Zeeman, Paschen-Back and Stark Effects Questions and Answers

Q: An atom is subjected to a weak uniform magnetic field \(\vec{B}\). The number of lines in its Zeeman spectrum for transition from \(n = 2, l = 1\) to \(n = 1, l = 0\) is

The Zeeman spectrum has 10 lines.In a weak field, spin-orbit coupling splits the n = 2, l = 1 level into 2P3/2 and 2P1/2, 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 2P3/2, 2P1/2 and 2S1/2). With the selection rule Δmj = 0, ±1:2P3/2 → 2S1/2: 6 lines (the mj = ±3/2 → &mp;1/2 transitions are forbidden);2P1/2 → 2S1/2: 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.

Q: For a singlet state of electronic system, the Landé splitting factor will be equal to :

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 \( S = 0 \).

Q: The ground state of an alkali metal is a 2S1/2 state. The difference in energy levels arising in the presence of a weak magnetic field B, given in terms of Bohr magneton \(\mu_{B}\) is :

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.

An atom is subjected to a weak uniform magnetic field \(\vec{B}\). The number of lines in its Zeeman spectrum for transition from \(n = 2, l = 1\) to \(n = 1, l = 0\) is

A.8
B.10
C.12
D.5

Exam Year(s)

GATE Physics - 2024,

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.

Discuss in Forum

For a singlet state of electronic system, the Landé splitting factor will be equal to :

A.3/2
B.5/2
C.1/2
D.1

Exam Year(s)

MH SET - 2018, MH SET - 2020, SET Exam - 2018, SET Exam - 2020,

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 \( S = 0 \).

Discuss in Forum

The ground state of an alkali metal is a ²S_1/2 state. The difference in energy levels arising in the presence of a weak magnetic field B, given in terms of Bohr magneton \(\mu_{B}\) is :

A.\(\mu_{B}B\)
B.\(2\mu_{B}B\)
C.\(4\mu_{B}B\)
D.\(6\mu_{B}B\)

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

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.

Zeeman, Paschen-Back and Stark Effects Questions and Answers

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