1. Identify the process involved

A free neutron undergoes $\beta^-$ decay. The standard nuclear reaction is:

where:

• $n$ = neutron
• $p$ = proton
• $e^-$ = electron
• $\bar{\nu}$ = antineutrino

2. Why this is the correct form

A neutron is electrically neutral. After decay:

• proton has charge $+1$
• electron has charge $-1$
• antineutrino has charge $0$

So total charge after decay is:

which matches the initial charge of the neutron.

Also, in neutron decay, the lepton produced with the electron is an antineutrino, not a neutrino, to conserve lepton number.

3. Check each option

• A: $n \rightarrow p + e^+ + \nu$

  This is incorrect because charge becomes:

  $$+1 + +1 = +2$$

  which does not match initial charge $0$.

• B: $n \rightarrow p + e^- + \bar{v}$

  This matches the known $\beta^-$ decay of neutron. Correct.

• C: $n \rightarrow p + e^+ + \bar{v}$

  Incorrect because charge is not conserved.

• D: $n \rightarrow p + e^- + v$

  Charge is conserved, but lepton number is not. In neutron decay, the emitted neutral lepton is an antineutrino, not a neutrino.

4. Final answer

Therefore, the correct nuclear process is:

So the correct option is B.