1

5. Atomic radiation processes

Einstein coefficients for absorption and emission

oscillator strength

line profiles: damping profile, collisional broadening, Doppler broadening

continuous absorption and scattering

2

Atomic transitions

absorption

hν + El Eu

emissionspontaneous Eu El + hν(isotropic)

stimulated hν + Eu El +2hν(direction of incoming photon)

3

A. Line transitionsA. Line transitions

Einstein coefficients

probability that a photon in frequency interval (ν, ν+dν) in the solid angle range (ω,ω+dω) is absorbed by an atom in the energy level El with a resulting transition El Eu per second:

atomic property ∝ no. of incident photons

probability ν є(ν,ν+dν)

absorption profile

probability ω є(ω,ω+dω)

probability for transition l u

Blu: Einstein coefficient for absorption

dwabs(ν, ω, l,u) =Blu Iν(ω) ϕ(ν)dν dω4π

4

Einstein coefficients

similarly for stimulated emission

Bul: Einstein coefficient for stimulated emission

and for spontaneous emission

Aul: Einstein coefficient for spontaneous emission

dwst(ν,ω, l ,u) = Bul Iν(ω) ϕ(ν)dν dω4π

dwsp(ν, ω, l,u) =Aul ϕ(ν)dν dω4π

5

Einstein coefficients, absorption and emission coefficients

Iν dF

ds

dV = dF ds

absorbed energy in dV per second:

dEabsν = nl hν dwabs dV − nu hν dwst dV

and also (using definition of intensity):

for the spontaneously emitted energy:

κLν =hν

4πϕ(ν) [nlBlu − nuBul ]

²Lν =hν

4πnu Aul ϕ(ν)

dwst(ν,ω, l ,u) = Bul Iν(ω) ϕ(ν)dν dω4π dwabs(ν, ω, l,u) =Blu Iν(ω) ϕ(ν)dν dω

4π

dEabsν = κLν Iν ds dω dν dF

Number of absorptions & stimulated emissions in dV per second:

nu dwst dVnl dw

abs dV,

stimulated emission counted as negative absorption

Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening

6

Einstein coefficients are atomic properties do not depend on thermodynamic state of matter

We can assume TE: Sν =²LνκLν

= Bν(T)

Bν(T) =nuAul

nlBlu− nuBul=nunl

AulBlu− nu

nlBul

From the Boltzmann formula:

for hν/kT << 1:nunl=gugl

µ1− hν

kT

¶, Bν(T) =

2ν2

c2kT

2ν2

c2kT =

gugl

µ1− hν

kT

¶Aul

Blu− gugl

¡1 − hν

kT

¢Bul

for T ∞ 0 glBlu = guBul

Relations between Einstein coefficients

nu

nl=gu

gle−

hνkT

7

Relations between Einstein coefficients

2 hν3

c2=AulBlu

gugl

µ1− hν

kT

¶

for T ∞ Aul =2 hν3

c2glguBlu

Aul =2h ν3

c2Bul

κLν =hν

4πϕ(ν)Blu

∙nl −

glgunu

¸

²Lν =hν

4πϕ(ν) nu

glgu

2hν3

c2Blu

only one Einstein coefficient needed

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.

glBlu = guBul2ν2

c2kT =

gugl

µ1− hν

kT

¶Aul

Blu− gugl

¡1 − hν

kT

¢Bul

2ν2

c2kT =

gugl

µ1− hν

kT

¶AulBlu

kT

hν

8

Oscillator strength

Quantum mechanics

The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation.

Eigenvalue problem using using wave function:

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:

Hatom |ψl >= El |ψl > Hatom =p2

2m+ Vnucleus + Vshell

V (t) = eNXi=1

E · ri = E · d d: dipol operator

[Hatom + V (t)] |ψl >= El |ψl >

with transition probability ∼ | < ψl|d|ψu > |2

9

Oscillator strength

flu: oscillator strength (dimensionless)

hν

4πBlu =

πe2

mecflu

classical result from electrodynamics

= 0.02654 cm2/s

The result is

Classical electrodynamics

electron quasi-elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion (damped harmonic oscillator):

x+ γx+ ω20x=e

meE

damping constant resonant

(natural) frequency

ω0=2πν0

ma = damping force + restoring force + EM force

the electron oscillates preferentially at resonance (incoming radiation ν = ν0)

The damping is caused, because the de- and accelerated electron radiates

γ =2 ω20e

2

3mec3=8π 2e2 ν203mec3

10

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:

nl: number density of absorbers

integrated over the line profile

oscillator strength flu is quantum mechanical correction to classical result (effective number of classical oscillators, ≈ 1 for strong resonance lines)

From (neglecting stimulated emission)κLν =hν

4πϕ(ν) nlBlu

σtotcl: classical

cross section (cm2/s)

ZκL,clν dν = nl

πe2

mec= nl σcltot

ϕ(ν)dν =1

π

γ/4π2

(ν − ν0)2 + (γ/4π )2[Lorentz (damping) line profile]

11

Oscillator strength

hν

4πϕ(ν) Blu =

πe2

mecϕ(ν) flu = σlu(ν)

absorption cross section; dimension is cm2

flu =1

4π

mec

πe2hν Blu

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

33/2πg

l5u3

¡1l2− 1

u2

¢−3g: Gaunt factor

Hα: f=0.6407

Hβ: f=0.1193

Hγ: f=0.0447

12

Line profilesLine 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

13

1. Natural damping profile

finite lifetime of atomic levels line width

NATURAL LINE BROADENING OR RADIATION DAMPING

τ = 1 / Aul (≈ 10-8 s in H atom 2 1): finite lifetime with respect to spontaneous emission

Δ E τ ≥ h/2π uncertainty principle

line broadening

ϕ(ν) =Γ/4π2

(ν − ν0)2 + (Γ/4π)2Lorentzian profile

e.g. Ly α: Δλ1/2 = 1.2 × 10-4 A

Hα: Δλ1/2 = 4.6 × 10-4 A

Δν1/2 = Γ / 2π

Δλ1/2 = λ2 Γ / 2πc

14

Natural damping profile

resonance line

Γ = Au1

Γu = ∑i<u Aui

Γl = ∑i<l Ali

Γ = Γu + Γl

natural line broadening is important for strong lines (resonance lines) at low densities (no additional broadening mechanisms)

e.g. Ly α in interstellar medium

but also in stellar atmospheres

15

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

ΔE(t) = h Δν = C/rn(t)

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 dispersion profile is obtained (=natural damping profile)

b) quasi-static approximation: applied when duration of collisions >> life time in level consider stationary distribution of perturbers

r: distance to perturbing atom

16

Collisional broadening – impact approximation

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 broadeningatom A perturbed by atom A’ of same species

important in cool stars, e.g. Balmer lines in the Sun

17

Collisional broadening

n = 4 quadratic Stark effect ΔE ∼ F2

field strength F ∼ 1/r2

ΔE ∼ 1/r4 (no dipole moment)

important for metal ions broadened by e- in hot stars Lorentz profile with γe ~ ne

n = 6 van der Waals broadeningatom A perturbed by atom B

important in cool stars, e.g. Na perturbed by H in the Sun

18

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 calculated in this way

W(F) dF: probability that field seen by radiating atom is F

we use the nearest-neighbor approximation: main effect from nearest particle

ϕ(∆ν) dν =W (F )dF

dνdν

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)

W(r) dr = [1−rZ0

W (x) dx] (4πr2N) dr

relative probability for particle in shell (r,r+dr) N: total uniform particle densityprobability for no particle in (0,r)

d

dr

∙W(r)

4πr2N

¸= −W (r) =−4πr2N

∙W (r)

4πr2N

¸

W (r) = 4πr2Ne−43 πr

3N

20

Quasi-static approximation

mean interparticle distance:

normal field strength: (linear Stark)

define:

from W(r) dr W(β) dβ:

r0 =

µ4

3πN

¶−1/3

F0 =e

r20

β =F

F0

W (β)dβ =3

2β−5/2e−β

−3/2dβ

W (β )∼ β−5/2 for β →∞ ∆ν ∼ β → ϕ(∆ν) ∼∆ν−5/2

Stark broadened line profile in the wings (not Δν-2)

note: at high particle density large F0

stronger broadening

21

Quasi-static approximation

complete treatment of an ensemble of particles: Holtsmark theory

+ interaction among perturbers (Debye shielding of the potential at distances > Debye length)

number of particles inside Debye sphere

Mihalas, 78

22

3. Doppler broadening

radiating atoms have thermal velocity

Maxwellian distribution:

Doppler effect: atom with velocity v emitting at frequency ν0, observed at frequency ν:

ν0 = ν − ν v cos θc

P (vx, vy , vz ) dvxdvydvz =³ m

2πkT

´3/2e−

m2kT (v

2x+v

2y+v

2z) dvxdvydvz

23

Doppler broadening

Define the velocity component along the line of sight: ξ

The Maxwellian distribution for this component is:

P (ξ) dξ =1

π1/2ξ0e−¡ξξ0

¢2dξ

ξ0 = (2kT/m)1/2 thermal velocity

ν0 = ν − ν ξc

if v/c <<1 (ν0 -ν)/ν <<1: line center

∆ν = ν0 − ν = −νξc= −[(ν − ν0) + ν0 ]

ξ

c≈ −ν0 ξ

c

if we observe at ν, an atom with velocity component ξ absorbs at ν0 in its frame

24

Doppler broadening

line profile for v = 0 profile for v ≠ 0 ν ν0

ϕR(ν − ν0) =⇒ ϕR(ν0 − ν0) = ϕ(ν − ν0− ν0ξ

c)

New line profile: convolution

ϕnew(ν − ν0) =∞Z−∞

ϕ(ν − ν0 − ν0ξc) P (ξ) dξ

profile function in rest frame

velocity distribution function

25

Doppler broadening: sharp line approximation

ΔνD: Doppler width of the line

ϕnew(ν − ν0) = 1

π1/2

∞Z−∞

ϕ(ν − ν0− ν0 ξ0c

ξ

ξ0) e−

¡ξξ0

¢2 dξξ0

ξ0 = (2kT/m)1/2 thermal velocity

Approximation 1: assume a sharp line – half width of profile function << ΔνD

ϕ(ν − ν0) ≈ δ(ν − ν0) delta function

ΔνD = 4.301 × 10-7 ν (T/μ)1/2

ΔλD = 4.301 × 10-7 λ (T/μ)1/2

26

Doppler broadening: sharp line approximation

ϕnew (ν − ν0) = 1

π1/2

∞Z−∞

δ(ν − ν0−∆νD ξ

ξ0) e−

¡ξξ0

¢2 dξξ0

δ(a x) =1

aδ(x)

ϕnew(ν − ν0) = 1

π1/2

∞Z−∞

δ

µν − ν0∆νD

− ξ

ξ0

¶e−¡ξξ0

¢2 dξ

∆νD ξ0

ϕnew(ν − ν0) = 1

π1/2 ∆νDe−¡ν−ν0∆νD

¢2Gaussian profile –valid in the line core

27

Doppler broadening: Voigt function

Approximation 2: assume a Lorentzian profile – half width of profile function > ΔνD

ϕnew (ν− ν0) = 1

π1/2 ∆νD

a

π

∞Z−∞

e−y2³

∆ν∆νD− y

´2+ a2

dy a =Γ

4π∆νD

ϕnew(ν− ν0) =1

π1/2 ∆νDH

µa,ν− ν0∆νD

¶

Voigt function (Lorentzian * Gaussian): calculated numerically

ϕ(ν) =1

π

Γ/4π2

(ν − ν0)2 + (Γ/4π)2

28

Doppler broadening: Voigt function

Approximate representation of Voigt function:

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

H³a, ν−ν0∆νD

´≈ e−

¡ν−ν0∆νD

¢2≈ a

π1/2¡ν−ν0∆νD

¢2

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 transitionsB. Continuous transitions

1. Bound-free and free-free processes

hνlk

consider photon hν ≥ hνlk (energy > threshold): extra-energy to free electron

e.g. HydrogenR = Rydberg constant = 1.0968 × 105 cm-1

absorption

hν + El Eu

emissionspontaneous Eu El + hν(isotropic)

stimulated hν + Eu El +2hν(non-isotropic)

ph

otoi

oniz

atio

n-

reco

mbi

nat

ion

bound-bound: spectral lines

bound-free

free-freeκνcont

1

2mv2 = hν− hc R

n2

31

b-f and f-f processes

Transition l u Wavelength (A) Edge1 ∞ 912 Lyman

2 ∞ 3646 Balmer

3 ∞ 8204 Paschen

4 ∞ 14584 Brackett

5 ∞ 22790 Pfund

Hydrogen

32

b-f and f-f processes

for hydrogenic ions

Gaunt factor ≈ 1

κb− fν =X

elements,ions

Xl

nl σlk(ν)

for H: σ0 = 7.9 × 10-18 n

νl = 3.29 × 1015 / n2

- for a single transition

- for all transitions:

κb−fν = nl σlk(ν)

Kramers 1923

extinction per particle

σlk(ν) = σ0(n)³νlν

´3gbf (ν)

Gray, 92

33

b-f and f-f processes

for non-H-like atoms no ν-3

dependence

peaks at resonant frequencies

free-free absorption much smaller

peaks increase with n:

σb-fn (ν) = 2.815× 1029Z 4

n5ν3gbf (ν)

late A

late B

hνn = E∞ − En = 13.6/n2eV

=⇒ ν−3 → n

6

Gray, 92

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:

Vega

Balmer

Paschen

Brackett

35

b-f and f-f processes : Einstein-Milne relations

Generalize Einstein relations to bound-free processes relating photoionizations and radiative recombinations

σlu(ν) =hν

4πϕ(ν) Bluline transitions glBlu = guBul

Aul =2h ν3

c2Bul

b-f transitions

κb−fν =X

elements,ions

Xi

σik(ν)

"ni − nenIon1

gi2g1

µh2

2πmkT

¶3/2eE

iIon /kT e−hν/kT

#

n∗i LTE occupation number

stimulated b-f emission

36

b-f and f-f processes : Einstein-Milne relations

²b−fν =X

elements,ions

Xi

σik(ν)2hν3

c2n∗i e−hν/kTspontaneous b-f emission

FREE-FREE PROCESSES

κf−fν = ne nIon1 σkk(ν)

h1 −e−hν/kT

istimulated f-f emission

²f−fν = ne nIon1 σkk(ν)

2hν3

c2e−hν/kT

σkk ∼ λ3: important in IR and Radio

T enters these expressions through the Maxwellian velocity distribution and Saha’sformula for ionization

37

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.

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

in general: κsc = ne σe

38

Thomson scattering

THOMSON SCATTERING: important source of opacity in hot OB stars

independent of frequency, isotropic

Approximations:

- angle averaging done, in reality: σe σe (1+cos2 φ) for single scattering

- neglected velocity distribution and Doppler shift (frequency-dependency)

σe =8π

3r20 =

8π

3

e4

m2e c4= 6.65× 10−25cm2

39

Rayleigh scattering

RAYLEIGH SCATTERING: line absorption/emission of atoms and molecules far from resonance frequency: ν << ν0

from classical expression of cross section for oscillators:

σ(ω) = fijσkl(ω) = fij σeω4

(ω2 − ω2ij)2+ ω2γ2

for ω << ωij σ(ω) ≈ fij σe ω4

ω4ij + ω2γ2

for γ << ωij σ(ω) ≈ fij σe ω4

ω4ij

σ(ω) ∼ ω4 ∼ λ−4

important in cool G-K stars

for strong lines (e.g. Lyman series when H is neutral) the λ-4 decrease in the far wings can be important in the optical

40

The H- ion

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)

only way to explain solar continuum (Wildt 1939)

ionization potential = 0.754 eV

λion = 16550 Angstroms

H- b-f peaks around 8500 A

H- f-f ~ λ3 (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

41

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

42

Examples of continuous absorption coefficients

Teff = 5040 K B0: Teff = 28,000 K

Unsoeld, 68

43

Summary

Total opacity

κν =NXi=1

NXj= i+1

σlineij (ν)

µni − gi

gjnj

¶

+NXi=1

σik(ν)³ni − n∗ie−hν/kT

´

+nenpσkk(ν, T )³1 − e−hν/kT

´+neσe

line absorption

bound-free

free-free

Thomson scattering

44

Summary

Total emissivity

line emission

bound-free

free-free

Thomson scattering

²ν =2hν3

c2

NXi=1

NXj=i+1

σ lineij (ν)gigjnj

+2hν3

c2

NXi=1

n∗iσik(ν)e−hν/kT

+2hν3

c2nenpσkk(ν, T)e

−hν/kT

+neσe Jν

45

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