 Solution

th-forbidden beta decay. Fermi-type and Gamow-Teller-type below refer to these spin-independent and spin-dependent contributions, including their higher-orbital generalizations.

For $^{129} I \to^{129} Xe$ , the nuclear spin changes by $2$ and parity is unchanged. An $l = 0$ lepton pair cannot carry $J \geq 2$ ; $l = 1$ would have the wrong parity. At $l = 2$ , Fermi transitions can carry $J = 2$ , and Gamow-Teller transitions can carry $J = 1, 2, 3$ , including the permitted transfers $2$ and $3$ . Therefore this is a second-forbidden, nonunique transition, with both Fermi-type and Gamow-Teller-type contributions permitted. The quantum numbers do not determine their actual relative matrix elements.
For $^{73} Kr \to^{73} Br$ , $Δ J = 1$ and parity is unchanged. Already $l = 0, S = 1$ carries $J = 1$ . The $l = 0$ operator for a Fermi transition has rank zero and cannot connect these unequal nuclear spins. This is an allowed Gamow-Teller transition. The decrease of atomic number makes the nuclear conversion proton-to-neutron; the angular momentum classification applies to either the energetically available beta-plus decay or electron capture branch.
For $^{158} Yb \to^{158} Tm$ , $0^{+} \to 2^{-}$ requires transfer $J = 2$ and a parity change. The lowest possible orbital order is $l = 1$ . The $S = 0$ pair for a Fermi transition then has only $J = 1$ , whereas the $S = 1$ pair for a Gamow-Teller transition can attain $J = 2$ . Thus this is a first-forbidden unique Gamow-Teller-type transition. Here unique means the maximal transfer $J = l + 1$ at the leading orbital order, as in a unique forbidden beta decay.
For $^{22} Mg \to^{22} Na$ , $0^{+} \to 0^{+}$ permits the scalar operator for a Fermi transition with $l = 0$ . The allowed Gamow-Teller transition operator has rank one, and the triangle rule excludes $0 \to 0$ . Interpreting the printed logarithm in the usual comparative half-life convention gives

f t = 1 0^{3.48} s = 3.02 \times 1 0^{3} s .

(2)

This small value supports the expected superallowed Fermi transition between isobaric analog states. The

0^{+} \to 0^{+}

quantum numbers alone would establish allowed Fermi character, but would not by themselves establish the large analog-state overlap implied by superallowed strength.

In the order listed, the concise classification is

1 : F + GT, second forbidden; 3 : GT, first forbidden unique; 2 : GT, allowed; 4 : F, superallowed .

(3)

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