Problems with the Uncertainty Budget of Half-Life Measurements

Chapter 20

Problems with the Uncertainty Budget of Half-Life Measurements Downloaded by PENNSYLVANIA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: November 16, 2006 | doi: 10.1021/bk-2007-0945.ch020

S. Pommé European Commission, Joint Research Centre, Institute for Reference Materials and Measurements, Retieseweg 111, B-2440 Geel, Belgium

The apparent tendency to underestimate the uncertainty of experimentally determined half-life values of radionuclides is discussed. It is argued that the uncertainty derived from a least-squares analysis of a decay curve is prone to error, as it does not sufficiently account for systematic deviations and medium frequency instabilities. As it is quite common for a series of activity measurement results to be autocorrelated, the prerequisite of randomness of data for common statistical tests to apply is not fulfilled. In this work, an attempt is made to provide an alternative data analysis method that leads to a complete and realistic uncertainty budget. A procedure is presented in which experimental uncertainties are subdivided into categories according to the rate at which they occur; i.e. at a low, medium or high frequency. A l l uncertainty components should be reported for traceability and evaluation purposes.

282

© 2007 American Chemical Society

In Applied Modeling and Computations in Nuclear Science; Semkow, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2006.

Downloaded by PENNSYLVANIA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: November 16, 2006 | doi: 10.1021/bk-2007-0945.ch020

283 Nuclear data evaluators are frequently confronted with the problem of deriving a recommended value and an associated uncertainty from a discrepant set of data [see references in (/)]. This issue is particularly apparent in the case of half-lives of radionuclides. A recent evaluation shows that, for the majority of the radionuclides, the spread of experimentally determined half-life values is larger than expected from the claimed accuracies (2). Lack of reliable data leads to recommended values based on a few discrepant data, requiring subjective judgement of the evaluator to identify possible outliers and to adjust unbalanced weighting among the accepted data. The situation is often aggravated by experimenters providing insufficient detail on how the half-life and its uncertainty were determined. The latter is clearly underestimated in many cases. Whereas in recent years more attention is being paid to traceability of results and comprehensiveness of the uncertainty budget, there is no common procedure on how to achieve realistic uncertainties, or on a concise but complete reporting style. One can identify two major ways to determine a half-life: by an absolute activity measurement of a known amount of radioactive material or by following the decay of a source. In this work we focus on the latter method, which is used for a major part of the radionuclides, excepting extremely long-lived ones. Besides experimental factors, such as the influence of impurities, background signals, detector stability, dead time, statistical fluctuations, etc. also the data analysis is considered as a source of error. In this work it is argued that the uncertainty obtained by least-squares fitting of an exponential to the data gives a false impression of accuracy in most cases. Presumably this is one of the reasons why claimed uncertainties on half-lives are often unrealistic. A n attempt is made to demonstrate the mechanism that hides certain types of instability and a method to take them into account in the uncertainty budget.

Basic Uncertainty Equations Two Activity Measurements Consider the basic case in which the radioactive decay of a source is measured twice under similar conditions, yielding an activity A\ and A in the first and second measurement respectively. The duration of the measurements is relatively short and their reference times differ by an amount T. A purely exponential decay curve is assumed, hence the half-life follows from the activity ratio R = A /A via In2 2

X

2

m

r

m

1 / 2

=r—.

1 / 2

\nR The corresponding relative uncertainty is then


(i)

284 σ(Τ )_ χ/1

Γ

1 / 2

1 σ(Λ) In Α

1 σ(Λ)

=

*

Λ

1 |g f4J Λ7·)( ^

2

σ

2

Α

2

^ '

where the relative uncertainty of R is the (squared) sum of the relative uncertainties of the activities A and A . Assuming the latter to be constant, the uncertainty on the half-life is inversely proportional with time (Eq. 2). Consequently, in order to reduce the uncertainty by a factor of two, one needs to double the decay time. This entails a linear relationship on a log-log scale, demonstrating that less and less is gained by continuing this experiment at great length. Eventually, more can be achieved by reducing the random and systematic uncertainties of the activity measurements. x


2

=

2

Additional Activity Measurements In practice, multiple measurements are performed throughout the campaign, spanning a period Τ between the first and last. Usually, an exponential curve is fitted to the experimental data, yielding a best estimate of two unknown parameters: the activity Ao at reference time and the half-life T . The goodness of the fit is often used as a criterion for the uncertainty. A possible procedure is the following (3): 1) calculate the χ -value, ensuring that it indeed assumes its expectation value (i.e., the number of degrees of freedom); 2) adjust the half-life so that χ increases by a value of one; 3) adopt as the standard deviation the square root of the amount by which the half-life was varied. Unfortunately, experience has shown that the resulting uncertainty may be unrealistic (4). m

1

2,

This procedure is only valid i f the assumed exponential shape is rigorously applicable to the data, the deviations being of a stochastic nature and not prone to systematic deviations or patterns. A n alternative way of uncertainty assessment will be derived now from some simplifying assumptions, starting from the basic Eq. 2. At this point we make a distinction between the different underlying sources of variation that make a data point deviate from the ideal decay curve. For convenience, they will be subdivided in low, medium and high frequency deviations. • High frequency deviations: these sources of uncertainties occur at a rate that is higher than or comparable to the measurement time of one data point. A typical example would be counting statistics: Poisson processes are characterised by an exponential distribution of the interval times between successive events. By extending the measurement, one improves the statistical uncertainty on the actual count rate. Also ultra-high frequency instabilities, such as electronic noise, are included in this category. They are partly cancelled, though, by the 'integrating' effect of the duration of the


285


measurement. Normal statistical treatments (like LSQ fitting) apply because of the preservation of randomness of the data. Medium frequency deviations: these instabilities show up (at least partly) as a trend in the residuals. They include the so-called 'seasonal effects' (e.g., on a weekly basis or changing with the seasons of the year). Their effect is greatly underestimated i f one simply relies on the goodness of the fit to the data. Moreover, a fit tends to minimise the residuals and partly covers up the true medium frequency effects. Typical examples include the interference by radioactive impurities, the reproducibility of the source-detector geometry and noticeable changes in the detector efficiency (e.g., through temperature, pressure, humidity, electronic flaws). Low frequency deviations: these sources of uncertainties occur at a rate that is lower than or comparable to the duration of the whole measurement campaign. They remain practically invisible in the residuals, as the fit will compensate for this trend, hence erase it erroneously. Common problems in this range are systematic errors in the background subtraction and counter dead-time correction, long-term drift of the counting efficiency and source degradation (e.g., oxidation of the source, mechanical wear, precipitation of active material in a solution).

Including Instabilities in the Uncertainty Budget Sensitivity to Instabilities Figure 1 shows hypothetical residuals of a fitted decay curve to data showing high, medium and low frequency instabilities, respectively. The black dots represent the deviations of the data from the 'true' decay curve, the open circles show the deviations of the data from the fitted decay curve and the solid curve is the difference between the fitted and true decay curve. The experimentalist that performs the L S Q fit perceives only the hollow points. Considering that a slope equal to zero corresponds to the true half-life, one sees that the fitted slope shows in principle no bias with random deviations (high frequency), contrary to the medium and low frequency deviations. Whereas the experimentalist may still observe some 'seasonal' variations (medium frequency), he is unaware whether the LSQ adjustment of the fit has covered up major long-term effects (low frequency), resulting in a wrong half-life value. As the residuals give insufficient indications of these effects, he has to evaluate an exhaustive list of associated uncertainty components in the uncertainty budget.


286


τ

τ

«

1

·

1

1

1

•

«

Γ

Γ

time (arb. unit)

Figure 1. True (black dots) and perceived (hollow circles) residuals from a fit of a decay curve through hypothetical data affected by high (top), medium (middle) and low (bottom) frequency instabilities. Systematic deviations are not fully observed by the experimentalist, as the LSQ fit tends to minimize them.

Simplified Solution for High Frequency Deviations Some simplifications are assumed for the sake of argument: the activity of a source, A (i = Ι,.,,Λ), is measured η times with an equal relative uncertainty, at regular intervals between time = 0 to T. The residuals of the fitted decay curve show only stochastic deviations due to high frequency deviations (e.g., Poisson counting statistics). These are directly visible, hence can be accounted for without systematic bias. The half-life of the decay curve, or alternatively the slope of a 'trend line' through the residuals, is primarily determined by the outer data points, respectively at the start and the end of the measurement campaign. If one doubles the number of measurements of A and A in Eq. 2, the uncertainty on t

x

2


287 7Ί/2 is reduced by a factor ofV2 . Data points taken half-way through the campaign (i.e., at t = 772), do not influence the slope of the fitted curve. They only make the curve shift as a whole in the vertical direction. By approximation, one can assume that the relative impact of a measurement on the fitted half-life is proportional to the time difference with the middle, i.e., ~ | / - 772 | /(772). Under the aforementioned conditions, and data points being spread somewhat evenly in time over the period T, the uncertainty on T is well approximated by: f - . Λ-1/2 σ(Α) 1^-11 (3) Downloaded by PENNSYLVANIA STATE UNIV on August 6, 2012 | http://pubs.acs.org Publication Date: November 16, 2006 | doi: 10.1021/bk-2007-0945.ch020

m

~w

Ά/2

Σ

More specifically, i f the data are spread equidistantly in time, the reduction factor (for η > 2) is approximately equal to 2i

n+\

n-l

(4)

resulting in a convenient uncertainty for the high-frequency component u u v v i l a n u ^ formula ι υ ι m u i u IVJ °(T )j2

Γ Τ

U2

Ά/2

~\λΤ\η

+• ]I l

(5) A

'

in which one should assign a value of 3 to η in the case that η < 3. Using Eq. 5, an evaluator as well as an experimenter can make a realistic estimate of the contribution of counting statistics to the half-life if three variables are available: the number of measurements, the duration of the campaign, and a measure of the uncertainty for a typical activity measurement.

Identification and Treatment of Medium Frequency Deviations Medium frequency deviations may show up as systematic trends in the residuals of the fitted decay curve. They are partly obscured by data scatter due to high frequency instabilities and by the tendency of the fit to minimize the deviations. Identification and quantification of medium frequency effects require a sufficient amount of observations, at comparably small time intervals and preferably with low short term uncertainties, as e.g., from the counting statistics. A n example is shown in Fig. 2, depicting residuals from a half-life determination of Fe (4). The uncertainty bars pertain to counting statistics and background correction only. One clearly observes additional uncertainty components attributed to source repositioning and medium-term detection instabilities. Grouped data refer to holiday periods in which the source was not repositioned. Clearly such sources of variation create a non-random spread of data, hence they invalidate the uncertainty on the half-life provided by the L S Q 55


288


fit. In the present example, the latter underestimates the uncertainty by about a factor of four. In the literature, one finds cases where medium frequency instabilities are clearly present, yet not properly taken into account.

decay time T(d)

Figure 2. Residuals from the fit of an exponential decay curve to the measured activity of a Fe source, relative to the overall standard deviation (4). 55

If the effects are less obvious than in Fig. 2, one can revert to an autocorrelation plot to check the randomness of the data. Such a plot is presented in Fig. 3 for the Fe data (4) and also for a set of Z n measurement data (J). Whereas the Z n data show no significant autocorrelation, this is obviously not the case with the Fe data. The impact of medium frequency instabilities on the half-life uncertainty is difficult to quantify. There's the possibility of applying Eqs. 3 or 5, yet this requires the choice of a suitable value η for the number of occurrences of the considered effects; e.g., when considering the contribution of the geometrical reproducibility one may set η equal to the number of times the source was repositioned. For a conservative approach, η = 1 is a safe assumption. 55

65

65

55

Taking into Account Low Frequency Deviations Residuals offer practically no help with identifying systematic errors and slowly varying measurement conditions. It is left to the experimenter's expertise to produce an exhaustive list of potential problems, such as systematic errors involved with background subtraction (aside from its random component), changing system dead time conditions, long-term detector instability, source


289


degradation, structural changes to the set-up, etc. Again, one may apply Eqs. 3 and 5 with η = 1. The independent contributions are summed quadratically.

Figure 3. Autocorrelation plot of the residuals of a decay curve fit to and Zn (5) activity measurement results, respectively.

Fe (4)

65

As an example, consider the error produced by applying a slightly incorrect correction for dead time of the extending type. The ratio R of the activities A and A at a time t and t respectively, is calculated from measured count rates r and r and a correction factor involving the characteristic dead time r. x

2

x

2i

x

2

R = £L = IL (Pi-P2* Pi r e

(6)

9

2

where p and p represent the true count rates (excluding count loss by dead time). Assuming that implicit use is made of an extending dead time Γ + Δ , in which Δ is the absolute uncertainty on r, one finds a relative uncertainty on the activity ratio of x

l

^ )

=

1

_ -rp -P M e

1

2

e f p i

_

p 2 M

l|

=

-In V

l n

\JL\

(7)

1

in which T and T represent the system real time and live time, respectively, and the ratio of both corresponds to the dead time correction factor used in the respective measurements. A similar analysis for non-extending dead time yields R

L

~TR

\T ~ R

A

_Δ

1-ρ Δ

τ

\-

R

P

L

2

JL]

2

1 Δ

τ

'

ΛΪ

2

-1 J


(8)

290 The propagation of these uncertainty components follows directly from insertion into Eq. 2.


An Aggregate of Uncertainty Components A hypothetical uncertainty budget for the activity ratio R is presented in Fig. 4. In the example, the high and medium frequency components were assigned a value of 5 % and 1 %(n = 1), respectively, while the low frequency uncertainty was assumed to grow proportionally with time, by an amount of 2 % per decay constant. The graph shows that the main source of uncertainty moves from high to medium and then to low frequency instabilities. The corresponding half-life uncertainty generally decreases with time, because of the propagation factor (λΤ)~ (Eq. 2). On the other hand, so does usually our grip on its size and origin. χ

1

•I -1 •I " 1 £

g

.

1

1

hypothetical case

high frequency medium frequency low frequency independent sum

-

4% 3%

. — r ^ -

τ ΟΟ

'

,

1

0.5

1

1

1

1

1

1

1

10

11

1

1.5

Γ 2.0

Τ/τ

Figure 4. A hypothetical example of the uncertainty budget of an activity ratio as a function of time, showing the contribution of high, medium and low frequency instabilities.

An Alternative to Fitting If the data points follow a single exponential decay curve, a fit is not required to determine the half-life. A n alternative approach is to compare every observation with all previous ones and calculate the corresponding half-life value via Eq. 1. One obtains a distribution, like in Fig. 5, which is a superposition of Cauchy distributions with common maximum but different widths. The top (or


291


weighted average value) corresponds to the most probable half-life value, in good agreement with the result from the fit.

half-life (days)

Figure 5. Distribution of half life values obtained via Eq. 1 using a combination of all activity measurements of a Fe source (4). 5 5

As data are being collected, one can follow the evolution of the calculated half-life. One should make sure that the deviations between final and intermediate results do not exceed the maximum boundaries allowed by the uncertainty budget. As an additional test, one can e.g., randomize or invert the order of analysis of the data points. Fig. 6 shows the evolution of the intermediate half-life values obtained for Fe. It includes one and two standard deviation boundaries (Eq. 5) for the high and medium frequencies combined, assuming the corresponding relative uncertainty on the activity to be 0.02 % (n = number of data analyzed) and 0.018 % (n = 3), respectively. Systematic errors and long term effects are not included, as they remain invisible. 55

Conclusions Whereas two activity measurements of a radioactive source may in principle suffice to determine the half-life of the decaying radionuclide, it takes considerably more effort to assess the uncertainty on the resulting value. Experience shows that the uncertainty derived merely from a least-squares fit of an exponential decay curve to a sequential series of activity measurements is often suspect. This is partly attributed to observations being correlated over time,


292


as the stability of the measurement process is to some extent disturbed by effects of various origin and duration. With respect to that, the analyst has to collect a sufficient amount of observations and check for their randomness, the latter being a prerequisite for the validity of statistical tests. Moreover, as long-term effects are mostly obscured by a fit, they have to be identified and scrutinized by other means. In this work, a simplified formula has been suggested to take the various effects and their rate of occurrence into account in the uncertainty budget.

T(d) 55

Figure 6. Intermediate results for the half life of Fe with time, analyzed in a forward and in a reverse time order. The uncertainty limits (k = 1 and 2) correspond to the independent sum of a 0.02% and 0.018 % high and medium frequency component, respectively.

References 1. 2. 3. 4. 5.

MacMahon, D.; Pearce, Α.; Harris, P. Appl. Radiat. Isot. 2004, 60, 275-281. Woods, M . J.; Collins, S. M . Appl. Radiat. Isot. 2004, 60, 257-262. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, N Y , 1969. Van Ammel, R.; Pommé, S.; Sibbens, G. Appl. Radiat. Isot., in press. Van Ammel, R.; Pommé, S.; Sibbens, G . Appl. Radiat. Isot. 2004, 60, 337339.


Problems with the Uncertainty Budget of Half-Life Measurements

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