Hydrogen energy level diagram

Energy levels and transitions of neutral hydrogen (H I): the 74 strongest classified lines below 2 µm from the NIST Atomic Spectra Database and the 45 levels they connect, each line with its vacuum wavelength, frequency and, for 65 of them, the reduced dipole matrix element derived from the NIST transition rate. Measured lifetimes, transition rates, hyperfine constants and isotope shifts from the literature are included, each with its citation.

Hover a line or a level to isolate it, click to keep it selected. Scroll to zoom, drag to move.

Download: diagram (PDF) diagram (SVG) transitions (CSV) levels (CSV) source code

Key transitions of hydrogen

λ, ν from NIST level energies unless a measured frequency for this isotope is available. A: measured value or NIST. Γ/2π: natural linewidth of the upper level, 1/(2πτ).

¹H hyperfine structure (I = 1/2)

Measured A, B constants; sources in the data file.

Isotope shifts

Shift = ν(first isotope) − ν(second isotope).

Closed and nearly closed transitions of hydrogen

A transition is closed when its upper level can decay only to the level it was excited from, so the atom keeps scattering photons. “closed”: among the levels NIST lists, no other level below the upper level has the opposite parity and |ΔJ| ≤ 1, so no other electric-dipole decay exists. Otherwise the leak is the sum of the branching ratios of the other decay lines of the upper level (or 1 − branching ratio where that of the line itself has been measured), and the number of photons is 1 / leak. “≥” and “≤”: at least one possible decay path has no listed rate, so the leak is a lower limit. These numbers are derived from the branching ratios, lifetimes and rates of this page, not measured as such; the tag names the weakest of the inputs, * marks theory and ≈ a model or semi-empirical calculation. Listed: lines that start on the ground level or on a metastable level and leak at most 10 %, unless more than three possible decay paths have no listed rate. Fine-structure levels only: hyperfine and Zeeman dark states and forbidden decays without a listed rate are not considered. Select a line in the diagram for the full list of its leak channels.

Listed H transitions below 2 µm

Reduced dipole matrix elements follow the convention A = ω³|d|²/(3πε₀ħc³(2J′+1)), with J′ the upper level. Tags show where a number comes from: measured, NIST compilation, high-accuracy theory, or a model calculation.

H energy levels

H⁺ ionisation limit 109678.77 cm⁻¹ = 13.59843 eV (91.175 nm)

Sources

• A. E. Kramida, At. Data Nucl. Data Tables 96, 586 (2010), Table 1 (hfs splittings in hydr…
• Brandt, Cooper, Rasor, Burkley, Matveev, Yost, Phys. Rev. Lett. 128, 023001 (2022), arXiv…
• Grinin, Matveev, Yost, Maisenbacher, Wirthl, Pohl, Haensch, Udem, Science 370, 1061 (2020…
• Hellwig et al., Essen et al. and other hydrogen-maser measurements, averaged in S. G. Kar…
• Kolachevsky, Matveev, Alnis, Parthey, Karshenboim, Haensch, Phys. Rev. Lett. 102, 213002 …
• NIST ASD
• NIST ASD level energies
• Parthey, Matveev, Alnis, et al., Phys. Rev. Lett. 107, 203001 (2011), arXiv:1107.3101
• Sommerfeldt et al., Phys. Rev. A 102, 042811 (2020), arXiv:2008.03950, Table II
• Sommerfeldt, Mueller, Volotka, Fritzsche, Surzhykov, Phys. Rev. A 102, 042811 (2020), arX…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• W. L. Wiese and J. R. Fuhr, J. Phys. Chem. Ref. Data 38, 565 (2009), Table 6 (exact hydro…
• de Beauvoir, Nez, Julien, Cagnac, Biraben, Touahri, Hilico, Acef, Clairon, Zondy, Phys. R…

All optical frequencies are hyperfine centroids (centre of gravity), as stated in the sources; 'value_text' repeats the number exactly as printed because 16-digit frequencies are at the limit of double precision. frequency_THz refers to 1H, frequency_THz_2 to deuterium; isotope shifts are nu(2H) - nu(1H). Non-schema additions: transition type 'two-photon' (1S-2S, 1S-3S, 2S-8D: E1E1 two-photon lines with no single-photon E1 amplitude), 'value_text', 'use_source'/'use_url', 'interval_MHz' in hyperfine blocks, top-level 'ionization_energy', 'conflicts' and 'additional_frequency_data' (measured frequencies that do not map onto a pair of levels drawn by the site: combinations, the 2S-4P centroid, levels 4S/6S/8S/12D, the 21 cm line). Hyperfine A for 1H (I = 1/2): the F = J+1/2 to F = J-1/2 splitting equals A (J + 1/2). For 2H (I = 1) and J = 1/2 the splitting is 3A/2. Lifetimes of n = 2-6 levels are exact hydrogenic theory (not measurements), obtained as 1/sum(A) from Wiese & Fuhr 2009; the 2S lifetime is the calculated two-photon value. The Lamb-shift and 2S-2P3/2 entries are rf intervals given in THz for schema uniformity.

Atoms with measured data

• Lithium-632 levels, 85 lines
• Lithium-732 levels, 85 lines
• Beryllium-917 levels, 26 lines
• Sodium-2336 levels, 106 lines
• Magnesium-2418 levels, 61 lines
• Magnesium-2518 levels, 61 lines
• Potassium-3934 levels, 84 lines
• Potassium-4034 levels, 84 lines
• Potassium-4134 levels, 84 lines
• Calcium-4035 levels, 61 lines
• Calcium-4335 levels, 61 lines
• Chromium-5256 levels, 87 lines
• Chromium-5356 levels, 87 lines
• Rubidium-8536 levels, 94 lines
• Rubidium-8736 levels, 94 lines
• Strontium-8440 levels, 70 lines
• Strontium-8640 levels, 70 lines
• Strontium-8740 levels, 70 lines
• Strontium-8840 levels, 70 lines
• Cadmium-11115 levels, 21 lines
• Cadmium-11315 levels, 21 lines
• Cadmium-11415 levels, 21 lines
• Caesium-13331 levels, 79 lines
• Barium-13744 levels, 89 lines
• Barium-13844 levels, 89 lines
• Dysprosium-16139 levels, 55 lines
• Dysprosium-16239 levels, 55 lines
• Dysprosium-16339 levels, 55 lines
• Dysprosium-16439 levels, 55 lines
• Erbium-16673 levels, 90 lines
• Erbium-16773 levels, 90 lines
• Erbium-16873 levels, 90 lines
• Thulium-16972 levels, 73 lines
• Ytterbium-17118 levels, 26 lines
• Ytterbium-17318 levels, 26 lines
• Ytterbium-17418 levels, 26 lines
• Mercury-19927 levels, 47 lines
• Mercury-20127 levels, 47 lines
• Mercury-20227 levels, 47 lines

All elements: periodic table