Free Neutron Decay

Question

I've read that free neutrons can decay into hydrogen, but it's rare because the energy from the decay usually sends the electrons away, unable to bind with the protons. But if trillions of free neutrons formed and then decayed around the same time in reasonably close proximity, could most of the decay result in hydrogen formation?

Answer 1

Let's just crank it out with:

$$ n \rightarrow H + \bar{\nu_e} $$

I'm going to set $m_{\nu}=0$, for ...reasons.

We have:

$$ p^{\mu}_n = (m_n, 0) $$

going to:

$$ p^{\mu}_H = (E', p') $$ $$ p^{\mu}_{\bar\nu_e} = (p', -p') $$

which is trivial in the space part, the time component says:

$$ E' + p' = m_n $$

or

$$ p'^2 + M_H^2 = (m_n-p')^2 = m_n^2 + p'^2 - 2m_np'$$

or:

$$ p' = \frac{m_n^2 - M_H^2}{2m_n} = 0.78145\,{\rm MeV}$$

while

$$ \Delta E = m_n-M_H = 0.78178\,{\rm MeV} = 1.00042 p'$$

which I think means there is not a lot of phase space for that neutrino.

Answer 2

Definitely, almost obviously.

Most free neutron decay would send the electron out, with only a rare subset having bound hydrogen.

However, at large enough numbers, then an electron that had already been flung out, could lose energy by Bremsstrahlung near a second proton, and then be slowed down enough to be captured by a third proton.