5. Atomic radiation processes

Transcription

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 1

2 Atomic transitions absorption hν + E l E u emission spontaneous (isotropic) E u E l + hν stimulated (direction of incoming photon) hν + E u E l +2hν 2

3 A. 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 E l with a resulting transition E l E u per second: dw abs (ν, ω,l,u)=b lu I ν (ω) ϕ(ν)dν dω 4π atomic property no. of incident photons probability ν є (ν,ν+dν) probability ω є (ω,ω+dω) absorption profile probability for transition l u B lu : Einstein coefficient for absorption 3

4 Einstein coefficients similarly for stimulated emission dw st (ν,ω,l,u)=b ul I ν (ω) ϕ(ν)dν dω 4π B ul : Einstein coefficient for stimulated emission and for spontaneous emission dw sp (ν, ω,l,u)=a ul ϕ(ν)dν dω 4π A ul : Einstein coefficient for spontaneous emission 4

5 dw st (ν,ω,l,u)=b ul I ν (ω) ϕ(ν)dν dω 4π dw abs (ν, ω,l,u)=b lu I ν (ω) ϕ(ν)dν dω 4π Einstein coefficients, absorption and emission coefficients I ν df dv = df ds Number of absorptions & stimulated emissions in dv per second: ds n l dw abs dv, n u dw st dv absorbed energy in dv per second: de abs ν = n l hν dw abs dv n u hν dw st dv and also (using definition of intensity): stimulated emission counted as negative absorption de abs ν = κ L ν I ν ds dω dν df κ L ν = hν 4π ϕ(ν) [n lb lu n u B ul ] for the spontaneously emitted energy: Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening ² L ν = hν 4π n u A ul ϕ(ν) 5

6 Relations between Einstein coefficients Einstein coefficients are atomic properties do not depend on thermodynamic state of matter We can assume TE: S ν = ²L ν κ L = B ν (T) ν B ν (T) = n u A ul n l B lu n u B ul = n u n l A ul B lu n u n l B ul From the Boltzmann formula: n u n l = g u e hν kt g l for hν/kt << 1: n u n l = g u g l µ 1 hν, B kt ν (T) = 2ν2 c 2 kt 2ν 2 c 2 kt = g u g l µ 1 hν kt B lu g u g l A ul 1 hν kt Bul for T 0 g l B lu = g u B ul 6

7 2ν 2 c 2 kt = g µ u 1 hν g l kt B lu g u gl A ul 1 hν kt Bul g l B lu = g u B ul Relations between Einstein coefficients 2ν 2 c 2 kt = g µ u 1 hν Aul kt g l kt B lu hν 2 hν 3 c 2 = A ul B lu g u g l µ 1 hν kt for T A ul = 2 h ν 3 c 2 A ul = 2h ν3 c 2 κ L ν = hν 4π ϕ(ν) B lu ² L ν = hν 4π ϕ(ν) n u g l g u B lu B ul n l g l n g u u g l 2hν 3 g u c 2 B lu 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

8 Oscillator strength Quantum mechanics The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation. Eigenvalue problem using using wave function: H atom ψ l >= E l ψ l > H atom = p2 2m + V nucleus + V shell Consider a time-dependent perturbation such as an external electromagnetic field (light wave) E(t) = E 0 e iωt. The potential of the time dependent perturbation on the atom is: NX V (t)=e E r i = E d i=1 [H atom + V (t)] ψ l >= E l ψ l > d: dipol operator with transition probability < ψ l d ψ u > 2 8

9 Oscillator strength The result is hν 4π B lu = πe2 m e c f lu f lu : oscillator strength (dimensionless) classical result from electrodynamics = cm 2 /s Classical electrodynamics electron quasi-elastically bound to nucleus and oscillates within outer electric field as E. Equation of motion (damped harmonic oscillator): ẍ + γẋ + ω 2 0x = e m e E ma = damping force + restoring force + EM force damping constant γ = 2 ω2 0e 2 3 m e c = 8π 2 e 2 ν m e c 3 resonant (natural) frequency ω 0 =2πν 0 the electron oscillates preferentially at resonance (incoming radiation ν = ν 0 ) The damping is caused, because the de- and accelerated electron radiates 9

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: integrated over the line profile Z κ L,cl ν dν = n l πe 2 m e c = n l σ cl tot n l : number density of absorbers σ tot cl : classical cross section (cm 2 /s) ϕ(ν)dν = 1 π γ/4π 2 (ν ν 0 ) 2 +(γ/4π ) 2 [Lorentz (damping) line profile] oscillator strength f lu is quantum mechanical correction to classical result (effective number of classical oscillators, 1 for strong resonance lines) From κ L ν = hν 4π ϕ(ν) n lb lu (neglecting stimulated emission) 10

11 Oscillator strength hν 4π ϕ(ν) B lu = πe2 m e c ϕ(ν) f lu = σ lu (ν) absorption cross section; dimension is cm 2 f lu = 1 4π m e c πe 2 hν B lu 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 f lu = 25 g /2 π l 5 u 3 l 2 u 2 g: Gaunt factor Hα: f= Hβ: f= Hγ: f=

12 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

13 1. Natural damping profile finite lifetime of atomic levels line width NATURAL LINE BROADENING OR RADIATION DAMPING τ = 1 / A ul Δ E τ h/2π ( 10-8 s in H atom 2 1): finite lifetime with respect to spontaneous emission uncertainty principle Δν 1/2 = Γ / 2π line broadening Δλ 1/2 = λ 2 Γ / 2πc e.g. Ly α: Δλ 1/2 = A ϕ(ν)= Γ/4π 2 (ν ν 0 ) 2 +(Γ/4π) 2 Hα: Δλ 1/2 = A Lorentzian profile 13

14 Natural damping profile resonance line Γ u = i<u A ui Γ = A u1 Γ l = i<l A li Γ = Γ u + Γ l 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 14

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/r n (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 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 15

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/r 2 ΔE 1/r 2 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 16

17 Collisional broadening n = 4 quadratic Stark effect ΔE F 2 field strength F 1/r 2 ΔE 1/r 4 (no dipole moment) important for metal ions broadened by e - in hot stars Lorentz profile with γ e ~ n e 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

18 Quasi-static approximation t perturbation >> τ = 1/A ul perturbation practically constant during emission or absorption atom radiates in a statistically fluctuating field produced by quasi-static perturbers, e.g. slowmoving 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 ϕ( ν) dν = W (F ) df dν dν we use the nearest-neighbor approximation: main effect from nearest particle 18

19 Quasi-static approximation nearest neighbor approximation assumption: main effect from nearest particle (F ~ 1/r 2 ) 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 Z r 0 W (x) dx] (4πr 2 N) dr probability for no particle in (0,r) relative probability for particle in shell (r,r+dr) N: total uniform particle density d dr W(r) 4πr 2 = W (r) = 4πr 2 N N W (r) 4πr 2 N W (r)=4πr 2 Ne 4 3 πr3 N 19

20 Quasi-static approximation mean interparticle distance: r 0 = µ 4 1/3 3 π N normal field strength: F 0 = e r 2 0 (linear Stark) define: β = F F 0 from W(r) dr W(β) dβ: note: at high particle density large F 0 stronger broadening W (β)dβ = 3 2 β 5/2 e β 3/2 dβ W (β ) β 5/2 for β ν β ϕ( ν) ν 5/2 Stark broadened line profile in the wings (not Δν -2 ) 20

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 21

22 3. Doppler broadening radiating atoms have thermal velocity Maxwellian distribution: P (v x,v y,v z ) dv x dv y dv z = ³ m 3/2 e m 2kT (v2 x +v2 y +v2 z ) dv x dv y dv z 2πkT Doppler effect: atom with velocity v emitting at frequency ν 0, observed at frequency ν: ν 0 = ν ν v cos θ c 22

23 Doppler broadening Define the velocity component along the line of sight: ξ The Maxwellian distribution for this component is: P (ξ) dξ = 1 π 1/2 ξ 0 e ξ ξ0 2 dξ ξ 0 =(2kT /m) 1/2 thermal velocity ν 0 = ν ν ξ c if we observe at ν, an atom with velocity component ξ absorbs at ν 0 in its frame if v/c <<1 (ν 0 -ν)/ν <<1: line center ν = ν 0 ν = ν ξ c = [(ν ν 0)+ν 0 ] ξ c ν 0 ξ c 23

24 Doppler broadening line profile for v = 0 profile for v 0 ν ν 0 ϕ R (ν ν 0 ) = ϕ R (ν 0 ν 0 )=ϕ(ν ν 0 ν 0 ξ c ) New line profile: convolution ϕ new (ν ν 0 )= Z ξ ϕ(ν ν 0 ν 0 ) P (ξ) dξ c profile function in rest frame velocity distribution function 24

25 Doppler broadening: sharp line approximation ϕ new (ν ν 0 )= 1 π 1/2 Z ϕ(ν ν 0 ν 0 ξ 0 c ξ ) e ξ 2 ξ 0 ξ 0 dξ ξ 0 Δν D : Doppler width of the line Δν D = ν (T/μ) 1/2 Δλ D = λ (T/μ) 1/2 ξ 0 =(2kT/m) 1/2 thermal velocity Approximation 1: assume a sharp line half width of profile function << Δν D ϕ(ν ν 0 ) δ(ν ν 0 ) delta function 25

26 Doppler broadening: sharp line approximation ϕ new (ν ν 0 )= 1 π 1/2 Z δ(ν ν 0 ν D ξ ) e ξ 2 ξ 0 ξ 0 dξ ξ 0 δ(a x)= 1 a δ(x) ϕ new (ν ν 0 )= 1 π 1/2 Z µ ν ν0 δ ξ ν D ξ 0 e ξ ξ0 2 dξ ν D ξ 0 ϕ new (ν ν 0 )= 1 π 1/ 2 ν D e ν ν0 νd 2 Gaussian profile valid in the line core 26

27 Doppler broadening: Voigt function Approximation 2: assume a Lorentzian profile half width of profile function > Δν D ϕ(ν)= 1 π Γ/4π 2 (ν ν 0 ) 2 +(Γ/4π) 2 ϕ new (ν ν 0 )= 1 π 1/2 ν D a π Z e y 2 ³ 2 ν ν D y + a 2 dy a = Γ 4π ν D ϕ new (ν ν 0 )= 1 π 1/2 ν D H µ a, ν ν 0 ν D Voigt function (Lorentzian * Gaussian): calculated numerically 27

28 Doppler broadening: Voigt function H Approximate representation of Voigt function: ³ a, ν ν 0 ν D e ν ν0 νd 2 line core: Doppler broadening a π 1/2 ν ν0 νd 2 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 28

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 29

30 B. Continuous transitions 1. Bound-free and free-free processes absorption hν + E l E u photoionization - recombination hν lk emission spontaneous (isotropic) stimulated (non-isotropic) E u E l + hν hν + E u E l +2hν bound-bound: spectral lines bound-free κ cont free-free ν consider photon hν hν lk (energy > threshold): extra-energy to free electron e.g. Hydrogen 1 2 mv2 = hν hc R n 2 R = Rydberg constant = cm -1 30

31 b-f and f-f processes Hydrogen Transition l u Wavelength (A) Edge Lyman Balmer Paschen Brackett Pfund 31

32 b-f and f-f processes - for a single transition κ b f ν = n l σ lk (ν) - for all transitions: κ b f ν = X X n l σ lk (ν) elements, ions l Gaunt factor 1 for hydrogenic ions for H: σ 0 = n σ lk (ν) =σ 0 (n) ³ νl 3 gbf (ν) ν Kramers 1923 Gray, 92 ν l = / n 2 extinction per particle 32

33 b-f and f-f processes Gray, 92 late A σ b-f n (ν) = Z 4 n 5 ν 3 g bf (ν) peaks increase with n: hν n = E E n =13.6/n 2 ev = ν 3 n 6 late B for non-h-like atoms no ν -3 dependence peaks at resonant frequencies free-free absorption much smaller 33

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 Paschen Vega Brackett 34

35 b-f and f-f processes : Einstein-Milne relations Generalize Einstein relations to bound-free processes relating photoionizations and radiative recombinations line transitions σ lu (ν) = hν 4π ϕ(ν) B lu g l B lu = g u B ul A ul = 2h ν3 c 2 B ul b-f transitions κ b f ν = X elements, ions " X σ ik (ν) n i n e n Ion 1 i g i 2g 1 stimulated b-f emission µ h 2 2πmkT n i 3/2 e Ei Ion /kt e hν /kt # LTE occupation number 35

36 b-f and f-f processes : Einstein-Milne relations spontaneous b-f emission ² b f ν = X elements, ions X i σ ik (ν) 2hν3 c 2 n i e hν/kt T enters these expressions through the Maxwellian velocity distribution and Saha s formula for ionization FREE-FREE PROCESSES stimulated f-f emission κ f f ν = n e n Ion 1 σ kk (ν) h1 i e hν/kt ² f f ν = n e n Ion 1 σ kk (ν) 2hν3 c 2 hν /kt e σ kk λ 3 : important in IR and Radio 36

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. in general: κ sc = n e σ 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

38 Thomson scattering THOMSON SCATTERING: important source of opacity in hot OB stars σ e = 8π 3 r2 0 = 8π 3 e 4 m 2 e c4 = cm 2 independent of frequency, isotropic Approximations: - angle averaging done, in reality: σ e σ e (1+cos 2 φ) for single scattering - neglected velocity distribution and Doppler shift (frequency-dependency) 38

39 Rayleigh scattering RAYLEIGH SCATTERING: line absorption/emission of atoms and molecules far from resonance frequency: ν << ν 0 from classical expression of cross section for oscillators: σ(ω) =f ij σ kl (ω)=f ij σ e ω 4 (ω 2 ω 2 ij )2 + ω 2 γ 2 ω 4 for ω << ω ij σ(ω) f ij σ e ω 4 ij + ω2 γ 2 ω 4 for γ << ω ij σ(ω) f ij σ e σ(ω) ω 4 λ 4 ω 4 ij 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 39

40 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 = ev λ ion = Angstroms H - b-f peaks around 8500 A H - f-f ~ λ 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

41 Additional continuous absorbers Hydrogen H 2 molecules more numerous than atomic H in M stars H 2+ absorption in UV H 2- 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

42 Examples of continuous absorption coefficients Unsoeld, 68 T eff = 5040 K B0: T eff = 28,000 K 42

43 Summary Total opacity κ ν = NX i=1 NX j=i+1 σ line ij µ (ν) n i g i n j g j line absorption + NX i=1 σ ik (ν) ³n i n ie hν/kt +n e n p σ kk (ν,t) +n e σ e hν /kt ³1 e bound-free free-free Thomson scattering 43

44 Summary Total emissivity ² ν = 2hν 3 c 2 + 2hν 3 c 2 NX NX i=1 j=i+1 σij line (ν) g i n j g j NX n iσ ik (ν)e hν/kt i=1 line emission bound-free + 2hν3 c 2 +n e σ e J ν n e n p σ kk (ν,t)e hν/kt free-free Thomson scattering 44

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 45