on the quantum mechanical pair distribution functionderived from the Lennard-Jones intermolecular potential, and it includes contributionsfrom 11 2 dimer bound states. Although dimer transitions contribute to the translationalcontinuum, no sharp spectral features are observed that can be attributed to dimer tran-sitions. These low temperature, low pressure spectra are the first experimental searchesfor dimer structure across the translational band.The 112 spectra show the sharp R(0) line of HD superposed on the translational bandat 89.2 cm', where the HD is present in the gas in natural abundance. The electricdipole moment of HD has been measured from the intensity of this line and it equals(0.81 + 0.05) x 10' debye.The low temperature 112 spectra provide an experimental test of the theory of collision-induced absorption and the model of the 11 2 intermolecular potential. Furthermore, sincethe present experimental conditions are close to those found in the atmospheres of theplanets Uranus and Neptune, these spectra also have relevance to astronomical observa-tions of 112 . This work demonstrates, in particular, the feasibility of using measurementsof the HD R(0) line and the underlying 11 2 translational band to obtain planetary D/Hratios.In addition to hydrogen, spectra of H 2-He mixtures, D2, N2, 02; and mixtures of N2,CH4 , and CO with Ar, have been obtained at 78 and 88 K over the spectral region 20-180cm'. The N2 and N 2-Ar spectra exhibit structure superposed on the collision-inducedtranslational-rotational band; this structure has not been previously observed and it isprobably due to dimer transitions.iiiTable of ContentsAbstract^ iiList of Tables vList of Figures^ viAcknowledgement^ vii1 Introduction 11.1 Collision-Induced Absorption ^ 21.2 Planetary Atmospheres 51.3 Hydrogen Dimers ^ 81.4 Hydrogen Molecular Clouds ^ 91.5 HD ^ 111.6 Synopsis 122 The Theory of Collision-Induced Absorption 142.1 Introduction ^ 142.2 The Integrated Translational Absorption Coefficient ^ 162.2.1^Formalism of the Invariant Binary Absorption Coefficient ^ 162.2.2^The Pair Induced Dipole Moment ^ 192.2.3^Remarks on the Quadrupole Induced Dipole Moment ^ 232.2.4^The Evaluation of the Binary Absorption Coefficient ^ 25iv2.2.5 The Pair Distribution Function and the Calculated TranslationalAbsorption Coefficient ^  292.3 The Spectral Density Formalism  372.4 The Theoretical Spectrum of Hydrogen ^  402.5 The Birnbaum-Cohen Lineshape  472.5.1 The Synthetic B-C Spectra^  493 Experimental Apparatus and Procedures^ 553.1 Introduction  ^553.2 The Low Temperature Far-infrared Absorption Cell ^ 573.2.1 The Cell Optical Design ^  583.2.2 The Cell Mechanical Design  603.2.3 The Cell Thermal Design^  633.2.4 Temperature and Density Measurements ^ 663.3 The Interferometer ^  693.3.1 The Optical System  693.3.2 Interferometer Scanning and the Infrared Source ^ 703.3.3 Beamsplitters and Filters  ^713.4 Transfer Optics and Windows ^  733.5 Electronics and Data Acquisition  743.6 Gas Handling Equipment^  763.7 Experimental Procedures  773.8 Liquid Nitrogen and Argon Transfers ^  773.9 Liquid Helium Transfers ^  783.10 Temperature Gradients and Turbulence ^  783.11 Interferometer Scanning and Chopping Rates  803.12 Optical Alignment and Turbulence ^  813.13 Cell Improvements Recommended for the Future^ 813.13.1 Mirror Adjustments^  813.13.2 Reduction of Turbulence  823.14 Ortho-Para Measurements ^  834 The Translational Absorption Band of Hydrogen^ 854.1 Introduction ^  854.1.1 On the Discrepancy Between the LF and HF Spectral Regions .^864.2 General Remarks on the AbsQrption Spectra ^  874.2.1 The Spectral Band and Interferogram Averaging ^ 874.2.2 Temperature, Density, and Spectral Accuracy  884.2.3 Typical Spectral Features ^  894.3 The Temperature Dependence of the 112 Spectrum ^ 894.3.1 The Spectrum Fitting Procedure ^  914.3.2 The Frequency of the Peak and the Bandwidth vs. Temperature^914.3.3 The Spectral Invariants vs. Temperature ^ 944.3.4 The B-C Parameters vs. Temperature  964.4 Hydrogen Dimers ^  964.4.1 Introduction and Previous Work^  964.4.2 The Predicted Dimer Structure in the Translational Band^984.4.3 Experimental Dimer Search ^  994.5 The 112 Composite Average Spectra at 25.5 and 36 K ^ 1014.5.1 The Removal of a Residual Baseline ^  1014.5.2 The Averaging Procedure ^  1034.5.3 The Composite Average Spectra at 25.5 and 36 K ^ 105vi54.6HDThe Low Temperature Spectra Compared to Previous Measurements^.4.6.1^The Spectral Invariants vs. Temperature ^4.6.2^The B-C Parameters vs. Temperature 1131151181215.1 Introduction ^ 1215.1.1^The D/H Abundance Ratio ^ 1225.2 The HD Spectra^ 1235.2.1^The Integrated Absorption Coefficient ^ 1255.2.2^Line Fitting ^ 1265.3 The HD Dipole Moment 1285.3.1^The Density Dependence of the Dipole Moment ^ 1285.3.2^The Measured HD Dipole Moment ^ 1295.3.3^The Density Dependence of the HD Linewidth and Frequency 1295.3.4^A Determination of the HD Concentration in D2 ^ 1315.4 Future Investigations of HD Dimers ^ 1316 Experimental Errors 1336.1 Temperature, Pressure, and Density Errors ^ 1336.2 Optical Errors ^ 1346.2.1^Spectrometer Calibration ^ 1346.2.2^The Reproducibility of Background Spectra ^ 1346.2.3^Sources of Repeatability Errors ^ 1366.2.4^Additional Sample Spectra Errors 1376.2.5^Pathlength Errors ^ 1386.3 Water Ice 1396.3.1^Ice Absorption ^ 139vii6.3.2 Phase Effects ^  1406.4 Detector Non-Linearities  1446.4.1 Interferogram Contrast ^  1456.4.2 A Computer Simulation of Detector Nonlinearity ^ 1466.4.3 Corrected Interferograms ^  1476.4.4 Detector Nonlinearity Tests  1486.4.5 Detector Nonlinearity Comments ^  1497 Other Spectroscopic Studies^ 1507.1 The Deuterium S(0) line  1507.2 Hydrogen-Rare Gas Mixtures ^  1517.3 Nitrogen-Argon Mixtures  1547.4 Methane-Argon Mixtures ^  1567.5 Carbon Monoxide-Argon Mixtures ^  1577.6 Oxygen ^  1598 Conclusion^ 161Appendices^ 164A The Absorption Cell Optical Tests^ 164Bibliography^ 167viiiList of Tables2.1 The Thermal de Broglie Wavelength of 112 vs. Temperature ^ 302.2 The Calculated Binary Absorption Coefficient at Various Temperatures ^ 332.3 The Binary Absorption Coefficient of the Translational Band of Normal 11 2 353.1 The Pathlength Limits Due to Diffraction ^  593.2 The Beamsplitters and Low-pass Filters  71A.1 The Spot Size as a Function of Pathlength due to Astigmatism^ 165A.2 The Ratio of the Far-Infrared Signal Maximum/Minimum as a Functionof Pathlength ^  166ixList of Figures1.1 The Far-infrared Spectrum of Normal 112 at 77K ^ 51.2 The Voyager Infrared Spectrum of Jupiter 71.3 The Voyager Infrared Spectrum of Uranus ^ 71.4 112 Dimer Spectral Features Near the 5(0) and S(1) Lines ^ 91.5 The Submillimeter Spectrum of Normal 112 at 36K 132.1 The Lennard-Jones intermolecular potential of 112 ^ 322.2 The pair distribution function of 112 at 20 K 322.3 The Expansion Coefficients of the Dipole Moment ^ 432.4 The S(0) rotational spectrum of H2 at 77 and 20K— measurements andcalculations ^ 442.5 The translational-rotational band of 112 at 300 and 77K—measurementsand calculations ^ 452.6 The B-C Synthetic Spectrum of 112 at 297K ^ 522.7 The B-C Synthetic Spectrum of 112 at 77K 522.8 The B-C Synthetic Spectrum of 112 at 25.5K ^ 533.1 A schematic diagram of the experimental apparatus ^ 563.2 The Cell Mechanical Design ^ 613.3 The Laser Spot Pattern on the Field Mirror ^ 623.4 The Folding Mirror Mechanism ^ 623.5 The Low Temperature Absorption Cell 643.6 The Outer Radiation Shield ^ 64x3.7 The Optical Path of the Michelson Interferometer ^  703.8 Examples of LF and HF Background Spectra  724.1 The LF absorption spectrum of 11 2 at 34.4 K ^  904.2 The LF absorption spectrum of I-12 at 25.0 K  904.3 The peak of the translational band vs. the square root of the temperature 924.4 The width of the translational band vs. the square root of the temperature 924.5 The half-width of the translational band lineshape vs. the square root ofthe temperature ^  934.6 cli tr vs. temperature for the LF 112 spectra  ^954.7 -yitr vs. temperature for the LF H2 spectra ^  954.8 7-1 and T2 vs. temperature for the LF 112 spectra  974.9 The B-C parameter S vs. temperature for the LF 112 spectra ^ 974.10 The predicted 11 2 translational emission spectrum ^  1004.11 The LF H2 translational absorption spectrum at 22.4 K  1004.12 The composite average spectrum at 36K, HF multiplier=1.00 ^ 1064.13 The composite average spectrum at 36K, HF multiplier=1.05 ^ 1074.14 The composite average spectrum at 36K, HF multiplier=1.10 ^ 1084.15 The composite average spectrum at 25.5 K, HF multiplier=1.00 ^ 1094.16 The composite average spectrum at 25.5 K, HF multiplier=1.05 ^ 1104.17 The composite average spectrum at 25.5 K, HF multiplier=1.10 ^ 1114.18 The lineshape function G(o) at 36 and 25.5 K ^  1144.19 a itr vs. temperature for the translational band of 112 ^ 1164.20 -yitr vs. temperature for the translational band of 112  1164.21 7-1 and T2 vs. temperature for the translational band of H2 ^ 1194.22 S vs. temperature for the translational band of H2 ^  119xi5.1 Absorption spectra of 112 and D2 at various densities and temperatures . 1245.2 The fit to the R(0) line   1275.3 The density dependence of the HD dipole moment and the R(0) linewidth 1306.1 The repeatability of the LF backgrounds ^  1356.2 The repeatability of the HF backgrounds  1356.3 The effects of window distortions ^  1386.4 The 'absorption spectrum' of HF backgrounds taken at the end, and thebeginning, of a low temperature run ^  1416.5 The absorption spectrum of ice II  1416.6 The absorption spectrum of ice I ^  1426.7 The phase diagram of water ice  1436.8 The spectrum of helium demonstrating the effect of ice ^ 1436.9 The interferogram contrast vs. the detector signal amplitude ^ 1457.1 The spectra of D2 near S(0) ^  1527.2 The spectrum of 112 and a H2-He mixture^  1527.3 The absorption spectra of N2 and N2-Ar mixtures ^ 1557.4 The absorption spectra of CH4 and CH4-Ar mixtures  1557.5 The sample spectrum of CO ^  1587.6 The spectrum of a 23/73 CO-Ar mixture ^  1587.7 The rotational spectrum of 02 ^  160xiiAcknowledgementI would like to thank my mother and grandmother, and the rest of my family and friends,for their continued support. I have also enjoyed the hospitality of the people of Canada.The expert assistance of Peter Haas and the Physics department machine shop, andStan Knowtek and the Physics department electronics shop was crucial to the develop-ment of the experimental apparatus. I am indebted to Alex Leung for executing thedetailed mechanical design of the low temperature multipass cell.Scientific discussions with C. Bulmer, M. Legros, N. Poulin, B. Price, and I. Shinkoda,have been invaluable in the course of my thesis work. I have benefited greatly, throughoutgraduate school, from discussions and collaborations with Mark Halpern. Irving Ozierhas contributed valuable suggestions to the direction of my experiments, to their analysis,and to the completion of this document.Finally, it has been a great privilege to work with Herbert Gush. I thank him for hissupport, his scientific guidance, and for the memorable experiences of working on trulybeautiful experiments.Chapter 1IntroductionHydrogen is the most abundant element in the universe. Cold dark regions of the galaxythat are shielded from ultraviolet radiation are expected to contain vast quantities ofhydrogen in its molecular form 11 2 . The atmospheres of massive planets are also anenvironment composed primarily of 112. Observational evidence for molecular hydrogen isnot, however, easily obtained since 11 2 is transparent to both visible and radio frequencyradiation. By contrast, the presence of atomic hydrogen in stars is indicated by theBalmer absorption lines and the emission of Ha radiation by solar flares. Interstellar spaceis also filled with a sparse gas of atomic hydrogen which is detected via the 1420 MHzhyperfine transition using radio telescopes. 11 2 can be detected if it exists in moderatelydense conditions, as collisions between 11 2 molecules result in the weak absorption ofinfrared radiation. The present investigation of the submillimeter absorption spectrumof cold hydrogen gas is relevant to the longstanding astronomical problem of detecting112.11 2 is also the simplest diatomic molecule, and for this reason many of its propertiescan be calculated from first principles. Historically, the quantum mechanical nature ofatoms and molecules has been elucidated through the study of hydrogen. The charac-terization of intermolecular forces, again via the study of hydrogen, is an important areaof research in molecular physics. The present experiments are directly relevant to thestudy of intermolecular interactions, particularly the van der Waals binding of two 112molecules at low temperatures.1Chapter 1. Introduction^ 2This thesis reports laboratory measurements of the submillimeter spectrum of lowtemperature , 30 K, low pressure (1-3 atm.) hydrogen gas over the wavenumber range20-320 cm' (A=0.5-0.031 mm). This spectral region covers the collision-induced trans-lational absorption band of 112. The phenomenon of translational absorption arises whentwo molecules collide, absorb a photon, and emerge from the collision with altered trans-lational energies. Only transitions between translational energy states contribute to thisabsorption band, and the rotational, vibrational, and electronic energy states of thecollision pair remain unaffected.In order to study the low frequency translational band, a long pathlength (up to 60m), cryogenic, multi-reflection absorption cell has been designed and built, and it hasbeen coupled to a Fourier transform interferometer. Spectra of hydrogen have been ob-tained at a higher spectral resolution than previous measurements, over new temperatureand pressure regimes. These low temperature and low pressure experiments approximatethe conditions found in planetary atmospheres, and the results are relevant to the inter-pretation of planetary spectra. Translational absorption is very weak, and in the presentexperiments radiation passes through an amount of gas equivalent to 1.4 km of 112 gas atSTP (1 atm. pressure at 0° C). Using the same spectrometer-cell apparatus, less detailedstudies have also been made of the far-infrared spectra of deuterium, oxygen, nitrogen,methane, carbon-monoxide, and mixtures of the latter gases with argon.1.1 Collision-Induced AbsorptionAn isolated hydrogen molecule in the electronic ground state has a center of symmetryand therefore has no permanent electric dipole moment; the derivative of the dipole mo-ment with respect to internuclear distance is also zero. As a result, 112 does not exhibita normal infrared rotational or vibrational spectrum. The 112 molecule does possess anChapter 1. Introduction^ 3electric quadrupole moment which gives rise to weak transitions between vibrational androtational energy states with the rotational selection rules AJ = 0, ±2, and vibrationalselection rules Av = 0, ±1, ±2, ... A sample of dense hydrogen gas is observed, however,to absorb radiation over a continuum of frequencies near the vibrational and rotationalfrequencies of the 11 2 molecule. This phenomenon is called collision-induced absorption,and it results from transient dipole moments induced in pairs (or clusters) of collidingmolecules by intermolecular forces. Collision-induced absorption is still weak, but it is thedominant mechanism responsible for the far-infrared spectra of compressed gases com-posed of: homonuclear diatomic molecules such as 11 2 and N2j other symmetric moleculeswith no permanent dipole moments such as CO 2 and CH4 , and rare gas mixtures such asHe-Ar and Ne-Ar. Collision-induced spectroscopy was initiated by Welsh and co-workers[1], and contemporary work in this field appears in recent reviews [2], [3].The collision-induced dipole moment arises via a long-range interaction where thequadrupolar electric field of one molecule polarizes the other and via a short-range over-lap distortion of the colliding molecule's electronic distributions. The induced dipolemoment is a function of the internuclear separations and relative orientations of the col-liding molecules, and as a result, molecular vibrational and rotational transitions becomeinfrared active. In the case of hydrogen, prominent fundamental vibrational lines Q i (0)and S 1 (0) occur near 4161 and 4498 cm -1 , respectively; prominent pure rotational linesS(0), S(1), and S(2) occur near 354, 587, and 814 cm -1 . The induced dipole momentis modulated by the translational motion of the colliding molecules and transitions be-tween translational energy states become possible. Translational energy transitions areresponsible for the broad linewidths of the collision-induced spectrum and they also giverise to a pure translational spectrum, a quasi-continuum extending from 0 to about 300-CM 1 .The integrated intensity of a collision-induced spectral line is found to vary as theChapter 1. Introduction^ 4square of the gas density (for moderate densities), where this demonstrates that onlycollisions between pairs of molecules are responsible for absorbing radiation. The the-ory of collision-induced absorption shows that the integrated intensity of the collision-induced spectrum (normalized to unit density and pathlength) depends on: the na-ture of the dipole moment induction process; the properties of the molecule such as itsquadrupole moment, polarizability, and size; and the intermolecular potential. The the-ory of collision-induced absorption was developed by Van Kranendonk and co-workers[4], [5], [6], [7], and it is discussed in chapter 2.Collision-induced lines and the translational band of 112 are very broad and differin this regard from sharp lines due to allowed dipole or quadrupole transitions. Thebreadth of these spectral lines can be interpreted in terms of the Heisenberg uncertaintyprinciple .A.E.At > h, where the duration of a room temperature collision is extremelyshort, 2 x 10' sec. and the range of energy transitions is very large, 300 cm-1.In general, the translational band and the rotational lines of most diatomic moleculesoverlap, but the rotational constant of 112 is large 60 cm'), and the rotational linesare widely spaced. If spectral linewidths are reduced by performing measurements on acold gas sample, the pure translational band band can be observed.The measurements of the rotational spectrum of 112 and H2-rare gas mixtures byKiss, Gush, and Welsh first indicated that the spectrum below 400 cm-1 includes con-tributions from translational absorption [9],[10]. Following these measurements, the highfrequency wing of the pure translational absorption spectrum of rare-gas mixtures wasobserved by Kiss and Welsh [11]. Figure 1.1 is the spectrum of 112 at 77K measuredby Bosomworth and Gush; it shows the first observation of the isolated translationalspectrum of 112 [8]. Notice that the absorption peak of the translational band is about 10times weaker than the S(0) peak, and that the region of the translational peak has not3S(0)HYDROGEN77.3°KSeparated TranslationalCorn anentChapter 1. Introduction^ 5200^400Frequency, a (cm- 1 )Figure 1.1: The far-infrared spectrum of normal 11 2 at 77.3 K measured by Bosomworthand Gush [8]. The collision-induced translational band extends from 0-400 cm -1 and itpartially overlaps the S(0) rotational line which peaks near 371 cm -1 .been measured with high precision. Subsequent measurements of the entire translational-rotational spectrum of 11 2 have been conducted over a wide range of temperatures [12],[13], [14], but measurements of the translational band remain imprecise due to its ex-tremely low absorption and the difficulties inherent in submillimeter spectroscopy. Inthe present work, efforts have been made to obtain precise spectra over the peak of thetranslational band. By using very low temperature 11 2 , the peak absorption is enhanced,and the translational band is virtually isolated from the S(0) line (see Figure 1.5).1.2 Planetary AtmospheresKnowledge of the collision-induced translational-rotational spectrum of 11 2 is essentialto the study of the atmospheres of the giant gaseous planets. The translational bandin particular is responsible for the thermal opacity of the cold planets Uranus and3.00(NT2.0E1.0b0Chapter 1. Introduction^ 6Neptune [15], [16]. Far-infrared spectra of the planets Jupiter, Saturn, Uranus, andNeptune have been obtained by the IRIS spectrometer onboard the Voyager spacecraft[17],[18],[19],[20],[21],[22]. Figures 1.2 and 1.3 show the spectra of Jupiter and Uranus.Many properties of a planet's atmosphere may be deduced from the Voyager spec-trum. Laboratory measurements of the H2 absorption spectrum are used to estimate, ateach frequency, the pressure at which the optical depth of the atmosphere equals unity.This is the atmospheric layer primarily responsible for the emission of radiation, and thebrightness temperature plotted in Figures 1.2 and 1.3 corresponds to the emitting layer'stemperature. As a result, the brightness temperature spectrum reveals the atmospherictemperature profile as a function of pressure (or height) [24],[25]. The chemical com-position of the atmosphere is found by comparing radiative transfer calculations to themeasured spectrum. The abundance of helium in particular is determined using the Heenhancement of the H2 collision-induced spectrum due to the H2-He overlap interaction(see Figure 7.2). In practice the atmospheric temperature profile and composition aredetermined using infrared spectra in conjunction with radio occultation data; this is par-tially due to the limited accuracy of laboratory 112 spectra. The Voyager spectra continueto be extremely valuable and a recent analysis of Jupiter's spectrum has determined theH2 ortho/para ratio as a function of height [26].Figure 1.2 shows the low frequency emission spectrum of Jupiter which is dominatedby the S(0) and S(1) collision-induced lines, and the sharp low frequency lines due toammonia. The brightness temperatures of the strongly absorbing S(0) and S(1) peaksare low because emission at these frequencies originates from high atmospheric regionswith low pressures (0.2-0.3 bar) and temperatures. The weaker translational band isemitted from deeper atmospheric layers with pressures of 1-2 bars, and consequentlywarmer temperatures.Chapter 1. Introduction150^145a, 130w 1251207115110200^300^400^500^600^700Wavenumber lcmFigure 1.2: The Voyager spectrum of Jupiter [23]. The planetary emission spectrum hasbeen converted to that of an equivalent blackbody per measured frequency interval. Thesharp lines below 300 cm' are due to ammonia. The boxes enclose the locations of I-1 2dimer features near the centers of the collision-induced lines S(0) and S(1).200^250^300^350^400Wavonumbw (cm -1 )Figure 1.3: The Voyager infrared spectrum of Uranus [21]. Since the planet's temperatureis 60 K, emission occurs primarily over the translational band and this band is furtherenhanced by the presence of helium. The dotted curves are theoretical spectra calculatedusing the atmospheric temperature profile obtained from radio occultation measurementsand they are scaled for different values of the helium mole fraction.Chapter 1. Introduction^ 8Figure 1.3, the emission spectrum of Uranus, is noisy above 350 cm' because lit-tle power is radiated at these frequencies by a 59.4K equivalent blackbody. This figureshows how the He enhancement of the H2 translational band is used to determine plane-tary helium abundances, quantities which were unknown prior to the Voyager missions.The molar helium abundances of the other gaseous planets are: Jupiter, 0.12; Saturn,0.04; and Neptune, 0.15 [22]. The emission spectra of Uranus and Neptune originate inatmospheric layers at pressures of 1-2 bar, and temperatures of 60K; note that thisenvironment is quite close to the conditions employed in the present experiments.1.3 Hydrogen DimersA pair of hydrogen molecules may become bound together as a van der Waals complexor dimer since the intermolecular potential is attractive over a range of intermolecularseparations. The binding energy of 112 dimers is very low, 2.5cm' 3.6K), anddimers form only if collisions involve low kinetic energies. Spectral structure due todimers exists only if the dimers have relatively long lifetimes, and hence, only if collisionsin the gas are infrequent. The 112 dimer possesses a small permanent dipole moment,provided both molecules are not in the J = 0 state, and rotational transitions betweendimer states are observed as sharp features superposed upon the broad vibrational androtational collision-induced lines.Dimer structure in the rotational band of 112 was first observed as small ripples nearthe centers of the S(0) and S(1) features of the Voyager spectrum of Jupiter (see Figure1.2). The low temperature, low pressure environment of planetary atmospheres, alongwith the extremely long optical path, provide good conditions for observing dimers.Subsequent laboratory measurements by McKellar have confirmed the identification ofthese ripples as dimer features [27], and this spectrum is presented in Figure 1.4.Chapter 1. Introduction^ 9400^500^600WAVENUMBER / cm-1Figure 1.4: The S(0) and S(1) lines of equilibrium 11 2 at 77K as measured by McKellar[27]. The ripples in the spectrum correspond to rotational transitions of the 1-1 2 dimer.Note the similarity of this spectrum to Figure 1.2, the spectrum of Jupiter.In Figure 1.4 the sharp features on either side of the S lines are due to dimer boundstate transitions, and the broader ripples on the high frequency side are due to bound-freetransitions. The 11 2 dimer spectrum near 5(0) has been observed at ti 20K using thespectrometer-cell apparatus described in chapter 3, and by McKellar [28]. On the basisof these observations, it is expected that dimer transitions should also appear in the lowfrequency region of the translational band. The search for dimer structure in this bandhas been one of the major efforts of the present work.1.4 Hydrogen Molecular CloudsGiant Molecular Clouds are cold (20-100 K) regions of the galaxy composed primarilyof hydrogen molecules with a density of ,-1000 cm', roughly 1000 times the density ofinterstellar space. Particularly dense regions within molecular clouds are believed to beChapter 1. Introduction^ 10the birthplace of stars. In the absence of strong 112 infrared transitions, the distributionof molecular clouds in the galaxy is generally traced indirectly by observations of the COAJ = 0 4-- 1 emission line at 115 GHz; where the CO/H2 abundance ratio is roughly10. In certain circumstances, however, molecular hydrogen is directly observable inemission or absorption. Interstellar 112 absorbs radiation in the far-ultraviolet, via elec-tronic transitions, but these observations require a UV background star and spaceborneinstruments. Infrared vibrational-rotational transitions are observed in emission fromphoto-dissociation regions where 112 molecules are excited by an ultraviolet radiationfield, or by shock waves. Observation of these 112 vibrational quadrupole lines is becom-ing an important astronomical technique for studying excited molecular gas. The S(2)pure rotational line of 112 has been observed in the Orion nebula, and the detection of theS(1) line has been recently reported [29]. Despite these direct observations, 112 withinmolecular clouds is largely undetected, and the distribution of interstellar molecular hy-drogen remains an important unsolved problem in astrophysics [30], [31], [32].The emission of low frequency radiation by H2 molecules is predicted to occur withincold molecular clouds [33]. Hydrogen molecules are presumably formed when hydrogenatoms recombine on the surfaces of dust grains. Newly formed H2 molecules are ejectedfrom the grain, but molecules may recondense on the grain, provided the temperature islow. If the gas density in the neighbourhood of the grain is sufficiently high, 112 moleculeswill interact and thereby radiate in the frequency range 0-60 cm'. The translationalemission band is predicted to exhibit structure due to free-bound H2 pair transitionsand resonances between free-free transitions (see Figure 4.10). The low temperature, lowpressure, low frequency 112 spectra described in chapter 4 are the only measurementsthat can be compared to this predicted translational spectrum.It was suggested that H2 translational emission might explain the anomalous inten-sity observed at frequencies above 12 cm" in Gush's 1978 spectrum measurement of theChapter 1. Introduction^ 11Cosmic Background Radiation [34], [35]. Subsequent measurements of the CBR spec-trum using a rocket-borne, liquid helium cooled, differential Michelson interferometer,performed by Gush, Halpern, and Wishnow, show that the CBR has a Planckian spec-trum characterized by a temperature of 2.736 ± 0.017 K, with no intensity excess at highfrequencies [36]. This is the most precise measurement of the CBR temperature to date,and it is consistent with the CBR temperature of 2.735+0.060 K, obtained by the FIRASspectrometer onboard the COBE satellite [37].The submillimeter spectrum of the galaxy has also been measured by the FIRASspectrometer and it is modelled by emission from two populations of dust [38]. Radiationfrom 20.4 K dust is the dominant component of the galactic emission spectrum and thereis a minor contribution from 4.77 K dust. No explanation is given for the origin of the lowtemperature dust, but it is conceivable that translational emission by molecular hydrogencontributes to the submillimeter spectrum of the galaxy.1.5 HDAll of the low temperature hydrogen spectra obtained during the present experimentsshow a sharp absorption feature superposed upon the translational band at 89.2 cm -1 .This feature is identified as the R(0) line of hydrogen deuteride, where the HD is presentin the hydrogen sample as a naturally occurring isotopic impurity. The isotopically sub-stituted molecule HD possesses a small permanent electric dipole moment, and thereforerotational transitions with selection rules AJ = +1 are infrared active. The intensity ofthe R(0) line is analyzed in chapter 5, and the dipole moment of HD is measured to be(0.81 ± 0.05) x 10' debye. This work has been previously reported by Wishnow, Ozier,and Gush [39].Even though the dipole moment of HD is quite small, and the HD/H 2 abundance isChapter 1. Introduction^ 12about 1/3165, the peak intensity of the R(0) line is about 1/10 the peak absorption ofthe collision-induced translational band. It is possible that HD rotational line emissionfrom cold molecular clouds is more intense than the H2 rotational quadrupole emissionspectrum. The R(0) and R(1) rotational lines of HD could potentially act as directtracers of interstellar molecular hydrogen [40].The D/H abundance ratio of planetary atmospheres is one of the important mea-surable quantities in cosmology. The D/H ratio, along with the other light elementabundances, is a probe of the radiation and particle energy densities of the early nude-°synthesis phase of the universe [41]. The deuterium abundance of the pre-solar nebulais believed to be preserved in Jupiter's atmosphere, and the cosmological D/H ratio isdeduced from this using a model of the evolution of the galaxy. The present H2 spectraqualitatively resemble the predicted high resolution spectra of the planets Uranus andNeptune [42]. An important aspect of these experiments is the demonstration that D/Hratios of planetary atmospheres may be determined by measuring the intensities of theHD R(0) line, and the underlying H2 collision-induced translational band.1.6 SynopsisFigure 1.5 is the submillimeter absorption spectrum of hydrogen gas at 36K (see Figure4.14). The broad collision-induced translational band extends from zero frequency to apeak near 50 cm-1, and it starts to overlap the sharply rising S(0) line near 270 cm'.The peak intensity of the translational band is enhanced with respect to Figure 1.1, andthe translational band is considerably better isolated from the S(0) line at 36K thanat 77K. The sharp feature at 89.2 cm' is the R(0) line of HD, and no other obviousspectral structures are observed.100.0^200.0Frequency cm -14.0'^ 35.97Chapter 1. Introduction^ 13 .Figure 1.5: The submillimeter absorption spectrum of normal 11 2 at 36K. Notice thatthe peak of the translational band is higher, and its overlap with the S(0) line less, thanFigure 1.1. The sharp line at 89.2 cm' is the R(0) line of HD. The gap in the spectrumis due to an absorption band of the crystal quartz detector filter. See Figure 4.14The goal of the present work has been to measure the the collision-induced 11 2 transla-tional absorption band with improved spectral resolution, sensitivity, and accuracy. Thiswork is the first investigation of the submillimeter 11 2 spectrum at low temperatures andpressures, and these measurements are consistent with the Poll and Van Kranendonktheory of translational absorption [7]. At the present time, these 11 2 spectra are the onlymeasurements which may be compared to theoretical predictions of dimer structure inthe translational band. In addition, these spectra are important to the understanding ofastrophysical environments composed of cold molecular hydrogen gas.Chapter 2The Theory of Collision-Induced Absorption2.1 IntroductionIn order to introduce terminology used throughout this work, a review of the theoryof collision-induced absorption is presented in this chapter. Aspects of this theory arenot well known outside the community of specialists in the field, and the present reviewdevelops the concepts which underlie the analysis and interpretation of the experimentaldata. Since comparisons are made between the predictions of theoretical models andmeasured spectra, the results of previous important experiments in this field are alsodiscussed.The theory of collision-induced absorption was developed by Van Kranendonk [4],[5].Following the first measurements of pure translational absorption, Poll and Van Kra-nendonk extended the theory to collision-induced translational absorption [43], [7]. Thistheory relates the integrated absorption coefficient of the translational spectrum to thephysical properties of the colliding molecules. Among these properties are the molecularquadrupole moments and polarizabilities, and the intermolecular potential. The notablefeature of this theory is that the integrated absorption of a collision-induced spectrum canbe calculated without the complexity of finding explicit dipole moment matrix elements.The integrated absorption coefficient depends upon both the model of the inductionprocess and the pair distribution function. Poll and Van Kranendonk adopt the so-called exp-4 model to describe the collision-induced dipole moment. The dipole moment14Chapter 2. The Theory of Collision-Induced Absorption^ 15arises from two induction processes: (1) the long-range quadrupolar interaction whichhas a 1/R4 dependence, and (2) the short-range overlap distortion interaction which isproportional to e-RIP, where R is the intermolecular distance and p is a range parameter.The pair distribution function is proportional to the probability that two molecules will befound at a certain distance apart, given a temperature and intermolecular potential. Thepair distribution function may be calculated either classically or quantum mechanically,where at high temperatures the classical value is valid, but at temperatures below 80 Kquantum mechanical effects must be taken into account.The spectrum bandshape, as distinguished from the integrated absorption, is calcu-lated by expressing the absorption coefficient at a given frequency in terms of a spectraldensity [44]. The spectral density is the Fourier transform of the autocorrelation func-tion of the electric dipole moment, and it is proportional to the probability of absorption.Theoretical spectrum calculations are based on expressing a global spectral density as asuperposition of spectral densities, each associated with a term of the expansion of thedipole moment operator in spherical harmonics. Recent calculations of collision-induced112 rotational lines utilize ab-initio models of the 112 intermolecular potential (includinganisotropic terms), and the induced dipole moment [45],[28]. The calculated spectruminvolves transitions between bound and free states of a pair of 112 molecules, and it isconsistent with the measured spectrum, including the dimer structure. At the presenttime, no comparable calculations of the 112 translational absorption spectrum have beenreported.In order to fit experimental data or model planetary atmospheres, it is useful toapproximate the spectral density by a lineshape which has a simple analytic form. Alineshape proposed by Birnbaum and Cohen [46], is often used to fit experimental absorp-tion spectra. This lineshape has only 3 free parameters and spectra calculated using thislineshape fit well the entire translational-rotational 112 spectrum over the temperatureChapter 2. The Theory of Collision-Induced Absorption^ 16range 77 to 300K [14], [13].2.2 The Integrated Translational Absorption Coefficient2.2.1 Formalism of the Invariant Binary Absorption CoefficientThis section reviews the work of Poll and Van Kranendonk [43], [7]. The absorptioncoefficient per unit pathlength measured experimentally isA(a) = 1 ( 1n 1:8(a)/^4(a))(2.1)The absorption coefficient is independent of the optical pathlength, but is a function ofthe density of the sample gas. a is the frequency in cm -1 , 1 is the absorption pathlength,./b(a) is the background spectrum, and Ma-) is the sample spectrum. The measuredintegrated absorption coefficient isa' =^A(a-)da = -c-1 J A(v)dv^ (2.2)The absorption coefficient associated with a given frequency v in Hz (or cm -1 ) isgiven by [47]41-2A(v) = 3hcV v iEjPi( i -Iii1 26( 1' vii)^(2.3)where V is the volume. Pi and Pf are the populations of the initial and final states, /ivis the matrix element of the dipole moment of the system between the states i and f,and vif is the frequency of a molecular transition.vif = (Ef — Ei )/h^ (2.4)The integrated absorption coefficient isa =^E(Pi Pf)ilLif rhvii^ (2.5)i<fChapter 2. The Theory of Collision-Induced Absorption^ 17where K = 870/3h2c and the summation takes place over all states Ei < Ef• Thetransition moment is equal to the sum of the squares of the components of the dipolemoment matrix element, that is,lizif = Kitif ).12 + 1( 114 )y + Kit f (2.6)The evaluation of the integrated absorption requires knowledge of the wavefunctions inorder to calculate the matrix elements of the dipole moment. However, the integratedabsorption coefficient can be written in the form of a trace which is invariant under uni-tary transformations, and it may thereby be evaluated in terms of an arbitrary completeset of functions. The summation in eqn. 2.5 may be unrestricted by noting that theequation is symmetric in i and f (that is, for Ef < Ei, vif is negative). If the factor hvifis written out in terms of energy,a = V ifand,=^ Pittifitfi-Ei)•V ifBecause p, is Hermitian and^= Elk, the above becomesa = vic (14H,111)where the brackets denote an ensemble average over the system in thermal equilibrium. His the Hamiltonian of the 2 molecule system, and in general, it includes the translational-rotational-vibrational motion of the molecules. [H, p] is the commutator of H and p.The invariant form is nowa = 11-Tr 1.13 p[H,Vwhere P is the density operator defined by(2.10)e-priP = T^r e-OH^(2.11)(2.7)(2.8)(2.9)Chapter 2. The Theory of Collision-Induced Absorption^ 18and f3 = 1IkT.The integrated intensity may be expanded in powers of the density,,^ ,a = aiP2 -r ce2P3 -r- • • • (2.12)where a l is the binary absorption coefficient, a2 the ternary absorption coefficient, etc...Only the binary absorption coefficient needs to be considered for low density gases; it is1 ical = -i —vTr {P2(12)/2(12)[K(1) + K(411(12)1} (2.13)where P2 (12) is the first term in the density expansion of the density operator, K(1)and K(2) are the kinetic energies of molecules 1 and 2, and i(12) is the dipole momentinduced by the interaction of the two molecules. Note that in eqn. 2.13 only the kineticenergy appears in the commutator; this occurs because the potential commutes with ii.Any interaction between rotational and translational terms in the Hamiltonian influencesonly the density operator.The expression of the binary absorption coefficient may be reformulated in terms ofthe relative motion of the colliding molecules. In a coordinate system where R c denotesthe center of mass position, and R the intermolecular separation, integration of eqn. 2.13over ii, yieldsK A3a1 = -i —2-z Tr le -91112[K, mil (2.14)K is the kinetic energy and H the Hamilitonian of the relative rotational and transla-tional motions of the molecules, where the Hamiltonian is assumed to be separable intotranslational and rotational parts. .1 = ( 2k T)1")1" is the thermal de Broglie wavelength,where m is the reduced mass of a pair of 11 2 molecules, m = m im2 /(mi +m2 ). .A is equalto (V/Zt ) 1 /3 , where Zt is the translational partition function [43]. Z is the rotationalpartition function of a single molecule,Z = ED(2J+ 1)e -sE(J)^(2.15)JChapter 2. The Theory of Collision-Induced Absorption^ 19E(J) is the rotational energy level for a 112 molecule given by,E(J) = B0J(J +1) — Do(J(J + 1))2 + Ho(J(J +1))3^(2.16)where, J is the rotational quantum number, and Bo = 59.3392, Do = 0.04599, Ho =5.2 x 10-5 (in cm') [14].The rotational partition function for equilibrium 112 involves values of the nuclear spindegeneracy, gj = 1, 3, for even or odd J respectively. In the case of normal hydrogen,the partition function is treated as two separate partition functions, one for each speciespara and ortho, and the spin degeneracy is considered to be 1. The fractional populationof a given rotational state is then weighted by the normal relative population of eachspecies: para=0.2508, ortho=0.7492 (see eqn 2.40).2.2.2 The Pair Induced Dipole MomentThe pair induced dipole moment p is a function of the distance fi between the moleculesand the orientations of their molecular axes r-i and r-2. In a space fixed coordinate system,the orientations of ii, r-i , and r-2 are denoted by the anglesit . (e, 0), Qi = (01, 01), Q2 = (92,2)The drawing below depicts molecules 1 and 2 and the angles 0 and 0 associated withii. The other angles are not drawn, but are determined by a parallel translation ofthe XYZ coordinate system to the centers of mass of molecules 1 and 2. The angularChapter 2. The Theory of Collision-Induced Absorption^ 20dependence of it can be expanded in terms of the spherical harmonics of CI, St i , and E2 2 .This expansion uses as a basis the complete set of eigenfunctions of J?, Jiz, Ji, J2z,L2 , Lz. J 1 and J2 are the angular momenta of molecules 1 and 2, and L is the relativeangular momentum of the pair. Evaluation of the matrix elements requires a change ofbasis to the eigenfunctions of 4, 4, L2 , J2, Jz , where J s = J 1 + J2 and J = J 1 +J 2 + L. The wavefunctions include the Clebsch-Gordan coefficients owing to this changeof basis. The wavefunctions are denotedip j-711;121,4( -1 , f21, 0 2 )where v indicates the spherical components 0,+1. The spherical components of relativeto the Cartesian components are=^= 4(A. ±Each component is written as an expansion over the complete set of functions(647 2 \ 1/2/iv =^E APtiA2AL; R )oit.A2LA (0,^c22)3 ) Al A2 AL(2.17)(2.18)The coefficients A(A 1 A 2 AL) are independent of the chosen coordinate system andthey classify the dipolar induction mechanisms in terms of the parameters A i A 2AL. Theexp-4 model describes two general catagories of dipole induction: short-range overlapdistortions of the molecule's electronic distributions, and long-range interactions betweenthe quadrupolar electric field of one molecule and the polarizability of the other. Themagnitude of the overlap dipole is proportional to e -R/P and that of the quadrupolarinduced dipole is proportional to R- 4 .The dominant cause of infrared absorption in the translational-rotational band of11 2 is the quadrupolar interaction, but the anisotropic overlap induced dipole momentand the 'interference' between the quadrupolar and overlap moments also contribute toChapter 2. The Theory of Collision-Induced Absorption^ 21absorption. He-Ar mixtures demonstrate translational absorption which is entirely dueto an angle independent overlap induced dipole moment; this mechanism does not causeabsorption in pure hydrogen gas due to the symmetry of 112-112 collisions. The collision-induced vibrational absorption spectrum of 112 is due to the derivative of the induceddipole moment with respect to internuclear separations. In this case, S and 0 branchesare due primarily to a quadrupolar induced dipole, and the Q branch is due to both asubstantial isotropic overlap induced dipole and a quadrupolar induced dipole moment[5].The most significant A()1A2AL) coefficients of the exp-4 model and its subsequentextensions are presented in the following list [48],[47]. Included in the list are briefdescriptions of the dipole moment induction mechanisms and the selection rules for tran-sitions (J" 4-- J'). The selection rules are derived from forming matrix elements of thedipole moment operator. The wavefunctions are written as triple products of spheri-cal harmonics (eqn 2.31), and the components of the dipole moment operator are alsoexpressed as triple products of spherical harmonics (eqn. 2.18). The matrix elementsinvolve integration over the angles SI, Sii, and 522, and the resulting nonzero terms satisfythe triangle conditions [49]A(JP14), A(JP24), A(L"LL').Other limitations on the allowed transitions result from the symmetry of the 112 moleculeand the 112 dimer [28]. The selection rules listed below apply to cases of large angularmomenta, and when only low J states are occupied the triangle rules determine theallowed transitions. For instance, if J1=1, J2=0, and L=0, then the allowed transitionsdue to the A(2023) induced dipole component are: AJi = 0, +2; AJ2 = 0; and AL = +3.Chapter 2. The Theory of Collision-Induced Absorption^ 221. A(2023) = —A(0223) = 0 aQ /R4 + 6e-R/Pquadrupole moment-isotropic polarizability + anisotropic overlap L=3,AJI = 0, ±2; AJ2 = 0; AL = ±1, ±3; or AJi = 0; AJ2 = 0, ±2; AL = +1, ±3.Allows a rotational transition in one molecule of a colliding pair, also pure transla-tional transitions. The only coefficient that includes both induction mechanisms.2. A(2021) = —A(0221) = 6e-R/P, anisotropic overlap L=1,AJi = 0, ±2; AJ2 = 0; AL = ±1; or AJi = 0; AJ2 = 0, +2; AL = +1.Allows a rotational transition in one molecule of a pair, pure translational transi-tions, and transitions between the dimer bound states L = 0,1.3. A(2233) = V8/15 -yQ/R4, quadrupole moment-anisotropic polarizability,AJi = 0, ±2; AJ2 = 0,±2; AL = ±1,±3.Allows simultaneous rotational transitions of both molecules, so called double tran-sitions. A(2211) is similar, but AL = +1.4. A(4045) = —A(0445) = -V-5- a0/1r, hexadecapole moment-isotropic polarizability,AJ1 = 0, ±2,±4; AJ2 = 0; AL =or AJi = 0; AJ2 = 0, +2, ±4; AL = +1, +3, +5.Allows single molecule rotational transitions up to AJ = 4, has a minor influenceon the high frequency wings of rotational lines.5. A(0001) = 4"e-R1P, isotropic overlap, AL = ±1.Applies to H2-He or rare gas collisions, allows pure translational transitions only.a is the mean polarizability, -y is the anisotropy of the polarizability, Q is the quadrupolemoment, 0 is the hexadecapole moment, and are strength parameters for the overlapinduced dipole moment. The relative strengths of these induced dipole components asfunctions of intermolecular distance are plotted in Figure 2.3.Chapter 2. The Theory of Collision-Induced Absorption^ 232.2.3 Remarks on the Quadrupole Induced Dipole MomentThe dipole moment induced in one molecule due to the quadrupolar electric field of theother isii, = a : E^ (2.19)This is a tensor product of a, the polarizability tensor, and E, the electric field. Theelectric field at a point in space is given byE = —VV^ (2.20)The potential V, at the center of mass of molecule 1, due to the charge distribution ofmolecule 2 may be written as a multipole expansionV(1) . E  1  I rn P" (cos 9)p drn=0 Rn+1(2.21)The situation is described by the drawing below. The molecules 1 and 2 are separated1 2by a distance R, p is the charge density of molecule 2, r and 0 are the coordinates of thevolume element dr with respect the center of mass of molecule 2, 0 is the angle between7. and R, and Pn (cos 0) are the Legendre polynomials. The integral is over the volume ofmolecule 2.Homonuclear diatomic molecules are neutral and symmetric, and the first non-zeroterm in the multipole expansion is the n=2 quadrupole term given below. If molecule 2Chapter 2. The Theory of Collision-Induced Absorption^ 24is in the J = 0 state, even this term is zero.1V(1) =^/7-2 (cos2e_) p dT (2.22)For a set of discrete charges the quadrupole moment is defined by [47]Q = (E eirP2(cos et))^ (2.23)where ei,ri, Oi are the magnitude, and coordinates of the ith charge with respect tomolecule fixed coordinates, and P2 is the Legendre polynomial. The brackets denote theexpectation value over the electronic ground state. The magnitude of the electric field ofmolecule 2 at the center of mass of molecule 1 is1E oc —R4The equation for it is simplified by choosing a coordinate system with the z axisoriented along molecule 1 (as drawn). In this coordinate system, the polarizability tensorfor a linear molecule is diagonal and may be expressed asaxx 1 —1= a 1 + —1 (2.24)CZZ 1 2This is a sum of an isotropic part and an anisotropic part, whereaxx = ayy = a1, azz = a ll^ (2.25 )and1a =^(2a± + all),^=(a11 — al)^(2.26)The components of^in a spherical basis, in a space fixed coordinate frame, areexpressed by rotations of Euler angles applied to the diagonal polarizability tensor, andinverse rotations applied to the components of the electric field [50]. it now has the formIL = [aI — -y1\4] : E^ (2.27)Chapter 2. The Theory of Collision-Induced Absorption^ 25where I is the identity matrix and M is a product of rotations applied to the secondterm in eqn. 2.24. The components of p, are in this way written in terms of the sphericalharmonics of two of the Euler angles.Equation 2.27 demonstrates that the dipole moment induced in molecule 1, due tothe quadrupole field of molecule 2, is a sum of two terms which are proportional to:a Q^7 QandR4^R4Note that the quadrupole moment of molecule 1 also induces a dipole moment in molecule2 and the net pair induced dipole moment is=^t12.^ (2.28)2.2.4 The Evaluation of the Binary Absorption CoefficientThe binary absorption coefficient, eqn. 2.14, is evaluated using the expression of thedipole moment in terms of the spherical harmonics, eqn. 2.18. The Hamiltonian for therelative motion of molecules 1 and 2 ist 2 2H V(R)2 2 ^2T2m R 1- 21 -1 (2.29)where I is the moment of inertia of the hydrogen molecule, and V(R) is the intermolecularpotential. The first three terms constitute the kinetic energy operator which may bewritten as a sum of the following four termsh2 a^a^h2^t 2^t 2K^ R2 ^L2 +2m aR OR 2mR2^21 1 2/ 2The eigenfunctions of the Hamiltonian have the form(2.30)o(R, S2, ci1, n2) = xkL(R)YLm(n)YA , (noY.T2m2 (02)^(2.31)Chapter 2. The Theory of Collision-Induced Absorption^ 26XkL(R) denotes the radial part of the wavefunction, where k is the radial quantum num-ber, and nm, YArni YJ2m2 are the spherical harmonics. The trace in eqn. 2.14 isevaluated by taking the sum, over the complete set of eigenfunctions 0, of the quantity< 1. /3 1.4K , it]kb > . Details may be found in [7],[43], and [51].The result below reflects the four terms of the kinetic energy operator and they areassociated with modulation of the dipole moment by changes in R,E2,n,, and f22.272 rL, JD° [-1 IAPI.A2AL; 412(L(L + 1) AiPti + 1) A2(A2 + 1))AiA2AL ° M1/4.(A1A2AL; R)I2]MR2X g(R) R2 dR^ (2.32)The prime denotes differentiation with respect to R. g(R) is the pair distribution functionand it is discussed in the following section. The above expression gives the completeintegrated binary absorption coefficient for the far-infrared spectrum of a, diatomic gas;it includes translational and rotational transitions.In general, the rotational and translational bands of diatomic gases are blended to-gether (i. e. for N2 and 02). The wavefunction of the colliding pair is separable into anangular and radial part only if the intermolecular potential is strictly isotropic. In thecase of 112, the energy difference between rotational states is so large that even if a smallanisotropy of the potential mixes rotational energy levels, the effect on the rotationalenergies is insignificant; in the present context the Hamiltonian is considered separable.Phenomenologically, an isotropic potential dictates that H2 molecules in a bound dimeract like free rotors, and the complex rotation is independent of the rotational state ofthe molecules. Treated in the greatest detail, this is not true, and some modern modelsutilize a potential with small anisotropic components.Chapter 2. The Theory of Collision-Induced Absorption^ 27The integrated binary absorption coefficient for the translational band alone is27r 2 E E LA, (AJ)LA,(AJ)^[ iii(A i A2 AL; 41 23mc A2AL A./^L(L R2+ 1) 1A(A i A2AL; R)1 21g(R) R2 dR^(2.33)where^L,(AJ) EP(J)C(J, J+AJ;00) 2^(2.34)JP(J) is the fractional population of the Jth state, and C(J, A, J+AJ; 00) are the Clebsch-Gordan coefficients. The translational band results from transitions where there is nochange in the net rotational energy of the colliding pair; this includes a subcategory ofdouble transitions where (J1 , J2) (.12, JO, and J1 J2. The present experimentalstudy involves temperatures below 77K; only the J = 0 and 1 rotational states arepopulated, and double transitions are irrelevant to the translational band. Finally, theintegrated absorption coefficient, including double transitions as the second term, is givenby [43]aitr = [4.2-1 + 413 + b2,7 ba3JC] L2(0)77 + [L2(0) 2 + 2L2 (+2)L2 (-2)] d2 ,777 (2.35)where a 1 , a3 , b, and d are dimensionless and are expressed in terms of the exp-4 inductionmodel. 77 has the dimensions of an absorption coefficient.a1 = (EdecT) -cri° =^, 10 -4a3 = ( 3 /ecr)e-'/P =^0.6 x 10 -4b =^(aQ lea')^=^5.0832 x 10 -4^(2.36)d = 4(-yQ/ecr 5) =^5.6703 x 10 -5n . (7re2a-3 /3mc) = 1.2089 x 10'cm5sec'a is the size parameter for a Lennard-Jones potential, here equal to 0.2928 nm=5.5331ao (Bohr radii), p is a range parameter for the overlap interaction, and p/a = trChapter 2. The Theory of Collision-Induced Absorption^ 28Physical quantities used in the above constants are: a = 5.4394 [52], Q = 0.48468eag[53], -y = 2.0354 [52], e = 4.8032 x 10' esu. The integrals /1,1-3, J,1C are writtenbelow, where x = R/o-.' --zE(x-i) (c7i^ P)2 L(L + 1)IL = 47r^e P^— +^x2 ) g(x) x2 do ,..7 = 336r i x-8 g(x) dx001C .= 327-4 fo e- `L'(x-1) (--a + —3 ) g(x) x-3 dxp x(2.37)The quantities L2(Z1J) are given by eqn. 2.34, where the relevant Clebsch-Gordan coef-ficients squared for rotational transitions of AJ = 0, ±2 are:C (J, 2, J; 00)2 = J (J -I- 1) / (2J — 1)(2J + 3)C(J, 2, J+2; 00)2 = (3/2)(J + 1)(J + 2)/(2J — 1)(2J + 3)^(2.38)C(J, 2, J-2; 00)2 = (3/ 2)J(J + 1)/(2J — 1)(2J + 1)The term L2 (+2)L2 ( —2) denotes the contribution to the translational band from collisionsin which both molecules undergo a rotational transition, but the net rotational energyof the pair is unchanged. The term L2(-2) therefore involves the population of theupper rotational state, and the value (J + 2) must be inserted for J in the coefficientC(J,2,J-2;00)2 above. The fractional population of a state J for normal hydrogen isZeven1P(J)even ------ ^(2J + 1) e-SEM(0.2508)1P(J),,dd =^(2J + 1) e-t3E(J)(0.7492)Z odd(2.39)The binary absorption may now be calculated, provided the pair distribution function isknown.Chapter 2. The Theory of Collision-Induced Absorption^ 292.2.5 The Pair Distribution Function and the Calculated Translational Ab-sorption CoefficientThe pair distribution function is proportional to the probability that two molecules inthermal equilibrium are separated by a distance R. It is normalized such that g(R)^1as R^oo. Classically, the pair distribution function isg(R) = e-13v(R)^(2.40)where V(R) is the intermolecular potential. For hydrogen gas, the Lennard-Jones inter-molecular potential12^ trcrV(R) = 4€ (--R-) — ((2.41)is adequate to represent the virial coefficients when e/k = 37K and (7=0.2928 nm. € isthe depth of the potential well and a is the radial distance at which the potential crosseszero.The integrated absorption coefficient is calculated using the classical pair distributionfunction in the expressions for the integrals 11,13 , J , 1C . The integrals are calculatednumerically and the integrated absorptions for various temperatures are presented inTable 2.2. The terms in eqn. 2.35 which dominate the integrated absorption are b2,7and ba3J, the quadrupole induced dipole moment, and the 'interference' between thequadrupole and anisotropic overlap dipole moment, respectively.At low temperatures the de Broglie wavelength of an individual 112 molecule becomescomparable in size to o-. The 'hard shell' model of repulsion between molecules is thereforeinvalid, and a classical treatment of the system is inappropriate. Poll and Miller calculatethe quantum mechanical pair distribution function for a Lennard-Jones potential [54]; itis given byg(r1,r2) = 2A6 E Ex:(ri,r2)e-oHx,i(ri,r2)^(2.42)i^LiChapter 2. The Theory of Collision-Induced Absorption^ 30where r 1 and r2 are the positions of two molecules, Pi is a normalized Boltzmann factor,H is the Hamiltonian of the translational motion, and x(r i , r2 ) form a complete set oftranslational wavefunctions. A = ( 2irmh2kT )1/2 is the thermal de Broglie wavelength [54].• The thermal de Broglie wavelength is the cube root of the quantum volume VQ whichrelates the chemical potential pc , to the number density n, of an ideal monoatomic gas[55]eme/kT = nVQ . (2.43)In the case of an ideal diatomic gas, the right hand side is divided by the rotationalpartition function.. for the hydrogen molecule is tabulated below for a few representativetemperatures.Thermal de Broglie Wavelength of I -12Temperature (K)Wavelength (A)3000.72000.9801.4401.9202.8Table 2.1: The thermal de Broglie wavelength of 11 2 as a function of temperatureThe wavefunctions are obtained by solving the one dimensional Schrodinger equation,/(/ + 1) 2mXiI(R) [k2^R2^—2-n V(R)} x(R) = 0^(2.44)where k 2 = 2mE/h 2 , m is the reduced mass of the 11 2 pair, E is the relative translationalenergy, and V(R) is the Lennard-Jones potential. The solutions of the Schrodinger equa-tion have discrete and continuous energy eigenvalues. The discrete eigenvalues correspondto two weakly bound states characterized by 1 = 0 and 1 = 1. Figure 2.1 shows the loca-tion of the two bound states, the two quasi-bound states 1 = 2, 3, and the Lennard-Jonespotential. The effective potential curves correspond to adding the rotational energy ofthe H2 pair to the intermolecular potential energy, and expressed in cm',2m,^hV(R)eff^(--Th v, kn.) + /(/ R21) ) 87r2mc^(2.45)Chapter 2. The Theory of Collision-Induced Absorption^ 31Poll and Miller have calculated the quantum mechanical pair distribution function ofhydrogen at various temperatures and tabulated the results [54]. In these calculationsthe symmetry of the wavefunction with respect to the coordinate exchange of the two 112molecules does not contribute significantly to the pair distribution function unless thetemperature is below 5 K. At these very low temperatures, Hy bound states dominate thepair distribution function, but their effect is negligible at temperatures above ,-- 150 K.Figure 2.2 shows the classical and the quantum mechanical pair distribution functions of112 using a Lennard-Jones potential at a temperature of 20 K.The figure shows that at large separations the pair distribution curves are equivalent,but the area under the classical curve is much greater than the area under the quantumcurve. The calculated absorption using the classical pair distribution function is thereforemuch larger than the absorption calculated quantum mechanically.The integrals Z., 13, J, 11 are evaluated numerically by interpolating the tabulatedquantum mechanical pair distribution functions of Poll and Miller [54]. The binaryabsorption coefficient for the translational band is then calculated at T = 20, 40, and 80K. Table 2.2 lists the contributions to the binary absorption coefficient from the variousterms of eqn. 2.35. Although the terms are calculated to high accuracy, they rely on theLennard-Jones intermolecular potential, and hence, the validity of these results is limitedby this approximation.102 —gefft) AT 20•1(Chapter 2. The Theory of Collision-Induced Absorption^ 323^5^7Intermolecular Distance (A)Figure 2.1: The energy levels of the (112)2 complex superimposed on the Lennard-Jonesintermolecular potential (/ = 0) and 'effective" potential curves for / = 1, 2, 3, from [56].Figure 2.2: The pair distribution function of 112 at 20 K from [54]. The relations for theclassical and quantum functions are given in the text, WK denotes the Wigner-Kirkwoodexpansion.Chapter 2. The Theory of Collision-Induced Absorption^ 33TemperatureCLASSICAL RESULTS20.00^40.00^80.00 200.00 300.00Ii Integral 0.2379E+03^0.1425E+03^0.1409E+03 0.2055E+03 0.2623E+0313 Integral 0.2696E+03^0.1624E+03^0.1619E+03 0.2393E+03 0.3075E+03J Integral 0.6196E+03^0.3286E+03^0.2719E+03 0.2976E+03 0.3324E+03K Integral 0.7338E+03^0.4094E+03^0.3703E+03 0.4738E+03 0.5706E+03L2 (0) 0.2997E+00^0.2997E+00^0.3003E+00 0.3166E+00 0.3221E+00L2(+2)L 2 (-2) 0.8981E-01^0.8981E-01^0.9080E-01 0.1277E+00 0.1653E+00The Five Terms of the Integrated Absorption CoefficientIi. 0.8620E-34^0.5161E-34^0.5116E-34 0.7866E-34 0.1021E-3313 0.3517E-34^0.2118E-34^0.2116E-34 0.3298E-34 0.4309E-34,7 0.5800E-32^0.3076E-32^0.2551E-32 0.2944E-32 0.3343E-32K 0.8108E-33^0.4523E-33^0.4100E-33 0.5531E-33 0.6775E-33Double trans. 0.2163E-34^0.1147E-34^0.9596E-35 0.1477E-34 0.2135E-34The Binary Absorption Coefficient0.6753E-32^0.3613E-32^0.3043E-32 0.3623E-32 0.4188E-32QUANTUM RESULTSTemperature 20.00^40.00^80.00Z. Integral 0.1420E+03^0.1257E+03^0.1395E+0313 Integral 0.1639E+03^0.1454E+03^0.1619E+03J Integral 0.2930E+03^0.2441E+03^0.2414E+03K Integral 0.3732E+03^0.3230E+03^0.3426E+03L2 (0) 0.2997E+00^0.2997E+00^0.3003E+00L2 (+2)L2 (-2) 0.8981E-01^0.8981E-01^0.9080E-01The Five Terms of the Integrated Absorption CoefficientZ. 0.5143E-34^0.4553E-34^0.5065E-3413 0.2138E-34^0.1896E-34^0.2116E-34.7 0.2742E-32^0.2285E-32^0.2265E-32K 0.4124E-33^0.3569E-33^0.3793E-33Double trans. 0.1023E-34^0.8522E-35^0.8520E-35The Binary Absorption Coefficient0.3238E-32^0.2715E-32^0.2724E-32Table 2.2: The calculated binary absorption coefficient for the translational band ofnormal I-I2 at various temperatures using the Poll and Van Kranendonk theory of colli-sion-induced absorption. The table shows the values of the integrals and various terms ofeqn. 2.35. The pair distributions functions are based on the Lennard-Jones intermolecu-lar potential; the classical cases are are given by eqn. 2.40 and the quantum mechanicalcases are obtained from interpolating the tables of Poll and Miller [54]. The binaryabsorption coefficient is in the units cm' sec -1 .Chapter 2. The Theory of Collision-Induced Absorption^ 34Comments on the Binary Absorption CoefficientComparisons can be made at this point between the calculated binary absorption co-efficient, aitr, and the measured experimental integrated absorption coefficient, a, byrecalling thataitr DJ-a^c f A(a)da= no2 J p2n2(2.46)n is the gas density (in mol. cm-2), which is equal to pno, where p is the density in amagatunits, and no is Loschmidt's number (2.687 x 1019 cm-3). An amagat is the ratio of thesample gas density to that of an ideal gas at 0° C and 1 atm. pressure. The measuredquantity A(o-)1p2 has the units cm-1 amagat-2; integrating it over the absorption band,and multiplying by c/n(2,, yields a result in the units cm5 sec. Table 2.3 comparesthe calculated binary aborption coefficients using classical and quantum mechanical pairdistribution functions to measurements at a number of temperatures.The translational and rotational absorption bands overlap, especially at high temper-atures, and the intensity of the translational band alone must be inferred from fitting theentire translational-rotational band. The absorption spectrum is fit by a superposition ofthe translational band and rotational line profiles, each with the appropriate populationweighting and a common lineshape function. The lineshape used by Bosomworth andGush [8], is a modified Lorentzian with an exponential tail, and that used by Bachet etal. [14] and Dore et al. [13] is the B-C lineshape, which is discussed later in this chap-ter. The integrated absorption between 37 and 22 K is obtained by fitting low frequencyspectra (LF 'group' spectra), described in chapter four, with the B-C spectrum over theregion 20-180 cm, and extrapolating to 1000 cm'.Table 2.3 shows that the integrated absorption measured at high temperatures, sub-stantially exceeds the calculated absorption. This excess is probably not attributable tothe inference of the translational band absorption from measurements of the entire band.Chapter 2. The Theory of Collision-Induced Absorption^ 35Temp. (K) TheoryClassical Quantum^(x10 -33Experimentcm5 sec')300 4.19 5.6005.1265.96'200 3.62 4.95'80 3.04 2.72 2.5003.5863.02'40 3.61 2.7237.2 3.0835.0 3.0434.4 2.9425.5 3.0525.2 3.2225.0 3.2324.4 3.3722.4 3.3820 6.75 3.24Table 2.3: The binary absorption coefficient of the translational band of normal 11 2 . Theclassical and quantum mechanical calculations are given in Table 2.2. The measurementsare: a, [8]; b and c are inferred from [12] and [13]; the low temperature measurements arefrom the low frequency spectra (LF group spectra) reported in this thesis, see chapterfour.Chapter 2. The Theory of Collision-Induced Absorption^ 36Bachet et al. compared measurements of the integrated absorption at 300 and 195 K tocalculations of the absorption due to only the quadrupolar induced dipole moment usinga variety of intermolecular potentials. The discrepancy between the measured and cal-culated absorptions was interpreted as the overlap contribution, where this contributionvaried widely in response to the choice of the intermolecular potential [14]. The presentlow temperature calculations use a quantum mechanical pair distribution function basedon the Lennard-Jones potential, and include the overlap and interference contributionsto the integrated absorption. Table 2.3 shows that the calculated and measured lowtemperature integrated absorption coefficients are consistent.The formalism of Poll and Van Kranendonk relates the integrated translational ab-sorption to the molecular quadrupole moment and polarizabilities, and the intermolecularpotential. Collision-induced absorption may be used to measure molecular quadrupolemoments, provided that the polarizability and intermolecular potential are accuratelyknown. Alternatively, the variation of the integrated absorption as a function of tem-perature is a probe of the intermolecular potential, since the molecular properties donot change with temperature. The discussion of the comparison between measured andcalculated binary absorption coefficients is continued in chapter four.Trafton extended the theory of collision-induced absorption by calculating the 112translational absorption bandshape [57]. This work stems from eqn. 2.3 where the ab-sorption at a given frequency is calculated from the matrix elements of the dipole moment.The wavefunctions are separated into an angular part 0, which is a product of spheri-cal harmonics, and a radial part x which is is obtained by numerical integration of theSchrodinger equation. The matrix elements of the dipole moment are then formed byintegrating the product of the wavefunctions and the dipole moment operator over all47r 23h V E(P1 - Pf )Ittif 1 28(wif - w)c.A(w) = 1<f(2.48)Chapter 2. The Theory of Collision-Induced Absorption^ 37space.(flitli) = 0; [I: X*f(R) X1(R) dRi cs dSl (2.47)Trafton's calculated 112 translational bandshape is comparable to the Bosomworthand Gush 77 K spectrum, the only quasi-isolated 112 translational band measurement atthat time. Integrating over the calculated 300 K translational band yields an absorptioncoefficient within 1.2% of Poll and Van Kranendonk's result. This calculation does notinclude wavefunctions associated with 11 2 bound states and therefore the bandshape isonly accurate at high temperatures.Trafton applied these bandshape calculations to modelling planetary atmospheres,and the collision-induced absorption of 11 2 was shown to be largely responsible for thethermal opacity of Jupiter's atmosphere [15],[16]. Prior to this demonstration, a largeinternal heat source was required to explain Jupiter's infrared emission (a small internalheat source may still be required).2.3 The Spectral Density FormalismModern theoretical treatments of collision-induced absorption utilize the spectral densityformalism to calculate the spectral bandshape [58], [59], [46], and the following discussionis from [44]. The spectral bandshape is related to the Fourier transform of the dipolecorrelation function, and as a result, a connection is made between the microscopic motionof molecules and the absorption spectrum of a macroscopic system.Equation 2.3 gives the absorption of radiation at a given frequency due to allowedelectric dipole transitions. It is repeated below in terms of w, the angular frequency(= 27-ca).Chapter 2. The Theory of Collision-Induced Absorption^ 38The matrix element g, refers to the net dipole moment of the entire system, and summa-tion takes place over all states Ef > E1 , and wif = (Ef — Ei)/h. The expression for A(w)is cast into a different form by: (1) assuming the system to be in thermal equilibrium, (2)replacing the delta function by its integral representation 8(wi1 —w) = f°:3 eq'if-4")tdt,and (3) changing the description of the time dependent dipole matrix elements from theSchrodinger representation to the Heisenberg representation [60]. The absorption coeffi-cient is now written27r A(w) = 3hcV(1 e-f3Pnw^e-""(1z(0)p,(t))dt (2.49)where the brackets denote an ensemble average of the expectation value of i40)/2(t) overall states.The dipole moment correlation function for collision-induced absorption at low den-sities isN2(1.0)11(t)) = 2(10)1t(t)) (2.50)where the dipole moments on the right side refer to the pair induced dipole moment, andN is the number of molecules in the optical path.The expression for absorption in terms of a spectral density function g(co,T) is27r2n2w(1 — e---°h,„)Vg(u), T)A(w) =  ^ (2.51)3hcwhere n = NIV is the molecular number density, T is the temperature, and g(c,),T) isthe spectral density function..9(co,T)^E iiti26(wi1 cv)^ (2.52)if= Tr .1 1:: e-iwt(p.(0)it(*dt^(2.53)The upper equation stems from eqn. 2.48 and the lower equation is from eqn. 2.49.g(cv,T) is a measure of the transition probability and it is the Fourier transform of thepair dipole moment correlation function as shown above.Chapter 2. The Theory of Collision-Induced Absorption^ 39The general significance of the spectral density formalism is that spectral lineshapescan be directly related to processes which affect the dipole correlation function. For ex-ample, the Fourier transform of a sinusoidally oscillating dipole moment decaying expo-nentially in time yields the Lorentzian lineshape. Conversely, the microscopic behaviourof a system, such as the inhibitions of molecular rotations in a liquid, may be obtainedby inverse Fourier transformation of a spectrum [49].The spectral moments of g(w) are defined by0.Mn = V icong(w)dw.1-0.(2.54)The spectral moments are related to the time derivatives of the dipole moment correlationfunction, and they also connect the theoretically derived and experimentally measuredspectra. In the equations below, the experimental quantities on the left are called spectralinvariants, and they are related to the spectral moments by [44]71 ,_ I ,..1,1(w ) duj = 3h I  40)/71 2 du)^720M^2 i co tanh( 12-'-')^3c^°, i A(w) 2_^271- 2 ,,,alj n2 elw = -3hc Ivi 1(2.55)(2. 56)The absorption coefficient per unit pathlength, A(w), retains the definition used through-out this chapter, eqn. 2.1. The experimentally measured absorption coefficient, per unitdensity squared per unit pathlength, is defined by eqn. 4.1 and equals A(w)/n2 .The spectral invariant a l , is the same as the integrated binary absorption coefficientmultiplied by 27r. The lineshape function is defined byG(o) A(u)/n2(2.57)cr(1 — e -hco 1 la )The lineshape function is the absorption coefficient divided by the frequency and thestimulated emission factors, and G(Q) is proportional to the probability of absorptionChapter 2. The Theory of Collision-Induced Absorption^ 40[48]. The spectral invariant is analogous to an integrated lineshape function; however,it stems from an absorption coefficient that involves a spectral density m(w) which is theFourier transform of a symmetrized pair dipole moment correlation function [44],[58]. Inthis formulation, the factor tanh(l"--") has the role of a stimulated emission term, but itis not exactly the same. The relation between G(cr) and -4(u.;) is demonstrated by theratio (neglecting constant factors)71(w) (tanh(hcal2kT))-1 hccr IkTe— (2.58)G(a) cx (1 _ e—hr- IkT)-1 —For hccr kT , the ratio is 1, but for low frequencies (below 10 cm-1 at T = 30 K)the ratio approaches 2. In order to remain consistent with previous experimental andtheoretical studies, the spectral invariants aur and ryitr for the measured translationalband are obtained in chapter four using the definitions above.The spectral density function for collision-induced absorption includes contributionsfrom the various A(A1A2AL) components of the induced dipole moment, the superpo-sition of which produces the measured spectrum. For a given component, the spectralinvariants, and ratios of spectral invariants, depend on such properties as the polarizabili-ties, quadrupole moment, overlap range and strength parameters, and the intermolecularpotential. Measured spectral invariants at various temperatures allow a test of thesemolecular parameters, or a test of the models used for the intermolecular potential andthe induced dipole moment [48], [59].2.4 The Theoretical Spectrum of HydrogenSchaefer [61], Schaefer and McKellar (SM) [45], [28]; and Meyer, Frommhold, and Birn-baum (MFB), [62], have produced the most sophisticated calculations of the absorptionspectrum of molecular hydrogen. These calculations include contributions to the spec-trum by free-free, free-bound, bound-free, and bound-bound 112 pair transitions. BothChapter 2. The Theory of Collision-Induced Absorption^ 41groups use an ab-initio induced dipole moment in the calculations instead of the exp-4 model. SM utilize an anisotropic intermolecular potential and the discussion belowgenerally follows this formulation.The absorption at a given frequency is given by eqn. 2.51, where the spectral densityisg(cd, T) = E Pr E Pt (r it i lizr,,,,Irt)1 2 S(w„ , wtt , — w)^(2.59)rr'^ttlP,. is the fractional population of a rotational state, Pt is the normalized Boltzmannfactor for translationpt = 0 e -pEtV(2.60)where ao is the thermal deBroglie wavelength. Transitions take place between rotationalstates denoted r, r' and translational states t,t'.The wavefunctions are obtained by solving the Schrodinger equation2m + H2 + V ( , rZ , R) = 0 . (2.61)H1 and H2 are the rigid rotor Hamilitionians of the individual 1-1 2 molecules; R is theintermolecular separation; and rl and r-2 point along the axes of the two molecules. Theintermolecular potential is written as a triple product of spherical harmonics analogousto the expansion of the dipole moment operator. The expansion coefficients are ob-tained from ab-initio calculations, and are scaled to fit the experimental data of secondvirial coefficients and molecular beam magnetic resonance transitions [28], [63]. Band-shape calculations performed with this potential demonstrate more spectral detail andare computationally more complex than those which use an isotropic potential. MFBemploy an isotropic potential; the wavefunctions are therefore separable, and only theradial Schrodinger equation needs to be solved [62].The wavefunctions obtained from solving the Schrodinger equation are correct for acolliding pair composed of two 1-1 2 molecules in different states. 11 2 molecules are bosonsChapter 2. The Theory of Collision-Induced Absorption^ 42and if the H2 dimer is treated as a quasi-diatomic molecule, the total wavefunction of adimer of para 112 is purely symmetric. The total wavefunction of an ortho dimer, with anuclear spin of 0, 1, or 2, is a sum of symmetric and antisymmetric wavefunctions, withweights 2/3 and 1/3 respectively.The wavefunctions are now applied to the induced dipole moment operator. Theinduced dipole moment operator has been calculated from first principles by treatingthe collision complex as a molecule and calculating the electronic distribution [63]. Theexpectation value of the complex dipole moment is dependent on the intermolecularseparation and orientations of the two 112 molecules, and the dipole moment operator isexpanded as a triple product of spherical harmonics, as in eqn 2.18. Figure 2.3 showsthe R dependence of the expansion coefficients D(Ai A2AL) which are associated with thesame dipole induction mechanisms as the previous coefficients A(A1)2AL). The ab-initioinduced dipole moment is very similar to the exp-4 model and the dominant D(2023)component has an approximately 1/R4 dependence.Finally, the spectral density is calculated by obtaining the matrix elements of thedipole moment for all dipole moment expansion coefficients. The spectral density in-cludes transitions between all initial and final rotational and translational energy states,where the states are weighted by the relevant population statistics and Clebsch-Gordancoefficients. A theoretical spectrum is then calculated from the spectral density.SM characterize the spectrum in terms of the contributions due to the free-free, free-bound, bound-free, and bound-bound 112 pair transitions. Each contribution is composedof a sum over all the dipole moment expansion coefficients D(Ai A2AL). Figure 2.4A showsSchaefer's synthetic spectrum of the entire translational-rotational band of equilibrium112 at 77K. The upper curve is the total absorption spectrum, and the lower curve isthe free-free contribution; the bound state contributions are reflected by the differencebetween them. Figures 2.4B and 2.4C show the calculated spectrum and McKellar'sChapter 2. The Theory of Collision-Induced Absorption^ 43Figure 2.3: The expansion coefficients of the ab-initio induced dipole moment [63].The coefficients and primary interaction mechanisms are: 2023,0223, quadrupole mo-ment-isotropic polarizability; 2021,0221, anisotropic overlap; 2233 and 2211, quadrupolemoment-anisotropic polar. ; and 4045,0445, hexadecapole moment-isotropic polar.recent measurements of the S(0) rotational line for equilibrium 11 2 at 77 K, and para 11 2at 20 K. The experimental measurements are the noisy solid curves, the total calculatedspectra are the smooth solid curves, and the dotted curves are the free-free contributions.Note that the 20 K spectrum is more intense and the dimer features are sharper than the77 K spectrum.The SM calculations correspond closely to the measured spectra. The inclusion ofthe anisotropic intermolecular potential accurately reproduces the details of the dimerstructure, although some deviations between the calculated and measured spectra oc-cur at frequencies higher than the S(0) transition frequency. In general, these graphsdemonstrate the validity of the theory of collision-induced absorption and the ab-initioH2 intermolecular potential.MFB classify the theoretical spectrum in terms of the dipole moment expansion coef-ficients, where each component involves a summation over all types of 11 2 pair transitions.12a20010077 K300 ^ 400 ^510 ^ 6.00 700^800IA11 til■ 1 7060".4cos— 4 0"c302010345^3t0^355^360 31.1.1 330^335(_I)365^370^375^360Chapter 2. The Theory of Collision-Induced Absorption.111,,..111.,■LiiiiI,■■■■■■■11.,..111,,I,■•■■■■1111“,111,11-330^335^340^345^350^355^360^355^370^375^380^385^390Figure 2.4:A: The calculated translational-rotational band of equilibrium 112 at 77K, [61]B: The S(0) line of equilibrium 112 at 77K, from Schaefer and McKellar [45]C: The 5(0) line of para 112 at 20K, from McKellar and Schaefer [28]The various curves are explained in the text.H2 — H2 297 Kfttos0223, 20230221, 2021• .4 \^total./^% /^2233 \.--,./‘■/ \\ \ ‘‘.^ .^I 500 ,000^1500^2000L4)^(cm-1) H2-H2 77.4K4.10%66IE0223, 2023'2_,^.."' Iota,\e-\r.. \^i^•,\0221, 2021^i^s I,x^.,; .,c‘ —\. / ,---../ - -..\ ,1^I i •••..^'1, ^r ^\^, 2233^ s./1 '' \ ........^/1 ^.^i^.^.^.^.^1^.. .., Chapter 2. The Theory of Collision-Induced Absorption^ 45500 1000(cm -1 )Figure 2.5:A: The translational-rotational spectrum of H2 at 297.5K. The measurements are fromBachet et al. [14], and the calculations are from MFB [62]. The components of thespectrum are described in Figure 2.3.B: The translational-rotational spectrum of equilibrium 11 2 at 77K. The measurementsare from Birnbaum [12], and the calculations are from MFB [62] .10 5rob0C13 10'6toE010AChapter 2. The Theory of Collision-Induced Absorption^ 46Figure 2.5 shows the decomposition of the entire 112 translational-rotational band in termsof the induced dipole components. The calculations are those of MFB [63], and the mea-surements at 295 and 77K are by Bachet et al. [14] and Birnbaum [12], respectively. Thespectra have been measured and calculated at low resolution (10 cm') and therefore thedimer structure does not appear. The D(2023) component of the induced dipole momentclearly dominates the 112 spectrum and only in the high frequency wings of the rotationallines is there a noticeable contribution from other mechanisms. In the 77K spectrum, atthe location of S(0), the minor dipole moment components contribute only 1 part in 100to the total absorption. Minor dipole contributions to the translational band are furtherreduced as the temperature is lowered, because double transitions are eliminated sinceonly the J=0 and 1 rotational states are populated. These graphs justify the assumptionused in previous experiments and in the analysis of the present low temperature spectra,that the measurements can be reasonably fit using the B-C lineshape function which isattributed to the D(2023) component of the induced dipole.A Comment on the Interference Between the Quadrupolar and Overlap In-duced Dipole MomentsAn apparent discrepancy exists between the results of the above bandshape calculationsand those of Poll and Van Kranendonk. In the calculation of the integrated binary ab-sorption coefficient, an 'interference' term between the quadrupolar and overlap induceddipole moments accounts for about 18% of the total absorption of the translational bandat 300 K (see Table 2.3). In the pure rotational band the interference effect is negativeand it results in an , 8% reduction of the integrated absorption [6]. The theoreticalbandshape calculations of SM and MFB, however, make no reference to an interferenceeffect between dipole induction mechanisms.The two points of view may be reconciled by realizing that the interference termChapter 2. The Theory of Collision-Induced Absorption^ 47results from expressing the integrated absorption in the form of the trace of Pp[K, it]over a complete set of states (eqn. 2.14). Evaluation of this inner product involves termsof the form p/2 and p,2 [7], [51]. The A(2023) dipole moment expansion coefficient isexpressed as the sum of two terms, the quadrupolar and overlap term. The square of thedipole moment operator will therefore yield cross products, or the 'interference' terms.On the other hand, the transition moments which constitute part of the spectraldensity, eqn. 2.59, involve terms of the form I (f 12 = f) f (i). Matrix elementsare calculated by numerical integration, where the wavefunctions and the dipole momentoperator are expanded in products of spherical harmonics. The total matrix elementis calculated, the result is then squared, and no cross products explicitly occur. Theintegrated absorption obtained from either formalism is the same, as both start from thesame premise and merely do the calculation differently.2.5 The Birnbaum-Cohen LineshapeDetailed quantum mechanical calculations of collision-induced absorption are computa-tionally complex, and yet it is obviously desirable to compare measurements to theoryand to other measurements. In order to make comparisons of this kind, an empiricallineshape function has been devised, and a spectrum is synthesized by superposing line-shapes positioned at molecular transition frequencies, where the lines are weighted bythe appropriate population statistics, frequency factors, and transition moments.The B-C lineshape function, suggested by Birnbaum and Cohen, is well suited tomodelling collision-induced spectra of homonuclear diatomic molecules [46]. They derivea spectral density function from the Fourier transform of a physically plausible pair dipolemoment correlation,function. This spectral density includes a lineshape that fits well theasymmetric collision-induced lines, and in addition, spectral features due primarily toChapter 2. The Theory of Collision-Induced Absorption^ 48the same induction mechanism can be represented by the same lineshape function.The MFB quantum mechanical calculations of low resolution translational-rotationalspectra of 112 (Figure 2.5) can be reproduced accurately by associating a B-C lineshapefunction with each component of the induced dipole moment [64]. The dominant A(2023)induced dipole component is well described by the B-C lineshape. The anisotropic overlapcomponent A(2021) is, however, better represented by another lineshape function. TheA(0001) isotropic overlap component which is significant in 11 2-rare gas collisions is alsomodelled more accurately by the alternate lineshape. An extended version of the B-C model combines both lineshape functions for each dipole component, and it is usedto reproduce the quantum mechanical calculations of H2 absorption over a wide rangeof frequency and temperature [64]. In addition, radiative transfer calculations of theatmosphere of Jupiter have used the B-C lineshape in modelling the Voyager infraredspectra.Many experimental measurements of 11 2 absorption have been analyzed by fittingthe observations with a single B-C lineshape function [14],[13]. The measurements arerepresented in this way by an 'effective' A(2023) component, where it encompasses theother induced dipole components. This procedure is used to analyze the present spectra.The following discussion is primarily from Bachet et al. [14].^The B-C lineshape, for a particular transition^-4 Ef is given by(T,/,„2+70 ^(Z) rif (CV_ )^7r= - e^ (2.62)1 + (w_ ri )2Terms within this equation are defined asw_^w — wifTo =-- f3h/2^ (2.63)= (1 + (w_ r_ 2) t) 1 /2 ,r2 , _2 \ 1/21 /^2 -r• 7o)^/71Chapter 2. The Theory of Collision-Induced Absorption^ 49and Ki(z) is the modified Bessel function of the second kind; it is approximated byKi(z)= (1 + r z z + 0.56232)1/2 e-z(2.64)2 z + 0.46671 zThe parameters Ti and T2 are decay constants of the pair dipole moment correlationfunction. The duration of a collision is approximated by (T1T2)1/2, and this is inverselyproportional to the width of a spectral line. The B-C lineshape successfully accounts forthe observed upward frequency shift of the rotational lines; for instance, the peak of theS(0) collision-induced line at 77 K, is at 371 cm', as compared to the Raman scatteringline at 354 cm'.2.5.1 The Synthetic B -C SpectraThe translational absorption band, in general, overlaps the S(0) and S(1) 112 rotationallines. At low temperatures these lines are well separated, but some overlap persists in thewings. In order to understand the absorption due to the translational band alone, themeasured spectra are fitted by synthetic spectra obtained from the B-C lineshape. Theintegrated intensity ait, and spectral invariant ryitr are thereby obtained for the isolatedtranslational band.The synthetic B-C spectra have three adjustable parameters: ri, T2, and S, where Tiand T2 are the B-C decay constants, and S is a linestrength. The absorption at a givenfrequency is given by eqn. 2.48, and it is rewritten below with the assumption that thespectral density is composed of a superposition of B-C lineshapes (double transitions arenot included) [14].^w(1 — e-'61hw) (E P(J) E cv, A, J'; 00)2 sLArLA(w -A(w) = 273h2nc2where w j ji denotes the frequency of a transition J' 4-- J.(2.65)Chapter 2. The Theory of Collision-Induced Absorption^ 50The summation over L, A denotes the inclusion of all the components of the induceddipole moment, where a linestrength S and a lineshape 11 are associated with each com-ponent. The 11 2 spectrum is so clearly dominated by the A(2023) component (see Figure2.5), that a reasonable synthetic spectrum is obtained using only this component. Thequadrupole induced dipole, in turn, dominates the A(2023) component of the dipolemoment, and the line strength for quadrupolar induction is given by2S = 247J ( c-1Q--)g(R)R2 dRo^R4 (2.66)An S obtained by fitting measured spectra should exceed that which is calculated aboveusing known values of a and Q, since the measured S encompasses the contributions toabsorption from: the quadrupolar, overlap, and interference interactions associated withthe A(2023) dipole component, and the other minor dipole components.Synthetic absorption spectra based on eqn. 2.65 are generated by the superpositionof a translational band and rotational lines according to A(w) = AT(w) + As(o)(w) + As(i)(w) + • • • (2.67)whereandAT(w)^2; C2 ca(1 — CST') > P(J)C(J, 2, J; ovsrp)— 7 h2n JAs(J)(w) = 23hc72n2w(1 — e -13tw1P(J)C(.7, 2, J+2; 00) 2 ,5T(ca — cvs(J))(2.68)(2.69)1' is the B-C lineshape, given by eqn. 2.62; P(J) is the fractional population of the Jthstate, given by eqn. 2.40; and the Clebsch-Gordan coefficients are given by eqn. 2.39.Figure 2.6 shows the synthetic spectrum of 112 at 297 K using the values of 7-1, r2and S obtained by Bachet et al. from fitting the measured spectrum (not shown) with aB-C spectrum. The temperature, T i., 72 (in units of 10 -14 sec), and S (in units of K A')Chapter 2. The Theory of Collision-Induced Absorption^ 51are written sequentially on Figures 2.6-2.8. Deviations between the measured spectrumat 297K and its synthetic counterpart are less than 0.5% of the peak absorption. Thedeviations occur in the high frequency wing of the rotational lines and stem from thefailure to include double transitions and the U(1) hexadecapole transition in the syntheticspectrum [14].Figure 2.7 shows the reconstructed synthetic spectrum of normal 112 at 77K, wherethe B-C parameters are obtained from measurements by Dore et al. [13]. Figure 2.8depicts a B-C synthetic spectrum of normal 112 at 25.5 K, where the B-C parametersare obtained from fitting the translational spectrum shown in Figure 4.16. These figuresdemonstrate the dramatic increase in peak intensities and line resolution which are ob-tained by measuring the H2 spectrum at low temperatures. The translational band peakat 25.5 K is stronger than at higher temperatures, but it is still lower than the intensityminimum between the S(0) and 5(1) lines. This fact further illustrates the very weakabsorption of the translational band.The spectral invariants al and obtained from the translational-rotational syntheticspectra differ only slightly from the measured quantities. At 300K, al and obtainedfrom Figure 2.6, differ from those measured by 0.5% and 1.0%, respectively. The differ-ence between the synthetic and measured spectral invariants at 77 K is: ai, 1.3%; 71,0.4%. The correspondence between the measured spectral invariants and those obtainedfrom fitted spectra demonstrates the usefulness of modelling the 112 spectrum using theB-C lineshape.Spectral invariants obtained from measuring the entire translational-rotational spec-trum do not depend upon a lineshape function. This is not the case for the translationalband alone, since this band must be extracted from the overall spectrum using some line-shape. The integrated absorption coefficients of the translational band listed in Table2.3 are inferred from the B-C parameters of previously measured 112 spectra between 300.^I^,^,^I^.^I^.^.^I^.^I^1^,^,^,^I^,^,500.0 1000.0 1 ' 10 0.01^10.01I^I^. 177.4010.10^4.70^212.00Chapter 2. The Theory of Collision-Induced Absorption^ 522.00.0^1^,297.004.44^2.39^300.00---Frequency cm''Figure 2.6: The B-C synthetic spectrum of H2 at 297 K using the values of 71 ,7-2 (x 10 -14sec), and S (K A6 ) of Bachet et al. [14].2.00.01^,^,0.0I^I^I^,^,^.^I^.,^I^I^1^i^I^I^I^I^.^1^,500.0 1000.0 1500.0Frequency cm - 'Figure 2.7: The B-C synthetic spectrum of H2 at 77K using the values of ri , T2 (x 10'sec), and S (K A6 ) of Birnbaum [13].25.5018.27^9.58^290.86V 1^IChapter 2. The Theory of Collision-Induced Absorption^ 532.00.00o 1.0InCXzI^I^I^I^I^I I I I0.01 I 500.0 1000.0^1500.0Frequency cm-'Figure 2.8: The B-C synthetic spectrum of 112 at 25.5 K using the values of r1,r2 (x10-14sec), and S (K A') from the present measurements (see chapter four).and 77 112. In the present low temperature work, the spectral invariants aur and ryitr areobtained from eqns. 2.56 and 2.55 using the B-C translational band which best fits themeasurements.The spectral invariants of a B-C synthetic spectrum are directly related to the B-C lineshape parameters. For instance the area under the B-C translational-rotationalspectrum, the spectral invariant al, may be written in terms of r1 , T2 and S [14]. Thespectral invariants of the translational band alone in terms of the B-C parameters are(not including double transitions):aitr = (27r2 )) (h/2)3hc ^T T E p(J)c(J, 2, J; 00)2S^(2.70)1 2^j7.20 E P(J)C (J, 2, J; 00)25= 3c jand the ratio of the two invariants isaltr^^7172(2.71)Chapter 2. The Theory of Collision-Induced Absorption^ 54The close correspondence between the measured low temperature 112 spectra and thefitted B-C synthetic spectra is demonstrated in chapter four. The B-C lineshape is usedin the analysis of the measured spectra, and the variation of the spectral invariants andthe parameters 7-1 , T2, and S with temperature is examined.Chapter 3Experimental Apparatus and Procedures3.1 IntroductionThe precise measurement of the translational band of hydrogen is difficult for severalreasons. Firstly, collision-induced absorption by hydrogen is very weak, and the intensityof a light beam is diminished significantly only by passing it through a large amount ofgas. This is accomplished by using either high density gas samples, or long absorptionpaths. Secondly, the translational band and the S(0) rotational line are quite broad,owing to the short duration of molecular collisions, and the two features are resolvedonly if the gas temperature is quite low. The observation of the 112 dimer spectrumrequires both low temperatures and low densities, otherwise the weakly bound pairs arebroken apart by energetic and frequent collisions. In these experiments, even though thepathlength is over 52 meters and the temperature is below 40 K, the 112 absorption isstill weak and the measurement requires high experimental sensitivity and stability.The configuration of the experiment is depicted in Figure 3.1. Infrared radiationemitted by a mercury arc lamp is modulated by a chopper. This light enters a dual-input dual-output Michelson interferometer, and is split into two beams by a Kaptonbeamsplitter. The reflected and transmitted beams propagate to either side of a movingmirror carriage and return to the beamsplitter where they recombine. The IR radiation isthen focussed on to the interferometer output aperture. A He-Ne laser enters the secondinterferometer input, and propagates through the interferometer to the second output55Gas HandlingInterferometerchopper ref.lock-inamp.chopper^mercury^beamsplitter Iarc < It--•I moving_____.1 mirrorsI^laserLI transfer optics1.4---Vold White CellL.Vacuum Tanklaser fringe det.Bolometer LiquidHelium16 bit^ A 1)-- computertriggers from fringesFigure 3.1: A schematic diagram of the experimental apparatus.Modulated radiation passes through the interferometer and the multipass cell to the bolometricdetector. The rectified detector signal, the interferogram, is digitized according to triggers generatedby the laser interference fringes, and recorded on magnetic tape. The cell is cooled by circulatinghelium vapour through the heat exchanger tubing. Absorption spectra are obtained from ratios ofsample spectra with respect to empty cell or He background spectra.Chapter 3. Experimental Apparatus and Procedures^ 57where interference fringes are detected. The transfer optics relay the infrared beam fromthe interferometer to the multipass absorption cell. The infrared beam is brought to afocus at the field mirror edge and then it fills the opposing folding mirror; this continuesback and forth up to a 60 m path. The light leaving the cell is focussed by the transferoptics on to a bolometric detector.The cell is cooled by the continuous flow of a cryogenic liquid or gas through theheat exchanger tubing attached to the cell outer wall. The cold cell is surrounded byradiation shields and the assembly sits within a vacuum tank. The optical cell is filledand evacuated via the gas handling plumbing. Four vacuum pumps (not shown) arenecessary for the evacuation of the interferometer and transfer optics, vacuum tank andcell, gas handling system, and the detector.The chopped signal from the bolometer is rectified using a lock-in amplifier. Theresulting signal, the interferogram, is sampled by a 16 bit A-D converter which is triggeredby the zero-crossings of the laser fringes, and then recorded on magnetic tape. Theinterferograms are averaged and Fourier analysed on the UBC mainframe computer.3.2 The Low Temperature Far-infrared Absorption CellThe objective of this study is to observe weak absorptions at low pressures, low temper-atures, and long wavelengths. Previous work on collision-induced absorption has utilizedlow temperature multipass cells [65], [56], but these cells have not been optimized forlong wavelength studies, where diffraction losses must be kept to a minimum. The cellbuilt for the present work is unique, in that it allows absorption to be measured overpathlengths from 4 to 60 m, at temperatures as low as 20 K, over a wavelength range from0.5 mm to the visible. The cell optical design and a computer model of the cryogenicsystem have been discussed previously [66]. In the following sections the cell optical,Chapter 3. Experimental Apparatus and Procedures^ 58mechanical, and thermal designs are reviewed.3.2.1 The Cell Optical DesignA multipass mirror cell or 'White cell' consists of three spherical mirrors, a field mirrorand two folding mirrors, all of the same radius of curvature [67]. The distance betweenthe mirrors is set to the radius of curvature and the pathlength is adjusted by changingthe angle between the folding mirrors. The advantage of an optical cell of this type is thatradiation from a divergent source propagates efficiently over long paths. In the presentcell, the Pyrex mirrors, made by Interoptics Ltd. of Ottawa, Ont., have a 1 m radius ofcurvature and are coated with gold on chromium. The field mirror is 20 cm wide, and 8cm high; the folding mirrors are 'D' shaped and 10 cm wide.A light beam is focussed at the field mirror edge such that the beam fills the opposingfolding mirror. The folding mirror then forms an image of the input spot on the fieldmirror. Radiation successively diverges to the folding mirrors, and is focussed again onthe field mirror, until it reaches the opposite edge of the field mirror and exits the cell.The optical path length L isL = (2n + 2)/ (3.1)where 1 is the distance between the mirrors, and n is the number of spots on the fieldmirror . Figure 3.3 shows the alignment laser spot pattern across the field mirror for a12 m pathlength.The cell has an f/10 optical system, a much 'faster' beam than other low temperatureWhite cells reported in the literature. The large diameter optics minimize the diffractionof long wavelength radiation and allow the cell optical system to be efficiently matchedto the f/5 exit cone of the interferometer. An image of the interferometer exit aperture,magnified by only a factor of 2, forms the input to the cell, and in so doing, many smallChapter 3. Experimental Apparatus and Procedures^ 59spots can be stacked across the field mirror.Long wavelength radiation is diffracted by the effective aperture of the folding mir-rors. The diffraction pattern caused by a circular aperture, at the focus of a convergingspherical wavefront, is an Airy pattern [68]. The Airy radius to the first dark ring isr = (1.22A/D)R, where R is the distance from the aperture, D is the aperture diameter,and A the wavelength. The longest optical path is determined by the size of the diffrac-tion spot and the number of spots which can fit across the 16.8 cm distance betweenthe entrance and exit field mirror slots. The table below gives the diffraction spot sizeas estimated from the Airy disk diameter for a point source input, and the maximumpathlength attainable for different frequencies.o- A spot size pathlength20 cm" 0.5 mm 1.22 cm 56 m30 cm' 0.33 mm 0.85 cm 80 mTable 3.1: The pathlength limits due to the diffraction spot diameterThe ultimate cell pathlength is limited by the optical aberrations of diffraction andastigmatism, both of which cause the broadening of spots on the field mirror. Visiblelight tests of these aberrations and infrared spot overlap tests are described in Appendix1. In practice the maximum pathlength which can be obtained with virtually no over-lap between neighbouring spots at the cell exit is 52 m and the overlap is tolerablefor pathlengths up to 60 m. Background spectra obtained with a 24/1m thick Kaptonbeamsplitter demonstrate that the spectrometer-cell system transmits radiation down tor-15cm" (Figure 3.8). However, this beamsplitter has low efficiency at long wavelengthsand an accurate measurement of the cell transmission below 20 cm' requires a wire gridpolarizing interferometer.Chapter 3. Experimental Apparatus and Procedures^ 603.2.2 The Cell Mechanical DesignFigure 3.2 shows the cell, vacuum tank, and transfer optics in cross-section. The positionof the field mirror F is fixed with respect to the cell end-flange lid by invar bars S. Inthis way the contraction of the cell body C with temperature does not affect the mirrorspacing. The folding mirrors D are mounted on a complex set of hinged plates suchthat one mirror may be tilted with respect to the other, and this tilt is preserved asthe mirrors fold symmetrically for pathlength adjustment. Figure 3.4 shows the foldingmirror mechanism. The pathlength can be changed and the mirrors aligned while the cellis cold using the adjustment rods A. The cell is evacuated and filled via the pumping lineP which is baffled to prevent room temperature radiation from entering the cell. Pyrexwindows W in the cell mechanical end-flange lid, inner radiation shield, and vacuum tanklid, allow inspection of the laser alignment spots on the field mirror. The infrared beamenters the cell through windows or field lenses L which are positioned near the edge ofthe field mirror on extension tubes attached to the optical end-flange lid.The cell body C is a stainless steel tube of 8.66" 0.D., 8.35" I.D., length 46", and 35liters volume. The cell exterior has been electroplated with copper 0.030" thick) toinsure good thermal conductivity. This copper coating is porous and to supplement it sixcopper strips, 0.020" thick and 1.750" wide, were soldered along the length of the cell.The heat exchanger tubing H is 50 ft. of 1/2" copper tubing soldered on to the cell body.This tubing runs the length of the cell, turns, and runs back upon itself, in nine helicalturns. The temperature of the cell is monitored by five platinum resistance thermometersfixed to the cell outer wall in the positions numbered 1,2,3,4,5 (no. 3 only works at roomtemperature). Eighteen 2.2 KS2 3 Watt resistors are fixed to the cell in three rings ofsix and these can be used to heat the cell if necessary. The cell emissivity is reduced bywrapping it with aluminized Mylar superinsulation. Figure 3.5 is a photograph of theFigure 3.2: The cell mechanical design.F, field mirror; D, folding mirrors; S, invar spacing bars; C, cell body; A, mirror adjustments; P,pumping line; W, viewing window; L, IR entrance and exit windows; 1,2,4,5, platinum resistancethermometers; H, heat exchanger; I and J, inner and outer radiation shields;K, Kevlar suspensionrings; V cell vacuum tank; T, transfer optics vacuum tank; 0, transfer optics; B, bolometer dewar.Figure 3.3: The laser spot pattern on the field mirror for an optical path of 12 m.Figure 3.4: The folding mirror mechanism. The design permits one mirror to be tiltedwith respect to the other and this tilt is preserved as the mirrors fold symmetrically forpathlength adjustment.62Chapter 3. Experimental Apparatus and Procedures^ 63absorption cell with the heat exchanger tubing and copper strips.The cell is cooled by the continuous flow of cryogenic liquids; helium, nitrogen, andargon have been used. The cryogen flows through a siphon, through the cell heat ex-changer H, and then around an inner radiation shield I and an outer shield J. Each shieldis made of polished copper and has one helix of copper tubing running along its length.Figure 3.6 shows the outer radiation shield. The entire assembly is suspended withinthe stainless steel vacuum tank V by Kevlar rings K which have 1/10 the thermalconductivity of stainless steel for an equivalent tensile strength. The cell and shields aremounted to the Kevar rings by stainless steel clips.The transfer optics vacuum tank T is isolated from the cell vacuum tank by polypropy-lene windows. Radiation is focussed into, and out of, the cell by mirrors 0 which aremounted on a plate fixed to the end-flange lid of the cell vacuum tank. Light from thecell is focussed by the output transfer optics into a detector dewar B placed on top ofthe transfer optics case.3.2.3 The Cell Thermal DesignThe Predicted Helium ConsumptionA detailed computer model of the predicted thermal performance of the cell is describedby Wishnow [66]; it is derived from a similar calculation by Gush used to estimate thehelium consumption of the cooled rocket-borne spectrometer [36], [69]. The calculationis based on balancing the conductive and radiative heat inputs to each cell componentwith the heat extracted by the helium vapour coolant. The set of equations is solvednumerically for the unknowns of the helium mass flow rate, and the temperatures of theinner and outer radiation shields, given a desired cell operating temperature.The predicted liquid helium consumption ranges from 3 to 23 liters/day to maintain64Figure 3.5: The low temperature absorption cell. The cell is electroplated with copperand then the longitudinal copper strips and the helical heat exchanger tubing are solderedon to it. The temperature is sensed by 4 platinum resistors mounted on the cell exterior.Figure 3.6: The outer radiation shield. The shield is made of polished copper and theheat exchanger tubing wraps around it.Chapter 3. Experimental Apparatus and Procedures^ 65the cell at 20 K. The prediction depends primarily on the assumed values of the celland shield emissivities since their surface areas are quite large. Values of emissivitiescorresponding to the low consumption rate are: 0.015 for superinsulation surroundingthe cell, 0.02 for polished copper shields, and 0.03 for the stainless steel vacuum tank[70]. The emissivities corresponding to the high consumption rate are 10 times the valueslisted above, and are considered worst possible cases.The Cell Thermal PerformanceIn practice, a liquid helium flow rate of about 1.5 liters/hour maintains the cell at ,-,, 25 K.Cooling the cell down from liquid nitrogen temperature requires a flow rate of 3 to 4 1/hrand consumes 25-30 liters. The net consumption of helium during an experiment is muchhigher than 1.5 1/hr because the sample gas is admitted to the cell at room temperatureand must be cooled. This generally requires a flow rate of ,-,, 4 l/hr and a wait of about1-2 hr; the net helium consumption is about 3 1/hr. Overall, an experimental run ofabout 30 hr duration involving cooling the cell, obtaining background spectra, fillingwith the cell with the sample gas and cooling, and then maintaining low temperaturesduring the collection of sample spectra and final backgrounds, will consume just over 100liters. The temperature uniformity and stability of the cell are discussed in the sectionon experimental procedures.The helium flow of 1.5 1/hr to maintain the cell at 25 K, vaguely corresponds to themaximum predicted flow rate. This correspondence suggests that the emissivities of thecell components are very much worse than those described in the literature; it is notclear why this is the case. A possibility is that the superinsulation is not thermally wellconnected to the cell, and thus the cold cell wall is not the shiny aluminized surface, butthe porous copper electroplated surface.The predicted helium consumption may not be accurate since the thermal model ofChapter 3. Experimental Apparatus and Procedures^ 66the cell does not include all the heat sources encountered in practice. Among these arethe conduction of heat to the cell by the sample gas which fills the cell pumping line. Theoptical end-flange lid is exposed to room temperature radiation via holes in the radiationshields which allow the infrared beam in and out of the mirror cell. Infrared filters forthese holes have been omitted in order to transmit as much radiation as possible, andthe alignment laser. Finally, the thermal conductivity of the Kevlar rings may have beenunderestimated since their resin content is unknown.3.2.4 Temperature and Density MeasurementsThe temperature of the sample gas is sensed by four platinum resistance thermometersattached to the cell outer wall as indicated in Figure 3.2. The resistors are type KGC-0108 from Omega Engineering, a = (R373 /R273 — 1)/100 = 0.003925, and they have anominal resistance of 100 fi at 0° C. This type of resistor is low cost (,--, $10) and has alarger a (and thus higher purity) than industrial platinum resistors, where a = 0.00385.The higher purity allows resistance ratios measured with these resistors to be comparedto tabulated ratios obtained from precision platinum resistors. Resistance is measuredusing a four-wire, low power, AC auto resistance bridge.The thermometer calibration utilizes the consistency of the ratio of resistance dif-ferences with temperature, known as Cragoe's Z function. For a given element, thequantityRT - RT0 Z = (3.2)RTi - RT0 2is the same for all resistors, where RT is the resistance at a temperature T, and RT, andliTi are the resistances at two calibration temperatures [70]. The assumption underlyingthe Z function is that all platinum resistors share the same dependence of resistance ontemperature and differences between resistors are due to impurities which do not varyChapter 3. Experimental Apparatus and Procedures^ 67with temperature. Calibration of these resistors requires measuring their resistances at273.15 and 4.222 K. The temperature associated with a given resistance is then obtainedby referring to a table of Z as a function of temperature for calibrated precision platinumresistors [71].The absolute accuracy of the temperature reading using the platinum resistors andthe Z function is better than 0.2 K over the entire temperature range 14 to 373 K; overthe range 14 to 40 K the accuracy is better than 0.1 K. The accuracy of the resistancetemperature readings was obtained by comparing them against 21 temperatures deter-mined using the following thermometric standards: the triple point and vapour pressureof normal hydrogen (13.956-20 K), the vapour pressure of oxygen (58-90 K), the ice point(273.15 K ), and boiling water (373.15 K).The platinum resistors sense the temperature on the outside of the cell and not theactual temperature of the gas within. The determination of the gas temperature is madeby a simple average of the four resistance temperatures, where two resistors are locatedon the cell body (no.'s 2 and 4) and one on each lid (no.'s 1 and 5). In practice, theaverage resistance is found and then the temperature is determined; any error related toaveraging resistance rather than temperature is less than 0.1 K. The cell temperature isnot exact since there is a temperature gradient across the cell and the temperature driftsin time during data collection.The sample gas pressure is measured using an Ashcroft K-3 strain gauge pressuretransducer. According to the manufacturer's specifications it has a accuracy of 0.2% ofthe full scale reading of 10 Volts at 60 psi. The sensitivity of the gauge is 310.3 Torn Volt,and thus the accuracy in the pressure reading is 6.2 Torr, or 0.5% at pressures usedin these experiments. This gauge suffered from instabilities in both the zero offset andsensitivity, and it has been discontinued by the manufacturer. Once the instabilitieswere recognized, the gauge readings at vacuum and atmospheric pressure were checkedChapter 3. Experimental Apparatus and Procedures^ 68prior to each day of data collection. In general the gauge has an error roughly twicethat specified, although a few cases of larger sensitivity changes may have occurred. Thegauge is mounted at the room temperature end of the cell pumping line.During data collection the cell temperature and gas pressure are recorded every 10—15 minutes and these readings are converted to densities using the following procedure.The volume occupied by 1 mole of a pure sample gas (not a gas mixture) in thermalequilibrium is obtained fromP= 1^B(T)^C(T) — +^+^+ • • •RT V^V2^V3(3.3)where P is the pressure reading in atm., R is the gas constant (82.056 cm3 atm./moleK), T is the temperature reading in K, V is the volume in cm3/mole, and B and C arethe 2nd and 3rd virial coefficients. The density of the gas in amagat units is a ratio ofthe gas density to that of an ideal gas at NTP and is given by(3.4)where Vo = 22413.6 cm3/mole. The hydrogen sample gas density is obtained using onlythe 2nd order virial coefficents obtained from tables [72]. The error in density from using2nd rather than 3rd order coefficients at a density of 16.0 Am. at 25 K is 0.02 Am. ; at35K it is 0.01 Am. If the ideal gas law is used to find the hydrogen density, the result is1.3 Am. too low at 25 K, and 0.5 Am. too low at 35 K.During an experiment the cell is filled with a hydrogen sample and sealed. Thetemperature, and consequently the pressure, is varied to obtain different conditions, butthe gas density remains constant provided the gas is not liquified. The densities obtainedfrom temperature and pressure measurements of a sealed 112 sample over the range 23—38 K have a maximum variation of 2%, and a standard deviation of 1%.Chapter 3. Experimental Apparatus and Procedures^ 693.3 The Interferometer3.3.1 The Optical SystemThe principles of Fourier transform spectroscopy have been widely reported, and interfer-ometer design and data analysis procedures are discussed in Bosomworth and Gush [73].The optical design of the present interferometer has been reported by Buijs and Gush[74], and its application to absorption spectroscopy has also been described [75]. Figure3.7 is a schematic diagram of the interferometer optical path. The interferometer is a highthroughput (Ml ,--, 1.6 x 10' cm' str), dual-input dual-output, Michelson interferometerwith a folded optical path such that a mechanical motion of 25 cm produces a 1 m opticaldelay. Novel features of the optical system include: two dimensional corner reflectors onthe moving mirror carriage, and a laser which is co-aligned with the infrared beam foraccurate measurement of the interferometer path difference. The corner reflectors reduceoptical alignment and delay errors due to mirror rotations as the carriage moves, andlong pathlength differences can be achieved without dynamic alignment techniques. ASpectra Physics 116A single mode He-Ne laser propagates through the interferometerand the laser interference fringes, detected by a PIN photodiode, trigger the sampling ofthe interferogram.The inset of Figure 3.7 shows the configuration of the beamsplitter assembly in whichthe glass laser beamsplitter is suspended by a three-legged spider mount over a holecut in the center of the infrared pellicle beamsplitter. The two beamsplitters are madeparallel by the use of an autocollimator. Alternatively, an alignment laser is reflected offthe Kapton beamsplitter and the assembly is translated to reflect the laser off the glassbeamsplitter. The glass beamsplitter is then adjusted so that the laser spot on a distantscreen does not move when the assembly is translated.Chapter 3. Experimental Apparatus and Procedures^ 70Figure 3.7: The optical path of the Michelson interferometer [75]. S, source; B, beam-splitter; M, moving mirror carriage; D, detector. Alternate source and detector postions:S and D. The center of B is a glass beamsplitter for the laser which monitors the pathdifference. The inset shows the configuration of the beamsplitter assembly.3.3.2 Interferometer Scanning and the Infrared SourceThe moving mirror carriage is driven continuously by a Teflon nut riding on a precisionthreaded rod which is turned by a PMI DC motor. The performance of the drivingsystem is not exactly the same for forward and reverse directions, and the interferogramsanalyzed are all forward direction scans. The interferometer is set up to obtain pathlength differences of -4 to +103 cm, but only symmetric interferograms of path difference±2.3 cm are acquired for the hydrogen spectra.The interferometer scanning is controlled by issuing ASCII commands to a Z80 mi-croprocessor system built by the physics department electronics shop. The experimentswere conducted using a mirror scan speed which produced 570 Hz laser fringes or a me-chanical carriage motion of 9.02 x 10 cm/sec. Further comments on the choice of scanspeed are found in the procedures section.Chapter 3. Experimental Apparatus and Procedures^ 71The far-infrared source for the absorption measurements is a Phillips HPK-125W highpressure mercury arc lamp operated at a constant DC current of about 0.85 A. The arclamp has proven to be very stable, and over the range 20-100 cm" it radiates 2-4 timesmore power than an approximately 1100 K (dull orange glow) blackbody oven sourceused in initial experiments. Over the range 120-180 cm' the arc lamp intensity is about1.25 times that of the blackbody source.3.3.3 Beamsplitters and FiltersThe spectrum of hydrogen has been measured over more than a decade in frequency usingtwo spectral bands. The table below lists the frequency range, beamsplitter thickness,and the cold low-pass filters for the low frequency (LF) and high frequency (HF) spectralregions. In order to maximize the signal during the LF experiments the interferometeroutput aperture is a slit, 3 x 7 mm. The HF experiments use a 3 mm diameter circularoutput aperture.Region freq.(cm') beamsplitter low-pass filter20 - 100 24 ,am Kapton 2 mm thick wedged TeflonLF 0.13 mm black polyethylene125 - 180 24 inn Kapton (same as above)(2nd order)IR labs 1 mm crystal quartz wedgedHF 50 - 320 8 ILM Kapton with 5/10 pm diamond powder0.13 mm black polyethyleneTable 3.2: The beamsplitters and low-pass filters used for the LF and HF measurements.Figure 3.8 displays empty cell background spectra obtained at 35 K which are repre-sentative of the two frequency regions. Features of note in the low frequency spectrumare: the peak of the beamsplitter efficiency at 55 cm -1 , the beamsplitter null at 1101^1^I^1^1 I^ 1^1^1^1^1 II^1^I^I^I^1 I^I I I I i^1ik V_——(1 11°1^1^I^1 5°14°1100.00^50.0I01^1^1^1^1^11200.00I^I^1^I^I^I 1250.001 ?^I00.00 1^i^1BACKGNDS. LOW FREQ. 1A29F.P, HI FREQ. 4M13F.P ResolutIon=0.241 CM.-' E. WIshnow2.00. 01.5• _inca)..1C0.5FrequencyFigure 3.8: Examples of LF and HF background spectra obtained at 35KThe heavy curve is a LF background and the light curve is a HF background. The null of the LFbeamsplitter efficiency is at 110 cm'. Teflon absorption bands occur near 45 and 55 cm" andthe cutoff is at 190 cm'. In the HF spectra, crystal quartz absorption bands occur at 133 and260-270 cm-1 and the cutoff is at 350 cm". 'Channel' spectra with a -,70 cm" period are due topolypropylene windows.Chapter 3. Experimental Apparatus and Procedures^ 73cm -1 , the Teflon absorption lines near 45 and 55 cm -1 and the cutoff at 190 cm'. Thehigh frequency spectrum shows the peak of the beamsplitter efficiency at X165 cm -1 , theabsorption bands of crystal quartz at 133 and 260-270 cm -1 , and the quartz cutoff at350 cm'. 'Channel spectra' are observed in both backgrounds with a ,70 cm' period,where this is due to six polypropylene windows, each 0.002" thick, in the spectrometer-celloptical path.Quartz has a lower cutoff frequency than sapphire 370cm -1 ) and it was used asthe low-pass filter in order to restrict the spectra to the translational band only. Duringthe course of the experiments, it was observed that the sensitivity of the absorption mea-surements was limited by the maximum absorption; therefore, eliminating the S(0) lineimproved the signal-to-noise of absorption over the much weaker translational spectrum.3.4 Transfer Optics and WindowsThe interferometer, including the source and chopper, is enclosed by a large vacuum tankwhich is evacuated to a pressure of 0.15 Torr by a Stokes mechanical pump. Althoughthis is not a particularly low pressure, it is stable and water vapour lines cancel consis-tently. The interferometer tank twists slightly with respect to the optical cell when it isevacuated, but a motorized mirror mount within the tank can be adjusted to compensatefor this motion.The transfer optics couple the f/5 interferometer output beam to the f/10 mirrorcell optics, and the mirror cell output to the f/5.5 detector input. The transfer optics,excluding elements in the interferometer tank, are mounted on a plate fixed to the coldcell vacuum tank lid which is contained within an independent vacuum chamber.The vacuum chambers of the interferometer, transfer optics, cold cell, and detectordewar are all separated by polypropylene windows. The advantage of isolated vacuumChapter 3. Experimental Apparatus and Procedures^ 74systems is that the detector can be dismounted for helium transfer, and repairs or alter-ations of the interferometer can take place while the cell is cold. The disadvantage ofthis system is the strong channel spectra present in Figure 3.8.Polypropylene is a good window material since it has a low index of refraction of1.48, and a piece 0.002" thick is better than 90% transmissive over the frequency range0-500 cm'. Polypropylene is less reflective and absorptive than an equivalent sheet ofMylar. The pressure at which a 1.75" diameter window 0.002" thick bursts has beenmeasured to be ,400 psi, and the bursting pressure has been observed to be roughlyinversely proportional to the window area. The largest window, 2.4" in diameter, safelysupports atmospheric pressure, and the absorption cell windows 0.875" in diameter areestimated to burst at 400 psi. (this has not been measured). The strength of the materialrises when cold, and the cell windows may support even larger pressures.At room temperature, helium diffuses through polypropylene making leak detectiondifficult, but at liquid nitrogen temperatures the windows are virtually impermeable.Hydrogen does not diffuse rapidly enough through these windows to show a measureabledecrease in the sample cell pressure.3.5 Electronics and Data AcquisitionThe detector is a germanium bolometer, serial number 777, made by Infrared Laborato-ries. The detector operates at roughly 1.6 K by pumping on the helium reservoir. Thesensitivity of this detector is measured to be approximately 5 x 105 Volts/Watt, oper-ating with a 4 Volt bias. The sensitivity is obtained from bench top measurements ofthe detector response to a Barnes blackbody source operating at 600° C. The detectorpreamplifier has a gain of 1000 leading to an overall sensitivity of 5 x 108 V/W. The noiseof the detector-preamp system measured at 104 Hz by an HP audio spectrum analyzer isChapter 3. Experimental Apparatus and Procedures^ 75about 48 µV/ Hz, leading to an N.E.P. of 9.6 x 10' W/ H z , where IR labs quotes avalue of 9.1 x 10' W/ Hz. The system spectral noise density and sensitivity diminishwith frequency due to the detector time constant, and at 162 Hz the noise is about 30-40µV/ Hz. The detector linearity has been examined and is discussed in the chapter six.During an experiment a typical detector signal is a 162 Hz rounded triangle waveof 2.5 to 5 V peak-to-peak amplitude. This signal passes through a voltage divider ofgain 0.198 to an Ithaco 391A lock-in amplifier with a filter time constant set at 4 msec.The lock-in output is filtered by a Stanford Research Systems low-pass amplifier with anadjustable cutoff which was set at 13.5 Hz and 20 Hz, for the LF and HF experimentsrespectively. The amplifier is AC coupled and is set to a gain of 10 dB yielding aninterferogram signal of 4 to 8 V when the interferometer path difference is zero (zpd),and 0 V far from zpd.The interferogram signal is sampled by a 16 bit A-D converter-computer system withan input range of ±10 Volts, and a data rate up to 1000 Hz. The A-D has a digitizingnoise of 1.3 bits at a sampling rate of 285 Hz; this corresponds to a spectral noisedensity of about 23 pV/JIZ, or about 1/2 to 3/4 of the detector noise. This A-Dconverter is under the control of the PDP Micro-11 computer (the A-D converter inthe Z80 interferometer controller has not been used since pickup from the Z80 clockcauses even worse digitizing noise). The interferogram is sampled at constant pathlengthdifference via A-D triggers generated by the zero-crossings of the He-Ne laser fringes. Inthese experiments every two fringes produces a trigger pulse, and the 'folding' frequencyis therefore 1/(2 x 2 x 632.9nm) = 3950 cm'.The digitized detector signal is read, and written to tape, via a high speed inter-rupt driven assembly language program running on the PDP Micro-11 computer. Theprogram searches for the interferogram maximum and scans symmetrically forwards andChapter 3. Experimental Apparatus and Procedures^ 76backwards across it. Asymmetric interferograms consisting of forward scans starting atthe scan reversing switch and extending some specified length may also be acquired.The interferograms are read from the tape on to the UBC mainframe computer andthey are averaged by aligning the maximum point of of each interferogram. Forward andreverse scans are averaged separately and the interferograms are not interpolated to findthe true zpd. The failure to interpolate prior to interferogram averaging has no effecton the spectrum for frequencies below 200 cm-1, and produces an error less than 1%for frequencies less than 300 cm'. Symmetric interferograms consisting of 32768 pointsacquired in the forward scanning direction are Fourier transformed and power spectraare obtained, where the value of the power spectrum at a frequency sigma isP(a) = (cosine coeff.(cr)2 + sine coeff.(o)2)112 (3.5)The optical path length difference of the interferogram corresponds to a resolution limitof 0.241 cm-' and the interferograms are not apodized since the broad features of thehydrogen spectrum are well resolved.3.6 Gas Handling EquipmentSample gas cylinders are attached to a manifold and gases are flowed through a containerof 300 gms. of Union Carbide molecular sieve 5A. The hydrogen sample gas and heliumbackground gas are both UHP grade, 99.999% pure, supplied by Linde. The only signif-icant impurity in the spectral regions studied is 1120 which is present in these gases ata concentration under 5 ppm. In the case of hydrogen, the sieve is placed in an ice bathso that ortho-para conversion does not take place, otherwise the sieve may be cooled ina liquid nitrogen bath. The gas is further dried by passing it through a coil of coppertubing immersed in liquid N2. In order to prevent pump oils from entering the cell, theChapter 3. Experimental Apparatus and Procedures^ 77gas passes through a final stainless steel cold trap which is located at the entrance of thecell pumping line.Observation of water ice absorption at 230 cm' demonstrates that the gas dryingprocedures are inadequate (see chapter six). In fact, the efficiency of molecular sievesdepends on the H2 O vapour pressure and for these very pure gases the sieve is inactive.Completely pure hydrogen for future experiments can be obtained by passing the gasthrough a palladium diffusion filter, but this will not work for other gases. A betterwater vapour trap using phosphorus pentoxide or a large surface area liquid nitrogentrap is desirable. Despite the fact that gases pass through a number of cold traps, theywarm to nearly room temperature before entering the cell. Modifying this system toprecool gases to liquid N2 temperature before they are admitted to the cell would saveboth time and liquid helium.3.7 Experimental ProceduresThe quality of the results obtained using the spectrometer-cell apparatus depends uponprocedures adopted during the experiments. Methods of transfering cryogenic liquidshave been developed which minimize audio noise in the interferograms due to vibrationsand turbulence. In addition, the source chopping rate and mirror scanning speed havebeen chosen to move the audio frequencies in the interferogram away from audio noisesources. Another procedural issue is the realignment of the cell optical path betweenbackground and sample spectra.3.8 Liquid Nitrogen and Argon TransfersThe cell is precooled (or operated at 78 K) using liquid N2 siphoned from a 250 literstorage dewar. The transfer is controlled by pressurizing the dewar with an externalChapter 3. Experimental Apparatus and Procedures^ 78cylinder of helium or by adjusting the dewar relief valve overpressure. Liquid is initiallytransferred at about 2.5 psi, pressure, and once nitrogen temperature is reached thetransfer pressure is reduced. The minimum transfer pressure for liquid N2 is about 1.5—2.0 psi, and for liquid Ar it is 3.0-3.5 psi, since its density is much greater than liquidN2. In the cases of liquid N2 and Ar coolants, liquid circulates through the cell heatexchanger and boils in the tubing causing vibrations. These vibrations can cause audiofrequency variations in the transmitted light intensity.3.9 Liquid Helium TransfersHelium circulates through the heat exchanger as a cold vapour and boiling is not a sourceof vibration. In general, background spectra at ,--, 30 K are quieter than those at 78 or88 K. However, maintaining a stable flow rate using liquid helium is somewhat difficult.An oscillation with a -- 20 sec. period appeared in the interferogram and it was coincidentwith pressure fluctations at the heat exchanger outlet. Apparently, liquid helium fromthe storage dewar enters the heat exchanger and rapidly boils, since the cell is , 30 K.The boiling generates a local high pressure region which temporarily blocks the liquidtransfer. The transfer resumes after the pressure blockage dissipates. These oscillationshave been corrected by insuring that the highest flow impedance is in the liquid transferline and not in the heat exchanger. A syringe needle has been put at the bottom of thetransfer siphon, in the liquid helium, and this restricts the liquid flow rate.3.10 Temperature Gradients and TurbulenceOnce the periodic flow fluctations are eliminated, subtle effects of cell temperaturenonuniformity appear. Despite the fact that the heat exchanger is a helix which turnsChapter 3. Experimental Apparatus and Procedures^ 79back upon itself, and that the cell is electroplated with copper, the point where he-lium first contacts the cell is the coldest region. A steady helium flow, necessary tomaintain low temperatures, produces a ,,, 2 K temperature gradient across the cell. Thetemperature gradient causes an ambiguity in measuring the gas temperature, and drivesconvection within the sample gas. The gas convection causes slow optical signal vari-ations, mostly below 0.5 Hz, where these variations are presumeably due to bulk gasmotions. The dominant problem caused by the gas turbulence is the inability to alignprecisely the cell optics and maximize the detector signal.Two schemes have been developed to reduce the cell temperature gradient. The firstsolution is to stop the helium flow during data collection (drift technique). The cold spotwarms up, in fact it now becomes the warmest point, and the temperature gradient isgreatly reduced. The turbulent noise in the signal is virtually eliminated; however, theoverall cell temperature rises during the ,- 1/2 hour data collection period. The secondsolution is to vapourize the liquid helium before it enters the heat exchanger and therebyeliminate localized cooling due to evaporating the liquid. An insert to the transfer siphonhas been built which contains a heated copper mesh to vapourize the liquid followed bya platinum resistor to sense the vapour temperature. The vapour flow rate is set by thehelium dewar overpressure, and the heater power determines the vapour temperature.The device permits steady helium transfers and stable cell operating temperatures, butit does not eliminate turbulence.Using the drift technique at about 25 K results in a maximum temperature gradient of0.75 K across the cell, but the cell rises 2.3 K in 30 min; at 35 K the temperature rises 1 Kin 30 min. The helium vapourizer technique results in a temperature gradient of about1.8 K where the average temperature drift is kept below 0.5 K in 30 min. At temperaturesbelow 25 K, the minimum temperature gradient obtained using the vapourizer is 1.3 K,where it is tuned so that the temperature rises about 0.8 K in 30 min. It is apparent thatChapter 3. Experimental Apparatus and Procedures^ 80the thermal conductivity of the cell is too low to allow a stable operating temperaturewithout a thermal gradient.3.11 Interferometer Scanning and Chopping RatesThe interferometer scanning speed and chopping rate are chosen to avoid including theaudio frequencies of vibration and turbulence in the interferogram signal. The interfer-ometer system employs a chopper and lock-in detection technique, rather than a rapidscan technique, in order to accurately define the infinite absorption level. An advantageof this system is that the frequency of detection may be raised well above low frequencyvibrations. Room and pump vibrations occur below about 30 Hz, and resonances of theoptical system and detector microphonics occur at higher frequencies. 162 Hz was chosenas the chopping frequency since this region is consistently free of electrical interferenceand transfer related noise.Once the chopping frequency is established, the scan speed of of the interferometeris chosen to place the audio frequencies of interest in the interferogram away from audionoise sources. In general, higher scanning speeds are desirable since this allows thecollection of many interferograms in a given measurement time, and turbulent noise canbe reduced by averaging. The scan speed setting (`10.0 slow') corresponds to a He-Nelaser fringe rate of 570 Hz, where the optical path length difference varies at a speed of 570Hz/15800 cm-1 =0.036 cm/sec. At this scan speed 20 cm -1 occurs in the interferogramat 0.72 Hz, higher in frequency than most of the gas turbulence which occurs from 0.05to 0.5 Hz; 350 cm-1 corresponds to an audio frequency of 12.6 Hz, and this is free fromelectrical interference at 180 Hz. The time constant of the lock-in amplifier is set at 4ms. and the low pass filter is set at 20 Hz so that the highest audio frequencies associatedwith 350 cm -1 are not attenuated by the electronics.Chapter 3. Experimental Apparatus and Procedures^ 813.12 Optical Alignment and TurbulencePressurizing the cell with the sample gas results in a small shift of the spots on thefield mirror and optical realignment is required. If ignored, the shift is large enough tomove from one exit spot to another when long paths are used. After the cell is filled,the correct path length must be found and the signal maximum obtained. The cell pathlength is reduced until 15 spots are counted visually across the field mirror, and then thefolding mirrors are adjusted by finding successive signal maxima until the correct path isfound. Long period turbulence in the gas makes finding the exact position of the signalmaximum difficult, and this causes an error in the absorption measurement. In addition,the location of the spots on the field mirror for sample spectra may not be exactly thatof the backgrounds, again causing an absorption error.3.13 Cell Improvements Recommended for the Future3.13.1 Mirror AdjustmentsThe screws which adjust one folding mirror with respect to the other are too coarse(1/4-20) and this contributes to optical alignment difficulties. This design error permitsa slight movement in the folding mirror's relative positions as the folding angle is changedand these adjustments should be replaced by screws with fine threads.If the folding movement was highly reproducible, a new measurement technique couldbe implemented. Short path measurements, with the absorption cell filled, could serveas backgrounds for long path measurements of the sample spectra, and the absorptioncoefficient can be obtained from the ratio of these spectra. The time between takingbackgrounds, filling the cell, and taking sample spectra would be greatly reduced. ThisChapter 3. Experimental Apparatus and Procedures^ 82pathlength altering technique reduces errors due to experimental drifts with time, elim-inates error associated with cell pressure changes such as window shape or water icereflectivity, and allows data to be collected more efficiently during a measurement run.However, new systematic problems arise since it is now necessary to precisely measurethe transmission of the cell as a function of pathlength (Note that the transmission willvary with temperature and depend on the amount of ice coating the mirrors).3.13.2 Reduction of TurbulenceTurbulence within the gas cell can make obtaining precise absorption measurementsdifficult. The following proposals raise the cell thermal conductivity and thereby reducethe cell thermal gradient, the driving source of the gas turbulence.The cell thermal conductivity would be improved by applying a thicker copper coatingto the cell exterior, either by a plasma spray technique, or by more electroplating. Thesimplest proposal is to drive a copper sleeve of about 0.030" wall thickness into the cellinterior. This sleeve will certainly improve the conductivity along the cell, but it will nothelp in cooling the end-flange lids. Alternatively, a thermally floating shield, a coppertube with end caps which encloses the optical path and the mirrors, could be suspendedfrom the invar bars in the cell. This shield, with holes for light to enter and exit themirror cavity and for the viewing port, would not be in contact with the cell walls, andwould reach some average temperature of the sample gas. The temperature gradientswithin the shield would certainly be less than the temperature gradient of the cell outerwalls. The sample gas temperature should be measured within the shield, as the presentresistors mounted on the exterior of the cell may not indicate the correct temperature.Finally, the cell body could be remade out of a copper tube designed to fit the presentend-flange lids. It is not certain, however, that the present indium vacuum seals wouldwork, even though copper and stainless steel have similar thermal expansion coefficients.Chapter 3. Experimental Apparatus and Procedures^ 833.14 Ortho-Para MeasurementsA cold gas composed entirely of 112 molecules in the J = 0 rotational state exhibitsno translational absorption spectrum. A 112 molecule in the J = 0 state is sphericallysymmetric, that is, the expectation value of any particular orientation of the molecularaxis is zero [76]. Therefore, there is no external electric field due to the molecule, andno quadrupolas induced dipole moment during binary collisions (even though the J = 0matrix element of the quadrupole moment in the molecule fixed frame is non-zero).Likewise, the anisotropic overlap induced dipole is zero since it also depends on thecolliding molecule's orientations. The present low temperature experiments have used'normal' hydrogen, with an ortho-para ratio of 0.7492:0.2508, so that a relatively strongtranslational absorption spectrum is observed. Ortho-to-para conversion of the 112 samplemay take place via adsorption on the cold cell walls or by passing the gas through thecold traps. In order to understand if conversion has taken place, samples of 112 gas aretaken during the absorption measurements and their ortho-para ratio is tested.A resistance bridge built by Dr. Walter Hardy compares the ortho-para ratio of asample gas to that of normal hydrogen [77]. The apparatus consists of two glass bulbpirani vacuum gauges which form two arms of a DC resistance bridge and are immersedin liquid N2. The method utilizes the difference in the specific heat at constant volumebetween para and ortho 112 at 77K(c.c,vorPatrh: = 0°.03028 in units cal deg' mole-'), andhence, the higher thermal conductivity of para with respect to ortho 112 [78]. A fixedcurrent of 650 mA passes through both pirani gauges, and the bridge current is nulledwhen both the sample and reference bulbs contain normal H2 at 40 Torr. The samplebulb is then evacuated and refilled to the same pressure with the 112 sample to be tested.If the sample is pure para 112, a bridge current offset results. This is because the paragas conducts heat from the pirani filament better than the normal gas, and as a result,Chapter 3. Experimental Apparatus and Procedures^ 84the filament is colder and its resistance is lower than the normal reference gauge. Thebridge current offset is nearly linearly proportional to the sample para concentration andit is calibrated by the above comparisons of normal to normal, and para to normal, H 2 .Excluding cases of extreme errors due to leaks into the sample gas flasks, all samplestaken from the absorption cell retained the normal ortho-para ratio to within the 1%precision of this apparatus.The normal hydrogen reference sample is taken from the same gas cylinder as the11 2 used in the absorption spectra. The ortho-para apparatus also has a built-in UHPhydrogen reference cylinder, but measurements conducted using this gas produced aconsistent current offset when compared to two other UHP hydrogen samples. In bothcases the offset corresponds to an ortho-para ratio of 0.69:0.31, and the built-in referencecylinder is believed to be contaminated.Chapter 4The Translational Absorption Band of Hydrogen4.1 IntroductionThe collision-induced translational absorption band of hydrogen has been measured overthe temperature range 21 to 38 K, and the pressure range 0.6 to 3 atmospheres, using anoptical path of 52 and 60 meters. 11 2 spectra are obtained over the wavenumber range 20to 320 cm' with considerably higher resolution (0.24 cm -1 ) and sensitivity than previousstudies. The new low temperature and low pressure experimental conditions allow thepure translational band to be observed virtually isolated from the 11 2 rotational lines.Previous studies using 1-3 meter light pipes have been limited to temperatures above77 K, resolutions greater than 2 cm', and have required densities of over 100 amagatto observe any appreciable absorption in the translational band [8],[12],[13],[14]. Thepresent low pressure, high resolution measurements permit details of the submillimeterspectrum of 11 2 to be examined for the first time. In addition, the low pressure, lowtemperature experimental conditions employed in the present study are close to thosefound in planetary atmospheres.The analysis of the measured hydrogen spectra is presented in this chapter. Thegeneral features of the spectra, and the variation of the best fit B-C lineshape parameters,are examined as a function of the temperature. The experimental conditions of a lowtemperature, low density gas allow the formation of 112 bound pairs, and the search fordimer transitions in the translational band is presented. Average spectra which combine85Chapter 4. The Translational Absorption Band of Hydrogen^ 86data from the low and high frequency spectral regions are obtained at temperaturesof 25.5 and 36K, and the spectral invariants of these 'composite' spectra are comparedto previous measurements and theoretical calculations.4.1.1 On the Discrepancy Between the LF and HF Spectral RegionsThe measurements of the hydrogen spectrum extend from 20 to 320 cm-1, a factor of 16in frequency. This frequency range is too great to be covered with a high signal-to-noiseratio by a single interferogram, because of the limited dynamic range of the bolometerdetector. For this reason the spectrum was split into a low frequency (LF) region, 20-180cm-1, and a high frequency (HF) region, 50-320 cm'; the beamsplitters and filters forthese spectral regions are given in Table 3.2. Since a special effort was made to findevidence for dimers, most observing time was spent on the low frequency region, and atotal of 4 independent LF experimental runs were undertaken. As a result considerabledata exists on which to form an estimate of the reproducibility of the spectra. TheHF region was studied during only one experimental run, and the cell was kept coldcontinuously for 30 days as different gas densities and temperatures were investigated.Upon subsequent analysis of the results, a discrepancy of r•., 10% between the inten-sities of the LF and HF absorption spectra has been observed in the region of overlap;this discrepancy is larger than the errors of internal consistency within the two sets ofmeasurements. It has not yet been possible to explain definitively this discrepancy, andthe HF experiments could not be repeated because of the substantial cost. In the regionof overlap the shapes of the two spectra are nearly the same, and an intensity mismatchof this kind might result from errors such as: incorrectly counting the number of passesin the cell, or a departure from a 'normal' ortho-para ratio of the HF gas sample. Theseexplanations, however, conflict with the written records made during the experiments.In the following analyses, the greatest weight is given to the low frequency dataChapter 4. The Translational Absorption Band of Hydrogen^ 87which is regarded as the most reliable. The effect of the higher frequency data on, forexample, the fitted line shape parameters turns out to be small, and despite this troublingdiscrepancy, the derived physical data is valid.4.2 General Remarks on the Absorption Spectra4.2.1 The Spectral Band and Interferogram AveragingThe LF spectra cover the 112 translational spectrum from the half-maximum on the lowfrequency side of the peak to the half-maximum on the high frequency wing of the band.The HF spectra cover the spectrum from roughly the peak of the translational band tothe low frequency wing of the S(0) rotational line. Examples of LF and HF absorptionspectra are shown in Figures 4.12-4.17; examples of LF and HF absorption spectra priorto dividing by p2 1 are shown in Figures 5.1 and 7.2, respectively; and the LF and HFbackground spectra are shown in Figure 3.8.The temperature dependence of the translational band of normal 11 2 is examined using28 LF spectra obtained over the temperature range 22-38 K, and the density range 7.65to 23.28 amagat. These studies are depicted in Figures 4.3-4.9. Low temperature LFspectra are also examined for evidence of spectral structure due to 11 2 dimers. TheLF spectra were concentrated around 25 and 35 K, and HF spectra were subsequentlyobtained near these temperatures in order to obtain complete spectral coverage overthe translational band. The composite average spectra, composed of both LF and HFspectra, are shown in Figures 4.12-4.17. The composite average spectra are used in thecomparison of the present results to previous experiments and the calculated integratedabsorption coefficient, Figures 4.19-4.21.Power spectra are obtained from the Fourier transformation of averaged symmetricinterferograms according to the procedures described in chapter three. In general, eachChapter 4. The Translational Absorption Band of Hydrogen^ 88'individual' spectrum is derived from 4-6 sample interferograms and 6-8 backgroundinterferograms. The interferograms are acquired in the interferometer forward scanningdirection. The duration of an interferogram is 2.5 minutes, and the time required tocollect the set is about 20-30 minutes.Absorption spectra per unit density squared, per unit pathlength are given by= 172i   ) (4.1)where p is the sample gas density in amagat, and / the pathlength. L, is the samplespectrum, and 4 is the background spectrum, an average of empty cell spectra takenbefore and after the sample data.4.2.2 Temperature, Density, and Spectral AccuracyThe measurement of the sample gas temperature and density is discussed in chapter three.In brief, the cell temperature measurements vary by less than ±1K due to either thermalgradients along the cell, or drifts in temperature during the data collection period. Theaverage temperature reported for a given spectrum is estimated to be accurate within±0.25 K, and the relative temperatures of the various spectra are stated to a precisionof 0.05 K. The gas pressure is measured to an accuracy of about 1%, and the absolutedensity is estimated to have an uncertainty of +1%.The accuracy of the absorption spectra is limited primarily by long term experimentaldrifts and an intensity mismatch between the LF and HF spectral regions. Within eachspectral region, the uncertainty of the 112 translational absorption band measurements isestimated to be ±2% of the peak absorption from 40-90 cm', +3% across the LF region,and ±5% across the HF region. The 112 spectra are free of water vapour lines, althoughwater ice contamination is observed above 200 cm". The issue of spectral accuracy iscovered more fully in chapter six.Chapter 4. The Translational Absorption Band of Hydrogen^ 894.2.3 Typical Spectral FeaturesTypical LF absorption spectra are presented in Figures 4.1 and 4.2, where the broadcollision-induced translational band is the dominant spectral feature, and contributionsfrom the S(0) line are negligible. The solid curve is the measured spectrum and thedashed curve is the best fit B-C synthetic spectrum. The filename; density (in amagat),temperature; B-C lineshape parameters T1 , T2 (x10 -14 sec) and S (in K A6 ); and spectralinvariants altr (cm 5 sec") and 'Yi ltr (cm5 sec) are given sequentially on the graph.The LF absorption spectra often droop below a B-C spectrum over the frequencyregion 90-130 cm'. This aberrant shape occurs at frequencies near the LF beamsplitternull, and an anomalously low absorption is also seen in the HF spectra below 65 cm".The aberrations occur in frequency regions where the background spectra have low in-tensity (see Figure 3.8), and they are partially caused by a residual baseline intensitywhich underlies the spectra. The subtraction of the residual baseline is discussed in thesection on the composite average spectra.Small distortions of the smooth bandshape near 45 and 55 cm -1 are related to theinexact cancellation of steep regions in the sample and background spectra due to theabsorption bands of the teflon cold filter. The topic of spectral distortions is discussedfurther in the dimers section.The sharp spike superposed on the translational band at 89.2 cm" is the R(0) lineof HD which is present in the hydrogen gas sample as a naturally occurring isotopic`impurity'. The analysis of the HD line is discussed in the following chapter.4.3 The Temperature Dependence of the H2 SpectrumThe set of LF absorption spectra illustrates the detailed temperature dependence of theH2 translational band for the first time. In previous higher temperature spectra, theChapter 4. The Translational Absorption Band of Hydrogen^ 906.05A29F.NC23.28 34.4013.42^8.98 233.560.1850E-31^0.2226E-570 .00. 0 40.0 80.0^120.0Frequency cm-1160.0 200.03A27F.NC^13.68^25.0017.59^10.36^279.980.2019E-31^0.3670E-576.04.007 40.0 80.0^120.0Frequency cm-'160.0 200.0Figure 4.1: An example of a LF absorption spectrum of H2 at 34.4 K. In this and thefollowing figure, the solid curve is the measured spectrum and the dashed curve is thebest fit B-C spectrum. The density and temperature; B-C parameters r1 , 72, and S; andan, and ryitr; are given sequentially on the graph.Figure 4.2: An example of a LF absorption spectrum of H2 at 25.0K. The format of thegraph is the same as that above.Chapter 4. The Translational Absorption Band of Hydrogen^ 91shape of the translational band has been obscured by the substantial overlap of the S(0)rotational line. The present 112 spectra show clearly the translational band shape, anddemonstrate the validity of fitting the spectrum with the B-C lineshape function.4.3.1 The Spectrum Fitting ProcedureThe measured spectrum is fitted by a synthetic spectrum characterized by the temper-ature and the B-C lineshape parameters ri, T2, and S, using a non-linear least squaresfitting routine. The synthetic B-C spectra are discussed in chapter two. The fit isweighted by a frequency dependent function which is the product of the absorption spec-trum (prior to dividing by p21), and a generic LF background spectrum. This weightingprocedure emphasizes fitting the peak of the absorption spectrum, between 45 and 60cm', and the region of maximum signal-to-noise of the spectrometer, between 60 and 80cm". The fit of the B-C synthetic spectra to individual spectra is illustrated in Figures4.1 and 4.2, and the fitting procedure determines the B-C parameters to better than1%. The best fit B-C synthetic spectra are then used in the following examination of thetemperature dependence of the translational band.4.3.2 The Frequency of the Peak and the Bandwidth vs. TemperatureThe absorption spectrum depends on the population distribution of the molecular trans-lational energy states. Figure 4.3 shows that the frequency of the translational bandpeak varies as T1/2, where this variation reflects the temperature dependence of the peakof a Maxwellian velocity distribution. Figure 4.3 may be linearly extrapolated to a peakfrequency of 90 cm' at 77.4 K where Birnbaum has measured a peak at approximately86 cm-1 [12].Figure 4.4 shows the dependence of the full width at half-maximum (FWHM) as afunction of the square root of the temperature. The translational band has a broad peak4.5 5.0 6.0 6.55.5IITemp. (K)80.01 October. 1992PEAK^vs. SORT T—6.68 10.9460.0Ua)a)11-—Ne 40.0eaa)20.06.0^6.5200.01 October, 1992DEL PK vs. SORT T63.70 12.02EU2 150.0LL-c100.0coa)50.04.5^5.0^5.5s/Temp. (K)Chapter 4. The Translational Absorption Band of Hydrogen^ 92Figure 4.3: The frequency of the peak of the translational band vs. the square root of thetemperature. In all figures of this section, + denotes a LF 11 2 spectrum, and ^ denotesa composite average spectrum; the y intercept and the slope of the fitted line are givenin the top left.Figure 4.4: The width (FWHM) of the translational band vs. the square root of thetemperature.41-+-Fr+1 October. 1992DEL G vs. SORT T1.63^9.06Chapter 4. The Translational Absorption Band of Hydrogen^ 93780.060.0cv1-440.020.04.5^5.0^5.5^6.0^6.5iTemp. (K)Figure 4.5: The half-width of the translational band lineshape function vs. the squareroot of the temperature. The lineshape function is given by eqn. 4.2.which broadens further as the temperature increases. The data are somewhat noisy asthe position of the high frequency half-maximum falls within the lower signal-to-noiseregion of the beamsplitter. The FWHM of the translational band is considerably broaderthan the FWHM of the rotational lines S(0) and S(1) since stimulated emission affectsthe spectrum more strongly at low frequencies. At 37K, the FWHM of the translationalband, the S(0), and S(1) lines are: 145, 83, and 81 cm' (based on a B-C spectrumextrapolation of the translational band measurements).The half-width of the lineshape function G(a) from zero frequency to the half-maximum frequency, or from the S(0) or S(1) free-molecule transition frequency to thehalf-maximum on the high frequency side, does not depend on the transition. G(cr) isgiven byA(cr) ^G(c)^cr(1 — e-hco- kT )(4.2)where for the present purposes, A(o) is the best fit B-C spectrum to the experimentaldata. Figure 4.5 shows the half-width of G(o) as a function of T1/2. This plot is lessChapter 4. The Translational Absorption Band of Hydrogen^ 94noisy than Figure 4.4, because the half-maximum of the lineshape function occurs ina spectral region with higher signal-to-noise than the half-maximum of the absorptionspectrum.The data points in Figures 4.3-4.5 are the results of 28 independent experiments.The scatter of the points is on the order of ±5%, where this is consistent with thereproducibility of the spectra (see chapter six).4.3.3 The Spectral Invariants vs. TemperatureThe variation of the spectral invariants altr and ryit, with temperature is shown in Figures4.6 and 4.7. altr is the integrated binary absorption coefficient in terms of w, eqn.2.56, andit is related to the physical properties of hydrogen by the Poll and Van Kranendonk theoryof collision-induced absorption as described in chapter two. ry itr is given by eqn. 2.55, andit is analogous to an integrated lineshape function (the relationship between G(a) andryitr is discussed in chapter two). The present results are the first investigations of altrand ryltr for the absorption spectrum of I-12 below 77K. The variation of the spectralinvariants with temperature is related to the pair distribution function and hence to theintermolecular potential. Figure 4.6 demonstrates that altr increases as the temperaturedecreases, where this effect is due to an increase in the lifetime of molecular collisions,and to an increase in the contribution of 1-1 2 bound pairs to the pair distribution function,as the temperature is lowered (see Table 2.2). A detailed evaluation of the implications ofthese new results for the theory of collision-induced absorption requires complex quantummechanical calculations, outside the scope of this thesis. The results should, however,stimulate new work in the field.The errors in the determinations of the spectral invariants are ti ± 5%, as accessed bythe variations of the data points in Figures 4.6 and 4.7. The larger deviations of alt, atlow temperatures are related to the fact that lower sample gas densities must be used to5 October. 1992GAMMA vs. TEMP++0+F0.020.0^25.0^30.0^35.0Temperature (K)5.040.0Chapter 4. The Translational Absorption Band of Hydrogen^ 953.02.5a) 0S.0.50. 05 October. 1992ALPHA vs. TEMP+ + ++ + + +11 +20.0^25.0^30.0^35.0^40.0Temperature (K)Figure 4.6: ait, vs. temperature for the LF H2 spectra. The variation of the integratedbinary absorption coefficient of the translational band with temperature. ait, is definedby eqn. 2.55.Figure 4.7: 7ltrvs. temperature for the LF H2 spectra. The variation of the spectralinvariant -yur of the translational band with temperature. yit, is defined by eqn. 2.54.Chapter 4. The Translational Absorption Band of Hydrogen^ 96avoid liquification, and thus the absorption measurement has a lower signal-to-noise thanat higher temperatures. Systematic errors due to turbulence and the formation of ice alsobecome more influential at low temperatures. -yitr is less noisy at low temperatures thanaltr because it emphasizes the low frequency portion of the spectrum; the spectral regionover which the spectrometer is the most sensitive, the 11 2 absorption is the greatest, andice is not absorptive.4.3.4 The B -C Parameters vs. TemperatureFigures 4.8 and 4.9 demonstrate the variation of the B-C lineshape parameters r1 , r2, andS for the translational band as a function of temperature. Previous experiments haveexamined the variation of these parameters for the entire translational-rotational bandover the temperature range 77 to 300 K, and the synthetic spectra which are fitted to thesemeasurements are therefore dominated by the S(0) and S(1) lines [13],[14]. Figures 4.8and 4.9 provide new information on the 11 2 absorption spectrum since the present studyconcentrates on the translational band and extends the temperature range of previousexperiments.4.4 Hydrogen Dimers4.4.1 Introduction and Previous WorkThe formation of bound states for a pair of 11 2 molecules is predicted by solving theSchrodinger equation for an intermolecular potential with an attractive part [43],[79],[80],[45]. The energy levels and intermolecular potential for the 1-1 2 dimer are shownin Figure 2.1, and transitions between rotational states are allowed according to theselection rules Al = ±1, ±3, where 1 describes the end-over-end rotation of the complex.Bound-bound and bound-free transitions of molecular hydrogen pairs were first observed1 October, 1992TAU 1 vs. TEMP^TAU 2 vs. TEMPX X x X x)0(xix x25.0a)20.015.0X••■•••10.044.—C1CtF.7.4^5.00. 05 October. 1192VS. TEMP_ ++4. + ++ +El + ++^+444..Chapter 4. The Translational Absorption Band of Hydrogen^ 9720.0^25.0^30.0^35.0^40.0Temperature (K)Figure 4.8: .The B-C lineshape parameters Ti and 7-2 vs. temperature for the LF 112 spectra. +denotes T1 , and x denotes T2.400.0300.0200.0ti)^25.0^30.0^35.0^40.0Temperature (K)Figure 4.9: The B-C lineshape parameter S vs. temperature for the LF 112 spectra.Chapter 4. The Translational Absorption Band of Hydrogen^ 98in the fundamental vibrational band at 20 K by Watanabe and Welsh [56]. Radiation isabsorbed at frequencies near those of the free 11 2 molecule when an 11 2 dimer transitionoccurs simultaneously with a vibrational and/or rotational transition of one of the dimermolecules. The matrix element of the electric dipole moment of these transitions is non-zero, and structure is observed superimposed upon the broad collision-induced vibrationaland/or rotational lines.Structure in the 11 2 rotational band was first observed in the far-infrared spectrumof Jupiter obtained by the Voyager spacecraft, Figure 1.2, and this was subsequentlyinterpreted as evidence of dimer absorption [81],[82]. Dimer transitions near 5(0) andS(1) have been observed in the laboratory by McKellar at 77K [27],[45], and in 11 2 andD2 at 20 K [28]; this work is shown in Figures 1.4 and 2.4. Dimer structure has also beenobserved superposed on the S(0) line of 11 2 and D2 using the present spectrometer-cellapparatus near 20 K, and a spectrum of the 5(0) region of D2 is shown in Figure 7.1.Pure 11 2 dimer bound state transitions are expected to appear at frequencies lowerthan 10 cm', below the frequency range of the present spectrometer. A new polarizinginterferometer, patterned after the rocket-borne instrument COBRA [36], [69], is underconstruction with the intention of observing 11 2 dimer bound state transitions.4.4.2 The Predicted Dimer Structure in the Translational BandThe existence of dimer structure in the translational band of low temperature 11 2 hasbeen predicted by Schaefer and Meyer [33], and the astronomical basis for this pro-posal is described in the introduction. Figure 4.10 shows the calculated low temperature112 emission spectrum [63]. The ripples superimposed upon the translational band aredue to continuum-bound dimer transitions (below 30 cm"), and resonances betweencontinuum-continuum H2 pair transitions. The structure of a 11 2 absorption spectrumwould qualitatively resemble that of the emission spectrum, but the absorption spectrumChapter 4. The Translational Absorption Band of Hydrogen^ 99would have a different frequency dependence and include stimulated emission. The searchfor dimer structure in the translational band of 112 has been a major focus of the presentlow temperature spectroscopic studies.4.4.3 Experimental Dimer SearchFigure 4.11 is the spectrum of hydrogen at 22.4 K, obtained using a gas density of 9.39amagat and a pathlength 60 m. The sample spectrum is an average of 4 spectra, eachconsisting of 6 interferograms, and the total observation time is 60 min. The backgroundspectrum is composed of an average of 2 empty cell and 3 helium spectra taken before,and 2 helium and 3 empty cell spectra taken after the 112 spectra. All background spectraare acquired at the same temperature as the 112 spectra, and the helium backgrounds areobtained at the same pressure as the 112 spectra in order to cancel the effects of windowdistortions. No 'channel' spectra are observed in the helium spectra, and using both theempty cell and helium filled backgrounds reduces systematic errors due to experimentaldrifts. A substantial effort has been made to obtain high quality low temperature spectra,and in order to do this, transfer noise, temperature gradients, and window distortionshave been kept to a minimum.The sample density and temperature have been determined on the basis of previousobservations of dimer lines near S(0), where dimer lines are observed to broaden signifi-cantly above -- 10 amagat, and dimer structure is enhanced as the temperature is lowered,particularly for densities below ,,, 4 amagat. The condition chosen for the present spectrais a compromise between temperature and density; the density must be high enough tocause measurable absorption, yet the sample should not liquefy. The peak absorptioncorresponding to data shown in Figure 4.11 is 23%, and the uncertainty in the absoluteabsorption of this spectrum is -- 4% of the peak.In the region of highest signal-to-noise, from 40-80 cm-1, no obvious structure isChapter 4. The Translational Absorption Band of Hydrogen^ 1008D6.0in1116 4DC2.0>.c Figure 4.10: The predicted 11 2 translational emission spectrum, from [30]. The structurebelow 30 cm' is due to free-bound transtions and that at higher frequencies is due toresonances between free-free transitions.6.0V520.0^40.0^60.0^80.0^100.0^120.0Frequency cm''Figure 4.11: The LF 11 2 translational absorption spectrum at 22.4 K. The filename,temperature, and density in amagat are written on the graph, and the the pathlength is60 m. Note the absence of obvious dimer features.Chapter 4. The Translational Absorption Band of Hydrogen^ 101observed in the spectrum. Likewise, over the region 20-40 cm", no structure is apparent,although the measured spectrum is noisier. The spectrum does not display any structurewhich might be associated with 112 dimers at a level greater than 1% of the translationalband peak. The evidence is that dimers make only a continuous contribution to the 112absorption spectrum under these experimental conditions, that is at T = 22.4 K andp = 9.39 amagat.4.5 The H2 Composite Average Spectra at 25.5 and 36 KComplete translational band absorption spectra have been obtained by averaging theLF spectra, covering 20-180 cm', and the HF spectra covering 50-320 cm'. TheHF spectra have been collected using the same procedure as the LF spectra, but areconcentrated near the temperatures of 25 and 35K. A weighted average of the LF andHF spectra is obtained for the temperatures 25.5 and 36 K, and the results are presentedin this section. A discrepancy of about 10% exists between the intensities of the LF andHF spectra, and an effort to eliminate this discrepancy by removal of a residual baselinein the raw spectra is discussed. The spectral invariants of the complete spectra areobtained and compared to previous experimental work and the calculations of chaptertwo.4.5.1 The Removal of a Residual BaselineWe believe that the 10% intensity discrepancy between the LF and HF spectra, overthe frequency range where they overlap, is caused by a detector non-linearity. The HFbackground spectra, with a larger bandwidth and thus higher interferogram signal, mayhave been reduced in intensity; this topic is examined further in chapter six. A residualbaseline in the raw spectra may be a consequence of the detector non-linearity, and^Chapter 4. The Translational Absorption Band of Hydrogen^ 102regardless of the cause, will perturb the absorption measurements.The Residual Baseline Intensity of the Background SpectraFigure 3.8 shows typical background spectra for the LF and HF cases, and a small residualintensity across the band, about 0.5% of the LF peak, is observed. The residual baselinehas roughly the same amplitude for both LF and HF backgrounds, and is most evidentin the following regions: 0-10 cm'; near the LF beamsplitter null; and at 135 cm'and 263-273 cm -1 , the crystal quartz absorption bands. In all of the above regions, it isreasonable to expect a background spectrum with zero intensity. Although the baselineintensity is a small fraction of the spectrum peak, it is greater than 10% of the HFspectrum intensity below 60 cm-1 .The Influence of a Baseline on the Absorption SpectrumAn absorption spectrum obtained from a sample and a background spectrum which bothinclude a residual baseline is anomalously low. Suppose a background spectrum /b(c)and a sample spectrum I3 (a) are both composed of a true part I, and a residual partI'. Suppose also that the sample transmits some fraction f of the background intensity,and that the residual part is independent of the transmitted intensity. The measuredabsorption isA = —ln ( Is + 1 = —ln ( f. 4 + ii)4 + P 4 + -11If the true background intensity is 1.0, the true transmission of the sample is 0.6, andthe residual part is 0.1, the measured absorption is 0.45; whereas the true absorption is0.51, a difference of 12%. In these experiments, the residual part is not the same for thesample and background spectra, and this illustration exaggerates the absorption error.(4.3)Chapter 4. The Translational Absorption Band of Hydrogen^ 103The Baseline RemovalThe residual baseline is removed separately from all the LF and HF sample and back-ground spectra. The procedure applied to each LF spectrum involves the following: fitthe spectrum over the region 10-25 cm', and find the minimum value of the fitted curve;fit the spectrum over the region 100-119 cm', and find the minimum of the fit; find theaverage value between 263-273 cm'; find the average value between 325-335 cm"; andset the point at 520 cm' to zero. Fit a line to the above 5 points and subtract this linefrom the LF spectrum. The procedure applied to each HF spectrum is: fit the spectrumover the region 20-50 cm', and find the minimum value of the fit; find the minimumvalue of the spectrum near 135 cm'; average the spectrum over the regions 263-273,410-420, 485-490 cm'; and set the point at 690 cm-1 to zero. Fit a line to the above 6points and subtract this line from the HF spectrum.After baseline removal both the LF and HF raw spectra are very close to zero overthe relevant spectral regions, but the corrected LF and HF absorption spectra are stillinconsistent. However, the baseline removed HF absorption spectra more closely resemblethe LF spectra over the translational band peak, 55-70 cm', and the droop of the LFspectra near 110 cm' is reduced. In regions of high signal-to-noise, the baseline removedabsorption spectra are nearly identical to the unprocessed spectra, as is expected.4.5.2 The Averaging ProcedureThe composite average translational band spectra are obtained by a two step procedure.First, individual absorption spectra obtained on the same day under nearly the sameconditions are averaged as a 'group'. Second, the groups are averaged according to theweighting procedure given by eqn. 4.4 below.The group average consists of 8 to 20 sample interferograms, and 14 to 24 backgroundChapter 4. The Translational Absorption Band of Hydrogen^ 104interferograms. Helium backgrounds have been used rather than empty cell backgroundswhen possible. In some cases, spectra obtained on a given day at moderate densities pbserve as backgrounds for high density spectra pa , and a new group spectrum is obtainedat a density of p' = — pi.The maximum temperature variation of the individual spectra which comprise the25.5 K composite spectra is +2/-1 K for the LF case, and +4/-3 K for the HF case.The maximum temperature variation of the individual spectra which compose the 36 Kcomposite spectra is +2/-1.5 K for the LF case, and +2.5/-3 K for the HF case. The36K composite spectrum is obtained from 3 LF and 8 HF group spectra, and the 25.5 Kspectrum is obtained from 5 LF and 7 HF group spectra (except Figure 4.17 which isobtained from 4 LF and 3 HF group spectra).The composite average spectrum at a frequency o isAavg(a) w1ti(a) wti(a)Ai(u)where the weighting function iswti(a)   (Bi(a)) NintiAmax Amax \ Nintmax •A i (cr), Bi(cr), and Nint i are: the absorption intensity (prior to dividing by p2 1), the back-ground intensity, and the number of interferograms, of a group spectrum i. Amax) Bmax)and Nintmaz are: the maximum absorption, the maximum background intensity, and themaximum number of interferograms which compose a group spectrum, within the set ofgroup spectra. The number of interferograms in a LF group is multiplied by m = (# HFgroup spectra/ # LF group spectra), in order to balance the contribution of the largernumber of HF groups in the average.Intermediate averages of LF average spectra and HF average spectra are also obtained,weighted by a similar procedure for each separate spectral region. The weighting function(4.4)(4.5)Chapter 4. The Translational Absorption Band of Hydrogen^ 105of eqn. 4.5 is averaged and normalized, and the result is then used to weight the fit of theB-C synthetic spectrum to the composite average spectrum.4.5.3 The Composite Average Spectra at 25.5 and 36 KThe comparison of the LF average spectra to the HF average spectra illustrates theintensity mismatch between these regions. The upper graphs of Figures 4.12-4.17 showthe LF and HF average spectra as a heavy and light line, respectively. This series offigures shows that the discrepancy between these spectral regions is reduced by simplymultiplying the HF spectra by a constant factor prior to averaging. The filename of theLF average spectrum and the average temperature, and the filename of the HF averagespectrum and the average temperature, are written on the graph.The lower graphs of Figures 4.12-4.17 show the composite average spectrum as asolid line, the dotted lines show the contributions of the translational band and the S(0)line to the best fit B-C synthetic spectrum denoted by the broad dashed line. Writtenon these graphs are: the filename; the temperature; the B-C lineshape parameters T1,r2(x10-14 sec), and S (in K A.6); the spectral invariants ait, (cm5 sec-1), and -yitr (cm5sec); and the residuals from the fit.In Figures 4.12 to 4.14, the 36K average spectrum is obtained by multiplying the HFspectra by the factors 1.0, 1.05, and 1.10, respectively. The lowest residuals between theaverage spectrum and the best fit B-C spectrum are found by using an HF multiplier of1.10, but this is only slightly better than a multiplier of 1.05. The composite spectrumreflects the intensity mismatch in the regions 100-120 cm' and near 135 cm-1, since theweights of the LF and HF spectra over these regions is zero, respectively.Figures 4.15 and 4.16 show the 25.5K LF and HF average spectra and the compositeaverage spectra obtained by multiplying the HF spectra by the factors 1.0 and 1.05. The25.5K average spectrum, like the 36K average spectrum, is better fit by a B-C spectrum106A4.00007 . 0100.0^200.0^300.0Frequency cm-10 . 0 1 0 0 . 0 200.0 300.035AV35.9714.02^9.41^248.860.1723E-31^0.2268E-570.2559E-14Frequency cm'Figure 4.12: (A) The LF and HF average spectra at 36K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.00). The LF HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 36K, using a HF multiplier=1.00. The solidcurve is the spectrum, the dotted curves are the B-C translational band and S(0) line,and the dashed curve is the net fitted spectrum. Written on the graph are: the filename;the temp; T2, and S; ait, and "Yitr; and the residuals from the fit.25r.01-X....._.••t2. 300.0200.0100.0107A100.0^200.0^300.0Frequency cm -135.05AV35.9713.93^9.17^249.040.1779E-31^0.2270E-570.1165E-14BFrequency cm -1Figure 4.13: (A) The LF and HF average spectra at 36K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.05). The LF & HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 36K, using a HF multiplier=1.05. The solidcurve is the spectrum, the dotted curves are the B-C translational band and S(0) line,and the dashed curve is the net fitted spectrum. Written on the graph are: the temp;71, T2 , and S; a it, and 'yitr; and the residuals from the fit.--—-^ei---300.06.035.10L 36.1235.10H 35.81I IIIIIIIIIIIILIIIIIII100.0^200.0Frequency cm-'I0.00.0II-11(111111i35.10AV35.9713.83^8.94 249.260.1841E-31^0.2272E-57E-146.0100.0^200.0Frequency crn-10.00.0 300.0eiFigure 4.14: (A) The LF and HF average spectra at 36K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.10). The LF & HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 36K, using a HF multiplier=1.10. The solidcurve is the spectrum, the dotted curves are the B-C translational band and S(0) line,and the dashed curve is the net fitted spectrum. Written on the graph are: the temp;Ti, T2, and S; an, and -yitr; and the residuals from the fit.108AB109eiA0 . 00 . 0^li ^ill^r^II^i^1^1^1^il25AV25.5218.31^9.94^289.80^0.2049E-31^0.3721E-570.1055E-136.0t1 0 0 . 0 200.0 300.0Frequency cm - 'Frequency cm - 'BFigure 4.15: (A) The LF and HF average spectra at 25.5 K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.00). The LF & HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 25.5K, using a HF multiplier=1.00. The solidcurve is the spectrum, the dotted curves are the B-C translational band and S(0) line,and the dashed curve is the net fitted spectrum. Written on the graph are: the temp;r1 , r2, and 5; a ltr and 7itr; and the residuals from the fit.0.00. 0 100.0^200.0^300.0^25.05AV^ t25.5218.27^9.58 290.860.2138E-31^0.3735E-57^I^0.9138E- 4^ ItiiiIiiI////// l. iit'SB110A100.0^200.0^300.0Frequency cm-1Frequency cm-'Figure 4.16: (A) The LF and HF average spectra at 25.5K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.05). The LF & HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 25.5K, using a HF multiplier=1.05. The solidcurve is the spectrum, the dotted curves are the B-C translational band and S(0) line,and the dashed curve is the net fitted spectrum. Written on the graph are: the temp;Ti, 72, and S; aur and -yitr; and the residuals from the fit.11.11^1^I.71^/........../".I I^I $^I^$ Ern06. %I0.01^1^1^1^1100.0 200.0 300.0Till^I^I^I^1^I ^I^I^I^I^I^$^I^$25A.10AV25.66^17.28^10.38^281.270.2007E-31^0.3592E-570.9028E-14BFrequency cm - '111III^I^I^I^II^I^I^I^11111^I^11^I^$25A.10L 25.2925A.10H 26.0211111111111100.0 200.0 300.0I^I^1 1"0.0Frequency cm - 'Figure 4.17: (A) The LF and HF average spectra at 25.5 K. The heavy line is the LFspectrum and the light line is the HF spectrum (x1.10). The LF HF filenames andavg. temps. are on the graph.(B) The composite average spectrum at 25.5K, using a HF multiplier=1.10. HF spectraare taken either at the beginning of the run or use a low density 11 2 background to avoidthe ice band (180-300 cm' in the previous 25 K figures). The solid curve is the spectrum,the dotted curves are the B-C translational band and S(0) line, and the dashed curve isthe net fitted spectrum. Written on the graph are: the temp; 7.1 , T2 , and S; altr and 'Yuriand the residuals from the fit.Chapter 4. The Translational Absorption Band of Hydrogen^ 112if the HF spectra are multiplied by a factor of 1.05. The composite spectra show anexcess absorption with respect to the B-C synthetic spectrum in the frequency region180-300 cm'. Note that the weighting of the fit emphasizes the translational bandpeak, so the high frequency excess does not strongly affect the fitted B-C spectrum. Themeasured spectrum has been contaminated by absorption due to very small amounts ofwater ice. The dip at 230 cm' corresponds closely to an ice phonon line at 229 cm-1,and in general the band corresponds to absorption by the crystalline phases of ice, I andII, and vitreous ice. The distortion of the 112 spectra due to ice is discussed further inchapter six.Figure 4.17 shows the average 25.5K spectrum obtained by multiplying a subset ofthe HF spectra by 1.10. The group spectra are selected from absorption spectra which areobtained using at least some helium filled backgrounds, or from hydrogen backgroundsat a lower pressure. This selection of group spectra partially avoids problems that areassociated with changes in the ice absorption or reflection due to cell pressure changes.The intensity variation of the LF group spectra which compose the 25.5 and 36 Kaverage spectra is ±4% and ±3% of the peak absorption, respectively. The variationof the HF group spectra which compose the 25.5 and 36K average spectra is similarto the LF case, provided the ice band is excluded. If the procedure of applying thesimple HF multiplier is correct, a reasonable estimate of the uncertainty of the averagespectra is ±5% of the absorption peak. However, the most conservative estimate ofthe uncertainty of the average spectra includes the value of the HF multiplier and istherefore +3/ - 10% of the peak intensity. The uncertainty of previous measurementsof the translational-rotational band of H2 can be interpreted as ±5% of the S(1) peakat 297K [14]; therefore, the present measurements of the translational band of H2 areabout 10 times more precise than previous studies.35.10AV^35.97113A0.00. 0 100.0^200.0Frequency cm-'300.0IIIlIII IIIIIi25A.10AV^25.66300.0100.0^200.0Frequency cm-1Figure 4.18:(A) The lineshape function G(or) for the 112 translational band at 36 K obtained fromthe spectrum of Figure 4.14 (HF multiplier=1.10). The filename and temperature arewritten on the graph.(B) The lineshape function G(o) for the H2 translational band at 25.5 K obtained fromthe spectrum of Figure 4.17 (HF multiplier=1.10). The filename and temperature arewritten on the graph.Chapter 4. The Translational Absorption Band of Hydrogen^ 114Figure 4.18 shows the lineshape function G(a) for the translational band of hydro-gen at 36 and 25.5 K. These graphs are derived from the composite average spectrawhich involve HF multipliers of 1.10, Figures 4.14 and 4.17. The lineshape function isproportional to the transition probability and the figure demonstrates the increase inthe probability of low frequency translational energy transitions as the temperature islowered. It is notable that the amplitude of the semi-empirical B-C lineshape functiondrops slightly at frequencies below , 12 cm'. The experimental data near 20 cm -1 israther noisy, but it does not indicate that the lineshape function drops at low frequencies.In fact, the large increase in the curves at low frequencies differs distinctly from the B-Clineshape; this is another interesting issue for future low frequency measurements.4.6 The Low Temperature Spectra Compared to Previous MeasurementsThe preceding figures present new measurements of the complete translation absorptionband of normal H2 at 25.5 and 36 K. The translational band of H2 has been measuredwith high sensitivity, at a spectral resolution 10 times higher than previous work, overa new temperature range. The low temperature measurements demonstrate quantummechanical aspects of the pair distribution function, and yield new information regardingthe hydrogen intermolecular potential.The spectral invariants a lts and "Yitr, and the B-C parameters for the compositespectra have been obtained from the best fit B-C synthetic spectra in accordance withthe procedure applied to the LF individual spectra. In Figures 4.19 through 4.22, theboxes denote the results of the composite average spectra (Figures 4.12-4.17), and theerror of the spectral invariants and the B-C parameters is indicated by the spread of theboxes. It is apparent that although the accuracy of the spectra is at worst ±10%, thevariation of the integrated absorption coefficient is only about ±4%. The uncertainty ofChapter 4. The Translational Absorption Band of Hydrogen^ 115the spectral invariants is not particularly sensitive to the discrepancy between the LF andHF spectra because the fit of the B-C synthetic spectra to the absorption measurementsis weighted most heavily over the low frequency spectral region.4.6.1 The Spectral Invariants vs. TemperatureThe spectral invariants for the translational band of 112 are compared to previous highertemperature measurements in Figures 4.19 and 4.20. The crosses denote the values of thetranslational band spectral invariants which are inferred from the B-C parameters of theentire translational-rotational band of normal H2 measured by Dore et al. [13], Bachetet al. [14], and Birnbaum [12]. The diamonds are the measurements of Bosomworth andGush, where the translational band has been extracted from the entire spectrum using amodified Lorentzian with an exponential tail [8]. The circles denote the integrated binaryabsorption coefficients calculated using the Poll and Van Krandendonk formalism and thePoll and Miller quantum mechanical pair distribution function at 20, 40, and 80 K; thepoints at 200 and 300 K are calculated using a classical pair distribution function. Boththe quantum and classical pair distribution functions are derived from the Lennard-Jonesintermolecular potential. Note that aitr differs from the integrated binary absorptioncoefficient of Tables 2.2 and 2.3 by a factor of 27r.Discussion of the Integrated Absorption CoefficientFigure 4.19 shows that at higher temperatures there is a discrepancy between the mea-sured and calculated binary absorption coefficients. The low temperature results form theclosest correspondence between measurements and theory, where this may be partiallydue to the advantage of measuring the translational band isolated from the rotationallines.^1 ^1^I^I^1^I ++0^+^+^+- _+co:8+ +000.64.0, 1.060.0I^I^IMO■■1+++ 47I^I^I^I^I^I^I0+Chapter 4. The Translational Absorption Band of Hydrogen^ 1160.0^50.0 100.0 150.0 200.0 250.0 300.0Temperature (K)Figure 4.19: altr vs. temperature for the translational band of H2. ^ are the compositeaverage spectra, 4- denotes data from Dore et al. [13], Bachet et al. [14], and Birnbaum[12], 0 are from Bosomworth and Gush [8], and o are calculations based on the theoryof Poll and Van Kranendonk (the values plotted differ from Table 2.3 by a factor of 27r)^20.0^40.0 60.0 100.0^200.0 300.0Temperature (K)Figure 4.20: 7i t, vs. temperature for the translational band of 11 2 . -yit, is defined byeqn. 2.55. ^ are the composite average spectra, and + denotes data from Dore et al. [13],Bachet et al. [14], and Birnbaum [12].Chapter 4. The Translational Absorption Band of Hydrogen^ 117It is possible, but unlikely, that the high temperature discrepancy is due to the highsample densities (over 100 amagat) used in previous experiments. The measured excessabsorption might then be due to tertiary collision-induced mechanisms. However, manyprevious experiments have verified that H2 absorption is dependent on p2, [12],[14], andspecifically, the p2 dependence of the translational band has been reported [8]. It isalso possible that previous high temperature experiments have been contaminated byimpurity absorption, by water vapour for example. In the low temperature experiments,all impurities in the sample gas are frozen out, and the possibility of excess absorptionis greatly reduced.The inferred translational band measurements of ait, in Figure 4.19 exceed the cal-culations using the classical pair distribution function by 33% at both 200 and 300 K.A similar discrepancy was found by Bachet et al., where alir measured over the entiretranslational-rotational band exceeded that calculated using a Lennard-Jones potentialby 13% at 195 K and 25% at 300K, [14]. However, they calculated the absorption coeffi-cient due to only the quadrupolar induced dipole moment and interpreted the discrepancyas the contribution of the overlap and 'interference' between the overlap and quadrupolarinduced dipole moments. The present calculations of the translational band, however,already include the overlap and interference terms (see Table 2.2).Unlike the high temperature results, the low temperature measurements of ait, arequite close to those calculated using a quantum pair distribution function based on theLennard-Jones potential. The variation of the integrated absorption as a function oftemperature is a probe of the intermolecular potential in the sense that low tempera-ture, slowly moving, molecules interact over relatively long ranges, whereas high speedmolecules can penetrate the potential to smaller distances. The comparison of measure-ments and Lennard-Jones calculations shown in Figure 4.19 leads to the conclusion thatChapter 4. The Translational Absorption Band of Hydrogen^ 118the long-range portion of the 112 intermolecular potential is well described by a Lennard-Jones potential, but the short-range repulsive part of the intermolecular potential is not.The present measurements could serve as a valuable test of the 112 ab-initio potentialwhich was used to model the low temperature S(0) and S(1) rotational lines (see Figure2.4).Figure 4.20 shows the variation of the spectral invariant l'itr as a function of tem-perature. The low temperature data points at 25.5 and 36K continue the roughly T'behaviour of -yit, from higher temperatures. The variation of the low temperature datapoints of Figure 4.20 is less than 4.19 because ryitr emphasizes the low frequency regionof the translational spectrum (see the above discussion of Figures 4.6-4.7). The varia-tion of the measured aitr and -yitr with temperature, displayed in Figures 4.19 and 4.20,generally resemble the variation of both the measured and calculated invariants of theentire translational-rotational band [62].4.6.2 The B-C Parameters vs. TemperatureFigures 4.21 and 4.22 show the variation of the B-C parameters with temperature, wherethe parameters have been obtained from fitting the composite average spectra, and frompreviously reported fits of measurements of the entire translational-rotational band. Thetime constants ri and T2 increase smoothly as the temperature is lowered, and the ratio71/7-2 is approximately 2 over the temperature range 25 to 300K. The duration of acollision, (7-1 7-2)1/2, also increases as the temperature decreases. The parameter S isproportional to the integrated absorption [14], and Figure 4.22, the variation of S withtemperature, resembles Figure 4.19.Figures 4.21 and 4.22 provide sufficient information to predict the entire translational-rotational absorption spectrum of 112, from 25 to 300K. The parameters 7-1, T2 , and Smay be obtained by interpolating the graphs, and the B -C synthetic spectrum is thenChapter 4. The Translational Absorption Band of Hydrogen^ 120calculated according to the prescription of chapter two. The B-C synthetic spectrumreproduces the measured collision-induced absorption spectrum of hydrogen, and thepredicted spectrum is valid over a wavelength range of two orders of magnitude. Theaccuracy of the predicted spectrum is limited primarily by the accuracy of the experi-mental spectra from which the B-C parameters are derived, and at low temperatures bythe failure to include dimer transitions in the model.Chapter 5HD5.1 IntroductionIn the course of measuring the low temperature translational band of 11 2 , a weak absorp-tion feature has been observed at a frequency of 89.23 ± 0.01cm -1 . This line is identifiedas the R(0) J = 1 4- 0 transition of hydrogen deutride which occurs naturally as anisotopic 'impurity' in ordinary hydrogen. The identification of this line is based on thecorrespondence of its frequency to previous measurements of (7=89.227950 cm -1 [83], andthe fact that any other candidate gas is frozen out at these low temperatures. In addition,a weak R(1) line at 178 cm -1 has been observed in the 77K 11 2 spectra. The observationof these lines was surprising, as we did not anticipate that the low concentration of HD in112 , combined with the small dipole moment of HD, would cause measurable absorption.The HD molecule in the electronic ground state possesses a small permanent electricdipole moment and rotational transitions are allowed according to the selection rulesAJ = ±1. The dipole moment originates from the fact that the molecular center of massand center of charge do not coincide. The nuclei of the molecule undergo a zero-pointvibrational motion, and if the electrons do not follow the nuclear motion exactly, anoscillating dipole moment occurs [84].The rotational spectrum of HD was first observed by Trefler and Gush [85], and theabsorption spectrum has subsequently been investigated in the pure gas and in gas mix-tures over a temperature range of 77 to 300 K [86],[87]. The above precise frequency121Chapter 5. HD^ 122measurement of the R(0) line was obtained using the radiation produced by nonlin-ear mixing of CO2 lasers [83]. Recent measurements of the intensities and frequenciesof higher J lines have been reported using laser diode spectrometers [88], and Fouriertransform spectrometers [89]. Theoretical aspects of the infrared spectrum of HD havebeen reviewed by Poll [90].5.1.1 The D/H Abundance RatioThe distribution of molecular hydrogen in cold regions of the galaxy can in principle betraced using astronomical observations of HD rotational line emission, as mentioned inthe introduction. The observation of HD lines can also be used to understand the D/Habundance ratio, one of the few measurable quantities in cosmology. Deuterium is formedduring the early nucleosynthesis phase of the universe and is subsequently consumed bystars. The D/H abundance ratio, along with the abundances of 'He,' He, and"Li, is deter-mined by the energy and particle densities of the early universe. Big Bang nucleosynthesiscalculations are consistent with measured elemental abundances only if the ratio of thenumber of baryons to photons is , 10-9. Therefore, the mass density of the universe inthe form of baryons, fiB < 0.2, is considerably less than the closure density, and this massdeficiency is presently seen as the strongest argument for the existence of non-baryonic'dark matter' [41].The measured D/H ratio varies, depending upon the astrophysical environment: onEarth, D/H=1.58 x 10-4 [91]; in Jupiter's atmosphere, rs, 3.5 x 10-5 [25]; and in theinterstellar medium, 1-2 x10-5 [41]. Present day D/H ratios must be interpreted using amodel of the chemical evolution of the galaxy to obtain the cosmological abundance ratio.The atmospheres of the giant gaseous planets are expected to preserve the abundanceratio of the pre-solar nebula, but D/H ratios measured spectroscopically using lines ofdeuterated methane (CH3D) require further interpretation via models of atmosphericChapter 5. HD^ 123chemistry. Planetary D/H ratios are also measured using visible and near-infrared HDvibrational overtone transitions, but these observations involve modelling the scatteringof short wavelength radiation by clouds and haze [42].Bezard, Gautier, and Marten have proposed observing the R(0) and R(1) lines of thegiant planets as a probe of the atmospheric deuterium abundance [42]. This measure-ment avoids the above problems, and the submillimeter HD lines are reasonably free ofobscuration by other molecular lines. The present results demonstrate the feasibility ofdetermining D/H ratios from planetary spectra using the intensities of the R(0) line andthe underlying 11 2 collision-induced translational band.5.2 The HD SpectraThe HD line is observed in all the H2 spectra, and the results presented in this sectionare obtained from LF H2 spectra, over a temperature range of 23 to 38 K, a densityrange of 5.5 to 23.3 amagat, and pathlengths of 52 and 60 meters. These HD spectraare a subset of the LF spectra of the I-12 translational band studies. One set of LFD2 spectra was acquired at temperatures near 28 K, at a density of 2.9 amagat, over a52 meter path (see Figure 7.1). Sample spectra obtained on same day under the sameconditions are averaged as a 'group' and each average consists of 8-20 interferograms,where the maximum optical path difference of an interferogram corresponds to a spectralresolution of 0.22 cm -1 . Background spectra are composed of empty cell spectra takenbefore and after the sample spectra. In order to estimate the uncertainty of the R(0)line intensity, the average sample spectra are also ratioed against background spectracomposed of a different set of empty cell spectra (generally backgrounds taken before thesample spectra).Some of the absorption spectra are shown in Figure 5.1, where the HD line at 89.2-a-b----Chapter 5. HD^ 1240.4A0.20 20^60^4----4._100CT, cmFigure 5.1: Absorption spectra of H2 and D2, the sharp line at 89.2 cm' is the R(0) lineof HD. The H2 densities, temperatures, and pathlengths are: a, 16.00 am, 37.2K, 52 m;b, 13.68 am, 25.2K, 52 m; c, 10.24 am, 25.5K, 52 m; d, 7.65 am, 23.7K, 60 m. Curve eis D2 at 2.93 am, 28.0K, 52 m.Chapter 5. HD^ 125cm -1 is observed on top of the 11 2 translational continuum. The HD lines are unresolved,that is, the pressure broadened linewidth is less than the width of the interferometerlineshape function. In spite of the small amount of HD in the absorption cell, the highsensitivity of the spectrometer-cell apparatus yields a detection of the line at a signal-to-noise of ti 15. The present spectra allow the measurement of the HD dipole momentunder very different temperature, pressure, and concentration conditions than previousmeasurements. In order to measure the dipole moment two assumptions are made: (1)the mixing ratio HD/H 2 of the gas sample is 1/3165, where this is based on the deuteriumabundance of deep ocean water [91] and all deuterium is in the form of HD; and (2) theHD lineshape is Lorentzian.5.2.1 The Integrated Absorption CoefficientThe dipole moment of a diatomic molecule is related to the integrated absorptioncoefficient ao (per unit density, per unit pathlength) of a rotational line byao^3hcZ8 7r nocro(e_E(J)/kT e—E(J+1)/kT)(j+1)1123 (5.1)where no is Loschmidt's number, cro is the frequency of the transition, and the equationincludes Boltzmann factors for the lower and upper state. p. is the magnitude of thepermanent electric dipole moment and it is related to the sum of the squares of thedipole moment matrix elements for transitions AJ ±1 by112  (J 4- 1)(2J + 1) =2 = kis! 2 + lily! 2 + !µs I 2^(5.2)where the dipole moment operator and the wavefunctions are written in spherical polarcoordinates [92]. Z is the rotational partition function (eqn. 2.40), and the rotationalenergy states are obtained from the HD rotational constants: B o = 44.665379; Do =0.0257502; Ho = 0.19039 x 10', all in cm -1 [89]. It should be noted that one of theChapter 5. HD^ 126reasons the R(0) line is observable in the present spectra is that virtually all the HDmolecules are in the ground state at these low temperatures.5.2.2 Line FittingFigure 5.2 shows an expanded section of the H2 absorption spectra that are interpolatedto a frequency step size 8 times smaller than Figure 5.1. The interpolated spectra areobtained by Fourier transforming interferograms lengthened by zero-filling. The areaunder the interpolated HD line is found by assuming that the measured lineshape is theconvolution of a Lorentzian absorption line with an instrumental lineshape function ofthe form sin x/x. From the convolution theorem, the interferogram component which isresponsible for the HD line is a cosine at an angular frequency coo with an exponentiallydecaying amplitude.G(t) . Cal° cos wot, —T < t < T= 0, It( > T^(5. 3)where t is the time delay between the two arms of the interferometer, and T is the timedelay associated with the maximum interferometer optical path difference. The Fouriertransform of G(t) is the lineshape g(w),2 g(w) = pug + az [a — CaT(a cos AceT — Au) sin AcoT)] (5.4)where Au) = co — coo. The HD spectral feature is described by the lineshape g(w), wherethe integrated area of g(w) is 27r. g(w) is fitted to the measured spectrum using anonlinear least squares fitting program with 4 free parameters: the height and slope ofthe H2 translational band baseline; the parameter a which is related to the Lorentzianlinewidth; and a line strength multiplier applied to g(w) from which the integrated areaA-Chapter 5. HD^ 1270^ 14^I^I8 ^1 88 92cr, cm-1Figure 5.2: The fit to the measured spectra over the R(0) line. The solid curves are theinterpolated curves of Figure 5.1, and the dotted curves are the fits obtained by assumingthe HD line is an instrument broadened Lorentzian superimposed on a linear continuum.Chapter 5. HD^ 128of the HD line is found [93]. The frequency of the HD line is input by hand to the fittingroutine.The results of fitting the measured spectra are shown in Figure 5.2 by the dashedline. The fit is good over the central maximum of the HD line and it even reproduces thefirst few ripples of the measured spectrum. The D2 spectrum at the bottom of Figure 5.2demonstrates that for narrow linewidths (low densities) the measured spectrum stronglyresembles a sin x/x curve which is given by g(w) if a 0 and x = AcvT.5.3 The HD Dipole Moment5.3.1 The Density Dependence of the Dipole MomentThe area under the HD line equals rmpaol, where r„, is the HD/H2 mixing ratio (1/3165),p is the sample density in amagat, ao is the integrated absorption coefficient and / is thepathlength. The dipole moment is then calculated from ao using equation 5.1, and theresults of 10 spectra are plotted in Figure 5.3 as a function of density. In previously re-ported experiments, the integrated absorption coefficient of the HD rotational spectrumwas shown to be density dependent, and the free-molecule dipole moment was derivedfrom the integrated intensity extrapolated to zero density [90]. ao is temperature depen-dent and since the present work involves HD line intensities which are not measured atthe same temperature, the dipole moment, which is independent of the temperature, isplotted as a function of the density.The density dependence of the integrated absorption is caused by an interferencebetween the permanent dipole moment and the collision-induced dipole moment [94],[95].There exists some controversy, however, as to whether the interference is constructive ordestructive. Previous work has shown that the integrated intensity of low J rotationallines decreases as a function of density at room temperature [88],[87]. At 77 K, theChapter 5. HD^ 129integrated intensity of R(0) increases slightly as a function of density [86],[87], but thepresent results show a slight decrease as a function of density. The R lines of the HDvibrational band have a marked density dependence, where these lines are described by aFano lineshape [96]. The HD pure rotational lineshape is, however, not strongly affectedby density and the Lorentzian approximation is valid [86].5.3.2 The Measured HD Dipole MomentThe uncertainty of the HD dipole moment is estimated by finding the dipole momentfrom a given sample spectrum using both background data sets, and the variation of thesemeasurements is represented by the error bars of Figure 5.3. This variation is largerthan the ±1% uncertainty in the density measurement, and it reflects the systematictransmission errors of the spectrometer-cell system when measuring weak absorptions.An additional error owing to the fitting procedure may exist for the highest and lowestdensity points. A least squares fit of a straight line to the data, weighted by the reciprocalof the square of the error bar height, gives the zero density intercept, A = (0.82 +0.04) x 10' debye. The negative slope of this line is caused entirely by the extremehigh and low density points, and the weighted average of A, excluding these points, is(0.78 + 0.02) x 10' debye. A value of A which includes both interpretations of the datais therefore (0.81 ± 0.05) x 10 -3 debye.The present value of the dipole moment of the R(0) transition is consistent with thebest theoretical calculation of 0.8463 x 10 -3 debye [97], and with the previous measure-ment at 77K by McKellar of (0.818 ± 0.026) x 10 -3 debye [86]. The present measurementis inconsistent, however, with the work of Ulivi et al. where p has the following valuesat the various temperatures: 0.719 ± 0.003 at 77K; 0.803 ± 0.012 at 195 K; 0.883 ± 0.028at 295 K (all x10' debye) [87]. No explanation is given in this latter work for theunexpected variation of p with temperature.---^ -Chapter 5. HD^ 1300.90E^II^I^11111^I^ii,... 016lEc.)0.080^0/0 113///^a//// El D_ /A/(b)^--o-^-_o / IIIIIILIII^L1o^10^20p,amagatFigure 5.3: The upper graph shows the density dependence of the HD dipole momentobtained from fitting the spectra. The dashed lined is the least squares fit to the data.The lower graph shows the R(0) linewidth (FWHM) as a function of density. The triangledenotes the deuterium spectrum. The dashed line is a linear fit to the hydrogen data,excluding the extreme density points.Chapter 5. HD^ 1315.3.3 The Density Dependence of the HD Linewidth and FrequencyFigure 5.3b shows the dependence of the HD linewidth on density. The data pointsare rather scattered across this plot, which is not surprising since the linewidths areinferred from unresolved spectral lines. A straight line fit to the data gives a broadeningparameter of 0.01 cm -1 amagat' and this compares favorably with the earlier result of0.006 cm-1 amagat' for HD in an unspecified mixture of 11 2 at 77K [87].A frequency shift of the R(0) line of -0.03 cm -1 is observed for the three highestdensity spectra (and the spectrum at 8.33 amagat). Previously measured frequencyshifts at 77K are: for HD mixed with .11 2 , +0.16 x 10'cm' amagat' [87], and for pureHD 9 x 10'cm' amagat' [86]. The largest frequency shift predicted for the highestdensity spectrum measured in the present work is therefore +0.004 cm', which is muchsmaller than the frequency step size of 0.030 cm'.5.3.4 A Determination of the HD Concentration in D2The R(0) line is also present in the D2 spectrum shown in Figures 5.1 and 5.2, wherethe line is much stronger and narrower than the 11 2 spectra. The concentration of HDin the D2 sample is measured using the integrated area of the R(0) line obtained fromthe fitting procedure and the HD dipole moment derived above. The HD/D 2 ratio ismeasured to be 1/339, in good agreement with the manufacturer's specification that theHD content is less than 1/250.5.4 Future Investigations of HD DimersUnder the low temperature conditions of these investigations, HD-H 2 dimers may form,and the dimer spectrum is predicted to show small 'satellite' lines near the R(0) line[98]. A measurement of the dimer spectrum requires obtaining high resolution spectraChapter 5. HD^ 132of samples with high dimer concentrations. A 1% HD/H 2 gas mixture at a densityof 3-6 amagat should exhibit 70-80% absorption at the R(0) peak using the presentspectrometer-cell apparatus. This measurement would consume only 1-2 atm-liters ofHD and the increased absorption should permit a more precise determination of the HDdipole moment. In order to observe the dimer spectrum it may be necessary to saturatethe R(0) peak and measure spectra of 10 to 50% HD/H 2 mixtures.Chapter 6Experimental ErrorsThe systematic errors in these experiments fall into 4 categories: (1) errors associatedwith measuring the sample gas temperature and density; (2) errors associated with opticalproblems such as the alignment of the cell mirrors, and cell window distortions; (3) errorsin the spectrum due to water ice impurities; and (4) errors due to detector nonlinearities.The dominant problem with the H2 absorption spectra reported herein is the 5-10%mismatch between the LF and HF spectra (see Figures 4.12-4.17). The errors of thefirst 3 categories are too small to be responsible for the intensity mismatch, and this isis presumed to be due to detector nonlinearities.6.1 Temperature, Pressure, and Density ErrorsThe method of measuring the sample gas temperature and density is discussed in theexperimental chapter. In brief, the error in measuring the gas temperature results fromusing resistors mounted on the exterior of the cell (not in the gas itself), where the cell isnon-uniform in temperature. The maximum temperature drift, or temperature gradientacross the cell, is +1 K during data collection. Averaging the temperatures of the 4 cellresistors, 3 times during a sample data collection run, yields a temperature measurementwith an estimated accuracy of ±0.25 K.The accuracy of the pressure transducer is two times worse than the manufacturer'sstated accuracy, or about ±1% for the 1-3 atm pressures used in these experiments.The pressure sensor is at room temperature, whereas the sample is at low temperatures,133Chapter 6. Experimental Errors^ 134however, no significant pressure gradient exists along this density gradient since thesensing tube is 1" in diameter.The gas density is obtained from the temperature and pressure measurements usingthe 2nd order virial coefficients and the error in the determination of the gas densityis +1.5% from the above considerations. The standard deviation of a set of densitymeasurements of a 16.00 amagat gas sample sealed in the cell, over the temperaturerange 25-35 K, is 0.15 amagat, or an uncertainty of ±1%.6.2 Optical Errors6.2.1 Spectrometer CalibrationThe spectrometer frequency calibration has been established to better than 0.0025 cm -1by measuring the frequencies of CO, 0 2 , and H 2 O lines. A spectrum of saturated COlines is presented in Figure 7.5 where the infinite absorption level of the LF spectra isdemonstrated to be accurate to better than 1% of the spectrum peak, for high resolutionspectra. The infinite absorption level for the lower resolution 11 2 spectra should be moreaccurate than the above case by a factor of 4, however, the residual baseline discussed inchapter 4 persists at a level of 0.5% of the spectrum peak.6.2.2 The Reproducibility of Background SpectraThe estimated accuracy of the translational absorption spectra is based on the repeata-bility of background spectra taken at the beginning and end of a data collection run,a span of about 16 hours. Figure 6.1 shows the natural logarithm of the ratio of twosuch backgrounds, where ideally this 'absorption spectrum' should equal zero at everyfrequency. The reproducibility is quite good and the difference between the LF back-grounds corresponds to a 2% transmission error across the LF spectral region (omittingI II ^I^I^I I^I^II I I1———————V20 00 I 40,00^1 66,00 I "I" I^I^r^"^P^.P° 1140 00 1"t' °Pbackgnd. comp.^13,14,15/2,3 Feb22 60 m.^Resolution=0 241 CM.. E Imre.0.20 TY Nenment.9. 19120. . 1992_______v„.„0^IliiiirlI^r I' °^P i^Fp co I^Ps° 9°141' bockgrod comp 1.2/22,23 Mor16 60 m^Resolutlen=0.241^CM..^ E. WI9finew0. 6. Errors^ 135Figure 6.1: An 'absorption spectrum' of LF backgrounds taken before and after thesample spectra which demonstrates the repeatability of the LF spectra.Figure 6.2: An 'absorption spectrum' of HF backgrounds taken before and after thesample spectra which demonstrates the repeatability of the HF spectra.Chapter 6. Experimental Errors^ 136the region of the beamsplitter null, 100-120 cm"), and a 1% transmission error between40-90 cm'.The repeatability of the HF background spectra is demonstrated in Figure 6.2. Overthe region 50-200 cm', the difference between the backgrounds corresponds to a trans-mission error of ti 3%, and over the region 200-320 cm -1 a conservative estimate ofthe transmission error is 5%. The background ratio near 135 and 263-273 cm -1 isanomalous owing to the absorption bands of the crystal quartz cold filter.The repeatability of successive 11 2 spectra, under the same conditions, is better thanthe background repeatability, and it corresponds to a 1% transmission variation acrossboth spectral bands. For this reason, 11 2 spectra at moderate densities are used asbackgrounds for 11 2 spectra at higher densities. The time interval between the high andmoderate density spectra is about 1 hour, and the transmission error of these spectra isestimated to be about +1.5% for the LF and HF cases.The minimum transmission of the 11 2 spectra at 25 K is 52%, where the absorptionis limited by the maximum allowable gas density before the sample liquefies. The mini-mum transmission of the spectra at 35 K is 33%, where absorption is restricted by the 3atm. pressure limit of the gas handling system. The uncertainty of the 11 2 translationalabsorption band measurements, based on repeatability considerations, is conservativelyestimated to be ±2% of the peak absorption from 40-90 cm -1 , +3% across the LF region,and +5% from 200-320 cm'.6.2.3 Sources of Repeatability ErrorsThe background repeatability errors are caused by a number of systematic problemswhich are listed below. The sensitivity of the bolometer drifts owing to the temperaturevariation of the helium reservoir, but this problem has been minimized by pumping onthe detector overnight before acquiring data. In a number of HF spectra, the detectorChapter 6. Experimental Errors^ 137sensitivity drifted due to a voltage drop of the preamplifier batteries. The intensity ofthe mercury arc source is stable and it is not a contributor to the repeatability error.Pressurizing the cell with the sample gas slightly affects the mirror alignment, and themultipass cell has to be realigned between initial backgrounds, sample spectra, and finalbackgrounds. As a result, the optical paths of the various spectra do not intersect exactlythe same regions of the absorption cell mirrors and windows and the cell transmissionchanges; this is a significant problem.The AC detector signal increases as the cell is cooled since the reflectivity of the mir-rors increases as the temperature decreases; however, efforts have been made to acquirespectra and backgrounds at nearly the same temperature. The problems of changes inthe cell optical path, and the changes in reflectivity with temperature become even moresignificant if ice coats the mirrors.6.2.4 Additional Sample Spectra ErrorsThe above suggestions as to the origin of transmission errors pertain to both backgroundand sample spectra; however, some errors occur only with the sample gas in the cell. Den-sity fluctuations in the optical path due to sample gas turbulence cause a low frequencyvariation of the transmitted light intensity. The fluctuations occur below 1 Hz, but thespectrum below 25 cm-1 may be perturbed. The addition of the fluctuations to thesample spectra may slightly lower the absorption spectra, but this effect is not obviouslynoticeable. Turbulence has been previously discussed in the experiment chapter.Pressurizing the cell with the sample gas can cause ripples in the absorption spectrumsince a slight change in the cell window's thickness or position results in the failure ofthe sample and background 'channel spectra to exactly cancel (see background spectraFigure 3.8). The window distortion effect is illustrated by the oscillating curve in Figure6.3 which is the LF spectrum of helium at 2087 Torr at room temperature over a path20 100 1^1 46, 00 I^i^1 60 . 00^I^1 80 1 0 1100.00^1120 .00 '4°P° 1160.00 Helium. 2087 Torr 295 K 3/1 Feb10; 2046 Torr 78 K 2/1 Feb18 52^Resolullonr0 241 GC'Chapter 6. Experimental Errors^ 1380.150.10 -Figure 6.3: The absorption spectrum of helium demonstrating the error due to windowdistortions. The oscillating curve is He at 294 K and 2087 Torr. The flat curve is He at78 K and 2010 Torr. The pathlength is 52 m.of 52 m. The zero crossings of this curve correspond to the peaks and troughs of thebackground spectrum, and the magnitude of the effect depends upon the gas pressure.The flat curve of Figure 6.3 is the spectrum of He at 78 K at a similar pressure, itdemonstrates that the windows are far less flexible at low temperatures (see also Figure6.8). Low temperature window distortions cause at most a ±1% transmission error, andthe deviation of this helium spectrum from zero absorption is another indicator of theaccuracy of the absorption measurements.6.2.5 Pathlength ErrorsThe cell mirrors are located on each other's centers of curvature and the error in the mirrorspacing is less than 1 mm. The mirror adjustment mechanism is designed to minimize anyspacing changes as the folding mirrors are rotated, and the mirror spacing is determinedby invar bars which change in length by only 5 parts in 10 4 as the temperature is lowered1-^I^1-^I T^-F^I^Z^I 1^I^11^I^IIS November, 1992Chapter 6. Experimental Errors^ 139to 20 K. It is possible that the number of spots along the field mirror has been miscountedresulting in a pathlength error of ±4 m, or 7%. This miscounting error might accountfor the LF/HF intensity mismatch, but the pathlength was adjusted with great care, andthe LF spectra measured over 52 and 60 m paths overlap appropriately. The pathlengtherror is estimated to be 0.1% of the optical path.The overlap of neighbouring spots at the cell exit aperture allows radiation propagatedover a path of n +4 meters to be detected along with the desired optical path of n meters.The intensity distribution at the cell exit is unknown, and hence the contribution to thedetected signal from pathlengths other than the desired one is also unknown. On thebasis of the cell optical tests discussed in Appendix 1, the contamination of the 60 msignal by other paths is estimated to be less than 5%; at 52 m the contamination isnegligible.6.3 Water Ice6.3.1 Ice AbsorptionA spectral feature at 230 cm-1 is noticeable in all the HF hydrogen spectra, particularlyin the 25.5 K spectra. In addition, a band of excess absorption is observed over thefrequency range 180-300 cm-1 in the 25.5 K spectra. These features are due to verysmall quantities of ice in the multi-pass cell, where this this conclusion is based on thecomparison of the present spectra to previously reported ice spectra [99],[100],[101].Over the ,-, 3-4 week course of a low temperature data collection run, the backgroundspectra are observed to change. Figure 6.4 shows the absorption spectrum of a back-ground at the end of a run with respect to a background near the beginning. Note thatnegative absorption indicates that the IR source intensity was higher in the later spec-trum. Figure 6.6 shows the far-infrared absorption spectrum of conventionally formedChapter 6. Experimental Errors^ 140ice I, where the broad peak demonstrates that it is an orientationally disordered phase.Figure 6.5 shows the multi-peaked absorption spectrum of ice II, an orientationally or-dered phase, which is formed under high pressures (2 Kbar) at -50° C. Starting at thehigh frequency end of the spectra, every feature of Figure 6.4 corresponds to a line ofice II, except the strong peak at 230 cm -1 and the dip at 180 cm -1 which correspond toice I features (an additional peak at 70 cm -1 is possibly due to ice IX). The frequencycorrespondence between these spectra is not exact, but frequency shifts are expectedsince Figure 6.4 was obtained at a much lower temperature.The source of the ice is probably the 5 ppm residual water vapour content of the UHPsample gases. The sample gas passes through a molecular sieve and liquid nitrogen trapsen route to the the cell, but appears that the water vapour is not entirely frozen out.A typical gas sample uses about 5-700 atm-liters of gas which are flowed into the cellover 1-2 hours, and the entire experimental run involves over 5000 atm-liters of gas. Ifall the the water vapour content of this gas enters the cell and uniformly coats the wallsand mirrors, then an ice film ti 25 nm thick is formed. The infrared beam reflects off themirrors up to 59 times and absorption by the ice film alters the background spectrum.In addition to the water content of the sample gases, leaks or outgassing of the vacuumsystem could coat the cell entrance and exit windows with an ice film. In fact, the pyrexviewing window on the inner radiation shield slightly fogs with ice over the course of anexperimental run, but this is presumably caused by minor vacuum leaks which occur asthe mirror adjustment rods are turned.6.3.2 Phase EffectsIce has more solid phases than any other known substance and Figure 6.7 shows the phasediagram of ice. The phase boundary between ice I and ice II is of particular interest here,since it may be extrapolated to zero pressure at zero temperature. Apparently ice II is17 1292II^IIIIIIIIIIIIIIIIIIIIIIIIIIII——__—III!————_9-00.^I^i^I^"I.". ,^i^1'J" I^, 1,50.,00 I^i^I^pooioo i^I^i^1250,.00 ,^i^i^1300,.00ICE backgnd^comp. 22,23 mar16/1,2 mar5 60^m.^Resoluflon.0.241^CM.-' E. WIshnow0.60.40.20 .0-0.2-0.41410^I I I^1^I250^200^ISO.^. 0Figure 6.4: The 'absorption spectrum' of HF backgrounds near 30 K over a 60 m pathtaken at the end, with respect to the beginning, of a low temperature run.Figure 6.5: The absorption spectrum of ice II, from [99].0.60.6Lo00.4350^300.......250^200^q0^100^50CyCTChapter 6. Experimental Errors^ 142Figure 6.6: The absorption spectrum of ice I, from [101].formed when the cell is pressurized to 1-3 atm at 20-30 K and it remains ice II, in ametastable state, even when the pressure is relieved [99]. Other forms of solid water existand ice forms in a vitreous phase and in a form with voids filled with entrained gases;the latter form is certainly a possibility in the present case.An interesting effect is observed when measuring the spectrum of helium at low tem-peratures. Figure 6.8 shows the spectrum of helium taken near the middle of a datacollection run (not too much ice in the cell). The He spectrum is expected to be thesame as an empty cell background, and the 2% transmission error reflects the repeata-bility error referred to above. If a straight line is drawn across this graph at the 2%level, a lump of excess absorption from 180 to 260 cm' is seen along with a negativeabsorption dip at 230 cm -1 . The shape of this excess absorption strongly resembles theexcess intensity of the 11 2 translational spectra at 25.5 K (Figures 4.15 and 4.16). Thenegative dip has reached over -13% in the latter stages of the experiment when observingHe at 3 atm pressure.The negative absorption is possibly due to a change in the ice film absorption orreflection which occurs as the phase boundary between ice I and II is crossed when thePRESUME/SWChapter 6. Errors 143Phase diagram of ice. The solid and long-dashedlines are directly measured stable and metastablelines respectively, and the short-dashed and dottedlines are extrapolated or estimated stable andmetastable lines respectivelyFigure 6.7: The phase diagram of water ice, from [97]. Nevem,Oor. 111112teptiliN4/4404.01,6M441141——————100—SO 00 r00 00 r50 OD FOO DOAbs co..^ffioliurn 24.6 K 978 Torr 60 rn^7,8 9/3.4,5 Mor9 Resolution=0.241^CM.-'^C - WistIn..Figure 6.8: The absorption spectrum of helium at 760 Torr and 25.5 K over a 60 m path.The ice features above 180 cm resemble the distortion of the H2 average spectra.Chapter 6. Experimental Errors^ 144cell is pressurized. The dip is not observed when comparing backgrounds taken at thebeginning and end of a day of data collection (see Figure 6.2), although a disturbance ofthe spectrum does occur here. Apparently, a change in the optical properties of the icefilm occurs and then reverts to the initial state, perhaps with a time delay, as the cellis pressurized and evacuated. The behaviour of low temperature ice films would be aninteresting project for further investigations.The 11 2 translational band average spectra are derived from helium, or lower density112 , backgrounds whenever possible to avoid the ice features. This procedure works wellfor the 36 K spectra, but the 25.5 K spectra exhibit ice absorption unless the spectra whichcompose the average are carefully selected. The ice features may be more prominent at25.5 K because these spectra have been obtained late in the run when there is more icein the cell, or perhaps the ice properties are particularly dependent on temperature. Itis not possible to simply remove the excess ice absorption of the spectra because thequantity of ice which coats the mirrors is unknown, and the absorption coefficient perunit length of ice II has not been reported [100].6.4 Detector Non -LinearitiesThe previous uncertainties are all substantially smaller than 10%, and it is believed thatthe discrepancy between the LF and HF spectra is caused by nonlinearities of the ger-manimum bolometer. The evidence for the detector nonlinearity consists of a measuredincrease in the interferogram contrast as a function of a decrease in the optical signalamplitude, and the correspondence between the 11 2 spectra and simulated absorptionspectra involving a hypothetical nonlinearity. Attempts have been made to measure thedetector nonlinearity directly, but these have not been successful, and the exact origin ofthe intensity mismatch between the LF and HF spectra remains unknown.■•■■Chapter 6. Experimental Errors^ 1450.800Es 0.790C.) 0.780r:/)2a)0.770a) 0.7600.7500.0^1.0^2.0^3.0^40DC level VoltsFigure 6.9: The interferogram contrast as a function of the detector signal amplitude.The circles are measurements, the triangles are estimates based on a nonlinearity constantof E = 0.5 x 10-6 and a contrast of Interferogram ContrastThe optical bandpass of the HF spectrum is roughly twice that of the LF spectra, and toobtain similar detector operating conditions for the two cases, the interferometer outputaperture and source intensity have been reduced for the HF spectra. Despite thesealterations, the amplitudes of the chopped detector signals of the HF spectra are 1.4-1.6times greater than the LF cases. The anomalously low absorption of the HF spectra maybe caused by a detector nonlinearity since the HF background spectra, in particular,would be reduced in intensity. Note that the signal processing electronics of the lock-inamplifier, low pass filter, and A-D converter are all tested to be linear over the signalrange of the LF and HF spectra.A detector nonlinearity would be most apparent in the interferogram peak height,since the detector signal is roughly doubled in amplitude as the interferometer passesthrough zpd. An experiment has been conducted after the hydrogen experiments toChapter 6. Experimental Errors^ 146examine the interferogram peak height as a function of the DC level, the lock-in outputsignal when the interferometer is far from zpd. The interferogram contrastpeak-DC levelcontrast = ^DC level(6.1)increases as the DC level decreases. Figure 6.9 shows the measured contrast as a functionof the DC level, where the signal is attenuated by shifting the output spot of the multi-pass cell across the field mirror edge. The effect demonstrated in Figure 6.9 may not beentirely related to a detector nonlinearity, but it is possibly due to the nonuniformity ofthe interference fringes across the beamsplitter.6.4.2 A Computer Simulation of Detector NonlinearityA computer simulation has been performed to test the hypothesis that the detector signalV., is related to the applied optical power P, and a gain g, byV. = g(P — 6P 2) (6.2)where e is the strength of a simple nonlinearity term. The triangles of Figure 6.9 simulatethe contrast as a function of the DC level for e = 0.5 x 10' (in 16 bit A-D units) andan assumed contrast of 0.80. The spectrum obtained from the Fourier transform of aninterferogram recorded by the above detector response consists of two terms,H(a) = (1 — M.) S(o) — €C(o) (6.3)where I. is the interferogram DC level, S(a) is the true spectrum, and C(Q) is the auto-correlation function of the true spectrum [102]. The simulated spectrum equals the truespectrum multiplied by a factor less than unity, and it is reduced further in intensity atlow frequencies where the autocorrelation function is largest.Chapter 6. Experimental Errors^ 147The measured cosine background spectra and simulated background spectra are bothslightly negative at high frequencies. At frequencies below 20 cm-', the simulated spec-tra are even more negative than at high frequencies, but the measured cosine spectraare positive; in this regard the simulation does not represent the measurements. Theabsorption of hydrogen is derived from power spectra, and at least for high frequencies,the residual baseline of the spectra discussed in chapter 4 may be related to squaring thenegative cosine coefficients.Absorption spectra derived from the simulated nonlinear spectra are lower in intensitythan the true absorption, and simulated absorption spectra droop significantly belowthe true absorption over spectral regions where the background has low intensity. Themeasured 112 absorption spectra exhibit a similar drooping over the region of the LFbeamsplitter null (see Figure 4.1) and below 65 cm-' in the HF spectra. The reproductionof this absorption droop suggests that the measurements have been affected by a detectornonlinearity.6.4.3 Corrected InterferogramsThe effect of the nonlinearity on the measured interferograms can be corrected providedthat e and the true interferogram contrast are known. By assuming e = 0.5 x 10-6 and acontrast of 0.8, corrected interferograms yield LF background spectra which are zero athigh frequencies, and LF absorption spectra which no longer droop near the beamsplitternull. However, the residual baseline of a corrected background spectrum is raised near100 cm-1, and below 10 cm-1, regions where it seems reasonable to expect zero intensity.If the e used to correct the LF spectra is now applied to the HF spectra, the 10%intensity discrepancy between the two spectral regions remains unaffected. An e fourtimes greater than the LF case is required to raise the HF high frequency cosine terms tozero. The HF absorption spectra obtained using this larger E have a higher absorption,Chapter 6. Experimental Errors^ 148but the droop of the HF spectra below 65 cm -1 is not affected by either correction.The correction of the interferograms using this simple model of the detector nonlin-earity has proven unsatisfactory since the same nonlinearity factor does not apply toboth frequency regions. The LF and HF spectra are not brought into coincidence by thiscorrection and the the low frequency portions of the HF spectra remain anomalously low.The removal of the residual baseline, however, does somewhat correct the low frequencydroop of the HF spectra, and this procedure is described in chapter four.6.4.4 Detector Nonlinearity TestsMeasurements of the nonlinearity of the detector have been conducted by changing thedistance r between the detector and a source. The nonlinearity was expected to appearas a diminished signal at small r with respect to a 1/r 2 detector signal dependence.A mercury arc source, with a water cooled shroud and a chopper, was mounted on anoptical bench slide and the distance to the detector was varied from 53,to 135 cm. Thechopper frequency was 162 Hz, the detector low-pass filter was crystal quartz, and thedetector signal amplitude was varied over the same range as the hydrogen experiments.The optical bench was enclosed by a plexiglass box and a polyethylene tent which werepurged with both nitrogen gas from a cylinder and the boil-off gas of liquid nitrogen.A nonlinearity has been repeatably detected, but as a larger signal than expected atclose distances. This nonlinearity is due to residual water vapour in the optical path,despite thorough nitrogen purging. In fact, over the , 24 hour course of the tests, thedetector signal at a given source distance continuously rose as the water vapour contentdiminished.The data has been fitted by a function which includes an e-k'' water vapour absorptionterm, and still the detector signal at small r exceeds the value of the fit. The anoma-lous high signal at small detector-source distances may be caused by fitting data whichChapter 6. Experimental Errors^ 149includes a small offset, so that the signal at an infinite source distance is not zero. Thissituation can arise if the detector signal is not zero even though the source is off.The measurement of the detector nonlinearity turned out to be surprisingly difficult.The present tests are too imprecise to observe the suspected nonlinearity.6.4.5 Detector Nonlinearity CommentsThe bench top tests measure the response of the detector to slow changes in opticalpower; however, as the interferometer passes through zpd, the optical power increasesand decreases rapidly. The frequency response of the bolometer depends upon a numberof temperature dependent properties such as the heat capacity of the absorber and thethermal conductivity of the wires to the helium reservoir. The sensitivity and timeconstant of the detector can therefore change as the bolometer warms in response toan increase in the optical power. In the present case, the 162 Hz chopping frequency iscomperable to the detector time constant, and the detector response to a rapid changeof optical power is complicated.A new test needs to be performed which measures the detector response at the chop-ping frequency, to rapid changes in optical power. The contrast tests previously men-tioned have done this, but not very precisely. If the source is chopped at 162 Hz, and thesource intensity is varied using 2 polarizers, and a third polarizer is spun at 2-4 Hz, thenthe amplitude modulation of the chopped detector signal can be examined as a functionof the source intensity. This proposal simulates the dynamic interferogram signal, andthe optical path and water vapour absorption would remain constant during such a test.Chapter 7Other Spectroscopic StudiesDuring the course of measuring the translational band of hydrogen, a number of otherproblems have been studied. The main emphasis of these projects has been the searchfor dimer spectral features in the far-infrared. Highlights of these projects and par-ticularly interesting absorption spectra are presented in this section. In general thesespectra demonstrate the capabilities of the spectrometer-cell apparatus when applied tothe measurement of weakly absorbing gases. It is also notable that these low temperaturespectra are not contaminated by water vapour lines, a common problem in far-infraredspectroscopy.The gases used in these studies were supplied by Linde and the grades and puritiesare: D2, CP, 99.5% ; H 2 , UHP, 99.999% ; He, UHP, 99.999% ; Ar, UHP, 99.999% ; Ne,UHP, 99.996% ; N2, Medical, 99.998% ; CH4 ,UHP, 99.97% ; CO, CP, 99.5% ; 0 2 , UHP,99.995%.7.1 The Deuterium S(0) lineThe translational absorption band of D2 is shown in Figure 5.1, and the frequency regionof the S(0) line (179 cm') from this data set is shown in Figure 7.1. The peak of theD2 translational band has the same absorption as the lowest point of Figure 7.1; thiscomparison demonstrates yet again the extremely weak absorption of the translationalband. The spectral structure due to D2 dimer bound state transitions superimposed onthe collision-induced S(0) line is evident, and it matches the structure found in previous150Chapter 7. Other Spectroscopic Studies^ 151low temperature D2 spectra reported by McKellar [28]. A marked increase in the intensityof the dimer lines is observed as the temperature is lowered only 10 degrees. D2 dimertransitions should also be present in the translational band below 20 cm-1, but observingthis spectral region requires a low frequency polarizing interferometer. In the early phasesof the present work, observations were also made of the S(0) frequency region of H2 (354cm-') from 20-40 K.7.2 Hydrogen-Rare Gas MixturesA pure monoatomic gas exhibits no collision-induced absorption at low densities. How-ever, adding a monoatomic gas to hydrogen, or other diatomic gases, will enhance themeasured absorption spectrum. The absorption of radiation by a gas mixture takes placevia transient dipoles induced in both H2-112 collisions and H2-rare gas collisions. 112-H2absorption is dominated by the quadrupole induced dipole, and the H2 quadrupole mo-ment acting on a polarizable rare gas atom will also induce a dipole moment. DuringH2-rare gas collisions additional absorption occurs due to an isotropic overlap induceddipole moment, where this mechanism is absent in pure 112-112 collisions due to theirsymmetry.The absorption coefficient for a gas mixture (per unit pathlength) at a given frequencyisA an, p2 + aFh -x pH2 Px (7.1 )where aH, is the absorption coefficient Of 112 (per unit density squared of 112), cen,-x isthe absorption coefficient of the 112-rare gas enhancement (per unit density 112, per unitdensity rare gas), and pH2 and px are the densities in amagat units of the 112 and raregas components of the sample [14[1]. There is an alternative definition of the mixtureabsorption coefficient in terms of the absorption coefficient, per collision pair per unit3 Noyeregnr, 1992I^I I I I I^1^I I^I I I^I I 1 I,^1165 i00150.00I I^I^I 1155.100 ,60.,00 1170.00 pnioo1^,^1180.001 1S(0) region of^D2, 2.93 ornogat, 23.1, 26.7,^28.9, 33.3 K. 52 m^Resolufion=0.241 CM.' E. WishnowFigure 7.1: The absorption spectrum of D2 near S(0). The density is 2.93 amagat, thepathlength is 52 m, and the curves from highest to lowest correspond to temperatures of:23.1, 26.7, 28.9, 33.3K. Note the strong dependence of the dimer lines on temperature.1521.20.800_00.40.00.40O.0In0.20.0elt)44/4"4"'""4441411"A1_40.00I I80.00I^_I1120.00 1160.00 1200.00H2 25.8 K 9.03 AM.: H2—HE 25.3 K 9.03 e14./8.63 AM. Resolution=0.241 CM.' E. WIshnowFigure 7.2: The absorption spectra of 11 2 and a H 2-He mixture for a path of 60 m. Thedensity and temperature of the lower 1-1 2 spectrum is: 9.03 amagat, 25.8K, and 8.66amagat of He is added to obtain the upper spectrum at 25.3K. The He enhancement ofthe H2 spectrum is mainly due to the isotropic overlap induced dipole moment.Chapter 7. Other Spectroscopic Studies^ 153density, of the 112 and H2-X sample components [103]. Using either definition, if the raregas atom has the same polarizability as 112, and no isotropic overlap induction occurs, a50/50 mixture at the same density as a pure 112 sample has 1/2 the absorption.If the overlap induced dipole moment is insignificant, the shape of the spectrum re-mains unchanged by the addition of the rare gas to the hydrogen sample. However, theA(0001) isotropic overlap component of the induced dipole moment allows pure transla-tion transitions only (and Q branch vibrational transitions), and it can strongly enhanceand alter the translational spectrum.The absorption of H2-Ar mixtures has been studied over a pathlength of 52 m at 78and 88K by flowing liquid nitrogen or argon through the heat exchanger tubing of themultipass cell. The LF spectrum of 112 and the LF spectrum of a 50/50 mixture of 112'Ar, where both samples are at 88K and 1.62 atm pressure, have similar bandshapes andintensities. The absorption maxima and frequencies are: 112, 0.052 at 65 cm-1; and 112-Ar, 0.054 at 65 cm'. The similarity of the spectra demonstrates that the polarizabilityof argon is roughly two times larger than 112, and that overlap induction is not significant.The LF absorption spectrum of H2-Ne mixtures has been observed using a 52 mpathlength at temperatures near 32K, and the peak of the spectrum is dramaticallyshifted. The gas pressure, absorption maxima and peak frequencies are: 112, 1.35 atm.,0.3 at 57 cm"; 112-Ne, 1.02 atm., 0.1 at ,100 cm-1. This simple comparison suggeststhat isotropic overlap must be significant in order to shift the the peak of the spectrum,and that the polarizability of neon is smaller than 112 (it is 1/2 that of 112). Bound statetransitions of H2-rare gas complexes are expected to occur in the very low frequencyregions of these spectra, but no obvious structure has been observed.The absorption of H2-He is the most significant mixture spectrum from an astrophys-ical point of view. The concentration of He in the gaseous planets has been establishedby modelling the the Voyager far-infrared spectra using laboratory measurements of H2Chapter 7. Other Spectroscopic Studies^ 154and H 2-He absorption spectra [12],[104]. 11 2 rotational line absorption is dominated bythe 112 -11 2 interaction with a small He enhancement, as the polarizability of He is 1/4that of 112 . Over the translational band, however, 11 2-He collisions strongly enhance theabsorption spectrum due to the isotropic overlap interaction.The lower curve of Figure 7.2 shows the HF spectrum of 11 2 at a density of 9.03 amagatat 25.8 K over a 60 m path, and the upper curve shows the enhanced absorption obtainedat 25.3 K by adding 8.66 amagat of He to the gas sample. The shift of the peak frequencyand the altered and enhanced bandshape are apparent and demonstrate primarily theeffect of the isotropic overlap interaction. Given the astrophysical importance of 11 2-Hemixtures and the unique low frequency measurement capability of the spectrometer-cellapparatus, this topic is worthy of further investigation.The observations of the translational spectra of 11 2-rare gas mixtures are consistentwith previous studies of the H 2-rare gas collision-induced rotational spectra by Kiss,Gush, and Welsh [9]. This earlier work also demonstrates that the isotropic overlapinduced dipole moment is significant for 11 2 mixed with He and Ne, but not with Ar. Thepresent observations are also consistent with the values of the polarizabilities reported inthe literature [52].7.3 Nitrogen-Argon MixturesFigure 7.3 shows the translational-rotational spectra of various N2 and N2-Ar mixtures at78 and 88 K over a pathlength of 52 m. These spectra are the highest sensitivity, highestresolution, and lowest density spectra of the translational-rotational band yet reportedand are the first to show structure superposed on this collision-induced band. It is notablethat the minima of the ripples occur at the frequencies of the N2 free molecule S lines.This structure suggests that a band of unresolved dimer rotational lines is associated with111 1111^111111111^111 1 11111^111111 111^111111111^111111111^111111111^11111111[1_---_--...........r^_____merieroolliPirP-1- -^44411lobeisrefte, CiLgi'9^iiiiiii"?•?9iiiitiPriiiiiil”riiiiiil^7?.?9^iiiiiIi91.?°Iiiiiilr°iL?111Abs.^coeff. N2 & N2-Ar 78 & 88 K 52 m.^Resolution=0.241 CM.' E. November, V992Figure 7.3: The absorption spectra of N2 and N2-Ar mixtures. The pathlength is 52 m,the mixing ratio N2/Ar, pressures and temperatures are: a, 100/0, 538 Torr, 78K; b,48/52, 1219 Torr, 88K; c, 25/75, 729 Torr, 88K; d, 51/49, 361 Torr, 88K; e, 50/50 267Torr, 78K. Note the dimer structure superposed on the translational-rotational band. November, 19vI IIIIIII I IIIIIIIII I IIIIIIIII J IIIIIIIII I IIIIIIIII I IIIIIIIII I IIIIIIIII I IIIIIIIIII\------9ab—____----) dt......._....----,___,,...^, \l'i)?°113??9,,,,,I??°,!,,f,IP?•?°,c,,,,H5T3?917??9,,,,,IPP,,,,,,In?9,,,,,Abs.^coeff. CH4^&^CH4-Ar (50/50) mixture 113.5 -^117 K, 60 m.^Resolution=0.241 CM.'^E. Wishnow0.60a.80.4Figure 7.4: The absorption spectra of CH4 and CH4-Ar mixtures. The pathlength is60 m, and the mixing ratio CH4/Ar, pressures and temperatures are: a, 50/50, 1629Torr, 115.5K; b, 100/0, 794 Torr, 113.5K; c, 50/50, 1138 Torr, 116K; d, 50/50, 822Torr, 117K. The sharp centrifugal distortion dipole lines are observed on top of thecollision-induced band.Chapter 7. Other Spectroscopic Studies^ 156each N2 S line, where these dimer bands are analogous to unresolved P and R branchesof the vibrational spectrum of a diatomic molecule. The maxima of such a spectrumwould then appear offset from the frequencies of the N2 S lines, as shown in Figure 7.3.At low frequencies the dimer ripples have a more complex shape; this indicates thatintermolecular interactions involving N2 molecules in low J states are anisotropic. Notethat the overall intensity of curve a is roughly twice that of curve b indicating that thepolarizability of Ar is roughly the same as N2, as is expected.The structure of these spectra is very similar to structure observed in recent measure-ments of the fundamental vibrational band of N 2-Ar. [105]. The broad collision-inducedrotational spectrum of N2 has been measured many times, most recently reported in[106], and recent theoretical analysis is given in [47]. The present high resolution mea-surements are potentially valuable in assessing the validity of recent theoretical spectra[107],[108],[109]. The study of low temperature N 2-Ar mixtures is also relevant to the in-terpretation of the spectrum of Saturn's moon Titan which has an atmosphere composedof 82% N2 and 15% Ar [110].7.4 Methane-Argon MixturesFigure 7.4 shows the far-infrared absorption spectrum of CH 4 and CH4-Ar mixtures atvarious pressures near 115 K over a pathlength of 60 m. This work was motivated bythe possibility that CH4-Ar dimer transitions might be observed at very low frequencies,or as 'satellite' lines on the wings of the sharp methane absorption lines. The presentspectra are among the highest quality, highest resolution, and lowest density spectra yetreported, but in spite of this, no immediately obvious dimer features are observed.The CH4 molecule has no permanent dipole or quadrupole moments, and the broadChapter 7. Other Spectroscopic Studies^ 157collision-induced baseline of Figure 7.4 results from the octopole and hexadecapole mo-ments of the CH4 molecule inducing transient dipole moments in CH 4 or Ar collisionpartners. Recently a low resolution, high density, experimental study of the CH 4-Arcollision- induced spectrum has been reported [103]. The sharp lines in Figure 7.4 arisefrom the centrifugal distortion dipole moment, where analysis of these lines has beenpreviously reported by Rosenberg and Ozier [111].7.5 Carbon Monoxide-Argon MixturesFigure 7.5 shows the far-infrared transmission spectrum of the spectrometer-cell system,where the cell contains CO at a pressure of 7.4 Torr at 294 K, and the absorption path is52 m. The electric dipole allowed rotational transitions of 12 C'6 0 are saturated, and thishigh resolution spectrum demonstrates the accuracy of the infinite absorption level andfrequency calibration of the spectrometer. This spectrum is obtained from one sampleand one background interferogram, where each requires about 60 min. integration time.Figure 7.6 shows an expanded portion of the far-infrared absorption spectrum of a CO-Ar 27/73 mixture at a pressure of 30 Torr at 78 K over a 52 m. path. This investigationwas motivated by McKellar's observation of CO-Ar dimer 'satellite' lines in the wings ofthe P and R lines of the CO fundamental vibrational band [112]. The present spectra areobtained from 4 sample interferograms, an observation time of 120 minutes. The 1 2c160rotational lines are saturated and the other lines, in order of diminishing intensity, aredue to 13C160 , 12C180 , and perhaps a 12 C170 line at 41.2 cm'. Although the spectrum isquite good, the signal-to-noise is insufficient to observe any definite dimer structure. Thedimer binding energy is likely to be dependent on the rotational state of the CO moleculeand investigation of very low frequency CO lines is probably necessary to observe dimerstructure.5 Nov.mber, 1592I I^I I I I I I I I I I I I^1 I I I^I I I^I^I^I——20.00^i3?.°?^I^I^1 4P.CT^I^51)4? I^I 5411.(1?^I^I^I 7?'°?^1^i ei"?^I^I 9P°Spectrum^of^CO, 294 K, 7.4^Torr, 52 m. Resolution=0.005^CM.-t E. WiehnowFigure 7.5: The sample spectrum of CO at 7.4 Torr and 294K over a pathlength of 52 m.The CO lines are saturated and this spectrum demonstrates the intensity and frequencycalibration of the spectrometer.3^vemb.r, 1992IIIIIIII^I1I.I^I^III^I^I^I._..^AI^I^I^I^I^II^I^IA.I^I^I^I^I^I^II^1^I^I^I^I^I^I^I^1^Il..^ .II^I^I^I^I^1^I^II1°1°°1^I^I^I^I^i^I^1^1.2i°9^iiiiiil^14i°°Iiiiiiil^16i°9^iiiiiillai°9^ i°'icliiiiiiiiAbs.^coeff.^CO—Ar^(27/73)^mixture^78^K,^52^m.^Resolution=0.010^CM."' E. WlshnowFigure 7.6: The absorption spectrum of a 23/73 CO-Ar mixture at 30 Torr and 78K overa pathlength of 52 m. The rotational lines of 12ci60, 13c160, and 12c180, are observed,in order of diminishing intensity. 7. Other Spectroscopic Studies^ 1597.6 OxygenThe low temperature spectrum of 02 is shown in Figure 7.7, where the temperature, pres-sure, and pathlength are: 88 K, 241 Torr, and 52 m. 0 2 demonstrates collision-inducedabsorption at high densities, but the sharp lines observed here are due to magnetic dipoleallowed transitions. The electronic ground state of the oxygen molecule has 2 unpairedelectrons with a net spin 5=1. Coupling between the spin and the molecular rotationN splits the rotational energy levels into triplets, where the total angular momentumJ=N+S. The selection rules for transitions within the triplets are: AN = 0, A J = +1,and these transitions are observed in the microwave region near 2 cm -1 . Transitionsbetween the triplets with the selection rules AN = 2, AJ = 0, +1 [113], are responsiblefor the far-infrared absorption spectrum shown in Figure 7.7.This spectrum demonstrates the performance of the spectrometer-cell system at highresolution, where the sample spectrum is obtained from 2 interferograms, requiring 90minutes of observation time. The central lines of the two lowest triplets are nearlysaturated, and this spectrum is notable for the absence of strong water lines whichcan easily overwhelm the 0 2 spectrum at room temperature. Since this spectrum wasobtained, a mercury arc source has been installed in the spectrometer and the signal-to-noise below 40 cm -1 has been improved by at least a factor of 2.The spectrum of the 0 2 collision-induced vibrational band exhibits dimer structuresimilar to the N2 dimer ripples of Figure 7.3 [114]. Again, evidence of dimer formationis expected to appear in the present spectrum, but it is not obviously apparent. Therotational spectrum of 02 has previously been studied at room temperature [113], butthe low temperature, magnetic dipole allowed rotational spectrum of 02 is presented herefor the first time.Chapter 7. Other Spectroscopic Studies^ 160149••n9Nor, t99211 1 1 11 1 I'"'I^" I"^I" 1 I 1 S I 1 I I 1 1 1 I' 1 1 I 1 1 11 1 I3.02.004-101.00.020. 1001^25.1001^1 0^135.00^I 10. 1001 I^15.[00, I I 50. 1001 I^15. 1001 I I 60.001 I^65. 00P`), I I 7 .,", I I Abs. coef. f. 02 88K, 241 Torr, 52 m.^Resolution=0.010 CM.' E. WIshnowFigure 7.7: The absorption spectrum of 0 2 at 241 Torr and 88K over a pathlength of 52m. The lines are due to magnetic dipole rotational transitions.Chapter 8ConclusionThe far-infrared absorption spectrum of normal hydrogen gas has been measured over thefrequency range 20-320 cm', the temperature range 21-38 K, at pressures from 0.6-3atmospheres. Since observations are performed on very low temperature gas samples, thecollision-induced translational band of 112 is better isolated from the rotational lines thanprevious higher temperature measurements; in fact, the translational band is isolated topractically the greatest extent possible. The 112 translational spectrum is measured usingnew temperature and pressure conditions at a spectral resolution 10 times higher thanprevious studies.These spectra comprise the best available measurements of an extremely weak ab-sorption phenomenon. By comparison, the weak collision-induced absorption peak of theS(0) line of normal 112 is over 15 times, and the 5(1) peak over 50 times, greater than thepeak of the translational band at these low temperatures. In addition, the present lowtemperature, low pressure spectra are the first experiments to search for dimer structureacross the translational band.Although dimer structure is not observed in these translational spectra (it may yetappear below 10 cm-1), the R(0) dimer line of HD is observed as a sharp feature super-posed on the translational continuum at 89.2 cm-1. The analysis of the integrated area ofthis line yields a measurement of the HD electric dipole moment of (0.81 ± 0.05) x 10'debye. This value corresponds to previous theoretical predictions and measurementsmade at higher temperatures and pressures.161Chapter 8. Conclusion^ 162The low temperature multi-reflection absorption cell developed for these experimentsperforms well, and optical pathlengths of 60 m are obtained even for wavelengths as longas 0.5 mm. The cell temperature instabilities and gradients are reasonably small andsample spectra are reliably acquired at very low temperatures. The absolute accuracyof the present spectra is comparable to previous measurements made in more convenienttemperature regions.The present spectra confirm the predictions of the Poll and Van Kranendonk the-ory of collision-induced absorption, and the measured integrated absorption coefficientis consistent with that calculated using the quantum mechanical pair distribution func-tion obtained from the Lennard-Jones intermolecular potential. The low temperatureresults provide a closer correspondence between measurement and theory than all pre-vious experiments over the temperature range 77-300 K. The temperature variations ofthe spectral invariants a it, and ryit, are extended to new temperature regimes, and newinformation is obtained regarding the H2 intermolecular potential. The detailed shape ofthe present spectra should also provide a valuable test of modern ab-initio models of theH2 intermolecular potential.The low temperature spectra demonstrate the validity of fitting the H2 translationalband with the B-C lineshape function, where previous experiments have emphasizedfitting the S(0) and S(1) rotational lines. The temperature variations of the B-C lineshapeparameters are also extended to new temperatures, and this work may be applied tomodelling the Hz component of planetary spectra at very low temperatures.Additional studies of H 2-rare gas, N 2-rare gas, and CH 4-Ar, CO-Ar, and 0 2 have beenundertaken in an effort to observe dimer structure in the far-infrared. Of these studies,the N2 and N 2-Ar spectra most clearly demonstrate structure across the translational-rotational band.The work presented in this thesis is also relevant to the longstanding astronomicalChapter 8. Conclusion^ 163problem of detecting interstellar molecular hydrogen. A particularly important feature ofthese experiments is the demonstration that the D/H ratio of planetary atmospheres canbe determined from the intensities of the HD R(0) line and the underlying 112 translationalband.Appendix AThe Absorption Cell Optical TestsTests of the cell optical system were made on an optical bench using visible laser lightand during the absorption measurements using far-infrared radiation. The visible testsestablish the cell pathlength limits due to the aberrations of diffraction and astigmatismand these limits correspond well to calculated aberrations. The cell performance in thefar-infrared is consistent with the limits determined from the bench top tests.The diffraction spot size does not grow with successive reflections [66]. This wasdemonstrated by a test in which the cell mirrors were aligned on an optical bench anda 3 mm diameter aperture was placed over one of the folding mirrors. Laser light wasintroduced to the system such that the beam passed through this aperture many times.The Airy pattern at the cell exit was magnified and photographed. It was observed thatthe patterns corresponding to passage through the aperture 1,2,3,4,5, and 9 times wereequivalent, and the same as predicted by the Airy disk diameter to better than 10%.Diffraction is an aberration which is wavelength dependent, and in the visible, wereno other aberration present, the maximum cell optical path would be 116 m, for aninput spot of 6 mm diameter. The cell image also suffers from the optical aberrationof astigmatism, where in this case, the image distortions accumulate with successivereflections.The amount of image distortion due to astigmatism was calculated by following theappendix of Horn and Pimental [115]. Rays which reflect from one side of the fieldmirror to the other have the largest angle of incidence with respect to a folding mirror.164Appendix A. The Absorption Cell Optical Tests^ 165The images on the field mirror associated with these paths therefore suffer the largestastigmatism as compared to smaller reflection angles. The distance between the sagittaland tangential focii is calculated by adding up the contributions to distortion from all ofthe reflection angles present in the multiple reflection cell. The diameter of the circle ofleast confusion isd . 2f (A.1)where s — t is the difference between the two focii and f is the f number of the beam.The results of calculating the distance between the sagittal and tangential focii andthe astigmatic circle of least confusion from a point source input are tabulated below.Included in the table are measurements obtained from bench top tests of the White celloptics. The tests were made by focussing a laser at the field mirror edge such that thebeam then fills the folding mirrors. Measurements of spot diameters were then madewith a ruler.spots path (m) s - t (cm) diam.(cm) measured diam.(cm)23 48 12.9 0.6525 52 14.0 0.7027 56 15.1 0.7629 60 16.2 0.81 0.8345 92 24.8 1.24 1.25Table A.1: The calculated and measured cell spot size as a function of pathlength dueto astigmatismThe diffraction spot diameter for a 30 cm -1 point source input to the multipass cellis 0.85 cm (see Table 3.1). For low frequency radiation, the table above shows thatdiffraction and astigmatism over long optical paths yield comparable distortions of theimage. The effect of astigmatic distortion may be slightly reduced by defocussing theinput image so that the spots on the field mirror are slightly elliptical, thereby allowing.9 — tAppendix A. The Absorption Cell Optical Tests^ 166a longer path before spot overlap. Obviously at higher frequencies image distortion isdue primarily to astigmatism.The actual performance of the cell is very similar to the limits imposed by astigmatismand diffraction. During the operation of the cell in the low frequency far-infrared (20-180 cm-'), the overlap of spots was examined by comparing the intensity of a signalmaximum to the neighbouring minimum as the pathlength is changed and spot imagessweep across the field mirror exit. Note that these tests were conducted when the cellwas at room temperature; the AC signal increases by about 10% when the cell is cold.The table below lists the number of field mirror spots, the corresponding pathlength,the ratio of a signal maximum to the neighbouring minimum, and the AC voltage of thedetector signal which is proportional to the optical intensity.spots path (m) ratio max/min VAC23 48 19.2 1.22725 52 9.4 1.19827 56 5.9 1.16429 60 3.5 1.11931 64 2.6 1.057Table A.2: The measured ratio of the cell far-infrared output signal maximum/minimumas a function of pathlengthThe table above is only an indicator of the spot overlap as it uses the intensity oflight at the midpoint between spots and not the contribution of the neighbouring spot atthe center of the output image. Therefore, the estimates above are worst cases of overlapcontamination, and the cell operates with virtually no pathlength ambiguity at 52 m andacceptable overlap up to --60 m.Bibliography[1] H. R. Welsh. In MTP International Review of Science-Physical Chemistry, Volume3, page 33, London, 1972. Butterworths.[2] G. Birnbaum. Phenomena Induced by Intermolecular Interactions NATO ASI Se-ries, volume 127. Plenum Press, New York, 1985.[3] A. Borysow, L. Frommhold, and. P. Dore. Int. Journal of Infrared and Mm Waves,8(4), 1987.[4] J. Van Kranendonk. Physica, 23:825, 1957.[5] J. Van Kranendonk. Physica, 24:347, 1958.[6] J. Van Kranendonk Z. K. Kiss. Can. J. Phys., 37:1187, 1959.[7] J. D. Poll and J. Van Kranendonk. Can. J. Phys., 39:189, 1961.[8] D. R. Bosomworth and H. P. Gush. Can. J. Phys., 43:729, 1965.[9] Z. J. Kiss, H. P. Gush, and H. L. Welsh. Can. J. Phys., 37:362, 1959.[10] Z. J. Kiss and H. L. Welsh. Can. J. Phys., 37:1249, 1959.[11] Z. J. Kiss and H. L. Welsh. Phys. Rev. Lett., 2:166, 1959.[12] G. Birnbaum. J.Q.S.R.T., 19:51, 1978.[13] P. Dore, L. Nencini, and G. Birnbaum. J.Q.S.R.T., 30:245, 1983.[14] G. Bachet, E. R. Cohen, P. Dore, and G. Birnbaum. Can. J. 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