Editing Beta decay (section)

==Types of beta decay transitions==
{{main|Beta decay transition}}
Beta decays can be classified according to the angular momentum ([[Angular momentum operator|{{mvar|L}}&nbsp;value]]) and total spin ([[Spin (physics)|{{mvar|S}}&nbsp;value]]) of the emitted radiation. Since total angular momentum must be conserved, including orbital and spin angular momentum, beta decay occurs by a variety of quantum state transitions to various nuclear angular momentum or spin states, known as "Fermi" or "Gamow–Teller" transitions. When beta decay particles carry no angular momentum ({{math|1=''L'' = 0}}), the decay is referred to as "allowed", otherwise it is "forbidden".

Other decay modes, which are rare, are known as bound state decay and double beta decay.

===Fermi transitions===
A '''Fermi transition''' is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin <math>S=0</math>, leading to an angular momentum change <math>\Delta J=0</math> between the initial and final states of the nucleus (assuming an allowed transition). In the non-relativistic limit, the nuclear part of the operator for a Fermi transition is given by
<math display="block"> \mathcal{O}_{F}=G_{V}\sum_{a} \hat{\tau}_{a\pm} </math>
with <math>G_V</math> the weak vector coupling constant, <math>\tau_{\pm}</math> the [[isospin]] [[Ladder operator|raising and lowering operators]], and <math>a</math> running over all protons and neutrons in the nucleus.

===Gamow–Teller transitions===
A '''Gamow–Teller transition''' is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin <math>S=1</math>, leading to an angular momentum change <math>\Delta J=0,\pm 1</math> between the initial and final states of the nucleus (assuming an allowed transition).
In this case, the nuclear part of the operator is given by
<math display="block"> \mathcal{O}_{GT}=G_{A}\sum_{a} \hat{\sigma}_{a}\hat{\tau}_{a\pm} </math>
with <math>G_{A}</math> the weak axial-vector coupling constant, and <math>\sigma</math> the [[Pauli matrices|spin Pauli matrices]], which can produce a spin-flip in the decaying nucleon.

===Forbidden transitions===
{{Main|Beta decay transition}}

When {{math|''L'' > 0}}, the decay is referred to as "[[forbidden transition|forbidden]]". Nuclear [[selection rule]]s require high {{mvar|L}}&nbsp;values to be accompanied by changes in [[nuclear spin]]&nbsp;({{mvar|J}}) and [[Parity (physics)|parity]]&nbsp;({{mvar|π}}). The selection rules for the {{mvar|L}}th forbidden transitions are:
<math display="block">\Delta J=L-1, L, L+1; \Delta \pi=(-1)^L, </math>
where {{math|1=Δ''π'' = 1 or −1}} corresponds to no parity change or parity change, respectively. The special case of a transition between isobaric analogue states, where the structure of the final state is very similar to the structure of the initial state, is referred to as "superallowed" for beta decay, and proceeds very quickly. The following table lists the Δ{{var|J}} and Δ{{var|π}} values for the first few values of&nbsp;{{mvar|L}}:

{|class="wikitable"
|-
! Forbiddenness !! Δ{{var|J}} !! Δ{{var|π}}
|-
|Superallowed ||0 ||{{no}}
|-
|Allowed ||0, 1 ||{{no}}
|-
|First forbidden ||0, 1, 2 ||{{yes}}
|-
|Second forbidden ||1, 2, 3 ||{{no}}
|-
|Third forbidden ||2, 3, 4 ||{{yes}}
|}