1. Spectral Line Broadening Hubeny & Mihalas Chap. 8 Gray Chap. 11
Natural Broadening
Doppler Broadening
Collisional Broadening: Impact, Statistical, Quantum Theories

2. Broadening of Absorption Profile
Natural – energy uncertainty due to finite lifetime
Doppler – thermal motion of gas
Pressure – perturbations in energy levels due to collisions (encounters) with charged particles [important in transfer equation]
Stellar rotation – Doppler shifts across disk
Stellar turbulence – Doppler shifts from motion [important in line synthesis]
Instrumental – projected slit of spectrograph [always important]

3. Natural Broadening Uncertainty principle
level j depopulated by spontaneous emission, rate Aji (Einstein coeff.)
Lifetime for j to i
Lifetime for all downward transitions
FWHM j i

4. Natural Broadening Damping constant Lorentzian profile
Small, important in low density gas

5. Doppler Broadening by Thermal Motion
Profile at Doppler shifted frequency by speed ξ
Integrate over Maxwellian velocity distribution along the line of sight

6. Doppler Broadening by Thermal Motion
Substitute
Then final profile has form
H(a,V) = Voigt profile

7. Voigt Profile Gaussian in core and Lorentzian in wings
IDL version: IDL> u=findgen(201)/ IDL> v=voigt(0.5,u) IDL> plot,u,v

8. Collisional Broadening: Classical Impact – Phase Shift Theory
Suppose encounter happens quickly and atom emits as an undisturbed oscillator between collisions but ceases before and after
Frequency content of truncated wave from FT
Power spectrum (observed)

9. Collisional Broadening: Classical Impact – Phase Shift Theory
Probability number occurring in time dT at T where T0 = average time between collisions
Mean energy spectrum is then Lorentzian profile damping constant Γ=2/T0

10. Collisional Broadening: Classical Impact – Phase Shift Theory
Frequency of collisions = 1/T0
Suppose collisions occur if particles pass within distance = impact parameter ρ0 N = #perturbers/cm3, v = relative velocity cm/s
Then damping parameter is

11. Weisskopf approximation
perturber is a classical particle
path is a straight line
no transitions caused in atom
interaction creates a phase shift or frequency shift given by

12. p exponents of astronomical interest
p = 2 linear Stark effect (H + charged particle)
p = 3 resonance broadening (atom A + atom A)
p = 4 quadratric Stark effect (non-hydrogenic atom + charged particle)
p = 6 van der Waals force (atom A + atom B)
Cp from experiment or quantum theory

13. Weisskopf approximation
Total phase shift
Atom
r(t)
t = 0
v=constant p ψp 2 π 3 4
π/2 6
3π/8
perturber path

14. Weisskopf approximation
Assume that only collisions that produce a phase shift > η0 are effective in broadening
Weisskopf assumed η0 =1 , yields damping
depends on ρ, T
Ignores weak collisions η < η0

15. Better Impact Model: Lindholm-Foley
Includes effects of multiple weak collisions, which introduce a phase shift Δω0 ; ΓLF > ΓW
Impact theory fails for: small ρ, large broadening time overlap of collisions nonadiabatic collisions p 3 4 6 Γ
2π2C3N
11.37 C42/3 v1/3 N
8.08 C62/5 v3/5 N
Δω0
9.85 C42/3 v1/3 N
2.94 C62/5 v3/5 N

16. Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation
Imagine atom sitting in a static sea of perturbers (OK for slow moving ions) that produces a relative probability of perturbing electric field and Δω
Close to atom, consider probability that nearest neighbor is located at a distance in the range (r,r+Δr) = W(r) dr
Corresponding frequency profile

17. Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation
Probability proportional to (1) % that do not occur at <r (2) increasing numbers at increasing distance
Differentiate wrt r

18. Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation
Consider frequency shifts relative to that for mean interparticle distance r0 #particles x volume for each = total volume
Insert into expression W(r)
Express with relative frequency shift

19. Statistical Theory for Collisional Broadening: Nearest Neighbor Approximation
Replace W(r) with W(Δω)
Probability that atom will experience a perturbing field to give a frequency shift Δω

20. Apply to Linear Stark effect p=2
Express in terms of normal field strength
Change of variables

21. Apply to Linear Stark effect p=2
Then probability in terms of field strength is [note missing minus sign in Hubeny & Mihalas]
Final expression for profile

22. Holtsmark Statistical Theory
Ensemble of perturbers instead of single
more particles, more chances for strong field
e- attracted to ions, reduce perturbation by Debye shielding
in stellar atmospheres density is low, number of perturbers is large, and Holtsmark distribution is valid

23. Hydrogen: Linear Stark Effect
each level degenerate with 2n2 sublevels
perturbing field will separate sublevels
observed profile is a superposition of components weighted by relative intensities and shifted by field probability function

24. Hydrogen: Linear Stark Effect
each component shifted by
profile is a sum over all components
density dependent shift (N)
statistical theory OK for interactions H + protons
impact theory ~OK for interactions H + electron, but electron collisions are non-adiabatic

25. Quantum Calculations for the Linear Stark effect of Hydrogen
unified theory for electron and proton broadening for Lyman and Balmer series: Vidal, Cooper, & Smith 1973, ApJS, 25, 37
IR series Lemke 1997, A&AS, 122, 285
Model Microfield Method (not static for ions) Stehle & Hutcheon 1999, A&AS, 140, 93

26. Summary
final profile is a convolution of all the key broadening processes
convolution of Lorentzian profiles: Γtotal=ΣΓi
convolution of Lorentzian and Doppler broadening yields a Voigt profile
convolution of Stark profile with Voigt (for H)
calculate as a function of depth in atmosphere because broadening depends on T, N (Ne)