Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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226 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe idea of delta scattering has served as a theoretical too! for introduction of the tracklengthestimator. 42It also has recently been applied in practical calculations in connectionwith correlated games/ 9 ' 4 0The second-moment equation of the game follows easily from the analog momentequation derived in previous chapters by simply substituting T sand C sin place of the analogkernels, and will not be detailed here. Similarly, the number of collisions to be played ina history will satisfy Equation (5.192) with the respective kernels. The heuristically obviousfact that delta scattering increases the number of collisions per history can also be seen onthe basis of Theorem 5.12.VI. PARTIALLY UNBIASED ESTIMATORSWe have seen in Section 5, V.A that a set of estimatorsS{f(P,P'),f,(P'),f. ;(P' ,P"),{nf f(P' ,P")}:..,} - S{f,f„f ,{nfj} (5.206)result in an unbiased estimate of the reaction rateRJdPiKP)f(P)in an analog game governed by the kernels T(P,P'), C 5(P',F'), and {C n(P',P")}^, if therelationdP'T(P 1P') f(P,P') + C 11(P')f a(P') + c s(P')JdP"C s(P'.P")f s(P',P")+ c r(P') 2 Q n(P') jdP"C n(P',P")nf n(P\P") --- |dP'T(P,P')f(P') = I(P) (5.207)holds. It has also been seen in Section 5.V. B that any nonanalog game that satisfies theweight generation rules of Theorem 5.8 is also unbiased with the same estimators. Estimatorsin the set (5.206) that satisfies Equation (5.207) were called partially unbiased estimatorsas any such set results in the same expected partial score in a flight followed by a collision.Obviously, the simplest partially unbiased set isS{f(P'),0,0,(0}}S{f,f u,f s,{nf„}^= ]} is a shorthand notation of the estimation procedure in which f(P,P')is scored when a free flight from P to P' is played, f a(P') is the score assigned to an absorptionat P', and f s(P',P") and f„(P',P") are the scores if a particle emerges at P" from a scatteringor from an n fold multiplication at P', respectively. [This means that if in an n-fold multiplicationthe secondaries emerge at P' (' u, P' (' 2), P" n), respectively, then the score fromthis event isas detailed in Section 5.III.A.JIn an analog game, we have a certain freedom in defining the separate scores; namely,the score assigned to a free flight can also be attached to the scores due to the different
227possible events in a collision, provided the expected score in a free flight + collision processremains unchanged. This idea is a simple illustration of the possible variety of the partiallyunbiased estimators and also a motivation for seeking transformation procedures that leavethe expected score unaltered. A simple example of the heuristic arguments above is that thebasic estimator S{f(P'),0,0,{0}} is equivalent to the estimator set S{0,f(P'),f(P'),{f(P')}*„.i}since both sets give the same score, f(P'), at every collision point P'. Note, however, thatthe two sets do not necessarily yield the same variance, as follows from the results of Section5.III.B.Let us introduce the following notationsf Es(P') = |dP"C s(P',P")f s(P',P") (5.208)andUP') = nJdP"C n(P',P")f„(P',P") (5-209)Evidently, f E5(P') is the expected score from a scattering at P' and it will be called theexpected scattering estimator. Similarly, f E[1(P') is the expected n-fold multiplication estimator.With these notations, Equation (5.207) becomesdP'T(P.P') f(P,p') + c a(P')f,(P') + c,(P')UP') + c,.(P') 2 q„(P')UP'>= JdP'T(P,P')f(P') = I(P) (5.210)Accordingly, the set S{f,f a,f Es,{f En}nFor convenience, in the following derivations let us denotei} that satisfies Equation (5.210) is partially unbiased.Po(P') = c,(P'),P,(P') = c s(P')P n + 1(P') = CXPOq n(P'), n = 1,2,... (5.211)andg o(P,P') = f.(P') + F(P,P'), G,(P,P') = FES(P') + X.P')g„ +1(P,P') = f E„(P') + f(P,P'), n = 1,2,... (5.212)With these notations, the set of estimators in Equation (5.210) is rewritten as 8{0,IgX 0Iand the Equation itself, which expresses the condition that the set be unbiased, is rewrittenas|dP'T(P,P') £ p,(P')g,(P,P') = KP) (5.2!3)•> i = 0where, according to Equation (5.211)ip,(p') = 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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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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