Anal. Chem. 1888, 6 0 , 1224-1230
1224
the absolute R, values of reference materials, more sophisticated equipment is needed, and hence, there are only very few reference materials for which these values have been determined (7). Thus the reported method is a useful tool for the analytical chemist to perform quantitative diffuse reflectance infrared Fourier transform measurements at low concentrations. ACKNOWLEDGMENT We dedicate this paper to Professor Dorfel on the occasion of his 60th anniversary.
LITERATURE CITED (1) Fuller, M. P.; Grlffiths, P. R. AM/. Chem. 1978, 50, 1906. (2) Kuehl, D.; Grlffhs, P. R. J . Chromatogr. Sci. 1979, 17, 471. (3) Kortum, 0.Reflectance Spectroscopy; Springer: HeMelberg, 1969. (4) Hecht, H. G. Modern Aspects of Reflectance Spectroscopy; Plenum: New York, 1968. (5) Kortum, G.; Braun, W. Z . Phys. Chern. (Munich) 1966, 4 8 , 282. (6) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969. (7) Richter, W. Appl. Spectrosc. 1983, 3 7 , 32.
RECEIVED for review August 7, 1987. Accepted January 27, 1988.
Analysis of Inorganic Powders by Time-Wavelength Resolved Luminescence Spectroscopy Edgar F. Paski and M. W. Blades*
Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1 Y6
A spectrometer for measuring time-wavelength resolved lumlnescence spectra was constructed. A gated Integrator was used for tlme resolutlon. W i d samples of CaWO, C a m , , SrWO,, and SrMoO, were exclied at room temperature by an exchner laser at 193 and 248 m. The substances examined showed fkst order decay khetlcs and the spectra were broad, featureless Gausslan bands. The iumlnescence spectral envelope was described by the parameters ilfetlme, peak maxima, peak half-width, and lntenslty factor. Data reduction was carried out by using a llnear algebra construct and slmplex optlmizatlon. The atgorlUm was evaluated wlih synthetic data. For twocomponent mixtures, overlaps greater than 0.3 half-wldths in the spectral domain and llfethne ratios greater than 1:1.3 were resolveii and parameter values were estlmated wlth an error of less than f2%.
Luminescence spectroscopy is a very useful and powerful technique for the analysis of materials, especially when a substance must be examined either remotely or nondestructively. In many situations, the object probed does not undergo permanent changes in its composition or form. These attributes car. be important when attempting to characterize a substance in hazardous environments, for examining precious objects such as cultural artifacts or gemstones, and for industrial process on-stream analysis. The luminescence observed from a substance is a function of many parameters such as species excited, chemical and physical environments of the species, emission and excitation wavelengths, and the temporal behavior of the emission. Thus, luminescence measurements are inherently high in information content on both the species present and their environment. An increase in the dimensionality of luminescence measurements affords a more complete utilization of information present in the luminescence signal, leading to a more comprehensive characterization of the luminescent material, for example, the separation of mixtures of polynuclear aromatic hydrocarbons by using a phosphorescence emission-excitation matrix ( I ) . The scope of multidimensional luminescence measurements in analytical chemistry has been reviewed by Warner ( 2 ) . 0003-2700/88/0360-1224$01 SO/O
Luminescent inorganic solids are complex electronic systems consisting of ions, molecules, or atoms or any combination of these entities held in fixed regions in space. Each constituent of the solid may interact with its neighbors and the energy levels present are dependent on both the constituents of the solid and the way in which the constituents are arranged and bound to each other. The luminescence from an inorganic solid may be strongly affected by the principal chemical species present, the type of bonding between species present, the way the species are arranged in space, trace impurities present, defects in the crystal lattice or glass, individual particle size in powders, temperature, pressure, and the presence of external electric or magnetic fields. The physics of luminescent inorganic materials has been reviewed recently by Blasse (3). Luminescence from an inorganic solid is a potentially rich source of information about the substance. Multidimensional luminescence measurements may provide a means to use this information to characterize certain attributes of an unknown substance. Alkaline-earth tungstates and molybdates with the scheelite structure are relatively simple and well understood luminescent inorganic solids (4-8). Generally, their emission spectra are broad, Gaussian-shaped bands ( 4 , 9 )devoid of fine structure at room temperature. Luminescence from the pure substances exhibits simple first order decay kinetics a t room temperature (4, 6). However, mixtures of any two of these substances produce a resulting emission spectrum that cannot be easily resolved into the individual constituents for an "unknown" mixture. In this study, the potential of timewavelength resolved luminescence spectroscopy for resolution of highly overlapped spectra originating from simple systems such as mixtures of alkaline-earth molybdates and tungstates is examined. EXPERIMENTAL SECTION A block diagram of the spectrometer is shown in Figure 1. A Corona Model PC400-HD2 computer with 10 megabyte hard disk and STB GRAPHIX PLUS I1 video board were used to control and acquire data from the spectrometer. Instrument control and data acquisition software was written in BASIC. Data reduction programs were run on a Perkin-Elmer 7500 series professional computer system using a Silicon Valley Systems FORTRAN 77 compiler. All calculations were done with REAL*8 variables to reduce truncation error effects. 0 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988 SAnPLE
A
U
EXClflER LASER
T
1-
0
DATA REDUCTION ALGORITHM The absorption and emission spectra of localized excitations in inorganic solids may be described by the configuration coordinate model (9). To a first approximation, the emission band profile for luminescence originating from a localized excitation in a solid substance may be described by a Gaussian function with the characteristic parameters of intensity, peak maxima, and peak width
S, = K , exp((m - V ) ~ / ( C W ~ ) )
CORONA C O M P U T E R
Flgure 1. Instrument block diagram.
A GCA/McPherson Model 270 monochromator with a 1200 line/mm holographic grating and a Model 700-51 scan controller were used. All spectra were collected with a 0.5-mm slit width. The Model 700-51 scan controller was highly susceptible to electromagneticinterference (EMI) from the excimer laser as well as other laboratory noise sources. To eliminate spurious changes in wavelength and scan direction caused by EMI, all electrical lines between the scan controller and monochromator were connected by an isolation relay only when scanning wavelength. The isolation relay was controlled by toggling a line on the Corona parallel printer port. Wavelength scanning was accomplished by connecting a second line on the parallel port to the external oscillator input of the scan controller. A machine language routine called by the instrument control program toggled this line at about 600 Hz for an appropriate number of pulses to scan from one wavelength to another. Sample luminescence was detected with a Hamamatau R955 photomultiplier tube (PMT) mounted in a McPherson Model EU-701-93PMT housing. The PMT socket was wired for fast response by using a base circuit described by Harris (10).All reported spectra have been corrected for system spectral response. System spectral response was determined with an Electro Optics Associates Model L-10 quartz lamp and a Model P-101 power supply using the method of Stair (11). The intensity of each laser pulse was monitored by a 1P28PMT and peak detector circuit. A luminescent screen of calcium tungstate powder .sandwiched between quartz and Pyrex disks was used to convert laser W radiation to visible and protect the photomultiplier from direct exposure to laser radiation. The peak detector circuit was built around a Burr-Brown 4085KG peak detector module with external buffer amplifiers and logic gates to protect the module and provide additional signal control. Fluke Model 413C and Kepco Model ABC 1500M power supplies were used for signal and reference photomultipliers, respectively. A Stanford Research Systems Model SR245 computer interface and Model SR250 gated integrator and boxcar averager were used to acquire intensity data and fire the laser. Triggering was done with an EG&G SGD-040Bphotodiode mounted near the sample holder and reference PMT. Communicationsbetween the Corona and the Model SR245 was by the system serial port; all communications cables had grounded shields to minimize pickup of external noise. Standard 50 Q coaxial cables with BNC connectors were used for all other signal and control lines. A Lumonics Model TE861T-3 excimer laser was used to excite the sample powders. Both ArF and KrF gas mixtures were used for 193- and 248-nm excitation, respectively. Standard operating conditions recommended by the manufacturer were used. Stable laser performance when using ArF was obtained only after inserting a V420PA40A varistor in the thyratron circuit directly between the thyratron grid and ground. Alkaline-earth molybdate and tungstate salts were prepared from analytical reagent grade materials by precipitation from aqueous solution (12). The precipitates were digested (13) for several days and then annealed for 4 h at 600 "C (14). Yields were greater than 98%. Sample powders were held in test tubes made from 7 mm x 1 mm wall quartz tubing.
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(1)
where v is the frequency (cm-l), K, is a constant, proportional to quantity of emitter, m is the peak maxima (cm-'), w is the peak half-width (cm-'), S, is the observed luminescence intensity at frequency v, and c is the constant relating half-width to standard deviation. In simple systems where trapping and energy transfer are either absent or insignificant, the time behavior of luminescence following pulsed excitation follows first-order kinetics. The instantaneous luminescence intensity observed at a time t after pulsed excitation is given by
Tt = Kb eXp(-t/?)
(2)
where Tt is the observed luminescence intensity a t time t , t is the time after excitation pulse, Kb is a constant, proportional to quantity of emitter, T is the lifetime of emitting species. When luminescence intensity is measured with a gated integrator having a time window width of At
Integrating (3) and collecting terms, the measured luminescence intensity is
Tt= KbT[exp(-t/T)
- exp(-(t
+ A~)/T)]
(4)
Therefore, the time-wavelength resolved spectrum for a species may be described by combining (1) and (4), then expressing the relationship in matrix form
[Dl = [SI[Tl
(5)
where [D] = i X j data matrix of luminescence intensities for i wavelengths and j times, [SI = i X 1concentration term and spectrum vector, and [TI = 1 X j' normalized time behavior vector. The ith and j t h values for the [SI and [TI vectors are given by
S, = C exp((m - v J 2 / ( c w 2 ) )
(6)
where S, is the spectral response value at frequency v, and C is a constant term, proportional to quantity of emitter TI= T[exp(-t,/T) - exp(-(t, + A ~ ) / T ) ] (7) where TI is the normalized intensity value at gate opening time t, and gate width At. If there are n emitting species present, and assuming that each emitting species acts independently and that the observed luminescence is a linear combination of emission from each species present, then the i X 1 [SI and the 1 X j [TI vectors may be replaced by i X n and n X j matrices, respectively. The parameters of concentration factor, peak maxima, peak width, and lifetime for each emitting species present in a mixture may be obtained by the following two-stage data reduction scheme. Stage I: Estimate the lifetime behavior for each emitter by the method of Knorr and Harris (15). (A) Guess n,the number of emitting species present and a lifetime value, T,, for each species. (B)Construct an estimated n X j normalized time behavior matrix [TE] using eq 7. (C) Compute an estimated spectral behavior matrix [SEI by multiplying the data matrix by the pseudoinverse of the
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11. JUNE 1. 1988
Table I. Parameter Assignments for Two-Component Mixtures B
A
lifetime, ps peak maxima. em-' peak half-width, cm-' intensitv
10 23000 6ooo
0.25 to 250
10
1 to 50
14wOto23000
m
F
estimated time behavior matrix using eq 8, then an estimated data matrix [DE] using eq 9, and finally the squared error, SQE, between [D] and [DE] with (10). [SEI = [D][TE]T([TE][TE]T)-l
(8)
[DE] = [SE][TE]
(9)
2
UJ
zc
0.1
UJ
YA
0
0.5 1 .o PEAK SEPARATION
1.5
Fbure 2. Maximum error in estimated parameters for both wmpo-
(D) Guess new values for lifetimes and repeat steps (B) and (C) using simplex optimization (16) to obtain a minimum for SQE. Stage I1 (E) From [SEI, compute estimates for the parameters C, rn, and w for each of the n components present. (F) Construct a new i X n estimated spectral behavior matrix [SEI using eq 6 and estimates for C, rn, and w for each component. (G) Construct a new n X j estimated normalized time behavior matrix [TE] using eq 7 and estimated values for T . (H)Compute an estimated data matrix [DE] using eq 9, and the squared error, SQE, from (10). (I) Guess new values for C,rn, w, and r; repeat steps (F) to (H) using simplex optimization to obtain a minimum for SQE. In step I one may set fixed values for any parameters for any components. This approach may be useful when a& tempting to characterize an unknown in the presence of known emitters or to examine small changes in selected parameters.
RESULTS AND DISCUSSION The data reduction algorithm was evaluated by examining artificial spectra created by using the system model given above with parameters similar to t h m of typical luminescent inorganic salts such as CaMoOl and CaW04 (8). Each artificial spectrum was corrupted by adding Gaussian noise (17)a t levels comparable to that of experimental measurements. All artificial spectra created were stored in data files having exactly the same format as laboratory-collected data to simulate aa closely as possible effects such as data truncation errors, measurement dynamic range, and the collection of only portions of a spectral envelope. Intensity values were stored in the fixed format "XX.XXX*, wavelength coverage was from 300 to 700 nm in 10-nm steps, and time coverage was from 1 to 20 ps in 1-ps steps with a window width of 1 ps. Algorithm performance for two-component mixtures was examined for the effects of spectral overlap in both time and wavelength domains, noise, guess values, and relative concentrations of the two components. Parameters assigned to component A were held constant while parameters assigned to component B were varied as summarized in Table I. For two-component mixtures, the principal question is how great can the overlap in peaks be before the algorithm fails to estimate the parameters of the two spectral envelopes a t an acceptable level of error. The performanceof the algorithm for overlaps in both the time and wavelength domains is shown in Figure 2. The horizontal axis is the peak separation in the spectral domain in units of peak half-widths. The vertical axis is the ratio of lifetimes of component B to component
nents A and B as a function of peak separation in both time and Wavelength domains. Synthetic data with 1% relative standard deviation noise. component B intensity = IO. and other parameters as in Table I. Guess values used were A = 12.3 ps and E = 123 ps.
(n
I
I
rdative standard deviation noise added and vertical axis indicating intensity: (a) lifetimesA = 10 ps, E = 25 ps; peak maxima A = 435 nm. E = 465 nm; (b) lifetimesA = 10 ps, E = 1 0s; peak maxima A = 435 nm. 8 = 714 nm. F ~ u3.I ~ Synhtic spectra VM I 1%
A, this format was chosen to cover a wide range of lifetimes. The shaded regions represent situations where the relative error between estimated and actual values lies within the stated bounds. Generally, as the separation between the two peaks increases, the error in estimated parameter values decreases to a minimum value and then rises as the peak separation increases. The increase in error as peak separation increases is due to having less of the component B spectrum included in the data collection window. At a separation of 1.5 halfwidths, less than half of the component B spectrum is within the data collection window used in the simulation experiment. In the time domain, a similar behavior is observed at extremes in lifetimes for component B. At short lifetimes, very little signal from component B is present in the data collection window used and this small signal is highly sensitive to
ANALYTICAL CHEMISTRY, VOL. 60. NO. 11. JUNE 1. 1988
a
a
b
1227
b
.I E
0 ,
LL
P t A K SEPARATION
o
3 5
PEAK SEPARATION
C
Fgun 4. Maximum error in estimated parameters for both components A and B (a)lifetimes, (b) peak htemlties. (c)peak maxim, (d) paak half-widths: synthetic data with 1 % relative standard deviation noise. component B intensity = 10. other parameters as in Table I: guess values used, A = 12.3 PS, B = 123 11s.
truncation errors. For cases where component B has a relatively long lifetime,there is little change in its intensity within the data collection window and any minor error in estimating the slope corresponds to a large error in the lifetime estimate. Parameters for both components present in the synthetic spectrum shown in Figure 3a were estimated with a maximum error of less than 2%. In Figure 3b, component A parameters were estimated within 1%of their actual values. However, the error in estimates for all component B parameters was greater than 10%despite the large separation from component A in both time and wavelength domains. Component B contributes very little to the overall spectmn,its peak maxima lies outside the data collection window and its rapid decay provides little time domain information. In two-comwnent mixtures, the estimated parameters for peak maxima and width consistently showed the least error regardless of peak separation, added noise, and the lifetimes or the relative intensities of A and B as can be seen in Figurea 4, 5, and 6. In Figure 4, the greatest overall error for the parameters for either component A or component B is shown; error levels for parameters for components A and B are shown in Figures 5 and 6, respectively. Estimates for component A parameters were generally very good as can he seen in Figure 5. Difficulties in estimating any of the component B parameters for cases where component B has short lifetimes are shown in Figure 6. The performance of the algorithm in noisy environments is shown in Figure 7, where the error shown is the maximum error found for any of the parameters for any component. Gaussian noise a t levels ranging from 0 to 5 % was added to artificial spectra; the intensity of component B was held constant a t 10,lifetime wexi values were 12.3 and 123 pa, and lifetime estimates from stage I were held constant during stage I1 in the data reduction algorithm. The overall trends in reliability of eatimated parameter values with degree of overlap in the spectral and temporal domains hold and, to a first approximation, the error level encountered is proportional to the noise level in the spectrum. A noise level of 1%relative
Figure 5. Maximum error in estimated parameters for component A oniy: (a) lifetimes. (b) peak intensity, (c) peak maxima, (d) peak half-wm: Synthetic data with 1 % relathre standard deviation noise, component B intensity = 10, other parameters as in Table I; guess values used. A = 12.3 PS, B = 123 PS.
a
h "
Fgure 6. Maximum error in estimated parameters for component B only: (a) lifetimes, (b) peak intensity, (c) peak maxima, (d) peak half-widths: synthetic data wim 1 % relative Standard deviation noise. component B intensity = 10. other parameters as in Table I: guess values used. A = 12.3 ps, B = 123 fis.
standard deviation is normally achieved with the apparatus described and was used in all synthetic spectra that follow. The algorithm appears to be relatively insensitive to initial guess values for component lifetimes as can be seen in Figure 8. Generally, the initial guess may be within a factor of 10 of the real value and the algorithm seems to perform better when lifetime guesses are on the high side.
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ANALYTICAL CHEMISTRY. VOL. 80. NO. 11, JUNE 1. I988
Figure 7. Maximum error in estimated parameters for both components A and B as a function of added noise: (a) no noise added, (b) 1% RSO. IC) 2% RSD. Id) 3% RSD. (e) 4% RSD. (1) 5 % RSD. Synthetic data with component B lntenstiy = IO. other parameters as In Table 1. Guess values used: A = 12.3 ps. B = 123 fis.
Flgure 0. Maximum error in estlmated parameters for both compcnents A and Bas a function of peak Intensity. Peak intensities used: (a) A = IO. B = 5 0 (b) A = 10, B = 20; (c) A = IO. B = 10: (d) A = IO. B = 5; (e) A = 10, B = 2; (1) A = IO, B = 1. Synthetic data wim 1% RSD noise. other parameters as in Table 1. Guess values used: A = 12.3 fis. B = 123 w.
a
THREE COMPONENT SYSTEM
C
i
i
1
nmt€T of cmcnentr g m s d
i
Flgm 10. plot of residual error vs number of components guess&: l~etlmes(ps)A = 10. B = 25, C = 2.5 p e a k separahs range (A-B) 0.2-0.8 haH-wktms. ( A X ) 0.8-1.2 half-wb3U-q 1% RSO noise: equal
peak Intensities.
Flgure 8. Maximum error in estimated parameters for both compcnents A and B as a functlon of lifetime guess values. Guess values (ps)used: (a) 12.3. 123 (b) 12.3, 1.23 (c) 12.3, 7.89 (d) 123, 1.23. Synthetic data wkh 1% RSD robe. component B htenslty = IO. othsr parameters as In Table I.
The effect of various amounts of component B present is ahown in Figure 9. The trend of increased error in estimated
parameter values for a given component as the amount of that component present decreases is illustrated here. This trend reinforces the notion that estimate errors are more a function of how large the signal is from a given component present rather than the degree of overlap in either time or wavelength domains. Synthetic mixtures containing three components were suceeBsfuuy separated into their individual components. The examples in Table I1 show typical performance of the algorithm for highly overlapped spectra. Generally, if the peak
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
Figure 11. Measured spectra for S ~ ( M O ~ , ~ ~ W(a)~ 193 , ~ ~nm ) Oex~: citation; (b) 248 nm excitation.
a
1229
separation was greater than 0.5 half-widths and the lifetime ratio between two components was greater than 2:1, peak envelopes of comparable volumes were successfully resolved as can be seen in comparing results for mixtures 1 and 3. The algorithm was also used to estimate the number of components present in an unknown mixture. This was done by running the algorithm using an increasing number of components in the initial guess and plotting the value for SQE against the number of components guessed present as in Figure 10. Normally, as the number of components guessed present increases, the value of SQE decreases rapidly until the correct number of componentspresent is reached and then decreases at a much slower rate or rises. The numerical instabilities associated with the pseudoinverse calculation are probably the reason for the observed rise in SQE when the number of components guessed exceed those present. This approach was successfully used to find the number of emitting species present in real powder samples and estimates of the parameters for each component present. The measured spectra for a crystalline powder containing both SrMoOl and SrW04excited at 193 and 248 nm are shown in Figure 11. Three significant emitting componentsare present: WO,2+, MOO?+, and a long-lived component from the quartz cell used to hold the sample powder. The spectra shown in Figure 12 are the difference between the measured spectrum of Sr(Mo0.05W0.95)04 excited at 248 nm and a spectrum computed from parameters estimated by the algorithm. Here, a plot of SQE vs number of components guessed has a distinct local minima at three components. A consequence of this rise in SQE when guessing four components present is seen in Figure 12d. This rise in SQE when a fourth component is guessed present for a system containing
b
C
excited at 248 nm: (a) one; (b) two; Figure 12. Difference spectra as a function of the number of components guessed for Sr(Moo,05Wo,g5)0,
(c) three; (d) four components.
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
Table 11. Comparison of Actual a n d Estimated Parameters for Three-Component Mixtures" halfwidth, cm-'
intensity
component
lifetime, p
maxima, cm-'
A
10.00 (10.05) 25.00 (25.23) 2.50 (2.48)
mixture 1 23000 (22964) 19400 (19382) 18200 (18204)
6000 (6023) 6000 (5999) 6000 (6008)
10.00 (10.10) 10.00 (9.87) 10.00 (10.07)
10.00 (10.02) 25.00 (25.09) 2.50 (2.50)
mixture 2 23000 (22989) 19400 (19403) 15800 (15838)
6000 (6019) 6000 (6017) 6000 (6013)
10.00 (9.97) 10.00 (9.95) 10.00 (10.01)
10.00 (10.22) 25.00 (1.85) 40.00 (33.62)
mixture 3 23000 (22734) 19400 (20398) 18200 (18673)
6000 (6208) 6000 (1E-8) 6000 (6101)
10.00 (10.90) 10.00 (0.72) 10.00 (9.95)
B C
A
B C
A
B C
"Estimated values are in parentheses. guess times, ps: 36.9, 12.3, 7.89.
:1% RSD noise added;
Table 111. Spectral Parameters Examined
for the Compounds
compound CaMoO, CaWO, SrMoOl SrW0,
lifetimes, 9.7 8.6 0.48 0.75
ps
Found
maxima, cm-I
halfwidth, cm-'
19209 24111 23924 19426
6952 6327 7421 7765
only three components is likely due to numerical instabilities present in the calculation of the pseudoinverse matrix in eq 8. Further refinements in the simplex optimization to allow for numerical instabilities may reduce this anomalous behavior.
Spectral parameters for the time-wavelength resolved spectra of some molybdate and tungstate salts prepared by precipitation from aqueous solution are given in Table 111. The results are within reasonable agreement with published data (4, 6, 7) with the exception of the calcium molybdate lifetime; sample contamination is suspect for the anomalously short value measured. The long-lived luminescence near 390 nm in the molybdate spectra is from the quartz test tube used to hold the sample powders. ACKNOWLEDGMENT The authors thank the Perkin-Elmer Corporation for donation of the Model 7500 professional computer system used in this work. Registry No. Sr(Moo,06Wo,96)04, 113810-64-3; CaW04, 779075-2; CaMoO,, 7789-82-4; SrWO,, 13451-05-3; SrMoO,, 13470-04-7. LITERATURE CITED Ho, C.-N.; Warner, I. M. Anal. Chem. 1982, 5 4 , 2486-2491. Warner, I. M.; Patonay, G.; Thomas, M. P. Anal. Chem. 1985, 5 7 , 463A-483A. Blasse, G. Mater. Chem. Phys. 1987, 16, 201-236. Treadaway, M. J.; Powell, R. C. J. Chem. Phys. 1974, 6 1 , 4003-401 1. Gurvitch, A. M.; Gutan, V. 6.; Meleshkin, 6. N.; Mikhailin, V. V.; Mikhalev, A. A.; Tornbak, M. I. J. Lumin. 1977, 15, 187-199. Tyner, C. E.; Drickamer, H. G. J. Chem. Phys. 1977, 6 7 , 4103-4115. Groenink, J. A.; Hakfoort, C.; Blasse, G. Phys. Status Solidi A 1979, 5 4 , 329-336. Grasser, R.; Scharmann, A.; Strack, K. R. J. Lumin. 1982, 27. 263-272. Tyner, C. E.; Drotning, W. D.; Drickamer, H. G. J. Appl. Phys. 1978, 47, 1044-1047. Harris, J. M.; Lytie, F. E.; McCain, T. C. Anal. Chem. 1976, 48, 2095-2098. Stair, R.; Schneider, W. E.; Jackson, J. K. Appl. Opt. 1983, 12, 1151-1154. Kotera, Y.; Seklne, T.; Yonemure, M. Z.Phys. Chem. 1961, 218, 197-203. Vogel, A. I.; A Textbook of Quantitative Inorganic Analysls, 3rd ed.; Longmans: London, 1961; Chapter 1. Kroger. A. Some Aspects of the Luminescence of Solids; Elsevier: New York, 1948. Knorr, F. J.; Harris, J. M. Anal. Chem. 1981, 5 3 , 272-276. Daniels, R. W. An Introduction to Numerical Methods and Optimizatlon Techniques; North-Holland: New York, 1978; Chapter 8. Press,W.; Flannery, B. P.; Teukolsky, S. A,; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1986; Chapter 7.
RECEIVED for review October 6,1987. Accepted February 5 , 1988. This work has been carried out with the support of the Natural Sciences and Engineering Research Council of Canada.
