Zeeman spectra and the Paschen–Back limit

For an electron, the magnetic moment has the opposite sign to angular momentum. Combining the magnetic energy with that sign gives a positive Zeeman Hamiltonian proportional to $L_z+g_sS_z$, with $g_s\simeq2$. Within an isolated LS-coupled J level, the projection theorem packages the expectation into $g_Jm_J$. The familiar Landé expression uses orbital factor one and spin factor two; anomalous magnetic moments and configuration mixing modify high-precision values.

This electronic weak-field approximation assumes the Zeeman matrix elements are small compared with separations to other fine-structure J levels. Degenerate magnetic substates are diagonalized along the field axis. For J equal to zero the first-order shift vanishes, and the displayed Landé quotient must not be evaluated at its zero denominator. Second-order shifts can remain. If hyperfine structure is resolved and the field is smaller still, F and a corresponding $g_F$ replace J and $g_J$.

In the hyperfine Paschen–Back regime, electronic Zeeman energy exceeds the hyperfine interval but remains small against fine structure. J remains useful while nuclear and electronic projections $m_I,m_J$ become approximate labels. The leading hyperfine remainder then contains a diagonal product of those projections. Nuclear magnetic energy is smaller because it involves the nuclear magneton, but precision work cannot simply discard it. The intermediate crossover requires diagonalizing coupled states with the same total projection.

The electronic Paschen–Back regime instead requires magnetic splitting larger than the spin-orbit scale within the relevant term. Orbital and spin projections $m_L,m_S$ are then useful, and the leading magnetic shift is $\mu_BB(m_L+2m_S)$. Residual spin-orbit contributions remain as perturbations. The field must still be small enough that neglected diamagnetic quadratic terms and mixing with remote configurations are controlled. This is a window between scales, not a limit of indefinitely increasing field.

For an optical transition, subtract the lower-level Zeeman shift from the upper-level shift and divide by h. A plot of one level alone therefore does not give the whole spectrum. In an electric-dipole transition, spherical polarization components select projection changes zero or plus or minus one relative to the field axis. Which components are detected also depends on viewing direction and polarization analysis. An allowed component can have zero intensity in a particular collection geometry.

In the electronic Paschen–Back approximation, electric dipole coupling preserves spin projection. The leading spin Zeeman contribution then cancels between upper and lower states, leaving orbital projection changes to organize a simpler pattern. Near the crossover, mixing changes both line positions and strengths, so a sudden switch of labels is only an asymptotic description. The vector lab illustrates continuous projection of a vector, whereas a real angular-momentum measurement has discrete allowed projections and a coupled atomic spectrum requires a Hamiltonian.