Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

352 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsthe kernel, we have assumed that the energy and direction of the fission neutrons areindependent of the energy and direction of the neutron that gave rise to the fission. Thisassumption is justified in most practical cases. Let z(r,E,r G) be a function that satisfies theequationz(r,E,r„) = JdP"z(r',E',r„)K s(P",P) +- X(EJrJ T(r (,->rJE) (6.100)withP = (r,E),P" == (r',E')Obviously, z(r',E',r„) is the collision density at P' = (r',E') due to a neutron that emergedfrom a fission at r„. Furthermore, let us define the functionS(r) = j dEi|i(P)c f(P)v(P)/k eff(6.101)S(r) is the density of fission neutrons emerging from around the spatial point f [Moreprecisely, it is the fission density in the hypothetical system where the collision density isi|/(P).] Let us multiply Equation (6.100) by S(f„) and integrate with respect to r Then itis seen that the resulting equation is identical to Equation (6.81) if we putOJ(P) - |dr'z(r,E,r')S(r') (6.102)Finally, let us denoteZ(r.r') = J dECf(P)v(P)z(r,E,r') (6.103)It follows from the interpretation of z(r,E,r') that Z(r.r') is the density of fission neutronsemerging from around r due to a fission neutron started from r'. An equation containingonly the fission density S and the kernel Z is obtained if Equation (6.102) is multiplied byc fv and integrated with respect to E. Doing so and taking note of Equation (6.101), we haveSir) = dr'Z(r,r')S(r') (6.104)keff JThis equation has the form of a conventional eigenvalue problem. Its solution is sought byiteration:S w (r) = jdr'Z(r.r')S ( "-' ) (r') (6.105)The iteration is equivalent to the method of successive generations since S (n)(r) representsthe fission source density in the n-th generation of the neutrons and is related to the(n — l)-th-generation fission source through the kernel Z(r.r'). LetS(n>= drS(n)(r) (6.106)