The 1D neutron diffusion equation with a plane source is Dd2

The 1D neutron diffusion equation with a (plane) source is

?Dd²?(x)/ dx2+ K²D?(x) = Q?(x)

where ?(x) is the neutron flux, Q?(x) is the (plane) source at x = 0 and D and K2 are constants.
(Hint: Apply Fourier transform. Solve the equation in transform space. Transform your solution back into x-space. )

Answer: ?(x) = (Q/2kD) e^?|Kx|

Solution

The neutronic sources Q g(r) represent the production of secondary neutrons from scattering (including neutrons from (n, xn) reactions) and fission reactions: Q g(r) = XG h=1 ? g ? h(r) ? h(r) + ? g(r) Keff XG h=1 (2) ? ? f h(r) ? h(r) where G = total number of energy groups ? g ? h(r) = macroscopic scattering cross section from group h toward group g ? g(r) = fission spectrum in group g ? ? f h(r) = product of the macroscopic fission cross section by the average number of neutrons emitted per fission in group h.