Anal. Chem. lg84, 56, 1404-1409
1404
Improved Analytical Performance in Multichannel Spectrometry by Compensation of Nonrandom Signal Fluctuations Avraham Lorber Nuclear Research Centre-Negev,
P.O. Box 9001, Beer-Sheva 84190, Israel
A varlant of the Internal reference method Is presented. Several Internal standards are measured simultaneously over a perlod of tlme. Thelr fluctuatlons are computed and then are multlplled wlth a projection matrlx, which Is constructed from the elgenvectors of a callbratlon data matrlx. The projectlon makes lt posslble to determlne nonrandom fluctuations In the analytlcal channels. The method Is appllcable when the factors that affect the Internal standards slmllarly affect the analytes. Fisher’s varlance ratio test Is used lo test whether the method can successfully be applied. Appllcatlons to Inductlvely coupled plasma emlsslon spectra show that the method succeeds In reduclng drlft from 10% to below 0.1 % wlth 0.1 % RSD. Implementatlon to multlchannel (or multlplex) analytical systems coupled to mlnl/mlcrocomputers is slmple.
Today multichannel (or multiplex) analytical instruments are commonly coupled to mini/microcomputers. This coupling has already produced major improvements in analytical performance. The resolution of overlapping spectra has been made possible by the use of the multidimensional data available from multichannel instruments and by the use of the computational power to isolate the components (1-3). Another potential area for improving the analytical performance of multichannel systems is in compensating for nonrandom fluctuations by computation techniques based on the internal reference method (4) and on the generalized standard addition method (5). The internal reference method (IRM) principle has been employed for over 60 years to compensate for nonrandom fluctuations (4), mainly in atomic emission spectrometry. The simple form of the line ratio of the analytical-internal reference channels pair went unaltered until recently. But difficulties which arose during its application to the inductively coupled plasma (ICP) emission source motivated some suggestions for alternative presentations of the IRM, Myers and Tracy (6), Belchamber and Horlick (7), and Lorber and Goldbart (8)suggested that by replacing the simple line pair ratio by a correlation of the analytical to the internal reference signal, superior compensation could be obtained. This method will be referred to as the analyte-internal reference correlated method (AIRCM). The improvement in S I N achieved by the AIRCM varies from 2-fold (7) to 10-fold (6) and in some cases even higher (9). This method, although superior to the simple line ratio method, does not use all of the available multichannel data nor does it use the full computational power. A method exploiting the latter is the generalized internal reference method (GIRM). In the GIRM several internal standard channels that respond differently to variations of the parameters of the analytical system are measured simultaneously. Since the fluctuations in analytical channels cannot be directly correlated with fluctuations in internal standard channels, instrumental parameter fluctuations serve to connect the two. Compensation is possible by calibrating the responses of in-
ternal standard and analytical channels to changes in instrumental parameters. Lorber et al. (9) found that when the GIRM was applied to ICP, a drift of 10% was removed and that the random noise was slightly amplified. Implementation of the GIRM necessitates the fulfillment of the following prerequisites: (a) considerable computation speed (computation of the GIRM for a single measurement requires 20 s on a Digital PDP 11/34 minicomputer); (b) a calibration procedure (calibration for the response of the spectral lines to the variation of each instrumental parameter), enabling each parameter to be varied independently of the others (this requirement cannot be met when noncontinuous of unstable analytical systems are considered); (c) the set of internal standard channels must comprise a nonsingular matrix of coefficients describing the response of internal standards to variations of instrumental parameters. In this paper a computational method free of these prerequisites is suggested and its validity is tested on data obtained from an ICP atomic emission source.
THEORY The assumption underlying the suggested method-the internal reference projection method (1RPM)-is that the factors which cause fluctuations of the internal standard channels also cause fluctuations of the analytical channels and act linearly. As is usual in analytical chemistry, calibration is done at the first step but, rather than a matrix of regression coefficients, the calibration step produces projection matrices. Temporal measurements of fluctuations in internal standard channels are then smoothed to remove random errors and are subsequently used to determine the fluctuations in analytical signals. In the following theoretical section, the construction of the projection matrices and the compensation procedure is described. Construction of t h e Projection Matrices. An array of data obtained from an experiment containing m rows and n columns may be presented in matrix notation as D. It is a well-known result in linear algebra that any real matrix may be decomposed into three matrices
D=USVT
(1) where U is an m X m orthonormal matrix (Le., Uw = I,, where UTdenotes the transpose and I,,, the identity matrix of dimension m),V is an orthonormal, square matrix of order n, and S is an m X n matrix. Its upper n X n matrix is a diagonal matrix (only the diagonal elements may differ from zero), with elements sl, s2,..., sk, ...,s , for which s1 I s2 L ... sk 2 ... s., These scalars are the nonzero singular values of the matrix D. The singular values are unique and are the square roots of the eigenvalues of the matrix DTD. The subscript k denotes the smallest significant eigenvalue. The lower ( m - n) x n part of S is a zero matrix. The decomposition presented in eq 1 is referred to as the “singular value decomposition” (SVD) (IO). This decomposition serves in constructing the projection matrices. It may be noted that because the SVD is considered as a further decomposition of other orthonormal decomposition techniques ( l o ) ,their eigenvectors may also serve to construct the projection matrices.
0003-2700/84/0356-1404$01.50/00 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
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The second step in the construction of projection matrices is to determine k , the number of significant eigenvalues. This step is called abstract factor analysis (AFA). Methods of determining k can be found in the literature (3). When k is determined, an approximation, D, to the data matrix, D, which is reproduced from the k significant eigenvalues and eigenvectors is obtained by
D i U S V T
(2)
Now U is a m X k matrix containing the first k eigenvectors that span the space of D DT,VT is a k X n matrix containing the first k eigenvectors that span the space of DTD, S is a diagonal matrix with the first k singular values as the diagonal elements. As AFA assumes that all nonsignificant eigenvalues and eigenvectors are due to random error, the matrix D is now free from randon errors. The projection matrices are obtained from the significant eigenvectors by
zc = V V T
ZR = 0 U T
(3)
The matrix ZR is the projection of the rows, and the matrix Zc is the projection of columns, which means
D=ZRD
D = D Zc
(4)
The noise rejection capability of projection matrices as presented in eq 4 is in force for any vector affected by the same factors from which the projection matrix was constructed (11). The relations presented by eq 4 are used by IRPM to smooth experimental errors and to find nonrandom fluctuations. A signal emerging from an analytical system is denoted by, Ii (i = 1, ...,s). There are p internal standard signals (i = 1, ...,p ) and s - p analytical signals (i = q, ..., s), here q = p + 1. The relative signal, liR,is the ratio of the instantaneous signal to its value at a previous measurement. The fluctuation in a signal, xi, is xi = IiR - 1. m sets of measurements of deliberate changes or accidental fluctuations serve as the data matrix for calibration. In decomposition techniques it is necessary for the number of rows to be greater than or equal to the number of columns, the data matrix should be arranged accordingly. Considering the case, m > p , a projection matrix, P = V VT,which is a p X p matrix, is constructed from the internal standards fluctuation data. In the case where p > m, the projection matrix, P, should be constructed by P = U UT. A vector of m fluctuations of ith analytical signal is added to the matrix of internal standards fluctuation. The augmented matrix is decomposed and the eigenvectors are used to construct the projection matrix, Z (this designation differs from the designation P in order to avoid confusion between these matrices) which is a (p + 1) X (p 1)matrix. Only the last row of the matrix, Z, which gives the linear combination of the internal standards that enables fluctuation to be predicted in the analyte, is stored. This decomposition and construction is repeated s-p times for each analytical signal. Compensation €or Nonrandom Fluctuations. The compensation procedure is as follows. Temporal fluctuations of p internal standards are multiplied according to eq 4 by the projection matrix ii = Px,in order to remove random noise from internal standards. The resulting vector, I,is therefore free from random noise. This step is needed in order to minimize error propagation in the next step. When the number of internal standards equals the number of factors affecting their fluctuations, the projection matrix is a unity matrix. Therefore, smoothing is impossible for this case. In the second step, detection of nonrandom fluctuations in analytical channels is accomplished by inserting the smoothed vector ii into the equation
+
P
xi =
CZiJ3”j/(l j=l
-qp+l)
(5)
0 0
5
10
15
20
25
30
1
TIME (minute) Flgure 1. Time dependence of spectral lines intensity for the two spectral lines serving as analytical channels.
The zij (i = q , ..., s) are the values of the last row of the projection matrix, Z, which was constructed from the data of ith analytical channel. The last step, compensation, is executed according to
here I+$ is the analytical calibration function, Ci is the concentration of the analyte i. When there is no change in the concentration of the analyte, the ratio lP/(l+ xi) should equal unity. Therefore, any variation in this ratio indicates a variation in concentration. The scheme for compensation presented here illustrates one possible way of using the projection matrices. Another possibility exists in decomposing the full m X s data matrix, D, and constructing an s X s projection matrix. Here, the detection of fluctuations in analytical channels will require an iteration procedure. The iteration will start by projecting a vector with s elements, whose first p elements are the fluctuations of internal standards and whose remaining s - p elements are set to zero. The last s - p elements of the resulting vector, together with the “experimental”fluctuations of internal standards, will serve as input for the subsequent iteration. Although computation time for performing iterations poses no difficulty, as the computation is very simple, we preferred a method that does not require iteration in order to avoid convergence-assurance problems.
EXPERIMENTAL SECTION The variations in the intensities of spectral lines examined in this study were measured by an ICP excitation source coupled to a photodiode array (1024 element) detector. Experimental facilities, procedure, and reasons for selecting spectral lines are described elsewhere (9). The calibration data consisted of 20 measurements in which the plasma parameters were varied to different degrees. Analytical data used to test the suggested compensation method consisted of 64 measurements of successive intervals, each of which is a signal integrated over a period of 30 s. The fluctuations in intensities found for Sc I1 424.68 and Cr I 425.44 nm spectral lines are presented in Figure 1. These two spectral lines served as analytical lines while all others served as internal standards. The singular value decomposition was performed by the SVDRS subroutine of Lawson and Hanson (IO). The data
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
Table 111. Variance Ratio Values for Various Sets of Internal Standards
Table I. Relevant Parameters of the Elements Chosen for This Study element and state Ga I Ar I Eu I1 Sr I1 Ca I s c I1 Cr I
excitation ionization wavelength, potential, potential, nm eV eV 417.21 419.83 420.51 421.55 422.67 424.68 425.43
3.07 14.74 2.95 2.94 2.93 3.23 2.91
6.00 15.76 5.64 5.69 6.11 6.54 6.76
E,,
+E,
eV
8.59 8.63
internal standards
variance ratio values for the analytes Eu Ar Ga Ca Sr Sc Cr
Eu, Ar, Sr, Sc 360 100 180 Ar, Cr, Ca, Ga 0.32 0.35 0.58 Eu, Ga, Sc, Sr 320 1.75 1.10 Eu, Ar, Ga, Ca 0.02 0.05 0.01
9.77
Table 11. Singular Values Obtained for the Data Matrix singular values internal standards 1 2 3 4 5 6 Eu, Ar, Ga, 0.802 0.353 0.224 0.034 0.004 Ca, Sr Eu, Ar, Ga, 1.015 0.362 0.229 0.042 0.005 0.004 Ca, Sr, Sc Eu, Ar,Ga, 0.894 0.379 0.226 0.034 0.005 0.003 Ca, Sr, Cr were processed by a Digital PDP 11/34 minicomputer. The program was written in Fortran IV.
RESULTS AND DISCUSSION Relationships among AIRCM, GIRM, and IRPM. The IRPM may be viewed as an extention of the AIRCM, by which the analyte signal is regressed to more than a single internal standard. However, multiple regression is allowed on condition that the number of independent variables equals the rank of the calibration data matrix. Therefore, not only is the IRPM presentation an extention of the AIRCM but it also allows the use of any number of internal standard channels. Moreover, the random errors in measuring the internal standard channels are removed in the smoothing step, which is achieved only by using more internal standards than the rank of the calibration matrix. IRPM may also be considered as a special case of the GIRM for situations in which the fluctuations are sufficiently small to allow a linear approximation (8). However, the IRPM usage is more convenient than the “linear” GIRM for the following reasons. (a) The fluctuations in the instrumental parameters do not need to be determined in IRPM. This fact simplifies the experimental design a t the calibration step and the instrumental parameters may be allowed to vary randomly. (b) While the GIRM is inapplicable when the fluctuations of the internal standard channels cannot be fully described by the influence of the instrumental parameters, the IRPM is still applicable. Rank Determination. Table I1 presents the singular values obtained by the decomposition of the data matrix. Many methods have been recommended for the purpose of rank determination (3). When these methods were applied to our data, conflicting results were obtained. Some methods determined the value three as the rank while others found the value four. We therefore constructed the projection matrices by considering both possibilities. The singular values obtained from the decomposition of data matrices in which a column for the data of analytical signal was included are also presented in this table. From the values it is obvious that the addition of analytical signals has no effect on the third and fourth singular values; therefore, analytical signals may be described well by the same factors that affect the internal standard signals. Although the comparison of the singular values may qualify as a rough indication for the adequacy of determining
the fluctuations in analytical signals by a set of internal standards, statistical tests on the adequacy of the set can be done by target factor analysis (TFA) (3, 11). Adequacy of a Set of Internal Standard Signals to Predict Fluctuations in Analytical Signals. The distribution of errors in multiplying a projection matrix by a vector was studied in conjunction with TFA (11). On the basis of the error distribution, statistical inference on the hypothesis that the vector lies within the same space of the projection matrix is given by Fisher’s variance ratio test. The hypothesis underlying IRPM is that the analytical channel fluctuations may be described by the same factors describing the fluctuations of internal standard channels. In terms of linear algebra, the hypothesis is that a vector of fluctuation data for an analytical channel lies within the space generated by the eigenvectors of the internal standards data. Therefore, the application of the variance ratio as developed by Lorber (11) can serve to test the adequacy of a set of internal standards. The projection is on the space spanned by the eigenvectors, U, which results in the projection matrix U UT. In cases where the number of internal standard channels is greater than the measurements, the test is not applicable (111,and a reduced set of internal standards should therefore be selected. Table I11 presents the variance ratio value for four sets of internal standards. Each row presents a different set of internal standards and includes the variance ratios for the other channels which served as analytical signals. The variance ratio is distributed in an F distribution with 13 and 3 degrees of freedom. The first row shows that when no atomic spectral line is present in the internal standards set, compensation for atomic lines is impossible. The second row shows that a set consisting of atomic lines and an argon spectral line adequately describes the fluctuations of ionic spectral lines. The third row shows that spectral lines of elements supplied from the solution cannot be correlated with the argon line. The fourth row shows that a combination of atomic, ionic, and argon spectral lines is the optimal internal standard set. The set in the fourth row was selected to compensate both for atomic Cr and for ionic Sc spectral lines. Characterization of Compensation Results. The projection matrix P which was obtained for the five internal standard spectral lines is presented at the top of Table IV. The last two rows are the z i j values obtained from the last row of the projection matrix into which ith analytical channel fluctuation data were incorporated. The projection matrices were constructed from three eigenvectors. The values in the table give an interesting insight into the relative importance of the various internal standards to the calculation of fluctuations in analytical signals. When the ratio of an off diagonal element to the diagonal element is examined, the low value indicates that the corresponding spectral lines are poorly correlated. The conclusions which can be drawn from the table are in close agreement with the conclusions drawn from the variance ratio values (Table 111). The residual fluctuations after compensation for the 64 measurements are presented in Figures 2 and 3. In addition to IRPM, the results of residual fluctuations by using AIRCM and GIRM for compensation are also presented in the figures.
ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
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Table IV. The Projection Matrixa Eu
Ar
Ga
Ca
Sr
Eu Ar Ga Ca Sr
0.4194 -0.0154 -0.0655 0.1239 0.4729
-0.0154 0.9987 0.0152 -0.0204 0.0210
-0.0655 0.0152 0.6797 0.4499 -0.1036
0.1239 -0.0204 0.4499 0.3662 0.1179
0.4729 0.0210 -0.1036 0.1179 0.5360
sc Cr
0.3156 -0.0579
0.0303 -0.0562
-0.1088
0.0491 0.2429
0.3603 -0.0842
0.3772
sc
Cr
0.4261 0.3125
a The upper five rows are the projection matrix for the internal standards. The last two rows are for the analytical spectral lines. The projection matrices were constructed from three eigenvectors.
Table V. Statistical Measures of Compensation Performance corrected compensation mean signal method fluctuation, % Cr
AIRCM GIRM IRPM[3] IRPM[4]
RSD, %
median fluctuation, %
no. of runs 7 43 27 27
-0.084 0.014 0.077 0.084
0.24 0.12 0.10 0.09
-0.07 0.002
0.18 AIRCM 0.004 GIRM -0.044 IRPM[3] IRPM[4] -0.059 Confidence intervals on 95% significance level are
0.10
0.20
0.10
0.11
0.000 -0.08
0.09
-0.05
sc
0.08 0.08
24 38 21 23
0.017% > mean > -0.017% 0.085% > RSD > 0.059% 42 > number of runs > 24
02
a2
01
0.1
0.0
0.0
-0 1
-0.1
- 0.2
a2 0.1
I
02
ae
on
I
z
0.1
2
0.0
El
z
--z ;c
0 -az
V
3
-a1
-0.1
3z
-0.2
U
LL
3
0.3
a2
0.3 0.1
0.2 0.0
0.1
-0.1
0.0
-02
TIME (minute)
Flgure 2. Time dependence of residual fluctuations after compensatbn for Sc I 1 424.68 nm. Compensation method: (a) AIRCM, (b) GIRM, and (c) IRPM.
AIRCM was performed on pairs having very similar physical parameters. For the Sc ionic line, the Sr ionic line served as internal and for cr atomic line, the ca atomic line served as internal standard. The statistical meaning of the residual fluctuations is summarized in Table V. The statistical measures are presented for AIRCM, GIRM, and IRPM by using three eigenvectors (IRPM[3]) and IRPM by using four eigenvectors (IRPM[4]).
-a3
- 0.L -0.5
TIM E (minutel
Flgure 3. Tlme dependence of restdual fluctuations after compensation for Cr I 425.44 nm. Compensation method: (a) AIRCM, (b) GIRM, and (c) IRPM.
When a claim for successful removal of all nonrandom noise is to be tested, the following three statistical measures should be examined: (a) mean, (b) standard deviation, and (c) ran-
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 8, JULY 1984
domness. The RSD of the electronic readout system was measured previously (9) to equal 0.07%. This value was used to obtain confidence intervals for the above three statistical measures on a 95% significance level. Randomness is measured here by the number of runs (12).The number of runs is the number of times that successive data points pass the median. The confidence intervals are also presented in Table
APPENDIX To illustrate the IRPM, we generated a data set by forming linear combinations of six measurements on four channels from two independent factors. The data matrix is 8.0 4.0
2.0 5.0 6.0 9.0 6.0
V. The mean residual fluctuation obtained by GIRM falls within the confidence interval, while the others are outside. However, when IRPM is compared with AIRCM it is obvious that IRPM succeeds in reducing nonrandom noise significantly. For Cr as analytical channel this is not obvious, but considering the large RSD and low number of runs, the low mean value obtained by AIRCM is of no significance. This result will be more significant when the fact that the AIRCM was applied to very similar line pairs is considered. These line pairs may be viewed as ideal pairs in terms of selection rules developed in the classical internal reference method (4). RSD for all methods is outside the confidence interval with very similar values, giving an error propagation of 50% when computations are performed. The randomness as measured by the number of runs shows that the GIRM values are very random and that the IRPM values are also random, while when AIRCM is used there is a danger that complete removal of nonrandom noise is not achieved even when the line pair is very similar. For all statistical measures IRPM[3] gives very similar values to IRPM[4]. It is clear from the above statistical analysis that the characteristics of residual fluctions for GIRM are preferable to those obtained by IRPM. However, when the flaws rkgarding the implementation of GIRM are considered, there is a clear advantage in using IRPM. Moreover, the mean residual fluctuation for IRPM is lower than 0.1% for both lines, which have very different physical parameters. This is very satisfactory when compared with precision and accuracy offered by current analytical instruments.
CONCLUSIONS The IRPM, like GIRM, takes full advantage of the vistas opened by coupling multichannel (multiplex) analytical system to computers. In contrast to GIRM, implementation of IRPM is straightforward. Selecting an appropriate set of internal standards will result in removal of nonrandom noise. The same set may serve to compensate for all types of spectral lines experienced in ICP. Therefore, accurate (not only precise) analytical results may be expected when there are no systematic errors during sample and standard preparation. Moreover, signals from the sample matrix may serve to detect and compensate for homogeneous errors (errors affecting all sample constituents to the same degree) in sample preparation. Determination of a signal in spectrometry requires background subtraction. In our system, a photodiode array is used and the signal magnitude is high; this therefore presents no problem. However, this is not always the case. The other extreme is a situation in which accurate background subtraction is the limiting case. Such situations occur when the background signal is comparatively higher than the analyte signal. IRPM may also be applied in such situations to determine background fluctuations rather than signal fluctuations. This case is of interest as it allows compensation without addition of any species to the sample. The background data at locations other thaii the analytical channels will serve as internal standard channels. In a forthcoming publication (13) the results of applying of IRPM to this case will be presented. ACKNOWLEDGMENT The author wished to express his thanks to M. Glouberman and U. Frohlichman for their help in executing this study.
8.0 4.0 5.0 4.0 6.5 6.0 9.0 6.0 10.5 0.0 3.0
6.0 3.0 4.5 6.0 7.5 3.0
The first three channels were taken as internal standard channels. The SVD for this matrix is 8.0 4.0 8.0 -0.4593 -0.2760 6.0 6.0
-0.4722 -0.5806 -0.2168
6.0 0.0
25.96 0 4.84)
(0
X
0.3398 -0.0371 -0.7539
-0.5883 (-0.7368
-0.4055 -0.6996 0.6264 0.2570)
The projection matrix is then readily obtained as P = V VT 0.8889 -0.2222 0.2222 -0,2222 0.5556 0.4444) 0.2222 0.4444 0.5556
0.5 58 3 -0.4055 -0.6996 -
=
(
--
0.7368 0.6254 0.2570
-0.5583 -0.4055 -0.6996 -0.7361 0.6254 0.2570
(
SVD for the complete data set which includes the fourth analytical channel is (2.0 8.0 4.0
8.0 5.0 3.0) 6.0 = (-0.2405 -0.4605 -0.2741 0.4789)
5.0 4.0 6.5 4.5 6.0 6.0 9.0 6.0 9.0 6.0 10.5 7.5 6.0 0.0 3.0 3.0 X
(2:*gg
i.85)
-0.5273 (-0.7224
-0.3505 0.1024 -0.4707 0.3419 -0.5807 -0.0346 -0.2200 -0.7530
x
-0.3627 -0.6264 -0.4450 0.6339 0.2727 -0.0443
and the projection matrix, Z, is 0,8000 -0.2667 0.1333 0.2667 0.5333 0.4000 0.1333 0.1333 0.4000 0.4667 0.2667 0.2667 0.1333 0.2667 0.2000 From the last row of this projection matrix numerical values for eq 5 are obtained x, =
0.26672, t 0.13332, 0.8
t
0.26672, --_
To illustrate the advantage of using more internal standards than the number of factors, the first row of the data matrix was perturbed by 0.1 to give: xT = (6.1,6.1, 8.9}T. When this vector is multiplied by the projectin matrix as 2 = Px,the smoothed vector, %, is obtained: fT= (6.04, 5.99, 9.01)T. Values of this vector are inserted into eq 5 and x4 is found to equal 6.015. Thus, we see that random noise of 1.5% is reduced to 0.25%. Compensation is accomplished by inserting this value into eq 6.
LITERATURE CITED (1) Ritter. G. L.: Lowrv. S. R.: Isenhour, T. L.;Wllklns, C. L. Anal. Chem. 1876,’48,591-596. (2) Saxberg, Bo E. H.; Kowalski, B. R . Anal. Chem. 1979, 5 1 , 1031-1038. . - - . . - - -. (3) Mallnowski, E. R.; Howery, D. G. “Factor Analysis In Chemistry”: Wiley: New York, 1980. (4) . . Barnett. W. B.: Fassel. V . A,; Kniseley, R. N. Spectrochlm. Acta, Part 38 1968,2338, 643-664. (5) Kalivas, J. H.; Kowalski, B. R. Anal. Chem. l9d2,5 4 , 560-565. .
I
Anal. Chem. 1984, 56,1409-1411 (6) Myers, S. A.; tracy, D. H. Spectrochlm. Acta, Part 8 1983, 388, 1227-1253.
(7) Belchamber, R. M.; Horlick, 0. Spectrochim. Acta, Part 8 1982, 378, 1037- 1046. (8) Lorber, A.; Goldbart, 2. Anal. Chem. 1984, 56, 37-43. (9) Lorber, A.; Goldbart, 2.; Eldan, M. Anal. Chem. 1984, 56, 43-48. (IO) Lawson, C. L.; Hanson, R. J. "Solving Least Squares Problems"; Prentice-Hall: Englewood Cliffs, NJ, 1974. (11) Lorber, A. Anal. Chem. 1984, 56, 1004-1010.
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(12) Bennett, C. A,; Franklin, N. L. "Statlstlcal Analysls in Chemistry and Chemical Industry"; Wlley: New York, 1961. (13) Lorber, A,; Eldan, M.; Goldbart. 2.; Harel, A., unpublished results, Dec 1983.
Received for review December 12, 1983. ~~~~~~d March 14, 1984.
Effect of Particle Size on Photoacoustic Signal Amplitude R. Stephen Davidson* and Doreen King
Chemistry Department, The City University, Northampton Square, London EC1 V OHB, United Kingdom
The relationshlp between partlcle size and photoacoustic (PA) signal ampiltude was examlned by the use of potasslum chromate and potassium ferricyanide. For both materials a decrease in partlcle slze resulted In an Increase in the PA signal amplitude. I f reflectance propertles of the materlal are important in determlnlng the PA signal amplitude, a decrease in particle slze should have led to a decrease In the PA signal amplitude. Obviously the Kubeika-Munk theory Is of llmlted use In quanttfylng PA slgnal amplltude measurements, and the results Illustrate the Importance of other factors such as surface area.
The PA theory, as developed by Rosencwaig and Gersho (1) is extremely complicated and, to simplify the theory relating to solid samples, expressions were devised for six special cases. Three of the special cases relate to optically transparent solids, and a further three relate to optically opaque solids (2). The theory has been found to be reasonably sound, when tested experimentally, but modification of the theory was necessary to include light scattering effects (3). A Kubelka-Munk analysis has been applied to correct PA spectra for light scattering effects (4),and the implications of the Kubelka-Munk analysis for the PAS theory have been considered for light scattering thermally thick samples (5). This paper does not deal with a detailed mathematical treatment of the PA effect but with the practical implications of the already developed equations. We have previously examined the relationship between sample concentration and PA signal amplitude for potassium dichromate (6). This compound was found to behave as a typical strong absorber, giving an increase in the apparent absorption with a decrease in particle size (4). The Kubelka-Munk equation
K / S = (1 - R)2/2R where K is the absorption coefficient, S is the scattering coefficient, and R is the reflectance can be used to predict the behavior of weak, as well as strong, absorbers in diffusely scattered light. Consequently, if such a theory is included in an explanation of the PA effect, then the relationship between the Kubelka-Munk theory and the predicted PA effect should be extended to include a study of weak absorbers. This, to date, has not been done for samples which have been ground, as opposed t o samples which have been adsorbed onto substrates (5) and which are relatively weak absorbers. Weak absorbers such as silica and alumina have been used as supports and their optical properties, e.g., reflectance properties, 0003-2700/84/0356-1409$01.50/0
upon the PAS spectra of adsorbed species have been assessed (5). Consequently, we now report a series of simple experiments to determine how the particle size of a light absorbing material affects the magnitude of its PAS signal.
EXPERIMENTAL SECTION A range of potassium chromate (BDH) and potassium ferricyanide (Koch-Light) particle sizes were obtained by first grinding the powders and then passing the powders through a set of sieves, to give a particle size range of 45 pm to >250 pm diameter. Powders of diameter 45 pm to 53 pm were also diluted by adding magnesium oxide (Fisons, average particle diameter 40 rm). PAS measurements were obtained as outlined previously (6). The magnitude of the PAS signal was not significantly affected by small variations in the weight of the sample contained in the sample tray. However, for all measurements, the sample tray waa filled to a constant volume without attempting to compact the powders. Reflectance measurements were carried out with a PerkinElmer Lambda 5 UV/visible spectrometer. This is a double-beam ratio recording instrument with a filter-grating monochromator in a Littrow configuration. It has a holographic grating of 1440 lines/mm and the light sources are deuterium and tungstenhalogen lamps. The extinction coefficients for the compounds used were measured in aqueous solution with a Perkin-Elmer 402 UV/visible spectrometer and are given below h
Cr,O,ZCrO,*Fe(CN),3-
362 370 420
E
x 103 3.5 2.5 0.8
RESULTS Figure 1 shows the effect of particle size on the PA signal amplitude for the powder samples of potassium chromate and potassium ferricyanide. For both materials there is a decrease in signal amplitude with an increase in particle size. In the case of potassium chromate there is a slight visual increase in the strength of the yellow color as the particle size increases. For potassium ferricyanide, however, there is a very marked visual change in the color of the powder as the particle size increases. For small particle sizes the powder has an orange/yellow color, whereas for large particle sizes the color is orange/red. Reflectance measurements showed that for both potassium chromate and potassium ferricyanide powders an increase in particle size led to an increase in the amount of light absorbed (Figure 2). Figure 3 shows the PA spectra of the powders diluted with magnesium oxide. A Kubelka-Munk treatment of these PAS resulta does not result in a linear relationship between the K/S 0 1984 American Chemical Society
