Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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134 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe collision kernel can be expressed in terms of wavelengths from the conditionC(X->X')dX' = C(E~->E')dE' (4.116)since these two quantities describe the same physical probabilities. From Equations (4.115)and (4.116) the collision kernel is(E') 2C(X-»X') - --' C(E-^E')eand now the integraljdXC(X-^X')is convergent.It can be proved 16that this transformation leads to the same sampling as the applicationof a distortion factor (E7E) 2in the energy representation.Hoogenboom 9 described the energy of the neutrons in lethargy (u) units:u = logE 0Ewhere E 0is a reference energy. In this presentation the relationdu ~dEH"automatically" introduces the ~E~ 1distortion and thusJduC(U-Hi')is convergent.E. SCORING IN THE ADJOINT MONTE CARLOIn the way described in the previous sections sequences of points in phase-space aregenerated and the densities of these points represent the values of particles entering or leavingcollisions at these points. The knowledge of these densities is the aim of the adjoint treatmentIf one wishes to use the value as importance function in the collision density simulation.However, if one uses the adjoint simulation solely, then relation between the value functionsand the physical quantity (reaction rate) to be calculated must be found.I. Reaction RatesTheorem 4.11 — Let Equation (4.36) describe the outcoming collision density of certainparticles and Equation (4.100) describe the value of particles leaving collisions. Then thereaction rate defined by Equation (4.49) can be expressed in terms of the value of theparticles leaving the collision asK - drdE X*(r,E)Q(r,E) (4.117)
Proof.Multiply Equation (4.36) by x*(r,E) and Equation (4.100) by \(*\E) anall terms over r and E. From comparison of the two equations one concludes:drdEx(r,E)P x(r,E) = J J drdE X*(r,E)Q(r,E)The left hand side integral is the definition of the reaction race, as given in fequat'on . IiThe new expression of the reaction rate is extremely plausible. If on 'expressed as:R = JJdrdEr aR« = JJdrdEx(r,E)^- r-'.R«> =[ j d r d E ^ r . E ) ^ ^8(r,E)if Q"' and QJ. 1 ' denote the set of sources and first-collision sources, respectively.The values R (,)are not statistically independent of each other, since all the estimates arebased on the same representation of the values.
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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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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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