Spectral Line Formation Classical picture of radiation Intrinsic vs. extrinsic broadening mechanisms Line absorption coefficient Radiative transfer in spectral lines

Spectral Line Formation-Line Absorption Coefficient Radiation damping (atomic absorptions and emissions aren’t perfectly monochromatic – uncertainty principle) Thermal broadening from random kinetic motion Collisional broadening – perturbations from neighboring atoms/ions/electrons) Hyperfine structure Zeeman effect

Classical Picture of Radiation Photons are sinusoidal variations of electro-magnetic fields When a photon passes by an electron in an atom, the changing fields cause the electron to oscillate Treat the electron as a classical harmonic oscillator: mass x acceleration = external force – restoring force – dissipative E&M is useful! (well…)

Atomic Absorption Coefficient N0 is the number of bound electrons per unit volume the quantity n-n0 is the frequency separation from the nominal line center the quantity e is the dielectric constant (e=1 in free space) and g=g/m is the classical damping constant The atomic absorption coefficient includes atomic data (f, e, g) and the state of the gas (N0), and is a function of frequency. The equation expresses the natural broadening of a spectral line.

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

The Classical Damping Line Profile

The Classical Line Profile Look at a thin atmospheric layer between t2 (the deeper layer) and t1 The line profile is proportional to kn At line center n=n0, and Half the maximum depth occurs at (n-n0)=g/4p In terms of wavelength Very small – and the same for ALL lines!

k = 2.5 x 104 per gram of neutral sodium An example… The Na D lines have a wavelength of 5.9x10-5 cm. g = 6.4 x 107 sec-1 The absorption coefficient per gram of Na atoms at a distance of 2A from line center can be calculated: Dn0-n = 1.7 x 1011 sec-1 and N = 1/m = 2.6 x 1022 atoms gm-1 Then k = 3.7 x 104 f and f=2/3, so k = 2.5 x 104 per gram of neutral sodium

The Abundance of Sodium In the Sun, the Na D lines are about 1% deep at a distance of 2A from line center Use a simple one-layer model of depth x (the Schuster-Schwarzschild model) Or krx=0.01, and rx=4x10-7 gm cm-2 (recall that kNa=2.5 x 104 per gram of neutral sodium at a distance of 2A from line center) the quantity rx is a column density

Natural Broadening From Heisenberg's uncertainty principle: The electron in an excited state is only there for a short time, so its energy cannot have a precise value. Since energy levels are "fuzzy," atoms can absorb photons with slightly different energy, with the probability of absorption declining as the difference in the photon's energy from the "true" energy of the transition increases. The FWHM of natural broadening for a transition with an average waiting time of Dto is given by A typical value of (Dl)1/2 = 2 x 10-4 A. Natural broadening is usually very small. The profile of a naturally broadened linen is given by a dispersion profile (also called a damping profile, a Lorentzian profile, a Cauchy curve, and the Witch of Agnesi!) of the form (in terms of frequency) where g is the "damping constant."

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

Line Absorption with QM Replace g with G! Broadening depends on lifetime of level Levels with long lifetimes are sharp Levels with short lifetimes are fuzzy QM damping constants for resonance lines may be close to the classical damping constant QM damping constants for other Fraunhofer lines may be 5,10, or even 50 times bigger than the classical damping constant

Add Quantum Mechanics Define the oscillator strength, f: related to the atomic transition probability Bul: f-values usually tabulated as gf-values. theoretically calculated laboratory measurements solar

Collisional Broadening Perturbations by discrete encounters Change in energy approximated by a power law of the form DE = constant x r-n (r is the separation between the atom and the perturber) Perturbations by static ion fields (linear Stark effect broadening) (n=2) Self-broadening - collisions with neutral atoms of the same kind (resonance broadening, n=3) if perturbed atom or ion has an inner core of electrons (i.e. with a dipole moment) (quadratic Stark effect, n=4) Collisions with atoms of another kind (neutral hydrogen atoms) (van der Waals, n=6) Assume adiabatic encounters (electron doesn’t change level) Non-adiabatic (electron changes level) collisions also possible

Pressure Broadening – DE=constant x r-n Type Lines Affected Perturber 2 Linear Stark Hydrogen Protons, e- 3 Self broadening or resonance broadening Common species Atoms of the same type 4 Quadratic Stark Most, esp. in hot stars Ions, e- 6 Van der Waals Most, esp. in cool stars Neutral hydrogen

Approaches to Collisional Broadening Statistical effects of many particles (pressure broadening) Usually applies to the wings, less important in the core 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

Damping Coefs for Na D

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

Thermal + Turbulence The average speed of an atom in a gas due to thermal motion - Maxwell Boltzmann distribution. The most probably speed is given by Moving atoms are Doppler shifted, and individual atoms will absorb light at slightly different wavelengths because of the Doppler shift. Spectral lines are also Doppler broadened by turbulent motions in the gas. The combination of these two effects produces a Doppler-broadened profile: Typical values for Dl1/2 are a few tenths of an Angstrom. The line depth for Doppler broadening decreases exponentially from the line center.

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 in computation 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.

Calculating Voigt Profiles Tabulated as the Hjerting function H(u,a) u=Dl/DlD a=(l2/4pc)/DlD =(g/4p)DnD Hjertung functions are expanded as: H(u,a)=H0(u) + aH1(u) + a2H2(u) + a3H3(u) +… or, the absorption coefficient is

Plot a Damped Profile