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

2. 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

3. 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…)

4. 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.

5. 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

6. The Classical Damping Line Profile

7. 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!

8. 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

9. 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

10. 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."

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. Damping Coefs for Na D

18. 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

19. 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

20. 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.

21. 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:

22. 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.

23. 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

24. Plot a Damped Profile