Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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172 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsis, the terms describing the contributions by the different secondaries, do not depend on i.Correspondingly, the summations reduce to multiplications by n and Equation (5.78) becomesWM 1(P) = jdP'T(P,P')[W'f(P,P') + c a(P')W%(P')+ c s(P') JdP' C 5(P' ,P") Wf)(P' ,P")+ c f(P') E nq n(P') | dF'C n(P' ,F') WX(P' ,P")]+ JdP'T(P,P')[c 5(P') JdFC 5(P',F)WM 1(P")+ C f(P') E nq„P'|dP"C„(P',P")W;;M 1(P")] (5.79)N= 1 Jwhere W n' is the weight of a particle that emerges from a multiplying collision of n secondaries.For the analog game (where the statistical weight of the particle is not altered in thesimulation), the first moment Equation (5.78) reduces toM 1(P') = JdP'T(P,P')[f(P,P') +c a(P')f a(P')+ c S(P')JdP"C S(P',P")f s(P',P") +C F(P') E nq„(P')JdP"C n(P',P")f n(P',P")]+ J dP'T(P,P') JdF'C(F^F)M 1(F') (5.80)where C(P',F') is the total collision kernel in Equation (5.74).The second moment of the score due to a particle starting from P with a weight of unityis again denoted by M 2(P). It is obtained from Equations (5.75) through (5.77) after multiplicationby s 2and integration. From Equation (5.75), we haveW 2 M 2(P) = JdP 1 T(P 1PO[W 2 F(P,?') + 2W'f(P,P')N ](P',W) + N 2(P',W)]withN r(P',W) = Jdss r -&(P',W',s)Taking into account the identityS S 1) 2 = 2 (s, - a,) 2 + EE (Si ~ a,)(s k- a k) + (s, - a,)a,i^k+ 2 EE (S 1- a,)a k+ E a? + EE a,a ki k i i ki^k
173and assuming again that the secondaries in a collision emerge with equal weights, simplealgebra shows thatN 1(F 1W) = c,(P')Wf,(P') + c s(P')JdP'C s(P',F')W"ff i(P',P") + M 1(P")]+ cXP') S nq n(P') I ClFC n(P',P")W;;[f n(P',P") + M 1(P")]andN 2(P',W) = c,,(P')(W a ) 2 f:;(P') f c,(P"> J~dP"C s(P',F)(W") ; 'f s(P',F)]L(P' .F)+ 2M 1(F)] + c s(P') £ n4 1(P')|dP"C,,(P',p'';Kw;;)4 rxp',p'')[f n(P'.F')+ 2M 1(P")] + £ f(P') i n(n - l)q„(P')[j dP"C n(P',P") W^f 11(P',F)+ M 1(F)]V + JdFC(P',F)(W) 2 M 2(P")The details of the derivation are given in Appendix 5B. After a short algebraic manipulationthe second-moment equation can be rewritten asW 2 M 2(P) == |dP'T(P,P')c a(P')l'W'f(P,P') + Wf 1(P')] 2+ E [ r J JdP'f(P,P')c s(P')JdP'C s(P',F)|W'f(P,P')+ W"f s(P' ,F)] 2 - r (W")' M r(P")+ X ( 2 ) |dP'T(P,P')c,(P') X nq n(P')x { IdFC n(F,F)IWf(FF) + W;';f„(P',F)] 2 - r (W^) r M r(P") |.+ jdP'f(P,P')6 f(P') 2 n(n - I)C] n(P')x(JdP"c n(P',p")w;;[f n(P',p") + M 1(F)]IdP-T(P 1F)Cj(P') 2 (n - i)q„(P')(W') 2 f 2 (P,P'j (5.Hi!The equation concerning the analog game follows from Equation (5.81) by setting (he we.i^iequal to unity and will not be detailed here.( . AN EQUIVALENT NONMULTIPLYING GAMEIt is remarkable that the results of this chapter weequation for the simulated quantity. In other words, in M M > i ,. .'.',-n > > > -Carlo simulation of the particle transport processes, knf>w Hitransport equation is not required.The transport equation < O I K I Ut . ( . >t< u.i
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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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