Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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242 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsLet us now consider the case of a volume source. Multiplying the equation by Q V(P)in Equation (5.247) and integrating with respect to P, the resulting equation readswhereM., = I 1+ — j dP J dTe- 1 — Idto'M.ir + Dio.co') (5.248)4"frv JV Jo 4TT JM, =T4'JTV JandlT idPM,(P) = R (5.249)= 4^V I D P I ' ( P )The last and most important approximation in our model is that M 1I -P") in Equation (5.248)is replaced by its average value M 1in Equation (5.249). Doing so, Equation (5.248) becomesM 1« I 14 CP 1M 1i.e.,M 1= I 1Z(I - cP c) (5.250)wherep=_L Jd PI ci're' T = IdP[I - e~ t r , w ]4irV Jv Jo 4TTV JV= — dr dw[l - e-" D fr^] (5.251)4LTV JV J4-HP cis the first-flight collision probability 8in V. More precisely, P, is the probability that aparticle started in V from an isotropic, uniform distribution suffers its next collision in V.The approximation above is called the separation assumption, referring to the fact that theaverage of a product is approximated by the product of the separate averages. It can be seenthat in the simple form above, the separation assumption Is equivalent to the assumptionthat the collision points are uniformly distributed over V. An analogous approximation wasintroduced through an invariance principle by Stuart 44for escape probability calculations.Note that the first-flight collision probability P cfor a body of a given shape is a uniquefunction of the optical mean chord lengthT = a? (5.252)in the body, 8whereI = 4v/Sis the geometrical mean chord length.
243In the case of the collision rate estimation, the source term, of the first-moment equationI, in Equation (5.248) equals the first-flight collision probability, I 1= P cand hence theexpected total score in Equation (5.250) readsM 1= R = P c/(1 - cPJIn the case of the surface source, Equation (5.246), the expected total score duincidentparticles isM 1= —^dPM 1(P)and it is similarly approximated asM « I 1+ CP 1M 1(5.254)whereand1 f f D(P)2-rrSIt can be seen 8that P Cis the probability that a neutron incident isotropically ion the surface of V will suffer its next collision in V, and it is related to tcollision probability P CasP c= T(I - P c)Inserting Equation (5.250) into the RHS of Equation (5.254), the expected total score dueto the surface source becomesM 1- I 1+ I 1TC(I - P c)/(1 - cP c)Again, for collision rate estimation, I 1= P C, I 1= P c, andM, « f(l ~ PcV(I - cP c) (5.256;In the expressions of the approximate total score, Equatiof the first-flight collision probability P Cappears, whose
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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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