LIBS Analysis Calculator
Stark and Doppler widths of a LIBS plasma line, electron density from a Stark width, the Saha ratio and the McWhirter LTE limit.
Doppler FWHM
0.005657 nm
v_p = 2037 m/s; with instrument 0.05032 nm
Stark FWHM
0.2000 nm
Δλ_S = 2w · nₑ / 10¹⁶ cm⁻³
Observed FWHM (Voigt, exact)
0.2128 nm
Gaussian 0.05032 nm ⊗ Lorentzian 0.2000 nm
Saha ratio n⁺/n⁰
80.19
ionised fraction 98.8 % (single ionisation)
McWhirter LTE limit
4.320e+15 cm⁻³
nₑ is 23.1× the limit: necessary condition met
Line profile
- Doppler + instrument (Gaussian)
- Stark (Lorentzian)
- Observed line (Voigt)
Δλ_D = λ₀ √(8 k_B T ln 2 / (m c²)), Δ_G = √(Δλ_D² + Δλ_inst²)
Δλ_S = 2w · nₑ / 10¹⁶ cm⁻³
n⁺nₑ/n⁰ = 2 (U⁺/U⁰) (2π mₑ k_B T / h²)^(3/2) exp(−χ / k_B T)
nₑ ≥ 1.6×10¹² √T ΔE³ cm⁻³ (T in K, ΔE in eV)
Electron-impact Stark broadening of an isolated line is Lorentzian with a width linear in nₑ; w is the tabulated electron-impact half-width at 10¹⁶ cm⁻³ for the chosen line and temperature (typically 0.001–0.1 nm; Griem's 1974 tables give w directly. Tables of measured full widths w_m at a density nₑ,m, such as the Konjević critical reviews, give w = (w_m/2)·10¹⁶ cm⁻³/nₑ,m), and ion broadening, a few percent for ionic lines, is neglected. Hydrogen Balmer lines follow the linear Stark effect instead (nₑ ∝ Δλ^(3/2)). The Saha ratio assumes LTE, single ionisation and no lowering of the ionisation energy; take U⁺/U⁰ from partition-function tables (the ground-term weights alone give ½ for Na, 2 for Ca and ⅙ for Al). The McWhirter criterion is necessary but not sufficient for LTE in a transient LIBS plasma.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Isolated-line electron-impact Stark width linear in nₑ (w from tables, ion broadening neglected), Doppler and instrument Gaussians in quadrature, exact Voigt; Saha in LTE for single ionisation without ionisation-potential lowering. Not for hydrogen Balmer lines.
- Konjević N., Lesage A., Fuhr J. R., Wiese W. L. (2002). Experimental Stark widths and shifts for spectral lines of neutral and ionized atoms. J. Phys. Chem. Ref. Data 31, 819–927. doi:10.1063/1.1486456
- Cristoforetti G., De Giacomo A., Dell'Aglio M., et al. (2010). Local thermodynamic equilibrium in laser-induced breakdown spectroscopy: beyond the McWhirter criterion. Spectrochim. Acta B 65, 86–95. doi:10.1016/j.sab.2009.11.005
- Griem H. R. (1974). Spectral Line Broadening by Plasmas. Academic Press, New York (tabulated electron-impact Stark widths).
- Griem H. R. (1997). Principles of Plasma Spectroscopy. Cambridge University Press (Saha equation, Stark broadening).
- NIST Atomic Spectra Database, ionization energies. physics.nist.gov
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