761
Anal. Chem. 1986, 58. 761-765
polarization-encoded and crossed-beam thermal lens systems. It is simple, free from interference noise, applicable to short as well as long path length samples, and does not necessitate expensive polarizing optics. In comparison to the second harmonic approach, it is superior in performance in terms of signal-to-noise ratio. To achieve better performance with the second harmonic technique, more effective fundamental signal filtering is needed in the lock-in amplifier. Although the propagation-encoded system is somewhat more elaborate to construct, once set up and aligned, it should either remain in alignment indefinitely (19) or need only minor adjustments. 0
2
4
6
B
O
2 4 TIME I M I N I
6
8
0
2
4
6
8
Flgure 2. Comparlsorl of the liquid chromatograms of 51 ng of 0 nitroaniiine obtained from (a) propagation-encoded,(b) polarization-
encoded, and (c) second harmonic thermal lens detection systems: laser power, 7 m W modulation frequency, 50 Hz; flow rate, 1 mL/min. system. The loss of SIN ratio is about a factor of 4. The sloping base line observed in the figure is, in part, a result of the noisy chopper employed. Because the installed notch filter has a very narrow frequency bandwidth, any drift in the modulated frequency would result in a variation in the residual (fundamental) background signal and, consequently, a drift in the base line. Our chopper tends to stabilize somewhat at higher frequencies. Therefore, as the chopping frequency was increased to 150 Hz,which is the upper limit set by the available matched capacitor pair in the notch filter, the base-line drift was alleviated substantially; i.e., the base-line stability approaches those shown in Figure 2a,b. However, the noise levels in both the single-laser/dual-beam systems appear to remain invariant, indicating that the dual-beam system is somewhat less susceptible to chopping noise. Presently, the principal noise in the propagation-encoded system arises from turbulence in the cell and flow pulsation (20,22). The former can be reduced through refinement in cell design. The latter can be alleviated by using surgeless pumps or applying the differential thermal lens technique (23). CONCLUSIONS The arrangement described here combines the best of both
ACKNOWLEDGMENT We thank A. Keith Jameson for helpful discussions. LITERATURE CITED (1) Gordon, J. P.; Leite, R. C. C.; Moore, R. S.;Porto, S. P. S.;Whinnery, J. R. J. Appl. Phys. 1985, 36, 3. (2) Dovichi, N. J.; Harris, J. M. Anal. Chem. 1980, 52,2338. (3) bovichi, N. J.; Harris, J. M. Anal. Chem. 1981, 53, 106. (4) Leach, R. A.; Harris, J. M. J. Chromafogr. 1981, 278, 15. (5) Leach, R. A.; Harris, J. M. Anal. Chem. 1984, 56, 1481. (6) Miyaishi, K.; Imasaka, T.; Ishibashi, N..Anal. Chem. 1982, 54, 2039. (7) Mori, K.; Imasaka, T.; Ishibashi, N. Anal. Chem. 1982, 54, 2034. (8) Buffett, C. E.; Morris, M. D. Anal. Chem. 1982, 54, 1824. (9) Buffett, C. E.; Morris, M. D. Anal. Chem. 1983, 55,376. (IO) Nolan, T. G.; Weimer, W. A.; Dovichi, N. J. Anal. Chem. 1985, 56, 1984. (11) Higashi, T.; Imasaka, T.; Ishibashi, N. Anal. Chem. 1984, 56, 2010. (12) Long, M. E.; Swofford, R. L.; Albrecht, A. C. Science 1978, 797, 183. (13) Alfheim, J. A.; Langford, G. H. Anal. Chem. 1985, 57,861. (14) Long, G. R.; Bialkowski, S. E. Anal. Chem. 1984, 56, 2806. (15) Nakanishi, K.; Imasaka, T.; Ishibashi, N. Anal. Chem. 1985, 57, 1219. (16) Jansen, K. L.; Harris, J. M. Anal. Chem. 1985, 57, 1698. (17) Pang, T. K. J.; Morris, M. D. Anal. Chem. 1984, 56, 1467. (16) Yang, Y. Anal. Chem. 1984, 56,2336. (19) Pang, T. K. J.; Morris, M. D. Appl. Spectrosc. 1985, 39, 90. (20) Yang, Y.; Hairrell, R. E. Anal. Chem. 1984, 56, 3002. (21) Berthoud, T.; Delorme, N.; Mauchien, P. Anal. Chem. 1985, 57,1216. (22) Dovichi, N. J.; Harris, J. M. Anal. Chem. 1981, 53,689. (23) Pang, T. K. J.; Morris, M. D. Anal. Chem. 1985, 57, 2153.
RECEIVED for review August 5,1985. Accepted November 22, 1985. This research was supported by grants from Loyola University of Chicago Summer Research Grant, Research Stimulation Fund, and Small Research Grant.
Pulsed Photothermal Refraction Spectrometry Using an Elliptic Gaussian Excitation Beam Norio Teramae,' Edward Voigtman, Jose Lanauze, and James D. Winefordner* Department of Chemistry, University of Florida, Gainesville, Florida 32611 The use of a one-dimensional heat source in pulsed photothermal refraction spectrometry has been studied both experimentally and theoretically. The one-dlmensional heat source is obtained by focusing the excltatlon beam with a cylindrical lens. The sensltlvlty Is shown to be superior to that obtained by uslng a two-dimensional heat source. A minlmum detectable absorptlvlty of 9 X IO-' cm-' was obtained for amaranth In methanol.
The application of lasers to analytical spectrometryhas been one of the most exciting advances of recent years (1).Whole 'On leave: D e p a r t m e n t of I n d u s t r i a l Chemistry, F a c u l t y of Engineering, T h e U n i v e r s i t y of Tokyo, Bunkyo-ku, T o k y o 113, Japan.
new areas have been developed that would not be possible with conventional light sources. Since the first detailed description by Gordon et al. (2), the thermal lens effect has developed into a powerful analytical technique for high-sensitivity absorption measurements (3-5). The effect is related to the spatial variation of the refractive index that results from the localized temperature increase caused by absorption of the excitation source. The variation in the refractive index can be detected by phase fluctuation spectrometry (6, 7), thermal diffraction spectrometry (8,9),thermal lensing spectrometry (10-19), photothermal deflection spectrometry (20-22), and photothermal refraction spectrometry (23,24). The optical setup for photothermal refraction is similar to the one for transverse photothermal deflection, but the detection method of this technique can be related to the one for thermal lens
0003-2700/66/0358-0761$01.50/00 1966 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
spectrometry. The theoretical background for the photothermal refraction technique has been described in the literature (23). The photothermal refraction signal increases as the spot size of the pump beam decreases. The signal intensity is also proportional to the excitation energy. Thus, it is expected that a higher signal would be obtained by using a laser source with higher power and by focusing the excitation beam. In practice, however, this is not always possible since too high a power density might damage the optical elements and lead to chemical and/or physical alteration of the sample. Therefore, it would be desirable to use a low-power-density source without a loss of sensitivity. In this paper, some experimental and theoretical results are presented about the photothermal effect based on a one-dimensional heat source. Comparison of the effect based on a one-dimensional excitation source with the one based on a heat source having a Gaussian intensity profile is also discussed. The sample used in this study was amaranth, which has been recently banned for use in foods, drugs, and cosmetics because the dye is suspected to be a teratogen. Therefore, trace determinations of the dye are important.
If we consider the case of thermal lensing detection, the region of interest is near the optical axis. Thus, we can neglect the exponential term depending on the y coordinate, and we can rewrite eq 1 as
Since a cylindrical lens transforms a two-dimensional beam into a “one-dimensional” beam, the heat conduction in the y direction can be assumed to be negligible near the optical axis. Green’s function for the one-dimensional heat conduction, G(x, x’, t? (cm-l), is given by
where D is the thermal diffusivity (cm2s-l) and is given by D = k / p C , where h is the thermal conductivity (J cm-l K-I), p is the density (g ~ m - ~and ) , C is the heat capacity (J g-l K-l). Temperature Rise. The temperature rise is given by
THEORY In the theoretical model for the photothermal effect, the pump and probe beams have been assumed to have Gaussian intensity profiles (25-28). However, the actual beam profile of laser beams do not always have Gaussian profiles. The beam quality of pulsed lasers is often very poor, and an elliptic Gaussian beam is obtained as an output from the laser cavity. The thermal lens effect with an elliptic Gaussian beam was discussed briefly by Fang and Swofford (26). Until now, a rotationally symmetric lens has been used to focus the excitation beam. Hence, a circularly symmetric heating region is formed within the sample. On the other hand, in the recent experiments on photothermal deflection spectrometry (29,301, a cylindrical lens has been used to focus the pumping beam in one dimension on a TLC plate. Focusing in one dimension was reported to give noisy results because of irregularities of the TLC plate (29). Heating a sample in one dimension is interesting, but no theoretical model has been reported. In this section, a theoretical analysis for the photothermal effect based on the one-dimensional heating is described. The mathematical expressions for the thermal lensing effect are well-documented for both the CW (2, 25, 26) and the pulsed (27,B) excitation cases. The theoretical analysis shown in this section is similar to the previous two-dimensional heating squrce models. The main differences between the present and previous models are in the expression of both the initial temperature distribution and the propagation function describing the temperature rise in an infinite medium. Heat Function and Green’s Function. If the absorbed energy within a sample dissipates completely as heat, the heat generated, Q ( x , y) (J s-l cmw3),per unit volume in the sample by the absorption of the elliptic Gaussian beam (w: >> w,’) is given by
where Psb is the total power absorbed in the sample per unit length, J s-l cm-l, w xand w ydenote the 1f e2 spot size of the beam in the x and y directions. Pabscan be approximated as Poa for the CW excitation and Ha for pulsed excitation, where a is the absorbance per unit length, Po is the power (J s-l) of the heating beam entering the sample, and H is the total effective heating energy (J)per laser pulse for the pulsed laser. The parameter H i s defined as H = J ~P(t) P dt, where t , is the laser pulse duration and P ( t ) represents the temporal shape of the laser pulse power.
In the case of pulsed excitation, it should be noted that the time integration over the duration of the heating beam is explicitly included in the function Q ( x 9. After substitution of eq 2 and 3 into eq 4,the integration about the x coordinate may be evaluated through use of the following relation (31):
1:
exp(-p2x2 + q x ) dx = ( & / p )
(5)
exp(q2/4p2)
Then,
1
AT(x, t) =
1
+ 8Dt’/wX2
The remaining integral is simplified by the transformation E = (w2+ 8Dt9-I, using the following relation (32):
(7) and also, the temperature rise is given by
AT(x, t) =
($)(:)I
L
(1 2+’):
X
]
(8)
where t, is the characteristic time constant and is given by t, = w,2/4D. By use of a series representation for the error function, erf, and ignoring the terms that do not contribute to the formation of the thermal lens, the temperature rise for CW excitation can be written as
AT(%, t ) =
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
:)I(
(z)( dT
- 1)
-
In the case of pulsed excitation, the temperature rise is obtained from eq 6 and is given by
Focal Length. Based on the paraxial approximation, and assuming that the temperature rise induced in the sample is very small, the focal length, f , of the thermal lens within the sample can be derived from the following equation (5):
f = -[1 (dn/dT) (d2T/d~2),=o]-1
(11)
where 1 is the thickness of the sample (cm). Thus, the focal length can be calculated by using eq 9-11. The focal length of the thermal lens excited by a CW laser is given by
can be evaluated in terms of focal length. By comparison of eq 12 with eq 14 and eq 13 and eq 15, it is seen that the thermal lensing effect of the one-dimensional heat source is weaker than that of the Gaussian heat source by a factor of 2w2/w,w, for the CW excitation and by a factor of w4/wX3wy for the pulsed excitation. Since wy is larger than w, and w, is almost equivalent to w, the thermal lensing effect based on one-dimensional heat source gives a lower sensitivity. In addition, it can be expected from eq 12 and 13 that the temporal variation of the signal for a one-dimensional heat source is slower than that for the Gaussian heat source. On the other hand, the one-dimensional heat source can be used for photothermal refraction spectrometry without a loss of sensitivity. The heat function for photothermal refraction based on a elliptic Gaussian excitation beam is given by eq 1. In this case, the exponential term depending on the y coordinate cannot be neglected, since the heated and probed region of interest is extended along the y axis. The temperature rise and the focal length were calculated for two cases. In the first case, the heat conduction in the y direction was taken into account. In this case, the problem to be solved is almost the same as the usual TEMm beam. The solution can be obtained easily, if we assume that the radical coordinate is separated into vertical ( x ) and horizontal (y) dimensions, that is r2 = x 2 + y2. Thus the focal length can be expressed by the same equation given by eq 17 of Dovichi’s theory (23). In the other case, the heat conduction of the y-direction was neglected. In this case, the differential inverse focal length for pulsed excitation can be expressed from eq 1 and 13,
akw,wyt, 2Hcu In the case of the pulsed laser excitation, the focal length is given by
(&)( dT
1+
f2( t )=
(&)( akw2
$)(1
+ tc/2t)-1
(14)
where w is the beam radius and represents the l / e 2 spot size of the beam and t, is given by t, = w2/4D. When the same kind of heat source is used, the focal length of the thermal lens excited by a pulsed laser is given by (27)
f ( t )=
(=)(
akw2tc
$)(1
+ 2t/t,)-2
(15)
The signal for the thermal lens for CW laser excitation can be written in the approximate form (4)
AI/I(t=m)= -2Z/f+) (16) where 2 is the cell position with respect to the beam waist position and LIZ = -I+.+ The signal in a pulsed thermal lens can be written in the approximate form (27) M / Z ( t = O ) = -2Z/f,t=o, (17) From eq 16 and 17, the signal of the thermal lensing effect
: ) 2
exp( -
5)
dy
... (18)
If we integrate eq 18 over the probe beam pass, from --oo to +a, then
-1= 2Ha f &kw,tc Comparison of Photothermal Effect of One-Dimensional Heat Source with That of Gaussian Heat Source. In the case of a heat source having a Gaussian intensity profile, the focal length of the thermal lens excited by a CW laser is given by (25, 16)
763
(g)(1+;)12 (19)
Thus, the final results for both cases were expressed by the same equations given by Dovichi et al. (23). Photothermal refraction based on an elliptic Gaussian excitation beam is expected theoretically to give the same sensitivity independent of the ellipticity. It should also be noted in the expression for the signal intensity for photothermal refraction that there is no term depending on the interaction length for the pump and probe beams. Thus, it is expected that the elliptic Gaussian laser beam with low power density could be used as an excitation source for the photothermal refraction spectrometry with no loss of sensitivity.
EXPERIMENTAL SECTION Apparatus. Figure 1 shows a block diagram of the experimental setup used. The pumping source for the dye laser was a Molectron UV-24 nitrogen laser that was operated at a repetition rate of 20 Hz. A quartz plate was placed between the nitrogen and the dye lasers and was used to split a small portion of the pumping beam to a fast photodiode (PDl), which sent a trigger to a boxcar averager. The dye laser used was a Molectron DL-I1 dye laser, and the laser dye was Coumarine 480 (7 x lo9 M ethyl alcohol solution). The dye laser beam was reflected by mirrors onto a cylindrical lens with a 50-mm focal length, which focused the excitation beam into a 1-cm-squarequartz cuvette cell containing the sample solution. The cylindrical lens transformed a two-dimensional beam into a “one-dimensional”beam. Thus, a split-shaped heating region was formed within a sample. A peak power of 0.38 mJ/pulse at the sample position was measured with a Molectron 53-05 SW pyroelectricjoule meter. A 0.5-mW He-Ne
764
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
I-r laser
Photc
s
laser
He-Ne laser
4 Lz ~~Sa"Ple
Mirror
I> Light
trap ,Filter
Mirror
(A)
. rigger
I
Recorder
I
IVI Oscilloscope
(6)
Figure 2. Photothermal refraction signals for the (A) one-dimensional heat source produced by a cylindrical lens and (B)two-dimensional heat source produced by a spherical lens. Sample used is amaranth of 6 ppm (1 X M) methanol solution.
Figure 1. Block diagram of the apparatus for pulsed photothermal
refraction.
laser, Aerotech OEM05P, was used as the probe beam. The probe beam was focused with a spherical lens (L2,focal length 125 mm), and was aligned so that the optical axis of the pump and probe lasers had a crossed configuration and the probe beam passed through the slit-shapedheating region. The intensity of the central portion of the probe beam was measured by a silicon photodiode (PD2), Hamamatsu 81188-06. The photocurrent from PD2 was amplified by a Thorn EM1 Model A1 current-to-voltageconverter. The converter output was amplified by a Keithley 103A nanovolt amplifier with a band pass filter (10-100 kHz). The amplifier output was acquired with a PAR 1641162 boxcar averager with 10-ps integrator time constant, 1-sfilter time constant, and gate duration of 0.5 1s. The output from the boxcar averager was recorded by a strip chart recorder. Reagents. Coumarine 480 was purchsed from Exciton Chemical Co., Inc. Amaranth (dye content 95%) was purchased from Aldrich. Methyl alcohol was reagent grade.
RESULTS AND DISCUSSION The output signal from the current-to-voltage converter was monitored by an oscilloscope. The rise time of the currentto-voltage converter was less than 20 bs, which was sufficient to measure the photothermal refraction signal. The transient signal of the pulsed photothermal refraction had a minimum value instantaneously after excitation, and then the signal gradually reached zero. The signal recovered within approximately 50 ms so that the repetition rate of the laser could be increased up to 20 Hz. To evaluate the detection sensitivity of the present technique, the dependence of the signal intensity on the sample concentration was studied using amaranth as a sample. The detection sensitivity of photothermal refraction based on a one-dimensional heat source was compared with the one of photothermal refraction based on a two-dimensional heat source with a Gaussian intensity profile. In this experiment, a cylindrical lens, L,, in the experimental diagram shown in Figure 1was replaced with a 50-mm-focal-lengthspherical lens. The optical alignment was readjusted to obtain a maximum signal. A slight vertical and horizontal translation of lens L1 was required for adjustment. As a result, an unexpected decrease of the signal intensity was observed. An example of the results is shown in Figure 2. In this figure, the signal on the left side was obtained by focusing the excitation beam with a cylindrical lens. The signal on the right side was obtained by focusing the excitation beam with a spherical lens.
I
I
-8
,
1 1 1 1 1 1 1 1
-7
, , ,111111 , -6
I
l , , , , ,
I
-5
I
I , , ,
,,I
-4
log [Amaranth](moles/liter)
Figure 3. Comparison of analytical calibration curves for amaranth in methanol obtained by using a one-dimensional heat source (A) and a two-dimensional heat source (6).
Both lenses have almost the same focal length (50 mm). I t can be easily seen from the figures that focusing the excitation beam in one dimension gives about 3 times higher signal intensity than the conventional focusing method. The analytical calibration curves of amaranth were obtained for both focusing methods. The concentration range measured was from 24 ppb (4X lo-* M) to 30 ppm (5 X M) for the focusing with a spherical lens and from 9.6 ppb (1.6 X M) to 30 ppm (5 X M) for the focusing with a cylindrical lens. The results are shown in Figure 3. In the former case, the curve was linear up to about 12 ppm. The minimum detectable absorptivity a t a signal-to-noise ratio of 21 was 2 X lo4 cm-l. As for the latter case where the excitation beam was focused in one dimension, a linear relationship was obtained up to ca. 4 ppm as shown in Figure 3a. Based on a limiting signal-to-noise ratio of 21, the minimum detectable concentration was 6.7 X lo4 M (4 ppb) for amaranth in methyl alcohol. This value corresponds to an absorptivity of 9 X 10" cm-l. The limit of detection for the one-dimensional heat source was 3 times better than that for the two-dimensional heat source, although theory predicts the same signal intensity for both heat sources. Unfortunately, the discrepancy between the theory and the experimental results cannot be made at the present stage, but the photothermal refraction based on
ANALYTICAL CHEMISTRY, VOL. 58, NO. 4, APRIL 1986
the one-dimensional heat source certainly gives good sensitivity. The detection limit for amaranth reported here compares favorably with the detection limit obtained by resonance Raman spectrometry (33,34). A detection limit of 24 ppb (8.2 x cm-l a t 488 nm) for amaranth was reported by resonance Raman spectrometry using a 200-mW (488 nm) argon ion laser (33). The limit of absorptivity for amaranth reported by means of laser-induced photoacoustic spectrometry (LIPAS) was found to be 2.4 X cm-l using a 300-mW (488 nm) argon ion laser (35). This value is of the same order as the detectivity obtained in this study. As for the comparison of the detection limit between pulsed and CW laser photothermal refraction, the detection limit for pulsed excitation reported here compares favorably with the detection limit for photothermal refraction based on a stable CW excitation; a detection limit of 4.8 X cm-l was obtained for cobalt chloride in methyl alcohol using a 4-mW He-Cd laser (24). As for thermal lensing spectrometry based on pulsed laser excitation, two analytical studies have been reported for the visible range. By use of a dye laser with an output power of 0.02 mJ/pulse at 3 Hz,a detection limit of 4.7 x lo4 cm-l was calculated for tetraphenylporphine in chloroform (16). When a dye laser (1mJ/pulse) pumped by a XeCl excimer laser was used, the minimum detectable absorptivity of 3 x cm-l was obtained for uranium(V1) in aqueous solution (18). Taking the pulse energy into account, these values can be estimated to be in the same order compared with the value obtained by the pulsed photothermal refraction presented here. In the experimental diagram shown in Figure 1,a nanovolt amplifier with a filter unit was used for signal processing. This amplifier is not an essential one. However, the use of the amplifier was required to eliminate a large de drift on the photothermal refraction signal. The use of a more stable laser system and a low-noise signal processing system might improve the sensitivity of photothermal refraction based on the onedimensional heat source. Registry No. Amaranth, 915-67-3.
LITERATURE CITED (1) Kliger, D. S., Ed. “Ultrasensitive Laser Spectroscopy”; Academic Press: New York. 1983.
765
(2) Gordon, J. P.; Leite. R. C. C.; Moore, R. S.; Porto, S. P. S.; Whinnery, J. R. J. Appl. Phys. 1985, 36,3-8. (3) Kliger, D. S. Acc. Chem. Res. 1980, 13, 129-134. (4) Harris, J. M.; Dovichi, N. J. Anal: Chem. 1980, 52,695-708A. (5) Fang, H. L.; Swofford, R. L. I n “Ultrasensitive Lasar Spectroscopy”; Kliger, D. S., Ed.; Academic Press: New York, 1983; 176-233. (6) Smith, D. C. I€€€ J. Quantum Electron. 1989, Q E - 5 , 600-607. (7) Davis, C. C.; Petuchowski, S. J. Appl. Opt. 1981, 20, 2539-2554 1981, 20, 4151. (8) Peiietier, M. J.; Thorsheim, H. R.; Harris, J. M. Anal. Chem. 1982, 54, 239-242. (9) Peiietier, M. J.; Harris, J. M. Anal. ch8m. 1983, 55, 1537-1543. (IO) Miyaishi, K.; Imasaka, T.; Ishibashi, N. Anal. chef??. 1982, 54, 2039-2044. (11) Carter, C. A.; Harris, J. M. Appl. Spectrosc. 1983, 37, 166-172. (12) Carter, C. A.; Harris, J. M. Anal. Chem. 1983, 55, 1256-1261. (13) Carter, C. A.; Harris, J. M. Anal. Chem. 1984, 56, 922-925. (14) Yang, Y. Anal. Chem. 1984, 56, 2336-2338. (15) Fujiwara, K.; Lei, W.; Uchikl, H.; Shimokoshi, F.; Fuwa, K.; Kobayashi. T. Anal. Chem. 1982, 5 4 , 2026-2029. (16) Mori, K.; Imasaka, T.; Ishibashi. N. Anal. Chem. 1982, 54, 2034-2038. (17) Mori, K.; Imasaka, T.; Ishibashi, N. Anal. Chem. 1983, 55, 1075-1079. (18) Berthoud, T.; Mauchien, P.; Omenetto, N.; Rossi, G. Anal. Chlm. Acta 1983, 153,285-269. (19) Long, G. R.; Bialkowski, S. E. Anal. Chem. 1984, 56, 2806-2811. (20) Jackson, W. B.; Amer. N. M.; Boccara, A. C.; Fournier, D. Appl. Opt. 1981, 20, 1333-1344. (21) Aadmodt, L. C.; Murphy, J. C. J. Appl. Phys. 1983, 54, 581-591. (22) Seii, J. A. Appl. Opt. 1984, 23, 1586-1597. (23) Dovichi, N. J.; Nolan, T. G.; Weimer, W. A. Anal. Chem. 1984, 56, 1700-1 704. (24) Nolan, T. G.; Weimer, W. A.; Dovichi, N. J. Anal. Chem. 1984, 56, 1704-1707. (25) Whinnery, J. R. Acc. Chem. Res. 1974, 7 ,225-231. (26) Fang, H. L.; Swofford, R. L. J. Appl. Phys. 1979, 50, 6609-6615. (27) Twarowski, A. J.; Kliger, D. S. Chem. Phys. 1977, 20, 253-258. (28) Bailey, R. T.; Cruickshank, F. R.; Pugh, D.; Johnstone, W. J . Chem. Soc., Faraday Trans. 2 1980, 76,633-647. (29) Chen, T. I.; Morris, M. D. Anal. Chem. 1984, 56, 1674-1677. (30) Masulima, T.; Sharda, A. N.; Lloyd, L. B.; Harris, J. M.; Eyring, E. M. Anal. Chem. 1984, 56, 2975-2977. (31) Gradshteyn, I.S.; Ryzhik, I.M. “Table of Integrals, Series, and Products Corrected and Enlarged Edition”; Academic Press: New York, 1980; p 307. (32) Gradshteyn, I.S.; Ryzhik, I.M. “Table of Integrals, Series, and Products Corrected and Enlarged Edition”; Academic Press: New York, 1980; p 337. (33) Higuchi, S.; Tanaka, J.; Tanaka, S. J. Spectrosc. SOC. Jpn. 1978, 27. 353-359. (34) Stobbaerts, R. F.; Van Haverbeke, L; Herman, M. A. J. Food Scl. 1983, 48, 521-525. (35) Oda, S.; Sawada, T.; Kamada, H. Anal. Chem. 1979, 57, 686-688.
RECEIVED for review July 15, 1985. Accepted November 1, 1985. This research was sutmorted bv Grant NIHGM11373-22.
