Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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328 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe integral Monte Carlo method can be easily extended to problems where the changein the reaction rate is due to simultaneous changes of several system parameters. In thiscase, however, it is necessary to estimate the derivatives of the reaction rate with respectto all the parameters involved, which requires either several independent differential MonteCarlo estimations or a correlated differential eame.II. DIFFERENTIAL MONTE CARLO: SENSITIVITY ANALYSISAnalysis of the effect of uncertainties in some characteristic parameters on given reactionrates, or determination of the sensitivity of reaction rates to small changes in the systemparameters, play an important role in nuclear design and operation. In complex systems,the only way of performing such an analysis is through Monte Carlo simulation. CorrelatedMonte Carlo presented in the previous Chapter is a possible tool for examining such effects.An alternative method is the estimation of the sensitivity of the reaction rates on some systemparameters that characterize the uncertainties (perturbations). To be more specific, let usconsider a reaction rate R depending on the parameter a. Assume that we are interested inthe variation of the reaction rate when the parameter varies around a given a value. If Rcan be expanded into a Taylor series around this value, the variation due to a change of Aain a is expressed asd"RR(a + Aa) - R(a) =- 5R(a) = ^ TT (Aa)" (6.44)where d" R/da" is the n-th derivative of R(a) at a. If the change Aa in the parameter a issufficiently small and R(a) is a suitably smooth function of a, then the first few terms onthe RHS of Equation (6.44) approximate well the change in the reaction rate.(6.45)Parametric derivatives of the reaction rate are estimated in differential Monte Carlo games.Use of the Taylor series form of the change in the reaction rate has the advantage over thecorrelated Monte Carlo technique of providing this change for arbitrary (but small) variationsof the system parameter(s), while correlated games determine the perturbation due to a givenvariation of the parameter(s) (cf. Section G). In other words, the derivatives are characteristicof the sensitivity of the reaction rate in question to the variation of the parameters. In fact,sensitivity is defined as the ratio of the fractional changes in the reaction rate and in theparameter when the parameter perturbation tends to zero, 44i.e.,Equations (6.44) and (6.45) concern the case when the reaction rate changes due to thevariation of a single system parameter. In the case of several parameters, the correspondingmultivariate Taylor series applies and the partial derivatives of R with respect to the parametersin question have to be estimated. For the sake of simplicity, we shall constrain ourderivation to the single-parameter case; extension of the considerations to several parametersis straightforward.An unbiased procedure of estimating the first derivative was first proposed by Mikhailov 57and independently by Miller' 9 and Takahashi. 81 The schemes were proposed for the calculationof reactivity changes in nuclear reactors due to small variations in system parameters.
129Hall 31 gave a constructive derivation of a multiparameter s*= • r , , j dc >« >'i c , . jprocedure. The relative merits of the correlated and diffeuur. , >«r < PRUU" 1culations are compared for special problems by Rief, 66_6!iatthat estimates parametric derivatives is given by Matthes."An unbiased game estimating the first parametric derivative is derived in Section i basis of measured reaction rates and calculated sensitivities. The considerateto estimation of higher-order derivatives in Section D. A simple analyrica' -i • •in Section E illustrates the procedure. Finally, the derivations are extended t< \> >> ' ,contribution functions which also depend on the system parameters. In die disthe game that estimates the parameter-dependent reaction rate at a given i> u»will be called the unperturbed game; the system at the given parameter vale 3 * 1 « r>turbed system. The differential game estimates the derivative of the uiiperate with respect to the parameter at its given value.A. ESTIMATION OF FIRST-ORDER DERIVATIVESLet us first consider a correlated unperturbed analog game and recall the second formof the generalized moment Equation (6.7) that gives the moment of a correlated score functionF(s) = F(S 1 5S 2):M{F(s)}(P„, 1) = (Fwhere pointed brackets stand forE W'(i)f(i) E w(i)f(i)(FJ...]) = E {dP! JdP 1... (dP„ JdP nJdPO 1 1(6.46)n T(Tv 15POC(POP 1)i=land for the sake of brevity, we putT(P 11JO +1)FJ.,.]f(i) = f(P,_. S,P0the contribution function assigned to a flight from P 1..., to PJ. It is tacitly assumed in Equation(6.46) thatEWe shall first assume that this contribution function is common to both [correlated) partkiesConsider a correlated game in which the unperturbed system corresponds to a parametervalue a and the analog particle scores S 2in this system. Let the perturbed system becharacterized by the parameter value a 4 Aa and let the nonanalog particle give the scores,. This means that the transition and collision kernels in the two systems differ due to the
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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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Page 290 and 291: 278 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
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Page 335 and 336: 323The weight factors associated wi
Page 337 and 338: 325It is seen from Equation (6.37)
Page 339: 327G. PARAMETRIC PERTURBATIONS: INT
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 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
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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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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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