Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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270 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsi.e., the average number of flights with the separation assumption isN = 1/(1 nP c)The product of the variance and the number of flights in the survival-biased analog gamebecomesE 1 == D 2 N - P c(l P c)/[d - enP c) 2 (l - e 2 nP c)(l - nP c)] + a (5.334)wherea = P ce 2 n(n - l)[(m (2)- M 2 ]/[(l - e 2 nP c)( 1 - nPj] (5.335)The second moment of the score in the equivalent nonmultiplying game followsEquations (5.322) through (5.324) asfromM 2(P) - 2M 1(P) - JDP'T(P 1P') + | vdP'T(P,P')|dP"e 2 n 2 C s(P',P")it 2(P")i.e., the average second moment in the separation assumption isM 2= (2M 1- P,)/(l - e 2 n 2 P c)Hence, the variance readsS 2 -- PfI - P c)/[(1 enP t) 2 (l - e 2 n 2 P c)]The mean number of flights satisfies the equationX(P) = 1 + JDP'T(P |dP'T(P,P')|dP"C 7P'^s(P',P")>f(P")i.e.,M = 1/(1 - P c)Finally, the inverse of the efficiency in the equivalent game reads= P C/[(1 - enP c) 2 (l 6 2 n 2 P c)Now the efficiency of the equivalent game is higher than that of the survival-biased analoggame if E 1 > %~ '. Notice that a in Equations (5.334) and (5.335) is nonnegative for nSs 1 sincem,„ = -' - IdP'4TTV 1 dP"C,!P',P")M,(P") -' - - dP' jdP"C s(P',P")M,(P")_4TTV J J '= MTThus, the efficiency relation above certainly holds if E2"' — a > 'to' 1 , i.e., if(1 - P c)/|(l - e 2 nP c)(l - nPj] > 1/(1 - e 2 n 2 P c)
27!After rearrangement, this relation becomes( n_ 1)(1 _ e2 n) > Owhich holds if1 < n < 1/eor,equivalently, if1 - c„ < c < 1/(1 - C 11)Numerical experiments show 3 -' that the gain in efficiency predicted byabove is actually realizable. Figures 5.4 and 5.5 present the efficiency ratinonmultiplying game and the survivabbiased multiplying analog game v-be*'" < v i • •collision rate in homogeneous slabs of different thicknesses, x, in a moaoetwi_ i. « isimulation. The absorption probability is c a= 0,2 and 0,3, respective! | > i i •i «•vof secondaries per collision, c = n(l — C 2)]. It is seen that the efficien » •
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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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Page 232 and 233: 220 Monte Carlo Particle Transport
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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
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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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