# Calculate the temperature of neutrons emerging from a reactor

## Homework Statement

A collimated beam of thermal neutrons emerges from a nuclear reactor and passes through a speed selector into a detector. The number of neutrons detected in a second with speeds in the range 4000 to 4010 m s−1 is twice as large as the number per second detected with speeds in the range 2000 to 2010ms−1. What is the temperature of the moderator in the nuclear reactor?

## The Attempt at a Solution

So the speed distribution is proportional to ##v^3e^{-\frac{v^2}{v_{th}^2}}## so my instinct was to write ##v^3e^{-\frac{v^2}{v_{th}^2}}|_{4000}^{4010}=2v^3e^{-\frac{v^2}{v_{th}^2}}|_{2000}^{2010}## but then i don't know how to solve this.

Many thanks

Last edited:

TSny
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So the speed distribution is proportional to ##v^3e^{-\frac{v^2}{v_{th}}}## so my instinct was to write ##v^3e^{-\frac{v^2}{v_{th}}}|_{4000}^{4010}=2v^3e^{-\frac{v^2}{v_{th}}}|_{2000}^{2010}##

The speed distribution should have a factor of ##v^2## instead of ##v^3##. However, the rate at which neutrons strike the detector would have the factor of ##v^3##.

What does ##v_{th}## represent? If it has dimensions of speed, then note that the argument of your exponential function is not dimensionless.

Can you explain the notation ##v^3e^{-\frac{v^2}{v_{th}}}|_{4000}^{4010}## as regards the interpretation of ##|_{4000}^{4010}## ?

The speed distribution should have a factor of ##v^2## instead of ##v^3##. However, the rate at which neutrons strike the detector would have the factor of ##v^3##.

What does ##v_{th}## represent? If it has dimensions of speed, then note that the argument of your exponential function is not dimensionless.

Can you explain the notation ##v^3e^{-\frac{v^2}{v_{th}}}|_{4000}^{4010}## as regards the interpretation of ##|_{4000}^{4010}## ?
Sorry that is meant to be ##v_{th}^2## I have missed of the ##^2##. That notation is meant to be putting in limits from 4000 to 4010

TSny
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That notation is meant to be putting in limits from 4000 to 4010
Limits of an integration? Did you perform an integration?

Limits of an integration? Did you perform an integration?
no, but i probably should have!!

TSny
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Yes. However, an interval of 10 m/s is quite small compared to 2000 m/s or 4000 m/s. So, you can get a decent answer by considering the integrand as constant over the interval of 10 m/s.

Yes. However, an interval of 10 m/s is quite small compared to 2000 m/s or 4000 m/s. So, you can get a decent answer by considering the integrand as constant over the interval of 10 m/s.
So can we approximate this as ##(4000)^3e^{-\frac{4000}{v_{th}^2}}\int_{4000}^{4010}{dv}=2(2000)^3e^{-\frac{2000}{v_{th}^2}}\int_{2000}^{2010}{dv}## ?

##T=\frac{m}{K_B}\Big(\frac{4000^2-2000^2}{\ln{\frac{4000^3}{2(2000^3)}}}\Big)^2##

TSny
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So can we approximate this as ##(4000)^3e^{-\frac{4000}{v_{th}^2}}\int_{4000}^{4010}{dv}=2(2000)^3e^{-\frac{2000}{v_{th}^2}}\int_{2000}^{2010}{dv}## ?

##T=\frac{m}{K_B}\Big(\frac{4000^2-2000^2}{\ln{\frac{4000^3}{2(2000^3)}}}\Big)^2##
Looks good. But did you drop a factor of 2 in the relation between ##v_{th}^2## and ##T##?

Looks good. But did you drop a factor of 2 in the relation between of ##v_{th}^2## and ##T##?
I did- was a typo though. Thank you for your help!! Very much appreciated