LA-10363-MS

A Sample Problem for

Variance Reduction in MCNP

LosN A T I O N A L L A B O R A T O R Y

AlamosLos Alamos National Laboratory is operated by the University of Californiafor the United States Department of Energy under contract W-7405-ENG-36.

An Affirmative Action/Equal Opportunity Employer

This report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither The Regents of the University of California, the United States Governmentnor any agency thereof, nor any of their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, or represents that its use would not infringeprivately owned rights. Reference herein to any specific commercial product, process, or service bytrade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by The Regents of the University of California, theUnited States Government, or any agency thereof. The views and opinions of authors expressedherein do not necessarily state or reflect those of The Regents of the University of California, theUnited States Government, or any agency thereof. The Los Alamos National Laboratory stronglysupports academic freedom and a researcher's right to publish; therefore, the Laboratory as aninstitution does not endorse the viewpoint of a publication or guarantee its technical correctness.

LA-10363-MS

LosAA

UC-32hued: October 1985

A Sample Problemfor Variance Reductionin IWICNP

Thomas E. Booth

IaliinmLosAlamosLosAlamos,NationalLaboratoryNew Mexico87545

A SAMPLE PROBLEM FOR VARIANCE REDUCTION IN MCNP

by

ThomasE. Booth

ABSTRACT

The Los AlamoscomputercodeMonteCarloNeutronPhotonusefulvariancereductiontechniquesto aidthe MonteCarlouser.

(MCNP) has manyThis reportapplies

manyof these techniquesto a conceptuallysimple but computationallydemandingneutrontransportproblem.

I. INTRODUCTION

This report is based on a series of four 50-minvariance reduction talks (MCNP Variance ReductionTechniques,video reels#12-15)given at the MagneticFusion Energy Conference on MCNP,* Los AlamosNational Laboratory,October 1983.It is an overviewofall variance reduction techniques in MCNP and not anin-depthconsiderationofany. In fact, the techniquesaredescribed only in the context of a single conceptuallysimple, but demanding, neutron transport problem,with only enough theory presented to describe the gen-eral flavor of the techniques. Detailed descriptions arein the MCNP manual.1

This report assumesa generalfamiliarity with MonteCarlo transport vocabulary such as weight, roulette,score,bias, etc.

II. VARIANCEREDUCTION

Variance-reducingtechniques in Monte Carlo calcu-lationscan often reduce the computer time required toobtain resultsof sufficientprecision.Note that precision

*VideotapesoftheentireconferenceareavailablefromRadia-tionShieldingInformationCenter,OakRidgeNationalLabo-ratory,OakRidge,TN 37830.The readerwishingto run thesampleproblemhereshouldreferto the appendixbeginningonpage67forinputfiledetailsmodifiedsincetheconferenceandafterthewritingofthisreport.

is onlyone requirement for a goodMonte Carlo calcula-tion. Even a zero variance Monte Carlo calculationcannot accurately predict natural behavior if othersources of error are not minimized. Factors affectingaccuracywereoutlinedby Art Forster, LosAlamos(Fig.1).**

This paper demonstrates how variance reductiontechniquescan increasethe efficiencyof a Monte Carlocalculation.Two user choicesaffect that ei%ciency,thechoiceof tally and of random walk sampling.The tallychoice (of for example, collision vs track length esti-mators) amounts to trying to obtain the best resultsfrom the random walks sampled. The chosen randomwalk samplingamounts to preferentiallysamplingim-portant particles at the expense of unimportantparticles.

A. Figureof Merit

The measure of efficiencyfor MCNP calculations isthe figureof merit (FOM) defined as

1FOM = G:r T

**Videoreel #l 1, RelativeErrors,Figureof Merit fromMCNPWorkshop,LosAlamosNationalLaboratory,October4-7, 1983.Availablefrom RadiationShieldingInformationCenter,Oak RidgeNational Laboratory,Oak Ridge,TN37830.

1

1. CODEFACTORS

PHYSICSANDMODELSDATAUNCERPAINTIESCROSS-SECTIONREPRESENTATIONERRORSINTHECODING

2. PROBLEM-MODELINGFACTORS

SOURCEMODELANDDATAGEOMETRICALCONFIGURATIONMATERIALCOMPOSITION

3. USERFACTORS

USER-SUPPLIEDSUBROUTINEERRORSINPUTERRORSvA~ANCEREDUcTIoNABUfjECHECKINGTHEOUTPUTUNDERSTANDINGTHEPHYSICALMEASUFtEMEIW

Fig. 1. Factorsaffectingaccuracy.

where am,= relative standard deviation of the mean andT = computer time for the calculation (in minutes). TheFOM should be roughly constant for a well-sampledproblembecauseO%ris (on average)proportional to Nl(N= number of histories) and T is (on average)propor-tional to N; therefore,the product remains approximatelyconstant.

B. GeneralComments

Although all variance reduction schemes have someunique features,a fewgeneralcomments are worthwhile.Considerthe problem of decreasing

(whereCJ2= history variance, N= number of particles,and p = mean) for fixed computer time T. To decreasea~,, we can try to decrease o or increase Nthat is,decrease the time per particle history-or both. Un-fortunately, these two goals usually conflictbecause de-creasing a normally requires more time per history be-cause better information is required and increasing Nnormallyincreasesa becausethere is lesstime per historyto obtain information. However, the situation is nothopeless.It is often possible to decrease o substantially

without decreasingN too much or increase N substan-tiallywithout increasingo too much so that

ca,,=

Pm

decreases.Many techniques described here attempt to decrease

a~r by either producing or destroying particles. Sometechniquesdo both. In general, (1) techniques that pro-duce tracks work by decreasing a (we hope much fasterthan N decreases),and (2) techniques that destroy trackswork by increasing N (we hope much faster than aincreases).

IIIILTHE PROBLEM

The problem is illustrated in Fig. 3, but beforediscuss-ing its Monte Carlo aspects, I must point out that theproblem is atypical and not real. I invented the sampleproblem so most of the MCNP variance reduction tech-niquescould be applied. Usually, a real probiem will notneed so many techniques. Furthermore,without under-standing and caution, variance-reducing techniquesoften increasethe variance.

Figure2 is the input file for an analog MCNP calcula-tion and Fig. 3 is a slice through the geometry at z = O.

2

3SAMPLE PROBLEM FOR MFE TALKSc TALLIES FOR PARTICLES WITH E>.OIMEV

i o (1 -21):-22 1 -2.03E0 -1 -3 23 1 -2.03E0 -1 -4 34

:789

1011121314151617181920

1 -2.03E0 -1 -5 41 -2.03E0 -1 -6i -2.03E0 -i -7 :1 -2.03E0 -1 -8 71 -2.03E0 -1 -9 81 -2.03E0 -1 -lo 91 -2.03E0 -1 -11 101 -2.03E0 -1 -12 111 -2.03E0 -1 -13 121 -2.03E0 -1 -~4 13i -2.03E0 -1 -15 141 -2.03E0 -1 -16 151 -2.03E0 -1 -17 16t -2.03E0 -1 -18 171 -2.03E0 -1 -19 18i -2.03E0 -t -20 190 -1 -21 20

21 1 -2.03E-2 -1 -22 21220 1 21 -2223022

:3456789

10111213141516171819202122

MODEcMl

SRCIS15PNPSINFIF4F5PooEOTOCLITNCTMEPROMPPRINT

CY 100PY oPY 10PY 20PY 30PY 40PY 50PY 60PY 70PY 80PY 90PY 100PY 110PY 120PY i30PY +40PY 150PY 160PY 170PY 180PY 2000PY 2010

0THE FOLLOWING IS

1001 -.010SCHAEFFER PORTLANO CONCRETE

8016 -.52911023 -.01612000 -.00213027 -.03414000 -.33719000 -.01320000 -.04426000 -.0146012 -.001

0 1.E-6 o 2 1.014 14

;25 .5 11000000 1 3R 15R2R02021200 2005 0 00 19R 1 0 0 ~ONLY CELL 21 CONTRIBUTES TO POINT.01 100100 1000 100001.0E123 0.0 0045-5 -5

DETECTOR TALLY

Fig. 2. InputfileforananalogMCNP calculation.

/CELL 21 CONCRETEp= 2.03E-2 ~/CC

F4 TRACK-LENGTHFLUX ESTIMATE

x

4

2000 cm

CONCRETEP= 2.Q3 91CC ~

e200

VOID

ELL 2(

~ 200+

4-

POINT DETECTOR CONTRIBUTIONS/ TAKEN ONLY FROM CELL.21

\>m

WANT FLUX HERE IF5 FLUX AT A POINT CELL22

VOID

CYLINDER 100-cm RADIUS. PARTICLES- CROSSING THIS SURFACE ARE -.. . -

(*PERFECT SHIELDS)

Fl (SURFACE CROSSING TALLY)

/3 CELL 2-19

POINT ISOTROPICNEUTRON SOURCE

25962 MeV50% 14 MeV

25% 2- 114MeV uniform

Fig. 3. Theproblem.

The primary tally is the point detector tally (F5)at the topof Fig.3, 200 cm from the axis of the cylinder(y-axis).Apoint isotropic neutron source is just barely inside thefirstcell(cell2)at the bottom of Fig.3.The sourceenergydistribution is 25%at 2 MeV, 50%at 14 MeV, and 25Y0uniformly distributed between 2 and 14 MeV. For thisproblem, the detectors will respond only to neutronsabove0.01 MeV.

Aperfectshieldimmediatelykillsany neutrons leav-ing the cylinder(exceptfrom cell 21 to cell 22). Thus, totally (F5),a neutron must

1. penetrate 180cm of concrete(cells2-19),2. leave the concrete (cell 19) with a direction close

enough to the cylinder axis that the neutron goesfrom the bottom of cell 20 (the cylindricalvoid) tothe top and crossesinto cell21,

3. collide in cell 21 (becausepoint detector contribu-tions are made only from collision/sourcepoints),and

4. have energyabove 0.01 MeV.These eventsare unlikelybecause

1.

2.

3.

4.

180 cm of 2.03-g/cm3 concrete is difficult topenetrate,there is only a small solid angle up the pipe (cell20),not many collisionswill occur in 10 cm of 0.0203-g/cm3concrete,andparticlesloseenergypenetrating the concrete.

Before approaching these four problems, knowledgeabout the the point detector technique can be applied tokeep from wasting time; only collisions in cell 21 cancontribute to the point detector. Collisionsin cells 2-19cannot contribute through the perfect shield, that is, zeroimportance region. Thus, the MCNP input is set (PDOcard, Fig. 2) so that the point detector ignorescollisionsnot in cell 21. If the point detector did not ignore col-lisionsin cells2-19, the followingwould happen at eachcollision.

1.

2.

3.

4.

The probabilitydensity for scattering toward thepoint detectorwould be calculated.A point detector pseudoparticle would be createdand pointed toward the point detector.The pseudoparticlewould be tracked and exponen-tiallyattenuated through the concrete.The useudoparticle would eventually enter theperfe&shield(cell1)and be killedbeca&e a straightline from any point in cells 2-19 to the point de-tector would enter the perfectshield.

There is no point proceedingwith these stepsbecausethepseudoparticlesfrom cells2-19 are alwayskilled;time issaved by ignoringpoint detector contributions from cells2-19.

IV. ANALOGCALCULATION

Inspection of Fig. 4, which is derived from MCNPsummary tables, shows that the analog calculation fails.Note that the tracksenteringdwindleto zero as they try topenetrate the concrete (cells2-19).This problem will beaddressed in more detail later, but first note that thenumber weightedenergy(NWE)is very low,especiallyincells 12, 13, and 14. The NWE is simply the averageenergy,that isNWE= JN(E)E dE

~N(E) dE

where E = energy and N(E) = number density at energyE.This indicatesthat there are many neutrons below0.01MeVthat the point detector willnot respond to. There isno sense followingparticles too low in energy to con-tribute; therefore, MCNP kills neutrons when they fallbelowa user-suppliedenergycutoff.

V. ENERGY AND TIME CUTOFFS

A. EnergyCutoff

The energycutoff in MCNP is a singleuser-suppliedproblem-wide energy level. Particles are terminatedwhen their energy falls below the energy cutoff. Theenergy cutoff terminates tracks and thus decreases thetime per history.The energycutoff should be used onlywhen it is knownthat low-energyparticlesare either ofzero importance or almost zero importance. A numberof uitfallsexist.

i.

2.

3.

Remember that low-energyparticlescan often pro-duce high-energyparticles (for example, fission orlow-energy neutrons inducing high-energyphotons).Thus, even if a detector is not sensitiveto low-energyparticles, the low-energy particlesmay be important to the tally.The energy cutoff is the same throughout theentire problem. Often low-energyparticles havezero importance in some regions and high im-portance in others.The answerwillbe biased(low)if the energycutoffis killingparticles that might otherwisehave con-tributed. Furthermore, as N+IXI the apparent er-ror will go to zero and therefore mislead the un-wary.Seriousconsiderationshouldbe givento twotechniques (discussed later), energy roulette andspace-energyweight window, that are always un-biased.

5

CELL

PROGR PROBL

2 23 34 45 56 67 78 8~ ~

10 1011 1112 12!3 !3f4 f415 1516 16i7 1718 1819 1920 2021 2122 22

TOTAL

TRACKSENTERING

4783217615639395112871708744313118

:o000000

10644

POPULATION COLL:

3931931593362205115

63401610

76200000000

6281

SIONS COLLISIONS* WEIGHT

(PER HISTORY)

139491505712510

7390421322191587

96130423033021e

1700000000

58985

3.5593E+O03.8421E+O03.1921E+O01.8857E+O01.0750E+O05.6622E-014.0495E-01z.45z2E-oj7.7571E-025.8688E-028.4205E-025.5626E-024.3378E-03o.

::o.0.0.0.0.

1.5051E+OI

NUMBERWEIGHTED

ENERGY

2.4144E-034.5943E-042.0566E-041.4450E-049.3995E-051.0022E-046.4696E-056.2827E-051.0691E-046.2272E-052.2207E-051.993+E-063.7686E-06o.0.0.0.0.0.0.0.

FLUXWEIGHTEO

ENERGY

4.7075E+O01.9643E+O01.2067E+O08.4454E-015.6654E-015.8205E-014.4866E-014.6476E-Oi5.1448E-012.4500E-011.1767E-016.6168E-047.9961E-04o.0.0.0.0.0.0.0.

AVERAGETRACK WEIGHT

(RELATIVE)

I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OO1.0000E+OOI.0000E+OOI.0000E+OO1.0000E+OOt.0000E+OOI.0000E+OOo.0.0.0.0.0.0.0.

AVERAGETRACK MFP

(CM)

5.8207E+O03.9404E+o03.3058E+O03.0062E+O02.7411E+O02.7733E+O02.5496E+O02.6046E+O03.0390E+O02.4143E+O02.t16tE+O01.919~E+O02.0823E+O0o.0.0.0.0.0.0.0.

ANALOG CALCULATION - NO VARIANCE REDUCTION TECHNIQUES

TALLY I TALLY 4NPS MEAN ERROR I=OM MEAN ERROR Fokl1000 0. 0.0000 0 0. 0.0000 02000 0. 0.0000 0 0. 0.0000 03000 0. 0.0000 0 0. 0.0000 03919 0. 0.000o 0 0. 0.0000 0

TALLY 5MEAN ERROR FOM

o. 0.0000 00. 0.0000 00. 0.0000 00. 0.0000 0

******%*******************************************************************OUMP NO. 2 ON FILE RUNTPE NPS = 3919 CTM = .61

Fig.4. Analogcalculation.

B. TimeCutoff

The time cutoff in MCNP is a single user-supplied,problem-widetime value.Particlesare terminated whentheir time exceeds the time cutoff. The time cutoffterminates tracks and thus decreasesthe computer timeper history.The time cutoffshouldonlybe used in time-dependent problems where the last time bin will beearlierthan the cutoff.

The sample problem in this report is time-independ-ent, so the time cutoff is not demonstrated here.

C. TheSampleProblemwithEnergyCutoff

Figure 5 gives the results of an MCNP calculationwith a O.01-MeVenergy cutoff. Note that the numberweighted energy is about 1000 times higher, so theenergy cutoff has changed the energy spectrum as ex-pected. Furthermore, note that about four times asmany historieswere run in the same time although thetotal number of collisionsis approximatelyconstant.

Despite more histories, fewer tracks enter deep intothe concrete cylinder. This may seem a little counter-intuitive until one remembers that the energy cutoffkills the typical particle that has had many collisionsand is below the energy cutoff, that is, the typicalparticledeep in the concrete.This decreasein the tracksentering is not alarming because we know that onlytracks with energy less than 0.01 MeV were killed andthey cannot tally.

The trouble with the calculation is that the largeamount of concrete is preventing neutron travel fromthe sourceto the tallyregion.The solutionis to preferen-tiallypush particlesup the cylinder.Four techniques inMCNP can be used for penetration,

1. geometrysplitting/Russianroulette,2. exponentialtransorm,3. forcedcollisions,*and4. weightwindow.

VI. GEOMETRY SPLITTING ANDRUSSIAN ROULETTE

Geometry splitting/Russian roulette is one of theoldest,most widelyused variance reduction techniques.As with most biasing techniques, the objective is tospend more time samplingimportant (spatial)cellsandless time sampling unimportant cells. The technique(Fig.6) is to

1. divide the geometryinto cells;2. assignimportances (In)to these cells;and

*Therewill not be an exampleusingforcedcollisionsforpenetrationproblemsbecauseit isawkwardtodoinMCNP.Infact,analterationto theweightcutoffgameisoftennecessary.

3. when crossing from cell m to cell n, computeV = In/l~. Ifa. v = 1,continue transport;b. v 1, split the particle into v = I./I~ tracks.

A. RussianRoulette(v < 1)If v 1)If v > 1, the particle is entering a more important

regionand is split into v subparticles.Ifv is an integer,this is easy to do; otherwise v must be sampled. Con-sidern < v < n + 1,then

Probability SplitWeightp(n) = n + 1 v wt, = wt/n sampledp(n+l) = v n wt, = wt/(n + 1) splittingThe sampled splittingschemeabove conservesthe totalweightcrossingthe splittingsurface,but the split weightvaries, depending on whether n or n + 1 particles areselected.

L Actually,MCNP does not use the sampled splittingscheme.MCNP usesan expectedvalue scheme:

Probability SplitWeightp(n) = n + 1 v wt, = wt/v expectedvaluep(n + 1)= v n v-n,= wt/v splittingThe MCNP schemedoes not conserveweightcrossingasplittingsurfaceat each occurrence.That is, if n particlesare sampled,the total weightentering is

wn. =~. wt wt,

However,the expectedweightcrossingthe surfaceis wt:Wtp(n) n ~ + p(n + 1)(n + 1) ~= wt.

7

8CELL

PROGR PR08L

2 23 34 45 56 67 78 89 9

10 1011 1112 1213 1314 1415 1516 16i7 171819 ;:20 2021 2122 22

TOTAL

NPS

1%3om4000500060007cummoo9000

100DO1100012000130ecii391m

TRACKS POPULATIONENTERING

15416 140044445 30982f97 1580

973 716467 331233 17j110 6556 4340 2420 15

8 73 200 :o 00 00 00 00 00 00, 0

23968 20076

TALLY tHEAN

::o.0.0.0.0.0.0.

::o.0.0.

ERROR0:0000O.0000

:%%O.00000.00000:0000o.OoooO.ocmoO.0000

::%%

%%%

FOMo0

:o000000000

27380156117830366117267654201861557842

:,00000000

57662

\

1.9802E+O01. I176E+O05.6057E-012.6210E-Oi1.2357E-015.4768E-023.0069E-021.3316E-021.i097E-025.5842E-033.0069E-035.7274E-04o.0.0.0.0.0.0.0.0.

\4.1425E+o0

NUMBERWEIGHTEO

ENERGY

2.2661E+O01.0718E+o08.8425E-017.9762E-017.2799E-018.0i05E-017.4618E-Ot9.0855E-015.8161E-015.31OOE-OI4.0663E-014.8527E-02o.0.0.0.0.0.0.0. \

FLUXWEIGHTEO

ENERGY

6.0L130E+o03.96S8E+O03.4412E+O02.9569E+O02.6838E+O02.6783E+O0

AVERAGETRAcK WEIGHT

(RELATIVE)I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+ooI.0000E+ooI.0000E+OO

2.4966E+002.6749E+O01.7610E+O0i.6936E+001.5i99E+O03.3019E-01o.0.0.0.0.0.0.0. I

1.00CK3E+O0I.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOI.0000E+OOo.0.0.0.0.0.0.0.

0

(o. 0.

COLLISIONS PER HISTORYHAS DECREASEO

AVERAGE ENERQYHAS INCREASISD

COLLISIONS COLLISIONS* WEIGHT

(pER HISTORY)

********e****.******.*********.*******.*.******.****************Q6***o****OUMP No. 2 ON FILE RUNTPF NPs = 13968 cm! * .6o

NOTES:

1) N INCREASED FROM 3919 TO 13968ENERGY CUTOFF-O.01 MeV 2) TRACKS STOP SOONER BECAIJSE OF ENERGY CUTOFF

31 PARTICLES NOT GETTING TO TALLY REGIONS

AVERAGETRACK MFP

(CM)6.9886E+O05.9464E+O05.6866E+O05.5260E+O05.4005E+O05.5811E+O05.5021E+O05.6290E+O04.9657E+O04.4423E+O04.6999E+O03.1147E+O0o.0.0.0.0.0.0.0.0. \

HAS INCREASED

LTOTAL NUMBER oFcoLLleloNs pRocESsEDABOUTTHESAME

TALLYMEAN

o.0.0.0.0.0.::o.0.0.0.0.0.

4ERROR

O.00000.00000.0000O.omo0.00CQO.0000

::%%

::%%o.OlxloO.olx)o0.000oO.ocm

FOMo0000000000000

TALLY 5MEAN

o.0.0.0.::o.0.0.0.0.0.0.0.

ERROR FOMO.0000 0O.moo oO.ofxlo o0.0000O.owo :O.0000 00.0000 00.0000 00.0000 0O.ocmo o0.0CKM3 o

0::%%0.000o :

Fig,5. EnergycutoffofO.01MeV.

I = IMPORTANCE EXAMPLE T2LE = 3j/

o

+

SPLITTING113

i 1f3113

11- 113orx

KILL t113

RUSSIAFlROULETTE

I\

\

14

ISPLITTING SURFACES

Fig.6. Geometrysplitting/Russianroulettetechnique.

T

The MCNP scheme has the advantage that all parti-cles crossing the surface will have weight wt/v.Furthermore, if

1. geometry splitting/Russian roulette is the onlynonanalogtechnique used and

2. all source particles start in a cell of importance ISwith weightw,, then all particlesin cellj willhaveweight

L

regardlessof the random walktaken to cellj.MCNPS geometry splitting/Russian roulette in-

troducesno variancein particleweightwithin a cell.Thevariation in the rzwnberof tracks scoringrather than avariation in particle weight determines the historyvariance. Empirically, it has been shown that largevariations in particleweightsaffecttalliesdeleteriously.Booth2 has shown theoretically that expected valuesplitting is superior to sampled splitting in high-variancesituations.

C. CommentsonGeometrySplitting/RussianRoulette

One other small facet deserves mention. MCNPnever splits into a void although Russian roulette maybe played entering a void. Splitting into a void ac-complishesnothingexceptextra trackingbecauseall thesplit particlesmust be tracked across the void and they

10

all make it to the next surflace.The stdit should be done.

according to the importance ratio of the last nonvoidcelldeparted and the first nonvoid cell entered (integersplittinginto avoid wastestime, but it does not increasethe history variance). In contrast, noninteger splittinginto a void may increasethe history variance and wastetime.

Finally, splitting generally decreases the historyvariancebut increasesthe time per history.

Note three more items:1.

2.

3.

Geometry splitting/Russian roulette works wellonly in problems without extreme angular de-pendence. In the extreme case, splitting/Russianroulettecan be uselessif no particlesever enter animportant cellwhere the particlescan be split.Geometry splitting/Russianroulette will preserveweightvariations.The technique is dumb in thesense that it never looks at the particle weightbeforedecidingappropriate action. An example isgeometry splitting/Russian roulette used withsourcebiasing.Geometry splitting/Russianroulette are turned onor off together.

D. Cautions

Althoughsplitting/Russianroulette is among the old-est, easiest to use, and most effective techniques inMCNP, it can be abused. Two common abusesare:

9

1. compensating for previous poor sampling by avery largeimportance ratio and doing the splittingall at once.

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problem is so bad without splitting that it is hard toguess how much splitting/Russian roulette has im-proved the efficiency.Contrast this improvement to the(questionable)FOM differenceof 27 (Fig.7) to 23 (Fig.8) between the factor of 2 splitting and the refinedsplitting. Usually one can do better with a variancereduction techniqueon the second try than on the first,but usuallyby not more than a factor of 2.

Quicklyreachingdiminishingreturns is characteristicof a competent user and a good variance reductiontechnique. Competent users can quickly learn goodimportancesbecausethere is a very broad near-optimalrange.Becausethe optimum isbroad, the statisticsoftenmask which importance set is best when they are all inthe vicinityof the optimum.

Now that a reasonablyflat track distribution has beenobtained, perhaps it is time to explain why one expectsthis to be near optimal. There are some plausibleargu-ments, but the real reason is empirical; it has beenobserved in many similar problems (that is, essentiallyone-dimensionalbulk penetration problems) that a flattrack distribution is near optimal. The radius of theconcrete cylinder is large enough (100 cm) that thecylinder appears much like a slab; very few particlescross its cylindrical surface at a given depth (c-oordinate) compared to the particle population at thatdepth. Indeed, if the radius were infinite, the cylinderwould be a slab and no particles would cross its cylin-drical surface.

Aplausibleargument for flat track distribution can bemade by considering an extremely thick slab andpossibletrack distributions for two cases. For too littlesplitting, the track population will decrease roughlyexponentially with increasing depth and no particleswill ever penetrate the slab. For too much splitting,theimportanceratiosare too large;the track populationwillincrease roughly exponentially and a particle historywill never terminate. In both cases, albeit for differentreasons, there are never any tallies. If neither an ex-ponentiallydecreasingpopulation nor an exponentiallyincreasingpopulation is advisable, the only choice is aflatdistribution.

Of course, there are really many more choices thanexponentiallydecreasing,flat, or exponentiallyincreas-ingpopulations,but track populationsusuallybehave inone of these ways because the importance ratios fromone cell to the next are normally chosen (at least for afirst guess) equal. The reason is that one cell in theinterior is essentiallyequivalent to the next cell,so thereis littlebasisto choosea differentimportance ratio fromone cell to the next. However, the cells are not quiteequivalent because they are different depths from thesource, so the average energy (and mean free path)decreaseswith increasingdepth. This is probablywhy itwasnecessaryto increasethe importance ratio from 2 to

2.15 in the deep parts of the sample problem. Note,however,that this is a small correction.

Returning to Fig. 8, note that the energy and meanfree path decrease with increasing depth, as expected.Not also that the higher splitting has decreased theparticlesper minute.

VII. ENERGYSPLITTING/ROULETTE

Energy splitting/Russian roulette is very similar togeometrysplitting/Russianroulette exceptenergysplit-ting/rouletteisdone in the energydomain rather than inthe spatialdomain. Note two differences.

1. Unlike geometry splitting/roulette, the energysplitting/rouletteusesactual splittingratios as sup-plied in the input file rather than obtaining theratios from importances.

2. It is possibleto play energysplitting/rouletteonlyon energydecreasesif desired.

There are two cautions.1. The weightcutoff game takes no account of what

has occurredwith energysplitting/roulette.2. Energysplitting/rouletteis played throughout the

entire problem. Consider using a space-energyweightwindowif there is a substantialspacevaria-tion in what energiesare important.

One can expect an improvement in speed usingenergyrouletteby recallingthat the problemran a factorof 4 faster with an energy cutoff of 0.01 MeV thanwithout an energy cutoff. Low-energy particles getprogressivelyless important as their energydrops, so itmight help to play Russian roulette at several differentenergiesas the energydrops. In the followingrun, a 50!40survivalgame was played at 5 MeV, 1 MeV, 0.3 MeV,0.1 MeV, and 0.03 MeV. The energies and the 50Y0survivalprobabilitywere onlyguesses.

The energy roulette (splitting does not happen herebecausethere isno upscatter)resultsare shownin Fig.9.Note that there were substantially(-50%) more tracksentering,approximatelythe same number of collisions,and three times as many particles run. The FOM looksbetter, but the mean (Tally 1)has increasedfrom 5.OE-7(Fig. 8) to 8.4E-7.This deserves note and caution, butnot panic,becausethe error is 18%1,so poor estimates inboth tally and error can be expected.Despite the previ-ous statement, the energy roulette looks successful inimprovingtallies 1and 4.

VIII. IMPLICIT CAPTURE AND WEIGHTCUTOFF

A. ImplicitCapture

Implicit capture, survival biasing,and absorption byweightreduction are synonymous.Implicit capture is a

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PROGR PROBL

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: 45 56 67 78 89

1:;: 1112 1213 1314 1415 1516 1617 1718 1819 1920 2021 2122 22

TOTAL

TRACKSENTERING

11938529250104827441441774114411441054112429343844337431243654274424838674927

152231283

107616

POPULATION COLLISIONS COLLISIONS NUMBER* WEIGHT WEIGHTEO

(PER HISTORY) ENERGY11744491846514469407338163780380338333803394840404oi739774059398239353749

1822430451

1283

130555

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1.9875E+O01.0727E+O05.7591E-012.7700E-Oi1.2674E-016.2377E-023.2883E-021.5241E-027.4165E-033.2429E-031.7202E-038.0545E-043.6893E-04i.7810E-047.9894E-053.6959E-051.6748E-056.2137E-06o.6.5663E-11o.

4.1643E+O0

1.9553E+O01.1897E+O08.6965E-017.9325E-017.8981E-017.9364E-Of7.5226E-017.2149E-017.0749E-017.5649,E-016.9620E-016.5721E-016.6088E-016.6189E-017.0947E-016.3603E-016.6484E-017.0403E-011.2396E+O01.7785E+O07.0419E-01

FLUXWEIGHTEO

ENERGY

5.8909E+O04.0414E+O03.3229E+O03.0387E+O02.9908E+O02.8062E+O02.5813E+O02.5273E+O02.4918E+O02.4433E+O02.3424E+O02.2911E+O02.2587E+O02.1957E+O02.1324E+O02.0865E+O02.0783E+O02.2236E+O03.2943E+O04.1554E+O02.7866E+O0

AVERAGETRACK WEIGHT

(RELATIVEI1.3213E+O01.6629E+O01.9476E+O02.0971E+O02. i426E+O02.2181E+O02.3164E+O02.3520E+O02.3750E+O02.3664E+O02.4325E+O02.4785E+O02.4924E+O02.5004E+O02.5102E+O02.5629E+O02.5527E+O02.4620E+O07.9810E-019. I074E-043.6840E-05

AVERAGETRACK MFP

(CM)6.8365E+O05.9855E+O05.6454E+O05.5575E+O05.6477E+O05.5340E+O05.4448E+O05.4441E+O05.4572E+O05.4819EiO05.3524E+O05.3043E+O05.3570E+O05.307iE+O05.3140E+O05.2996E+O05.3146E+O05.45iiE+O01.0000+1237.3805E+021.0000+123

SUBSTANTIALLYIMPROVED

TALLY 1NPS MEAN ERROR FOM1000 ~.2j356E-07 .3667 2720003000400050006006700080009000

1000011000li427

6.41539E-075.70257E-076.76150E-076.76891E-076.73265E-076.89040E-076.86656E-077.04305E-077.25099E-077.00085E-077.32339E-07

.2949

.2393

.1863

.1679

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:11671093

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26273232323435363636

/34

TALLY 4MEAN

1.50695E-131.06818E-131.02834E-131. I1525E-131.13338E-131.14934E-131.14298E-131.17711E-131. IBI17E-131.22116E-131.18453E-131.22412E-13

ERROR.325!.2461

1982:1622.1465.1312.1198.1095

::%.0948.0946

7 TALLY 5FOM MEAN

34 i.06148E-i66.85473E-17

% 6.49562E-i743 6.79i32E-f742 6.89970E-1743 6.74936E-1745 6.74760E-1746 6.8725iE-i747 6.97952E-~744 7.09365E-i745 6.88873E-1742 7.21438E-1715LAs-rTN4E

ERROR.3673.2928.2387.1910.1686.1510.1359.1245.1161.1114.1060.1049

tFOM

2727273131333436363536

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27

ABSTRACTI. INTRODUCTIONII. VARIANCE REDUCTIONA. Figureof MeritB. General Comments

Fig. 1IIII. THE PROBLEMFig. 2Fig. 3IV. ANALOG CALCULATIONV. ENERGY ANDTIME CUTOFFSA. Energy CutoffFig. 4B. TimeCutoffC. TheSampleProblem with EnergyCutoff

VI. GEOMETRY SPLITTING ANDA. RussianRoulette (v < 1)B. Splitting (v > 1)Fig. 5Fig. 6C. Comments on Geometry Splitting/RussianRouletteD. CautionsE. TheSample Problem with GeometryF. Discussionof ResultsFig. 7Fig. 8.

VII. ENERGYSPLITTING/ROULETTEVIII. IMPLICIT CAPTURE AND WEIGHT CUTOFF A. Implicit CaptureFig. 9B. WeightCutoffFig. 10C. WeightCutoff and Implicit CaptureAppliedto the

IX. FORCED COLLISIONSA. CommentsFig. 11Fig. 12B. CautionC. Forced Collisions Applied to the Sample Problem

X. DXTRANFig.13Fig.14A. DXTRAN Viewpoint #1B. DXTRAN Viewpoint #2C. The DXTRAN ParticleFig. 15.D. Inside the DXTRAN SphereE. Terminology - Real Particle, PseudoparticleF. CommentsG. CAVEATSH.DXTRAN Applied to the Sample ProblemI. DiscussionFig. 16.XI. TALLY CHOICE, POINT DETECTOR VERSUS RING DETECTORFig. 17.Fig. 18Fig. 19

XI. TALLY CHOICE, POINT DETECTOR VER-SUS RINGDETECTOR