What Factors Affect Neutron Flux in a Finite Medium?

[damyoro]

Deaar all
good morning

I am very interested to the flux in a slab of extrapolated thickness a, containing distributed sources of neutron. A I have an example in which the source is given as s(x)=S(x+a/2) where S is a constant and x distance from the center of the slab.
You mentioned in one of your post that you would do it but I could find.
I would really appreciate any help of reference from you
Thank you in advance

 

[damyoro]

I wanted to add the reference of the book : Introduction to Nuclear reactor theory by John R. Lamarsh

CHp5 Problem
a)By using the diffusion equation for a planar source located at x', show that the diffusion kernel for an infinite slab of thickness a is given by

G(x,x^' )=L/(Dsinh(a/L)) {█(sinh 1/L (a/2-x)sinh 1/L (a/2+x^' ),x>x',
sinh 1/L (a/2+x)sinh 1/L (a/2-x^' ),x<x'
b) using this kernel calculate the flux in a slab containing uniformly distributed souces emitting S neutrons/cm3.sec

 

[daveb]

What have you tried so far? This looks to be a homework problem.

 

[damyoro]

You are right and thank you for your reply
This is my try
general solution is C=A*e^(-x/L)+Be^(x/L)
And I use as boundary conditions φ(a/2)==0
and as source condition
J(a/2-x')=J(x') +1
What I found is the following
Ф+(x,x^' )=-(Le^((x-a/2)/L))/2D[cosh x^'/L-cosh((x^'-a/2)/L) ] +(Le^((-x+a/2)/L))/(2*D[cosh x^'/L-cosh((x^'-a/2)/L) ] )

I really tried hard but until now I could not get it right.
One of my problems is to find the conditions for determining the constants.

thank you in advance