A Proposed Dynamic Pressure and Temperature Primary Standard.

Diatomic gas molecules have a fundamental vibrational motion whose frequency is affected by pressure in a simple way. In addition, these molecules have well defined rotational energy levels whose populations provide a reliable measure of the thermodynamic temperature. Since information concerning the frequency of vibration and the relative populations can be determined by laser spectroscopy, the gas molecules themselves can serve as sensors of pressure and temperature. Through measurements under static conditions, the pressure and temperature dependence of the spectra of selected molecules is now understood. As the time required for the spectroscopic measurement can be reduced to nanoseconds, the diatomic gas molecule is an excellent candidate for a dynamic pressure/temperature primary standard. The temporal response in this case will be limited by the equilibration time for the molecules to respond to changes in local thermodynamic variables. Preliminary feasibility studies suggest that by using coherent anti-Stokes Raman spectroscopy we will be able to measure dynamic pressure up to 108 Pa and dynamic temperature up to 1500 K with an uncertainty of 5%.

1. Introduction

With modern laser diagnostic techniques, it is possible to characterize the pressure (P) and temperature (T) of a gas at the molecular level. The measurement times for these techniques are such that the response to changes in T and P is limited only by the fundamental relaxation and transport processes of the molecular system. This provides the basis for a new approach to the calibration of transducers used in the measurement of dynamical P and T. The essence of dynamic calibrations is the determination of the time dependent response of the transducer, which requires, at a minimum, the application of a stimulus with known time dependence, i.e., a “standard” dynamic source.

If one were to rely on conventional sensors (whose response functions are not a priori known) to characterize the dynamic source, an inescapable circularity emerges from the preceding paragraph. Approaches to solution of this problem have traditionally [1] relied on some form of calculable source. In essence, this is a source some properties of which can be determined from accurate measurements, for example of quasi-static values of P and T and time rate of change of position, and whose time dependent P and T is then derived from an appropriate theoretical prescription, e.g., from hydrodynamics for sound propagation or fluid mechanics for shock waves (with appropriate equations of state for isentropic or adiabatic expansions). It must be recognized that every theory relies to some degree on idealizations and that any laboratory realization of a dynamic source is non-ideal. Thus, sources of dynamical P and T cannot be accurately known from theory alone; measurement of the “standard” source always is required.

Ideally, in the maintenance of national standards one seeks to relate the measured quantity to a constant of nature, maintained, for example, in the energy levels of an isolated atom or molecule. We are proposing this type of approach for the development of a “standard” source for dynamic P and T. The essence of our approach is to combine the very best in calculable generators, fast transducers, and high-speed digital data acquisition systems with a new, fundamental measurement approach. The latter relies on the use of laser-based diagnostic techniques, developed over the past 10 years, to determine the P and T of the dynamic system. The unique characteristics of the optical techniques are:

T and P are derived from measurement of the optical transitions between the atomic or molecular energy levels of the constituents of the dynamic source, i.e., the atoms or molecules are the fundamental transducers of the local P and T environment

optical measurements can be accomplished with a single laser pulse of nanosecond duration, with the consequence that the “response time of the transducer” reduces to the equilibration time for the atoms or molecules (in the interaction region) to respond to changes in the local thermodynamic variables

optical measurements can be accomplished within harsh environments by means of transmitted or reflected laser beams and, for multiple beam techniques, spatial resolution within the source volume can be defined by the regions of overlap of these beams, e.g., mm3 dimensions.

Our approach relies mainly on the use of nonlinear Raman spectroscopies, since these have consistently been shown to provide useful diagnostic spectra in very short times with high spatial resolution[2]. The spectrum determined with these nonlinear Raman approaches is the simplest, best understood, and most highly characterized of any optical diagnostic technique. Comparisons of spectra observed for systems in known (static) states of P and T with the predictions of theory provide a high degree of certainty in the use of these data for P and T measurement.

The purpose of this paper is to describe the nonlinear Raman optical measurements that can provide the new primary standard for dynamical P and T. This description will include information on the P and T dependence of the spectrum and a brief consideration of the important elements of a measurement system which can be applied to a dynamic source. For the purposes of this discussion we do not consider the dynamic source in any detail; however, the information we present is considered applicable to a suitably designed shock tube source. The state-of-the-art in optical diagnostics is now at a point where accurate measurement of such a “standard” dynamic source is possible. Accuracy limits of the order of 5% for the metrologically significant range of Pup to 108 Pa and T up to 1500 K appear achievable.

In the following, we begin with an operational description of the use of nonlinear Raman spectroscopy for P and T measurement, drawing from the already established data base on the T and P dependence of observed spectra. We will then outline the elements of a measurement system for a dynamic source. Areas needing significant instrumental development are included in this discussion. Some questions with regard to the P and T dependence of the Raman spectrum which need further fundamental research also are highlighted. The presentation style is intended to be descriptive rather than rigorous; for completeness, more detailed information on the T and P dependence of nonlinear Raman spectra is included in the Appendix.

2. Nonlinear Raman Optical Diagnostics

The proposed approach to measurement of the T and P of the dynamic source is coherent anti-Stokes Raman spectroscopy (CARS)[3]. In its most simple realizations this technique uses two lasers, termed the pump and the Stokes beams, whose frequency difference is selected to be in resonance with a pure vibrational transition of a diatomic gas molecule contained in the source medium. The nonlinear interaction of the electric fields of these lasers with the molecules of the source medium generates a third, laser-like beam, termed the anti-Stokes beam, which carries the information about the molecular system, in particular about its T and P.

A useful arrangement of these beams, which provides a high degree of spatial resolution, is shown in figure 1. In this configuration, termed BOX-CARS, the interaction volume is defined by the region of overlap of the two pump beams, k0 (derived from one laser source), and the Stokes beam, k. Sample volumes of millimeter and submillimeter dimension are remotely accessible in this arrangement.

Figure 1

The approximate geometrical arrangement of the pump, (subscript 0), Stokes (subscript s), and generated anti-Stokes (subscript as) beams in a CARS experiment. The sample region is at the intersection of the crossing beams. The phase matching condition for (folded) BOXCARS [11] is indicated.

The information about the local T and P environment of the molecules in the interaction volume is determined from the spectrum of the generated anti-Stokes beam (designated by kas in fig. 1). The spectrum is obtained by measuring the power of the anti-Stokes beam as a function of the frequency difference between the pump and Stokes lasers. Considering for the moment a static system, a spectrum can be obtained by measuring this power as we change the frequency difference between a tunable narrowband Stokes-laser and a fixed frequency narrowband pump-laser. Since this is simply a power measurement, we can essentially eliminate the use of traditional spectroscopic instruments (e.g., prism or grating spectrometers) and retrieve an undistorted measure of the information imparted by the molecular system. The narrowband Stokes and pump laser sources can readily be made of essentially delta-function-like bandwidth for this application.

The special conditions for static systems and narrowband lasers, described in the last paragraph, have been achieved in the laboratory in order to determine the fundamental molecular response, i.e., its spectrum, under known conditions of T and P. This has been accomplished for certain ranges of these variables and for a few selected molecular systems [4]. We will illustrate the basics of spectroscopic T and P determinations by describing spectra derived from these studies.

3. Temperature Dependence of CARS Spectra

In figure 2 we show CARS spectra of pure N2 as a function of T with the pressure held fixed at 1.0 atm. The horizontal axis is the frequency difference between the pump and Stokes lasers and the vertical axis is the (calculated) power in the anti-Stokes beam (in an unspecified arbitrary unit). These spectra are referred to as vibrational Q-branch spectra, because the optical transition involves a change only in the vibrational quantum state (quantum number ν) and no change in rotational state (i.e., no change in the rotational quantum number, J) [5]. The relevant states and modes of motion are schematically illustrated in figure 3.

Figure 2

Calculated CARS spectra for the N2 vibrational Q branch as functions of T for fixed P (= 1 atm). The horizontal axis is the frequency difference between the pump and Stokes lasers. The vertical axis is a measure of the CARS power. Although the absolute units of this power are arbitrary, the relative magnitudes as a function of T are accurately represented. Selected transitions and bands of the complete spectrum are indicated. In the bottom panel, the spectral region from Q(16) thru Q(0) is shown on an expanded frequency scale.

Figure 3

The solid curve schematically represents the potential energy of the ground electronic state versus internuclear separation. Vibrational and rotational energy levels (quantum numbers ν and J, respectively) also are indicated (not to scale). A molecular transition, Q(2), associated with the vibrational Q branch is indicated by the arrow.

Returning to figure 2, we see that there are many maxima in the power as a function of frequency difference. Each of these arises from a pure vibrational transition which originates in a different rotational state J. The vibrational frequency depends, to a small degree, on the rotational state because the rotation of the molecule results in a slight stretching of the bond length producing a small change in the forces binding the molecule and a concomitant change in the vibrational frequency. The vibrational frequency (we use the traditional spectroscopic unit cm−1, 1 cm−1≃30 GHz, in the figure) is approximately 2329.91 cm−1 for the J=0 rotationless state; the value decreases approximately according to 0.01738J(J+l), which places the J=10 transition at ≃2328.00 cm−1, less than a 0.1 % change. The strength (integrated area) of an individual transition is a function of the population difference between the initial (ν =0, J) and the final (ν = l, J) states of the transition. This population dependence in the relative strengths of the transitions as a function of the rotational level, J, is the basis for temperature determination, since for systems in thermodynamic equilibrium the state populations are functions only of the temperature. The vibrational Q-branch spectrum is very useful for measuring T because there are essentially no corrections to apply in order to relate the strength of a fully resolved transition to the population differene and therefore to the T [6]. As is seen in figure 2, the transitions for higher-J states increase in strength with increasing T; this simply mimics the population shifts to higher energy states associated with increasing T.

Strictly speaking, the temperature determined from the relative populations of the rotational levels should be called a “rotational temperature.” In like manner, a “vibrational temperature” can be determined from a measurement of the relative populations of the vibrational levels. We observe in the higher-T spectra in figure 2 that there are spectral maxima for transitions labeled ν = l→ν = 2. This is a vibrational Q branch which originates in the first excited vibrational level, ν = 1, and terminates in the second excited level, ν = 2. At sufficiently high T there is a significant population in the ν = 1 state and this transition becomes observable. This transition is totally analogous to that discussed above which initiated in the vibrational ground state, ν = 0. Comparison of the integrated areas between the 1→2 and 0→1 transitions gives a measure of the “vibrational temperature.” The rotational temperature of the vibrationally excited state also can be determined. The assumption of local thermodynamic equilibrium can thus be tested, since all levels should yield the same thermodynamic temperature in the equilibrium situation.

4. Pressure Dependence of CARS Spectra

First we note that the integrated area of the entire spectrum is a function of the number of molecules with which the intersecting laser beams interact. This feature often is used as a means of measuring species concentration in a diagnostic environment [2,3]. Each molecular species, e.g., N2, O2, CO, H2, etc., has a separate vibrational resonance because the resonant frequency is a sensitive function of the binding forces and the masses of the atoms comprising the molecule. Thus, intercomparison of the areas of these different Q branches can be used as a measure of the relative numbers of each molecule in the sample volume. Because it is very difficult to make accurate measurements of absolute intensity, it has been found that the absolute intensity of a transition is not a good measure of the density or pressure of a sample.

Fortunately, there are good measures of the pressure of a sample which can be recovered from the Q-branch spectrum of some diatomics. We illustrate these by spectra of the Q branch of D2, which has been extensively studied in our laboratory. Figure 4 presents CARS spectra calculated from results of these experimental studies. The features of primary interest in this figure are the resonance frequencies and widths of the transitions. Figure 4 illustrates the important fact that these transitions change their width (broaden) and their resonance frequency (shift to lower values) with increasing pressure. In the pressure range shown, this broadening and shifting are linear with pressure. For the Q(1) transition of D2, the broadening rate is 0.0012 cm−1/atm (we use the conventional half width at half peak height as our measure of width) whereas the shift rate is —0.0019 cm−1/atm, i.e., the line shifts more rapidly than it broadens. We see also that the transitions in the D2 Q-branch spectrum remain isolated, non-overlapped, up to 100 atm. We thus identify a very appealing approach to pressure measurement in that it is tied to the measurement of the frequency positions and widths of molecular transitions.

Figure 4

Calculated CARS spectra for the J = 0 and J = 1 transitions of the vibrational Q branch of pure D2 at T = 295 K. The zero of intensity for each P is shifted by an amount proportional to P. The dotted vertical lines indicate the resonance frequency of the Q(1) line at each P. The solid sloping line drawn through the vertical lines is thus an indication of the linear with P shift of the resonance frequency. At P = 0 the Q(0) line is at 2993.57 cm−1 with the higher J transitions at lower frequencies approximately by the amount 1.056J(J + 1).

The situation described above for the D2 Q branch should be contrasted with the observations of pressure broadening and shifting in the Q branch of N2. The broadening rates of the N2 transitions are typically 30–40 times larger than that of the D2 Q(1) transition. A shifting rate for a N2 transition is typically 10 times smaller than its broadening rate, thereby making accurate frequency determination less certain with increasing pressure. Additionally, we note that the interline spacing in the N2 Q branch is approximately l/60th that of D2. As a consequence, a transition such as Q(10) would broaden to overlap most of the other transitions shown in figure 2 at a pressure of 100 atm at room temperature. This overlap of transitions leads to important changes in the appearance of the spectrum which are discussed briefly in the Appendix. At this point it suffices to say that the N2 Q branch involves a complicated spectral distribution function which has the consequence that pressure determinations would generally be less reliable than those derived from the approach based on the Q branch of D2.

To this point we have restricted consideration to static systems and very high resolution measurements with narrowband lasers. We have displayed the results of careful measurements under known, static conditions of T and P which demonstrate that temperature can be determined by the measurement of the relative intensities of molecular transitions and that pressure can be determined from the positions and widths of molecular transitions. We turn now to discuss briefly the measurement approach required to characterize a dynamic system.

5. Single Shot Diagnostic Measurements

First we point out that we are interested in very rapid measurements, in the range of microseconds or less, in order to characterize the full bandwidth of transducers. Next we note, from the previous discussion, that a relatively large amount of spectral information must be acquired in order to characterize T and P. These requirements of time and spectral range (at relatively high resolution) result in the selection of a technique in which all the spectral elements are measured at once, i.e., a single shot measurement. Single shot measurements are the strong suit of nonlinear Raman diagnostics because we can employ pulsed laser sources and obtain high optical power levels which, by virtue of the inherent nonlinear response of these techniques, results in the generation of strong, readily detected, CARS signals.

The most straightforward approach to single shot measurement is to replace the scanned, narrowband Stokes laser, considered above, with a broadband Stokes laser whose frequency width is sufficient to provide the entire Raman spectrum of interest in a single shot. For example, the entire 60 cm−1 region in figure 2 would be measured. We note that a similar spectral bandwidth would be required for temperature measurement via the D2 Q branch because of the larger interline spacing in the D2 Q branch, cf. figure 4. The measurement of the spectrum then requires multiple detectors coupled to an instrument which disperses different frequency components in the anti-Stokes beam to the separate spatial locations of the detectors. This combination of a spectrometer and detector we refer to as a multichannel system.

6. Instrument Development

The basic tools for these single shot measurements do exist. The necessary laser systems have been realized in the research laboratory. Adaptation of these to specific diagnostic situations will be required. Similarly, a large variety of spectrometer/detector systems are available and this should allow for reasonably straightforward realization of the specific systems required for our application. Because of the requirements for large spectral ranges and high resolution, noted above, some multiplexing of multichannel systems will be necessary. More details on the instrumentation are discussed in the following.

6.1 Lasers

The basic requirements for the laser sources are well within the state-of-the-art of laser technology. The combination of a single-frequency, pulsed Nd:YAG oscillator with a series of amplifiers and a frequency doubling crystal will provide an adequate Raman pump laser. This laser also will serve as a pumping source for amplifying the required broadband dye laser for the Stokes beam. The oscillator for this Stokes laser will require special design, tailored specifically for the transitions selected for optical characterization of the dynamic source.

6.2 Spectrometer/Detector System

This system also will be tailored to the transitions chosen. It will be unique in that it will have a high resolution (0.001−0.05 cm−1) system for frequency and linewidth determination along with the more conventional low resolution (0.1−0.5 cm−1) channel for temperature determinations. A high resolution multichannel system has not yet been attempted in diagnostic applications. Our ability to selectively add a component such as D2 to serve as a pressure transducer in the working medium allows us to consider this approach. Incorporating a significant fraction of N2 in this medium allows for a number of independent checks on the temperature and on the state of thermodynamic equilibrium. Depending on the pressure range, we may be able to determine the temperature from the D2 spectrum. An approach under consideration is to simultaneously measure the Q branch of D2 for pressure determination and the pure rotation S branch (transitions within a vibrational state involving a change of +2 in the rotational quantum number only) of N2 to determine the temperature [7].

The discussion in the last few paragraphs indicates one area of the research necessary to develop a standard source of dynamic P and T. There are additional fundamental studies required to assure accurate knowledge of the T and P dependence of CARS spectra and ultimately to enable the selection of a measurement approach optimized for application to the dynamic source. We turn now to a brief discussion of these.

7. Fundamental Studies

The most important fundamental question to be answered in this work is the dependence of the spectra of our specific dynamic system on T and P. What is unique and new about the system we propose is that it is by necessity a mixed gas system, for example D2 contained at low concentrations in N2. Except for our recent studies (see ref. [4] and below), there is relatively little work on mixed gas systems. Additionally, we are interested in these systems over quite large ranges of P and T.

There are a number of motivations for the selection of a mixed gas system. First, no one molecule presents all the necessary spectral characteristics which allow an adequate characterization of both T and P. Of the many single component systems considered, D2 comes closest to fulfilling this requirement. D2 provides a good measure of P because it has very narrow, isolated lines. This characteristic makes it difficult to simultaneously measure the relative strengths of all the Q-branch transitions to obtain an accurate measure of T. Furthermore, it is much harder (i.e., it takes much longer) to establish thermodynamic equilibrium for D2 than it is for N2 because of the much larger gaps between the energy levels in D2. Large departures from thermodynamic equilibrium can degrade the performance of a dynamic source and the accuracy of the spectroscopic measurements themselves. The use of pure D2 has a number of other limitations; for example, a shock tube dynamic source would be very limited in dynamic range by the use of such a light gas. We note also that the cost of running a pure D2 system would be prohibitive, as also might the potential safety hazards associated with pure D2 at high pressures or high temperatures.

The use of a mixed gas system is potentially an advantage for pressure measurements. This statement is derived from our observation that mixed gas systems such as D2:X, with X = He, Ar, and N2, have a larger shift to width ratio than pure D2. There remain some fundamental questions in this regard, however, because we have observed line shape asymmetries in the spectrum of D2:Ar and H2:Ar [8]. We also know that these asymmetries are functions of T. These observations need to be fully understood in order to reliably calibrate P; furthermore, these studies must be extended to the D2:N2 system. An adequate spectral model and the pertinent molecular parameters must be determined and tested over the complete range of P and T of interest for the dynamic source.

8. Characterization of the Dynamic Source

8.1 Accuracy Levels in T and P

The accuracy level for dynamic pressure measurements using the proposed optical/molecular approach is estimated to be approximately 5%. This estimate is based on our ability to measure the widths and shifts of the D2 Q-branch transitions. For the mixed gas systems proposed as standards, the lines shift at least as rapidly than they broaden. Thus our accuracy goals, imply that we can locate the line center to approximately l/10–l/20th of the line width and that we can make line-width measurements at approximately the 5% level. Achieving these measurement accuracies in a single shot experiment is one of the development goals of this work.

Accuracy in temperature measurements depends on our ability to measure the relative intensities of spectral features associated with different molecular quantum states. In practice, this comes down to comparing an observed spectrum to predicted spectra such as those shown in figure 2. In a single shot experiment this comparison must include the effects of a finite-resolution, spectroscopic instrument function. The accuracy of these comparisons for single shot measurements can be of the order of 5% [3,4]. This is the target level of accuracy for temperature measurements on the dynamic source. The simultaneous and independent measurement of pressure will help eliminate systematic errors in either of these measurements.

8.2 Temporal Evolution

It is important to recognize that the nonlinear laser diagnostic techniques which allow us to obtain a 10 ns “snap-shot” of the local T and P environment of a molecule, can not easily be extended to provide a continuous-in-time measurement (“movie”) of the evolution of T and P in a dynamic source. These techniques do provide a primary standard which can be used to calibrate measurement approaches which yield good relative measures of T and P with rapid and continuous temporal response. Until this point, we have had no primary standard which could assure the accuracy of these measurements. The primary standard would be used to accurately pin the P and T values at representative points in the temporal evolution of the source during each calibration run. Further, the nonlinear optical techniques would be used to hold the absolute calibration of the source over long periods of use.

9. Conclusions

We have described a new approach to the measurement of T and P of a dynamic source. The measurement is based on the fundamental properties of molecules, specifically on the energies and populations of the vibrational and rotational levels of diatomic gas molecules. The nonlinear optical technique, coherent anti-Stokes Raman spectroscopy (CARS), has been proposed as the method for acquiring the information from the molecular system. This technique has the advantages of very rapid (nanosecond) measurement times, small sampling volumes, and a well understood and verified T- and P-dependent spectrum. The development of this measurement system to provide a reference standard for dynamic calibrations of T up to 1500 K and P up to 108 Pa with a 5% accuracy appears highly feasible.