Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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244 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsIn the case of a general monoenergetic moment equation, the separation assumption willbe used in the following formdPCA(P) JdP'T(P,P')A(P,P') JdP"C(P',P")M(P")dr dw dTe ~ T A(r,w;r + Dw,w)c(r + Dw) M (5.257)Where we put X = V, a x= 1/4-rrV for a volume source and X = S, a x= 1/2-rrS for asurface source. A(P,P') is some given function, c(P) is the mean number of secondaries percollision at P, and M stands for the total score moment in questionM =4TTV JdPM(P)C. ON THE QUALITY OF THE APPROXIMATIONThe form of the separation assumption as introduced above suggests that it may onlybe used in monoenergetic, isotropic transport in bodies of more or less regular shape.Although the assumption might be generalized to more realistic cases, such a generalizationdoes not seem to offer any advantage, for two reasons. First, in complex problems, theevaluation of the integrals on the RHS of Equation (5.257) would be almost as laborious asthe application of more precise deterministic solutional schemes. Second, the approximationsproposed here are primarily intended for semiquantitative (and usually comparative) analysisof various Monte Carlo methods and, except for very special cases, such an analysis canbe performed on elementary transport problems with satisfactory results.Therefore, when judging the quality of the approximation, the most meaningful questionis, how well does Equation (5.253) approximate the exact collision rate of monoenergeticparticles in a homogeneous medium with isotropic scattering? In other words, what can besaid about the difference of the exact and approximate solutions'? Extensive numerical testsshow 31that the approximate and Monte Carlo values of collision rates and related quantitiesare in very good agreement for a not-too-high number of secondaries per collision (c < 1),and the approximation consistently underestimates the exact collision rates.Although no general proof has so far been found, the following arguments make it morethan probable that underestimation is an inherent feature of the approximation. Let usintroduce the following notationsM 1(r) =-- ~4irJdCoM 1(P)andI ,Or) = •— Idw IdP'T(P.P') = —- ldc*[l e" < r D < H "4ir J •'• 4TT JThus, from Equation (5.251), the first-flight collision probability readsPc == ~ jdrl,(r) (5.258)
245Integrating the first-moment equation (5.247) with respect to co we have, withr - Ir' - r/to + RM,(r) = I,(r) + — dcocr dr'-r|e "" cr|r ''"" r| cM ](r 4-wlr'-r|)4TT J JV 'or substitutingdwd[r'—r| = dr'/|r'--r| 2the moment equation becomesM 1Or) = I,(R) + c dr'G(r,r')M,(r') (5.259)whereG(r,r') = G(R',R) = — e-'^'-^/lr'-r| 24TTAccordinglyJ 1Cr) --= Jdr'G(r,r') - [dr'Gtr' r) (5.260>The solution of Equation (5.259) is written in the Neumann series fromM,(r) = £ c*~'I k(r) 15.261;whereI K + 1(R) = DR'G(R,R')I K(R') (5,262)Finally, denoting P 1= P candJvP k= 7 VJ ydrl t(r) (5.263)it follows from Equations (5,261) and (5.263) that the. total expected score becomesM 1= ^ E-'P k(5.264:k= iNow this quantity is approximated in Equation (5.253) asM 1« 2'P"
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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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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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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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