Nuclear Data Section
International Atomic Energy Agency
P.O.Box 100, A-1400 Vienna, Austria
Memo CP-D/680 (Rev.3)
Date: 14 April 2014
To: Distribution
From: N.Otsuka, S. Kopecky, P Schillebeeckx
Subject: LEXFOR entry for covariances in the AGS format
Reference: Memo CP-D/564(Rev.)
IRMM and NDS have recently completed the compilation of neutron transmission data together with and the covariance information. The data resulted from measurements at the GELINA facility (EXFOR 23077) [1]. Two new headings are proposed
Dictionary 24 (Data Headings)
TOF-MIN Lower boundary of time-of-flight
TOF-MAX Upper boundary of time-of-flight.
In order to fulfill the action A51 of the 2010 NRDC meeting, addition to the LEXFOR
“
Covariance
” is proposed below.
Definition
For a measured quantity at two points σ i
and σ j
(e.g., cross section at two incident energies; i , j =1,2,…, m ), covariance between them are defined as cov

 i
,
 j



 i
  i
 
 j
  j

  i
 j
  i
 j
(1)
If the cross section depends on p parameters (source of uncertainties) { x k
}
( k =0,1,2,…, p ),
 i
  i
 k p 

0
 
 x i k i
 x i k  x i k
,

(2)
Eq.(1) can be rewritten as cov(
 i
,
 j
)
 k , l p 

0
 
 x i k i cov
 x i k , x l j
  
 x l j j  
0
 i
 
0
 j
  ij
 k , l p 

1
 
 x i k i cov
 x i k , x l j
  
 x l j j
.
(3) where k,l = 0 gives the uncorrelated uncertainty Δ
0
σ i
=(∂σ i
/∂ x
0
)Δ x i
0
.and δ ij
is the
Kronecker’s delta.
Δσ ij
…
=cov(σ i
,σ j
) (i=1,…n) forms the following covariance matrix .

Compact Expression of Covariance Matrix by Cholesky Decomposition
Eq.(3) is expressed as
V

MM t 
S

V

S t

(4)
( V
=cov(σ i
,
σ j
), M
={Δ
0
σ i
}, S
α
={∂σ i
/∂ x
The p × p matrix V
α
=cov( x i k , x j l i k
}, V
α
=cov( x i k
, x j l
))
) is positive definite and symmetric, and therefore there is a matrix L which satisfies V
α
= LL t
( Cholesky decomposition ):
V

MM t 
S

LL t
S t


MM t 
D

D t

, (5) where D
α
= S
α
L is a m × p matrix. The ij -th component of the Eq.(5) is cov(
 i
,
 j
)
 
0
 i
 
0
 j
  ij
 k p 

1
 k
 i
  k
 j
. (6)
( D
α
={Δ k
σ i
}, k
=0, 1,2,…, p and i
=1,2,…, m ) . We may regard the vector {Δ
0
σ i
} as the total uncorrelated uncertainty and the vector {Δ k
σ i
} ( k =1..
p ) as the k -th fully correlated uncertainty. They may be coded under ERR-1 etc. with the correlation property flag U (uncorrelated) or F (fully correlated).
These vectors allow us to keep the full covariance information with (1+ p ) m elements instead of m ( m +1)/2 elements, and effective to archive the full covariance information of high-resolution time-of-flight spectra (e.g., transmission, capture yield) where m is huge [1,2].
Example: Compilation of AGS vectors published in Table 1 of [3].
REACTION (79-AU-197(N,TOT),,SIG,,AV)
...
ERR-ANALYS ERR-2, ERR-3 and ERR-4 are AGS vectors.
(ERR-T,,,P) Total uncertainty
(ERR-1,,,U) Uncorrelated uncertainty due to counting statistics
(ERR-2,,,F) Background model (dK/K=3%)
(ERR-3,,,F) Normalization (dN/N=0.25%)
(ERR-4,,,F) Sample areal density
STATUS (TABLE) Table 1 of Eur.Phys.J.A49(2013)144
...
EN-MIN EN-MAX DATA ERR-T ERR-1 ERR-2 ERR-3 ERR-4
EV EV B B B B B B
4000. 5500. 20.10 0.22 0.12 0.11486 -0.14226 0.04020
5500. 6500. 21.30 0.23 0.15 0.09843 -0.14226 0.04259
6500. 8000. 20.15 0.20 0.12 0.07380 -0.14226 0.04030
8000. 10000. 17.31 0.20 0.12 0.06558 -0.14226 0.03462
References
[1] B. Becker et al., “Data reduction and uncertainty propagation of time-of-flight spectra with AGS”, J. Instrum.
7 (2012) P11002.
[2] N. Otuka, A. Borella, S. Kopecky, C. Lampoudis, P. Schillebeeckx, “Database for time-of-flight spectra including covarainces”, J. Kor. Phys. Soc. 59 (2011)1314.
[3] I. Sirakov et al., “Result of total cross section measurements for
197
Au in the neutron energy region from 4 to 108 keV at GELINA”, Eur. Phys.
J.A
49 (2013)144.

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