Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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164 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsFor r = 1, after making use of the identities in Equations (5.51) and (5.55) Equation (5.50)the first moment equation becomesWM 1(P) = |dP'T(P,P')[W'F(P,P') + c a(P')W a f a(P')fdP"C(P',P")W"f s(P',P")]fdP'T(P,P')JdP"C(P' ,P")W"M,(P") (5.56)where, for the sake of brevity, we introduced the notationM 1(P) =M{s}(P,l)for the expected score due to a starter of unit weight at P. It again should be stressed that,although nothing has been said so far about the statistical weights W', W\ and W" thatappear in the equations, they are assumed to be known. Their generation rule will grant theunbiasedness of the nonanalog game, i.e., that the expected score be the same as in theanalog game for which it substitutes. Consequently, the only unknown in Equation (5.56)is the expectation M 1(P) and it is entirely determined by the equation.In an analog game governed by the kernels T(P,P') and C(P',P"), the statistical weightof a particle does not change during the simulation, i.e.,W" = W' = W a = WThus, denoting the contribution functions that are applied in the analog game by f(P,P'),f a(P'), and f s(P',P"), respectively, and the expected score due to a starter of unit weight byM 1(P), Equation (5.56) for an analog game takes on the formM 1(P) • JdP'T(P,P')[f(P,P') + c a(P')f a(P')+ JdP"C(P',P")f s(P',P")] + JdP' T(P,P')jdP"C(P', F)M 1(P") (5.57)Note that with the simplest choice of the contribution functions, such as f a= f s= 0 andf(P,P') — f(P'), Equation (5.57) reduces to Equation (5.13) derived by heuristic arguments.Equations (5.56) and (5.57) provide us with the tools necessary for the characterization anddiscussion of different nonanalog and analog Monte Carlo schemes. In Chapter 5.V, theseequations are investigated and conditions established with respect to the statistical weightsand contribution functions under which the nonanalog game provides an unbiased estimateof the reaction rate in Equation (5.2).As for the second moment of the score, let us insert F(s) = s 2 into Equation (5.50).By taking advantage of the identities in Equations (5.51) through (5.55), the equationbecomesW 2 M 2(P) =JdP'T(P,P')c a(P')|W'f(P,P')+ WT 3(P')] 2 + 2 Qj|dP'f(P,P')JdP"C(P',P")[W'f(P,P')+ W't(P',P")] 2 --''(W")'M r(P") (5.58)
165where again we have putM 2(P) =M{s 2 }(P,S)for the second moment of the score due to a starter of unit weight at P. No surprise, thesecond moment satisfies an equation which contains also the first moment. Therefore, thesolution of Equation (5.58) still may be more difficult than the determination of the expectedscore. In what follows, we do not attempt to solve this equation analytically, instead. Equation(5.58), together with Equations (5.59) and (5.60) below, will be used in Chapters 5,Vi and5. VIII. for the analysis of the relative magnitude of the variance of games with differentcontribution functions and different nonanalog kernels.For an analog game, Equation (5.58) readsM 2(P) -- dP'T(P,P')c a(P')lf(P,P') + f a(P')] 2+ V Q JdP'T(P,P')[dP"C(P',P")|f(P,l > ') -i- f s(P'.P")rrM,.(P"j+ JdP'T(P,P')|dP"C(P',P")M 2(P") (5.59)It can be seen easily that the equations describing the second moment of thebe cast into a form which contains the variance as an unknown. For the sakeillustrate this statement with the analog case. Denoting the variance of theunit weight starter byD-YP) = M 2(P) -M-(P)and subtracting M 2 (P) from both sides of Equation (5.59), we obtainD 2 (P) -- D 0(P) + Df(P) -2R 01(P)+ JdP'T(P,P')JdP"C(P',P")D 2 (P") (5.60)whereDl(P) = jdP'T(P,P')c a(P')[f(P,P') + f a(P')] 2- {jdP'T(P,P')c a(P')[f(P,P') + f a(P')]} 2Df(P) = dP'T(P,P') dP"C(P',P")|f(P,P'l f f„(P\P") + M,(P")f- {JdP'T(P,P')[dP"C(P',P")[f(P,P') ~i f.(P',P") -< M 1(F)IPandR 01(P) = j|dP'T(P,PX(P')[f(P.P') + f„(P')]}x \ |dP'T(P,P') JdP"C(P',P")(f(P,P') + i s(P',F) + M 1(P";!-
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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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