Where is this extra energy in the Hydrogen Atom coming from?

The mass of hydrogen is 938.783073804(56) MeV/c^2
The mass of a proton is 938.272088163(35) MeV/c^2
The mass of an electron is .510998949(97) MeV/c^2
The mass defect is thus   -.000013308(84) MeV/c^2
The ionization energy is   .000013598(43) MeV      (with relativistic and nuclear corrections)


Data from NIST and CODATA. Hydrogen should be equal in mass to a proton plus an electron minus the ionization energy, but acccording to the above, a hydrogen atom is 0.290 electronvolts heavier/more energetic than that would suggest, and this difference is statistically significant.

The ionization energy takes into account the kinetic energy of the electron, and I would assume the kinetic energy of the nucleus in the center of mass frame is zero, so where is this extra energy coming from? For reference, this difference is about 3 times larger than the maximum predicted mass of the electron neutrino, so not an insignificant clerical error!

• "I would assume the kinetic energy of the nucleus in the center of mass frame is zero" How is that consistent with conservation of momentum? Oct 16 at 1:25
• Because the electron in the ground state is distributed spherically symmetrically about the nucleus. Any kinetic energy in the nucleus would just be the kinetic energy of the atom as a whole, no? Oct 16 at 2:09
• You mean you are considering the nucleus to be stationary because of its mass compared to the electron, and so it has zero kinetic energy. Oct 16 at 2:31
• Yes, but even we did not neglect the nucleus's KE, it would be on the order of the ratio of electron to proton mass times the electron's KE (13.6 eV), which would be only about 0.007 eV, clearly not the source of the 0.3 eV that's "missing". Oct 16 at 3:02
• Please link to source page where these numbers are given if possible, there may be some relevant information on how these were determined. Oct 16 at 4:41

Your mass defect result 13.3 eV is indeed too low, it should be around 13.6 eV. Most of this discrepancy (0.3 eV) is most probably not real, otherwise basic textbooks would have to rewritten. CODATA gives proton and electron masses and those numbers should be quite reliable. They do not give hydrogen atom mass so it is from some other source which is most probably not guarranteed to be consistent with CODATA.

It is difficult to measure mass of neutral hydrogen atom to similarly high accuracy as the mass of the proton, because the atom is electrically neutral and quite unstable (forms $$H_2$$ molecules or ions). I do not think digits in 938.783073804 MeV/c2 after 6th decimal are certain to be correct.

To investigate this further I would go after how this number was determined, or how in general hydrogen atom H1 mass is best determined. Direct measurement of mass is probably problematic, resulting in low accuracy result. It may be that the more accurate way is actually the indirect way, assuming atom mass = mass of proton + mass of electron - ionization energy (known from spectroscopy).

• Indeed, the source for that number was incorrectly attributed to NIST, and I was not able to find the true source. Using the AME 2020 atomic mass evaluations, I get that the actual mass defect is 13.62(92) eV. I am happy with that level of agreement and I will consider the "issue" resolved. 0.03 eV is almost exactly the thermodynamic energy of a particle at room temperature anyway so that's about as close of an agreement as I should expect. Oct 16 at 8:06
• > "0.03 eV is almost exactly the thermodynamic energy of a particle at room temperature anyway so that's about as close of an agreement as I should expect." Why? Measurements can be done at much lower temperatures that the room temperature. Oct 16 at 12:51
• All I meant by that was that I'm not surprised that the errors in measurement for a neutral body are on the order of their thermal motion. No doubt the measurements were made in a cold trap but any number of factors could lead to an error on the order of thermal energy. Oct 16 at 15:21