P. Jacquinot, Rep. Progr. Pbys., 23, 267 (1960). P. Connes, Rev. Opt., 38, 157 (1959). T. Dohi and T. Suzuki, Appl. Opt., I O , 1359 (1971). F. A . Jenkins and H. E. White, Fundamentals of Optics", 3rd ed., McGraw-Hill, New York, 1957, p 300. (7) J. F. James and R. S. Sternberg, "The Design of Optical Spectrometers", Chapman & Hill Ltd., London, 1969, p 189. (3) (4) (5) (6)
(8) R . J. Bell, "Introductory Fourier Transform Spectroscopy", Academic Press, New York, 1972, p 287. (9) D. J. Johnson, F. W. Plankey, and J. D. Winefordner, Can. J. Spectrosc., 19, 151 (1974).
for review June 2 y lg7'. Accepted August l 2 ~ 1975. This work supported by AF-AFOSR-74-2574.
Errors in Absorbance Measurements in Infrared Fourier Transform Spectrometry because of Limited Instrument Resolution Robert J. Anderson' and Peter R. Griffiths* Department of Chemistry, Ohio University, Athens, Ohio 4570 1
Calculations have been performed of the errors introduced in the measurement of peak absorbances by Infrared Fourler transform spectrometry as a result of the flnlte spectral resolutlon of the Instrument. A Lorentzian line shape and either triangular or no apodization has been assumed. A simple expression Is presented for the minlmum error expected in the former case. For comparison with conventlonal Infrared spectrometers, similar calculations were made assuming a triangular slit function. Except for large peak absorbances, comparable errors are calculated for trlangular apodization and a triangular slit function. Errors wlthout apodization are much smaller and become large only when resolution is numerically greater than line width. Spectra of the 2231 cm-' band of benzonltrile in CCI4 solutions have been measured and confirm the results of the calculations.
The measurement of an absorption or emission spectrum with a spectrometer of finite resolution distorts the spectrum to a greater or lesser extent depending on the resolution of the instrument. The differences between true and apparent intensities which are caused by the finite spectral resolution of an instrument will be termed resolution errors in this work. They arise because the intensity experimentally observed at any wavelength is actually the intensity of the spectrum averaged over the spectral bandpass of the spectrometer. Such errors have been of particular importance in infrared spectrometry where comparatively weak sources and insensitive detectors have often necessitated use of rather large spectral bandpasses. A number of authors have discussed the effects of limited resolution on spec1ra obtained with conventional grating and prism spectrometers. Dennison ( I ) , Ramsay ( 2 ) , and others (3-5), have calculated the distortion of single absorption lines by various spectrometer slit functions. Either Lorentzian (1-4) or Gaussian ( 4 , 5 ) line shapes and triangular (1-3, 5 ) , Gaussian (1, 3, 4 ) , or Lorentzian ( 4 ) (or, more accurately, Cauchy) slit functions were assumed. Experimental investigations of resolution errors found with liquid samples and conventional spectrometers have been reported by Ramsey ( 2 ) and others (6-9). A general rule of thumb (2, 7, 8,101 appears to have emerged from this work which states that for conventional spectrometers the full On sabbatical leave from Department of Chemistry, Ithaca College, Ithaca, N.Y. 14850. Author to whom reprint requests should be sent.
width a t half height ( F W H H )of the absorption band (plotted on a linear absorbance scale) should be greater than five times the spectral slit width of the instrument to maintain acceptable distortion. The reader is referred to the literature cited for further details. A review of theoretical and experimental work on shapes and intensities of infrared absorption bands has been published by Seshadri and Jones (11). The review by Nielson et al. (3) treats absorption by gases. Despite the increasing use of infrared Fourier transform spectrometry (FT-IR), no systematic study has been published concerning resolution errors as they occur in FT-IR. We here report the results of theoretical and experimental investigations of such errors. Calculations of resolution errors for single absorption lines measured by FT-IR using either unapodized or triangularly apodized interferograms have been performed. The results are compared with similar calculations for a conventional spectrometer with a triangular slit function. Experimental FT-IR spectra of liquid samples have been measured and the observed resolution errors are compared with those calculated. Finally, some additional practical considerations regarding the choice of an apodization function are discussed. THEORETICAL CALCULATIONS
Theory. The resolution of a spectrometer is quantitatively described by its instrument function or instrument line shape (ILS) (11, 1 2 ) , u(v,vi), which describes the response of the instrument to radiation of frequency u when the instrument measures a t a frequency ui. For conventional spectrometers this is commonly called the spectral slit function, but for FT-IR we prefer the generalized terminology. The ILS would be observed experimentally by measuring the spectrum of an infinitely narrow emission line. The ILS functions considered in this work depend only on the difference between u and vi and can be written u(v - vi). If the intensity of radiation of frequency u incident on the spectrometer is given by Z ( v ) , the apparent intensity recorded by the spectrometer a t frequency vi is given by the convolution of Z ( u ) with the ILS function
For absorption measurements, the intensity of radiation incident on the sample, Z,,(u), which is transmitted to the detector is I ( u ) = I,(u) exp ( - k ( v ) l ) where k ( u ) is the absorption coefficient and 1 is the optical depth. If Z,(v) is taken
ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975
2339
the width of the line being measured. p is defined below for each of the ILS’s considered:
A
p = p =
(V-V,I
Figure I.Sinc ( A ) , sinc2 shapes.
(4, and triangular (C) instrument line
R is defined in the text. The scales are identical for all curves
to be constant over the spectral bandpass of the instrument, the apparent transmittance a t frequency ui becomes (11:
Ta(ui)=
a(u - ui) exp (-h(u)l)du
Jw
Jm
U(U
- vi)dv
For the present calculations, the absorption feature is assumed to have a Lorentzian shape ( 1 1 ) ,i.e.
where Atpeakis the true maximum absorbance of the line, uo is the frequency of maximum absorbance, and y is the half width a t half height of the line, Le., y = FWHH12. Such a contour is expected for a single line whose shape is determined by pressure broadening (11,13),and has been shown to provide a moderately good representation for a single absorption band of solutions ( 2 , l l ) . The ILS of a Fourier transform spectrometer depends on the way in which the interferogram is apodized prior to Fourier transformation. If the interferogram is unapodized, the ILS is a sinc (sin x/x) function with the explicit form (14,15) u(u - vi)
= 2 sin (2?rA(v
- u i ) ) / ( 2 ~ A (-u v i ) )
(3)
where A represents the maximum optical retardation used in obtaining the interferogram. When triangular apodization (hereafter referred to simply as apodization) is used, the ILS is a sinc2function: U(U
- vi) = sin2 ( n A ( v - q))/(?rA(v-
(4)
This would also be the ILS of a dispersion spectrometer operating in the diffraction limit (1, 11).If the dispersion instrument is operated with equally wide entrance and exit slits, as is typical with most infrared spectrometers, the ILS is approximately a triangle (1,2,11): U(V
- Vi) u(u
=
1 - J u - Yil/W
- Vi)
=0
Iu
Iu
- Vi1 < w
- Vi] > w
(5)
where w is the width in frequency units of the slits. The sinc, sin$, and triangular ILS’s are shown in Figure 1. It is useful to define a resolution parameter, p , which specifies the resolution of the spectrometer as compared to 2340
l l ( A - 2 ~=) R I F W H H w/(2y) = RIFWHH
sinc, sinc2 triangular
(6)
(7)
In Equation 6, R has been substituted for l/A and is normally taken as the nominal resolution inherent in the interferogram ( 1 4 ) .A substitution of R for w has been made in Equation 7 to simplify later discussions. For the triangular function, R represents the distance in frequency units from the center, vi, of the instrument bandpass to the point where the ILS goes to zero (See Figure 1.). For the sinc2 fuhction, R is the distance from the center of the bandpass to the first zero in the ILS, and the major lobe of the function is bounded by vi f R. For the sinc function, R is the distance from the center to the second zero and bounds the first positive and negative lobes. Though R as defined above is intended to be a measure of instrumental resolution, it should be recognized that care must be exercised in discussing the apparent resolution of spectra obtained with such different ILS’s. Thus the measured spectrum of two equal sharp emission lines a distance 2R apart would be resolved to the base line midway between the lines for each ILS. However the observed spectrum of two such lines R apart would show no dip between the lines for the triangular ILS, a dip of 19% for the sin$ ILS (the Rayleigh criterion of resolution), and would be resolved to the base line for the sinc ILS though the intensity of each line would be significantly distorted by overlap with the first negative lobe of its neighbor. Lines separated by 0.5 R would not be resolved using any of the instrument line shapes. As a result of our definition of R , the F W H H of the sinc ILS is less than that of either the sinc2 or triangular ILS for a given value of R. This tends to favor the sinc function when comparing the photometric accuracy of measurements made with different ILS’s but the same R , and it may properly be asked why, for example, we have not defined R as the F W H H for each ILS. While this would remove the bias in favor of the sinc function, it would in turn bias the results in favor of the sin$ ILS. If R were defined as the F W H H , two narrow lines separated by R would show a dip between the lines for the sin$ function but would show no dip for either the sinc or the triangular ILS. In fact a n y choice of definition for R tends to favor one ILS over another. The advantage of our definition is that it facilitates a comparison of resolution errors when a given interferogram is transformed with and without apodization, while the results for a triangular ILS can be readily compared to the results calculated for a sinc2 ILS of similar
FWHH. Substitution of Equation 2, either Equations 3, 4, or 5, and Equation 6 or 7 into Equation 1 allows computation of apparent line shapes by means of successive numerical integrations for selected values of vi on the curve. In this manner, apparent maximum absorbances (A;eak), apparent line widths, apparent integrated absorbances, etc., can be calculated. This is obviously a computer time consuming procedure which must be repeated for each value of A&ak and p of interest. Fortunately, for practical analytical work, it is AEeakwhich is of primary importance (8, 11) for Beer’s law calculations etc., and the remainder of this work will therefore be limited to a discussion of A;eak. Setting vi = u0 in Equation 1, defining y = ( u - ui)/R,and introducing the definition of p , the expression for the apparent minimum transmittance becomes:
ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975
.
-3 -2
O
I
-2
2
I
Figure 2. Logarithm of apparent peak absorbance vs. logarithm of true peak absorbance for a triangular ILS Curves A through F correspond to values of the resolution parameter p of 0, 0.5, 1, 3, 10, and 25
where the limits of integration are 0 to infinity for sinc and sin$ ILS’s and 0 t o 1 with the triangular ILS, and o(y) = 4 sin (27ry)l(27ry)
sinc
~ ( y =) 2 sin2 ( x y ) / ( ~ y ) ~ sinc2 ~ ( y=) 2(1 - y )
O
I
loa
log Abeak
triangular
A;eak is calculated as -log Tgeak. Calculations. The integration of Equation 8 was performed numerically for selected values of the parameters Abeak and p, and the results in the form of graphs of log A;eak vs log Abeak for various values of p are presented in Figures 2, 3, and 4. Computer programs were written in Fortran IV and calculations were done in double precision. A 32-point Gaussian-quadrature algorithm (16) was used with the triangular ILS. The integrations with sinc and sinc2 functions, which extend to infinity, were performed by summing up three areas, each calculated with 32-point Gaussian-quadrature, and which together accounted for transmittance out to the 28th positive lobe in the ILS, plus an area extending from there to infinity calculated from asymptotic expressions for the integral for large y. The computations were facilitated by computing, not the average transmittance as indicated in Equation 8, but rather the average absorption (1 - T ) , which contains smaller contributions from large y , and then subtracting from 1. The calculations are believed accurate to a t least 1 X in transmittance. Results of Calculations, The results for the triangular ILS presented in Figure 2 are the simplest to interpret. For small values of Akeak and all values of p , there is a linear portion with slope equal to one. This represents a region in which the Beer-Lambert law would be obeyed experimentally (assuming Abeak is proportional to concentration and path length) since A;eak is proportional to Abeak.Note, however, that the apparent peak absorbance is not in general equal to the true peak absorbance. The vertical distance on the graph from a given curve to the line for p = 0 represents the log (Abeak/A;eek)for that value of p and Abeak and thus indicates the magnitude of the resolution error. For values of p greater than about 1, a break occurs in the curves at Abeak approximately equal to 0.7 and a second h e a r portion occurs with a slope of indicating a square root dependence of the apparent absorbance on the true absorbance. This is consistent with earlier work using a triangular ILS in which it was shown analytically that if p and Abeak are large and Abe,,k/4p2 < 1, then A;eak is proportional to (Abeak)”2/P (1, 3 ) .
I
2
‘bwk
Logarithm of apparent peak absorbance vs. logarithm of true peak absorbance for a sinc2ILS Curves A through G correspond to values of t h e resolution parameter, p , of 0,0.1,0.5,1,3,10,and25 Flgure 3.
The results for the sinc2 ILS shown in Figure 3 are in many ways quite similar to those in Figure 2. For all values of p there is an initial linear (Beer-Lambert law) region of slope 1, and the resolution errors in this region are at least approximately the same as for the triangular ILS, though more will be said of this later. For large p , a break in the curves occurs, once again a t Abeak about 0.7, and a square root region (slope = 1/2) ensues. The magnitude of the resolution errors in this region too is the same as in Figure 2. With the sinc2 function, however, a third region of very small slope occurs at larger values of A&ak. For p greater than 1, this happens when AEeak exceeds approximately 1, and for smaller p it begins a t greater values of A;eak. The third region with its large resolution errors is due to the presence of the secondary lobes in the ILS (corresponding to y > 1) which cause the instrument to respond to radiation of frequencies substantially removed from vi. Some 90.3% of the area under the sin$ curve is contained in the central lobe and 99.0% is within the ten lobes on either side of center. Thus, with a broadband source, 9.7% of the instrumental response is due to radiation of a frequency more than R removed from the center of the bandpass and 1%is from more than 10R away. Conversely, if the peak transmittance were measured for a band which was 100% absorbing between vi - R and vi +- R and was 100% transmitting elsewhere, and a broadband source were used, the result would be 9.7% T. The results for the sinc2 ILS can be understood in terms of the above discussion. As the peak absorbance, Akeak,of the absorbing line centered in the central lobe of the ILS increases from zero, A;,,kWill increase much as with the triangular ILS until almost all the intensity which would have been passed by the central lobe is absorbed. (Figures 7 and 8 of reference 3 are useful for visualizing how an instrument with a triangular ILS responds to increasing values of Abeak.)At this point, the secondary lobes of the ILS are still passing intensity not absorbed by the wings of the line. As indicated in the preceding paragraph, this corresponds to about 10% T (AEeak= 1) when the line is narrower than the central lobe ( p < 1) or to a greater absorbance if the line is wide enough to cover more than the central lobe. As Abeak increases further, Aapeak increases only very slowly as the wings of the Lorentzian line absorb intensity from the wings of the ILS. The results for the sinc ILS shown in Figure 4 differ substantially from those in the preceding figures. As with the other two functions, there is an initial linear region with slope of 1, but with the important difference that the resolution errors are much smaller in magnitude. For example,
ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975
2341
Table I. Coefficients Used in Estimating Resolution Errors -alia'
a1 P
0.05 0.1 0.25 0.5 0.75 1.0 3.0
10.0
sinc
sinc'
triangular
sinc
sinc'
triangular
1.000 1.000 1.000 0.998 0.985 0.957 0.649 0.270
0.984 0.968 0.920 0.841 0.765 0.695 0.380
0.998 0.993 0.962 0.878 0.787 0.705 0.368 0.137
0.000 0.000 0.000 -0.005 -0.020 -0.032 0.078 0.351
0.009 0.018 0.042 0.075 0.105 0.134 0.306 0.470
0.000 0.000
0.142
0.002 0.020 0.054 0.093 0.309 0.481
-3 -2
log Abeo,
Figure 4. Logarithm of apparent peak absorbance vs. logarithm of true peak absorbance for a sinc ILS Curves A through F correspond to values of the resolution parameter p of 0, 1, 3, 10, 25,and 50
for p = 0.5 and Atpeak = 0.5, the ratios Of A:eak/AbeakfOl'triangular, sinc2, and sinc ILS's are 0.869, 0.802, 1.0002. Note that with some values of p and Abeak,the apparent absorbance with a sinc ILS may be greater than the true absorbance while this is never the case with the other functions. For large p (greater than lo), the break at Abeak equals 0.7 occurs followed by a square root region though once again the magnitude of the resolution errors is not as great as with the other two functions. As with the sinc2 function, a third region occurs for large A;eak, in this case characterized by a rapid increase in ABpeak with Abeak. In fact for large enough Abeak,the apparent transmittance decreases to zero and then becomes negative. The explanation is analogous to the case of the sin$ function. Behavior is qualitatively similar to the triangular ILS until most of the intensity has been absorbed from the central lobe. In this case, however, the next most important region of the ILS contributing to instrumental response is the two negative lobes on either side of center. Their effect is to make the averaged transmittance negative. As Abeak increases even more, T;eak becomes positive again. Alternate Analysis. Before considering these results further, it is advantageous to discuss an alternate method of carrying out the integration in Equation 8 in which the exponential is expanded in a power series, and the series is integrated term by term. The result is a power series in Abeakand can be written:
The coefficients t, are given in integral form by: tnb) =
(-1n n!
J
&)dy (1 4 p 2 y V
+
where the limits of integration and definitions of u(y) are as with Equation 8. For the triangular ILS, the integrals can be evaluated analytically ( 3 ) to give explicit expressions for the t,. The first three are:
Other coefficients can be calculated numerically for specific values of p . With the sinc ILS only t o = 1 can be given explicitly. By expanding the logarithm in the definition of Ageak, substituting for T&ak from Equation 9, and collecting like powers of Afpeak,a power series expression for A;eak is obtained and can be written: m
A;eak =
al
a2 =
[tan-;
t z = (In
+
(2p)- In (1 4p2) 4P2 tan-' (2p)
]
2342
*
ANALYTICAL CHEMISTRY, VOL. 47,
NO. 14,
= -tJln
(0.5t:
(10)
10
- tz)/ln 10
(UlAbeak - A;eak)/alAbeak
4P For the sinc2 ILS only t o and t l can be given explicitly as:
to= 1
an(P)(Abeak)'
(12)
Because the series do not converge rapidly, they are useful for calculating resolution errors only if p and Abeak are less than 1. The coefficients a1 and a2, however, provide useful estimates of resolution errors. For small values of Abeak(the initial linear portion of the curves in Figures 2, 3, and 4 ) , the ratio A;eak/Akeak, which represents the resolution error, is given by a l . For the triangular and sinc2 ILS's it is clear from Figures 2 and 3 that errors are smallest in the linear region. Thus, for a dispersion instrument with a triangular ILS or an FT-IR spectrometer using triangular apodization, the minimum error expected in the measurement of the peak absorbance of a Lorentzian line is given by a 1 and simple closed form expressions are presented for this. Because deviations from the initial linear part of the curves may be either positive or negative with the sinc ILS, the same general statement cannot be made in this case, Le., smaller errors may actually be found under some conditions outside the linear region than is predicted by a l . Representative values for a 1 calculated from Equations 10, 11, and 12 or by numerical integration for the sinc ILS are presented in Table I. As an example, for p = 0.5, the maximum values of A;eak/Abeak expected with triangular and sin$ ILS's are 0.88 and 0.84. The estimate of the ratio as given by a1 for a sinc ILS is 0.998. These can be compared to the exact ratios for p = 0.5 and Abeak= 0.5 given earlier. In some types of chemical analysis, the presence of a resolution error may not matter as long as Beer's law is followed experimentally, i.e., as long as A;eak is well represented by alAbeak.As Abeak increases and deviations from linearity occur, the fractional deviation is given by:
to = 1 t l = -In 10
n=l
The first two a, are:
N
(-adal)Abeak
The quantity -a2/a1 is tabulated in Table I. As an example, for p = 0.5 and Abeak = 0.5, the predicted percentage deviations for triangular, sinc2, and sinc ILS's are 1.0, 3.8, and -0.25%. Discussion of Theoretical Results. On the basis of these calculations, it is concluded that resolution errors in FT-IR can be expected to be considerably smaller if the in-
DECEMBER 1975
m-1
crn-'
Figure 5.
Spectra of benzonitrile/CCI4solution measured at nominal 2 cm-' resolution ( A ) not using apodization and (B) using triangular apodi-
zation terferogram is not apodized than if it is triangularly apodized. Errors with a conventional spectrometer with a triangular ILS will be somewhat smaller than FT-IR using triangular apodization (assuming equal values for p ) but much larger than FT-IR without apodization. These conclusions are supported both by the absolute errors as represented by the coefficient a1 or found in the graphs, and also by the deviations from the Beer-Lambert law as calculated using -azlal. An experimental test of the conclusions is reported in the next section. One final point regarding resolution errors in FT-IR should be made before describing the experimental results. One goal of data analysis when seeking photometric accuracy is to remove the effects of instrumental distortions as far as possible. With regard to resolution errors, the goal would be to remove the effects of the ILS from the observed spectrum by deconvolution. The convolution of a true spectrum with an ILS is equivalent to multiplication of the Fourier transform of the true spectrum by the transform of the ILS followed by transformation back to frequency space ( 1 7 ) . In FT-IR then, if the interferogram is assumed to have been collected without distortion, deconvolution of an apodized spectrum (for simplicity the term apodized spectrum will be used to mean a spectrum obtained from a numerically apodized interferogram and similarly for unapodized spectrum) is accomplished by dividing the apodized interferogram by the apodization function which simply regenerates the unapodized interferogram and associated spectrum. T o put it differently, deconvolution of the unapodized spectrum requires dividing the original interferogram from zero to maximum experimental retardation by 1 and dividing it from there to infinite retardation by zero, clearly an impossibility. The higher spatial frequency components in the interferogram are simply not present, and no rigorous method of deconvolution can regenerate them (18) though some efforts at extrapolating to obtain their value have been made (19). A spectrum obtained without apodization, therefore, represents the maximum deconvolution possible, and for this reason alone one might expect the unapodized spectrum to be the more accurate. EXPERIMENTAL MEASUREMENTS
The System. The conclusions of the previous section rest on assumptions inherent in the mathematical model, e.g., the assumption of a Lorentzian line shape, which are only approximately true, especially for liquid samples. In
order to test the validity of the calculations for real systems and also to make a general comparison of the relative usefulness of apodized and unapodized FT-IR spectra for analysis, a large number of absorbance measurements were performed on a model system, a solution of benzonitrile in carbon tetrachloride. The band a t 2231 cm-l is nearly ideal for these measurements for a number of reasons. First, it is well isolated, the nearest major absorption being several hundred wavenumbers away. Second, it appears to be essentially a single band. Though not quite symmetric it conforms quite well to a Lorentzian line shape. Third, its width is narrow enough so that significant resolution errors are observed for typical instrumental resolutions but not so narrow as to preclude obtaining an accurate measurement of the true peak absorbance. Fourth, it lies in a spectral region reasonably clear of atmospheric absorption. Fifth, solvent absorbance is minimal a t this frequency. All quantitative measurements reported were performed on this band. Experimental. All measurements were carried out on samples from a single solution of benzonitrile in carbon tetrachloride, 6.644% w/w. Fifty milliliters of solution was prepared by weighing 5.0859 g benzonitrile (Eastman Organic Chemicals) and 71.4637 g CCl4 (Baker Analyzed Reagent) into a volumetric flask using an analytical balance. Spectral measurements were made over a four-week period, and the absorbance of the solution a t 2231 cm-l measured under identical conditions on the first and last day was identical within experimental error. Sealed cells of 54.1, 81.8, and 207.4 Hm pathlength were constructed by amalgamating metal spacers (20) of appropriate thickness between KCl windows (Wilks Scientific Corp.). A demountable cell with KBr windows (Barnes Engineering Co.) was used to provide a pathlength of 479.8 wm. Pathlengths were determined by counting fringes (20). Spectra were measured on a Digilab FTS-14 Fourier transform spectrometer in single beam mode and ratioed against a stored spectrum of air obtained separately. Apodization of interferograms was performed by the FTS-14 software, an important point which will be discussed later. The spectrometer was purged for water vapor but not COS. From 100 to 400 scans were signal averaged to obtain high signal-to-noise ratio in the spectra, and all calculations were carried out in double precision. Spectra obtained at 0.5, 1, 2, 4, 8, and 16 cm-' nominal resolution (as given by FTS-14 instrument settings and defined specifically below) were 1, 2, 4, 4, 4, and 8 times zero filled (21) to avoid errors due to an insufficient number of data points across the
ANALYTICAL CHEMISTRY, VOL. 47, NO. 14, DECEMBER 1975
2343
Table 11. Results of Measurements on the 2231 cm-' band of Benzonitrile/CCl, Solution Presented as the Ratio A;eak/Akeak -.
t
Apeak
0 . 5 ~cm-I
Calcd
1.0 cm-'
Exptl
Calcd
Exptl
2.0 cm-I
Calcd
4.0 cm-'
Exptl
Calcd
Exptl
0.537 ? .005b Unapodized 1.000 0.995 1.000 0.998 1.000 0.991 1.000 1.001 Apodized 0.971 0.980 0.943 0.965 0.888 0.913 0.786 0.836 0.797 i- .008C 1.000 1.000 1.004 1.000 0.999 1.001 1.006 Unapodized Apodized 0.964 0:987 0.931 0.940 0.868 0.907 0.758 0.832 1.96 i .12d Unapodized 1.000 . . . 1.000 0.987 1.000 0.990 0.997 0.964 Apodized 0.890 0.950 0.816 0.895 0.716 0.797 0.591 0.700 4.7e1f Unapodizedg ... ... ... . . . 0.0 0.0 0.0 0.002 Apodized ... ... ... . . . 0.38 0.53 0.32 0.44 a , ,unapodized 1.000 . . . 1.000 . . . 1.000 . . . 0.997 . . . a,,apodized 0.979 . . . 0.959 . . . 0.916 . . . 0.830 . . . Nominal spectrometer resolution. b 54.4-pm pathlength. C 81.8-pm pathlength. pathlength. f479.8-pm pathlength. g Values are transmittance.
P
Apparent peak absorbance of the 2231 cm-' band of benzonitrile/CCI, solution as a function of the resolution parameter p Flgure 6.
The pairs of curves labeled A to C correspond to measurements with 207.4, 81.8, and 54.1 pm pathlength cells using triangular apodization (A)and not using apodization ( 0 )
peak. Except for measurements of peak width, spectra were plotted on a linear transmittance scale, and the transmittance of the peak at 2231 cm-l was obtained by the baseline method. With the isolated line and flat base line, this method worked well. From the variation in transmittance observed in the numerous replicate samples run, it is believed that the reported values are good to about 0.003T. T o isolate changes in absorbance caused solely by differences in resolution or apodization, several spectra were normally computed from a single interferogram. For example, apodized and unapodized spectra of 2, 4, 8, and 16 cm-l resolution were obtained from a single interferogram. Any differences observed were therefore caused only by the length of the interferogram or its apodization. The size of the data system (256K) precluded calculating all spectra from a single interferogram, but a 4 cm-' resolution spectrum could be computed along with any other and some check for sample-to-sample variation was possible in this way. Because of these procedures, relative absorbances measured in the same cell but with different resolutions or 2344
8.0 cm-I
Calcd
1 6 . 0 cm-'
Exptl
Calcd
Exptl
0.965 0.609
0.978 0.680
0.717 0.378
0.716 0.460
0.985 0.579
1.012 0.665
0.719
0.353
0.717 0.441
0.0004g 0.444
-0.006g
0.563
0.907 0.268
0.913 0.366
0.002 -0.004 -0.004 -0.037 0.25 0.39 0.17 0.29 0.940 ... 0.725 ... 0.667 ... 0.438 ... d 207.4-pm pathlength. e Calculated from
apodizchions are known somewhat more precisely than is indicated by the absolute error estimate given above. In all, approximately 100 spectra were obtained from some 25 samples. Experimental Results. The spectrum from 2350 to 1400 cm-' of the benzonitrile/CC14 solution in the 81.8-wm pathlength cell is presented in Figure 5. The nominal resolution is 2 cm-l. The true FWHH of the 2231 cm-' band was measured from 0.5 cm-' resolution unapodized spectra plotted on an absorbance scale and is 7.49 f 0.05 cm-'. The true widths of the bands a t 1492 and 1448 cm-I are more difficult to measure since they are not completely isolated but are approximately 2.75 and 2.25 cm-I respectively. The exact value of R = l / A a t any nominal resolution of the FTS-14 can be calculated from the number of points in the interferogram, the location of zero path difference in the interferogram (PKLOC in Digilab software), and the distance between data points as determined by the wavelength of the reference He-Ne laser. For nominal instrument resolutions of 0.5, 1, 2, 4, 8, and 16 cm-l, the value of R is 0.4846, 0.9738, 1.967, 4.013, 8.361, and 18.25 cm-l implying values of the resolution parameter p for the 2231 cm-' band of 0.0647, 0.1300, 0.2626, 0.536, 1.116, and 2.436 respectively. The values of p for the 1492 and 1448 cm-l bands a t 2 cm-l resolution are 0.71 and 0.87. In comparing the spectrum in Figure 5A (unapodized) with that in 5B (apodized), an interesting difference is seen. In the former, the 1492 and 1448 cm-' bands appear stronger than the 2231 cm-l band, and in the latter they appear weaker. In fact, the true intensities are essentially as observed in 5A. The intensities as seen in the apodized spectra have been significantly distorted by resolution errors, particularly so for the two narrower lines, and this has led to the apparent reversal in intensities. This effect is, of course, well known with dispersion instruments (8).A spectrum of the sample obtained on a Perkin-Elmer 621 grating spectrometer with mechanical slit widths adjusted according to the instrument manual to produce a 2 cm-l spectral slit width was essentially identical to that of 5B. The complete quantitative results of the measurements on the 2231 cm-' band are presented in Figure 6 and Table 11. The former is a plot of A&,k vs. p for both unapodized and apodized spectra obtained with the three shortest pathlength cells. The dashed line interpolations between the last two points on curves for the results for unapodized spectra represent calculated values. Table I1 presents the true peak absorbance for each pathlength, the resolution
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lma
2wo
yo0
cm-1
Figure 7. Spectrum of benzonitrile/CCI4 solution measured at 4 c m - I nominal resolution without apodization
errors (Akeak/Abeak)as calculated from Equation 8 using those values of Abeak with the appropriate value of p as given above, the resolution errors as estimated by al, and the experimentally determined resolution errors. The values of Abeakwere obtained from Figure 6 as the extrapolation of the measured peak absorbances to zero p . Both the apodized and unapodized spectra extrapolate to the same value of Abeakwithin experimental error and the two results were therefore averaged to obtain the reported value. The error limits given assume an uncertainty in the true peak transmittance of f0.003.The values of Abeak for the three shortest pathlengths follow Lambert's law within that stated uncertainty. The longest pathlength produced a calculated true peak absorbance of 4.7 which, of course, could not be measured experimentally. Discussion of Experimental Results. As seen from Table 11, the agreement between the experimental results for the unapodized spectra and the calculations is excellent, within experimental error in almost every case. As predicted, resolution errors are very small unless p exceeds 1.
For the apodized spectra, the agreement between calculated and experimental resolution errors is not as good. The qualitative predictions of the previous section are, however, borne out. As is clearly seen in Figure 6, resolution errors found in the apodized spectra are much larger than those in the unapodized. Measurable errors are found even a t 0.5 cm-l resolution, the maximum resolution of the FTS-14. The ratio Aapeak/Abeakdecreases with Abeak as well as p as predicted. As a consequence, plots of A&ak vs. cell pathlength exhibit distinct curvature. In addition, there is good agreement between the minimum errors as estimated by a 1 and the experimental results. Beyond this, however, quantitative agreement between the theory and experiment is not found. In general, as is seen in Table 11, the experimental resolution errors are not as large as predicted. For example, for Atpeak= 0.797 and p = 0.5357, the predicted ratio of A:eak/Abeak is 0.758 while the experimental ratio is 0,822, a difference of 7%. The differences are smaller for small p and ALaxand increase with either factor, particularly the former. The deviations between the theoretical and experimental results for the apodized spectra are apparently due to the manner in which apodization is handled by the FTS-14 software. For reasons associated with phase correction of the interferogram ( 2 2 ) , apodization never begins a t zero optical path difference (ZPD) but always slightly beyond.
cm-1
Figure 8. Scale expanded spectrum of the 2231 cm-' band of benzonitrile measured at 1 c m - ' nominal resolution without apodization. True peak absorbance is 1.96
Thus the true apodization function is actually trapezoidal rather than triangular, and the true ILS is not strictly sinc2. The distance between ZPD and the point at which apodization begins is independent of resolution and represents about 4% of the interferogram at 4 cm-l resolution, less at higher resolution (smaller R ) and more a t lower resolution. Since the true apodization is intermediate between triangular and no apodization, the observed resolution error would be expected to be less than that calculated assuming a sinc2 ILS. The deviation between the theoretical and experimental results should be greatest for low resolution spectra as is found. T o verify this interpretation, the following experiment was performed. Two apodized spectra were measured at 8 cm-I resolution using the 81.8-pm pathlength cell. The first was obtained in the normal manner. With the second, half the data points prior to ZPD were skipped before data collection for the interferogram was begun. The result of this is to cut in half the distance between ZPD and the start of apodization bringing the apodization function closer to true triangular. (At the same time a small increase in A occurs which lowers p by about 4%.) It was observed that the ratio of A:eak/Atpeakwent from 0.665 f 0.008 in the first case to 0.625 in the second. The calculated value is 0.579. Since cutting in half the distance between ZPD and the start of apodization halves the deviation between theory and experiment, it appears that essentially all the deviation is due to this single factor. Agreement between the theoretical calculations and experiments performed with a true triangular apodization function would undoubtedly be excellent. One of the advantages of FT-IR spectrometers which has been mentioned in the past is the accuracy with which the ILS is known (23). The excellent agreement between our calculations and experimental results supports this statement. It is of interest to compare resolution errors found in FT-IR with those found using dispersion instruments. The calculated errors for a dispersion spectrometer based on a triangular ILS were somewhat smaller than those calculated and found for FT-IR using triangular apodization. It must be recognized, however, that the true ILS of a dispersion spectrometer is difficult to determine and is only ap-
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Figure 9. Single beam spectra of the 2347 cm-' COP band measured at 1 cm-I nominal resolution ( A ) without apodization and (B)with triangular apodization
proximately represented by a triangle (11). Experimental work seems to indicate that actual resolution errors are larger than those calculated assuming a triangular ILS. For example, Russell and Thompson in their work with dispersion instruments found that for p = 0.5, Ageak/Abeakis approximately 0.82 (7). This is closer to the value of 0.80 calculated for p = 0.5 and Atpeak = 0.5 for a sinc2 ILS than the value of 0.87 calculated for a triangular ILS. It seems then that resolution errors found with triangular apodized FT-IR and conventional dispersion spectrometers will not differ greatly. CHOICE OF APODIZATION FUNCTION From the results presented above, it is clear that there are distinct advantages in not apodizing the interferogram. There are, however, certain disadvantages which must be recognized before deciding whether to apodize or not. First of all, there is the problem of the prominent side lobes of the ILS which are reflected in the spectra when measuring a band whose width is less than R ( p > 1).An example of this is shown in Figure 7 , an unapodized spectrum taken at 4 cm-l resolution in the 81.8-pm cell. The 2231 cm-I band is wider than 4 cm-' and appears normal, but the 1492 and 1448 cm-l bands are narrower and numerous wiggles appear near the base of those lines. Since spectra of 2 cm-l resolution are easily obtained in FT-IR, and bands narrower than this are rare in solutions, this is probably a minor problem for liquid samples. It is also possible under some conditions for the sinc ILS to introduce wiggles near the peak absorption of a band. This can occur for strong bands when R is appreciably less than the band width. For the 2231 cm-I band, such wiggles were observed only at 1 and 2 cm-l resolution. An extreme example is shown in Figure 8 which is a scale-expanded spectrum of the 2231 cm-' band measured a t 1 cm-l resolution in the 207.4 pm cell (Abeak = 1.96). Unless care is taken, such effects might lead to the erroneous conclusion that a peak is split. As far as quantitative measurements are concerned, the additional uncertainty introduced in the determination of Ageak is small compared to the resolution errors of the sinc2 ILS. Finally, it should be noted that compensation for atmospheric C02 and water vapor absorption in ratio-recorded spectra is more difficult with unapodized spectra than with apodized. An example of this is seen in Figure 5 . The disturbance in the base line of the unapodized spectrum to the high frequency side of the 2231 cm-l band is caused by atmospheric COz. These two spectra were calculated from the same interferograms with only the apodization function changed. We have noticed such effects in our spectra before, and the reason for them is now apparent. 2346
With the FTS-14 and most mid-infrared Fourier transform spectrometers, a ratio-recorded spectrum is obtained by separately measuring the spectrum of the sample and reference, e.g., air, and then dividing the former intensities by the latter with the computer to obtain the transmittance spectrum. If the apparent COz (or H20)absorption is different in one spectrum than the other, e.g., because of noise or changing COa content in the air, uncompensated absorption appears in the final spectrum. The strongest lines of the COS spectrum as observed in our spectrometer have very large true peak absorbances. They are also narrow, of the order of 0.1 cm-l at atmospheric pressure which corresponds to a p of 20 at R = 2 cm-l. When such lines are observed in unapodized single beam spectra, they appear very strong. The calculation of a transmittance spectrum therefore involves calculating a t the COn frequencies the ratio of two intensities near zero, a procedure which is extremely sensitive to noise or small differences in CO2 absorption in the two spectra being compared, and uncompensated absorption often appears in the transmittance spectrum. When apodized spectra are used, however, large resolution errors occur for the COz lines. This prevents the apparent intensities from going to zero at the COa absorptions and even minimizes differences in apparent intensities which are caused by real differences in absorption in the two spectra. Taken together, these effects make effective compensation likely. The difference between apparent atmospheric COz absorption as seen in unapodized and apodized single beam spectra is shown in Figure 9. These are typical spectra a t a nominal resolution of 1 cm-I of the 2347 cm-l band of COz as it occurs in the optical path of the spectrometer. Absorption by CO2 containing the 13C isotope is seen to the low frequency side of the main band. The apodized spectrum of C02 is more regular and much less intense than the unapodized spectrum. The intensity in the latter goes to zero or even becomes negative at some lines. It is also interesting to observe the marked effect of the ILS in the unapodized spectrum. The oscillation observed on the high frequency side of the spectra extending out of the figure is not due to absorption lines but is an artifact of the ILS. Apparently, at this resolution, the secondary lobes from the various lines in the R branch reinforce each other to produce the pronounced effect, It is almost absent a t 0.5 cm-I resolution and at 2 cm-1 the oscillations on the R branch side are smaller but oscillations appear on the P branch side where the vibration-rotation lines are more widely spaced. CONCLUSION Within the past year, three mid-infrared FT-IR spectrometers have been introduced for which the maximum
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resolution available is 2 cm-l. Even with instruments capable of higher resolution, it is common to work with 2 to 4 cm-l resolution to decrease scanning and computational time. In solution spectra, bands with FWHH of 2 to 5 cm-' are not uncommon (8,11,24). For a 4 cm-' wide band measured with R = 2 cm-l, the results of this work indicate that the measured peak absorbance will be at least 16% low if triangular apodization is used. In this case, a considerable increase in accuracy is possible if the interferogram is not apodized. Triangular apodization is only one of many forms possible (12). Apodization somewhere between triangular apodization and no apodization at all would introduce smaller resolution errors than the former but not have the disadvantages of the latter.. Such a function might be an excellent compromise for analytical work. LITERATURE CITED (1) D. M. Dennison, Phys. Rev., 31, 503 (1928). (2) D. A. Ramsay, J. Am. Chem. SOC.,74, 72 (1952). (3) J. R. Nielsen, V. Thornton, and E B. Dale, Rev. Mod. Phys., 16, 307 (1944). (4) H. J. Kostkowski and A. M. Bass, J. Opt. SOC.Am., 46, 1060 (1956). (5) S. Brodersen, J. Opt. SOC.Am., 44, 22 (1954) (6) R. N. Jones, D. A. Ramsay, D. S. Keir, and K. Dobriner, J. Am. Chem. SOC.,74, 80 (1952). (7) B. A. Russell and H. W. Thompson, Spectrochim. Acta, 9, 133 (1957). (8) H. J. Sloane, Appl. Spectrosc., 16, 5 (1962).
(9) J. Morcillo, J. Herranz, and M. J. de la Cruz. Spectrochim. Acta, 15, 497 (1959). (10) W. J. Potts. Jr. and A. L. Smith, Appl. Opt.. 6, 257 (1967). (11) K. S. Seshadri and R. N. Jones, Spectrochim. Acta, 19, 1013 (1963). (12) P. R. Griffiths, "Chemical infrared Fourier Transform Spectroscopy", Wiley Interscience, New York, 1975. (13) H. A. Lorentz, K. Ned. Akad. Wet. Proc., 8 , 591 (1906). (14) P. R. Griffiths, C. T. Foskett, and R. Curbelo, Appl. Spectrosc. Rev.. 6, 31 (1972). (15) D. C. Champeney, "Fourier Transforms and Their Applications." Academic Press, New York, 1973, p 20. (16) IBM Svsternl360 Scientific Subroutine Packaae. Version Ill. Subroutine DQG32. (17) G. Horlick, Anal. Chem., 43(8), 61A (1971). (18) G. Horlick, Anal. Chem., 44, 943 (1972). (19) R. L. Kirlin and A. M. Despain, Air Force Cambridge Research Laboratories. ReDort No. AFCRL-69-0039 11968). (20) W. J. Piice, in "Laboratory Methods in infrared Spectroscopy", 2nd ed., R . G. J. Miller and B. C. Stace, Ed., Heyden and Sons, Ltd., London, 1972, p 97. (21) P. R. Griffiths, Appi. Spectrosc., 29, 11 (1975). (22) L. Mertz. InfraredPhys., 7, 17 (1967). (23) T. Hirschfeld, Paper 307, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, Ohio, 1975. (24) L. W. Herscher, Spectrochim. Acta, 15, 901 (1959).
RECEIVEDfor review May 19,1975. Accepted September 5, 1975. The authors gratefully acknowledge the financial support provided by the National Science Foundation Grant GP-38728X. This work was presented in part at the 30th Symposium on Molecular Structure and Spectroscopy, Ohio State University, June 20, 1975.
Substituted Benzophenone as Fluorometric Reagent in Automated Determination of Nitrate B. K. Afghan and J. F . Ryan Analflical Methods Research Section, Canada Centre for Inland Waters, P. 0. Box 5050, 8 6 7 Lakeshore Road, Burlington, Ontario, Canada L7R 4A6
Substituted benzophenones, under appropriate reaction conditions, react to form strong fluorescent species with a number of Ions such as boron, vanadium, chromium, and nitrate. The proposed method utilizes 2,2'-dlhydroxy-4,4'-dimethoxybenzophenone as a new and sensitive fluorometric reagent for the determination of nitrate. The procedure to eliminate possible Interferences from high concentrations of chloride, sulfide, and humic acid substances Is also incorporated In the automated method. The proposed method has been applied to a wide variety of natural waters and sediments. The analysis may be performed at a rate of 20 samples per hour. The method can be used to detect nitrate as low as 5 wg/llter nitrate-nitrogen. The proposed method has also been compared with the most wldely used colorimetric method for the determination of nitrate.
It is generally accepted that the majority of substituted benzophenones tend to produce phosphorescence instead of fluorescence (1-3). The main reason for this is that the majority of these compounds possess the lowest excited singlet state of (n,r'+) character; therefore, intersystem crossing to triplet manifold is usually very efficient. It is also well established that certain environmental factors such as substitution, solute-solvent interaction, nature of the catalyst, etc., can alter these compounds and the nature of transition, energies, and intensity of luminescence ( 4 ) .
Acetophenone and related compounds are known to react in concentrated sulfuric acid, and produce polymeric species (5). Therefore, it is possible to alter the transition probabilities of these molecules by changing the reaction conditions, suitable substitution, solvent, etc., and produce species which may result in fluorescence instead of phosphorescence. In fact, in our laboratories, it was found that benzophenone, generally considered to produce phosphorescence, was made to produce strong fluorescence when dissolved in concentrated sulfuric acid. In addition to that, it was also found that the fluorescence of various substituted benzophenones was markedly affected by the nature and position of substituents in the benzophenone molecule. It was further observed that the fluorescence of some substituted benzophenones in the presence of trace quantities of other ions such as boron, nitrate, chromate, and vanadate was enhanced considerably. Therefore, it was decided to systematically investigate these compounds as possible fluorometric reagents for the analysis of various constituents in water, waste waters, and sediments. The fact that these reactions can be carried out in a media with relatively high acid concentration offers an additional advantage, particularly when analyzing sediments or other solid samples. If these reagents can be optimized to selectively determine various contaminants, they will prove advantageous in the routine analysis of sediments, since the error due to contamination will be minimized considerably. During the analysis of sediments, the majori-
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