Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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28 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsFIGURE 2.7.A particle night crossing boundaries of zones of different materials.denotes the number of particles entering a collision with coordinates dP about P and itsname is incoming density, or simply collision density.From the above definitions it is clear that none of these functions is a "density function"since generally the normalization conditionsJ X(P) dP = 1andJi|/(P)dP = 1are not satisfied.The incoming density is closely related to the flux. Let us recall the definition of theflux (fluence, by the rigorous ICRU terminology):(r,E)dadEis the number of particles entering an infinitesimal sphere of radius dr. cross-sectional areaof da = Tr (dr) 2with direction and energy dE about E. The expected path length ofthese particles is (see Equation 2.14)4--- - dpThus the expected number of particles entering collisions in the infinitesimal sphere(i.e. the collision density) isi|i(r,E)dr dE = a(r,E) < d( >
29therefore:4,(P) cr(r,,E) cp(P) (2,24)The inverse relationCj)(P)(T(r,,E)(2.25)is valid only if cr ^ 0, which condition clearly reflects the very simple physical fact that invacuum the flux is still a reasonable quantity whereas no collisions can happen if no materialis present.G. QUANTITIES TO BE DETERMINED: REACTION RATES, RESPONSES,SCORESGenerally, the aim of a Monte Carlo calculation is the estimation of the value of a.physical quantity or values of several quantities. For the sake of simplicity we restrict ourdiscussion to the determination of a single quantity. Extension of the considerations toparallel examination of several quantities is straightforward.These quantities can represent a great number of physically interpretable data varyingfrom the number of collisions in a space element to leakage probabilities, detector responsesor doses absorbed in certain regions of the core of a reactor or even in organs of ananthropomorphic phantom. Just to preserve generality of the discussion all these quantitieswill be called either as receptor responses or reaction rates.A common feature of these responses is that they can be formulated as weighted integrals(or functionals) of one of the collision densities.Thus a response (or reaction rate) is most generally formulated as:(2.26)orThe f and T 4, weight function are derived from the physical connection between thecollision density and the quantity to be determined. The integrations are extended, to theregion of interest, or, by other words the f weight functions have to vanish outside the rangeof interest. Since the subscript of f is trivial from the type of collision density used in theintegrals of (2.26) it is generally omitted.Naturally if R is calculated not for a finite range but only for a point, e.g., for r G, thenthe weight function contains a Dirac-delta component:f(P) = f,(E)8(r -rjthenSimply, if the number of collisions are to be computed for a phase space domain P 1f(P)—
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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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