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Anal. Chem. 1985, 57,2537-2540
Compensation for Background Variation by Generalized Background Subtraction Avraham Lorber," Zvi Goldbart, and Alon Hare1 Nuclear Research Centre-Negev, P.O.Box 9001, Beer-Sheva 84190, Israel
Accurate subtraction of the background slgnai at the anaiytlcai wavelength is a critical step In the determination of the true analytical signal. A mathematical treatment of multlple point data that removes the background's contribution to the data is derived. Each data point tf/Is cQrreded by sublractlng the term, b/Cbfi/Cb,2,where b/ is the background value and the summatlon is taken over all points. This background Subtraction method is applied to the determination of law amounts of uranlum in phosphorlc acid by inductively coupled plasma atomlc emission spectrometry in the presence of spectral interferences. Several spectral points In the close vicinity of U I I 385.466 nm are used to resolve the overlapped spectra. Complete spectral resolution and a 4-fold Improvement of the detection limit were achieved by implementing the method introduced here.
method, we present here its application to the determination of trace levels of uranium in phosphoric acid by inductively coupled plasma atomic emission spectrometry (ICP-AES).
THEORY Problem Formulation and Notation. Boldface capital letters are used for matrices, e.g., X, superscript T for transposed matrices, e.g., XT,boldface lower case characters for vectors, e.g., x, and xl,and small characters for scalers, e.g., x,. Superscript denotes the pseudoinverse, e.g., . ' X The pseudoinverse (4) is used to solve a system of linear equations and differs from the inverse by making possible the solution of nonsquare and singular matrices. There are m (i = 1, ..., m) data points for each sample (sensors, wavelengths, or time intervals), n 0' = 1, ..., n) calibration solutions, qnd p ( k = 1,...,p ) analytes. The matrix D which is of the order m X n is the matrix of instrumental responses. Each instrumental response signal, d, consists of analyte contribution, s, and background contribution, b,,,, or
+
J ,
The evaluation of the background contribution to the gross analyte signal is a critical step in quantifying analyte concentration. For single point data, the blank subtraction and calibration curve methods are used to remove the background from the measured response. In the traditional calibration curve approach, a response vs. concentration of analyte plot is generated by empirically observing responses of several concentrations. When the curve is linear, the output of data regression represents the intercept (background) and the slope (sensitivity). If accurate recovery of concentration information is desired, the sensitivity and the background in the sample should be equal to their values in the standard. The same sensitivity and background values are found when the standard matches the sample with respect to major concomitaces. This fact presupposes an approximate knowledge of the desired result and imposes a limitation on the method. The availability of one-dimensional data on a sample (Le., multiwavelength, chromatogram) allows the quantitation of a multicomponent mixture, even though each measured point is not fully selective for an individual component. The quantitation approaches are the generalization of the traditional calibration curve method ( I ) and the standard addition method (2,3).The feature of background value determination by the calibration curve method was incorporated in the multivariable generalization of the method by Brown et al. (1). They constructed a nonzero intercept model by adding a vector of ones to the concentrations matrix. Compensation for variation in the background values between the sample and the standard is usually accomplished by measuring the background values on both sides of the spectral line. Thus, the availability of several data points removes the need for a close match between the sample and the standard. However, this feature was not incorporated in the multivariable generalization of the calibration curve method. In this paper a new principle of background subtraction for one-dimensional data, the generalized background subtraction (GBS) method, is presented. With the GBS method, compensation for variation in the background value is possible. In order to demonstrate the usefulness of this mathematical
J ,
= 'LJ +
dZj
bCJ
(1)
The vector b is the vector of background at m data points. The background magnitude may vary from sample to sample, but its spectrum is constant. Thus, we may write
biJ = b,a,
(2)
here a, is the constant that accounts for the variation of the background vector from one calibration solution to other. The vector a is a vector of TI elements of variations in the background vector. From these observations we may write
or in matrix notations as
D = S + baT
C is the matrix of concentrations of analytes in the calibration solutions and is of order p X n. The matrix F of order m p connects the concentrations to analyte responses by
0003-2700/85/0357-2537$0 1.5070 0 1985 American Chemical Society
X
(4)
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985
or
Inserting this value into eq 4 results in
D - baT = FC Solution of eq 4 is not trivial, however, there are two possibilities to solve it: (a) Assume that the there is no variation in the background value during the calibration step. This is equivalent to vector a being a vector of ones; thus eq 4 becomes
D - bjT = FC
(5)
here j is a vector of n elements, all of them are ones. This allows determination of the background vector a t the calibration step, and background subtraction at the determination step. (b) Determine the background vector before the calibration step. This allows compensation for background variation during the calibration step also. Solution of the Constant Background Model. Equation 5 contains two unknown quantities b and F. To solve it two matrix equations are needed. The equation may be separated into two equations by alternative and more complicated formulation (4)as has been shown for the generalized standard addition method (3). Thus, eq 5 contains enough degrees of freedom that allow determination of both unknown quantities. For isolation of b, the equation is right multiplied by j. The multiple jTj equals n, and b equals
b = (D - FC)j/n
(6)
Inserting this value into eq 5 results in
D - DjjT/n = FC - FCjjT/n
(7)
The mean of the responses for each sensor is defined by n
ai= Cdi,/n
(8)
j=1
Thus, the vector of mean responses, d, is computed by
d = Dj/n
(9)
The zero mean matrix of instrumental responses, D, is computed as
D =D
- djT =
D - DjjT/n
(10)
With similar considerations, the zero mean matrix of concentration, C, is given as
C = C - CjjT/n
(11)
Now, it is observed that eq 7 is equivalent to
D = FC
(7)
that may now be solved to find F as
F = DC+
(12)
Now, the background vector is found by inserting eq 12 into eq 6 b = d - DC+e (13) By the use of the model of eq 5, only a constant background may be subtracted; thus, variation in the background of an unknown sample cannot be compensated for by this model. Compensation is possible, however, by varying background model. Solution of the Varying Background Model. For isolation of a, eq 4 is left multiplied by bT. After rearrangement we obtain
aT = bT(D- FC)/bTb
(14)
D - -D bbT = FC - -FC bbT bTb bTb which may be written as
here I is the unity matrix. In order to understand the result in eq 15, it is important to understand the properties of the matrix I - bbT/bTb.This matrix is a projection matrix (5) that has two properties. I t is a symmetric matrix, and when multiplied by itself, the resulting matrix is the matrix itself. The matrix also acts as a filter that removes background from a data vector. It may be seen that when it is multiplied by any multiple of the background vector, the resulting vector is the zero vector. In terms of linear algebra, multiplication of the matrix F by the projection matrix (I - bbT/bTb)cause the resultant matrix to be orthogonal to the vector b. Therefore, the mathematical procedure used by the GBS method is not merely background subtraction but, rather, orthogonalization to the background. The transformation of the data presented by eq 7 is to right multiply both sides of eq 5 by the projection matrix I - jjT/n. In eq 15 the transformation is made by left multiplying both sides of eq 4 by I - bbT/bTb.It is true that in scalar algebra multiplying both side of an equation by a constant that differs from zero has no effect on the resulting values. In matrix equations this is also true, but on condition that multiplying the matrix that multiply the equation by its inverse will result in a unit matrix. For projection matrices, the inverse is the projection matrix itself. As has been mentioned, the multiplication of a projection matrix by itself results in a matrix that equals the projection matrix. The transformations presented in eq 7 and 15 are therefore real transformations. Quantitation of an Unknown Sample. The quantitation step for both models is according to
(I - bbT/bTb)d, = (I - bbT/bTb)Fc,
(16)
here d, is the vector of instrumental responses of the unknown sample and c, is the vector of determined concentrations. In the case of the constant background model in the calibration step, the determined F and b are inserted in this equation.
EXPERIMENTAL SECTION Experimental facilities and operating condition are listed in Table I. In this study the direct reading spectrometer was operated as a scanning monochromator. The scan was done by moving the entrance slit to scan a spectral region of 0.14 nm across the U I1 385.466 nm spectral line. For each measurement, 11 data points were acquired. The integration time was set to 5 s for each data point. Stock solutions were prepared by dissolved pure metals or reagents (Specpure grade, Johnson-Matthey) in dilute acids (Suprapur grade, Merck) and deionized distilled water. A uranium calibration solution was prepared as 200 mg/L in 5.5 M phosphoric acid. Solutions of the interfering elements Cr, V, Fe, and Mg were prepared in the concentrations of 1,1,5, and 10 g/L, respectively, in diluted acids. The background data vector was determined by running pure water and served to remove the background contribution from all calibrations and unknown samples by multiplying each data vector by I - bbT/bTb. Because each element is contained in a separate solution, there was no need to perform matrix inversion to determine F. There was no sample treatment. Quantitation of an unknown sample is according to eq 16. The equation is solved by the QR decomposition method (6). Subroutines were written in FORTRAN IV and added to the original software that operates the JY48 spectrometer. Several modifications in the original program that allow for real time compensation were also incorporated. The QR decomposition
ANALYTICAL CHEMISTRY, VOL. 57, NO. 13, NOVEMBER 1985 2539
Table I. Experimental Facilities and Operations Conditions aerosol geneator spray chamber plasma power supply gas flows sample feed observation zone spectrometer grating slits resolution photomultipliers
Meinhard concentric glass nebulizer, type T-230-A2, with average uptake rate of 2.3 mL/min Plasma-Therm, Scott type 2.5 kW, 27.12 MHz rf generator, Plasma-Therm, normally operated at 1.25 kW plasma gas, 15 L/min arson; aerosol carrier gas, 1 L/min argon, no intermediate gas was used peristaltic pump, Gilson Minipuls 2 15 mm above the top of the induction coil direct reading spectrometer; Jobin-Yvon Model JY48, with 44 standard analytical channels and four external channels Jobin-Yvon holographic grating ruled with 2550 grooves/mm, giving spectral coverage from 160 to 410 nm entrance slit, 25 pm; photomultiplier slits, 50 pm 0.0195 nm standard type, Hamamatsu R300
Table 11. Summary of Data on Interfering Elements and Uranium
element
wavelength, nm
sensitivitp
Cr
385.422 385.479 385.529 385.44 385.396 385.412 385.454 385.465 385.537 385.47
88 30 46 4 5 1 2 1200 300 20000b
Fe Mg
U V CN
amt in samples, mg/L 100
3000 2000 1-100
200
“Sensitivityis defined as counts/(mg/L). *This value is counts. and solution were computed by the HFT and HS1 subroutines of Lawson and Hanson (6). A Digital Equipment Corp. PDP 11/23 minicomputer operated under the RSXll operating system was used.
RESULTS AND DISCUSSION The samples were obtained from a pilot plant for the recovery of uranium from wet phosphoric acid (7). The process is based on the adsorption of the uranium on chelating resins. The content of uranium in the acid after uranium chelation varies in the range 1-100 mg/L. An analytical method with a detection limit lower than 1 mg/L was therefore necessary. The most prominent spectral line of uranium, U I1 385.958 nm (8),was severely overlapped by the Fe 1385.991 nm and 385.921 nm spectral lines. The iron in the solution (3g/L) gave a signal which was equivalent to 200 mg/L uranium. Resolution of such a severe interference was impossible for the required detection limit. The U I1 385.466 nm spectral line, which is not severely interfered with by Fe, was therefore chosen. Figure 1 depicts the spectrum, in the close vicinity of U I1 385.466 nm, of pure water, a blank solution that contained all concomitants but no uranium, and a standard solution that contained 20 mg/L uranium as well as all other concomitants. The blank solution raised the background by 16% compared to the pure water. Aspiration of a solution of 32% phosphoric acid revealed that 10% of this variation is caused by lowering the uptake rate. Imbert and Mermet (9) documented a 60% decrease in uptake rate for a solution of 32% phosphoric acid. The remaining 6% background enhancement is due to line wings of the concomitances (10). All the elements present in the solution were examined and it was found that only Cr, Fe, Mg, and V interfered with U I1 385.466 nm. The spectral positions, sensitivities, and concentrations of the interferents in the sample are given in Table 11. The interference from iron was not previously documented. Also, a spectral line, whose magnitude was larger by 20% than the continuum background in pure water, was observed. After a series of examinations, it was assigned to a molecular band of CN. Only
Table 111. Deviation of Background Signals (%) Caused by Varying Plasma Parameters and Remaining Deviation after Compensation by the IRPM
varying incident power to plasma varying flow rate
obsd
corrected
-12 15
-0.2 0.3
-7 8
0.2 0.1
Table IV. Comparison of Results Obtained by ICP-AES and Delayed Neutrons (mg/L) ICP-AES delayed neutrons 3 6 22 25 35
5 8 22 26 36
ICP-AES
delayed neutrons
74 120 141 141
74 121 140 141
Mg, Cr, and CN spectral lines directly overlap with U I1 385.466 nm. The CN interference may be considered as background and thus corrected by the generalized background subtraction. The Mg 1385.454nm and Cr 385.479 nm may be corrected only by information from their spectral lines that have unique response with respect to uranium. Therefore, the spectral range was chosen so that it would scan a spectral region of 0.14 nm across 385.466 nm. Although Fe and V interferences do not directly contribute to the uranium signal, they must be taken into account because they contribute within the scanned spectral region. The magnitude of the background signal in the scanned spectral region was found to be 110000 counts. the “detection limit” is defined as the concentration associated with a signal whose magnitude is twice (or three times) the noise in the background signal. A convention widely accepted by ICP users (11)states that the noise equals 1% of the background signal. Applying this convention to our data results in a 2 mg/L detection limit. From these figures it is clear that deconvolution of the overlapping data should be accompanied by accurate background subtraction. The fact that the background determined from a run of pure water could be used for background subtraction from samples containing 32% phosphoric acid (there was 16% difference in the background values) clearly demonstrates the success of the background subtraction. Table 111 presents data on the efficiency of the background compensation. It may be seen that even with a large variation of the plasma parameters, the background subtraction is accurate. A relative standard deviation of 0.25% in the background value was found which results in a detection limit of 0.5 mg/L for uranium, which is a 4-fold improvement. There was no degradation in the compensation performance with time. Table IV compares results obtained by this method to results obtained by delayed neutron activation analysis. It
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 13,NOVEMBER 1985
chelating resins or solvent extraction) or the direct coupling of HPLC to ICP-AES. By use of the GBS method, better detection limits and selectively were obtained without any sample treatment. Thus, we have demonstrated that improved analytical performance can be obtained by exploiting all the information from the available data.
CONCLUSIONS
v
385.537 'r
Two alternative methods for solving the multivariate calibration equations are presented. The selection between the alternatives depends on the following situations: (a) Precise background values are available beforehand. Precision is enough because compensation for inaccuracies is possible. (b) The background values are constant for all calibration solutions. The analyst must decide which model is a suitable description of his experiment. In the quantitation step, the background vector, which was determined either by separate means or by the multivariate calibration, is used to construct a projection matrix. Multiplication of the data vector by this matrix results in removal of the background contribution to the signal. The background removal is effective when there is no variation in the ratios of the elements of the background vector. When this condition is not fullfilled, compensation is possible by using the generalized internal reference method (12). Registry No. U, 7440-61-1; phosphoric acid, 7664-38-2.
LITERATURE CI*ED
85.L30
WAVE LENGTHhm)
385.5
Flgure 1. The spectrum in the close vicinity of U I1 385.466 nm in (a) pure water, (b) a blank solution that contained all concomitants but no uranium, and (c) a standard solution that contained 20 m g l l uranium as well as all other concomitants.
may be seen that good agreement is achieved between both methods. At the low range there is a bias of 2 mg/L between the methods. Alternative approaches for obtaining better selectivity for uranium are separating the uranium from the matrix (by
(1) Brown, C. W.; Lynch, P. F.; Obremski, R. J.; Lavery, D. S. Anal. Chem. 1982. 54. 1472. (2) Saxberg, B.-E. H.': Kowalski, B. R. Anal. Chem. 1979, 5 I v 1031. (3) Lorber, A. Anal. Chem. 1985, 57, 952. (4) Campbell, S. L.; Meyer, C. D. "Generalized Inverses of Linear Transformations"; Pitman: London, 1979; pp 32-41, (5) Lorber, A. Anal. Chem. 1984, 56, 1004. (6) Lawson, C. L.; Hanson, R. J. "Solving Least Squares Problems"; Prentice-Hall: Englewood Cliffs, NJ, 1974. (7) Ketzinel, 2.; Volkman, Y.; Hassid, M. "Research on Uranium Recovery from Phosphoric Acld in Israel", IAEA TC-491, 1983. (8) Boumans, P. W. J. M. "Line Coincidence Tables for Inductively Coupled Plasma Atomlc Emission Spectrometry"; Pergamon Press: Oxford, 1980. (9) Imbert, J. L.; Mermet, J. M. Analusis 1984, 12, 209. ( I O ) Boumans, P. W. J. M.; Vrakking, J. J. A. M. Spectrochim. Acta, Par? B 1984, 398, 1291. (11) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A. (12) Lorber, A.; Eldan, M.; Goldbert, 2. Anal. Chem. 1985, 57,851.
RECEIVED for review March 25, 1985. Accepted July 3, 1985.
