# How do you measure the mass of the electron very precisely?

This week I showed my high school students that if you add the mass of a proton and the mass of an electron the result is higher than the mass of a hydrogen atom, because of the binding energy being negative. I used very precise measurements of the masses, and some student asked how these measurement are actually done. I did some research and found out how you can do it with Penning traps, but in the case of an electron you actually use a carbon ion with only one electron, as it is described here : https://www.mpg.de/7961020/electron-mass

It seems to me that if you use an electron bound to a nucleus, there is binding energy involved. If you get rid of it by quantum physics calculations, then you cannot use this result to show that adding the masses of a proton and an electron gives a mass lower than the mass of an hydrogen atom. It would be a circular argument.

Is there another way to determine the mass of an electron very precisely (not the way Millikan did), without any binding energy involved ?

• The "standard answer" might be "if it stinks, it's chemistry; if it wriggles it's biology; if it doesn't work, it's physics" . Doing any highly accurate measurement in physics is very difficult. Try measuring the speed of light - easy to a couple sig figs, but after that you need an evacuated tube to remove the effect of nonuniform atmosphere, etc., followed by phase vs. group velocities, and on and on. Apr 16, 2021 at 14:11
• Actually where did you find the precise measurements?Actually the mass of the electron can be seen how much volt you need to apply to a particle accelerator to get pair production. Personally I would be suprised if you can measure the weight of a hydrogen atom precise enough. Apr 16, 2021 at 19:00
• You use a very, very small scale.
– Mark
Apr 16, 2021 at 20:35

The experiment you mentioned measures the electron mass in an indirect way: It does not measure the mass of the electron, but instead the magnetic moment (see footnote). This is a property that can be predicted by theory (QED) very precisely. The biggest uncertainty in the predicted magnetic moment comes from the uncertainty in the electron's mass (more precisely: the mass of the unbound electron). So the experimenters turned it around: Since the measurement is more precise than the theoretically predicted value, they can make constraints on the electron's mass. They performed additional experiments to test other predictions by QED, in order to be confident that QED works in this regime. I worked on one of the follow-up experiments that did these further QED tests.

The most accurate direct mass-measurement of the electron that I am aware o (it has been a few years!) is a direct cyclotron-frequency comparison between carbon-6+ and a free electron in the same trap, but here you still have to account for the binding-energy of the carbon-ion.

But this is not a problem, because the binding energy can be measured independently through Laser- or X-Ray spectroscopy (or they can be calculated with QED).

Footnote: The measurement is a $$g$$-factor measurement, similar in spirit to the $$g-2$$ measurement that has recently been published for the muon. In the muonic case, there seems to be a discrepancy between the QED and measurement. No such discrepancies have yet been found for the electron, but of course, it makes it even more exciting to take a look at the electron again!

• Your answer is very interesting. Does it mean that there is no way to show the existence of binding energy by measuring masses ? I guess this would be another question, but it is what motivated my question in the first place. My goal was to show my students experimentally that the mass of an atom is not simply the sum of the masses of protons, neutrons and electrons that constitute it. Apr 16, 2021 at 13:29
• @Physicsteacher It's simply a question of how to measure the mass of each particle to sufficient accuracy and precision to be able to "see" the binding energy contribution. Apr 16, 2021 at 14:08

Millikan's experiment determined just the charge on the electron. If you want a more accurate determination, there was this question How did scientists manage to measure the charge of electron so precisely?

The specific charge of an electron $$\frac{e}{m_e}$$ can be found by seeing how much they are deflected by a given magnetic field. The speed of the electrons is found from the known accelerating voltage.

Then, since the charge is known, the mass of the electron can be deduced.

Depending how accurate you want it, there is also a way here https://en.wikipedia.org/wiki/Electron_rest_mass#Determination that determines the electron mass from the Rydberg constant.

• Thanks for your answer. I should have been more specific : I am not interested in determining the mass of the electron this way, because the result is not sufficiently precise in order to show a difference between the mass of the hydrogen atom and the sum of the masses of an electron and a proton. Apr 16, 2021 at 9:00
• Are you thinking that Millikan's experiment isn't accurate enough or involves the binding energy somehow? (also an edit to the answer posted a link to give a more accurate determination of the charge) Apr 16, 2021 at 9:03
• Is e/m measured very precisely ? You would need it to obtain a very precise value for the mass, when knowing the charge. I could not find any information on the greatest precision achieved. Apr 16, 2021 at 13:22
• not sure about the accuracy of e/m sorry, if mass spectrometry isn't accurate enough, perhaps the 'cross fields' method might be an improvement. Otherwise the Rydberg constant method mentioned might be the way to go...all the best with it Apr 16, 2021 at 15:07