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5. Atomic Radiation Processes 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 A.A. LineLine transitionstransitions Einstein coefficients probability that a photon in frequency interval in the solid angle range is absorbed by an atom in the energy level El with a resulting transition El Eu per second: abs dω dw (ν, ω,l,u)=Blu Iν(ω) ϕ(ν)dν 4π atomic property ~ no. of incident probability for probability for photons absorption of with photon with probability for absorption transition l u profile Blu : Einstein coefficient for absorption 3 Einstein coefficients similarly for stimulated emission st dω dw (ν,ω,l,u)=Bul Iν (ω) ϕ(ν)dν 4π Bul : Einstein coefficient for stimulated emission and for spontaneous emission sp dω dw (ν, ω,l,u)=Aul ϕ(ν)dν 4π Aul : Einstein coefficient for spontaneous emission 4 st dω abs dω dw (ν,ω,l,u)=Bul Iν (ω) ϕ(ν)dν 4π dw (ν, ω,l,u)=Blu Iν(ω) ϕ(ν)dν 4π Einstein coefficients, absorption and emission coefficients Number of absorptions & stimulated emissions dF dV = dF ds Iν in dV per second: abs st ds nl dw dV, nu dw dV absorbed energy in dV per second: abs abs st stimulated emission counted dE = nl hν dw dV nu hν dw dV ν − as negative absorption and also (using definition of intensity): abs L dEν = κν Iν ds dω dν dF Absorption and emission coefficients are a function of Einstein coefficients, occupation numbers and line broadening hν κL = ϕ(ν)[n B n B ] ν 4π l lu − u ul L hν for the spontaneously emitted energy: ²ν = nu Aul ϕ(ν) 4π 5 L hν L hν κ = ϕ(ν)[n B n B ] ²ν = nu Aul ϕ(ν) ν 4π l lu − u ul 4π Relations between Einstein coefficients Einstein coefficients are atomic properties do not depend on thermodynamic state of matter L ²ν We can assume TE: Sν = L = Bν (T) κν n A n A B (T)= u ul = u ul ν nu nlBlu nuBul nl Blu Bul − − nl nu gu hν From the Boltzmann formula: = e− kT nl gl 2 nu gu hν 2ν for hν/kT << 1: = 1 ,Bν (T)= kT n g − kT c2 l l µ ¶ 2 2ν gu hν Aul kT = 1 2 gu hν c gl − kT Blu 1 Bul µ ¶ − gl − kT ¡ ¢ glBlu = guBul for T ∞ 0 6 2 2ν gu hν Aul kT = 1 2 gu hν glBlu = guBul c gl − kT Blu 1 Bul µ ¶ − gl − kT ¡ ¢ Relations between Einstein coefficients 2 2ν gu hν Aul kT kT = 1 c2 g − kT B hν l µ ¶ lu 3 2 hν Aul gu hν 2 = 1 c Blu gl − kT for T ∞ µ ¶ 3 Note: Einstein coefficients atomic 2 hν gl quantities. That means any Aul = 2 Blu c gu relationship that holds in a special thermodynamic situation (such as T 2h ν3 A = B very large) must be generally valid. ul c2 ul hν g κL = ϕ(ν) B n l n ν 4π lu l − g u · u ¸ only one Einstein 3 L hν gl 2hν coefficient needed ²ν = ϕ(ν) nu 2 Blu 4π gu c 7 Oscillator strength Quantum mechanics The Einstein coefficients can be calculated by quantum mechanics + classical electrodynamics calculation. Eigenvalue problem using using wave function: p2 Hat om ψl >= El ψl >Hat om = + Vnucleus + Vshell | | 2m Consider a time-dependent perturbation such as an external iωt electromagnetic field (light wave) E(t) = E0 e . The potential of the time dependent perturbation on the atom is: N V (t)=e E ri = E d d: dipol operator · · i=1 X [Hat om + V (t)] ψl >= El ψl > | | 2 with transition probability < ψl d ψu > 8 ∼ | | | | Oscillator strength hν πe2 The result is Blu = flu 4π mec 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): 2 e x¨ + γx˙ + ω0 x = E ma = damping force + restoring force + EM force me the electron oscillates preferentially at resonance damping (incoming radiation ν = ν0) constant resonant The damping is caused, because the de- and 2 2 2 2 2 (natural) accelerated electron radiates 2 ω0 e 8π e ν0 frequency γ = 3 = 3 3 mec 3 me c 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: 2 n : number σ cl: classical L,cl πe cl l tot integrated over the line profile density of cross section κν dν = nl = nl σtot m c absorbers (cm2/s) Z e 1 γ/4π ϕ(ν)dν = [Lorentz (damping) line profile] π (ν ν )2 +(γ/4π)2 − 0 oscillator strength flu is quantum mechanical correction to classical result (effective number of classical oscillators, ≈ 1 for strong resonance lines) L hν From κ = ϕ ( ν ) n l B lu (neglecting stimulated emission) ν 4π 10 Oscillator strength hν πe2 absorption cross section; ϕ(ν) Blu = ϕ(ν) flu = σlu(ν) dimension is cm2 4π mec 1 mec flu = hν Blu 4π πe2 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 5 3 2 g 1 1 − f = 3/2 5 3 2 2 : Gaunt factor lu 3 π l u l − u g ¡ ¢ Hα: f=0.6407 Hβ: f=0.1193 Hγ: f=0.0447 11 LineLine profilesprofiles 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 -8 t = 1 / Aul (¼ 10 s in H atom 2 1): finite lifetime with respect to spontaneous emission Δ E t ≥ h/2π uncertainty principle Δν1/2 = Γ / 2π 2 Δλ1/2 = Δν1/2λ /c line broadening -4 e.g. Ly α: Δλ1/2 = 1.2 10 A -4 1 Γ/4π Hα: Δλ1/2 = 4.6 10 A ϕ(ν)= 2 2 π (ν ν0) +(Γ/4π) − Lorentzian profile 13 Natural damping profile resonance line excited line 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 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 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) Γe ~ ne n = 3 resonance broadening atom A perturbed by atom A’ of same species important in cool stars, e.g. Balmer lines in the Sun Γ ~ n ΔE ~ 1/r3 e e 16 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 broadening atom A perturbed by atom B important in cool stars, e.g. Na perturbed by H in the Sun Γe ~ ne 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) Integral equation for W(r) differentiating differential equation relative probability for particle in shell probability for no particle in (0,r) (r,r+dr) N: total uniform particle density Differential equation Normalized solution 20 Linear Stark effect: Quasi-static approximation mean interparticle distance: normal field strength: F r 2 note: at high particle density large F define: β = = 0 0 F0 { r } stronger broadening from W(r) dr W(β) dβ: 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), 2β ∞ y3/2 W (β)= e− ysin(βy)dy Chandrasekhar (1943, Phys.
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