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Theoretical calculation of the standard deviation in atomic emission spectroscopy
Sprcaochimien Acta. Vol. Printed in Great Britain
Theoretical
0584-8547/81/040385.~/0 @1981PeqmonFw%L%l.
36B, No. 4. PP. 385 to 392, 1981.
calcdation
of the standard deviation
in
atomic
emission
sp--PY E.
D.
PRUDNIKOV
Earth’s Crust Institute, State University, Leningrad, 199164, U.S.S.R. (Receiwd 28 Februmy
1980; in revised form 16 October 1980)
AMxaet-In atomic emission spectroscopy (ABS), a simple formula is obtained for the calculation of the limits of detection which takes into account the blank samples values. An expression is proposed for the standard deviation due to apparatus parameters, blank value and concentration of an element. ‘I’hese results are applied to practical problems.
NOMENCLATURE atomic weight of element, atomic mass unit transition probability, s-t coefficient for calculating the signal value. A-’ cm-’ photomultiplier factor related to gain per stage, dimensionless, coefficient for the calculation of the r.m.s. fluctuation A2 m-2
=-I
d.1. of the
background
intensity,
nm-’
intensity of excitation source contimmm, W cmm2sr-l nm-’ standard deviation of BA, W cme2 ST-* run-’ relative concentration of element, % relative limit of detection, % relative limit of detection for confidence probability of 0.997 (criterion 3 s), % blank concentration of ‘element, % coefficient for the calculation of the flicker fluctuation, A2 UP, srm2s-l effective aperture of the spectral instrument, cm2 average diamata of the flame region focused on the entrance slit of the spectml instnmmnt, cm coe&Gnt of photodetector thermal noise, A2 exponential function, d.l. electronic charge, C excitation energy of state y J focal length of the collimator, cm solution flow rate, ml min-’ frequency response band width, Hx statistical weight of ground state, d.1. statistical weight of state u, d.l. slit height, cm dark current of the photodetector, A signal value for element quantity q, A signal value for unit amount of element (q = l), A signal value for blank sample, A Boltxmann’s constant, J K-t numerical factor chosen according to the confidence level desired (a value 3 for &,,
E. D.
386
SC %l SC7 %dd,*
PRUDNIKOV
calibration standard deviation methodical standard deviation standard deviation due to analyst additive standard deviation multiplicative standard deviation measured standard deviation fluctuation standard deviation relative value of standard deviation, d.l. excitation source temperature, K transmission factor of optics, d.1. volume of sample solution, ml volume flow rate of gases, ml min-’ slit width, cm weight of sample, g nebulization efficiency, d.1. atomization efficiency, d.l. photomultiplier sensitivity, A W-’ instability factor of the excitation and measurement, d.Z. spectral slit width, nm frequency at the centre of the line, s-l
IN ATOMIC EMISSION spectroscopy (AES) a theory has been developed for estimating detection limits. In this context a paper of MANDEL’SHTAM [l] and a paper of WINEFORDNER and VICKEWS [2] have to be mentioned. Those authors used different approaches for the theoretical calculation of the detection limit, while only the boundary conditions for the analysis of the limit quantity were considered. It should be pointed out that usually one takes into account the value of the useful signal and the random errors from the excitation source, spectroscopic instrument and recording system. According to KAISER [3-S], a statistical theory of the detection limits ought to take into consideration also the fluctuations due to the blank sample and the useful (net) signal. Criteria for the detection of the minimum analytical signal have been discussed by ZIL'BERSTEIN [6] and BOWMANS [7]. It is clear, however, that random errors and therefore the standard deviation play a most important role in the determination of the concentration of an element. There exist empirical data that describe a functional dependence of the standard deviation on the concentration [8-101. These results have been summarized in a monograph L63. The standard deviation was proposed to be built up from two terms* S=
Js2ti*t+
CL.
(1)
Here stir is not related to the value of the measured signal while the second term is proportional to the signal.
* A glossary of symbols used in this article and their definitions is given in the Nomenclature. [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
S. L. hfANDEL'S.tlTAM,Spectmchim. Acta 33B, 577 (1978). J. D. WI~RDNER and T. J. VI-, Anal. Chem. 36, 1939 (1964). H. KAISER, Spectrochim. Acta 3, 40 (1947/1949). H. KAISER, Optik 21, 309 (1964). H. KAISER, Z. Anal. Chem. 209, 1 (1965). KH. I. SIL'BERSTEIN, (Editor) SpectraalAnalysis of the Pure Substances. Khimia, Leningrad (1971); Hilger, Bristol (1977). P. W. J. M. BOUMANS.Specrrochim. Acta 33B, 625 (1978). V. V. NALIPUIOV, V. V. NEDLERand M. P. MEN’SHOVA, Zauod. Lab. 27, 861 (1961). KH. I. SIL’BERSTEIN, N. G. POLIVANOVA and 2. G. FRATKIN,Zh. Pri’kl.Spectrosk. 11,204 (1969). G. EHRLICH,H. SCHOLZEand R. GERBATXH,Spectrochim. Acta 24B, 641 (1969).
Theoretical cakularion of the standard deviation in atomic emissionspectroscopy
387
According to NALIMOV[ 111, the dependence of the .random errors on the concentration can be approximated only by a linear function: s = b,c + a,,
(2)
where bl and a, are coefficients. The following expression deviation was derived by BOUMANS [12, 131:
for the relative standard
(24 for a system dominated by source fluctuation (proportional) noise. We use the following approximate formula for the standard deviation and relative standard deviation in flame emission spectrometry [ 141 s = (sLcsc2+ $)l’2, s, =
(2b)
(s:e.+$)1’2.
Here the standard deviation depends on the element concentration. Furthermore the above expressions are not connected with the fundamental parameters of AES. In the present paper, we shall attempt to calculate theoretically the standard deviation in AES due to the apparatus parameters, the blank value and the concentration of an element. We shall consider the simple formula for the limit of detection of AES given in paper [15] n, = @+$+J2, (3) L=
g 2TfwH. 0f
This function n,,, = f(L) is similar to the experimental Actually, according to statistical theory [3-51, the proportional to the random error. On the other hand, proportional to the concentration of the element and spectral instrument [ 1,2]. 2.
dependence s, = f(c). value of the detection limit is the value of the useful signal is the optical conductance of the
THEORY
2.1 General expression for the limit of detection We use the formula of WINEFORJXXR and VICKERS for the limit of detection in AES [2,16]. With the aid of the 3 s criterion as used by KAISER[3-5J formula (3) may be rewritten as n, = aI.-‘(d + bL + ~tL~)l’~, (4) where Cl=
5.7 >;:10Ug, exp (EJkT) %&&ldflY
’
b = 2e,BMAfyB, Ah,,
c, = y2AB;AA;Af, d = 2e,BMAfi,. [ll] [lq] [133
[lb] [15] [163
V. V. NALIMOV,Application of Mathematical Statisticsin Analysis of Substance p. 431. M. Fismatgiz (1960). P. W. J. M. BOWMANS,Spectrochim.Acta 3lB, 147 (1976). P. W. J. M. BOWMANS,2. Anal. Chem. 299, 337 (1979). E. D. PFUJDNIKOV and Y. S. SHAPKDJA, Vestn. Leningrad. Uniu. No. 24, p. 46 (1978). E. D. PRUDNIKOV,Spectrochim. Acta 34B, 293 (1979). E. D. PRUDNKOV. Zh. Anal. Khim. 27, 2327 (1972).
388
E. D.
PRUDNIKOV
Here aL_’ is determined by the emission intensity of a single atom, bL is the level of the shot noise of the excitation source background, cfL* is the flicker noise of the excitation source and d is the shot noise due to the thermal electron emission of a photocathode. Let the blank value be equal to qbl. Then according to WINEFORDNER and VICKERS[2], expression (4) gives the absolute quantity (in grams) and we may write: i,=3xLx6~1023~
.
aA
’
il%l,
(5)
+A2i:q:J1’?
s,,, = (&?,f&fhfi$&,~
Let us introduce
lbl =
the reciprocal sensitivity (S) of the analysis
s=$= i1
aA
(6)
18x 1023L.
we also put k, = 2e,BMAf ki=d+bL+qL2
As in references [6] and [12] the combined fluctuations background fluctuations (sbg) and blank fluctuations (sbi)
in the signal consist of
s = (St,+ s;J1? If we take the KAISER’S3 s criterion, random blank error may be introduced
(level of confidence=0.997) into equation (4) as follows
[3-71, then the
qmin;0.997 = 3S(k + kdS-‘qi,, + A2S-2q;,)“2.
(7)
The simple expression (7) for qmin:0.*7 shows the dependence of the detection limit of AES on instrumental parameters and on the blank value. This relationship may be considered to be the most general expression for the limit of AES detection. 2.2 Expressions for the standard deviation in AES It is known [2-6) that the limit of detection according to the following formula
may be calculated by KAISER'S criterion
qmin= y (8)
3s %nin ~0.997 =
7
11
=
3ss.
Using equations (7) and (8), we may write an expression deviation in AES for the quantity 9.
for the absolute standard
si = (k + k,S-‘q, + A2S-2q:)“2.
(9)
Thus, we have obtained a simple formula for the standard deviation in AES and the dependence of this standard deviation on the general parameters of the spectral instrument and the concentration of an element. Expression (9) includes the total amount of the element in the excitation source qo =q+qbl-
(10)
standard deviation of the blank, sbl, and analyte plus blank, &+bl, are obtained by use of equations (5) and (9).
The
sbl = (kdS-‘qbl +A2S-2q;,)“2, Si+bl= (k +s;,+ k,S-‘q +A2S-2q2)1’2. This is the most general expression for the absolute standard deviation in AES.
(11)
Theoretical calculation of the standard deviation
From (ll),
in atomicemissionspectroscopy
389
we may obtain an expression for the relative standard deviations i, = iiq = S’q; s, = sSq-‘, s,, = (kSzq-* + k&j-l + A*)l’*, Srt+u
(12)
(k +sk)S*+k,s+A2 lR
=
4*
[
I.
4
Therefore, we may calculate the relative and absolute standard Using (12), one can also investigate the limit of detection qmin;0.997
S ,_
q-*0;
;
deviations
0.33 for k, = 3,
=
= 3S(k.j + &)“’
0.33 =
In the following section, we shall compare these theoretical data. 3.
hACIK.AL
for AES.
(13)
results with experimental
hF%ICATlON
3.1 Comparison of theoretical and experimental results Let us consider empirical expressions (1) and (2) and the theoretical expressions (9), (11) and (12). An additive error is due to the thermal noises of the photodetector, the shot and flicker noises of the excitation source background and the shot and flicker noises of the blank signal. The multiplicative error depends on the shot and flicker noises of the measurable useful signal. Expressions (2a) and (2b) are similar. Our theoretical results permit us to define this expression more precisely
si = J s,,* c*+s;~c+s;.
(14)
Here s,, is defined by an instability of the element excitation and by the measuring error; +c is the measured signal shot noise and s, is the additive error of expression (1). Now we consider expressions (11) and (12) for the two limit concentrations
q-*0 = (/q + &)l’* = constant
si +bl
= S&i + sV2 sr,+bl 4
’
maximum value
I
(15)
q-,ol Si+bl = S,i+b, =
AS-‘q - q A = constant
1
The theoretical results (15) co&m the empirical graphical dependences [6]. We shall calculate the standard deviation for a concrete method of AES in the following section. 3.2 Calculation
of the standard deviation for flame AES
We calculate the standard deviation of the flame determination of lithium in rocks and minerals. The analytical method is published in paper [17] where we used the double-grating monochromator DFS-12 and an air/acetylene flame [16]. We introduce the relative percent concentration of the element in the sample. Let us take a 0.1 g weight and a 100 ml final volume .of the sample. Then we may write [2,16] TV,%298 ’
=
v,
10%
3OOF,(rSW,
= k,q%.
(16)
Taking into account these relations, we obtain from (4)-(6), (12) and (16) the following [17] Y. S.
SHAFKINA
and E. D.
FWIDNIKOV,
Vesm. Leningrad. Univ. No. 24, p. 142 (1969).
E. D.
390
PRUDNLICOV
expression for the standard deviation:
%=
(
(17)
where s =3.16X 10”go exp (E,,/kT)A ~oguA,d,dWF>2TfU
’
ki = 2e,BMA& + 2eCBMA@~A&$DIF)2TfWH wH2,
+ r2AB:AAfAf(DlF)“T; k, = 2e,BMAf, k, =
TV,n
T/298 V s
3F&3Ws
’
The value of the individual instrumental parameters for the 670.8 nm line of lithium may be found in papers [2,15,16]: exp X= 2.72x; go=2; E, =2.96X lo-l9 J; k = 1.38 x 1O-23 J - K-l* T=2500K; A “7; v, = 0.45 x 1o15 s-l; 8,=4; A,(= 0.35 x 10’s_‘; (ip 2 1 cm; y =3.5x104AW-‘; D/F-0.187; T,=O.3; W=O.Olcm; e, = 1.6 x lo-l9 C; B = 1.3; M = 106; Af = 1 Hz; H=4cm; id = 10T9 A; B, = hs,=6~lO-“Wcm-~sr-’ 6 x 10m9W cmT2 sr-’ nm-‘; Ah, =O.l run; A=O.Ol; nm-‘; V,=7200 cm3 min-‘; ‘tmm =1.2; V,=lOOml; F,=3mimin-‘; a=O.l; /3=0.1; W,=O.lg; C%. Then: S=2.6x10m9; k,=1.2~10-~~; k,=4.16~10-‘~; kc =2.4x 1O’l; A=O.Ol and ST‘= 3*g xc2lo-
ls + 2.6 x 10-l’
112 +
C
3.9x lo-“+2.6x C2
1o-4
10-6+1
(18)
C
The obtained theoretical dependence is shown in the diagram (Fig. 1). In the same figure, we give the experimental instrumental errors for the flame spectrometric
0.30I-
0.20,b
,O.IC
1, I.0 -7
-6
-3
C ,,.,0.997
-4
-3
-2
-,
0
I
2
cg cx
Fig. 1. Theoretical () and experimental (O--O--O) dependence of the relative instrumental standard deviation on the concentration of lithium in a flame spectrometric detennination.
Theoretical calculation of the standard deviation in atomic emission spectroscopy
391
determination of lithium. These results are published in paper [14]. Our theory is confirmed by the experimental data. The differences between experimental and theoretical data are due to the inexactitudes of the individual apparatus parameters. A graphical method may be used to find the detection limit (see Fig. 1). For the sample, the relative instrumental limit of lithium detection is (theoretically) equal to c,,,+~.~~ = 1.7 x 10e7%. For the solution, we have cmin;o.997= 1.7 X lo-” g ml-‘. The experimental = 4.5 x 10e7% for the sample and 4.5 x lo-l2 g ml-’ for the values are equal to ctininz0.997 solution. The calculation of the standard deviation takes the blank value into account. It is difficult to predict the blank sample theoretically. Therefore, we take an experimental blank value, which is equal to cbl = 1.1 x lo-‘% [14]. Then the theoretical blank error part may be calculated from formulas (11) and (12) s&S2k2 1 kdS2k,zCb, A2S2k2C2 * ‘2 ( Sk, + s2;: b’ As a result we may write
.
=o,olr.gx
lo-“c~1.5
x 10-10+2.6~10-6+
1)“’
s‘i*bl = O.Ol(
si+bl=(l.gX
1.9 x lo-lo c* +2.6;10-6+ lo-“+2.6x
l)l’*,
(19)
10-‘“~+10-4~2)‘“.
In Fig. 2, we compare the theoretical results with the experimental material taken from the paper [14]. The difference between them increases with respect to the pure instrumental error. This fact may be explained using the blank error dependence on the uncontrollable analytical factor [6]. According to our theory, we have &,,:o.p97 = 4.5 x 10e7%. Experimentally cmin:0.997= 4.5 x 10G6%. Here, the limit of detection is determined by the blank error. -I
--6
-7
theor. -6
eari. -3
Crn,“,0.997
-4
-3
-2
-,
0
I
Lg CK
Fig. 2. Theoretical and experimental dependence of the absolute and relative standard deviatheoretical; tions on the lithium concentration in a flame spectrometric determination: Q -0 -0 experimental for the relative standard deviation; x -- x -- x experimental for the absolute standard deviation.
E. D. PRUDNIKOV
392
We replace the theoretical error. According to [14]
blank error in expression (19) by the experimental
‘bL, = 1.5 x 10-6% (
S,‘&,
=o~ol(2.25~~10-x+2.6~10-6+1)1’2
blank (19a\
Here we get a good agreement between experimental and theoretical data (see the broken curve in Fig. 2). We have shown above that the standard deviation in AES may be calculated. A study of the relative standard deviation gives us more information than the absolute standard deviation (see Fig. 2). The precision of the calculations may be improved by using more accurate values of the spectral apparatus parameters. The blank error decreases if special analytical methods are applied [6]. 3.3 Standard deviation and other errors in a concentration determination We have considered the instrumental and blank random errors of a concentration determination. It is necessary to take also into account the calibration error (s,), the methodical error (s,) and the analyst error (s,) [14]. The calibration error characterizes principally the instrumental factor of the analysis. This value may be calculated theoretically. For three standard samples, the calibration error value is equal to s, = sJ1.7. The methodical error is determined by the concomitants in the matrix. It is di&ult to predict these matrix effects theoretically. The concomitants in the matrix may be only determined experimentally. The analyst error correlates with the methodical error. One can scrutinize these errors jointly. These errors determine the error value for high concentrations of the elements. It is necessary to emphasize that the methodical and calibration errors determine also the systematical error of the analysis caused by matrix effects and by erroneous calibration. The systematical error can be decreased by perfecting the methodical basis of the analysis. Therefore, the limit of detection depends mainly on the instrumental and blank errors. Other errors influence more the precision of high concentration determinations. One cannot calculate theoretically all errors. However, it is possible to give the general expression for the total standard deviation of AES. We utilize the experimental data [ 141 and define more precisely the coefficients in the expression (19). We may write finally for the flame AES determination of lithium
4.
CONCLUSIONS
The standard deviation is a most important factor in analysis. In this article simple expressions have been derived for the limit of detection and for the standard deviation in AES. The standard deviation of the flame emission spectrometric determination of lithium has been calculated. A comparison of the theoretical and experimental data was made and it was confirmed that a calculation of the standard deviation gives more information than the limit of detection. These results are interesting for the theory and practice of AES. They also lead to a better estimate of the standard deviation and limit of detection in AES [2-61. This is important also in view of the rapid development of inductively coupled plasma AES (see e.g. [13,18-211).
Acknowledgements-The manuscript was prepared for publication at the Free University of West-Berlin during a scientific mission of the author. The author thanks Prof. Dr. H. BRADACZEK, Institute of Crystallography, Free University West-Berlin for the opportunity to finish the manuscript of this article. [18] V. A. F-EL, Science 202, 183 (1978). [19] R. M. BARNES.C.R.C. Crir. Reu. Anal. Chem. 7, 203 (1978). [ZO] S. GREENFIELD,H. McD. MCGEACHM, and P. B. SMITH, Talama 23. 1 (19761. [21] W. J. BOYKO.P. N. KELIHER,and J. M. MALLOY.Anal. Chem. 52, 53R (1980).
Theoretical
0584-8547/81/040385.~/0 @1981PeqmonFw%L%l.
36B, No. 4. PP. 385 to 392, 1981.
calcdation
of the standard deviation
in
atomic
emission
sp--PY E.
D.
PRUDNIKOV
Earth’s Crust Institute, State University, Leningrad, 199164, U.S.S.R. (Receiwd 28 Februmy
1980; in revised form 16 October 1980)
AMxaet-In atomic emission spectroscopy (ABS), a simple formula is obtained for the calculation of the limits of detection which takes into account the blank samples values. An expression is proposed for the standard deviation due to apparatus parameters, blank value and concentration of an element. ‘I’hese results are applied to practical problems.
NOMENCLATURE atomic weight of element, atomic mass unit transition probability, s-t coefficient for calculating the signal value. A-’ cm-’ photomultiplier factor related to gain per stage, dimensionless, coefficient for the calculation of the r.m.s. fluctuation A2 m-2
=-I
d.1. of the
background
intensity,
nm-’
intensity of excitation source contimmm, W cmm2sr-l nm-’ standard deviation of BA, W cme2 ST-* run-’ relative concentration of element, % relative limit of detection, % relative limit of detection for confidence probability of 0.997 (criterion 3 s), % blank concentration of ‘element, % coefficient for the calculation of the flicker fluctuation, A2 UP, srm2s-l effective aperture of the spectral instrument, cm2 average diamata of the flame region focused on the entrance slit of the spectml instnmmnt, cm coe&Gnt of photodetector thermal noise, A2 exponential function, d.l. electronic charge, C excitation energy of state y J focal length of the collimator, cm solution flow rate, ml min-’ frequency response band width, Hx statistical weight of ground state, d.1. statistical weight of state u, d.l. slit height, cm dark current of the photodetector, A signal value for element quantity q, A signal value for unit amount of element (q = l), A signal value for blank sample, A Boltxmann’s constant, J K-t numerical factor chosen according to the confidence level desired (a value 3 for &,,
E. D.
386
SC %l SC7 %dd,*
PRUDNIKOV
calibration standard deviation methodical standard deviation standard deviation due to analyst additive standard deviation multiplicative standard deviation measured standard deviation fluctuation standard deviation relative value of standard deviation, d.l. excitation source temperature, K transmission factor of optics, d.1. volume of sample solution, ml volume flow rate of gases, ml min-’ slit width, cm weight of sample, g nebulization efficiency, d.1. atomization efficiency, d.l. photomultiplier sensitivity, A W-’ instability factor of the excitation and measurement, d.Z. spectral slit width, nm frequency at the centre of the line, s-l
IN ATOMIC EMISSION spectroscopy (AES) a theory has been developed for estimating detection limits. In this context a paper of MANDEL’SHTAM [l] and a paper of WINEFORDNER and VICKEWS [2] have to be mentioned. Those authors used different approaches for the theoretical calculation of the detection limit, while only the boundary conditions for the analysis of the limit quantity were considered. It should be pointed out that usually one takes into account the value of the useful signal and the random errors from the excitation source, spectroscopic instrument and recording system. According to KAISER [3-S], a statistical theory of the detection limits ought to take into consideration also the fluctuations due to the blank sample and the useful (net) signal. Criteria for the detection of the minimum analytical signal have been discussed by ZIL'BERSTEIN [6] and BOWMANS [7]. It is clear, however, that random errors and therefore the standard deviation play a most important role in the determination of the concentration of an element. There exist empirical data that describe a functional dependence of the standard deviation on the concentration [8-101. These results have been summarized in a monograph L63. The standard deviation was proposed to be built up from two terms* S=
Js2ti*t+
CL.
(1)
Here stir is not related to the value of the measured signal while the second term is proportional to the signal.
* A glossary of symbols used in this article and their definitions is given in the Nomenclature. [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo]
S. L. hfANDEL'S.tlTAM,Spectmchim. Acta 33B, 577 (1978). J. D. WI~RDNER and T. J. VI-, Anal. Chem. 36, 1939 (1964). H. KAISER, Spectrochim. Acta 3, 40 (1947/1949). H. KAISER, Optik 21, 309 (1964). H. KAISER, Z. Anal. Chem. 209, 1 (1965). KH. I. SIL'BERSTEIN, (Editor) SpectraalAnalysis of the Pure Substances. Khimia, Leningrad (1971); Hilger, Bristol (1977). P. W. J. M. BOUMANS.Specrrochim. Acta 33B, 625 (1978). V. V. NALIPUIOV, V. V. NEDLERand M. P. MEN’SHOVA, Zauod. Lab. 27, 861 (1961). KH. I. SIL’BERSTEIN, N. G. POLIVANOVA and 2. G. FRATKIN,Zh. Pri’kl.Spectrosk. 11,204 (1969). G. EHRLICH,H. SCHOLZEand R. GERBATXH,Spectrochim. Acta 24B, 641 (1969).
Theoretical cakularion of the standard deviation in atomic emissionspectroscopy
387
According to NALIMOV[ 111, the dependence of the .random errors on the concentration can be approximated only by a linear function: s = b,c + a,,
(2)
where bl and a, are coefficients. The following expression deviation was derived by BOUMANS [12, 131:
for the relative standard
(24 for a system dominated by source fluctuation (proportional) noise. We use the following approximate formula for the standard deviation and relative standard deviation in flame emission spectrometry [ 141 s = (sLcsc2+ $)l’2, s, =
(2b)
(s:e.+$)1’2.
Here the standard deviation depends on the element concentration. Furthermore the above expressions are not connected with the fundamental parameters of AES. In the present paper, we shall attempt to calculate theoretically the standard deviation in AES due to the apparatus parameters, the blank value and the concentration of an element. We shall consider the simple formula for the limit of detection of AES given in paper [15] n, = @+$+J2, (3) L=
g 2TfwH. 0f
This function n,,, = f(L) is similar to the experimental Actually, according to statistical theory [3-51, the proportional to the random error. On the other hand, proportional to the concentration of the element and spectral instrument [ 1,2]. 2.
dependence s, = f(c). value of the detection limit is the value of the useful signal is the optical conductance of the
THEORY
2.1 General expression for the limit of detection We use the formula of WINEFORJXXR and VICKERS for the limit of detection in AES [2,16]. With the aid of the 3 s criterion as used by KAISER[3-5J formula (3) may be rewritten as n, = aI.-‘(d + bL + ~tL~)l’~, (4) where Cl=
5.7 >;:10Ug, exp (EJkT) %&&ldflY
’
b = 2e,BMAfyB, Ah,,
c, = y2AB;AA;Af, d = 2e,BMAfi,. [ll] [lq] [133
[lb] [15] [163
V. V. NALIMOV,Application of Mathematical Statisticsin Analysis of Substance p. 431. M. Fismatgiz (1960). P. W. J. M. BOWMANS,Spectrochim.Acta 3lB, 147 (1976). P. W. J. M. BOWMANS,2. Anal. Chem. 299, 337 (1979). E. D. PFUJDNIKOV and Y. S. SHAPKDJA, Vestn. Leningrad. Uniu. No. 24, p. 46 (1978). E. D. PRUDNIKOV,Spectrochim. Acta 34B, 293 (1979). E. D. PRUDNKOV. Zh. Anal. Khim. 27, 2327 (1972).
388
E. D.
PRUDNIKOV
Here aL_’ is determined by the emission intensity of a single atom, bL is the level of the shot noise of the excitation source background, cfL* is the flicker noise of the excitation source and d is the shot noise due to the thermal electron emission of a photocathode. Let the blank value be equal to qbl. Then according to WINEFORDNER and VICKERS[2], expression (4) gives the absolute quantity (in grams) and we may write: i,=3xLx6~1023~
.
aA
’
il%l,
(5)
+A2i:q:J1’?
s,,, = (&?,f&fhfi$&,~
Let us introduce
lbl =
the reciprocal sensitivity (S) of the analysis
s=$= i1
aA
(6)
18x 1023L.
we also put k, = 2e,BMAf ki=d+bL+qL2
As in references [6] and [12] the combined fluctuations background fluctuations (sbg) and blank fluctuations (sbi)
in the signal consist of
s = (St,+ s;J1? If we take the KAISER’S3 s criterion, random blank error may be introduced
(level of confidence=0.997) into equation (4) as follows
[3-71, then the
qmin;0.997 = 3S(k + kdS-‘qi,, + A2S-2q;,)“2.
(7)
The simple expression (7) for qmin:0.*7 shows the dependence of the detection limit of AES on instrumental parameters and on the blank value. This relationship may be considered to be the most general expression for the limit of AES detection. 2.2 Expressions for the standard deviation in AES It is known [2-6) that the limit of detection according to the following formula
may be calculated by KAISER'S criterion
qmin= y (8)
3s %nin ~0.997 =
7
11
=
3ss.
Using equations (7) and (8), we may write an expression deviation in AES for the quantity 9.
for the absolute standard
si = (k + k,S-‘q, + A2S-2q:)“2.
(9)
Thus, we have obtained a simple formula for the standard deviation in AES and the dependence of this standard deviation on the general parameters of the spectral instrument and the concentration of an element. Expression (9) includes the total amount of the element in the excitation source qo =q+qbl-
(10)
standard deviation of the blank, sbl, and analyte plus blank, &+bl, are obtained by use of equations (5) and (9).
The
sbl = (kdS-‘qbl +A2S-2q;,)“2, Si+bl= (k +s;,+ k,S-‘q +A2S-2q2)1’2. This is the most general expression for the absolute standard deviation in AES.
(11)
Theoretical calculation of the standard deviation
From (ll),
in atomicemissionspectroscopy
389
we may obtain an expression for the relative standard deviations i, = iiq = S’q; s, = sSq-‘, s,, = (kSzq-* + k&j-l + A*)l’*, Srt+u
(12)
(k +sk)S*+k,s+A2 lR
=
4*
[
I.
4
Therefore, we may calculate the relative and absolute standard Using (12), one can also investigate the limit of detection qmin;0.997
S ,_
q-*0;
;
deviations
0.33 for k, = 3,
=
= 3S(k.j + &)“’
0.33 =
In the following section, we shall compare these theoretical data. 3.
hACIK.AL
for AES.
(13)
results with experimental
hF%ICATlON
3.1 Comparison of theoretical and experimental results Let us consider empirical expressions (1) and (2) and the theoretical expressions (9), (11) and (12). An additive error is due to the thermal noises of the photodetector, the shot and flicker noises of the excitation source background and the shot and flicker noises of the blank signal. The multiplicative error depends on the shot and flicker noises of the measurable useful signal. Expressions (2a) and (2b) are similar. Our theoretical results permit us to define this expression more precisely
si = J s,,* c*+s;~c+s;.
(14)
Here s,, is defined by an instability of the element excitation and by the measuring error; +c is the measured signal shot noise and s, is the additive error of expression (1). Now we consider expressions (11) and (12) for the two limit concentrations
q-*0 = (/q + &)l’* = constant
si +bl
= S&i + sV2 sr,+bl 4
’
maximum value
I
(15)
q-,ol Si+bl = S,i+b, =
AS-‘q - q A = constant
1
The theoretical results (15) co&m the empirical graphical dependences [6]. We shall calculate the standard deviation for a concrete method of AES in the following section. 3.2 Calculation
of the standard deviation for flame AES
We calculate the standard deviation of the flame determination of lithium in rocks and minerals. The analytical method is published in paper [17] where we used the double-grating monochromator DFS-12 and an air/acetylene flame [16]. We introduce the relative percent concentration of the element in the sample. Let us take a 0.1 g weight and a 100 ml final volume .of the sample. Then we may write [2,16] TV,%298 ’
=
v,
10%
3OOF,(rSW,
= k,q%.
(16)
Taking into account these relations, we obtain from (4)-(6), (12) and (16) the following [17] Y. S.
SHAFKINA
and E. D.
FWIDNIKOV,
Vesm. Leningrad. Univ. No. 24, p. 142 (1969).
E. D.
390
PRUDNLICOV
expression for the standard deviation:
%=
(
(17)
where s =3.16X 10”go exp (E,,/kT)A ~oguA,d,dWF>2TfU
’
ki = 2e,BMA& + 2eCBMA@~A&$DIF)2TfWH wH2,
+ r2AB:AAfAf(DlF)“T; k, = 2e,BMAf, k, =
TV,n
T/298 V s
3F&3Ws
’
The value of the individual instrumental parameters for the 670.8 nm line of lithium may be found in papers [2,15,16]: exp X= 2.72x; go=2; E, =2.96X lo-l9 J; k = 1.38 x 1O-23 J - K-l* T=2500K; A “7; v, = 0.45 x 1o15 s-l; 8,=4; A,(= 0.35 x 10’s_‘; (ip 2 1 cm; y =3.5x104AW-‘; D/F-0.187; T,=O.3; W=O.Olcm; e, = 1.6 x lo-l9 C; B = 1.3; M = 106; Af = 1 Hz; H=4cm; id = 10T9 A; B, = hs,=6~lO-“Wcm-~sr-’ 6 x 10m9W cmT2 sr-’ nm-‘; Ah, =O.l run; A=O.Ol; nm-‘; V,=7200 cm3 min-‘; ‘tmm =1.2; V,=lOOml; F,=3mimin-‘; a=O.l; /3=0.1; W,=O.lg; C%. Then: S=2.6x10m9; k,=1.2~10-~~; k,=4.16~10-‘~; kc =2.4x 1O’l; A=O.Ol and ST‘= 3*g xc2lo-
ls + 2.6 x 10-l’
112 +
C
3.9x lo-“+2.6x C2
1o-4
10-6+1
(18)
C
The obtained theoretical dependence is shown in the diagram (Fig. 1). In the same figure, we give the experimental instrumental errors for the flame spectrometric
0.30I-
0.20,b
,O.IC
1, I.0 -7
-6
-3
C ,,.,0.997
-4
-3
-2
-,
0
I
2
cg cx
Fig. 1. Theoretical () and experimental (O--O--O) dependence of the relative instrumental standard deviation on the concentration of lithium in a flame spectrometric detennination.
Theoretical calculation of the standard deviation in atomic emission spectroscopy
391
determination of lithium. These results are published in paper [14]. Our theory is confirmed by the experimental data. The differences between experimental and theoretical data are due to the inexactitudes of the individual apparatus parameters. A graphical method may be used to find the detection limit (see Fig. 1). For the sample, the relative instrumental limit of lithium detection is (theoretically) equal to c,,,+~.~~ = 1.7 x 10e7%. For the solution, we have cmin;o.997= 1.7 X lo-” g ml-‘. The experimental = 4.5 x 10e7% for the sample and 4.5 x lo-l2 g ml-’ for the values are equal to ctininz0.997 solution. The calculation of the standard deviation takes the blank value into account. It is difficult to predict the blank sample theoretically. Therefore, we take an experimental blank value, which is equal to cbl = 1.1 x lo-‘% [14]. Then the theoretical blank error part may be calculated from formulas (11) and (12) s&S2k2 1 kdS2k,zCb, A2S2k2C2 * ‘2 ( Sk, + s2;: b’ As a result we may write
.
=o,olr.gx
lo-“c~1.5
x 10-10+2.6~10-6+
1)“’
s‘i*bl = O.Ol(
si+bl=(l.gX
1.9 x lo-lo c* +2.6;10-6+ lo-“+2.6x
l)l’*,
(19)
10-‘“~+10-4~2)‘“.
In Fig. 2, we compare the theoretical results with the experimental material taken from the paper [14]. The difference between them increases with respect to the pure instrumental error. This fact may be explained using the blank error dependence on the uncontrollable analytical factor [6]. According to our theory, we have &,,:o.p97 = 4.5 x 10e7%. Experimentally cmin:0.997= 4.5 x 10G6%. Here, the limit of detection is determined by the blank error. -I
--6
-7
theor. -6
eari. -3
Crn,“,0.997
-4
-3
-2
-,
0
I
Lg CK
Fig. 2. Theoretical and experimental dependence of the absolute and relative standard deviatheoretical; tions on the lithium concentration in a flame spectrometric determination: Q -0 -0 experimental for the relative standard deviation; x -- x -- x experimental for the absolute standard deviation.
E. D. PRUDNIKOV
392
We replace the theoretical error. According to [14]
blank error in expression (19) by the experimental
‘bL, = 1.5 x 10-6% (
S,‘&,
=o~ol(2.25~~10-x+2.6~10-6+1)1’2
blank (19a\
Here we get a good agreement between experimental and theoretical data (see the broken curve in Fig. 2). We have shown above that the standard deviation in AES may be calculated. A study of the relative standard deviation gives us more information than the absolute standard deviation (see Fig. 2). The precision of the calculations may be improved by using more accurate values of the spectral apparatus parameters. The blank error decreases if special analytical methods are applied [6]. 3.3 Standard deviation and other errors in a concentration determination We have considered the instrumental and blank random errors of a concentration determination. It is necessary to take also into account the calibration error (s,), the methodical error (s,) and the analyst error (s,) [14]. The calibration error characterizes principally the instrumental factor of the analysis. This value may be calculated theoretically. For three standard samples, the calibration error value is equal to s, = sJ1.7. The methodical error is determined by the concomitants in the matrix. It is di&ult to predict these matrix effects theoretically. The concomitants in the matrix may be only determined experimentally. The analyst error correlates with the methodical error. One can scrutinize these errors jointly. These errors determine the error value for high concentrations of the elements. It is necessary to emphasize that the methodical and calibration errors determine also the systematical error of the analysis caused by matrix effects and by erroneous calibration. The systematical error can be decreased by perfecting the methodical basis of the analysis. Therefore, the limit of detection depends mainly on the instrumental and blank errors. Other errors influence more the precision of high concentration determinations. One cannot calculate theoretically all errors. However, it is possible to give the general expression for the total standard deviation of AES. We utilize the experimental data [ 141 and define more precisely the coefficients in the expression (19). We may write finally for the flame AES determination of lithium
4.
CONCLUSIONS
The standard deviation is a most important factor in analysis. In this article simple expressions have been derived for the limit of detection and for the standard deviation in AES. The standard deviation of the flame emission spectrometric determination of lithium has been calculated. A comparison of the theoretical and experimental data was made and it was confirmed that a calculation of the standard deviation gives more information than the limit of detection. These results are interesting for the theory and practice of AES. They also lead to a better estimate of the standard deviation and limit of detection in AES [2-61. This is important also in view of the rapid development of inductively coupled plasma AES (see e.g. [13,18-211).
Acknowledgements-The manuscript was prepared for publication at the Free University of West-Berlin during a scientific mission of the author. The author thanks Prof. Dr. H. BRADACZEK, Institute of Crystallography, Free University West-Berlin for the opportunity to finish the manuscript of this article. [18] V. A. F-EL, Science 202, 183 (1978). [19] R. M. BARNES.C.R.C. Crir. Reu. Anal. Chem. 7, 203 (1978). [ZO] S. GREENFIELD,H. McD. MCGEACHM, and P. B. SMITH, Talama 23. 1 (19761. [21] W. J. BOYKO.P. N. KELIHER,and J. M. MALLOY.Anal. Chem. 52, 53R (1980).