Hoyle's calculation of 7.65 MeV level of carbon nucleus

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Discussion Overview

The discussion revolves around the calculation of the energy level of the carbon nucleus, specifically the 7.65 MeV level, in the context of the triple alpha process. Participants explore the mass values of helium-4 and beryllium-8, their implications for energy calculations, and the effects of ionization at high temperatures.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant calculates the mass of helium-4 and beryllium-8, concluding it results in a discrepancy with the expected 7.65 MeV level, prompting a request for clarification on the calculation.
  • Another participant provides updated mass values for helium-4 and beryllium-8, yielding a different energy calculation of 7.4135 MeV before considering kinetic energy.
  • A participant notes that at 100 MK, helium-4 must be considered fully ionized, suggesting the use of the mass of the alpha particle instead of the atomic mass including electrons.
  • Another participant agrees that beryllium will also be fully ionized and discusses the neglect of electron binding energies in the calculations, arriving at a revised energy level of 7.37 MeV, though still not matching the expected value.
  • One participant questions whether the mass of Carbon-12 would be less than 12 amu due to full ionization.
  • A participant asserts that electrons are accounted for in their calculations and references a 1954 paper that identifies the energy level of the C-12 nucleus at approximately 7.68 MeV above ground level, which relates to the resonance strength in the context of the mass sum of beryllium-8 and the alpha particle.

Areas of Agreement / Disagreement

Participants express differing views on the correct mass values and the implications for energy calculations, with no consensus reached on the correct approach or final energy level. The discussion remains unresolved with multiple competing calculations and interpretations.

Contextual Notes

Participants highlight the importance of considering ionization effects and the inclusion of binding energies in their calculations, which may affect the outcomes. There are also references to specific historical papers that may influence the understanding of the energy levels discussed.

damosuz
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Mass of helium-4 (4.0015 u) plus mass of beryllium-8 (8.0053 u) gives 12.0068 u, which is 6.35 MeV above fundamental level of carbon-12. For triple alpha process T= 100 MK, which corresponds to a kinetic energy of 2 x 3/2 kBT = 0.03 MeV for the particles. This amounts to 6.38 MeV, way below 7.65 MeV. I would like to know what is wrong with my calculation.
 
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From wiki, Helium-4 is 4.002602 amu's and Beryllium-8 is 8.00530510 amu's. Doing some back of the envelope calculations, I get 7.4135 MeV prior to including kinetic energy.
 
4,002602 u is helium-4 including mass of electrons, but at 100 MK, helium-4 must be completely ionized, so I guess you should take mass of alpha particle...
 
damosuz said:
4,002602 u is helium-4 including mass of electrons, but at 100 MK, helium-4 must be completely ionized, so I guess you should take mass of alpha particle...

Ah, a valid point. I'll have to see what I can find on the matter.
 
Beryllium will be fully ionized as well.
If we neglect binding energies of electrons (<1 keV), we don't have to care about them - we remove 2+4 electrons from the left side and 6 electrons from the right side of the reaction, which does not change anything.
4.002602 u + 8.005305 = 12.007907 u or 7.37 MeV above 12.00000. Hmm, still wrong, but much closer.
 
Wouldn't Carbon-12's mass be less than 12 amu since it will be fully ionized too?
 
Electrons are included on both sides in my calculation. Removing them does not change the difference, it just makes calculations more complicated
I found the answer in the original 1954 paper, page 130:
The important energy level of the C12 nucleus in the present problem is one very recently identified by Dunbar, Pixley, Wenzel, and Whaling (1953). This level occurs at about 7.68 mev above ground level, which corresponds to a value of E_R of about 0.31 mev (It will be recalled that E_R is measured relative to the sum of the masses of Be8 and alpha-particle, this being about 7.37 mev above the ground level of C12
The resonance is just close enough to be strong, but not that close to the sum of masses.
 

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