Alternative Method for the Calculation of the Detection Limit in Emission Spectrography Sir: Detection limits in spectrochemical trace analysis were calculated in reference to blank values taking into account their distribution. In the original work of Kaiser ( 1 ) which has been further elaborated by Boumans ( Z ) , one assumed normal distribution of the blank signals even after transformation of the measured absorbances A of the recorded analytical lines. The detection limit was derived from the intersection of the calibration line and the noise level line. The former one has been obtained by plotting the logarithmic intensities for the different standards in function of the logarithmic concentrations, the latter one has been determined from the blank signals, taking into account their distribution. Kaiser ( I ) accepted 3 0 as criterion of significance. The log-I values were normalized to the background and corrected for blank contribution. Hubaux and Vos ( 3 ) described a method for the calculation of the detection limit in view of the statistical concepts of error of the first and second kind for analytical procedures involving calibration lines. This method has been elaborated in this paper for spectrochemical analysis which involved the investigation of the kind of distribution of the measured absorbances. Indeed, as stated by Behrends ( 4 ) , the distribution of the measured analytical signals is not always a normal one. Furthermore, the distribution after transformation of the measured absorbances must be calculated. For the investigation of the distribution of the measured absorbances, we applied the Xz-test and the “Skewedness and Excess test” (6) on 40 records of yttrium ion lines with absorbances near 05A, 15A, and 3A. Therefore, a standard solution containing 5 ppm yttrium in 6 N hydrochloric acid has been prepared and excited by the graphite spark (7). Results are given in Table I. The reference values (6) were S and E test: 0,374 skewedness (S),0.733 excess (E), X2-test 95%, 5.991. From these results, we concluded that a logarithmically normal distribution describes the dispersion on the measured absorbances very well. Indeed, the dispersion on the A- values is due to a complex of independent factors such as plate inhomogenity, variations in plasma conditions, sample inhomogenity, densitometer instability, etc.; in this situation, a logarithmically normal distribution is known to fit very well. In spectrochemical trace analysis, transformations to obtain a linear characteristic to 0.05 A are often used. We applied Seidel transformation to the measured absorbances; this gives good results within a narrow range of wavelengths (8, 9). Accepting logarithmically normal distribution for the measured absorbances A, we calculated the distribution parameters pw,Hw, and L W for the Seidel function.
The results (pw, Hw, and L w in function of varying standard deviation and mean values of log A ) are partly represented in Table I1 which may be extended for any qOg and PIog. The procedure for the determination of the detection limits can be summed up as follows: Record the absorbances of the analytical line at X, for 15 replications of three standard mixtures, prepared independently. Also measure the A a t A, for 15 blank samples. Calculate ulog and plog for blank and standards. Calculate OW, Hw,and LW or derive from Table I1 for the blank and for the standard mixtures. Finally plot pw, Hw, and Lw us. log C for the three standards. The detection limit is obtained from the intersection of the higher confidence limit a t 99.86% confidence level for the blank, with the lower confidence level line a t 99.86% confidence level for the standards. This is illustrated by Figure 1. The true calibration line is obtained from the plot pw us. log C for the three standards. This method allows calculation of the detection limit of a spectrochemical analysis procedure from the Pw-values of a number of standards and a series blank record. It is a function of the blank value, the calibration line for a standard series, and the distribution of the P w values for blank and standards. The detection limit also depends on the photographic emulsion and the transformation used. The proposed calculation method is illustrated in the following example: Standard solutions containing 2 ppm, 5 ppm, and 10 ppm europium in 6N HCl Suprapur were prepared, 1-ml solutions were dried in a graphite electrode (Schunk and Ebe) and excited by hollow cathode excitation under argon a t 2 Torr and 70 mA discharge current (10). Fifteen records were made for each standard. Confidence levels were taken on 95% confidence level. Instead of pw, Hw, and Lw, we plotted log I (derived from the emulsion characteristic), us. log C. The detection limit was calculated for Eu 4129 A as described in (1-3) (1st method) and as stated in this paper (2nd method). From a comparison of the results in Figure 2 , it can be seen that the calculation of the detection limit value, because of the distribution of measured signals, reflects much better the analytical properties of the applied spectrographic method. Nomenclature ( 5 ) . A = measured absorbance of an analytical line Pw = Seidel value of a measured absorbance A, P w = log (10” - 1) p = mean value of normally distributed signals H , L = confidence limits on 99.86% (3 u ) confidence level for normally distributed signals; H, higher confidence limit; L , lower confidence limit qOg = standard deviation for the logarithmical values of logarithmically normally distributed signals
-
Table I. Results of Statistical Analysis of Measured Absorbances Skewedness (S) and Excess (E)test
X2-test
log (lo*-1)
log A
A 109
A
0.5A 1.5 A 3.OA 188
0.4125 5.421 1.016
log A
0.421 6.855 1.186
(10A-1)
S
E
0.293 5.624
0.936 0.344 0.232
1.813 1.255 0.478
...
ANALYTICAL CHEMISTRY, VOL. 47, NO. 1, JANUARY 1975
S
0.095 -0.219 0.085
E
S
E
0.103 0,826 0.369
0.356 0.262
0.511 1.169
...
...
Table 11. Computed Mean Values gw,Confidence Limits p ~p~, on 95% Confidence Level and Standard Deviation uw for the Seidel Values of Different Absorbances with Different Dispersion 0.200
0.100
ii l o g
OW
LW
-1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.2270 0.2345 0.2443 0.2571 0.2738 0.2960 0.3257 0.3653 0.4186 0.4900 0.5852
-1.225 -1.122 -1.018 -0.912 -0.806 -0.697 -0.587 -0.473 -0.356 -0.234 -0,106
0.176 0.337 0.515 0.718 0.954 1.234 1.573 1.991 2.511 3.162 3.982
HW
-0.581 -0.465 -0.346 -0.221 -0.089 0.052 0.206 0.376 0.568 0.789 1.049
-1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.1123 0.1156 0.1199 0.1255 0.1328 0.1424 0.1549 0.1717 0.1940 0.2240 0.2846
-0.912 -0.806 -0.697 -0.587 -0.473 -0.356 -0.234 -0.016 0.030 0.176 0.065
-0.234 -0.106 0.030 0.176 0.337 0.515 0.718 0.954 1.234 1.573 1.878
-0.585 -0.471 -0.354 -0.231 -0,102 0.035 0.138 0.346 0.528 0.735 0.745
-1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.1691 0.1744 0.1812 0.1901 0.2017 0.2169 0.2371 0.2640 0.3000 0.3484 0.4133
-1.070 -0.965 -0,859 -0.752 -0.642 -0.530 -0.415 -0.296 -1.171 -0.039 0.102
-0.039 0.102 0.255 0.423 0.613 0.832 1.088 1.395 1.771 2.236 2.818
-0.583 -0.469 -0.351 -0.227 -0.097 0.042 0.193 0.358 0.544 0.757 1.006
-1.00 -0.90 -0.80 -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00
0.0560 0.0576 0.0597 0.0624 0.0660 0.0706 0.0766 0.0846 0.0953 0.1096 0.1289
-0.752 -0.642 -0.530 -0.415 -0.296 -0.171 -0.039 0.103 0.255 0.423 0.613
-0.415 -0.296 -0.171 -0.039 0.102 0.255 0.423 0.613 0.823 1.088 1.395
-0.586 -0.473 -0.356 -0.233 -0.105 0.031 0.178 0.339 0.518 0.722 0.960
WW
Figure 1. The detection limit conception (1) Blank signals and their distribution. (2) Upper confidence line at 99.86% confidence level for standards. (3) Lower confidence line at 99.86% confidence level for standards. (4) True calibration line
Ni1
iqf
0.150
0.050
piog = mean value f o r the logarithmical values o f logarithmically
n o r m a l l y d i s t r i b u t e d signals
uw = standard deviation for the Seidel transformed values o f loga r i t h m i c a l l y n o r m a l l y d i s t r i b u t e d signals = mean value for t h e Seidel transformed values o f l o g a r i t h m i cally n o r m a l l y d i s t r i b u t e d signals Hw,L w = confidence l i m i t s o n 99.86% confidence level f o r Seidel transformed values o f logarithmically n o r m a l l y d i s t r i b u t e d signals log C = logarithmical concentration ( i n pprn) l o g I = logarithmical i n t e n s i t y o f a n analytical l i n e l o g C D = detection limit P w , = decision limit A, = wavelength o f a n analytical l i n e pw
.i
1
?
0
~-~D iog
c
Figure 2. Comparison of the proposed detection limit calculation method with that described by Kaiser ( 7) and Bournans (2) Hollow cathode excitation, p = 2 Torr f 0.1 argon, j = 70 mA, exposure time 493 seconds, Eu 4129 A line. 1st Method, detection limit, 1.5 ppm. 2nd Method, detection limit 2.1 ppm. Hand L, 9 5 % confidence levels.
ACKNOWLEDGMENT Numerical results were programmed by Ir. A. Cogghe and calculated on the Siemens 4400 ordinator of the State University of Ghent. We thank L. Pszonicki, B. Strzyzewska and S.Sterlinski of the Institute for Nuclear Research, Warsaw, Poland, for their kind interest in our work.
LITERATURE CITED ( I ) H. Kaiser, Fresenius'Z. Anal. Chem., 149, 46 (1956). (2) P. W.J. M. Boumans, 16th Coll. Spectrosc. Int. Heidelberg 1971, Prepr., Adam Hilger, London, 1971 Vol (I, p 247. (3) A. Hubaux and G. Vos, Anal. Chem., 42, 849-855 (1970). (4) K. Behrends, Fresenlus' 2.Anal. Chem., 235, 391 (1967). (5) IUPAC Committee, "Nomenclature, Symbols and Usage in Analytical Atomic Spectroscopy," Part I, published 1967. (6) D. Himmelblau "Process Analysis by Statistical Methods," J. Wiley and Sons, New York, N.Y., 1969. (7) J. A. C. Broekaert, Bull. SOC.Chim. Belges, 1973, 561-569. (8) M.Honer jager-Sohm and H. Kaiser, Specfrochim. Acta. 2, 396 (19). (9) L. H. Ahrens and S. R. Taylor, "Spectrochemical Analysis." Pergamon Press, London, Paris, 1961. (10) J. A. C. Broekaert. 17th Coll. Spectrosc. Int., Firenze 1973, Acta, Vol Ill, p 149-154.
F. M. Bosch J. A. C. Broekaert Laboratory for Inorganic Technical Chemistry State University of Ghent Grote Steenweg Noord 12 B-9710 Zwijnaarde (Belgium)
RECEIVEDfor review November 13, 1972. Resubmitted October 23,1973 and June 5,1974. Accepted August 27,1974.
ANALYTICAL CHEMISTRY, VOL. 47, N O . 1, JANUARY 1975
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