[Special Relativity]Slow Neutron Capture

In summary, the formation of a helium-4 nucleus from a helium-3 nucleus and a slow neutron releases energy in the form of a gamma-ray. Using the conservation of energy equation, it can be calculated that the energy of the gamma-ray is approximately 1.1e-20 J. However, this approach does not take into account the effects of relativity. To account for this, one can solve for the energy and momentum equations simultaneously. It is also important to verify that the kinetic energy of the helium-4 nucleus is negligible in this process.
  • #1
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Homework Statement



A helium-4 nucleus (rest mass 4.00151 u) is formed when a helium-3 nucleus (rest mass 3.01493 u), stationary in the laboratory frame, captures a slow neutron (rest mass 1.00867 u), in the following reaction.
3He+n→4 He+γ
Determine the energy (in MeV) of the γ-ray in the laboratory frame. (1 u ≡ 1.66 × 10−27 kg.)


Homework Equations



Given Above

The Attempt at a Solution



As far as I can see classically this would be a doddle. With Conservation of Energy and momentum to be taken into consideration before and after. However without the initial velocities of the neutron or the recoil velocity of the He-4 Nucleus I am not sure how to calculate the energy of the gamma ray. I'm also pretty unsure where the relativity comes into this.

The neutron is slow, so not traveling relativistically, the initial nucleus is at rest. The gamme ray will travel at c, which makes it a pain all in all that i can tell since working out the Lorentz factor is impossible when V=C.

Thanks for any input.
Adam
 
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  • #2
The slow neutron is essentially at rest. In other words, its kinetic energy is negligible compared to its rest energy. You can assume the He-4 nucleus moves non-relativistically and solve for the energy of the photon. Then go back and check if the He-4 nucleus does indeed move non-relativistically.

If that dubious sort of logic doesn't sit well with you, you can solve the relativistic conservation of energy and momentum equations simultaneously. If you go with this approach and it's not working out for you, post your work here so we can see where you're getting stuck.
 
  • #3
Thanks for your help. This way my attempt:

Considering Consv. of E

E[tex]_{n}[/tex]+E[tex]_{He3}[/tex]=E[tex]_{He4}[/tex]+[tex]_{\gamma}[/tex]

Since the neutron is a slow neutron its Kinetic Energy is negligable compared with its rest energy. Hence:

E[tex]_{n}[/tex]=m[tex]_{n}[/tex]*c[tex]^{2}[/tex]
E[tex]_{He3}[/tex]=m[tex]_{He3}[/tex]*c[tex]^{2}[/tex]
E[tex]_{He4}[/tex]=m[tex]_{He4}[/tex]*c[tex]^{2}[/tex]
E[tex]_{\gamma}[/tex]=Unknown

Thus:

E[tex]_{\gamma}[/tex]=m[tex]_{n}[/tex]*c[tex]^{2}[/tex]+m[tex]_{He3}[/tex]*c[tex]^{2}[/tex]-m[tex]_{He4}[/tex]*c[tex]^{2}[/tex]

=c[tex]^{2}[/tex]*u(1.00867+3.01493-4.00151)
=c[tex]^{2}[/tex]*u(0.02209)

Therefore:

E[tex]_{\gamma}[/tex]=3e8*1.66e-27*0.02209 = 1.1e-20 J

My issue with this is, there is no relativity involved. I have treated it purely classically, but since nothing but the gamma-ray is traveling relativistically I don't understand my mistakes.

Thanks again for any input.

P.S The powers are meant to be subscripts, for some reasons the latex _{blah-de-blah} command isn't working. Apologies
 
  • #4
When you used E=mc2, you used relativity. What you found was that some rest mass disappeared in creating He-4, and the missing mass was converted to the energy of the photon. The concept that mass and energy are equivalent is not part of Newtonian physics.

You forgot to square c when calculating the energy of the photon, and you should verify that your answer is consistent with the assumption that the kinetic energy of He4 is negligible. (Find its momentum and use that to calculate its kinetic energy.)
 
  • #5
Oh why thank you, I had not noticed that I had only used c.
And thanks for the input. Being 3rd Year now they expect our answer to have the rigour of textbook apparently, so ill make sure to add some explanation of what you said about mass and energy being equivalent just being a relativistic principle.

Ill also make sure to confirm the negligability of the KE before filling it in.

Thanks again =]]
 

1. What is slow neutron capture in the context of special relativity?

Slow neutron capture, also known as the s-process, is a nuclear reaction that takes place in stars and is responsible for the creation of heavier elements through the capture of slow-moving neutrons. This process is influenced by special relativity due to the fact that the neutrons must be moving slowly enough to be captured by atomic nuclei.

2. How does special relativity affect the rate of slow neutron capture?

Special relativity plays a crucial role in the rate of slow neutron capture because it dictates the speed at which neutrons must be moving in order to be captured by atomic nuclei. This is due to the fact that special relativity states that as an object's velocity increases, its mass also increases, making it more difficult for the neutron to be captured by the nucleus. Therefore, the slower the neutron is moving, the higher the probability of it being captured and leading to the creation of heavier elements.

3. Can slow neutron capture occur without the influence of special relativity?

No, slow neutron capture cannot occur without the influence of special relativity. This is because special relativity dictates the speed at which neutrons must be moving in order to be captured by atomic nuclei, which is essential for the s-process to take place. Without the influence of special relativity, the rate of slow neutron capture would be significantly different and the creation of heavier elements would be affected.

4. How does the energy released during slow neutron capture relate to special relativity?

The energy released during slow neutron capture is related to special relativity through the famous equation E=mc². This equation shows that mass and energy are equivalent and can be converted into each other. In the context of slow neutron capture, this means that the increase in mass due to special relativity also leads to an increase in energy, which is released during the process.

5. How does the study of slow neutron capture contribute to our understanding of special relativity?

The study of slow neutron capture plays a crucial role in our understanding of special relativity. By observing the effects of special relativity on the rate of slow neutron capture, scientists can gain a better understanding of the fundamental principles of this theory. Additionally, studying the energy released during the process can also provide further insights into the relationship between mass and energy predicted by special relativity.

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