Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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64 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhereas the variance can he empirically estimated by the square of the standard deviation(s).n - 1 E (m - Ad2 (3.37)Equation (3.30) is an unbiased estimate of D(cp), i.e.M(s 2 ) " D 2The standard deviation of the mean(sji) is an unbiased estimate of D/'Vn:S (m-, - fx) 2 (3.38)Vn V n ( n ~ 1) .«This formula is hardly applicable in an actual calculation, since evaluation of the subtractionson the K(IS assumes the storing of all |x rs up to the end of the run. A moreconvenient formulation is achieved by elementary transformations:nis2 (P-. - M-) 2 = 2 (P-? - 2 P-.M< + p-) 2= Eft + n — ( X P-i= S ft 2 - - ( X ^This latest formulation drastically reduces the store requirement: one has to store onlythe sums of the scores and their squares.Now, the empirical standard deviation is computed as:1n(n - 1)(3.39)Frequently, the relative standard deviation, or coefficient of variation, is given. It isdefined by the relations,. =and hence can be computed ass,. =n - 1S Pin.(2 (i,,) 2 n JDuring the simulation of the history of even a single particle, the score may be obtained
65as a sum of many contributions. If the i-th particle undergoes m interactions, with !,.contributions, each, then the score isniNaturally, for the statistical evaluation the score ((X 1) is to be handled as an independentestimate, its composition from the f Mcontributions is not of interest.C. THE EFFICIENCYFrom the computational point of view, it is not the variance itself but the computer timerequired to reach a given variance that should be reduced by the introduction of nonanaloggames. A modification of the analog game will certainly change the average time requiredby the individual simulations.It follows from Equation (3.38) that if the standard deviation of the mean (the quantitywhich characterizes the statistical uncertainty) is limited to s u>than the minimum numberof simulations isif s 2is the variance of the method. If the average time per simulation is denoted by t, thena computer time of(3.40)is consumed during the total game.If one compares techniques, the error limit (s c) is fixed, thus the time in Equation (3,40)is proportional to product of the variance square and the average time per simulation. Theinverse of this product is called the efficiency:better.In the comparison of two techniques, that with higher efficiency is considered to be theA detailed analysis and comparison of the efficiencies of a wide range of techniques isthe task of Chapter 7.APPENDIX 3A: ENERGY SELECTION FROM THEKLEIN-NISHINA FORMULAAs has been stated in Section 3.1.D the differential ;is generally approximated by the Klein-Nishina formula, dr rfor initially free and at rest target electrons.
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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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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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