Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

234 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationswhich gives the expected value of the score over the next flight from P. In a homogeneousmedium, if IfP 1) = f for D 1< L and zero otherwise, the expectation estimator scoresP 1XP)fflOn the other hand, if d in Equations (5,224) and (5.225) is the distance to the boundaryof the region, inside which the reaction rate is to be estimated, from the starting point Pinside the region, then 1'(D 1) = 0 if D 1> d and the transformed estimator scoresf(D) =f, :,(P,P')I(P)/[ 1 -0eififD =s dD > d(5.227)i.e., it gives a contribution independent of the site of the next collision point, provided thiscollision takes place inside the region of interest, and the score is zero if the particle leavesthe region. Combined with a nonanalog game which forces the particle to stay inside theregion, this estimator is used in the expected leakage probability method proposed byKschwendt and Rief. 20This method will be reviewed in Section 5.VIII.D.Let us now consider a function opposite to that in Equation (5.244), i.e., letX(D 1D 1) = a, if D ? dand zero otherwise. Then the transformation yieldsI(P)e T(d) if D 5= df(D) = f E2(P,P') = {0 if D < d(5,228)This estimator scores only if the free flight from P is longer than d. According to Theorem5.18, any normalized linear combination of the estimators in Equations (5.227) and (5.228)is also partially unbiased, i.e., for arbitrary function a(P), the estimatorf sl(P,P') = a(P)f E1(P,P') + [1 - a(P)]f E2(P,P')is also partially unbiased. This estimator has the explicit formf SI(P,P') = Ia(P) x(P 111P') + [1 - a(P)]x(P\P d)[e T