Are there experiments where the charge to mass ratio for protons have been determined in the same way as in Thompson’s experiments (where he determined the charge to mass ratio of electron) but with protons instead of electrons? Which experiments are performed to determine the charge to mass ratio of proton? Maybe somebody has pdfs of Thompson’s papers where he was reporting about the charge to mass ratio of proton and hydrogen ions.
[Physics] In which experiments the charge to mass ratio of proton was determined
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The one-word answer is yes.
You are also correct that the neutron is not just a proton and electron living together. The process of merging a proton and electron proceeds via the weak force. Specifically, an up quark in the proton exchanges a W boson with the electron. The W boson carries a unit of positive charge from the quark to the electron. In that process the up quark (charge +2/3) is converted to a down quark (charge -1/3) so that the proton (uud) becomes a neutron (udd). The negatively charged electron is converted into a neutrino. This is one important point left out in your question. The full reaction is $p+e^-\to n+\nu_e$.
There is a general principle of quantum field theory called crossing symmetry that roughly states that for any process I can exchange what I call initial and final particles. So you are correct that neutron decay $n\to p+ e^- + \bar\nu_e$ implies that the process $p+e^-\to n+\nu_e$ can also happen.
This process does also happen in nature. It is one mode of radioactive decay of nuclei. Some nuclei with a sufficiently large number of protons can become more stable by absorbing one of their electrons and converting one proton into a neutron. This can happen because electron orbitals have a small but non-zero overlap with the nucleus, so that they "sometimes come into contact with" the protons.
This process can also happen artificially as you suggest. In fact, it seems that accelerators used in medical facilities produce neutrons as a by-product, exactly as you suggest, and this is apparently a difficulty that must be dealt with, see this paper.
In general, because the mass difference between the proton and neutron is about an MeV, in any system including protons and electrons at a temperature of order an MeV or higher, there will necessarily be populations of both neutrons and protons connected to each other by such processes, with relative amounts determined by the relevant Boltzmann factors. This should include systems where thermal fusion is taking place.
However the actual process of producing helium from hydrogen, as far as I understand, does not depend on capturing an electron on a proton to form a neutron. In stellar nucleosynthesis, two protons merge to form deuterium. That is, in the process of merging, one proton is converted into a neutron by the emission of a positron and a neutrino. Helium-2 (two protons) is highly unstable, so that this proton-to-neutron conversion producing stable deuterium is more important.
It is most definitely the magnitude of charge that matters, reversing the charge all it does is make a mirror image circle, so since in your experiment, you are measuring inherently sign-less/directionless parameters like radius, time, and magnetic field, speed etc. You would get a positive value for e/m. However, when you do report the value of $e/m$, you must report it with a sign, i.e negative. So, the ratio should be negative but, from the experiment As we are dealing with directionless quantities, you get a positive number as an output for the experiment, but when someone asks for the specific charge, not the magnitude then you report it with a sign.
Best Answer
I am not sure if Thompson ever determined the charge-to-mass ratio of a proton, but currently, the most precise measurements of the charge-to-mass ratio of a proton still use a magnetic field like Thompson, but rely on measuring (cyclotron) frequencies rather than deflection. As frequencies are the quantities that can be determined most accurate (see the Nobel lecture of Hänsch: “Never measure anything but frequency!” ), the result is much more precise.
The charge-to-mass ratio ($q/m$) is obtained from the cyclotron frequency of a proton in an magnetic field $B$. The cyclotron frequency is associated with the motion of a charged particle in the plane perpendicular to the magnetic-field direction and is given by
$\nu_c=\frac{qB}{2\pi m}$.
See for instance Phys. Rev. Lett. 74, 3544 (1995).