5. Atomic radiation processes
advertisement
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 hν + El Æ Eu emission spontaneous Eu Æ El + hν stimulated hν + Eu Æ El +2hν (isotropic) (direction of incoming photon) 2 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 El with a resulting transition El Æ 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 ν є (ν,ν+dν) probability ω є (ω,ω+dω) absorption profile probability for transition l Æ 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 Æ 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: 2ν2 gu kT = c2 gl for T Æ ∞ 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 2 hν3 A ul gu = c2 B lu gl for T Æ ∞ 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 ω0=2πν0 e E me 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 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: integrated over the line profile Z πe2 L,cl = nl σ cl κν dν = nl tot me c 1 γ/4π 2 ϕ(ν )dν = 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 absorption cross section; dimension is cm2 hν πe 2 ϕ(ν) B lu = ϕ(ν) flu = σlu (ν ) 4π mec flu = 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 Hα: f=0.6407 Hβ: f=0.1193 Hγ: 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 Æ line width NATURAL LINE BROADENING OR RADIATION DAMPING τ = 1 / Aul Δ E τ ≥ h/2π (≈ 10-8 s in H atom 2 Æ 1): finite lifetime with respect to spontaneous emission uncertainty principle Δν1/2 = Γ / 2π Δλ1/2 = λ2 Γ / 2πc line broadening Γ/4π2 ϕ(ν ) = (ν − ν0)2 + (Γ/4π)2 Lorentzian profile e.g. Ly α: Δλ1/2 = 1.2 × 10-4 A Hα: Δλ1/2 = 4.6 × 10-4 A 13 Natural damping profile resonance line Γu = ∑i<u Aui Γ = Au1 Γl = ∑i<l Ali Γ = Γ u + 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 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 Æ line shifts, asymmetries & broadening ΔE(t) = h Δν = 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 Æ lifetime shortened Æ 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 Æ consider stationary distribution of perturbers 15 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 Æ 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 16 Collisional broadening n=4 quadratic Stark effect ΔE ∼ F2 field strength F ∼ 1/r2 Æ ΔE ∼ 1/r4 (no dipole moment) important for metal ions broadened by e- in hot stars Æ Lorentz profile with γe ~ 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 Æ 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 Æ field at location of absorbing or emitting atom statistical frequency of particle distribution Æ probability of fields of different strength (each producing an energy shift ΔE = h Δν) Æ field strength distribution function Æ 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 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 W(r) dr = [1 − W (x) dx] (4πr 2N) dr 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 19 Quasi-static approximation mean interparticle distance: r0 = normal field strength: F0 = define: β= from W(r) dr Æ W(β) dβ: W (β)dβ = W (β ) ∼ β −5/2 for β → ∞ µ 4 πN 3 ¶−1/3 e r 20 (linear Stark) F F0 note: at high particle density Æ large F0 Æ stronger broadening 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 Δν-2) 20 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 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 ν0, observed at frequency ν: v cos θ ν0 = ν − ν c 22 Doppler broadening Define the velocity component along the line of sight: ξ The Maxwellian distribution for this component is: P (ξ) dξ = 1 π 1/2ξ 0 e ¡ ξ ¢2 − ξ0 dξ ξ0 = (2kT /m)1/2 ν0 = ν − ν if v/c <<1 Æ (ν0 -ν)/ν <<1: ∆ν = ν0 − ν = −ν ξ c thermal velocity if we observe at ν, an atom with velocity component ξ absorbs at ν0 in its frame line center ξ ξ ξ = −[(ν − ν0) + ν0 ] ≈ −ν0 c c c 23 Doppler broadening line profile for v = 0 ϕ R (ν − ν0 ) Æ =⇒ profile for v ≠ 0 ν Æ ν0 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 24 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 ΔνD: Doppler width of the line ΔνD = 4.301 × 10-7 ν (T/μ)1/2 ΔλD = 4.301 × 10-7 λ (T/μ)1/2 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 25 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 26 Doppler broadening: Voigt function Approximation 2: assume a Lorentzian profile – half width of profile function > ΔνD 1 Γ/4π 2 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 27 Doppler broadening: Voigt function Approximate representation of Voigt function: ³ H a, ν −ν0 ∆νD ´ ≈e ≈ π ¡ ν−ν 0 ¢2 − ∆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 28 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 B. Continuous transitions 1. Bound-free and free-free processes absorption hν + El Æ Eu photoionization - recombination emission hνlk spontaneous Eu Æ El + hν stimulated hν + Eu Æ El +2hν (isotropic) (non-isotropic) bound-bound: spectral lines bound-free free-free consider photon hν ≥ hνlk κνcont (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 30 b-f and f-f processes Hydrogen Transition l Æ u Wavelength (A) Edge 1Æ∞ 912 Lyman 2Æ∞ 3646 Balmer 3Æ∞ 8204 Paschen 4Æ∞ 14584 Brackett 5Æ∞ 22790 Pfund 31 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 l σ lk (ν ) = σ 0(n) σ0 = 7.9 × 10-18 n nl σlk (ν) Gaunt factor ≈ 1 ³ ν ´3 l ν gbf (ν ) Kramers 1923 Gray, 92 νl = 3.29 × 1015 / n2 extinction per particle Æ 32 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 ν-3 dependence peaks at resonant frequencies free-free absorption much smaller 33 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 34 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 35 b-f and f-f processes : Einstein-Milne relations ²b−f ν spontaneous b-f emission = X elements, ions X i 2hν3 ∗ − hν/kT σik (ν ) 2 ni e c T enters these expressions through the Maxwellian velocity distribution and Saha’s formula for ionization FREE-FREE PROCESSES κνf−f ²νf−f stimulated f-f emission = ne nIon 1 = ne nIon 1 h i −hν/k T σk k (ν) 1 − e 2hν3 −hν /kT σk k (ν) 2 e c σkk ∼ λ3: important in IR and Radio 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: κsc = ne σe 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: σe Æ σe (1+cos2 φ) 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: ν << ν0 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 λ-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 Æ 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 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
