Emission, absorption and term analysis

Hermiticity of H₀ makes every eigenvalue Eₙ real, so the bound spectrum is a set of points on one real axis. Energy conservation for atom-plus-field ties each absorbed or emitted photon to a pair of those points, hcσ = Eᵤ − Eₗ, so a spectrograph reading of σ = 1/λ_vac fixes how far apart the pair is and says nothing about where on the axis it sits. Which pairs appear strongly is decided by two things that are not energies: the dipole matrix element ⟨l|d|u⟩ must be non-zero — where it vanishes the line is suppressed to far weaker magnetic-dipole or quadrupole strength, not made impossible — and the initial level must be populated. So the raw data form a graph, not a list of energies: levels are vertices, observed lines are edges, and each edge carries a measured difference with an uncertainty. Term analysis is the inverse problem on that graph — recover vertex values from edge differences — and every property of the answer follows from that structure.

Apply H₀ + cI to an eigenket: (H₀ + cI)|n⟩ = (Eₙ + c)|n⟩. The kets are the same kets and every eigenvalue moves by the same c, so every difference Eᵤ − Eₗ, and with it every predicted wavenumber, is unchanged. The map from level scheme to line list thus has a kernel with one dimension for each connected component of the level graph, and the inverse is fixed only up to one additive constant per component. Spectroscopy handles this by convention. Writing Tₙ = −Eₙ/hc puts the zero at the ionisation limit, so bound terms are positive and the ground term is the ionisation energy: 41449.5 cm⁻¹ for sodium. The other common choice puts the zero at the ground level and quotes every term as an excitation. Nothing in the line list chooses between them. The zero is imported — from the limit of a Rydberg series, or from a measured photoionisation threshold — and that import is extra physics, not extra lines.

Because each line is a difference, the signed wavenumbers around any closed loop of levels must sum to zero — the Ritz combination principle, formally the same statement as Kirchhoff's loop rule for potentials. Hydrogen: 82259.2 + 15233.0 = 97492.2 cm⁻¹ closes Lyman-α, Balmer-α and Lyman-β. The number of independent closure conditions is fixed by the graph: with V levels, L observed lines and C connected components it is L − V + C. Three mutually connected levels give 3 − 3 + 1 = 1 test; a scheme of 20 levels joined by 60 lines gives 41. Run those tests before fitting anything. A closure that fails by many times the combined uncertainty means a blended line, a misassignment, a second species in the discharge, or air wavelengths used where vacuum ones were needed.

Assemble the model as σ = A T, with A the L × V incidence matrix carrying +1 in the lower-level column and −1 in the upper. Weight by W = diag(1/uₖ²) and minimise (σ − AT)ᵀW(σ − AT); the normal equations are (AᵀWA)T̂ = AᵀWσ. That matrix is singular, and the previous section says why: every row sums to zero, so A⋅1 = 0 and the all-ones vector spans the null space, leaving rank V − C. Fix the gauge by pinning one term per component, or call numpy.linalg.lstsq, which returns the minimum-norm solution and so silently imposes ΣT = 0. The covariance (AᵀWA)⁺ is finite only in the difference directions, which is the algebra telling you to quote Tᵢ − Tⱼ and not Tᵢ. Judge the fit by χ² on L − (V − C) degrees of freedom, and inspect the residual on each line.

Which edges you actually observe depends on which vertices are occupied. In a discharge or arc the upper levels are populated, nᵤ/nₗ = (gᵤ/gₗ)exp(−hcσ/k_BT), and with k_BT = 0.695 T in cm⁻¹ a 5000 K source puts 3 × exp(−16967.6/3475) ≈ 2.3% of sodium's ground population into 3p — enough to light the D lines and, more importantly, to light lines between excited levels. Drop to a 300 K vapour cell and that ratio falls to about 10⁻³⁵: nothing is up there, and the only transitions available are absorptions out of the ground level. That has a structural consequence. A pure absorption spectrum is a star graph centred on the ground level, with L = V − 1, so L − V + C = 0: it carries no closure test at all. You can assign it, but you cannot check it. The redundancy that validates a term scheme comes from emission lines joining excited levels to each other.

A term scheme is the eigenvalue list of H₀ and nothing more. It carries no eigenvectors, so degeneracies, level compositions and g factors must come from elsewhere — intensities, Zeeman patterns, selection rules. Unresolved structure collapses silently: read sodium's 3p as one term and you get its centre of gravity, 16967.6 cm⁻¹, with the 17.2 cm⁻¹ fine-structure splitting hidden inside, and every difference built on it inherits that error. Two systematic traps deserve naming. Wavelengths tabulated in air need converting: nₐᵢᵣ ≈ 1.000277 near 589 nm, so σ_vac = σₐᵢᵣ/n shifts a 17000 cm⁻¹ line by 4.7 cm⁻¹, thousands of times a modern uncertainty and a guaranteed closure failure. And Doppler and pressure shifts move the line centre, so the difference you fit belongs to the free atom only after those are corrected.