Atomic field shifts and spectral differences
| English | Español |
|---|---|
| Landé factor | Landé factor |
| Stark splitting | Stark splitting |
A decision before an answer
- The same magnetic field needs different quantum labels when its energy scale crosses the spin-orbit splitting.
- Your goal: Calculate orbital and spin magnetic shifts in the appropriate coupling regime.
Choose magnetic quantum labels
- An electron has a negative magnetic moment: μ_L=−μ_B L/ℏ and μ_S≈−2μ_B S/ℏ. With B along positive z, the interaction −μ·B therefore gives H_Z=μ_B B(L_z+2S_z)/ℏ. μ_B≈5.788×10⁻⁵ eV/T. In an uncoupled basis with orbital projection m_l and spin projection m_s=±1/2, the shift is μ_B B(m_l+2m_s).
- Use this uncoupled rule when the magnetic interaction is large relative to the relevant spin-orbit splitting but still small relative to orbital energy gaps, so quadratic diamagnetic and orbital restructuring effects can be neglected. Large and small are comparisons of energies, not universal Tesla thresholds. In the opposite weak-field regime, m_l and m_s are not independently conserved labels for an LS-coupled eigenstate.
In the uncoupled electron approximation, m_l=1,m_s=−1/2 and B along +z give a linear magnetic shift:
m_l+2m_s=1−1=0. This statement uses the uncoupled regime.
Resolve a weak-field multiplet
- When LS coupling dominates a weak magnetic perturbation, J=L+S and m_J label the multiplet. The first-order shift is μ_B B g_J m_J, with g_J=1+[J(J+1)+S(S+1)−L(L+1)]/[2J(J+1)] in the approximation g_s=2. L,S,J here are dimensionless quantum numbers.
- For L=1,S=1/2, J=3/2 gives g_J=4/3 and shifts −2,−2/3,+2/3,+2 times μ_B B. J=1/2 gives g_J=2/3 and shifts ±μ_B B/3. Do not substitute J=0 into the singular formula; a J=0 state has no first-order vector projection shift. Intermediate fields require a coupled Hamiltonian rather than mixing weak-field and uncoupled formulas.
An upper level shifts by +2u and a lower level by −u. An allowed emitted photon shifts by:
Photon energy is E_upper−E_lower, so the change is 2u−(−u)=3u.
Subtract the two level shifts
- A photon’s energy is the upper atomic energy minus the lower energy. Its field-induced change is therefore ΔE_γ=ΔE_upper−ΔE_lower. Count allowed transitions using the stated selection rules as well as the two level patterns. Several transitions can share one frequency; the number of split states is not automatically the number of spectral lines.
- In a spin-neglected orbital model, an upper L=1 triplet has shifts m_l μ_B B with m_l=−1,0,+1, and a lower L=0 state has zero shift. Electric-dipole Δm_l=0,±1 permits three photon shifts −μ_B B,0,+μ_B B. This normal-triplet model does not describe every real atom or fine-structure multiplet. Polarisation and observation direction can change which components are detected.
With μ_B=5.788×10⁻⁵ eV/T, B=2 T and weak-field LS-coupled labels L=1,S=1/2,J=3/2,m_J=1/2, the shift is (4/3)(1/2)μ_B B=7.7173×10⁻⁵ eV. In a different, uncoupled regime, m_l=1,m_s=−1/2 instead gives zero linear shift. A normal orbital L=1→0 triplet at 2 T has photon shifts −1.1576×10⁻⁴,0,+1.1576×10⁻⁴ eV. Separately, a degenerate dipole pair with d=2 e·nm and F=10⁵ V/m has shifts ±0.0002 eV, a 0.0004 eV splitting.
For L=1,S=1/2,J=1/2, the Landé factor in the g_s=2 approximation is ____ (use a fraction).
1+[3/4+3/4−2]/(2·3/4)=2/3.
Use electric-field symmetry and degeneracy
- In a uniform electric field F along z, an electron’s perturbation is +eFz for the stated electrostatic potential convention. A nondegenerate parity eigenstate has ⟨z⟩=0, so its first-order diagonal Stark shift vanishes, although a quadratic shift can remain. Exactly degenerate opposite-parity states instead allow an off-diagonal dipole matrix element.
- In an original two-state model, suppose the dipole matrix is [[0,d],[d,0]], with real d in charge×length units. The perturbation F times this matrix has eigenvalues ±dF and opposite superpositions of the two states. Changing a basis phase changes the off-diagonal sign but not the splitting 2|dF|. Actual near-degenerate atoms require comparison with fine structure and other small splittings before an exactly degenerate approximation is adopted; this supplied matrix is not a universal hydrogen coefficient.
Declare the coupling regime, keep the negative electron moment sign, and subtract lower-level shifts. Zero diagonal dipole expectation does not exclude degenerate mixing or quadratic shifts.
Which answer fits this case?
Calculate orbital and spin magnetic shifts in the appropriate coupling regime
A nondegenerate parity eigenstate’s zero first-order Stark shift guarantees no electric-field response at any order.
Off-diagonal dipole couplings can produce a quadratic shift; degeneracy changes the leading-order treatment.
Keep the distinctions
- Landé factor 朗德因子 — Projection factor relating a weak-field LS-coupled magnetic shift to g_J m_J.
- Stark splitting 斯塔克分裂 — Electric-field-induced separation of atomic energy levels, with symmetry and degeneracy controlling leading order.
- Calculate orbital and spin magnetic shifts in the appropriate coupling regime.
- Find photon-energy shifts from allowed upper-minus-lower level changes.
- Contrast parity cancellation with degenerate electric-field splitting.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.