Strontium-87 energy level diagram

Energy levels and transitions of neutral ⁸⁷Sr (Sr I, nuclear spin I = 9/2): 40 levels and 70 lines below 2 µm, each with its vacuum wavelength, frequency and, for 68 of them, the reduced dipole matrix element. Level energies come from the NIST Atomic Spectra Database; transition rates are 57 NIST, 8 measured, 4 semi-empirical. Neutral strontium is also the starting point for loading Sr⁺ ion traps by photoionisation.

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

Key transitions of ⁸⁷Sr

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).

⁸⁷Sr hyperfine structure (I = 9/2)

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

Isotope shifts

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

Closed and nearly closed transitions of ⁸⁷Sr

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.

All ⁸⁷Sr 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, a semi-empirical calculation (Kurucz line list, used only where no measurement or NIST value exists; typically within a factor 2), or a model calculation.

⁸⁷Sr energy levels

Sr⁺ ionisation limit 45932.20 cm⁻¹ = 5.69487 eV (217.712 nm)

Sources

• Bushaw et al., Z. Phys. D 28, 275 (1993) [as quoted in Stellmer & Schreck, Phys. Rev. A 9…
• Courtillot et al., Eur. Phys. J. D 33, 161 (2005)
• Grundevik, Gustavsson, Lindgren, Olsson, Rosen, Z. Phys. A 311 (3) (1983)
• Heider & Brink, Phys. Rev. A 16, 1371 (1977) [as quoted in Beloy, Derevianko, Johnson, 'H…
• Heinz et al., Phys. Rev. Lett. 124, 203201 (2020)
• Kluesener et al., Phys. Rev. Lett. 132, 253201 (2024)
• Kluge & Sauter, Z. Phys. 270, 295 (1974) [as quoted in Heinz et al., Phys. Rev. Lett. 124…
• Kurucz, R. L., Can. J. Phys. 89, 417 (2011); semi-empirical line list gf3800z.pos of 2012…
• Margolis et al., 'The CIPM list Recommended values of standard frequencies: 2021 update',…
• Muniz, Young, Cline, Thompson, 'Cavity-QED determination of the natural linewidth of the …
• NIST ASD
• NIST ASD level energies
• Nicholson et al., Nat. Commun. 6, 6896 (2015)
• Okamoto, Aoki, Torii, Phys. Rev. A 113, L060803 (2026)
• Okamoto, Aoki, Torii, Phys. Rev. A 113, L060803 (2026) [as quoted in Pucher, Kristensen, …
• Onishchenko et al., 'The frequency of the ultranarrow 1S0-3P2 transition in 87Sr', arXiv:…
• Pucher, Kristensen, Kroeze, '88Sr Reference Data', arXiv:2507.10487v3 (2026)
• Puljic, Cipris, Aumiler, Ban, Santic, Phys. Rev. A 111, 032809 (2025)
• Safronova, Porsev, Safronova, Kozlov, Clark, Phys. Rev. A 87, 012509 (2013)
• Stellmer & Schreck, Phys. Rev. A 90, 022512 (2014)
• Yasuda & Katori, Phys. Rev. Lett. 92, 153004 (2004)
• Zhang, Tiwari, Ganesh, Singh, Phys. Rev. Research 6, 043106 (2024)
• zu Putlitz, Z. Phys. 175, 543 (1963), as summarized in Boyd PhD thesis (2007) [as quoted …

Conventions. (1) Isotope-shift keys 'a-b' mean nu(a) - nu(b), copied in the sign convention each source prints: '87-88' etc. from Pucher / Mickelson / Trautmann, '88-84' etc. from Stellmer & Schreck, Zhang, Patel. 87Sr shifts refer to the hyperfine centre of gravity. (2) Hyperfine B_MHz is the electric-quadrupole constant (called Q in the Boyd / Stellmer papers). (3) Fields beyond SCHEMA.md: other_values (competing measurements), reduced_matrix_element_ea0 (convention of the cited source; Pucher uses <J||er||J'> = sqrt(2J'+1)|d| with J' the upper level, Safronova uses the standard symmetric reduced element; the two agree numerically for these lines), line_strength_au, natural_linewidth_*, cascade_branching_to_3P, repump_efficiency, frequency_THz_other_isotopes, value_Hz_str / rel_unc for clock frequencies beyond double precision, unc_plus / unc_minus, type 'M2', and the top-level 'rydberg' block. (4) A_s for 679 / 688 / 707 nm are experimental 3S1 lifetime times THEORETICAL branching; the 5s4d 3D infrared A values and branchings are pure theory (Safronova 2013). (5) Conflicts are kept side by side in other_values: 707 nm 88Sr frequency of Pucher et al. (423.91634 THz) is about 2.8 GHz off the NIST Ritz value and looks like a sign error; 1P1 lifetime 5.216(14) vs 5.234(8) vs 5.263(4) ns; 1P1->1D2 rate 5.3(5)e3 (direct) vs 3.9(1.5)e3 (Hunter) vs 9.25(40)e3 s^-1 (theory); 461 nm frequency 650 503 815(6) vs 650 503 775(10) MHz; 87Sr 3P2-3D2 isotope shift, where Trautmann et al. p

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-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