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Derivation of an uncertainty propagation factor for half-life determinations
Journal Pre-proof Derivation of an uncertainty propagation factor for half-life determinations S. Pommé, T. De Hauwere PII:
S0969-8043(19)30816-4
DOI:
https://doi.org/10.1016/j.apradiso.2020.109046
Reference:
ARI 109046
To appear in:
Applied Radiation and Isotopes
Received Date: 30 July 2019 Revised Date:
7 December 2019
Accepted Date: 15 January 2020
Please cite this article as: Pommé, S., De Hauwere, T., Derivation of an uncertainty propagation factor for half-life determinations, Applied Radiation and Isotopes (2020), doi: https://doi.org/10.1016/ j.apradiso.2020.109046. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Derivation of an uncertainty propagation factor for half-life determinations
S. Pommé‡1, T. De Hauwere1,2
1
European Commission, Joint Research Centre (JRC), Geel, Belgium 2
TU Delft, Delft, The Netherlands
An analytical equation is derived for the uncertainty propagation factor for a half-life determination from a least-squares fit to equidistant activity measurements performed with identical relative uncertainties. The obtained formula applies to a purely random statistical uncertainty component. It is equivalent to the solution published by Parker in Nucl. Instr. Meth. A 286, 502. A more general equation for weighted least-squares fitting is derived and presented in a compact manner. It is used as a benchmark to verify the applicability of Parker's solution to non-equidistant data with unequal uncertainties.
‡
[email protected]; tel. +32 (0)14 571 289
1
Highlights • Derived uncertainty propagation factor for half-life from activity measurements. • Special formula for equidistant measurement data with equal relative uncertainty. • Derived compact general formula for weighted fit and used it as benchmark. • Special formula works well with mild deviations from conditions. Keywords: Half-life; Uncertainty; Propagation; Activity; Nuclear data
2
1 Introduction The exponential-decay law (Rutherford, 1900; Semkow, 2007) is the foundation of the common measurement system for radioactivity (Judge et al., 2014). It relies on the invariability of the nuclear decay constants – or, inversely, the corresponding half-lives – which appears to apply rigorously within the boundaries of metrological accuracy (Emery, 1972; Hahn et al., 1976; Norman et al., 1995; Pommé et al., 2016a, 2017a,b,c,d, 2018a,b; Pommé, 2019; Pommé and De Hauwere, in press). As a result, nuclear decay can be used as a clock (Dalrymple, 1994; Currie, 2004; Nir-El, 2004, Pommé, 2015; Pommé et al., 2014, 2016b). Half-life values of radionuclides are measured by various techniques, depending on their order of magnitude (Pommé, 2015). Half-lives in an intermediate range are measured by repeated activity measurements over periods of time, thus establishing a decay curve to which an exponential decay function is fitted. Generally, least-squares fitting routines are applied to determine the slope of the decay curve and the ensuing uncertainty is obtained from the covariance matrix (Bevington and Robinson, 2003). The latter procedure often gives rise to significant underestimation of the half-life uncertainty, since the model doesn't account for systematic deviations due to metrological imperfections (Pommé, 2007, 2016). Medium- and long-term instability and non-linearity in the experiment propagate with a larger factor into the half-life uncertainty than the purely random variations (Pommé et al., 2008; Pommé, 2015). In state-of-the-art half-life measurements, the short-term random variations – such as counting statistics – may represent only a small fraction of the total uncertainty budget. For example, in a
99m
Tc half-life measurement (Pommé et al.,
2018c), the uncertainty component due to random variability in more than 2000 measurements added very little to the total, since the attainable accuracy is limited by the linearity of the ionisation chamber, and possibly to an even larger extent by the chemical composition of the source influencing the probability for internal conversion of the 2.17 keV transition. On the other hand, the short-term reproducibility issues (including counting statistics for source and background) were the major uncertainty component for a
55
Fe half-
life measurement, in spite of the presence of long-term, systematic effects due to pulse pileup, background subtraction and gas pressure variations (Pommé et al., 2019). Correct propagation of uncertainty components is an important issue. An approximate formula was proposed by Pommé et al. (2008) to propagate random uncertainty components from n equally spaced activity measurements over a time period T to the fitted half-life T1/2:
3
σ (T1/2 ) T1/2
≈
2 λT
2 σ ( A) n +1 A
(1)
The equation can also be used to evaluate medium and low-frequency instabilities, if the parameter n receives a new meaning as the number of periods covered by a cyclic effect on the experiment. For low-frequency effects, the value n=1 is selected by default. A more rigorous equation for the propagation factor of random uncertainties was published by Parker (1990), as part of a procedure to measure the half-life of a source by gamma-ray spectrometry, relative to a reference radionuclide with a well-established, long half-life. For a series of n measurements of activity ratios of both nuclides, R, assuming a constant measurement uncertainty, σ(R), Parker derived the following uncertainty formula:
σT = 1/2
2 3 T1/2 ( n − 1) n σR 2 ln 2 T n ( n − 1)
(2)
On the basis of this equation, the following alternative to Eq. 1 was recommended specifically for high-frequency random variations in a review paper (Pommé, 2015)
σT
1/ 2
T1 / 2
≈
2 λT
3( n − 1) σ A n ( n + 1) A
(3)
No mathematical derivation was published to justify its validity. In this paper, it will be demonstrated how Eq. 3 is found starting from the uncertainty equations for linear leastsquares fitting and transforming them to the case of exponential decay. 2 Derivation of Equation 3 2.1
Linear least-squares fit
In principle, the best fit of an exponential function to a nuclear decay curve can be obtained by means of the maximum likelihood estimation method using the Poisson distribution to account for the probability for the observed number of decays. For sufficiently high amounts of counts, a Gaussian distribution is an adequate approximation and the maximum likelihood search is conveniently replaced by a least-squares fit (Kendall and Stuart, 1961). Whereas a non-linear fit can be easily performed to obtain the decay constant from the slope of the
4
exponential, it is advantageous to study the transform to a linear problem to derive the uncertainty propagation factor in Eq. 3. A variable y is considered which relates linearly with a variable x, through the free parameters a and b representing the intercept and the slope, respectively: y = a + bx
(4)
In the context of this paper, y represents the logarithm of the activity, y=lnA, and x corresponds with time, t. Consequently, the intercept refers to the initial activity, a=lnA0, the slope is related to the decay constant, b=–λ, and the focus of this work is on the uncertainty of the least-squares estimate of b. The uncertainty on the xi values is assumed to be negligible, σx=0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi, propagated by the slope b. In this paper, the 'spread' of the nominal xi and yi values, will be quantified by their 'variance', e.g. var( x ) = ∑ ( xi − x ) 2 / n and cov( x, y ) = ∑ ( xi − x )( yi − y ) / n , which should not be confound
with their measurement uncertainty (i.e. var( x) ≠ σ x2 ). Minimisation of the weighted sum of the squares of the deviations between measured and fitted values,
y − a − bxi χ = ∑ i σi i =1 n
2
2
(5)
through ∂χ / ∂a = 0 and ∂χ / ∂b = 0 yields analytical equations for the estimated values of 2
2
a and b (Kendall and Stuart, 1961; Bevington and Robinson, 2003); They can be reduced to ⌢ ⌢ a = y − bx and b = ∑ ( xi − x )( yi − y ) / ∑ ( xi − x ) 2 = cov( x, y ) / var( x ) . 2.2
Uncertainty of the slope
Ignoring systematic errors which would introduce correlations between uncertainties, the 2 2 2 uncertainty of the slope can be derived from σ b = ∑σ i (∂b / ∂yi ) . In the special case of
constant uncertainties, σi = σ y , this results in (Bevington and Robinson, 2003)
5
σ = 2 b
nσ y2 n n∑ x − ∑ xi i =1 i =1 n
2
2 i
=
σ y2 n
∑ (x i =1
=
i
(6)
−x ) 2
σ y2
(7)
var( x )
The Eq. 6 is valid independently of the statistical distribution of the y data. If a normal ⌢ distribution applies to y, than it also applies to the estimated intercept a and slope bˆ , since they are sums of normal random variables. Eq. 7 facilitates the interpretation: the variance of the slope decreases when the uncertainty on the y values is small and many data are taken over a wide range of x values, or alternatively when the uncertainty of the mean of the y values is small compared to the spread of the x data. If the standard uncertainty of the y data is not known a priori, it can be estimated from the spread of the residuals to the linear fit, through n
σ y2 ≈ s 2 =
∑ ( y −a − bx ) i =1
i
2
i
(8)
n−2
in which n-2 is the number of degrees of freedom of the linear fit. The 'variance' of xi can be calculated directly from the sum of all ( xi − x )2 terms divided by n. 2.3
Equidistant measurements
Applying an additional condition that the xi data are equidistant by an amount ∆x, such that xi=i∆x, one can simplify Eq. 6 further. For a set of n data, the total distance between the extremes, x1 and xn, is T = ∆x ( n − 1) and the mean value is x = ∆x (n + 1) / 2 . The term in the denominator of Eq. 6 equals n +1 ( xi −x ) = ( ∆x ) ∑ i − ∑ 2 i =1 i =1 n
n
2
2
2
6
2 n 2 n +1 = (∆x) ∑ i − n i =1 2 2
n(n2 −1) n(n + 1) = (∆x) =T2 12 12(n − 1) 2
(9)
in which use was made of the Faulhaber-Bernouilli formulas (Knuth, 1993): n
∑i = i =1
n(n + 1) and 2
n
∑i i =1
2
=
n(n + 1)(2n + 1) 6
(10)
Combining Eqs. 6 and 9, one obtains
σb =
2.4
σy T
2
3( n − 1) n( n + 1)
(11)
Transform to half-life
When transforming a non-linear fit of an exponential curve into a linear fit through y=lnA, the uncertainties of the measured quantities must by modified by
σy =
d ln A σ σA = A dA A
(12)
which relates the standard uncertainty of y to the relative uncertainty on the activity values A. The relative uncertainty of the slope equals the relative uncertainty on the decay constant (or the half-life), i.e.
σλ 2 = λ λT
σb / b = σλ / λ . Combining this with Eqs. 11 and 12, one obtains 3( n − 1) σ A n( n + 1) A
(13)
which is, according to the Gauss-Markov theorem (Wooldridge, 2013), the best linear unbiased estimator (BLUE) under the conditions of equidistant time between measurements and the same relative uncertainty in each activity value. This result is equivalent to the Eq. 2 proposed by Parker (1990).
7
3 Weighted least-squares fit 3.1
Uncertainty of the slope
Since the case of a constant uncertainty in y and fixed spacing in x is quite particular, it is of interest to consider the general case in which σi and ∆xi vary for each data point. The uncertainty on the xi values is assumed to be negligible, σx=0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi. Ignoring systematic errors which would introduce correlations between uncertainties, the uncertainty of the slope obtained from an inverse-variance weighted least-squares fit – i.e. minimising χ2 in Eq. 5 – results in (Bevington and Robinson, 2003) n
σ b2 =
1
∑σ i =1
2 i
n xi − ∑ ∑ 2 2∑ 2 i =1 σ i i =1 σ i i =1 σ i n
xi2
n
1
(14)
2
The interpretation suggested by Eq. 7 for constant σi can also be looked for in Eq. 14 for weighted least-squares fitting. A weighted mean of the xi and yi values and a weighted 'variance' of the xi data – xw = ∑ pi xi , yw = ∑ pi yi and var( x ) = ∑ pi ( xi − xw ) , respectively 2
– can be calculated using normalised inverse-variance weights pi = 1 / σ i2
∑ 1/ σ
2 j
. The
variance of the weighted mean, σ y2w , equals 1 / ∑ 1 / σ 2j . The Eq. 14 can be rearranged to obtain a similar result as in Eq. 7: n
σ b2 =
i =1
n
i =1
n
2 i
1
σ y2
=
w
xi − xw σ i2 i =1 n
∑
2 i
n xi − ∑ ∑ 2 2 i =1 σ i i =1 σ i
xi2
∑σ =
1
∑σ
1
2
n
1
∑σ i =1
2
n 1 ∑ 2 i =1 σ i
2
σ y2
w
var( x)
(15)
2 i
The variance on the slope of a weighted fit is proportional to the variance of the weighted mean of the yi values and inversely to the weighted spread of the xi data.
8
3.2
Transform to half-life
As derived from Eq. 12, when transforming a non-linear fit of an exponential curve into a linear fit through y=lnA, the standard uncertainties of the y values equal the relative uncertainties on the activity values, σ y = σ A / A . The weighting factors to be applied are
(
therefore pi = σ Ai / Ai
) ∑ (σ −2
Aj
)
/ Aj
−2
.
Combining this with Eq. 15, one obtains
σλ 1 = λ λ
n ∑ σ Ai / Ai i =1
(
∑ (σ n
i =1
=
1
λ
Ai
/ Ai
)
−2
−1/2
) ( t − t ) ∑ (σ −2
2
i
w
n
i =1
Ai
/ Ai
)
−2
σ Aw 1 var(t ) Aw
(16)
in which ti is the reference time of the measurement i expressed in the same time unit as 1/λ, t w = ∑ pi ti is the weighted mean of the reference times of the measurements, using the
weighting factors pi based on the relative uncertainties of the activity values, var(t ) = ∑ pi (ti − t w )
σ A / Aw = w
( ∑ (σ
Ai
2
/ Ai
represents
)
)
−2 −1/ 2
the
spread
of
the
reference
times
and
is the relative uncertainty of the weighted mean activity. In
Fig. 1, the meaning of the variables involved is depicted for a hypothetical example. Explicit calculations via Eq. 16 involve the following steps: (1) calculate the sum of the inverse square of the relative uncertainties on the activity values, (2) normalise these values individually by their sum to obtain the weighting factors pi, (3) calculate the uncertainty of the weighted mean activity as the inverse square root of the normalisation factor, (4) calculate the weighted mean time tw, and the 'variance' of t, and (5) apply Eq. 16.
4 Case studies 4.1
Few measurements
The uncertainty propagation formula for the half-life obtained from the ratio of two activity measurements with the same relative uncertainty is (Pommé, 2015)
9
σT
1/2
T1/ 2
=
2 σA λT A
(17)
The same result is obtained from the approximation in Eq. 1 for n=3 and from the BLUE in Eq. 3 for n=2 and n=3. The third point in the middle of the decay curve does not influence the slope, only the amplitude. For large n, the BLUE is proportional to
3 / n , whereas the
approximation in Eq. 1 is somewhat lower, ~ 2 / n . 4.2
Systematic error
The reduction of the propagation factor by n −1/ 2 (Eqs. 1-3) applies to stochastic effects which randomly vary with each measurement, not to systematic errors which affect subsequent data in a correlated manner. Non-linearity in detector output or changes in source integrity, geometrical and environmental conditions may occur at any phase of the project. Live-time corrections for pileup and dead time may introduce a systematic error at the start of the measurement campaign, not so much at the end. On the other hand, systematic errors in background subtraction may affect the measurements at the end of the campaign, while being negligible at the start. It is suggested to apply n=1 in Eq. 1 to propagate an average value of each systematic uncertainty component at the start and end of the campaign, which effectively leads to a correct propagation:
σλ 2 σA / A+0 1 σA = = . This effect cannot be λ λT 2 λT A
reproduced with Eq. 3. 4.3
Variable uncertainties
The condition of constant relative uncertainties on the activity measurements is difficult to fulfil throughout the measurement campaign. Whereas statistical accuracy can be maintained by measuring the source at increasingly longer time intervals during the decay, the statistical influence of background subtraction gains importance as the activity of the source diminishes. When using Eqs. 1-3 in practice, it has been suggested to take an average σ A / A value of these uncertainty components at the beginning and end of the campaign, since the most extreme measurement data carry the largest weight. The adequacy of this approach can be tested by comparing the results of Eq. 3 with that of the general equation Eq. 16. In Table 1 a hypothetical set of n=5 data is shown, for which the uncertainty increases progressively with time. When applying the above procedure, the
10
approximating uncertainty from Eq. 3 still agrees quite well with the exact value from Eq. 16. The situation is more complex when the uncertainties vary more randomly from one measurement to another. Large differences in relative uncertainties σ A / A can diminish the 'effective' number of measurements n that define the slope. The uncertainty estimations through Eqs. 1-3 should then be focussed on data with comparably small uncertainties. 4.4
Non-equidistant measurements
Measurements may also deviate from the prerequisite that they are performed at regular time intervals. In Table 2 a hypothetical set of n=5 data is shown, for which the time spacing increases progressively with each measurement. In spite of the large variations in ∆t, the approximate uncertainty calculated from Eq. 3 remains surprisingly close to the correct value from Eq. 16. Larger variations may occur when a majority of data would be taken in the middle of the measurement campaign, which would lead to an underestimation of the half-life uncertainty by Eq. 3. This could be solved by decreasing the 'effective' number n of measurement data. Conversely, if the measurements would be mainly taken at the beginning and end of the campaign, Eq. 3 would overestimate the real uncertainty. The data at the extremes have the highest impact on the overall uncertainty.
5 Conclusions The Eq. 16 presents a rigorous formula for calculating the random uncertainty component for a half-life determination by means of a weighted least-squares fit of an exponential function to a decay curve obtained by consecutive measurements of activity with well-established relative uncertainties on the activity values. It does not account for systematic errors, which introduce correlations in the uncertainties. The Eq. 3 is a mathematical reduction of Eq. 16, which rigorously applies under the conditions that the time intervals between the measurements and the relative uncertainties of the activity values are constant. The validity of Eq. 3 can be extended to cases where relative uncertainties in the activity and/or time differences between measurements show mild variations. Large variations in uncertainty as well as extreme grouping of data towards the centre, start or end of the measurement campaign affect the number data 'effectively' contributing to the uncertainty of the fit. Medium and long-term systematic errors can be propagated with Eq. 1.
11
References Bevington, P. R., Robinson, D. K., 2003. Data reduction and error analysis for the physical sciences. Third Edition. (McGraw-Hill, New York, USA). Currie, L. A., 2004. The Remarkable Metrological History of Radiocarbon Dating [II]. J. Res. Natl. Inst. Stand. Technol. 109, 185–217. Dalrymple, G. B., 1991. The Age of the Earth. California: Stanford University Press, ISBN 08047-1569-6. Emery, G. T., 1972. Perturbation of nuclear decay rates. Annu. Rev. Nucl. Sci. 22, 165–202. Hahn, H.-P., Born, H.-J., Kim, J. I., 1976. Survey on the rate perturbation of nuclear decay. Radiochim. Acta 23, 23–37. Judge, S. M., Arnold, D., Chauvenet, B., Collé, R., De Felice, P., García-Toraño, E., Wätjen, U., 2014. 100 Years of radionuclide metrology. Appl. Radiat. Isot. 87 27–31. Kendall, M.G., Stuart, A., 1961. The advanced theory of statistics. Vol. II. Inference and relationship. Griffin & Co., London. Knuth, D. E., 1993. Johann Faulhaber and sums of powers. Math. Comp. 61, 277–294. Nir-El, Y., 2004. Dating the age of a nuclear event by gamma spectrometry. Appl. Radiat. Isot. 60, 197–201. Norman E. B., Sur, B., Lesko K. T., Larimer, R.-M., DePaolo D. J., Owens T. L., 1995. An improved test of the exponential decay law. Phys. Lett. B 357, 521–525. Parker, J. L., 1990. Near-optimum procedure for half-life measurement by high resolution gamma-ray spectroscopy. Nucl. Instrum. Meth A. 286, 502–506. Pommé, S., 2007. Problems with the uncertainty budget of half-life measurements. In: Applied Modeling and Computations in Nuclear Science. T.M. Semkow, S.Pommé, S.M. Jerome, and D. J. Strom, Eds. ACS Symposium Series 945. American Chemical Society, Washington, DC, 2007, ISBN 0-8412-3982-7, 2007 pp. 282-292. Pommé, S., Camps, J., Van Ammel, R., Paepen, J., 2008. Protocol for uncertainty assessment of half-lives. J. Radioanal. Nucl. Chem. 276, 335–9. Pommé S., Jerome S. M., Venchiarutti, C., 2014. Uncertainty propagation in nuclear forensics. Appl. Radiat. Isot. 89, 58–64.
12
Pommé, S., 2015. The uncertainty of the half-life, Metrologia 52, S51–S65. Pommé, S., 2016. When the model doesn’t cover reality: examples from radionuclide metrology. Metrologia 53, S55–S64. Pommé, S., et al., 2016a. Evidence against solar influence on nuclear decay constants. Phys. Lett. B 761, 281–286. Pommé, S., Collins, S.M., Harms, A., Jerome, S.M., 2016b. Fundamental uncertainty equations for nuclear dating applied to the
140
Ba-140La and
227
Th-223Ra chronometers. J. Env.
Rad. 162–163, 358–370. Pommé, S., et al., 2017a. On decay constants and orbital distance to the Sun—part I: alpha decay. Metrologia 54, 1–18. Pommé, S., et al., 2017b. On decay constants and orbital distance to the Sun—part II: beta minus decay. Metrologia 54, 19–35. Pommé, S., et al., 2017c. On decay constants and orbital distance to the Sun— part III: beta plus and electron capture decay. Metrologia 54, 36–50. Pommé, S., Kossert, K., Nähle, O., 2017d. On the claim of modulations in 36Cl beta decay and their association with solar rotation, Solar Phys. 292:162 (pp.8). Pommé, S., Stroh, H., Altzitzoglou, T., Paepen, J., Van Ammel, R., Kossert, K., Nähle, O., Keightley, J. D., Ferreira, K. M., Verheyen, L., Bruggeman, M., 2018a. Is decay constant? Appl. Radiat. Isot. 134, 6–12. Pommé, S., Lutter, G., Marouli, M., Kossert, K., Nähle, O., 2018b. On the claim of modulations in radon decay and their association with solar rotation, Astroparticle Physics 97, 38–45. Pommé, S., Paepen, J., Van Ammel, R., 2018c. Linearity check of an ionisation chamber through 99mTc half-life measurements. Appl. Radiat. Isot. 140, 171–178. Pommé, S., Stroh, H., Van Ammel, R., 2019. The
55
Fe half-life measured with a pressurised
proportional counter. Appl. Radiat. Isot. 148, 27–34. 13
Pommé, S., 2019. Solar influence on radon decay rates: irradiance or neutrinos? Eur. Phys. J. C 79, 73 (pp. 9). Pommé, S., De Hauwere, T., 2020. On the significance of modulations in a time series, Nucl. Instr. Meth. A, in press. Rutherford, E., 1900. A radio-active substance emitted from thorium compounds, Philos. Mag. Ser. 5 Vol. 49 (No. 296), 1–14. Semkow, T. M., 2007. Exponential decay law and nuclear statistics. Applied Modeling and Computations in Nuclear Science (ACS Symposium Series vol 945) ed. T. M. Semkow et al. (Washington, DC: ACS/OUP) pp 42–56. Wooldridge, J. M., 2013. Introductory Econometrics: A Modern Approach. 5th ed. Mason, OH: South-Western Cengage Learning, ISBN-13: 978-1-111-53104-1
14
Table 1: Uncertainty calculation of a decay constant (λ=1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the measurement uncertainty, the approximate uncertainty from Eq. 3 deviates only by 6% from the exact uncertainty from Eq. 16, if in Eq. 3 use is made of the average value of the relative uncertainty at the beginning and end of the measurement campaign, <σΑ/A>i=1,5. ti (s)
σ i = σ A Ai
weight pi
pi (ti − t w ) 2
tw
1
1%
0.37
0.60
σy
2
1.2%
0.26
0.02
var(t )
1.27
3
1.4%
0.19
0.10
<σΑ/A>i=1,5
1.6%
4
1.8%
0.11
0.34
σλ/λ (Eq. 16)
0.477%
5
2.2%
0.08
0.56
σλ/λ (Eq. 3)
0.506%
i
2.27
w
0.61%
15
Table 2: Uncertainty calculation of a decay constant (λ=1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the time gap between measurements, the approximate uncertainty from Eq. 3 deviates only by 3% from the exact uncertainty from Eq. 16. Larger differences are found if measurements are extremely focussed to the centre or to the beginning and end of the campaign. ti (s)
σ i = σ A Ai
weight pi
pi (ti − t w ) 2
tw
1
5%
0.2
5.4
σy
2
5%
0.2
3.5
var(t )
5.46
4
5%
0.2
1.0
<σΑ/A>i=1,5
5%
8
5%
0.2
0.6
σλ/λ (Eq. 16)
0.410%
16
5%
0.2
19.2
σλ/λ (Eq. 3)
0.422%
i
6.2
w
2.24%
16
Figure Captions Fig. 1: Example of a linearization by y=lnA of a decay curve of n=5 activity measurements with variable relative uncertainties σy=σi=σ(Ai)/Ai obtained over a time interval T. The uncertainty on the slope of the fitted line is proportional to the uncertainty of the weighted mean of y, σ yw = σ Aw / Aw , and inversely proportional to the spread of the measurement reference times t,
var(t ) .
17
Fig. 1
15
tw
t var(t )
σy
12
y=ln(A)
9
w
yw
slope = -λ
y
6
σy=σA/A 3
n=5 T
0 0
1
2
3
4
5
6
x=t
18
Derivation of an uncertainty propagation factor for half-life determinations
S. Pommé‡1, T. De Hauwere1,2
1
European Commission, Joint Research Centre (JRC), Geel, Belgium 2
TU Delft, Delft, The Netherlands
Highlights • Derived uncertainty propagation factor for half-life from activity measurements. • Special formula for equidistant measurement data with equal relative uncertainty. • Derived compact general formula for weighted fit and used it as benchmark. • Special formula works well with mild deviations from conditions. Keywords: Half-life; Uncertainty; Propagation; Activity; Nuclear data
‡
[email protected]; tel. +32 (0)14 571 289
1
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Author statement Stefaan Pommé is the main author, responsible for the conceptualization and writing of the paper. Tota De Hauwere is a student and trainee, who assisted with part of the work in the context of training.
S0969-8043(19)30816-4
DOI:
https://doi.org/10.1016/j.apradiso.2020.109046
Reference:
ARI 109046
To appear in:
Applied Radiation and Isotopes
Received Date: 30 July 2019 Revised Date:
7 December 2019
Accepted Date: 15 January 2020
Please cite this article as: Pommé, S., De Hauwere, T., Derivation of an uncertainty propagation factor for half-life determinations, Applied Radiation and Isotopes (2020), doi: https://doi.org/10.1016/ j.apradiso.2020.109046. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Derivation of an uncertainty propagation factor for half-life determinations
S. Pommé‡1, T. De Hauwere1,2
1
European Commission, Joint Research Centre (JRC), Geel, Belgium 2
TU Delft, Delft, The Netherlands
An analytical equation is derived for the uncertainty propagation factor for a half-life determination from a least-squares fit to equidistant activity measurements performed with identical relative uncertainties. The obtained formula applies to a purely random statistical uncertainty component. It is equivalent to the solution published by Parker in Nucl. Instr. Meth. A 286, 502. A more general equation for weighted least-squares fitting is derived and presented in a compact manner. It is used as a benchmark to verify the applicability of Parker's solution to non-equidistant data with unequal uncertainties.
‡
[email protected]; tel. +32 (0)14 571 289
1
Highlights • Derived uncertainty propagation factor for half-life from activity measurements. • Special formula for equidistant measurement data with equal relative uncertainty. • Derived compact general formula for weighted fit and used it as benchmark. • Special formula works well with mild deviations from conditions. Keywords: Half-life; Uncertainty; Propagation; Activity; Nuclear data
2
1 Introduction The exponential-decay law (Rutherford, 1900; Semkow, 2007) is the foundation of the common measurement system for radioactivity (Judge et al., 2014). It relies on the invariability of the nuclear decay constants – or, inversely, the corresponding half-lives – which appears to apply rigorously within the boundaries of metrological accuracy (Emery, 1972; Hahn et al., 1976; Norman et al., 1995; Pommé et al., 2016a, 2017a,b,c,d, 2018a,b; Pommé, 2019; Pommé and De Hauwere, in press). As a result, nuclear decay can be used as a clock (Dalrymple, 1994; Currie, 2004; Nir-El, 2004, Pommé, 2015; Pommé et al., 2014, 2016b). Half-life values of radionuclides are measured by various techniques, depending on their order of magnitude (Pommé, 2015). Half-lives in an intermediate range are measured by repeated activity measurements over periods of time, thus establishing a decay curve to which an exponential decay function is fitted. Generally, least-squares fitting routines are applied to determine the slope of the decay curve and the ensuing uncertainty is obtained from the covariance matrix (Bevington and Robinson, 2003). The latter procedure often gives rise to significant underestimation of the half-life uncertainty, since the model doesn't account for systematic deviations due to metrological imperfections (Pommé, 2007, 2016). Medium- and long-term instability and non-linearity in the experiment propagate with a larger factor into the half-life uncertainty than the purely random variations (Pommé et al., 2008; Pommé, 2015). In state-of-the-art half-life measurements, the short-term random variations – such as counting statistics – may represent only a small fraction of the total uncertainty budget. For example, in a
99m
Tc half-life measurement (Pommé et al.,
2018c), the uncertainty component due to random variability in more than 2000 measurements added very little to the total, since the attainable accuracy is limited by the linearity of the ionisation chamber, and possibly to an even larger extent by the chemical composition of the source influencing the probability for internal conversion of the 2.17 keV transition. On the other hand, the short-term reproducibility issues (including counting statistics for source and background) were the major uncertainty component for a
55
Fe half-
life measurement, in spite of the presence of long-term, systematic effects due to pulse pileup, background subtraction and gas pressure variations (Pommé et al., 2019). Correct propagation of uncertainty components is an important issue. An approximate formula was proposed by Pommé et al. (2008) to propagate random uncertainty components from n equally spaced activity measurements over a time period T to the fitted half-life T1/2:
3
σ (T1/2 ) T1/2
≈
2 λT
2 σ ( A) n +1 A
(1)
The equation can also be used to evaluate medium and low-frequency instabilities, if the parameter n receives a new meaning as the number of periods covered by a cyclic effect on the experiment. For low-frequency effects, the value n=1 is selected by default. A more rigorous equation for the propagation factor of random uncertainties was published by Parker (1990), as part of a procedure to measure the half-life of a source by gamma-ray spectrometry, relative to a reference radionuclide with a well-established, long half-life. For a series of n measurements of activity ratios of both nuclides, R, assuming a constant measurement uncertainty, σ(R), Parker derived the following uncertainty formula:
σT = 1/2
2 3 T1/2 ( n − 1) n σR 2 ln 2 T n ( n − 1)
(2)
On the basis of this equation, the following alternative to Eq. 1 was recommended specifically for high-frequency random variations in a review paper (Pommé, 2015)
σT
1/ 2
T1 / 2
≈
2 λT
3( n − 1) σ A n ( n + 1) A
(3)
No mathematical derivation was published to justify its validity. In this paper, it will be demonstrated how Eq. 3 is found starting from the uncertainty equations for linear leastsquares fitting and transforming them to the case of exponential decay. 2 Derivation of Equation 3 2.1
Linear least-squares fit
In principle, the best fit of an exponential function to a nuclear decay curve can be obtained by means of the maximum likelihood estimation method using the Poisson distribution to account for the probability for the observed number of decays. For sufficiently high amounts of counts, a Gaussian distribution is an adequate approximation and the maximum likelihood search is conveniently replaced by a least-squares fit (Kendall and Stuart, 1961). Whereas a non-linear fit can be easily performed to obtain the decay constant from the slope of the
4
exponential, it is advantageous to study the transform to a linear problem to derive the uncertainty propagation factor in Eq. 3. A variable y is considered which relates linearly with a variable x, through the free parameters a and b representing the intercept and the slope, respectively: y = a + bx
(4)
In the context of this paper, y represents the logarithm of the activity, y=lnA, and x corresponds with time, t. Consequently, the intercept refers to the initial activity, a=lnA0, the slope is related to the decay constant, b=–λ, and the focus of this work is on the uncertainty of the least-squares estimate of b. The uncertainty on the xi values is assumed to be negligible, σx=0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi, propagated by the slope b. In this paper, the 'spread' of the nominal xi and yi values, will be quantified by their 'variance', e.g. var( x ) = ∑ ( xi − x ) 2 / n and cov( x, y ) = ∑ ( xi − x )( yi − y ) / n , which should not be confound
with their measurement uncertainty (i.e. var( x) ≠ σ x2 ). Minimisation of the weighted sum of the squares of the deviations between measured and fitted values,
y − a − bxi χ = ∑ i σi i =1 n
2
2
(5)
through ∂χ / ∂a = 0 and ∂χ / ∂b = 0 yields analytical equations for the estimated values of 2
2
a and b (Kendall and Stuart, 1961; Bevington and Robinson, 2003); They can be reduced to ⌢ ⌢ a = y − bx and b = ∑ ( xi − x )( yi − y ) / ∑ ( xi − x ) 2 = cov( x, y ) / var( x ) . 2.2
Uncertainty of the slope
Ignoring systematic errors which would introduce correlations between uncertainties, the 2 2 2 uncertainty of the slope can be derived from σ b = ∑σ i (∂b / ∂yi ) . In the special case of
constant uncertainties, σi = σ y , this results in (Bevington and Robinson, 2003)
5
σ = 2 b
nσ y2 n n∑ x − ∑ xi i =1 i =1 n
2
2 i
=
σ y2 n
∑ (x i =1
=
i
(6)
−x ) 2
σ y2
(7)
var( x )
The Eq. 6 is valid independently of the statistical distribution of the y data. If a normal ⌢ distribution applies to y, than it also applies to the estimated intercept a and slope bˆ , since they are sums of normal random variables. Eq. 7 facilitates the interpretation: the variance of the slope decreases when the uncertainty on the y values is small and many data are taken over a wide range of x values, or alternatively when the uncertainty of the mean of the y values is small compared to the spread of the x data. If the standard uncertainty of the y data is not known a priori, it can be estimated from the spread of the residuals to the linear fit, through n
σ y2 ≈ s 2 =
∑ ( y −a − bx ) i =1
i
2
i
(8)
n−2
in which n-2 is the number of degrees of freedom of the linear fit. The 'variance' of xi can be calculated directly from the sum of all ( xi − x )2 terms divided by n. 2.3
Equidistant measurements
Applying an additional condition that the xi data are equidistant by an amount ∆x, such that xi=i∆x, one can simplify Eq. 6 further. For a set of n data, the total distance between the extremes, x1 and xn, is T = ∆x ( n − 1) and the mean value is x = ∆x (n + 1) / 2 . The term in the denominator of Eq. 6 equals n +1 ( xi −x ) = ( ∆x ) ∑ i − ∑ 2 i =1 i =1 n
n
2
2
2
6
2 n 2 n +1 = (∆x) ∑ i − n i =1 2 2
n(n2 −1) n(n + 1) = (∆x) =T2 12 12(n − 1) 2
(9)
in which use was made of the Faulhaber-Bernouilli formulas (Knuth, 1993): n
∑i = i =1
n(n + 1) and 2
n
∑i i =1
2
=
n(n + 1)(2n + 1) 6
(10)
Combining Eqs. 6 and 9, one obtains
σb =
2.4
σy T
2
3( n − 1) n( n + 1)
(11)
Transform to half-life
When transforming a non-linear fit of an exponential curve into a linear fit through y=lnA, the uncertainties of the measured quantities must by modified by
σy =
d ln A σ σA = A dA A
(12)
which relates the standard uncertainty of y to the relative uncertainty on the activity values A. The relative uncertainty of the slope equals the relative uncertainty on the decay constant (or the half-life), i.e.
σλ 2 = λ λT
σb / b = σλ / λ . Combining this with Eqs. 11 and 12, one obtains 3( n − 1) σ A n( n + 1) A
(13)
which is, according to the Gauss-Markov theorem (Wooldridge, 2013), the best linear unbiased estimator (BLUE) under the conditions of equidistant time between measurements and the same relative uncertainty in each activity value. This result is equivalent to the Eq. 2 proposed by Parker (1990).
7
3 Weighted least-squares fit 3.1
Uncertainty of the slope
Since the case of a constant uncertainty in y and fixed spacing in x is quite particular, it is of interest to consider the general case in which σi and ∆xi vary for each data point. The uncertainty on the xi values is assumed to be negligible, σx=0, or otherwise absorbed into the uncertainties of the corresponding yi values, σi. Ignoring systematic errors which would introduce correlations between uncertainties, the uncertainty of the slope obtained from an inverse-variance weighted least-squares fit – i.e. minimising χ2 in Eq. 5 – results in (Bevington and Robinson, 2003) n
σ b2 =
1
∑σ i =1
2 i
n xi − ∑ ∑ 2 2∑ 2 i =1 σ i i =1 σ i i =1 σ i n
xi2
n
1
(14)
2
The interpretation suggested by Eq. 7 for constant σi can also be looked for in Eq. 14 for weighted least-squares fitting. A weighted mean of the xi and yi values and a weighted 'variance' of the xi data – xw = ∑ pi xi , yw = ∑ pi yi and var( x ) = ∑ pi ( xi − xw ) , respectively 2
– can be calculated using normalised inverse-variance weights pi = 1 / σ i2
∑ 1/ σ
2 j
. The
variance of the weighted mean, σ y2w , equals 1 / ∑ 1 / σ 2j . The Eq. 14 can be rearranged to obtain a similar result as in Eq. 7: n
σ b2 =
i =1
n
i =1
n
2 i
1
σ y2
=
w
xi − xw σ i2 i =1 n
∑
2 i
n xi − ∑ ∑ 2 2 i =1 σ i i =1 σ i
xi2
∑σ =
1
∑σ
1
2
n
1
∑σ i =1
2
n 1 ∑ 2 i =1 σ i
2
σ y2
w
var( x)
(15)
2 i
The variance on the slope of a weighted fit is proportional to the variance of the weighted mean of the yi values and inversely to the weighted spread of the xi data.
8
3.2
Transform to half-life
As derived from Eq. 12, when transforming a non-linear fit of an exponential curve into a linear fit through y=lnA, the standard uncertainties of the y values equal the relative uncertainties on the activity values, σ y = σ A / A . The weighting factors to be applied are
(
therefore pi = σ Ai / Ai
) ∑ (σ −2
Aj
)
/ Aj
−2
.
Combining this with Eq. 15, one obtains
σλ 1 = λ λ
n ∑ σ Ai / Ai i =1
(
∑ (σ n
i =1
=
1
λ
Ai
/ Ai
)
−2
−1/2
) ( t − t ) ∑ (σ −2
2
i
w
n
i =1
Ai
/ Ai
)
−2
σ Aw 1 var(t ) Aw
(16)
in which ti is the reference time of the measurement i expressed in the same time unit as 1/λ, t w = ∑ pi ti is the weighted mean of the reference times of the measurements, using the
weighting factors pi based on the relative uncertainties of the activity values, var(t ) = ∑ pi (ti − t w )
σ A / Aw = w
( ∑ (σ
Ai
2
/ Ai
represents
)
)
−2 −1/ 2
the
spread
of
the
reference
times
and
is the relative uncertainty of the weighted mean activity. In
Fig. 1, the meaning of the variables involved is depicted for a hypothetical example. Explicit calculations via Eq. 16 involve the following steps: (1) calculate the sum of the inverse square of the relative uncertainties on the activity values, (2) normalise these values individually by their sum to obtain the weighting factors pi, (3) calculate the uncertainty of the weighted mean activity as the inverse square root of the normalisation factor, (4) calculate the weighted mean time tw, and the 'variance' of t, and (5) apply Eq. 16.
4 Case studies 4.1
Few measurements
The uncertainty propagation formula for the half-life obtained from the ratio of two activity measurements with the same relative uncertainty is (Pommé, 2015)
9
σT
1/2
T1/ 2
=
2 σA λT A
(17)
The same result is obtained from the approximation in Eq. 1 for n=3 and from the BLUE in Eq. 3 for n=2 and n=3. The third point in the middle of the decay curve does not influence the slope, only the amplitude. For large n, the BLUE is proportional to
3 / n , whereas the
approximation in Eq. 1 is somewhat lower, ~ 2 / n . 4.2
Systematic error
The reduction of the propagation factor by n −1/ 2 (Eqs. 1-3) applies to stochastic effects which randomly vary with each measurement, not to systematic errors which affect subsequent data in a correlated manner. Non-linearity in detector output or changes in source integrity, geometrical and environmental conditions may occur at any phase of the project. Live-time corrections for pileup and dead time may introduce a systematic error at the start of the measurement campaign, not so much at the end. On the other hand, systematic errors in background subtraction may affect the measurements at the end of the campaign, while being negligible at the start. It is suggested to apply n=1 in Eq. 1 to propagate an average value of each systematic uncertainty component at the start and end of the campaign, which effectively leads to a correct propagation:
σλ 2 σA / A+0 1 σA = = . This effect cannot be λ λT 2 λT A
reproduced with Eq. 3. 4.3
Variable uncertainties
The condition of constant relative uncertainties on the activity measurements is difficult to fulfil throughout the measurement campaign. Whereas statistical accuracy can be maintained by measuring the source at increasingly longer time intervals during the decay, the statistical influence of background subtraction gains importance as the activity of the source diminishes. When using Eqs. 1-3 in practice, it has been suggested to take an average σ A / A value of these uncertainty components at the beginning and end of the campaign, since the most extreme measurement data carry the largest weight. The adequacy of this approach can be tested by comparing the results of Eq. 3 with that of the general equation Eq. 16. In Table 1 a hypothetical set of n=5 data is shown, for which the uncertainty increases progressively with time. When applying the above procedure, the
10
approximating uncertainty from Eq. 3 still agrees quite well with the exact value from Eq. 16. The situation is more complex when the uncertainties vary more randomly from one measurement to another. Large differences in relative uncertainties σ A / A can diminish the 'effective' number of measurements n that define the slope. The uncertainty estimations through Eqs. 1-3 should then be focussed on data with comparably small uncertainties. 4.4
Non-equidistant measurements
Measurements may also deviate from the prerequisite that they are performed at regular time intervals. In Table 2 a hypothetical set of n=5 data is shown, for which the time spacing increases progressively with each measurement. In spite of the large variations in ∆t, the approximate uncertainty calculated from Eq. 3 remains surprisingly close to the correct value from Eq. 16. Larger variations may occur when a majority of data would be taken in the middle of the measurement campaign, which would lead to an underestimation of the half-life uncertainty by Eq. 3. This could be solved by decreasing the 'effective' number n of measurement data. Conversely, if the measurements would be mainly taken at the beginning and end of the campaign, Eq. 3 would overestimate the real uncertainty. The data at the extremes have the highest impact on the overall uncertainty.
5 Conclusions The Eq. 16 presents a rigorous formula for calculating the random uncertainty component for a half-life determination by means of a weighted least-squares fit of an exponential function to a decay curve obtained by consecutive measurements of activity with well-established relative uncertainties on the activity values. It does not account for systematic errors, which introduce correlations in the uncertainties. The Eq. 3 is a mathematical reduction of Eq. 16, which rigorously applies under the conditions that the time intervals between the measurements and the relative uncertainties of the activity values are constant. The validity of Eq. 3 can be extended to cases where relative uncertainties in the activity and/or time differences between measurements show mild variations. Large variations in uncertainty as well as extreme grouping of data towards the centre, start or end of the measurement campaign affect the number data 'effectively' contributing to the uncertainty of the fit. Medium and long-term systematic errors can be propagated with Eq. 1.
11
References Bevington, P. R., Robinson, D. K., 2003. Data reduction and error analysis for the physical sciences. Third Edition. (McGraw-Hill, New York, USA). Currie, L. A., 2004. The Remarkable Metrological History of Radiocarbon Dating [II]. J. Res. Natl. Inst. Stand. Technol. 109, 185–217. Dalrymple, G. B., 1991. The Age of the Earth. California: Stanford University Press, ISBN 08047-1569-6. Emery, G. T., 1972. Perturbation of nuclear decay rates. Annu. Rev. Nucl. Sci. 22, 165–202. Hahn, H.-P., Born, H.-J., Kim, J. I., 1976. Survey on the rate perturbation of nuclear decay. Radiochim. Acta 23, 23–37. Judge, S. M., Arnold, D., Chauvenet, B., Collé, R., De Felice, P., García-Toraño, E., Wätjen, U., 2014. 100 Years of radionuclide metrology. Appl. Radiat. Isot. 87 27–31. Kendall, M.G., Stuart, A., 1961. The advanced theory of statistics. Vol. II. Inference and relationship. Griffin & Co., London. Knuth, D. E., 1993. Johann Faulhaber and sums of powers. Math. Comp. 61, 277–294. Nir-El, Y., 2004. Dating the age of a nuclear event by gamma spectrometry. Appl. Radiat. Isot. 60, 197–201. Norman E. B., Sur, B., Lesko K. T., Larimer, R.-M., DePaolo D. J., Owens T. L., 1995. An improved test of the exponential decay law. Phys. Lett. B 357, 521–525. Parker, J. L., 1990. Near-optimum procedure for half-life measurement by high resolution gamma-ray spectroscopy. Nucl. Instrum. Meth A. 286, 502–506. Pommé, S., 2007. Problems with the uncertainty budget of half-life measurements. In: Applied Modeling and Computations in Nuclear Science. T.M. Semkow, S.Pommé, S.M. Jerome, and D. J. Strom, Eds. ACS Symposium Series 945. American Chemical Society, Washington, DC, 2007, ISBN 0-8412-3982-7, 2007 pp. 282-292. Pommé, S., Camps, J., Van Ammel, R., Paepen, J., 2008. Protocol for uncertainty assessment of half-lives. J. Radioanal. Nucl. Chem. 276, 335–9. Pommé S., Jerome S. M., Venchiarutti, C., 2014. Uncertainty propagation in nuclear forensics. Appl. Radiat. Isot. 89, 58–64.
12
Pommé, S., 2015. The uncertainty of the half-life, Metrologia 52, S51–S65. Pommé, S., 2016. When the model doesn’t cover reality: examples from radionuclide metrology. Metrologia 53, S55–S64. Pommé, S., et al., 2016a. Evidence against solar influence on nuclear decay constants. Phys. Lett. B 761, 281–286. Pommé, S., Collins, S.M., Harms, A., Jerome, S.M., 2016b. Fundamental uncertainty equations for nuclear dating applied to the
140
Ba-140La and
227
Th-223Ra chronometers. J. Env.
Rad. 162–163, 358–370. Pommé, S., et al., 2017a. On decay constants and orbital distance to the Sun—part I: alpha decay. Metrologia 54, 1–18. Pommé, S., et al., 2017b. On decay constants and orbital distance to the Sun—part II: beta minus decay. Metrologia 54, 19–35. Pommé, S., et al., 2017c. On decay constants and orbital distance to the Sun— part III: beta plus and electron capture decay. Metrologia 54, 36–50. Pommé, S., Kossert, K., Nähle, O., 2017d. On the claim of modulations in 36Cl beta decay and their association with solar rotation, Solar Phys. 292:162 (pp.8). Pommé, S., Stroh, H., Altzitzoglou, T., Paepen, J., Van Ammel, R., Kossert, K., Nähle, O., Keightley, J. D., Ferreira, K. M., Verheyen, L., Bruggeman, M., 2018a. Is decay constant? Appl. Radiat. Isot. 134, 6–12. Pommé, S., Lutter, G., Marouli, M., Kossert, K., Nähle, O., 2018b. On the claim of modulations in radon decay and their association with solar rotation, Astroparticle Physics 97, 38–45. Pommé, S., Paepen, J., Van Ammel, R., 2018c. Linearity check of an ionisation chamber through 99mTc half-life measurements. Appl. Radiat. Isot. 140, 171–178. Pommé, S., Stroh, H., Van Ammel, R., 2019. The
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Fe half-life measured with a pressurised
proportional counter. Appl. Radiat. Isot. 148, 27–34. 13
Pommé, S., 2019. Solar influence on radon decay rates: irradiance or neutrinos? Eur. Phys. J. C 79, 73 (pp. 9). Pommé, S., De Hauwere, T., 2020. On the significance of modulations in a time series, Nucl. Instr. Meth. A, in press. Rutherford, E., 1900. A radio-active substance emitted from thorium compounds, Philos. Mag. Ser. 5 Vol. 49 (No. 296), 1–14. Semkow, T. M., 2007. Exponential decay law and nuclear statistics. Applied Modeling and Computations in Nuclear Science (ACS Symposium Series vol 945) ed. T. M. Semkow et al. (Washington, DC: ACS/OUP) pp 42–56. Wooldridge, J. M., 2013. Introductory Econometrics: A Modern Approach. 5th ed. Mason, OH: South-Western Cengage Learning, ISBN-13: 978-1-111-53104-1
14
Table 1: Uncertainty calculation of a decay constant (λ=1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the measurement uncertainty, the approximate uncertainty from Eq. 3 deviates only by 6% from the exact uncertainty from Eq. 16, if in Eq. 3 use is made of the average value of the relative uncertainty at the beginning and end of the measurement campaign, <σΑ/A>i=1,5. ti (s)
σ i = σ A Ai
weight pi
pi (ti − t w ) 2
tw
1
1%
0.37
0.60
σy
2
1.2%
0.26
0.02
var(t )
1.27
3
1.4%
0.19
0.10
<σΑ/A>i=1,5
1.6%
4
1.8%
0.11
0.34
σλ/λ (Eq. 16)
0.477%
5
2.2%
0.08
0.56
σλ/λ (Eq. 3)
0.506%
i
2.27
w
0.61%
15
Table 2: Uncertainty calculation of a decay constant (λ=1 s-1) hypothetically obtained from a least-squares fit to 5 data on an exponential decay curve. In spite of the non-linear increase of the time gap between measurements, the approximate uncertainty from Eq. 3 deviates only by 3% from the exact uncertainty from Eq. 16. Larger differences are found if measurements are extremely focussed to the centre or to the beginning and end of the campaign. ti (s)
σ i = σ A Ai
weight pi
pi (ti − t w ) 2
tw
1
5%
0.2
5.4
σy
2
5%
0.2
3.5
var(t )
5.46
4
5%
0.2
1.0
<σΑ/A>i=1,5
5%
8
5%
0.2
0.6
σλ/λ (Eq. 16)
0.410%
16
5%
0.2
19.2
σλ/λ (Eq. 3)
0.422%
i
6.2
w
2.24%
16
Figure Captions Fig. 1: Example of a linearization by y=lnA of a decay curve of n=5 activity measurements with variable relative uncertainties σy=σi=σ(Ai)/Ai obtained over a time interval T. The uncertainty on the slope of the fitted line is proportional to the uncertainty of the weighted mean of y, σ yw = σ Aw / Aw , and inversely proportional to the spread of the measurement reference times t,
var(t ) .
17
Fig. 1
15
tw
t var(t )
σy
12
y=ln(A)
9
w
yw
slope = -λ
y
6
σy=σA/A 3
n=5 T
0 0
1
2
3
4
5
6
x=t
18
Derivation of an uncertainty propagation factor for half-life determinations
S. Pommé‡1, T. De Hauwere1,2
1
European Commission, Joint Research Centre (JRC), Geel, Belgium 2
TU Delft, Delft, The Netherlands
Highlights • Derived uncertainty propagation factor for half-life from activity measurements. • Special formula for equidistant measurement data with equal relative uncertainty. • Derived compact general formula for weighted fit and used it as benchmark. • Special formula works well with mild deviations from conditions. Keywords: Half-life; Uncertainty; Propagation; Activity; Nuclear data
‡
[email protected]; tel. +32 (0)14 571 289
1
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Author statement Stefaan Pommé is the main author, responsible for the conceptualization and writing of the paper. Tota De Hauwere is a student and trainee, who assisted with part of the work in the context of training.