Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Neutron Induced
Cross Section Measurements
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies
Trieste, Italy, 19 – 30 May 2008
peter.schillebeeckx@ec.europa.eu
Joint Research Centre (JRC)
IRMM - Institute for Reference Materials and Measurements
Geel - Belgium
http://irmm.jrc.ec.europa.eu/
http://www.jrc.ec.europa.eu/
1
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Neutron cross section measurements (Part 1)
• Data reduction and uncertainties (Part 2)
– Experimental observables
– Definitions and terminology
– Fitting a mathematical model to experimental data
– Basic operations
– AGS
– Example
ISO Guide, “Guide to the expression of uncertainty in
measurement”, Geneva, Switzerland, ISO,1995
Mannhart, “A Small Guide to Generating Covariances of
Experimental Data”, Juni 1981, PTB-FMRB-84
F.H. Fröhner, Nucl. Sci. Eng. 126 (1997) 1 -18
2
Experimental observables
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Experimental observables
– Transmission
– Reaction yields
• Definitions and terminology
• Fitting a mathematical model to data
• Basic operations
• AGS
• Examples
3
Transmission
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
SAMPLE IN
Total
BKG fit
BKG points
2
10
1
10
0
10
-1
10
Texp  NT
-2
3
10
4
10
5
10
'
'
Cin
 B in
C 'out  B 'out
6
10
TOF / ns
Total
BKG fit
BKG points
2
10
1
10
0
10
-1
10
-2
10
3
10
4
10
1.0
0.8
0.6
0.4
0.2
0.0
0
20000 40000 60000 80000
Neutron Energy / eV
5
10
TOF / ns
Transmission
10
SAMPLE OUT
3
10
Response / (1/ns)
3
10
Response / (1/ns)
4
6
10
Transmission : data reduction
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
5
Sample in
Sample out
(1) Dead time correction
(1) Dead time correction
(2) Background subtraction
(2) Background subtraction
'
'
Cin
 Bin
Texp  NT
C'out  B 'out
RRR
Resonance shape analysis
Texp (Tn )   R T (Tn , E n ) e tot (En ) dE n
URR
Correction for resonance structure
n2
 n tot
ntot 
e
e
(1 
var  tot  ...)
2
 Texp    e  ntot 
Reaction yield : data reduction
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
6
Reaction measurement
Flux measurement
(1) Dead time correction
(1) Dead time correction
(2) Background subtraction
(2) Background subtraction
  Cr'  Br'
Yexp  Nr
r C'  B'
RRR
Resonance shape analysis
Yexp (Tn )  Nr  RT (Tn ,En ) r (En ) Yr (En ) dEn
URR
Correction for self-shielding
and multiple scattering (Fc)
 (n,r )  Fc  Yexp 
Dead time correction (1/2)
232
t
tk  ti  t
1
DTC ( t i ) 
si
1
To
ti-1 ti ti+1
si 
Time
ti
 Nj
j t k
Th(n,) at 15 m
7
Before Correction
10
2
1
10
1.15
DTC
1.10
DTC
t1 t2 t3 t4
Response (Counts / ns)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
To is the number of bursts
1.05
1.00 4
10
10
5
10
6
Time - Of - Flight / ns
System dead time for on-line data processing (time-of-flight + amplitude):
– Acqiris DC282 (digitizer) = 350 ns
– CAEN N1728B (digitizer) = 560 ns
– Conventional system DAQ2000 = 2800 ns (without amplitude similar to digitizers)
Moore, Nucl. Instr. Meth. 169 (1980) 245
Mihailescu et al., submitted to NIM A
Dead time correction (2/2)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
8
20
Response (Counts / ns)
232
Th(n,) at 15 m
Before
After (/ 100)
100
15
10
100000
10
200000
300000
200000
300000
20
1
0.1
100000
15
200000
300000
Time - Of - Flight / ns
10
100000
Background correction (1/3)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Response (counts / ns)
10
9
0
Total
Background filters
Average Values
Fit
Black Resonance Filters
Element
-2
10
Ag
5.19
W
18.83
Co
132.00
Na
2850.00
S
-4
10 1
10
2
10
3
10
4
10
5
10
6
10
Neutron Energy / eV
Detection Principles
Energy / eV
102710.00
Background correction (2/3)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
TOF - spectrum / (1/ns)
10
10
Black Resonance Filters
1
Exp.
Fit.
Black Resonances
10
10
10
0
Element
-1
-2
102710
10
2850
132.00
18.83 5.19 eV
-3
1
10
100
Time-Of-Flight / s
1000
Energy / eV
Ag
5.19
W
18.83
Co
132.00
Na
2850.00
S
102710.00
Background measurement (3/3)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
BR- filters
11
detector
Definitions and terminology
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Experimental observables
• Definitions and terminology
– Probability and statistical quantities
– Error and Uncertainty
– Propagation of uncertainties
• Fitting a mathematical model to data
• Basic operations
• AGS
• Examples
12
Probability and statistical quantities
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
13
P(x) theoretical probability density function (PDF) of x
P(x,y) theoretical probability density function of (x,y)
• Mean
 x  x   x P( x) dx
• Variance
 2x  ( x   x )2   x   x 2 P( x ) dx
• Covariance
 2xy  ( x   x )( y   y )   x   x  y   y P( x, y ) dxdy
• Correlation

( x, y ) 
 2xy
x y

Observed statistical quantities
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
14
n observations x1, …, xn of variable x
m observations (x1,x1) …, (xn,yn) of variable (x,y)
• Sample mean
1 n
x   xi
n i 1
• Sample variance
s 2x 
• Sample covariance
s 2xy
• Sample correlation


2
1 n
x

x
 i
n  1 i 1


1 n n

  xi  x yi  y
n  1i 1j 1
r( x, y ) 
s xy
sx sy

Function of variables : linear function
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
15

n observations x1, …, xn of variable x
• Linear function
z = f(x1,…,xn)
k
z  f(x i )   ai x i
i1
• Mean
k
 z  f(x i )   ai i
i1
• Variance
 2z 
f ( x i )   f 2

 a i2  2xi    a i a j  xix j
i
i j
Function of variables : non-linear function
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Taylor expansion
16
z  f ( x i )  f ( i )  
i
(1st order !)
f
x i
(x i  i )

z  f ( x i )  f ( i )   gi ( x i   i )
gi 
i
• Mean
 z  f ( x i )  f ( i )
• Variance
 2z 
f ( x i )   f 2

f
x i

 gi2  2xi    gi g j  xix j
i
i j
Matrix notation
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Linear function
z  f (x)  A x
– Mean
z  A x
– Variance
Vz  A Vx AT
• Non-linear function
z  f ( x)  f ( x )  G x ( x   x )
(1st order Taylor !)
– Mean
 z  f ( x )
– Variance
V z  G x V x GTx
17
fi
aik 
x k
gx,ik
fi

x k
sensitivity matrix
design matrix
Error and Uncertainty (ISO Guide)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Uncertainty (dispersion (or width), always > 0)
“Uncertainty is a parameter associated with the result of a
measurement, that characterizes the dispersion of the values
that could reasonably be attributed to the measurand”
• Error (difference between two quantities, can be + or -)
“Error is the result of a measurement minus
a true value of the measurand”
18
Error
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
'
'
Cin
 Bin
Texp  NT
C'out  B 'out
• Systematic error
Mean that would result from an infinite number of measurements of the
same measurand carried out under repeatability conditions minus a true
value of the measurand.
The expectation value of the error arising from a systematic effect = 0
• Random error
Result of a measurement minus the mean that would result from an
infinite number of measurements of the same measurand carried out
under repeatability conditions.
The expectation value of a random error = 0
19
Systematic error (ISO Guide)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• If a systematic error arises from a recognized effect, the effect can be
quantified and a correction or correction factor should be applied.
• The uncertainty on the correction factor should not be quoted as a
systematic error.
• One should not enlarge the uncertainty to compensate for a systematic
effect that is not recognized!
20
Evaluation of uncertainty components
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
21
• Type A evaluation
– Method of evaluation of uncertainty by the statistical analysis of series of
observations
– Is obtained from an observed frequency distribution
• Type B evaluation
– Method of evaluation of uncertainty by means other than the statistical analysis of
series of observations.
– Is obtained from an assumed probability density function (subjective probability) .
– The evaluation is based on scientific judgement
 Theoretical distribution
 Experience from previous measurement data
 Data provided in calibration or other certificates
• Propagation of uncertainties  combined uncertainty
V f  Gx V x Gx
T
gx,ik 
fi
x k
non-linear problem : approximation!
Example : Counting experiment
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
22
• Type A evaluation
– Perform repeated measurements and record the number of
events in time t
– Calculate the uncertainty from the standard deviation of the
observed frequency distribution
1 n
s 2x 
 x i  x 
2
n  1 i 1
• Type B evaluation : Poison statistics
– The observed quantity is distributed as a Poison distribution
– The uncertainty is defined by the standard deviation of a Poison
distribution
2x   x
How to quote uncertainties
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
23
• Evaluation of uncertainty components
– Type A : statistical analysis of repeated measurements
– Type B : scientific judgement
• All uncertainties are statistical !
(uncertainty due to counting statistics, uncertainty on the normalization, …)
(correlated or not-correlated uncertainties)
• Standard (ux) or expanded uncertainty
s
( x  u x ) with u x  x
n
– Standard uncertainty
:
– Expanded uncertainty
:
p > 1
e.g.
p = 1.96

p = 1  p = 0.68
 

P  x  x  p
p
n

p = 0.95
Fitting a mathematical model to data
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Definitions and terminology
• Experimental observables
• Fitting a mathematical model to data
– Least squares – Maximum likelihood
– Generalized least squares method
• Basic operations
• AGS
• Examples
24
Central limit theorem
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
25
Central limit theorem:
The sum of a large number of independent and identically-distributed
random variables will be approximately normally distributed
If x1, …, xn are independent and identically distributed with mean x
and variance 2 then :
for large n :
the distributi on of x 


x
i
n
P x  x ,  x dx 
i
is the normal distributi on with parameters ( x ,
 1  x   2 
1
x  
exp   
dx



2  x 
2x


with  x 

n

)
n
Central limit theorem
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
26
General: based on maximum entropy principle
Fröhner, Nucl. Sci. Eng. 126 (1997) 1
For observables x  ( x1,..., xn )
with covariance matrix
Vx
which are identically distributed with mean  x


P x  x , V x dx 
1
 1

1
exp   ( x   x )T V x ( x   x ) dx
det( 2V x )
 2

Fitting a mathematical model to data
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
27
Problem: n data points (x1,z1), …, (xn, zn) and a model function z  f ( x, a) that
in addition to the variable x also depends on k parameters a1,…,ak with n>k.
Solution:
(1) Least squares : find the vector a such that the curve fits best the given data in
the least squares sense, that is, the weigthed sum of squares is minimized:
1
2 (a)  (zexp  f ( x, a))T V Z (zexp  f ( x, a))
(2) Maximum likelihood: find the vector a that maximizes the likelihood function.
When the PDF is the normal distribution :


 1

1
P a z, V z da  exp   ( z exp  f ( x, a))T V z ( z exp  f ( x, a))  da
 2

Least squares method is equivalent to Maximum likelihood
Generalized least squares
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
28
Input from experiment : zexp and Vz, xexp and Vx
• Linear model
• Non-linear model (1st order Taylor)
zm  f ( x, a)  Ga a
V  ( V z  Gx V x G )
T
x
1
ga,ij 
fi
a j
zm  f ( x, a)  f ( x, ao )  Ga (a  ao )
gx,ij 
fi
x j
V  ( V z  Gx V x Gx )
1
a  (Ga V Ga )1(Ga V zexp )
T
T
1
V a  (Ga V Ga )1
T
T
1
1
(a  ao )  (Ga V Ga )1Ga V ( z exp  f ( x, ao ))
T
T
1
V aao  (Ga V Ga )1
T
solved by iteration
Van der Zijp (proceedings ESARDA)
Basic operations
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Definitions and terminology
• Fitting a mathematical model to data
• Basic operations (+ , - , x ,  )
– Background
– Normalization
– Fit correlated data
• AGS
• Examples
29
Background
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
30
Example: y vector dimension 2  z = f(y1, y2, b)
z  y b
Vy uncorrelated uncertainties from counting statistics
 2
 y1
: Vy = 
0

0 

 2y 
2 
2
Vb uncertainty on background (not correlated with Vy) : Vb =  b
 1 0  1
Vz = 

0
1

1


 2
 y1
 0

 0

0
 2y
0
2
0 1

0   0

 b2    1

Vz  A Vx AT
a ik 
0
1
 1
fi
x k

 2   2
b
 y1
 2
b



 2y   b2 
2

 b2
Background
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
31
(y1-b, y2-b) = (z1,z2)  (z1+z2, z1-z2)
 2   2
b
y
V( z 1 , z 2 )   1 2
 
b



2
2
 y  b
2

 b2
  2   2  4 2
b
y
y
V( z 1  z 2 , z 1  z 2 )   1 2 2 2
  
y1
y2

 2y   2y 
1
2 
 2y   2y 
1
2 
 2z  z   2y   2y  4b2
1
2
1
2
 2z  z   2y   2y
1
2
1
2
Background
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
32
(y1-b, y2-b) = (z1,z2)  (z1+z2, z1-z2)
Covariance

2
z1  z 2

2
z1  z 2
Only diagonal terms

    4
2y1  2y2  2b2

 
2y1  2y2  2b2
2
y1
2
y1
2
y2
2
y2
2
b
Propagation of uncertainties : covariance
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
33
(y1-b, y2-b) = (z1,z2)  (z1+z2, z1-z2) =(v1,v2)  (v1+v2)=2z1
Covariance
 2   2  4 2
b
y
y
 22z  Vv 1  v 2  1 1  1 2 2 2
  
1
y1
y2

 2y   2y 
1
2 
 2y   2y 
1
2 
Only diagonal terms
 22z  2 2y  2 2y  4b2  2 2z  2 2z
1
1
2
1
2
1
1


4( 2y   b2 )  4 2z
1
1
Normalization
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
34
Example: y vector dimension 2  z = f(y1, y2, N)
z Ny
Vy uncorrelated uncertainties from counting statistics
 2
 y1
: Vy = 
0

0 

 2y 
2 
2
VN uncertainty on normalization (not correlated with Vy) : VN = N
N 0 y 1 
Vz  

0 N y 2 
not -linear
 2
 y1
 0

 0

0
 2y
0
2
0  N


0  0

2   y 1
N

0  2 2
2 2
N


y
N
1
y1
 
N
 y1 y 2  2
N

y2  
2

y1 y 2 N

2 2
2 2
N  y  y 2 N
2

Normalization
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
35
(Ny1, Ny2) = (z1,z2) =  (z1z2, z1/z2)
2
N2  2  y 2  2

y
y

1 2 N
1
N
y
1

V( z 1 , z 2 )  
2
2
2
2
2
 y y 
N  y  y 2 N 
1 2 N
2



y12 2 2 
4
2
2
4
2
2
2
2
2
2
2
2
N y   N y   4N y y 

2 y1
1 y2
1 2 N N  y1  2 N  y 2


y2

V( z 1z 2 , z 1 / z 2 )  
2
2
2
y


y1 2 2
y1 2
2
2
1


N y 
N y

y
2
2
4
1
2
2


y2
y2 y2


2
 2z z  N4 y 22  2y  N4 y12  2y  4N2 y12 y 22 N
1
2
 2z / z
1
2
1
2
 2y
y
 1  1  2y
2
y 22 y 24
2
Normalization
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
36
(Ny1, Ny2) = (z1,z2) =  (z1z2, z1/z2)
Covariance
 2z z 
1
2
 2z / z 
1
2
2
N4 y 22  2y  N4 y12  2y  4N2 y12 y 22 N
1
2
 2y
y
1
 1  2y
2
y 22 y 24
2
Only diagonal terms
2
N4 y 22  2y  N4 y12  2y  2N2 y12 y 22 N
1
 2y
2
2
2
y

1
 1  2y  2 N
2
y 22 y 24
N2
y12
y 22
Fit correlated data:
common “offset “ uncertainty
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
37
z1 and z2 are observables of the same quantity K which are deduced from the
experimental data (y1,y2) affected by a common “offset error” with uncertainty b
and mean =0.
The covariance matrix for (z1,z2)= (y1-0,y2-0) is :
V( z1,z 2 )
 2   2
b
y
 1 2
 
b



2
 b

 b2
 2y
2
where y1 and y2 are the uncorrelated uncertainty components of y1 and y2 (e.g.
due to counting statistics).
The best value K is obtained by minimizing the expression:
 2 (k )  ( z exp  K ) T V (z1 ,z
1
2)
( z exp  K )
Fit correlated data:
common “offset” uncertainty
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Solution :
 2 (k )  ( z exp  K ) T V (z1 ,z
The best value k:
1
K
38
2)
z 1 2y  z 2  2y
2
1
 2y   2y
1
with uncertainty :
( z exp  K )
 K2 
 2y  2y
1
  b2
2
 2y   2y
1
2
2
(weighted average)
Fit correlated data:
common “normalization” uncertainty
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
39
z1 and z2 are observables of the same quantity K which are deduced from the
experimental data (y1,y2) affected by a common “normalization” with uncertainty N
and mean N = 1.
The covariance matrix for (z1,z2) = (N y1, N y2) with N = 1 is :
V( z1,z 2 )
 2  y 2  2
1 N
y
 1
 y 1y 2  2
N



2 2
 y 2 N

2
y 1y 2  N
 2y
2
where y1 and y2 are the uncorrelated uncertainty components (e.g. due to
counting statistics).
The best value k is obtained by minimizing the expression:
 2 (k )  ( z exp  K ) T V (z1 ,z
1
2)
( z exp  K )
Fit correlated data:
common “normalization” uncertainty
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Solution :
40
 2 (k )  ( z exp  Z)T V (z1 , z ) ( z exp  Z)
1
2
The best value K:
with uncertainty :
K
z 1 2y  z 2  2y
2
 2y
1
 2Z 

 2y
2
( weighted average)
1
 (y1  y 2 )
2
2
N
2
 2y  2y  ( y 12  2y  y 22  2y )  N
1
2
2
1
2
 2y   2y  ( y 1  y 2 ) 2  N
1
2
Only in case (y1 – y2)2N2 is small K approaches the weighted average.
 Known as : ”Peelle’s Pertinent Puzzle”, see Fröhner NSE 126 (1997) 1
 Due to “non-linearities”
Fit correlated data:
common “normalizaton” uncertainty
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
41
Solution : treat the normalization as a separate observable
Observables: (N’exp, y1, y2) with covariance matrix V(N’, y1, y2)
The function is : (N’exp, y1,y2) = (N’,KN’, KN’) = f(N’,K)
V(N',y1,y 2 )
 2
 N'
 0

 0

0
 2y
1
0
(note N’ = 1/N)

0 
0 

 2y 
2 
(N’, K) are determined by minimizing:
 2 (N' , K )  ((N' exp , y 1, y 2 )  (N' , N' K, N' K ) T V (N1',y ,y
1
2)
((N' exp , y 1, y 2 )  (N' , N' K, N' K ))
Example
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
42
Two measurements (z1, z2) of variable Z each with 10% uncorrelated
uncertainty and 20% uncertainty on the calibration factor N:
z1 = 1.5
z2 = 1.0
Vz1,z 2
0.15 2  (1.15 x0.20 ) 2

2
 1.0 x1.5 x0.20
0.1125 0.06 


0.05 
 0.06
1.0 x1.5 x0.20 2 

0.10 2  (1.0 x0.20 ) 2 
Best value for Z ?
Weighted average (only uncorrelated terms)
: 1.154  0.083
Least square without covariance
: 1.154  0.083
Least squares with covariance
: 0.882  0.218
Least squares with calibration factor as parameter
: 1.154  0.254
Least squares without covariance + adding uN afterwards
: 1.154  0.245
Summary : Z = N (C – B) (1/2)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
43
• Uncertainties
– VC : diagonal, uncorrelated uncertainties due do counting statistics
Poison (Type B), or preferably by repeated measurements (Type A)
– B : background subtraction introduces correlated uncertainties
– N : normalization
 introduces correlated uncertainties
• N and B can be considered as due to systematic effects. The value for
B and N are the systematic corrections. These corrections have their
uncertainties (or better covariance matrix)
• Not correcting for N and/or B implies that the value Z can not be
quoted !
Summary : Z = N (C – B) (2/2)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Be careful with large uncertainties on the normalization.
• Quote all the components involved in the data reduction process
• Separate as much as possible the components which create
correlated uncertainties
• Include the normalization in the model
44
General : Zi = f(x1,…,xn, c1,…,cm)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
45
• xi
: VX
only uncorrelated uncertainties (counting statistics)
• ci
: Vc
correlated uncertainties (normalization, background, …)
 x1 
 . 
 
 . 
 
 . 
 xn 
Y 
 c1 
 . 
 
 . 
 . 
 
c m 
fi
gik 
Yk
 2
 x1
 0

V Y   ...

 0
 0

0
...
0
...
0
...
...
...
0
...  2x
n
...
0
 2x
0
2
0 

0 

... 

0 
V c 

V Z  G TY V Y G Y
Experimental observables
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
T = f( NT,td, Cin, Bin, Cout, Bout)
46
Yr = f( Nr, r, td, Cr, Br, C, B)
• C
counting statistics (not correlated)
• td
utd from least square fit of time interval distribution
• B
least square fit (correlated uncertainties)
• NT
uNT  0.5 % (alternating sequences of “in-out” measurements)
Borella et al., Phys. Rev. C 76 (2007) 014605
• r
depends on method
• Nr
depends on method (related to r)
(n,) for total energy principle with PHWT uNr  2%
Borella et al. Nucl. Instr. Meth. A577, (2007) 626
AGS: Analysis of Generic TOF-Spectra
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Reaction yield
  Cr'  Br'
Yexp  N
r C'  B'
47
Transmission
C’
dead time corrected counts
B’
background contribution
N
normalization factor
T exp  N
'
'
Cin
 Bin
C'out  B 'out
Histogram operations + Covariance information (AGS)
Yexp + covariance
Texp + covariance
input
Reaction models : Resonance parameters + covariances
Cross sections + covariances
Uncertainty propagation in AGS
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
48
Z : function of vectors Z1,Z2, and parameter vector b:
Z = F(b, Z1, Z2)
Uncertainty propagation results in:
T
T
 F 
 F 
 F
 F 
 F 
 VZ 
  
VZ    Vb    
1
 b 
 b 
 Z1 
 Z1 
 Z 2
AGS


 VZ
2

 F

 Z 2
T

  ...

VZ  S Z S Z  D Z
T
correlated part
dimension: n x k
uncorrelated part
diagonal : n values
n : length of vector Z
k : number of common sources of uncertainties
Uncertainty in AGS : Z1 = F(a1,Y1)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
49
Z1 = F( a1 , Y1 )
VZ1
 F 
 Va
 
1

a
 1
T
 F 
 F 
 F 

  
 VY 

1

a

Y

Y
 1
 1
 1
T
Z1, Y1
a1
: dimension n
: dimension k1
Va1 covariance matrix
(symmetric & positive definite)
VY1
Va1 = La1 La1T (Cholesky decomposition)
 F  only diagonal terms


 Y1 
La : lower triangular matrix
only diagonal terms : DY1 = VY1
D Z1
 F 
 L a
S a1  
1
 a1 
 F 
 F 
 D Y 

 
1

Y

Y
 1
 1
2Y1u
VZ1  Sa1 Sa1  DZ1
T
dimension: n x k1
T
n values (diagonal)
Example : Z = a Y
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
50
Z=aY
 F 
 F 
VZ    Va  
 a 
 a 
Va   a  a
T
 F 
 F 

 VY 

 Y 
 Y 
VY
T
Z, Y
a
: dimension n
: dimension 1
only diagonal terms : DY = VY
 Z  only diagonal terms


 Y 
F
Y
a
DZ  a2 DY
Sa  Y a
VZ  S a S a T  D Z
dimension: n x 1
n values (diagonal)
a 2 2Y u
Uncertainty in AGS : Z = F(b, Z1, Z2)
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
51
Z, Z1, Z2
b
Z = F( b , Z1 , Z2)
Vb = Lb LbT
 Z 
Sb    L b
 b 
VZ1 = Sa1 Sa1T + DZ1
VZ2 = Sa2 Sa2T + DZ2
dim Sa1 = n x k1
dim Sa2 = n x k2
T
T
 F 
 F 
 F
 F 
 F 
 VZ 
  
VZ   Vb    
1
 b 
 b 
 Z1 
 Z1 
 Z 2
VZ  S b S b

S Z   Sb


T
 F 
 S a S a T
 
1
1

Z
 1
 F 

 S a
1
 Z1 
 F

 Z 2

 S a
2

: dimension n
: dimension kb
T
 F 
 F

  
 Z1 
 Z 2

 S a S a T
2
2





 F

 Z 2

 VZ
2




T
 F

 Z 2



T
T
 F 
 F 
 F
 D Z 
  
 
1
 Z1 
 Z1 
 Z 2
T
 F 
 F 
 F
 D Z 
  
D Z  
1
 Z1 
 Z1 
 Z 2

 D Z
2


 D Z
2

VZ  S z S z  D Z
T
dimension: n x k
k = kb + k1 + k2
n values (diagonal)
 F

 Z 2



 F

 Z 2



T
T
AGS, File format
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
52
Storage and calculus
Covariance matrix
n2 elements (e.g. 32k x 8 bytes
2n2 mult. & sum. /step (20 steps
 8 Gb !)
 8x1010 flops!)
AGS representation
n (k+1) elements (32k, 20 corr.
n (k+1) mult. & sum./step (20 steps
X Z Dz
Sz
 5 Mb)
 7x103 flops)
AGS commands
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
ags_mpty
ags_getA
ags_getE
ags_getXY
ags_addval
ags_avgr
ags_func
ags_idtc
ags_divi
ags_mult
ags_lico
ags_ener
ags_fit
ags_fxyp
ags_edit
ags_list
ags_putX
ags_scan
Write only commands
Create an empty AGS file
Import spectra from another AGS file
Import/interpolate evaluated data from an ENDF file
Import histogram data from an ASCII file
Read/Write commands : Operations on spectra
Add a constant value to all Y-values of a spectrum
Average Y values per channel
Calculates the Y values for a special function
Determine the dead time correction of a TOF-spectrum
Divide a spectrum by another
Multiply a spectrum with another
Linear combination of n spectra with n constants
Build energy from TOF X-vector
Non-linear fit of spectra
User-programmed function
Read Only commands
Edit constants and scalers attached to a spectrum
List Y values of spectra with common X values
Export final result to an ASCII file
Scan the contents of an AGS file
53
AGS, Script for transmission data
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
54
# create ags-file
ags_mpty TRFAK
# read sample out
scaler=TOout,CMout
ags_getXY TRFAK /SCALER=$scaler /FROM=spout.his /ALIAS=SOUT
# read sample in
scaler=TOin,CMin
ags_getXY TRFAK /SCALER=$scaler /FROM=spin.his /ALIAS=SIN /LIKE=A01SOUT
# dead time correction
dtcoef=DTCOEF
ags_idtc TRFAK,A01SOUT /DTIME=$dtcoef /LPSC=1
ags_idtc TRFAK,B01SIN /DTIME=$dtcoef /LPSC=1
# normalize to central monitor and divide by bin width
ags_avgr TRFAK,C01SOUT /CMSC=2
ags_avgr TRFAK,D01SIN /CMSC=2
#calculate background contribution
ags_func TRFAK /FUN=f01 /PARFILE=PAROUT /ALIAS=SBOUT /LIKE=A01SOUT
ags_func TRFAK /FUN=f01 /PARFILE=PARIN /ALIAS=SBIN /LIKE=A01SOUT
#subtract background
ags_lico TRFAK,E01SOUT,G01SBOUT /ALIAS=SOUTNET /PAR=1.0,-1.0
ags_lico TRFAK,F01SIN,H01SBIN /ALIAS=SINNET /PAR=1.0,-1.0
#create transmission factor
ags_divi TRFAK,I01SOUTNET,J01SINNET /ALIAS=TRFAK
'
'
Cin
 Bin
T '
C out  B 'out
Calculation of transmission factor
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Sample in
10
Cout
Bout
Texp  N
10
2
1
10
100
Time / s
'
'
Cin
 B in
C 'out  B 'out
1000
10
4
10
3
10
2
1.0
Transmission factor
10
Sample out
Bin
4
3
5
Cin
Counts
10
5
Counts
10
55
0.8
0.6
0.4
0.2
0.0
1
10
100
1000
1
10
100
Time / s
1000
Output AGS_PUTX
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Bin / Bin : 10.0 %
Bout / Bout : 5.0 %
N / N
: 0.5 %
56
TT
CV
==DD
ZZ
Z Z++SSSS
XL
800
1600
2400
3200
.
.
.
16000
16800
17600
18400
19200
20000
20800
21600
.
.
.
964000
972000
980000
988000
XH
1600
2400
3200
4000
.
.
.
16800
17600
18400
19200
20000
20800
21600
22400
.
.
.
972000
980000
988000
996000
Z
0.999
0.999
0.999
0.999
.
.
.
0.899
0.818
0.701
0.594
0.501
0.504
0.581
0.707
.
.
.
0.999
1.037
1.001
1.010
Z
0.79E-2
0.86E-2
0.92E-2
0.97E-2
.
.
.
1.30E-2
1.24E-2
1.15E-2
1.06E-2
0.98E-2
1.00E-2
1.09E-2
1.22E-2
.
.
.
5.91E-2
6.09E-2
6.01E-2
5.92E-2
Zu
0.59E-2
0.67E-2
0.73E-2
0.78E-2
.
.
.
1.07E-2
1.02E-2
0.93E-2
0.84E-2
0.76E-2
0.77E-2
0.85E-2
0.98E-2
.
.
.
3.75E-2
3.89E-2
3.80E-2
3.77E-2
DZ
S
Zu2
Bin
Bout
N
0.35E-4
0.45E-4
0.54E-4
0.61E-4
.
.
.
1.15E-4
1.04E-4
0.86E-4
0.71E-4
0.57E-4
0.59E-4
0.73E-4
0.97E-4
.
.
.
14.06E-4
15.13E-4
14.46E-4
14.23E-4
0.14E-2
0.18E-2
0.21E-2
0.24E-2
.
.
.
0.51E-2
0.53E-2
0.54E-2
0.55E-2
0.56E-2
0.57E-2
0.58E-2
0.60E-2
.
.
.
3.98E-2
4.04E-2
4.05E-2
3.96E-2
-0.08E-2
-0.10E-2
-0.12E-2
-0.13E-2
.
.
.
-0.25E-2
-0.24E-2
-0.21E-2
-0.18E-2
-0.15E-2
-0.16E-2
-0.19E-2
-0.23E-2
.
.
.
-2.18E-2
-2.31E-2
-2.23E-2
-2.20E-2
0.50E-2
0.50E-2
0.50E-2
0.50E-2
.
.
.
0.45E-2
0.41E-2
0.35E-2
0.30E-2
0.25E-2
0.25E-2
0.29E-2
0.35E-2
.
.
.
0.50E-2
0.52E-2
0.50E-2
0.50E-2
AGS Output -> Covariance matrix
Time-of-flight / ns
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Time-of-flight / ns
57
Example : 232Th(n,) in URR
CRP IAEA “Th-U fuel cycle”
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
4
10
'
C
'
B
10
10
10
3
2
Y,exp  N
'
1 C B
N
r

C'
10
'
B
2
'
  C'  B 
r C  B
'

10
1
'
B
'
2850
B’ = b0 + b1
132.00
TOFb2
B 0
(n,  )  Fc Y ,exp
18.83 5.19 eV
0
1
10
100
1500
Time-Of-Flight / s
A. Borella et al. NSE 152 (2006) 1-14
10
1000
'
B 1
0
1
'
B1
B' =aoB'0 + a1B'1
10
100
Time-Of-Flight / s
232Th(n,)
1000
500
0
0
B0
'
C
1
102710
10
Y ,exp
Black Resonances
 n mbarn
TOF - spectrum / (1/ns)
10
58
3
50
100
Neutron Energy / keV
150
Example : 232Th(n,) in URR
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
59
Correlated uncertainties due to background
in capture data
Emin Emax


,u
keV keV
mb
(%)
(%)
4
6

6 1107.9 0.49 0.17 1.00 0.60 0.57 0.58 0.55 0.53 0.62 0.58 0.56 0.61 0.58 0.54 0.51
1.00 0.55 0.56 0.53 0.51 0.60 0.57 0.55 0.60 0.56 0.53 0.49
8 934.2 0.44 0.19
8
10
845.1 0.43 0.21
1.00 0.54 0.51 0.49 0.58 0.55 0.52 0.57 0.54 0.51 0.47
10
15
749.1 0.38 0.15
1.00 0.52 0.50 0.59 0.56 0.54 0.59 0.55 0.52 0.49
15
20
638.7 0.39 0.18
1.00 0.48 0.57 0.54 0.52 0.56 0.53 0.50 0.47
20
30
571.3 0.36 0.14
1.00 0.55 0.52 0.50 0.54 0.51 0.48 0.45
30
40
490.3 0.32 0.18
1.00 0.61 0.59 0.64 0.61 0.57 0.54
40
50
429.6 0.31 0.19
1.00 0.56 0.61 0.58 0.54 0.51
50
60
382.9 0.33 0.22
1.00 0.59 0.56 0.52 0.49
60
80
311.4 0.30 0.18
1.00 0.61 0.57 0.54
80 100
242.5 0.33 0.22
1.00 0.54 0.51
100 120
217.8 0.33 0.23
1.00 0.48
120 140
201.6 0.33 0.24
1.00
A. Borella et al. NSE 152 (2006) 1-14
Example : 232Th(n,) in URR
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
60
dead time, background + normalization (1.5%)
Emin Emax


,u
keV keV
mb
(%)
(%)
4
6

6 1107.9 1.70 0.17 1.00 0.85 0.85 0.86 0.86 0.87 0.88 0.88 0.88 0.89 0.89 0.88 0.88
1.00 0.88 0.89 0.89 0.89 0.91 0.91 0.91 0.92 0.92 0.91 0.91
8 934.2 1.64 0.19
8
10
845.1 1.63 0.21
1.00 0.89 0.89 0.90 0.91 0.91 0.91 0.92 0.92 0.91 0.91
10
15
749.1 1.61 0.15
1.00 0.90 0.91 0.93 0.92 0.93 0.93 0.93 0.93 0.92
15
20
638.7 1.61 0.18
1.00 0.91 0.92 0.92 0.93 0.93 0.93 0.93 0.92
20
30
571.3 1.59 0.14
1.00 0.93 0.93 0.93 0.94 0.93 0.93 0.93
30
40
490.3 1.56 0.18
1.00 0.95 0.95 0.96 0.95 0.95 0.95
40
50
429.6 1.56 0.19
1.00 0.95 0.96 0.95 0.95 0.95
50
60
382.9 1.55 0.22
1.00 0.96 0.96 0.95 0.95
60
80
311.4 1.55 0.18
1.00 0.96 0.96 0.96
80 100
242.5 1.56 0.22
1.00 0.96 0.95
100 120
217.8 1.55 0.23
1.00 0.95
120 140
201.6 1.55 0.24
1.00
A. Borella et al. NSE 152 (2006) 1-14
Importance of covariance data
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
61
Application of reaction model to deduce model parameters:
REFIT, SAMMY

resolved resonance parameters
FITACS

average resonance parameters
Minimize 2 as a function of parameters a (resonance parameters)
1
 2 (a)  ( z exp  zM (a))T V z,exp( z exp  zM (a))
Covariance of parameters a
1
V a  (Ga V z,exp Ga )1
T
Covariance of calculated ZM
Vz,M  Ga V a Ga
T
Average resonance parameters from data in URR
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
Zexp
:
Model :
a
:
<>
average capture cross section
Generalized SLBW in URR
0 and 1
average radiation width for  = 0 , 1
Zexp = 1.6% and  = 0.0
Zu = 1.6 %
Zc = 0.0 %
0
1
En / keV
5
50
100
62
1.18 % 1.00
0.95 %
- 0.74
1.00
Zexp = 1.6% and  = 0.9
Zu = 0.5 %
Zc = 1.5 %
1.81 % 1.00 0.88
0.97 %
1.00
,M (%)
0.48
0.33
0.35
Sirakov et al., Annals of Nuclear Energy, 2008doi:10.1016/j.anucene.2007.12.008,
,M (%)
0.88
0.80
0.97
Summary
Workshop on Nuclear Reaction Data for Advanced Reactor Technologies – Trieste, 19 – 30 May, 2008, P. Schillebeeckx
• Error ≠ uncertainty
– Random error
: expectation value = 0
– Systematic error (correction factor) : expectation value = 0
• All uncertainties are statistical
• Evaluation of uncertainties
– Type A : statistical analysis of repeated measurements
– Type B : scientific judgment
• Importance of covariance information
• Quote all components involved in the data reduction
process which create correlated uncertainties
• AGS, reduction of TOF-data with propagation of correlated
and uncorrelated uncertainties including full reporting of
the reduction process
63