Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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476 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsequation system is easily obtained:dxvk(\) Ks(x) = dk'l 1,(k'|x)P*(k'.x)+ 2 (PKr)(X) aVr^k'bOP^k'.x)(7.98)The boundary conditions in Equation (7.97) yieldkt(0) = 1, k;(X) = B v, 0(7.99)In Equation (7.98), the following notation has been introduced:Uk'jx) = jdkk"t(k',k|x) = — { dkk"T(k'.k|x,y)| y(7.100)andr„(k'|x) = |dkk"r(k',k]x) = — F dkk"R(k',k|x,y)| y(7.101)Equation (7.98) is not yet suitable for further manipulations because in its RHS it containsthe moments of the functions t and r, and not only the moments of P ' . In order to proceed,we introduce a further assumption concerning the transition and reflection probabilities.Let us note that for integer values of the particle numbers k and k', the probabilitieshave the formA(k',kjx,y) = A')A k (x,y)[l - A(x,y)] for A - T or R (7.102)whereA(x,y) = (i.l|x,y)for A = T or RThe moments of t and r follow from Equations (7.100) and (7.101), with the aid of Equation(7.102), as0(k'|x) - 0, a,(k'|x) ----- k'a(x) (7.103)anda 2(k'|x) = 2k' 2 a(x)A(x,x) + k'2(x)[l - 2A(x,x)] (7.104)where a = r or t and a(x) is related to the derivatives of T and R in Equations (7.92) and(7.93) asa(x)da (1,1 x) - —• A(x,y)dyfor a = t or r, and A = T or R. Now, since T(x,x) =1 and R(x,x) = 0, we have from
47'?Equations (7.103) and (7.104).t,(k'|x) = k't(x),U(k'|x) = K'(2k -- I)T(X)andr,(k'jx) =r,(k'ix)k'r(The relations above hold for integer values of ! ^ ' ,tjar noninteger values of k'. 'Then introducing (,'s, (,i,(7.98), we obtainddxX(x) Ki(X) = t(x)K-f(x) + r(x)K](x)for the first moments and-- - 2X(x)dxK 2(X) == t(x)|2Ki(x)Kv(X)r(x)K; (x)[l + 2Kf(X)]for the second moments. Equations (7.107) and (7.108) . > >- >•of simulation particles propagating in positive and neg;x in the slab. What we are really interested in is the \ U 1 1, n >.,.variance of the number of real particles leaving the slab « . iparticle at x has a weight equal to 1/I(x), the v-th monat X, i.e., the vth moment of the score, is related to the mot. A
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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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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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Page 441 and 442: 42*3Similarly, for the fourth momen
Page 443 and 444: 43 sOn the other hand, multiplying
Page 445 and 446: 43.¾This will be proven in the lem
Page 447 and 448: 435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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Page 482 and 483: Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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Page 496 and 497: 484 Monte Carlo Panicle Transport M
Page 498 and 499: 486 Monte Carlo Panicle Transport M
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Page 525 and 526: 513INDEXAAbsorptiondefined, 25photo
Page 527 and 528: 515score probability in general tim
Page 529: 517path stretching, 447—455splitt
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