, -,

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LA-UR -78-772~’$sf!!

kJ33+-3TITLE: THE SHIELDING FACTOR METHOD FOR

CROSS SECTIONS: MINX/SFHINX AND.

~Ro~ucING EFFECTIVE

THE CCCC INTERFACE SYSTEM

AUTHOR(S): R. E. MacFarlane, C. R. Weisbin and N. C. Paik

SUBMITTED TO: RSIC Multigroup Cross Section SeminarOak Ridge, TennesseeMarch 14-16, 1978

JI alamoslc Laboratory

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The Loo Alamoe Scientific Laboratoryrcqueetc that thepubllrher identify this ●rticle as work performed underthe mmplceu of the USERDA.

oftho University of CaliforniaLOS ALAMOS, NEW MEXIC087645

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DEVELOPhl~NT ADhflNIW~MTIONcolWRACTw. 7405-ENG, 36

THE SHIELDING FACTOR METH9D FOR PRODUCING EFFECTIVE CROSS SECTIONS:MINX/SPHINX AND THE CCCC INTERFACE SYSTEM

R. E. MacFarlaneLos Alamos Scientific Laboratory, University of California

Theoretical DivisionP O Box 1663, Los Alamos, NM 87545

C. R. WeisbinOak Ridge National Laboratory

P O Box X, Oa!:Ridge, TN 37830

!?.C. PaikWestinghouse Advanced Reactor Division

P O Box 158, Madison, PA 15663

ABSTRACT

The Shielding Factor Method is an economical designer-oriented method for producing the coarse-group space and energyself-shielded cross sections needed for reactor-core analysis.Extensive experience with the ETOX/lDX and ENDRUN/TDOWN systemshas made the SFM the method of choice for most US fast-reactordesign activities. The MINX/SPHINX system was designed toexpand upon the capabilities of the older SFM codes and to in-corporate the new standard interfaces for fast reactor crosssections specified by the Committee for Computer Code Coordina-tion. MINX is the cross-section processor. It generates multi-group cross sections, shielding factors, and group-to-grouptransfer matrices from ENDF/B-IV and writes them out as CCCCISOTXS and BRKOXS files. It features detailed pointwise reso-nance reconstruction, accurate Doppler broadening, and an effi-cient treatment of an+.sotropicscattering. SPHINX is the space-and-energy shielding code. It uses specific mixture and geome-try information together with equivalence principles to constructshielded macroscopic multigroup cross sections in as many as 240groups. It then makes a flux calculation by diffusion or trans-port methods and collapses to an appropriate set of cell-averagedcoarse-group effective cross sections. The integration of MINXand SPHINX with the CCCC interface system provides an efficient,sccurate, and convenient system for producing effective U):OSSsections for use in fast-reactor problems. The system has al.wproved useful in shielding and CTR applications.

INTRODUCTION

The complexity of a typica? reactor core makes it impractical tosolve the neutron transport problem with full space and energy detail.Tor this reason designers normally use effective cross sections aver-aged over relatively coarse energy groups and space regions. The Shield-ing Factor Method (SFM) is an economic~.1method for producing fhese ef-fective cross section that was originally developed in Russia. Devel-opment of the SFM in the US has been chiefly for the fast”-reactorpro-gram, and extensive experience has been accumulated with the ETOlC2/lDX3and ENDRUN’’/TDOWNscode systems. More recently the SFM has receivedincreased attention for thermal power reactor analysis with the dev~l-opment of EPRI-CELL and EPRI-CPM for the electric utility industry.SFM’code systems are traditionally divided into two parts: the crosssection processor (e.g., ETOX) and the space-energy collapse code (e.g.,lDX). The MINX/SPHINX system follows this pattern. It was designed toexpand upon the capabilities of the older SFM codes and to incorporatethe standard interface formats for fast reactor codes specified by theCommittee for Computer Code Coordination (CCCC).7 The MINXa cross sec-tion prrtessor generates a library of multigroup cross sections, shield-ing factors, and group-to-group transfer matrices from ENDF/B-IVg eval-uated nuclear data and writes it out as CCCC ISOTXS and BRXOXS files.The SPHINX1° space-energy code uses specific mixture and geometry in-formation together with equivalence pr?.nciplesand a diffusion or trans-port flux calculation to construct effective coarse-group cell-averagedmacroscopic cross sections in CCCC format.

The MINX/SPHINX system is in routine use on both IBM and CDC equip-ment. Comparisons with the older SFM codes show generall= good a ree-

{ment. Co~~arisons with independent codes such as ETOE-21 /MC2-2,F* VIM,13and GGC-5 give confidence that the MINX/SPHINX system is suitable forthe routine analysis of large fast-reactor cores.

THEORY OF THE SHIELDING FACTOR METHOD

The goal of the SFM is to define effective cross sections for somerange of ener~y (E in group g) and some region of space (~ in volume v)that preserve macroscopic overservables such as reaction rate. Clearly,

(1)

iwhere Ux is the cross section for isctope i and reaction x at E and ~,and 0 is the neutron scalar flux at that energy and position. Similarexpressions can be constructed to preserve the group-to-group scatter-ing rates and the transport cross section.

Unfortunately, the flux needed for Eq. (1) is not known before thefact; in fact, it is one of the quantities being sought in the analysis.In addition it is very complex, being full of sharp dips and peaks causedby resonances in the cross sections. However, experience has shown thatit is possible to separate the variation of the flux into a part that isrelatively smooth with respect to energy group and spatial zone size anda remaining resonance part. The variations in the smooth part can bedetermined by a multigroup flux calculation, but the intra-group fluxmust be selected by model.

The class of codes represented by MC* and GGC-5 determines thismodel flux by making a detailed flux calculation for a simplified homo-geneous system. This is an expensive procedure. The SFM codes, on theother hand, assume that the Intragroup flux can be modeled as

W(E)$(E) ‘~ ‘ (2)

where W is a smooth function of energy reflecting the fission and scat-tering sources into E and Z is the total macroscopic cross section.Formally, Eq. (2) gives the $lux for an infinite homogeneous system sat-isfying the narrow resonance approximation. However, heter~geneous sys-tems can be included using equivalence princples.ls Extension to wideand intermediate-width resonances is also possible.16

Furthermore, in evaluating the numerator of Eq. (l), it ts assumedthat the important effect is the interaction between a resonal’cein uand the dip in the flux caused by that resonance (self-shielding). Tfiereaction rate becomes

\

U:(E)W(E)dE ,

(J:+ +(E)

g

where

11 z j60 ‘y”

i j*‘jut ‘

(3)

(4)

and where pi is the number density for isotope i in the homogeneous m:!x-ture. The simplification comes from assuming that 00 is constant in 8.The result is a single parameter, the “background cross section per atom,”which can be used to characterize self-shielding. The cross sectionsproduced by MINX are computed using

.

. ,,.?

.

1u:

— WdE

0’U. + (J:

Xg “

I

●

1WdE

o +0:

8°

The results are tabulated as cross sections for U. = ~ and shielding,.

(5)

(6)

for several values of 00. SPHINX then computes U; using Eq. (4). Inheterogeneous systems an additional “escape cross section per atom” isadded. The corresponding shielded cross section is then obtained byinterpolating the f-factors for this Uo. Temperature is handled in thesame way.

This approach makes a composition-independent cross-section librarypossible. The economy of the SFX results from being able to use thislibrary many times.

THE MINX PROCESSING CODE

MINX was designed to combine and improve upon the resonance capa-bilities of ETOX2 and ENDRUN” and the anisotropic scattering capabilitiesof ETOG17 and SUPERTOG.la It is a modular code that uses paging tech-niques and variable dimensioning to make it possible to process the com-plex evaluations found in ENIF/B-IV.9 The normal flow through the codeis pictured in Fig. 1.

First, detailed pointwise cross sections are generated from ENDF/Bresonance parameters and cross sectioris using the method of RESEND.19The energy grid is suitable for linear interpolation to within a userspecified accuracy. The results are written out as a “pointwise-Eh~F”(PENDF) tape suitable for printing, plotting, or further processing,

These pointwise cross sections are then accurately Doppler broadenedto any desired temperatures using the method of SIGMA-1.2U This approachhas the advantage of correctly broadening smooth cross sections, back-grounds, and multi-level representations. Since broadening is a smooth-ing process, the results are thinned to a user specified accuracy andwritten out as PENlll?tapes. Examples uf the number of points producedin reconstruction and Doppler thinning are given in Table 1. Althoughthis highly accurate process $s expensive, it only has to be done oncefor a particular evaluation. Many subsequent averaging runs with differ-ent parameters can be made using the one temperature-dependent PENDF

1 MINX--fnaln prqram 1

II

Fig. 1. Structure of MINXcode illustrating functionalblocks and data flow. ETOPLis

tape.’Carlo

not a part of MINXe

Table 1. Results of MINXReconstruction and Doppler

Points PointsbatNuclide at O Ka 2100 K**AL

c 404 404

Fe 8798 5033235U

7209 2660238U

50372 6683

ResonanceBroadening

CP Seconds(CDC7600)

68.1

302.8

492.3

3483

238Ua0.5% reconstruction except l.O% for .b0.1% thinning tolerance.

The ttape can also be reworked for use by

This procedure will not workonly statistical knowledge of thepointwise cross sections vs T andENDF/B distributions of resonance

continuous-energy Monte

in the unresolved energy range whereresonances is available. Effectiveuo are produced by averaging over thewidths using methods based on ETOX.2

?fultigroupcross sections are computed using Eq. (5) and appropriategeneralizations. The group structure and smooth weight function arechosen by the user. The energy integrations are performed by adaptivequadrature starting from the union grid of the functions in the inte-grands. The nature of the PENDF cross section grid assures that allfeattres are well-represented. Fission yields are averag$d to preserveWf a~d slowing-down parameters are averaged to Ereserve Ua and CO .The transport cross section is computed as a -- Vae where ‘currents

weighting is used for the total cross sectio~.

Elastic and discrete-inelastic scattering both obey two-body kine-matic3 ● MINX usually performs the resulting complex integrals over energyand angle with a semi-analytic method22 based on an expansion in thelaboratory system. The analytic integrals are obtained by a recursionrelation, and the single energy integral is performed adaptively to auser specified tolerance. When this is not appropriate (e.g., lightisotopes and near thresholds) MINX au~omatically changes to a directnumerical integration in the center-of-mass frame.

Group-to-group cross sections for continuum reactions are evaluatedusing analytic integratf,onsover secondary energy and the standard adap-tive quadrature for initial energy. Fission chi vectors by isotope are

producedincident

The

by averaging the ENDF/B spectrum appropriate to a specifiedenergy.

final step is t~ format the results of the multigroup averagingmmdule into the CCCC-1117 ISOTXS (cross sections and matrices) and - -BRKOXS (shielding factors) files.

THE SPHINX SPACE-ENERGY CODE

SPHINX combines an extended version of the resonance treatment oflDX9 with the one-dimensional diffusion theory flux calculation of lDXor the one-dimensional transport flux transport of ANISN.23 It iS mod-ular in structure and uses the flexlble POINTR system2° of dynamicstorage allocation. The entire code was assembled in accordance withthe CCCC specifications for code compatibility.’S2S The use of CCCCinterface files makes communication with other CCCC-compatible codessuch as TWOTRANZ! and VENTURE27 straightforward.

The basic structure of SPHINX is shown in Fig. 2. The variousexecution paths through che code are administered by the ZEUS CONTROLmodule using input data from the CCCC standard and code-dependent inter-faces listed in Table 2. The fundamental cross-section data, intermedi-ate results, and final answers are transmitted using the CCCC files de-scribed in Table 3.

0MIUN

.Vttut

ISOTX9● Rmoxs

L2?k-$L

‘G-T-GI

mbl?vuslomMODULE

small t L**,,● t,blam 1 Veeb19m(61b I(UDTO110t.ol, v t mm,,,,W*UP8)I b,. ”,.)

I M?umM TO ZBV& tOMW.OL J

Fig. 2. Execution paths through SPHINX.

Table 2. SPHINX ControJ.Files

Name Standard Description

ZEUSIAXSRINPSKODXIANISINZNBTDNFPRINTGEODSTNDXSRF

ZNATONSEARCHSNCONS

FIXSRC

anonononononoyesbyes

yesyesyes

Modular control inputResonance module inputDiffusion module inputTransport module inputZone atomic densitiesPrint controlGeometry descriptionNuclide density and cross

section parametersZone nuclide atomic densitiesCriticality search parametersSn constants

yes Volume and surface sources

aSee Ref.bSee Ref.

I----

Table

Name

10 for detailed specifications.

7 for detailed specifications.

ISOTXSGRUPXSBRKOXSRTFLUXATFLUXRCURNTACURNTRAFLUXMFLUXRZFLUXPWDINT

3. SPHINX Standarda Interface Files

Description

Nuclide-ordered cross section dataGroup-ordered cross section dataResonance self-shielding dataRegular scalar fluxAdjoint scalar fluxRegular currentAdjoint currentRegular angular fluxAdjoint angular fluxRegular zone-averagedPower densities

flux

aSee Ref. 7 for detailed specif~cations,

,.

The first step in most problems is to useprepare effective self-shielded cross sectionsfi.edcomposition and geometry. The background

the resonance module toappropriate to the speci-crass section an is computed

for each-group and nuclide using Eq. (4): An additional escap= term canbe added for one of the seven options: (1) cylindrical cell using Sauer’sapproximation28 for the Dancoff factor in a hexagonal lattice, (2) cy-lindrical cell using Sauer’s approximation to the Dancoff factor in asquare lattice, (3) symmetric slab cell,(5) isolated rod,

(4) asymmetric slab celll~(6) cylindrical cell with the Bell approximation to

the Dancoff factor, and (7) symmetric slab cell with the Bell approxima-tion. S.Olf-shieldingfactors are then computed at a. by Langranian in-terpolation. Effective cross sections are defined as in lDX except thatprovision is made for an elastic group-to-group matrix. The results arewritten in ISOTXS format for communication with the flux modules.

On option, the code then branches to the diffusion module. The cal-culation is identical to that in lDX except that input is in ISOTXS for-mat and cross-section storag: has been modified to allow for up to 240groups and for several additional partial reaction types (i.e, n2n, na,nd, ...). The cross sections are then collapsed to a subset group struc-ture using the computed flux and written out in ISOTXS format.

The optional transport module uses the S method to obtain the flux.EThe method is identical to ANISN except for t e ISOTXS intetface capa-

bility ● When the flux has bean obtained, cross sections are collapsed toa subset group structure and zone-averaged using either volume or fluxweighting. This provides a capability for cell homogenization.

The use of standard files provides many possible paths. For exim-ple, the flux from a diffusion calculation is easily available as an in-put guess for a subsequent transport calculation using already shieldedcross sections.

LIBRARIES AND UTILITIES

SPHINX is normally used with onc of the existing multigroup librar-ies generated by MINX. LIb-IV30 is a 50-group 101-isotope library gen-erated from ENDF/B-IV. The library includes all the general purposeevaluations from ENDF/B-IV plus the two copper isotopes and the ninelumped fission products from ENDF/13-111. All materials wsre run at 300,900, and 2100 K using 4 to 6 a. values with decade steps. Scatteringmatrices are given to p3. The weight function consists of a 1.4 MeVfission spectrum joined at 820.8 keV to a l/E snape which, in turn, joinsto a 0.025 eV Maxwellian at 0.10 eV. The library also contains delayedneutron yields and spectra for seven isotopes generated in CCCC DLAYXSformat using NJOY.sl

VITAMIN-C32 is a 171-group library with 36 isotopes chosen for im-for temperature, u , Legendre order, and weight function are similar to

PLIB-IV, except tha a velocity exponential fusion peak has been attachedin the 14 MeV range.

These Iibrari(lsrequire severalutility codes tn ozder to knit them

Into a system with MINX and SPHINX.First, BINXS3 is a code for convert-ing back-and-forth between binary andBCD modes for transmission of ISOTXS,BRKOXS, and DLAYXS files between labor-atories. LINX33 is a code for addingnew isotopes to an existing CCCC cross-section library. Finally, CINX34 is acollapse code that can be used to gen-erate a subset library tailored to aparticular set of problems. As anexample> CINX has been used to producza 126-group subset of VITAMIN-C espe-cially designed for LMPBR core andshield analysis.3s Figure 3 illus-trates how these codes and librariescombine to form a complete system.

Oser Input

‘BEwl s

Ihruy o MIMI

~tirmolnLIUXBINX

Cccc

*umsrenus o

CINX

@

Cccc

mmMllurt o SPHINX dirti uses~try

-0 Cccc

DLz3-ddiffusion

Fig. 3. Cutline of CCCC inter-

face system for generating multi-group constants for fast reactoranalysis.

CODE VALIDATION

The MINX/SPHINX system has F m tested in a variety of ways.32 3sOne ongoing project is a comparison of various processing codes beingcarried on by a committee of industrial and national laboratories (theDOE Code Evaluation Working Croup). In order co minimize confusing com-plications, this group as analyzed a simple homogeneous composition typ-ical of a large fast-breeder reactor core. The current results for someimportant parameters are given in Table 4. Larger differences exist be-tween the fluxes and various cross sectjors. !.tthe present time, forthis type of problem, the chief causes of these differences seem to be:(1) group structure and weight function, (2) elastic removal treatmentand (3) unresolved self-shielding. In any case, tl,enumbers in Table 4are less than the uncertainties associated with tl-,ebasic evaluated dataand with other design conservatism. They imply that the MINX/SPHINXsystem is accurate for routine fast-reactor design.

,.

.

Table 4. Comparis~lnof Various Codes for aBuckled Homogeneous Fast Reactor Model

Parameter.—

‘keff(C28/F49C28/F25T49/F25l’28/F25:F40/F25:F41/F25

1.

2.

3.

4,8

5*

6,

7*

8,

ARD ORNL LASL

MC -2 50 g/SPHINX 126 g/SPHINX 50 g/lDXValue Z cliff Z cliff % cliff

1.0040 0.10 0031 0.19

0.1585 -0.06 -0.88 -0.26

0.1447 -’0.35 -0.76 -0.35

0.9132 -0.33 0.12 -0,12

0.0206 0.44 1.12 0.68

0.1806 0.17 0.55 0.33

1.29& 0.29 0.36 0.33

REFERENCES

LASLETOX/lDXZ cliff—-

o.i~-0.32-0.48-0.23-0.44-0.44-0.15

GE50 @mwN% d$ff

0.170.000.28().22

4.271.550.30

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..

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—

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