Anal. Chem. 1984, 56,37-43
7440-48-4; Ni, 7440-02-0; Cu, 7440-50-8; Zn, 7440-66-6; Ga, 7440-55-3; ’ 7440-65-5; Zr’ 7440-67-7; Nby 7440-03-1; Mo’ 7439-98-7; La, 7439-91-0; Ce, 7440-45-1; Dy, 7429-91-6; Yb, 7440-64-4; W, 7440-33-7; T1, 7440-28-0; lithium metaborate, 13453-69-5.
LITERATURE CITED (1) Ronan, R. J.; Kunselman, G. I n “Methods and Standards for Environmental Measurement”; Kirchoff, W. H., Ed.; US. Natlonal Bureau of Standards: Washington, DC, 1977; Special pub. 484, pp 107-111. (2) McQuaker, N. R.; Kluckner, P. D.; Chang, 0. N. Anal. Chem. 1070, 51, 666-895. (3) Waish, J. N.; Howle, R. A. Mln. Management 1080, 43, 967-974. (4) McLaren, J. W.; Berman, S. S.; Boyko, V. J.; Russell, D. S. Anal. Chem. 1081, 5 3 , 1802-1806. (5) Skogerboe, R. K.; Urasa, I. T. Appl. Spectrosc. 1078, 3 2 , 527-532. (6) Johnson, 0. W.; Taylor, H. E.; Skogerboe, R. K. Appl. Spectrosc. 1070, 3 3 , 451-456. (7) Burman, J.-0.; Pouter, C.; Bostrom, K. Anal. Chem. 1078, 5 0 , 679-680. (8) Bankston, D. C.; Humphrls, S. E.; Thompson, G. Anal. Chem. 1070, 51. 1218-1225. (9) Sinex, S. A.; Cantillo, A. C.; Helz, G. R. Anal. Chem. 1080, 5 2 , 2342-2346. (10) Cantiilo, A. Y. Ph.D. Thesis, University of Maryland, Coiiege Park, MD 1982. (1 1) Sinex, S. A.; Helz, G. R. Envlron. Sci. Techno/. 1082, 16, 620-625. (12) Suhr, N. H.; Ingamells, C. 0. Anal. Chem. 1068. 3 8 , 730-734. (13) Ingamelis, C. 0. Anal. Chem. 1068, 38. 1228-1234. (14) Moody, J. R.; Lindstrom, R. M. Anal. Chem. 1077, 49, 2284-2267. (15) Dean, J. A., Ralns, T. C., Eds. ”Flame Emission Atomic Absorption Spectrometry: Components and Techniques”; Marcel Dekker: New York, 1971; Vol. 2. (16) Nygaard, D. D. Anal. Chem. 1070, 51, 681-884. (17) Decker, R. J. Spectrochim. Acta, Parts 1060, 356, 19-31. (18) Davis, J. C. “Statistics and Data Analysis in rhoiogy”; Wiiey: New York, 1973.
37
(19) Fabbi, B. P.; Espos, L. F. Geol. Surv. Prof. Pap. (U.S.) 1076, No. 840, 89-93. (20) Katz. A.; Grossman, L. Geol. Surv. Prof. Pap. (US.) 1076, No. 840, 49-57. (21) Machavek, V.; Rubeska, I.; Sixta, V.; Suicek, 2. Geol. Surv. Prof. Pap. (US.)1076, No. 840, 73-77. (22) Manheim, F. T.; Hathaway, J. C.; Flanagan, F. J.; Fletcher, J. D. Geol. Surv. Prof. Pap. ( U . S . ) 1076, No. 840, 25-28. (23) Thomas, J. A.; MountJoy, W.; Huffman, C. Geol. Surv. Prof. Pap. (US.)1078, No. 840, 119-112. (24) Walker, G. W.; Flanagan, F. J.; Sutton, A. L.; Bastron, H.; Berman, s.; Dinnin, J. I.; Kenkins, L. B. Geol. Surv. Prof. Pap. (U.S.) 1076, No. 840, 15-20. (25) Weigand. P. W.; Thoreson, K.; Grlffin W. L.; Heier, K. S. Geol. Surv. Prof. Pap. (US.)1076, No. 8?0, 79-81. (26) Dybczynskl, R.; Suschny, 0. Final Report on the Intercomparlson Run SL-1 for the Determlnation of Trace Elements in a Lake Sediment Sample”; Internatlonai Atomic Energy Agency: Vienna, 1979; IAEA/ RL/64, 79-10306. (27) Faiiey, M. P. Ph.D. Thesis, University of Maryland, College Park, MD, 1979. (26) Quisefit, R. D.; DeLaBatie, R. D.; Faucherre, J.; Malingre, G.; Vie Le Sage, R. Geostands. Newsl. 1070, 3 , 181-184. (29) Bothner, M. H.; Aruscavage, P. J.; Ferrebee, W. M.; Baedecker, P. A. Estuarine Coastal Mar. Scl. 1060, IO, 523-541. (30) Slnex, S. A.; Helz, 0. R. Environ. Geol. ( N . V . ) 1981, 3 , 315-323. (31) Meggers, W. F.; Coriiss, C. H.; Scribner, B. F. ”Tables of Spectral-Line Intensitles”; U.S. Natlonal Bureau of Standards: Washington, DC, 1975; Mongraph 145. (32) Abbey, S.,Geostands. News/. 1080, 4 , 163-190. (33) Giadney, E. S.; Goode, W. E. Geostands. Newsl. 1081, 5 , 31-64. (34) Certificate of Analysis, 1961; National Research Council, Chemistry Dlvlsion, Montreal Road, Ottawa, Canada. (35) Certificates of Analysis, 1978 and 1982; U.S. National Bureau of Standards, Washington, DC.
RECEIVED for review September 29, 1982. Resubmitted September 12, 1983. Accepted September 27, 1983.
Generalized Internal Reference Method for Simultaneous MuItichanne1 Analysis Avraham Lorber and Zvi Goldbart* Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheua 84190, Israel
The appllcablllty of the Internal reference method Is llmlted by the need to find a reference channel that will mimic the fluctuatlons of the analyticai channel. A generailred Internal reference method (GIRM) Is suggested, that enables compensatlon of nonrandom fluctuatlons In analytical channels, regardless of the parameters affecting them. I n the generalized method several Internal standard channels, that respond differently to variations of the parameters of the analytical system, are measured simultaneously. The criteria for selecting Internal standards and the calcuiatlon procedure are described. The GIRM Is applicable to systems that are flicker noise llmlted and that perform multichannel (or multlplex) measurements simultaneously.
Inductively coupled plasma (ICP) atomic emission spectrometry (AES) is now one of the most important analytical methods for the accurate determination of minor elements and features very low limits of detection for trace elements. Both precision and limits of detection are signal to noise limited. Noise characteristics for optical spectrometry sources were investigated by Winefordner et al. ( I ) . Two major types of noise were considered shot noise and fluctuation noise (flicker noise). Shot noise is due to the statistical nature of photon arrival at the detector and electron emission in the detector 0003-2700/84/0356-0037$01.50/0
and amplifying electronics. Shot noise follows the Poisson distribution with the standard deviation proportional to the square root of the number of events. The second type of noise is the low-frequency noise called fluctuation (flicker) noise. This arises from changes caused by slow drift of analyte supply rate and light source drift. Analysis of the limiting noise in the ICP source (2, 3) showed that the source flicker noise contribution was dominant and limited the precision to 1% This limit applies to other optical spectrometry methods as well (1). A well-known method to reduce the nonrandom fluctuations is the internal standard method (ISM) (now known as the internal reference method) (4),which has been applied successfully in direct current (DC) arc spectrometry. Extensive theoretical and experimental studies of ISM in analytical emission spectrometry were performed by Barnett, Fassel, and Kniseley (5,6). Their aim was to set criteria for a systematic procedure for choosing the right internal standard for each analytical line. Their list of criteria is copied here for convenience of discussion: (1) The internal standard’s concentration should be negligibly low. (2) The internal standard should be very pure compared to the elements being determined. (3) Both internal standard and analyte lines should have the same excitation energies. (4) When photographic methods of recording are used, both
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0 1983 American Chemical Society
38
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY lQ84
lines should be of approximately the same wavelength. (5) The volatilization rate of the internal standard element and the analyte element should be similar, (6)The ionization energies of the internal standard and the analyte element should be similar. (7) The atomic weights of the two elements should be roughly the same. (8) Both lines should be free of self-absorption. (9) Both lines should be of approximately the same intensity, especially when photographic methods are used. Conditions 1,2, 8, and 9 can easily be achieved; the other conditions constitute the ideal case. The internal reference method has been used widely in spark and DC arc analytical emission spectrometry. These sources are characterized by time dependence of the analyte feed rate to the excitation zone and temporal and spatial fluctuations. The advantage of the IRM in these cases is in meeting criterion 5 mentioned above, when appropriate match of line pair is achieved. Other criteria for selecting the internal standard are of less importance as the mass transfer rate from sample to excitation zone is responsible for most intensity changes and source dependent noise. The reproducibility of a DC arc signal can be improved by this method from 40% deviation to lo%, and to 2% for the spark source. Barnett et al. (5,6) checked the validity of the IRM for the ICP and found only limited success. The absence of a valid model for the source and the inapplicability of local thermodynamic equlibrium (LTE) conditions were the reasons for the unsuccessful application of IRM to the ICP. The problem of the applicability of the IRM to ICP was discussed a t the second ICP conference at Noordwijk (7). It was noted there that many lines that were chosen showed negative response. To overcome this limitation it was suggested having one standard line for each analytical line. A recent note by Belchamber and Horlick (2) concluded that as a standard, one must choose the parameter that mimics in as many ways as possible the fluctuation behavior of the analyte. It is suggested there, that this parameter may be a spectral line, flow, pressure, or acoustic emission. It is the aim of this work to present a mean for generalizing the internal reference concept and using it for drift correction. The generalized method may be applied without regard to the five restrictions (criteria 3 to 7) mentioned above. The method applies to all simultaneous multichannel methods and not only to emission spectrometry.
ROTA T I 0 N
r
PLASMA
PARAMETERS
CORRELATION CALCULATED EXPERIMENTAL
ANALYTICAL
ANALYTICAL
SIGNALS
SIGNALS I
*
1 1 1 1 f
COMPARISON
-
111111
Flgure 1. Mnemonic diagram illustrating the GIRM.
at the time in which the analytical calibration function has been determined). To illustrate the GIRMs calculation procedure, we assume that the fluctuation of ith spectral signal, yo is described by a linear sum of fluctuations of r ( r Ip ) system parameters, dj. The assumption is valid both for the internal standard and for the analytical signals. Therefore we can write for the internal standard signals the set of equations given by (1). uij
+ upZd2 + ... + Uljdj + ... + ul,rdr ~1 uz,ldl + uz,2d2 + ... + Uzjdj + ... + uz,rdr = yz ul,ldl
THEORY In contrast to the traditional internal reference method that cannot be applied to systems violating criteria 3 to 7 (mentioned in the former section),the generalized internal reference method (GIRM) uses information on the variations of signals gathered simultaneously from channels having different physical parameters, to correct for variations in signals. The calculation procedure that includes three steps is described schematically in Figure l. The temporal characteristics of p internal standards signals (signals that are not affected by change of analytical sample concentration) serve as sensors to detect variation of the instrumental parameters. Rotation of the fluctuation data from the spectral line space to the instrumental parameter space is performed in the first step. In the second step the determined instrumental parameters are used to find the fluctuations of the analytical signals (signals produced by a species whose concentration is to be determined). In the last stage the calculated variations in the analytical signals are compared to the measured analytical signals variations to obtain information on the concentrations. When both calculated and measured variations are equal, the variation is attributed merely to system parameter variation from the “set point” (the system’s condition
up,ldl
+ up,,dz + ... + Upjdj + ...+ up,,dr = y p
are the linear coefficients describing the variation of the ith internal standard signal due to fluctuation in the jth instrumental parameter. In matrix notation eq 1 is written as Ud = y (2) where U is a p X r matrix, d is the vector of instrumental parameter fluctuations with r elements, and y is the internal standard fluctuation vector with p elements. In cases that (p> r) a least-squares solution of the matrix equation is given by d = (Uw)-lUTy (3) The calculated value of any analytical signal is computed by an equation similar to eq 1 with the aid of the vector d. The linear model of the GIRM presented in eq 1-3 may serve as a possible model for a rough approximation as will
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984 SYSTEMS
SYSTEM PARAMETERS
SAMPLE
UPTAKE AEROSOL
I
t
TRANSPORT P R OC E S S E S
ANALyTEl
D E NSI TY
-
\ N E B U L I Z I N G GAS
1
FLOW
RATE
b % - G L T l
/ER
1
J
I
Flgure 2. Block dlagram illustrating the compound system concept and the alternative system parameters representation. Inductively coupled plasma is used as a model.
be shown later. However, the actual relations describing the signal fluctuation cannot be described as a summation of the system parameter fluctuations. In order to form a suitable basis to develop the method, one must define some terms and analyze the compound system involved. System Consideration. The ICP, which serves as a popular light source for atomic emission spectrometry is used to demonstrate the concept of the compound system and its analysis. The compound system of the ICP, shown in Figure 2, transforms the sample, which is the input signal, to an output light signal. The light detection and electronic readout system is not considered, as it is assumed that its main coptribution to the noise is shot noise which is negligible. Each system is characterized by an individual transfer function that is the ratio of the output signal to the input signal. The transfer function is time dependent but ita value is controlled by external system parameters shown at the right side of the figure. A change in the transfer function of one system will result in the proportional change of its output signal which is the input signal for the following system. The transfer function of the compound system will therefore be the product of the transfer functions of the individual systems. In terms of system theory the impulse response of a compound system h,(t) fully characterizes a linear system. Provided that the impulse response is known, the output signal f ‘ ( t ) can be calculated from the input signal X ( t ) with the convolution interal (8)
[ ( t )= jmhc(T)X(t 0 - 7 ) d7
h.,(t)*x(t)
(4)
If the impulse response hi(t) of each of the individual system processes is known, the impulse response of the complete linear system of three components can be expressed as
[ ( t )= h3(t)*(h,(t)*[h,(t)*x(t)l~ = h3(t)*h,(t)*hl(t)*X(t) (5)
The systems that comprise the compound transfer function are more practically represented by system parameters that are measurable quantities, while the impulse response of the systems cannot usually be measured, especially during an analytical run. The analytical functions describing the systems in terms of system parameters are usually unknown. However, we can describe the transformation of the input signal to the output signal by system parameters directly when the fol-
39
lowing requirememts are fulfilled: (a) all system parameters that affect the signal have been included; (b) the changes in system parameters are mutually independent; (c) each functional relationship,gj, describing the dependence of the output signal on the ratio, lj, of the present jth system parameter value to its value at the “set point”, can be found experimentally. We observe that a change in one parameter changes the output signal, which is the input signal for the following parameter change. The transfer function of the compound system is therfore the product of the individual functional relationship values. In order to omit time dependence, we notice that the time lag between the entrance of the sample and the exit of radiation is short when compared to the integration time. The output signal therefore represents the sample which enters the system within the same time interval. The time dependence of the response function can be replaced by a serial integration index, m. To further simplify the notation we notice that the exact number of a measurement in a sequence is of no importance. The ratio of the present measurement value to the value at the point referred to as the “set point”, is the quantity of importance. Thus, the integration index is omited. The preceding assumptions and observations allow us to write the dependence of the output signal on system parameters as
where the product is taken over the r system parameters and x and f‘ are integrations of the input and output signals, respectively, taken over an arbitrary period. The explicit presentation of the plasma parameters in eq 6 seta a limitation on the precision that may be obtained from the method. In order to achieve high precision, the precision in measuring system parameters, when calibrations are performed, should be at least as high as that in measuring analytical signals. This is not the case in instrumental analytical systems (if gas flow rates are assigned to system parameters, they can be measured within a relative error of 1 % , which is 2 orders of magnitude higher than a 12 bit A/D converter resolution). Therefore, the system parameters fluctuation measurements should be replaced by sensors that have the same precision as thcae of analytical signals, The replacement is done by an alternative definition of the IRM. Defining the Internal Reference Method. “The internal reference method (IRM) is a method of correction for system fluctuations by measuring reference signals that emerge from the same source as analytical signals”. The term signal will be limited to the output signals of the system, I, (i = 1, ...,s). Deviation in signal is expressed by a relative signal, IiR,defined as the ratio of the instantaneous signal to its value at the ”set point”. Reference signal is an internal standard signal that serves as a scale for measuring the values of the system parameters at a specific integration period. The reference signal, IrefR, is a product of r components, each of which, IrefJR, varies due to the jth parameter while all other parameters are held constant according to
Ir,f,jRE IrefR(Zj) (all Zk #
Z j held
constant)
(7a)
and the total reference signal is
With this definition of the internal reference signal, it is obvious that the components of the reference signal replace
40
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984
the instrumental parameters and the compound reference signal described by eq 7b acts in the same way as the compound transfer function in eq 6, thus providing a means to avoid the presentation of the parameters explicitly. For a single internal standard, the internal standard signal serves as the reference signal. The IRM Calculation Procedure. The calculation procedure is as described in Figure 1: p internal standard signals are measured 0,1 r ) and used to calculate the components IrefJRof the reference signal by solving the set of equations (8a). The f i s t equation in this set is equivalent to eq 7b. The
In the following section the sets of eq 8,9, and 10 will be solved for three degrees of complexity as follows: (a) solving eq 10 assuming I*: = IrefR;(b) solving eq 9 and 10 assuming the response of I*: to all components of the reference signal is the same; (c) the general case, solving all three equations. Schemes for Solution. Line Ratio Internal Reference Method (LRIRM). This is the well-known IRM, which is the simplest case of the full method. Let eq 8 be I*: = IrefRfor all analytical signals, regardless of the fluctuation in system parameters. Inserting it into eq 10 results in
Ci = +i(I?/IrefR)
(11)
This solution shows the same dependence for all analytical lines, as the reference line has only one component. The physical interpretation is that the change in analytical signal is related to the compound system and cannot be related to any specific system or to its corresponding system parameter. Analyte-Internal Reference Correlated Method (AIRCM). This method assumes that each calculated analytical signal is correlated to the reference signal by its specific equation
rest of the equations that describe the functional relationships are determined by
fi, '
IijR(lj) = f i j ( I r e f i R ) (all lk # lj held constant)
(8b)
The components of the reference signal can now be used to calculate the "calculated analytical signal"-I*iR for analytical signals i (i = q, ..., s). The calculated analytical signal is the signal that is obtained if the analyte concentration is held constant. The set of equations for the calculated analytical signals is given in eq 9. When there is no change in the
I*lR = f i (IreP) (12) Using this assumption we still observe a compound system that cannot be divided into systems and that reacts with the same response to all system parameters. Equation 12 is superior to eq 11,as it permits a specific functional relationship, f i , of the ith relative analytical signal to the reference signal. These functions must be found experimentally and may even have a negative response. There are two experimental methods to find the functional relations. The first is to collect random analytical and reference signals and correlate them by a common statistical regression. The second is to perform a predetermined change in system parameters. Each method has its advantages and limitations. The first method is superior to the second in that the measured changes are representative of the fluctuations that usually occur in the compound system. It is limited, however, as there is no guarantee that the fluctuations that have,occurred represent the fluctuations in the measuring interval. The advantage of the second method is that the coefficients are determined by taking measurements throughout the whole range of possible fluctuations. This approach may cause inaccurate determination of coefficients as the weight it applies to each change in any parameter is the same, while the first approach represents the right weight proportions. Generalized Internal Reference Method (GIRM). In GIRM, eq 8, 9, and 10 are solved by assuming that the functions f , ,can be written as a power series
y,, = alJxl + biJx12+ ... (9)
where yl,. is the relative fluctuation in the ith signal caused by variation of the jth instrumental parameter and xI is the fluctuation of the jth component of the reference signal. The fluctuation terms are connected to the relative signal by y1 = I*: - 1 and xI = I*,,# - 1. Each equation in the set of equations (8) then becomes r
IiR
= ll(1 j=l
concentration of the analyte, the ratio IiR/I*iRshould equal unity. Therefore, any variation in this ratio indicates a variation in the concentration Ci
ci = +i(IiR/I*iR)
(10)
where $, is the analytical calibration function. Considering the concentration as the input signal to the system, one observes that eq 10 acts in the same way as eq 6.
(13)
+ aijxj+ bijx:j2 + ...)
Equation 14 is a nonlinear equation. However, by assuming small fluctuations we may neglect all terms of order higher than 1 and obtain
yi
zaijxj j=l
which connects the fluctuation in the signal to fluctuations in the components of the reference signal linearly. This equation is the same as eq 2 except that the instrumental
,
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984
parameters are now replaced by xi. The set of eq 8 and 9 now becomes eq 16. The reasons for deriving the linear approx1
1
a2.1
a2.2
a ,2
timization problems can be found in ref 10. The determination of the coefficients in eq 14, can be done in one of the following ways: variation of each system parameter separately; variation of all system parameters simultaneously. The experimetnal design for the first approach is easier, but the second method has an advantage of having more statistical information for each parameter change (for the same number of experiments). It is easier to examine the benefits of each method when using the linear approximation. Equation 16 can then be written for a set of m experiments as
AX = Y %,2
*
.
as,j
(16) imation are the following: It presents a simple method for solving the GIRM and a first approximation for a more accurate solution of eq 8 by nonlinear methods. It serves to describe new criteria for selection of internal standards and to estimate the error propagation involved in the method. We can describe eq 16 by means of matrix notation as
Ax =
(2)
(17)
where A is the s X r matrix composed of ai, components, x is the vector of r components of the fluctuations of the reference signal, and y1 is an upper subvector of measured fluctuations in the internal standard signals. y2 is a lower subvector of (s - p ) elements, of the calculated analytical signal fluctuations. Considering the upper p X r submatrix Al, which corresponds to the coefficients of internal standard signals and the lower (s - p ) X r submatrix A2, which corresponds to the coefficients of the analytical signals, the solution of the GIRM method is performed according to the following steps: The vector x is found by solving the equation
A1x = Y1 which has a least-squares solution
x = (AlTA1)-lAlTy1
(18) (19)
The vector y2 is then simply YZ = A ~ ( A I ~ A I ) - ~ A I ~ Y ~
(20)
More accurate solutions of eq 8 are obtained by taking into account higher orders of the term x,. For example, by adding second-order terms to eq 14,we get r
r
The higher order terms are added to the right-hand side of eq 16 and the system is solved iteratively by one of the methods described in ref 9. The linear system (eq 16) serves as an initial approximation. A generalized approach is the solution of the nonlinear system (eq 8) by one of the methods of unconstrained optimization. We solve the objective function
where wi values are the weights. The objective function minimizes the least-squares sum of the differences between the calculated values of the relative internal standard signals, I*iR,and its measured value, IiR.The sum is taken over the internal standard signals. A first estimate for IrefiR can be obtained by eq 19. Methods for solving unconstrained op-
41
(23)
where X is the r X m matrix for m changes in the r components of the reference signal and Y is the s X m matrix for m changes in s signals. The problem,ofsolving eq 23 is similar to the problem of finding the concentrations of several components in a set of multicomponent solutions with an array of detectors, each of which is interfered by all accompanying components. Such a problem was analyzed by Kowalski et al. (11, 12) for the generalized standard additon method (GSAM),who found that the first method is preferable as its error propagation is minimal. This will, of course, be even more important for our nonlinear system. Therefore, it is highly recommended to vary one parameter for determination of each coefficient.
DISCUSSION In the simple internal reference method, since the mathematical relationship is a simple ratio, the urulesnfor selecting an appropriate standard are mainly physical. In the GIRM, since the signals are correlated to system parameters, the physical criteria are not important, except that the signals of the standard channels must be large enough to avoid shot noise limited precision. Instead of physical criteria we have the following mathematical demands: The rank of the coefficient matrix should be sufficient to avoid singularity, and the number of internal standards must be equal or greater than the number of considered parameters. The error in internal standard signals should be limited to a level that will be stated in the following section. The standard channels should be selected for minimum error propagation. Rank Deficiency in the GIRM. It has been stated that for r instrumental parameters we must have p internal standard signals so that p 1 r. More precisely, p should be larger than or equal to the rank (A). A matrix whose rank is smaller than r is singular. This can happen for the following reasons: (a) The rows of the matrix A are multiples of other rows. This is the case when some internal standards respond in the same manner to instrumental parameter variations, so that they behave according to the criteria for internal standards mentioned by Barnett et al. (5). This can also occur if one internal standard is correlated to another internal standard. (b) The rows of the matrix are linear combinations of each other. This case is very unlikely to happen and has not yet been observed in cases of analytical applications. (c) A column of the matrix is a multiple of another one. This happens when two instrumental parameters affect the same system and thus only one column is needed to define the system change; the change in signal cannot be related to either of the instrumental parameters alone. This can also occur if the signal is affected similarly by two different system parameters, not necessarily the same system. Id) A column of the matrix is a linear combination of other columns. In this case the number of systems affecting the signal is exceeded by the number of assumed instrumental parameters. To avoid singularity and permit solution of the
42
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984
matrix, one system parameter should be excluded. With this principle the analytical system can be analyzed for the minimum number of systems that can defiie the analytical output. In principle, the rank of the matrix A can be calculated by the Gaussian process of elimination. However, experimental errors destroy the eleggnce of this process, and statistical methods of decision must be brought for rank determination. A few methods have been developed and applied to the determination of the number of species in multicomponent systems (13-17). The application of rank (factor) analysis to chemistry has been reviewed by Malinowski and Howery (18). Error Propagation in the GIRM. Solving eq 16, one finds two extreme cases of accuracy: In the first case (the “well conditioned” case), each coefficient of the equation is related to one plasma parameter and thus the coefficient matrix (except the internal reference row) is diagonal. This leads to a solution in which each equation is solved independently. The uncertainty in determination of each variable is, therefore, not affected by the others. The opposite case (the “ill Conditioned” case) leads to a singular matrix that cannot be inverted in order to find a solution. Therefore, the error in solving it tends to infinity. Between those two extremes lies the general case in which the errors in the experimental data are propagated. An example for error propagation is quoted here (12). 3.56 -1.92 (-1.92 2.44)
(::) (-:;) =
(24)
The solution of this sytem is x = (-4.0, 3.0)T as is easily verified. Suppose the right-hand side, b, is perturbed by the error vector 6b = (0.3, 0.4)T which produces a relative error IlSbll/llbll of 2% (11.11 r the Euclidian norm of the vector, defined by the square root of the sum of the squared elements of the vector). The corresponding solution for the perturbed vector R = x 6x is; R = (-3.7, 3.4)T which corresponds to a relative solution error of 10%. Thus an error amplification factor of 5 results. This fact should be considered when a set of internal standard signals is selected. The set should give the minimum error amplification possible. There are two methods for determining the condition of a matrix (19). One is an analytical method that determines the “condition number”, that can be calculated by
+
cond (A) =
(xl/xk)1/2
(25)
where XI is the largest eigenvalue of the matrix ATA and is the smallest significant eigenvalue. The second method is the “sensitivity analysis”. Errors due to the Extraction of the Signals from the Background, The measured gross signal, IG,,is composed of IG, =
IC,+ I d
+
18
(26)
where I,, is the signal due to the concentration of species i, is a background signal that is dominated by drift noise (source background), and I , is a background signal that is dominated by shot noise (electronic components in the detection system). Let us assume no drift components and deal only with shot dominated noise. If we require that the relative error not exceed el, we get the condition
Id
(27) is1
This is the ideal case. When an attempt is made to solve eq 8, one begins with I,,, which includes the drift noise in the background, I d . For this case we have the condition P
e2
> s d * [ c ( I d / I c , ) 2 / @- r)11’2 i=l
(28)
where S d is the drift component value in the background. The signal from the drift dominated background may be corrected by considering it as an analytical signal, given that its correlation to the various instrumental parameters is known. This calculated value of the background is inserted in calculating I,. Successive approximations will result in a background that is free of drift. In order that convergence will occur, the following condition should be satisfied:
e2 >> el
(29)
The criterion for stopping the iteration is llYl”+l - Y1”ll
= el
(30)
where yl”is the value of the vector of internal standard signals at the nth iteration. The final result will be free of drift. This iteration procedure can be applied to problems of analyzing small signals in high background spectra, in which the background noise is dominated by drift noise.
CONCLUSIONS The GIRM takes full advantage of the multidimensional data obtainable in the multichannel analytical system, in order to improve the performance of analytical instruments. This is an alternative to the common approach of improving the stability of the analytical instrument components. Application of the GIRM to existing analytical signal sources results in the following benefits: The characterization of nonrandom noise can be accomplished (20), to find the contribution of various components of an instrumental system. Systems of different firms can thus be compared and improvement in components performance can be quantitatively evaluated. This concept can be further extended to detect possible differences between sampling techniques and between different sample preparation methods. The precision of simultaneous multielement analysis can be improved (21) up to the limit of shot noise. In high background spectra, the GIRM can be used to determine precisely the background at the analytical channels during the data collection period. The limit of detection, which is inversely proportional to the background noise, can thus be improved. The method as presented here does not include correction for spectral interelement interferences, though extention of this case is possible by the generalized standard addition method (11) or by multivariate calibration. The motivation for generalizing the internal reference method was due to problems encountered in applying the method to the ICP source as mentioned above. However, other multichannel systems also may be considered. Of particular interest is the application of the GIRM to isotopic dilution mass spectrometery (IDMS). Successful application will result in reducing the number of isotopes (acting as internal standards) in multielement analysis and precision improvement. In a following publication the suggested method will be applied to an ICP source (21). ACKNOWLEDGMENT The authors wish to thank A. Levin and S. Adar for carefully reading this manuscript. LITERATURE CITED (1) Winefordner, J. D.; Avnl, R.; Chester, T. L.; Fltzgerald, J. J.; Hart, L. P.; Johnson, D. J.; Plankey, F. W. Spectrochm. Acta, Part B 1976, 318, 1-19. (2) Beichamber,R. M.; Horllck, G. Spectrochlm. Acta, Part 8 1982, 378. 71-74. (3) Boumans, P. W. J. M.; McKenna, R. J.; Bosveld, M. Spectrochlm. Acta, Part8 1881, 368, 1031-1058. (4) Boumans, P. W. J. M. “Theory of SpectrochemicalExcitation”;Hllger & Watts: London, 1966.
43
Anal. Chem. IQ04, 56,43-48 (5) Barnett, W. B.; Fassel, V. A,; Kniseiey, R. N. Spectrochlm. Acta, Part (8) (7) (8) (9)
(IO) (11)
(12) (13) (14)
6 1066, 2 3 6 , 843-884. Barnett, W. B.; Fassel, V. A.; Kniseiey, R. N. Specffochim. Acta, Part B 1970. 256. 139-181. Greenfield, S.’ ICP Inf. Newsl. 1078, 4 , 199-220. Duursma, R. P. J.; Smlt H. C.; Maessen, F. J. M. J. Anal. Chim. Acta 1081, 133, 393-408. Carnahan. B.; Luther, H. A.; Wliks, J. 0. “Applled Numerical Methods”; Wlley: New York, 1989. Murray, W. ”Numerical Methods for Unconstrained Optimization”; Academic Press: London, 1972. Saxberg. Bo E. H.; Kowalski, B. R. Ana/. Chem. 1070, 51, 1031-1038. Jochum. C.; Jochum, P.; Kowalskl, B. R. Anal. Chem. 1981, 5 3 , 85-92. Wallace, R. M. J . Phys. Chem. 1960, 64, 899-901. Wallace, R. M. J.; Katz, S. M. J . R y s . Chem. 1964, 68, 3890-3892.
..
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~
(15) Katakis, D. Anal. Chem. 1965, 3 7 , 878-878. (18) Hugus, 2. Z.,Jr.; El-Awady, A. A. J . Phys. Chem. 1971, 7 5 , 2954-2957. (17) Wernimont, G. Anal. Chem. 1967, 39, 554-557. (18) Malinowski, E. R.; Howery, D. G. ”Factor Analysis in Chemistry”; Wlley: New York: 1980. (19) Rice, J. R. Matrix Computation and Mathematical Software”; McGraw-Hill: New York, 1981. 120), Lorber. A.: Goldbart.. 2. Sosctrochim. Acta. Part 6 1983, 386, 211 . (Supplements). (21) Lorber, A.; Goldbart, 2.; Eidan, M. Anal. Chem. 1984, 56, 43-48.
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RECEIVED for review September 20,1982. Accepted September 2, 1983.
Correction for Drift by Internal Reference Methods in Inductively Coupled Plasma Simultaneous Multielement Analysis Avraham Lorber, Zvi Goldbart,* and Michael Eldan Nuclear Research Centre-Negev, P.O. Box 9001, Beer-Sheua 84190, Israel
The internal reference method (IRM) Is a common method for the correctlon of nonrandom fluctuatlons in atomic emlssion spectrometry (AES). A generallred form of the IRM (GIRM) Is presented In thls paper, whlch enables the detectlon of variations in discrete plasma parameters, through internal standards. An Inductively coupled plasma (ICP) system was used to evaluate the GIRM. Changes wlth tlme In net ilne intensity, of various spectral Ilnes, over 30-min periods were recorded by a photodiode array system and then corrected for the dflft by the GIRM. It was found that the remalnlng nolse exceeded that of the detectlon system (0.07 % ) by 30% to 100% only, whereas wlthout thls treatment the drlft was a 100 times hlgher than that of the detection system.
There are three common approaches to the stabilization of output signals that are affected by nonrandom noise: (a) the noise characteristics of the signal generating system are improved; (b) the physical parameters of the system are kept constant by a measuring and feedback system; (c) the signal is corrected by computational techniques only. The first approach was applied to an ICP system and succeeded in reducing noise to less than 0.5% by using an improved nebulizer and stabilized flow (1). The application of the second approach to ICP-AES was proposed by Ohls and Koch (2) and was tested by Dorn (3). A certain degree of improvement in the stability of the argon signal was achieved at the cost of considerable complication of the measurement and control system and the requirement that the accuracy in measuring the variable parameters of the analytical system be at least as high as that in measuring the analytical signals. This requirement becomes especially rigorous in cases where small changes in the experimental variable parameters cause amplification of the instability of the analytical signals. This approach is especially important and common for cases in which the deviations in the output signal are caused by changes in system parameters such as chemical processes, which are irreversible. The third approach has become more attractive in recent years, with the rapid advance and the miniaturization of 0003-2700/84/0356-0043$01.50/0
computers and the tendency to incorporate them in modern analytical apparatus. In a recent note Belchamber and Horlick (4) suggested that changes in one variable parameter-the pressure in the spray chamber-are representative of most of the plasma parameters. Correlating the measured changes of this parameter to changes in the analytical signal allows its correction. This treatment has the same limitation as the second approach, with respect to the accuracy in measuring the plasma parameter. Others have suggested that the drift of the output signal is a simple function of time and thus can be corrected either by prior calibration (5) or by measuring a number of standards before and after the analytical run (6). This treatment can be applied only to sources for which this assumption is valid. A well-known method for correcting nonrandom fluctuations in AES is the IRM (7). A spectral line with physical parameters similar to those of an analytical line is selected as a reference, and it is assumed that both lines are affected to the same degree by various parameters, so that the ratio of their signal intensities is constant. This method does not provide the sought for correction of output signal for the relatively new emission sowces such as ICP and direct current plasma, and therefore its use is limited Mainly to atomic emission sources such as DC arc, sparks, and glow discharge. The reason that the IRM does not hold for ICP is that the observed changes in the internal reference line do not necessarily mimic those in the analytical lines (8). The IRM can be used for plasma sources that are in local thermodynamic equilibrium even though the line pair is not similar, on condition’that the temperature change is known (7). Barnett et al. (9) checked experimentally and concluded that an ICP source is dominated by non-LTE mechanisms. Three versions of the IRM were presented in detail in a separate paper (IO). In the present work the compatibility of the three versions was tested experimentally.
THEORY The line ratio internal reference method (LRIRM) correlates variations in concentration C,of the ith element in a sample to variations of the ratio IiR/IrefR,where IiR is the relative intensity (=intensity at present measurement divided 0 1983 American Chemical Society
