Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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70 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsSelect p,, p 2, P 1, and p,P 1> 0.15536*yesComputeR = p,* p,ComputeR=I- logp,p, > exp(-R) >P 1> "vR f>yesR = R- logp„* Exactly: ———, where e is the base of the natural logarithm.0.5 + eFIGURE 3B. 1. The flow chart for selecting the energy from the Maxwellian distribution.where p ;-s are independent samples from the Gaussian distribution. The fact that the selectionof normally distributed random numbers is rather time consuming makes this process inefficient.2. Selection from a Gamma DistributionSelection schemes can be based on recipes developed for the gamma distribution. Thegeneral gamma distribution is given byP(R) = FT"; R T r ~ 1 e" X x1 (Tl)which turns into the form of Equation 3B.1 if X = 1 and if) = 3 A 2. The selection schemegiven by McGrath et al. 3sis presented in Figure 3B.1.B. NEW ENERGY SELECTION FROM THE DIFFERENTIAL THERMALNEUTRON CROSS-SECTIONStill preserving the free gas model with no interference between the scattering atomsand no chemical binding effect the differential scattering cross section in the laboratorysystem can be approximated as follows(j E / m, f m, / K 2 X 2 I ,„ „CT(E 0-^E.uf) = -- /— ~ exp -- ;,- E 0- E B.2^ 4-rr V E 0y] 2*. 2 1 2KTC 2 \ ° 2m,/ Jwhere m, is the mass of the scattering nucleus, m is the mass of the neutrone -- 2m(E + E 11-- 2uVEE ( )).and
71Following Erikson's results' 3the expression of Equation (B.2) can be transformed sothat the resulting PDF (the normalized cross section) consists of the product of two independentprobability variables: p and q. The new probability density function ish(p.q) = CpexpC-q 2 );B..7where C is a normalization constant,P = Vl - 2Z(x + Z 2q = [Xp + (Z 2 - l)/pimX = --m tThe new variables p and q have physical meanings only if their range consists of two sets 1 ":R(p,q) - R,(p,q) + R 2(p,q) (B.4)whereR,(p,q) = jp.q: -a < q « a, 0 andR 2( P,q) = {p,q: a < q, (H - i) ^
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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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120 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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