INTEGRAL AND DIFFERENTIAL DATA FITTING EXERCISES RELEVANT TO
NEUTRON PROPAGATION AND REACTOR PHYSICS EXPERIMENTS.
CORRELATION COEFFICIENTS
A. Gandini
University of Rome “La Sapienza”
1. Introduction
In the seventies at ENEA/Casaccia there was an important commitment in relation to methods for
exploiting the experimental data relevant to integral quantities measured in critical facilities, the so called
‘a posteriori’ information along with Bayes’ terminology, with the objective of ‘adjusting’ neutron group
cross section data, the ‘a priori’ information, adopting statistical inference methods. Such commitment
stemmed on the possibility of rapidly calculating the sensitivity coefficients required making use of the
Generalized Perturbation Theory (GPT) methodology, on development in the same period [1,2]. Such
differential data adjustment approach was also proposed in some cases for exploiting experimental data
measured in shielding benchmarks [3].
As mentioned above, the adjustment methodology is based on statistical inference methods and have been
first proposed by Linnik [4]. This methodology, adapted to the nuclear domain, is illustrated in Reff [5]
and [6]. In reference [5] the nuclear data do be adjusted were assumed to be in a multi-group form, which
was considered an unavoidable (at that time) limitation. Its application for the definition of optimal multigroup cross section set libraries, as well as appropriately assessed covariance matrices based on integral
experiments on experimental facilities, has been however pursued by different laboratories [see comments
on this and equivalent methodologies in Ref. 7]. In particular, in the 70s and 80s at CEA it was decided to
use integral experiments to assess multi-group cross-section uncertainties by using adjustment procedures.
The library ERALIB1 [8] used with the ERANOS code system [9] is the result of this methodology .
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1. A. Gandini, “Generalized Perturbation Theory (GPT) Methods. A Heuristic Approach”, Advances Nucl.Sci.Techn., Vol 19, Plenum Press, 1987
2. A. Gandini et al, “Analysis and Correlation of Integral Experiments in Fast Reactors with Nuclear Parameters”, Int. Conf. Physics Fast Reactors, London 1969
3. A.K. McCracken, NEA Specialist Meeting on SensitivityStudies and Shielding Benchmarks. Paris, 22 November 1977.
4. Y.V. Linnik, "Method of Least Squares and Principles of the Theory of Observation", Pergamon Press, London, 1961 [transl. from Russian, original edition 1958]
5. A. Gandini, "Nuclear Data and Integral Measurements Correlation for Fast Reactors. Part 1: Statistical Formulation",CNEN Re. RT/FI(73)5 (1973)
6. A. Gandini, M. Salvatores, “Nuclear data and Integral Measurements Correlation for Fast reactors. Part 3: The Consistent Method”, CNEN Rep. RT/FI(74)3 (1974)
7. “Assessment of Existing Nuclear Data Adjustment Methodologies”, Document NEA/NSC/WPEC/DOC/(2010)429.
8. E. Fort, et al., "Improved Performances of the Fast Reactor Calculation System ERANOS-ERALIB1", Annals of Nuclear Energy, vol. 30 (2003)
9. G. Rimpault, et al., “The ERANOS Code and Data System for Fast Reactor Neutronic Analyses,” Proc. Physor 2002 Conference, Seoul (Korea), October 2002.
In reference [6] an extension of the methodology was then proposed in which the differential data to be
adjusted are not the multi-group cross sections but the very nuclear parameters from which the multigroup data files themselves are constructed.
For a long time this proposal has had limited applications [10] until year 2010, when an initiative between
the American INL and BNL Laboratories started an important collaboration in which this approach is
being pursued [11], with experts from BNL taking care of the sensitivities of group cross sections with
respect to reaction model parameters (such as those determining neutron resonances, optical model
potentials, level densities, strength functions, etc.) and experts from INL taking care of the sensitivities of
integral quantities with respect to group cross-sections. These sensitivities are then combined so that the
sensitivity of the generic integral quantity R with respect to the nuclear parameter p k is defined as
R
R s j

p k
j s j p k
where R represents quantities such as keff, reaction rates, reactivity coefficient, etc., and sj a multigroup
cross section (the j index accounting for isotope, cross section type and energy group).
Applying the adjustment approach would allow then to improve directly the basic a priori data and their
associated covariance matrix.
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10. A. D’Angelo, A. Oliva, G. Palmiotti, M. Salvatores, and S. Zero, “Consistent Utilization of Shielding Benchmark Experiments,” Nuclear
Science Engineering 65, 477 (1978).
11. G. Palmiotti, M. Salvatores, H. Hiruta, M. Herman, P. Oblozinsky, M. T. Pigni, “Use of Covariance Matrices in a Consistent (Multiscale) Data
Assimilation for Improvement of Basic Nuclear Parameters in Nuclear Reactor Applications: from Meters to Femtometers”, Int. Conf. on
Nuclear Data for Science and Technology, Jeju Island, Korea (2010).
2. Propagation experiments
As a practical example of application of the consistent method, in reference [11] the case of the 23Na isotope is
considered. For this case propagation experiments of neutrons in a medium dominated by this specific isotope have
been considered. These kinds of experiments were specifically intended for improving the data used in the shielding
design of fast reactors. Two experimental campaigns taken from the SINBAD database [12] have been used in this
practical application: the EURACOS campaign (at Ispra), and the JANUS-8 campaign (at Winfrith).
The main purpose of the EURACOS experimental campaign was that of studying the neutron deep penetration in
homogeneous materials found in the construction of advanced reactors (i.e., Na and Fe). Thus, the analysis of these
experiments can be effectively utilized for the sodium cross section improvement. The neutron source of the
EURACOS system is made by the Al-U plate, having 80 cm in diameter and 1.8 cm in thickness. This source is
driven by TRIGA Mark II reactor located next to the irradiation facility. Measurements with activation detectors with
reactions 32S(n,p) and 197Au (n,g) were carried out at various distances from the source in order to analyse fast and
epithermal neutron attenuations. Analysis was performed by the Monte Carlo code MCNP5 and EMPIRE cross
section data [13].
The JANUS Phase 8 experiments were performed at the ASPIS facility. The neutron source was generated by Al-U
plate, which is driven by the NESTOR reactor, a 30kW light water cooled, graphite and light water moderated reactor,
located right next to this experimental facility. The distance between this source plate and the last detector point is
approximately 300 cm. A ~18 cm thick zone of mild steel plates precedes the sodium thanks. Again the computational
analysis was carried out with MCNP5 with EMPIRE cross section data. The neutron attenuation of several different
detectors were analysed, in particular for the following reaction rates:
32S(n,p)32P, 103Rh(n,n’)103mRh, 197Au(n, g)198Au, and 55Mn(n,g)56Mn.
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12. SINBAD-2009.02: Shielding Integral Benchmark Archive and Database, Version February 2009, RSICC DATA LIBRARY DLC-237, ORNL
13. M. Herman, R. Capote, B.V. Carlson, P. Oblozinsky et al, “EMPIRE: Nuclear Reaction Model Code System for Data Evaluation”, Nucl. Data Sheets
108 (2007) 2655-2715
For what concerns the experiments, a set of reaction rate slopes (one for each detector in the two experiment
campaigns) was selected. The selection was based, on:
- low experimental and calculation uncertainty,
- good reproduction of the neutron attenuation for the energy range to be characterised by the corresponding
detector,
- complementarity of information and
- good consistency among the C/E values.
A 41 group energy structure was adopted specifically to better describe the resonance structure of the 23Na.
The ERANOS code was used to calculate the multigroup sensitivity for the selected reaction rate slopes.
Once these sensitivity coefficients are evaluated, together with the definition of the covariance matrix
associated with the nuclear parameters, and before proceeding to the adjustment step, the degree of
complementarity of the information contained in the integral experiments considered was evaluated by
calculating the correlation coefficients, rmn , between measurements m and n [14], defined by the expression
rn, m 
SnT D Sm
( SnT D Sn )( SmT D Sm )
where and Sn and Sm are the sensitivity coefficient arrays for experiments n and m, respectively, and D is the
covariance matrix.
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14. A. Gandini, "Uncertainty Analysis and Experimental Data Transposition Methods Based on Perturbation Theory" in Handbook of Uncertainty
Analysis, Y. Ronen Ed., CRC Press, Boca Raton, Florida, 1988.
For the purpose of illustration below is reproduced the table illustrating the calculated correlation factors
among different detector reaction rate slopes in the two experiments.
As it can be observed, there is relatively little correlation among the selected slopes insuring, in this way,
a good complementarity of the information to be considered in the adjustment exercise
3. Reactor Physics and Criticality Safety Experiments
The interest in high-quality integral benchmark data is increasing as efforts to quantify and reduce
calculational uncertainties accelerate to meet the demands of next generation reactor and advanced fuel
cycle concepts. The International Reactor Physics Experiment Evaluation Project (IRPhEP) and the
International Criticality Safety Benchmark Evaluation Project (ICSBEP) continue to expand their efforts
and broaden their scope to identify, evaluate, and provide integral benchmark data for method and data
validation. Benchmark model specifications provided by these two projects are used heavily by the
international reactor physics, nuclear data, and criticality safety communities [15]. To note that in the
IRPhEP integral data base are included also experiments relevant to the TAPIRO reactor as described in
Ref [16], where we can read:
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15. J. Blair Briggs, Lori Scott, Enrico Sartori, Yolanda Rugama, “Integral Benchmarks available through the International Reactor Physics
Experiment Evaluation Project and the International Criticality Safety Benchmark Evaluation Project”, Int. Conference on Reactor Physics,
Nuclear Power: A Sustainable Resource, Casino-Kursaal Conf. Center, Interlaken, Switzerland, Sept. 14-19, 2008
16. NEA-1764 IRPhE-TAPIRO-ARCHIVE [see webpage: http://www.oecd-nea.org/tools/abstract/detail/nea-1764]
“The TAPIRO reactor, located in the ENEA Casaccia Centre near Rome, is a highly enriched uranium fast
neutron facility. The nominal power is 5 kW (thermal) and the core centre neutron flux is 4E+12/cm 2/s.
The reactor has a cylindrical core (12.6 cm diameter and 10.9 cm height) made of 93.5 % enriched
uranium metal in a uranium-molybdenum alloy which is totally reflected by copper. The copper reflector
(cylindrical-shaped) is divided into two concentric zones: the inner zone, up to 17.4 cm radius, and the
outer zone up to 40.0 cm. Radius. The height of the reflector is 72.0 cm. The reactor is surrounded by
borate concrete shielding about 170 cm thick. The maximum depth available for the epithermal column is
160 cm, reserved for filter/moderator materials.
The graphite column extends to the external reflector boundary where a sector of the outer copper
reflector has been removed and then characterised by a very hard neutron spectrum. Along the column the
spectrum gradually softens up to thermal values.
- Different materials can be interposed, such as Unat, Pb, Fe, etc. to reproduce spectrum transition
conditions at interface points between regions with different compositions.
- Activation foils can be used for activation analysis with threshold energies in the fast, intermediate and
epithermal regions”
Experiments to study the neutron propagation in Na/Fe mixture media at different energy spectrum
conditions have been extensively made in cooperation with CEA [17].
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17. D. Calamand, A. Desprets, H. Rancurel, R. Vienot, J.C. Estiot, J.P. Trapp, M.L. Bargellini, L. Bozzi, M. Martini, P. Moioli, D. Antonini, A. De
Carli, V. Rado, "Results of Neutron Propagation in Steel Sodium Mixtures with Various Source Spectra on HARMONIE and TAPIRO", 5th
International Conference on Shielding, Knoxville, USA, 18-22 April 1977.
4. Correlation between different reactor systems
Consider that the experimental information available is contained in the measurements of integral quantities
Q A , (  =1, 2, …, L) obtained form an experimental facility (system A). This information has to be transposed
to a set of quantities QB,m (m=1, 2, …, M) relevant to a reference system (system B). Let’s define the (LxJ)
and (MxJ) sensitivity matrices SA and SB, with elements, respectively,
s A , j 
s B,mj 
p o , j Q A ,
Q cal
A ,
p j
(1)
p o , j Q B,m
Q cal
B ,
p j
(2)
which can be calculated via GPT methods.
The transposition of the experimental information to the set of quantities Q B,m can be made using different
statistical inference method, for instance the Lagrange’s multipliers. As a result we can obtain adjusted
estimates of the quantities QB,m or, in a more convenient form, estimates of the relative quantities
y B,m 
cal
Q B,m  Q cal
B,m
Q cal
B,m
(3)
where Q B,m represent the corresponding integral quantities calculated with the ‘a priori’ differential
information.
Vector ~y B of the new estimates ~y B,m , defined as
~y
B, m 
~
Q B,m  Q cal
B, m
Q cal
B, m
,
(4)
results
~
y B  S BCpSTA (CA  SCpSTA )1 y ex
A
(5)
where CA is the dispersion matrix of the integral experimental data and where y ex
A is a
vector of elements
y ex
A ,

cal
Q ex
A ,  Q A ,
Q cal
A ,
(6)
Let us now consider the extreme case in which we wish to evaluate a single quantity QB
relative to the reference system, on the basis of the information contained in a single
measurement, Q ex
A ,1 , more or less correlated with QB. By following the above procedure we
obtain its relative correction estimate,
~y
yex
B
 rB,1 A ,1
B
A ,1
.
(7)
where rB,1 is to the correlation coefficient
rA,1 
S TA,1D S B
( S TA,1D S A,1 )( SBT D S B )
(8)
cal
while  A ,1 and  B are the a priori errors associated with Q cal
A ,1 .and Q B , respectively
From this equation we can clearly see how, as expected, the relative correction ~y B increases
proportionally with the correlation coefficient rB,1 , with the a priori error B and with the
ratio y ex
A ,1 /  A ,1 .
4.1. Example. Correlation between GUINEVERE and ELSY Reactors
For the correlation analysis between the facility GUINEVERE [18] and the reference design ELSY [19], in a
recent work [20] two integral quantities have been considered for analysis:
- Lead void effect
- Effect of reactivity due to the increase of the capture cross section dell'U238.
The correlation coefficients calculated using the ERANOS code are given below.
keff/keff
keff/keff
Correlation coefficient
(Pb void effect)
0.39
U238
(change of Sc )
0.93
There results a good correlation between the two systems in relation to the reactivity effect due to the U-238
capture cross-section change (which has relevance with respect to the Doppler effect). The poor correlation
between the two systems in relation to the lead void effect can be explained considering that the
predominant effect in GUINEVERE system is related to the neutron leakage while in ELSY the prevailing
effect concerns the hardening of the neutron energy spectrum.
We note that, while in the case of different integral quantities measured within an experimental facility it is
desirable that the correlation between them be as small as possible to assure the complementarity of the
information, in the case this information has to be transferred to a new system, it is desirable the opposite,
i.e., that the correlation coefficients be as close as possible to unity.
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18. G. Bianchini, M. Carta, F. Pisacane, “Set-up of a Deterministic Model for the Analysis of the GUINEVERE Experience”, PHYSOR
Malambu 2010 – Advances in Reactor Physics to Power the Nuclear Renaissance, Pittsburgh, May 9-14, 2010.
19. V. Sobolev, E., H. Aït Abderrahim "Preliminary Fuel Pin, Hexagonal Assembly and Core design for ELSY-600", ANS/826/07-15
(SCK·CEN)
20. L. Cretara, A. Gandini “Metodologia per l’analisi delle quantità integrali misurate in insiemi critici e loro correlazione rispetto ai reattori di
riferimento. Applicazione all’esperienza GUINEVERE in rapporto al reattore ELSY”, University Report (to be published)