Lecture Notes on Radiation Transport for Spectroscopy

Transcription

1 Lecture Notes on Radiation Transport for Spectroscopy ICTP-IAEA School on Atomic Processes in Plasmas 27 February 3 March 217 Trieste, Italy LLNL-PRES This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-7NA Lawrence Livermore National Security, LLC

2 Radiation Transport Outline Definitions, assumptions and terminology Equilibrium limit Radiation transport equation Characteristic form & formal solution Material radiative properties Absorption, emission, scattering Coupled systems LTE / non-lte Line radiation Line shapes Redistribution Material motion Solution methods Transport operators 2-level & multi-level atoms Escape factors

3 Basic assumptions Classical / Semi-classical description Radiation field described by either specific intensity I ν or the photon distribution function f ν Unpolarized radiation Neglect index of refraction effects (n 1, ω>>ω p ) è Photons travel in straight lines Neglect true scattering (mostly) Static material (for now) è Single reference frame

4 Radiation Description Macroscopic - specific intensity I ν energy per (area x solid angle x time) within the frequency range (ν,ν + dν) de = I v ( r,ω,t )( n Ω )dadωdv dt de = energy crossing area da within (dωdv dt) Microscopic - photon distribution function de =!! ( hv) f ν r, p,t spins I v = 2 hv3!! f c 2 ν ( r, p,t ) ( ) d 3! xd 3! p h 3 p = hv Ω c

5 Angular Moments th moment J ν = 1 4π I dω ν = energy density x c/4π 1 st moment H ν = 1 4π ni dω ν = flux x 1 /4π 2 nd moment K ν = 1 4π nni dω ν = pressure tensor x c /4π For isotropic radiation, K ν is diagonal with equal elements: K ν = 1 3 J ν I (P = 1 3 E) In this case, radiation looks like an ideal gas with = 4/3 γ

6 Thermal Equilibrium Intensity: Planck function Distribution function: Bose-Einstein B v = 2 hv3 c 2 1 e hv kt r 1 Energy density E rad ( T r ) = 4π c ( ) 4 B v T r dv = at r For T e =T r and n H = 1 23 cm -3 : f ν = 1 e hv kt r 1 E rad = E matter T 3eV LTE (Local Thermodynamic Equilibrium) - par8cles have thermal distribu8ons (T e, T i ) photon distribu8on can be arbitrary

7 Radiation Transport Equation 1 I v c t + Ω! i I v = α v I v + η v α v η v = absorption coefficient (fraction of energy absorbed per unit length) = emissivity (energy emitted per unit time, volume, frequency, solid angle) Boltzmann equation for the photon distribution function: 1 f ν c t + Ω! i f ν = f ν t coll I v = 2hν hν c 2 f ν The LHS describes the flow of radiation in phase space The RHS describes absorption and emission Absorption & emission coefficients depend on atomic physics Photon # is not conserved (except for scattering) Photon mean free path λ ν = 1/α ν

8 Characteristic Form Define the source function S ν and optical depth τ ν : S ν = η ν /α ν = B ν in LTE dτ ν = α ν ds Along a characteristic, the radiation transport equation becomes di v = I dτ v + S v I v ( τ v ) = I v v ( )e τ v + τ v ( e τ v τ ν ) Sv ( τ ν ) d This solution is useful when material properties are fixed, e.g. postprocessing for diagnostics Important features: Explicit non-local relationship between I ν and S ν Escaping radiation comes from depth τ ν 1 Implicit S ν (I ν ) dependence comes from radiation / material coupling Self-consistently determining S ν and I ν is the hard part of radiation transport τ ν

9 Limiting Cases Optically-thin τ ν <<1 (viewed in emission): I v = τ v S v d τ ν = dx η v d x η v dx I ν reflects spectral characteristics of emission, independent of absorption Optically-thick τ ν >>1: I v τ ( ) v = e τ v τ v ( ) ( τ ν ) Sv τ ν d τ ν S v ( τ ν ) I ν reflects spectral characteristics of S ν (over ~last optical depth) Negligible emission (viewed in absorption): I v ( τ v ) = I v ( )e τ v, τ = dx α ν d x α v v dx I ν / I ν () reflects spectral characteristics of absorption

10 Example Flux from a uniform sphere T = 2 ev, ρ =.1 g/cc, LTE 25 Δr =.1 cm flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

11 Example Flux from a uniform sphere T = 2 ev, ρ =.1 g/cc, LTE 25 Δr =.1 cm flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

12 Example Flux from a uniform sphere T = 2 ev, ρ =.1 g/cc, LTE 25 Δr = 1. cm flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

13 Absorption / emission coefficients Macroscopic description energy changes Energy removed from radiation passing through material of area da, depth ds, over time dt de = α ν I ν dads dωdν dt Energy emitted by material de = η ν dads dωdν dt Microscopic description radiative transitions Absorption and emission coefficients are constructed from atomic populations y i and cross sections σ ij : α ν = i< j σ ν, ij (y i g i y j ), η ν = 2hν 3 g j c 2 i< j σ ν, ij g i g j y j from T. Mehlhorn

14 Bound-bound Transitions Probability (per unit time) of Spontaneous emission: A 21 Absorption: B 12 Stimulated emission: B 21 J J A 21, B 12, B 21 are Einstein coefficients g 1 B 21 = g 2 B 12, A 21 = 2hv B c Two energy level system 2 g 2 ΔE = hv n 2 1 g 1 n 1 Line profile φ(ν) dν =1 φ v measures probability of absorption Transition rate from level 1 to level 2 R 12 = B 12 J, J = J v φ( v)dv (assuming that the linewidth << ΔE)

15 Cross section for absorption σ ( v) = hv o 4π B φ v 21 ( ) = πe2 mc f φ v 12 ( ) f 12 = oscillatorstrength πe 2 mc =.2654 cm2 /s Oscillator strength f 12 relates the quantum mechanical result to the classical treatment of a harmonic oscillator Strong transitions have f ~1 Sum rule (# of bound electrons) j f ij = Z Absorption and emission coefficients: πe 2 α ν = n 1 mc f φ v 12 ( ) 1 g n 1 2 g 2 n 1 η ν = 2hν 3 c 2 n πe 2 2 mc f φ v 12 ( ) absorp8on s8mulated emission spontaneous emission For now we are assuming that absorption and emission have the same line profile

16 Bound-free absorption Absorption cross section from state of principal quantum number n and charge Z ν σ bf = ng bf ν 3 hν kte 1 e cm2 Gaunt factor hv = threshold energy Free-free absorption Absorption cross section per ion of charge Z Z 2 σ ff = g ν 3 ff n e 1 e T e hν kte cm2 The term hν kte 1 e Gaunt factor accounts for stimulated emission

17 Scattering Interaction in which the photon energy is (mostly) conserved (i.e. not converted to kinetic energy) Examples: Scattering by bound electrons Rayleigh scattering - important in atmospheric radiation transport Scattering by free electrons Thomson / Compton scattering 2 σ T = 8π e 2 3 m e c 2 = 6.65x1 25 cm 2 hν m e c 2 Note: frequency shift from scattering is ~ Doppler shift from electron velocity is ~ hν / m e c 2 2kT / m e c 2 For most laboratory plasmas, these types of scattering are negligible Note: X-ray Thomson scattering (off ion acoustic waves and plasma oscillations) can be a powerful diagnostic for multiple plasma parameters (T e, T i, n e )

18 Radiation transport equation with scattering (and frequency changes): 1 I v c t + Ω i I v = α v I v + η v + σ v d ν d Ω 4π R( ν, Ω ;ν,ω) ν ν I (1+ f ) + R(ν,Ω; ν, Ω )I (1+ f ) v ν v ν s8mulated scagering The redistribution function R describes the scattering of photons (ν,ω) è (ν,ω ) Neglecting frequency changes, this simplifies to 1 I v c t + Ω d Ω i I v = (α v + σ v )I v + η v +σ v I 4π v ( Ω )g Ω ( i Ω ) (α v + σ v )I v + η v +σ v J v for isotropic scagering Scattering contributes to both absorption and emission terms (and may be denoted separately or included in α ν and η ν )

19 Effective scattering Photons also scatter by e.g. resonant absorption / emission Upper level 2 can decay a) radiatively A 21 Two energy level system 2 g 2 n 2 b) collisionally n e C 21 ε n e C 21 / A 21 The fraction of photons are destroyed / thermalized 1 g 1 n 1 The fraction (1-ε) of photons are scattered è energy changes only slightly (mostly Doppler shifts) è undergo many scatterings before being thermalized Note: ε 1 is the condition for a strongly non-lte transition and is easily satisfied for low density or high ΔE!

20 Population distribution LTE: Saha-Boltzmann equation Excited states follow a Boltzmann distribution Ionization stages obey the Saha equation y i y j = g i g j e (ε i ε j )/T e ε i = energyof state i T e = electron temperature N q N q+1 = 1 2 n e U q h 2 U q+1 2πm e T e NLTE: Collisional-radiative model Calculate populations by integrating rate equations 3/2 e (ε q+1 ε q )/T e N q = y i e (ε i ε i q U q = g i e (ε i ε i q q )/T e q )/T e number densityof chargestateq partitionfunctionof chargestateq dy dt = Ay R ij = A ij = C ij + R ij + (other) ij C ij = n e σ ij (ν)j(ν) dν hν, R = ji σ ij (ν) J(ν) + 2hν 3 c 2 vσ ij (v) f (v)dv e hν kt dν hν

21 Summary of absorption / emission coefficients Summed over populations and radiative transitions: πe 2 α ν = y i mc f φ v 2 ij ij ij η ν = 2hν 3 πe 2 y c 2 j mc 2 ij In NLTE, the source function has a complex dependence on plasma parameters and on the radiation spectrum: S ν = S ν (n e,t e ; y i (J ν )) ( ) 1 g y i j g i g j f ij φ ij v + σ bf g j y ij (ν) + n e σ ff ij (ν) (1 e hν /kt e ) i ( ) + σ bf ij (ν) + n e σ ff ij (ν) In LTE, absorption and emission spectra are complex but the source function obeys Kirchoff s law: S ν = B ν (T e ) e hν /kt e

22 Opacity & mean opacities opacity = absorption coefficient / mass density κ ν = α ν ρ Rosseland mean opacity: emphasizes transmission includes scattering appropriate for average flux 1 T = 2 ev, ρ =.1 g/cc, LTE Planck mean opacity: emphasizes absorption no scattering appropriate for energy exchange 1 κ R = 1 db dν ν κ ν dt db dν ν dt κ P = dνκ ν B ν dν B ν absorption coefficient (1/cm) db/dt photon energy (ev) 25 3

23 Example Hydrogen (T e = 2 ev, n e =1 14 cm -3 ) absorption coefficient absorption coefficient (cm -1 ) total bound-free free-free n=3 n=2 n=1 absorption coefficient (cm -1 ) LTE NLTE photon energy (ev) photon energy (ev)

24 Hydrogen, again (T e = 2 ev, n e =1 14 cm -3 ) emissivity source func8on 1 8 6x1 11 emissivity (erg/cm 3 /s/ev/ster) source function (erg/cm 2 /s/ev/ster) LTE NLTE photon energy (ev) photon energy (ev)

25 Flux from a uniform sphere revisited T = 2 ev, ρ =.1 g/cc, LTE / NLTE 25 Δr =.1 cm 2 LTE NLTE flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

26 Flux from a uniform sphere revisited T = 2 ev, ρ =.1 g/cc, LTE / NLTE 25 Δr =.1 cm 2 LTE NLTE flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

27 Flux from a uniform sphere revisited T = 2 ev, ρ =.1 g/cc, LTE / NLTE 25 Δr = 1. cm 2 LTE NLTE flux (cgs) 1 optical depth photon energy (ev) photon energy (ev) 15 2

28 Flux from a uniform sphere - Summary Black-body emission requires large optical depths Large optical depths è high 2 15 LTE NLTE r = 1. cm radiation fields è LTE conditions Boundaries introduce nonuniformities through radiation fields T r, eff 1 5 r =.1 cm r =.1 cm The radiation field will also change the material temperature..2.4 r/ r Conditions do not remain uniform in the presence of radiation transport and boundaries

29 Example Hydrogen Ly-α Ly-α emission from a uniform plasma T e = 1 ev, n e = 1 14 cm -3 Moderate optical depth τ ~ 5 Viewing angles 9 o and 1 o show optical depth broadening 1 cm 9 o 1 o 1.x1-9 9º 1º specific intensity (cgs) optical depth optical depth photon energy w.r.t. line center (ev) photon energy w.r.t. line center (ev) photon energy (ev) 1.22

30 Example Hydrogen Ly-α Ly-α emission from a plasma with uniform temperature and density T e = 1 ev, n e = 1 14 cm -3 Self-consistent solution displays effects of 9 o Radiation trapping / pumping Non-uniformity due to boundaries 1 cm 1 o specific intensity (cgs) 6x self-consistent 9º 1º uniform 9º 1º n=2 fractional population 3.x self-consistent uniform photon energy w.r.t. line center (ev) position (cm).8 1.

31 Coupled systems or What does Radiation Transport mean? The system of equations and emphasis varies with the application For laboratory plasmas, these two sets are most useful - LTE / energy transport : Coupled to energy balance de m = 4π α dt ν (J ν S ν ) dν α ν =α ν (T e ), S ν = B ν (T e ) Indirect radiation-material coupling through energy/temperature Collisions couple all frequencies locally, independent of J ν Solution methods concentrate on nonlocal aspects NLTE / spectroscopy : Coupled to rate equations dy dt = Ay, A = A (T, J ) ij ij e ν S ν = 2hν 3 S ij, S ij a + b J ij c 2 Direct coupling of radiation to material Collisions couple frequencies over narrow band (line profiles) Solution methods concentrate on local material-radiation coupling Non-local aspects are less critical

32 Line Profiles Line profiles are determined by multiple effects: Natural broadening (A 12 ) - Lorentzian Collisional broadening (n e, T e ) - Lorentzian Doppler broadening (T i ) - Gaussian Stark effect (plasma microfields) - complex φ(ν) = Γ 4π 2 ( ) 2 + ( Γ 4π ) 2 ν ν Γ = destruc8on rate φ(ν) = 1 H (a, x) Δν D π 1 Gaussian, Voigt and Lorentzian Profiles a = Γ 4πΔν D Δν D = ν c H (a, x) = a π e y2 (x y) 2 + a 2 2kT i m i dy φ, line profile factor Lorentzian Voigt, a=.1 Voigt, a=.1 Voigt, a=.1 Gaussian (ν-ν o )/

33 ψ ν Redistribution The emission profile is determined by multiple effects: collisional excitation è natural line profile (Lorentzian) photo excitation + coherent scattering photo excitation + elastic scattering è ~absorption profile Doppler broadening This is described by the redistribution function Complete redistribution (CRD): ψ ν = φ ν R(ν, ν ) R(ν, ν ) dv = φ( ν ), ψ (ν) = R(ν, ν ) J( ν )d v φ( ν ) J( ν )d v Doppler broadening is only slightly different from CRD, while coherent scattering gives R(ν, ν ) = φ(ν)δ (ν ν ) A good approximation for partial redistribution (PRD) is often R(ν, ν ) =(1 f )φ( ν )φ(ν)+ f R II (ν, ν ) where f (<<1 for X-rays) is the ratio of elastic scattering and de-excitation rates, includes coherent scattering and Doppler broadening R II

34 Hydrogen Ly-α w/ Partial Redistribution Ly-α emission from a plasma with uniform temperature and density T e = 1 ev, n e = 1 14 cm -3 Optical depth τ ~ 5 9 o Voigt parameter a ~.3 1 cm 1 o 6x1-9 5 PRD 9º 1º 3.5x PRD CRD uniform specific intensity (cgs) CRD 9º 1º uniform 9º 1º n=2 fractional population photon energy w.r.t. line center (ev) position (cm).8 1.

35 Material Velocity The discussion so far applies in the reference frame of the material Doppler shifts matter for line radiation when v/c ~ ΔE/E If velocity gradients are present, either a) Transform the RTE into the co-moving frame - or b) Transform material properties into the laboratory frame Option (a) is complicated (particularly when v/c è 1) - see the references by Castor and Mihalas for discussions Option (b) is relatively simple, but makes the absorption and emission coefficients direction-dependent

36 Al sphere w/ uniform expansion velocity T e = 5 ev ρ = 1-3 g/cm 3 R outer = 1. cm R inner =.9 cm optical depth areal flux v/c = v/c =.3 v/c =.1 v/c =.3 optical depth areal flux (erg/s/hz/ster) photon energy (ev) photon energy (ev) 22 24

37 Al sphere w/ uniform expansion velocity 1 v/c = v/c =.3 v/c =.1 v/c =.3 areal flux (erg/s/hz/ster) He-α Ly-α photon energy (ev) 18

38 2-Level Atom Rate equation for two levels in steady state: n 1 (B 12 J 12 + C 12 ) = n 2 (A 21 + B 21 J 12 + C 21 ) J 12 = φ 12 ( v)dv, C 12 = g 2 J v Two energy level system 2 g 2 n 2 g 1 e hν kt C 21 ΔE = hv Absorption / Emission: α ν = hν ( 4π n B n B )φ 12 (ν), η ν = hν 4π n A 2 21φ 12 (ν) 1 g 1 n 1 Source Function: S ν = n 2 A 21 n1 B 12 n 2 B 21 = (1 ε)j 12 + εb ν, ε 1 ε = C 21 A 21 1 e hν kt ( ) S ν is independent of frequency and linear in J - solution methods exploit this dependence

39 A Popular Solution Technique For a single line, the system of equations can be expressed as di v dτ v = I v + S v I v = λ ν S v (J ) S v = a ν + b ν J, J = 1 4π where the lambda operator and the source function depend on the populations through the absorption and emission coefficients. A numerical solution for λ ν J dω λ ν I v φ( v)dv uniquely specifies the entire system. Since depends on J through the populations, the full system is non-linear and requires iterating solutions of the rate equations and radiation transport equation. λ ν The dependence of on J is usually weak, so convergence is rapid.

40 Solution technique for the linear source function: I v = λ ν a ν + b ν J J = 1 4π J = 1 Λ dω 1 1 4π Λ = 1 4π λ ν a ν + b ν J φ( v)dv dω dω λ ν a ν φ( v)dv This linear system can (in principle) be solved directly for and in 1D this is very efficient The (angle,frequency)-integrated operator includes factors, so 1 1 Λ amplifies J (radiation trapping) Efficient solution methods approximate key parts of NLTE local frequency scattering LTE (linear in ΔT e ) non-local coupling λ ν b ν φ( v)dv Λ Λ J 1 e τ ν

41 For multiple lines, the source function for each line can be put in the two-level form ETLA (Extended Two Level Atom) or the full source function can be expressed in the following manner: S v = η ν c + α ν c + J Solve for each individually (if coupling between lines is not important) or simultaneously. (Complete) linearization expand and solve as before. l S ν = S ν (J l ) + l Multi-Level Atom η ν l (J l )φ ν l α ν l (J l )φ ν l l S ν J l ( J l J l ) S ν in terms of ΔJ l Partial redistribution usually converges at a slightly slower rate.

42 Numerical methods need to fulfill 2 requirements 1. Accurate formal transport solution which is conservative, non-negative 2 nd order (spatial) accuracy (diffusion limit as τ >> 1) causal (+ efficient) Many options are available each has advantages and disadvantages 2. Method to converge solution of coupled implicit equations Multiple methods fall into a few classes Full nonlinear system solution Accelerated transport solution Incorporate transport information into other physics Optimized methods are available for specific regimes, but no single method works well across all regimes

43 Method #1 Source (or lambda) iteration 1. Evaluate source function 2. Formal solution of radiation transport equation 3. Use intensities to evaluate temperature / populations iterate to convergence Advantages Simple to implement Independent of formal transport method Disadvantages Can require many iterations: # iterations ~ τ 2 False convergence is a problem for τ >> 1

44 Hydrogen Lyman-α revisited Source iteration (green curves) approaches self-consistent solution slowly Linearization achieves convergence in 1 iteration S ij = a + b J ij, J ij = J ν φ v dν ç linear func8on of J 6x º self-consistent uniform 3.x1-8 self-consistent uniform specific intensity (cgs) itera8on # n=2 fractional population itera8on # photon energy w.r.t. line center (ev) position (cm).8 1.

45 Method #2 Monte Carlo Formal solution method 1) Sample emission distribution in (space, frequency, direction) to create photons 2) Track photons until they escape or are destroyed Advantages Works well for complicated geometries Not overly constrained by discretizations è does details very well Disadvantages Statistical noise improves slowly with # of particles Expense increases with optical depth Iterative evaluation of coupled system is not possible / advisable Semi-implicit nature requires careful timestep control Convergence A procedure that transforms absorption/emission events into effective scatterings (Implicit Monte Carlo) provides a semi-implicit solution Symbolic IMC provides a fully-implicit solution at the cost of a solving a single mesh-wide nonlinear equation

46 Method #3 Discrete Ordinates (S N ) Formal solution method Discretization in angle converts integro-differential equations into a set of coupled differential equations (provides lambda operator) Advantages Handles regions with τ<<1 and τ>>1 equally well Modern spatial discretizations achieve the diffusion limit Deterministic method can be iterated to convergence Disadvantages Ray effects due to preferred directions angular profiles become inaccurate well before angular integrals Required # of angles in 2D/3D can become enormous Discretization in 7 dimensions requires large computational resources Convergence Effective solution algorithms exist for both LTE and NLTE versions [6] e.g. LTE synthetic grey transport (or diffusion) NLTE complete linearization (provides linear source function) + accelerated lambda iteration in 2D/3D (Note: this applies to all deterministic methods)

47 Method #4 Escape factors Escape factor p e is used to eliminate radiation field from net radiative rate y j B ji J y i B ij J = y j A ji p e Equivalent to incorporating a (partial) lambda operator into the rate equations è combines the formal solution + convergence method Advantages Very fast no transport equation solution required Can be combined with other physics with no (or minimal) changes Disadvantages Details of transport solution are absent Escape factors depend on line profiles, system geometry Iterative improvement is possible, but usually not worthwhile

48 Escape factor is built off the single flight escape probability p e = 1 4π dω e τ ν φ( v)dv e τ ν Iron s theorem this gives the correct rate on average (spatial average weighted by emission) 1 J S = p e Note that the optical depth depends on the line profile, plus continuum Asymptotic expressions for large optical depth: Gaussian profile Voigt profile p e 1 4τ ln τ π ( ) p e 1 3 a τ Evaluating p e can be complicated by overlapping lines, Doppler shifts, etc. Many variations and extensions exist in a large literature

49 References Stellar Atmospheres, 2 nd edition, D. Mihalas, Freeman, Theory of Stellar Atmospheres: An Introduction to Astrophysical Nonequilibrium Quantitative Spectroscopic Analysis, I. Hubeny and D. Mihalas, Princeton, 214. Radiation Hydrodynamics, J. Castor, Cambridge University Press, 24. Kinetics Theory of Particles and Photons (Theoretical Foundations of Non- LTE Plasma Spectroscopy), J. Oxenius, Springer-Verlag, Radiation Trapping in Atomic Vapours, A.F. Molisch and B.P. Oehry, Oxford University Press, Radiation Processes in Plasmas, G. Bekefi, John Wiley and Sons, Computational Methods of Neutron Transport, E.E. Lewis and W.F. Miller, Jr., John Wiley and Sons, 1984.