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Lineshifts in the Fundamental Band of CO: Confirmation of Experimental Results for N2 and Comparison with Theory
Journal of Molecular Spectroscopy 196, 290 –295 (1999) Article ID jmsp.1999.7866, available online at http://www.idealibrary.com on
Lineshifts in the Fundamental Band of CO: Confirmation of Experimental Results for N 2 and Comparison with Theory Caiyan Luo, R. Berman, Adriana Predoi-Cross, J. R. Drummond, and A. D. May Department of Physics, University of Toronto, Toronto, Canada M5S 1A7 E-mail: [email protected] Received January 26, 1999; in revised form March 25, 1999
We have used a three-channel version of a tunable difference frequency laser spectrometer to measure the collisionally induced lineshifts at room temperature for 26 lines of the fundamental band of CO perturbed by nitrogen. Each lineshift was obtained directly by comparing the line center positions of two simultaneous recordings, one for a pressure-shifted line, and the other for the same line in pure CO line at very low pressure. The experimental results are found to be in complete agreement with earlier measurements and confirm that shifts as small as 3 MHz may be measured in such a system. Our results are compared with theoretical calculations. The part of the shifting coefficient antisymmetric with respect to a change in sign of the line number m, is in disagreement with the calculations. © 1999 Academic Press INTRODUCTION
Measurements of lineshifts of the fundamental band of CO are quite difficult, since they are generally much smaller than the corresponding collisional widths. Relative to the number of papers published on the widths, there are very few papers on the measurements of the lineshifts in CO. There are only a total of 19, if one includes measurements of the fundamental, the first and the second overtone bands, and a variety of perturbers (1–19). Of interest to atmospheric physics is the broadening and shifting of the fundamental of CO perturbed by N 2. Here the shifts are very small and it is not surprising that meaningful measurements of the shifts have only been made in the last few years using laser diodes (11, 14, 17, 18), difference-frequency spectrometers (15, 16), or high-resolution Fourier transform spectrometers (10, 12, 14). Even with these advanced instruments, the reported error bars often reach 100% and there is some disagreement between various authors. With the increasing level of sophistication in remote sensing instrumentation, there is a need for ever more accurate spectroscopic data relevant to the atmosphere. Nowadays, it is more or less routine to measure broadening coefficients that are accurate at the 1% level or better. Since the shifts are typically a factor of 10 smaller than the widths, for a consistent treatment, any model or code for atmospheric absorption which uses widths measured to the 1% level would require shifts to be known at the 10% level. Such accuracy is obtainable with laser diodes or difference-frequency spectrometers, the latter being preferred for their ability to study many lines within a band. In a recent publication (16), we reported measurements of the shifting coefficient for 26 lines of the fundamental band of CO perturbed by N 2. One of the objectives of the present study was
an independent confirmation of those results. The second objective was to compare our results with theoretical calculations of the shifts. EXPERIMENTAL DETAILS
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Our two-channel tunable difference-frequency spectrometer has been described in detail by Duggan et al. (20). Briefly, the visible beam from a frequency-stabilized dye laser (Coherent 699-29) was mixed, in an antireflection-coated LiIO 3 crystal, with the beam from a frequency-stabilized single-frequency argon laser. In our system it is possible to generate a difference frequency from 2.5 to 5.5 mm. The resolution of the spectrometer is determined by the combined jitter of the two lasers and is better than 2 MHz (7 3 10 25 cm 21). The frequency changes in both the argon ion laser and dye laser were measured using a frequency-stabilized helium–neon laser and a temperaturestabilized confocal Fabry–Perot interferometer. Essentially, the interferometer provides the relative frequency scale during a scan over a spectral profile, and in the past the frequencystabilized helium–neon laser has been used to relate the frequencies of the line centers, measured at different pressures. We have changed this latter technique. In the present setup, the infrared beam was split into three beams using wedged germanium beamsplitters. One beam (approximately 32 of the incoming IR radiation) was sent through an absorption cell containing the CO–N 2 mixture under study. As usual, a second beam was sent directly to an IR detector for intensity normalization. The third beam was sent through a second absorption cell containing pure CO at 0.2 Torr (26 Pa). The signal from this cell provides a reference frequency for each scan, at or very close to the frequency of the free molecule. This reference
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FIG. 1. (a) Wide-profile, absorption curve for the R(18) line of CO perturbed by 92 kPa of N 2; narrow profile, same line for pure CO at 0.026 kPa. (b) Expanded section of profiles to illustrate the center positions as determined by a fitting routine. The measured shift exceeds the resolution limit of the spectrometer by a factor of 20.
cell technique is not new (10, 11, 14). Its use has two distinct advantages. First, it allows us to verify that the shift is proportional to the pressure. While this has always been assumed to be valid, recent Raman observations (21) have shown that it need not be so. Second, if the shifts are proportional to pressure, then a single careful measurement at high pressure can be used to determine the shifting coefficient. Further experimental details now follow. All of the LN 2-cooled InSb detectors were contained in dewars equipped with Brewster angle windows. The signals from each detector were sent to lock-in amplifiers, and then processed by a computer. The length of the experimental cell was 107.1 cm, and the length of the reference cell was 15 cm. Calcium fluoride windows were mounted on both cells at the Brewster angle. The pressure of the gas in the cell was measured using a MKS 120A pressure head with 0.05% absolute error. For each spectral line, a baseline was taken when the cell was empty. A typical noise in the baseline was 1 part in 500. The baseline was fitted using a third-order Chebyshev polynomial. All measurements were made at a fixed temperature within 1 K of 298 K.
The frequency for each line center was obtained by fitting the experimental spectra with different lineshape models. Which model was used is not important here. What is important, from the point of view of measuring the shifts, is that the reference line was assumed to be symmetric about the peak. On the other hand, the spectral profiles were decomposed into a strong symmetric component and a weak asymmetric component, assumed to arise from line mixing (14, 22). It is the frequency shift of the strong component that is reported here. This is equivalent to the frequency shift of the “center of gravity” of the line. ANALYSIS AND RESULTS
Figure 1a shows a typical pair of absorption profiles recorded simultaneously. The top trace is the R(18) line at 92 kPa, of a 0.1% mixture of CO in N 2. The bottom trace in Fig. 1a is the corresponding reference trace. In Fig. 1a the shift is too small to detect by eye and is considerably smaller than the width of the high-pressure line. Figure 1b shows the central portion of Fig. 1a on a very expanded frequency scale. Here the
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individual experimental points are visible. The dashed vertical lines locate the centers as reported by the fitting routine. Their relative displacement is the shift at this pressure. Although small, the negative shift is readily measured. Figure 2 shows a plot of the shift, D, of the R(18) line as a function of pressure. The “1 3 s” error bars are those obtained by repeating the measurement, at each density, a total of 10 times. Also shown in the figure is the best fit straight line to the measured shifts. Within the error limits, the shift was proportional to the pressure. Bouanich et al. (15) did detect small zero density offsets, which were sufficiently small to ignore in determining the shifting coefficients. (The slope of the line in Fig. 2 is the lineshift coefficient, d 0.) We have chosen to ignore any possible small zero density offset and treat the shift as being proportional to the pressure. The shifting coefficients for the rest of the lines were determined from measurements at a single high pressure near 70 kPa. These are listed in Table 1, for many lines in the P and R branch. The accuracy is comparable to the accuracy reported in (15) for the shifting coefficients of the first overtone of CO perturbed by N 2. Those results were obtained with a system very similar to the Toronto spectrometer. The results of the present study are plotted in Fig. 3a as a function of the line number, m, where m 5 2J for the P branch, and J 1 1, for the R branch. Also shown in the figure are our previous results (16). In that earlier experiment it was assumed that the conventional 1s error bars for the measured shifts were twice s (statistical), the error bars reported by the fitting routine when fitting a single spectral profile. The present results show that the 1s errors were correctly estimated since the two sets of results are essentially in complete agreement. The very small, systematic difference (less than the width of the error bars) evident in Fig. 3a may be attributed to the 3 K difference in temperature at which the two experiments were carried out. This would require the shifts near 300 K to become slightly more negative with increasing
FIG. 2. Shift of the R(18) line of CO as a function of pressure. The straight line fitted without the low (zero) pressure point is seen to pass well within experimental error, through the origin.
TABLE 1 Shifting Coefficients in Units of 10 23 cm 21 atm 21 for CO Perturbed by N 2 at 298 K
Note. The experimental error bars are about 0.1 3 10 23 cm 21 atm 21.
temperature. This would be in mild conflict with the less precise results of Drascher et al. (18), unless N 2, like O 2, shows a slightly more negative shift with increasing temperature near 300 K (see (18)). With an average shifting coefficient of 2 3 10 23 cm 21/atm and an estimated 1s accuracy of 0.1 3 10 23 cm 21/atm, the typical accuracy of the present results is 5%. This accuracy and the accuracy reported in (15) for the first overtone are certainly adequate for present day models of atmospheric absorption by CO. Figure 3b compares the present results with the lineshift coefficients for the fundamental band as measured by Fourier transform infrared spectroscopy (FTIR). While Fourier transform spectroscopy has the ability to measure many lines in a band, as does difference frequency spectroscopy, it is evident from the figure that technical problems (phase errors) make it difficult to measure very small shifts using FTIR spectroscopy. Note, the error bars for our results are less than half of the smallest error bars shown in Fig. 3b. Nevertheless, within the stated error bars, the FTIR results are in agreement with the present results. Figure 3c compares the present results with the few lines measured using laser– diode spectroscopy. Similar comments may be made about Fig. 3c as were made about Fig. 3b. However, as is well known, diode lasers do not lend themselves readily to the measurement of many lines in a band. It is usually the laser that dictates the lines that may be studied. We believe the present results illustrate that difference-frequency spectroscopy (with some averaging) may be used to determine IR shifts accurate to 10 24 cm 21 (3 MHz) and shifting coefficients of 10 23 cm 21/atm to 10% or better, for many lines in a band.
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theory of Robert and Bonamy (25) extends Herman’s calculations to higher orders by neglecting terms to an odd power in the interaction and assuming that higher order, even terms, are expressible as the square, cube, etc. of the second-order term. Now it is a simple matter to decompose both the theoretical and the experimental shifts (as a function of m) into a symmetric component, d s , and antisymmetric component, d as . Figure 4b compares the experimental values with the theoretical values of Bouanich et al. This comparison indicates that it is the antisymmetric part of the shifting coefficient (squares in Fig. 4b) that is most in error, the computed variation with line number being much greater than the observed variation. Consequently it is the angle dependent (anisotropic) part of the potential that seems to be wrong. This poses a serious problem since it is the anisotropic part of the interaction that plays a major role in determining the width of the lines, and the widths were correctly predicted by the potential used by Bouanich et al. (15). The isotropic part of the potential does not enter into the width and its strength had already been chosen by Bouanich et al. to bring the shifts, overall (primarily d s ) into agreement with experiment. This suggests that the problem does not lie in the choice of potential but perhaps in the theoretical treatment (25) itself. We now consider several resolutions to the conflict raised in the last paragraph. It is natural to inquire if the problem is FIG. 3. (a) Shifting coefficient as a function of line number, m; open points, present results; solid points, the results of Sinclair et al. (16). (b) A comparison with present results, open circles; with FTIR measurements, solid points from Ref. (14); solid diamonds from Ref. (10), and solid triangles from Ref. (12). (c) A comparison of the present results, open circles; with diode measurements, solid squares from Ref. (17); solid triangle from Ref. (11), and solid diamond from Ref. (14).
COMPARISON WITH THEORETICAL CALCULATIONS
Figure 4a shows a plot of the present results and the calculations of Bouanich et al. (14). The potential parameters in the calculations had previously been optimized to fit overall, the measured shifts across the entire band (14). We see that the agreement between the calculated and measured shifts is only semiquantitative. We now try to identify the reason for the low quality of fit between theory and experiment. In an earlier calculation using a less developed version of the Anderson–Tsao–Curnutte theory (23, 24) than that (25) used by Bouanich et al., Herman (26) established a number of rules concerning the widths and shifts of P- and R-branch lines. Neglecting rotation–vibration interaction, the shift, d iso, coming from the isotropic part of the interaction, arises in first order and is symmetric in line number [ d iso(2m) 5 d iso(m)], while the shift, d aniso, coming from the anisotropic part of the interaction, arises in second order and is antisymmetric in line number [ d aniso(2m) 5 2 d aniso(m)]. The calculations of Bouanich et al. satisfy these symmetry rules as they must since the
FIG. 4. (a) A comparison of the present results, open circles; with theoretical results from Ref. (14), solid circles. (b) As in (a) but decomposed into d s (triangles), a part symmetric with respect to a change in sign of the line number, m, and an antisymmetric part, d as (squares). Open symbols from present experiment, closed symbols from Ref. (14).
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FIG. 5. and (15).
LUO ET AL.
A plot of the Herman factor as a function of the absolute value of the line number; circles from present results, squares from Bouanich et al. (14)
created by the neglect of rotation–vibration interaction, a feature common to many of the theories. If the isotropic forces stretch the molecules, thus lowering the vibrational frequency of an anharmonic oscillator, there will also be a change in the rotational constant B and an associated shift in the lines that is asymmetric in m. However, following the simple calculation given in (27), it is possible to show such an effect is totally negligible for the CO molecule. There is a combined test of the theoretical treatment of the widths and the symmetric part of the shifts, d s . Herman’s third rule (see page 272 in (26)) may be written @ d 0 ~2m! 1 d 0 ~m!#@ g 0 ~2m! 1 g 0 ~m!# 3/ 2 5 2 d s @ g 0 ~2m! 1 g 0 ~m!# 3/ 2 5 H,
[1]
where d 0 and g 0 are the shifting and broadening coefficients and H the “Herman” constant. This was developed using a rigid cutoff radius b 0 , rather than curved trajectories. While a more modern treatment may lead to H not being totally independent of m, we may still anticipate that, if not the value of H, then at least the functional form of Eq. [1], is insensitive to the details of the calculation. Thus we speculate that a comparison of H ca1 to H exp is a combined test of the theoretical handling of the first-order shift and the width, a second-order result. In Fig. 5 we show a plot of H versus m as determined from the present measurements of the shifts and the experimental widths of Sinclair et al. (16). 1 We see that the experimental value of H is almost constant within the rather large 1 The present measurements of the widths are in agreement with those of Sinclair et al. (16). They will be presented in a separate publication dealing with the temperature dependence of the broadening coefficients.
error bars and more important that the measured and the theoretical values of H are in agreement except for low values of m. The latter point only reflects the fact that the calculated symmetric component does not accurately reproduce the experimental results for low values of m, a problem associated with the exact form of the short-range forces. It thus appears that the theoretical treatment of the shift coming from the isotropic part of the interaction, and of the widths, is reasonable. This brings us back to the treatment of the asymmetry in the shifts as a function of line number. As Herman points out (26), a contribution from the angle-dependent interactions to a shift is zero in Anderson’s original theory, where commutivity of the time development matrices was assumed. This is a subtle point and presumably must be handled carefully in any theory, either of infinite or finite order in the interaction. Unfortunately, the asymmetry of the shifts as a function of line number is not a unique test of the importance of noncommutivity. As shown by Boulet and Robert (28), an infinite order calculation based on the assumption of commutivity can also lead to asymmetry in the shifts. Thus the problem of commutivity and order of interaction are mixed, both contributing to the asymmetry in the shifts. Perhaps the relative importance of the two contributions (in any particular case) can only be estimated by carrying out exact calculations to third or even fourth order in the interaction. At least, the experimental evidence suggests the treatment of commutivity and the extension beyond second order should be reexamined. Such questions play an insignificant role in the widths and gives one hope that the agreement between calculated and computed widths was not just fortuitous.
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LINESHIFTS IN CO–N 2
CONCLUSIONS
We have used a three-channel version of our differencefrequency spectrometer to confirm earlier measurements of the frequency shifts in the fundamental band of CO perturbed by N 2. When taken in conjunction with the measurements of Bouanich et al. (15), it establishes that frequency-shifting coefficients, precise to 60.1 3 10 23 cm 21/atm may be measured using difference-frequency laser spectroscopy. For consistency, this accuracy is required if CO–N 2 absorption profiles are to be reconstructed for atmospheric modeling. We have shown that there is some disagreement between the measured and calculated shifting coefficients and have tentatively identified the theoretical treatment of the shifting coming from the anisotropic part of the interaction between CO and N 2 molecules as being the source of the problem. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council of Canada, COMDEV, Bomem Inc., the Atmospheric Environment Service, the University of Toronto Research Fund, the Canadian Space Agency, and the Industrial Research Chair in Atmospheric Remote Sounding from Space. We thank J.-P. Bouanich for providing us with numerical values of the calculated broadening and shifting coefficients for CO–N 2. A.D.M. acknowledges a number of fruitful correspondences with C. Boulet.
REFERENCES 1. D. H. Rank, P. Eastman, B. S. Rao, and T. A. Wiggens, J. Mol. Spectrosc. 10, 34 –50 (1963). 2. J-P. Bouanich, M. Larvor, and C. Haeusler, C. R. Seances Acad. Sci. Paris B 269, 1238 –1241 (1969). 3. J-P. Bouanich, M. Larvor, and C. Haeusler, C. R. Seances Acad. Sci. Paris B 270, 396 –399 (1970). 4. J-P. Bouanich, M. Larvor, and C. Haeusler, C. R. Seances Acad. Sci. Paris B 270, 1220 –1223 (1970). 5. J-P. Bouanich, J. Quant. Spectrosc. Radiat. Transfer 12, 1609 –1615 (1972).
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6. J-P. Bouanich and C. Brodbeck, J. Quant. Spectrosc. Radiat. Transfer 13, 1–7 (1973). 7. J.-P. Bouanich, Can. J. Phys. 61, 919 –922 (1983). 8. A. J. Mannucci, J. Chem. Phys. 95, 7795–7805 (1991). 9. F. Thibault, J. Boissoles, R. Le Doucen, R. Farrenq, M. Morillon-Chapey, and C. Boulet, J. Chem. Phys. 97, 4623– 4632 (1992). 10. A. Hamdouni, A. Barbe, J.-J. Plateaux, V. Langlouis, V. Dana, J.-Y. Mandin, and M. Badaoui, J. Quant. Spectrosc. Radiat. Transfer 50, 247–255 (1993). 11. N. Anselm, K. M. T. Yamada, R. Schieder, and G. Winnewisser, J. Mol. Spectrosc. 161, 284 –296 (1993). 12. J.-Y. Mandin, V. Dana, M. Badaoui, A. Barbe, A. Hamdouni, and J.-J. Plateaux, J. Mol. Spectrosc. 164, 328 –337 (1994). 13. F. Thibault, R. Le Doucen, J. P. Bouanich, and C. Boulet, J. Mol. Spectrosc. 171, 576 –578 (1995). 14. J.-P. Bouanich, F. Rachet, and A. Valentin, J. Mol. Spectrosc. 178, 157–164 (1996). 15. J.-P. Bouanich, D. Bermejo, J. L. Domenech, R. Z. Martinez, and J. Santos, J. Mol. Spectrosc. 179, 22–31 (1996). 16. P. M. Sinclair, P. Duggan, R. Berman, A. D. May, and J. R. Drummond, J. Mol. Spectrosc. 181, 41– 47 (1997). 17. B. Sumpf, J. P. Burrows, A. Kissel, H-D. Kronfeldt, O. Kurtz, I. Meusel, J. Orphal, and S. Voigt, J. Mol. Spectrosc. 190, 226 –231 (1998). 18. T. Drascher, T. F. Giesen, T. Y. Wang, N. Schmucker, R. Schieder, G. Winnewisser, P. Joubert, and J. Bonamy, J. Mol. Spectrosc. 192, 268 –276 (1998). 19. V. M. Devi, D. C. Benner, M. A. H. Smith, and C. P. Rinsland, J. Quant. Spectrosc. Radiat. Transfer 60, 815– 824 (1998). 20. P. Duggan, P. M. Sinclair, M. P. LeFlohic, J. W. Forsman, R. Berman, A. D. May, and J. R. Drummond, Phys. Rev. A 48, 2077–2083 (1993). 21. S. H. Fakhr-Eslam, G. Sheldon, J. R. Drummond, and A. D. May, Spectral Line Shapes, in press; S. H. Fakhr-Esalm, Ph.D. thesis, University of Toronto, 1999. 22. P. W. Rosenkranz, IEEE Trans. Antennas Propag. AP-23, 498 –505 (1975). 23. P. W. Anderson, Phys. Rev. 76, 647– 661 (1949). 24. C. J. Tsao and B. Curnutte, J. Quant. Spectrosc. Radiat. Transfer 2, 41–91 (1962). 25. D. Robert and J. Bonamy, J. Phys. 40, 923–943 (1979). 26. R. Herman, Phys. Rev. 132, 262–275 (1963). 27. A. D. May, V. Degen, J. C. Stryland, and H. L. Welsh, Can. J. Phys. 39, 1769 –1783 (1961). 28. C. Boulet and D. Robert, Chem. Phys. Lett. 60, 162–168 (1978).
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Lineshifts in the Fundamental Band of CO: Confirmation of Experimental Results for N 2 and Comparison with Theory Caiyan Luo, R. Berman, Adriana Predoi-Cross, J. R. Drummond, and A. D. May Department of Physics, University of Toronto, Toronto, Canada M5S 1A7 E-mail: [email protected] Received January 26, 1999; in revised form March 25, 1999
We have used a three-channel version of a tunable difference frequency laser spectrometer to measure the collisionally induced lineshifts at room temperature for 26 lines of the fundamental band of CO perturbed by nitrogen. Each lineshift was obtained directly by comparing the line center positions of two simultaneous recordings, one for a pressure-shifted line, and the other for the same line in pure CO line at very low pressure. The experimental results are found to be in complete agreement with earlier measurements and confirm that shifts as small as 3 MHz may be measured in such a system. Our results are compared with theoretical calculations. The part of the shifting coefficient antisymmetric with respect to a change in sign of the line number m, is in disagreement with the calculations. © 1999 Academic Press INTRODUCTION
Measurements of lineshifts of the fundamental band of CO are quite difficult, since they are generally much smaller than the corresponding collisional widths. Relative to the number of papers published on the widths, there are very few papers on the measurements of the lineshifts in CO. There are only a total of 19, if one includes measurements of the fundamental, the first and the second overtone bands, and a variety of perturbers (1–19). Of interest to atmospheric physics is the broadening and shifting of the fundamental of CO perturbed by N 2. Here the shifts are very small and it is not surprising that meaningful measurements of the shifts have only been made in the last few years using laser diodes (11, 14, 17, 18), difference-frequency spectrometers (15, 16), or high-resolution Fourier transform spectrometers (10, 12, 14). Even with these advanced instruments, the reported error bars often reach 100% and there is some disagreement between various authors. With the increasing level of sophistication in remote sensing instrumentation, there is a need for ever more accurate spectroscopic data relevant to the atmosphere. Nowadays, it is more or less routine to measure broadening coefficients that are accurate at the 1% level or better. Since the shifts are typically a factor of 10 smaller than the widths, for a consistent treatment, any model or code for atmospheric absorption which uses widths measured to the 1% level would require shifts to be known at the 10% level. Such accuracy is obtainable with laser diodes or difference-frequency spectrometers, the latter being preferred for their ability to study many lines within a band. In a recent publication (16), we reported measurements of the shifting coefficient for 26 lines of the fundamental band of CO perturbed by N 2. One of the objectives of the present study was
an independent confirmation of those results. The second objective was to compare our results with theoretical calculations of the shifts. EXPERIMENTAL DETAILS
290
Our two-channel tunable difference-frequency spectrometer has been described in detail by Duggan et al. (20). Briefly, the visible beam from a frequency-stabilized dye laser (Coherent 699-29) was mixed, in an antireflection-coated LiIO 3 crystal, with the beam from a frequency-stabilized single-frequency argon laser. In our system it is possible to generate a difference frequency from 2.5 to 5.5 mm. The resolution of the spectrometer is determined by the combined jitter of the two lasers and is better than 2 MHz (7 3 10 25 cm 21). The frequency changes in both the argon ion laser and dye laser were measured using a frequency-stabilized helium–neon laser and a temperaturestabilized confocal Fabry–Perot interferometer. Essentially, the interferometer provides the relative frequency scale during a scan over a spectral profile, and in the past the frequencystabilized helium–neon laser has been used to relate the frequencies of the line centers, measured at different pressures. We have changed this latter technique. In the present setup, the infrared beam was split into three beams using wedged germanium beamsplitters. One beam (approximately 32 of the incoming IR radiation) was sent through an absorption cell containing the CO–N 2 mixture under study. As usual, a second beam was sent directly to an IR detector for intensity normalization. The third beam was sent through a second absorption cell containing pure CO at 0.2 Torr (26 Pa). The signal from this cell provides a reference frequency for each scan, at or very close to the frequency of the free molecule. This reference
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FIG. 1. (a) Wide-profile, absorption curve for the R(18) line of CO perturbed by 92 kPa of N 2; narrow profile, same line for pure CO at 0.026 kPa. (b) Expanded section of profiles to illustrate the center positions as determined by a fitting routine. The measured shift exceeds the resolution limit of the spectrometer by a factor of 20.
cell technique is not new (10, 11, 14). Its use has two distinct advantages. First, it allows us to verify that the shift is proportional to the pressure. While this has always been assumed to be valid, recent Raman observations (21) have shown that it need not be so. Second, if the shifts are proportional to pressure, then a single careful measurement at high pressure can be used to determine the shifting coefficient. Further experimental details now follow. All of the LN 2-cooled InSb detectors were contained in dewars equipped with Brewster angle windows. The signals from each detector were sent to lock-in amplifiers, and then processed by a computer. The length of the experimental cell was 107.1 cm, and the length of the reference cell was 15 cm. Calcium fluoride windows were mounted on both cells at the Brewster angle. The pressure of the gas in the cell was measured using a MKS 120A pressure head with 0.05% absolute error. For each spectral line, a baseline was taken when the cell was empty. A typical noise in the baseline was 1 part in 500. The baseline was fitted using a third-order Chebyshev polynomial. All measurements were made at a fixed temperature within 1 K of 298 K.
The frequency for each line center was obtained by fitting the experimental spectra with different lineshape models. Which model was used is not important here. What is important, from the point of view of measuring the shifts, is that the reference line was assumed to be symmetric about the peak. On the other hand, the spectral profiles were decomposed into a strong symmetric component and a weak asymmetric component, assumed to arise from line mixing (14, 22). It is the frequency shift of the strong component that is reported here. This is equivalent to the frequency shift of the “center of gravity” of the line. ANALYSIS AND RESULTS
Figure 1a shows a typical pair of absorption profiles recorded simultaneously. The top trace is the R(18) line at 92 kPa, of a 0.1% mixture of CO in N 2. The bottom trace in Fig. 1a is the corresponding reference trace. In Fig. 1a the shift is too small to detect by eye and is considerably smaller than the width of the high-pressure line. Figure 1b shows the central portion of Fig. 1a on a very expanded frequency scale. Here the
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individual experimental points are visible. The dashed vertical lines locate the centers as reported by the fitting routine. Their relative displacement is the shift at this pressure. Although small, the negative shift is readily measured. Figure 2 shows a plot of the shift, D, of the R(18) line as a function of pressure. The “1 3 s” error bars are those obtained by repeating the measurement, at each density, a total of 10 times. Also shown in the figure is the best fit straight line to the measured shifts. Within the error limits, the shift was proportional to the pressure. Bouanich et al. (15) did detect small zero density offsets, which were sufficiently small to ignore in determining the shifting coefficients. (The slope of the line in Fig. 2 is the lineshift coefficient, d 0.) We have chosen to ignore any possible small zero density offset and treat the shift as being proportional to the pressure. The shifting coefficients for the rest of the lines were determined from measurements at a single high pressure near 70 kPa. These are listed in Table 1, for many lines in the P and R branch. The accuracy is comparable to the accuracy reported in (15) for the shifting coefficients of the first overtone of CO perturbed by N 2. Those results were obtained with a system very similar to the Toronto spectrometer. The results of the present study are plotted in Fig. 3a as a function of the line number, m, where m 5 2J for the P branch, and J 1 1, for the R branch. Also shown in the figure are our previous results (16). In that earlier experiment it was assumed that the conventional 1s error bars for the measured shifts were twice s (statistical), the error bars reported by the fitting routine when fitting a single spectral profile. The present results show that the 1s errors were correctly estimated since the two sets of results are essentially in complete agreement. The very small, systematic difference (less than the width of the error bars) evident in Fig. 3a may be attributed to the 3 K difference in temperature at which the two experiments were carried out. This would require the shifts near 300 K to become slightly more negative with increasing
FIG. 2. Shift of the R(18) line of CO as a function of pressure. The straight line fitted without the low (zero) pressure point is seen to pass well within experimental error, through the origin.
TABLE 1 Shifting Coefficients in Units of 10 23 cm 21 atm 21 for CO Perturbed by N 2 at 298 K
Note. The experimental error bars are about 0.1 3 10 23 cm 21 atm 21.
temperature. This would be in mild conflict with the less precise results of Drascher et al. (18), unless N 2, like O 2, shows a slightly more negative shift with increasing temperature near 300 K (see (18)). With an average shifting coefficient of 2 3 10 23 cm 21/atm and an estimated 1s accuracy of 0.1 3 10 23 cm 21/atm, the typical accuracy of the present results is 5%. This accuracy and the accuracy reported in (15) for the first overtone are certainly adequate for present day models of atmospheric absorption by CO. Figure 3b compares the present results with the lineshift coefficients for the fundamental band as measured by Fourier transform infrared spectroscopy (FTIR). While Fourier transform spectroscopy has the ability to measure many lines in a band, as does difference frequency spectroscopy, it is evident from the figure that technical problems (phase errors) make it difficult to measure very small shifts using FTIR spectroscopy. Note, the error bars for our results are less than half of the smallest error bars shown in Fig. 3b. Nevertheless, within the stated error bars, the FTIR results are in agreement with the present results. Figure 3c compares the present results with the few lines measured using laser– diode spectroscopy. Similar comments may be made about Fig. 3c as were made about Fig. 3b. However, as is well known, diode lasers do not lend themselves readily to the measurement of many lines in a band. It is usually the laser that dictates the lines that may be studied. We believe the present results illustrate that difference-frequency spectroscopy (with some averaging) may be used to determine IR shifts accurate to 10 24 cm 21 (3 MHz) and shifting coefficients of 10 23 cm 21/atm to 10% or better, for many lines in a band.
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theory of Robert and Bonamy (25) extends Herman’s calculations to higher orders by neglecting terms to an odd power in the interaction and assuming that higher order, even terms, are expressible as the square, cube, etc. of the second-order term. Now it is a simple matter to decompose both the theoretical and the experimental shifts (as a function of m) into a symmetric component, d s , and antisymmetric component, d as . Figure 4b compares the experimental values with the theoretical values of Bouanich et al. This comparison indicates that it is the antisymmetric part of the shifting coefficient (squares in Fig. 4b) that is most in error, the computed variation with line number being much greater than the observed variation. Consequently it is the angle dependent (anisotropic) part of the potential that seems to be wrong. This poses a serious problem since it is the anisotropic part of the interaction that plays a major role in determining the width of the lines, and the widths were correctly predicted by the potential used by Bouanich et al. (15). The isotropic part of the potential does not enter into the width and its strength had already been chosen by Bouanich et al. to bring the shifts, overall (primarily d s ) into agreement with experiment. This suggests that the problem does not lie in the choice of potential but perhaps in the theoretical treatment (25) itself. We now consider several resolutions to the conflict raised in the last paragraph. It is natural to inquire if the problem is FIG. 3. (a) Shifting coefficient as a function of line number, m; open points, present results; solid points, the results of Sinclair et al. (16). (b) A comparison with present results, open circles; with FTIR measurements, solid points from Ref. (14); solid diamonds from Ref. (10), and solid triangles from Ref. (12). (c) A comparison of the present results, open circles; with diode measurements, solid squares from Ref. (17); solid triangle from Ref. (11), and solid diamond from Ref. (14).
COMPARISON WITH THEORETICAL CALCULATIONS
Figure 4a shows a plot of the present results and the calculations of Bouanich et al. (14). The potential parameters in the calculations had previously been optimized to fit overall, the measured shifts across the entire band (14). We see that the agreement between the calculated and measured shifts is only semiquantitative. We now try to identify the reason for the low quality of fit between theory and experiment. In an earlier calculation using a less developed version of the Anderson–Tsao–Curnutte theory (23, 24) than that (25) used by Bouanich et al., Herman (26) established a number of rules concerning the widths and shifts of P- and R-branch lines. Neglecting rotation–vibration interaction, the shift, d iso, coming from the isotropic part of the interaction, arises in first order and is symmetric in line number [ d iso(2m) 5 d iso(m)], while the shift, d aniso, coming from the anisotropic part of the interaction, arises in second order and is antisymmetric in line number [ d aniso(2m) 5 2 d aniso(m)]. The calculations of Bouanich et al. satisfy these symmetry rules as they must since the
FIG. 4. (a) A comparison of the present results, open circles; with theoretical results from Ref. (14), solid circles. (b) As in (a) but decomposed into d s (triangles), a part symmetric with respect to a change in sign of the line number, m, and an antisymmetric part, d as (squares). Open symbols from present experiment, closed symbols from Ref. (14).
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FIG. 5. and (15).
LUO ET AL.
A plot of the Herman factor as a function of the absolute value of the line number; circles from present results, squares from Bouanich et al. (14)
created by the neglect of rotation–vibration interaction, a feature common to many of the theories. If the isotropic forces stretch the molecules, thus lowering the vibrational frequency of an anharmonic oscillator, there will also be a change in the rotational constant B and an associated shift in the lines that is asymmetric in m. However, following the simple calculation given in (27), it is possible to show such an effect is totally negligible for the CO molecule. There is a combined test of the theoretical treatment of the widths and the symmetric part of the shifts, d s . Herman’s third rule (see page 272 in (26)) may be written @ d 0 ~2m! 1 d 0 ~m!#@ g 0 ~2m! 1 g 0 ~m!# 3/ 2 5 2 d s @ g 0 ~2m! 1 g 0 ~m!# 3/ 2 5 H,
[1]
where d 0 and g 0 are the shifting and broadening coefficients and H the “Herman” constant. This was developed using a rigid cutoff radius b 0 , rather than curved trajectories. While a more modern treatment may lead to H not being totally independent of m, we may still anticipate that, if not the value of H, then at least the functional form of Eq. [1], is insensitive to the details of the calculation. Thus we speculate that a comparison of H ca1 to H exp is a combined test of the theoretical handling of the first-order shift and the width, a second-order result. In Fig. 5 we show a plot of H versus m as determined from the present measurements of the shifts and the experimental widths of Sinclair et al. (16). 1 We see that the experimental value of H is almost constant within the rather large 1 The present measurements of the widths are in agreement with those of Sinclair et al. (16). They will be presented in a separate publication dealing with the temperature dependence of the broadening coefficients.
error bars and more important that the measured and the theoretical values of H are in agreement except for low values of m. The latter point only reflects the fact that the calculated symmetric component does not accurately reproduce the experimental results for low values of m, a problem associated with the exact form of the short-range forces. It thus appears that the theoretical treatment of the shift coming from the isotropic part of the interaction, and of the widths, is reasonable. This brings us back to the treatment of the asymmetry in the shifts as a function of line number. As Herman points out (26), a contribution from the angle-dependent interactions to a shift is zero in Anderson’s original theory, where commutivity of the time development matrices was assumed. This is a subtle point and presumably must be handled carefully in any theory, either of infinite or finite order in the interaction. Unfortunately, the asymmetry of the shifts as a function of line number is not a unique test of the importance of noncommutivity. As shown by Boulet and Robert (28), an infinite order calculation based on the assumption of commutivity can also lead to asymmetry in the shifts. Thus the problem of commutivity and order of interaction are mixed, both contributing to the asymmetry in the shifts. Perhaps the relative importance of the two contributions (in any particular case) can only be estimated by carrying out exact calculations to third or even fourth order in the interaction. At least, the experimental evidence suggests the treatment of commutivity and the extension beyond second order should be reexamined. Such questions play an insignificant role in the widths and gives one hope that the agreement between calculated and computed widths was not just fortuitous.
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LINESHIFTS IN CO–N 2
CONCLUSIONS
We have used a three-channel version of our differencefrequency spectrometer to confirm earlier measurements of the frequency shifts in the fundamental band of CO perturbed by N 2. When taken in conjunction with the measurements of Bouanich et al. (15), it establishes that frequency-shifting coefficients, precise to 60.1 3 10 23 cm 21/atm may be measured using difference-frequency laser spectroscopy. For consistency, this accuracy is required if CO–N 2 absorption profiles are to be reconstructed for atmospheric modeling. We have shown that there is some disagreement between the measured and calculated shifting coefficients and have tentatively identified the theoretical treatment of the shifting coming from the anisotropic part of the interaction between CO and N 2 molecules as being the source of the problem. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council of Canada, COMDEV, Bomem Inc., the Atmospheric Environment Service, the University of Toronto Research Fund, the Canadian Space Agency, and the Industrial Research Chair in Atmospheric Remote Sounding from Space. We thank J.-P. Bouanich for providing us with numerical values of the calculated broadening and shifting coefficients for CO–N 2. A.D.M. acknowledges a number of fruitful correspondences with C. Boulet.
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