Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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108 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsV. SCORINGA. GENERAL FORMULATION OF THE REACTION RATESIn Chapter 4.IV., methods for finding the r, E and w coordinates representing eitherthe ingoing or the outcoming densities were described. For practical purposes generally notthe collision densities themselves but a functional of one of them, a reaction rate or receptorresponse is to be determined. The general formulation of such functionals was given inEquation (4.27) and the recipe for a Monte Carlo estimate was described in Theorem 4.8.Theoretically. most reaction rates or responses can be derived either from the ingoingor the outcoming densities and it is up to the user to choose the most suitable approach.Accordingly, there are (wo basic definitions of any reaction rate R:andR = JdPf x(P) x(P) (4.49)R = JdPf +(P)Vi-(P) (4.50)where f xand f,j, are the pay-off functions related to the respective collision densities.The actual forms of f and f Alare determined by the physical quantity to be estimated.However, from the elementary requirement that the result R must be the same, whether itis derived from Equation (4.49) or Equation (4.50) a relation between the two pay-offfunctions can be derived in the following way:Let us substitute Equation (4.35) into Equation (4.50):R = JJ drdEf +(r,E) jdr'T(r'-~>r|E)X(r',E) (4.51)If now we interchange the order of integration over r and r'' and then the symbols r and r'themselves Equation (4.51) becomes:R = JJ dr dE X(r ,E) Jdr' T(r->r' JE) f +(r' ,E) (4.52)From comparison of Equations (4.52) and (4.49) the relationf x(r,E) - Jdr'T(r--r'!E»f.(r'.E) (4.53)is obtained.B. ESTIMATION OF MORE IHAN ONE RESPONSEThe same set of random walk simulations can be used for the estimation of many differentresponses or reactor rates.One may be interested in the flux integrals in two or more different regions, or incalculating differential quantities; e.g., in determining the fluxes for different energy intervals.In such cases a set of responses is to be determined by the use of a set of pay-offfunctions:R' -dPff (P) Uj(P)
109ordPf;(P) x(P)The crucial point here is that all estimations are based on the same re]collision densities and therefore the results R (1) will not be statistically md^ ^i ,Kother. This technique, therefore, can be regarded as a use of the correlated ' jm; li>>v i >in Section 4.II.F. When talking about the "result" of a Monte Carlo game it- a- c < oihtiis.ndata obtained from all collisions and the user provides a supplementaryquantity which interests him from the collision data.C. MONTE CARLO ESTIMATION OF THE RESPONSESThe Monte Carlo estimation of a functional of the function
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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
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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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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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484 Monte Carlo Panicle Transport M
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486 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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