Error Analysis

1

Error Analysis

Analytical

02 October 2013

2

Penfold-Leiss Cross Section Unfolding

• Measure Yields at: where,

• The solution can be written in two forms:

nEEEE ,,, 21 niEE ii ,2,1

i

jjji

E

th ii EEENdkkkEnEY i

1

)(),,()(),()(

1

1

1 i

jjiji

iii NyN

3

NY

nnnnnn NNN

NNN

y

yy

2

1

,2,1,

22,21,

11,

2

1

0000

• Or, Matrix form:

YN 1

4

Statistical Errors

5

Statistical Error Propagation (1)

• With:

1 NB

YB • Then:

TBdYBd 22

• Note:

ii

i

yydy 1

bgiii yydy 2 In case of

background Subtraction

01

ijij

ij

NNdN

ii ydy

6

ny

yy

dY

00

0000

2

1

2

• Where:

0),cov(),var(

ji

iii

yyyyy

221

22212

12121

2

),cov(),cov(

),cov(),cov(),cov(),cov(

nnn

n

n

d

dd

d

7

Statistical Error Propagation (2)

ii

i

i

i

yydyd 1

22

1

1

1

1

1

1

222

2 ),cov(1 i

j

i

k

i

lillkikjiji

iii NNdNdy

Nd

For mono-chromatic

beam

Although,

0),cov(

,0),cov(

ji

ji yy

8

Statistical Error Propagation (Wrong)

YN 1

• Then:

2122 dYNd

• This is equivalent to:

1

1

222

2 1 i

jjiji

iii dNdy

Nd

Wrong

Wrong

9

Absolute Systematic Errors

10

Systematic Error Propagation (1)• For absolute beam energy uncertainty of δE (= 0.1%):

)()()(

iij

iijiij

ij

ij

ENENEEN

NdN

)()()(

ii

iiii

i

i

EyEyEEy

ydy

Ei (MeV) dyi/yi (%) dσi/σi (%)

7.9 17.4 12.6

8.0 12.3 10.5

8.1 10.0 9.1

8.2 8.6 7.1

8.3 7.6 6.3

8.4 6.8 5.8

8.5 6.1 5.2

This is the cross section dependence on energy

• Accounted for dNij due to energy error when calculating dyi

11

• With:

1 NB

YB • Then:

TBdNdYBd 2222

022.0017.0018.0018.0018.0018.0018.00017.0020.0021.0021.0021.0021.000020.0023.0025.0025.0025.0000025.0029.0031.0031.00000033.0039.0041.000000050.0058.0000000100.0

/ ijij NdN

iE

12

222

21

222

221

211

2 000

nnnn dNdNdN

dNdNdN

dN

• Where:

2

22

21

2

00

0000

n

2

22

21

2

00

0000

ndy

dydy

dY

13

Systematic Error Propagation (2)

21

1

2

1

1

1

1

1

1

222

2 ),cov(1

iii

i

jjij

i

j

i

k

i

lillkikjiji

iii

dNdN

NNdNdyN

d

14

Other Absolute Systematic Errors

2222

22 )( iii yTT

RR

IIEdydy

Beam Current, δI/I 3%

Photon Flux, δφ/φ 5%

Radiator Thickness, δR/R 3%

Bubble Chamber Thickness, δT/T 3%

Bubble Chamber Efficiency, ε 5%

• Then:

22

2ijij NdN

Simulation

15

ResultsI. Radiator Thickness = 0.02 mmII. Bubble Chamber Thickness = 3.0 cmIII. Number of 16O nuclei = 3.474e22 /cm2

IV. Background subtraction of 18O(γ,α)14C V. 17O(γ,n)16O: Still to do

16

ElectronBeam K.

E.

BeamCurrent

(µA)

Time(hour)

yi dyi (no bg)

dyi/yi (no bg,

%)

dyi (with bg)

dyi/yi(with bg,

%)7.9 100 100 545 23 4.2 134 24.6

8.0 100 20 581 24 4.1 77 13.3

8.1 80 10 852 29 3.4 60 7.0

8.2 20 10 634 25 3.9 40 6.3

8.3 10 10 812 28 3.4 39 4.8

8.4 4 10 746 27 3.6 36 4.8

8.5 2 10 763 28 3.7 32 4.2

11004.611137.912187.112435.112663.112875.112075.2012217.112852.112406.212908.212370.312801.30012086.312692.412097.612369.712540.800012258.612514.913236.113494.1000013539.213858.313013.50000013439.613782.900000014267.3

eeeeeeeeeeeee

eeeeeeeee

eeeee

e

N

17

ElectronBeam K. E.

Cross Section (nb)

Stat Error(no bg, %)

Stat Error(with bg, %)

7.9 0.046 4.4 24.5

8.0 0.185 6.0 20.7

8.1 0.58 6.3 14.7

8.2 1.53 8.2 13.8

8.3 3.49 9.1 13.3

8.4 7.2 10.6 13.8

8.5 13.6 12.2 14.8

ElectronBeam K. E.

Cross Section (nb)

Sys Error(Energy, %)

Sys Error(Total, %)

7.9 0.046 17.4 19.5

8.0 0.185 18.1 21.5

8.1 0.58 18.4 22.9

8.2 1.53 18.9 24.9

8.3 3.49 19.7 27.2

8.4 7.2 20.6 30.0

8.5 13.6 21.6 33.0

18

ElectronBeam K. E.

GammaEnergy (MeV)

ECM (MeV)

Cross Section

(nb)

SE1 Factor(keV b)

Stat Error(%)

Sys Error(%)

7.9 7.85 0.69 0.046 62.2 24.5 19.5

8.0 7.95 0.79 0.185 48.7 20.7 21.5

8.1 8.05 0.89 0.58 41.8 14.7 22.8

8.2 8.15 0.99 1.53 35.5 13.8 24.9

8.3 8.25 1.09 3.49 32.0 13.3 27.2

8.4 8.35 1.19 7.2 28.8 13.8 30.0

8.5 8.45 1.29 13.6 26.3 14.8 33.0

19

12C(α, γ)16O S-Factor Statistical Error: dominated by background subtraction from

18O(γ,α)14C (depletion = 5,000)

Systematic Error: dominated by absolute beam energy (δE = 0.1%)

SE1(300 keV) = 74±21 keV b

20

Relative Systematic Errors

21

Systematic Error Propagation (1)• For absolute beam energy uncertainty of δE (= 0.1%) and zero

relative beam energy uncertainty:

)()()(

iij

iijiij

ij

ij

ENENEEN

NdN

)()()(

ii

iiii

i

i

EyEyEEy

ydy

Ei (MeV) dyi/yi (%) dσi/σi (%)

7.9 12.5 12.6

8.0 10.8 10.5

8.1 9.3 9.1

8.2 8.0 7.1

8.3 7.0 6.3

8.4 6.3 5.8

8.5 5.6 5.2

This is the cross section dependence on energy

• Accounted for dNij due to energy error when calculating dyi

EE 8.70

iEEi 0

22

• With:

1 NB

YB • Then:

TBdNdYBd 2222

022.0017.0018.0018.0018.0018.0018.00017.0020.0021.0021.0021.0021.000020.0023.0025.0025.0025.0000025.0029.0031.0031.00000033.0039.0041.000000050.0058.0000000100.0

/ ijij NdN

iE

23

221

2212

1212

1

2

nnn

nn

n

dydydydydy

dydydydydydydydydydy

dY

• Where:Note: Correlation Coefficient =1

jiijji

iii

dydyyydyyy

),cov(),var( 2

221

22212

12121

2

),cov(),cov(

),cov(),cov(),cov(),cov(

nnn

n

n

d

dd

d

24

222

21

222

221

211

2 000

nnnn dNdNdN

dNdNdN

dN

2

22

21

2

00

0000

n

25

Systematic Error Propagation (2)

21

1

2

1

1

1

1

1

1

2

1

1

22

2

),cov(

21

iii

i

jjij

i

j

i

k

i

lillkikjij

i

jjijii

iii

dNdN

NNdN

dNdydyN

d

0),cov(

,0),cov(

ji

ji yy

26

Other Relative Systematic Errors

2222

22 )( iii yTT

RR

IIEdydy

• Then:

22

2ijij NdN

Simulation

Beam Current, δI/I 3%

Photon Flux, δφ/φ 5%

Radiator Thickness, δR/R 3%

Bubble Chamber Thickness, δT/T 3%

Bubble Chamber Efficiency, ε 5%

27

ElectronBeam K. E.

Cross Section (nb)

Sys Error(Energy, %)

Sys Error(Total, %)

7.9 0.046 12.5 15.3

8.0 0.185 10.2 13.5

8.1 0.58 8.3 12.2

8.2 1.53 7.0 11.4

8.3 3.49 6.0 10.7

8.4 7.2 5.3 10.5

8.5 13.6 4.7 10.1

Relative Systematic Errors Results

Note: Relative systematic errors do not get amplified in PL Unfolding

28

12C(α, γ)16O S-Factor Statistical Error: dominated by background subtraction from

18O(γ,α)14C (depletion = 5,000)

Relative Systematic Errors

SE1(300 keV) = 74±21 keV b