5. Atomic radiation processes
Einstein coefficients for absorption and emission
oscillator strength
line profiles: damping profile, collisional broadening, Doppler
broadening
continuous absorption and scattering
1
Atomic transitions
absorption
h + El  Eu
emission
spontaneous
Eu  El + h
stimulated
h + Eu  El +2h
(isotropic)
(direction of incoming photon)
2
A. Line transitions
Einstein coefficients
probability that a photon in frequency interval (, +d) in the solid angle
range (,+d) is absorbed by an atom in the energy level El with a
resulting transition El  Eu per second:
dwabs (ν, ω, l, u) = B lu I ν(ω) ϕ(ν )dν
atomic property
 no. of incident
photons
dω
4π
probability  є
(,+d)
probability  є
(,+d)
absorption profile
probability for
transition l  u
Blu: Einstein coefficient for absorption
3
Einstein coefficients
similarly for stimulated emission
dwst (ν, ω, l , u) = Bul Iν (ω) ϕ(ν)dν
dω
4π
Bul: Einstein coefficient for stimulated emission
and for spontaneous emission
dwsp (ν, ω, l, u) = A ul ϕ(ν)dν
dω
4π
Aul: Einstein coefficient for spontaneous emission
4
dwst (ν, ω, l , u) = Bul Iν (ω) ϕ(ν)dν
dω
4π
dwabs (ν, ω, l, u) = B lu I ν(ω) ϕ(ν )dν
dω
4π
Einstein coefficients, absorption and emission coefficients
Iν
dF
dV = dF ds
Number of absorptions
& stimulated emissions
in dV per second:
nl dw abs dV, nu dwst dV
ds
absorbed energy in dV per second:
stimulated emission counted
as negative absorption
dEνabs = nl hν dw abs dV − nu hν dwst dV
and also (using definition of intensity):
dEνabs = κL
ν Iν ds dω dν dF
κL
ν =
hν
ϕ(ν) [ nl B lu − nuBul ]
4π
for the spontaneously emitted energy:
Absorption and emission coefficients are a
function of Einstein coefficients, occupation
numbers and line broadening
²L
ν =
hν
n A ϕ(ν )
4π u ul
5
Relations between Einstein coefficients
Einstein coefficients are atomic properties  do not depend on thermodynamic
state of matter
We can assume TE:
B ν (T ) =
²L
Sν = νL = Bν (T )
κν
Aul
nuAul
n
= u
nl B lu − nu B ul
nl Blu − nnul B ul
gu − hν
nu
=
e kT
nl
gl
From the Boltzmann formula:
for hν/kT << 1:
gu
2ν2
kT
=
c2
gl
for T  ∞
nu
gu
=
nl
gl
µ
¶
hν
1−
,
kT
¶
µ
hν
1−
kT Blu −
0
gu
gl
A
¡ ul
1−
hν
kT
2ν2
Bν (T ) = 2 kT
c
¢
B ul
gl Blu = guB ul
6
gu
2ν2
kT =
2
c
gl
µ
¶
hν
1−
kT Blu −
gu
gl
A
¡ ul
1−
hν
kT
¢
gl Blu = guB ul
B ul
Relations between Einstein coefficients
2ν 2
gu
kT
=
c2
gl
for T  ∞
2 hν3
A ul gu
=
c2
B lu gl
Aul
µ
hν
1−
kT
¶
2 h ν 3 gl
=
B
c2 gu lu
2 h ν3
A ul =
Bul
c2
hν
ϕ(ν ) B lu
κL
=
ν
4π
²L
ν =
¶
µ
hν A ul kT
1−
kT B lu hν
·
¸
gl
nl −
n
gu u
hν
gl 2hν3
ϕ(ν) nu
Blu
4π
gu c2
Note: Einstein coefficients atomic
quantities. That means any
relationship that holds in a special
thermodynamic situation (such as T
very large) must be generally valid.
only one Einstein
coefficient needed
7
Oscillator strength
Quantum mechanics
The Einstein coefficients can be calculated by quantum
mechanics + classical electrodynamics calculation.
Eigenvalue problem using using wave function:
Hatom |ψ l >= El |ψl >
H atom
p2
+ Vnucleus + Vshell
=
2m
Consider a time-dependent perturbation such as an external
electromagnetic field (light wave) E(t) = E0 eiωt.
The potential of the time dependent perturbation on the atom is:
V (t) = e
N
X
i=1
E · ri = E · d
d: dipol operator
[Hatom + V (t)] |ψl >= El |ψ l >
with transition probability
∼ | < ψ l |d|ψ u > |2
8
Oscillator strength
hν
πe 2
B =
f
4π lu
mec lu
The result is
flu: oscillator strength (dimensionless)
classical result from electrodynamics
= 0.02654 cm2/s
Classical electrodynamics
electron quasi-elastically bound to nucleus and oscillates within outer electric field as E.
Equation of motion (damped harmonic oscillator):
ẍ + γ ẋ + ω20 x =
damping
constant
γ=
2 ω20 e2
3 mec3
2 2
=
8π e
3 me
ν20
c3
resonant
(natural)
frequency
0=20
e
E
me
ma = damping force + restoring force + EM force
the electron oscillates preferentially at resonance
(incoming radiation  = 0)
The damping is caused, because the de- and
accelerated electron radiates
9
Classical cross section and oscillator strength
Calculating the power absorbed by the oscillator, the integrated “classical”
absorption coefficient and cross section, and the absorption line profile are
found:
Z
integrated over the line profile
ϕ(ν)dν =
2
πe
cl
= nl σtot
κL,cl
ν dν = nl
me c
γ/4π
1
π (ν − ν0 )2 + (γ/4π)2
nl: number
density of
absorbers
σtotcl: classical
cross section
(cm2/s)
[Lorentz (damping) line profile]
oscillator strength flu is quantum mechanical correction to classical result
(effective number of classical oscillators, ≈ 1 for strong resonance lines)
hν
L
ϕ(ν ) nl B lu
κ
=
From
(neglecting stimulated emission)
ν
4π
10
Oscillator strength
hν
πe 2
ϕ(ν) B lu =
ϕ(ν) flu = σlu (ν )
4π
mec
flu =
absorption cross section;
dimension is cm2/s
1 mec
hν B lu
4π πe 2
Oscillator strength (f-value) is different for each atomic transition
Values are determined empirically in the laboratory or by elaborate numerical atomic physics calculations
Semi-analytical calculations possible in simplest cases, e.g. hydrogen
flu =
25
g
3/2
5
3
π l u3
¡1
l2
−
¢
1 −3
u2
g: Gaunt factor
H: f=0.6407
H: f=0.1193
H: f=0.0447
11
Line profiles
line profiles contain information on physical conditions of the gas and
chemical abundances
analysis of line profiles requires knowledge of distribution of opacity with
frequency
several mechanisms lead to line broadening (no infinitely sharp lines)
- natural damping: finite lifetime of atomic levels
- collisional (pressure) broadening: impact vs quasi-static approximation
- Doppler broadening: convolution of velocity distribution with atomic profiles
12
1. Natural damping profile
finite lifetime of atomic levels  line width
NATURAL LINE BROADENING OR RADIATION DAMPING
 = 1 / Aul
 E  ≥ h/2
(≈ 10-8 s in H atom 2  1): finite lifetime with respect to
spontaneous emission
uncertainty principle
1/2 =  / 2
1/2 =   / 2c
line broadening
e.g. Ly : 1/2 = 1.2 × 10-4 A
Γ/4π
1
ϕ(ν) =
π (ν − ν0 )2 + (Γ/4π)2
Lorentzian profile
H: 1/2 = 4.6 × 10-4 A
13
Natural damping profile
resonance line
u = i<u Aui
 = Au1
l = i<l Ali
 =  u + l
natural line broadening is important for strong lines (resonance
lines) at low densities (no additional broadening mechanisms)
e.g. Ly  in interstellar medium
but also in stellar atmospheres
14
2. Collisional broadening
radiating atoms are perturbed by the electromagnetic field of neighbour atoms,
ions, electrons, molecules
energy levels are temporarily modified through the Stark - effect: perturbation
is a function of separation absorber-perturber
energy levels affected  line shifts, asymmetries & broadening
E(t) = h  = C/rn(t)
r: distance to perturbing atom
a) impact approximation: radiating atoms are perturbed by passing particles at distance r(t).
Duration of collision << lifetime in level  lifetime shortened  line broader
in all cases a Lorentzian profile is obtained (but with larger total Γ than only natural damping)
b) quasi-static approximation: applied when duration of collisions >> life time in level
consider stationary distribution of perturbers

15
Collisional broadening
n=2
linear Stark effect ΔE » F
for levels with degenerate angular momentum (e.g. HI, HeII)
field strength F » 1/r2
 ΔE » 1/r2
important for H I lines, in particular in hot stars (high number density of free
electrons and ions). However , for ion collisional broadening the quasi-static
broadening is also important for strong lines (see below)
n=3
resonance broadening
atom A perturbed by atom A’ of same species
important in cool stars, e.g. Balmer lines in the Sun
 ΔE » 1/r3
16
Collisional broadening
n=4
quadratic Stark effect E ∼ F2
field strength F ∼ 1/r2
 E ∼ 1/r4
(no dipole moment)
important for metal ions broadened by e- in hot stars  Lorentz profile with e ~ ne
n=6
van der Waals broadening
atom A perturbed by atom B
important in cool stars, e.g. Na perturbed by H in the Sun
17
Quasi-static approximation
tperturbation >> τ = 1/Aul
 perturbation practically constant during emission or absorption
atom radiates in a statistically fluctuating field produced by ‘quasi-static’ perturbers,
e.g. slow-moving ions
given a distribution of perturbers  field at location of absorbing or emitting atom
statistical frequency of particle distribution
 probability of fields of different strength (each producing an energy shift ΔE = h Δν)
 field strength distribution function
 line broadening
Linear Stark effect of H lines can be approximated to 0st order in this way
18
Quasi-static approximation for hydrogen line broadening
Line broadening profile function determined by probability function for electric field caused by all
other particles as seen by the radiating atom.
W(F) dF: probability that field seen by radiating atom is F
dF
ϕ(∆ν ) dν = W (F )
dν
dν
For calculating W(F)dF we use as a first step the nearest-neighbor approximation:
main effect from nearest particle
19
Quasi-static approximation –
nearest neighbor approximation
assumption: main effect from nearest particle (F ~ 1/r2)
we need to calculate the probability that nearest neighbor is in the distance
range (r,r+dr) = probability that none is at distance < r and one is in (r,r+dr)
Zr
Integral equation for W(r)
2
we find solution by differentiating
W(r) dr = [1 − W (x) dx] (4πr N) dr
 differential equation
0
relative probability for particle in shell (r,r+dr)
N: total uniform particle density
probability for no particle in (0,r)
d
dr
·
·
¸
¸
W
(r)
W(r)
= −W (r) = −4πr 2N
2
4πr N
4πr 2 N
4
W (r) = 4πr 2Ne − 3 πr
3
N
Differential equation
Normalized solution
20
Quasi-static approximation
mean interparticle
r0 =
distance:
normal field strength:
define:
F0 =
µ
4
πN
3
e
r 20
¶−1/3
W (r)dr = e
− rr
0
3
r3
d{
}
r0
(linear Stark)
F
r0 2
β=
={ }
F0
r
note: at high particle density  large F0
 stronger broadening
from W(r) dr  W(β) dβ:
W (β)dβ =
W (β ) ∼ β −5/2
for β → ∞
3 −5/2 −β − 3/2
β
e
dβ
2
∆ν ∼ β →
ϕ(∆ν ) ∼ ∆ν −5/2
Stark broadened line profile in the wings,
not Δλ-2 as for natural or impact broadening
21
Quasi-static approximation – advanced theory
complete treatment of an ensemble of particles: Holtsmark theory
+ interaction among perturbers (Debye shielding of the potential at distances >
Debye length)
Holtsmark (1919),
Chandrasekhar (1943, Phys. Rev. 15, 1)
2β
W (β) =
π
Z
∞
−y 3/2
e
y sin(βy)dy
0
Ecker (1957, Zeitschrift f. Physik, 148, 593 & 149,245)
4/3
2βδ
W (β) =
π
number of particles
inside Debye sphere
Mihalas, 78
2
g(y) = y 3/2
3
Z
Z
y
T 1/2
D = 4.8
cm
ne
4π 3
δ=
D N
3
∞
e−δg(y) y sin(δ 2/3 βy)dy
0
∞
(1 − z −1 sinz)z −5/2 dz
Debye length, field of ion vanishes
beyond D
number of particles
inside Debye sphere
22
3. Doppler broadening
radiating atoms have thermal velocity
Maxwellian distribution:
³ m ´3/2
m
2
2
2
P (vx , vy , vz ) dvx dvy dvz =
e − 2 kT (vx+ vy +vz ) dvx dvy dvz
2πkT
Doppler effect: atom with velocity v emitting at frequency 0, observed at
frequency :
v cos θ
ν0 = ν − ν
c
23
Doppler broadening
Define the velocity component along the line of sight: 
The Maxwellian distribution for this component is:
P (ξ) dξ =
1
π 1/2ξ 0
e
¡ ξ ¢2
−
ξ0
dξ
ξ0 = (2kT /m)1/2
ν0 = ν − ν
if v/c <<1  (0 -)/ <<1:
∆ν = ν0 − ν = −ν
ξ
c
thermal velocity
if we observe at , an atom with velocity
component  absorbs at 0 in its frame
line center
ξ
ξ
ξ
= −[(ν − ν0) + ν0 ] ≈ −ν0
c
c
c
24
Doppler broadening
line profile for v = 0
ϕ R (ν − ν0 )

=⇒
profile for v ≠ 0
  0
ξ
ϕR (ν0 − ν0) = ϕ(ν − ν0 − ν0 )
c
New line profile: convolution
ϕ new (ν − ν0) =
Z∞
−∞
ξ
ϕ(ν − ν0 − ν0 ) P (ξ) dξ
c
profile function in rest
frame
velocity distribution
function
25
Doppler broadening: sharp line approximation
ϕ new (ν − ν0) =
1
π1/2
Z∞
−∞
ξ ξ
ϕ(ν − ν0 − ν0 0 ) e−
c ξ0
¡ ξ ¢2
ξ0
dξ
ξ0
D: Doppler width of the line
D = 4.301 × 10-7  (T/)1/2
D = 4.301 × 10-7  (T/)1/2
ξ0 = (2kT /m)1/2
thermal velocity
Approximation 1: assume a sharp line – half width of profile function << D
ϕ(ν − ν0 ) ≈ δ(ν − ν0 )
delta function
26
Doppler broadening: sharp line approximation
new
ϕ
(ν − ν0 ) =
1
π1/2
Z∞
−∞
δ(ν − ν0 − ∆νD
¡ ¢
ξ − ξξ0 2 dξ
)e
ξ0
ξ0
δ(a x) =
ϕnew(ν − ν0) =
1
π1/2
1
δ(x)
a
¶ ¡ ¢2
Z∞ µ
ξ
ν − ν0
dξ
ξ
e − ξ0
δ
−
∆νD
ξ0
∆νD ξ0
−∞
¡ ν−ν 0 ¢ 2
1
−
ϕ new (ν − ν0) = 1/ 2
e ∆ν D
π
∆νD
Gaussian profile –
valid in the line core
27
Doppler broadening: Voigt function
Approximation 2: assume a Lorentzian profile – half width of profile function > D
Γ/4π
1
ϕ(ν) =
π (ν − ν0 )2 + (Γ/4π)2
1
a
ϕnew (ν − ν0 ) = 1/2
π
∆νD π
Z∞
−∞
2
³
∆ν
∆νD
e −y
dy
´2
− y + a2
a=
Γ
4π∆νD
¶
µ
1
ν
−
ν
0
ϕ new(ν − ν0 ) = 1/2
H a,
π
∆νD
∆νD
Voigt function (Lorentzian * Gaussian):
calculated numerically
28
Doppler broadening: Voigt function
Approximate representation of Voigt function:
³
H a,
ν −ν0
∆νD
´
≈e
≈
¡ ν−ν 0 ¢2
−
π
∆ν D
a
¡
¢2
1 /2 ν−ν 0
∆ν D
line core: Doppler broadening
line wings: damping profile
only visible for strong lines
General case: for any intrinsic profile function (Lorentz, or Holtsmark, etc.) – the
observed profile is obtained from numerical convolution with the different
broadening functions and finally with Doppler broadening
29
Voigt function
normalization:
Z∞
H(a, v) dv =
√
π
−∞
usually α <<1
max at v=0:
H(α,v=0) ≈ 1-α
~ Gaussian
~ Lorentzian
Unsoeld, 68
30
B. Continuous transitions
1. Bound-free and free-free processes
absorption
h + El  Eu
photoionization - recombination
emission
hlk
spontaneous
Eu  El + h
stimulated
h + Eu  El +2h
(isotropic)
(non-isotropic)
bound-bound: spectral lines
bound-free
free-free
consider photon h ≥ hlk
cont
(energy > threshold): extra-energy to free electron
e.g. Hydrogen
R
1
mv2 = hν − hc 2
2
n
R = Rydberg constant = 1.0968 × 105 cm-1
31
b-f and f-f processes
Hydrogen
Transition l  u
Wavelength (A)
Edge
11
912
Lyman
21
3646
Balmer
31
8204
Paschen
41
14584
Brackett
51
22790
Pfund
32
b-f and f-f processes
κb−f
= nl σlk (ν )
ν
- for a single transition
- for all transitions:
f
κb−
ν
=
X
elements,
ions
for hydrogenic ions
for H:
X
nl σlk (ν)
l
σ lk (ν ) = σ0(n)
Gaunt factor ≈ 1
³ ν ´3
l
ν
gbf (ν )
Kramers 1923
Gray, 92
0 = 7.9 × 10-18 n
l = 3.29 × 1015 / n2
extinction per particle 
33
b-f and f-f processes
Gray, 92
late A
σb-f
n (ν)
29
= 2.815 × 10
Z4
g (ν )
n5ν 3 bf
peaks increase with n:
hνn = E∞ − En = 13.6/n2 eV
=⇒ ν
−3
6
→n
late B
for non-H-like atoms no -3
dependence
peaks at resonant frequencies
free-free absorption
much smaller
34
b-f and f-f processes
Hydrogen dominant continuous absorber in B, A & F stars (later stars H-)
Energy distribution strongly modulated at the edges:
Balmer
Vega
Paschen
Brackett
35
b-f and f-f processes : Einstein-Milne relations
Generalize Einstein relations to bound-free processes relating photoionizations and
radiative recombinations
line transitions
σlu(ν) =
hν
ϕ(ν ) Blu
4π
gl Blu = guB ul
2 h ν3
A ul =
Bul
c2
stimulated b-f emission
b-f transitions
κb−f
ν
=
X
elements,
ions
X
i
"
σik(ν ) ni −
nenIon
1
gi
2g1
µ
h2
2πmkT
n∗i
¶3/2
i
e EIon /k T e−hν /kT
#
LTE occupation number
36
2. Scattering
In scattering events photons are not destroyed, but redirected and
perhaps shifted in frequency. In free-free process photon interacts with
electron in the presence of ion’s potential. For scattering there is no
influence of ion’s presence.
in general: sc = ne e
Calculation of cross sections for scattering by free electrons:
- very high energy (several MeV’s): Klein-Nishina formula (Q.E.D.)
- high energy photons (electrons): Compton (inverse Compton) scattering
- low energy (< 12.4 kEV ' 1 Angstrom): Thomson scattering
37
Thomson scattering
THOMSON SCATTERING: important source of opacity in hot OB stars
8π 2
8π e 4
−25
2
σe =
=
6.65
×
10
cm
r0 =
3
3 m2e c4
independent of frequency, isotropic
Approximations:
- angle averaging done, in reality: e  e (1+cos2 ) for single scattering
- neglected velocity distribution and Doppler shift (frequency-dependency)
38
Rayleigh scattering
RAYLEIGH SCATTERING: line absorption/emission of atoms and
molecules far from resonance frequency:  << 0
from classical expression of cross section for oscillators:
σ(ω) = fij σkl (ω) = fij
ω4
σe
(ω 2 − ω2ij )2 + ω 2γ 2
ω4
for ω << ωij
σ(ω) ≈ fij σe 4
ωij + ω 2γ 2
ω4
for γ << ωij
σ(ω) ≈ fij σ e 4
ωij
σ(ω) ∼ ω 4 ∼ λ−4
important in cool G-K stars
for strong lines (e.g. Lyman
series when H is neutral) the
-4 decrease in the far wings
can be important in the
optical
39
What are the dominant elements for the continuum opacity?
- hot stars (B,A,F) : H, He I, He II
- cool stars (G,K): the bound state of the H- ion (1 proton + 2 electrons)
The
H-
ion
only way to explain solar continuum (Wildt 1939)
ionization potential = 0.754 eV
 ion = 16550 Angstroms
H- b-f peaks around 8500 A
H- f-f ~ 3 (IR important)
He- b-f negligible,
He- f-f can be important
in cool stars in IR
requires metals to provide
source of electrons
dominant source of
b-f opacity in the Sun
Gray, 92
40
Additional continuous absorbers
Hydrogen
H2 molecules more numerous than atomic H in M stars
H2+ absorption in UV
H2- f-f absorption in IR
Helium
He- f-f absorption for cool stars
Metals
C, Si, Al, Mg, Fe: b-f absorption in UV
41
Examples of continuous absorption coefficients
Unsoeld, 68
Teff = 5040 K
B0: Teff = 28,000 K
42
Summary
Total opacity
κν =
N
N
X
X
i=1 j= i+1
+
N
X
i=1
µ
¶
g
i
σline
(ν
)
n
−
nj
i
ij
gj
³
σik(ν ) ni − n∗i e −hν/k T
´
³
´
−hν /kT
+nenpσ k k(ν, T ) 1 − e
+ne σe
line absorption
bound-free
free-free
Thomson scattering
43
Summary
Total emissivity
N
N
2hν 3 X X line
gi
²ν = 2
σij (ν ) nj
c
gj
line emission
i=1 j =i+1
N
2hν 3 X ∗
+ 2
ni σ ik (ν)e −hν/k T
c
bound-free
i=1
2hν3
+ 2 nenpσk k(ν, T )e− hν/kT
c
free-free
+neσ e Jν
Thomson scattering
44
Summary
•Modern model atmosphere codes include:
millions of spectral lines
all bf- and ff-transitions of hydrogen helium and metals
contributions of all important negative ions
molecular opacities
45
Concepcion 2007
complex atomic models for O-stars (Pauldrach et al., 2001)
46
NLTE Atomic Models in modern model atmosphere codes
lines, collisions, ionization, recombination
Essential for occupation numbers, line blocking, line force
Accurate atomic models have been included
26
elements
149 ionization stages
5,000 levels ( + 100,000 )
20,000diel. rec. transitions
4 106 b-b line transitions
Auger-ionization
recently improved models are based on Superstructure
Eisner et al., 1974, CPC 8,270
AWAP 05/19/05
47
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Concepcion 2007
Recent Improvements on Atomic Data
• requires solution of Schrödinger equation
for N-electron system
• efficient technique:
R-matrix method in CC approximation
• Opacity Project
Seaton et al. 1994, MNRAS, 266, 805
• IRON Project
Hummer et al. 1993, A&A, 279, 298
accurate radiative/collisional data
to 10% on the mean
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Confrontation with Reality
Photoionization
Nahar 2003, ASP Conf. Proc.Ser. 288, in press
Electron Collision
Williams 1999, Rep. Prog. Phys., 62, 1431
 high-precision atomic data 
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Improved Modelling for Astrophysics
e.g. photoionization cross-sections for carbon model atom
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Przybilla, Butler & Kudritzki 2001b, A&A, 379, 936