Optimal Allocation of Nuclear Detector’s Time for Radioactive Samples
M. S. ALJOHANI
Department of Nuclear Engineering
King Abdulaziz University
P.O. Box 80204, Jeddah, 21589
SAUDI ARABIA
Abstract: - Accuracy in measuring radioactivity of a sample is directly proportional to the time allocated for that sample. With scarcity of time, laboratories are faced with huge number of samples to be analyzed in a limited span of time. Such a situation was encountered during Chernobyl accident when many nuclear laboratories were flooded by foodstuffs and other samples for radioactivity analysis. This paper offers a methodology that arrives at the optimal allocation of nuclear detector’s time for a given number of samples, under time constraints. The methodology is based on minimizing the sum of associated standard deviations of the net counting rate of samples. This is done with the assumption that the background radiation is the same for all samples. For validation, an analytical solution was devised with the background radiation assumed negligible. Results found using this methodology were compared to those found analytically; both produced identical results.
Keywords: Minimization of standard deviation, Optimal detection time, Counting statistic.
1 Introduction
Laboratories are sometimes overwhelmed with large number of radioactive samples coming for measurement and are constrained by time limits. Such a situation was encountered during Chernobyl accident when many nuclear laboratories were flooded by foodstuffs and other samples to be analyzed [1-4]. This time is usually dictated by the supplier of these samples and/or by the laboratory conditions. Since radioactivity usually differs from one sample to another, the measurement time needed for each sample has to be different. The challenge becomes in how to optimize the allocation of measurement time among these samples.
The measurement of nuclear radiation involves a phenomenon that is stochastic in nature. The longer the time allocated for measurement the higher the accuracy of results. However, with scarcity of time one cannot measure indefinitely. Achieving the optimal allocation of time to these samples can be obtained by minimizing the total associated standard deviation of all samples.
In this paper, (n) samples are used; each sample has a different radioactivity level. The objective is to minimize the sum of associated standard deviation of samples. It is apparent that minimizing the associated standard deviation would result in a nonlinear objective function,
thus, requiring a nonlinear programming algorithm to solve it.
The solution found using the proposed nonlinear model was verified by analytically solving the system of equations assuming the background radiation is negligible. This verification was used in two examples were introduced. The results found in the examples for the verification stage are identical to results found using the proposed nonlinear model.
2 Definition of Variables
= Counts due to both radioactive sample and background.
= Counts due to background only while testing sample .
= Counting rate due to the radioactive sample without background.
= Counting rate due to background while testing sample .
= Measurement time of radioactive sample with background.
= Measurement time for background only while testing sample .
= Total time given to test all (n) samples.
= Associated standard deviation of net counting rate for sample .
= The relative error for sample .
3 Theory
Assuming that we have only one radioactive sample, the principle of error propagation is applied to minimize statistical uncertainty. Let’s consider the measurement of the net counting rate from a long-lived radioactive sample in the presence of a more or less constant background. The net counting rate due to the radioactive sample excluding background [5] is:
= - (1)
When error propagation technique is applied, one gets:
= + (2)
Or
= (3)
It is known that and . Substituting into equation (3) yields:
= (4)
With + = , and equation (4) can be expressed as:
= (5)
Assuming that (n) samples are available for measurement, the total associated standard deviation for all samples is:
(6)
Substituting (5) into (6) one gets the objective function to be minimized, that is:
Minimize
= ++ ……+ (7)
Laboratory environment is usually unchanged while working on samples, implying that counting rate due to background can be assumed the same for all samples. Therefore, =is measured only once and equation (7) becomes:
Minimize
= + + ……+ (8)
Equation (8) is the objective function that needs to be minimized with the following constraint:
(9)
One might require a specific relative error for some sample when there are drastic differences in rates. In this case, an additional constraint should be added for each requirement.
where
(10)
Caution must be taken when choosing a value for so as not to end up in a region of infeasibility.
If however, equal relative error is required for all samples, then constraints involving all samples should be added; in which case, there is no freedom left to allow for optimizing the objective function. In other words, these equality constraints will transform the problem to a deterministic one involving (n) unknowns with (n) equations that can be solved analytically. For example if we require that all samples to have equal relative error then the following set of equalities should be used.
(11)
The above set of equations (11) would need equation (9) to close the system resulting in (n) unknowns with (n) equations that can be solved analytically. This system has made the objective function obsolete (there is no room left for the objective function to vary and be optimized).
4 Validation
The aim of this section is to come up with a method to validate the numerical model. The method chosen solves the problem analytically under the assumption that the background radiation in equation (8) is negligible (i.e. =0), and then becomes:
(12)
Differentiating in equation (12) with respect to , then equating the derivatives to zero, and with some algebraic manipulation one gets:
= , = , … , = (13)
This relates measurement times to counting rates when the background radiation is negligible (i.e. =0). The set of equations (13) makes (n -1) equations and (n) unknowns. One more equation is needed to close the system; the equation that fulfills this purpose is the constraint equation (9). The system of equations in (13) plus that of equation (9) can now be solved analytically and results can be compared with the results found using the proposed numerical nonlinear model.
Without a loss of generality, equations (13) above can be rewritten as:
= (for =1, 2, 3…) (14)
Substituting for in the constraint equation (9) yields:
= 1 +++ …+ (15)
Since is the only unknown variable in equation (15) one can solve for . The values for all (=2, 3...) can then be found by plugging in the value of in equations (14). To show the above steps, two examples will be presented. The first example will be solved using the proposed nonlinear model. For verification purposes, a second example is solved using both, the proposed nonlinear model and the analytical method. The background radiation in the second example is assumed negligible (i.e. =0) to facilitate for verification.
5 Examples
The examples are presented below with their corresponding solutions:
Example 1:
The total time allotted for testing the radioactivity of 10 samples is = 1000 minutes, given that the counting rate due to only background is = 50 counts / minutes. The counting rates in counts per minute due to both, sample and background are given below:
100 / 200 / 150 / 120 / 180220 / 300 / 250 / 290 / 230
The above example was solved using the nonlinear model presented in this paper. The minimum value for the objective function is = 16.2737 and the optimal allocations of the 1000 minutes allotted to the 10 samples are:
67.21 / 86.74 / 78.11 / 71.94 / 83.49 / 89.78100.34 / 94.00 / 99.14 / 91.23 / 138.01
Example 2:
This example was solved using both the proposed nonlinear model and validated using the analytical method. The total time allotted for testing 6 samples is = 3000 minutes. Counting rates per minute are given below; background is assumed negligible (i.e. = 0):
220 / 520 / 1050 / 622 / 117 / 876Using the proposed nonlinear model, the minimum value for the objective function is = 5.92751; the optimal allocation of the 3000 minutes allotted to 6 samples is:
383.3905 / 510.6851 / 645.5077542.1480 / 310.6196 / 607.6491
The same example was solved using the analytical method, the minimum value for the objective function is = 5.92275; and the optimal allocations of the 3000 minutes allotted to 6 samples are:
383.3896 / 510.7016 / 645.4994542.1210 / 310.6190 / 607.6695
6 Discussion
It should be noted that the objective function in equation (8) along with the constraint equation (9) constitute a nonlinear system; both are functions of ,. Fortunately, the objective function is a monotonically decreasing function of , and the constraint equation (9) is purely linear. Therefore, the solution found is a global minimum. This is illustrated by comparing the results found from the nonlinear system with those found using the analytical one. The great compatibility between the two results exemplifies the effectiveness of the methodology presented in this paper.
References
[1] Abdul-Majid S., Abulfaraj W. H., Al-Johani M. S., Mamoon A., Abdul-fattah A. F. and Abubakar K. M., Radiation Monitoring of Imported Food to Saudi Arabia after Chernobyl Accident, Radiation Protection Practice. Vol. 2, Seventh International Congress of the International radiation Protection Association, Sydney, Australia, 1988, p.1090.
[2] Mamoon A., Abdul-fattah A. F., Abulfaraj W. H., Abdul-Majid S., Al-Johani M. S., and Abubakar K. M., Monitoring of Radioactivity in Imported Foodstuffs Experience Gained and Recommendations,. Radiation Protection Practice. Vol. 2, Seventh International Congress of the International radiation Protection Association, Sydney, Australia, 1988, p.1094.
[3] Abdul-fattah A. F., Abulfaraj W. H., Abdul-Majid S., Food Analysis and Radioactivity in Saudi Arabia, Final Report No. 96-407, Faculty of Engineering, King AbdulAziz University, Jeddah , Saudi Arabia, 1988.
[4] Abulfaraj W. H., Abdul-Majid S. and Abdul-Fattah A. F., Radiation Monitoring of Imported Food to Saudi Arabia after Chernobyl, Transactions of the American Nuclear Society 54, 39 1987.
[5] Glen F. Knoll, Radiation Detection and Measurement, John Wiley & Sons Inc. 1979.