22.54 Neutron Interactions and Applications (Spring 2004)
Problem Set No. 4
Due: March 30, 2004
Problem 1
(a) Consider Monte Carlo sampling of the distance that a neutron travels before it suffers
an interaction for which the macroscopic cross section is Σ . How does one determine this
distance?
(b) Repeat (a) for determining the polar and azimuthal angles θ and ϕ for a spherically
symmetric (isotropic) distribution.
Problem 2
Explain importance sampling using the example of calculating the average of a random
variable A using the canonical ensemble. What is the connection with the Metropolis
method? Give a proof of the Metropolis method after stating clearly what it is.
Problem 3
Write out the Neutron Transport Equation and define the various quantities in this
balance. Interpret each term in this equation. Discuss what assumptions, if any, are
involved. What is the major difference between this equation and the Boltzmann
transport equation in the kinetic theory of gases? Derive the energy-dependent diffusion
equation from the Neutron Transport Equation.
Problem 4
Derive the escape probability for a neutron to go from energy E' to E (E' > E) as given in
Eq.(10.14) of the Lecture Notes.