Transcript Workshop on DECAY DATA EVALUATION

Discrepant Data. Program LWEIGHT
Edgardo Browne
Decay Data Evaluation Project
Workshop
May 12 – 14, 2008
Bucharest, Romania
Statistical Analysis of Decay Data
• Relative g-ray intensities
• a-particle intensities
• Electron capture and b intensities
• Recommended standards for energies and
intensities
• Statistical procedures for data analysis
• Discrepant data
1.
Relative g-ray Intensities
Ig = A × e(Eg)
A ± DA … Spectral peak area
e± De … Detector efficiency
Several measurements with Ge detectors:
Ig1= A1 × e1, Ig2 = A2 × e2, …
e1, e2, … are determined with standard calibration
sources, thus they are not independent quantities.
Best value of Ig is a weighted average of Igi. A
realistic uncertainty Dig should not be lower than the
lowest uncertainty in the input values.
Same criterion applies to g-ray energies.
Precise half-life values are important for g-ray
calibration standards
The IAEA Coordinated Research Programme (CRP) gives:
dT1/2/T1/2  0.00144 T1/2 /T1, where
T1 is the maximum source-in-use period for a given
radionuclide (15 years or 5 half-lives), whichever is shorter.
Then the contribution to the uncertainty in the radiation
intensity calibration using this radionuclide will not exceed
0.1%.
Example: 133Ba - T1/2 =10.57± 0.04 y - T1 = 15 y, then
dT1/2/T1/2 = 0.00144 x 10.57/15 = 0.0010,
Experimental value is 0.04/10.57 = 0.0039.
The contribution to the uncertainty is >0.1%.
A = A1(434) + A2(614) + A3(723)
The areas of the individual peaks are not
independent of each other.
DO NOT use A1(434), A2(614), and A3(723) to
determine T1/2(434), T1/2(614), and T1/2(723),
respectively, and then average these values to
obtain T1/2.
Use “A” to determine T1/2.
2.
a-particle Intensities
Ia = A × e
A ± DA … Spectral peak area
e ………..Geometry (semiconductor detectors)
e is the same for all a-particle energies.
Best value of Ia is a weighted average of Iai.
Uncertainty is the external (multiplied by c)
uncertainty of the average value.
Same criterion applies to a-particle energies, but
because of the use of standards for energy
calibrations, a realistic uncertainty should not be
lower than the lowest uncertainty in the input values.
3.
Electron Capture and b Intensities
Most electron capture and b intensities are from
g-ray transition intensity balances.
Ib,e
IN
OUT
Ib or Ie = OUT - IN
4.
Recommended Standards for
Energies and Intensities
Recommended standards for g-ray energy calibration
(1999), R.G. Helmer, C. van der Leun, Nucl. Instrum.
and Methods in Phys. Res. A450, 35 (2000).
Update of X Ray and Gamma Ray Decay Data
Standards for Detector Calibration and Other
Applications, IAEA-Report, Vienna 2007.
Recommended Energy and Intensity Values of Alpha
Particles from Radioactive Decay, A. Rytz, Atomic
Data and Nuclear Data Tables 47, 205 (1991)
I strongly suggest reading the following
paper
Decay Data: review of measurements, evaluations
and compilations, A.L. Nichols, Applied Radiations
and Isotopes 55, 23 (2001).
5.
Statistical Procedures for Data
Analysis
Averages
Unweighted
x(avg) = 1 / n  xi
sx(avg) = [ 1 / n (n – 1)  (x(avg) – xi)2]1/2 Std. dev.
Weighted
x(avg) = W  xi / sxi2 ; W = 1 /  sxi-2
c2 =  (x(avg) – xi)2 / sxi2 Chi sqr.
cn2 = 1 / (n – 1)  (x(avg) – xi)2 / sxi2 Red. Chi sqr
sx(avg) = larger of W1/2 and W1/2 cn. Std. dev.
Discrepant Data
•Simple definition: A set of data for which cn2 > 1.
•But, cn2 has a Gaussian distribution, i.e. it varies
with the number of degrees of freedom (n – 1).
•Better definition: A set of data is discrepant if cn2 is
greater than cn2 (critical). Where cn2 (critical) is such
that there is a 99% probability that the set of data is
discrepant.
c2n (critical)
[n=N-1]
n
c2n (critical)
n
c2n (critical)
----------------------------------1
2
3
4
5
6
7
8
9
10
6.6
4.6
3.8
3.3
3.0
2.8
2.6
2.5
2.4
2.3
11
12
13
14
15
16
17
18 - 21
22 - 26
27 - 30
> 30
2.2
2.2
2.1
2.1
2.0
2.0
2.0
1.9
1.8
1.7
1 + 2.33  2/n
Limitation of Relative Statistical Weight
Method (Program LWEIGHT)
For discrepant data (c2n > c2n(critical)) with at least
three sets of input values, we apply the Limitation of
Relative Statistical Weight method. The program
identifies any measurement that has a relative
weight >50% and increases its uncertainty to reduce
the weight to 50%. Then it recalculates c2n and
produces a new average and a best value as
follows:
• If c2n  c2n(critical), the program chooses the
weighted average and its uncertainty (the larger of
the internal and external values).
• If c2n > c2n(critical), the program chooses either
the weighted or the unweighted average,
depending on whether the uncertainties in the
average values make them overlap with each
other. If that is so, it chooses the weighted
average and its (internal or external) uncertainty.
Otherwise, the program chooses the unweighted
average. In either case, it may expand the
uncertainty to cover the most precise input value.
Simple Example
X=
500±1
1000±100
n=N - 1
X(avg)= 500 ± 5
cn =25,
2
c 2 (critical) =6.6
n
Data are discrepant
We change to 500±100 (Same statistical weights). Then
X(avg)= 750 ± 250
44Ti
Half-life
T1/2$REF HALF-LIFE
99Wi01 60.7 1.2
98Ah03 59.0 0.6
98Go05 60.3 1.3
98No06 62.0 2.0
90Al11 66.6 1.6
83Fr27 54.2 2.1
44Ti
Half-life (LWEIGHT)
44Ti Half-life Measurements
INP. VALUE INP. UNC.
R. WGHT chi**2/N-1 REFERENCE
.607000E+02 .120E+01
.141E+00 .826E-01
99Wi01
.590000E+02 .600E+00 MIN *.563E+00* .479E+00
98Ah03
.603000E+02 .130E+01
.120E+00 .163E-01
98Go05
.620000E+02 .200E+01
.507E-01 .214E+00
98No06
.666000E+02 .160E+01
.792E-01 .348E+01
90Al11
.542000E+02 .210E+01
.460E-01 .149E+01
83Fr27
No. of Input Values N= 6 CHI**2/N-1= 5.76 CHI**2/N-1(critical)= 3.00
UWM
:.604667E+02 .164796E+01
unweighted average
WM
:.599288E+02 .450317E+00(INT.) .108057E+01(EXT.)
weighted average
INP. VALUE INP. UNC.
R. WGHT chi**2/N-1 REFERENCE
.607000E+02 .120E+01
.161E+00 .563E-01
99Wi01
.590000E+02 .681E+00
*.500E+00* .487E+00
98Ah03
* Input uncertainty increased .114E+01 times *
.603000E+02 .130E+01
.137E+00 .663E-02
98Go05
.620000E+02 .200E+01
.580E-01 .188E+00
98No06
.666000E+02 .160E+01
.907E-01 .334E+01
90Al11
.542000E+02 .210E+01
.526E-01 .156E+01
83Fr27
No. of Input Values N= 6 CHI**2/N-1= 5.63 CHI**2/N-1(critical)= 3.00
UWM
:.604667E+02 .164796E+01
unweighted average
WM
:.600634E+02 .481846E+00(INT.) .114378E+01(EXT.)
weighted average
LWM
:.600634E+02 .114378E+01
Min. Inp. Unc.=.600000E+00 LWEIGHT value
LWM has used weighted average
and external uncertainty
Recommended value: 60.0 (11) y
I strongly suggest reading the following
paper
M.U.Rajput, T.D.Mac Mahon, Techniques for
Evaluating Discrepant Data, Nucl.Instrum.Methods
Phys.Res. A312, 289 (1992).