/H _O / NASA Technical Memorandum 105999 Laser Rayleigh and Raman Diagnostics for Small Hydrogen/Oxygen Rockets Wilhelmus A. de Groot Sverdrup Technology, Inc. Lewis Research Center Group Brook Park, Ohio and Frank J. Zupanc National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio Prepared for the SPIE International Symposium on Lasers, Sensors, and Applications sponsored by the Society of Photo-Optical Instrumentation Engineers Los Angeles, January 16-23, 1993 (_']A_&-TM- I059o9) LASER RAYLEI GH N93-17995 A_J_ #,_MAN _[AGN_STICS FOR SMALL IW A HYiJ° C,GE'_/_ XYGE N P,_CK_TS (NASA) Uncl as G3/20 0142864 Laser Rayleigh attd Raman Diagnostics For Small llydrogen/Oxygen Rockets Wilhelmus A. deGroot Sverdrup Technology, Inc. Lewis Research Center Group Brook Park, Ohio 44142 and Frank J. Zupanc National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 ABSTRACT Localized velocity, temperature, and species concentration measurements in rocket flow fields are needed to evaluate predictive computational fluid dynamics (CFD) codes and identify causcs of poor rocket performance. Velocity, temperature, and total number density information have been successfully extracted from spectrally resolved Rayleigh scattering in the plume of small hydrogen/oxygen rockets. Light from a narrow band laser is scattered from the moving molecules with a Doppler shifted frequency. Two components of the velocity can be extracted by observing the scattered light from two directions. Thermal broadening of the scattered light provides a measure of the temperature, while the integrated scattering intensity is proportional to the number density. Spontaneous Raman scattering has been used to measure temperature and species concentration in similar plumes. Light from a dye laser is scattered by molecules in the rocket plume. Raman spectra scattered from major species are resolved by observing the inelasticaily scattered light with a linear array mounted to a spectrometer. Temperature and oxygen concentrations have been extracted by fitting a model function to the measured Raman spectrum. Results of measurements on small rockets mounted inside a high altitude chamber using both diagnostic techniques are reporled. 2. INTRODUCTION Improving the performance of on-board rockets yields rewards in terms of extended mission life, or enhanced mission goals through a reduced propellant ioad.l Computational fluid dynamics (CFD) codes are routinely used both to evaluate rockets which are under development and to establish the effect of design changes on existing rockets. In some cases an acceptable prediction of rocket performance can be made with the current level of predictive technology. For high and medium thrust rockets, predictions of the global parameters such as thrust and specific impulse, show good agreement with measurements. Using these codes for predicting the performance of certain low thrust rockets, however, leads to inaccurate predictions, the specific causes of which are not identified. 2,3hnproved predictions for these rockets can only be made based on a belier understanding of the physics of small rockets, a refined analysis of the individual rocket, and an increase in numerical capabilities. Such developments are aided by the availability of localized fluid dynamic and thermodynamic data upon which these improved codes can be verified. The rockets tested in this investigation were gaseous hydrogen/oxygen rockets mounted inside a high altitude simulation chamber. The absence of ambient light inside the high altitude chamber combined with the low luminosity of the water vapor rich plume provided ideal conditions for spectroscopic probing by lasers. Laser Doppler anemometry requires the use of particles. The large gradients inherent in small rockets as well as the difficulties in introducing particles precluded the application of this technique to small rockets. This prompted the development of a technique based on Rayleigh scattering. This technique yields velocity and in some cases the temperature and number density of the molecules in a high velocity plume. 4For Rayleigh scattering a narrow band laser beam (pulsed or cw) was directed through the plume. The laser light scattered from the moving molecules was Doppler shifted and collected at two locations. The magnitude of the observed Doppler shift is dependent on the relative angles between the incident and scattered light and the gas velocity. Since the relative angle between the incident and collection optics was known, the mean velocity of the observed gas volume was determined from the magnitude of the Doppler shifts measured at each location. Inadditiontoa meangasvelocitye,achmoleculpeossessietss own thermal motion which depends on the gas temperature and the molecular mass. When the gas is in thermal equilibrium, the molecular velocity obeys a Maxwellian distribution, which results in a Gaussian profile for the scattered laser light. _Therefore, in the absence of other significant broadening parameters, the degree of broadening for a single species can directly be related to the temperature. Accordingly, by resolving the line shape of the scattered light, and relating this to the reference laser line shape, it was possible to directly estimate the molecular velocity and translational temperature. Furthermore, since the scattering intensity for a single species is linearly proportional to the molecular number density, total number density could be estimated for the mixture by comparing the total scattered intensity and the intensity scattered by a reference gas. Spontaneous Raman scattering was implemented for measuring temperature and species number densities. Light from a pulsed dye laser was focused in the rocket plume. Both Rayleigh and Raman scattering processes occur inside this focal volume. Raman scattering occurs when light photons exchange energy with rotational and/or vibrational internal molecular energy modes. Incident photons that lose energy to a molecular energy system are shifted to a lower spectral frequency. This spectral shift depends on the scattering species since these energy losses are discrete and equal to a change in molecular energy quantum. Thus each observed shifted spectral line indicates the presence of a specific species, characterized by the magnitude of the shift from the incident laser line. For a set of molecules in thermal equilibrium, the population over all possible energy levels can be described by a Boltzman distribution. Since quantum jumps are equally likely from all initial energy levels, the distribution of the Raman scattered photons over the different possible shifted lines for each species reflects this Boltzman distribution. Light scattered from the probe volume was collected and spectrally separated. Measurement of the intensity distribution of this light over the possible spectral range for the measured species yielded the Boltzman distribution from which the temperature was determined. The overall intensity of this distribution depends linearly on the incident laser intensity and on the number density of the scattering species. Because the laser intensity was known, the number density was directly determined from the scattered intensity. Both techniques are impeded by the presence of ambient light. Raman scattering is a naturally weak process which requires minimum ambient and stray light to achieve acceptable signal to noise ratio (SNR). Rayleigh scattering is three orders of magnitude stronger and is therefore less sensitive to experimental conditions. However, for the measurements reported in this paper, the combined effects of low scattering number densities and the need for highly resolved spectra resulted in low signal levels. Judicious placement of pinholes, baffles and beam stops was therefore required to achieve an acceptable SNR. The limited number of windows of the high altitude simulation chamber were blocked to remove all ambient light, and extensive use was made of optical fibers for accessing the chamber. Velocily and temperature measurements based on spectrally resolved Rayleigh scattering were successfully made on a lab model gaseous hydrogen/oxygen rocket engine with a nozzle area ratio of 30:1. The nozzle was modified for the Raman measurements since the number densities in the exit plane of the full scale rocket were prohibitively low. A reduced area ratio nozzle was installed which made number density and temperature measurements feasible in the exit plane. The results obtained _vith both techniques were compared to predictions by existing CFD codes. Results are reported in this paper. 3. ROCKET TEST FACILITY The rockets tested were designed for space operation. To simulate a space environment, tests were conducted in a high altitude simulation chamber. 6 This chamber is 0.91 m in diameter and 1.81 m in length. The chamber is evacuated by means of a supersonic diffuser which is driven by a two stage venturi ejector stack. The 0.102 m diameter diffuser fully captured the rocket plume. The diffuser perfornmnce was optimized by locating the diffuser entrance from 0 to 0.10 m behind the rocket exit plane. When optical access was required, a trade-off between the diffuser distance and optical access was necess_lry. Under optimal conditions, chamber pressure was maintained at 1.3 kPa, simulating conditions at a height of 35 kin. During tests when the plume was not captured sufficiently, the pressure gradually rose. Tests were usually aborted after the pressure exceeded 6.9 kPa. A detailed description of the high altitude simulation facility and the optical access is given in Ref. 6. The rockets, shown schematically in Fig. 1were in the 110 N thrust class. 2They were designed to operate with gaseous hydrogen and oxygen with an oxygen-to-fuel ratio of 3.0-5.0. The propellant was supplied by high pressure trailers. A combination of upstream regulators and calibrated sonic venluris controlled the propellant flow rates to the thruster which werecontinuousmlyonitoredT.he 0.025 m diameter, 0.102 m long combustion chamber converged into a nozzle with a tl_roat diameter of 0.0128 m and exit area ratio of 33:1. Gaseous hydrogen was used to regeneratively cool the combustion chamber and nozzle. The hydrogen entered wall passages through a channel in the nozzle exit plane and propagated upstream through the chamber walls toward the combustion chamber. The hydrogen flow was then split by means of calibrated flow passages. Part of the hydrogen was directed towards tile core flow of the injector and part was diverted to a sleeve which distributed the fuel in a thin film over the combustion chamber wall. The amount of fuel diverted for fuel fihn cooling was regulated between 50% and 75%. The thruster injector was a stacked platelet injector. Oxygen was injected upstream of a spark plug and flowed past the spark plug tip towards the combustion chamber. At the spark plug tip, hydrogen was injected through radial passages in the stacked platelets into the oxygen flow and was entrained in the recirculation zone created behind the blunt spark plug tip. After initially igniting the oxygen rich mixture, the spark plug functioned as a flame holder, creating turbulent mixing and maintaining combustion. The oxygen rich product mixture entered the combustion chamber and gradually consumed the hydrogen fuel film at the wall. Several conditions were favorable for applying Rayleigh scattering to the plume. The choice of propellants led to a plume with extremely low luminosity in the wavelength range of interest for Rayleigh scattering, providing an acceptable signal-to-noise ratio. Furthermore, the low pressure in the exit plane guaranteed that the mean-free-path between molecules was much larger than the scattering interaction wavelength. Accordingly it could be assumed that the scattering occurred in the "collisionless" regime, where uncorrelated scattering of molecules could be assumed. This greatly simplified thc scattering analysis. 7 The Rayleigh scattering analysis assumed water to be the single scattering species. Because the overall mixture ratio was fuel rich it is reasonable to assume that the plume was fuel rich. Water molecules have a much larger Rayleigh scattering cross section than hydrogen and the much lower molecular mass of hydrogen causes its scattering contribution to be spread over a much wider spectral range. Therefore water vapor was assumed to be the dominant scattering species. Due to the fuel film cooling however, there was a considerable variation in local mixture ratio, gradually varying from an oxygen rich core to a fuel rich perimeter region. As previously stated, water vapor was the dominant scatterer in those regions that were fucl rich. Oxygen contributed significantly to the total scattering for measurements in the core. This made it impossible to estimate the temperature and total number density. The extremely low luminosity of the hydrogen-oxygen combustion products in the plume was also favorable for Raman scattering. As with the Rayleigh scattering, the low density made a simple, collision free analysis possible. However, the same low density made it difficult to detect Raman scattering in the exit plane during the 30 second test times. Therefore. for Raman scattering experiments, the nozzle length was reduced to an area ratio of 1.86:1. A simple one-dimensional nozzle analysis showed the pressure in this exit plane to be about 0.1 MPa., theoretically providing a sufficient number density for Raman scattering diagnostics. The redesigned test rocket wall was water cooled. The injected hydrogen which provided film cooling was lower in temperature than the hydrogen injected in the regeneratively cooled rocket. As a result, the fuel film injection velocity was different than during the Rayleigh experiments. H2 Boundary Layer . j! 02 ,H2 Mixing .... / Layer 02 H2 H2 - Regeneratively Cooled Wall H2 Fig. I. Regeneratively cooled hydrogen-oxygen thruster flow field with fuel fihn cooling. 4.SPECTRALLY RESOLVED RAYLEIGH SCATTERING Raylcigh scattering occurs when an incident electromagnetic wave and a molecule interact without exchanging energy Three conditions have to be satisfied for a relatively simple treatment of both theory and application. The wavelength of the incident wave should be much larger than the size of the scattering molecule. If this is satisfied, a simple oscillating dipole is induced inside the molecule, which creates its own electromagnetic field and subsequent radiation (scattering). The dipole oscillates with the frequency of the incident wave, and consequently the scattered wave is of the same frequency.. For a simple "eollisionless" treatment of Rayleigh scattering theory, the scattering number densities have to be low enough to ensure that the collisional mean free path of the molecules is much larger then the interaction wavelength. If that is the case, the incident wave will interact with the molecules in an uncorrelated manner 7and the resulting Rayleigh scattering is characterized by a simple Gaussian shape. Finally, the scattering volume should consist of a single component gas or the mixture composition should be known. When awave is incident on a single component gas volume, the scattered intensity depends only on the molecular number density, and can be used for measuring number densities. Since the strength of the dipole that is created varies with type of molecule, the scattered intensity varies with type and number of scattering molecules. In addition, because the bandwidth of the Gaussian shape depends on the molecular weight, the contribution of each species to the final profile is a Gaussian of widely varying shape. It is therefore generally difficult to measure molecular number densities in a gas mixture if the gas composition is not known apriori. The directionality of Rayleigh scattering depends on the polarization of the incident wave. For a linearly polarized incident wave, the dipole oscillates one-dimensionally and the radiation shows a toroidal distribution, with the strongest scattering perpendicular to the oscillating dipole. It is therefore important to detect the Rayleigh scattered radiation perpendicular to the electric field vector. 4.ITheory The random velocities of the individual molecules of a thermalized gas volume at rest follow a Maxwellian velocily distribution. Because of this, the Rayleigh spectrum that results when an incident plane wave of single frequency is scattered by this gas volume exhibits a Gaussian shape centered around the incident wavelength. This shape possesses a bandwidth (FWHM) which is directly proportional to the square root of the gas temperature. For a gas volume with mean velocity not equal to zero, the peak position of the observed spectrum is shifted from the incident wavelength. The shift depends on the gas velocity and the relative orientation between the incident and observed light, as illustrated in Fig. 2a for a backscatter and Fig. 2b for a for_'ard scatter geometry. A detailed derivation shows that the power scattered by a single component gas in the direction of observation given by "scattered" vector ks into solid angle d_ in frequency, interval df is:4 Ps(f) dfd_ = I0n V_ _d-cr I sin :Z S(f) dfdf2 (1) _hcrc l,, is the incident irradiancc m W/m 2. n the molecular number density in m-3, and W_ is the scattering volume. The differential Rayleigh scattering cross section is given in m2/sr, and Z is the angle between the incident electric field _ector k,, and k. The frequency distribution function S(f) is the normalized spectrum of Rayleigh scattered light. This normalized spcctnml can be obtained by integrating the Maxwellian velocity distribution over the velocity space. This gives: 4 /-= S(f) df - 2 "v_, e-(2,rf, l_-fi)2/a2K2 df (2) aK 4 where a = (2_¢T/m) _/2 is the "most probable speed" of the molecules in m/sec, and K is the scattering vector magnitude, where the scattering wave vector can be obtained from the relation I_ = ks - ko- The thermodynamic constant K"is Boltzmann's constant (1.38 × 10-23J/K), T is the gas temperature in Keh, in, and m is the molecular mass in kg. The vector _ indicates the mean velocity whose magnitude is given in m/sec. Eq. 2 represents a Gaussian vclocit_ distribution with a bandwidth (FWHM) of 0.265 a K Hz. k \' ko !S ks ...',.. ....K=ks-ko V \ \, RADIAL kS AXIAL K= ks-k o Fig. 2.Vector diagram of Rayleigh scattering: (a) backscatter and (b) forescatter. For a single component gas with known molecular mass, the temperature can be extracted from the bandwidth of the observed scattered light. The mean velocity of the molecules can be found from the location of the peak of the Gaussian profile with respect to the incident wavelength as is shown in Fig. 3. The number density can be estimated from the total integrated Rayleigh scattered light. To obtain a density measurement, a reference measurement is necessary, where the total integrated Rayleigh scattering is measured with identical optical geometry on a gas with known number density and known differential scattering cross section. The number density measured is linearly proportional to the ratio of the total integrated Rayleigh scattering powers for both cases. The absolute number density can be found from the reference number density and ratio of differential scattering cross sections. For a multi-component system with unknown composition, the total scattered Rayleigh profile can be assumed to be made up of the sum of the contributions from each individual species. If all species possess the same mean velocity, each contribution consist of a Gaussian lineshape located at the same spectral location with different spoctml widths, dependent on the scatterers' molecular mass. The determination of temperature and number density from such a profile rcquircs additional information. However, in many cases, a single species dominates the scattering process and a simple singlc componcnt curve fit can yield the desired data. 4.2 Application. Plumes of hydrogen-oxygen rockets consist predominantly of water vapor. Both pulsed and continuous _ave (c_l strategies were employed to generate Rayleigh signals. The cw technique utilized the 514.5 nm line of a cw Argon-lon laser. The pulsed strategy utilized the second harmonic oulpul from an injection seeded Nd:Yag laser (_. = 532 rim, 150 MHz bandwidth). Both approaches satisfied the condition that the wavelength of the incident wave was much larger than the 1nm effective diameter of the watermolecule. The light was vertically polarized and the detection direction, given in Eq. 1as the vector ks, was in a horizontal plane. This implies that the angle 2' was 90°, which maximized the scattered power in the detection direction. The optical geometry for the cw strategy is shown in Fig. 4. Various steering optics directed the laser beam through the optical port at 60°, parallel to the optical table mounted underneath the thruster, and between the diffuser and thruster. A 1000 mm focal length lens was used to focus the beam to a diameter of 180 _tm in the near-field region of the thruster. Since stray laser light could impede the measurement by entering the detection system, extensive use was made of baffles, masks, high quality optical windows and beam stops. For the pulsed strategy, the laser beam was directed straight across the exit plane, with the detection from various angles. t40O, Laser ]___ Velocity 1200 1000 u_ 80_3 t-- Z Z S Density/ ,_ 7J 0 6()l, ( c 400 _o¢_ Temperatur_ t_ 200 10 20 30 40 50 60 70 80 90 lO0 BiN Fig. 3. Typical spectral scan of Rayleigh scattering over two times the free spectral range. Two fiber optic collection probes were positioned at angles 0s = 31° and 149° (see Fig. 2) for fore- and backscatter, respectively, to determine two components of the velocity. Each probe consisted of two lenses, one lens to collect scattered light and a second one to refocus it into a fiber, and a 10 m long fused silica fiber with a diameter of 1000 Ixm which gave an effective collection f# of 4.55. This geometry resulted in an adequate collection solid angle without introducing aperture broadening effects caused by the variation of collection angle over the collection optics. The fiber optic guided the scattered light through lhe ailiiude chamber bulkhead to an optical table, where the fiber oulpul was collimated ,with an achromatic doublet and passed through a single pass scanning Fabry-Perot interferometer. The free spectral range of this interferomelcr was 21.4 GHz, sufficient to resolve the bandwidth (FWHM) of a nominal Doppler broadened Rayleigh spectrum of water (T=I000 K. X=514.5 nm, and 0s=31°), which was approximately 6 GHz. The filtered light leaving the Fabry-Perot was focused through a 1000 lam aperture; collimated; passed through a multilayer dielectric interference filter to eliminale luminous background contributions; and detected with a 20% quantum efficiency, Gallium-Arsenide photomultiplier tube (PMT). A single threshold photon counter was used to measure the resulting Rayleigh scattering signal. Piezoelectric spacer elements were used to vary the Fabry-Perot mirror spacing over two adjacent orders of interference, thereby tuning the delecled PMT signal through the Rayleigh spectrum. A typical spectral scan is displayed in Fig. 3. Temperature and velocity were extracted from the measured Rayleigh spectrum by curve fitting a model function based ou Eq. 1to this spectrum. It was necessary to expand the model function to include the instrument response function. This instrunlent function was experimentally deternlined for both the unshifted, unbroadened argon-ion laser line, and the Nd:Yag second harmonic output. A small portion of laser light, reflected from the altitude chamber window was collected. attenuated with neutral density filters and coupled into a reference fiber. This fiber guided the light into the test section where it was mixed with the Rayleigh scattered light and collected with the collection probes. This reference signal could be turned on and off with the placement of a shutter on the collection side of the reference fiber. Excellent agreement was obtained between the experimentally measured and calculated reference spectra. For number densily measurements, calibration spectra were obtained by traversing the measurement probe volume across the rocket exit plane while the facility was purged with anitrogen gas of known temperature, density, and scattering cross section. High temperature spectra in the rocket plume were obtained at the same radial locations by traversing the probe volume along the path of the laser bcanl during a rocket fire. Use of finite diameter optical fibers reduced the spectral resolution of the interferometer due to the increase in solid angle from the light collimated through the instrument. 7This limited the sensitivity of the temperature measuremenl. Eq. I shows that the Doppler width was proportional to the magnitude of the scattering vector. This vector was much larger in the backseatter geometry, which caused the Doppler broadening to be much greater in the backscatter direction, surpassing the sensitivity of the instrument even at low temperatures (a theoretical limit of 245 K). As a result, temperature measurements could only be made in the backscatter geometry. NOZ'ZL_p_ _k_._ B - __ AFFLES ALT'r]ET_f E /# _ _________ __L[ ............ CHAMBER g117.1 ARGON IO..N.......... REFERENCE / - F/Jrd LASER FIBER /1" I L2 L3 ...__ , A F COLLECTION _ ] FIBER FABRY-PEROT Fig. 4. Diagram of Rayleigh scattering optical arrangement with cw laser source. An error analysis based on bias errors and the maximum likelihood parameter estimation procedure used in the curve fitting process, showed that the typical uncertainty in the mean velocity estimate was 200-300 m/s, independent of the magnitude of the mean velocity. Temperature determination was slightly less accurate, primarily due to the sensitivity of the estimation to noisy spectral profiles. Typical temperature uncertainty was 100-150 K. Number density estimates for a single component gas typically fall within 2-5% of the mean value. 5. VIBRATIONAL-ROTATIONAL RAMAN SCATFER1NG Raman scattering is a process that occurs simultaneously with Rayleigh scattering when a plane electromagnetic wave is incident on a molecule with vibrational and rotational degrees of freedom. Because the internal molecular energy field changes during molecular vibration, the dipole induced by an incident wave changes accordingly. If the frequency of the incident wave is much higher than the molecular vibrational frequency, this change only becomes perceptible across many oscillations of the incident electric field. As a consequence, Rayleigh scattering which results from a constant oscillating dipole will be the dominant feature of the scattering spectrum. But a modulation of the dipole occurs with a period equal to the vibrational period of the molecule which results in the appearance of a much weaker spectral line shifted away from the Rayleigh frequency by a magnitude equal to the vibrational frequency. A similar spectral line appears due to the fluctuation in induced dipole during a molecular rotation. In this case, the period of change is half the period of one rotation since the molecular susceptibility towards dipole generation is equal for a molecule rotated 180°. The lines are therefore shifted away from the incident frequency by twice the rotational frequency. A quantum mechanical description of this phenomenon is based on the exchange of energy between an incident photon and rotational-vibrational molecular energy modes. Since a molecule can only exist in discrete vibrational and rotational energy states, it absorbs enough of the energy of an incident photon Io cause a transition to a upper energy state. The rcsulting "scattered" photon is less energetic and appears in the spectnnn shifted to a Iowcr frcqucncy. Or in the case th:_t themoleculiesalready in an upper state, it might release enough energy to the photon to return to a lower energy state. It then scatters a more energetic photon which appears in the spectrum shifted to a higher frequency. Wave function calculations s shot' that vibrational transitions are only allowed between adjacent levels, which means a change of one vibrational quantum number. A vibrational transition to a upper energy level, given by a quantum change of Av = +1, is referred to as Stokes scattering. A vibrational transition to a lower level, indicated by quantum change Av = -1, is referred to as anti-Stokes scattering. Similar calculations show that rotational transitions are only allowed between every other level, thus corresponding to a rotational quantum number change of two. Scattering caused by rotational transitions given by quantum jump AJ = +2 are referred to as S-branch scattering. A change of AJ = -2 yields the O-branch and the pure vibrational transition AJ = 0 gives the Q-branch. The energy of the scattered photon, and accordingly the spectral location at which it appears, depends on the amount of energy that has been exchanged with the molecule. This is equal to the difference between the two energy levels. Since these differences are unique for each molecule, the spectral line location can be used to identify the specific species that cause the scattering. 5.1 Theory Tile distribution of the molecules over all possible energy levels of a gas volume in thermal equilibrium is described by the Boltzmann distribution. Since each of these molecules contributes to the scattered light, many lines appear in the spectrum that are shil_ed from the incident light. Vibrational and rotational motions of the molecules interact, with the magnitude of interaction dependent on the specific levels. Thus the amount of energy that is needed for a pure vibrational transition (Q-branch) of a single molecule, depends on the initial rotational level of that molecule. As a consequence, each rotational level in a pure vibrational transition appears at a different wavelength in the spectrum. The strength of each of these lines is directly proportional to the number of molecules in that specific rotational energy mode. The resulting distribution reflects the initial rotational distribution, which is an accurate indication of the rotational temperature. Because the differences in line locations are small, high resolution is needed to resolve the distribution. As the result of poor resolution and line broadening, they usually show up in the spectrum as one single line. Spectral lines caused by simultaneous rotational-vibrational transitions (O- and S-branches) lie further apart. However, these transitions have a lower transition probability, given by the Placzek-Teller coefficients. Accordingly, they are easier to resolve but more difficult to detect. At higher temperatures, more vibrational levels become populated. As a result, upper vibrational transitions (v=l---_2, v=2---_3) cause new series of lines to appear, each representing the full rotational distribution of the particular vibrational state. The relative strength of each vibrational series reflects the molecular number density in that initial vibrational energy mode. Con_parison of the different vibrational series line intensities gives the vibrational temperature. Comparing the total integrated scattering intensity over all transitions to that of a reference gas at known number density and temperature yields the species number density. For a single component gas in thermal equilibrium, the total power scattered in the direction of observation in solid angle d_ as the result of one allo_vable transition can be given as:9 0: v'.J'--_.v'.J" whcrc I,_ _sthc incident irradiancc in W/m 2. n the molecular number density in m"3, and V_ is the scallering volume in cross c,,on, ,gdi v,re nlW,nhmcs2h,s ,r . ,soc,,onSv.,.rcOcc, Lu_LI 0, v'.J' _,_v',J" Ihc Boilzmann distribution of the molecular population in its initial undisturbed state. This function only depends on the lemperalurc, and is given by: