GATE PH 2013 • Q12
MCQ • 1 Mark
Section 9: Nuclear and Particle Physics
Nuclear Models & Decays
Subtopic: Beta Decay & Fermi Theory, Gamma Decay & Selection Rules
Question Statement:
In the β decay process, the transition 2+ →3 +, is
Options:
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(A) allowed both by Fermi and Gamow-Teller selection rule
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(B) allowed by Fermi and but not by Gamow-Teller selection rule
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(C) not allowed by Fermi but allowed by Gamow-Teller selection rule
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(D) not allowed both by Fermi and Gamow-Teller selection rule
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.12)**
**1. Nuclear Beta Decay Selection Rules**:
- In nuclear $\beta$-decay, the transition can proceed via Fermi or Gamow-Teller mechanisms:
- **Fermi Transition**: Lepton spins couple to $S = 0$ (singlet), carrying zero orbital and spin angular momentum:
$\Delta J = 0, \quad \Delta \pi = \text{no} \quad (0^+ \to 0^+ \text{ allowed})$
- **Gamow-Teller (GT) Transition**: Lepton spins couple to $S = 1$ (triplet), carrying 1 unit of angular momentum:
$\Delta J = 0, \pm 1 \quad (0 \to 0 \text{ forbidden}), \quad \Delta \pi = \text{no}$
**2. Application to 2+ -> 3+ Transition**:
- Initial state: $J_i^\pi = 2^+$
- Final state: $J_f^\pi = 3^+$
- Angular momentum change:
$\Delta J = |J_f - J_i| = |3 - 2| = 1$
- Parity change:
$\Delta \pi = \pi_i \times \pi_f = (+1) \times (+1) = +1 \implies \text{no parity change}$
- Since $\Delta J = 1$ (non-zero) and $\Delta \pi = \text{no}$, Fermi matrix element $\langle 1 \rangle = 0$, while Gamow-Teller matrix element $\langle \vec{\sigma} \rangle \neq 0$.
- Thus, the transition is a **pure Gamow-Teller allowed transition**.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **Option (B)** (`pure Gamow-Teller allowed`).
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