Francium energy level diagram

Energy levels and transitions of neutral francium (Fr I): the 77 strongest classified lines below 2 µm from the NIST Atomic Spectra Database and the 52 levels they connect, each line with its vacuum wavelength, frequency and, for 75 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.

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Key transitions of francium

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).

²¹²Fr hyperfine structure (I = 5)

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

Isotope shifts

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

Closed and nearly closed transitions of francium

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 Fr 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.

Fr energy levels

Fr⁺ ionisation limit 32848.87 cm⁻¹ = 4.07274 eV (304.424 nm)

Sources

• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Allegrini, Arimondo, Orozco, J. Phys. Chem. Ref. Data 51, 043102 (2022) [arXiv:2209.11423…
• Aubin, Gomez, Orozco, Sprouse (Opt. Lett. 28, 2055 (2003) according to the supplement Tab…
• Aubin, Gomez, Orozco, Sprouse, Phys. Rev. A 70, 042504 (2004), as given in Sansonetti, J.…
• Aubin, Gomez, Orozco, Sprouse, Phys. Rev. A 70, 042504 (2004), as given in the supplement…
• Gomez, Orozco, Perez Galvan, Sprouse, Phys. Rev. A 71, 062504 (2005) [arXiv:physics/04120…
• Grossman, Fliller, Orozco, Pearson, Sprouse, Phys. Rev. A 62, 062502 (2000), as given in …
• Grossman, Fliller, Orozco, Pearson, Sprouse, Phys. Rev. A 62, 062502 (2000), as given in …
• Kalita et al., Phys. Rev. A 97, 042507 (2018) [arXiv:1710.07604], Table II (measured in i…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• Lassegues et al. (CRIS collaboration, ISOLDE), arXiv:2604.12467v2 (2026), "Energies and l…
• NIST ASD
• NIST ASD level energies
• Sansonetti, J. Phys. Chem. Ref. Data 36, 497 (2007) Table 6 (Bauche et al. 1986)
• Sansonetti, J. Phys. Chem. Ref. Data 36, 497 (2007) Table 6 (Duong et al. 1987)
• Simsarian, Orozco, Sprouse, Zhao, Phys. Rev. A 57, 2448 (1998), as given in Sansonetti, J…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…
• UDel Portal for High-Precision Atomic Data and Computation (Safronova group), Fr I tables…

NIST level energies refer to 212Fr (Sansonetti 2007 Table 5); several of them (6d, 8s, 7d, 9s, 9p-12p ...) carry the Biemont et al. 1998 reference and are partly Ritz-formula predictions - see conflicts. Every lifetime, hyperfine constant and frequency carries the isotope it was measured in: Stony Brook MOT work is 210Fr (I=6), ISOLDE work mostly 212Fr (I=5) and 221Fr (I=5/2), TRIUMF 206-213 and 221. measured_energy_cm (extra field on levels) holds level energies above 7S measured in a given isotope (hyperfine centres of gravity as tabulated). rme_J_ea0 is |<J||D||J'>| in e a0 in the symmetric convention of the Safronova papers / UDel portal (A = 2.02613e18 |d|^2 / ((2J'+1) lambda^3[Angstrom])). The UDel portal does carry neutral Fr (element key "Fr1": 512 distinct matrix elements, 383 rates, lifetimes for 7p-14p, 8s-14s, 6d-13d, 7 hyperfine constants; data service https://atom.ece.udel.edu/graphql behind https://www1.udel.edu/atom/); only the 7s-7p matrix elements and 7p lifetimes there are experimental (Simsarian et al. 1998), everything else is all-order theory and is marked "theory". Portal A_s / branching were computed by the portal with NIST energies, so rates into or out of 8s, 9p, 10p inherit the NIST energy errors listed under conflicts (small effect, below 0.2 % in omega^3). Isotope shifts are nu(first) - nu(second). The 7S-8S and 7S-9S entries are two-photon lines added for their measured frequencies; type "M1" marks them as E1-forbidden. Extra fields beyond the sc

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