Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

225Equation (5.205) determines the relation between the expected scores in the two games doeto a flight-collision event pair. Thus, if the partial score in the delta-scattering game satisfiesEquation (5.205), the game is unbiased.In the special case of the simplest collision estimator, i.e., ifI 1(P) =jdP'T(P,P')f(P')andI 6(P) =fdP'T 6(P,P')f B(P')Equation (5.205) yieldsfdP'T(P,P')F(P') = jdP'T s(P,P')F 8(P') +• J dP.TsCP.P.^P.) j dP'TfP, ,P')F(P'>Multiplying Equation (5.204) by f(P') and integrating with respect to P', it is seen thatf fi(P') = f(P')[l -q S(P')]satisfies condition (5.205). The result is heuristically obvious: the probability of an analogcollision at P' is 1 — q 8(P'); therefore, in any collision (analog or delta scattering), thescore is equal to the analog score times the probability of an analog collision. It is equallyobvious that if the contribution function in the analog game is additive, i.e., iff(P,P') = f(P,P,) + f(P,,P')(this is the case with the track-length estimator), thenf 8(P,P') = f(P,P')This can again be proven by multiplying Equation (5.204) by f(P,P'), integrating over P',and comparing the result to Equation (5.205).A special application of delta scattering can substantially simplify the simulation. If theartificial cross section CT 0(P) is chosen such that the modified cross-section CT 5(P) is independentof the position of the particle, then free flights can be simulated without regard topossible crossings of boundaries between two different media. 11 - 45 This may make the trackingmuch faster, since there is no need to calculate geometrical and optical distances betweencollision points and region boundaries. Note, however, that the gain in computing time bythis trick again is deteriorated by the loss due to the increased number of collisions. Tcminimize this loss, it is advisable to choose