( ) ( ) d 3! xd. ( )= 4π c ( ) Radiation Transport for Spectroscopy. Lecture Notes on Radiation Transport for Spectroscopy.

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1 LLNL-PRES-668 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Liermore National Laboratory under Contract DE-AC-7NA7. Lawrence Liermore National Security, LLC Lecture Notes on Radiation Transport for Spectroscopy ICTP-IAEA School on Modern Methods in Plasma Spectroscopy March 6-, Trieste, Italy Radiation Transport for Spectroscopy Outline Definitions, assumptions and terminology Equilibrium limit Radiation transport equation Characteristic form & formal solution Material radiatie properties Absorption, emission, scattering Coupled systems / non- Line radiation Line shapes Redistribution Material motion Solution methods Transport operators -leel & multi-leel atoms Escape factors Basic assumptions Radiation Description Classical / Semi-classical description Radiation field described by either specific intensity I or the photon distribution function f Unpolarized radiation Neglect index of refraction effects (n, ω>>ω p Photons trael in straight lines Neglect true scattering (mostly Static material (for now Single reference frame Macroscopic - specific intensity I energy per (area x solid angle x time within the frequency range (, + d de = ( r,ω,t ( n Ω dadωd dt de = energy crossing area da within (dωd dt Microscopic - photon distribution function de = h f r, p,t spins = h c d xd p f ( r, p,t h p= h Ω c Angular Moments th moment J = π I dω = energy density x c/π st moment H = π ni dω = flux x /π nd moment K = π nni dω = pressure tensor x c /π For isotropic radiation, K is diagonal with equal elements: K = J I (P= E In this case, radiation looks like an ideal gas with γ = / Intensity: Planck function B = h c e h kt r Energy density E rad T r = π c For T e =T r and n H = cm - : Thermal Equilibrium B T r d = at r Distribution function: Bose-Einstein f = e h kt r E rad =E matter T ev "#"#$%&'(*+,$-./&,%'+,6 7&+8%'*9&:*;*+,&'-9;+8$/9"( * <( 7$;$/-9;+8$/%&/*&+;+&+. α η Radiation Transport Equation c t + Ω i = α +η = absorption coefficient (fraction of energy absorbed per unit length = emissiity (energy emitted per unit time, olume, frequency, solid angle Boltzmann equation for the photon distribution function: f + Ω i f c t = f t 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 consered (except for scattering Photon mean free path λ =/α coll = h h c f Characteristic Form Define the source function S and τ : S = η /α = B in dτ = α ds Along a characteristic, the radiation transport equation becomes This solution is useful when material properties are fixed, e.g. postprocessing for diagnostics Important features: Explicit non-local relationship between I and S Escaping radiation comes from depth τ Implicit S (I dependence comes from radiation / material coupling d = I dτ + S ( τ = e τ + τ ( e τ τ S ( τ d τ Self-consistently determining S and I is the hard part of radiation transport

2 Limiting Cases Optically-thin τ << (iewed in emission: Example Flux from a sphere T = ev, ρ =. g/cc, τ = S d τ = η d x η dx dx I reflects spectral characteristics of emission, independent of absorption =+>?@??A%, Optically-thick τ >>: - τ ( = e ( τ τ S ( τ d τ S τ I reflects spectral characteristics of S (oer ~last τ - - Negligible emission (iewed in absorption: = e τ,τ = dx α d x α dx τ - I / I ( reflects spectral characteristics of absorption Example Flux from a sphere T = ev, ρ =. g/cc, Example Flux from a sphere T = ev, ρ =. g/cc, =+>?@A%, =+>A?@?%,.. Absorption / emission coefficients Macroscopic description energy changes Energy remoed from radiation passing through material of area da, depth ds, oer time dt de = α I dadsdωddt Energy emitted by material de = η I dadsdωddt Microscopic description radiatie transitions Absorption and emission coefficients are constructed from atomic populations y i and cross sections σ ij : α = σ,ij (y i g i i<j y j, η = h g i y g j c j g j σ,ij i<j B+$,(@C*'$+/ Probability (per unit time of Spontaneous emission: A Absorption: B Stimulated emission: B A, B, B are Einstein coefficients Bound-bound Transitions J J g B =g B, A = h B c Line profile φ measures probability of absorption φ( d= Transition rate from leel to leel R = B J, J = J φd (assuming that the linewidth << E Two energy leel system g E =h n g n Cross section for absorption σ= h o π B φ πe = mc f φ Oscillator strength f relates the quantum mechanical result to the classical treatment of a harmonic oscillator Strong transitions hae f ~ Sum rule j f ij =Z (# of bound electrons Absorption and emission coefficients: πe α = n mc f φ g n g n η = h c n πe mc f φ f = oscillatorstrength πe mc =.6 cm /s &9$+78$/D98,'&;*-*,99$/ 97$/;&/*$9*,99$/ For now we are assuming that absorption and emission hae the same line profile Bound-free absorption Absorption cross section from state of principal quantum number n and charge Z σ bf = h e kte ng cm bf E&/;B&%;$+ Free-free absorption Absorption cross section per ion of charge Z The term Z σ ff =.69 8 g ff n e e T e h e kte E&/;B&%;$+ h = ;+*9$'-*/*+F. h kte cm accounts for stimulated emission

3 Scattering Interaction in which the photon energy is (mostly consered (i.e. not conerted to kinetic energy Examples: Scattering by bound electrons Rayleigh scattering - important in atmospheric radiation transport Scattering by free electrons Thomson / Compton scattering σ T = 8π e m e c =6.6x cm h m e c Note: frequency shift from scattering is ~ h/m e c Doppler shift from electron elocity is ~ kt/m e c For most laboratory plasmas, these types of scattering are negligible Note: X-ray Thomson scattering (off ion acoustic waes and plasma oscillations can be a powerful diagnostic for multiple plasma parameters (T e, T i, n e Radiation transport equation with scattering (and frequency changes: c t + Ωi = α +η + σ d d Ω π R(, Ω ;,Ω I (+ + R(,Ω;, Ω I (+ f + f 98,'&;*-9%&G*+/F The redistribution function R describes the scattering of photons (,Ω (,Ω Neglecting frequency changes, this simplifies to c t + d Ω Ωi = (α +σ +η +σ I π ( Ω g Ωi ( Ω (α +σ +η +σ J B$+9$;+$7%9%&G*+/F Scattering contributes to both absorption and emission terms (and may be denoted separately or included in α and η Effectie scattering Photons also scatter by e.g. resonant absorption / emission Upper leel can decay a radiatiely A b collisionally n e C The fraction ε n e C /A of photons are destroyed / thermalized The fraction (-ε of photons are scattered energy changes only slightly (mostly Doppler shifts undergo many scatterings before being thermalized Note: ε Two energy leel system g is the condition for a strongly non- transition and is easily satisfied for low density or high E n g n : Saha-Boltzmann equation Population distribution Excited states follow a Boltzmann distribution Ionization stages obey the Saha equation N q U q = N q+ n h e U q+ πm e T e : Collisional-radiatie model Calculate populations by integrating rate equations dy dt =Ay / e (ε q+ ε q /T e A ij =C ij +R ij + (other ij y i y j = g i g j e (ε i ε j/t e R ij = σ ij (J( d h, R = ji σ ij ( J(+ h c i q ε i =energyofstatei T e =electrontemperature N q = y i e (ε i ε q /T e numberdensityofchargestateq U q = g i e (ε i ε q /T e partitionfunctionofchargestateq i q C ij =n e σ ij (f(d e h kt d h Summary of absorption / emission coefficients Summed oer populations and radiatie transitions: πe α = y i mc f φ ij ij gy i j + σ bf g j y ij (+n e σ ff ij ( ( e h/kt e ij i η = h πe g y i c j f mc g ij φ ij + σ bf ij (+n e σ ff ij ( e h/kt e ij j In, the source function has a complex dependence on plasma parameters and on the radiation spectrum: S = S (n e,t e ; y i (J In, absorption and emission spectra are complex but the source function obeys Kirchoff s law: S = B (T e Opacity & mean opacities opacity = absorption coefficient / mass density Rosseland mean opacity: emphasizes transmission includes scattering appropriate for aerage flux Planck mean opacity: emphasizes absorption no scattering appropriate for energy exchange db d κ = dt κ R db d dt dκ B κ P = db absorption coefficient (/cm. κ =α ρ T = ev, ρ =. g/cc, db/dt Example Hydrogen (T e = ev, n e = cm - Hydrogen, again (T e = ev, n e = cm - absorption coefficient emissiity 9$+%*B/%8$/ absorption coefficient (cm n= n= total - bound-free n= free-free -. absorption coefficient (cm emissiity (erg/cm /s/ev/ster source function (erg/cm /s/ev/ster 6x..

4 Flux from a sphere reisited T = ev, ρ =. g/cc, / Flux from a sphere reisited T = ev, ρ =. g/cc, / =+>?@??A%, =+>?@A%, Flux from a sphere reisited Flux from a sphere - Summary T = ev, ρ =. g/cc, / =+>A?@?%, " Black-body emission requires large s " Large s high radiation fields conditions " Boundaries introduce nonities through radiation fields T r, eff r =. cm r =. cm r =. cm " The radiation field will also change the material temperature... r/ r.6.8. Conditions do not remain in the presence of radiation transport and boundaries Ly-α emission from a plasma T e = ev, n e = cm - Moderate τ ~ Viewing angles 9 o and o show broadening Example Hydrogen Ly-α cm 9 o o Example Hydrogen Ly-α Ly-α emission from a plasma with temperature and density T e = ev, n e = cm - Self-consistent solution displays effects of Radiation trapping / pumping Non-ity due to boundaries cm 9 o o.x -9 6x -9.x -8 specific intensity (cgs º º specific intensity (cgs 9º º 9º º n= fractional population position (cm.8. Coupled systems or What does Radiation Transport mean? The system of equations and emphasis aries with the application For laboratory plasmas, these two sets are most useful - Line Profiles Line profiles are determined by multiple effects: Natural broadening (A - Lorentzian Collisional broadening (n e, T e - Lorentzian Doppler broadening (T i - Gaussian Stark effect (plasma microfields - complex Γ π φ(= + ( Γ π Γ >-*9;+%8$/+&;* / energy transport : Coupled to energy balance de m = π α dt (J S d α =α (T e, S = B (T e Indirect radiation-material coupling through energy/temperature Collisions couple all frequencies locally, independent of J Solution methods concentrate on nonlocal aspects / spectroscopy : Coupled to rate equations dy dt = Ay, A = A (T, J ij ij e S = h S c ij, S ij a+b J ij Direct coupling of radiation to material Collisions couple frequencies oer narrow band (line profiles Solution methods concentrate on local material-radiation coupling Non-local aspects are less critical φ(= D π H(a,x a= Γ π D D = c H(a,x= a π e y kt i m i dy (x y +a, line profile factor " " " # " $ " % Gaussian, Voigt and Lorentzian Profiles Lorentzian Voigt, a=. Voigt, a=. Voigt, a=. Gaussian " & ( - o /

5 Redistribution The emission profile ψ is determined by multiple effects: collisional excitation natural line profile (Lorentzian photo excitation + coherent scattering photo excitation + elastic scattering ~absorption profile Doppler broadening This is described by the redistribution function R(, R(, d=φ(, ψ (= R(, J( d φ( J( d Complete redistribution (CRD: ψ =φ Doppler broadening is only slightly different from CRD, while coherent scattering gies R(, =φ(δ ( A good approximation for partial redistribution (PRD is often R(, =( f φ( φ(+ f R II (, where f (<< for X-rays is the ratio of elastic scattering and de-excitation rates, includes coherent scattering and Doppler broadening R II specific intensity (cgs 6x -9 Hydrogen Ly-α w/ Partial Redistribution Ly-α emission from a plasma with temperature and density T e = ev, n e = cm - Optical depth τ ~ Voigt parameter a ~. PRD 9º º CRD 9º º 9º º n= fractional population cm.x o..6 position (cm o.8 PRD CRD. Material Velocity Al sphere w/ expansion elocity The discussion so far applies in the reference frame of the material Doppler shifts matter for line radiation when /c ~ E/E If elocity gradients are present, either a Transform the RTE into the co-moing frame - or b Transform material properties into the laboratory frame Option (a is complicated (particularly when /c - see the references by Castor and Mihalas for discussions ( * >H??*I J>A? 6K FL%, K M $;*+ >A@?%, M //*+ >?@N%, areal flux /c = /c =. /c =. /c =. Option (b is relatiely simple, but makes the absorption and emission coefficients direction-dependent areal flux (erg/s/hz/ster Al sphere w/ expansion elocity -Leel Atom Rate equation for two leels in steady state: areal flux (erg/s/hz/ster. /c = /c =. /c =. /c =. He-α Ly-α Two energy leel system n (B J +C =n (A +B J +C g n J = J φ d, C = g e h kt C g E =h Absorption / Emission: α = h ( π n B n B φ (, η = h π n A φ ( Source Function: S = n A ε n B n B =( εj +εb, ε =C e h kt A S is independent of frequency and linear in J - solution methods exploit this dependence g n A Popular Solution Technique For a single line, the system of equations can be expressed as d dτ = + S = λ S (J S = a + b J, J = π dω φd where the lambda operator λ and the source function depend on the populations through the absorption and emission coefficients. A numerical solution for J uniquely specifies the entire system. Since λ depends on J through the populations, the full system is non-linear and requires iterating solutions of the rate equations and radiation transport equation. The dependence of on J is usually weak, so conergence is rapid. λ Solution technique for the linear source function: = λ a + b J J = π dω J = Λ π dω Λ= π dω λ a + b J φd λ a φd This linear system can (in principle be soled directly for and in D this is ery efficient The (angle,frequency-integrated operator Λ includes factors, so Λ amplifies J (radiation trapping Efficient solution methods approximate key parts of local frequency scattering (linear in T e non-local coupling λ b φd Λ J e τ

6 For multiple lines, the source function for each line can be put in the two-leel form ETLA (Extended Two Leel Atom or the full source function can be expressed in the following manner: η c + η l l (J l φ l S = α c + α l l (J l φ J l Sole for each indiidually (if coupling between lines is not important or simultaneously. (Complete linearization expand S in terms of J l S S =S (J l + ( J l J l J l and sole as before. l l Multi-Leel Atom Numerical methods need to fulfill requirements. Accurate formal transport solution which is conseratie, non-negatie nd order (spatial accuracy (diffusion limit as τ >> causal (+ efficient Many options are aailable each has adantages and disadantages. Method to conerge 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 aailable for specific regimes, but no single method works well across all regimes Partial redistribution usually conerges at a slightly slower rate. Method # Source (or lambda iteration Hydrogen Lyman-α reisited Source iteration (green cures approaches solution slowly Linearization achiees conergence in iteration. Ealuate source function. Formal solution of radiation transport equation. Use intensities ities to ealuate temperature re / populations ;*+&;*;$ %$/:*+F*/%* 6x -9 S ij = a+b J ij, J ij = J φ d 9º #'/*&+B/%8$/$B J.x -8 Adantages Simple to implement Independent of formal transport method Disadantages Can require many iterations: # iterations ~ τ False conergence is a problem for τ >> specific intensity (cgs ;*+&8$/O n= fractional population ;*+&8$/O position (cm.8. Method # Monte Carlo Formal solution method Sample emission distribution in (space, frequency, direction to create photons Track photons until they escape or are destroyed Adantages Works well for complicated geometries Not oerly constrained by discretizations does details ery well Disadantages Statistical noise improes slowly with # of particles Expense increases with Iteratie ealuation of coupled system is not possible / adisable Semi-implicit nature requires careful timestep control Conergence A procedure that transforms absorption/emission eents into effectie scatterings (Implicit Monte Carlo proides a semi-implicit solution Symbolic IMC proides a fully-implicit solution at the cost of a soling a single mesh-wide nonlinear equation Method # Discrete Ordinates (S N Formal solution method Discretization in angle conerts integro-differential equations into a set of coupled differential equations (proides lambda operator Adantages Handles regions with τ<< and τ>> equally well Modern spatial discretizations achiee the diffusion limit Deterministic method can be iterated to conergence Disadantages Ray effects due to preferred directions angular profiles become inaccurate well before angular integrals Required # of angles in D/D can become enormous Discretization in 7 dimensions requires large computational resources Conergence Effectie solution algorithms exist for both and ersions [6] e.g. synthetic grey transport (or diffusion complete linearization (proides linear source function + accelerated lambda iteration in D/D (Note: this applies to all deterministic methods Method # Escape factors Escape factor p e is used to eliminate radiation field from net radiatie rate y j B ji J y i B ij J =y j A ji p e Escape factor is built off the single flight escape probability p e = π dω e τ φd e τ Equialent to incorporating a (partial lambda operator into the rate equations combines the formal solution + conergence method Adantages Very fast no transport equation solution required Can be combined with other physics with no (or minimal changes Disadantages Details of transport solution are absent Escape factors depend on line profiles, system geometry Iteratie improement is possible, but usually not worthwhile Iron s theorem this gies the correct rate on aerage (spatial aerage weighted by emission J S = p e Note that the depends on the line profile, plus continuum Asymptotic expressions for large : Gaussian profile Voigt profile p e τ ln τ π p e a τ Ealuating p e can be complicated by oerlapping lines, Doppler shifts, etc. Many ariations and extensions exist in a large literature

7 References Stellar Atmospheres, nd edition, D. Mihalas, Freeman, 978. Radiation Hydrodynamics, J. Castor, Cambridge Uniersity Press,. Kinetics Theory of Particles and Photons (Theoretical Foundations of Non- Plasma Spectroscopy, J. Oxenius, Springer-Verlag, 986. Radiation Trapping in Atomic Vapours, A.F. Molisch and B.P. Oehry, Oxford Uniersity Press, 998. Radiation Processes in Plasmas, G. Bekefi, John Wiley and Sons, 966. Computational Methods of Neutron Transport, E.E. Lewis and W.F. Miller, Jr., John Wiley and Sons, 98. Laser Absorption For narrow band optical laser, can neglect emission but must consider index of refraction n n= ω p n < for n e aboe a critical density: crit n e = m ω e πe ω cm - for.6 micron laser Refractie effects - cured photon paths modified absorption / emission inariant phase space quantity is I/n instead of I c t n + ni I n + dn ds i n n = nα n + nη n>/;:*%;$+ D planar geometry Laser incident on plasma w/ increasing n e ϑ = initial angle of incidence w.r.t. normal light propagates to position n e = cos ϑ n e crit inerse bremsstrahlung absorption along path resonant absorption (p-polarization at turning point - dependent on local density gradient, Boltzmann analysis Radiation transport effects with plasma transport Plasma transport model explicitly treats ion and (ground state neutral atoms Excited states are assumed to be in equilibrium on transport timescales: g i gi, gi, nx = fxng + fxni, fx = fx ( ne, Te Transport model uses effectie ionization / recombination and energy loss coefficients which account for excited state distributions, e.g. ni nn + ( nivi = Pn i n Pn r i, + ( nnvn = Pn i n + Pn r i t t Tabulated coefficients are ealuated with a collisional-radiatie code in the optically thin limit Detached diertor simulations exhibit large radiation effects Specifications: L= m, n= m -, q in = MW/m, β=. Flux Q r q out CR UR RMS%$''9$/&'6+&-&8:*;&'*9"$78%&''.6;/ T#(S%$''9$/&'6+&-&8:*,$-*'QL+&-&8$/;+&/97$+; URS7&+&,*;*+V*-;&'*9 P + S+&-&8:*WX $; S7&+8%'*WX "#$%&#'(-*9%+78$/$B;*-*;&%*--:*+;$++*F$/+*,&/9/%&/F*-< P"#*'&#'(-*;&'9$B;*7&+8%'*&/-7$Q*+&'&/%*%&/F*-+&,&8%&''.@