# How far apart are neutrinos at rest?

If I cool down Hydrogen then I can make it into a metal, where the spacing of the atoms will be somewhat around twice the Bohr radius which as we know is inversely proportional to electron mass. $$r_n=\frac {n^2 \hbar ^2}{Zk_ee^2m_e}$$

However, in the Bohr radius there is also a factor which is proportional to electron charge, while neutrinos are not charged, so maybe the analogy is meaningless.

Nevertheless one can attach significance to the fact that the Bohr radius is inversely proportional to the electron mass , which is about 1800 times smaller than the proton mass. In particular, this big difference in mass is why we think of the nucleus as being small and stationary, and why chemical bonds are made with electrons and not protons.

I was wondering if this inverse dependence on mass would somehow extend to neutrinos in their rest state as well. Is there a sense in which the lightness of neutrinos contributes to a property of having longer-range interactions or taking up a greater "volume" in their rest state, by analogy with electrons?

• Does this answer your question: physics.stackexchange.com/questions/89202/… – Dale Apr 30 '20 at 10:43
• "If I cool down Hydrogen then I can make it into a metal". At low temperatures hydrogen is a molecular solid. To make it a metal requires very high pressure. – my2cts Apr 30 '20 at 10:59
• "maybe the analogy is meaningless". It is meaningless because neutrinos do not form atoms. – my2cts Apr 30 '20 at 11:01
• "inversely proportional to the electron mass" This is meaningless as long as the electron mass is a constant. You can make muonic hydrogen, where the electron is replaced by a muon. It doesn't last for long as the muon decays. – my2cts Apr 30 '20 at 11:08
• I'm curious about how you'd slow down neutrinos, and how you'd observe such low energy neutrinos. Neutrinos are fermions, so constrained neutrinos have to obey the Pauli exclusion principle. But constraining them isn't easy! – PM 2Ring Apr 30 '20 at 12:20

The analogy in numbers is meaningless because the electron and the proton form a bound state under the electromagnetic attraction. But it leads into wondering whether there could be a bound state of two neutrinos under the weak or the gravitational force.

There is a relevant question here, whether neutrinos can form bound states, which is answered in the negative for the Standard Model, due to the large mass of the Z. It would need extensions of the standard model.

As for gravitational bound states of neutrinos, they are explored in this question and answer. The conclusion of the answer is that it would be very improbable for such an "atom" to form due to the very small value of the gravitational coupling.

Neutrinos form a noninteracting Fermi gas. The density can have any value. To achieve a neutrinos at rest experimentally would require cooling them to incredibly low temperature. This is the consequence of their very small mass. At the same time there is no cooling mechanism. So to achieve this experimentally is very much impossible.