Why Does the Proton-Electron Mass Ratio Diverge from Direct Division?

[neilparker62]

In the CODATA table of physical constants, there are very precise numbers given for mass of electron and mass of proton. And an even more precise number for proton electron mass ratio. But when you divide the mass of proton by the mass of electron, you don't get the same number as the proton electron mass ratio The numbers start diverging at about the 8th significant figure.

I'm asking because I need a very precise value of reduced electron mass. So I'm not sure whether to use the standard formula:

me * mp / (me + mp)

or one with the proton electron mass ratio:

mp / (1 + mp/me)

They should be the same but they're not and it makes quite a big difference in calculated results.

 

[mfb]

Which units do you use for the calculation, and which units do you need for the result? The different parameters have different, often correlated sources of uncertainties. For nonlinear combinations (like ratios), this can lead to a best value that is not the ratio of the individual best values. Unless you get information about those correlations, you'll have to be conservative with the uncertainty estimate.

I would check "me / (me/mp + 1)", but this is just a guess.

You can also ask CODATA if they have the reduced mass somewhere, or check the corresponding publications - it could even be an input to the CODATA fit.

 

[neilparker62]

It's an atomic transition energy calculation. So output unit would be energy. Convert to frequency via Planck's constant. Inputs: reduced mass of electron x c^2 x terms with fine structure constant.

Sounds like a good idea to contact CODATA for a value of the electron's reduced mass.

Thanks for the suggestion.

 

[mfb]

Inputs: reduced mass of electron x c^2 x terms with fine structure constant.

This product is much better known than the individual components. 6*10^-12 uncertainty - expressed as 1/m, but as the speed of light is fixed the frequency has the same relative uncertainty.

Electron mass in MeV and Planck constant in eV*s are nearly 100% correlated: Comparison

 

[neilparker62]

Are we saying that if QED effects in certain energy levels somehow cancel out, then the Rydberg formula with 1/n^2 - 1/(n+1)^2 is still valid? I see a lot of literature where scientists seem to be doing something like this in order to calculate ever more accurate values of the Rydberg constant. Correlation coefficient on Balmer series frequencies (H I outermost level of each primary quantum number) vs 1/n^2 - 1/(n+1)^2 is 1 at the level of accuracy my PC can manage. But the resultant gradient from the regression analysis does not yield the CODATA value of the Rydberg constant. It is miles out.

 

[neilparker62]

Have to correct myself. The gradient of the above regression yields the Rydberg constant Rh as:

10 967 776.9

The 'super accurate' value given by CODATA is

10 973 731.568 508(65)

but that is for R (infinity).

It's a pity they don't give Rh as well and then we could easily determine reduced mass. I'll ask them for it!

 

[neilparker62]

I would check "me / (me/mp + 1)", but this is just a guess.

Thanks, I'm using that formula and it seems to work fine.