Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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132 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsasThe differential cross section given in Equation (3.9) can be rewritten in terms of energyE' E e e ,a(E->E') = _ + + / i + - 17E E E' V E E '2(4.112)where e is the electron rest mass energy (m ec 2~ 0.511 MeV), the value of the constant kis not relevant at the moment. From the energy ratio relation given in Equation (3.8), thelimiting values of E (the energy of a real photon before a real collision which is the energyof a pseudo-photon after a pseudo collision) areE sj E
If the scatterer atom is hydrogen, then A a = 0, i.e., the integral given in Eq(4.114) is again logarithmically divergent.These two examples clearly demonstrated that the fac(4,110) for the normalization of the adjoint collision kernphoton and in neutron transport calculations.The methods of solution are basically the same for the two types ofmethod is based on the fact that there is always an upper limit for energieis no use in following the history of any pseudo-particle whose energysource (the real, physical source). Let us denote the maximum source tthe new energy E can be selected from the distorted kernelC(E-^E') = •C(E->E'),0,if E E.ifENow, for the normalization the integral becomes:jdEC(E-*E') = jdEC(E-»E')and this integral is convergent for all finite E M's.A disadvantage of this method is that the form of the new collision kernel is problemdependent. A change of the source leads to change of the kernel,A better approach is the distortion of the collision kernel In. the whole energy rangeindependently of the highest source energy. If the distored kernels are defined asC(E->E') = |-EC(E-+E')then the normalization factor ofdEC(E-^E')is bounded. Proofs are given in the literature both for photons 16 and for neutrons. 9It follows from Theorem 4.10 that if the EVE distortion is applied, the weight has tobe multiplied by E/E' (E > E') after every pseudo-collision. At first sight it may seem thatnow the weight factor may become unbounded. This problem is solved again by the practicallimitation that no pseudo particles collided to above E Mare followed further.Several techniques developed for sampling the adjoint collision kernels are describedand compared in two papers of DeMatteis and Simonini. 6 - 7for neutron and photon transport,respectively.Finally, we should like to call the attention of the reader to the fact thaintegrals like that given in Equation (4.110) may depend on the choice of(This statement is not a physical nonsense, since the quantity C* has no physi. U
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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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182 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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