ANALYST, NOVEMBER 1987, VOL. 112 1541

Calibration in Atomic Absorption Spectrometry

Paul Kabaila Department of Statistics, Latrobe University, Bundoora, Victoria 3083, Australia Thaung Lwin* CSIRO Division of Mathematics and Statistics, Private Bag 10, Clayton, Victoria 3168, Australia and Phil Strode CSlRO Division of Mineral Chemistry, P. 0. Box 124, Port Melbourne, Victoria 3207, Australia

A practical procedure for calibration in atomic absorption spectrometry is described, the basic statistical problem is formulated and an approximate solution is provided. A computer program, developed to carry out this procedure, is also described. This program has been tested in practice and the results of typical examples are shown. The program uses three to ten standards and includes a number of novel features such as the estimation of curve-fit and drift errors, determination of the limit of detection and a procedure to identify an outlying standard. The error in the working standards is also considered. Keywords: Nun-linear calibration; atomic absorption spectrometry; errors in working standards

Calibration in atomic absorption spectrometry (AAS) must be carried out against standards of known concentration whenever unknown samples are to be determined. As the shape of the calibration curve depends on the particular elements or instrument parameters used, great care must be taken in choosing the family of curves used to determine the calibration curve. In addition, information on the precision of the estimates is needed to establish the reliability of the results.

Although many commercially available atomic absorption instruments have inbuilt curve calibration facilities, these procedures are found to be unsatisfactory for the various reasons explained recently by Tyson. 1 The major drawbacks of these inbuilt procedures include the following: (i) the determination of precision is from replicate measurements only and does not account for curve-fit and drift errors, (ii) very few outlier detection procedures are available to detect the outlying standards and (iii) often the forms of the calibration curves employed are not readily available and for the available family of curves employed there is no clear suggestion as to which curve or curves could be used effectively in wide-ranging situations.

Although Tyson' sets out the shortcomings of the current inbuilt calibration procedures he does not present an alterna- tive. Here we present a comprehensive procedure using a family of curves which we have found experimentally to represent most accurately the range of curves obtained in practice. The precision of the analysis is determined account- ing for such errors as curve fit, drift and the error in the working standards used. In addition, the limit of detection is determined and a procedure is used to identify outlying standards. The family of curves used in the curve-fitting procedure are also compared with two of the available widely used curves. Although the computational requirements of such a procedure are high, the program, in BASIC, will run in a mini- or microcomputer with 48K of RAM, Such a minicom- puter could be connected on-line to an atomic absorption instrument, by means of a suitable interface.

Formulation of the Non-linear AAS Calibration Problem Calibration Curve Calibration curves in AAS are well known to be, in general, initially straight lines becoming progressively curved at higher

absorbances. The factors contributing to this non-linearity are many and varied and include broadening of the resonance lines due to too high a lamp current, lack of resolution due to too wide a slit width and stray light or the obscure compound decomposition process in the flame. Although it is possible to control the instrumental parameters leading to curvature, it is seldom possible to overcome curvature entirely, particularly at absorbances over 0.4 where the curvature is greater. As the processes leading to non-linearity of the calibration curve are complex, the modelling of these processes would be extremely difficult at the present state of knowledge.

A family of curves is required that is wide enough to cover the full range of elements and instrument parameters con- sidered in AAS. Data from a large number of calibration experiments carried out, over a period of time, using routine flame AAS in the analytical laboratory of the CSIRO Division of Mineral Chemistry were examined. These include a large proportion of the elements that can be determined by AAS and many different operating conditions. These empirical curves exhibit the following significant features: (i) local linearity, (ii) monotonicity and (iii) downward concavity. Further ionisation and kinetic flame reaction effects may result in a more complexly shaped curve with points of inflection, and even regions of negative slope.1 However, we exclude this last possibility from consideration, as we have not experienced such a curve in our laboratory. It may be due to a set of instrument parameters we have not yet applied.

A number of families of calibration curves have been proposed. Gabler et a1.2 and Margoshes and Rasberry3 suggested a class of the form

x= 00 + 81Y + . . . + 0,YP where X and Y represent concentration and absorbance. The symbols 0,,8*, . . . 8,, and p are the parameters of the curve. Mitchell et al.4 suggested the use of a series of straight lines based on contiguous standards enclosing the sample. The regression line which gives the most narrow confidence band around the predicted concentration is used to calculate the sample concentration. Schwartzs suggested curves of the form

Y = el + e2x + e3x2 Y = el + e2x + e3 i n x Y = 8,+ e , x + . . . + edrp

Barnett6 proposed the family

* To whom correspondence should be addressed. x = (91Y + 83Y2)/(82Y - 1) . . . . (1)

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1542 ANALYST, NOVEMBER 1987, VOL. 112

which has been provided as an inbuilt routine for commer- cially produced spectrometers from Perkin-Elmer. Limbek et a1.7 proposed the family

x = y/(el + e2y + e3yz) . . . . (2)

which has been the basis of a commercial routine in spectrometers produced by Varian and also by Baird Atomic.

All these calibration curves define a function, say y(Xi7Q), depending on a set of parameters Q when X is set at Xi . Various techniques of fitting these curves have been employed. A comparison of some of the above curves and the related curve-fitting procedures have been made by Tysonl and by de Galan et al.8 The general conclusion arrived at in these papers is that once an appropriate curve has been fitted by the least-squares method, the estimation of unknowns is reasonably satisfactory. The necessity for weighting in the least-squares program was noted by some workers4; however, to our knowledge the existing curve-fitting procedures either use an unweighted least-squares method or only approximate the weighting.

With most of these curves, there is the possibility that the constraints of monotonicity and concavity could be violated. As for higher degree polynomials, a further disadvantage is that they require an undesirably large number of standards to estimate the many parameters involved, in addition to the requirement for the operator to supply the degree of the polynomial in advance.

Neither the literature cited above, nor the inbuilt programs mentioned earlier, provide a comprehensive error analysis-a fact emphasised by Tyson.1

In this paper we consider a single family of curves which was found to satisfy our requirements. It is defined as Y = y ( X , Q ) which is a solution for y in the equation

x = el + e2y + 83Ye4 (--03 < el < m; e2 > 0; e3 3 0; e4 > 1) (3)

At X = Xi , the quantity Yi = y (X i , @) is the theoretical mean of the absorbance readings obtained from the same stock solution having a true concentration of Xi .

Errors in Working Standards

Arelatedproblemwhichcompounds the main problemis that of theerrorsin the workingstandards. Toovercome this problemit is necessary to extend the Berkson model9 to the non-linear instance. This requires a knowledge of the previous data regarding the nature of the errors.

Error Variance of Absorbance

A third complicating feature is that the absorbance values have heterogeneous standard deviations for different means at different levels of concentration. To allow for this, a mean - variance relationship is needed. This aspect has been dis- cussed by Roos,lO but an error analysis incorporating such a feature was lacking.

Probability Model for the Data

A probability model is needed for the analysis of errors involved both in the absorbance measurements and the concentration values of the working standards. This is now described.

The data available in a calibration experiment are of the form

[ x i , (Yij, . . ., YiM); i = 1, . . ., n] . . . . (4) Here, xi is the nominal value of the concentration of the ith standard and Yii is the jth replicate absorbance reading corresponding to the ith standard. The total number of standards is denoted by n and the number of replicate absorbance readings per standard is denoted by rn.

Let X i be the true (unknown) concentration and xi be the nominal concentration of the ith standard. Let y(X,,Q) be the

value of Y given by the curve in equation (3) when X = X,. Here the curve represented in equation (3) is assumed to be the calibration curve appropriate for the data. We note that one could choose any form of curve and proceed with the development of the probability model and the analysis of errors. In particular the curves in equation (1) or (2) could be chosen to achieve similar results.

Each observation Y,, is assumed to have a theoretical mean y(X,,Q), which is the value of the calibration curve at the ith standard, and a theoretical variance oL2, the value of which depends on y(X,,Q). Therefore, under the condition that X , is the true concentration level of the ith standard, we have

* * ( 5 ) } . . . . WL,IXL) = Y(XL7Q)

Var(Y,IX,) = u,2

where E( Y,,lX,) denotes the theoretical mean or expected value and Var(Y,lX,) denotes the variance of YL, given this condition. The quantity o,2 can take various forms as i changes. The particular form chosen for our purpose is

denoting that the conditional standard deviation ui of Yij is a linear function of the value of the calibration curve at the ith standard. The true concentration level Xi of the ith standard is an unknown quantity and is assumed to be a random variable distributed around the nominal value xi with a variance reflecting the error of measurements in making up a standard of a given nominal value. Thus the mean and variance of Xi are denoted by

E(XJ = xi

Var(Xi) = t i 2

where ti2 is assumed to depend on the level of the standard. A particular form chosen for ti2 is

so that the standard deviation of Xi is proportional to the nominal level x i .

The procedure of fitting a chosen curve y(Xi,Q) and chosen variance function oi2 to the data [equation (4)J is discussed in the following section. This fitting procedure forms the basis of a practical solution which also provides a related error analysis and a method of determining the limit of detection. We stress here that our main contribution is a proper error analysis program and the provision of a method for determining the limit of detection. Both these features seem to be lacking in the available published techniques.

zi2 = k2x.2

Practical Solution to the Problem An exact solution to the problem of AAS calibration is computationally involved and is not necessary for practical purposes. The following section provides a curve-fitting procedure which is easy to implement and provides satisfac- tory estimates of the standard errors and the parameters involved, while using only the facilities available in typical laboratory micro- or minicomputers.

Curve-fitting Procedure By examining a large number of analytical curves, it was found empirically that the form of the calibration curve given by equation (3) is appropriate for the range of integer values (2, . . ., 8) for e4. Therefore, O4 was taken to be fixed when estimating 81,02,e3,84 of the curve given by equation (3). A simultaneous estimation of (a1,a2) in equation (6) is also needed to obtain the appropriate weighting in a least-squares procedure for determining the curve [equation (3)]. The procedure is summarised below. (i) For the ith standard, the

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ANALYST, NOVEMBER 1987, VOL. 112 1543

observed mean and variance of the m replicate absorbance readings can be obtained by the equations

m

] = I Y, = 2 Y,,/m . . . . * (7)

m

I = , S , 2 = 2 (Y,, - Y,)~/(KV - 1) . . . . (8)

By fitting a straight line to the graph of obsezved standard deviation S, versus the corresponding mean Y,, initial esti- mates of al and a2 are obtained as the intercept and the slope of the line. These are used to obtain conditional maximum likelihood estimators &, and &2 by maximising the likelihood function of the observed sample variances (S12, . . . , Sn2) with their theoretical counterparts taken as

012 = (a1 + a2 F,)2 (i = 1, . . ., n) . . (9) (ii) By taking in equation (3) to be one of the integers (2, . . ., S), the weighted sum of squares of the deviations (WSS) given below is minimised with respect to 4, 8 2 and O3

n

I = I wss = 2 w,(0,2)[2,(!3)]2 . . . . (10)

where the deviations (in concentration) of the standards from the fitted curve are given by

z,(g = X, - (0, + e2 Ft + 0JLe4) (i = 1, . . ., n ) (11)

w,(o?) = 1/Var [Z,(@)] . . . . (12)

and the weights are given by

The weights are calculated using the approximate equation

Var [Z , (Q)] = a,2 [pi(Q)]2/m + k2~,2(1 + ~ 2 ) ~ . . (13) where p l ( @ ) is the slope of the curve [equation (3)] at Y = F, and o12 is estimated from equation (9) using the maximum likelihood estimates &, and &2 in place of al and a2.

The curve fitting was carried out without the constraints O 2 > 0 and e3 2 0 and then a check was made as to whether these constraints were satisfied. These are the constraints necessary for the monotonicity and concavity of the curve. In practice, the constraint O2 > 0 is always satisfied at the minimum of the sum of squares of deviations. When the analytical curve is close to linearity, it will occasionally happen that O3 < 0 at the minimum. In such an instance, we fit a straight line which is equivalent to fixing €I3 at zero.

An estimante of the parameter @ obtained by this method is denoted by 8.

Determination of Concentration From Sample Absorbance

Using the above estimate of (3, one can readily estimate the concentration of a new sample from its absorbance value. Let, Y01, . . ., YO, be the replicated readings of the absorbance of the sample, and also let

r yo = , 2 Yo,/r . . . . . . (13)

J = 1

Then an estimate of the concentration is

If we have more than one new sample, then we apply equation (14) repeatedly with 8,) replaced by the corresponding average absorbance of any new sample under consideration.

Errors in Working Standards The calculations indicated in the previous sub-sections require a knowledge of the errors in the working standards. For the ith standard with nominal value x i , the error in xi is expressed by the standard deviation kxi where k is taken as a constant. The

determination of k rests on the availability of previous data on the magnitude of errors in the sequences of operations (such as weighing and volumetric measurements). Such data for AAS preparations are available in Weir and Kofluk.11 Based on calculations for Cu data we have taken k = 0.002 for the error analysis.

Statistical Analysis of Data from a Calibration Experiment

Calculation of the Limit of Detection

The absorbance readings from the blank (standard with zero concentration of analyte) are used to determine the standard deviation at zero concentration. If any unknown sample has an absorbance reading of less than 0.1 A then six additional readings of the blank are taken after the entry of the last sample to improve the reliability of the estimate. The limit of detection is then defined as

Limit of detection = y(O;@) + 360 . . (15)

where y(O;@) is the estimated absorbance of zero concentra- tion and a0 is the estimated standard deviation. If the samples have been diluted, the limit of detection is multiplied by the dilution factor. This procedure is based on IUPAC recom- mendations. 12

For an iron determination the limit of detection was calculated, by this method, to be 0.16 mg 1-1. Fig. 1 shows a chart recorder plot of a 0.16 mg 1-1 iron solution under the same conditions as those used for the limit of detection determination. Fig. 1 demonstrates that the limit of detection calculated by our method differs significantly from the blank and, we believe, provides a more realistic value than that quoted by the instrument manufacturers.

Estimation of Instrumental Drift Error

Usually m/2 absorbance readings of each standard are taken before the sample(s) is (are) read and the remaining m/2 readings are taken after, with rn being an even number not less than 4. Drift is measured by taking the difference between the average absorbance readings of the standards before and after the samples are read. Let Di be an estimate of the instrumental drift at the use of the ith standard. Then Di is calculated as

Dj = F j ( l ) - y!’) . . . . . . (16)

where the superscripts (1) and (2) refer to “before” and “after” sample absorbance readings, respectively. The stan- dard error in measuring the drift in the ith standard is

DE = [Var(Di)]J = 2ai/m4 . . . . (17) The final analytical curve is fitted to the average of all the standard absorbance readings. The above procedure assumes linear drift. However, as the drift is usually small, acceptable estimates of the sample concentrations are obtained. Large values of drift are observed occasionally, however, when a partial blocking of the nebuliser or spray chamber has occurred; in such an instance the results are rejected.

0.03 1 1

a 0

-0.01 ’ I I I I

0 8 16 24 Ti me/s

Fig. 1. which corresponds to the calculated limit of detection

Chart recorder plot of an Fe solution, the concentration of

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Calculation of Curve-fit Error This is the standard error of the fitted value of the calibration curve at the nominal value of the concentration of each standard. For the ith standard, this is calculated as

CFE = {Varb(xi,@]}J . . . . (18) wher? Varb(xi,@] is the variance of the fitted curve value y(xi,8_), and can be evaluated as shown by Kabaila et a1.13

Calculation of Over-all Error

The measurement error of absorbance values of the ith standard is given by the standard error

M E = ailmb . . . . . . (19) while the drift error is given by equation (17) and the curve-fitting error by equation (18). The over-all error is then calculated as

OE = [(ME)2 + (DE)2 + (CFE)2]& . . In practice, estimates are used instead of the corresponding parameters to obtain estimates of the standard errors, which are expressed as a percentage of the concentration estimate.

Outlier Detection

The dilute metal ion solutions used as standards in AAS can change in concentration with time, owing to such effects as hydrolysis, evaporation and adsorption on to the walls of the containers. Although this problem can be overcome by making up fresh standards, it is seldom practicable and a procedure to readily detect and identify an outlying standard would reduce the need for this. A comparison of the determined and known standard concentration values, however, will not usually identify an outlier, as the curve will often fit as poorly to the standards on either side of the outlier as to the outlier itself. The procedure described by Gentleman and Wilk14 was used to locate an outlying standard in the analytical curve.

In this procedure one standard is sequentially dropped and a curve calibration is carried out on the remaining standards. In this way a quantity is obtained and displayed (Table 1) for each standard, which we refer to as the coefficient of fit.

If SSI = sum of squares of deviation (in concentration) of standards from the analytical curve using data from all standards and SS2 = sum of squares of deviations (in concentration) of the standards from the analytical curve using data from all but the jth standard, then we calculate

Coefficient of fit for the jth standard = SS1/SS2 . .(21)

A large coefficient of fit indicates, qualitatively, that the jth standard is probably an outlier. For the procedure used here to work most effectively, four or more evenly spaced standards should be used.

Curvature Factor This quantity is defined as the ratio of the slope of the curve at the highest standard to the slope at the lowest, i.e.,

A numerical value from equation (22) of greater than 1 is taken to indicate curvature. This quantity is provided merely to give information about the degree of non-linearity of the Calibration curve. This is useful to the analyst as a larger than normal degree of curvature, for a particular element and set of instrument parameters, can indicate problems with the choice of those parameters.

Table 1. Example of a computer printout of the standards and calibration curve data and results. Initial estimate of the measured standard deviation of absorbance = 0.0060. Curvature factor = 1.2

Ob- served absor- bance 0.0050 0.1218 0.2323 0.3418 0.4548 0.5543 0.6455 0.7463 0.8398

Stan- dard

concen- tration/ mgl-*

0.00 2.50 5.00 7.50

10.00 12.50 15.00 17.50 20.00

Esti- mated

concen- tration/ mg 1-1 0.021 2.506 4.938 7.442

10.124 12.569 14.879 17.506 20.014

Dif- ference,

Yo

+0.26 -1.25 -0.78 +1.23 +0.55

+0.03 +0.07

-0.81

Errors

Meas- ured

RSD, %

3.8 2.3 1.7 1.4 1.3 1.2 1.1 1.1

Data to reveal a poorly fitting standard:

Drift Curve maxi- RSD, mum,

% %

1.8 0.2 1.2 0.6 0.9 0.7 0.7 0.5 0.6 0.2 0.6 0.3 0.6 0.1 0.7 0.7

Over- all

RSD, Yo

4.4 3.1 2.6 2.1 1.7 1.6 1.4 2.0

Standard/ Coefficient of fit Standard/ Coefficient of fit mg I-’ (the lower the better) mg 1-l (the lower the better)

0 1 .o 12.5 1 .o 2.5 1.0 15 1.8 5 1.0 17.5 1 .o 7.5 1.0 20 1.0

10 1.2

Statistical Analysis of Data from a Prediction Experiment

In an AAS prediction experiment the concentration of samples is determined from their absorbance measurements. This can be achieved by repeating the procedure given under Determination of Concentration from Sample Absorbance for each sample. For a typical sample, the average of the new readings on absorbance is denoted by yqand the correspond- ing unknown by Xo, and its estimate by X o . An analysis of the errors in 8, can be made along similar lines to those under Statistical Analysis of Data from a Calibration Experiment. The drift error is not measured now. The measurement error or replicate error and the curve-fit error can be calculated as described in the next section. The total error is then calculated as in equation (21) without drift error being included.

Replicate Error

The theoretical variance of the absorbance of a sample whose unknown concentration is Xo, is

002 = [a1 + a2y(X,,€I)]2 . . . . so that the theoretical replicate error of the average absor- bance Yo is

adr . . . . . . . . (24)

The quantity ao2 can be directly estimated by So2, the observed variance of rn replicate absorbance values. However, this is not satisfactory when rn is small. Assuming that the variance function remains unchanged in a prediction experiment one can use the estimates kL1 and k2 from the section Curve-fitting Procedure and obtain a more realistic estimate of oo2 as

ao2 = (kcl + kzyo)2 . . , . . . (25 )

Curve-fit Error

For a typical sample with an unknown concentration Xo, the curve-fit error is given by

Varly(Xo,&)] . . . . . . (26)

for which an approximate expression is readily derivable along the lines of the derivation of a curve-fit error for a standard.

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Table 2. Example of a computer printout of the estimated concentra- tions and precision values for a number of unknowns. Element determined: Cu. Limit of detection = 0.1210 mg 1-1 = 0.0210 absorbance

Concentration/ Sample No. mg 1-1 RSD, Yo

1A 1B 1c 2A 2B 2 c 3A 3B

6.4992 3.0 10.2515 2.3

<o. 1210 8.4096 2.6 2.1462 5.2

0.9788 10.8 17 3172 1.7

<0.1210

Table 3. Determination of Cu in nominal standards. Data from a calibration experiment

Absorbance Cu concentration/

mg 1-1

0 2.5 5.0 7.5

10.0 12.5 15.0 17.5 20.0

Run 1 0.003 0.118 0.228 0.336 0.448 0.549 0.642 0.740 0.827

Run 2 0.006 0.125 0.234 0.343 0.457 0.557 0.646 0.751 0.842

Run 3 0.002 0.120 0.232 0.340 0.453 0.549 0.639 0.744 0.843

Run 4 0.005 0.124 0.235 0.348 0.461 0.562 0.655 0.750 0.847

Table 4. Parameter estimates

Estimate

Maximum likelihood Weighted least squared Parameter (ML)/mg I-’ mg 1-1

a1 0.001965 k 0.000465 a2 0.007424 k 0.001530

0* 20.62 k 0.6096 03 3.92 It 0.4865 @‘I 2.00 f 0.3102

8, -0.0620 k 0.0216

Development of a Program Rather than to just carry out a curve calibration and determine unknown samples, we have designed a computer program to provide as much information as possible and to assist the operator to carry out the most precise determination possible and to assess better the reliability of the results obtained.

We have designed the program to correspond with the normal working sequence of an AAS determination. The absorbance readings can be entered as they are read and thus the program could be easily adapted to run with the atomic absorption instrument on-line to a computer.

The standard concentrations and absorbances are entered initially (two absorbance readings for each standard) and an initial estimate of the calibration curve is carried out. From the display at this stage (Table 1) the analyst can elect to continue or re-run the calibration curve with new standards or with different instrument parameters, The sample numbers and absorbances are then entered (in duplicate if required) followed by the standard absorbances again (two readings for each standard). The precision data for the calibration curve (Table 1) and the sample results (Table 2) are then displayed (and/or printed). Other values required by the program (e.g., dilution factor, sample masses) are entered when requested.

The relative standard deviations (RSD) displayed are for approximately the 95% confidence limits. To avoid a “divide- by-zero” error the RSD is not calculated if a standard has a concentration of zero. A dilution error (0.2% RSD) and weighing error (0.1% RSD) are added to the over-all sample RSD (where applicable).

An Example We now provide an analysis of the data in Table 3 by applying the technique mentioned under Statistical Analysis of Data from a Calibration Experiment. The parameter estimates are given in Table 4, together with their standard errors. These parameter estimates are used in evaluating the concentration estimates in Table 1, which also provides a summary of the data and the final analysis, including the curve-fitting error, drift error and over-all error. Column 1 of Table 1 gives the average absorbance for each standard. Columns 2 and 3, respectively, provide the nominal concentrgion and esc- mated concentration using equation (14) with Yi in place of Yo when calculating for the ith standard. For each standard, the standard deviation of four absorbance values is calculated and is expressed as RSD (i.e., a percentage of the mean absorbance) in column 5. The curve-fitting error for the ith standard is calculated according to equation (18); it is then expressed as a percentage of the concentration estimate in column 6. As a check on the formula for the curve-fitting error, we used a bootstrap method.15

The drift error for the ith standard is calculated according to equation (17) and is given as a percentage of the concentration estimate in column 7. The over-all error for each standard is calculated as in equation (20) and given in column 8 as a percentage of the concentration estimate.

After the analysis of the data from the calibration experi- ment, we proceeded with the estimation of concentration in the eight samples of the prediction experiment given in Table 2. One absorbance reading is obtained for each sample and the corresponding estimate of concentration is calculated using equation (14). The standard error of the concentration estimate is calculated and given as a relative standard error ( i e . , a percentage of the estimate). Further, the limit of detection is calculated by using equation (15) and is given in terms of absorbance units in addition to the corresponding concentration units.

Alternative Curve Fitting Using Two Other Commonly Used Curves

In this section, we consider a comparative study involving the family of curves given by equation (3) and the two alternative families of curves given by equations (1) and (2), respectively. Other types of curves discussed under Formulation of the Non-linear AAS Calibration Curve are not of wide use and, therefore, are not included in the comparison.

It should be stressed again here that the general probability model given under Probability Model for the Data and the general methods of fitting a selected calibration curve and a selected variance function as outlined under Curve-fitting Procedure are valid for any choice of a specific calibration curve or variance function. Also, it should be noted that the curve-fitting procedures given in Bennett6 and Limbek et af.7

which use equations (1) and ( 2 ) , respectively, do not consider a weighted least-squares method. Therefore, it is possible to select either equation (1) or (2) and combine it with the general curve-fitting technique as was carried out in the section Practical Solution to the Problem for the family of curves resulting from equation (3). This should improve the fitting of the two alternative curves from equations (1) and ( 2 ) .

In the following comparison, four elements representing a broad range of AAS calibrations are considered. Table 5 shows typical calibration data sets for Cu, Fe, Pb and Al. These data sets were used to apply each of the three curves from equations (1)-(3). No weighting procedure was used in fitting these curves. The published algorithms using the curves from equations (1) and (2) as given by Bennett6 and Limbek et al. ,7 respectively, do not suggest any weighting so that, to have a comparison on an equal footing, we have ignored weighting in each instance. The concentration of a number of

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~~~ ~~

Table 5. Calibration data used for comparison of calibration methods

Copper Iron Lead Aluminium

Concentration/ Absorbance Concentration/ Absorbance Concentration/ Absorbance Concentration/ Absorbance mg 1-1 mg 1-1 mg 1-1 mg 1- I

0 0.004 0 0.009 0 0.007 0 -0.001 5.0 0.232 5.0 0.373 20.0 0.244 50 0.317

10.0 0.455 10.0 0.670 40.0 0.454 100 0.537 15.0 0.646 15.0 0.892 60.0 0.632 150 0.697 20.0 0.840

Table 6. Concentration estimates obtained for each calibration method on each element using the calibration data in Table 5

Estimates obtainedimg 1-1

Concentration of standard/ For our method, For method 1, For method 2,

mgl-1 equation (3) equation (1) equation (2)

7.5 7.5 7.7 7.5 12.5 12.6 12.7 12.7 17.5 17.5 17.7 17.6

2.5 2.5 2.6 2.6 7.5 7.5 7.7 7.6

12.5 12.6 12.8 12.7

10.0 9.9 9.5 9.5 30.0 29.9 29.3 29.3 50.0 49.6 48.9 48.8

25.0 25.3 24.8 24.8 75.0 75.2 75.6 75.6

125.0 125.7 125.8 125.8

Copper-

Iron-

Lead-

Aluminium-

standards (not used in the calibrations) was estimated for each element and for each curve. The results so obtained are compared with the known concentrations of these “samples” in Table 6. The estimates obtained are better in each instance for equation (3) than either of the two alternative calibration procedures.

All the data sets given here do not indicate a significant blank. For data sets with a significant blank, the fitting of curves from equations (1) and (2) require the blank value to be subtracted first. This is a disadvantage as the procedure of subtracting the blank increases the replicate error variance of the absorbance values thus obtained. On the other hand equation (3) can handle a significant blank without a necessity for subtraction.

We have found the family of curves [equation (3)] given in this paper to be at least equal to, if not better with respect to the applicability to AAS, than those given by Bennett6 or Limbek et aZ.7

Discussion and Conclusion The precisions determined for the samples are calculated entirely from the standard absorbance measurements. As the matrix of the standards can have an effect on the precision, it should be matched to that of the samples if the precision obtained from reading the standards is to be applicable to the samples. Also, the order in which the standard absorbance measurements are taken is important. For instance, if each duplicate measurement is taken immediately after the origi- nal, then the precision result will probably be unrealistically low when applied to the samples.

The absorbance readings for a given element and set of instrument parameters can be enhanced or suppressed by the presencc, in solution, of non-analyte chemical components. This interference can be due to chemical processes in the flame or to changes in the viscosity and its effect on nebulisation and can lead to errors in the determination of the

sample concentrations. We do not believe it is practicable to quantify these matrix interferences using physical modelling as the interferences are complicated and not completely understood. However, the results of prior experience could be stored on a computer storage device and then recalled, when desired, to correct the concentration determination and/or bias the choice of calibration curves and estimation of precision. We have not attempted to do this, however, and believe that a simpler and more appropriate solution is to make up the standards with a matrix as closely matched to the samples as possible and then to adjust the instrument parameters to minimise any interferences.

The display of the various error contributions to the over-all precision has proved very useful in determining which errors contribute greatly to the over-all precision. For example, we long believed that the drift error with a double-beam instrument would be less than that with a single-beam instrument. However, the drift errors obtained on each type of instrument in this laboratory have not been significantly different.

The procedure proposed in this paper employs a family of curves different to those available in the literature. It is found to be at least equal to, if not better than, those widely used in practice. In addition, we have employed a general curve- fitting technique for the proposed family. This general curve-fitting technique can be applied to any chosen family of curves. A detailed analysis of the various sources of errors involved in an AAS calibration experiment is also provided and demonstrated for a Cu data set. The procedure we have described has been used extensively in the CSIRO Division of Mineral Chemistry Analytical Laboratory for many years and has proved to be satisfactory.

A copy of the program, in BASIC, is available on request.

References 1. 2.

3.

4.

5. 6. 7.

8.

9. 10. 11.

12.

13.

14. 15.

Tyson, J. F., Analyst, 1984, 109, 313. Gabler, R. G., Brown, R. E., and Haynes, J . G. , Int. Lab. , 1971, MayIJune, 8. Margoshes, M., and Rasberry, S . D . , Anal. Chem., 1969, 41, 1163. Mitchell, D. G., Mills, W. N., Garden, J. S. , and Zdeb, M., Anal. Chem., 1977, 49, 1655. Schwartz, L. M . , Anal. Chem., 1979,49, 2062. Barnett, W. B., Spectrochim. Acta, Part B , 1984. 39, 829. Limbek, B. E . , Rowe, C. J. , Wilkinson, J . , andRouth, M . W., Am. Lab., 1978, 89. de Galan, L., van Dalen, H. P. J . , and Kornblum, G. R. , Analyst, 1985, 110, 323. Berkson, J., J. Am. Stat. Assoc., 1950, 45, 164. Roos, J. T. H., Spectrochim. Acta, 1973, 288, 407. Weir, D. R., and Kofluck, R. P.. At. Absorpt. Newsl., 1967,6, 24. IUPAC, “Compendium of Analytical Nomenclature”, Pergamon Press, Oxford, 1978, p. 114. Kabaila, P., Lwin, T., and Strode, P., Technometrics, submit- ted for publication. Gentleman, J. F., and Wilk, M. B., Biometrics, 1975, 31, 387. Efron, B., Ann. Statist., 1979, 1.

Paper A511 85 Received May 20th, 1985 Accepted June 2nd, 1987

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