served when the ionic strength was kept almost constant by saturating the solutions with tetraethylammonium bromide. Hydrolysis Rates. Results are shown in Table 111. These need no interpretation as regards selection of a suitable medium for quantitative polarographic determination of phthalic anhydride. The hydrolysis reaction in a given medium is of first order, overall with respect to the anhydride, since one of the reactants, H20 is in large excess and its concentration remains practically constant. Calculated half lives are listed in Table I11 and are in close agreement with the observed ones. Determination of Phthalic Anhydride in Synthetic Samples. Solutions of the anhydride, 50 to 100 ppm, in Solvent A,
0.12M HCl, were prepared from the stock solution. To these solutions was added a mixture of benzoic and phthalic acid in 500-fold excess relative to anhydride. The solutions were polarographed, and from the limiting currents of the waves, 0.95, and by the standard addition method, the anhydride content of the solutions was obtained with a relative standard deviation of + 5 Z. Neutral solvents such as Solvent B, have not been used; they also may prove to be suitable, provided pH effects are under control. N
RECEIVED for review November, 14 1969. Accepted March 16, 1970.
Detection Limits for Gamma-Ray Spectral Analysis V . C . Rogers Departments of Ph ysics and Chemical Engineering, Brigham Young University, Provo, Utah 84601 GENERAL EXPRESSIONS FOR the limits of detection and quantitative determination, as presented by Currie ( I ) , are given by the following equations : Lc = kauo L D = Lc f k p D
(1)
(2)
LQ = kQUg (3) where Lc, the critical level, is the number of counts an observed signal must not exceed in order to yield the decision “not detected” with a probability of 1 - a. The detection limit, L D ,is the true mean of a signal whose distribution of possible outcomes intersects LC such that a given fraction, 1 - p, of them will be greater than Lc. The determination limit, LQ,is defined as the true mean of a signal whose relative standard deviation is l/kQ. The quantities UO, U D , and U Q are the standard deviations associated with zero, LD, and LQ net counts, respectively, and k, and ka are abscissas of the standardized normal distribution corresponding to the probability levels, 1 - a and 1 - 0. Currie ( I ) has obtained expressions for Lc, LD,and LQ when the variance of the net signal is given in the following form : o ~ ( S= ) S
+ B + B/m
(4)
where S = true mean of net signal
B m
= =
true mean of background number of observations of background
A
+ ‘id (a, + a_,> A + ( n 2 - ‘/d(an + a_,) ai - (n
= t=
-n
u Z ( A )=
(1) L. A. Currie, ANAL.CHEM.,40, 586 (1968). (2) D. F. Covell, ibid., 31, 1785 (1959). (3) S. Sterlinski, ibid., 40, 1995 (1968).
A ai n
net area of gamma-ray photopeak counts in ith channel = number of channels on either side of peak channel (channel 0) 1 = total number of channels in peak = =
2n
+
The end channels, -n and + n , are usually chosen to be the points of inflection (2, 3) or, particularly in the case of spectra obtained with Ge(Li) detector systems, they are chosen to be at the continuum background level ( 4 , 5 ) . Equations 5 and 6 can easily be extended to the case where a different number of channels are used on each side of the peak channel. The result is N
A =
i=l
and .*(A)
=
A
+ ahr)
a, - ’/?N(al
+ ‘ I N N - 2)(a1 + aN)
(5)
(6)
(7)
(8)
where channels 1 and N are the end channels of the peak and N , the number of channels in the peak, is three or greater. It should be noted that when A is zero, Equation 8 is the expression for UO. Substituting Equation 8 into Equations 1 to 3 and setting k, = kp = k , yields the following expressions for Lc, L D ,and LQ.
+a ~ ) ] ” ~ k 2 + k[N(N - 2)(a1 +
LC = l/Zk[N(N - 2)(a1
However, in many nuclear activation analysis methods, the standard deviation of the net signal is not explicitly expressed by Equation 4. For example, the net area and standard deviation for gamma-ray photopeak analysis using the summing-channels method is given by (2, 3): n
where,
LD =
LQ = 2‘ 2 ( 1
QN)]~’*
+ aN)]‘“> + [1 + N ( N - 2)(a1 kQ2
(9)
( 1 0) (11)
It is observed that Equation 10 can be written as: LO
k2
+ 2Lc
(12)
a relationship which is also obtained by Currie for cases in which Equation 4 is applicable. (4) 0. U. Anders, ANAL.CHEM., 41, 428 (1969). ( 5 ) J. Hamawi and N. C. Rasmussen, Mass. Inst. of Tech. Report MITNE-107, Cambridge, Mass., Sept. 1969. ANALYTICAL CHEMISTRY, VOL. 42, NO. 7, JUNE 1970
807
Table I. Working Expressions for Lc, LDand LQ. LC
LD
m = l 2) (ai UN)]"'
+
0.823 [N(N 1.163 (NB)l/' 2.706 1.645 [N(N - 2) (ai 2.706 2.326 (NB)lI2
+
+
+ UN)]'"
(k,
ks = 1.645, kQ = 10) m = 4 0.411 [ ( N 6) ( N - 2) (ai ~N)I'" 0.582 [(N 6)B]1/2 2.706 0.823 [(N 6) ( N - 2) (UI UN)]~/' 2.706 1.163 [(N 6)B]Ii2 =
+ +
+
++
+
+ +
J-Q
In order to compare the results given in Equations 9 to 11 with the expressions resulting from the use of Equation 4, the total background under the peak is defined as
B = '/z(N - 2) (ai
+a3
Equations 9 through 11 become
(1 3)
The factor ( N - 2) is used in the definition rather than N because channels 1 and N do not contribute counts to the net area (5). Using Equation 13, the quantities Lc, L D , and LQ are given as follows:
and ,
r
(14) (15) (16)
The dependence of the detection limits upon the number of channels in the peak is observed from these expressions. If the calibration factor, keV per channel, is changed, but everything else remains the same including the total background counts B, then the limits of detection are increased because channels 1 and N contain fewer counts, and, hence, there is a larger uncertainty associated with the linear background. Statistical uncertainties associated with gamma-ray spectra are usually reduced with the use of spectrum-smoothing techniques. Hamawi and Rasmussen (5) have examined the effect of spectrum-smoothing upon the standard deviation of the peak area. They use a modified Gaussian function in determining the peak area and establishing the detection limits. In the summing-channels method, if the uncertainties associated with al and UN are greatly reduced, then the uncertainty in A is also reduced significantly. As a simple example, suppose the counts in channels 1 and N , at the continuum background, are obtained by averaging over m adjacent channels on each side of the photopeak. Then
dl = average background counts in channel 1 obtained from averaging over channels 2 - m to 1 d N = average background counts in channel N obtained from averaging over channels N to N m -1
+
808
ANALYTICAL CHEMISTRY, VOL. 42, NO. 7 , JUNE 1970
Caution should be exercised in using Equations 9 to 11 and 18 to 20. They are valid only if the background can be fit to a straight line, and if the energy dependence of the background is accounted for in obtaining 61 and 6 ~ .For these reasons the number of channels used in the averaging procedure should not contain parts of other photopeaks, and should not be located on prominent Compton edges. Clearly, nonlinear functions can also be used to fit the background (6). Expressions for Lc, Lo, and LQ, using CY = p = 0.05, kQ = 10 and m = 1 and 4, are listed in Table I. It is interesting to observe that in the limit m-+a (welldetermined background in channels 1 and N) the above expressions for Lo, L D , and Lo approach the same values as they do for the case using Equation 4 with m-m as listed in Reference 1. That is
LC
-+
L D+ k 2
k d B
+2
(21)
k 6
RECEIVED for review February 2, 1970. Accepted March 12, 1970. (6) P. Quittner, ANAL.CHEM., 41, 1504 (1969).
