# Calculate velocity of the fastest neutron inside a 96Mo nucl

## Homework Statement

Calculate the velocity of the fastest neutron in a 96Mo nucleus and, based on this, explain whether or not we are safe to consider such nucleons in a non-relativistic way. Hint: first
calculate the Fermi energy.

## Homework Equations

Fermi energy from Fermi gas model: https://en.m.wikipedia.org/wiki/Fermi_gas

## The Attempt at a Solution

Calculating the Fermi energy using Wolfram:
http://m.wolframalpha.com/input/?i=(hbar^2/(2*1.674929*10^-27kg))((3*pi^2((6*10^43)))/1m^3)^(2/3)

Fermi Energy = 30.3946 MeV

Fermi Momentum = √(2mE)
m: mass of neutron
E: Fermi energy

Fermi Momentum = √(2*939.565*30.3946) = 238.988 MeV/c

Velocity = Fermi Momentum / mass
= 238.988 MeV/c / 939.565Mev/c^2
= 0.2547c

So the fastest neutron in 96Mo is travelling at .2547 * speed of light. Provided my calculation is correct, I'm not sure whether we are safe to calculate them in a non-relativistic way though?

haruspex
Homework Helper
Gold Member
the fastest neutron in 96Mo is travelling at .2547 * speed of light. Provided my calculation is correct, I'm not sure whether we are safe to calculate them in a non-relativistic way though?
Ultimately it depends on what you are going to calculate and how accurately you need the answer. What does it give for the Lorenz factor? That might be a clue.
https://en.m.wikipedia.org/wiki/Relativistic_speed

haruspex
Homework Helper
Gold Member
I calculated the lorentz factor and got 1.03. I'm not sure if this means we are safe to consider the neutrons as non-relativistic though.

http://m.wolframalpha.com/input/?i=1/sqrt(1-(0.2547^2/1^2))
Yes, that's what I got. Borderline, I'd say. I can only repeat, it depends what question you are trying to answer, which perhaps says it is not safe to ignore it.

When you say 'borderline' what do you mean?

https://en.m.wikipedia.org/wiki/Fermi_gas#Relativistic_Fermi_gas

This says "For particles with energies close to their respective rest mass we have to use the equations of special relativity"

Considering the neutrons fermi energy is 30.3946 MeV and it's rest mass is 939.565Mev/c2 we could say its energy isn't close to it's rest mass.

haruspex