Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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196 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthe mean number of secondaries per collision. The solution of this equation is easily obtainedascjl - e-"- c ) , ]/(l - c) if c < 1c,t if c = l(5.125)The stationary expected absorption rate follows from the solution Equation (5.125) by lettingt tend to infinity, and it is seen that the solution tends to a finite value with increasing timeif c < 1, but it is unbounded at c = 1. This is in accordance with Theorem 5.1 since theconditions of the Theorem are met only if c < 1. Finally, we note that in view of the resultsin Section 5.1I1.C, the solution for c < 1 in Equation (5.125) also defines the expectedabsorption rate for c > 1.V. ANALYSIS OF THE FIRST-MOMENT EQUATIONWe have established the moment equations that describe the expectation of variouspowers of the total score. There remains, however, a number of open questions. First, it isto be clarified what kind of contribution functions result in estimates of the required reactionrate, Equation (5.2). Second, it is not yet clear how to choose the statistical weight of aparticle in order to keep a nonanalog game unbiased, i.e., in order to ensure that a nonanaloggame does result in the same expected score as the analog game it corresponds to. Thesequestions will be answered in Sections A and B of this chapter. We consider here the firstmomentequation of such multiplying analog and nonanalog games in which the secondariesof a multiplying event are indistinguishable. The considerations below can be easily generalizedto the case of distinguishable secondaries. We remind the reader that the statisticalweights of the split fragments in a game with splitting are essentially arbitrary (except forthe conservation rules in Theorems 5.4 and 5.6) and they do not follow from the requirementof unbiasedness. Selection of the split weights will be investigated in Section 5.VIILI.For easy reference, let us recall the first-moment equation for a multiplying game. Theequation concerning an analog game is given in Equation (5.80) asM 1(P) = I 1(P) + JdP'T(P,P')JdP"C(P',P")M 1(P") (5.126)whereI 1(P) = JdP'T(P 1P') [f(P,P') +C 8(PX(P')+ C 5(P') JdP" C 8(P' ,P") f s(P' ,P")+ c f(P') 2 nq n(P')JdP"C n(P',P")f n(P',P")j (5.127)andC(P',F') - c s(P')C s(P',P") + C 1(P') i nq„(P')C„(P',P") (5.128)
197[Note that we have omitted the absorption term c a(P')8(P' — P), which leads out of thesimulation and plays no role in Equation (5.127)]. The nonanalog first-moment equation.,from Equation (5.79), isM 1(P) = I 1(P) + dP'T(P,P') dF'C(P',P")M,(F') (5,129)withI(P) = dP'T(P,P') W - W a .— f(P,P') + — fiP')W W "W" -a s(P')|dP"c s(P',p")^f s(P',P")andc f(P') 2 nq„(P') I dP"C„(P',P") f„(P',P") (5.130)W" °° , w"C(P',P") = c,(P')C„(P',P") — + off') 2 nq n(P')C n(P',P") (5.131)A. UNBIASED ESTIMATORSIt has been seen in Chapter 5.1 that starting the particles from the real physical sourcedensity in an analog game, the expected score due to a particle that departs from point Psatisfies the equationM 1(P) = I(P) + dP'T(P,P') ,P')/c dP"C(P',P")M,(P")(5.132)withI(P) =jdP'T(P,P')f(P')(5.133)where f(P) is the weighting function in the reaction rate integralR =dPi[i(P)f(P)to be estimated. It was also seen in Section 5.1.A that Equation (5.132) determines theexpected score in such a game, where a history consisting of the collision points PJ,,P 2,...,P nyields a total score ofM-(P) = E f(?DIn Section 5.1.1) and subsequent chapters, we have constructed a general Monte Carlosimulation in which every event that happens to a particle may contribute to the total scoreThe first moment of the score in a general analog game Is given in Equations (5,126) and(5.127). Comparing these equations to Equations (5.132) and (5.133), it is apparent that inthe simplest game, the contribution functions (or estimators) are defined asf(P,P') = f(P'); f a(P') = f s(P',P") = f„(P',P") = 0
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Library of Congress Cataloging-in-P
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III. Statistical Considerations •
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IV. Further Generalizations — 183
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Appendix 6A:Unbiased Estimation of
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2 Monte Carlo Particle Transport Me
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Chapter 2INTRODUCTIONWhen we starte
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thus,As it is proved" — and is so
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9B. THE PROBABILITY MIXING METHODTh
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11It is easy to prove 15by summing
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13but the most probable values:x, =
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ISand give a random sign (by the us
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18 Monte Carlo Particle Transport M
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100 Monte Carlo Panicle Transport M
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102 Monte Carlo Particle Transport
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25!) Monte Carlo Particle Transport
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305Chapter 6SPECIAL GAMESIn the maj
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3©7A. CORRELATED MOMENT EQUATIONSI
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309Equation (6.1) will be referred
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311The second moment of the score d
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313andC S(P',P")/C S(P',P") = y(P',
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315introduced in Equation (6.1) for
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317andL„(P 0,P;,...,Pi + 1) = L n
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319whereB„(P„,P; P:, ,) = A*(P
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E. EXAMPLES AND SPECIAL TECHNIQUESL
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323The weight factors associated wi
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325It is seen from Equation (6.37)
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327G. PARAMETRIC PERTURBATIONS: INT
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129Hall 31 gave a constructive deri
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331kernels, which remain unchanged
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333and from Equation (6.51)w';,,(i)
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335According to Equations (6,55) th
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337Minimization is performed by set
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339and letdsdaLet us consider a cor
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341whereis the length of the flight
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and from Equation (6.79)I / ds \ 2
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345Let us assume that we are intere
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347Introduction of this factor also
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3492. Let i|/ n (P) denote the solu
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3SJNow, if ( "> > 0, then k = i an
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353andk„ = S l "VS'" •" (6. if;
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3SSLet k„ denote the normalizatio
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3572. It is then proved that with t
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and start new histories until N new
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361The application of source iterat
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363we haveD 2 [k] = < 2 (k, - k,) )
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365neutron density, S 0 for every
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367if the same relation holds for t
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369Then the multiplication factors
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371in the unperturbed system, the w
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373The perturbation of the effectiv
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375where P" = (r',E'). Multiplying
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377iterate of the derivative of the
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579where P* = (r*,w*,E) = (r*,E*).
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381This estimator, unlike f, in Equ
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383tends to some stable attracting
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3§SIo the case of the next-event e
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387LDFIGURE 6.1.Geometry in scoring
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389Let us denoteThen Equation (6.18
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391is to be increased in order to o
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393Now, since bl(P) = g(P) is bound
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395is scored at every collision wit
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397and its expectation isR — R 1n
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3994. If O > 0 m, then the particle
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401plays a distinguished role), and
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403Theorem 6.7 — If x and s denot
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this a priori information, the best
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40?Thus, the bias introduced into t
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409In view of Equation (6.226) and
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41.1This means that if we use the k
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413Then the whole-sample and rare-s
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41.5According to the results of Sec
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41?(Note that vD 2 [|i|v] is indepe
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419withkOn the other hand, § is an
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421Thus, the expected ratio reads
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423St follows from Equations (6.265
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425Wc start from the observation th
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427Nowi'6.77K)kj„,and therefore(V
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42*3Similarly, for the fourth momen
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43 sOn the other hand, multiplying
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43.¾This will be proven in the lem
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435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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6. Brown. F. B. and Martin, W. R.,
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67. Rief, H. and Fioretti, A., Mont
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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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a ( X506 Monte Carlo Particle Trans
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513INDEXAAbsorptiondefined, 25photo
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515score probability in general tim
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517path stretching, 447—455splitt
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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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