Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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126 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsvector is very fast. Here, the a ymatrix elements can be computed in a one generation MonteCarlo calculation.Let us now investigate the case where a small perturbation changes the fission neutroncollision density to \*, the fission kernel to K*(P',P) and the effective multiplication factorto k* -k* =dPdP'xT(P')K*(P',P)dPxT(P)If the differences are small we may approximate xt asXt(P) = X 1(P) + SXr(P)andJdP Xt(P) JdPxXP)1dP8 Xf(P)dP Xf(P)By these approximation the difference between the two multiplication factors is:8k eff= k! ff- k rff= j -1• F JJdPdP' Xl(P)[K*(P',P) -K 1(P',P)]JdPXf(P)ff / d P 8 X f ( P ) ff ]4 j JdPdP'8 Xf(P)K f(P',P) ~ - J JdPdP'[ X f(P) + 8 Xf(P)JK f(P',P) (4.97)JdPx 1(P)If the second and third terms of Equation (4.97) may be neglected (§x < x) then theequation reduced toJJdPdP' Xf(P',P) K f(P',P)8k ef= j (4.98)J dP Xf(P)where K.tP'.P) = Kt(P',P) - K(P',P).In this approximation 8k can be obtained in parallel with k cfeffin a correlated MonteCarlo, since only the transport kernels of Equation (4.96) and (4.98) differ from each other.Methods to evaluate Equation (4.97) with all terms are reviewed by Bernnat.'Criticality is discussed in detail in Chapter 6.III.VIL ADJOINT MONTE CARLOInvestigation of the adjoint transport equations plays an important role in modern MonteCarlo research. Equations adjoint to the transport (collision density) equations can be intra-
duced and discussed in various ways. Basically, one may consider the adjoint to eithintegral or the integro-differential transport equation. Some confusion may arise if onenot take into account that conversion from integro-differential to integral forms ancversion to adjoint do not commutate."The adjoint to an equation can be found purely mathematically. Instead, we appthe problem from a rather physical point of view. First, the value of the particle is d
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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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176 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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484 Monte Carlo Panicle Transport M
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486 Monte Carlo Panicle Transport M
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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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