Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

110 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsThe score from an entire history is the sum of the contributions:fx x= S r„ x(4.59)or2 (4-55)If the f xand C 4, pay-off functions are correctly chosen then the expected values of the(4.54) and (4.55) estimates must be the same:W - W = RAfter following j = 1,2, . . . ,n histories each scoring p, XJand/or u.^ , the average ofthe scores is an unbiased estimator of R:ij=in j „ iD, EXAMPLES OF PAY-OFF FUNCTIONSAn extremely wide variety of physical quantities may be determined by Monte Carlo.In the following subsections several representative examples are shown and discussed.1. Collision Density and Reaction Rate IntegralsThe simplest question to raise in a Monte Carlo game is: how many collisions occured,i.e. how many particles entered collision, in a certain volume element (or domain) F? Theanswer is trivially:R = dr dE.).(r.E) = drdEh r(r)cj>(r,E) (4.56)rwhere h r(r) =1, if reT0, otherwiseThus from a comparison of Equations (4.50) and (4.56) the pay-off function is:f +(r,E) = h r(r)If only a part of the spectrum is of interest, e.g., one wishes to count the number ofcollisions which the particles enter with energy between E 1and E 2, then the correspondingpay-off function becomesf 1((r,E) =h r(r)h, ;(E)where h, =0,if E 1< E < E 2otherwise