Institute for Reference Materials and Measurements (IRMM)Geel, Belgium

http://www.irmm.jrc.be http://www.jrc.cec.eu.int

230th ACS NM, Washington D.C., Aug 2005

Problems with the uncertainty budget of half-life measurements

S. Pommé

230th ACS NM - T1/22

Half-life Measurements

• Half-life determination by following the decay of

a radioactive source

• The problem of data discrepancy; examples

• Autocorrelation causes underestimation of the

uncertainty

• Example: the decay of Fe-55

230th ACS NM - T1/23

456

457

458

459

460

461

462

463

464

465

Vaninbrouckx Lagoutine Schrader Unterw eger Martin

Day

s

Half-life of 109Cd

a selection of the ‘best’ data444

454

464

474

1950 1965 1968 1968 1981 1982 1982 1996 2004

Day

s

230th ACS NM - T1/24

751

752

753

754

755

Lagoutine Dietz Houtermans Unterw eger

Day

s

Half-life of 134Cs

230th ACS NM - T1/25

310

311

312

313

314D

ay

s

Half-life of 54Mn

a selection of the ‘best’ data

270

280

290

300

310

320

Da

ys

230th ACS NM - T1/26

970

980

990

1000

1010

1020

Houte

rman

s

Lago

utine

Hoppe

sM

orel

Karm

alitsy

n

Schöt

zig

Van A

mm

el

Day

s

Half-life of 55Fe

a selection of the ‘best’ data

230th ACS NM - T1/27

Why are uncertainties underestimated?

** by analogy with ‘It’s the economy, stupid!’ by James Carvil, advisor of Bill Clinton in the elections of 1992

230th ACS NM - T1/28

24

25

26

27

28

29

30

31

32

33

0 50 100 150 200 250 300 350

T-T0 (days)

acti

vity

(kB

q)

Example: decay of 55Fe

fitted decay curve

measured activity

T1/2=1005.0 d ± 1.4d

230th ACS NM - T1/29

Procedure ‘uncertainty from fit’

1) calculate the 2–value, ensuring that it assumes its expectation value = the number of degrees of freedom

2) adjust the half-life so that 2 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

Bevington, P.R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill Book Company, New York, USA, 1969

230th ACS NM - T1/210

Residuals 55Fe add uncertainty component of 0.0175%

-5.0

-2.5

0.0

2.5

5.0

0 50 100 150 200 250 300 350T-T 0

de

via

tio

n /

sig

ma

-5.0

-2.5

0.0

2.5

5.0

0 50 100 150 200 250 300 350T-T 0

de

via

tio

n /

sig

ma

2/=3.5

2/=1.0

230th ACS NM - T1/211

0

50

100

150

200

250

300

dev iation / sigma

#d

ata

po

ints

Check of distribution of residuals

BEFOREintroducing extra

uncertainty component

0

50

100

150

200

250

300

dev iation / sigma

#d

ata

po

ints

AFTER

2/=3.5

2/=1.0

230th ACS NM - T1/212

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)

<T

1/2

> (

d)

=> fit underestimates uncertainty!

230th ACS NM - T1/213

Why?

230th ACS NM - T1/214

Residuals 55Fe: blow up

80 100 120 140-5.0

-2.5

0.0

2.5

5.0

(mea

sure

men

t - fi

t) /

stan

dard

dev

iatio

n

decay time T-T0 (days)

uncertainty only including counting statistics=> does not fully account for spread of data

230th ACS NM - T1/215 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

Autocorrelated data are not random

autocorrelation plotof the residuals

230th ACS NM - T1/216

Alternative Procedure

• Subdivide instabilities according to frequency (low, medium, high)

• Apply new (simplified) uncertainty formula:

n= the number of occurances of the effect

n=1 for medium and low frequencies

• Add all components as square sum

A

A

nTT

T )(

1

2)(

2/1

2/1

230th ACS NM - T1/217

high

true deviation experimentalist view fitted slope

medium

Re

sid

ua

ls (

arb

itra

ry u

nit)

low

X Axis (arb. unit)

Subdivide instabilities according to frequency

noise, counting statistics, …

=> random effects

geometrical reproducibility, ‘seasonal’ effects,

short-lived impurity, …

dead time, detector/source degradation,background subtraction, …

230th ACS NM - T1/218

T1/2 from combination of all data

Fe-55

0

5000

10000

15000

20000

25000

30000

955 965 975 985 995 1005 1015 1025 1035 1045 1055

T1/2

fre

qu

en

cy

)/ln(

)2ln(

2121 AATT /

T1/2

230th ACS NM - T1/219

Realistic uncertainty

Fe-55

990

995

1000

1005

1010

1015

1020

0 30 60 90 120 150 180 210 240 270 300 330time (d)

<T

1/2

> (

d)

230th ACS NM - T1/220

230th ACS NM - T1/221

Residuals Ba-133

NIM A390 (1997) 267-273

data scatter exceeds uncertainty=> unidentified HIGH FREQUENCY component

230th ACS NM - T1/222

Residuals Cs-134

NIM A390 (1997) 267-273

MEDIUM FREQUENCY instability=> fit underestimates uncertainty

230th ACS NM - T1/223

Residuals Ce-144

NIM A390 (1997) 267-273

sign of LOW FREQUENCY deviation=> fit tends to hide it

230th ACS NM - T1/224

Residuals Y-88

NIM A390 (1997) 267-273

only a few measurement data=> LACK OF INFORMATION

230th ACS NM - T1/225

Residuals Cd-109

NIM A390 (1997) 267-273

OVERESTIMATION of measurement uncertainty=> overestimates uncertainty of half-life?

230th ACS NM - T1/226

Residuals 99Tcm

electrometerrange

switching

systematicdeviation

for short times

ARI 60 (2004) 317-323

230th ACS NM - T1/227

Conclusions

• perform many measurements with good statistical accuracy and carefully study the randomness of the residuals

• identify and quantify 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 EASY !

230th ACS NM - T1/228

3500

4000

4500

5000

5500

6000

6500

Flynn Hegedues Lloret Vaninbrouckx

Day

s

Half-life of 93mNb

230th ACS NM - T1/229

4920

4930

4940

4950

4960

Schrader Martin Unterw eger Kuzmonko Woods

Day

s

Half-life of 152Eu

a selection of the ‘best’ data

4700

4750

4800

4850

4900

4950

5000

2004 1998 1997 1992 1986 1983 1983 1980 1978 1972 1990

Day

s

230th ACS NM - T1/230

Residuals Co-57

NIM A390 (1997) 267-273

decreasing accuracy with time=> extension of experiment becomes ineffective

230th ACS NM - T1/231

Residuals Ba-133 and Eu-152

NIM A390 (1997) 267-273

‘independent’ measurements showing correlations=> ‘medium term’ detector instability