Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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502 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThen we set the space-dependent parts of the expected scores in the homogeneous and realsystems equal to each other (instead of requiring the equality of the leakage rates, as in theprevious approach). Thend|xM,(x,p,) =M,(x)i.e.which yields the biasing parameter asber = AThen the stretched cross section has the form
In all the approximations above, the stretched cross section is expressed asa(x,|A,E) = cr(x,E)[ 1 — FXB(X,E)Lwhere b(x,E) is one of the quantities in Equation (7.175), (7.178), or (7.181)transition kernel then has the formT(x -> xV.E) = 7-7 cr(x',fx,E)expj - J dta(t,E)[I -- u.b(t.E)l/(jiS-T(x —» x'|p,,E)| 1 - p,b(x',E)|exp dtb(t,E)rj(t,E) ( I . f C./where T(x -* x'||x,E) is the unbiased (analog) transition kernel. Assume that the analogcollision kernel is written in the formC(|x,E -» u2,E'jx') = c(x',E)CK(E -» E'|X',{JL)C P.((JL U.'JX'TLE') (74F)withdE'C E(E —» E'|x',|x) = J d|x'CJ>-> |x'|x',E,E') = 1This form of the collision kernel poses no limitation on the game since any collision kernelcan be written in the form of Equation (7.183). Indeed, by settingC E(E-» E'|x,(x) = Jdix'C(L*,,E -»• u7,E'jx)/c(x,E)andC^l -> p/|x,E,E') = C(|x,E~» u7,E'|x) j J du7C(p.,E u7,E'|x)we have the factorization in Equation (7.183). The biased kernel is given by Equation (7. ? 72)as, . I CT(X',E') , c(x' ,E')C(H 1E -+ |x',E'|x') = c(x',E')C E(E -> Ejx.FJL) - I7F7T7-7T = C t: t7.'[x.fx);2 a(x',(i,',E')211 - M-'tKx'JE')'= C(^1Eix'.E'jx') —- C ( X ; E } , - lc(x',E)C|x((i-* m-IX'.E.E'J]2[1 —M- Kx ,E )J(7.184;Finally, let the analog source be written in the formQ(x,|x,E) = Q 0(X 5E)Q 1x(SxIx 5E)
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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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Page 464 and 465: 452 Monte Carlo Particle Transport
Page 482 and 483: Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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Page 496 and 497: 484 Monte Carlo Panicle Transport M
Page 498 and 499: 486 Monte Carlo Panicle Transport M
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Page 525 and 526: 513INDEXAAbsorptiondefined, 25photo
Page 527 and 528: 515score probability in general tim
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