Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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110 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe score from an entire history is the sum of the contributions:fx x= S r„ x(4.59)or2 (4-55)If the f xand C 4, pay-off functions are correctly chosen then the expected values of the(4.54) and (4.55) estimates must be the same:W - W = RAfter following j = 1,2, . . . ,n histories each scoring p, XJand/or u.^ , the average ofthe scores is an unbiased estimator of R:ij=in j „ iD, EXAMPLES OF PAY-OFF FUNCTIONSAn extremely wide variety of physical quantities may be determined by Monte Carlo.In the following subsections several representative examples are shown and discussed.1. Collision Density and Reaction Rate IntegralsThe simplest question to raise in a Monte Carlo game is: how many collisions occured,i.e. how many particles entered collision, in a certain volume element (or domain) F? Theanswer is trivially:R = dr dE.).(r.E) = drdEh r(r)cj>(r,E) (4.56)rwhere h r(r) =1, if reT0, otherwiseThus from a comparison of Equations (4.50) and (4.56) the pay-off function is:f +(r,E) = h r(r)If only a part of the spectrum is of interest, e.g., one wishes to count the number ofcollisions which the particles enter with energy between E 1and E 2, then the correspondingpay-off function becomesf 1((r,E) =h r(r)h, ;(E)where h, =0,if E 1< E < E 2otherwise
IllSimilarly, solid angles of interest can be "cut out" by a window function h,„.The use of the counterpart pay-off function f can inform us about the number of particlesleaving collisions in a certain phase-space domain.The rate of a certain kind of interactions is also easily determinable. The number ofinteractions type i (having a cross section CT 1) is trivially:drdE -^—f i|i(r,E)The corresponding pay-off function is obvious from this formula.The flux integral can also be handled as a reaction rate. From the elementary connectionbetween the ingoing density and the flux, the flux integrated over a domain T is expressedasR = | dr j dE
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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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160 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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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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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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