Line intensities and Collisional-Radiative Modeling H. K. Chung - PowerPoint PPT Presentation

Line intensities and 
 Collisional-Radiative Modeling H. K. Chung (many slides from Y . Ralchenko & J. Seely presentations at ICTP-IAEA School in 2017) http://indico.ictp.it/event/7950/other-view?view=ictptimetable https://www-amdis.iaea.org/Workshops/ICTP2017/ May 8 th , 2019 Joint ICTP-IAEA School on Atomic and Molecular Spectroscopy in Plasmas Trieste, Italy 1

Spectroscopic observables of matter states INTRODUCTION 2

Experimental X-Ray Spectra What are the spectral lines? Can we determine the plasma temperature and density? Other plasma properties? Unexpected discoveries? 3

Spectral Line Intensity (optically thin) Einstein coefficient or transition probability (s -1 ) Upper state density (cm -3 ) ( almost ) purely I N A h strongly depends = ⋅ ⋅ ν atomic parameters ij j ij ij on plasma conditions Photon energy (J) Energy emitted due to a specific transition from a unit volume per unit time j E ij i 4

5 fields of expertize to constitute plasma spectroscopic analysis INGREDIENTS OF SPECTROSCOPIC ANALYSIS 5

1) A Complete Set of Atomic Data BOUND-BOUND TRANSITIONS Energy levels of an atom Continuum A 3 A 1 → A 2 +hv 2 Spontaneous emission A 1 +hv 1 ↔ A 2 + hv 1 +hv 2 Photo-absorption or emission B 1 A 1 +e 1 ↔ A 2 +e 2 Collisional excitation or deexcitation Ground state of ion Z+1 BOUND-FREE TRANSITIONS A 1 B 1 +e → A 2 +hv 3 Radiative recombination B 1 +e ↔ A 2 +hv 3 Photoionization / stimulated recombination B 1 +e 1 ↔ A 2 +e 2 Collisional ionization / recombination A 2 B 1 +e 1 ↔ A 3 ↔ A 2 +hv 3 Autoionization / Dielectronic Ground state of ion Z Recombination (electron capture + stabilization) Atomic Physics Codes: FAC, HULLAC, LANL, GRASP-2K 6

2) Population Kinetics Modeling The key is to figure out how to manage the infinite set of levels and transitions of atoms and ions into a model with a tractable set of levels and transitions that represents a physical reality! (Completeness + Tractability + Accuracy) N max N max dn i n W n W ∑ ∑ = − + i ij j ji dt j i j i ≠ ≠ RR DR 2 W B J n C n W A B J n D n ( ) n = + + β + γ = + + + α + α + δ ij ij ij e ij ij e ij ji ij ji ji e ji e ji ji e ij A ij Spontaneous emission B ij Stimulated absorption B ij Stimulated emission C ij Collisional excitation D ij Collisional deexcitation γ ij Collisional ionization α ijDR Dielectronic recombination β ij Photoionization (+st. recom) α ijRR Radiative recombination δ ij Collisional recombination 7

3) Radiation Transport Radiation field carries the information on atoms in plasmas through population distributions • Radiation intensity I(r,n,v,t) is determined self-consistently from the coupled integro-differential radiation transport and population kinetic equations 1 [ c ( / t ) ( n )] I ( r , n , , t ) ( r , n , , t ) ( r , n , , t ) I ( r , n , , t ) − ∂ ∂ + ⋅ ∇ ν = η ν − χ ν ν • Emissivity η ( r,n ,v,t) and Opacity χ ( r,n ,v,t) and are obtained with population densities and radiative transition probabilities ⎡ ⎤ ( ) 3 2 * h / kT h / kT 2 h / c ( g / g ) n ( ) n e ( ) n n ( , T ) e − ν − ν ∑∑ ∑ ∑ η = ν α ν + α ν + α ν ⎢ ⎥ i j j ij i i e ν κ κ κκ ⎣ ⎦ i j i i > κ * h / kT [ n ( g / g ) n ] ( ) ( n n e ) ( ) − ν ∑∑ ∑ χ = − α ν + − α ν i i j j ij i i i ν κ i j i i > h / kT n n ( , T )( 1 e ) − ν ∑ + α ν − e κ κκ κ 8

4) Line Shape Theory for Radiation Transport • Line shape theory is a theoretically rich field incorporating quantum-mechanics and statistical mechanics Line shapes have provided • successful diagnostics for a vast range of plasma conditions – Natural broadening (intrinsic) – Doppler broadening (T i ) ν – Stark broadening (N e ) ν ij – Opacity broadening Ground state of ion Z – Resonance broadening (neutrals) 9

5) Particle Energy Distribution • Time scales are very different between atomic processes and classical particle motions : separation between QM processes and particle mechanics Is this a valid assumption? • Radiation-Hydrodynamics simulations – Fluid treatment of plasma physics • Mass, momentum and energy equations solved – Plasma thermodynamic properties – LTE (Local Thermodynamic Equilibrium) (assumed) • PIC (Particle-In-Cell) simulations – Particle treatment of plasma physics • Boltzmann transport and Maxwell equations solved – Electron energy distribution function – Simple ionization model (assumed) 10

First, identify lines and then obtain line intensities using a kinetics code and determine the temperature and density of the plasma emission region. 11

Statistical Distributions of Electronic Level Population Density 3 Representative Models POPULATION KINETICS MODELS 12

(1) Thermodynamic equilibrium ● Principle of detailed balance – each direct process is Photons balanced by the inverse ● radiative decay (spontaneous+stimulated) ↔ photoexcitation Atoms Electron ● photoionization ↔ photorecombination s Ions ● excitation ↔ deexcitation ● ionization ↔ 3-body recombination ● autoionization ↔ dielectronic capture 13

TE: distributions Four “systems”: photons , electrons , atoms and ions ● Same temperature T r = T e = T i ● We know the equilibrium distributions for each of them ● – Photons: Planck – Electrons (free-free): Maxwell – Populations within atoms/ions (bound-bound): Boltzmann – Populations between atoms/ions (bound-free): Saha 14

TE: energy scheme Maxwell Continuum Energy Ionization Saha energy Bound states Boltzmann N g E E ⎛ ⎞ − Boltzmann: 1 1 exp 1 2 ⎜ ⎟ = − ⎜ ⎟ N g T ⎝ ⎠ 2 2 e 15

Planck and Maxwell ● Planck distribution ● Maxwell distribution 3 2 E 1 2 E ⎛− ⎞ ( ) B E 1 / 2 ( ) = f E dE E exp dE = ⎜ ⎟ ⎜ ⎟ 2 2 E / T M h c e 1 1 / 2 3 / 2 T T − π ⎝ ⎠ e e 16

Saha Distribution A Z (+ e) ↔ A Z+1 + e (+ e) I 3 / 2 Z 1 N + g 2 mT 1 Z π − ⎛ ⎞ T Z 1 2 e e + = e ⎜ ⎟ Z 2 N g h N ⎝ ⎠ Z Z+1 Z e E E − i 0 − T g g e ∑ = e Z Z , i i Which ion is the most 3-body ionization abundant? recombination I Z 1 N + ( ) Z 1 ~ 10 1 >> dielectronic = autoionization Z T N capture e 17

Local Thermodynamic Equilibrium • (Almost) never complete TE : photons decouple easily…therefore, let’s forget about the photons! • LTE = Saha + Boltzmann + Maxwell • Griem’s criterion for Boltzmann: collisional rates > 10*radiative rates [ ] 3 1 / 2 3 14 ( [ ] ) ( [ ] ) 7 N cm 1 . 4 10 E eV T eV Z − > × Δ ∝ H I (2 eV): 2 × 10 17 cm -3 e 01 e C V (80 eV): 2 × 10 22 cm -3 • Saha criterion for low T e : [ ] H I (2 eV): 10 17 cm -3 5 / 2 1 / 2 3 14 6 ( [ ] ) ( [ ] ) N cm 1 10 I eV T eV Z − > × ∝ C V (80 eV): 3 × 10 21 cm -3 e z e 18

LTE Line Intensities No atomic transition data (only energies and statistical ● weights) are needed to calculate populations I N E A g E A E E ⎛ ⎞ Intensity ratio Δ Δ − ● 1 1 1 1 1 1 1 exp 1 2 ⎜ ⎟ = = − ⎜ ⎟ I N E A g E A T Δ Δ ⎝ ⎠ 2 2 2 2 2 2 2 e Or just plot the intensities on a log scale: ● Boltzmann plot g I N A E i AE exp( E / T ) = ⋅ ⋅ = − i e G ln( / I g AE ) E / T ln( ) G = − − i i e Aragon et al, J Phys B 44 , 055002 (2011) 19

Saha-LTE conclusions • Collisions >> radiative processes – Saha between ions – Boltzmann within ions • Since collisions decrease with Z and radiative processes increase with Z, higher densities are needed for higher ions to reach Saha/LTE conditions – H I: 10 17 cm -3 – Ar XVIII: 10 26 cm -3 ASD: can calculate Saha/LTE spectra!!! 20

Deviations from LTE Partial LTE (PLTE) for high excited states • Radiative processes are non- negligible – LTE: coll.rates (~n e ) > 10*rad.rates • Non-Maxwellian plasmas Radiative (~n -3 ) • Unbalanced processes Collisional (~n 4 ) • Anisotropy • External fields • … 21

(2) The other limiting case: 
 Coronal Equilibrium Low electron densities! Aug 21, 2017 22

Coronal Model • High temperature, low density and optically thin plasmas (J v = 0) • Excitations (and ionization) only from ground state… x • …and metastables x x • Does require a complete set of x collisional cross sections x • Do we have to calculate all direct and inverse processes?.. 23

Coronal Model Rates (N e2 ) << Rates (N e ) << Rates • (spontaneous) – 3-body recombination not important – Collisional processes from excited levels dominated by spontaneous radiative decays – Left with collisional processes from ground levels x and radiative processes from excited levels Atomic processes: • x x – Collis. ionization (including EA), x – Radiative recombination (including DR) – Collisional excitation x – Radiative decay (including cascades) Ions basically in their ground state • Ionization decoupled from excitation • 24