Discover the Velocities of Nucleons | Proton, Neutron and More

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In summary, the conversation discusses the velocities of the proton and neutron in Deuterium/Hydrogen2, as well as how to calculate the velocities of nucleons for arbitrary atoms. It is mentioned that the uncertainty principle and the potential well affect these velocities, and that for the deuteron, experimental numbers are readily available. For an arbitrary nucleus, estimates must be made using the nuclear radius and potential. The conversation also delves into the concept of a degenerate fermi gas and how it relates to the momentum and velocity of nucleons. A rough estimate for the average speed is given using the fermi momentum.
  • #1
jaketodd
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Thanks everyone for your help over the years...

Much appreciated if someone will tell me:

A) The velocities of the proton and neutron in Deuterium/Hydrogen2

B) How to calculate the velocities of nucleons for arbitrary atoms
 
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  • #2
Do you mean how fast they're moving around...? Is that a meaningful concept for nucleons? They are confined in a very small volume, and by the uncertainty principle, the uncertainty in their momentum/velocity will be very large.
 
  • #3
For the deuteron it's easy to find the experimental numbers. It's a triplet S state with binding energy 2.22 MeV. The best fit for the potential well has depth 38.5 MeV, meaning the total kinetic energy is 38.5 - 2.22 = 35.7 MeV. (Do you really want the velocity, or is that good enough?)

For an arbitrary nucleus it's harder to make a good estimate. The nuclear radius is roughly r = r0A1/3 where r0 = 1.25 f. The potential is somewhere between harmonic oscillator and square well with a depth typically 50 MeV. If you want the total KE or the average KE don't just take the lowest level in this well, remember the nucleons will occupy the well states up to some highest level.
 
  • #4
Assume the nuclear matter (protons and neutrons as indistinguishable) forms a degenerate fermi gas. The fermi momentum is determined by:

[tex]
A = 2 \, \frac{4 \pi R^{3}}{3} \, \frac{4 \pi k^{3}_{F}} {3} \, \frac{1}{(2\pi)^{3}} = \frac{4}{9
\pi} (k_{F} R)^{3} \Rightarrow R = \left(\frac{9\pi}{4}\right)^{\frac{1}{3}} \frac{A^{1/3}}{k_{F}} = R_{0} \, A^{1/3}[/tex]
[tex]
R_{0} = \left(\frac{9\pi}{4}\right)^{\frac{1}{3}} \frac{1}{k_{F}} \Rightarrow k_{F} = \left(\frac{9\pi}{4}\right)^{\frac{1}{3}} R^{-1}_{0}
[/tex]
Using the emprical result [itex]R_{0} = 1.2 \, \mathrm{fm}[/itex], we get:
[tex]
k_{F} = 1.6 \, \mathrm{fm}^{-1}
[/tex]
The momentum corresponding to this is:
[tex]
p_{F} \, c = \hbar \, c \, k_{F} = 316 \, \mathrm{MeV}
[/tex]
which is one third of the rest energy of a proton (neutron). That is why one should use relativistic equation:
[tex]
p = m \, c \, \beta, \gamma, \ \gamma = (1 - \beta)^{-1/2}, \; \beta = v/c
[/tex]
Then, use the fact that you have a FD distribution in momentum space to find the distribution in velocity space. From this distribution you can find the most probable speed, the average speed and the root mean square speed, for example. A rough estimate, however is to simply use the fermi momentum:
[tex]
\beta_{F} = v_{F}/ c = \frac{p_{F}/(m \, c)}{\sqrt{1 + (p_{F}/(m \, c))^{2}}} = 0.32
[/tex]
 
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  • #5
Thanks all you guys and/or gals! =)
 

FAQ: Discover the Velocities of Nucleons | Proton, Neutron and More

What are nucleons?

Nucleons are particles found in the nucleus of an atom. They include protons and neutrons, which are held together by the strong nuclear force.

What is the velocity of nucleons?

The velocity of nucleons can vary depending on the specific nucleon and the environment it is in. In general, nucleons in the nucleus of an atom have velocities of about 10% of the speed of light.

How are the velocities of nucleons measured?

The velocities of nucleons can be measured using specialized equipment such as particle accelerators or detectors. Scientists can also use mathematical models to estimate velocities based on other known factors.

What factors can affect the velocities of nucleons?

The velocities of nucleons can be affected by factors such as the nuclear force, the presence of other particles, and the shape and size of the nucleus. They can also be affected by external forces, such as electromagnetic fields or collisions with other particles.

Why is studying the velocities of nucleons important?

Studying the velocities of nucleons is important for understanding the structure and behavior of atomic nuclei. It can also provide insights into nuclear reactions and the formation of elements. Additionally, studying nucleon velocities can help us better understand the fundamental forces at work in the universe.

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