Nuclear Data Section
International Atomic Energy Agency
P.O.Box 100, A-1400 Vienna, Austria
Memo CP-D/704
Date:
To:
From:

20 May 2011
Distribution
N. Otsuka

Subject:

Coding of partial uncertainties and correlation matrices

Though compilation of detailed uncertainty information has been often stressed in theNRDC Meeting [1], the EXFOR Format is still not satisfactory to include all available
information from experimentalists. For example, we have only one heading ERR-S for
uncorrelated (statistical) uncertainty.
In this memo, I will present some examples of error analysis information available
from experimentalists, and propose the format to keep experimental information forevaluators and programmers in a computer readable manner. I limit the scope of thispaper to the energy dependent cross sections (excitation function) of a reaction, i.e.,
only energy-energy correlation of a given reaction is discussed. In addition to thiscorrelation, there has been other experimental information of correlations (e.g.,reaction-reaction correlation in fission spectrum averaged cross sections provided
from KURRI etc., correlation between Legendre coefficients from PTB etc.). But I
will not treat these correlations in this memo to simplify discussion and avoid drasticchanges in the EXFOR Formats. Hereafter “correlation” always means correlation ina quantity between two incident neutron energies (cross energy correlation).
An example of a recent experiment - C. Sage et al. (2009, 2010) at Geel [2,3]Below is an example of 241Am(n,2n)240Am energy dependent activation cross section
derivation with the 27Al(n,α)24Na monitor cross section (i: index of incident neutronenergy point, k: type of correction).

σ Am (i) = σ Al (i )C k ,Am (i )S Am (i ) [ I ε f Σ (i ) f r (i ) n(i ) Φ 0 (i )]Al∏S Al (i ) [ I ε f Σ (i ) f r (i ) n(i ) Φ 0 (i )]Am k C k ,Al (i )Definitions of parameters:
σ Am (i) : 241Am(n,2n)240Am cross section
σ Al (i) : 27Al(n,α)24Na cross section
S(i): number of counts
I: gamma-ray emission probability (constant)ε: peak efficiency for emitted gamma (constant)f r (i): 1-exp(-λt r (i)) where t r (i) is irradiation timef Σ (i): (1/λ) Σ exp(-λt d (i))[1-exp(-λt m (i)] where t d (i) and t m (i) are starting and measuring
timen(i): number of target nuclei in the sample
Φ 0 (i): neutron flux
C flux (i): correction due to flux fluctuation

C low (i): correction due to low-energy secondary neutron

Creating the full covariance matrix of the results of measurements we may present thecomponent of the uncertainties in accordance with their correlation properties. It is
usually classified to (1) uncorrelated (short energy range correlation; SERC), (2) fullycorrelated (long energy range correlation; LERC), and (3) partly correlated (mediumenergy range correlation; MERC) [4]. These are also introduced at the beginning of
the ENDF-6 Formats Manual Chapter 33.
The numbers of count S are uncorrelated for both isotopes. The decay gamma
intensity I and detector efficiency ε are fully correlated for both isotopes because one
gamma line was used for one reaction. The monitor cross section σ Al was adopted
from a standard library with its correlation matrix, which takes a value between -1 and
1 for each element.
Special care has to be directed to the numbers of nuclei n for the samples used in theexperiment. The authors used different Al samples at each neutron energy i, andtherefore it should be treated as uncorrelated assuming that there is no systematic bias
in the results of the sample thickness measurement. On the other hand, the authors
used 4 Am samples at neutron energy i=(1,6,9), (2,3,4), (5,7) and 8. Within eachgroup, they are strongly correlated while it can be assumed as uncorrelated between
neutron energies belonging to two different groups. (This example shows us thatdetails of samples are important for correlation property analysis.)
Correlation properties considered by authors are tabulated below. The uncertainties ofparameters not listed in this table were neglected by authors.
Source

σ Al

S Am

S Al

I Amn Aln Am

ε Al /ε Am[f Σ f f ] Am

PropertyPUUFUPFF

C low,Am /C low,AlU

U: Uncorrelated (SERC), F: Fully correlated (LERC), P: Partly correlated (MERC)
These are defined under ERR-ANALYS in EXFOR 23114 as follows:
ERR-ANALYS (ERR-T) Total uncertainty.
(MONIT-ERR) Uncertainty of monitor cross section.
(ERR-S) Counting statistics uncertainty for Am .
(ERR-1) Counting statistics uncertainty for Al.
(ERR-2) Uncertainty of gamma-rays intensities .
(ERR-3) Uncertainty of Al target nuclei number(ERR-4) Uncertainty of Am target nuclei number(ERR-5) Uncertainty of ratio of detection efficiencyfor Al to Am one.
(ERR-6) Uncertainty of time factors for Am.
(ERR-7) Uncertainty due to correction for presence oflow-energy breakup neutrons.

In comparison with above the table, we realize inconsistency between the heading andits correlation property:
1) The heading ERR-1 (defined as the 1st systematic uncertainty in the EXFORdictionary) is applied to an uncorrelated (statistical) uncertainty ΔS Al because
only one heading ERR-S is available for uncorrelated (statistical) uncertainties
in the current EXFOR Formats.
2) The headings ERR-3 (defined as 3rd systematic uncertainty in the dictionary)and ERR-7 (defined as 7th systematic uncertainty in the dictionary) are applied
to uncorrelated uncertainties Δn Al and Δ(C low,Am /C low,Al ).

Also correlation properties (LERC or MERC) are not indicated by the compiler.
“Correlation Factors” in the EXFOR Formats
The EXFOR Formats Manual defines a data field for “correlation factors”. What is“correlation factors”? Its definition has not been clear from the EXFOR Formats
manual. This field has been used by only two entries (EXFOR 10921 [5] and 12869[6]), which report neutron cross sections measured by the ANL Dynamitron. Partialuncertainties and (macro) correlation matrix are tabulated in [5]. Partial uncertainties
are defined under ERR-ANALYS of EXFOR 10921.001 as follows:
ERR-ANALYS (ERR-T) total error
Partial errors and their sources:(ERR-S) statistical error(ERR-1,,,1.0) Gamma-ray detection efficiency.
(ERR-2,,,0.0) Irradiation geometry.
(ERR-3,,,1.0) Uranium deposit,mass,and isotope content.
(ERR-4,,,1.0) Extrapolation correction for fissions andcorrection for finite thickness of deposit.
(ERR-5,,,1.0) Correction for neutron absorption in Cusample.
(ERR-6,,,1.0) Error in correction for neutron
scattering by the sample and fission chambercomponents.
(ERR-7,,,0.5) Neutron source characteristics.
(ERR-8,,,0.0) Correction of fissions for neutron
background and neutron scattering due totantalum cup.

Three values 1.0, 0.5 and 0.0 are coded as “correlation factors” in EXFOR 10921, andthey are corresponding to “c(100): 100% correlated”, “c(50): 50% correlated” and“uc: uncorrelated” uncertainties in Table III of the article. Thanks to the guidance ofDr. D.L. Smith, I could reproduce the (macro) correlation matrix tabulated in Table
IV of the article from the coded partial uncertainties by the following formula
12Vij = ∑ K ijq eiq e jq
q =1,C ij = Vij / ViiVij
where 12 partial uncertainties are considered. The coefficient K ijq =1.0 for i=j, 0.0, 0.5
or 1.0 for i≠j, while e iq and e jq are partial per-cent uncertainties for the i-th and j-thneutron energy bin, respectively. Similar equations are seen in p.130 of [7].
In general, we can compile correlation properties within the current format when theauthors assume a constant “correlation factor” for off-diagonal element K ijq (=1.0 fori=j, =constant for i≠j) for each source q. Use of 3 values (1.0, 0.5 and 0.0) seems tome a good solution to avoid unnecessary complication (named as Occam’s Lazor
assumption by Dr. Smith).
However, the current EXFOR format is not sufficient when authors provided variousvalues between -1 to 1 as K ijq . (Micro) correlation matrices considered by Sage et al.
[3,4] show such examples. See Appendix B of [4]. We should be able to compile
detailed correlation properties when authors provide them by an appropriate
methodology.

Proposal
In order to improve this situation, I propose to apply ERR-1, ERR-2 etc. for any partial
uncertainties regardless of their correlation property (unless there is a more specificheading, e.g., MONIT-ERR), and indicate correlation properties in the 4th field by
U: Uncorrelated (random, statistical)
F: Fully correlated
P: Partially correlatedC: Correlated (when information is insufficient to determine F or P).
(blank): Correlation property unknown
Example 1: EXFOR 23114
ERR-ANALYS (ERR-T,,,P)
Total uncertainty(MONIT-ERR,,,P) Monitor cross section(ERR-1,,,U)
Number of counts (Am)(ERR-2,,,U)
Number of counts (Al)(ERR-3,,,F)
Gamma intensity (Am)(ERR-4,,,U)
Sample mass (Al)(ERR-5,,,P)
Sample mass (Am)(ERR-6,,,F)
Efficiency ratio (Al/Am)(ERR-7,,,F)
Decay correction (Am)(ERR-8,,,U)Secondary neutron correction (Am/Al)

Example 2: EXFOR 10921
ERR-ANALYS (ERR-T,,,P) Total uncertainty(ERR-1,,,U) Counting statistics and reproducibility(ERR-2,,,F) Gamma-ray detection efficiency.
(ERR-3,,,U) Irradiation geometry.
(ERR-4,,,F) Uranium deposit,mass, and isotope content.
(ERR-5,,,F) Extrapolation correction for fissions andcorrection for finite thickness of deposit.
(ERR-6,,,F) Correction for neutron absorption in Cu
sample.
(ERR-7,,,F) Uncertainty in correction for neutron
scattering by the sample and fission chambercomponents.
(ERR-8,,,P) Neutron source characteristics.
(ERR-9,,,U) Correction of fissions for neutron
background and neutron scattering due totantalum cup.

If authors give (micro) correlation matrices for partial uncertainties defined with theflag P, it can be given under the keyword COVARIANCE. Such coding samples areappended to this Memo.
The heading ERR-1, ERR-2... will be defined as 1st, 2nd ... partial uncertainties in thedictionary. The heading ERR-S can be kept where only one uncorrelated (statistical)
uncertainty is considered.
I thank M. Herman, and P. Obložinský, A. Plompen, V. Pronyaev, P. Schillebeeckx
and D.L. Smith for their instruction and discussion.
References[1] For example, M. Lammer, O. Schwerer (ed.), INDC(NDS)-0141, p23, 1982.
[2] C. Sage, V. Semkova, et al., Phys. Rev. C81(2010)064604.
[3] C. Sage, Thèse de Doctorat, Universitè de Strasbourg (2009).
[4] For example, W. Mannhart, INDC(NDS)-0558, D.2.1, 2011 (in print).
[5] G. Winkler et al., Nucl. Sci. Eng.76(1980)30.
[6] P. Guenther et al., Nucl. Phys. A448(1986)280.

[7] D.L. Smith, “Probability, statistics, and data uncertainties in nuclear science andtechnology”, American Nuclear Society, Inc., Illinois, USA (1991).
Appendix: Coding samples (where COVARIANCE gives free text information)
EXFOR 23114.002 (in preparation) [2-3]Because partial correlation matrix for monitor cross section is given, evaluators canrevise the total correlation matrix when they renormalize cross sections by a new
standard cross section!!
BIB
REACTION(94-AM-241(N,2N)94-AM-240,,SIG)...
ERR-ANALYS (ERR-T,,,P)
Total uncertainty(MONIT-ERR,,,P) Monitor cross section(ERR-1,,,U)
Number of counts (Am)(ERR-2,,,U)
Number of counts (Al)(ERR-3,,,F)
Gamma intensity (Am)(ERR-4,,,U)
Sample mass (Al)(ERR-5,,,P)
Sample mass (Am)(ERR-6,,,F)
Efficiency ratio (Al/Am)(ERR-7,,,F)
Decay correction (Am)(ERR-8,,,U)Secondary neutron correction (Am/Al)
COVARIANCE (EN,MEV)
8.34 9.15 13.33 16.1 17.16 17.9 19.36 19.95 20.61(COR,ERR-T,PER-CENT) Total correlation
10035100
374210038435310040455758
100414557598410021243031393910030344445585951100
2022293040423965
100(COR,MONIT-ERR,PER-CENT) Partial correlation due to standard100
43 10000 100006 10000912 100
001112 100 100
00111140
40 10000111140
40 100 100
100
0111140
40 100 100 100(COR,ERR-5,PER-CENT) Partial correlation due to sample mass
100
0 100
0 100 100
0 100 100 100
0000 100
100
0000 100
0000 1000 1000000000 100
1000000 100
00 100COMMON
ERR-3
ERR-4
ERR-5
ERR-6
PER-CENT
PER-CENT
PER-CENT
PER-CENT
1.2
0.1
0.3
3.0
ENDCOMMON
DATA
EN
DATA
ERR-T
MONIT-ERR ERR-1
ERR-2
ERR-7
MEV
MB
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
8.34
96.8
6.5
1.9
5.0
1.0
0.9
9.15
162.9
5.7
1.9
4.0
1.0
0.6
13.33
241.8
4.6
1.6
2.5
1.0
0.4
16.1
152.4
4.6
2.
2.1
1.0
0.6
17.16
116.1
4.4
2.
1.5
1.0
0.6
17.9
105.7
4.4
2.2
1.30.7
0.7
19.36
89.5
8.2
3.1
6.3
2.0
0.6
19.95
102.1
5.8
4.1
1.4
1.0
0.6
20.61
77.9
8.8
5.4
5.7
1.6
0.6

ERR-8
PER-CENT

0.3
0.3
0.3
0.3
1.3
1.4
1.4

ENDDATAENDSUBENT

EXFOR 10921.002-003 [5]SUBENT
BIB
REACTION((29-CU-63(N,A)27-CO-60,,SIG)/(92-U-238(N,F),,SIG))...
ERR-ANALYS (ERR-T,,,P) Total uncertainty(ERR-1,,,U) Counting statistics and reproducibility(ERR-2,,,U) Irradiation geometry(ERR-3,,,F) Gamma-ray detector efficiency(ERR-3,,,F) Uranium deposit, mass, isotope content(ERR-4,,,F) Extrapolation correction for fissions andcorrection for finite thickness of deposit(ERR-5,,,P) Neutron source characteristics 7Li(p,n)(ERR-6,,,P) Neutron source characteristics D(d,n)(ERR-7,,,P) Correction for activity induced in thesample by neutron background(ERR-8,,,U) Correction of fission for neutron background andneutron scattering due to Ta cup,
7Li(p,n) source(ERR-9,,,U) Correction of fission for neutron background andneutron scattering due to gas-cell components,
D(d,n) source(ERR-10,,,F) Correction for neutron absorption in the coppersample(ERR-11,,,F) Correction for neutron scattering by the sampl
and fission chamber components
COVARIANCE (EN,MEV)
3.560 3.800 4.065 4.361 4.656 4.954 5.120 5.185...
(COR,ERR-T,PER-CENT) Total correlation
1002100
41710051937100...
(COR,ERR-5,PER-CENT)
100501005050100505050100...
(COR,ERR-6,PER-CENT)
100501005050100505050100...
(COR,ERR-7,PER-CENT)
100501005050100505050100...
ENDBIBNOCOMMON
DATA
EN
EN-RSL-HW DATA
ERR-T
ERR-1ERR-2
ERR-3
ERR-4
ERR-5
ERR-6
ERR-7
ERR-8
ERR-9
ERR-10
ERR-11
ERR-12MEV
MEVNO-DIM
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
PER-CENT
3.560
0.044
7.27E-05 50.
50.
3.0
1.5
1.0
0.5
1.5
1.8
2.2
3.800
0.08
2.204E-04 11.5
10.5
3.0
1.5
1.0
0.2
1.5
1.8
2.2...
ENDDATAENDSUBENTDistribution:
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