Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

(1 - - -ft) 2— aeexlciX'c - fl) (1 - c(C 6)The moment equations take on the formMilM(x) = I(x) + dx'e "'•" 1 CM(X')(CI)Differentiating Equation (Cl) with respect to x, we haved d f xdxdxM,(x) = — I(x) - cM(x) + J dx'e-- (x '" • 0 CM(X')AI(x) - I(x) + (1 - c)M(x)(C.2)dxand the boundary condition follows from Equation (Cl) asM(X) = l(X)(C.3)The solution of Equations (C.2) and C.3) is easily obtained asdM(X) = I(X)e ( '- c X X - x ) - J < dx'e ' p - (1 ••CHx' - x) 7 l(x') - I(x')dx= I(x) + c j dx'e-"-^' T(x')(CA)xIn the special case whenl(x) - ax'e 0 "(CS)the solution readsci(i ••• I)X 1 "M(x) - ae , , x x' +(1 - c - if) (1 - c - d) 2 (1 - c - WSpecifically for i = 0 [when I(x) = ae*"], the moment in the straight-aheadreadsapproximatiorM(x) = ae flx [l - •d - ce- ( 1 -- i , , < x •''!/(I r» (CJ)REFERENCES1. Amster, H. J. and Djomehri, M. J-, Prediction of statistical error in Monte Carlo transport calculations,Nucl. Sci. Eng., 60, 131, 1976.2. Amouyal, A. and Benoist, P., Nouvelle Methode de Determination dc Facieur d'Utiiisation Thcrrakji'.;d'une Cellule, Rapport CEA No 517, 1956.3. Booth, T. E. and Amster, H. j . , Prediction of Monte Carlo errors by a theory generalized to treat tracklength estimators, Nucl. Sci. Eng., 65, 273, 1978.