Probability distribution for hard sphere elastic collisons

[Jack_O]

I'm researching neutron moderators and I want to model how many collisions are required for fast neutrons to be moderated to thermal temperatures and the distribution of energy absorbed by the moderator during the process. I have quickly worked out the %age velocity reduction for the simple head on 1d elastic collisions:

http://dl.dropbox.com/u/4550021/1d%20elastic.xls

(100% for H1, 67% for D2 and 15% for C12, i.e. light water, heavy water and graphite moderators) For graphite i can then say the largest energy absorbed is 1-(1-0.15)² because KE is proportional to v²(e.g. for 1MeV neutron first C12 would receive 278KeV). To cool a 1MeV neutron to 0.1eV with head on graphite collisions would be (log0.1-log10⁶)/log(1-0.15)=99 collisions.

But I'm struggling to come up with a more realistic 3d model because I can't find a probability distribution for sphere collisions.

Basically I'm looking for a graph with radius of a sphere from 0 to 1 on the x-axis and probability of impact (of an incident sphere) on the y axis.

 

[eljose79]

Hello,

Your research on neutron moderators sounds very interesting. Neutron moderation is an important aspect in nuclear reactors, so it's great that you are looking into it.

To model the number of collisions required for fast neutrons to be moderated to thermal temperatures, you can use the concept of mean free path. Mean free path is the average distance a neutron travels before colliding with a moderator. This can be calculated using the cross-section of the moderator material and the density of the moderator.

For a more realistic 3D model, you can use Monte Carlo simulations. This involves randomly generating the positions and velocities of neutrons and tracking their interactions with the moderator. By repeating this process many times, you can obtain a probability distribution for the number of collisions required for neutron moderation.

For the distribution of energy absorbed by the moderator during the process, you can use the concept of energy transfer in elastic collisions. This involves calculating the energy transferred from the neutron to the moderator in each collision and summing these energies over all collisions. This will give you a distribution of energy absorbed by the moderator.

I'm not sure if there is a specific graph that shows the probability of impact for a sphere collision, but you can use the concept of cross-section to calculate the probability of a neutron colliding with a moderator of a certain size.

I hope this helps with your research. Good luck!