Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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364 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswe have a conservative estimate of the varianceD 2 [kJ =£ V 2 [l + 2p,/(l - d)]/(n - m + 1)Yet two problems remain to be solved when we estimate the variance according to thisexpression. First, the dominance ratio, d, is not easy to estimate. Second, numerical experimentsshow 25that in certain cases this approximation is overconservative, i.e., it resultsin spuriously high estimates of the variance. Gast and Candelore 25propose an alternativeformula with empirically fitted constants:D 2 [k] = V 2 [l + 10p 2 /(l - p,)]/(n - m + 1) (6.125)Equation (6.125) can be directly estimated in the iteration procedure as follows. An estimateof the generationwise variance V 2is obtained asnV V (k, - k) 2 /(n - m + 1)j"«'and the correlation coefficient p, is estimated asn- 1Pi = S (kj - k)(k J + 1~ k)/[V 2 (n - m - 2)]j = mTest calculations show 25that for small systems the correction term [second term in bracketsin Equation (6.125)] is negligible and only becomes essential if the multiplying materialextends over larger regions.E. A ONE-STEP SCHEME: ACCELERATION OF THE ITERATIONIt was shown in Sections 5.V.C and 5.VIII.F that importance sampling may, in principle,result in estimation schemes of fixed-source reaction rates with zero variance. The sametechnique can also be implemented in the source iteration procedure to obtain an exactestimate of the effective multiplication factor. Most naturally, zero-variance schemes ineigenvalue estimations are no more realizable in practice than those in fixed-source problems.Nevertheless, approximations to these schemes may substantially reduce the variance of theestimated eigenvalue.Importance sampling resulting in zero-variance criticality calculations was first proposedby Goad and Johnston. 29They combined the method with a technique similar to the ELPmethod (cf. Section 5. VIII. D), as a result of which the starters are not allowed to escapefrom the system. An alternative scheme proposed by Kalos 38is given below. It is based onthe method of importance sampling without statistical weights.Consider the fission-neutron density formalism of the third method in Section A. Thespatial distribution of the fission neutrons, S(f), is governed by Equation (6.104)S(r) = ----- dr'Z(r,r')S(r') (6.126)k efr Jwhere Z(r,r') is the density of the fission neutrons emerging from around f due to a fissionneutron started from r'. In the course of the source iteration, the n-th-generation fission
365neutron density, S 0 for every r such thatdrZ.ir.rd > 0i.e., V is definitely positive at every point that can serve as a starting point of fission neutronscontributing to the generation of next-generation fission neutrons. LetSrXr) = S(r)V(r)andZir.r'l = V(r)Z(r,r')/V(r') (6.129)Then Equation (6.126) can be rewritten asSr(r) = jdr'Z(r,r')£f(r') (6.130)Equation (6.130) is analogous to Equation (6.126) and defines an iterative procedure similarto that in Equation (6.127):cf (n) (r) = jdr'Z(r,r')y ( n -"(r') (6.131)By analogy to Equation (6.128), an estimate of k eiTfrom the n-th and (n — l)-th generationsreadsk„ = Jdr^ryJdrcT^ ')(r') (6.132)Now, if the importance function V(f) is the solution of the equationV(r) = — jdr'V(r')Z(r',r) (6.133)
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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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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
Page 325 and 326: 313andC S(P',P")/C S(P',P") = y(P',
Page 327 and 328: 315introduced in Equation (6.1) for
Page 329 and 330: 317andL„(P 0,P;,...,Pi + 1) = L n
Page 331 and 332: 319whereB„(P„,P; P:, ,) = A*(P
Page 333 and 334: E. EXAMPLES AND SPECIAL TECHNIQUESL
Page 335 and 336: 323The weight factors associated wi
Page 337 and 338: 325It is seen from Equation (6.37)
Page 339 and 340: 327G. PARAMETRIC PERTURBATIONS: INT
Page 341 and 342: 129Hall 31 gave a constructive deri
Page 343 and 344: 331kernels, which remain unchanged
Page 345 and 346: 333and from Equation (6.51)w';,,(i)
Page 347 and 348: 335According to Equations (6,55) th
Page 349 and 350: 337Minimization is performed by set
Page 351 and 352: 339and letdsdaLet us consider a cor
Page 353 and 354: 341whereis the length of the flight
Page 355 and 356: and from Equation (6.79)I / ds \ 2
Page 357 and 358: 345Let us assume that we are intere
Page 359 and 360: 347Introduction of this factor also
Page 361 and 362: 3492. Let i|/ n (P) denote the solu
Page 363 and 364: 3SJNow, if ( "> > 0, then k = i an
Page 365 and 366: 353andk„ = S l "VS'" •" (6. if;
Page 367 and 368: 3SSLet k„ denote the normalizatio
Page 369 and 370: 3572. It is then proved that with t
Page 371 and 372: and start new histories until N new
Page 373 and 374: 361The application of source iterat
Page 375: 363we haveD 2 [k] = < 2 (k, - k,) )
Page 379 and 380: 367if the same relation holds for t
Page 381 and 382: 369Then the multiplication factors
Page 383 and 384: 371in the unperturbed system, the w
Page 385 and 386: 373The perturbation of the effectiv
Page 387 and 388: 375where P" = (r',E'). Multiplying
Page 389 and 390: 377iterate of the derivative of the
Page 391 and 392: 579where P* = (r*,w*,E) = (r*,E*).
Page 393 and 394: 381This estimator, unlike f, in Equ
Page 395 and 396: 383tends to some stable attracting
Page 397 and 398: 3§SIo the case of the next-event e
Page 399 and 400: 387LDFIGURE 6.1.Geometry in scoring
Page 401 and 402: 389Let us denoteThen Equation (6.18
Page 403 and 404: 391is to be increased in order to o
Page 405 and 406: 393Now, since bl(P) = g(P) is bound
Page 407 and 408: 395is scored at every collision wit
Page 409 and 410: 397and its expectation isR — R 1n
Page 411 and 412: 3994. If O > 0 m, then the particle
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
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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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