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 Incoming links: Isobaric analog state

Past exam of the particle and nuclear physics course of the University of Cambridge / 2015 / Paper 1 / 4 / v / Solution 2026-10-20

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The state marked X in F is the

1/ 2^{-}

analogue at

11.20 MeV

excitation. Its absolute nuclear energy relative to O is

2.25 + 11.20 = 13.45 MeV

. Hence

Q_{X} = 16.26 - 13.45 - 0.511 = 2.299 MeV .

(1)

It is a same-spin, same-parity allowed beta decay between isobaric analog states. The large analogue Fermi overlap makes it a superallowed analogue transition, in the broad analogue-state terminology. Since

J = 1/2

rather than zero, this classification does not exclude a Gamow-Teller contribution.

For a transparent numerical estimate, take the Fermi part to dominate both this branch and the reference F-ground-state decay, neglect Coulomb and electron-mass corrections in their phase-space ratio, and use the Sargent rule. For

T_{z} = (N - Z) /2

, Ne has

T = T_{z}^{magnitude} = 3/2

and goes from

T_{z} = - 3/2

- 1/2

in its analogue multiplet. The isospin ladder strength in analogue beta decay is therefore three. The F→O ground-state mirror transition in

T = 1/2

has Fermi strength one. Thus the analogue beta-decay branching estimate gives

λ_{X} ≃ \frac{3}{95 s} (\frac{2.299}{1.739})^{5} = 0.128 s^{- 1}, B_{X} ≃ (0.15 s) λ_{X} = 0.0191 ≃ 1.9%.

(2)

This percent-level result is an estimate under the stated assumptions, not a unique consequence of the level diagram. More generally, if

f

denotes an allowed phase-space integral and

S = g_{V}^{2} ∣ M_{F} ∣^{2} + g_{A}^{2} ∣ M_{GT} ∣^{2}

, then

B_{X} = \frac{τ _{N e}}{τ _{F}} \frac{f _{X}}{f _{F}} \frac{S _{X}}{S _{F}} .

(3)

The diagram fixes the Fermi strengths but not both axial strengths. Including them and proper Positron phase-space functions changes the numerical result. Assuming equal total allowed strengths instead of the Fermi-only ratio three would give approximately

0.64%

; that alternative makes a different nuclear-structure assumption. The rate enhancement from removing the forbiddenness explains why the X branch can be appreciable despite its much smaller endpoint than the Ne ground-state branch.

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Past exam of the particle and nuclear physics course of the University of Cambridge / 2019 / Section A / 3 / Solution 2026-10-18

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Use the leading lepton-orbital classification of a beta decay transition. For lepton-pair orbital angular momentum

l

, Fermi transitions have pair spin

S = 0

and total lepton angular momentum

J = l

; Gamow-Teller transitions have

S = 1

and

J = ∣ l - 1∣, \dots, l + 1

. Addition of angular momentum requires

∣ J_{i} - J_{f} ∣ \leq J \leq J_{i} + J_{f},

(1)

while the nuclear parity change requires

π_{f} / π_{i} = (- 1)^{l}

. The least allowed

l

gives the order:

l = 0

means an allowed beta decay, and

l = n > 0

n

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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Superallowed beta decay Created 2026-10-18 Updated 2026-10-20

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An especially strong allowed beta decay, conventionally including pure Fermi transitions between

0^{+}

isobaric analog states. Those states have closely related spatial and spin structure, so the isospin ladder operator has a large overlap. A

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

assignment alone fixes the allowed Fermi character, but does not establish analog-state overlap or a large rate; a small comparative half-life supplies additional evidence.

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