# He-3 atom recoil

When a T atom decays into He-3 atom ground state, what is 1) the kinetic energy of its recoil energy and 2) the theoretical uncertainty of that energy?

The beta decay energy of a triton is something like 18 500 eV. When (most of time) a free electron is emitted, its energy can acquire any continuous values - 100 eV or 110, or 100,10 or any value in between, because the rest is distributed to a neutrino, with flavour of antielectron.

The usual result is He3+ cation, hydrogenlike atom. Sometimes the emitted electron ionizes the existing electron on the way out, leaving an He-3 nucleus.

But surely an emitted electron does not have a free choice of energy on the end below a few tens of eV?

A He atom has no excited states within 20 eV of ground state.

So what is the probability that a T atom does not emit a free electron, but decays into a bound state of He-3 atom, giving all the rest of 18 500 eV to the neutrino? What is the probability that the bound state is specifically ground?

A ground state of He-3 is stable. It follows that its energy and momentum has no uncertainty whatsoever and can be measured to arbitrary precision.
A ground state T atom is long-lived. With lifetime in the order of 10ˇ8 s, the theoretical decay width of T should be around 10ˇ-23 eV
What is the present experimental uncertainty of triton decay energy, that 18 keV?? I guess bigger than 10ˇ-23 eV, but how big by last best estimates?

Suppose that a He-3 atom recoiled by neutrino emission could have its energy and momentum measured with a great precision, and the energy and momentum of original T were also known to a great precision, e. g. because of smallness.

How precisely could the rest mass of the roughly 18 500 eV neutrino be calculated from the measurement of the energy of the recoiled ground state He-3 atom? I. e. how does the uncertainty of recoil atom energy map into uncertainty of neutrino rest mass, perhaps depending on what the rest mass is?

mfb
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But surely an emitted electron does not have a free choice of energy on the end below a few tens of eV?
If that refers to the energy after it left the nucleus, why not? The 18500 eV value takes into account that the electron has to leave the nucleus.

What is the present experimental uncertainty of triton decay energy, that 18 keV??
Of the order of 1 eV or less. That is not the problem.

How precisely could the rest mass of the roughly 18 500 eV neutrino be calculated from the measurement of the energy of the recoiled ground state He-3 atom? I. e. how does the uncertainty of recoil atom energy map into uncertainty of neutrino rest mass, perhaps depending on what the rest mass is?
Forget it. You get a neutrino at 18.5 keV. It is extremely ultra-relativistic at that point as we know all neutrino masses are at most of the order of 1 eV. The energy-momentum relation is so close to a massless particle that you are orders of magnitude away from a realistic measurement. The uncertainty principle will ruin every attempt to measure differences that small.

To measure a neutrino mass, you need the opposite case: slow neutrino, fast electron, only then the rest mass has a relevant influences. That is what KATRIN will do, for example.

vanhees71
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Just to add a few things regarding the tritium beta decay measurements of neutrino mass:

The first thing you need is a very high-resolution spectrometer that allows you to measure the electron energy to a high precision. This is definitely a bottleneck. At some point sub-eV (around 0.02 eV) you would also need a tritium source with a temperature lower than 300 K. Random thermal movements of the source will tend to erase the features that you want to see. The nuclear recoil can be taken into account when computing the end-point as long as you know the mass well enough.

With KATRIN, we are essentially reaching the bound for how high precision you can put into a spectrometer (the rooftops in Karlsruhe would not allow a larger one). There is an alternative which is currently being investigated called Project 8 http://www.project8.org

vanhees71
mfb
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You would also need a source of tritium atoms (instead of molecules), otherwise you get all the mess with rotational and vibrational excitations - something that is a large issue for KATRIN.
PTOLEMY is another idea to get more precise than KATRIN, also with RF measurements of electrons, but with a different setup.

If that refers to the energy after it left the nucleus, why not? The 18500 eV value takes into account that the electron has to leave the nucleus.
In other words, the electrons that remain bound in He-3 atom are accompanied by neutrinos with extra 20 eV or so over and above the decay energy?
The uncertainty principle will ruin every attempt to measure differences that small.
Uncertainty principle established to be around 10ˇ-23 eV? But I agree there would be practical problems.
To measure a neutrino mass, you need the opposite case: slow neutrino, fast electron, only then the rest mass has a relevant influences.

You can get a slow neutrino, but you cannot get a stationary neutrino with known no momentum, like with electron. The direction of the neutrino is an extra unknown to solve for.

mfb
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In other words, the electrons that remain bound in He-3 atom are accompanied by neutrinos with extra 20 eV or so over and above the decay energy?
Should be.
You can get a slow neutrino, but you cannot get a stationary neutrino with known no momentum, like with electron. The direction of the neutrino is an extra unknown to solve for.
That's why you measure the spectrum close to the end-point.
Uncertainty principle established to be around 10ˇ-23 eV? But I agree there would be practical problems.
I don't see the relevance of that number.
Calculate the recoil you get with a neutrino mass of 0 eV and with a neutrino mass of 1 eV. Then calculate how cold the atom has to be to be able to see such a difference, and then calculate the necessary trap size to get such a low ground-state momentum. My guess: larger than the earth.