Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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14 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthus the CDF isP(x) ' p(x)dx = X 1 1 ' 1Joi.e., the direct inverse distribution method can be applied, a random sample x is obtainableby Equation (2.2):p = x" M x = p"">*-o (2.9)This method is not fast enough in most computers since the execution of the a boperationis quite slow if b is not an integer. Therefore, if n is an integer, the following method canreplace the execution of Equation (2.9).Theorem 2.6 — Let x (the sample) be the maximum of n + 1 random numbers, thenthe x is a representative sample from Equation (2.8).Proof: The probability, that a random number p, is equal or less than x c(0 =£ x„ =£ 1) isjust X 0. The probability that all random numbers from a set of n + 1 independent element isless than x„ isN t 1P(Pi " x„, p 2x„,...,P 11+1X 0) = n (p, x L))i = iN +1=n x ()=1x 01=i•In the method proposed here, a set of n + 1 random numbers has to be generated. Inmost computers the generation of the elementary random numbers is much faster than theexecution of Equation (2.7), however, especially for large values of n. it might be advisedto check the actual computer times before the selection of one of the two procedures.If x is equidistributed on (a,b) then the transformation /. - - (x - a)/(b - a) should firstbe applied.F. SAMPLING FROM THE NORMAL DISTRIBUTIONFrom the positive half of the Gaussian distribution,p(x) = - exp(-x 2 /2) (2.9)"Y TTa simple rejection method is given in Reference 15:let: x' - "1OgP 1y = - iogp 2accept x' if, and only if
ISand give a random sign (by the use of a third random number) to x':x =s-x+ 1.if p, ^0.5- 1.if p 3< 0,5The efficiency of this procedure is fairly high:exp( - 1/2) =-= 0,76Approximate Gaussian PDF sampling can be rctheorem: according to which the sum of a large n " it 1 j * .! •>distributed random variables (with finite variance) is t >'i i> >uiiis• j n lb i.the X 1variables are uniformly distributed on (0,1), then ih> a i. - : n I ,JKwith a variance ofV'12then x = Xf„ i x, is asymptotically (n -» 00 ) normally distributed with a residual expectedvalue ofna„ = an = -and a variance ofs„ = Vn-sTherefore, if samples from a normal distribution with a - 0, s ~ I (Equation 2.9) haveto be generated then the(2.10)rule is to be applied, with n -*
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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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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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