Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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334 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsthenSeveral other special cases are reported, for example, in References 67 to 69.As a consequence of the additive weight generation rules, it may happen that thestatistical weight of a particle becomes negative, a phenomenon preferably avoided inordinary nonanalog games.Another consequence of the additive weight generation rules is that in an analog differentialgame (which is played according to the analog kernels), particles temporarily havingzero statistical weight must not be eliminated from the simulation (as is done in an ordinarygame). Such particles may obtain nonzero weights in a later stage of the simulation. In otherwords, statistical weights in an analog differential game determine the actual contributionsof the particles to the score, but are not measures of the "size" of the particles, which isthe case in an ordinary nonanalog game.A further pecularity of the differential games is that in a nonanalog differential game,multiplicative and additive weights are present in parallel. They are to be generated independently,according to the respective generation rules. The score is to be multiplied by theproduct of the two weight factors, but all the weight-dependent biasing schemes like splittingand Russian roulette are to be applied according to the multiplicative weight value. (Notethat the additive weight is sometimes considered simply as part of the contribution function.In our formalism, this weight is a direct descendant of the multiplicative weight of anordinary nonanalog game, while the definition of the contribution function is unchanged.Therefore, we think it more logical to call it a weight rather than a contribution function.)In what follows, we discuss briefly the nature of the moment equation. Since the specificform of the weight generation rules has not been exploited in the derivation of the momentequation in Chapter 5.II, the relations in Equations (5.56) and (5.58) apply here too. Thus,the r-th moment of the differential score satisfies the equationwhere W ( nis the weight of the particle at P, WJ 0and W" nare related to W ( 1 )according toEquations (6.55) through (6.58), and we have omitted a from the arguments of the kernels.In spite of the formal equivalence of the ordinary nonanalog and differential moment equations,they are essentially different. This difference is due to the fact that the former, aftermaking use of the (multiplicative) weight generation rules, does not contain informationabout the previous history of the particle, while the latter will always "remember", due tothe additive weight factors. This property can be best visualized through the first-momentequation. From Equation (6.59) with r = 1(D(6.60)
335According to Equations (6,55) through (6.58)WJ n_{ +w"'(P,P')W 1~ ' W ( 1 )andw;,, W«"(P,P') + Wj))(P',?")W "(I) ~+W YY(1>still containing W (1), the. weight accumulated during the history precedingP. In contrast to that in an ordinary nonanalog game, the corresponding weigfrom Theorem 5.8 asW'—W= w(P,P')and= w(P,P')w s(P',P")independently of W.It is interesting to note that an alternative form of the differential first-moment equationcan be derived from the analog unperturbed moment equation. Let us write the ordinaryfirst-moment equation (5.57) into the following formM{s}(P) = J dP'T(P,P')f(P,P') +|dP'T(P,P')JdP"C(P',P")M{s}(P")Since, ( ds 1 ddifferentiation of the first-moment equation above yieldsM j^-J(P) = J dP'T(P,P')w (,) (P,P')f(P,P')+ JdP'T(P,P')JdP"C(P',P")[w (l, (P,P') + w^CP'.P'OJMisKP")+ |dP'T(P,P') JdP"C(P',P")M|-^|(P") (6.6 i)This equation is clearly different from Equation (6.60) and is similar in form to the momentequations investigated in Chapter 5, i.e., it is indeed of the Fredholm type. This, however,is reached at the expense that the source term of the equation contains the expected scorein the ordinary game. On the other hand, Equation (6.61) suggests an alternative methodof estimating parametric derivatives. This method is analogous to the perturbation sourcemethod discussed in Section 6.LE and will not be repeated here.
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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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9B. THE PROBABILITY MIXING METHODTh
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25!) Monte Carlo Particle Transport
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Page 296 and 297: 284 Monte Carlo Particle Transport
Page 317 and 318: 305Chapter 6SPECIAL GAMESIn the maj
Page 319 and 320: 3©7A. CORRELATED MOMENT EQUATIONSI
Page 321 and 322: 309Equation (6.1) will be referred
Page 323 and 324: 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: 333and from Equation (6.51)w';,,(i)
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 and 376: 363we haveD 2 [k] = < 2 (k, - k,) )
Page 377 and 378: 365neutron density, S 0 for every
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
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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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