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Spectral Line BroadeningHubeny & Mihalas Chap. 8
Gray Chap. 11
Natural Broadening Doppler Broadening
Collisional Broadening: Impact, Statistical, Quantum Theories
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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]2
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
3
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ΔEΔt ≥ h
€
Δt =1
A ji
€
Δt =1
A jii< j
∑
€
hΔνΔt ≥h
2π⇒ Δν ≈
1
2πΔt
j
i
Natural Broadening
• Damping constant • Lorentzian profile
• Small, important in low density gas
4
€
Γ= A jii< j
∑
€
φ ν( ) =Γ/(4π 2)
(ν −ν 0)2 + (Γ/(4π ))2
Doppler Broadening by Thermal Motion
• Profile at Doppler shifted frequency by speed ξ
• Integrate over Maxwellian velocity distribution along the line of sight
5
€
φ ν −ξν 0
c
⎛
⎝ ⎜
⎞
⎠ ⎟=
Γ/(4π 2)
(ν −ν 0 −ξν 0
c)2 + (Γ/(4π ))2
€
φ ν −ξν 0
c
⎛
⎝ ⎜
⎞
⎠ ⎟W (ξ)
−∞
+∞
∫ dξ =Γ/(4π 2)
(ν −ν 0 −ξν 0
c)2 + (Γ/(4π ))2−∞
+∞
∫ 1
πe−(ξ /ξ 0 )2 dξ
ξ0
Doppler Broadening by Thermal Motion
• Substitute
• Then final profile has form
• H(a,V) = Voigt profile
6
€
ΔνD =ξ0ν 0
c, V =
ν −ν 0
Δν D
, y =ξ
ξ0
, a ≡Γ
4πΔν D
€
φ(V ) =a
π 3 / 2Δν D
e−y 2
(V − y)2 + a2−∞
+∞
∫ dy ≡1
π Δν D
H(a,V )
Voigt Profile
• Gaussian in core and Lorentzian in wings
• IDL version:IDL> u=findgen(201)/40.-2.5IDL> v=voigt(0.5,u)IDL> plot,u,v
7
€
H(a,V ) ~ e−V 2
+ aπV 2
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)
8€
E(ω) =1
2πE(t)e iω tdt =
E0
2π−∞
+∞
∫ e i(ω −ω 0 )t
−T / 2
+T / 2
∫ dt
=E0
2π
2sin[(ω −ω0)T /2]
(ω −ω0)
€
I(ω) = E⋅ E * =E0
2
2π
sin2[(ω −ω0)T /2]
[(ω −ω0) /2]2
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
9
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W (T) = e−T /T0dT /T0
€
I(ω) =E0
2
2πT0
sin2[(ω −ω0)T /2]
[(ω −ω0) /2]20
∞
∫ e−T /T0dT
= E02 1/π
(ω −ω0)2 + (1/T0)2 →Γ/(2π )
(ω −ω0)2 + (Γ/2)2
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
10
€
1
T0
= Nπρ 02v
€
Γ=2Nπρ 02v
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
11
€
Δω =Cp
r p
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
12
Weisskopf approximation
€
r(t) = [ρ 02 + v 2t 2]1/ 2
13
Atom
perturber path
t = 0 v=constant
r(t)
• Total phase shift
€
η(ρ) =Cp
dt
r p =−∞
+∞
∫ Cp
dt
[ρ 2 + v 2t 2]p / 2 =−∞
+∞
∫Cp
vρ p−1ψ p
€
ψ p = πΓ [(p −1) /2]
Γ [p /2]
p ψp
2 π
3 2
4 π/2
6 3π/8
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
14
€
ρ0 =Cpψ p
η0v
⎛
⎝ ⎜
⎞
⎠ ⎟
1
p−1
€
ΓW = 2πNvCpψ p
v
⎛
⎝ ⎜
⎞
⎠ ⎟
2
p−1
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 broadeningtime overlap of collisionsnonadiabatic collisions
15
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I(ω) =Γ/(2π )
(ω −ω0 − Δω0)2 + (Γ/2)2
p 3 4 6
Γ 2π2C3N 11.37 C42/3 v1/3 N 8.08 C6
2/5 v3/5 N
Δω0 0 9.85 C42/3 v1/3 N 2.94 C6
2/5 v3/5 N
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
16
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I(Δω)d(Δω) ∝W (r)[dr /d(Δω)]d(Δω)
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
17
€
W (r) dr = [1 − W (x)dx0
r
∫ ] 4πr2N dr
€
d
dr
W (r)
4πr2N
⎡ ⎣ ⎢
⎤ ⎦ ⎥= −4πr2N
W (r)
4πr2N
⎡ ⎣ ⎢
⎤ ⎦ ⎥
⇒ W (r) = 4πr2N exp(−4
3πr3N)
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
18
€
(VN)4
3πr0
3 = V ⇒ r0 =4
3πN
⎛
⎝ ⎜
⎞
⎠ ⎟−1/ 3
€
W (r) = 4πr2N exp(−4
3πr3N) = 3
r
r0
⎛
⎝ ⎜
⎞
⎠ ⎟
21
r0exp(−
r
r0
⎛
⎝ ⎜
⎞
⎠ ⎟
3
)
€
Δω =Cp
r p ⇒ r =Cp
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ p
⇒r
r0=
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ p
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
19
€
3r
r0
⎛
⎝ ⎜
⎞
⎠ ⎟
2dr
r0= 3
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
2
pd
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
1
p= d
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p ⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
€
W (r) dr = 3r
r0
⎛
⎝ ⎜
⎞
⎠ ⎟
2dr
r0exp(−
r
r0
⎛
⎝ ⎜
⎞
⎠ ⎟
3
) = exp[−Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p] d
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p
Apply to Linear Stark effect p=2
• Express in terms of normal field strength
• Change of variables
20
€
F0 =e
r02 = e
4
3π N
⎛
⎝ ⎜
⎞
⎠ ⎟
2
3= 2.6eN
2
3
€
β ≡F
F0
=r0r
⎛
⎝ ⎜
⎞
⎠ ⎟2
=Δω
Δω0
, x ≡Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p=
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
2
x = β−
3
2 ⇒ dx = −3
2β
−5
2
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
21
€
W (r) dr = exp[−Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p] d
Δω0
Δω
⎛
⎝ ⎜
⎞
⎠ ⎟
3
p= exp[−x] dx
W (x) dx =W (β) dβ =3
2β
−5
2 exp(−β−
3
2 ) dβ
€
I(Δω) d(Δω) =W (β)dβ
d(Δω)d(Δω) =W (β)
d(Δω)
Δω0
Holtsmark Statistical Theory• Ensemble of perturbers
instead of single• more particles, more
chances for strong field• e- attracted to ions,
reduce perturbation byDebye shielding
• in stellar atmospheres density is low, number of perturbers is large, and Holtsmark distribution is valid
22
Hydrogen: Linear Stark Effect
• each level degeneratewith 2n2 sublevels
• perturbing field will separate sublevels
• observed profile is a superposition of components weighted by relative intensities and shifted by field probability function
23
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-adiabatic24
€
I(Δλ )d(Δλ ) = IkWF
F0
⎛
⎝ ⎜
⎞
⎠ ⎟dF
F0k
∑ = IkWα
Ck
⎛
⎝ ⎜
⎞
⎠ ⎟dα
Ckk
∑
α ≡Δλ
F0
, F0 = 2.60eN 2 / 3
€
Δλk =CkF
Quantum Calculations for theLinear Stark effect of Hydrogen
• unified theory for electron and proton broadening for Lyman and Balmer series: Vidal, Cooper, & Smith 1973, ApJS, 25, 37
• IR seriesLemke 1997, A&AS, 122, 285
• Model Microfield Method (not static for ions)Stehle & Hutcheon 1999, A&AS, 140, 93
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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)
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