Hoyle's calculation of 7.65 MeV level of carbon nucleus

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The discussion centers on the calculation of the energy level of the carbon nucleus, specifically the 7.65 MeV level. Initial calculations using the masses of helium-4 and beryllium-8 yield a discrepancy, with results showing 6.35 MeV and later adjustments bringing it closer to 7.37 MeV. Participants note the importance of considering ionization at high temperatures, as helium-4 and beryllium-8 would be fully ionized at 100 MK. A reference to a 1954 paper reveals that the relevant energy level for carbon-12 is approximately 7.68 MeV above the ground state, indicating the resonance is significant but not precisely aligned with the mass sum. The discussion highlights the complexities of nuclear binding energy and mass calculations in astrophysical contexts.
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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