Reading Real Spectra with the NIST Database

Everything computed in this course has a measured counterpart. The hydrogen levels, the fine structure, the alkali quantum defects, the term symbols, the selection rules, and the Einstein coefficients are all tabulated, for essentially every atom and ion, in the NIST Atomic Spectra Database. Reading that data is a skill: the tables are compact, their conventions are specific, and a measured spectrum becomes physics only once its lines are assigned to tabulated transitions. This lesson reads the database as the quantitative record against which the theory is checked.

Levels: energies and term symbols

The Levels table lists the stationary states of an atom or ion. Each row carries a configuration, a term symbol, a total angular momentum $J$, and an energy.

• Energy in wavenumbers. Level energies are tabulated in ${\mathrm{cm}}^{-1}$, the spectroscopic unit $\tilde{\nu }=E/hc$. The ground state is set to $0{\mathrm{cm}}^{-1}$ and every excited level is quoted as the wavenumber of the photon that reaches it from the ground state. The conversion is $1\mathrm{eV}=8065.544{\mathrm{cm}}^{-1}$, and a transition's wavelength is the reciprocal difference of two levels.
• Term symbol. Each level is labelled ${}^{2S+1}{L}_J$ in Russell–Saunders notation: the left superscript is the spin multiplicity $2S+1$, the letter encodes the total orbital angular momentum $L$ ($S,P,D,F$ for $L=0,1,2,3$), and the right subscript is $J$. A configuration such as $3p$ in sodium produces the two levels ${}^2{P}_{1/2}$ and ${}^2{P}_{3/2}$, split by the spin–orbit interaction.
• Ionization limit. The table ends at the ionization energy, the wavenumber at which the series of levels converges; states above it are in the continuum.

The additivity of wavenumbers is the Ritz combination principle: if two lines share a common level, the sum or difference of their wavenumbers is itself the wavenumber of a third line in the spectrum. This is the tool that builds a level scheme from a raw list of lines before any of the levels are known individually.

Lines: wavelength, transition rate, and line strength

The Lines table lists transitions. Each row ties a pair of levels to an observed wavelength and to the atomic quantities that set the line's intensity.

• Wavelength and wavenumber. The transition is quoted by air or vacuum wavelength $\lambda$ and by wavenumber $\tilde{\nu }={\tilde{\nu }}_{\text{upper}}-{\tilde{\nu }}_{\text{lower}}$. Above $2000\mathring{\mathrm{A}}$ the database quotes air wavelengths by convention, so a measured vacuum wavelength must be converted by the refractive index of air before comparison.
• Transition probability $A_{ki}$. The spontaneous-emission Einstein coefficient from upper level $k$ to lower level $i$, in ${\mathrm{s}}^{-1}$. The sum over all downward channels gives the level's decay rate; its reciprocal is the lifetime${\tau }_k=1/{\sum }_i{A}_{ki}$.
• Oscillator strength $f_{ik}$ and line strength $S$. Dimensionless characterizations of the transition strength, related to $A_{ki}$ by

with $g_i,g_k$ the lower and upper statistical weights.¹ The line strength $S$ is the square of the reduced dipole matrix element and is symmetric between the two levels; $A$, $f$, and $S$ carry the same physical content weighted by different powers of the transition frequency.

The intensity a line actually shows in a source combines the atomic $A$-coefficient with the population of the emitting level. In a source at temperature $T$ the emitted line intensity is

with $Z$ the partition function. A weak line can come either from a small $A$ or from a sparsely populated upper level; separating the two requires knowing the source conditions, which is why quantitative spectroscopy reports $A$-values, not raw intensities.

The air conversion is not a rounding detail. The refractive index of air varies with wavelength, temperature, and pressure, and the difference between a vacuum and an air wavelength at $589\mathrm{nm}$ is $0.16\mathrm{nm}$ — a quarter of the $0.6\mathrm{nm}$ doublet splitting and thousands of times the wavelength precision of a laser measurement. Matching a measured line to a catalog entry therefore requires knowing which convention each is quoted in, and converting one to the other with the tabulated index before comparing.

The Grotrian diagram

The tables are read most naturally as a picture. A Grotrian diagram places the levels on a vertical energy axis, groups them into columns by term (by $L$ and by multiplicity), and draws each tabulated transition as a line connecting its two levels. Selection rules are visible directly: allowed electric-dipole transitions connect columns differing by $\Delta L=\pm 1$ with $\Delta J=0,\pm 1$, and no allowed line runs within a column.

Reconstructing the diagram is the first step in assigning an unknown spectrum. Given a levels table, plot the levels, draw only the transitions the selection rules permit, and the pattern of allowed lines predicts which wavelengths should appear. The sodium example makes the mechanism concrete: the $3p\to 3s$ transition is the famous $D$ doublet, two lines at $589.0$ and $589.6\mathrm{nm}$ separated by the $3p$ fine-structure splitting of $17{\mathrm{cm}}^{-1}$, both visible as adjacent rows in the levels table.

Assigning a measured spectrum

A recorded spectrum is a list of wavelengths and relative intensities with no labels. Turning it into physics means matching each line to a tabulated transition.

The procedure rests on two facts. The Ritz combination principle links lines that share a level, so the raw wavelength list already constrains the level scheme before any element is guessed. And the tabulated $A$-values predict relative intensities, so a candidate assignment is checked not only on line positions but on whether the strong lines are the ones the catalog says should be strong. A correct assignment reproduces both the wavelengths and their intensity ordering.

Computed versus tabulated positions

The database is also the scoreboard for theory. A line position computed from a model — a Bohr energy, a fine-structure formula, a quantum-defect fit — is compared with the tabulated wavenumber, and the residual measures where the model stops. The residuals form a ladder that mirrors the structure of the course.

Each row is a lesson in this course made quantitative. The Bohr model's residual is the fine structure; the fine-structure formula's residual is the Lamb shift and the hyperfine structure; the quantum-defect fit absorbs core penetration into an empirical ${\delta }_{\ell }$ whose $\ell$-dependence is itself the physics.

Hydrogen as the benchmark

The residual ladder does not stop at three rungs. In hydrogen the tabulated transition frequencies are known to a precision that outruns every term in the fine-structure formula, and the comparison of theory with data becomes a determination of fundamental constants rather than a check of a model. The two-photon$1S\to 2S$ frequency, measured against a frequency comb locked to a cesium clock, is one of the most accurately known frequencies in nature. Because the gross-structure prediction is $\tilde{\nu }=R_{\infty }(1-1/4)$ up to fine-structure, QED, and finite-nuclear-size corrections, the measured frequency combined with the calculated corrections yields the

Rydberg constant to twelve significant figures.² The same fit is sensitive to the proton charge radius through the finite-size shift of the $S$-states, and the tension between the value extracted from hydrogen spectroscopy and the value from muonic hydrogen — the proton-radius puzzle — was a decade-long discrepancy visible only because the atomic data are this precise. Reading the database at this level is no longer bookkeeping: the last digits of a tabulated hydrogen line are where atomic physics tests the Standard Model.

Using the database as a working tool

The database turns the course's calculations into checkable predictions. A level scheme comes from the Levels form, read as term symbols and wavenumbers; a line list comes from the Lines form, read as wavelengths tied to $A$-values and line strengths; a Grotrian diagram assembles the two into a picture whose allowed transitions are the selection rules made visible; and an unknown spectrum is assigned by the Ritz principle and scored against tabulated intensities. The residual between a computed and a tabulated position is the running measure of how much atomic physics a given model contains — the same residual that, chased to its smallest values in hydrogen, turns atomic spectra into the most stringent test of QED and, through the optical clock, into the definition of the second.