Zeeman, Paschen-Back and Stark Effects Questions and Answers

Answer: B

The Zeeman spectrum has 10 lines.

In a weak field, spin-orbit coupling splits the n = 2, l = 1 level into ²P_3/2 and ²P_1/2, and the anomalous Zeeman effect splits each level into 2j + 1 sublevels with different Landé g-factors (g = 4/3, 2/3 and 2 for ²P_3/2, ²P_1/2 and ²S_1/2). With the selection rule Δm_j = 0, ±1:

• ²P_3/2 → ²S_1/2: 6 lines (the m_j = ±3/2 → ∓1/2 transitions are forbidden);
• ²P_1/2 → ²S_1/2: 4 lines.

Because the g-factors differ, all 6 + 4 = 10 lines have different energies. 5 or 8 undercount the allowed transitions, and 12 counts all 4 × 2 + 2 × 2 pairs without the selection rule.

Answer: D

The Landé factor is 1.

\[ g = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)} \]

In a singlet state \( S = 0 \), so \( J = L \) and the fraction vanishes: \( g = 1 \). Such states show the normal Zeeman effect, with the splitting of the classical orbital value.

3/2, 5/2 and 1/2 need a non-zero spin; for example the ³P₁ level has g = 3/2. None of them is possible when \( S = 0 \).

Answer: B

The splitting is \(2\mu_{B}B\).

For a \(^{2}S_{1/2}\) state, L = 0 and S = J = 1/2, so the Landé factor is \(g_{J} = 2\). In a weak field each level shifts by \(\Delta E = g_{J}m_{J}\mu_{B}B\), with \(m_{J} = \pm\frac{1}{2}\):

\[ \begin{aligned} E_{\pm} &= \pm\frac{1}{2} \times 2\mu_{B}B = \pm\mu_{B}B \\ E_{+} - E_{-} &= 2\mu_{B}B \end{aligned} \]

\(\mu_{B}B\) is the shift of each sublevel from the original level, not the difference between them. \(4\mu_{B}B\) and \(6\mu_{B}B\) would need larger g or J values.

Answer: D

In a weak field the D₁ line splits into 4 lines and the D₂ line into 6.

This is the anomalous Zeeman effect. The Lande g factors are 2 for \({}^{2}S_{1/2}\), 2/3 for \({}^{2}P_{1/2}\) and 4/3 for \({}^{2}P_{3/2}\). Because upper and lower g factors differ, every allowed transition (\(\Delta m_{J} = 0, \pm 1\)) has its own frequency.

For D₁ (\(m_{J} = \pm\frac{1}{2}\) to \(\pm\frac{1}{2}\)) there are 4 allowed transitions. For D₂, the upper level has four sublevels (\(m_{J} = \pm\frac{3}{2}, \pm\frac{1}{2}\)), and the allowed transitions to the two lower sublevels number 6, since \(\pm\frac{3}{2} \rightarrow \mp\frac{1}{2}\) would need \(\Delta m_{J} = 2\).

Three lines each is the normal Zeeman pattern, which applies only to singlet transitions, not to these doublets.