Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

346 Monte Carlo Particle 'Transport Methods: Neutron and Photon Calculationsdifferent Aa values. In a parametric study of reaction rate perturbation, a single differentialgame may be substituted for a large number of distinct correlated games.In a simple reaction rate perturbation, if R" denotes an estimate of the v~th derivativeof the reaction rate, then^ - (Aa)"SR = R(a + Aa) - R(a) « J\ R ( "» -is an estimate of the reaction rate perturbation, where N is the number of terms kept in theTaylor series expansion. Numerical experiments show that two terms in the expansion alreadygive satisfactory estimates in typical perturbation calculations. 67 '*Note that in order to give an a posteriori estimate of the variance of the result, it is alsonecessary to estimate the covariance matrix M{(d s7daJ(d!f/da^)} of the final scores in thedifferential game. If V denotes an estimate of the (v,u,) element of the covariance matrix,then the variance of the estimated reaction rate perturbation due to the parameter changeAa isIII. CRITICALITY CALCULATIONSOne of the most important quantities characterizing a nuclear reactor is the effectivemultiplication factor, k eff, a measure of how far the fissile system is from the critical state.If k eff= 1, the system is critical; if k eff< 1, it is subcritical; and if k eff> 1, the system isin a supercritical state. This factor is an eigenvalue-type quantity appearing in the transportequation in the form4.(P) = I dP'>(P") K 5(F 1P) + -- K,(P".P)jL k ef,(6.81)whereK S(P",P) =[dP'cJP")C 5(P'JP J T(PJP)andK 1(PJP) = |dP'c f(P") 2 nq n(P")C„(P",P')T(PJP)J11 = 1while c sand c fare the respective scattering and fission probabilities, and C 5and C 1are therespective probability densities of the neutron's coordinates after a scattering or fission. q nis the probability that n neutrons are produced in a fission and T is the transition kernel.All these quantities were defined in Chapter 5.Equation (6.81) is a homogeneous equation, i.e., it describes a system with a steadystateneutron flux (collision density) without an external source. If the system characterizedby the kernels in Equation (6.81) is critical, then the equation has a solution with k ctf= 1.In the opposite case, the system can be hypothetically altered to critical by changing thenumber of neutrons produced in a fission by an appropriate factor. This factor is just l/k cfr.