Institute for Reference Materials and Measurements (IRMM)Geel, Belgium
http://www.irmm.jrc.behttp://www.jrc.cec.eu.int
BIPM, Uncertainties Workshop, 17 Sept 2008
A protocol for uncertainty assessment of half-lives
S. Pommé
BIPM, 17 Sept 20082
Half-life Measurements
• Half-life determination by following the decay of a radioactive source
• The problem of data discrepancy; examples
• New procedure for uncertainty calculation
BIPM, 17 Sept 20083
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31
32
33
0 50 100 150 200 250 300 350
T-T0 (days)
activ
ity (k
Bq)
Example: decay of 55Fe
fitted decay curve
measured activity
T1/2=1005.0 d ± 1.4d
BIPM, 17 Sept 20084
Half-life and uncertainty from least-squares fit
991
994
997
1000
1003
1006
1009
1012
1015
1018
0 30 60 90 120 150 180 210time (d)
<T1/
2> (d
)
=> fit underestimates uncertainty!
BIPM, 17 Sept 20085
Residuals 55Fe: blow up
80 100 120 140-5.0
-2.5
0.0
2.5
5.0
(mea
sure
men
t - fi
t) / s
tand
ard
devi
atio
n
decay time T-T0 (days)
uncertainty only including counting statistics=> does not fully account for spread of data
Van Ammel et al., ARI (2006)
BIPM, 17 Sept 20086 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
Autocorrelated data => not stochastic
autocorrelation plotof the residuals
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high
true deviation experimentalist view fitted slope
medium
Res
idua
ls (a
rbitr
ary
unit)
low
X Axis (arb. unit)
High, medium, low frequency instabilities
noise, counting statistics, …
=> random effects
geometrical reproducibility, ‘seasonal’ effects,
short-lived impurity, …
dead time, detector/source degradation,background subtraction, …
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Intermediate conclusion
• fitted parameters are only meaningful if the model rigorously applies to the data
• hence the model should include all possible medium and long-term instabilities
• common statistical tests (uncertainty!) require randomness of data; they do not apply to autocorrelated data
• the uncertainty derived from a fit is unreliable if these conditions are not fulfilled
BIPM, 17 Sept 20089
BIPM, 17 Sept 200810
751
752
753
754
755
Lagoutine Dietz Houtermans Unterw eger
Day
s
Half-life of 134Cs
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456
457
458
459
460
461
462
463
464
465
Vaninbrouckx Lagoutine Schrader Unterw eger Martin
Day
sHalf-life of 109Cd
a selection of the ‘best’ data444
454
464
474
1950 1965 1968 1968 1981 1982 1982 1996 2004
Days
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‘Trend analysis’ of residuals
short-term
medium-term
long-term
BIPM, 17 Sept 200813
Residuals Ba-133
NIM A390 (1997) 267-273
data scatter exceeds uncertainty=> unidentified HIGH FREQUENCY component
BIPM, 17 Sept 200814
Residuals Cs-134
NIM A390 (1997) 267-273
MEDIUM FREQUENCY instability=> fit underestimates uncertainty
BIPM, 17 Sept 200815
Residuals Ba-133 and Eu-152
NIM A390 (1997) 267-273
‘independent’ measurements, yet positively correlated=> instability is not stochastical
BIPM, 17 Sept 200816
Residuals Ce-144
NIM A390 (1997) 267-273
sign of LOW FREQUENCY deviation=> fit tends to minimise it
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Residuals 99Tcm
electrometerrange
switching
systematicdeviation
for short times
ARI 60 (2004) 317-323
BIPM, 17 Sept 200818
Residuals Y-88
NIM A390 (1997) 267-273
only a few measurement data=> LACK OF INFORMATION
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An alternative procedure
S. Pommé, American Chemical Society Press, 2007
• Quantify all sources of instability
• Determine their rate of change
• Apply uncertainty propagation
• Sum all components independently
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ACS symposium series 945
An alternative data analysis method thatshould lead to a realistic uncertainty
budget
BIPM, 17 Sept 200821
J. Radioanal. Nucl. Chem. 276 (2008) 335
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consider START and STOP
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0 50 100 150 200 250 300 350
T-T0 (days)
activ
ity (k
Bq)
STOP(A2,T2)
START(A1,T1)
)A/Aln(2ln)TT(T21
121/2 −=
22
22
21
12
1/2
1/2A
)A(A
)A(λT1
T)T( σ
+σ
=σ
T=T2-T1
BIPM, 17 Sept 200823
Uncertainty Propagation
• Quantify all sources of uncertainty, σ(A), and their ‘frequency’ of occurrence, n
• Apply uncertainty propagation formula:
n= the number of occurrences of the effect
conservative value n=1 for medium and low frequencies
• Sum all components independently
A)A(
1n2
T2
T)T(
2/1
2/1 σ
⎭⎬⎫
⎩⎨⎧
+λ≈
σ
BIPM, 17 Sept 200824
Hypothetical case Fe-55
type σ(A)/A n factor σ(T1/2)/T1/2
high 1% 100 0.71 0.706%medium 1% 5 2.90 2.897%
low 1% 1 5.02 5.018%
λT=0.2T1/2=1005 d
T=290 d
BIPM, 17 Sept 200825
Realistic uncertainty
Fe-55
990
995
1000
1005
1010
1015
1020
0 30 60 90 120 150 180 210 240 270 300 330time (d)
<T1/
2> (d
)
BIPM, 17 Sept 200826
Literature: lack of information
• most papers on measurement of T1/2 contain insufficient information for a proper review
• sometimes result is given without description of experimental design
• too succinct reporting style even by reputed reference laboratories
• lack of transparency, traceability
• => need to redo undocumented experiments=> incomplete report is lost information
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What is essential information?
• description of the experiment
• any circumstance that is considered relevant for a traceable account of how the half-life value and its uncertainty were calculated
• result and uncertainty budget
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What was measured and how?
• activity as count rate, current, ratio, …; which part of the decay (particle, energy range)?
• how many sources? which was their initial activity?
• which (type of) detector was used? at which efficiency?
• number of measurements performed?
• time period of measurement campaign?
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How were the data analysed?
• method of linearization – e.g. by least squares fit: mention free and fixed
parameters, explicitly state which statistical weights were assigned to the data!
• which corrections were performed?– radioimpurity correction
– background subtraction
– differentiate between stochastic uncertainty components and possible ‘systematic’ components (long-term component!)
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Residual plot
• always present a detailed residual plot!
• avoid ‘cleanup’ of ‘outliers’; rather indicate them by a different symbol than the data that were used for the linearization process
• perform many measurement with excellent statistical accuracy, in order to increase the chance of observing and quantifying medium-term instabilities
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Exhaustive uncertainty budget
• assemble a table with all identifiable sources of uncertainty
• separate components that should be visible in the residuals from the invisible ones
• the estimated amplitude of the short- and medium-term components should cover all visible deviations in the residuals
• be generous when estimating the (mostly invisible) effects of long-term instabilities– investigate thoroughly systematic effect by dead time
– study and discuss possibe non-linearity of detector response
BIPM, 17 Sept 200832
Final Conclusions
• perform many measurements with good statistical accuracy and carefully study the non-stochastical part of the residuals
• identify and quantify short, medium and low frequency instabilities separately and apply a conservative propagation factor to T1/2
• report in sufficient detail, for traceability
Half-life measurements are NOT TRIVIAL !