# Maxwell-boltzmann distribution on detector

Hi, here's something seemingly rather simple that has puzzled me a bit.

If i have a collection of thermal neutrons. They will follow the Maxwell-Boltzmann distribution. I guess this is a pretty good approximation for the energy spread of thermalized neutrons in water if one ignores absorption etc. Because water heavily scatters neutrons and they would become thermal after just a few tens of cm.

But now, my point is, if i were to have such a pond of water with neutrons. And if i stuck a small neutron detector into that pond. Then this neutron detector would not get a passing flux which would follow the M-B distribution now would it?

Why not? well, because the faster neutrons would move much more around the pool, and thus would "reach" the detector more often.

As a thought experiment, if you have a box with two particles bouncing of the walls inside the box, one particle going at 0.9c and one going at 0.01c. Then a detector in the box would see the faster one much more often, right? So should i normalize a M-B distribution to sqrt(E) to get the actual distribution which I were to expect if I had a neutron source in water?

the whole point being that the M-B distribution is a population distribution. Whereas what a detector sees has to do with what particle in the flux pass it.

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Bill_K
What you say sounds right to me.

Mute
Homework Helper
You're making some assumptions which may or may not be true depending on the detector or your setup. For example, in your thought experiment, your detector doesn't absorb the particles - it lets them go, allowing them to be measured multiple times. What if your detector absorbs the particle?

Even if your detector lets particles go, in your pond example, what are the odds that a fast particle is measured more than once? If the pond is reasonably large, I would guess the odds are low that the particle would make it back to the detector to be detected again.

Also, where is the detector assumed to be in relation to the neutrons? If the neutrons have been given enough time to diffuse through the pond, then when you put your detector in the pond, there are going to be very many slowly moving neutrons close to the detector that will get measured, along with some fast moving neutrons. However, if the fast moving neutrons aren't likely to be measured more than once, then by virtue of the fact that there are many more slower neutrons, I would say the slower neutrons will be measured much more often, and you stand a good chance of seeing the MB distribution because in the region where your detector is the population of neutrons follows the MB distribution and you aren't likely to measure the fast neutrons more than once.

Now, maybe if you put all of the neutrons in a small box and your detector lets the neutrons go after detection, then the faster neutrons might be measured multiple times, skewing the measured distribution. So, for this reason, you probably wouldn't use a detector that let's particles go when it's likely you would be overcounting particles. (Not to mention a further possible complication that the detector changes the neutron's speed after detection!)