Calculate the temperature of neutrons emerging from a reactor

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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?


Homework Equations




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
 
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Answers and Replies

  • #2
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}## ?
 
  • #3
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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
 
  • #4
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?
 
  • #5
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Limits of an integration? Did you perform an integration?
no, but i probably should have!!
 
  • #6
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.
 
  • #7
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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.
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##
 
  • #8
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##?
 
  • #9
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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
 

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