Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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444 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsdenotes the variance of the score due to a starter from x, then Equation (5.98) with thestraight-ahead scattering kernels becomesD 2 (x) = e- ( X - x ' - M 2 (x) + c F dx'e «' °M((x) + - | dx'e" < x '- s ) D 2 (x')Jx n Jxc f x= I(x) + - dx'e- , V l ) D 2 (x') (7.3)n AIf we insert the expression of M 1(x) in Equation (7.2) into the source term of Equation (7.3).we obtain!(x) = [ e-
445and on the optical thickness X. Analytical minimization of Equation (7.7) is cumbersomebut it can be easily minimized by simply calculating Q(n) at successive integer values of itIn Table 7.1, optimum splitting ratios are given for selected values of c and X.TABLE 7.1CX 0.1 0.2 0.3 0.4 0.5 0.62.5 1 1 1 1 1 [3.0 2 1 1 1 1 14.0 2 2 2 1 1 15.0 2 2 2 1 1 110 3 ? 2 2 2 115 3 3 2 2 2 120 4 3 2 ? 230 4 3 3 2 2 240 5 3 3 2 2 250 5 4 3 3 2 2The numerical results obtained from the mode! are certainly not realistic; however, theyreflect the main tendencies expected from more rigorous models. Thus, it is seen that theoptimum number of split fragments per collision increases with increasing absorption (decreasingc) and also with increasing thickness of the slab. It is also observed that the optimumsplitting ratio varies very slowly as the thickness of the slab is increased and therefore thereal optimum is very likely a noninteger value of n. In other words, collisionwise splittingcan only be changed in rough steps and does not seem to be a sufficiently fine tool ofefficiency maximization. (One could object that a noninteger splitting ratio can also berealized by allowing for two possible outcomes of every splitting with appropriate probabilites. It can, however, be seen that even in this case, collisionwise splitting is less flexibleand efficient than geometrical splitting.)Geometrical (or surface) splitting can also be optimized analytically in the straightaheadmodel. Let us consider again the transmission of particles through a slab of thicknessX in which the mean number of secondaries per collision is equal to c. Analytical treatment,of the problem is especially simple in purely absorbing media. Efficiency of surface splittingin this case was investigated in Reference 15. However, in most practical cases, optimizationof splitting is related to deep-penetration calculations with mild absorption and thereforeresults from a purely absorbing model are not very enlightening. The straight-ahead scatteringmodel offers a simple tool for investigating the geometrical splitting procedure.Let a splitting surface be situated at x = x», i.e., assume that a particle is split into nfragments whenever it crosses this surface. With the notations of Section 5.IV.B, this meansthat the only nonvanishing splitting probability is g n(x) and it is nonzero at x = x, only:§k.n if x = x.0 otherwiseFurthermore, the weights of the split fragments areW (l)„ = 1/n, (i = 1,2, n)
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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
Page 405 and 406: 393Now, since bl(P) = g(P) is bound
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Page 409 and 410: 397and its expectation isR — R 1n
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Page 413 and 414: 401plays a distinguished role), and
Page 415 and 416: 403Theorem 6.7 — If x and s denot
Page 417 and 418: this a priori information, the best
Page 419 and 420: 40?Thus, the bias introduced into t
Page 421 and 422: 409In view of Equation (6.226) and
Page 423 and 424: 41.1This means that if we use the k
Page 425 and 426: 413Then the whole-sample and rare-s
Page 427 and 428: 41.5According to the results of Sec
Page 429 and 430: 41?(Note that vD 2 [|i|v] is indepe
Page 431 and 432: 419withkOn the other hand, § is an
Page 433 and 434: 421Thus, the expected ratio reads
Page 435 and 436: 423St follows from Equations (6.265
Page 437 and 438: 425Wc start from the observation th
Page 439 and 440: 427Nowi'6.77K)kj„,and therefore(V
Page 441 and 442: 42*3Similarly, for the fourth momen
Page 443 and 444: 43 sOn the other hand, multiplying
Page 445 and 446: 43.¾This will be proven in the lem
Page 447 and 448: 435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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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
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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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