synthesis methods and variational methods in nuclear .../67531/metadc... · mex.) . nucl scl frig.,...

Compiled: August 1970

Distributed: September 1970

LA-4461-BI8

UC-80, REACTOR

TECHNOLOGY

TID-4500

LOS ALAMOS SCIENTIFIC LABORATORYof the

University of CaliforniaLOS ALAMOS • NEW MEXICO

Synthesis Methods and Variational Methods

in Nuclear Reactor Theory, Physics, and

Computational Methods: A Bibliography

Compiled by

Jean Furnish

-LEGAL NOTICf-This report m prepared ** an account of worksponsored by the United Statei Government. Neitherthe United SUtes nor the United Sttteti Atomic EnergyCommJalon, nor any of their employ««, nor any oftheir contractors, subcontractors, or their employees,makes any warranty, express or implied, or assumes anylegal liability or responsibility for the nccuraey, com-pletenesf or usefuliWH of any information, apparatus,product or process disclosed, or represents that Its usewould not infringe privately owned rights.

DISTRIBUTION OF THIS DOCUMENT IS

SYNTHESIS METHODS AKB^VAKxATlbHAL METHODS INNUCLEAR REACTOR THEORY, PHYSICS,ANDCOMPUTATIONAL METHODS: A BIBLIOGRAPHY J

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Voluffi* 23 (1969)

c2Ut NUMERICAL STUlHRi OK COMPJNEll SPACK-T1MESYNTHESIS. Yaslnsky, J. 0. (WesUnghousc Electric Corp..W«*t Mifflfo, Pa.). Nucl. Scl. Knp... M: 1S8-68(NOV. 19C6).

The ability in obtain arrurale ivlutioiu to tlme-tlepewk ntgroup diffusion problems by simultnnenuiily synthentxlng in bothUM x and t illmeiiuion.s IN demonstrated numerically. Tho n»ten-tlfil of thu combined »;|inic-tlnit (tyittheals method becomcH appar-ent from several cuniparlaoiui of synthesis aoluliona with exact(la a finite iliffort-nce Hi-mr) Icof.roup, two-dimensional, limc-dopendi'rit diffuMun Mutation* Ills' twu different reactor ucoimlrli-H.U reterunceii. (auto)

C4443 ASYMMKTR1C DISCONTINIIITIKS IN SVVTHESISTECHNIQUES FOR 1N1TIAL-VALUK PROHLKMS. Becker,Martin (Hcnaselacr Polytechnic Inst., Troy. N. Y.J. Nucl.Sol. Eng., 34: 343-4(Dec. 1968).

Toe buaea for uae of asymmetric discontinuities at cverdeter-mtnod lnl->rt:iccn are rtlsciwscrf. Funclloo.it notation in csed toremove symmetry conaidrrntinna. Thr oxprcssluns for neutronflux and adjointH arc shown lo IKIHSCSS propcrtlcx <•> asymmetricdlncontinultlos; trial fum-llon cxp:ini«!ona arc determined for treat-ment of thitHO. (M. I-.S.)

tin SPECTRUM SYNTHESIS WITH SPATIALLY DISCON-TINUOUS BASIS SPECTRA. Vaugbaa, E. U.: ROM. P. F.sHauiimecfat, D. P. (Atomics Intermnttooa',. Caooga Park, Calif.).Traa*. Am«. Nud. Soc.. l i t M»-»«8OT. M«».

From International Catenae* on «b» Cwtructtve UseaofAtomic Energy, Washington, D. C. 8aa CONF-681101.

METHODS OF RESOLVING INTERFACE AMBIQU-m E S IN WSCONTltNUOUSVARIATlONAL METHODS. Ifeckiir,Martin (Raamelaer Potytecbalo teat., Troy, N. Yo). Tirana.A m r . Nod. Soc, U : 52»(Noc. 1*5»).

From toUntatkmal CoR^ercooa on tbs CoaatrwUrc Uaea ofAtomic Emergy, WaaUogtoa, D. C. Sa« CONF-681101.

4431 V/UUATIONAL DERIVATION OF DiSCRETE-ORDI-NATE-UKE APPROXIMATIONS. NatalaoH, N.; Gdbard, E. M.(Btttts Atomic Power Lab., W«*t UlfOla, PaJ. TraH. Amer.Haoi. Sao., 11: 69S-lffior. 1M().

From btcnsBttonal Conforawe on the ConatmcttT* USM ofAtamio Eaar^, WaaftiECtoa. D. C. S M CONF-S81101.

•r-™. SPACE-DEPENDENT DYNAMIC BEHAVIOR OF FAS1SEACTORS USING THE TIME. DISCONTINUOUS SYNTRESUMETHOD. Keaaler.C. fOeccnaohaft fuar KmforadMnc mbH.Ktttonabe, CtrJ. Traoa. Amer. Nod. Sue, lln HtCMov. W6fl).

From totenational Conference oa the Cooctmettre Uaes ofAtomic Energy, WaaWngtoB, D. C. See CONF-6U101. V.

B4443 ON THE USE OF DIFFERENT RADIAL TRIAL FUNC-TTOHS IN DIFFERENT AXIAL ZOUES OF A NEUTRON FLUXSYNTHESIS COMPUTATION. Wachapreaa. E. L. (KnoliaAtomte Power Lab.. Scbenectaoy.N.Y.). NucL Scl. Eng., 34:«2-^D«5C. IMS).

Uae of radlat trial (iuictlont In different axial zonea to avoidftac dlac«tlBuitiea la dlactwaed. Combining coefficient* are de-fined for uu varlona uiitl xonca; axial oeutroa teafcoge I* de-fined is Urtia of RiiUrU notation. (M.L.S.)

F4441 THE CALCULATION OF NEUTUON FI.UX RATIOS INCRITICAL SYSTEMS UY TIIR INIHHKCT VAKIATIONAI.METHOD. Poinraiilnjt, G. C. (l.oa AlamoH .Scientific Lub., N.Mex.). Nucl. Scl. frig., 34: 308-12(l)cc. L9QS).

A dlacuKHlon lu K'V'.n u( the usr of the Imllivcl varintlonalmethod for KunoraliiiK Iho trial r»;<ullunK midiHi to compute avarlatlonal caUnnitc of a homoguneoUH functional of Ihu solu-tion to an uiKcnvalnc equation. It Is shown th.it one UHU of ibomethod lvadH to no dlfflculUen, whcrcaH another use nlvu.i mean-lngleea rcHulU. In this tatter Instance, tho method of weightedresiduals can bo used lo Kenciato the nucuxHury trial functlom.With the trial {unctions known, the v-.ul.itlonal cslimatc of Uwfunctional of intercut follows by quadrature. Jauth)

C7412 (BKL-12911) ITERATIVE SOLUTIONS TO REAC-TOR EQUATIONS. Goldstein. Rubin t>*rookh»vFn NaUonnlLab., Upton. N. Y.). Mar. 1937. lOp. (CONF-670S01-1Q.Dep. CFSTI.

From International Conference on Research Reactor Utiliza-tion and Reactor Mathematics, Mexico City, Me*.

Recursion formulas for obtaining Itprntivt- notations to ireacforkinetics equations nrc dVrivcd. I'oniparisonsi of ilor.ilivc andvarlaticnal tcchnlqwa for kinctk's equations solution." arc dia-Chsacd. Application of thr Iterative process to the solution ofnCin-BcIf-adjoint etjuullons of the nrulru!i rfnoainii- :tlisor|ption-type >.» described. (M.L.s.)

Pt337 ANALYTICAL APPROACH TO CONTINUOUS REACTORREFUELING. Yaaukawa, S. (Japan Atomic Energy ResearchInsL. Tokyo). Nucl. Sci. Eog.. 35: l-13(Jan. 1969).

The analysis of the three-dimensional continuous refueling laapproximated by toe variational method. The axial flux distribu-tion is treated by the trial function and expressed by tho analyticexpression, using the elliptic function. Introducing the perturba-tion technfejus In the flux expansion, the higher order cross-sectionexpansion correction of the axial flux distribution is achieved byusing tfco elliptic function aa the base function in the flux expan-sion. It la shown that the group constanta, averted by the fluxand its square, can bo cxpreasod by the simple ration.il function.As a special example, tho nature of the eigenvalue uniler continuousunidirectional refueling is »hc a, and It is pointed out that noinecaution Is required U 11K? fuel burnup in evaluated liy the rcactivity-erea aietliod. (aulh)

Volume 23 (1969)

DH34t ON THE USE OF DYNAMIC COORDINATE FUNCTIONSIN BURN-UP-SYNTHKHIS-CALCULAT!ONS. Nathschlneger, P.{Wagner, J. Onstllut fuer neaktorteehnikc Sslborsdorf, Austria).Atomkernenorgie, 14: 5-7 (Jan,,-Feb. 1969).

A dynamic coordinate mothoJ in usod to optimize computer limefor burnup synthesis calculation*. Tho Nuwton-Coatrs quadraturefarmulnn are used to compute the matrlK eUimontH In the Uirnupsynthesis varlatlonnl codes; eigenvalues aro dotormlnod by thovector iteration method. Error function* are given for assooslngUM accuracy of the calculation*. (M.L.S.)

1940S (AI-AEC-12721. rpp 69-81) NUCLEAR SAFETY.Vaugb&n. E. U. (Atomics International. Canoga Park, Calif.).

Improvements In present state kinetics codes for use In fastreactor safety calculations are described. The spectrumeyn-thesis method In the one-dlmenslonal s u « c case is exploredand tested by comparison with multlgroup diffusion theory. Cal-culations are madcWtth the XMAS>RAUM-PLEASER system onthe two-region fast critical assembly ZPR-IH-48. (D.C.L)

H13474 <AERE-n-6773(Vol.4).j.p 1098-1123) OPTIMISATIONOF NUCLEAR RKACTOIC SHIEIDINCi CtiAllAC TE1HSTICS ONTHE BASIS OF VAIUATIONAL METHODS. Abagyan. A. A.; Du-binin, A. A.; Kaiuuxli, Yu. A.: Orlov, V. V.; Petrov, E. E.; Pia*o,V. Ya. (Goaudanttvemi)'! Komltet po Ispol'covanlyu Atomnoi Ea-•rgli SSSR, Mosoow. FUfko-Enerfetlcheskil Instltut).

Methods of optimizing ahleld composition and skip) are con-sidered. The opthnlr.atlun of nhield composition Is carried outby mean* of adjoint lunctlon* and prrtiirlu'tlun theory formulas.On >hi. banls uf perturlatlun tlwory forniutua the con>:e|« of sshield material cf{|<-t*nry funrltun with respect todlltvrvut shlaM-ln( characteristics I* formulated. Shielding material efficiency(Unctions reveal the degree of sensitivity of various (Meldingcharacterlatlc* to th- variation of shield composition. Expcrl-mentally measured elficiency functions fur some shielding ma-terials are preacntcd and are compared with calculations. Onthe basis of the concept of shield material efficiency functionsthe criteria of optimum shield composition for different casesIn classical and non-classical approximations of vnriationalcalculus are Riven. Shield shape optimization Is carried out onthe basis of the Eulcr equation. The attenuation of radiation inshielding layers is described by the beam analysis method. Theequation obtained for the function describing the form of theshielding Is solved by the method of successive approximations.(22 references) (auth)

C13574 (RP1-328-U2. pp 132-51) REACTOR PHYSICS:THEORETICAL. (Rensselaer Polytechnic Inst., Troy, N. Y.).

The problem of re-entrant hoi* effects on tUme-of-fllght ex-periments was studied; good correlation between theoreticalpredictions and experimental results for thermal spectrum ex-periments was obtained. Single-region and two-region fast

spectrum experiments war* also analyzed. Aa aa outgrowth ofthis work, « systematic comparison was made of th* coautoalyused definitions tt svxrsg* scattering angle*; It was coaele&athat algntficantly different values of th* aagl* e n fca octtiaad.Analytic and numerical trsatmsata of tte time-Jep*n4*«t flui !•pulled bat systems were studied. Analytic expression* w e n ob-tained for fast reactor flux and importance spectra in Unas cfsimple cross-section ratio eorrelattoa*. The appHcntioe ofOreen's function modes to th* analysis of space-tiM MatHe*•bowed Hat, at teast la a aarica of test cases, spatially dlaooartlnuoua Graea'a function modes are quit* successful a pradletiBgpower shifts. The discontinuous variaticaal method crproach to•pace-energy problems waa studied, with emphasis o» *i» lor-mulatlOB of tlme-dlacontlnuoua initial-value problcau. it waa

. shown (hilt, by using asymmetric discontinuities, th* a«*d forstaggered discontinuities may be eliminated, it waa also abeamthat use of asymmetric discontinuities la coBjisteat with skaforward bla* of She time-like variable In aa laiUal-val-j* probiaa.(D.C.W.)

FIf 133 EXrREMUM VARIATIONAL PRINCIPLES FOK THEMONOENKRQETIC NEUTRON TRANSPORT EQUATION WITHARBITRARY ADJOINT SOURCE. BusHk, A. J. (BetUs AtomicPower Lab., West Mlfflln, Pa.). Nucl. Sd. Eng., 35: 303-18(Mar. 1?C9).

A self-adjoint posltlvr-dcflnltr variation*! principle is |>rc*rhta>whtch leads to upper ai»l lower bounds for f>*,<>), where j>',<i) isan Integral nvnr position and angular direction of the proikiet ofth» one-velocity iieutntn triumport flux, 4>, and an arbltraiy a«flouitBtmrii:, K". The Kuler eiiuatlon of the functional l:< a m-w torm ofDie urn—velorlly Ikilti'iiionn neutrun ti-R»::port equation In which ths(lipcmlcnl vurlulile In I;III—hail Uie Mini of tf>aml <•'. vkw <>' lathi- ailjoint flux. Whi-n a trial fimMlon cunMiHtlnK of an expansionIn i.pln rli al Imrmimli'.-i I.1, lib.il, un<- ulitalnn us u l w r l ' lr.,ini<l for^ • » the quantify (US,,*(P-N'; S,»-<i;s,. *(P-N"; S,»>. where6,(r,O> - [S<r.l» • S«(r.-a)|/2. S,(r.nj - [S<r.f»-SNr.-H) | /2 .«<P-N';S|) !• an odd P- N appruximntlon to a problem with thesame cross sections as the original problem, but with nourre St;0(P-N";S}) is an even 1'- N approximation to a problem withsource S], and U Is the operator that lakes a function flr.il) IntoKr.-O). (auth)

B.C23t9S ^APL-P-3803) VAKIATIONAL MKTIIOU8 AND NEU-TRON FLUX SYNTHESIS. WachspresH, K. L. fCnolls AtomicPower Lab., Schencctndy, N. Y.). Jan. 13, 1969. 2Sp. (CONF-690401-1). Dep. CFSTI

From Conference en the Kffectivc Use of Computers In theNuclear Industry, KnoxviUe, Tcnn.

A review of synthesis techniques for calculating neutron fluxesin a reactor la given. Muthcdn of Rclec'llng base functions for »yn-thasls computations In the time domain are described; mathemati-cal properties of these base functions tire outlined. Methods ofchoosing the combining (unctions for definition of the functionalare prescribed; criteria for and conRtrajntn on these choices aredelineated. The concept of thu trial function is explained; Its rolela synthesis computations Is defined. The varlstlonal formallcmIs developed for direct and adjoint applications. Variation*! equa-

Volume 23 (1969)

Uoos ara constructed for discrete raulUgroup diffusion problems;error spproximationa are Included. Application of tbe variation*!equations to spatial synthesis of the thermal spectrum probisra iaditcuMiei. The method of diacontisftMua trial functlona for nultl-chsnnet synthesis formulation ia detailed. 42 references. (M.L.8.)

F23ff» (WAPD-TM-771) EXTREMIUM VARIATIONALPRWClPMiS FOR THE MONOLNEM5ET1C TRANSPORT EQUA-TION WITH ARBITRARY ADJOINT SOURCE JLWBR DEVELOP-KEHT PROGRAM). Bualik, A. 1. (Bettif Atomic Power Lab.,Pittaburgh, Pa.). F«b. 1968. Contract AT-ll-l-GEN-14. 48p.Dep. CFSTI.

A variations! method ia riven for etolermlninj tto lower and•pper bounds for the monuenergetic transport equation with ar-bitrary adjoint aource. Tfce functional la chosen aa a foirm of UMcae-vcloclty BolUmanii transport equation; apherlcal haramnlcaexpansions are uaed *s trial functlona. Three typea of boundaryconditions (b.c.) »re considered: vacuum b.c , symmetric b.c«nd antisymmetric b.c. Mathematical properties of to* functionalare deduced; ststlonarity criteria are applied to the functionalfor e!i three typea of bxmdary condition*. Lower end upper bujndaore obtained for tbe functional. A numerical example ia included.16 references. (M.L.S.)

2*1011 A GKNERALIZATION O F THE FINITE IMFFERBNCEAPPROXIMATION TECHNIQUE APPLIED TO A NODAL MODELOK SPACK-DKI>ENDI:NT NUCLEAR REACTOR KINETICS, A I -couffe. Raymond Kdmond. Seattle, Unlv. of Washinstoa, IMS.162p.

Thesis.A nodal :>pproxima!fcm of the tuuij-depeadent, one dimensional

neutron dlifiision e<|u:illon Including the affects of delnyod eeutroe*and simple temperature feedback, has been developed. Thle sp-proxlmnliim Is based on n generalization of the conventional finitedifference u-dmli|Ue in thai one Is able to supply Information or thislcchn(i|uc about tlw exjirotiHi solution In addition to a boundary andinitial conditlui!. Thv :i|>j>ioxliniitlt>n U derived from a variatlontlprlncipk- ami i.-,, tliercfore, consistent with tbe previously derivedHynlhculii mi-UHxl :iml nay be viewed us a subclass of such amethod. 'Ilu- aUilljulu:) of the parllcuiur method developed are -duacrtbcil. (Diissur. Atwtr.)

24169 (FEM4S) JlEACTIVITY JNTEORALS AND THEIRUSE FOR THE SOLUTR1N OF CERTAIN PROBLEMS IN REAC-TOR PHYSICS. Stumbur, E. A. tOoaadsrstvennyl Komltet poIspol'xo*an!ljnj AtomooS Enerjil SSSR, ObBiMk. FUlto-Energeti-cheakil Instllut). 1968. 51p. (In RussUa). Dep.

The work set* forth the concept of Integrals Involving reactivity,specific fum.tlonals of solutions to tie Bottsmann kinetic aqua-tion, which nre simply connected with the k«f( of an arbitrarygeometry r<»ctor. Invariant Integral relations are t.ubllshed forthe Boltznunn equattot) In tl.« case of critical and noncrltlcai sy»-tema. The "method of compenealed perturballona." which re-sults in a vurtmlon equation for tbe Intrgrala Involving reacti>-ayit, Introducnd. Tlw developed theory to uaed to solve some prob-lem* in renetor nhyalcs having to do with the Interdependence ofthe eirlticsl al«g and compoeiUoi of a reactor or the reflector sice,eto. Home Bontrasla of thc«retlcal reaulta and experlmmtal dataare dlscuswud. Tho work consklu.va further poeslbllillus for de-

veloping thu methods here aot fortli In order to solve other prob-lems la renctor phyalca. (tr-suth) (JPl«!

BM i l 1 AN ANOMALY ARISING IN THE COLLAPSED-GROUPFLUX SYNTHESIS APPROXIMATION. Adams, C. H.; SUeajr,W. M. Jr. (Qaner*: Electric Co., Sohenectedy, N, Y.K NucUSoU Eig., M: 444-7(.June 1949).

Ah anomaly encountersd in a flux-weighted ccilapsed-groupsynthesis problem is described. Tbe difference between group de-pendent and coUapsed-group synthesis is defined. In tbe collapacd-ctlccla^ons the expnnsion functions are used in the dual role ofweighting function ahapeti. The reactor model to which this pro-cedure la applied is shown; the group cross sections, used in thecalculation*, are defined. Results of numerical evaluation arocompared, graphically, for group-dependent and collapsod-groupcalculations; this vompaiHeon shows the anomaly. Cause* for theanomaly are analyzed. Ul references) (M.L.S.)

A2M17 SPACE-ANGLE SYNTHESIS. Swtbel. Harry E.(BatteUe-Nortbwest, RichUnd, Wash,): Bowes. Bradley. Nuc!.ScL Eng., Si: 436-8{Jime 1989).

Application of the apace-syntheaia critical slab a id oubcrUtcalfixed source problems Is described. Spatial functions are chosenfor varlatlonal synthesis solution to the Boltzmann transport equa-tion and the angular functions are determined. Results of numericalevaluation of the Boltzmann equation solutions are tabulated.(M.L.S.)

' C33104 METASTATIC METHOD IN NUCLEAR REACTORCORE KINETICS CALCULATIONS. OalaUc A. (CNEK. Rome).Nuci. Sci. Eng., 37: 30-40(July 1989).

A qu^ "-static method is proposed for evaluating spatial effect*on nuclear .eactor kinetics. Tbe neutron flux shapo la calculatedapproximately us an asymptotic solution of the two-group space-time diffusion equations, where delayed neutron behavior I* in-cluded. Two iterative procedures urv alternatively used aoconS-UK to lha amount of reactivity involved. The first one operate*until prompt cilllcalily In reached. The uccond procedure re-places tho first one as soon us the renclor «oee supcrpromptertti-csl. Tbe main Icuture of thu approach adopicd la the pox»ibUUyof tiolecllng an initial guess such that conver»tenc« Is reached atUs first Deration. The matter Is then rcduci-d lo HOIVIHK trxoeigenvalue problems. Theoretical and nunn-rlcnl comparlMiniiwith Henry'« ndiaballc model oulliiic the main role uf prrlurliedadjoint fluxes and correct neutron-flux sh:i|x- (the urcoml n rntooly for auporproinnlcrtticnl excursions) in di-finlni; the (.iinern-tion time and reactivity. When compared v h the exuet notation,results uf sample problem* show subutantlal accurncy In the fluxshape and amplitude. In uubpromjitcritlcal excursion*, on!y (hesynti>?sls method l« as accurate as the inrl.iKlatlc ens and yieldserrors of few percent at the flux peak. In Ike reactivity rangeabove prompt critical, differences between the exact result* andtbe nMtaatatlc ones ore unessential 0 references, (auth)

Volume 23 (1969)

B34f32 (KAPL-P-3811) ANOMALY ARISING IN THE COL-LAPSED-GROlll' FLUX KVNTHKSIS APPROXIMATION. A.laros,C. H.; Stacoy. W. M. Jr. (Knolls Atomic IViwur Lnb., Schnnec-ta.1y. N. V.). June 17, 19C9. Contract W-31-109-Eng-52, 10p,(CONF-f.90fi09 12). Dep. CFSTt.

From tSth Annual Meellng of the American Nuclear Rcclely,Seattle, Wash.

An anomaly iii'lnlng friim u fhix- wi'lghlcd eoll!ipf.ril-f>r<iu|isynthesis (Mlrul;iiloii fur mi ariinihnly• iixl'li'ti cylinilrfteal run:IH di.siuHM-d. Ite.sull.i limn unlni; fxpannluii iunclliMiu n» witgUttngihapcs arc described; thece remiltH arc eompored with actualvalues. Behavior of Hit- rali'iilalcri flukes is analyzed; reasons(or the behavior u'f thi< flux anomaly arc t'X|>lalnrri. 11 references.(M.L.S.)

34942 SOLUTION OF THE SPACE-TIME DEPENDENTNEUTRON KINETICS EQUATIONS FOR A REFLECTED SLABREACTOR. McFadden. James Hugh. Ames, Iowa, Iowa SiUleUniv., 1968. lOOp.

Thesis.A modal analysis technique is used to solve the spacc-ttme and

space-frequency dependent neutron diffusion equations for a re-flected slab reactor. A two-energy-group model !s employed anddelayed neutron effects are neglected In the solutions of the timedependent equations. The neutron flux is expanded In & series ofknown space functions multiplied by the corresponding time coef-ficients. The space functions are a form of the Green's functionmodes and the K nntorovich variational method1 is used to determinethe time coefficients. A coupled mode method Is applied to twotime dependent problems. Approximate solutions arc determinedby various sets >f trial functions and are compared with an exactsolution as determined by a finite difference code. The frequencyresponse of the reactor to a localized driving; /unction also Is In-vestigated. Solutions are obtained with four sets of trial functions,of which two sets include coupled modes, and are compared atvarious detector positions in the fuel and reflector regions,Guides an: given for the selection of trial function!! bawd onthe iocntinns in the -incloj where the frequency response !Pto be determined. CTSS)

B37716 ANOMALIES IN VAHIATIDNAL FLUX SYNTHESISMETHODS. Froehlich, Relmar (Qulf Gfn&ral Atomic fnc,.San Diego, Calif.*. Trans. Amer. Nucl. Sac. 12: lSO-ipune1969).

From 15th Annual Meeting of the American Nuclear Society,Seattle, Wash. See CONF-C90C09.

E3772* REACTION RATES AS WEIGHTING FUNCTIONS INFAST REACTOR SPACE-ENERGY SYNTHESIS. Neunold, R. J.(Babooek and Wticox Co., Lyachburg, Va.); ott, K. O. Trans.Attm. Nucl. Soe.. 12: 214-15(June 1969).

From 13th Annual Meeting of the American Nuclear Society.Seattle, Wash. See CONF-690S09.

A373F32 GKNKnA.U7.KD NODAL APPROXIMATION TO THENEUTRON TRANSPORT EQUATION. Sttn.stior.). n. O.j Uobkln,M. A. (I)nlv. of Wnshlngton, ScaiUc). Trans. Amer. NucJ. Soc.,13; 21G-l'/(Juno 1969).

from 16th Annual Meeting of the Amertc&n Nuclear Society,Seattle, Wain. See COKF-690609.

A"37742 ANALYSIS OF FUEL-BLOCK WORTH MEASURE-MENTS FOR A SMALL FAST ASSEMBLY IN A TI1HKE-DIMKN-SIONAL CONFIGURATION. Sargle. D. A.; Cohen. S. C.: Moore,R. A. (Gulf General A .mic Inc.. Kan Dlsgo, Calif.). Nucl. Pel.Eng., 37: S62-70(Aug. J9C9),

A number of fuel-block reactlvlty-worth measurements wereperformed In Core No. 1 of the thermionic critical experiment.The assembly is bare iinr.' neutruniciiliy hoinogcnt<ous, hut thegeometry Is esspntlnlly Ihrt-e-tlimrnslonal an.! Ihn dli!)en,ifcinsar» small. A synthcllc l>an:ij>orl |i«rturlialion nn-thod Is intro-duced for tba Piialyais of the fuel-block worths, The agreementbetween experiment and aitilyHiH based upon this method IK goon.A useful extension of IIH> IIWUHHI would l>e a r"taxallon of the fir^t-order perturbation rcstrictiun. 1<; rufcrences. (aulhi

B3771* A VARIATIONAL ITERATIVE METHOD FOR THESOLUTION OF THE ENERGY-DEJ'ESDKNT DIFFUSION EQUA-TION. Busllk. A. J. (Westinghojse Electric Corp.. WestMlMlin. Pa.). Trans. Amer. Nucl. Soc., 12: 152-3(June 1M9).

From 15th Anr.ual Meeting of ihu American Nuclear Socioly,Seattle. Wash. See CONF-690609.

B3*717 ANOMALY ARISING IN THK COLLAPSED-GROUPFLUX SYNTHESIS APPROXIMATION. Adams. C. H.; Staccy,W. M. Jr. (Qcneral Electric Co.. SchenectuUy. N. Y.). Trasw.Amer. Nu l. Soc.. 12: lSl-2(June 10C9).

From 15th Annual Meeting of the American Nuclear Society,Seattle. Wash, See CONF-C90C09.

(CCNF-630001-, pp 3-lSj REVIEW OF COMPUTA-TIONAL METHODS FOR SPACE-TJME KINETICS. Henry, f. ?.(Mrtss«cliuBetts Inst. of Tecti., Cambridge).

.itpproxlmatlon tcchnlque»i for reactor problems are discussedwiS» -cfen»nc3 to computer processing. The few group diffusiontiwt> made! la outlined. Methods used In solving the apace-timeequ* ic «re described briefly; these Snclude: finite differencing,Hme - nthests, and eptce-timc synthssla. 2!? references. (M.I..S.)

Volume 23 (1969)

F43205 FURTHER REMARKS ON THE CALCULATION OFRATIOS IN CIUT1CAL SYSTEMS. Pomranlng, O. C. (GulfGeneral Atomic Inc., San Diego, Call!.). J. Nucl. Energy, 22:631(Ct t.1968).

A comparison l i made of the two varUtloml principles of Lewins|N8A 20: 180S9) and PomranEng |NSA 21: 320251 for calculating tharatio of reaction rate* In a critical ayatem. The simplify of theprinciples from a computational point of view ia -snaidered. (UK)

B,G4612S (WAPD-TM-891) WANSY: A PROGRAM TO SOLVETHE STATIC TWO-DIMENSIONAL GROUP DIFFUSION EQUA-TIONS BY SYNTHESIS METHODS. Yaslnnky, J. B. (BcttlaAtomic Power Lib., Pittsburgh, Pa.}. July 1969. Contract AT-U-1-Q£N-14. 64p. Dep. CFST1.

WANSY is .t syatheals program that combine* ono-dimenaional(rial factions obtained from normal, adjoint or direct InputWANDA solutions via axlally dependent mixing coefficients to yieldtwo-dimensional Cms. shapes. The trial functions may ba axiallytttsOootlMMue (different sets of trial functions may be uaod In dif-fn*at axial regions), and different numbors of trial functions mayb*>«*ed in different axial regions. The synthesis approximationmay be greup-dspondent, partially or totally gruup-colliipsod, nndweight fasjottasi scaling factors may be Input to improve collapsedSroap synthesis solultoa*. Th» rynthesls approximations and re-paired lapat a n described. 13 references, (auth)

E.A4f I2§ «PACE-EN:WOY FLUX SYNTHESIS IN TRANSPORTTHEORY. Lttositeld, V. 3. (Brookhaws National l.*b., Upton,* . * . ) . N«eLSel.Es«,,lTt 423-42<8ept. lt<9).

T*» efficacy of the overlapping group method In fast-reactor•Mfcrsfe is tavastifatad aad tested oa an Idealized fast-reactoraoMfltarrttea. A Ml transport-theory treatment la adopt«d andIk* cverlsflHag group equstloae aro derived by the Indirect us**f a variatfeacl priadpie. A amber of refinements to the basic• a t ho* h«v» bam utsm'jMd aad s«rv« to damoaatrat* that with• ftdletoiB flheimt of vnriattoeal fmctioaal aad trial functions itto pacsaMe to abtaia accurate eetlnutea aot oaly of the reactivityatd other Sat*grai«4uaatitl««ixit also of the detailed flu*. Theseiaetade: leaving bet* the spaee/aagle aad energy dependence atthe trlel ftwetfoas to be determined by the variation*! principle,toooraarattag dtscwitlnaoua trial ftuwtioas, and the use of a newvarteUoael prladple for criUcallty problems that leada to eatl-

i of homogeneous functions!* of the unknown fhue. 24 refer-. (with)

mentation approximation are UBC-O* as trial functions, leading to aset of differential equations and boundary conditions which can beinterpreted UH the annloga of clllier thi: Mark or M«i«Iiak proce-dures used in Wio auherlcal harmonica approximation. 4 refcr-SEKWS. tot$>

472*6 (CNM-R-2(VoM).pp2S0-6G) GENERAUZED VARI-ATION A L PRINCIPLBS FOR KEACTOfl ANALYSIS. Pomranlng,G. C. (Brookhaven National Lab., Uplon, N. Y.),

Varlatlonal prlnclplsa are compared for derivation and accuracy.The basic Ingredient in tho dor3v«tlonn 1B a particular uso ofLagrsnge multlpllora. UelDic the kicbniquo of postulating iho prln-olplea and then proving the accuracy. It la shown that more gen-eral varlatlonal principles can bo e«Jiibllnhu(l. 10 rofurumuia.(auth)

C47302 «CNM-B-2(VoU).pp 994-1014) A COMPARISON OFSOME NUMERICAL METHODS FOR SOLVING THE EQUATIONSOF REACTOR DYNAMICS. Szeligowskl, John; Hetrick, Dark!(Arizona Univ.. Tucson).

The numerical at>luiion of the equations of reaotor dynamicsIs frequently complicated by tho presence of one response timethat la very short compared to the overall tima-responae of t*ieayatem (e.g., slow transients in a faat reactor). The number ofUam steps in a standard method (e.g., Adams, Kunge-KutU) maybecome prohibitively large. Some computational methods thaietroumvent this llmllniion aro reviewed nnd compared. Th«aathoda fall into two general claHitoa: thomi basiid on oxpnnNlonRia a small paramotoir, and tho«a wtlng lnu<gral equations in wliluhthe aCawly varylnp factors in tbu Integrandn aro' appraximalnd bykaowa fimctioBal forms. Ton former, which Includu the "pr°>«l>t->anp approstimaiion," may bo ctassifliid an alntcular nerturtuitlonexpaaakMui; those arc typically roatricted to a limited range ofsou*) paramoler and humw will fail for a non-atatiuaaiy Hyatmnwhenavar some Umo-dcpondcnt coefficient strays outsidr a pre-scribed limit (e.g., a *low rnmp rencttvity Input itppronchinRprompt critical). Integral methods do not have this latl'-r limita-tion; the methods treated may bo classified as iterative, varla-tlonal, and lnrgent-eigenvalue methods. Some typical rxnmptoaoi slow transients In fast roactors demonstrato reduction of Ihowunbor of tfmo atepa by a factor of the order of 10*. 19 rcfor-

(autt)

<CNM-R-»(V6U>, pp. tti-4t) VARIAT10NAL DE-VELOPMENT OF «N THEORY. Cchreiner, Sbkldoa; Ssleagut,D. f. (CalHorala IWv., Berkeley).

A* alUrnctive dertvatior of the flN equationa .or UMS one vulocltytraassirt arcMeaj la plane geometry Is deacrtjed by using avsriatisaal spa reach. This permits ntaiaiag the angular ceg-

h i l d l t l b M b t l dp p g g g

ttosj picntira of the mttiiron aagular dlatrlbuMon. but leadsto SB waaaAlgwa raanltlsr the esilrc »et of equstiooa aa wellM tha howHiary surf oaaUmity oondittens to ba lsjposed at tater-faeM. The required variatioaal formalieat ta ovUisxtd. The directsad adjoint aagultr flux diatrlbutiOM topiled by tha angular seg-

B C4T451 CANL-V410. pp S87-442) RSACTOR COMPUTA-TION METHODS ANTJ THEORY. (Argonno National Lab.. IIL).

REACTORS—aautroB diffusion equations for two-dlm«nslonalcylindrical, use of discontinuous variations! syatbee:>taobaJqiM for aolutlon static muHtgroup

—kinetic equations for, formulations of microscopic multi-group ercss sections for solution to

—aautrom diffusion calculations for, use of grnup-space-dspsodsnt bucklings to simulate coovUajto system trans-formatloea for

—kiaatle equations for, error analyses of liakad-diifereoce.

6

Volume 23 (1969)

BEACTORS, WATER COOLED—neutron diffusion equations fortwo-dlmenslonai cylindrical use of discontinuous variation*!synthesis technique for solution Btattc multigroup

—kinetic equations for, formulations of microscopic multi-group croso sections for solution to (M.L.S.)

045*54 (GA-9423) 3C AH A L: A SINGLE CHANNEL SYN-THESIS DEPLETION CODE WITH TRIANGULAR MESH IN THEHORIZONTAL PLANK. Taylor, K. C; Malakhof, V.; Leighton,S. C. (Gulf Genera! Atomic. Inc., San Diego, Calif.). July 1,196S. Contract AT(04-i)-633. i43p. Dep. CFST1.

The digital computer code SCANAL is described. SCANAL Isdesigned to synthesize a three-dimensional power distributionfrom GAUGE and RELOAD FEVSR analyses and to utUIre theCOPE code to calculate complete distributions of core moderatorfuel, and coolant temperatures. The program is written for theUKTVAC 1108. (C references) (D.C.C.)

G62103 (CEA-R-3713) SPACE SYNTHESIS: AN APPLICA-TION OF SYNTHESIS METHOD TO TWO AND TMItEK DIMEN-SIONAL. MULTIGKOUP NEUTHON DIFFUSION EQUATIONS.Nguyeo-Ngsc, Hoan (Commtswirtat a 1'EnergiD Atomique, Sa-elay (France). Centre d'Ktudes Kuclcaircs). May 1069. 68p.On French). Dep. CFST1 (U. S. Sales Only).

In order to reduce computing time, two- and three-dimensionalmultigroup neutron diffusion equations in cylindrical, rectangularOC, Y), (X, Y, 7.) and hexagonal geometries are solved by themethod of synthesis using an appropriate variations! principle(stationary principle). The basic Idea is to reduce the number ofindependent variables by constructing two or thrce-dlmensiontt!solutions from coluttons of fewer variables, hence the name "syn-thesis method." Whatever the geometry, one Is led to solve a sys-tem of ordinary differential equations with matrix coefficients towhich one can apply well-known numerical methods: Chebyshev'spolynomial method, Gaussian elimination. Numerical resultsfurnished by synthesis programs written lor the IBM 7094, theIBM £60-75, and tho CDC 6600 computers, are confronted withthose which are given by programs employing the classical finitedifference method, (auth;

12443 (AI-AEC-12820) SPECTRUM SYNTHESIS IN FASTREACTOR ANALYSIS. Vaughan, E. U.; Rose, P. F.s Hausknecht,D. F. (Atomics International, Canoga Park, Calif.). June 30,1869. Contract AT <Q4-3)-701. 106p. Dep. CFSTI.

The synthesis method has been applied to fast reactor calcula-tions In the form oi" spectrum synthesis, in which the neutrondistribution in a region Is represented approximately as a linearcombination of a few predetermined energy spectra (the basisspectra), the coefficients being space-dependent functions the(mixing functions). Equations to determine the mixing functionsare obtained in several ways which Include the standard varla-tional and Galerl In methods, and the group balance method inwhich the original multigroup equations are replaced by condi-tions oi neutron balance in a number oi coarse groups equal tothe number of basis spectra. The equations neve been east intoa form suitable for solution by the one-dimensional, few-group.

&ffuslon-thaory code: RAUM, which has been generalised byrelaxing the restriction* that the matrix of diffusion eovfficlentsbe diagonal end the icatterit£ traiwfer matrix be triangular.Codes have been written to translate nraltferoup croM-*«ctloalibraries into the matrlce* required by RAUM, and also tc con-struct synthesized cpectr* from the RAUM output. Tkta can btdone eveni'when different regions ua* different b^ia spectra,providled the group balance method la used, with the same eo*r*egroupn In all regions. A number of calculations uetag this syn-thesis icheme are reported, mostly on the critical assemblyZPR-iMB, with or without partial voiding of aodiwo, Ming atwo-fold or fhreci-fold spectral basis consisUng of typical spec-tra, found In a cnlculKilon with the ooe-dlmenslonal multigroupdiffusion-theory code CAESAH. Comparison of the RAUM acdCAESAR calculations indicate* that the multiplication factor*generally agree to a few tenths of a parccnt, and that powardistributions, aisd spectra at Individual points (especially t» thecore), usually Agree we!3 alto. But these favorable bidlcaUons•re qunllflcd by occasional failures, in which HAUM gives veryunphyslcal resiiltn. Such failiirvH can bo avoided by proper choiceof the ix»rse groups for the group balance method, and evidencets presented that hud choice can be recognized by fHludyii« th«RAUM input matrices ihey yield, without the m-cd for k completeRAUM calculation, (auth)

^2462 (KFK-781A) NUMERICAL SOLUTION OF THE PO-SITION-DEPENDENT DYNAMIC EQUATIONS FOR FAST BREEDERREACTORS USING A VARIATIONAI, PRINCIPLE. Kessler, G.(Keroforscmingszentrum, Karlsruhe (West Germany), Institutfuer RcaktoreRiwIcMung). Aug. 1968. 123p. (In Germac%(EUR-S9S7). Dep.

The Simo-, posltlon-i and onorKy-douttndont multiRfoup neutrondiffusion oquttlons are formulated for fast breeder reactors;boundary conditions are specified. Heat transfer equation* andboundary conditions for fut-l-to-coulant heat flow arc given. Thefeedback equations for the fuel, clodding coolant, and structuralmaterials aro limed, Methods for aoltjtion to the position depen-dent kinetic equations arc dlscuKsed; the Kantorovllch varlatlonalprinciple is described in detail. Tim Hyuturas o{ d£tf«rential equa-tions for the time funetinn of the nonstationnry multigroup neutronflux and precursor* aro formulated. AiHiilcatlun or the KaninrovitchvarlationtJ principle to tlie solution* of ihu^e cquittiimn Is douilud.The caleulatlonul model used as a sample problem for illuHlnitlngthe yavlaUonat solution is deaeriUtl. Ncutronica inramuters arotabulated. Remits of power dlKtrl»«i|o«i3 reactivity, and ntmtronflux calculations are described; data arc shown grapnlcaHy. «1references, 4M.L.5,)

VOIUBM 22 (1968)

VARlATfONAL ESTIMATE OF TUB DIFFUSION CO-EFMCIKNT IN A LATTICE WITH CAVITIES. DsTOOght, Jacquesfflsir. tfhre, Rrtsjcsis). ttuUsesik, £0t 119-21 (Sspt. 19#?>.

The theory of Bcnotat is modiricd by ualag a deflaltloa of thediffusion coefficient taking Into account importance weighting. Itis shows that in the one group model a quadratic weighting by thefine structure flux exist* Instead of • linear one, and that themacroscopic variation of Importance across the reactor makesthe absorption correction or Bcsolst disappear. The BolUsuaaequation for the flux perturbed by an Imposed gradient Is thrownInto a scir-adjoint form by a method of Vladlmirov. A variation!expression of the diffusion coefficient Is given. The expressionfowsd ia closely related to Dlrichiet principle, sad involves onlydifferential expressions. It Is shows that diffusion theore trialfractions in a ton-capturing medium lead to the "electrostatic"approximation. By direct extension of • method ot Carter, thevariational expression found is ahowa to be identical to the onederived above. It Is shows, that the relative variation of the dif-fusion coefficient, obtained by tbe introduction of cavities, iaeoual to the relative variation of potential energy obtained bythe introduction of a dielectric cavity la a conaUat electric field.Henceforth, classical expressions iS*/r!ved by tvlya and Szego,give as explicit value for tbe diffusion coefficient ia the presenceof targe cylindrical cavities of arbitrary cross-sections. Elec-trical inaiogues arc briefly discussed, (auih)

C)M2 (KAFL-P-3392) TIME SYNTHESIS: A STUDY OFSYNTHESIS MODES AND WEIGHTING FUNCTIONS. Rydin,Roger A. iKnoils Atomic Power Lab., Scbeoectady, N. ¥.) .Nov. 7,1967. Contract W~31-109-eog-S2. ISp. (CONF-67U02-31). Dep. CFSTI.

From 15th Conference on Remote Syatema TechnolOfy aadAtom Fair, Chicago, 111.

The Time-Syntheala Method haa been ueed to Inveatlgate theeffect* of the choice of trial functions and weighting function* ontbe results of (yntltesis calculations. It waa found thct the time-dependent total power In a reactor could be adequately predicted .using any of the methods of weighting considered (region-balance,flux, or adjoint), but that the apatlal flux distribution was moredifficult to synthesize. In particular, region-balance weighting didnot give very good spatial results, flux weighting gave batter re-sults which were sensitive to the group spectra, and adjoint weight-ing gave tbe best results. A singularity In the total removal matrixwaa observed during the course of tbe transient In one of the ad-Joint weighted problems. The synthesis results were sensitive tothe choice and number of trial functions used. In particular, theasymptotic flux shape, corresponding to the final state of the re-actor, was not the best mode to use because of delayed neutronholdback effects. Similarly, the addition of transition modes be-tween the Initial and final afcttea were beneficial in Improving thetime-dependent flux shapes calculated early in the transient. Quail*tttlve criteria were developed for choosing synthesis modes basedupon the Initial eigenvalue separation of the core, the time scaleof the transient relative to the delayed neutron half-lives, sad thetotal magnitude of tbe perturbation. (mOki

VARIATIOHAL FUNCTUNALS FOR SPACE-TIMENEUTRONICS. Stecey. Weston M. Jr. (Knoi)s Atotsto PowerLab.. Sebenectady, N. Y.). Nucl. Set. Eng., 30: 448-f3(lM7).

Two variational principle*, which may be used as the basis ofmodal expansion :ipproxlmalions for treatment of space-tlci* de-pendence of neutron balance equations for dvscrfclns; reactors,are given. Time-dependent muUlgroup Pf equation* are writtenIn a variation*! matrix nutation; expansion functhmu which satisfyspalls! and temporal boundary conditions are obtained, A tlroe-IMegratcd approximation Is then uacd to obtain equations in spa-tial notation only. 12 references. (M.L.S.)

APPUCATION OF TIME-SYNTHESIS TECHNIQUESTO COUPLEB-CORE-TYPE REACTORS. Yaslnsky, 3. B.(Bettls Atomic Power Lab., West Mlfflln, V».U Trans. Amer.Hncl. Soc., 10: 570-1 (Nor. 1967).

From ISth Conference on Remote Systems Technology andAtom Fair, Chloago, 111,, Hot. 5-9,1967. Ses CONF-671102.

C.DSffit APPUCATION OF VARIATIONAL SYNTHESS TOOPTIMAL CONTROL. Staoev. Weaton M. Jr. (Oseeral EIsc-trio Co., Sohsswctady, N. Y.). Tran*. Aroer. Nucl. Soc., 10s6»4(No It6?>

From 18a Confereooe on F.stuote Systems Technology asijAtom Fair, Chloago, ni., Nov. 5-*, 1967. Sse CONF-671103.

B.GW2 MULTICROUP, TWO-DIMENSI(WAL SYNTHESISCALCULATIONS FOR FAST POWER REACTORS. Hutchus,B. A.l Kelley, M. D.; Cyorey, 0. L. (General Electric Co.,Sssayvale. Calif.). Trana. Aner. Had. Stic., 10: 626-7JNov.

Ttom 15th Confereooe on Remote Systems TechnoloRy andAtom Fair, Chtoago, m., Nov. 5-9,1967. See CONF-6TU02.

C4144 TIME SYNTHESIS: A STUDY OF SYNTHESIS MODESAND WEIGHTING FUNCTIONS. Rydin, R. A. (General Eleo-tirlo Co., Schenectady, N. Y.). Trana. Amer. Nucl. Soc., 10:509-70 (KciV. 1967).

Frsm ISth Conference on Remote Systems Technology andAtom Falir, Chicago, HI., Nor. 5-9.1967. Sea CONF-67I102.

F.C•IS7 SEMIDIRECT LEAST-SQUARES VAIUATTONALMETHODS ANS INITIAL-VALUE PROBLEMS. Bseker, Marttaffsassslatr Polytechnic h i t . , Troy. M. YJ. Traas. Amer. Nucl.Soo.,10i tSO-lKtar.lNT).

From ISth Confereaoe on Remote Systems Technology aadAtom Fair. Chicago, OJ., Mar. 5~», 19C7. Ss« CONF-sniOa.

Volume 22 (1968)

F7*24 VAKIATIONAL PMNC1PLE OF THE NEUTRONTRANSPORT THEORY AND PSEyDOENTROPY FRC9UCTION. Helmis, G. (Instltut fuer Magnetohydrodynamac. it/an,Ger.). Monatsber. Deut. Akad. Wiss., Berlin, 9: 186.91(1967).(In German).

The relation between tbe variations! principle of reactor theoryand the principle of the minimum pseudoentropy production waaderived by assuming a very simple model of the processes In thereactor. Usually, however, tl are is a qualitative relation betweenthe two principles according to the usual model. (J.S.R.)

1213* (BNWL-m. pp 2.1-S> REACTOR THEORY ANDCODE DEVELOPMENT. (Battelle-Norlhwost, Rlchland, Wash.Pacific Northwest Lab.).

Mghtoon iRotopi* for which additional and more accurate datehavo beon added to the BNW Master Cross Section Library arelisted. An nngulnr (tynthosls approximation muthod applied toanalyzing a thin critical slab <e outlined, and eigenvalue* tadtotal fluxes aro presented for several different thicknesses.(H.D.R.)

C142ft (RPI-328-100, pp 140-87) REACTOR PHYSICS:THEORETICAL. (Renssslaer Polytechnic Inst., Troy. N. Y.).

Using nuclear data intended for fast reactor large oxide oores,the Inverse moderating ratio and the flssion-to-absorptloa ratiowere correlated and the adjoint spectrum was evaluated. A pre-liminary analyiis of a large carbide core was also made using thesame data. The cross section correlations and the comparison ofanalytic and multigroup adjoint spectra are shown for the carbUesystem. The possibility of using czmldirect least-itwiare* varu~tional methods for initial-value problems in fast reactor anklyslsis examined. Static tilt test problems using Green's function mode*and variatioaal methods were performed for test reactors coasist-ing of bare slabs of varying thicknesses containing moBoenergttioneutrons. Directional and scalar fluxes as a function of hale depthare presented in an examination of re-entrant bole perturbation af-fects In water. The tiros dependent equation for the slowly dowmof a fast neutron pulse in a finite hydrogenous medium H I studied.The problems of Doppler broadening of p-wave and s-wave neutronresonances were formulated and examined. A theoretical atadjr oftime moments for thermal neutrons waa initiated using a singlerelaxation time model. Results obtained using • model tor theanalysis of time dependent neutron transport are prsssnssi. Theaccuracy of the importance-shape method for detemlaisg tfcageometric tackling in reflected reactors Is investigated. UU>Jt.)

A14305 STRATEGY FOR THE APPLICATION OF SPACE-ANGLE SYNTHESIS TO PRACTICAL PROBLEMS IN NEUTRONTRANSPORT. Natelson.M. (Bettli Atomic Power «*sb.. WestMifflln, Pa,). Contract AT(U-l)-gen-H. Kuol. Set. Eag., 31:325-36(1908). (WAPD-T-1986).

A strategy Is proposed for the application of spaee-angle syn-thesis (SAS) to the finding of solutions for practical nuclear reac-tor neutron transport problems. A simple SAS approximation isderived. Trial functions for tbe approximations are to be createdfor etch mesh point used In describing • set of similar problems

which are to be solved. The strategy Is conceraed wltfc <$oastruetiagproblems that are simpler than, but representative of, .diet set ofproblems finally to be solved. St Is from trwisport eotaiiete ofthese representative problems fast the e^S trial functions ere tobe formed. This strategy and the simple SAS approximation areapplied succepifalty to several sets of similar problems forwhich diffusion theory is inadequate, faatn)

C14309 ANALYSIS OF PULSED FAST-NEUTRON SPECTRAIN MULTIPLYING ASSEMBLIES. Jenkins, i. D.; Dsitcb, V. B.(Rensselaer Polytechnic Inst., Troy. N. YJ. Nucl. Set. Eng.,.31: 2M-33UM8U

A simple model Is developed to describe the time depends*neutron spectrum In pulsed systems whose decay may be domi-nated by either a fundamental or a peeudofmdaiAmital mode. 9achsystems Include a large class of fast multiplying assemblies andthermal nonmultiplylng assemblies. The simple mode providesqualitative understanding of the role played by the fundamentalor pseadofundsmental mode In the kinetic evofcslan of the ttme-dependsnt neutron flux and, when optimised by t vartatloaalprinciple, gives excellent quantitative descripUoH of the fhocfor a wide range of systems. Trial functions are presented wfctefc,when adjusted with a suitable varlatlonal principle, provide • foodestimate of the shape and decay rate of the dominant raprodaelagmode of such systems. The method works well for systems wherea fundamental mode exists and is also applicable la the rangewhere pseadofundamental mode behavior is observed. ElgeaAmc-tlon eigenvalue solutions are obtained for tha fast mnltlplytngsystem GODIVA and these, together with similar solutions forberyllium, provide a basis of comparison for the variationalmethods. The investigation show* that care should be exercisedin associating reactivity and period parameters with far sabcriH-eal ayatema because tbe flux shape is changing substantially andthe major regenerative mode Is not Isolated when tbe eigenvalueassociated with this mods lies in the continuum. In the farther sub-critical region an example shows a complete lack of a single domi-nant mode. 19 references, (auth)

F . --• • "

14I1» ON THE PERTURBATION FORMULA FOR RATIOSIN CRITICAL SYSTEMS. Ribarlc. M. (teat. Jctaf Stefan.Ljubljana. Yugoslavia). J. Nucl. Energy, Jl: SM-tMfOac.W«7).

Usaohev (1964) derived by physical reasooing a perturbationformula for computing the changes In the frequency ratios ofvarious processes In a critical reactor caused by a change in itacomposition. Lowlns »988) and Pomranlns « M » constructedanalogous varlatlona! principles for this purpose. Fischer (1*67)Inferred an equivalent parturbattiXi formula from tho/symptotlcbehaviour of tho pertvrbsd critical reactor. Usaohevhs perturbnUoaformula Is derived mathumattcaliy via Brillouia-Wig>er pertur-bation formula of qunntum mechanic*, tautk)

B.EI41IS FLUX SYNTHESIS BStNO GREEN'S FUNCTION INTWO-DEMEKSIONAL GROUP DIFFUSION EQUATIONS. Ko-bayatU.KaUukt fCyoto Unlv,). Nucl. Set. Eng., Jl: »l-101(1H»>.

Volume 22 (1968)

The group diffusion aquations 1> two dimensions are solved byaawemiist; the separation of variables ssctlonaUy. Using oee-cUacMiMal Grain's functions, the two-dimensional diffusioneqaatioafl ere transformed Ssto two seta el oae-dimensicnalthree-pot* difference equations at flae-awsh points. Assumingthat the separation of variables of x and y coordinates is positl-VSa Sn a co*r;« aiesh in a reactor, the two sets of one-dimensionaldifference equations »r» solved by the alternating direction Itera-tion wetfcod. Sampfet ealculaUons for !MU-HjO thermal reactorsshow feat this method gives fairly good results with few coarse andfine noshes and the computation time can be considerably reducedcompared with the usual finiU difference method. > references,(ants)

F.B14329 USE OF DUAL VARIATIONAL PRINCIPLES FOR THEESTIMATION OF ERHOK IN APPROXIMATE SOLUTIONS OFaarVSKXt PROBLEMS, Yaelaaky. '• B.; Kaplan, S. (BettlsAtonic Power Lab,, West Mifflln, Pa.). Hue!. Set. En*;.. 31:

»C«)A s»sthod for using reciprocal vsrlatloasl problems to develop

flgare* of mtrlt for approximate Mlutions of diffusion problemsla presented. The theory of the reciprocal problems is describedla both a continuous and discrete context. Connections with theI—ttncl of Slobodyomiky are discHsucd. A »tr:iti>grm is presentedfor t i r t f " t the method to the (iron-self-adjoint) group-dlffuiloncase. Limitations of the method are discussed and numerical ex-asaples given. It is concluded that the method is useful in one-,two-, and perhaps in small three-dimensional problems but isprobably computationally not practical for full-blown, detailed,thr*e-dimeasioncl calculations. • references, (auth)

F1Mt8 VARIATtONAL PRINCIPLES FOR CALCULATION OFARBITRARY RATJO OF PROCESSED IN CRITICAL. ASSEMBLIES.DwtTSdi, S. R. (Bhabha Atomic :ic«arch Centre, Trombay,Mia). J. Nucl. Energy, 22: 123-5(Fe4>. 1968).

k to shewn that the principles of Lewias and Pomranlng are thssuns to the first ord«r, sod that Pomranlng'a principle can boderived from tiui of Leilas. (UK)

C1MM GENERALIZED FORMULATION OF POINT NUCLEABREACTOR KINETICS EQUATIONS. Becker. Msxtte (Reauae-laar Polytechnic lost., Troy, 3 . Y.). Contract AT(3fr-3)-3M.Nucl. Sci. Ei«., 31: 458-64(1906).

The most general current formulations of the point reactorkinetics equations permit the flux shape function to be time de-pendent. This permissibility has led to the development d aclaaa of space-time analyses referred to aa adiabattc or qvaci-static. The use of Ume-iNtkperaJcne Importance weighting, bow-ever, can lead to difficulties, as i t shown in an example. Pointkinetics equations are derived from a variations! principle Insuch a way as to permit timc-tfependent importance shape func-tions. "Extra" terms due to the explicit time dependence of theshape functions appear, and normalization conditions are obtainedby which these terms can be eliminated. Additional differencesfrom conventional form appear if one efcooses to use different im-portance shape functions for flux and precursor equations, butthese differences can be neglected for many cases of practicalinterest, (with)

B.G1 t t » APPLICATION OF SYNTHESIS TECHNIQUES TO THECALCULATIONS OF THREE-DIMENSIONAL REACTIVITY-COEFFICIENT DISTRIBUTIONS, Bear, 4. L.; Judge. F. D.;Vsoerus, E, R. (Gennral Electric Co., Scheoectady. N. YJ.Contract W-Sl-108-eng-£2. Nucl. Sd. Eng., 31: 349-53(Feb,IMS). CKAPL-P-S312).

Application of synthesis techniques to calculation of three dl-msnsloasl reactivity coefficients is discussed. An expression usingthis synthesis method Is formulated. Comparisons between syn-thesized calculations and diffusion theory calculations are presentedgraphically. (M.L.S.)

B1432» ANOMALIES ARISING FROM THE USE OF AD-JOINT WEIGHTING IN A COLLAPSED GROUP-SPACE SYN-THESIS MODEL, Yaslnsky, J. B.: Kaplan, S. (WestinchouaeElectric Corp., West Mifflln, Pa.). Nucl. Sci. Eng.. 31s 3M-E(Fab. I«6«).

The use of the adjoint weighting procedure, aa used In a collapsedgroup-space aynthesls model. Is discussed. Trst problems sreasad to show how use of weighted adjoint* dn«troya the utility ofthis oeUspssd-group sycthsals. The use of adjoint weighting tech-niques Is compared to use of Galcrkin weighting. (M.L.S.)

1MM SYNTHESIS METHOD OF UNCERTAINTY AKALYSUIN NUCLEAR RKACTOK THKRMAL DESIGN. Feaech, Htarildueron. Henri M. (Massaolswetta Inst. of Tech., CmiateWga).Nucl. Sd. Eng.. 31: S05-13(19e8).

The principal methods of core design uncertainty aaalyata ar»crltlcaily reviewed. The overcanscrvatlam of the DetanaiaisUsMethod, which alms at cmwrlag that the deaiga Umlta ftsaast toexceeded In (he most loaded channel «or at the moat loaded spot),leada to a probabilistic approach la which the probability of saoaan event Is evaluated. Recent work In this direction Is diseases*.It Is emphsalccd, however, that a probabilistic reliability evalua-tion must cover the whole core, and not only its most heavilyloaded element. The Synthesis Mellaod fulfills this rwtalriauntwithout demanding the use of computers. The Synthesis Methodalso allows ths use of a realistic space-depeadeat reliakUilTcriterion. The various methods ssder review are oonparad astheir application to a fast gao-coolsd veactor core. The powerlevels corresponding to a given reliability are calculated sad DMSynthesis Method is seen to be more conservative than the classi-cal Statistical Method and less conservsUvs than ths ectaralsiatt*Method. 16 references. <autk)

10

Volume 22 (1968)

B22*13 tANlr-7419) REACTOR DEVELOPMENT PROGRAM.Progress Report, January 1MB. (Argoam National Ub. , CU).Feb. 27. 1968. Contract W-Sl-109-^ng-SB. 166p. Dtp, CF8TU

REACTORS—excursion* in, coherence effects In aupcir protaptcritical, (T); neutron flux eynthetia In, variation*! methodfor, (T); core deformation under iopulsiv* loadiag. aaalysteof cylindrical, (T)

NEUTRONS—flux synthesis In reactors, varlaHonal methodfor. <T)

CMIS* (KAPL-P-3425) SPACE-TIME REACTOR KIKETIC9DEVELOPMENT AT KAPL. Stacey, Weaton M. Jr. (KnollaAtomic Power LrJ>., Scbenectady, N. Y.). Mar. 1MB, ContractW-31-U*-e&g-52. 42p. (CONF-6804M-1). Dep. CFSTI.

From Conference on Induatrlal Neada aad Academic Re-search la Reector KineUca, Upton, N. Y.

Effort* la tka development of matfaoda for tha calculation oftransient power distribution* in operating reactor cores dariaga variety of plausible tranaianta ara reviewed. The davelop-meat of Xenon apatlal transient analysis and space-Mine fluxsynthesis la examined. Studies on deflning nodal coupllisg coef-ficients and on. the lnfiuenos of ipatial effeeta on the kineticsof low aource reactor startup* are alao discussed. (H.D.R.!

B25140 INTERFACE CONDITIONS FOR FEW-GROUP NEU-TRON DIFFUSION EQUATIONS WITH .'LUX-ADJOINT WEIGHTEDCONSTANTS. Busllk. A. J. (BetUa Atomic Power Lab., WestMifflln, Pa.). Contract AT(11-1)-Gen-U. Nucl. Sci. Sag.. 32:233-40(1068). (WAPD-T-2072;.

Few-group diffusion equation* are derived from rarlatioaalprinciples. It la shown that by proper choice o! trial (uncUon itle possible to derive a few-grot* theory In which Interface bound-ary condition! of continuity of few-group fluxea and currents areobtained, even when tbe few-group conatants are obtained by fin-adjoint weighting. Tbe analyeli la facilitated by tbe m* ot feee-tlonals that Incorporate the interface condition of flux continuityby mean* ot Lagranfb multiplier*. Two functionate arc used togive two variant* of tbe theory. Both functional* have ta EeJerequation* tbe F-l approximation to the tlme-lcdcpeadent, elfai-value form of the cnercy-dependent transport equation.. la addi-tion, the current and Dux interlace boundary condltioBW am partof the complement of Euler condition* of the funcUoaala. Tfcefunctional* admit trial function* disccntlnuou* In apeca aad energy.Tbe two functional* differ is that on* CM both fiux and curraatarcumenta, wbereaa the other haa only flux ariumeata, aad yieldathe P-l equatlona in aecond-order diffusion form, (aotis)

oowpi!*lnc an ©verity of ocdee HRO, Batttlle Kevtaed Tkermoa,nad Km, ha* bees ooded to provide a code pmoksge for aae la ra-actor anatyai*. Modlfieatioaa have bam ande to tte HAMMERcode to make tt more compatMe wiO otiMr eodta la «** at PHL.(aoth)

CM i l l (KAPUM-474I) STUDY OP THSMVLTKBAMHBI.SYNTHESIS METHOD FOR SPACE-TIME NEUT«O»acXlWeaton M. Jr. (KnoUa Atomic Power Lab.. ffcaanaLladiNov.1967. ContractW-31-10f-E«,-52. • £ D ^ I c W T I .

A mulllchannel apace-tlmc ayntheaia model for the calcalatieaof nonaeparablc reactor translenu ia developed froaj a niUHaaalprinciple which admlta expaaaloa function* taak are itiai aalHiaauaaiIn apace and time. In each of many apalial regiaaa. tae flaat doriafeach Interval ot time la expanded la known ftactkMw nfairlllaawith unknown cupaneloa coefticleata. Tke accuracy of tka me***,and l u auperiorlty with reapect to th* coavaatlowd i lw i i ihaiaulapace-time aynthcaU method, are ik.nonatratrd 8>y aevwml icat example*, (auth)

M » I APPLICATION OF TniE-SYNTHESIS TECHNIQUESTO COUPLED COSE REACTOR3. Yaatasky. J. B. (BettiaAtomic Power Lab., Waat Mtmin, Pa.). Kucl. Ssi. E«a>, 32:42S-9(1M8).

The appUcatioa to coupled core reactor problems of tbe BMHI-mode time aycmeals approximation ualag trial fuactkma HUr)witch are eaally obtaiaed la deacribed. (D.C.O

CANONICAL AND INVOLUTORY TRANSFORMA-TIONS Of VARIATONAL PROBLEMS INVOLVING HIGHERDERIVATIVES. Kaplw.S. (W«atl«e>iKia* Elertrtc Corp..Waat MttfUa, PM. J. Math. Aari. Appl.. 82: 4C-tS<Apr. 1SM|.

Bcetuee of their value In thia type of proWaaj it is clearly *e-•irabl* to extend tba notion of cawaaicml aad iavuWtary traaaformation to variational proMema tavolviBK higher derlvaUvaa.Although thia axtenakw la atraigbtfomard H iteoa eat K M la hareadily vtllanlc l% the IHcraUr*. Thua an aayaatMaa, • ( theproeeaa la given here. Only proMrma conUlalag aecaaU atiivevUvee ara conaldered; however, talr saaa aaeasa aaMcieBt to in-dicate tha pattern far ppobtciae wit* kighar dartvaUvaa. (W.D.MJ

274S1 <BNWL-634,pp 2.1-8) REACTOR THEORY ANDCODE DEVELOPMENT. (Battelto-Northweat, RSchland, Waah.PaoUic Northwest Lab.).

A spaoa-aagle ayntbeaia approximation of She transport equationhaa been developed which ahowa Improved accuracy over tbe PIapproximation. The problem considered U aa iafiaite hoaMgeacou*slab with a aaiform aource. Tbe BUteUe Revlasd THermoa eod*haa bam modified by dM addittoa of aa improved e*.ramt eakiua-tkm routine, flasion croac aeetioa averaglag both mleroMoptoaUyper isotope aad maeroaooptoaUy for the cell, and tte capability tohandle ap to eight mixture*. A oomputtr program entitled HTII.

A3M34 SPACB-ANGLE SYNTfiESJS. Zwlbel. H.S.Memorial bat., RlchUad, Waah.), Bowaa. » . Traaa. Aaaar.MwL loe., 11: mUvm lfM).

FMaa UXk AasMal Meeting of ha* AMarieait NHelaar laetaty,Toronto. Sea CONF-MOMl.

11

Volume 22 (1968)

B.€>M00 A SYNTHESIS METHOD FOR CALCULATING TEM-PERATURE. POWER, AND BOWING REACTIVITY COEFFI-CIENTS OF FAST FOWER REACTORS. Blomberg, Pehr E.(Argoane National Lab.. Idaho FaUs, Idaho). Trans. Aawr. Nucl.Soc.. 11: 20S-9 (JUM 1968).

From 14th Annual Meeting of the American Nuclear Society,Toronto. S M CONK-680601.

F3313f EXTIIEMUM VARIATIONAL PRINCIPLES FOR THEMONOKNERGKTIC TRANSPORT EQUATION WITH ARBITRARYADJOINT SOURCE. Uusllit, A. 4. (WeatiagbouSQ ElectricCorp., West Mimio, Pa.). Trana. Amer. Nucl. Soc.. 11: 315(June IMS).

From Mth Annual Meeting of tha American Nuclear Society.Toronto. See CONF-680S01,

B3292J NUMERICAL STUDY OF SINGLE-CHANNEL FLUXSYNTHESIS. Adama, Cyrua H.; Hydin, Roger A.; Stacey, WestonIt. Jr. (Knolls Atomic Power Lab., Schenectady, N. Y.). Trana,Amer. Nucl. Soc., 11: 16J-70(June IMS).

Fmm 14th Annual Mee*.ii* of the American Nuclear Sudoty,Toronto. So» CONF-6MC0I.

E32C2i SPACE-ENERGY SYNTHESIS TECHNIQUES FORFAST REACTOR CALCULATIONS. Hurley, T. E.j Wljliameon,J. W. (Webtinghous* Electric Corp., Madison, Pa.l. Trana.Amer. Nucl. Soc,, 11: 174-5(June 1MB).

From 14th Annual Meeting of the American Nuclear Society.Toronto. See CCNF-C8O6O1.

3 3 U f VARIVTIONAL MULTICHANNEL SPACE-TIMESYNTHESIS MODEL FOfl THE ANALYSIS OF NONSEPARABLETRANSIENTS. Staeey. Weaton M. Jr. (Kaolls Atomic PowerbO>.,Sc!tMWCtat!y.N. Y.). Trans. Amer. Nucl. Soo.. 11: 178-1

( From 14th Aan>£! Meettivg e? the American Hoclear Society,Toronto. See CONF-fflSSQX.

C32SW COMBINED SPACE-TIKE SYNTHESIS MODEL: NU-MERICAL COMPARISONS WITH EXACT TWO-GROUP TWO-DIMENSIONAL TRANSIENT SOLUTIONS. Yasiasky, J. B.; Lynn.L. L.I Kaplan, S.; Porscblsg, T. A. (BetUi Atomic Powsr Lab.,West Mlffls«, Pa.*. ITraas. Amer. NucL Soc., 11: m-S(Jiine9M9M)

Frees i4'tb Anual Meettag of the Amertcaa Nuctoar Society.Taneto. See CONF-««0<01.

ELEMENTARY SYNTHETIC APPROACH TO THREE-DIMENSIONAL TRANSPORT 3>£RTURBATJON THEORY. Sar-gla. D. A.; Cohen. S. C. (Oulf General Atomic Inc., Saa Diego,Callf.(. Traae. Amor. Nucl. Soc., l i t 173-4Wu«o ISM).

From 14th Annual Meeting of the American Nuclear Society,Toronto. See CONF-WCSOt.

F3S202 COMPLEMENTARY VARIATIONAL PRINCIPLES ANDTHEIR APPLICATION OF NEUTRON TRANSPORT PROBLEMS.RNoraatag, Q. C. (General Dynamic! COIJI., General AtomicDlT..Sa» Diego, Calif.). J. Math. Phyt. (N. Y.), 8: 20M-2108(Oct. 1M7).

Sereral variaUonal principle! are developed which give upperand lower bounds for the linear functional (8,*), where * i s theeotation.of the in&omogeneous equation H# « S with H a aeif-ad}oint. positive-definite, linear operator. Seme of the principle*bound this functional only with reapect to small or local varia-tion*, wberecs others give bounds for arbitrary varUUcns. Sev-eral of the remits obtained coincide with those of other authorswidely scattered throughout the literature, and we show that theseprinciples have a common origin. Other results given ere raw.Examples of the use of these principles are taken frotr the fieldof neutron transport theory, and both the linear Bolteminn ortransport equation and the diffusion equation are used. One In-teresting result is that certain "exact" values of the extrapote»«dandpqlnt for the Milne problem, which have been reported tn theliterature, fall, due to numerical inaccuracies, outside the boundscomputed here, (auth)

B.GM919 CALCULATION OF NONSEPARABLE REACTORGEOMETRIES USING THE EXPANDED FLUX SYNTHESIS METH-ODS. SchaefOer, Hermann (Techniscbe Hochschule, Stuttgart!.Kuk'eonlk, 11: 91-S(May 1968). 0n German).

Multigroup diffusion theory is used to formulate matrix cgsrea-oioos for the neutron flux. These expressions arc then usod in theexpanded flax syntbeats technique to describe two- and taree-dlineaslonet special reactor configurations in rectangular coordi-aatas having partial abeoriwr plate*. (M.L.S.)

B.CI M M EXPANDKD FLUX SYNTHESIS METHOD. Schaeffler,Hermann (Techntsche Hoehenhule, Stutlgart). NuUeolk . i l :M-t(May 1M8). fln German).

MaltlgnMp tk*ory Is used as the basis for the kteetlcs equationsfor cylindrical reactors having discrete control rods. The effectsof these control rods on the neutron flux are determined direct*;as waaiidid dux synthesis technique Is used to reduce the numberof unknowns In Uw final iiyalrm of differential equations. An tsi-amnla problem ia given. (M.I..S.)

12

Volume 22 (1968)

C3f»33 APPLICATION OF VARIATIONAL SYNTHESIS TOTHE OPTIMA!, CONTROL OF l\ POINT RKACTOIt MODEL.Stncty, Wcston M. Jr. (Knolls Atomic Power Lab., Schenectady,N. Y,!« Nud. Scl. Eng.. 33: 257-00(1968).

The techniques of varlsltousil synthesis arc applied to obtaina solution for the optimum reactivity control of a point reactormodrl, with tcnipctaturo feedback, subject to a general type ofoptimizing function. An Integral equation formulation cf the neu-tron dynamics allows the solution of a two-point boundary valueproblem that Is frequently associated with the differential equationformulation of the optimal control problem to fas replaced witha system of algebraic equations. (suthS

A4M03 NEW DERIVATION OF DISCRETE OHMNATE AP-PROXIMATIONS. Kapl;in, Stanley (Unl». of Southern Cali-fornia. Los Angelas). Nuel. Scl, Ens., $*• 76-B2(Oct 1W8).

It Is shown that the Idea of space-angle synthesis may be usedto provide a new framework for the derivation of discrete ordinal*methods. It la also shown that within this framework on* c*scleanly and systematically Incorporate various stratsge&u) timedat overcoming the ray effects that pKague discrete ordfast* netaodaIn two- and three-dimensional problsma. 1$ references, (aata)

A3t*3t VARIATIONAL APPROACH TO THE SELECTION OFTHE DIRECTION SETS IN THE DISCRETE S. APPROXIMATIONTO NEUTRON TRANSPORT THEORY. Jauho, Pekka; KalU, HeiWd(Technical Univ. of Helsinki). Nucl. Scl. Eng., 33: 251-4(1988).

The discrete S, method for solving the csutron transport prob-lems is analyzed by discrete ordinates method; the directionalneutron flux la approximated by a finite Lumber of neutron rayswith certain weights. The scalar flux is treated by S» aohitloa;discrete directions and weights are treated as free parameters.Two varlationa! formulations are used: the integral transportequation; and self-adjoint varictlonal formulation for monoaner-getlc Bolttmann equation by Toivaners. (auth)

B60883 ITERATIVK-VARIATIONAL METHODS FOR NUMERI-CAL SOLUTION TO I-LI.II'TICAI. PROMLKMS1N KKACTOHTHEORY. Aibertonl, Sergio; LuneUt, MasslmlUanos Macglon!.OabrlelU (Application] a Rlcerchn SclenUflehe. Mllaaf.pp217-82 a< Flstca del Iteallore. Rome, Conslglto Nazlonale delleRloerche. 1968. (In Italian).

From Couforenvo on Phyalc* of Reactors. Milan. See CONF-

Mcthoda for ltcrntivc-varlatlon.il techniques for rotation of u-dlmennional diffusion niu.-itluns for hrliTORimtHjn syKlems arcdescribed. U«c of 1MB u-t-hnlquc as a basis for a romiiutcr pro-gram Is dSscusHml. (M.I..S.)

C4J55P AI'I'HOXlMATt: .SOLUTION TO SPACE DKPENDENTKINETICS PKORLKMS. Suda, Nutwhido (Osaka Univ.). J.Nucl. Scl. Techno!. (Tokyo), 5: 377-8(Ju!y 1966).

An approximation U-chnlqut' for wiving Bpnii- dependent kineticsproblems Is given. A synthesis tociinlinic Is used for developing a.one-group model In which the delayed nculroA effect Is Ignored.Tiro tranalcnt flux distribution In a uniform infinite slab core isJitii niim-<l; ihewi rcsulitf arv compared with the exact solution.(M.L.S.)

VARIATIONAL MULTICHANNEL SPACE-TIME SYN-THESIS METHOD FOR NONSEPARABLE REACTOR TRANSIENTS.Stacey, Weaton M. Jr. (Knoll) Atomic Power Lab., Schenectady,N. Y.). Nucl. Scl. Eng., 34: 4S-S6(Oct. 1966).

A multichannel space-time syntheala model for the calculationof aoaseparable reactor transients is developed from a variationalfunctional which admits expansion functions UuM are discontinuousla space and time. In each of many spatial regions, the flux duringeach interval of time is expanded in known functions of positionwith unknown expansion coeftlstenls. The accuracy of the method,sad Its superiority with respget to the conventional single-channelspace-time synthesis method, are demonstrated by severs! nu-merical examples. 22 references, (auth)

13

Volume 21 (1967)

c11*3 ENERGY OPTIMAL XENON SHUTDOWN.Umbut. ftttmrr. Woedoock. Gordon; Babb. Albert (Univ.cfWasMagton. Seattle). 20p. (OON7-66060S-S2).ORAU. Gstelta. AED-CONF-6S-303-103.

Froo American Nuclrsr Society Meeting, Dearer.PeatryaglA's optimum theory 1* applied to tin problem of

determining a flux shutdown program that limits Xe poison-ing to a reactor while using the minimum nuckar energyoo fins integral. We determine the complete adjoint solutionand synthesize typical optimal trajectories. Tbs analysisis carried through for both the unrestricted problem and theproblem wber* the Xe density ia restricted within the con-trol period to some maximum acceptable poisoning. Muchof the analysts is similar to the problem where the controltime rather than the control energy is to be optimised. Ourmethod of eolutl"» is new In utilizing a more straightfor-ward optimum theorem and giving the foil solution on theboundary of restriction. It is briefly indicated how the sameanalysis otn be used to give a more rigorous proof of optl-mallty in the time optimal problem, (auth)

F7*21 <AD-<nlSO24) COMPI.EMKKTAHY VARIATIONALPRINCIPLES IN NKUTUON DIFFUSION THEORY. Arthurs, A. M.(Wisconsin Univ., Mudinon. Theoretical Chemistry Inet.).Aug. 24, 196S. Contract DA-U-0S2-URD-2O59. 13p. (W1S-TCI-1B4). CFSTI $3.00 cy. $0.65 mn.

Complementary variations) principles HiwxiaU.il with neutrondiffusion in solids arc presented. The resulting formulas areused to Jerivc new expreKsions which provide upper and lowerbounds lor the absorption probability, (auth)

B?*3f (WAPO-TM-C10) VARIATIONAL PRINCIPLE FOB THENEUTRON DIFFUSION EQUATION USING DISCONTINUOUS TRIALFUNCTIONS. Bualik, A. J. (Bettts Atomic Power Lab., Pitts-burgh, Pa. Oct. 1966. Contract AT(U-1)-Gea-14. 20p. Dep.mn. CFSTI $1.00 cy, 90.50 nu.

A variational principle far the neutron diffusion equation whichpermits the use of discontinuous trial fluxes Is presented. Theprinciple is based en the method of Lagranfte multipliers. Appli-cations of the principle to problems of axial synthesis are given,sad the principle U compered to that of Selengut and Wachspress.(luth)

C7452 SOLUTION OF THE SPACE-TIME NEUTRON-GROUPDIFFUSION EQUATIONS BY A TIME-DISCONTINUOUS SYNTHE-SIS METHOD. Yaslnsky, J. B. (WcsUnsnouse Elestrlc Corp.,West MlffUn, Pa.). Trans. Amer. Nuel. Boe., ft 4M-T(Oct.-Nov. 19SC).

B7f H APPLICATION OF SYNTHESIS TECHNIQUES TO THECALCULATIONS OF THREE-DIMENSIONAL REACTIVITY COEF-FICIENT DISTRIBUTIONS. Bear, J. L.; Judge, F. D,; Venerus,E. R. (General Electric Co., Schenectady, N. Y.). Trans. Amer.Nucl. Soc., S: 4T0-l(Oct.-Nov. 1S66).

B.G7*12 FLUX SYNTHESIS USING MODIFIED GREEN'S FUNC-TION MODES. Hooper, R. j . ; Becker, M. (General Electric Co.,Scbensctady, N. Y.). Trans. Amer. Nucl. Soc., 9: 471i|Oct.-Nov.19S6).

C10132 (WAPD-TM-641) TEST OF THE TIME SYNTHESISAPPROACH FOR THE SOLUTION OF KEACTOH KINETICSPROBLEMS. Bewick, J. A.; Kaplan, S. (Ueltis Atomic PowerLab., Pittsburgh, Pa.). Oct. I960. Contract AT(U-1)-Gen-14. 33p. Dep. mn. CFSTI $3.00 cy, $0.65 mn.

A series of numerical experiments are reported In which timedependent diffusion problems arc aolvcd approximately by various,versions of the modn! analysis and time synthesis methods, (auth)

B12140 (OEAP-4922) BISYN: A TWO DIMENSIONAL SYN-THESIS PROGRAM. Greebler. P.; Keltey. M. D.; Davis, R. A^Keck, C, A.; Duncan, W. A. (General Electric Co., San Jose,Calif. Advanced Products Operation). July 15,1965. ContractAT(04-3)-189. ISOp. Dep. mn. CFSTI $3.00 cy, $0.65 maw

A two-dimensional diffusion program (BISYN) for core analysisof fast reactors is described In which the two-dimensional fluxesend adjoints are synthesized from one-dimensional solutions usinga nonlteratlve technique. (D.C.W.)

F17211 GENERALIZATION OF THE VARIATIONAI. METHODOF KAHAN, IUDEAU. AND ROUSSOPOULOS, II. A VARIA-TIONAL PRINCIPLE FOR LINEAR OPERATORS AND ITS AP-PLICATION TO NEUTHON-TRANSPOHT THKOItY. KorUin, Mor-ton D. (Princeton Univ., N. J.); llrookn, Harvey. J. Math. Phya.(N. Y.)t R; 53-6(Jan. )!)f»7).

Addltionul [i|ipllcniii>nN of a generalized form of Ui» VHrlatlonslmolhui! of Kalian, Hideuu, and HfMiHHGpoulnH are preHimlod. Equa-tions used In neutron transport theory, ouch as the spherical har-monics operator form of the Boltzmann equation, aro derivedfrom the generalised variations! functional; and an interpretationof these operator equatlonn in term* of flux- and source-generat-ing operstors is suggested, A relation betwocn this variations!method and the variational method of Llpymann and Schwinger isestablished, and it iff ahown that the least-square varlatloaal func-tional of Becker for linear equations can be derived from a gener-alized variational functional, iauth)

14

Volume 21 (1967)

G, BS?«3 (KAPL-M-658S) SOME MATHEMATICAL PROPER-TIES OF THE MULTICHANNEL VARIATIONAL SYNTHESISEQUATIONS AND TWO-DIMENSIONAL SYNTHESIS NUMERICALSTUDIES. Wachspress, E. L. (Knolls Atomic Power Lab.,Schenectady, N. Y.), Dec. S2. 1S66. Coatract W-31-109-eng-52.Up. Dep. mn. CFSTI $3.00 cy, $C.6S mn.

It Is proved that the matrix approximating the group diffusionoperator <-V . W • A) in the multichannel variations! synthesismockup is positive definite. Some numerics] studies of a. highlyDonseparable reactor are described, (auth)

F21383 LIMITATION OF THE ROUSSOPOULOS VARIA-TIONAI. PRINCIPLE? Pomranlng, G. C. (General Atomic Div.,General Dynamics Corp., San Diego, Calif.). Nucl. Sci. Eng.,28: 150-2(Apr. 1967).

It i» shown that, in the application of the Roussopoulos varia-tional principle, for a certain class of trial functions the designa-tion of the error term as a second-order term is questionable,leading to doubts about the usual interpretation of the variations!procedure. t'D.C.W.)

F213»S COMPLEMENTARY VARIATIONAL PRINCIPLESIN NEUTRON DIFFUSION THEORY. Arthurs, A. M. (Ua!*. ofWisconsin, Madison). Proc. Roy. Soc. (London), Ser. A, 398:97-101 (M»r. 28. 1967),

Complementary variational principles associated with neutrondiffusion in solids arc presented. The resulting formulas are usedto derive new express-ons which provide upper and lower boundsfor the absorption probability, (auth)

F. A3MM CANONICAL AND INVOLUTORY TRANSFORMATIONSOF THE VABIATIONAL PROBLEMS OF TRANSPORT THEORY.Kaplan, S.; Darts, Jams* A. (Westinghouse Electric Corp.,West Mifflln, Pa.). NccL Sci. Eng.. 28: 166-76(May 1M7).

Tbe notion* of canonical and involutory transformation* fromthe calculus of variation* are applied to neutron transport prob-lem*. It 1* shown that the variation*] formulation* of Vladimirov,Selengui, Pomranlng and Clark, and Davit are related throughtxanCfcnnation* of this type. It 1* pointed out that tbe pair offunctional* Identified through the involutory transformation arereciprocal in the sense that the minimum of one is the maximumct the other. Tbe irapllc&Ucn* of this fact for the developmentof apprextmaties methods are tliacussed. (auth)

F,C27642 <AD-634B43» NEUTRON IMPORTANCE AND VARIA-TIONAL FORMULATION IN REACTOR PHYSICS. Nozawa, Rei-kichi (Uppsala Univ. (Sweden)). Feb. 25, 1966. ContractAF 61(052)-B74. 48p. (AKL-6G-0120). CFSTI $3.00 cy, $0.65me.

The neutron Importance is studied as a physical concept fora time-dependent system considering the case where successivedetection* are made as well as the final detection. Successivedetections make the importance equation inhomogeneous. Basedon tbe physical interpretation of neutron importance, a new for-mulation of variational principles is proposed in a manner ap-plicable to time-dependent systems. The proposed variationalfuoctionals are to be understood not as Lagrangian densities a*in classical or quantum theory of fields but as total detectiondate in experiments. As an application, a method of Green'sfunctions is developed to improve successively a trial flux. Themodification factor for the source term in the diffusion equationis derived in a most elementary way. (auth)

G.B21731 VARIATIONAL METHOD FOR MULTIGROUP NEU-TRON FIELD SYNTHESIS IN MULTIZONE REACTORS. KhrO-mov, V, V.; Slesarev, I. s.i Kuztnln, A. M. pp 11-32 ofInzhencrno-Fizickeskie Voprosy Yadcrnykh Reaktorov. Yu-rovoi, L. N. (ed.). Moscow, Atomlzdat, 1SS6. (In Russian).

Thn neutron field in & nuclear reactor can be described with adsgree of accuracy ty a multi-group set of kinetic equations wblcliare written in veetoi -matrix form with a P|-*pproximatlon to avoidtodlouR cn!culat!on». A functional J enn he computed on the baolsof tho neutron diffusion equations and the boundary condition*. Th»functional J is varied with reaped to vector function* which satisfythe diffusion equation. The dovlation of tho functional from theexact value Is a second-order difference compared with tho devia-tion in tho average vector function from Ihn function which I* asolution of tho problem. Tho mulligroup net of «quatlonH I* reducedto a set of few-group ciniition.i by the use of the variation*! princi-ple and l>« the luw sensitivity or the funrtttmal to error In tho vectorfunction.*. | | js assumed thai thu s|«ee ami <<iKTKy vum|x>nunt» ofthe neulrun llux ami ul the licutrun worths aro partially Mcparable.Tho Kolmi n-i Kivr :i|ipruxim»ti! viiluvs of !nli'i;ral fluxc.H and m;u-tron wonli.. In vari-ms rt'itctor /.om-s in a furllcular unergy nub-space. The vifootlvi- nHiltipliuallon factor of 0\<s reactor is deter-mined. (TTT)

AJttfli AKCLE-SPACE SYNTHESIS: AN APPROACH TOTRANSPORT APPROXIMATIONS. Kaplan, S.; Daris, J. A.; Na-telsoo, M. (WesUngbcKUie Electric Corp., West Ulfflin, Pa.).NucliScl. Eng.,28: 364-7S(Juno 1967). (WASD-T-1891).

A method I* proposed for solving the transport equation is whichtha angul-v dependence is treated by expansion In specially tailoredtrial functions. The working equations of the method are derivedand simple numerical example* presented, (auth)

G3Mfl <ANL-?310» REACTOR PHYSICS DIVISION ANNUALREPORT, JULY 1.1965 TO JUNG SO. 1966. (Arfowe NationalLib., m.). D«c. 19S6. Contract W-31-109-enf-38. 44*p.Dap. CFSTI.

Brief dMcription* of work on ike following subject* are pra-•Mited: flMloa propertlsa and cros* lection data including fastDMtroa soattsriBg studita, elastic neutron ccatUriBf from ele-•Mats o/ Istonnadiat* weight, elastic neutron acatttria*; fromMg cad H, teat neutron scaUsring from •aclci in lh* HUM r«glo«A - ft-ISO, the inttractloa of fast neutron* with Uw 1SS, 1M, and1M Inotopea of W. a awrch tor ftuctuaUona ie Urn fiMlon cross

IS

Volume 21 (1967)

section o | m U , neutron flux measurement* In toe 10-200 kaV re-gion, (d,n* atripplng reactions, fast neutron total cross aecUonausing a moimenergeUc source and an automatic facility, fast neu-tron energy degradation through the (n.yn*) proceaa, unitary model*of nuclear resonance reactions, the ?*Cf fission neutron spectrumfrom 0.003-15.0 MeV, direct and absolute measurements of »?T-age yield of neutrou in the thermal fission of ^ U and spanb. eousfission of **Ct, sponuneou* fiction half-lives of •wCra and >4iO Cm;thermal reactor physics including High Conversion <Hi-C) criticalexperiment, Hi-C uniform lattice ttriculattons, initial critical ex-periment* of the EBWR Pu recycle program, measurement ofcapture-to-fission ratios ot l"Pu and MVa in the Pu loading ofthe EBWR, control rod evaluation for thermal and intermediatereactors, amall reactivity measurementa in the Argonr-e ThermalSource Reactor (ASTR), neutron beam spectra extracted from theHigh Flux Irradiation Reactor, Arrjnne Advanced Research Reactor(AARR) critical experiments-preface, AARR critical experiments-control blade worths, AARR critical experiments-prompt neutronlifetime measurements by the Rossi-alpha technique, AARR criticalexperiments-Cd ratio measurements, AARR critical experiments-activation snd power distribution measurements, AARR critical ex-periments-void and material reactivity worths and temperaturecoefficients, AARR critical experiments-beam tube experiment*,AARR critical experiments-sunup source requirements andinstrument reaponse, AARR calculations—preface, AARR cal-culations-analysis of the critical experiment*, AARR calcula-tions-general reactor physics design analysis, AARR calcu-lations-reactor physics characteristics of the ITC, AARRcalculations-factors in optimization o! experimental fluxes,AARR calculatione-shield dealgr analysis, AARR calculations-analyses of hypothetical accidents; fast reactor physics includingthe neutron energy spectrum in a dilute UC-fuelod fast criticalassembly, neutron spectra in depleted U, calculation* of Ha-voidcoefficients in large fast neutron carbide cores In assembliesMo. 2 and 3 of ZPR-6, calculation* of the effect of thin slab het-erogeneities on the non-leakage reactivity component of Na void-ing, non-linearity in the spectral component of Na void effectas a function of Na content, effect of parameter uncertainties onNA void effect and critical size of fast reactors, Doppler-effectmeasurement* on a dliute carbide fast assembly-ZPR-6 assemblyHo. 4Z, measured physics parameters in a zoned fast UC core-ZPR-IV assembly No. 42. analysis of the uncertainties in theinterpretation of zone loaded experiments, measurement of thespatial distribution of the importance of fission neutrons In ZPR-Cassembly No. 455, standard deviation of Ion chamber current mea-surements In ZPR-6 assembly 42, measured reactivity removalrates in ZPR-6 attembly No. 47., the Argonne National Laboratoryof ZPR-3 assembly No. 4(1, critical assembly comparison calcula-tions using new cross section dnta, comparative neutronic charac-teristics of metal, oxide and carbldu K1IK-II driver fu.-la, the effectof fuel and blanket chunges on the EUR-1I flux, FAHKT Core I fuelIrradiation program and reference dcHlgn. twenty-six croup cross•eotlon set for W-bsncd rocket Hyutvma, physics measurementsla fast W rocket reactor critical experiments, measurement of'pace-dependent material worths In uovera! ZPH-9 assemblies,rocket critical asttumUlcu amilynle, physics measurement* In unoperettas; fast breeder power reactor, further neutronic studiesof the 1000 MW(c) metal-fueled fast breeder reactor, renctorphysics calculations for a 10,000 MW(th) Cast Na-coolrd breeder,fast breeder reactors for water desalting, criteria for the densityof monitoring points In large reactors; fast reactor safety Includ-ing capabilities of the present TREAT facility core «s a fast fluxloop meltdown facility, meltdown experiments using the Mark I In-tegrel Na loop, analyses of single pin loop meltdown experiments,

properties of Irradiated UOj pins prior to TRKAT facility tran-sients, phttographic (net reactor safety e.\|>erlmcnls on Irradiatedoxide pins ut the TREAT facility, transient ln-plle tests on UO.-Wcermet rocket fuel samples, design of the Mark II integral TREATfacility Na 1 K>p, calibration mockup for the large loop test sectionfor the TREAT facility, transient response of 6tand-o(f pressuretransducer assemblies on the TREAT facility integral Na loops,extensible multi-purpose vacuum glove box, experimental resultsand Improvements in the fast neutron hodoscope, the exact three-dimensional solution for thermoelastic stresses and displacementsIn finite and infinite tubes, transient vaporization of Na in reactorcoolant channels, convectlve heat cr mass transfer with phasechanges, theoretical prediction of tbermodynamic and transportproperties of metal vapors, equation cf state of reactor materialsat high pressures and temperatures, a modified equation of statefor hydrodynamic calculations in the AX-1 numerical program,properties of refractory ceramics at extremely nigh temperatures(UC liquid expansion), modification of the high temperature Wf'jament furnace, failure pressures of thick-walled doubly-rein-forced concrete eel'.*, maximum permissible body fcurdens ofPu isotopes and resulting release criteria, fast reactor meltdownaccident analysis code, PREAX; experimental physics techniquesand facilities Including a low geometry or counting chamber, abso-lute determination of fission rates in *MrJ and !*U and capturerates in 1MU by radiochemical techniques, precision fission ratemeasurements by fission track counting, solid-state Comptonspectrometer for measurement of reactor y spectra, feedbackstabilization of nuclear counting channels, signal splitting Intofast and slow channels, design and construction of an improvedMn bath counting system, iow fiux measurement of WIU epicad-mlum capture-to -fission ratio, reactor response to an oscillatingneutron source, neutron fluxes required for activating probematerials, a code to permit fission product decay correctionswithout the use of a reference foil, determination of the k-con-stant for the Dy substitution method, additional calculations ofthe activation of spheres in a nonlsotropic neutron flux, use ofa small digital computer in data analysis and control of criticalfacilities, a Ge(Li) detector system for the measurement of y-rays following inelastic neutron scattering, a multi-angle fastneutron time-of-flight system, multiple angle detector apparatusfor neutron elastic scattering and polarisation measurements,multiple scattering correction, automated computed control ofa fast neutron laboratory; reactor computation methods and theoryIncluding the Argonne Reactor Computation (ARC) system, theARC system glossary, the Multigroup Constants Code CMC*),modification of THERMOS to generate transfer cross sections,

. generation of multigroup cross sections using a coupled MC*-THERMOS code, variation of thermal cress sections with bucklingin consistent Pi and Bl calculations, development of a code tostudy fuel management, AMC-A Mor.le Carlo code, developmentand analysis of Monte Carlo methods, quasiatatic treatment ofapace dependent reactor transients, space depei. >nt kinetic cal-culation* using the WIGLE code, reactor systems analysis andhybrid computers, computation of the coupled error function bycontinued fractions, treatment of source discontinuities in thesolution of the diffusion equation, revision of the bulk shieldingcode MAC for the CDC-3600 computer, codes for analysis of•lactic scattering angular distributions, multilevel cross sectionsfor a fissionable Isotope, the effect of interference on the reso-nance integral mixtures of Th and U, the effect of randomncason group cross sections, tin chemical binding effects on the7**onsnce line shapes of l"U in a UOj lattice, equivalence be-tween homogeneous and heterogeneous resonance intcgralH incylindrical geometry, effect of the fluctuations in collision den-

16

Volmae 21 (1967)

sity on fast reactor iX>|iple. effect calculations, an approximatecalculation of upace dcpcnd.nl flux using a variaUoiiial principle,neutron-wave analysis; miscellaneous including energy Bpectrumof fruit cosmic-rny neutrons near Boa level, a CO; system fordirect conversion of nuclear energy to coherent laser light, the-ory of plfisma oscillations—generation of thermionic RF energyand Interaction* with DC, circulating shield reactor for npiicepower, and Improvised tthutter design for the JANUS reactor,A total of 609 references is lifted throughout the report. (M.L.S.)

i O

30191 SYNTHESIS OK THREE-DIMENSIONAL FM1X SHAPESUSING WSCONTINIIOIIS SETS OF TRIAL FUNCTIONS. Yasln-shy, J. n.: Kaplan, S. fWentinghouse Electric Corp., West Mtff-lin. Pa.). Nucl. Sel. ling.. 28: 426-37(Juno 10C7).

The method of flux synthesis !« extended In a systematic way toallow the possibility of umng different .sets of trial functions indifferent axial zones. The Ji.'e.H.siiry equations are derived Insome detail mid numerical examples are [resented. The resultsof these examples are very satisfactory and suggest, therefore,that the synthesis procedure can be made much more useful andpowerful by extending it in this way. In a more genera! contextthey suggcHt that the basic notation of deriving discontinuoua-type approximation methods from an appropriate varlatlonal prin-ciple is a valid and very effective idea, (auth)

C31950 (WAPD-TM-C3S) THE MtM.TIMODK SYNTHESIS AP-PROXIMATION FOR SPACE-TIME HKACTtiK DYNAMICS: ANAPPRAISAL OF FINITE DIFFERENCING MKTIHHlS. Clarke,W. G.; Margolis, S. G. (Butlis Atomic Power Ijih.. PlttHburgh,Pa.). May 1967, Contract AT(1 l-IV-CJcti-14. *6p. Oep.CFSTI.

An appraisal of possible finite rflffrriwfiiK MII'IIIIMIH for themultimode time synthesis approximation to Kiuice.-tlmo ruitctorkinetics with emphasis on accuracy, »cil>ililv, computer economy,and programming simplicity is picNeiile.l. A KOK'i'KAN IV digitalprogram lor the CDC-CG00, MUMKIN-'.', « « uwvl in iwrform•hese numerical studies and is iIcKerilml. Thy proi'.rim Is u««ito solve the multimodc kinetics cquatloux In UK" ul« • ucu uf reac-tivity feedback for up to nine trial Imirllinw mill Hiv il-lnyedneu-tron groups. The program description IncliidcH un i vinijilo ofhow multimodc kinetics paraim'tcrn can lie Kciicralnl by arith-metical operations on the synthi'Hln cdltn nf mi uxlming ono dimen-sional neutron diffusion prc ;ram for up to

( Hsional neutron difu p ;100 m e s h po in t s . Specia l icaUin-H o( Hiemated data link for roni|Hiting !iie niiilliiiip , i' and A and n UH linii|ue lo i tchi i ' ic ilintinuouN d e l s of lri;d fii!icci«n>i ( i iUi l

four i-.irir>iKr;uii ln

^v groups nndiutlo un auto-

rather difficult to Interpret and to extrapolate with n muUlgrocptheoretfcsl mode!. A mucSi simpler model Involving only few po-rtmotors has keen tried out which would atill give a good rcprc-•eatatioo ai the physical processes. This simple model is baledon a two-group iorrMUum, in which the groups overlap, derivedfrom the diffusion equation by a variations) procedure. Compari-son between conventional multigroup and Cwo-ovcrlnpplng-groupformallssw gives excellent agreement. It in hoped that with thehelp of this method the Interpretation of multizoncd critical cx-perlmcntn might be greatly cased. It is also suggested that syn-thesis inethotls in general be used to solve some special problemsin fast reactor physics. A list of 21 references ii included, (antti)

F35*07 THE USE OF NOETHER*S THEOREM IH REACTORPHYSICS. Tavel, M. A.; Clancy. B. E.; Pbmranlng,O. C.(Brookharcn National Lab., Upton, H. Y.). Contract AT(30-J>-Gcn-16. Nucl. Set. Bnt>, 39: 58-66(July 1967). (DNL-10687).

A use of the variation*! method which has been neglected fareactor theory Is discussed. This Is the invarlance theorem ofE. Noether which ha* been widely utttlzed In other area* of mathe-matical physics. Following a derivation of the theorem, Its UMto obtain solutlonn of the time-independent diffusion equation Isdemonstrated. The theorem In used to constrict a complete anal-ogy between the time-dependent diffusion process and classicalmechanic*. Conservation lawn arise In ihc construction oT thisanalogy and their |-jsniliie application in diHrussed. An aa-'egybetween the neutron diffusion vpiutlon and the Ume-depeodeaiSchrocdlngcr equation In also given. Several suggestions fcrgeneralizations of Nocthcr's theorem for use in reactor theoryarc made, (auth)

A3SH0O ANGULAR SYNTHESIS APPROXIMATION APPLIEDTO THIN CRITICAL SLABS. Zwlbel, H. S. (BatteMe-PacificNorthwest Lab., Richland, Wash.). Trans. Ainar. Nucl. Soc.,10: 213-14 {June 1967).

From 13th Annual Meeting of the American Nuclear Society,San Diego, Calif,, June 11-15,1967. See CONF-6TC/602.

A39203 STRATEGY FOR THE APPLICATION OF SPACE-ANGLE SYNTHESIS. Natelson, M. (Bettia Atomic Power Lab.,West Mifflto, Pa.). Trans. Amer. Nucl. Soc., 10: 171-2(Juw

)ili- klm-Uii. pnrumvtorsO.UIUII'AIH wllh dlecon- From 13th Annual Meeting of the American Nuclear Society,

San Diego, Calif., Juae 11-15,1967. See CONF-67060a.

F35554 (ANL-7320, pp 439-46) APPLICATION OF SPACE-ENERGY SYNTHESIS TO THE INTERPRETATION OF FASTMULTIZONE CRITICAL EXPERIMENTS. Storrer, e\ I Chaumont,J. M. (Commissariat a I Energle Atomlque, Ccdaracne (France).Centre d'Etudes Nuclealres).

bfornutljn gained from rauHiztined critical experiments U

B3S205 RHTIAL STUDIES OF COLLAPSED GKOUP-SP/LCESYNTHESIS: ANOMALIES ARISING FROM THE USB OFADJOINT WElGHTTNa. Yssinjky. J. B.; Kaplan, S.; Jaaoa.R. H. (Bettts Atomic Power Lab., West Mlfflln, Pa.). Trua.Amer. Nuci. Soc., 10: J72-3;June 1967)O

From 13th Annual Meeting of the American Nuclear Socttr/,San Diego, Calif., June 11-15,1987. Sse CONF-87060*.

17

VOIUM 21 (1967)

<•*•»» CANONICAL AND INVOLUTORY TRANSFORMATIONSOP VARIATIONAL PROBLEMS INVOLVING HIGHER DERIVA-TTVSS. Kaplaa.S. (Betas Atomic Power Lst.,W«st Miffll*.1*J. Trans. Amor. Noel. Sac., 10: 175-6 (Jus* MC7).

From 1 3 * Annual MMttaf of (he American Nuclear Society,Ska Diego. Calif.,. June 11-1S.1M7. SM CONP-ST0««.

THE EXTRAPOLATED ENDKHNT FOR THE MILNEPROBLEM, Pomranlng, O. C. (General Atomic Di»., OeneralSsmralM Corp., San Diego. Calif.); Lathrop, K. D. Nucl. Sol.EMU,. 29: 305-8(Aug. 1967).

Upper and lower bound* on the extrapolated endpolat for ahomogeneous source-free half-space with absorption tad Uotroplcscattering are calculated by relating the extrapolated endpolntto • linear functional and applyii g varlatlonal method*. (D.C.W.)

C382*9 VARIATIONAL PRINCIPLE FOR THB DETERMINA-TION OF PSEUDOFUNDAMENTAL BECAY EIGENVALUES.M t a , 4. D.: EM**. P. B. (SUMMfesr Myteetale Iajfc..

From 13th ABMMI i***** of &• Ames-Joe* Nuclear Society,Saa Cisgo. Calif.. June 11-15.1887. Sea CONF-S7M02.

B3 t | I t PERTURBATION MKT1IOO THAT UTIUZB8 THEVARIAT1GNAL PRINCIPLES ASSOCIATED WITH TI1K NEUTRON-DtFFUaON fEQUATION. Turley. Richard S. (low* State UBIT.,Amen). Tnuu. Amcr* Nucl. Soe,, 10: m-8(*uie itvt).

From 19th Aamwi MeeUwK cj the American Nudmr Society,•ka Mego, CaUf., JIHM 11-1S. 10S7. Seo CONF-C70602.

C98S3S rOHPUTATIONAL MSTHOOS FOR SOLUTION OFMKAUSTIC SPACE-TIME DYNAMICS PROOLKMS. Kaplan, 8.(BaMla Atomlo Power Ltb.o WMt Mlffilii, It.). TraM. Amor.Muei. Me., l t t SDSUUM 1»67>.

fnm 13tk ASMAI MMUBC of <k« Anwrlou Nuclear 8ocl«tjr,•W Dttfla. Calif., JvMi l - lS . lMT. SM CONF-670403.

VARIATIOKAL CALCULATICH OF COMPLEX MODESOF WtNOH SPATIAL oaaLJAIfKW. Baallk«A.J.:WelnM>l4jh.W.A. «8ettUAtosloPow«-Jt*b..WertMlHlln,PaJ. T mAaHr.Nuol.8oc., 10: »T-t<AM im).

F*em lltti AJWWU MMUag of (ha American Nuclear Society,U i A n 1 6 W 7 8MCONF-«70e02.

F40S6S A DERIVATION OF VARIATiONAL PRINCIPLES FORINHOMOGENEOUS EQUATIONS. Pomraning, G. C. (Brook-haven National Lab., Uptoe, N. Y.). Nucl. Sd. Eng., 2>: 220-88(Aug.l»87). <BNL-10514)

It ta shown that variatlonal principle! need not be postulatedsad then correctness proved; they can. In fact, be derived, makingtheir use more a matter of routine .than Ingenuity. A Lagrangemultiplier technique L. used to derive a aecond-order variationalprinciple lor estimating an arbitrary functlooal of the solution to

. as inhomogeneous equulion. The relation of this principle to afunctional Taylor series cxpantilon and to elementary perturbationtheory Is established. A normiilisutioa indviivndoRt aecund-ordurvarialtoaal principle for an arbitrary functional Is derived whichreduces to the Suliwlnger principle If Oio functional 1B linear. Twohigher-order varUtional principles are derlvud and are shown toba generalizations of the principles of Kostln and Rroukc. TheLagrange multiplier technique Is applied to the inbtinwRenoousSturm-Djlouville eauallon. which leads to a sccund-ordnr varia-tteoal prlnclpls for estlmatlBg an arbitrary fuociiona! which allowstrial fanotkNis that are not contlnuoua and do not satisfy the bound-ary conditions. This ittncltonal is'of the type ougKOSted by Busilkpins boundary teri3W. The differences between a. va.rUtlonal prin-ciple which can only be used to estimate a functional of Intcreataad oae which also acts as a Lagranglan are discussed. (au!h>

C4MCf THE APPLICATION OF THE CALCULUS OF VARIA-TIONS AND THE METHOD OF GREEN'S FUNCTION TO THESOLUTION OF COUPLED CORE KINETICS EQUATIONS.Carter, Nsal; Daaotukjr, Richard (Iowa State Uiu>., Ames),pp 249-69 of Couplad Reactor Kinetics. Checem, C. G. xKoshlar, W. H. («ds.). Coltea* Station, Tex., Texas A and MPress. 1967.

From Amsrioan Nuclear Society, Coupled Reactor KineticsConfaeKW, Collet* Stetton, Tex., Jan, 23-24,1947. See CONF-«70i07.

Solution to tfw coupled ctre kinetics equations Is presented Inthe form of a space-time modal analysis in terms of neutron flux.The flux is expanded in a series of space dependent functionshaving time dependent expansion coefficients. Sample calculations,utilising the method, are given. 4 relisrences. (M.L.S.)

BN W CALCULATION 6F THE SODtUM-VOtD EFFECT BYFLUX SYNTHESS. TnvaiU, A.J Helm, F. (ArgOMw NattowdLab., 014. TWM.A»ar.H«sl.a9en10i tl*-*<fim*inn.

Froa I t * Awaal MseHin of the Asierioan Waolear Soohrty.Saa Die«o. CaUt, Jaw U - » , 1 M 7 . See CONF-tTOtOt,

C.4 1 7 1 B SOLUTION OF THB SPACE-TIME NEUTRON GROUPDIFFUSION EQUATIONS BY A TIME-DISCONTINUOUS SYN-THESIS METHOD. Ysstnsky. 3. B. (WsaUagkaus* ElectricCorp., WMt MUnia,Pa.K Nmcl. Soi. Sag., Mi19«7)

A varlstlonal priaelple. which has as Ks stationary ooadttloas

18

Volume 21 (1967)

the direct and adjoint time-dependent croup diffusion equation!,it modified to admit time-disconUouous approximating function*.This extended principle la uaed to develop a ajmtheaia approxi-mation for the time-dependent group diffusion equation* whichpermits the use of different sets of trial functions at differenttimes during s transient analysis. The necesaary equations arederived in detail, and two numerical examples are presented. -Tlese examples show that the time-discontinuous synthesis methodIn capable of constructing accurate apace-time neutron fluxea,which vary smoothly in time, from spatial trail (unctions which arediscontinuous la time. In addition, these examples display thepotential of the new time synthesis for yielding computationallyless expensive solutions than are possible with the time-contin-uous synthesis procedure, (nuth)

F42M4 (WAPD-TM-6S9) CANONICAL AND INYQLUTORYTRANSFORMATIONS OF VARIATIONAL PROBLEMS INVOLVINGSECOND DERIVATIVES: APPLICATION TO BEAMS AND PLATES.Kaplan, S. (Bettis Atomic Power Lab., Pittsburgh, Pa.). May1967. Contract AT(11-1)-Gen-M. 54p. Dep. CFSTI.

The canonical and Involutory transformations of CM calculus ofvaristions are applied to variational problems in which the iet»-grand of the functional involves the second derivative of the arjtt-

. ment function. One and two dimensional examples, i.e., the vari-ations! problems governing the deflection of beams and plate*, areworked out In detail, (with)

C4**5* APPROXIMATE SOLUTION TO THE TIME-DEPEN-DENT MULTIGROUP NEUTRON-DIFFUSION EQUATIONS USINGA RESTRICTED VARIATIONAL PRINCIPLE. Robiaaon, JamMCondi. KnoxriUs, Tena, Univ. of Tennessee, UM. llSp.

Thesis.For gems reactors now in operation or being considered, it

has become apparent that a time-dependent mathematical n*u-tronlc node! which permits a shift in the neulron energy and/orspatial spectrum Is required for the dynamic study of thtMsystems. Models which describe (he behavior of the neutron*in space, energy, and time are well-known, e.g., the multlgroupneutron diffusion equations, but the solution to the resultant actof coupled nonlinear differential-integral equations must be ap-proximate. The classical approximation Is to Assume that theneutron oneiity and spatial spectrum* are Independent of tlmfl.The purpose of this study was to develop an approximate solu-tion for the tiine-rlcpumlont muUlgroup neutron diffusion equa-tion* with temperature-dependant parameters, thereby retainingn methan'sm to uvrinlt a shift in the *pall*l or energy neutronspectrum durliiK a transient. The nonlinear equations whichdeacrlbc the nuuuoiiic slate of the system are linearized la adiscrete fashion, i.e., the nuclear parameters are assumed con-slant over some spatial region and increment in time. Ac thecalculation progresses In time, the nuclear parameter* ar* ad-justed us lilctuieii lr:>v.\ the ncutronlc and Ihormodynamlc stateof Hie syHtom. An approximate solution Id obtained for the actof linearized neutron diffusion wjuiiliona by Hssumlng the neu-

tron flux can b* represented by linear combination el kaowaspatial modos with time varying coefficient*. T)M equation*which determine the time v«rylt« coefficients a n g«»*rated ifapplying a restricted variation*! principle. The spatial MOdMchosen to represent the flux'aru the firec** function Made*.The delayed neutron precursor* nro explicitly take* lato ac-count by assigning tfe* precursor equations aa eqvstloa* efconstraint to the functional wfcow Enter anaatlea* tachtda Matsystem aquatic**. Upon the application of th« variation! aria-ciple, the adjoint operator and adjoint fraction fl«c a n Intro-duced Into the analysis. Tbt solution to the adjoint *qaaUoB* laaa equally difficult as eh« solution of the diffusion •O.JBUSM!therefore, a modal expanaloa i t M i w o d for UM «dJ«M that.With the asoumed solution for the forward and adjoiat fhvt. tfettrestricted variallonat p inclpl* leads to a coupiad act ot UwsrdKfersntlal equalton* which deUrmlna UM time-varying ooafn-clonts appearing la the assumed solutions. The wtabsr of aaokequation*, in ganvral, ia th« product ot tiw mmbar of imodM b?tha number ot group* appearing la the maltlgroap fxawllMuThe number of equation* can b* radoced by assuulag that dif-ferent eMrgy group* can ba described by th« Mas tint* varyiagcoefficients. To demonstrate Uw validity of th* appratianl*solution, MvesmI transients, which wer* peifonn«d on ite TBKAT'reactor at Argonn* National Uiboratorlaa, •rs iiwlalsil Tb*analytical result* ar* presentsd graphically in Uw fcrat tt *ow*rtraces which ar* comparad with the *xp*rlm*ntal power trac**.Th* analytical model U composed of f Sv* *Mrgy groap*. «• tothre* cpatiai mod**, and six delayed neutron groaps. Ta* re-quired date to carry out a simulation ar* tb* neotroa aMdtlgraaaparameter*, delayed neutron fraction*, precursor decay coaataaU,and nuelid* *p*cineaUaii*. From the resuKs, It I* «cweladei UnUthe modal I* adequate tor abort-term transient* le taenMtl *y*-tema. Recommendttlou are act forth for aaeM wrteaatpna 41th* model to other proiijems of Interest la rsacior dyaamte*.(Di*s*r. Abetr.)

19

Voluw 20 (1966)

F113* AN ANALOGY BETWEEK THE VARIAT1OKALPRINCIPLES OF REACTOR THEORY AND THOSE OPCLASSICAL MECHANICS. Kaplan, 3. (WeattoghouMEUctric Corp., Wast Mlffiin, Pa.). Nucl. Sot. Sag., 38:234-7(HOY. IMS). (WAPD-T-1118).

A formal parallelism U shown to exlit between two clas-sical variatioaal principles governing tha time behavior ofmechanical system* mad iwo principles rslatlng to the A-mode eigenvalue problem of neutroa group dlffuikxi theory.By identifying the space variable with the time varfcble andiipe.ce derivatives (gradients snd dlvergeneee) with tinederivatives, the "ueual" variation*! principle of diflusioatheory is shows to be analogous to Hamilton's principleand the diffusion equations are analogous to the Lagraagaequations. Hamilton's canonical equations are then anal-ogous to the diffusion aquations in first-order form, andthe analog of the principle involving the canonical integralis a principle closely related to one proposed recently bySelesgut and Wachspress. (auth)

B114* BILINEAR AVERAGING FOR DIFFUSION THE-ORY PARAMETERS [Thesis]. Pttterla. Thome* Arthur.Madison, Wis., Volt, «J Wisconsin, IMS. IMp.

Bilinear averaging of group parameters lor use ia dif-fusion-theory calculations is evaluated. This prooedueaverages energy-dependent cross sections over both ne»-tron flux and adjoint spectra. Both perturbation theory aaa• varlatiooal principle are employed to obtain raultigroaedu^tsion theory with bUlnear averaged parameters M anappjroxlmaUon to the continuous energy-depecdeat P-lequations. These derivations snow that with blliMar aver-aging eigenvalues accurate to ftrst-oider errors in thefluxes &»ed to average the cross sections can be obtains a.It Is also shown that bilinear averaging I'epressnti' a con-sistent approximation to the energy-dependent equtttcmsfor perturbmtisa theory calculations. Extensira numericalcalculations ware aade to compare flux and bilinear aver-aging procedures for dinusion fteory calcalatioae. A lewperturbation theory calculations ware also earrtod out.<TCO)

A2*lt VARIATIONAL VACUUM BOUNDARY CONDI-TIONS FOR A PH APPROXIMATION. Dwrls, Jamas A.WiaUnghouaa Electric Co., Pittsburgh). Trues. Amer.Mael. •oc., •: 4H-S(Nov. IMS).

„„ A NUMERICAL COMPARISON OF HIGHERORDER VAWATIONAL PRINaPLES. Schreiner. Sheldon(Untv. of California, Berkeley): Seleogui, D. S. Trans.Amar. Nucl. Soc., es 48S-«Qio*. 1965).

A2 t f t TRANSPORT SYNTHESIS. Davit. James A.;Kaplsn, Stanley (Weatla^wuse Klectric Corp.£ PitUburgh).Trans. Amer. Nucl. Soc., 8: S09(No<r. 1965).

FU31 AN ANALOGY BETWEEN THE VARIATKWALHBIKCIPLES OF REACTOR THEORY AND THOSE OFCLASSICAL MECHANICS. Knplan, S. {Westlaghouseelectric Corp., Pittsburgh). Ttras. Amar. N«cl. Soc . »:

SKN IMS).

MM FEW LONG-TIME-INCREMENT DEPLETIONCALCULATIONS FOR APPROXIMATING BY SYNTHESISA MANY SHORT-TIME-INCREMENT DEPLETION CALCU-LATION. Flanagan. Charles A. (Weatiaghona* KlectrtcCorp., PitUburgh). Trans. Amar. Nwl . Soc.. S: S l«- l t•tor. 1MB).

AMM (WAPD-T-1M9) TRANSPORT SYNTHBHkDads, Jantes A.; Kaplan* Stanley (BctUs Atomic PowerL«b^ Pittsburgh, Pa.). Sept. IMS. Contract AT(U-1>-Gen-14. lBp. <CONF-<illQl-4S). Dap. mas CISTI$1.00 ey. |0.S0 mn.

From 194a Confvreoce on Remote Systems T dMSUtogy,Washtagto*. D. C

A FORTRAN program (THA^SY) Is developed to obtainsynthesised soiutiom to the S. at S« equations for s miae-•nergeUc x-y problem. Vac TUANSY synthesis la based mthe discrctc-ordiiwtc method. The purpose U the syatheelsis to lnvustljpite lbs feaniltillty of synlhiiiteMg soUttaM lahigh-order approximation* to the transport «ajuatl«n.(T.F.H.)

I.G

HIGHER ORDER VARIATIOWAL PRINCIPLES.• J ^ r * . D. 8. (General Electric Co., SdMaectady, N. Y.).Trans. Amer. Nucl. Soc.. I: «6(Nov, 1M6J.

Z = S L _ _ - — " * - — " » VAWATWKAL HULTKMANNtLSYMTHKMS WITH DISCONTINUOUS TRIAL FUNCTIONS.Wsohapraa«.It.L4»€akar,M. Kaoils Atamta PawaiLabM SehaMMtady, N. r j . Nor.iM*. CeatnetW-M-lW-eac-tt . t tn. Oca. majOF»TI»S4* ay. S«J* si

A vazittioaal tecaMdnttan of malttnhannal Byiwssis

20

Volume 20 (1966)

is developed for neutron diffusion computations throughuse of a functional which admits discontinuous trial func-tions. Detailed equations are given for a particular ctotoeof trial functions. A variattonal depletion formalism I*described for conducting reactor life stadias with a multJ-channel synthesis calculation model. (auQt)

F8333 CALCULATION OF FLUX DISTRIBUTION ANDEIGENVALUE OF ARBITRARILY SHAPED REACTORS BYMEANS OF THE CA LCULUS OF VARIATIONS. Unger, H.Atomkeraenergie, 10: 413-K;pfor.-Dec. 1965). (In Ger-man).

An investigation was made on the offect o! the reactorahapc on (he essential characteristics of the roactor,e.g., specific power ami surface heat flux density. Theproblem cunxlHtcd of calculating and comparing theneu?.n>n flux distribution ami crUtcalily of bar*, homo-geneous slmplo-bonietvc! reactors of arbitrary shapein order lo obtain thu reactor form associated with theoptimal roactor. A method Is presented for tho solutionof the problem by moans of a variation^ calculation.I B S )

T .B•»S7 TI1K APPLICATION OK VAIliATIONAL METH-ODS TO NKIJTIION niKKMAUZATlON PROBLEMS(ThcNiN|. Kaxsiiinau, Kobert B. Guimmvilis, Kla., Univ.of Florida. 1»(iS. 17-tp.

MCIIIIHIM IMKI»I on Iliu vuriaUonul muthud, VM, and thegeneralized vuriutfonal method, CVM, are applied to theciilculalion of the thnrmaltwiUon pnrametnrn u» and C,AHuucialvd with live ;iaym]>lolic time ciK«nvnluc throughthe power aeries A - u« > l^B* -C(li* fur ih« pulsed neu-tron experiment lo crystalline media. A procedure Is glvoa.for determining only the lowest or nsymploilc eigenvaluewithout recourse to parametric tnutheKls using sclocUtdva!ucn of tho geometric buckling, n l . The trla? functionsectuctcd lo rc|>rcsrnt the energy sjHX-trum arc the Laguerrepolynomials of order 1 weighted by the energy MaxwelUmn.lluwcvcr, the Kcner.-sl development of the analysis is notrestricted to these functions. Any suitable complete setof polynomials could be utilised. Several examples incor-poratlnx low ordered trial functions are given to provio*a detailed description of the mechanics of adapting the VMand the CVM to the problem and also to allow an analyticcomparison of the parameter* obtained from the two math-ods. General equations from which thermalicatloa. param-eters can be obtained using high ordered trial functions aredeveloped for both the VM and the GVM. Also Included aretabulations of pertinent Laguem weighted Integral* {up to18 polynomials) of energy-dependent scattering cro#a-sectioa parameters obtained from the Maturing kernelj$<E'-» £) selected to represent the scattering loteraoiioaethat may oesur in the particular medium. The calculationof the various integrals and bonce the thermalisation p*»rametera is transformed to a diecret* energy representa-

tion. The scattering kernels used in the calculations forthe crystalline media beryllium and graphite are based oathe energy exchange model formulated by Parks, ssd taoieaof the kerne! data for the selected energy points «s*d la tWsstudy are Included. The influence of the energy «Mh stseupon the values of the thermaUzation parameters la iavesti-gated and discuMed. Also discussed U the convargMO*behavior of the parameter* obtained using both tfce VM sadthe GVM. The values of the thermaliiatton paiirielSTS 1^and C, obtatneil In thic sturty for barylllum and graphiterespectively era: 1.33 x 10* em'/sec and «.4? x 1«* em4/sec; 2.15 x 10* em'/sec and l«.0« x 1O> cm*/»ec. (Disaerta-tlon Abstr.)

G •. :: :

123U fWAPD-T-1737) AN EVALUATION OF ATHKKE-DIMKN&1ONAL FLUX SYNTHESIS METHOD AS ANUCLEAH DESIGN TOOL. Flanagan, C, A.j Smttb, F. E.jBogar. O. F.; llutherford, C. II. (BcttU Atomic PowerLab., West Mifflln, Pa,!. Oct. 1964. Contract AT(11-1)-

t^iVso m?" <CONF"664-100>- OtP.mn.OFSntl.fltFjjjrn American Nuclear society Media* San Fraacltoo,

NO'f *D6C JL9Q4

Reacerch is described towards evaluating the «ae of thewclghlcc! residual flux synthesis misthud In thrcc-dlmea-•ional nuclear dealgn calculations. The cvaluatloR includescomparison of synthcais calculation* with direct three-dimensional calculations, a comparison of rewlta ironsynthesis calculations with experiment, and finally ma ex-amination o: the practicality of applying Uw method todepletion calculstlonj involving ooMiderable feometrlscomp? roily, (aulhp

EVALUATION OF VARIATIONAL METHODACCURACY. Erykalor, A. N. A t Encrff. (USSR), Its462-3(Nov. 1965). (In Rasslaa).

A »Wiy is made of a variations! method for dot«railalasTcritical reactor dimensions and neutron dhttrUMUou, wiftwhich the upper and lower llmlta of the «ige*va!u« of tsxhermitian operator can be calculated. The high accwaoyof the eigenvalues obtained usiag simple approxiauUoaa isdemonstrated for two examjdes. The accuracy to coaajMleafor a single-group diffusion equation for the case of coaacaatcross aection and coMinusoidal flux distribution and thecase of constant diffusion constant and cosinusoidal ab-sorption and fission cross-section distribution. (M.J.T.)

B

!•*** CCONF-«60204-4) VARIATIONAL METHOD* WTHE CALCULATION OF SPATIALLY DEPENDENT SWoTTRA. Fraacls, N. C. (Knolls Atomic Power Lab., Sobt-nectadr, N. Y.). U»«S|. Contract W-Sl-lOS-asf^S. Np.,l>ep. BOB. CFSTI Vt.00 ej. 53.50 aw. • ^ T . • • v.""»»

21

Voluw 20 (1966)

From Americas Nucteaz Society Meeting, Saa Diego,Calif.

A dlscusaion U presented of variations] methods forC U M In wliich the quantities studied a n stationary witkrespect to tiie variable of interest. Resonance capture Isnot studied. After an introduction of tbs spatially indepen-dent problem, a specific thermal-energy-range BptUallydependent neutron problem to discuss*!, Tba method Isthen applied to calculations ol croup conatasW far a wittl-group Monte Carlo method, muUigroup space-energy prob-lems is sSab geometry, and the use of overlapping group*U diffusion theory. fT.r.H.)

B1M33 A VAR1AT3ONAL APPROACH TO THE EFFECTOF BOUNDARY PERTURBATIONS WHEN THE MULTI-GBOUP DIFFUSION EQUATIONS APPLY. Splafe*, N.(Australian Atomic Energy Commission. Lucas Heights,New South Wales}. Up. (CONF-650602-71). ORTNS.Gnelin, AED-CONF-65-12S-107.

From American Nuclear Society 11th Annual Heating,Gattwburg, Teun.

Variation*! principles are given which allow perturba-tions of l&e surface defining the boundary of a region inwhich the multi-croup diffusion equations apply. The per-turbation may take the form of a change in boundary condi-tions or t change 5n the boundary shape. The principlesare useful for determining the reactivity worth of controlrods when She shape of the rod makes a direct calculationdifficult. In addition, from a single fir* and adjoint fluxcalculation, the worths of a host of control rods of slightlydifferent compositions (boundary conditions) and slightlydifferent shapes and sices can be determined. The appli-cation of the principle to the output from finite differencecomputer codes is discussed, (auiii)

G174K NUMERICAL SOLUTION OF THE DIFFUSIONEQUATIONS WITH THE KANTOROV1CH METHOD. GalH-goal. I. (CCR. Euratom, Ispra, Italy); GlorceUl, M. En-•rgia Mucl, (Milan), 13: 73-Bl(Feb. ISM).

Tie time-independent two dimensional Betatron diffusionequations are solved with a variettosa) method (Kantoro-vich's method). This awlhod is particularly convenientwhen the aeutrca flux is a smooth function aloof one ofthe two directions. A comparison between ftis methodand the finite Ofteresce method i s mads by means ofsome aumerical examples, (auth)

FIMi t A VARIATIONAL PRINCIPLE FOR RATIOS INCRITICAL SYSTEMS. Lewlns.J. (Univ. of Washington,•isttle). J, Nuel. Energy. Pt. A and B. » : l«l-3(F*b.IMC).

A variations! principle from which arbitrary ratios in

critical systems can be calculated in a stationary mannerto presented. This principle enables the perturbation ex-pression to be derived concisely using a compact operatornotation In a generalized X variable (position and velocity)(hat incorporates the usual exact and approximate modelsof reactor theory. (B.G.D.)

1S4M PARTICULAR SOLUTIONS OF THE ENERGY-DEPENDENT BOLTZMANN EQUATION. Mueller, Karl-Heinx (CCR EURATOM, Ispra. Baly). NuHeonik.S:38-4C(Jaa. 1»M).

The €arleisaa-Kupr*dse method for solving a coi$!sdset of integral equations with sectional holomorphic ker-nels combined with a method based on a variation proce-dure, of removing the energy from the field of free vari-ables, allows the construction of particular solutions of theneutron transport equation. The nuclear data eater Ihe cal-culation only after they are weighted and integrated over theenergy variables <E,E'). Thus, the procedure allows theuse of cross sections In a numerical form (e.g.. ihe ossalenergy group representation). An approximate solution canbe found from a linear system of algebraic equations, fauth)

F2!tS7 A NUMERICAL STUDY OF THE METHOD OFWEIGHTED RESIDUALS. Pomraninj. G. C. Nucl. Set.EMJ..24: 2»l-301(M«r. MM).

The varlational method and regioa-balaaea method,both special CBSM of the more general method of weightedresiduals, are each used as the formalism to develop aspatial expansion of the diffusion equation for two p w M w .These are (1) a spatially dependent spectrum problem torthe purpose ot computing the ssHf-shleldinc la the " T Oresonance xnd <2J 9 simple one-dimenstoaal elgenvihsproblem, to bath Instances numerical results indicatethat $h* varlational method is more accurate than theregUm-balance method. Ot particular interest i s the varia-tions] spatial -expansion approach to the eigenvalue prat. law.This may be a useful method for deriving a set of dfffir—asequations for the mulUKroup diffusion equation la * r t Itshould lead to an accurate rc|>reiu.'Matlo« of the flux witha relatively small number ot mesh points, (awth)

B21391 VARIATIONAL ITERATrVR METHODS FORELLIPTICAL PROBLEMS IK THE THEORY OF NUCLEARKEACTOns. AKtertoal. 3.; Lwwlll. M.: Maga>wi, O.(Unlv.. Milan). Atti Hcmisar. Mat, Fis. UnJv. Modean,M: 1M-3MM1SU). dnMattas}.

A vartatioMi iterative method la derived ?«r the soSntioaof clifptioal prablema w nuclear reactor th«sry. The Mrt-attonal base* for the rcsotwtkm of (ransmieeiea sr«hl»wiare givm. The tteratlve-varlaUowJ teohntqiiB «a«d for tt*numerical sUutlon is exMHiaMl. Nuswrieai emmpiea arethen given. (J3.R.)

22

Volume 20 (1966)

G223tf VARIATION PRINCIPLES FOR THE CRITICALFLUX CURVES OF REFLECTED REACTORS. BreUw-huber, L. (Technlsche Hcehscbula. Gnu). Z. Aagsw.Phys.. 80: 2»7-71(19M). (In German).

New minimum principles and stationary expressions a ngiven and discussed for the boundary coisditioas • « 0 and8*/8n + (l/d) • » 0 for the equation A# » 3** » 0. It i ssuggested to fulfill conditions at boundaries and interfacesonly in ihe f o m of natural boundary eoadUiou. Tbua, tk«use of discontinuous trial functions is encouraged, yieldinggreater flexibility and better numerical approximations Inthe mean. Boundary conditions are, nevertheless, satisfiedin the limit, (lath)

F24302 (GA-6273) A NUMERICAL STUDY OF THEMETHOD OF WEIGHTED RESIDUALS, Pomrsalag,G. C. (General Atomic. San Diego. Calif. JoknJ«y Hop-Una Lab. for Pure and Applied Science). Mar. M. I N I .Contract! ATiO4-3)-187. 40p. Dep. n a . CFSTI J1.00 cy.$0.50 ma.

The varlational method and the region-balance method,both special cases of the more geacral method of weightedresiduals, are each used as the formalism to develop tspatial expansion of the diffusion equation for two prob-lems. Tiese are a spatially dependent spectrum problemfor the purpose cf computing the self-shielding In the st*Pnresonaaee and a simple one-dimensional eigenvalue prob-lem. In b o * Instancss. the numerical result* indicate thatthe varlatioaal method is more accurate than the rtgioa-b»tana> netted. Of particular interest is the variaitoMlspatial, expansion approach to the eigenvalue problem. Thismay be a useful method for deriving a set of differenceequations for the multlgroup diffusion equation In that Itshould lead to an accurate representation of the flux with arelatively small number of mesh points, (auth)

FM714 IMPROVED CALCULATION METHODS FORRATIOS APPEARING IK REACTOR STATICS. Lewfns.MBny (0m». of Washington. SMttle). Nud. Energy.W-WMar. UM).

VartaMoMl and perturbation expressieae are developedfrom wale* to ealenlale ratto* of Merest in either criticalsystems or in sub-eritical sycteme having a source. Theemral adjoint (unctions employed as* related to the physi-cal concept of the taportaace. The ewyhasls la placed ontte calculation of the adjoint functioae by iterative methodsthat are already standard in reactor cedse. The eaprea-•tons given can improve ike accursey of results whilerequtttng aubtiaatiaUy less eomputing tinM than conven-tional direct calculations, (auth)

C.A2123* (NAA-SU-imi) SPACE-TIME FLUX SYN-THESIS METHODS FO1I THE APPROXIMATE SOLUTIONOF TIME-OKPENDENT DOLTZMANN NEUTRON TRANS-

PORT EQUATION. LMOO. V. (Atomics latemaUonal.Caaoga Park. Calif J. May 28. IMC. Co*roct ATai=1)-On»>«. 37p. Don. ma. CFSTI $tM «y, $9M a n .

Space-time fiux synmeeU methods for the sporo«1»s>tsoluMon of the time-dupsndenl BoHxwwm neatMn trunsnortequation are fcrmulated. The variatlonal and the OalerUoteca«io>ie are used. The epace-dtpendeni part of the nata-tion la otealned with the OTr-11 program—an SnHjne ae>lutka of the traaoport equation. Temperatort faintathaffects arc considered in the formulation. The fomMtetiatpresented here U adapted to the description of rapid tran-sients roUawtng a luge reactivity input, but the mewed canbe easily modified to cover ether reactor problems wherea space-time flux deacrlptfcm Is necessary. It should beiweful whenever the situation requires tta transport apart*.lmatioa for the description of the neutron flux, (auth)

AM i l * VARIATICNAL VACUUM BOVNSART CONDI-TlOm FOR A fy APPROKBSATION. Davis. Janws A.Hud. Sci. Eag.. XS: IM-WltM). fWAPD-T-lMt)

Approainule vacuum boundary condlticaa (or a Pj*ap-proximaUor are obtained by variatiooal nuthods. T»outatkwary principles are proposed, O K having what iscalled "odd" Marshak eondttionc as its natural boundaryeontittioM, and the otter saving "even" Murshak coadt-Uoae as its natural boundary condtttons. The principlesare valWI for arbitrary geometry. The odd Marshall cen-dUlonc are seen to be writable for a» odd-order PM«P-pronlaMtioa and the even Marshall conditions for an even-order PH approidmation. The odd Harshak conditions areprecisely tec conditions obtained by Vladtmtrov from ansilnmum princh>le in which certain restrictions areImposed on the scurce and scattering. The prescat treat-aMnt contains no s«eh reatricUons.

AM H 2 SELF-ADJOINT VARIATIONAL PRINCIPLEFOR DSII1VING VACUUM AND INTERFACE BOUNDAKYCONDITIONS IN THE SPHERICAL HARMONICS METHOD.Toivanen, Tims, Ntecl. Sd . Eag.. X: JTS-Mtfuly lf*»J.

By the technique of solMtmg Out total directional tkutinto even and odd porttoM m analp, the etsDnnary manscnrritrttc Bottsmann cquatkm wKh arbHrary colllaienkernel aM with arbHrary external dlrcelinnul aourae efa Ri-ncral ffstmuAry is symmctHacd to a »"lf-a«(|»li* IMMUThe continuity and boundary condMliMS for Uw madthujsi-lf-adjoint integiit-dUfercntlal equation are e«nHeKlyconstructed. A varlatluna! iirinclple is thrn set un by de-vising n self-adjoint Lagranaian whoue Minimum propertyis «qulvalrnt ic the symmetrised BoMsmann equation withthe associated continuity and houndnry coadHtone. Thedvvt>lu|icd varlational |>rlnci|itc contains no arbKrarineesand la used for deriving unique variational boundary wdltloas for the P t approKlmatlon of the spherical hamnten methed. H Is shown, for a geeeral geometry, thatapplying the aemUirect variations! method with an aagie-independent trial function yleMa, wMhcot any physical rea-soning, the correct P| differential equation and the outre •(ponding no- return-cur rent boundary condition, (auth)

23

Volune 20 (1966)

VAR1ATK>NAL METHOD FOR PROMPT NEU-TRON KINETICS. Koehler, Walter $5. (Kernforscnunx*-Mntnim, Karlsruhe, Gar.). Nukleonik, 8: 203-15(May1966).

It U abown that the Eulcr-LaRrange equation* of a varl-atianal principle are tho miiltiKraup diffusion fquationnfor prompt neutrons sad their adjoint equation*. The Kan-iorovlch method is used to get approxImHte solutiona forthe nonKuparahlc flux. The method is applied to a stepchange in configuration of the fast core of a eoupltd faat-thermal reactor. (*ulh)

AM4M NUMERICAL EXAMPLES OF SPACE ANGLESYNTHESIS. Nateleot, M.i Kaplan, S. (Bettla AtomicPower Lab., West MUttln, Pa.). Trans. Amer. Nucl.Soo.n 8s 197-8<June 1966).

G,B34f£3 SYMTIISSSS OF THREE-DIMENSIONAL FLUXSIIAPKS USING DISCONTINUOUS SETS OF TRIAL FUNC-TIONS. YssliMky, J. I).; Kaplan, S. (BetUs Atomic PowerLab., West MtJflln, Pa.), Trans. Amer. Nuci. Soo., »:198-9(June 0S{

BH4W DETERMINATION OF EIGENVALUES OF THEDIFFUSION EQUATION BY VARIATIONAL TECHNIQUES.BotsUd, J. W,; Ssleofute D. S. <U»5v. of California,Berkeley). TniM. Am*r. Nod. fas., 9: 19a-SCAme 19C6).

A144?! VARIATIONAL ESTIMATES OF DIFFUSIONAND EXTRAPOLATION LENGTHS WITH P, SCATTERING.HWTSOB n. H.; Seleafut, D. S. (Unir. of CalUonla,Berkeley). T H U S . Amer. Nucl. Soe.. 9J 193-4(*me 1966).

A, F34492 UNIFYING APPROACH TO VARIATIONALFORMULATIONS OF THE TRANSPORT EQUATION.Kaplan, Stanley; Davis, James A. (Bsttla Atomic PowerLab., Watt Mlfflta, Pa J . Trass. Amer. Hod. Soc., 9:

9 19S8).

B34t54 VARIATIONAL PiliNCIPLE FOR THE NEU-TRON-DIFFUSION EQUATION USING DISCONTINUOUSTRIAL FUNCTIONS. Buallk.A.J. (Bettls Atomic PowtrLab., West MSfflln, Pa.). Trans. Amer. Nuol. 9oo., 9:199 (JUM 196S).

G,B36143 (WAPD-T-188S) A VARIATIONAL PRINCIPLEFOR THE NEUTRON DIFFUSION EQUATION USING DIS-CONTINUOUS TRIAL FUNCTIONS. Busllk, A. J. (BstttaAtomic Power Lab., Pittsburtfi. Pa.). Dec. 1965. Coa-tract AT-ll-l-GEN-14. 14p. (CONF-660606-13). Dep.am. CFSTI $1.00 cy. $0.54 ma.

From American Nuclear Society Minting. Dsawt.Seleagut and Wachiprmi have developed a varlatloaal

principle for tb» neutron dUIUslon equatloa which permitsthe use of discontinuous trial functions. Another variations!principle is described which la useful with a wider claaa oftrial function*. (M.O.W.)

B

*44W VARIATIONAL TECHNIQUES APPLIED TOTHE INFINITE-MEDII'U SLOWINO DOWN OF NEUTRONS.Hawk, Thomas L.; Fflnech, Hrart (Matisacbusette fest.of Tech., Cambridge). Traas. Amer. Nucl. Soc., 9: 195-«(June 1966).

A

34«fi SPACE ANGLE SYNTHESIS: AN APPROACHTO TRANSPORT APPROXIMATIONS. Kaplan, Stanley;Davis, Jamec A. (BetHs Atomic Power Lab., WestMiffltn, Pa.). Trans. Amer. Nucl. Soc., 9: 196-7(June

403M (WAPD-T-1890) ANGLE-SPACE SYNTHESIS:AN APPROACH TO TRANSPORT APPROXIMATIONS.Kaplan, S.; Davla, J. A.; Natelaon, M. {BetUa AtomicPower Lab., Ptttaburgh. Pa.). May 1966. ContractAT(11-I)-Gen-14. 49p. (CONF-68060S-I4). Dep. ran.CFSTI $3.00 cyE «0.S0 mn.

From American Nuclear Society Meeting, Denver.Three new ideas are combined to produce an analytical

approach to solution of the neutron transport equation inrealistic multidlitiongionul reactor geometries. The methodla of the exoanaion approach type in which the flux la repre-sented by a linear combination of known function* of anclewith mixing coefficients which are unknown function* ofspace. The mechanics of the new approach are developedin detail and some initial numerical examples art deatcribed.(H.U.hi

24

Volune 20 (1966)

F4149* VAR1ATIONAL DESCRIPTION OF DBSIPATTVEPROCESSES. Pomranlag, O. C. (General Atomlo DiT..General Dynamic* Corp., San Diego, Calif J. i. Nuol.Energy: Pt. A and B, 20: 617-M(Aug. 1966).

The Rouascpouios variatUraal formalism Is applied toequation* with a first-order time derivative. The result-ing functional is a very general varlational characteiiza-tion of diaslpatlve processes in that It admits trial func-tions which do not satisfy the equation of interest, do notsatisfy the boundary conditions, and are not continuous. Itin shown that the previous functional* presented la theliterature for this type of problem are special ease* ofthis more general functional. The use of this variations!principle to estimate a rather general das* of character-istics of interest is discussed. It is also pointed out thatthis general functional is not unique and arguments aregiven to deal with this non-uniqueness. The variatloaaldescription of the Sturm- Uouvllle equation Is consideredin this same generality. This leads to a generalization,(to a more complete class of admissible trial functions) ofthe classical Raylelgh quotlsat for estimating eigenvalue*,(auth)

G4*967 SYNTHESIS METHODS IN REACTOR ANALYSIS.Kaplan, S. (Weatlnghouae Electric Corp., West Miffila.Pa.). Advances Nucl. Scl. Technol.. 3: 233-66(S966).(WAPD-T-1810).

A collection of method* for solving various problem* lareactor theory ia described. The unifying idea Is to redae*the number of independent variable* by constructing aa ap-proximate solution in the form of • linear combination ofknown functions of one or mttrm of thn variable*, with thecoefficient* of combinatic. being function* oaly of the re-maining variables. One of the Important applications of thedencribrd approximation methods la solving the aeutroagroup diffusion equation* la three dtmeaaloM with greatapatlal detail. (H.D.R.)

25

Voluae 19 (1965)

F,C .W?M TIME-DEPENDENT VAR1ATIONAL PRINCI-PLES FOR NONCONSERVATIVE SYSTEMS. Jeffrey LewiBS(Royal Engineers, Glasgow). Nucl. Scl. Eng., 20: S17-20(Dec. 1964).

Two variutional principles are discusied for time-(lcpcmlcnt problems in reactor physics. The first U astationary expression for the meter reading at a giventime, the second a stationary expression for the Integralof the meter reading up to a given time. Both the prin-ciples, unlike conventional lajjranKians extended to time-dependent noncon.se rvativo systems, have the advantage ofrcquirinK trial functions tu be exact only at one end of thetime interval of ink-rest. Either may be generalised toaccount fur nonlincarittes. Thi- second principle reduce*to the first by making a suitable identification, while thefirst principle in turn reduces to a well-known and power-ful variations! principle for the steady state, (uuth)

F2713 THK PIUNCII'LKS AND APPLICATIONS OFVAIUATIONAL MKT1IODS. Martin Becker. Cambridge,Massachusetts, Massachusetts InHlitute of Technology, 196130p. 95.00.

Fundamental variations! calculus techniques are de-scribed. As an example, these are applied to the calcula-tion of fuel depletion tn nuclear reactors. (T.F.H.)

A . H3IH ANALYTICAL REPRESENTATION OF NEUTRONFLUX D) PLATE-LAYERS. Karl-Hetaz Mutller (CCB-EURATOM, Ispra, Italy). Nukiconik, 6: 271-4(Sept. 1M4).

)The direct method of variation calculation, applied to the

Boltzmann equation for neutron transport, Allowed the con-struction of analytical expressions, by means of which the•eutron distribution in the reactor could be described indearly arranged ways. For the one-dimensional platelay«rs, three useful ways for the solution of shieldingquestions are shown and performed, (tr-auth)

1147 VAIUATIONAL ANALYSIS OF A NON-LINEARKINETICS PROBLEM. David S. Selengut (Knolls Atomic\«wer Lab., Schenectady, N. ¥.) . Trans. Am. Nucl. Soc.,7: 258-7(Nov. 1964).

DSM2 APPLICATION OF MULTICHANNEL SYNTHE-SIS TO TWO-DIMENSIONAL DEPLETION CALCULATIONS.W. H. Turner, F. C. Merrfman, and E. C. Hansen (KnollsAtomic Power Lab., Schencetady, N. Y ). Trans. Am.Nucl. Soc.. 7: 290(Kov. 1904).

DSMS LEAST SQUARES VARIATIONAL ANALYSIS OFTHE BURNUP PROBLEM. Martin Becker (Knolls AtomicPower Lab., Schenectady, N. Y.) and Henri Fenech. Trans.Am. Nusl. Soc., 7: 290-1 (Nov. 1964).

G.D543* (WAPD-TM-46H) TNTO2— A THREE-DIMEN-SIONAL NKUTltON FLUX SYNTHESIS AND DEPLETIONCALCULATION COMPUTEIi PKOCKAM. C. A. Flanagan,It. J. Breen, O. J. Mariowu, ami A. J. VigUotU (Wastlng-huiise Electric Corp. BelUs Auunlc Power Lab., We«tMifflln, Penna.). AUK. >9«- Contract AT-I1-1-GEN-14.l«>3p. IVp.(nin); $4.00(cy), 2«mn) OTS.

TNTO2 Is a Ihrcc-diincnslnnal neutron flux synthesis anddepletion code for liiilco 20(10 made up liy TNT2A. whichdetermines thu Integrals of DM product* ml two-dimensionaltrial at*: woight functlosia, TNT2B, which performs the de-pletion calculations, and TNT2C, which performs ra-depla-Uan calculations for doUllod power distributions. (K.E.U.)

«3»3 GENERAUZAT5ON OF THE VARIATWNALMETHOD OF KAHAN, RIDEAU, AND KOUSSOPOULOS ANDITS APPLICATION TO HF.UTRON TRANSPORT THEORY.Morton D. Kostin (Princeton Univ., N. J.) and HarrayBrooks, i. Math. Pbya. (N. Y.). 5: 1691-1700(Dee. 1964).

The variatlonal method of Kahan, Rideau, and Roussop-oulos (KRR) frequently used In neutron traasport theoryto estimate weighted averages la axtcrded and generalised.In the KRR variaMonul method a first variation In lha trialfunctions produces a second variation in the estimate ofthe weighted average. Two generalized varlallonal (uac-tionals which depend on trial operators instead of trialfunctions are given. A first variation In the trial operatorsproduces an nth variation in the estimate of ttva weightedaverage when an nth order generalized variatlonal functionalis used. Both perturbation theory an! the Klill variaUonaimethod are derived as special cases of the generalizedvariation*! ewthod. Several uxamplus including calculationsof transport equation spatial momenta using diffusion equa-Uon soluUons as trial opurators arc studied with gocd r*>ifults. (aulh)

I24IS {EUR-2174.0 SOLUJilONK NUMERICA DIALCUNI PROBLEMI BIUIMENSiONALl DELLA DIFFU-EIONE NEUTRONICA COL METODO VARIAZiONALB DlRITZ. (Numeri<T-"l Solution of Some Two-DlmcasloaalDlffnaiua Neutronlcal ProbUms by the VarlatiOMl RIUMethod). I. Calligani (European Atomic Energy Comnw-nlty. Joint Nuclear Research Center, ispra, lUIy and

26

Volume 19 (1965)

European Atomic Energy Community. European ScientificData Processing Center, Ispra, Italy). 1964. 82p. Dep.(mi).

The approximate solution* of the homogeneous problemand of the inhomogeneous problem of the multlgroup neu-tron diffusion theory may be obtained by searching for thefunctions which render suitable stationary functional*.These functions are calculated by the Rite method. Somenumerical experiments are given in order to compare theRite method with the finite difference method on two-di-mensional diffusion problems, (auth)

B1H3S A MODIFIED KANTOROWITSCH MKTHODj((R THE DIFFUSION THEOHETICAL CALCULATION OFHKACTOrtS WITH COMPLICATED GEOMETRIES. PART

. ||. Hermann Schaefrier (Tcchnische Hochschulc, Stuttgart).SuMiimft, C: 39S-4t>5(Dec. 1064). (In German)

the modified method was discussed for some specialnttd rod positions. It was applied to a reactor with re-mlcrlnc angles and expanded to some further geometries.Ntinerica! results were given for the eigenvalues and com-pircd with values from the literature, (tr-auth)

14803 (CONK-051-2'.) LEAST SQUARES VARIA-TION A L ANALYSIS <>!• NIK HURNUP PROBLEM. MartinBecker (Knull.s Atomic Power Lull., Sehcneetaily, N. Y.)and Henri Fcnuch (Massachusetts Inst. of Tech., Cam-bridge), [1964]. Contract W-31-109-cng-S2. I4p. Dop.(mn)j $l.00(cy), l(mn) OTS.

From American Nuclear Sociuly Meeting, San Francises,Nov.-Dec. 1964.

A homogeneous, bare-slab reactor is considered thathas a single fuel and is controlled by a variable uniformpoison. One-group theory is used to describe the neutronbehavior. Thu fuel bumup in this reactor is calculated byleast-squares variations! techniques. The FEVER com-puter code is used for the calculations. (T.F.H.)

G.B ... . - . _21*38 (CONF-6S0S01-4) VARIATIONAL SYNTHESISWITH DISCONTINUOUS TRIAL FUNCTIONS. E. L. Waca-•prass and M. Becker (Knolls Atomic Power Lab., Scbe-nectady, N. YJ. Feb. 1965. Contract W-31-109-eng-S2.22p. Dep.(mn); |1.00{cy), l(mn} CFSTI.

From the Application of Computing Methods to ReactorProblems, Argonne, 111.

The multichannel flux synthesis method for treating neu-tron-diffusion problems in several dimensions i* consid-ered. In this method, regions are represented by nodes,and nodal difference equations are derived with the aid oftbe lower-dimensional results. By suitably choosing afunctional and trial function*, nodal difference equationsare derived from variation*! principles. Discontinuoustrial functions are used. Methods for obviating the diffi-culties presented by these discontinuities are discussed.(T.F.H.)

B17234 A MODIFIED KANT JROWITSCH METKOD FOBTHE DIFFUSION THEORETICAL CALCULATION OF RE-ACTORS WITH COMPLICATED GEOMETRIES. PART 1.Hermann Schacffler (Tcehnisehe Hoehschulc, Stuttgart).Nuklconik, 6: 3R4-95(Dcc. 19(34). (In German)

For reactors with complicated geometry, as 1B present,for example, on account of any control elements (rods orplates) introduced in thu reactor, the diffusion equation t»not soluble analytically. An approximation method must heused. The modified Kantorowitsch method makes it possibleto calculate* a larne class of such nun-separable geome-tries. It differs from tin- approximation methods used Inthe literature In that it makes it possible to obtain veryaccurate results at rul.-ilivuly low expenditure. The calcu-lation time amounts, in the miMliflcri method, to only afraction of the tin*' ncri'SHary for other methods. It IspuKttihlc to calculate iiinipUealtcl proliUmn In practicablecalculation lime. The method «le|H-nds on a new formula-tion by means of whish Ihi- variation problem equivalent 1"

IW- differential equation problem can bo solved anproxi-m.iti'l.v. The method was demonstrated on the example ofa lure cylinder reactor with central partially insertedi.hick control rod and a reactor with re-entering angles.

lh)

G11*54 SOLUTION OF A TWO-DIMENSIONAL NEUTRONDIFFUSION PROBLEM BY FLUX SYNTHESIS. PeterKUian (AEG-Kernenergleanlsgv'n, Frankfurt am Main).Hukleonlk, 6: 340-4(Nov. 1064). On German)

The calculation of the two-dimensional distribution ofIke thermal neutron flux In the large cell of a water-mod-erated reactor was reduced with the help of a variationmethod to the calculation of the one-dimens«o»>al flux dis-tribution. The position-dependent superposition of a secondone-dUnenslonal asymptotic flux gives f. good approximationof the flux distribution. Iterations Improved the result* only•lightly, (tr-auth)

C273W (WANL-TNR-133) VARl-QUIR—A TWO-DIMENSIONAL TIME-DEPENDENT MULTI-GROUP DIF-FUSION CODE. J. W. Rieae and G. Collier (WestinghouseElectric Corp., Pittsburgh, Pa. Astronuclear Lab.).Sept. 1963. 93p.

A vartntlonal approximation is developed to solve thetime-dependent neutron diffusion equations. The method laincorporated into a computer code, which allow* up to 4energy groups and 6 precursor!). Some sample resultsare presented. <suth)

27

Volume 19 (1965)

B27407 A MORE STRAIGHTFORWARD USE OF VARl-AT7ONAL PRINCIPLES WITH BOUNDARY CONDITIONS.Harvey Amster (Univ. of California. Berkeley). Nucl. Set.Eng., 22: 354-9(1965).

A previous treatment (Nucl. Sci. Bag., It: 147-54(1963))of one-group diffusion theory for a homogeneous slab Isreviewed for the special situation in which scattering isisotroplc and no neutrons enter the medium outside Itssurface*. (C.E.S.)

A27406 THE TREATMENT OF BOUNDARY TERMS INA VARJATIONAL PRINCIPLE CHARACTERIZING TRANS-PORT THEORY. G. C. Pomraning (General Atomic Div.,Ueneral Dynamics Corp.. San Diego, Calif.). Nuc}. Scl.Eng.. 22: 259-61(1965).

A varlatlonal treatment (H. Amster. Nuc). Scl., Eng.,22: 255 (1965)) of the diffusion theory boundary conditionsis considered. This treatment is purported to be morecorrect than a variational treatment by Pomraning andClark (Nucl. Sci. Eng.. 1C: 147 (19G3)>. It is pointed cutthat the treatment of the diffusion theory (or higher orderPM> boundary conditions is, by necessity, arbitrary (evenwithin the framework of the variational methoJ) and thatthe two treatments are different in their aims. (C.E.S.)

A27409 ANSWKK TO POMRANING'S REBUTTAL ONVARIATIONAL BOUNDARY CONDITIONS. Harvey Amstor(Univ. cf California, Berkeley). Nucl. Scl. EnK., 22: 262(IMS).

Rebuttal to the arguments given in "The Treatment ofBoundary Terms In a Variational Principle CharacterizingTransport Theory" (Nucl. Scl. ling., 22: 259 (1905)) in pre-sented. (C.E.S.)

C292M THE APPLICATION OF TWO VARIATIONALTECHNIQUES TO THE ANALYSIS OF THE PULSED*NEUTRON EXPERIMENT. Robert B. Rasmuail. O. R.Dalton, and M. J. Ohanian (Univ. of Florida, GainMvUM).Trans. Am. Nucl. Soc., 8: 274(Hay « • « .

C29S29 COMPARISON OF THREE VARIATIONALMETHODS APPLIED TO A KINETICS PROBLEM. H.Fentch and V. Orphan (Massachusetts Inst. of Tech., Cam-bridge). Trans. Am. Nucl. Soc., 8: 229-4<May 1965).

A31430 STUDY ON AN ANALYTICAL REPRESENTA-TION OF THE NEUTRON FLUX IN PLATE LAMINATIONS.Karl-Heinz Mueller (CCR Euratom, Ispra. Italy). Nuk-leonik, 6: 271-4(1964). (In German) (EUR-876.d)

The direct variation~calculus method, if applied to theBoltzmann neutron transport equation, enables analyticalexpressions to be formulated by means of which the neutrondistribution in She reactor can be clearly described. Forthe iinidimensional plate lamination, three methods whichlend themselves readily to the solution of shielding ques-tions are adduced and applied, (auth)

A

?'?** (NUS-R-436) THE ASYMPTOTIC ANUULAR•ftPENDENT LEAKAGE Sl-ECTHUM OF THERMALNEUTRONS. KladiiSk. R, (Nuklcarni Inptilut "Jozeffitcfan," Ljubljana (Yugoslavia)?. Aug. 1964. 19p.Dep.(mn).

The application of the variational method in velocity-dependent transport theory is described for the case offt pulsed infinite slab. The extrapolated endpoint and thebuckling as functions of the slab thickness are calculatedfrom a given transcendental equation. The expression furthe angular and energy distribution of leakage neutrons isimproved by one iteration of the integral transport equa-tion. The results arc discussed for three different scat-tering models: mooatomic gas, Neikln water, and theEgelgtnff model. It is found that Nelkin's model gives abetter estimate for the integral constants, whereas theEgelstaff's law is more suitable for the differential spec-trum calculations, (auth)

C33207 ON THE INCLUSION OF BOUNDARY TERMSIN TIMS-DEPENDENT SYNTHESIS TECHNIQUES. BeckerMartin (General Electric Co., Schenectady, N. Y.). Nucl.'Scl. Enj., 22: 3S5-6(July 1965).

7a connection with the application of synthesis techniquesto various time-dependent problems, a variation^ prin-ciple waa previously presented for linear ilme-depcndentfroup-tUffuaion theory. However, the principle ia not ats-flmaiy with respect to arbitrary variations In the func-tions involved. It is shown that this difficulty can beavoided by Inclusion of appropriate boundary terms inthe functional. (D.C.W.)

C47*41 JANL-7050. pp 159-77) VAR1ATIONALMETHODS FOR THE SOLUTION OF PROBLEMS OF RE-ACTOR KINETICS. Gsllig«nl. Ilio (European Atomic

28

Volume 19 fl965)

Energy Community, lspra (Italy), Joint Nuclear ResearchCenter).

The solution of the time-dependent multigroup, multlre-glon diffusion equations may bo obtained by searching forthe functions that render stationery a particular functional.These functions are determined numerically by Kantoro-vich'a method and by Ruler's method. With the Krntorovichmethod the flux and the adjoint flux are expanded in a antof spaco-dependunt trlai functions with time-dependesti: co-efficients. Those coefficients arc determined by solvingwith a "one-step method" n system of ordinary differentialequations. The Euler method consist* in searching for thefunctions that render stationary the above functional in theclass of pioccwlse linear functions. These functions satisfya large system of ordinary differential equations and m-edetermined numerically by three different methods, the"implicit" method, the "Implicitly explicit" method, andthe Saul'vcv "alternatine" method. A comparison uctwevnthe Kantorovlch method and the Eider method is made bymeans of some numerical examples, (autt)

I

29

Volum 18 (1964)

tM A NEW ASYMPTOTIC DIFFUSION THEORY.0. C. Pomraniog and M. Clark, Jr. (Massachusetts but.of T*ci., Cambridge). Nuel. Sci. Eng., S»: 22?-33(Oct.IMS).

Tfcs vari«tional formaliam ia used to derive from theKaweMrgetic Boltzmaim equation a diffusion theory wishthe asymptotic transport diffusion coefficiert. By consid-ering an interface between two media as the limiting case•f a medium with continuously varying properties, the•atMdaiy conditioM are found to be continuity of currentMri a specMed discontinuity in the scalar flux. The vari-•Uoaal formalism gives toe linear extrapolation dlstancoft a pure scattsrer accurato to within one-half percent.NMMrical comparisons with i^MSicnl <P-I) diffusiontheory for a coll calculation indicate that the variationalejiNusien theory Is significantly more accurate; the ac-tttracy n|»pi>ara to be comparable with that of the P-3»etbwl. (with)

F11W ON THE DERIVATION OF A VARIATIONALPRSMCIPLC FOR UK EAR SYSTEMS, D. S. Selengul(KaoUt Atomic Power Lab,, iSchenectady. N. Y.). Nucl.Set. Bug., IT: JiO-lHOct. IMS). (TID-18620)

A specified physical system, such as a reactor, whosesta&r is described by a function of the relevant phase space•eenUactM is coMider*5. A derivation la outlined thateoablMt no* to lake a given linsar theory and immediately<*teto a Iagr.«»eian whosa stathmary proparty Is equivalentto tte eepatioM of toe theory and that constitutes a varla-Umii ipnncipl* to tbe cetimatUm of an arbitrary linearitoetiAiul of the aside of the system. <C.E.S.)

Cf«S9 4CONF-1B7-2T) APPLICATION OF THE VAR-IATSONAL. METHOD TO TJIE CALCULATION OF THETIME DKPENDENCE OF THE NKUTRON t'LUX IN SMALLPULSED 81MIS, CYUNDKiiS AND SPHKHES, F. D. JUOKCand P. II. Suited (RuniwoliK-r PutyU'rimic Inst., Troy,N. Y.), Indl. Contract W-31-10tt-en«-S2, Sop.

From Aroorlcan Nucluar Society Mtctlixc. Now York,Now. 1M3.

Tbe vitilaltonal mcllKMl 1» ;i|)|>lk'<l to the monounuriictlcUm« dependent tnuiKport cquutluu U> olrtaln a simple re-lation for t3w asymptotic decuy constunt in small pulsedasuembllus, Tbe results Imllcato Unit flat trial functionamay lie a reasonable representation of the flux distribu-tions ta the thin slab limit. Tills approach in superior toniMiy of the usual transport approximations, (with)

B1S13I (EUR-532.e) A MULTI-DIMENSIONAL, MULTI-GROUP, MULTIREGION, NEUTRON DIFFUSION CODE(ITERAT1VE-VARIATIONAL APPROACH). M. Lunellt andG. Maggioiti (MUan. Universita). 1954. 170p.

The program LOUISE m, in FORTRAN for tbe IBM 7094,calculates the crUteallty constant and critical fluxes dmultiregion (up to 25) reactors in the multlgroup (up to 10)approximation. Calculations can be performed in up to 7dimensions, snd an iterative-variational technique is used.The diffusion differential equation* considered are in-cluded, together with a (ample problem and the FORTRANlisting. LOUISE HI has given results In agreement withWII1RLAWAY, while requiring e much shorter executiontime, (auth)

B3 U I A VAMATIOKAL PWNCIPLE FOR HETEXOG-SW00S niBONANCE CAPTURE. Paul P. Gut (G*neratStectrto Co., Haafoird, Wash.). Trans. Am. Nucl. Boo., 6:

IMS).

D . B3XM APPLICATION OF flYNTHEfIS APPROXIMA-TION! TO TNKEB-inMENKONAL DEPLETION CAtOU-LATKINI AND TO C1CI.L TOEORY. V. Kaplaa sad O. J.MKTlcart (W«*WaghowM EW«"ic Corp., Httabursh).Trias. An. Nuel. loo., *: ?S4-8(Nor. IMS).

B,G1II4S (WAPD-TM-377) EQUATIONS AND PRO-GRAMS FOR SOLUTIONS OF THE NEUTRON GROUP DIF-FUSION EQUATIONS BY SYNTHESIS APPROXIMATIONS.8. Kaplan, O. J. Marlowe, and W. K. Cadwell (We«t!nghouMElectric Corp. Bettls Atomic Power Lab., Pittsburgh).Dec. IMS. Contract AT<ll-l)-Gen-14. 7lp.

A method for constructing approximate solutions to theIkroe-dimenilozal group diffusion equations uslsig onlyone- and two-dimensional programs is presented. Thecomputer programs that beve been developed to carry outthis ooutructlon are described, and Instructions are givenfor their use. (auth)

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Volume 18 (1964)

c15154 APPLICATION OF SYNTHESIS TECHNIQUESTO PROBLEMS INVOLVING TIME DEPENDENCE. S. Kap-lan, O. J. Marlowe, and J. Bewick (Westlngbouse ElectrioCorp., Pittsburgh). Nucl. Sci. Eng., 18: 163-76(Feb. 1864).(WAPD-T-158S)

A method described previously for syntheslilne thre«-dtmcnslonal flux distributions is extended to reactor ki-netic* problems and to lifetime studies. The method isoutlined, and sosne numerical example* are presented. Theresults of these show the method ID be a practical way ofsolving time-dependent reactor problems with a detailedapatial model, (auth) -

A 'IMSt AN IMPROVED FREE-SURFACE BOUNDARYCONDITION FOR THE P-S APPROXIMATION. C. C.Ponnning (General Electrio Co.. PieuaafcM, Calif.).Hud. Sci. Bag., 18: 5M-M(Apr. 1H4).

Variatlonal calculus Is used to obtain en appro«intate'ree-surface boundary condition for UM P» approodstaUoato the monoenergetfc neutron tranaport equations. Slabpometry la considered for simplicity. (C.E.S.)

sources, »ad the critical rod problem. For arbitrary,axially syaunetrlo sources inslie or oatiaide the rod. aknowledge of th» uncollided flux Is suffietaat U> determineDM escaps probability from the rod la term* of theeestandard problemi. (auth)

B154*8 (EUS-217.d(Pt. 1 and ID) USER LOSUNGSVER-FAHKEN FUR DIE EOLTZMANN-GLEICHUNG. TEtL IAND n. (The ApplicaUon of the Direct Method of theVasrlttlonal -Calculus. Pt.rt I and. U). K.-H. Mueller(Eurpoean Atomic Energy Community. Joint NuclearResearch Center, Iepra, Italy). 1964. 30p.

A functional is given in which the Pj approximation tothe Boltzmann equation of neutron dynamics is In the formof Euler equations. The requirement for the related ex-treme values la thus equivalent to the differential equationsystem of the P ( approximation. The direct method of cal-culating variations can be used to solve the variation prob-lem. An exclusive system of selection functions, suitablefor concrete reactor problems, provides a clear and sim-ple method for calculating criticality and neutron flux. Theresult of each of these three procedures is called reducedBoltzmanti-eq. This matrix is an Integro-dlff.eq. In (1)velocity and time, (2J space and time, and (3) time. Asuccessive application of transformation (2) after (1) or(1) after (2) la possible. Examples Illustrate the essentialsteps of the consideration, (auth)

MS73 VARIATIONAL BOUNDARY CONDITIONS FOBTHE SPHERICAL HAUMONICS APPROXIMATION TO THENEUTRON TRANSPORT EQUATION. G. C. Pomraoiag(General Electric Co., Pleasanton, Calif.). A u . Pfcys.(N. Y.), 27: U3-215(Apr. 1M4).

It 1* shown that the moaoencrgetlc neutron transportequation aed the associated boundaty conditlocs «aa becharacterized by a LagrangUn. A proper choice of thetrial function for this Lagrtngiaa leads to the widely usedspherical harmonica approximation as the fiitsr Tngnsgnequations. A set of boundary conditions for the sphericalharmonic equations U the result of the. logical applicationof the vartatlonal mothod. Thesn variatioMl boundary con-ditions appear to be tdgnlflcantly more accurate than theboundary condition* presently In general use. For exam-ple, the UA« of the variation*! boundary cowdtttoM r'. afree surface reduces the error (comiiared wllh UM> bwMd-ary conditions prenently used) in the linear extrapolationdlilance for the Milne proWoni ly several factors. Inparticular, the P-l (diffusion) approxlihallon yield* a v«l«*»of 0.7071 (In units of mean freo paths) and the P-S approaUcation yields a value of 0.71U, both cosaparing quit* fav-(jably with the exact value of 0.7104. (auth)

AIMS* NEUTRON TRANSPORT IN CYLINDRICALRODS. K. Bingham Cady and Melville Clark, Jr. (Massa-chusetts Inst. of Tech., Cambridge). Nucl. Sci. Eag. 18:4«l-S07(Apr. 1964).

A calculations! method for Boltzmann's one-velocity,liotropic scattering transport equation in developed forcylindrical rods. The starting point i* Pelerl's Integralequation, and the technique m&y be interpreted as a mo-ments method or a variatlonal method. Numerical resultsin the form of graphs arc given for a set of standard prob-lems. These problems include volume sources, surface

AM532 SPATIAL EXPANSION OF THE TRANSPORTEQUATION, G. C. Pomranlnic «nd M. Clark. Jr. (Mnasa-snusella fast, of Toch., Cambridge). J. N«tel. Kaeigy,Pt. A fc D, IB: 10!-209(Apr. 1904).

The solution of (no monoonorgotlc lr«Mport equalloK inslab geometry la expanded in term* of known *nMtid ftwe-tlons and unknown oninilar eoofflclcnts. Two general for-

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Voluae 18 (1964)

naliama-a vmrii l aada iMethod-are duvetoprd to obtain the *u«caaary eqwatloaa forthe angular cuefReiunta. The latter forattUaM la empha-ciaad alaoe it always yields aewtroa oonaervatioa. Farther,U ia shuwa that thia formalin*) traata exaetly My IncidentIkw boundary eotidlUoa, inciting tba dlaaeaUauoaa vacaum .bouadary condition. For amall ayatenu with a highly peakeddtroelioMl Owe, a epotial mt^anaion of tho directional fluxia integer powers of s t i e apatlai co-ordtoate) ia ahowa tojriaM wrtrewsly good result*. Tana, a jwlynoaWal atpawleaforaw a ooeptaawM te tha wiMy aised oagalar erpaarinaa,each aa She P-N (aphorieal hanaoaie) Method, whtoh arca m * aoearats for large ayatoaw wlta aa alatoat iaotropkidtraetioaai that. To aatpaaslx* tha lad that tint fonaailaaMdaratopad a i* anpHcabU to aay apatial waaaatoa taaoUcaa,tha pmotrattoa W a ooraaal l»aaia) proUeai ia oaaaidandwith asppaaatlal axpaaaioa liuaeUooa. Tha aaalyata ia aaowato radaea to tba a u e t traaaport raaalt ia all kaowa llaUta.

•I N I * A VAMATKNAL PHOCCDORB FOR CALCU-IATINO HKMt-naBor, Fiw-aaoup, «PATUU.T D I -PCNDKHT fKCTltA. P. A. OaabraUaro (KaoUa A t w kVwmr lak., Ihhaaaetadjr. N. YJ. 1'raaa. A a . Maol. laa. ,1: l«-U(Jaaa UM) .

LXAaT-SqUAKKt VAIOATKHAI. MCTWOS.Martai tmdme mi Waaurt Faaaeh NaiiaatajaaWa laat. *Tooh., CaiNbriofeaK Tcaaa. Aat. Maet. faa., 1: U-U(Jcao1N4).

A VAHIATIOHAL TCCHMQUC TO* SPACE-TMS NBUTKW UrnJHON. J. W. Rtooo (WaaUaghwiaaKUetrto Cara., Ptuabargh). Traaa. A M . Nael. Saa., It Xt-ittfaaa l t* i ) .

ADMM APPLICATION OF THK DHUCCT MZTIIOD OFVANIATKMIAt CA1.CMI.US TO T1IB BOUT7.MAMN BQUA-11OH. K, M. Maellar (EaratoM, laara, Italy). Traaa. AM.Nael. fco.. ?: 9-iatfaaa 1N4).

G2MM APPUCATION OF FLUX SYtTHICaV TOPARAMETRIC-OPTIMIZATION STUDIES. N. J. Cariaa, Jr.(Waatlmhouaa Electric Corp.. PUtabargh). Traaa. AM.Hue1.. Soc., 7: lSOaaa 1H4).

A907tl ON THE VARIAT1ONAL METHOD APPUED TOTHE MONOENERGETIC BOLTZMAKN EQUATION. JajwaxR. Mika daat. of Naelaar Kaaearch, Waraaw). Nael. Sei.I n j . . 1»: 977-IUuly MM).

Tha aaparattoa of tha aoa-aaU-adJoiataoaa froM tacmonoanargtUc BotUauum cojuatioa ia dtacaaaad. The proparboundary tcrma that yield tho bouadary oondiUoaa corra-apoadiaa; to tha adjoint Bolumaan ciojuatloa ara darivod.(C.S.S.)

B.EWW (COMF-44«-?> A VA1UATIONAL PROCEDURErOR CALCULATING HIGH-ENERGY FEW-GROUP"MTMLLY DEPENDENT SPECTRA. P. A. OatbralUroKaoUa AtOMic Power Lab., Schenectady. N. Y.). (Jaae'H4J. CoatractW-31-lM-cnc-SZ. 14p.

r«wi Aaaarican Nuclear Society lath Anmul Meeting.'Mladetphla. June 1H4.

A bat Method that provide* the accuracy comparable*• awltlffnMp Methods waa developed for reactor calcula-Uoaa la the 1« Mov to 0.125 av range. The eacrjty raqptk alvldftd iato thrao diacrcte lethargy groapa in which a**Ur«irt aat of two overlappiiiR apectra ia aaed tor each"oap. Raautta aro compared with IBM TM muUlcroup" a » calcalatiuaa. (K.K.U.) — • - •

C , A32721 A VARIATKJNAt APPROACH TO THE TIME-DEPEKDENT SLAB. R. Kladalh OCeraforaclaiagasantrttM,Karlaruaa, dor.). Nwkleonik, •: 147-S3(May 1M4>.

Tba aaymptotic belwvior of a neutron pula« iajoeted iatoaa ialinUe alab of flaito thickness is discussed. For illua-irattoa, a moaeUaetic transport of neutroaa wish iaotropfcacatteriag waa chosen. Two integrals ot the traaaportacawtloa aad tha varUUonal Method are used ia order toobtaia tha depaadeace of the extrapolated ead point upoathe elsb thickaoaa. The extrapolated tad poiat x, ia ob-tained aa tha largest positive rout of a givea transcendentalequation, ft U aaggeated that the aaymptolic Mode mayexist withla tha alab ao nwttor how thia tha slab ia. Tha•xprescioa for the aaguUr diatribaUoa of tha leaking a*a-troaa waa iMproved by one itoralioa of tha integral traaa-port eqaaUes). Tbe reaxlts akow a eharaatarlaUe aagalarpaakiag of tba aautroaa leakiag fr«M alaba of eatramalysmall thickn (avth)

32

Volua* 18 (1964)

l i f l i (HW-SA-3190) CONTROL OPTIMIZATION BYVAKIATiONAL CALCULUS. Albert L. Ruiz (GeneralK!rctrfc Co. Hartford Atomic Products Operation, Rlch-iMd, Wsnf-.). Sept. 10. 1963. Contract AT<45-l>-1350.

The use of dynamic programming to synthesize an optimalrimtrol system whcr: Ihc performance index is expressed as«* Integral leads to ress.Ua that have been shown to ben|ulv«)eat to the use of the calculus of variations with natu-ral boundary conditions. With this approach, the extensionto other types of boundary conditions Ite not obvious. On theother hand, the use of the calcuhui of variation)! leada to re-sults for all types of bountliiry conditions, as long as thevariables are continuous. For a linear process and qua-drntk: performance index, the derived control law Is thosame lor all cases of admissible boundary conditions, andthe feedback factors and gains arc calculated from sets offirst order differential equations that ore of the same gen-eral form in each case, although differing in detail. For anonlinear process or a nonquadratic performance index, themethod gives a symbolic expression for the optimal controllaw. The evaluation of this expression for particular prob-lems depends on one's ability to solve, exactly T approxi-mately, a nonlinear differential equation. ' ath)

(APED-4342) SPATIAL EXPANSION OF THETRANSPORT EQUATION. C. C. Pomranlng and M. Clark,Jr. (General Electric Co. Vallecitos Atomic Lab.,Plcasanton, Calif.)- Mar. 25, 1964. 39p. (63-APF.-16)

The solution of the monoenergetic transport equation Inslab geometry is expanded In terms of known spatial func-tions and unknown angular coefficients. Two general for-malisms, a variation*! method and a moment conservationmethod, are dcvcloiwd to obtain the necessary equations forthe angular coefficients. The latter formalism Is em-phasized since it always yields neutron conservation. Fur-ther, It Is shown that this formalism treats exactly anyincident flux boundary condition, Including the discontinu-ous vacuum boundary condition. For small system* witha highly peeked directional flux, a spatial expansion of thedirectional flux in Integer powers of i (the spatial coor-dinate) Is shown to yield extremely good results. Thus afX>lynonr<al expansion forms a complement to the widelyused angular expansions, such as the P-N (spherical har-monic) method, which are most accurate for large systemswith an almout Uotrupic directional flux. To emphasizethe fact that the formalisms developed are applicable toany spatlitl expansion functions, the penetration (of a normalbeam) problem is considered with exponential expansionfunctions. The analysis Is shown to reduce to the exacttransport result In all known limits, (auth)

B. E42S97 (TID-7O5O4p.m-7S» FAST GROUP FITTEDCONSTANTS IN FEW-GROUP THEOHY. P. A. Ombrellaro(Knolls Atomic Powar Lab., Bchenectaily. N. YJ.

Two achssAM for calculating fasl few-group constants forcor* depletion calculations are described. The fitted croM-seciioa scheme expresses the fast group conrtsnte 4a termsof Affective microscopic cross sections, which am obtainedby fitting the aisnplc group-constant formulas to tiw groupconstants obtained from a muttlgroup program such asMUFT. The second scheme uses a vartsttonal method tosolve the energy-dependent diffusion equations based onthe assumption (hat, for a given composition, the fluxspectrum ts adequately given at. a linear combination oftwo flux base spectra, and similarly for the current spec-trum, where the bases are chosen to represent soft andnurd spectra. The energy range from 10 Mev to 0.625 .eirIs represented In terms of lethargy, end few-group ran-,slants are obtained for a. three-, two-, and one-grouprepresentation, (R.E.U.)

B , E42MS (TID-705O4MM-51I)) A TWO-MODE VARIA-TIONAL PBOCEDURE FOR CALCULATING THEKMAL-MrFUMON THEOHY PARAMETERS. P. A. OatbreSUr.(Knolls Atomic Power Lab., fcheasctady, N. YJ.

TSM approximation of variations in tfc* thermal groapconstants with depletion by a two-mod* varicUoaal proce-dure ts discussed. The procedure was programmed InFORTRAN for (DM 704 (SPG), and the results war* com-pared with group constants obtained from DM SOFOCATEprogram. (R.E.U.)

G , B42M7 (TID-7050(p.6ST-e?9» SINGLE-CHAMNILSYNTHESIS. 1. H. Leonard (WesdnghouM Electric Core.BettU Atomic Power Lab., Pittsburgh).

The development of three-dimeIMIOMI flux and powerdistributions in a reactor by combining one- and two-dimensional calculations for different portions of Ike re-actor core JB discussed. Synthesis (combination) Involvinga single axial function is known as single-channel syntbeata,and is discusied. (R.E.U.)

33

Volume 18 (1964)

G . B4292S (TJD-705C(p.678-710)» MULTICHANNEL 8YN-THESC. E. L. Wachsprtsa (Knoll* Atomlo Power Lab.,Schenectady, N. Y.).

The development of three-dliueMlone! flux aad powerdistributions in a reactor by combining one- tad two-dimensional calculation* for different portion* of the re-actor core to discussed. Synthesis involving slmultaasowcomputation of several flux dlstributioM parallel to the caxis is discussed. (H.E.U.)

G.B42f2f (TID-7O!3O{p.7IO-2S]) VAR1ATJOKAL SYNTHE-SIS. S. Kaplan (Westinghouse Electric Corp. BettlaAtomic Power Lab., Pittsburgh).

The development of three-dimenaional flux and powerdistributions in a reactor by combining one- and two-dimensional calculations for Afferent portions of the re-actor cot* is discussed. It is assumed that the radial fluxdistributions are not Independent of the axial flux distribu-tions and that they vary continuously with the axial posi-tion. A set of trial radial distributions, representative ofthe possible extreme radial distributions, are combined bymeans of a variattoMl principle applied to the flux distri-butions over the entire core to give the optimum combina-tion of these trial radial flux distributions as a function ofaxial poaitlon, thus yielding a continuoua distribution offlux in three dimensions. (R.E.U.)

34

VOIUM 17 (1963)

F3741 WiNKSR OP THK MARK MILLS AWAKD—ACKNEHALI'/KD VAMATIOHAL MKTliOD FOK REACTORANALYSIS. Alfred L. Muwery, Jr. and Raymond L. I n t .ray (North CHIUIIIU Wale Coll., Haleifh). T « a s . A M .Nucl. Soc., 5: M7(N<>v. 19M).

C3*20 SPACi: AND TIMK .SYNTIIKSIS itY THE VAIll-AT1ONAL SflKTIKH). S. KupliU) (WcKtirjulMHiM- Kh-clrii-Corp., l>ittsburKli), TninH. Am. Nucl. Sw., 5: 41I-i::(Nov.1902).

F3121 THK cam-tnuc.rioN ov APPUOXJMATI-:T1IKOH1KS »Y VAMAT1ONAI. Ml TIIOUS. D. S. Scl.iiKUl(Knoits Atomic Power Lab., ScbrnecUdy, N. Y.). Tranc.Am. Nucl. Soc.. S: 413>14(Nov. 1962).

E40*3 VARIATIONAL METHODS IN NEUTRON THER-MALIZATION. D. S. Sclengut (Knolls Atomic Power Lab..Scheneetady, N. Y.). p.162-86 of "Nautron Physic*."New York, Academic Press, 1962.

The general formalism and the estimation of adjoint traitfunctions of the variaUonal methods in neutron tnermallza-Uon are treated. Some typical problems In neutron ther-raaUcatlon are discussed: the determination of the neutronspectrum in an Infinite medium with * uniform source; thespectrum of a decaying pulse of neutrons as a function ofUme in a finite Mock of material and the associated ther-jnallsatlon time constant; and the rcthermulir.tlion cro»«section and asymptotic decay length at large dintanccs looma localized static source of thermal neutrons. The varla-tlonal method provides a unified mcnns of calculation thenequantities in torms of :t few parameters with direct physi-cal interpretations. (W.D.M.)

E . BS29C <JC\PL-2220) A VARIATIONAL PROCEDUREFOR CALCULATING FAST FEW-GROUP CONSTANTS.P. A. OmbreHaro and F. D. KcOerlghi (Knolls AtomicPower Lsb., Schenectady, N. Y ). Auf. 10, lMt . Contrast

W-SI-lOt-w^-it.A varlatfeMl procedure tor eatonlaUiis; fat — r o Inr-

group ooactanU ia dtserfeed. For « cltaa i

d acoonMaa; to P-Ia ltaeartheory I* each group of a tow group scheme i

oombiaattoa of two bees flux spectra, to be tgroup fku to be obtalaad, aad aa a llaear eosAtaattoa eftwo base curreat spectra, the group curreat to be ebetfaei.The coefficients for eombiaiag the baa* spectra are pre-vided by the theory aad depeod only oa the coaceatratls—of thM component eUmsbU of the medium. Oace the flex satcurrent spectra la each group arc calculated, the group eea-ataata tor the mediuat can be caaily calculated fros* IOux-e-iectra wslghted library mtcroaeopfc cross iGroup ooastanta calculated in this nuumsr agree welt withthose obtained from the MUFT S program, iatsth)

C96*3 (WAPD-T-15M) SPACE AND TIMC SYNTHE-SIS BY THE VARIATIONAL METHOD. S. Kapkai «WeaUat>houaa Electric Corp. BetUs Atonic flower Lab., Pitta-burgh). O c t l M 2 . Contract AT<11-1>-OM-14. t*p.

ApproxtmaUon aMtkoda being devekved for the sotstteaof few-group dlffuatoa equatloae for core aeaisa* ranilrlagaolutlom to large-alae prrblsata are dsacrflwd. Thesemethods use the process of synthesis by coejstnsetlag ap-proximate solutions to muhl-dimeasloaal probleats out efsolutions to fewer dlmeasioarl prableass. Both apace sadtime synthesis by the varlattaaal muthod are deserlbed.(M.C.G.)

A OENERAUSSS VAWATTONAL MXTHODton BBACTOR ANALYSB. Alfrert L. Mowery. 3r, ami•ayamd L. Murray (North Oiroliaa State Coll..awl. Sal. E«g.. M: 4«t-19KD»e. iNt ) .

stekod (OVM) Is deecrlbsd. The aaalysia Is beetd oa tfcewtadoaal approach aad ta aa outgrowth ot UnreetiiatieMhifcahyperdrola netbod. U weMiice. Urn GVM cwaeietaeJ eaaataeriBg the trial fuaottos* Mat appear ayssmatHeaTiywimeratleally) la a peaitlvt-aeiMldeflatte varlatiaial•rtaeiple aa tadepeadeai functtoaa. h proposltloN la proved(••MMMttraU eJHwrally that the Approximate elfftavarae

35

Volune 17 (1963)

•Mataed from the CVM la «t least • • accurate »• theIMMtrie average of the associated approximate eigen-••tat. Atao, • conjecture U proposed that the accuracy of*• gMMralised vartaUonal eigenvalue ia comparable to**t of • varlattonal r«ault employing a trial function In--•waorattBK the dimensionality of S»th associated trialtactiOM. The appltcatlon of the CVH to the pcrturbatton-<UtaUaMl method yield* mul tc that establish tie method.^ SmeralUed method completes the perturbatlon-••rtaUonal method by providing th* even-order approximate'Mvito. For Illustration, the CVM Is employed to solve afcr* realtor wttk a grey control sheet. Uslag Rarlelgh-*M**«UmU*d cosine aeries and opttmtisd pyramidhfriloss as associated solutions, the gencrallied varl-*<*al elfeavclue accuracy indicates the effective eombt-•*»» of me dlawMteaiitles of the associated trial• «u«h)

a f ute equation that is in general nonlinear. By lineariz-infe these equations and using the tranafer function formal-Ism, the transformed equation of state for the modal coef-ficients for a generalized power coefficient is obtained.Illustrative calculations were made for a simple second-order temperature and xenon poisoning power coefficient.A stability criterion was derived for the spatial flux dis-tribution of an annular cor* model. Reasonably goodagreement was obtained between the calculated stabilitymargin and period of the spatial flux oscillations observedIn the Shipplngport Pressurised Water Reactor. Also cal-culated was the response of the spatial flux distribution Inan annular cor* to both sup and harmonic reactivitychanges. 34 references, (with)

(KAPL-2217) THE SPACE-TIME NEUTRONKINETICS BY A VARIATIONAL METHOD WITH APPLICA-T*JN TO POWER REACTOR DYNAMICS. D. E. DoughertyIKaeUs Atomic Powor Lab., Schenccuidy. N, Y.>. Dec. 26.Mtt. Contract W-Sl-109-eng-52. STp.

*t ««e of the acmidircct varlatlonal method, the time and•*•*• dependent neutron kinetic equations were reduced to• •y«*at of iategt-o-diftcruntial equations in the timo*w«l«, Th» slate vector of these equations definen the"•"-dependent coefficients for a set ot space mouos givenh « ntedal expannion of the neutron flux. Tin- accuracy cJ'**• method of analysis IK arbitrary, end solution* are***UaMe for reactor con%urullon» as complex &K theWrtent state of the art allows vtoady-Htutc diUermlnations,'• avoDd the* difficult tank ot determining orthoKoniil cigun-"t*^Um» for a modal fxpnnNiiin in a complex Ruomolry,'"•Wthoicunal Crci'n's Fdmiion muk'H are 'fevcloju'd from"approximate solution of thu lnlrgrat form of iho multi-Conn klwitlc equ;itk>ns. Sinct< an exact Htttady-Ktato nolu-"•9 «f tao ooo-gruup diHuniun inotlv! Is available for a**** Iwo-roglon ttlub rtwtor, UIIH «ulutlun wan vompurtHi*H* Uw results ulitalnud l>y both crlhonunul ami nonurU»>K-OMI medal oxpanaioiki. The basis of comparison was theasyaaptotlc region bucbline for a critical (steady-atate) two-region reactor, following a perturbation of the reactormaterial properties in either of 'he cor* regions. A two-term exsaMion by Greeks Function modes was clearlysuperior to an expatwlon In which the first two orthogonalaoiba for asymmetric perturbations ot lbs vore propertieswsr* used. Combining power coefficient dynamics with thereduced space-dependent neutron kinetic equations yields

c24037 (HW-76128) PHYSICS RESEARCH QUARTERLYREPORT, OCTOBER-DECEMBER 1962. (General Elec-tric Co. Hanford Atomic Products Operation, Richland,Wash.). Jan. 15. 1953. Contract AT(45-l)-1350. 69p.

An expression for the variaticnal optimum kinetic re-sponse representation of reactors was obtained, and Gaus-sian quadrature methods integrated over a logarithmic gridw«re used to numerically integrate the Egelstaff S-function.The efforts of small differences in the concentration of theP u M Isotope In Pu-Al fuel roda on criticality were studiedusing a cylindrical lattice array with two cones. Criticalapproaches and exponential measurements were used Inthe experiments. Extrapolation lengths as a function offissionable materials and calculated and measured num-bers of rods required for criticalily were obtained. RevoltsIndicate that the critical muss of Pu cither varie* morestrongly with Pu** concentration than the calculations showor that a systematic error I* present. The P-3 computeritrogram waa used to calculate thermal flaxen In latticeswith Al -Pu fuel cluster* for both poisoned and unpoisoncdcases. Values of IJ thermal and f, the thermal utilisation,were calculated. Similar calculations were performed ORa lattice fueled with 1'uOj-UO,. SfiuiiUvily of UH> calculatedflux travrrse to perturbations in Input jKjrametrm was alsoInvestigated. A scries of PCTK experiments wns mi.ily7.edusing a multl -energy transport group cotk- to derive valuesof \\m for Al-Pu fueled moderated lattices nnrl to pndlctreaction rules for various rrsonamrc (Irtrt'lnrH ihrmiKhouta In Uce celi. Cadmium r»t!o» ill i i l l buumlarirs nrre de-rived and were found lo fall short of mvnHitrvd VHIIU>S. Re-sults of sn experiment In whluh lit i-od cliiHlcra at liothhigh and low cxpomire Al-Pu fwt wi-rc lu-ntt-d wi-rv ana-lysed lo obtain values 7ur (he iliffcivncr in Uu> ehnnce In

36

Volute 17 (1963)

the neutron multiplication factor na a function of tempera-ture, ami values of lite effective ri'donancf Integral a* afunction of trm|H-raturf> wen» i'.ilciil»tf<l. Mc.tnureinrntaof the reactivity forflldicnlH of AI-IM fuel rods wi-rv madein the I'CTIl to verily the relative !>u content • • Klvcn bychemical analysta and to Invcntlx-it*' the uniformity of Pttconcenlrntlon. The momentum dintrlbutlon of (t particlesfrom She clccay of 1*"*" wan measured; two componentswith relative intensities or 14 and 50% were identified.Internal-conversion electrons from lit'" I. ami M sheilawere observed for an 88.4 fccv y transition also. A ratioof the intensity of the L to M electrons of 3 t 0.3 was ob-tained. A value of zero was selected for the quantum num-ber k of the Lu"fc" level. Critical mass measurementswere continued on plulonlum nitrate solutions In a 14-inchdiameter sphere to determine the effects of concrete re-flectors on the critical I ty of the vessel and the effect of anair saP between these reflectors and the core. The resultsverify that a 30-inch layer of concrete is a better reflec-tor >)i:m water; the critical concentration was about 10 to127 '. «s than when the sphere is reflected with water. Theeffect of an air gap between the core and reflector variedwith Pu concentration and total nitrate. Experiments witha water-reflected sphere were also performed. (D.C.W.)

2S2M VARIAT10KAL CALCULATIONS OF THE EX-TRAPOLATION LENGTH OF SOME FUEL ELEMENTS,WITH APPLICATION TO THE THERMAL UTILIZATIONFACTOR. P. Basso, B. Mtmtagaini, and V. PUipaoli(AOIP Nucleare. MMan). EnsrgU Nuel. (Milan). 10: 1ST-4«(MiylM3). (fa English)

The application <>f ths results of a previous work COB-eerning ths extrapolation length of fuel and absorber d e -ments to the calculation of the thermal utilisation factorIs dismissed. Sone calcclcUoas were performed to checkthe accuracy of the various appraUmaUoaa used, fa ellcases examined, the method of homogeaislrf the elementproved to be sufficiently accurate, as far as the extrapo-lation length. Is concerned. When this quantity Is deter-mined, SB elementary formula gives explicitly the thermalutilisation factor (in the one-group theory).

{•tcgivdjffertatial Boltsmaaa SMMUOB Is Investigated, fti s showa that rendering the Lagmglaa staUeaarjr wMli re-epect to asMll changes la Uw directloMi fhsi aad atjels*

. direoUoaal fkm Is eqeivaleat to sotvlag the BoHsstua. sa latf)olrtBoltxaMM*iBailons. Toptss disomsssd iaahiss fceMe of varUMoaal weight fwwtloas. the laetaatoa e f l i u U i r ytows la the faaetkwal, the iaterpretaUos ef a varlsHei liepttBMUSi for a aoaseif-adljciat operator, sad the oeoeadvariattaa. It Is sfeowa that, for the geasral trial rsasMoaeassatbte as4 within a special restricted trial fsaettoa ea-esmble. the variatlonal method i s a saddle poiat prladple.The formalism developed Is applied to the angular«stoniapoljraomlalsof thedirectioMJfmz. (aria)

2M13 THE VARIATKMIAL METHOD APPLIED TOTHE MONOEHERGST1C BOLTZMANN EQUATION. PAKTn. Q. C. Pomraniog and M. Clark, Jr. (Massaohttsstisfast, of Tech., Canbridge). Nttel. Scl. Eag.. U : lH-«4tfime IMS).

The mo»ora«rgeac Megro-dtffereDtisi TloltTia— caw-Uoa with an asWtrary Bcsttering kernel i s traaelsrsMd toa self-adjoint form aad the eorrsepoadlAg Lagvaaglaa. writ-ton. R is atown that tUstnnsfcnBaUoa retails ia a Iocs ofOs contlimiey {neutron conservation) iaforawUoa osaUlaedby the Boltsmana eqtwUon. TU3c lnformattoa ia reoovemdby writtng tfca directional {lux as OR sum of aa evea aadodd function (ia angle) and conaldertag a snlf-adtowt La-graagiM for only one portion (even or odd) of the dtreoUonalflux. This procedure Is showa to be •quivaleat to aisaiatlaatea nonself-adjolatmsn from the Bottsmaaa operator. Far-ther, It i s showa that this self-sdjeint principle Is an ex-trsBnuo principle If the mean muaber of sscondartea percollision is less t h u one. This aelf-ad)olitt fonaolisat i sapplied to the angular expnslon of the dlreetioaml fhatwhich resulto ia an Improved diffusion theory. Numericalremits for the linear extrapolation distaaoe and dlffoaioacoefficient a n compared with the classical (P - 1 ) difhstoatheory, (auth)

A2M12 THE VARIATIONAL METHOD APPLIED TOTHE MONOENERGETIC BOLTXMANN EQUATION. PARTI. O. C. Pomraning and M. Clark, Jr. (Massachusettsfast, of Tech.. Cambridge). Nuel. Sol. Eag., If: 147-84(June .H9).

The varlaUonal method as applied to the monoenergette

A.B.C21437 (KAPL-2900-I0) REACTOR TECHNOLOGYREPORT NO. «J— PHYSICS. (Kaolls Atomic Power L*h.tScheneetaiSy, N. Y.». May 1M3. Contract W-31-1W ewg-

A geometrically simple homogeneous assembly (sHA,described herein) was dualsned and bulk to provMe data

37

Voluoe 17 (1963)

pets M erlttoellty and flux distributions tor comparison& each quantities obtained theoretically from few-group* •wttlgroup treatment*. Since She calculated teformattonMead* wry little oa geometric or other mathematicaljoirtir-t*— these comparisons trill be highly useful Ina* evslwitloa of group coMtaate produced f rom baste mi-,to*r4eta. For example, the diffusion length tor thermallaargy Matrons sail be meaeured or calculated from thso-rttteal mo£ela far the scattertag of tharmd aeatroae.pud agreement Istwaaa the fro method* ta reported herehr ms owes of room-temperature enter aad polyethylene;tawed, the agreement to food for water over • wide raneeef tomasrstsros, bat falls for hot polyethylaM. Ike r s -Kltvlty eoefflelaat of alumlaam has consistently been eal-gmtfsd to be larger thaa meeaured. TbeoreUcally derived—daiiiatinas lor the MUFT tape alumtaum data atlll faUto produce aa aoceptable eosfficiaat, but Improvements tato model tor reactor oalcttlattona havs reduced the dls-ajrsawsat from SO to *B%. Tas prsjpsrattofi of MulttgroopM B (am* • • that tor MUFT, above) fcss typically been awry tabortous affair, tavoMmj many weighted average*«f microscopic mselear data. To auks thto a faster andMire strictest prooeca, the aemputer progrttan CADAVERhM besa developed to perform required averaging for bow•iastta awl laelasUc scattering awl smooth capture of M -tnas. Further work to taereaae the accuracy of MUTT«as performed to provide reeoaaaoe aetf-akleldtag factorsmr aktmlmna aad slroo«Mm. I k s factors that are vappMsdicsawtfcr decreased reeoaaaoa abaorpUoa reaalttngtnm neoaaaee scattsrlag. The use of flax awl poweraMribatlOBS la reactor cores results from a atftustoa'mtsry appmek to aautroa tramtportc aa alteraatlre tottittsdeecribed. Eayloylag such directly meaningful aadMcrrelatad misailtlss u rsflsctwa aad tnasmlssioa prob-rtOttte* sad aatdtrBCttoari aeatroa cwtrsats la slab

y. mathsmatteal reUttoaa caa be established tor•rttioaljty aad tor dtoadvaatsge faetora. Pile oscillator>»8salmiaj ha*a bass recognised as beiag more seasWvs*ea static Methods tor dstsrwtolag reeoasace lategrtls.Aa sspaefailly seasttlv* oscillator systom was iastallsd lafee Thermal Test Reactor (TTR); preltmtaarr awaaurs-aHMrta w«^ hafnium samples Indicate that aa accuracy ofs.Wlf Is M^Uabte from data aocumulatloa over aoas•Haute period at : » w power, m i s pisa wry mlaafi samples or on vsry weakly absorbing ma-terials. Oa She theoretical side, devekmmeat was begun oa• arogram, MOST, which serves to compare experimentaltrees-aectioa data (mlcroseopic) with the resuMs of quan-tum meehanleal eateulatloiw aad to provide vshws ef theo-fstleal aaelear paraiaetora v*lc)i yield tost Ms to the•moored data. A description Is given of a technique•hereby MOST caa alao be applied to the aoMloa of rawer( •ami aad complicated equations. Currant best values for

fsst-Mutron scattering croaa-sections in aluminum andHrconhim, and resonance pci J.. .stere in zirconium, were•stabllabed through theoretical calculations and re»lo* of-•eent data. The new Information thus developed was In-«orporated in MUFT data tapes. Eattmites on the extent to**ieh calculated critlcality valuea arc affected by chances"> wwlsar data ware obtaln»d, Reactivity changea wereI*6«la3ed for asts of calculations for w»t«r-mortera'«l »y»-*sau eoatelnlng either airconlum or aluminum; thesa*ssages were produced by altering elastic or tnelrittlc scat-'•riag la the metal. Ten percent changes in large portlona°* * e data were observed to Introduce rou^iily a CSX°»Uge to k. The ulxth edltktn of the chaH of the NHCIMOS•»* IssHsd. The deUllocI liohuvlor of neutron dlatrlbutbmii,** M ar3y eml/or spatial ite|>endi;nce, w»a examined with anumber of techniques. Special distributions of thermal neu-tron* la hydrogen-moderated fuel-bearing assemblies weremeasured and calculated for a variety of mixtures and tem-perature conditions. Vsry carefully performed choppermeasurement* were calculated aatlafactorily with a boundhydrogen kernel for systems wtlh low absorption ratea. Forsystems more highly loaded with fuel, a free hydrogenkernel produces best agreement with experiment; thia un-sattsfaetory situstlon is not understood. Space-dependentspectra st cpithermal enargtea can be calculated with aucbprograms as P1HG or TET, but only at rather high coat Incomputer time. Work was done to learn whether the appll-sattoa of vsrlsttonal techniques to Pl-typ* equations couldrsdaos conputer time without loss la accuracy. TheMOSKBAT program, developed la u s e s stadias, has aotbees proven to be e satisfactory successor, sltaoagh it isassrly as good as P1MG. Several latttoss were examinedtndetaU with both VIUQ aad TIT. to determine whetherthe low-order angular approxtmatioa la the former pro-duced slfaWeaat errors la the sptthsrmal spectra (andgroup (Mestanta). Although tnfiaite-meiUam spectral cal-e«lsttoas would i s insoenmte for the latUoss eoaaidared,Iks reeatts wMh a simple PI aawotropy proved to be suffi-ciently dose to the double PS available from TET results.Several formulatkms were dsvelopsd tor Iks calomlattoa ofapproTrtmata niacknees coeffJclente. Baagtng from a oiffu-stoa theory approach to uee of Integral theory variations!sxpressioM tor slab transmlssloa, refleetioa aad absorp-Vsm, flw mstkods wars relatively poor or good; Iks latteryielded results that differed by 1% or loss from exact val-• M tor s slab of say selected optical thickness aad soettsr-iag proparttea. CatoateMoas aad experiments are reportedtor the soetterlag cross seetioa, sad for the energy distri-

Agrstmsat bttwsaa the msasursd sad calculated spectraldaw at late low temperature adds ooafldoMW la las thso-rettoal awdsl employed for the scatteriaj.. Ths analysis ofMlewiag*>wn«xperimeateoosd«c«

38

Volume 17 (1963)

trona has long been viewed as a passible way of verifyingtheoretical scattering kernels. Only recently, however,have sufficiently accurate experiments been performedto bolster this view. Although the results available are notyet conclusive, those presented here show improved agree-ment between theory and experiment. A very accurate y«tsimple method of determining the absorption cesffictentand reflected angular nW- Station for a seml-tnfinlte planeslab with an arbitrary ratio or scattering; to absorption isdescribed. Results obtained with this method show a non-linear variation of the emergent flux with the cosine of theancle of emergence. When this is a strong effect, even adouble PI approximation may lead to an unsatisfactory fluxdescription. A varlatlonal method for nonlinear reactorkinetic problems Is reported, (auth)

E.B' T057 A VARIATIONAL PROCEDURE FOR CALCU-LATING FAST GROUP CONSTANTS. P. A. Ombrellaroand F. D. Federighi (Knolls Atomic Power Lab., Schenec-tady, N. Y.). Nud. Scl. Eng., 16: 343-56(Aug, S.963).

A variatlonal procedure for calculating fast energy fewgroup constants is described. For a given medium, themethod permits one to express the flux and current solu-tions of the Boltznumn equation, treated according to P - lalowlng down theory in each group of a few group scheme,as a linear combination of base flux apectra to obtain toegroup flux and as a linear combination o! base currentspectra to obtain the group current. The coefficient! forcombining the base spectra are provided by the theory anddepend only on the concentrations of the component ele-ments of the medium. Once the flux and current spectrain each group are calculated, the group constants for themedium can be easily calculated from base flux apectiaweighted library microscopic cross sections. Group con-stants calculated in this manner agree well with those ob-tained from the MUFT V program, (auth)

F3W11 A STUDY OF THE VARIATIONAL PRINCIPLE)OF NUCLEAR REACTOR PHYSICS. Paul Wesley Dicknon.Jr., Thesis, Raleigh, N. C , North Carolina State Coll.,1M2. 203p.

The application of the two main usts of the variation*!principle to reactor physics is explored. The two uses arcfirst, to derive exact equations of motion from a stationaryLagranglan and to study the Implication* of the Lagranglii^egarding constants of the motion, and second, to obtain ip-proximate solutions to non-linear differential equations.

Attention is restricted to diffusion theory, and hence theremits are applicable only to reactors for which diffusiontheory is adequate. Lagrangtans for one- and two-groupdiffusion theory as well as one for the one-group theorydeluding delayed neutron effects are presented and provedto be correct. From these Lagranglans the correspondingreactor Hamiltonians are found and uaed to determine the"conjugate momenta" and some constant* of the motion forreactor diffusion theory. The Implications or Interpreta-tions resultant from these constants and the conjugatemomenta m* discussed. The Hamilton-Jacobi equationsfor the one-groujt Hamlltonian were derived and shown to

• yield the expected result*. Time-dependent perturbationtheory In. shown to be applicable to reactor problems todetermine flux functions a* a function of time. V*riaUo»ltheory Is compared to the perturbation theory when car-ried, to higher orders for the one-group case. Two-groupvarlattonal theory ia discussed in general terms. A modi-fied form of vartadonal theory which will delineate a lovrrbound to an eigenvalue waa applied In general form to re-actor' problems. The standard upper bound varlationul the-ory and its lower bound form were then applied, using aone-parameter trl.il function, to many specific cased who*?upper bounds are known from perturbation theory. Two-parameter ons-group variatlonal theory was considered.It was found that a two-parameter trial function yieldsquite accurate results even for heavily loaded reactors,IT the poison Is not too greatly concentrated. The sprrlnlcase of the reflected reactor was considered, and a cor-rection to the perturbation theory determination of tt/kcwaa derived. A two-purameU-r variatlonal technique »•'*applied to thin name probiom and the corrc-clion rcuuir"1

for reflected reactors la discussed. Two nu-tluxls of attu*.using the fourth-order differentia! equation and the iii-ilrt*equation, on the two-group vartatkmal problem are pre-sented. Solutions for several one- are! two-parameter trial(uactlons were determined for general poison distributions.An analytical solution for a perturbed reactor system wasobtained, and the exact answer was compared with the var-Ittlonal answers for differential trial functions to detnon-itrate the validity of the method. (Diaaertation Abatr.)

39

VOIUM IS (1962)

F47* (UC:U.-Trnn8-l6!» ON SOME VAIUATIONALPHINCIPLKS »; THE THEORY OF OPKIIATIONAL EQUA-TIONS. M. M. Vainlicrg. Trannlateti by H. P. Kramer(Univ. of California Radiation Lab., Herkelcy) from U»»pckhi Mat. N«uk, 7: No. 2. 197-300(1952). 9p.

Theorem A was derived as a apcciul caae of tbe Lyuster-nlk proposition for conditional extremal* of functtonals, fortwo real fuactionala defined on a real Banach apace S anddlfferentlable at the point x,. Theorem B la derived la tinreal Htlbert space H for a given operator. F(30 «• frad fix),and a completely continuous operator. Ate), showing thatthe equation: fix --- AJF6c), has in each sphere || x || s r oftbe space H at least two different solutions corresponding toreal values of tbe parameter /», where p is a real number.Theorem C is derived as » proof for Theorem B for thecaae when the completely continuous operator Is self-adjoint. It Is noted that an application of Theorems B andC results in the existence theorems for characteristic func-tions for integral equations of the Hammerstein sod Ltch-tenatera type. (3.O.G.)

G743 ASYMETRIC CORE EXPERIMENTS AND THEIRANALYSIS BY A THREE-DIMENSIONAL CODE AND TWOFLUX SYNTHESIS TECHNIQUES. R. J. Rosiiberry andT, F. Ruane (Knolls Atomic Power Lab., Sohfinsctady,K. Y) . Trans. Am. Nuclear Soc., 4: No. 2. 286-<(Nov.

G.B741 SYNTHESIS OF THREE-DIMENSIONAL FLUXSHAPES. S. Kaplan (WesUnghouM Electric Corp., Pitts-burgh)- Trans. Am. Nuclear Soc., 4: No. 2, M7-6(Nov.1961).

3MS (KAPL-M-NCF-3) NON-LINEAR EQUATIONVAIUAT1ONAL METHOD. N. C. FrancU (Knolls AtomicPower Lab., N. Mox.). July 10, 13C1. Contract W-31-10B-Eng-52. op.

A nonlinear variatlonal method Is presented fur the in-BomogeMous ease of power reactor physics problem*.(D.t-C.)

FV57S (HW-68284) THE VARIATIONAL METHODAND REACTOR PHYSICS. C. W. Llndenmeier (GeneralElsstric Co. Hanford Atomic Products Operation, Rich-Sand, Wash.). Jan. 28, 1961. Contract [ATH5-D-1350].18p.

Some of the less familiar mathematical methods requiredin developing a variations! fortwlism for reactors are de-sorlbed and applied to obtain t)« muiti-group approxima-tion. Linear spaces, adjoint spaces, linear operator, adjointoperators, Green's function, physical interpretation of theadjoint, variatlonal methods, variatlonal functional in thediffusion, trial functions, the multi-group approximation,And overlapping and non-overlapping trial spectra are dis-cussed. (M.C.G.)

F,CMI3 (KAPL-2000-16) REACTOR TECHNOLOGYREPORT NO. 19—PHYSICS. (Knolls Atomic Power Lab.,Schenactady, N. Y.). Dec. 1961. Contract W-31-I09-Eog-62. 192p.

Nuclear Cross Sections. Tbe resonance integrals of Mn,Hf, aed Nb were measured. The mass spectrometer fila-ment enrichment technique was used to determine the8m1** effective activation cross section. The Ow(n.p) crosssection wae measured as a Junction of neutron eneirgy from12.6 to 16.6 Mev. The Fu(p,av} motion was used to cali-brate a l-Mev Cockcroft-Waiton accelerator used to ac-oalarate deuterons far producing neutrons. ABACUS, acomputer program ioz eroas-section calculations, is de-scribed. ABACUS was used to calculate the optical! modelparameters of toe anguSar distributions of neutrons scat-tered from Zr" at 0.25 to 7 Mev. Method* are describedfor calculating elastic and inelastic differential cross sec-tions, angular momentum coupling coefficients, and com-pound nucleus effects in neutron capture. Results are givenfor the development of a satii factory potential 'unction inthe direct interaction model. Photor.euUon production ?al-cuUtiobs for Be* and C u WOITB extended to an energy rangeEy «• 3 Mev. Computational work on the croon section ofthe thermal neutron scattering by water and polyethylenets described. Neutron Spectra and Fluxes. The thermalneutron scattering kernels were used in calculating neutronspectra in both infinite homogenaous media and latUces.Computer programs for the calculation of neutroa epeclraa n described. e.g.. TET, TRANSWAKRUM, and TRAM.The application ti variatlontl Uchnlque* to linear systemsla discussed. Few-Group Parametera: Crltlcallty. A

40

Volume 16 (1962)

two-mode variationai procedure was developed for calcu-lating fast-group diffusion theory parameters. In the vsrl-stional calculation of thermal spectra, it is shown that theadvantages of symmetrizatlon of the scattering kernel areoil. Methods are given for compilation of spectrum-everaged cross sections. The temperature dependenceoi the thermal diffusion length in paraffin was measuredand used to thrive transport cross sections. The crosssections for mission products were examined in detail withrespect to their poisoning effect on reactor cores, and it isconcluded that, while uncertainties in current data givesmall urrore, the assumption that stable products areformed directly in the fission process can lead to slgn'N-oant errors. Variation calculation of the reactivity k isdiscussed. Reactor Kinetics. The method for calculatingthe effect of neutron population fluctuations on rouotordesign was extended to account for the actual nonzerolifetime of prompt neutrons. A seml-dircot variationaimethod Is given for solving the spnee-timo neutron multi-group kinetic equations for reactor analysla. (D.L.C.)

G,B12413 MULTICHANNEL FLUX SYNTHESIS. £. L.Wachspress, fi. D. Burgess, and 3. Baron (Knolls AtomicPower Lab., Schenectady, N. V.). Nuclear Sci. and Eng.,12: 381-9(Mar. 1962).

A procedure is described for calculatis; neutron fluxesat nodes of a three dimensional grid with few points, inplanes perpendicular to & selected axis. Difference equa-tions relating fluxes within each plane are determined inadvance from detailed two-dimensional studies. This"multichannel synthesis" model may be used for param-eter studies an& dynamic analysis. Comparison of a rep-resentative calculation with results of a detailed three-dimensional computation is given, (auth)

A VAIUATIONAL PRINCIPLE FOR NONLINEARSYSTEMS. Jeffery Lewins (Staff Coll., Camberley, Surrey,Eng.). Nuclear del. and Eng., 12: 10-14<Jan. 1962).

The equations describing a reactor system are «sa>6-times nonlinear and do not admit a solution for the neutrondensity that is separable into a function of time only Slid afunction of the remaining variables. An appropriate varltn-tlonal principle is given by demanding that the calculationof the observable nature of the reactor Is Insensitive to iiw

value employed for the density, thus obtaining an equationfor the optimum distribution of detectors to meamsrc tbaobservable behavior. This optimum weighting function Unot Identical with the conventional adjoint function or Im-portance in the nonlinear range but the conventional treat-ment of linear systems Is found to be a special ca*e of n ,genera! principle. It is shown that the approximate treat-ment of nonlinear pystems as eigenvalue systems is funda-mentally unsound, (auth)

E.BIMQ9 DIFFUSION APPROXIMATION AND VAIUA-TlliNAL EXPRESSION FOR THE THERMAL NEUTRON vI>IST!UBUTION IN SPACE AND ENERGY. Akinao Khimixu(Nippon Atomic Industry Group Co. Ltd., (Japan)). NipponISniahlryoku Gakkaislii, •!: lGl-fi(M:tr. iar,2). (In English)

The diffusion approximation for the motion nf neutronswhich move in the energy space as well as in the ordinary•Ifire is derived from the dilfuaion approximation for thenmtlfin of mono-energetic neutrons. Under this approxi.iM-""ii, ths varintional expression for the space— energy disiri-Imlliin of thermal neutrons is derived. The functional in thisrMirrsslon becomes minimiini for the actual ilistrilnilion.The derived viirialimial method is applied t» tin- problem"I (he thermal neutron s|xctrum in a hflcrogcncnus l.iltice.Imilli)

A19733 AN EXPRESSION FOR THE NEUTRON BLACK-NESS OF A FUEL ROD AFTER LONG IRRADIATION.Hisao Yamakoshi (Transportation Technical Research Inst.,(Japan)). Nippon Genshiryoku Gakkaiphl. 4: 244-50(Apr.1962). On Japanese)

A varlatiorial expression is derived for the neutronblackness ff, of a fuel rod which has inhoraogeneous distri-bution of nuclear elements depending on the distance fromits axis. The following assumptions are made: scatteringwithin the rod is isotropic in the coordinate system anddoes not change the energy of an entering neutron; theincident neutron current is uniform on the surface of therod and is isotropic in solid angle; and monochromatictreatment is allowed. The evaluation of the variationaiexpression for H i>> jslng the solution of transport equationfor neutrons in the fuel rod as a trial function shows thatwhen a lot of P"13' accumulates in the vicinity of the rodsurface and Vvi remains near the axis of the rod, theblackness 0 in the energy 0.3 ev is Increased much moredue to the scattering than in the case of the homogenizedrod. The valueii of P for the homogenized rod are com-pared. Both results agree fairly well as the ratio of ab-sorption to scattering cross section increases, (auth)

41

Volume 16 (1962)

a1WI5 SOME NEW JMKT1IOUS OF FLUX SYNTHESIS.S. Kaplasi (Wthiu^houtH! Klcctric Corp., PltUburicM.Nuclear flei. and En*., 13: 22-31 (May 2W2). (WAPD-T-1374)

A varMtkMal principle la applied to develop a method foreoaatntsttaK approximate solution* to the lhm.ii dimensional,few grcup dilfiwlon equations UNIIMC only the solutions ofone and twa dimciwionul prut>ti<mii. Thnw additional re-lated molted* art.' described and a numerical example pre-sented which Is rcpEvwsntstiyc of several that were studied.It i« feuad that the- new mvthiHla KIVO conHkJcrulily Improvedri'MiltH compared witli I!H- cunvvnlicHuil HynUicutx nn<tliod.fcuih)

FZ4372 GENERALIZED VARIATIONAL METHOD.M. D. Kostin and H. Brook/* (Harvard Univ.. Cambridge,M*M.}. Trans. Am. Nucltar 8oc., S: No. 1, 41Uun*1MJ).

B.EXMSt A VAIUAT5ONAL PROCEDURE FOR CAI.CU-LATING FAST GROUP DIFFUSION THEORY PARAM-ETERS. P. A. Ombrtllaro and F. D. Foderigal (KnollxAtonlo Power Lab., l\ck»mc>julj, tt. Y.). Traaa. Am.Naclaar Boo., 8: No, I, «3-4UuDe 1968).

CWT$ THE SPACE-TIME NEUTRON KINETIC EQUA-TIONS OBTAINED BY THE SEM1DIRECT VARIATiONALm-nXfo. D. E. Doituerty <Knoll« Atomic Pow«r Lab.,S-hm«ta<Jy. K. Y.) and C. N. 8b«n. Nnelasr Sei. awE EftJ..l;i: 141-IWww lt62S.

Vuitg a a«midircct variattonal nwtkod, the time-diveadcnt ooefflelcnta of a modal expansion of U* nevtroalli»ea are (lven by the Eluler-Laitrange equattona obialnedfront a variational principle for the multisroup kiiMttc«|usUont. m order to avoid the afficult taak of dotermia-inK orthogonal eigenfunctloM for a modnl expanilon la arontpje* eooE>«tr7, an approximate aolutlon of the kineticftjMtSen* by lae method of the Green'a function remita In* ael of readily calculated ipace modes. Theae nKrlea cani iw Sw adapted to pcfturbatlona In the diffusion parameter•imit which a priori information la available, viun)

AM343 VARIATIONAL MONTZ OARLO CALCULA-TIONS OF FLUX DEPRESSION IK FOILS. F. R. Nakaeht,•i. Coldcteln, and M. II. Kaloa iCoIumbla Univ., Not- YorkMft VnStcd Nuclear Coi?., White Plains, N. Y.). Trans.Am. Nuclear Soc., S: No. 1. M-30une IMS).

. _ A COMPARISON OF A SELF-ADJOINT VARIA-TIONAL MKTIIOD TO M-GHOUP THERMAL SPECTRUMSALCULATIOKS OF IIETKKOGENEOUS ARRAYS. S. L.•Mfler <WrstlaghoMae Wecirlc Corp.. Pittabu«sW. Trass.Aw. NacJear Soc., B: No. 1, 9T-»tfiw It61).

C24843 8YKTHE8HI APPROXIMATIONS IN THE TIMEDIRECTION. 1. A. Biiwfek, A. F. Henry, and S. KaplanOVeaUnjbaua* Eleotrlc Corp., Pittsburgh). Trans. Am.NueUar Soo., 8: No. 1. M7-8Uua« 1M2).

B.C34M3 (MND-C-3SM-3) APWRC-SYNFAR-92, A PIAND D3N THEORY. FORTRAN-U CODE FOR STATIC ANDDYNAMIC SYNTHESIS OP TWO-DIMENSIONAL FLUX ANDREACTIVITY. T, M. OISM (Marito Co. Nuelear Wv.,BalUmore). M&ylMI, Contract (AT(3«-i)-M31i. T?p.

A computer code, APWKC-8YNFAR-0I, was proframmedla FORTRAr? H for syntihesls compaction of static flux andreactivity, or of stable period and corresponding flux shapela XY or RX geometry <m the IBM TOW. It also allowsafreet eowpatatlon ef »«* M M quaatitlos 5n eiie-eiimea-tkwal sBksrical gsonKtry. RwMmg time la 12 mfeMrtea orlass. (W.C.O.)

{MND-C-S«MH) APWRC-SYBURN, ArORTRAK-II PROGRAM FOR SYKTHtSlVCKD TWO-DIMEN-SIONAL n OR DSN BUKNUP CALCULATIONS. E. A.Calbetk and T. M. Olssn (Martin Co. Nuclear DSv..iaHHwm), JuaelMt. Contract ATOC-t)-«31. tip,

Ti» reactor jpkysles atHtnm code, APWRC-fSYBURN.was dushmsd fsr the IBM fWO mxi wrttlmi in FORTRAN>9I.k was develegwd for oae-dlisaeMioaal regLmmim dr toi-•lrrahrlae determmatiea «f iaotes* canemtrateeM durli*rtaoter twn*», ia«hMlwi eftecia td rod or other centre!»lgea»al»a varlatleii. wrmvmm core »ve>ra«trt r»*aS«*s.starta far ariMie«iett *tyMhc*iiUMl axial OWMW arob-fem. lystoal zmmkm ttm» in it t» 1 Mlim«*«. (M.C.O.)

42

Volume 16 (1962)

A

2$$49 THE REACTIVITY OF A CENTRAL AW GAPIH A BARE REACTOR. H. Haeggblom (AktlebolagH Atom-energl, Sweden). p.85-S4 of "Physics of Fast and inter-mediate Reactors. Vol. II." Vienna, Intatnatlo.ua AtomicEnergy Agsncy, 1982. On English?

Two methods a n presented lor calculation of the reac-Civity equivalence of a central gap in a bara reactor withrectangular croM section. The Krat ia a perturbation the-ory; the second ia on application of variation caloulua. 'Huteigenvalue la aolvesi for the tranaport equation In integraltorn. The principal difference between the two DMthoda Uin the assumption for the flax shape. In the perturbationtheory, it i* aaaumed that the flux t* unperturbed tad hatdie form A coa » x/L. The reactivity ac a function of thegap wldch ia expressed in closed form in one and two energygroup*, iMit the expression la valid only for reactivitychanges lest than about 8%. The variation catculua la de-veloped only for one energy group, but the accuracy is ap-proximately independent of the gap width. Tbi» in obtainedby varying the expression for the flux in order to optimisethe functional integral. The results are compared withthose obtained by ChemieSi and Kaplan by diffusion theoret-ical mntbed*. both)

om IB f jaotora far design awps*i* ere *~**ertaid. TheIjtrtfctd* relate the nature aad amatlrt faatere* ef themtlarli* fcaraal to the oaaraeterttttoa ef the a—Uia dto-

>•--•' —'leering rtgaatleafw. JafaimaMiatonpWVa MJWX M i N N i Mat WMT

t ere walaaUd by thto awthed. The first anWeanshavtor <af taa i

> a atoaaaMwuty to aa toftol

— var

(to eai#Hearahia preblem far tta eatewlatiea ef tte deeay <IK MM Han dsnadiiat dtotrftailM. The eeetad s r * l im to<B» talwiiattoa 9t the aaeetnim ef nmlrias to aa toJteito•wdhna geaaratad ay a uniform eeuroe j f Oaatoa neatreae,ttt M aa* in wilaalaiiag thermal groa» isnstials far ipSKisd MMdcto of a raaator. The

SMIKT to a ayalMi wMk

nUreaetordeatgii. (N.W.K.)

E.KM4W 4WAPD-BT-2SU-24)) THE DISCRIPTIOH 0?THE THERMAL NEUTRON SPATIALLY DEPlNDKJfTSPECTMOM BY MEANS OF VARIAT10NAL PMNOPLB.A. jr. Busllk (WeaHng^ouse Electric Corp. Bettls AtoaiiePower Lab., Plttebargh).

The appUcnUon of the calculus of variation* to thedetermination of the thermal neutron spaoe aad energydistribution ia dMcribcd, and it la ahownhow thin absot*-ing region* treated by blsckness theory can be iaehtdMi*the method. The general method is then applied to theproblem of s s absorber adjacent to a large feel bearingregion. Results are compared with the M nwSUgroupsolution of (he thermal space energy probtom by the dlgMcomputer code SLOP-1. In addition, a wanaer of taeetpfrating these resultB Into a one-group design method iapresented, (auth)

2f7l3 (TrD-t27M» VARIATIONAL MF.TIIOW8 IN NEU-TRON THRRMAUZAT1ON. I>, B. telewut (Kaelle AtomicPower Latt>., He(Nmectady, N, Y.). Mny 1MI. Coatraet

. - . . : . ,Mitt (WAPD-T-HSt> A COMPARSOM OF A WKIX-WlOINT VARIATIONAL METHOD TO at-OKOVP THKM-^LSPEP C"WTEM8. 8. L. flMiflar (W»«U«a*Wf t KtoMrfe Cety."•lie Atomki Power 1/A., Pittobai«h). Apr. \9U. Om-*«*«tAT(l]-t)-0RN-t4. Mp. .-.-

A aelf-MlJoInt vartalioiwt prlnotote retotwl to Uu< ihrriinl,*»«>»• etwunl wptfrtruw uraMaM by WeHi yarn toatoi ."Whit t M-craip therawl icdw wwl a isw-tiwwp wwhi torMeat oPM-dinctHwloMl hctoragcMMHW Mrray«. A dwiarlpUnt .«T the theoretical atetMd to giv«a toetedtog ewne efc~servatiOM • ( the dttferwwM beWow la* aalf ai jsl imethod «T »oeilk aad Uw Mxwelf-ad|etet•erted by Ctiam* and rederighi. Aparmatetera, actiyatto* ahapet, aid 1fraottoM are presented far three tjyea «farrays: abwertwr-fttaf aeito, akeorhoir-niMand abeerher-fcxl-neaftisl edte. <toM eetm sswjarliaiin «f tbe".MiMd wnater deaslty mothed to the vartotteaal atedst arealM iaoMad. ReemiU iwMeMed the atUMy tad pMaf ef theaaosies ef the aaif-adjstot vartaUoaal priMtoio to |!ridtottog the iatnertam a«cl«er penmetor* «f atotl-Dsiuel-abeerber soaihlnatleae at »ethItmpsratotree. (M.C.O.)

fh»t

VariaUoMl nwiltwU for cnlculaUag Uwnttal INK dtotrlbu-

Volxme 16 (1962)

SMJW (WAPD-T-UM» SYNTH ESB APPHOXIKA-TKXiS IN THE TIME DIRECTIONS. J. A. Bawick, A. F.Haarjr. and S. Kaplan <W»aUagfceiM» Elactrlc Corp.Saetia Atomic P»w«r LA.. PtttabtMghh Apr. INS. Con-tract AT<U-U-G«a-M. Up.

ApplieatiOB of an approKi*alloa pzvoadan In wfcicfc tharotation to appiMimatod by • liasar eonUnaUon ot luoc-U«ea whiek a n known, to Master prabtoiM involviaf UMU S K varlabt* (s dMerlbad, Tk* •pprwdMM Mtatto* hatma tono •tc^^.t) - T,fc) * ( UtfAH . . . . immjw*)wktn •fcjr.d « N spatial rt»pM appmprial* So varimwran M «f t. Dl«atrmtkiu a n givaa comaponcKaf to

O.R.DJ

F3127} <NP-ll?S4(yoi.ll>(Sw!e.IV)) «OME RKMAMCSON TUB USE OF VAUfATKWAl. METflOOl IN REACTOREXPCIUMKNTATION. llalgtt ChrtotoMtm (ChrMlaaMlfttvim-DK Instllult for Vl<k-rwtka|)u oet AndHfrlhi'l, nvrRt^Hnut. IV of ADVANCED C(»UH«K ON JN-COKK tNVTRI}.MENTATION KOil WATKK iStMll.KI) KKACTDK8, OR.UANI/.KII UV THK NKTIIKIU.ANlW-NOHWKltlAN KEAC*TOIt .SCHOOL AT BNSTITUTT F<Ht ATOMKNKRQI,KJKU.KH, NCNtWAY, 2i»l AUGUST- tMl tIKPTEMBEll1361. VOMJMKU. 4Tp.

II IM notoil thai traiwfcr ftiiwtioiw an.' itaod la relate by*ami output in ayateaw wick aa ruaoi**1*. T!M IMMIO MTMboliind tran»Iur te«Uo« Utchulqwa ara r«vlew»d nad UM><Kg*»rUMc« oC tratwktat cooddkins in rouoUtr «xp«riaw»>taUtm in po(n(»u oat. The theorottoal (duadathM for tatvavioitH oxpcrimontal tvohnlquM ia pwiaaateJ. Planwalau.

procadaraa ara iweludod. (J.H.D.)

44

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Abagyan, A. A.22:13474.

Adaas, C. H.22-28611, 34932, 37717; 22:32825.

Albartonl, S.22:5086.1; 22:22387.

Alcouff*. X. I.21:24011.

Aastcr, H.12:27407, 27409.

Argomte Rational Laboratory2J:49651; 22:22653; 21:30091.

Arthurs, A. M.21:7621, 21395.

Babb, A,21:1193.

Baron, S.1£:12413.

Basso, P.12.128230.

Baeslls-ltorthwast32:12139, 27481.

Bsar, J. L. .22.:16328i 21:7911.

Bccbar, M.23:4430, 4443; 21:6157, 18606; 2fi:M33; 12:2713,3563, 14803, 21638, 33207; 12:28537.

Bavisk, J. A.21:10132; li*15154j 14:24543, 30029.

Bloabarg, F. E.22(32800.

Sogax, C. F.2£:12368.

Bolstad, J. W.2£?34490.

Bows, B.22118617.

Braan, Et. J.J£:5436.

Br«lt««hub*r, L.22:22399.

Brooks, I .21:17281; 12t8383; lfc:24372.

Bttrgsss, X. D,14812413.

•usllfc, A. J.22:19133, 23999, 37716} 2it2514O, 33139; 21<7639,38558; 2£:34954, 36M3; 16:28416.

Ca*«U, V. %.U J 1 5 1 4 8 .

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Ghatawnt, J. M.21:35554.

Chrlsemsen, 3 .lfi:31273.

Clancy, B. I .21:35697.

Clark, M., Jr.lft:994, 18358, 28532, 40466; H»28412. 28412.

Clarka, H. C21:31950.

Cohan, 8. C.23.137742; 22:32981.

Colbath, IS. A.1&2444

CollUr, G.19:27397.

CurUe, M. J . , Jr.1£:28968.

Iteltch, t. B.22:14309; 21S38209; 12:7659.

Daltoo, G. X.12129206.

Daoofsky, K.21:40609.

DSYIS, J. A.21t254O4, 29841; 2P.t2978, 2990, 6264, 301W, 34493.34495,40360. <

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Carter, V,21:40*09.

s t , J.22:1586.

01ekM«, P. V., Jr.UtM511.

SoMglMrty, 0. B.11:21317; lft:22975.

MUata, A. A.21:13474,

OMWM, W. A.21:12140.

Dwlv*4i, 8. X.22:18598.

Brykalov, A. R.29.:U369.

r«4«riakt, r. n.17.t5298, 38057j 14:24400. :..

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Galligant, I .2£:17656; 12:12455, 47945.

Gatt, ?, P.J£:3121.

Galbard, I . M.22:4431.

Coldatalttc H.14:24362.

Goldctain, t .22:7412.

Graablar, p.21:12140.

Gu*ron, B.2£:18608.

Gyoray, G. L.Si : 6012.

Haaggbloa, H.14:26540.

Haaford Atoaic Produces Operation12:24037.

Hanwn, I . C.12:3562.

Bauaknacht, D. P.22:3913, 52443.

Bank, T. L22:34494.

Bala, P.£1:38682.

Halala, 6.22.17626.

Bamry, A. P.12:40424; I£t24543, 30029.

Haaryaon, B. B,2£:34491.

Batrick, D.22:47302.

Butehi.na, B. A.

Jaaoa, S. H.21:36205.

Jauho, P.22:39*36.

Janklna,, J. D.22:U3O«; 21(3*209.

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22:13474.Kick, C. A.

21:12149.Rell«y, H. D.

22:6012; 21:12140.liiaalar, 6.

22:4440, 52462.Vhroaov, V. V.21:21731.

Ktllan, P.1£:31654.

Kladnik, K.12:33196; li:32721.

far'il* Atoaie Powar Laboratory12:28437; lfi:9653.

Ksbayaahl, K.22:14316.

Koahlar, H. B.22:30852.

Ksstln, M. D.21:17281; 12:8383; 14:24372.

Kutaln, A. M.21>21731.

Uneaflald^ M. J.22:45129.

Lathrcp, K. D.21:40364.

t«lghton, 8. C.22:49654.

Laonard, J. H.2W

, 16329, 32033, 32S30, 4UM; 2itJUJ132,» l , 29404, 2H41. 31205, 3UM, M535, « W 4 jtlLM, 2990, JW31, 6S64. J4492, 34495, 3449*,53, 34967, 40360} jftt3120, UlAi, W154, 4»I» t

, J.21:1X93; 22:16059, 24714; 19_:1848; 14:14254.

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bieo, V.22:26235.

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22:32630.

M*Mloni, C.22:50863; 2ft:2a387j iftslSlSl.

Malakfaof, T.22:29654.

Mwsoll*, f. a.21<33195O.

Karlowi, 0. J.12:5436; l£i3220, 15146, 15154.

NePadaVM, J. « .22:34942.

NarrlWB, P. C.12:3562.

46

Mlka, J. R.18:30701.

Montagni.nl, B.J2»28230.

Moor*, X. A.22:37742.

Mowtry, A. L., Jr.11:3745, 11972.

Mueller, K.-H.22:18068; 12:3196, 31430; 12:15468, 28966.

Murlcy, T. B.22.:32»26.

Murray, R. L.17.83745, 11972.

Kakache, F. R.lg:24362.

«iiC«l«m, M.22:14303; 21:29841, 38203; 2£(34496, 49360.

Nattlson, M.22:4431.

Nathtshlatgcr, P.22:11348.

Neuhold, R. J.22:37726.

Hguyan-Kgoe, B.22:52103.

Nozawa, R.21:27663.

Ohanion, M, J.12:29206.

Olsen, T. M,16.:24943, 24944.

Oabrallaro, ?. A.12:28536, 32714, 42597, 42605; H»5298, 38057;l&:24400.

Orlov, V. V.22:13474.

Orphan, V.12:29529.

Ott, K. 0.22:37726.

fatrov, I. 8.22(13474.

Fiarpaolf, V.11:28230.

mtarle, T. A.22:1140.

PoMraulnc, G. C.22:4445, 43205, 47296; 21>33202; 21:21383, 35607,40364, 40365; 22:21957, 24302, 41409; 12:27408;U s 9 W . 18859, 24573, 28532, 40466; 11:28412,28413.

Poraehlng, T. A.22:32830.

Fnpfco, V. Ta.22tl3474.

taaalnaa, R. B.12s29206.

, R. B.

lacaotlaar Tolytacfentc Institut*2 2 l 3 22:14296.

Ribarle, H.21:14310.

Ri*M, J. W.12:27397; 12:28539.

RsblnMn, J. C.21t46656.

RoUstB, M. A.21>37732.

ROM, P. P.22:3913, 52443.

ReMbarxy, 1. J.14:743.

Kuans, T. F.16:743.

Rui«, A. L.32

Rutbarfo£<d, C. H.21:12368.

Rydin, R. A.22.1^882, 6146, 32825.

Sarglt, D. A.23.137742; 22:32981.

Sehaaffltr, H.22:39919, 39920; 12:17234, 17235.

Schralner, S. *22*47295; 29.(2980.

StlMgue, ». S.22*7295; 20.(2979, 2980, 34490, 34491? 12:3547;14tU80; 11:3821,, 4043; 1£29713

nMDf G* Mm16.:2297S.

Shtadtu. A.1&IU409.

SbnSlK, 8. L.1£:2436B, 30028.

Slcaarrv, 1. 8.£1(21731.

flttltk, F. I .

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*; D. 6.£3.(37732.

Stoirrsr, P.jtl(33554,

8ti«fcwr, I . A.j|2(26109.

47 V

Siida, B.22:42558.

SsrtbsX, H. S.21:28617.

Saisligovrtki, 321:47302.

Tavel, M. A.21:35607.

Tolvanen, 7.2£:30332.

Trawlli , A.21:38682.

Traylor, R. C.22:^9654.

Turlay, R. E.11:38210.

turner, S. B.

Ung«r, H.Z0;8353.

Valnb«rg, M. M.

Vaughan, E. D.21:3913, 13405, 52443.

Vtawrua, B. R.22:16328; 21:7911.

Vlgtlottl, A. J.3

Waehepcasa, E. L.23.:4442 > 23998; 2J.: 174.53 j 2&8G38; 12:21638;1842928 lg12413

rc J.22:11348.

W«inr«lch, W. A.

Willia«»cci, J. W.22:32826.

Woodcock, G.21:1193.

» B.3

Yaslcvky, J. B.2 i t2 l l8 , 45128; 22.:5982, 14320, Hi329v 30351,32830; £1:7652, 30191, 3S205, 42756; £0.:34953.

tattdsnm, 8.11:11337,

Yurovol, t . H.21:21731.

Zwibal, H. S.ii32434; 21:33180.

48

synthesis methods and variational methods in nuclear .../67531/metadc... · mex.) . nucl scl frig.,...

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