The Interplay Between the Statistical Correlationsof -ray Emission Probabilities and

Efficiency Calibration *

Ruy M. Castro 1,2, Vito R. Vanin 3, Paulo R. Pascholati 3, Nora L. Maidana 3

1 IEAv - CTA, São José dos Campos, Brasil2 DMAF - UNITAU, Taubaté, Brasil3 LAL - IFUSP, São Paulo, Brasil

14th International Conference on Radionuclide Metrology and its Applications ICRM2003 - Dublin, Ireland

Applied Radiation and Isotopes 60 (2004) 185-190

http://dx.doi.org/10.1016/j.apradiso.2003.11.014

* Presented in:

Contents

•Are the data correlated?

•Efficiency Calibration

•Example

•Decay parameters

Are the data correlated?

REFRelative P (P1408keV = 100 %)

121 keV 244 keV 344 keV 411 keV

1 145,0 ± 4,1 39,4 ± 1,3 128,2 ± 3,6 10,14 ± 0,54

2 138,5 ± 6,4 36,2 ± 1,8 128,2 ± 5,9 10,32 ± 0,51

3 132,9 ± 4,0 35,8 ± 1,0 128,2 ± 3,8 10,77 ± 0,38

4 144,6 ± 4,7 36,4 ± 1,2 128,2 ± 4,2 10,59 ± 0,27

5 141,0 ± 4,0 36,6 ± 1,1 127,2 ± 1,3 10,71 ± 0,11

6 140,6 ± 2,8 35,8 ± 0,6 128,2 ± 2,6 10,55 ± 0,22

7 136,9 ± 1,3 36,2 ± 0,3 127,1 ± 0,7 10,84 ± 0,07

8 136,7 ± 0,7 36,5 ± 0,4 126,9 ± 0,9 10,73 ± 0,10

Analyzing the 152Eu published data~ 32 experiments with the most intense transitions

Are the data correlated?

RI

RI RINI

30

25

20

15

10

5

010 11

RI411 keVRef

30

25

20

15

10

5

0-3 -2 -1 0 1 2 3

NI411 keV

Are the data correlated?

= 0.6

30

25

20

15

10

5

0-3 -2 -1 0 1 2 3

NI411 keV

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

NI 34

4 ke

V

Are the data correlated?

Some other cases:

Are the data correlated?

AreaF P

- Efficiency Area - Peak area F - Factor

•Activity•Normalization Constant•Reference area

P - Emission probability Usually:

• Area, F and P are independent• Area is uncorrelated• P are correlated for a multi gamma source, but their correlation coefficients are generally unknown

Efficiency

Efficiency Calibration

0

ln lniN

ii b

Ea

E

Eg – Gamma Energy

Eb – Reference Energy

Typical efficiency curve

For energies above 150-200 keV

Efficiency Calibration

1

2

ln

ln

ln n

Y

0 1ln ln ba a E E

Using a Matrix notation:

For example:

1

2

1 ln

1 ln

1 ln

b

b

n b

E E

E EX

E E

0

1

ap

a

Y X p

Efficiency Calibration

Fitting Procedure

11 1ˆ T Tp X V X X V Y

Problem: How to obtain V?

Variance in YWhat is V?due to: peak area, factor and emission probability

11ˆ

TpV X V X

Y Area Y F Y PV V V V

PF

AreaY lnln

Efficiency Calibration

Variance in Y due to Area: VY-Area

,

ln iArea i j

j i

AreaDArea F P

1

2

1 0 0

10 0

10 0n

Area

AreaD

Area

1

2

2

2

2

0 0

0 0

10 0n

Area

Area

Area

Area

V

TY Area Area Area AreaV D V D

1

2

2

1

2

2

2

0 0

0 0

0 0 n

Area

Area

Y Area

Area

n

Area

V Area

Area

Efficiency Calibration

Variance in Y due to F: VY-F

,

ln iF i j

i

AreaDF F P

1 1 1

1 1 1

1 1 1

F F F

D F F F

F F F

2F FV

TY F F F FV D V D

2 2 2

2 2 2

2 2 2

F F F

F F F

Y F

F F F

F F F

V F F F

F F F

Efficiency Calibration

Variance in Y due to P: VY-P

,

ln iP i j

j i

AreaDP F P

1

2

1 0 0

10 0

10 0

P

n

P

PD

P

1

2

n

2

1 2 1 n

2

2 1 2 n

2

n 1 n 2

cov , cov ,

cov , cov ,

cov , cov ,

P

PP

P

P P P P

P P P PV

P P P P

TY P P P PV D V D

1

2

n

2 1 2 1 n

1 1 2 1 n

2 2 1 2 n

2 1 2 2 n

2 n 1 n 2

n 1 n 2 n

cov , cov ,

cov , cov ,

cov , cov ,

P

P

Y P

P

P P P P

P P P P P

P P P PV P P P P P

P P P P

P P P P P

P

Efficiency Calibration

Fitting Procedure

11 1ˆ T Tp X V X X V Y

11ˆ

TpV X V X

Quality of the fitting: adjT

adj YYVYY 12

Interpolation:pXY

intint ˆT

pY intintintXVXV ˆ

Example

Typical efficiency calibration

• 1 multi gamma-ray source - 152Eu(partially know covariance)

• 2 105 Bq Activity• 20 cm source-detector distance• 2 h of measurement (live time)

Pile-up rejection and dead time correction

0

ln lniN

ii b

Ea

E

REF

AreaArea P

Relative efficiency

Example

Efficiency curve

Example

Calibration Data Energy I Correlation Matrix(keV) 344 411 778 1089 1299344 0,26689(13) 1 0,10 0,32 0,15 -0,07411 0,02229(3) 0,10 1 0,31 0,72 -0,08779 0,1303(8) 0,32 0,31 1 0,45 -0,20

1089 0,01712(3) 0,15 0,72 0,45 1 -0,121299 0,01612(8) -0,07 -0,08 -0,20 -0,12 1

Energy I Correlation Matrix(keV) 121 244 4445-2 44415-9 867 964 1085 1112 1212 1408121 0,2875(6) 1 -0,68 -0,70 -0,70 -0,69 -0,75 -1 -0,65 -0,62 -0,53244 0,0773(9) -0,68 1 0,76 0,76 0,62 0,69 0,68 0,62 0,59 0,52

4445-2 0,00316(4) -0,70 0,76 1 0,59 0,74 0,75 0,69 0,62 0,59 0,5944415-9 0,0263(3) -0,70 0,76 0,59 1 0,73 0,74 0,69 0,62 0,59 0,58

867 0,0422(5) -0,69 0,62 0,74 0,73 1 0,74 0,68 0,61 0,57 0,49964 0,1448(16) -0,75 0,69 0,75 0,74 0,74 1 0,74 0,66 0,62 0,531085 0,1004(12) -1 0,68 0,69 0,69 0,68 0,74 1 0,64 0,61 0,531112 0,1339(17) -0,65 0,62 0,62 0,62 0,61 0,66 0,64 1 0,55 0,491212 0,01432(20) -0,62 0,59 0,59 0,59 0,57 0,62 0,61 0,55 1 0,511408 0,208(3) -0,53 0,52 0,59 0,58 0,49 0,53 0,53 0,49 0,51 1

Example

Variance Matrix

Variance Matrix (x 106)244 344 411 444 778 867 964 1085 1089 1112 1212 1299 1408

244 153 7 7 92 7 92 96 102 7 98 104 7 94

344 7 11 7 7 8 7 7 7 7 7 7 7 7

411 7 7 41 7 9 7 7 7 9 7 7 6 7

444 92 7 7 122 7 90 85 86 7 82 87 7 87

778 7 8 9 7 66 7 7 7 12 7 7 1 7

867 92 7 7 90 7 179 104 103 7 99 102 7 91

964 96 7 7 85 7 104 141 105 7 99 104 7 91

1085 102 7 7 86 7 103 105 163 7 104 110 7 98

1089 7 7 9 7 12 7 7 7 86 7 7 6 7

1112 98 7 7 82 7 99 99 104 7 192 106 7 97

1212 104 7 7 87 7 102 104 110 7 106 332 7 111

1299 7 7 6 7 1 7 7 7 6 7 7 163 7

1408 94 7 7 87 7 91 91 98 7 97 111 7 227

VY-P (x 106)

244 344 411 444 778 867 964

244 136 0 0 85 0 86 89

344 0 0 0 0 1 0 0

411 0 0 2 0 3 0 0

444 85 0 0 92 0 83 79

778 0 1 3 0 38 0 0

867 86 0 0 83 0 140 97

964 89 0 0 79 0 97 122

Example

6,22 red

Results:With

covariances

a0 = 0,157(4)

a1 = -0,796(4) = 0,55

Without covariances

a0 = 0,1537(23)

a1 = -0,800(4) = 0,54

2,12 red

Example

Results: Uncertainties

with covariances without covariances

Ratio

0 500 1000 15000,8

1,0

1,2

1,4

1,6

Ratio

Energy (keV)

Decay parameters

Decay Scheme

Peak Areas

PF

Area

FPArea + corrections

P and correctionsdependents of the decay parameters

N

iif

0 1

1

0

1j

iij

Decay parameters

Decay Scheme

13

13301 1

fP

Constraints

13

13133

01

1330101 11

EffEFA total

Probability& area

Conclusion

• The emission probabilities of a multi gamma source are correlated

• The emission probabilities and efficiencies are dependents.

• Should use branching ratios and feeding fractions instead of gamma intensities.

References

• Ruy M. Castro , Vito R. Vanin , Otaviano Helene , Paulo R. Pascholati , Nora L. Maidana , Mauro S. Dias and Marina F. Koskinas, The Interplay Between the Statistical Correlations of -ray Emission Probabilities and Efficiency Calibration, Applied Radiation and Isotopes 60 (2004) 185-190http://dx.doi.org/10.1016/j.apradiso.2003.11.014

• Vito R. Vanin and Otaviano Helene, Covariance Analysis Within the Framework of the Least-squares Method in Update of X Ray and Gamma Ray Data Standards for Detector Calibration and Other applications, International Atomic Energy Agency, Viena (2007)