Absorption Anomalies in Ratio and Subtraction FT-IR Spectroscopy

Absorption Anomalies in Ratio and Subtraction. FT-IR Spectroscopy. David D. Weis and George E. Ewing*. Department of Chemistry, Indiana University, ...
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Anal. Chem. 1998, 70, 3175-3183

Absorption Anomalies in Ratio and Subtraction FT-IR Spectroscopy David D. Weis and George E. Ewing*

Department of Chemistry, Indiana University, Bloomington, Indiana 47405

Subtraction and ratioing of strong absorption bands in Fourier transform infrared (FT-IR) spectroscopy produces anomalous absorption errors. One source of error is the instability in the wavenumber scale of the FT-IR spectra. The possible causes of this error are explored. The thermal expansion and contraction of the cavity of the HeNe reference laser from a typical commercial instrument was found to produce changes in the laser wavenumber of (0.034 cm-1. Changes of this size are shown to introduce errors into the wavenumber scales of FT-IR spectra which are sufficient to produce the observed anomalies. The dependence of the error on instrumental and spectroscopic parameters is explored. Solutions to the problem are proposed. Often the goal in Fourier transform infrared (FT-IR) spectroscopy is to observe a small absorption obscured by a large background absorption. An approach that is used to circumvent this problem is spectral subtraction or spectral ratioing, in which a carefully scaled background spectrum is numerically removed by subtraction or division. Most commercial FT-IR systems provide software routines to accomplish this goal. Ideally, all that remains is the desired absorption. Unfortunately, the removal of background absorption is not completely effective. Two examples, which serve to introduce the anomalies, are the subtraction of water vapor spectra and the ratioing of spectra of polystyrene films. The first example concerns a gas-phase sample. Spectral subtraction of water vapor is crucial in many applications involving FT-IR investigations of atmospheric chemistry,1,2 where, prior to subtraction, water vapor produces the dominant features in a simple survey scan. Figure 1A shows the absorption spectrum of water vapor. Figure 1B shows the difference in absorbance when a subsequent spectrum of water vapor (which has been scaled and offset slightly) is subtracted. The spectra were acquired in immediate succession under almost identical conditions, yet a residual absorption containing both positive and negative features remains. The residual signal greatly exceeds the noise level in absorption-free regions (e.g., near 4000 or 3400 cm-1). Furthermore, the absorption anomaly shows a rough correlation with the water vapor absorption spectrum. The residuals are clearly unlike that of (random) noise. We consider next a condensed-phase sample. Figure 2A shows a single-beam spectrum of a polystyrene film. Figure 2B shows (1) Peters, S. J.; Ewing, G. E. J. Phys. Chem. 1996, 100, 14093-14102. (2) Weis, D. D.; Ewing, G. E. J. Geophys. Res. 1996, 101, 18709-18720. S0003-2700(97)01174-8 CCC: $15.00 Published on Web 07/03/1998

© 1998 American Chemical Society

Figure 1. Water vapor. (A) FT-IR absorbance spectrum of water vapor. (B) Scaled absorbance subtraction spectrum of water vapor at 4 cm-1 resolution.

the residual absorption when the base 10 logarithm of the ratio of two consecutive single-beam spectra of the same polystyrene film is taken to produce an absorbance spectrum. Again, a large residual absorption remains that is dominant in regions of strong absorption. Here, the absorption anomaly has a structure that closely resembles the first derivative of the single-beam spectrum. An often overlooked source of error in FT-IR spectroscopy is the reproducibility of the wavenumber scale. Only a few papers addressing the subject have appeared in the past 25 years. Hirschfeld3,4 derived approximate equations for the maximum error observed in absorbance subtraction and absorbance ratio spectra. Additionally, he derived criteria conditions for the wavenumber reproducibility required for a particular measurement. Brown et al.5 used simulated absorption bands and calculated the difference spectra for various amounts of wavenumber error between the two peaks. The difference spectra were (3) Hirschfeld, T. Appl. Spectrosc. 1975, 29, 524-525. (4) Hirschfeld, T. Appl. Spectrosc. 1976, 30, 550-551. (5) Brown, C. W.; Lynch, P. F.; Obremski, R. J. Appl. Spectrosc. 1982, 36, 539544.

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Figure 2. Polystyrene film. (A) FT-IR single-beam spectrum. (B) An absorbance spectrum obtained by taking the base 10 logarithm of the ratio of two consecutive single-beam spectra. Features due to CO2 have been marked with an asterisk.

found to contain residuals that resembled the first derivative of the parent peak. Many instrumental limitations introduce error into the wavenumber scale of an FT-IR spectrum, including beam divergence, mirror misalignment, and mirror drive instability.6 Another source of error in the wavenumber scale is the instability of the wavelength of the reference laser. This effect was touched on briefly by Hirschfeld7 and later mentioned by Griffiths and de Haseth.6 (Hirschfeld incorrectly described the instability as discrete shifts in wavelength; we will show that the instability is continuously variable.) An exhaustive Chemical Abstracts search and a review of a decade of Applied Spectroscopy and Analytical Chemistry revealed no further treatment of this phenomenon. Although the wavelength instability is only on the order of 2 parts in 106, we will show that this effect is sufficient to introduce residuals consistent with those shown in Figures 1B and 2B. What we wish to accomplish in this paper is an assessment of the contribution of reference laser wavelength instability to absorption anomalies. We begin with a brief description of the instrumentation used in this report. This is followed by our observations of both the absorption anomalies and the wavelength instability of a typical reference laser. Next we review the relevant principles of both FT-IR spectroscopy and laser operation. We consider the possible causes of the observed reference laser instability. Finally we explore the effects of laser instability on simulated absorption features and suggest methods for reducing the magnitude of the absorption anomalies. (6) Griffiths, P. R.; de Haseth, J. A. Fourier Transform Infrared Spectrometry; Wiley-Interscience: New York, 1986. (7) Hirschfeld, T. In Fourier Transform Infrared Spectroscopy: Applications to Chemical Systems; Ferraro, J. R., Basile, L. J., Eds.; Academic Press: New York, 1979; Vol. 2, pp 193-242.

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EXPERIMENTAL SECTION FT-IR Spectroscopy. The spectra shown in Figures 1 and 2 were obtained using a Magna 550 FT-IR spectrometer (Nicolet Instrument, Madison, WI), which is a typical research-grade medium resolution instrument. The other FT-IR spectrometers in our laboratory, a Nova-Cygni 120 (Mattson, Unicam, and Cahn, Madison, WI) and an IFS-66 (Bruker Instruments, Billerica, MA), demonstrate similar anomalies. All of these instruments use a Michelson-type interferometer or some variant. Central to the operation of each is a HeNe reference laser. The interference fringes of the reference laser are used to determine to high precision the position of the moving mirror of the interferometer. The spectra shown in Figures 1 and 2 were obtained by averaging 100 scans at 4 cm-1 resolution using triangular apodization. A liquid nitrogen-cooled type B mercury cadmium telluride (MCT-B) detector was used. Our spectrometers operate under a purge of compressed air that has been passed through a purification system (Balston 75-62 FT-IR purge gas generator, Haverhill, MA) to remove CO2 and H2O. The pressure inside the spectrometer is approximately 2 mbar higher than the ambient pressure and was found to fluctuate by only about (0.2 mbar over a 20 min period, as measured with a capacitance manometer (MKS Instruments Baratron 122AA, Andover, MA). Room-temperature spectra of water vapor (Figure 1) were obtained using a 91 cm stainless steel cell with wedged Si windows (Infrared Optical, South Farmingdale, NY) which contained N2 that had been bubbled through distilled water. A background single-beam spectrum of the dry N2-flushed cell was recorded 16 min prior to the acquisition of the water vapor spectra. After the introduction of water vapor, two absorbance spectra were obtained by taking the base 10 logarithm of the ratio of the single-beam of the clean cell and the single-beam of the water vapor. One of these spectra is shown in Figure 1A. Two spectra were collected in immediate succession, each requiring approximately 60 s of acquisition time. The two absorbance spectra, A(νj) and A′(νj), were then subtracted to obtain a difference spectrum, where

D(νj) ) A(νj) - (kA′(νj) + b)

(1)

and k and b are adjustable parameters determined by using a nonlinear least-squares regression program (Sigma Plot for Windows, Version 2.0, Jandel Scientific, San Rafael, CA) to minimize ∑D(νj)2. Room-temperature spectra of a 1.5 mil (38 µm) thick polystyrene film (Nicolet Instrument, Madison, WI) were obtained by placing the film inside the sample compartment of the spectrometer. The film was securely clamped into place near the focal point of the IR beam. To ensure that the film temperature did not change due to IR heating, the film was left in the IR beam for roughly 24 h prior to collecting spectra. Single-beam spectra were then collected in immediate succession, each requiring approximately 60 s of acquisition time (a single-beam spectrum is shown in Figure 2A). To estimate the peak-to-peak absorbance noise in the polystyrene spectra, two single-beam spectra of a neutral density filter were ratioed, and the base 10 logarithm was taken to obtain an absorbance spectrum. The neutral density filter was made of two

pieces of 400 mesh stainless steel wire cloth sandwiched together. The filter had approximately the same transmittance as the strong bands in the polystyrene film (∼3%) in the same spectral region (3100-2900 cm-1). Analysis of the HeNe Reference Laser. A Fabry-Perot interferometer (FPI) was used to measure the wavenumber stability of the HeNe reference laser from our Magna 550 FT-IR spectrometer. The HeNe reference laser was removed from the spectrometer and placed on a bench in the laboratory. The laser had been running uninterrupted for over 24 h to allow the laser tube to reach a stable temperature. The operational principles of the FPI are well known, and a description may be found in Fowles.8 Briefly, laser light was collimated with a lens, directed through the FPI, and focused onto the pinhole aperture of a detector (Hamamatsu R487, Bridgewater, NJ). The FPI, manufactured by the instrumental service shops of Indiana University, consisted of a fixed mirror and a moving mirror mounted on a piezoelectric crystal. The FPI was scanned by applying a sawtooth voltage signal of 18.5 Hz from a programmable ramp generator (Burleigh Instruments, Fishers, NY) to the piezoelectric crystal. The signal from the detector was digitized at 50 kHz and stored on a personal computer. The free spectral range of the FPI, 0.10 cm-1, served as a wavenumber scale for the inteferometric data. While it was not possible to obtain an absolute wavenumber calibration, the wavenumber differences could be determined. A position in the middle of the FPI data range was arbitrarily assigned a relative wavenumber value of zero. All other data were referenced to this zero wavenumber position. During data acquisition, the ambient air temperature 15 cm from the laser tube was found to be invariant within the precision limits ((0.1 °C) of our digital thermometer (Hanna Instruments HI8564, Woonsocket, RI). RESULTS AND DISCUSSION Subtraction and Ratioing of FT-IR Spectra. Figure 1A shows one of the absorption spectra used for scaled absorbance subtraction. Two water vapor spectra were subtracted using scaled absorbance subtraction as described by eq 1. The difference spectrum D(νj) is shown in Figure 1B. The near-unity value of k (0.9984) and the small size of b (1.24 × 10-4) used in this subtraction indicates that these two spectra are a very good match. The peak-to-peak error in this region of the spectrum is 1 × 10-3. The noise in an absorption-free region of the spectrum in which the single-beam irradiance is roughly the same as that in the water absorption region (4600-4300 cm-1) has a peak-to-peak noise of 2 × 10-4. The anomaly in the water vapor region is thus a factor of 5 larger than the peak-to-peak noise. Figure 2A shows the single-beam spectrum of a polystyrene film. Two single-beam spectra were acquired in immediate succession to obtain the absorbance spectrum shown in Figure 2B. In this case, the residual absorbance is 1 × 10-2 peak-topeak. Furthermore, the absorption anomaly closely resembles the first derivative of the single-beam spectrum, which is indicative of a wavenumber scale error.5 To estimate the noise level in this spectrum, two single-beam spectra of a neutral density filter were taken to obtain an absorbance spectrum. The filter had approximately the same transmittance as the strong bands in the (8) Fowles, G. R. Introduction to Modern Optics, 2nd ed.; Dover: Mineola, 1989.

Figure 3. Schematic of an idealized FT-IR instrument.

polystyrene film (∼3%) in the same spectral region (3100-2900 cm-1). The noise level in the filter spectrum was 2 × 10-3 peakto-peak, which is a factor of 5 smaller than the residual absorbance in the polystyrene spectrum. FT-IR Principles. We begin by reviewing the relevant principles of FT-IR spectroscopy (a thorough treatment may be found in Griffiths and de Haseth6). Figure 3 shows a schematic diagram of an idealized FT-IR instrument. This system consists of a infrared source, a Michelson interferometer, a detector, and a reference laser. In order for the data in an FT-IR experiment to be collected and stored electronically, the interferogram must be sampled at known, discrete mirror positions. This is accomplished by means of the reference laser. Each mirror position corresponding to a laser interference fringe (either a maximum or a minimum of laser interference) is like a tick mark on a ruler. To take advantage of the computationally efficient fast Fourier transform, the recorded interferogram must contain 2N points, where N is a positive integer. For a moving mirror initially at the point of zero path difference, the path difference of the ith laser fringe (where i ) 1, 2, ..., 2N), xi, is given by

xi ) i/νjref

(2)

where νjref is the reciprocal of the reference laser wavelength. Thus, the spacing between the discrete points in the interferogram is 1/νjref. Figure 4A shows the analog signal (solid curve) produced by the detector as the mirror moves, given an ideal spectrometer and monochromatic source. This interference pattern may be sampled at discrete intervals given by eq 2 and shown as the filled circles in Figure 4A. For clarity’s sake, only a portion of the interferogram is shown in Figure 4. For even a medium-resolution spectrum (e.g., 4 cm-1), thousands of points must be sampled. The Fourier transform of the interferogram will be a discrete spectrum containing 2N-1 distinct points. The wavenumber of the jth point (where j ) 1, 2, ..., 2N-1) in the spectrum is given by

νjj )

j 2

( )

N-1

νjref 2

(3)

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Figure 4. Sampling of interferometric data from a monochromatic source (s). (A) Interferogram sampled at 1/νj 0ref intervals (b). (B) h ) intervals (0). (C) Reconstruction of sampled data assuming that all sampling was done at 1/νj0ref (dashed Interferogram sampled at 1/(νj0ref + δ lines added as a guide to the eye).

line (such as this monochromatic source) is determined by the apodization function. The Fourier transform of the triangularly apodized interferogram of a resolution-limited line produces a band defined by6

[

A(νj) ) A h 0L

sin(π(νj - νj0)L) π(νj - νj0)L

]

2

(4)

where A h 0L is the absorbance at the band center, νj0, and L is the maximum optical path difference, given by

L ) 2N-1/νjref

(5)

corresponding to a displacement of the moving mirror in Figure 3 of L/2. The resolution (full width at half-maximum) is proportional to L-1. For triangular apodization, the full width at half-maximum is Γ ) 0.89/L.6 (While the peak absorbance goes up with L, the normalization factor A h 0L of eq 4 ensures that the integrated absorbance, A h 0 ) ∫A(νj) dνj, is invariant with resolution.) Reference Laser Instability. As can be seen from eqs 2 and 3, the wavenumber precision and accuracy in FT-IR depend on the precision and accuracy of νjref. If there is an error in the wavenumber of the reference laser, δh , then there will be an accompanying error, j, in the wavenumber scale of the spectrum:

j ) νjδ h /νjref

(6)

It is worth noting that j is not constant, but rather depends on νj, the infrared wavenumber. However, for a small wavenumber range, j may be treated as a constant. 3178 Analytical Chemistry, Vol. 70, No. 15, August 1, 1998

Let us now consider two measurements of the monochromatic infrared source, one using νjref ) νj0ref and one using νjref ) νj0ref + δ h , where νj0ref corresponds to an assumed (fixed) wavenumber of the reference laser (νh0ref ) 15 798 cm-1 for a HeNe laser). Using eq 2, we can see that the interference pattern shown in Figure 4A will be sampled over two distinct sets of path differences, xi and xi′, respectively. These locations are shown in Figure 4A as filled circles and in Figure 4B as open squares. In both cases, the recorded interferogram is simply a list of detector signal values. To perform the Fourier transform, it is necessary to assign a mirror position to each value in the list. This is done by using eq 2, where the index i comes from the sequence of the list of stored detector signals. If the wavenumber of the reference laser is assumed to be invariant (i.e., νjref ) νj0ref in both cases), which is the case for commercial FT-IR software, then the second interferogram becomes distorted, as is shown in Figure 4C. In the case of the interferogram actually sampled at intervals of 1/νj0ref (the filled circles in Figure 4C), the sampled data faithfully reproduce the original interferogram. The interferogram sampled at 1/(νj0ref +δ h ) (the open squares in Figure 4C) does not reproduce the original interferogram because the sampling interval was assumed to be 1/νj0ref. The result is that the second interferogram appears to have a shorter wavelength than the first. This is analogous to mistaking an unlabeled ruler marked off in 1/10 in. ticks for one marked off in 1/8 in. ticks. One would be led to believe that an object 8 in. long was actually 10 in. long. (In Figure 4, we have greatly exaggerated the size of δ h for clarity. Realistic values of δ h would yield imperceptible sampling interval changes in this figure.)

assessment of the size of δh and its influence on FT-IR spectra will follow. There are two requirements for lasing to occur at a particular wavelength.8 First, the laser cavity must be an integral multiple of the desired wavelength in order for a standing wave to be established, or

νjref ) m/2d

(7)

where m is an integer and d is the length of the laser cavity. Second, the gain must exceed the losses at the desired wavelength. When both of these conditions are satisfied, lasing will occur. Typically, several longitudinal modes exceed the gain threshold for lasing. Neighboring (∆m ) 1) modes are separated by ∆νjref as given by

∆νjref ) 1/2d

Figure 5. Absorption anomalies caused by changes in reference laser wavelength. (A) Two absorption bands obtained from the interferograms in Figure 4 sampled at 1/νj0ref (b) and at 1/(νj0ref + δ h) (0). (B) Subtraction of the two spectra from panel A. (C) The two spectra when the correct values of the laser wavenumber are used. These two spectra cannot be subtracted because their wavenumber scales are incompatible.

When these two interferograms are Fourier transformed, they will both have the same wavenumber scale given by eq 3 (since it is assumed that νjref ) νj0ref), but the distorted interferogram will produce a spectral band which appears to be shifted to a higher frequency (if δ h were negative, the band would appear to shift to lower frequency). This is shown in Figure 5A, where the filled circles correspond to the Fourier transform of the interferogram recorded with νjref ) νj0ref and the open squares correspond to the Fourier transform of the interferogram recorded with νjref ) νj0ref + δ h . When the difference between these two spectra is taken, a large anomaly remains (Figure 5B) which arises from the apparent frequency difference between the two bands. If the correct values of νjref are used in eq 2, then there is no distortion of the sampled interferograms. Upon Fourier transformation, two spectra (shown in Figure 5C) having different wavenumber scales, νjj and νjj′, given by eq 3 are produced. As can be seen in Figure 5C, these spectral bands do overlap; now, however, they possess distinct and incompatible wavenumber scales. Since the wavenumber scales are different (i.e., νjj * νjj′), subtraction is not appropriate. While it may be possible to align the two wavenumber scales using some interpolation scheme, such as zero-filling, this will introduce its own anomalies into the difference spectrum. Helium-Neon Reference Lasers. We now turn to an examination of the wavenumber instability of FT-IR reference lasers utilized in typical medium-resolution FT-IR instruments. An

(8)

A HeNe laser found in most commercial FT-IR spectrometers is neither monochromatic nor is its wavelength stable over time. For the 633 nm transition of Ne, the full width at half-maximum of the Doppler-broadened band is roughly 0.047 cm-1.9 Taking d ) 15 cm as a typical laser cavity length, the spacing of the laser modes is 0.033 cm-1, which indicates that lasing may be occurring on one, two, or three modes, depending on the positioning of the modes relative to the gain profile. In addition to generating more than one mode, eqs 7 and 8 show that the output characteristics of a laser, νjref and ∆νjref, can vary as the cavity length of the laser changes due to temperature fluctuations. The change in length of the laser tube, ∆d, as a function of temperature change ∆T is given by

∆d ) R∆Td

(9)

where R is the coefficient of linear expansion.10 For ∆d , d, the change in the laser wavenumber, δh , is given by

δ h)

m (R∆T) 2d

(10)

A typical value of R for glass is approximately 5 × 10-6 cm °C-1 (borosilicate glass, such as Pyrex, has a coefficient of linear expansion of about 1 × 10-6 cm °C-1, and ordinary glass is about 9 × 10-6 cm °C-1 10). For d ) 15 cm at νjref ) 15 798.2 cm-1, the integer value of m is 473 964. For a temperature change of only 0.4 °C, δh is 0.03 cm-1, which is comparable to the mode spacing of the laser. Thus, small temperature changes in the glass laser tube can cause a phenomenon known as mode sweeping in which νjref shifts with temperature changes. We have been able to observe mode sweeping with the HeNe reference laser used in our Magna 550 FT-IR spectrometer. A Fabry-Perot interferometer was used to measure the intensity and relative wavenumber of the various modes of this laser. A series of measurements is shown in Figure 6. Moving from the top of the figure to the bottom, the laser output frequency shifts from conditions in which a single mode dominates at 0.00 cm-1 (9) Optics Guide 5; Melles Griot: Irvine, CA, 1990. (10) Halliday, D.; Resnick, R. Fundamentals of Physics, 3rd extended ed.; Wiley: New York, 1988.

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treated as if it was measured at a mirror position of 1/νjref away from the previous measurement. To not become bogged down in the instrumental details, we will assume for the remainder of the paper that the laser output consists of a single, variable mode. Changes in the Refractive Index of Air. Next we consider the possibility that changes in the purge gas might induce a change in νjref as a consequence of a change in the refractive index of air, ∆n, due to fluctuations in temperature, composition, or pressure. Such an error would have the form

δ h ) ∆nνjref

Figure 6. Changes in the intensity and relative wavenumber of the longitudinal modes of a typical HeNe reference laser over time. The spectra were recorded at 8 min intervals.

to dual modes of near equal intensity at -0.010 and +0.024 cm-1. The observed mode spacing value ∆νjref of 0.034 cm-1 is consistent with our estimate based on the laser cavity length. The changes shown in Figure 6 occurred over a period of 24 min. Mode sweeping was much more rapid (on the order of seconds) when the laser was initially turned on, and the sweeping remained rapid for several hours thereafter. By simply blowing air with a fan onto the laser after it had warmed for over 24 h, it was possible to sweep through the entire free spectral range in less than 1 min. It is difficult to anticipate what the thermal history of the laser would be with the laser actually enclosed inside the spectrometer, bathed in the flowing air purge. Measurements of ambient temperature are not useful without a thorough understanding of the heat transfer between the laser and its environment since it is the temperature of the laser tube itself which is critical. Figure 6 shows that the output of this HeNe laser varies over a range which is about the same size as the mode spacing; thus, we may take δh to be (0.034 cm-1. Using eq 6 with νj ) 3650 cm-1 and νj ) 0.034 cm-1, we arrive at an error in the infrared wavenumber scale of j ) 0.008 cm-1. Figure 6 also shows that, under certain conditions, the laser output consists of two separate modes. This would produce two sets of laser interference fringes. Without a detailed understanding of the optics and electronics of a particular FT-IR spectrometer, it is not clear how the data acquisition would be affected. If the modes were resolvable by the interferometer, a slow time constant for the laser detector might not permit detection of the two separate fringes. If the two fringes could not be resolved, then a single fringe would be observed by the laser detector. In either case, it is not clear at what mirror position the interferogram would be sampled or whether this position would even be reproducible. If the optical system and electronics were able to distinguish and respond to both laser modes, then the resulting interferogram would be severely distorted, since each measured data point would be 3180 Analytical Chemistry, Vol. 70, No. 15, August 1, 1998

(11)

The pressure and temperature dependence of the refractive index of dry, CO2-free air was measured by Barrell and Sears.11 Using 20 °C and 1.013 bar (760 Torr) as a reference point, ∆n is approximately 7 × 10-8 for a pressure change of 0.2 mbar (see the Experimental Section) at the wavelength of the HeNe laser. This change in the refractive index leads to δh of approximately 0.001 cm-1, which is much less than δh due to mode sweeping. Similarly, a temperature change of 0.4 °C, which is more than adequate to cause substantial mode sweeping, gives ∆n of approximately 4 × 10-7 for a δh of approximately 0.006 cm-1. To obtain δh values for refractive index changes comparable to values for mode sweeping, the pressure fluctuations must be on the order of (10 mbar and the temperature fluctuations must be on the order to (2 °C. These fluctuations in pressure and temperature are much larger than the ones observed in our laboratory (see the Experimental Section). Since our instruments are not purged with a high-purity gas, the possibility remains that fluctuations in the partial pressures of CO2 or H2O could cause a change in the refractive index of the purge gas. Newbound12 has measured the dependence of the refractive index of air on the amount of CO2 and H2O present. The partial pressure of CO2 at sea level is approximately 0.3 mbar.13 We can calculate the value of ∆n for a 0.03 mbar spike of CO2 in our purge gas, which might be representative of a partial transient failure of our purge system. For such a spike, ∆n is approximately 4 × 10-9, which leads to a negligibly small δh of 6 × 10-5 cm-1. The relative humidity in our laboratory is typically 50%. Assuming a spike of water vapor in our purge with a relative humidity of 10%, ∆n is approximately 1 × 10-7, which leads to a δh of 0.002 cm-1. Thus, the fluctuations in pressure, temperature, or composition which might occur in our purge system would produce changes in νjref, through changes in n, that are orders of magnitude smaller than those caused by mode sweeping. Effect of Reference Laser Instability on a Simulated Feature. In this section, we will explore the effect of laser wavenumber instability on the subtraction of a single, resolutionlimited absorption band. The absorption anomaly can be simulated as outlined in the sections on FT-IR Principles and Reference Laser Instability: two wavenumber scales, νjj and νjj′, are generated from eq 3 using νjref ) νj0ref and νjref ) νj0ref + δ h , respectively. Absorption bands are then calculated using each scale, A(νjj) and A′(νjj′), using eq 4. Finally, a single wavenumber scale, νjj, (11) Barrell, H.; Sears, J. E. Philos. Trans. R. Soc. 1940, 238A, 1-64. (12) Newbound, K. B. J. Opt. Soc. Am. 1949, 39, 835-840. (13) Wayne, R. P. Chemistry of Atmospheres, 2nd ed.; Oxford Science: Oxford, 1991.

Figure 7. Simulated FT-IR absorption band for a resolution-limited feature. νj0 ) 3648.7 cm-1, A h 0 ) 1.35 cm-1, and L ) 0.2593 cm.

is used for both lists of absorbance values, A and A′. The result will be two bands such as those shown in Figure 5A, and the difference spectrum, D(νjj) ) A(νjj) - A′(νjj′), will have a form similar to the difference spectrum in Figure 5B. The peak-topeak residual is given by the difference between the maximum and minimum values of D(νj). The functional form of D(νj) or of the peak-to-peak residual is cumbersome, but it is a simple matter to construct a computer program which calculates D(νj) and determines its maximum and minimum values. It is worth noting that the above discussion is only accurate in the limit of small δh errors and moderate values of N. For large values of δh or N, i.e., for a large laser error or a large number of laser fringes, the maximum mirror path difference, L, will be significantly different in the two spectra. If there is an error of δh , the error in L will be 2Nδh . It can be shown with a more rigorous treatment utilizing the Fourier transform of a cosine wave that there is substantial band distortion for very large values of δh or N. (To obtain distortions at medium resolution, e.g., ∼4 cm-1, δh must be roughly 2 orders of magnitude larger than the observed value.) We can now explore how the difference spectrum, D(νj), depends on the parameters δh , νj0, A h 0L, and Γ. A simulated h 0 ) 1.350 absorption band centered at νj0 ) 3648.7 cm-1 with A cm-1 and L ) 0.2593 cm is shown in Figure 7, which corresponds to the intense absorption feature of water vapor marked with an asterisk in Figure 1A. Figure 8A shows the dependence of the peak-to-peak error on the error in wavenumber of the reference laser, δh . The results in Figure 8A show that, in the limit of realistic values for δh , the peak-to-peak error is directly proportional to the laser error. For δh of 0.034 cm-1, the expected peak-to-peak error in D(νj) is 2.5 × 10-3 absorbance units. Figure 8B shows the dependence of the peak-to-peak error on the location of the band center, νj0, for a fixed laser error of 0.034 cm-1. Again, a linear relationship between the peak-to-peak error and the band center exists. Figure 8B shows that D(νj) will be larger for highfrequency features than for low-frequency features. This is not surprising, given the form of eq 6. Figure 8C shows how the peak-to-peak error in D(νj) depends on the peak absorbance, A h 0L, for a fixed laser error of 0.034 cm-1. As the absorption by the line increases, the peak-to-peak error in D(νj) increases linearly. Not surprisingly, weakly absorbing bands produce a smaller error in the difference spectrum than intense bands. The dependence of the peak-to-peak error in D(νj) on the nominal spectrometer resolution for a fixed laser error of 0.034 cm-1 is shown in Figure

Figure 8. Dependence of the peak-to-peak error in spectral subtraction. (A) Effect of laser error for νj0 ) 3648.7 cm-1, A h 0 ) 1.35 cm-1, and L ) 0.2593 cm. (B) Effect of band center for A h 0 ) 1.35 cm-1, L ) 0.2593 cm, and δ h ) 0.034 cm-1. (C) Effect of peak absorbance for νj0 ) 3648.7 cm-1, L ) 0.2593 cm, and δ h ) 0.034 cm-1. (D) Effect of resolution for νj0 ) 3648.7 cm-1, A h 0 ) 1.35 cm-1, and δ h ) 0.034 cm-1.

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8D. In this case, the integrated absorbance, A h 0, rather than peak absorbance, A h 0L, was held constant. Here, the peak-to-peak error is strongly dependent on the resolution of the spectrometer. Reducing the resolution from 1 to 4 cm-1 reduces the peak-topeak error by 2 orders of magnitude. A further 4-fold decrease in resolution to 16 cm-1 brings about only an additional order of magnitude decrease in the peak-to-peak error. This treatment ignores overlap that would occur as the resolution decreases when there are multiple bands in the spectrum. Comparison of Theory and Experiment. The parameters which defined the simulated resolution-limited line shown in Figure 7 were selected to make the line consistent with the resolution-limited water vapor lines in the 4000-3400 cm-1 region shown in Figure 1A. The difference spectrum in Figure 1B has a peak-to-peak error of 1 × 10-3. Since the lines are at the same resolution and have similar band shapes, positions, and absorbance maxima, the data in Figure 8A can be used to estimate the size of δh in Figure 1B. A peak-to-peak error of 1 × 10-3 corresponds to δh of about 0.015 cm-1, which is within the range of the maximum fluctuations of (0.034 cm-1 for the reference laser. Thus, the peak-to-peak error in Figure 1B is consistent with the instability of the reference laser. A similar analysis of the data in Figure 2B is much more difficult because the polystyrene features are highly overlapping and not resolution-limited and, hence, cannot be faithfully simulated. Can errors of this size appear in the short time scales represented by Figures 1 and 2? Equation 10 and Figure 6 show that very small temperature fluctuations of ∼0.1 °C are adequate to significantly alter the laser performance. The laser output can change in two ways, as shown in Figure 6: a dominant laser mode can drift, or a second mode can appear. Either of these effects could easily produce δh ≈ 0.01 cm-1. Solutions. In this section, we propose two approaches which may be taken to reduce the residuals in ratio and subtraction spectroscopy. Figure 8D shows the resolution dependence of the peak-to-peak error in absorbance subtraction. Clearly, the error can be substantially reduced simply by reducing the resolution of the spectrum if the interfering absorption feature is resolutionlimited and the desired feature is not resolution-limited. This is a situation frequently encountered in systems containing both vapors and condensed materials.1,2 In the case of the subtraction of the spectrum of water vapor, this approach should be quite effectivesthe water vapor features will become broader and less intense while naturally broad features would not be affected. Figure 9 shows the subtraction of the two water vapor spectra at nominal resolutions of 4 and 16 cm-1. This is the same subtraction as is shown in Figure 1B with the exception of the resolution. The peak-to-peak error at 16 cm-1 is 2 × 10-4, a factor of 5 improvement over the peak-to-peak error of 1 × 10-3 obtained at 4 cm-1. The calculated improvement shown in Figure 8D is over an order of magnitude upon decreasing the resolution from 4 to 16 cm-1. The simulation used for Figure 8 is for a single absorption band and thus overlooks the overlap of water bands that will occur as the resolution is decreased. Changing resolution will not work for polystyrene, however. Since the polystyrene features are not resolution-limited, reducing the resolution will not change their widths or intensities. It is possible to decrease the resolution to the point where polystyrene 3182 Analytical Chemistry, Vol. 70, No. 15, August 1, 1998

Figure 9. (A) Scaled subtraction spectrum of water vapor at 4 cm-1 resolution (from Figure 1B). (B) Scaled subtraction spectrum of water vapor at 16 cm-1 resolution.

bands are resolution-limited, but this would require a resolution of about 30 cm-1. Resolutions this low would most likely distort the desired features in the spectrum. A second approach is to use a more stable reference laser. Frequency-stabilized HeNe lasers are commercially available, though their cost is in the thousands of dollars rather than the hundreds of dollars for unstabilized models. A typical frequencystabilized HeNe laser (Spectra-Physics 117A, Mountain View, CA) is specified to maintain δh to (1 × 10-4 cm-1 over an 8 h period. This is a reduction in δh of over 2 orders of magnitude. Using the simulated line shown in Figure 7, δh of 1 × 10-4 cm-1 causes a peak-to-peak error of only 7 × 10-6. This degree of stabilization is more than adequate to reduce peak-to-peak errors, such as those shown in Figures 1B and 2B, to a negligible value. CONCLUSIONS We have found large absorption anomalies in the subtraction and ratioing of spectra containing strong absorption bands. One cause of these anomalies is the parts-per-million errors in the wavenumber of the reference laser, which can introduce large errors in ratio and subtraction spectra by introducing error into the wavenumber scale of the spectra. These small errors in the laser wavenumber can be attributed to expansion and contraction of the laser cavity as the temperature of the laser cavity changes. The observed changes in the wavenumber of a HeNe reference laser from a typical commercial FT-IR spectrometer are equal to the mode spacing of the laser. Additionally, the laser was found to vary between single- and dual-mode lasing. It is unclear what effect dual modes will have on an FT-IR spectrum. Assuming single-mode operation of the reference laser, we have calculated the theoretical peak-to-peak error in the subtraction of two resolution-limited absorption bands. The calculated peak-to-peak error is consistent with the observed peak-to-peak

error in the subtraction of two absorption spectra of water vapor. In the case of ratioing or subtracting resolution-limited features, theory predicts that simply reducing the resolution can significantly reduce the size of the peak-to-peak residuals. Reducing the resolution on experimental data does, indeed, decrease the size of the peak-to-peak residuals but not by the same factor as predicted by theory. The discrepancy is due to the fact that real features overlap as the resolution decreases. We predict that, in the absence of other sources of wavenumber error, the use of a frequency-stabilized reference laser will reduce the peak-to-peak error to a negligible value. While reference laser instability is only one of many sources of error in the subtraction and ratio FT-IR spectroscopy, we believe that it makes a significant contribution. We would like to encourage commercial FT-IR manufacturers to offer a stabilized reference laser as an option for their instruments. The effectiveness of this modification can easily be evaluated by repeating the subtraction and ratio tests described in this paper. The stabilized laser option would improve the manufacturers’ instrument per-

formance in demanding ratio and subtraction applications. ACKNOWLEDGMENT This research was made possible by a grant from the National Science Foundation (NSF ATM96-31838). D.W. acknowledges fellowships from Eli Lilly and Co. and from the Pharmacia-Upjohn Co. We thank Prof. J. Reilly for useful discussions and for the loan of his Fabry-Perot interferometer. R. Sporleder and R. Landgrebe provided the data acquisition system for the HeNe wavenumber measurements. The authors particularly appreciate the thoughtful comments of Prof. P. Griffiths, who read an early draft of this paper. We also acknowledge useful discussions with A. Pratt and S. Oas at Nicolet Instrument and T. Johnson at Bruker Instruments.

Received for review October 22, 1997. Accepted May 15, 1998. AC971174A

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