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

24

25

26

27

28

29

30

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

BIPM, 17 Sept 20087

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, …

BIPM, 17 Sept 20088

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

BIPM, 17 Sept 200811

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

BIPM, 17 Sept 200812

‘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

BIPM, 17 Sept 200817

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

BIPM, 17 Sept 200819

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

BIPM, 17 Sept 200820

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

BIPM, 17 Sept 200822

consider START and STOP

24

25

26

27

28

29

30

31

32

33

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

BIPM, 17 Sept 200827

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

BIPM, 17 Sept 200828

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?

BIPM, 17 Sept 200829

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!)

BIPM, 17 Sept 200830

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

BIPM, 17 Sept 200831

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 !