Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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266 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsmoment of the score in the analog game is given in Equations (5.295) through (5.299) aftersetting all the weight factors to unity, i.e.,M 2(P) = I dP'T(P,P')[P(P(P') + 2KP 1PO^ 1(P') + 30(F)]I dP'T(P 1P') dP"C(P',P")!M 2(P") Mf(P")J (5.319)with^ 1(F) = c a(P') f a(P') + c S(P')JdP"C s(P',P")[f S(P',P") + M 1(P")]+ C 1(P') 2 nq„(P') fdP"C„(P',P")[f n(P',P") + M 1(P")] (5.320)andTf 2(P') = c a(P')f 2 (P') + c S(P')JdP"C S(P\P")[f S(P'.P") + M 1(P")] 2+ C 1(P') X nq N(P') fdP"C„(P',P")[f N(P',P") + M 1(P")] 2 (5.321)ii = i J+ c f(P') X n(n - Dq 1 1(POLJDFC N(POR)[UPOP") + M 1(F)]The second-moment equation for the equivalent game follows from Equation (5.92) asJl 2(P) = JdP'T(P,P')[f 2 (P,P') + 2f(P,P0|,(P') + I 2(P')]+ JdP'T(P,P')c(P')JdFC(P',F)[M 2(P') - M 2 (F)J (5.322)whereJf 1(P') = JdFC(POP") [f*(P',F) + M 1(P")] (5.323)J 2(PO - c(P')JdP"C(P',P")[f*(P',F) + M 1(F')] 2 (5.324)andc(P') = C 8(P') + C 1-(PO 2 nq n(P') (5.325)H= ithe mean number of secondaries per collision in the analog game. L(P' ,P") is the contributionfunction in a collision from P' to P", and the equivalent nonmultiplying game is unbiasedif L satisfies Equation (5.88) with unit weights, i.e., ifdP"C(P',P")f*(P',P") = c a(P')f a(P') + c s(P') JdF C 8(POF) f s(P',P")+ C 1(P') V nq„(P') I dP"C n(POPOf n(POP") (5.326)
267An Immediate consequence of this equation is that the quantities SF(P') and J 1(P') in Equations(5.320) and (5.323), respectively, are equal. [Note that they represent the expectedfinal score due to a particle entering a collision at P'. This quantity was denoted by N 1(P)in Chapter 5.111.] Introducing the notationA(P) = M 2(P) -Ji 2(P)the difference of the variances in the analog and equivalent games, it satisfies the equationA(P) = J dP'T(P.P')[Ss(P') -JS 2(P')]+ JdP'T(P 5P')[I - c(P')]JdP"C(P',P")[M 2(P") - Mf(P")]+ dP'T(P,P')c(P') dP" C(P',F)A(P")Now, A(P) > 0 if c(P') < 1 and J 2(P') > J 2(P'). In order to keep the derivation as shortas possible, we do not proceed further in full generality, but we assume thatf a(P') = O 5f s(P',P") = f n(P',P") - f c(P',P")in the analog game, i.e., we consider a game in which no score is assigned to an absorptionand the scores in a collision do not depend on the type of reaction in the collision. Thissimplification is also justified by the fact that the great majority of the commonly usedestimators are such indeed. Thenf*(P',P") =f t(P',P")satisfies Equation (5.326) and^ 2(P') - J 2(P') -- [1 - c(P')]JdP"C(P',P")M 2 (P")+ c f(P') E n(n - l)q n(P')|JdP"C„(P' 5P")[f c(P',P") + M 1(P")])^ (5.327)is positive If c(P') < 1. Thus, we have the following theorem.Theorem 5.22 — The variance of an analog game, with the contribution functions f(P,P')and f c(P',P") is not lower than that of the equivalent nonanalog game with the same contributionfunctions ifc(P') ^ 1A number of comments are proper here.1. Since SF — J 2in Equation (5.327) usually is definitely positive for c(P') < I, it isexpected that the variance of the equivalent game is lower than that of the analoggame even if c(P') is slightly greater than unity.
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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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Page 228 and 229: 216 Monte Carlo Particle Transport
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Page 317 and 318: 305Chapter 6SPECIAL GAMESIn the maj
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Page 321 and 322: 309Equation (6.1) will be referred
Page 323 and 324: 311The second moment of the score d
Page 325 and 326: 313andC S(P',P")/C S(P',P") = y(P',
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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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