Match the order of \beta - decays given in the left column to appropriate clause in the
right column. Here X(I\pi) and Y(I\pi) are nuclei with intrinsic spin I and parity 𝜋.
1. X (
1
2
+
) → Y (
1
2
+
)
i) First forbidden \beta-decay
2. X (
1
2
-
) → Y (
2
+
)
ii) Second forbidden \beta-decay
3. X(3+) → Y(0+)
iii) Third forbidden \beta-decay
4. X(4-) → Y(0+)
iv) Allowed \beta-decay
Options:
(A) 1 - i, 2 - ii, 3 - iii, 4 - iv
(B) 1 - iv, 2 - i, 3 - ii, 4 - iii
(C) 1 - i, 2 - iii, 3 - ii, 4 - iv
(D) 1 - iv, 2 - ii, 3 - iii, 4 - i
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.12)**
**1. Theoretical Principles of $\beta$-Decay Selection Rules**:
- Nuclear transitions $X(I_i^{\pi_i}) \to Y(I_f^{\pi_f})$ are classified by spin change $\Delta I = |I_i - I_f|$ and parity change $\Delta\pi = \pi_i \pi_f$:
- **Allowed**: $\Delta I = 0, 1$ (Fermi: $\Delta I = 0$, Gamow-Teller: $\Delta I = 0, 1$, with $0 \to 0$ forbidden for GT), **$\Delta\pi = \text{No}$** (parity unchanged).
- **First Forbidden**: $\Delta I = 0, 1, 2$, **$\Delta\pi = \text{Yes}$** (parity changes).
- **Second Forbidden**: $\Delta I = 2, 3$, **$\Delta\pi = \text{No}$** (parity unchanged).
- **Third Forbidden**: $\Delta I = 3, 4$, **$\Delta\pi = \text{Yes}$** (parity changes).
**2. Step-by-Step Evaluation of Each Transition**:
- **1. $X(1/2^+) \to Y(1/2^+)$**:
- $\Delta I = |1/2 - 1/2| = 0$, Parity: $(+) \to (+)$ (No change).
- Classified as **Allowed $\beta$-decay** $\implies$ **1 matches (iv)**.
- **2. $X(1/2^-) \to Y(5/2^+)$**:
- $\Delta I = |5/2 - 1/2| = 2$, Parity: $(-) \to (+)$ (Change).
- $\Delta I = 2$ with parity change is **First forbidden (unique)** $\implies$ **2 matches (i)**.
- **3. $X(3^+) \to Y(0^+)$**:
- $\Delta I = |3 - 0| = 3$, Parity: $(+) \to (+)$ (No change).
- $\Delta I = 3$ with no parity change is **Second forbidden** $\implies$ **3 matches (ii)**.
- **4. $X(4^-) \to Y(0^+)$**:
- $\Delta I = |4 - 0| = 4$, Parity: $(-) \to (+)$ (Change).
- $\Delta I = 4$ with parity change is **Third forbidden** $\implies$ **4 matches (iii)**.
- Matching set: **1 - iv, 2 - i, 3 - ii, 4 - iii**.
**3. Official Verification**:
The verified official IIT Master Answer is **Option (B)**.
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