20.   Spectral Line Shapes, Widths, and Shifts

 

• Doppler Broadening

  Doppler broadening is due to the thermal motion of the emitting atoms or ions. For a Maxwellian velocity distribution, the line shape is Gaussian; the full width at half maximum intensity (FWHM) is, in Å,
   

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  T is the temperature of the emitters in K, and M the atomic weight in atomic mass units (amu).
   

• Pressure Broadening

  Pressure broadening is due to collisions of the emitters with neighboring particles [see also Atomic, Molecular, & Optical Physics Handbook, Chaps. 19 and 57, ed. by G.W.F. Drake (AIP, Woodbury, NY, 1996)]. Shapes are often approximately Lorentzian, i.e., I(λ) ∝ {1 + [(λ - λ₀)/Δλ_1/2]²}^-1. In the following formulas, all FWHM's and wavelengths are expressed in Å, particle densities N in cm^-3, temperatures T in K, and energies E or I in cm^-1.

  Resonance broadening (self-broadening) occurs only between identical species and is confined to lines with the upper or lower level having an electric dipole transition (resonance line) to the ground state. The FWHM may be estimated as
   

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  where λ is the wavelength of the observed line.  f_r and λ_r are the oscillator strength and wavelength of the resonance line; g_k and g_i are the statistical weights of its upper and lower levels. N_i is the ground state number density.

  For the 1s2p ¹P - 1s3d ¹D₂ transition in He I [λ = 6678.15 Å; λ_r (1s^{2  1}S₀ - 1s2p ¹P) =  584.334 Å; g_i = 1; g_k = 3; f_r = 0.2762] at N_i = 1 × 10¹⁸ cm^-3: Δλ^R_1/2 = 0.036  Å.

  Van der Waals broadening arises from the dipole interaction of an excited atom with the induced dipole of a ground state atom. (In the case of foreign gas broadening, both the perturber and the radiator may be in their respective ground states.) An approximate formula for the FWHM, strictly applicable to hydrogen and similar atomic structures only, is
   

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  where µ is the atom-perturber reduced mass in units of u, N the perturber density, and C₆ the interaction constant. C₆ may be roughly estimated as follows: C₆ = C_k - C_i, with C_i (k) = (9.8 × 10¹⁰) (α_d R²_i (k) α_d in cm³, R² in a₀²). Mean atomic polarizability α_d ≈ (6.7 × 10^-25)  (3I_H/4E*;)² cm³, where I_H is the ionization energy of hydrogen and E*; the energy of the first excited level of the perturber atom. R²_i (k) ≈ 2.5 [I_H/(I-E_i (k))]², where I is the ionization energy of the radiator. Van der Waals broadened lines are red shifted by about one-third the size of the FWHM.

  For the 1s2p ¹P - 1s3d ¹D₂ transition in He I, and with He as perturber: λ = 6678.15 Å; I = 198 311 cm^-1; E^* = E_i = 171 135 cm^-1; E_k = 186 105 cm^-1; µ = 2. At T = 15 000 K and N = 1 × 10¹⁸ cm^-3: Δλ^W_1/2 = 0.044 Å.

  Stark broadening due to charged perturbers, i.e., ions and electrons, usually dominates resonance and van der Waals broadening in discharges and plasmas. The FWHM for hydrogen lines is
   

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  where N_e is the electron density. The half-width parameter α_1/2 for the H_β line at 4861 Å, widely used for plasma diagnostics, is tabulated in the table below for some typical temperatures and electron densities [33]. This reference also contains α_1/2 parameters for other hydrogen lines, as well as Stark width and shift data for numerous lines of other elements, i.e., neutral atoms and singly charged ions (in the latter, Stark widths and shifts depend linearly on N_e). Other tabulations of complete hydrogen Stark profiles exist.