The $D_1$ and $D_2$ lines of Sodium ($3^2P_{1/2} \rightarrow 3^2S_{1/2}$ and $3^2P_{3/2} \rightarrow 3^2S_{1/2}$) will split on the application of a weak magnetic field into
Options:
(A) 4 and 6 lines respectively
(B) 6 and 4 lines respectively
(C) 3 and 3 lines respectively
(D) 2 and 4 lines respectively
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2007 Q.8)**
**1. Anomalous Zeeman Effect Principles**:
- **Field & Concept**: *Section 6: Atomic and Molecular Physics* (`Zeeman, Paschen-Back and Stark Effects`)
- In a weak magnetic field, each energy level splits into $(2J + 1)$ Zeeman sublevels characterized by $M_J$.
- Selection rules for electric dipole transitions: $\Delta M_J = 0, \pm 1$ (except $M_J = 0 \rightarrow M_J = 0$ when $\Delta J = 0$).
**2. Number of Splitting Components**:
- **For $D_1$ line ($^2P_{1/2} \rightarrow ^2S_{1/2}$)**:
- Upper state $J = 1/2 \implies M_J = \pm 1/2$ (2 levels).
- Lower state $J = 1/2 \implies M_J = \pm 1/2$ (2 levels).
- Allowed transitions ($\Delta M_J = 0, \pm 1$): $2 \times 2 = 4$ distinct Zeeman lines.
- **For $D_2$ line ($^2P_{3/2} \rightarrow ^2S_{1/2}$)**:
- Upper state $J = 3/2 \implies M_J = \pm 3/2, \pm 1/2$ (4 levels).
- Lower state $J = 1/2 \implies M_J = \pm 1/2$ (2 levels).
- Allowed transitions: exactly 6 distinct Zeeman lines.
**3. Official Verification & Conclusion**:
- **Official Master Answer**: **Option (A)**: 4 and 6 lines respectively.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.