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The Classical Damping Constant. For a classical harmonic oscillator, The shape of the spectral line depends on the size of the classical damping constant For n - n 0 >> g /4 p , the line falls off as ( n - n 0 ) -2 Accelerating electric charges radiate. and
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The Classical Damping Constant • For a classical harmonic oscillator, • The shape of the spectral line depends on the size of the classical damping constant • For n-n0 >> g/4p, the line falls off as (n-n0)-2 • Accelerating electric charges radiate. • and • is the classical damping constant (l is in cm) The mean lifetime is also defined as T=1/g, where T=4.5l2
Collisional Broadening • Collisions with atoms of another kind (neutral hydrogen atoms) • Self-broadening - collisions with neutral atoms of the same kind • Perturbations by static ion fields (Stark effect broadening) • Adiabatic (electron doesn’t change level) and non-adiabatic (electron changes level) collisions with fast-moving electrons
Approaches to Collisional Broadening • Perturbations by discrete encounters • More important in line cores • The frequency shift depends on the separation • r-2 for perturbations by an ion or electron (linear Stark effect) • r-4 if perturbed atom or ion has an inner core of electrons (i.e. with a dipole moment leading to the quadratic Stark effect) • r-3 from a resonance effect when neutral atoms of the same species interact (e.g. hydrogen) • r-6 for van der Waals broadening (perturbed by a neutral atom of another species, esp. hydrogen)
Approaches to Collisional Broadening • Statistical effects of many particles (pressure broadening) • Usually applies to the wings • Some lines can be described fully by one or the other • Know your lines! • The functional form for collisional damping is the same as for radiation damping, but Grad is replaced with Gcoll • Collisional broadening is also described with a dispersion function • Collisional damping is sometimes 10’s of times larger than radiation damping
Doppler Broadening • Two components contribute to the intrinsic Doppler broadening of spectral lines: • Thermal broadening • Turbulence – the dreaded microturbulence! • Thermal broadening is controlled by the thermal velocity distribution (and the shape of the line profile) where vr is the line of sight velocity component • The Doppler width associated with the velocity v0 (where the variance v02=2kT/m) is and l is the wavelength of line center
More Doppler Broadening • Combining these we get the thermal broadening line profile: • At line center, n=n0, and this reduces to • Where the line reaches half its maximum depth, the total width is
Combining the Natural, Collisional and Thermal Broadening Coefficients • The combined broadening coefficient is just the convolution of all of the individual broadening coefficients • The natural, Stark, and van der Waals broadening coefficients all have the form of a dispersion profile: • With damping constants (grad, g2, g4, g6) one simply adds them up to get the total damping constant: • The thermal profile is a Gaussian profile:
The Voigt Profile • The convolution of a dispersion profile and a Gaussian profile is known as a Voigt profile. • Voigt functions are tabulated for use on computations • In general, the shapes of spectra lines are defined in terms of Voigt profiles • Voigt functions are dominated by Doppler broadening at small Dl, and by radiation or collisional broadening at large Dl • For weak lines, it’s the Doppler core that dominates. • In solar-type stars, collisions dominate g, so one needs to know the damping constant and the pressure to compute the line absorption coefficient • For strong lines, we need to know the damping parameters to interpret the line.
