Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

340 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand its derivative isAdaLLda 211(a) /11(a)da 11(a)/11(a)Therefore, the statistical weight in a second-order differential game, according to Equation(6.70), is; H(a) /11(a) W'da 2d(Aa) 2 l A a -°Successive use of the arguments above shows that the generalized moment equation of afunction of the r-th parametric derivative of the score has the formM G(£')} (P "' !) = {G % w < r ' (l)f(l) " G 2 w; r )o)f(i)where WJ r)(i) is the particle's weight after the i-thflightw; r)(i) = ~- w; r. 0(i) + Wi 0(I)WO.. ..G) = 7/7-7. wo)dad(Aa)'(6.71)Specifically, for the second derivative, the weight generation rule becomesWJ 2)(O = X -f~ 2IQg 0T(P,. ,,PJIa) + 2 TTl IQg 6C(POPJa) + [WO 0(I)] 2 (6.72)j„ i ua j = i uaand for the third derivativeWO/0) = 2 £:, 1Og 6T(P^ 1 5PJIa) + 2 1Og 6C(POPJa)j=, da 3+ 3WO 1(OWO,© + [WO)O)J 3It is stressed again that, apart from the differences in the weight generation rules, there isno difference between the estimation procedures of an ordinary reaction rate and its parametricderivatives. Therefore, in principle, the derivatives up to any order can be estimated alongwith the reaction rate itself in a single game.E, A SIMPLE EXAMPLEFor the sake of illustration, let us consider the estimation of the derivative of the collisionrate with respect to the total cross section in an infinite homogeneous medium of total crosssection IT and survival probability (mean number of secondaries per collision) c. Let thetransport in the medium be monoenergetic and isotropic. Then the kernels of the simulationareT(P,_ ,,PO ~aexp(-