Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

226 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe idea of delta scattering has served as a theoretical too! for introduction of the tracklengthestimator. 42It also has recently been applied in practical calculations in connectionwith correlated games/ 9 ' 4 0The second-moment equation of the game follows easily from the analog momentequation derived in previous chapters by simply substituting T sand C sin place of the analogkernels, and will not be detailed here. Similarly, the number of collisions to be played ina history will satisfy Equation (5.192) with the respective kernels. The heuristically obviousfact that delta scattering increases the number of collisions per history can also be seen onthe basis of Theorem 5.12.VI. PARTIALLY UNBIASED ESTIMATORSWe have seen in Section 5, V.A that a set of estimatorsS{f(P,P'),f,(P'),f. ;(P' ,P"),{nf f(P' ,P")}:..,} - S{f,f„f ,{nfj} (5.206)result in an unbiased estimate of the reaction rateRJdPiKP)f(P)in an analog game governed by the kernels T(P,P'), C 5(P',F'), and {C n(P',P")}^, if therelationdP'T(P 1P') f(P,P') + C 11(P')f a(P') + c s(P')JdP"C s(P'.P")f s(P',P")+ c r(P') 2 Q n(P') jdP"C n(P',P")nf n(P\P") --- |dP'T(P,P')f(P') = I(P) (5.207)holds. It has also been seen in Section 5.V. B that any nonanalog game that satisfies theweight generation rules of Theorem 5.8 is also unbiased with the same estimators. Estimatorsin the set (5.206) that satisfies Equation (5.207) were called partially unbiased estimatorsas any such set results in the same expected partial score in a flight followed by a collision.Obviously, the simplest partially unbiased set isS{f(P'),0,0,(0}}S{f,f u,f s,{nf„}^= ]} is a shorthand notation of the estimation procedure in which f(P,P')is scored when a free flight from P to P' is played, f a(P') is the score assigned to an absorptionat P', and f s(P',P") and f„(P',P") are the scores if a particle emerges at P" from a scatteringor from an n fold multiplication at P', respectively. [This means that if in an n-fold multiplicationthe secondaries emerge at P' (' u, P' (' 2), P" n), respectively, then the score fromthis event isas detailed in Section 5.III.A.JIn an analog game, we have a certain freedom in defining the separate scores; namely,the score assigned to a free flight can also be attached to the scores due to the different