tar bases were identified : quinoline and tetrahydroquinolines (cycloalkanopyridines). Cycloalkanopyridine types have been reported in coal (11) and similar structures have been found in petroleum (12, 13). The combined quantity of quinoline and cycloalkanopyridines amounted to about 3 % of the nitrogen-base concentrate. The anilines, which make up about 33% of the tar-base concentrate, are of interest because their presence has not been reported in raw shale oil naphtha (2, 4). They may be assumed to be produced in the hydrocracking process by rupture of the heterocyclic ring-for instance, 2-ethylaniline and 2propylaniline might result from the reductive cleavage of indole and quinoline which are known compounds in shale oil. This is supported by hydrogenation of petroleum frac(11) Clarence Karr and Ta-Chuang Lo Chang, J. Znst. Fuel, 31, 522 (1958). (12) C. F. Brandenburg and D. R. Latham, J. Chem. Eng. Data, 13,391 (1968). (13) H. V. Drushel and A. L. Sommers, ANAL.CHEM.,38, 19 (1966).
tions that contain known bases (IO, 14). Inspection of the aniline data in Table 111 shows aniline and ring-substituted alkylanilines through CB. The substitution pattern for the alkylanilines shows that 90% of the aniline molecules have substitution CY to the location of the amino group, 15 p substitution, and 25% y substitution. The preponderance of CY substitution is even more notable for the C1 to C Banilines than for the C1to C 3pyridines. RECEIVED for review September 10, 1969. Accepted November 17, 1969. Work done under a cooperative agreement between the Bureau of Mines, U. S. Department of the Interior, and the University of Wyoming. Reference to a company or product name does not imply approval or recommendation of the product by the U. S. Department of Interior to the exclusion of others that may be suitable.
(14) G. K. Hartung, D. M. Jewell, 0. A. Larson, and R. A. Flinn, J. Chern. Eng. Data, 6,477 (1961).
Features of the Modified Covell Method for Computation of Total Absorption Peak Areas in Complex Gamma-Ray Spectra Slawomir Sterlinski Institute of Nuclear Research, Department of Analytical Chemistry, Warsaw, Poland
This article describes results of an investigation of the features of the modified Covell method for comparison of the total absorption peak areas in nondestructive activation analysis. Principles of the method have been described in an earlier publication. Either a small computer or an ordinary calculator can be used. The method proposed gives a better evaluation of the precision of the peak area in a complex gamma-ray spectrum than the traditional method of Covell. The predominance of this method over that of Covell is demonstrated especially in the case of a high ratio of the height of baseline to the peak height-i.e., for small peaks appearing on a high background.
IN RECENT YEARS a number of elegant, often very sophisticated, algorithms based on the least squares method or on the stripping method have been published to analyze complex gamma-ray spectra. The principle of the algorithms is an assumption that a library of standard spectra is at hand. In connection with this, the programs written for these methods are possible and profitable only on large computers. Unfortunately, many small activation analysis laboratories either have no possibility of using such computers, or their access to them is very limited ( I ) . Algorithms of methods have also been described in which the objects of the mathematical analysis are total absorption peaks (TAP) in gamma-ray spectra. In these methods, no library of standard spectra is involved, appreciably reducing the requirements in regard to the computation technique; nevertheless, the execution of such procedures as the mathe-
matical smoothing of apparatus spectra, the control of TAP, or the examining of the first derivative of TAP is impossible without the use of a computer. If an activation analysis laboratory has neither a computer nor access to one, then the elaboration of a small number of spectra must be performed by sirnple technical means. Because of this, the development of simple methods of analysis of gamma-ray spectra is of real value. The method discussed, a comparison of TAP area in nondestructive activation analysis, was proposed in an earlier publication (2). The principle of this method is based on an appropriate modification of the known formula of Covell(3). The basic advantages of the method are the rapidity and simplicity of calculations and a better evaluation of precision of the TAP area in a complex gamma-ray spectrum as compared to Covell’s method. The present method can be applied (like Covell’s method) using only an ordinary calculator or a small computer. During the last two years the new method has been used in a laboratory practice for the interpretation of scintillation spectra of gamma radiation. A few new features of the method have been observed based on experimental data. In order to know all features of the method, systematic numerical studies were carried out. By means of a computer the TAP was generated for the various peak half-widths and the various ratios of the height of the baseline to the peak height. The calculated precision of the peak areas was compared to both Covell’s method and the present method.
(1) K. Liebscher and H. Smith, ANAL.CHEM., 40, 1999 (1968).
(2) S1. Sterlidski, ANAL.CHEM.,40, 1995 (1968). (3) D. F. Covell, ibid., 31, 1785 (1959). ANALYTICAL CHEMISTRY, VOL. 42, NO. 2, FEBRUARY 1970
151
By substituting Equation 1 and after summation we have (2) k-I
For k = 1 Equation 3 may be written in a simple way, assuming 1 1 n, and Sl2 S ,
+
n
Sn = nao
+ C (n - 2i + '/z>(at + a-t). a=1
(4)
For numerical calculations in the present paper Equation 3 was transformed into the form S,k
(n
k-1
- k f 1)
,
2 3
c
at
-(-k-)
f
+
i
>$b# 10
160
w,
170
190
190
,"x . 220
I
200
210
, 230
Channel number
Figure 1. Total absorption peak of the gamma radiation of lMLaat 1600 keV A thallium activated sodium iodate crystal 3'' X 3" was used. The solid line represents a Gaussian curve fitted to the experimental data contained in the interval xo f W by a maximum probability method (8). The points which do not lie on the Gaussian curve are marked with full points
This approach, though sophisticated in a sense, allowed relative universality to establish features of the method examined. BMIC MATHEMATICAL FORMULAE OF THE METHOD
The idea to modify Covell's method originated when it was observed that in his method the variance of the value N, calculated according to Equation 1 n-I
where ai is the number of counts in channel i, and n is the number of channels used to calculate N, covered on the left and on the right from a channel with maximum number of counts, ao, is seriously affected by accidental fluctuations of the counts in extreme channels, a, and a,. The weights of the terms a, anda, equal (n - 1/2)2; on the other hand, the weights of the other terms are 1 (2,3). To reduce this disproportion in statistical weights of the variance N , it was necessary to choose weights for the terms at in a different way. The change of weights for at, however, cannot be done only from a viewpoint of the variance; a linear combination of the type
c wtat, where w~denotes weight, must be proportional to the n
i=l
TAP area. This condition is fulfilled by the expression Sazl which is obtained by summing N for the successive N from k to k 1, then
+
Sat = N(n = k )
+ N(n = k + 1) +
,
..
+
N(n = k 152
0
+ 0. (2)
putting k I = n. It may readily be observed that for k = n, S,, transforms into N a n d for k = 1, S,I transforms into S,. The counts in the neighboring channels are independent. Eckhoff's results indicate that the correlation between the counts in different channels of a gamma-ray spectrum may be assumed to be zero (4). Therefore, the variance of S,I, can be expressed as Vsnb= (n
-k
+ 1)2
z=
k-1
- ( k - 1)
at f
n -I,
i-0
Taking k = 1 or k = n in Equation 6 we obtain directly the variances Vs, or V,. PROCEDURE FOR NUMERICAL STUDIES Generation of TAP. In a scintillation gamma-ray spectrum the TAP has a shape very similar to a Gaussian distribution (5, 6). Hence, in many papers it is assumed that the TAP can be described by the formula of Gauss (7-10). Deviations from the Gaussian form in the TAP occur in its furtherest and most extreme regions (Figure 1). Deviations in the right-hand peak region result, according to Heath, from certain optical effects in the scintillator, while those on the left-hand peak region result from the tailing contribution due to Compton scatter. In some analytical applications, for a very accurate description of TAP, a modified Gaussian curve ( 5 ) or a modified hyperbolic secant function (11) is introduced. In the present paper, an assumption was made that for numerical studies those channels will be used which are contained in the interval xm i 3u-i.e. at a distance not more (4) N. D. Eckhoff, Nucl. Instrum. Methods, 68, 93 (1969). ( 5 ) R. G. Helmer, R. L. Heath, L. A. Schmittroth, G. A. Jayne, and L. M. Wagner, ibid., 47, 305 (1967). (6) R. L. Heath, R. G. Helrner, L. A. Schmittroth, and G. A. Cazier, ibid., 47, 281 (1967). (7) L. Daddi and V. D'Angelo, Int. J. Appl. Radiat. Isotopes, 19, 407 (1968). ( 8 ) M. Ciampi, L. Daddi, and V. D'Angelo, ATucl.Instrum. Methods, 66, 102 (1968). (9) L. Daddi and V. D'Angelo, ibid., 42, 134 (1966). (10) A. K. Gribkov and L. D. Soshin, Pribory i Tekhika Eksperimenta, 6, 41 (1968). (11) B. R. Kowalski and T. L. Isenhour, ANAL.CHEM.,40, 1186 (1968).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 2, FEBRUARY 1970
than 3 standard deviations, u, from the maximum peak, x,. Because u z = W2/8In 2, where W is the half-width of the peak, the limit of the established interval may be expressed as x , & 1.274 W (Figure 1). From Figure 1 and 2 it can be observed that the righthand side of the TAP may be in the limits established and sufficiently well described by a Gaussian curve y ( x ) = y o exp [-2.772 ( x
- X,)~/W~],
/
(7)
where the independent variable x is a continuum expressed by a channel scale. The value Win Equation 7 is also expressed by channels. To describe an experimental point it is convenient to introduce a discreet variable, xi,corresponding to the i-th channel (8, 9). Then the value y ( x ) for the i-th channel may be written y ( x z ) = y o exp 1-2.772 (xi
- X~)~/W~I.
I
I
be introduced to Equation 8. For this purpose, use has been made of the properties of the function tanh(x) (11). Introducing functions A y o [l
- tan h(Bxi
+ C)],
(9)
+ n.
190
A[I
15
1.OOO
1.083 1.014 1.002 0.999 0.999 1.999 0.999 0.999
1.ooO 1.ooO 1.ooO 1.ooO
1.ooO 1.OOO 1.OOO
1.ooO 1.ooO
216
- X,)~/W~]+
- tanh(Bxt + c)])
+ mxi + n
(11)
w>
for n = 1,2,3, . . . . , entier (1.274 a n d k = 1 , 2 , 3 , . . . . , n. The first expression in Equation 12 equals 1 for k = 1; but, for k = n, it represents a ratio of the relative standard deviation N to the relative standard deviation S, (because N). The main idea in the calculation of this exS,, pression was to find answers to the questions: what value k , when n equals a constant, gives the minimum relative standard
Table I. Comparison of the Relative Standard Deviations S n k and S, for R
1.007 0.995 0.996 0.997 0.998 0.999 0.999 0,999 1.ooO 1.00(!
110
(10)
(12) W. Zimmermann, Rev. Sci. brstntrn., 32, 1063 (1961).
3
ZI
1CJ
When calculating N , S,, or S n k , the last term in Equation 11 disappears, but for the variances of these values only the ratio R occurs. The coefficients n and m do not appear in the open form. For each combination of the parameters R and W the following expressions were calculated
-
2
205
a, = (P/1.064 W)(exp[-2.772(xf
This shape of the baseline is a principle of Covell’s method and, indirectly, of the modified method. When the baseline is curvilinear, then the results of the comparison of the appropriate TAP areas in the spectra of a sample and a standard show a systematic error (2).
1 nk 1 1.ooO 2 1.ooO 3 1.ooO 4 1.ooO 5 1.ooO 6 1.OOO 7 1.ooO 8 1.OOO 9 !.ooO 10 1.ooO 11 1.OO0 12 1.ooO 13 1.ooO 14 1.ooO
xrn =?00.7Chnnnels
Mo
fsS
The ratio of the height of baseline to the peak height, R, is defined for the coordinate x = x,, or R = b(x,)/yo. Calculation Program. It was assumed that the peak area P equals a constant. Because P = 1.064 yo W (7), when changing W the peak height was yo = P/1.064 W. The expression for the terms af,according to Equations 8, 9, and 10, is of the form
the tailing contribution to the TAP arising from Compton scatter, can accurately be described (within the established limits). The parameters A , B, and C were established based on the 1600 keV peak due to the radionuclide 140La. The peak was measured at varying amplifications-i.e., at varying values of W ( W = 5.3, 7.7, 10.4, 13.8, 15.2, and 18 channels). The parameter A is in a first approximation, an independent of the half-width of the peak (in channels) for a given crystal and peak. On the other hand, the parameters B and C are functions of W. The relationship between B and C and W was found by the least-squares method. Baseline. A straight-forward shape of the baseline in the region of TAP was assumed. b(x,) = rnx,
185
Figure 2. The test of normality according to Zimmermann (12) for the peak from Figure 1. The points which lie on the straight line are described by the Gaussian curve and are the open circles in Figure 1
(8)
The mathematical description of the extreme left-hand side
=
I80
Channel number
of the TAP (in limits accepted above) needs a correction to
P(xJ
I ~
o.(
=
50 and W = 12 Channels
4
5
6
7
8
9
10
11
12
13
14
15
1.179 1.055 1.023 1.012 1.007 1.005 1.004 1.004 1.003 1.003 1.003 1.003
1.288 1.112 1.059 1.037 1.026 1.020 1.017 1.014 1.013 1.012 1.011
1.409 1.184 1.108 1.074 1.056 1.045 1.038 1.033 1.030 1.027
1.545 1.269 1.171 1.123 1.096 1.080 1.069 1.062 1.055
1.697 1.368 1.246 1.185 1.148 1.125 1.109 1.097
1.866 1.482 1.334 1.258 1.211 1.180 1.158
2.055 1.610 1.435 1.342 1.285 1.245
2.262 1.752 1.548 1.437 1.367
2.488 1.908 1.672 1.542
2.730 2.075 1.805
2.986 2.251
3.252
ANALYTICAL CHEMISTRY, VOL. 42, NO. 2, FEBRUARY 1970
153
Table 11. Comparison of the Relative Standard Deviations Nand S, for a Number of Values of R, for n = entier (1.274 W)
R W" 3 6 9
12 15 Peak
a
0 0.1 0.995 0.904 0.913 1.225 1.436 0.936 0.960 1.623 0.985 1.791 half-width in channels.
1 1.230 1.855 2.342 2.748 3.104
10 1.346 2.123 2.711 3.198 3.622
50 1.360 2.155 2.755 3.252 3.684
Table 111. Comparison of the Relative Standard Deviation S,(PVS,~I~/S,) for a Number of Values of the Peak Half-Width, W, for n = entier (1.274 W)
R 0 0.1 1.42 3 1.31 1.40 6 1.28 9 1.26 1.39 12 1.25 1.38 15 1.24 1.37 Peak half-width in channels.
W@
(I
Channel number, n Figure 3. Relationship between the relative standard deviation S, and n for W = 15 channels and R = 0, 1, 10, and 50
1 2.27 2.26 2.25 2.25 2.24
10
6.02 6.00 5.99 5.98 6.97
50 13.20 13.18 13.12 13.04 12.98
deviation S n k , and for what values of R and W does the modified method provide better precision than Covell's method. The second and third expressions in Equation 12 give the dependence of the relative standard deviations S , and N on n-Le., the number of channels used to calculate 2n 1.
+
RESULTS AND DISCUSSION
Comparison of the Relative Standard Deviation SnXand S,. The relative standard deviation S n k ( 6 n k ) for a given TAPi.e., fixed values of R and W-depends on n and IC. For n equals a constant, the value 6,k increases with k, reaching its maximum for k = n. Strictly speaking, the 6,k has a minimum for k = 2 or 3, but the minimum is insignificant and appears only for small values of n. In Table I the values 6 , k / 6 , for R = 50 and W = 12 channels are presented. From the practical point of view we may write
6, _
50
400
150
Figure 5. Gamma-ray spectrum of thermal neutron-irradiated graphite in conditions adapted for europium determination 2.078;
Irradiation time, 2 hours; cooling time, 3 hours; scintillation counter with Na I / l l crystal, 3 inches X 3 inches
l5
( x i - %)2Wi, where x = N (or S,) and W = 14 i - 1 1!vN (or l / V d b Calculated by Equations 5 and 6 according to measurement 2.
x2/14
I
Channel number
0.247 0.021: 0.335
,
" t
0.232 1.667
=
1s3
k'*5 T
1713
h489
Theoretical relative standard deviationb
i
$87
Modified method Sn
45 1
0.513 1.905
1 -
10 channels:
A231
Relative standard deviation x2/14"
a
=
'O0@
.
and W 4, then the precision S, is at least twice as better as the precision for N . For R = 50 and W = 15 channels, the precision S , is approximately 3.7 times better than the precision for N. Only in an ideal case, for a pure peak (R = 0) or when R is very small, does Covell's formula afford better precision for the evaluation of TAP area than Equation 4. Influence of Peak Half-Width on 6,. Peak half-width, W, expressed in channels depends, for a given crystal and peak, on the amplification of a spectrometer. It was of interest, whether the precision S, (or the relative standard deviation, 6,) depends on the value W. The answer to this question is given in Table 111. It can be seen that for a given R,the value 6, only slightly depends on W. The changes observed result, rather, from the fact that n = entier (1.274 W) and for small values of W the difference 1.274 W - n is, in per cent, larger than for larger values of W.
Experimental Examples. A sample with complex gammaray spectrum has been chosen as an example. For the determination of traces of europium in graphite, lszEu peak at 344 keV was used because 15*Smand le7Wpeaks interfere with the two highest lb2Eu peaks at 40 keV and 122 keV (Figure 5). The small l5*Eu peak at 344 keV appears on the high Compton continuum of 24Naand la7W. It is then the case where counting statistics are very poor. The ratio of the height of baseline to lS2Eu peak height (344 keV), R, equals 5.5. From the theoretical conclusion derived above, we can foresee that in this case the modified method should give a precision at least two times better than Covell's method. Fifteen measurements, under the same conditions, of the graphite sample were carried out and the results calculated both by Covell's method and by the modified method (Table IV). The relative standard deviations for N and S, were 51.3% and 2 3 . 2 x respectively. The calculated values of N and S , have no additional experimental errors because the value of x2/14 lies within the confidence interval.
RECEIVED for review July 1, 1969. Accepted October 20, 1969.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 2, FEBRUARY 1970
155
