(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the neutron density distribution function at any point in a monodirectional beam of monoenergetic neutrons moving along the x-axis is given by

$$n(x, \mathbf \omega) = \frac n {\pi} \delta( \mu -1)$$

where ##n## is the neutron density, ##\delta( \mu -1)## is the Dirac delta function, and ##\mu## is the cosine of the angle between ##\mathbf \omega## and the x-axis.

2. Relevant equations

##\int_{\Omega} n(x, \mathbf \omega) d \Omega = n##

3. The attempt at a solution

I simply checked that integrating over the solid angle gives the total neutron density:

$$\int_{\Omega} \frac n {\pi} \delta( cos \theta -1) sin \theta d \theta d \phi $$

## \mu = cos \theta \rightarrow d \mu = -sin\theta d \theta## and by the properties of Dirac's delta function the above integral reduces to

$$\frac n {\pi} \frac 1 2 2 \pi = n$$

I think this is a valid proof, but I'm not very sure how to "derive" the expression in the first place. For example: why the Dirac's delta function is expressed in terms of ##cos \theta## instead of just ## \theta## ?

Thanks

Ric

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# Neutron density of a beam

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