Li Ionization Energy from Rydberg Series

What this research found

The first ionization energy of neutral lithium was recovered from published spectroscopy alone, by extrapolating the measured 2s to np Rydberg series out to the limit its levels converge on as the principal quantum number grows without bound. Thirteen tabulated levels spanning n = 3 to 15 were fitted with a first-order Rydberg-Ritz law, giving a series limit of 43486.81 ± 0.10 cm⁻¹, equivalently 5.391677 eV. That reproduces the accepted NIST value to within 0.31 cm⁻¹, or 7.1 parts per million, and the analysis shows a constant quantum defect cannot deliver the same accuracy.

• The extrapolated limit is 43486.81 ± 0.10 cm⁻¹, equivalently 5.391677 ± 0.000013 eV, against an accepted NIST ionization energy of 43487.12 cm⁻¹. The 0.31 cm⁻¹ gap is 7.1 parts per million, a 3.0 sigma difference on the combined error budget.
• Treating the quantum defect as constant is statistically inadequate. That two-parameter model returns a reduced chi-square of 187 and a limit of 43485.72 ± 2.24 cm⁻¹, while adding the leading energy-dependent Ritz term drops the reduced chi-square to 9.2 and shrinks the statistical uncertainty more than fiftyfold, to 0.042 cm⁻¹.
• The np quantum defect is small, averaging 0.0468 ± 0.0013 across n = 3 to 10, with a fitted asymptotic value of 0.04716 ± 0.00006. That is consistent with a p electron held largely outside the compact closed 1s shell by its centrifugal barrier, and the n = 2 series origin sits below the plateau at 0.0407, the signature of deeper core penetration.
• The extrapolation does not depend on where the fit window is placed. Refitting over every combination of four lower bounds and eight upper bounds moves the first-order limit by less than about 0.1 cm⁻¹, while the constant-defect limit drifts by more than 8 cm⁻¹ and only creeps toward the accepted value once high-n levels dominate.
• Levels near the threshold add noise rather than information. Because the defect inferred from a single level depends on its shrinking gap to the limit, the fixed 0.1 to 1.0 cm⁻¹ tabulation resolution is amplified into erratic swings above n of about 20, exceeding 0.5 by n of about 37, which sets the practical upper edge of any useful window.

How it was done

Term energies for the neutral-lithium principal series were retrieved from the NIST Atomic Spectra Database, version 5.12, and levels carrying the 1s-squared np configuration and 2P-odd term were isolated, spanning n = 2 to 42. The two fine-structure components of each level were combined into a degeneracy-weighted centre of gravity, and the Rydberg constant was held fixed at its finite-mass value for lithium-7, 109728.73601 cm⁻¹, derived from the CODATA 2018 infinite-mass constant. Two nested models were then fitted by weighted non-linear least squares to the thirteen levels in the n = 3 to 15 window: a two-parameter constant-defect law and a three-parameter first-order Rydberg-Ritz law that lets the defect vary weakly with energy. Window choice was treated as a modelling decision and probed by refitting across a grid of lower and upper bounds, with the resulting 0.094 cm⁻¹ spread added in quadrature to the covariance-based statistical error to form the quoted uncertainty. Results were written up as a 13-page report with residual and sensitivity diagnostics.

Data sources

• NIST Atomic Spectra Database, version 5.12 (2024) - Li I energy levels for n = 2 to 42
• Radziemski, Engleman and Brault, Physical Review A 52:4462 (1995) - Fourier-transform emission spectroscopy underpinning the NIST Li I term values
• CODATA 2018 recommended values of the fundamental physical constants (Tiesinga et al., Reviews of Modern Physics 93:025010, 2021)
• Seaton, The Quantum Defect Method, Monthly Notices of the Royal Astronomical Society 118:504 (1958), and Quantum defect theory, Reports on Progress in Physics 46:167 (1983)