(3) J. 0. Rasmuson, V. A. Fassel, and R. N. Knlseley, Spectrochlm. Acta, Part B, 28, 365 (1973). (4) V. B. Fedorus, T. Ya. Kosolapova, Yu. B. Kuzma, and L. N. Kugai in “Tugoplavkie Karbidy”, €. V. Sarnsonov, Ed., Naukova Durnka, Kiev, 1970 p 244. (5) I. A. Etinger, L. F. Malceva, L. A. Sobranskaya, E. N. Marmer, V. I. Kindisheva, V. A. Nikolaeva, and T. V. Dubovik, Ref. 4, p 260. (6) J. Y. Marks and G. G. Welcher, Anal. Chem., 42, 1033 (1970). (7) R. E. Sturgeon and C. L. Chakrabarti, Anal. Chem., 48, 667 (1976). University, Ottawa, paper presented at the 4th I.C.A.S. Toronto, Canada 1973. (8) K. Fujiwara, H. Haraguchi, and K. Fuwa, Anal. Chem., 44, 1895 (1972). (9) K. Fujiwara, H. Haraguchi, and K. Fuwa, Chem. Left., 1973, 461. (10) B. V. Lvov and N. A. Orlov, Zh. Anal. Khlm., 30, 1661 (1975).
(11) I. RubeSka,.Methodesfhys. Anal., 3, 61 (1971). (12) D. T. Coker and J. M. Ottaway, Nature (London), 227, 831 (1970).
Ivan Rubegka
Geological Survey of CzeFhoslovakia 170 00 Prague 7, Kostelni 26 Czechoslovakia
RECEIVEDfor review December 17,1975. Accepted May 26, 1976.
Difficulties with Determining the Detection Limit with Nonlinear Calibration Curves in Spectrometry Sir: The detection limit is an important criterion with which to judge an analytical technique and may be defined in many ways that are summarized with pertinent references in a recent paper ( I ) . Ideally, the detection limit should indicate the smallest concentration which can be detected with a reasonable degree of confidence under specified conditions. Usually this means the concentration which yields an analytical signal which is some confidence factor (normally 1,2, or 3) times the standard deviation for measuring the blank signal. Alternately, the detection limit may be defined as the concentration which yields a signal-to-noise ratio (S/N) equivalent to some confidence factor (again usually 1, 2, or 3). The higher the confidence factor in either definition, the more conservative the definition and the more confident one can be that the measured signal is due to analyte in the sample rather than random measurement errors. The confidence factor under certain conditions may be related to the “z” or “t” statistic which allows a confidence level to be assigned. The first definition stated above may be formally written as
s1 = st - s b = zUs=o
(1)
if the z statistic is the confidence factor and where St = total signal (analyte plus blank) at the detection limit, S1 = analytical signal at the detection limit, s b = blank signal, and US=O = standard deviation for measuring SI,a t the detection limit. Practically US=O = f i U b where Ub is the standard deviation for a blank measurement. For this definition, it is assumed that enough measurements are made to obtain a good estimate of the standard deviation. If not, the t statistic may be employed. Note that the analytical signal is taken as the difference between the total signal and the blank signal and usually a reagent blank is used to establish the mean of the blank signal. Our and other researchers’ data indicate that the latter point requires more scrutiny because calibration curves may level off near the detection limit, particularly when dealing with ultratrace concentrations. Thus, the extrapolated intercept of the calibration curve is not zero but some finite positive value. Hence, we propose that where this behavior is exhibited that the extrapolated blank signal (signal at zero analyte concentration) instead of the measured blank signal mean be used for s b in the detection limit definition. For identification purposes, the detection limit obtained by letting S b equal the measured blank signal is denoted the theoretical detection limit while the value obtained by letting S b equal the extrapolated blank signal is denoted the practical detection limit. Four types of calibration curves are identified in Figure 1. The dotted line represents a confidence factor times the
standard deviation for the blank measurement. Hence, the intersection of the dotted line and the calibration curve is the theoretical detection limit. In Figure la, the theoretical and practical detection limits are identical. As long as the lower slope is used in Figure lb, the theoretical and practical detection limits are equal. Figures ICand d, show the two cases in which the theoretical and practical detection limits greatly differ (in some cases as much as two orders of magnitude). Calibration curves of other shapes are also possible and the above discussion should apply as long as the extrapolated blank signal is positive. We have observed nonlinearities in the calibration curves near the detection limit for molecular fluorescence and chemiluminescence measurements. Figure 2 is a graph of typical data obtained for fluorescence measurements of quinine sulfate (QS) in 0.1 N HzS04. The analytical signal plotted has been corrected for the blank fluorescence, and all analyte solutions were prepared by serial dilution of a stock quinine sulfate solution with exactly the same solution from the same bottle used for the blank measurement. The reasons for this behavior are not clear but it may be due to equilibria involving the analyte at low concentrations and sorption and desorption of the analyte and contaminants from the sample containers and the sample cell. If the blank solution is first transferred to a scrupulously cleaned beaker and then to the sample cell, the blank signal increases. Curry et al. ( 2 )encountered this leveling off effect for fluorescence measurements at concentrations below 3.1 x M for fluorescein and below 1.6 X M for Rhodamine B which they attributed to chemical interferences. Perchalski et al. ( 3 )show analytical curves for QS that are similar to that shown in Figure 2. The apparent nonzero intercept is particularly critical for QS since the detection limit of this chemical is used as a figure of merit for molecular fluorescence instruments. Many manufacturers of spectrofluorometers report a detection limit for QS of one part-per-trillion (ppTr) or less based on the traditional interpretation of the detection limit definition. The detection limit of the American Instrument Co.’s AmincoBowman Spectrophotometer is stated as 0.5 ppTr. However in their Compendium 2392-12-1, it is stated that solutions prepared by serial dilution appeared to be valid only down to pg/ml. Their conclusion is that solvent impurities which were not significant at higher concentrations display marked fluorescence or quenching effects. Farrand Optical Co. ( 4 ) concurs with these findings. Hence, reported detection limits can be misleading since a point is reached at which further dilution of the analyte does not significantly decrease the analytical signal even though the S/N is much greater than 2. For this case, one cannot ac-
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o
conc.
o
(a)
conc. (b)
3
3-1
I
1
I
I
I
I
,
I
2
3
4
5
6
7
LOG o
conc. (C)
o
conc.
(dl
,
‘
8
( C O S 3 XIO). p p T r
Flgure 2. Molecular fluorescence calibration curve for quinine sulfate in 0.1 N H2S04
Figure 1. Four possible calibration curve shapes
tually prepare an analyte solution which yields a signal one to three standard deviations greater than the measured blank signal. Clearly, such behavior is disturbing and more work is required to find the cause of this behavior. Although not usually stated, it is implied that one should be able to see a difference in concentration equal to the detection limit near the detection limit. In other words, if the detection limit for a technique is 1ppTr one should be able to distinguish 2 ppTr from 1 ppTr with some reasonable confidence. Clearly, this is not the case for some of the reported detection limits for quinine sulfate. Defining the blank signal ( s b ) as the extrapolated value rather than the measured value gives a more realistic assessment of the actual minimum concentration to which the technique may be applied. The use of this interpretation of the detection limit definition with the data in Figure 2 is discussed below. The slope of the log-log plot is 1 at higher QS concentrations as expected but is 0.3 at lower concentrations which is difficult to explain although others have reported such behavior for molecular fluorescence measurements (2, 3 ) . At 0.1 ppTr QS, the S/N was measured to be 7 and, hence, the detection limit would traditionally be expected to be less than 0.1 ppTr. The theoretical detection limit cannot easily be calculated here because the QS fluorescence signal is always several standard deviations above the measured blank signal. The practical detection limit is obtained from the calibration data and Equation 1 by finding the concentration at which the fluorescence signal is 2 ~ ( z 5taken to be 2) times the standard deviation of the blank (Ob = 9) above the extrapolated blank signal (sb= 60). The procedure yields 0.7 ppTr as the practical detection limit which corresponds to about the concentration in Figure 2 where the calibration curve levels off and which is much greater than the detection
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limit expected from the S/N a t 0.1 ppTr. The practical detection limit indicates the minimum concentration of QS that can be realistically measured in our particular system. In summary, the definition of the detection limit must be carefully used for cases in which calibration curves level off at low concentrations. It is suggested that the blank signal should be taken as the extrapolated value and not the measured value for a blank solution. Use of the extrapolated blank signal will ensure that the detection limit reflects not only the precision and noise characteristics of the measurement system but the sample handling and blank problems. If the calibration curve is linear with a zero intercept, then the theoretical and practical detection limits are equivalent. However, as analytical techniques such as luminescence spectroscopy and nonflame spectroscopy are pushed to allow measurement of extremely small concentrations, this may not always be the case.
LITERATURE CITED (1) J. D. Ingle, Jr., J. Chem. Educ., 51, 100 (1974). (2) R. E. Curry, H.L. Pardue, G. E. Mieling, and R. E. Santini, Clin. Chem. (Winston-Salem, N.C.), 19, 305 (1973). (3) R. J. Perchalski, J. D. Winefordner, and B. J. Wilder, Anal. Chem., 47, 1993
(1975). (4) Howard Madlin, Farrand Optical Co., personal communication, 1976.
J. D. Ingle, Jr.* R. L. Wilson Department of Chemistry Oregon State University Corvallis, Ore. 97331
RECEIVEDfor review April 5,1976. Accepted June 14,1976. Acknowledgment is made to the NSF (Grant No. MPS7520055) and the Oregon State Research Council for partial support of this research.
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
