Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

479importance function. Accordingly. b,(x) in Equations (7.112) and (7.1 Kpendent of I(x).In order to optimize the efficiency of the game, it remains to estimate the iof collisions to be played in a history due to a starter. Let s(x)dx be the prsimulation particle at x suffers a collision within dx about x. Then the expecollisions due to a starter isL = J dxs(x)[K )(x) + K , (X)IIn view of Equation (7.109), it can be rewritten asL = | dxs(x)I(x)[M)(x) + M 1(X)]The quality factor of the game isQ = D ; (X)L?3>- 2 (X) dxb .,(X)^(XVI(X')Xdxs(x)[M)(x) •+M 1(X)]I(X)Elementary variational considerations show that Q is minimum ifI 2 (x) = const. fF-(x)b +(x)/{s(x)[M{(x)+- M 1U)J}Finally, inserting b+(x) from Equation (7.112) into this expression, the optimum importancefunction readsP(x) = const. SP(x)[--t(x)M-;(x) + r(x)M)(x)]/{s(x)[M ; U) 4- M 1(X)!} (7 116;The constant factor is to be chosen such that 1(0) = 1.Let us note that making use once more of the explicit form for the first-moment Equation(7.110), an alternative, expression of the optimum importance FUNCTION can be derived itfollows from Equations (7.110) and (7.115) thatf 2 (x)[-t(x)Mt(x)4- r(x)M)(x)]" d— M)U) - 2EU)MIU)dx'(PCx)= A[ ? F ( x ) M1 (x)]dxi.e., the optimum importance function becomesF(x) = const. — [^2(x)M;{x)]/{s(x)|M J(X) 4 M , (x)\\ i7.;H4dxThe expressions of I(x) in Equations (7.116) and (7.11.7 , < ont,u,i i i>. b > tnot known before a Monte Carlo game is started. In pract,.,a> ici ! i/ao>m ol n>, 'p'cs'iiprocedure, approximate and estimated values must replace the functions in Hx;. SUCH approximationsusually are obtained from preliminary runs. In the next section, recipes arcproposedfor possible uses of results of preliminary runs in this particular scheme.