Lineshape modeling for collisional-radiative calculations

Transcription

1 Lineshape modeling for collisional-radiative calculations Evgeny Stambulchik Faculty of Physics, Weizmann Institute of Science, Rehovot , Israel ICTP IAEA Workshop on Modern Methods in Plasma Spectroscopy March 23 27, 2015 Trieste, Italy

2 Introduction Collisional-radiative (CR) modeling is widely used to diagnose laboratory and astrophysical plasmas through interpreting measured spectra. Lineshape analysis is an invaluable tool for plasma diagnostics. It allows for nonintrusively inferring plasma properties. In addition, line broadening affects the radiation transfer and, hence, the level populations in non-optically-thin plasmas. Therefore, failure to include line broadening in CR calculations may result in severe degradation of their diagnostics power. However, accurate lineshape calculations are rather time-consuming; inlcuding them directly in CR calculations is unrealistic. Computationally effective approximate methods of lineshape modeling that retain a reasonably good accuracy are required.

3 Introduction (cont.) What is a reasonably good accuracy? Factor 2 or better in FWHM. It should usually be sufficient for level population dynamics and qualitative overall spectra. However, FWHM alone is not always sufficient. Opacity: Line wings are important, especially in astrophysics; radiation energy flow.

4 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

5 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

6 Processes that affect lineshapes Natural broadening (spontaneous radiative decay, autoionization, including the Auger effect): w ul (time of life of u ) 1 +(time of life of l ) 1 ; relevant phenomena Lorentzian shape Doppler broadening associated with thermal or non-thermal radiator motion: w ul v c ω0 ul ; Gaussian shape for thermal Electromagnetic fields Microfields due to the motion of electrons and ions (including impact processes like excitation etc) Macroscopic fields (external or due to collective plasma phenomena, e.g. waves) This is the complex one... and focus of this talk.

7 Stark effect :: Isolated and hydrogenlike lines E ij distance between dipole- talking levels; V ij = d ij F perturbation due to electric field F. Simple estimates assuming quasistatic picture is valid for perturber at a distance r; binary approximation: w st δe = δe(r) P(r)dr; P(r)dr = 1 4πr 2 dr 4πN p r 2 dr V 1 V 1 Quadratic effect (V ij E ij ): Linear effect (V ij E ij ): δe(r) = V 2 / E F(r) 2 1/r 4 δe(r) = V ij F(r) 1/r 2 w st N p r 2 r 4 dr N pr 1 min N p; Short-range (impact) collisions w st N p r 2 r 2 dr N pr max N 2/3 p ; Long-range collisions

8 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

9 Broadening of isolated lines According to [Baranger, 1958], a u l transition assumes Lorentzian shape with FWHM defined by w = N e vf(v)dv σ uu (v) + σ ll (v) + f u (v) f l (v) 2, 0 u u where F(v) is the (Maxwellian) electron velocity distribution, σ ik (v) is impact cross section from i to k, and f k (v) is elastic scattering amplitude. w w in + w el l l The inelastic part: w in N e vf(v)dv σ uu (v) + σ ll (v) or 0 u u l l w in = σ uu N e v v + σ ll N e v v u u l l

10 Broadening of isolated lines (cont.) Typically (except for near-threshold energies), w el < (or )w in. w w in = σ uu N e v v + σ ll N e v v u u l l Compare to the natural broadening: w = (time of life of u ) 1 + (time of life of l ) 1 relevant phenomena Generalizing to other impact mechanisms (ionization, recombination of any kind), w depop. rate of u + depop. rate of l all mechanisms all mechanisms N.B.: All these rates are calculated anyway in CR models.

11 Broadening of isolated lines (cont.) One caveat If the partial width w ik of level i due to level k becomes comparable to E ik σ ik N e v v E ik, the isolated line assumption is no longer valid. Continuing using the Baranger formalism will bring unphysical results (overestimated broadening). A simple solution: w ik = min { σ ik N e v v, E ik } (A better approach will be discussed below).

12 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

13 Standard theory of line broadening In the standard theory (ST), the ions are usually given a [quasi]static role, while electrons are dynamic (impact): I(ω) = 1 π Re Tr df i W(F i ){d [iω ih s (F i ) + φ e (F i )] 1 d} av 0 Alternatively, if applicable, ions may be treated in the impact approximation, like electrons (φ e φ e + φ i ): I(ω) = 1 π Re Tr{ d[iω ih 0 + φ e + φ i ] 1 } av Intermediate cases?

14 Ion dynamics Significant disagreements between theory and experiments in shapes of Lyman and Balmer α and β. Hints for ion motion. H β [Kelleher and Wiese, 1973] Ly α [Grutzmacher and Wende, 1977]

15 Ion dynamics (cont.) Ratio between the quasi-static Stark width and the typical frequency of the microfield fluctuations: R = w st w dyn. w st ( d u d l )F 0 /, R 10 1 T = 1 ev Balmer β, protons Lyman α, protons w dyn v r = (kt/m p) 1/2 ( ) 4π 1/3 3 N p 0.1 Lyman α, electrons N e (1/cm 3 )

16 Computer simulations :: description The closest to ab initio calculations; since [Stamm and Voslamber, 1979]. The shape of a spectral line is calculated in three steps: The perturbing fields are simulated using the Particle Field Generator (PFG), by calculating the motion of a finite number of interacting electrons and ions (of a few types). Using this field as a perturbation, the emitter oscillating function is calculated by the Schrödinger Solver (SS). The power spectrum of the emitter oscillating function is evaluated using the Fast Fourier Transformation (FFT) method, giving the spectral line profile. We use a specific implementation [Stambulchik and Maron, 2006].

17 Computer simulations :: Scheme PFG (Particle Field Generator) N-body simulation + External fields SS (Schrödinger Solver) Line-shape calculation Û(t) <D (t)> FFT (Fast Fourier Transform)

18 Computer simulations :: Method The Hamiltonian of the atomic system: H = H + V(t). The perturbation V(t) is due to the plasma electric field (simulated by the PFG) and external electric and magnetic fields. We solve the Schrödinger equation idψ(t)/dt = HΨ(t) using the time-development operator U in the interaction representation: idū(t)/dt = V(t)Ū(t).

19 Computer simulations :: Method (cont.) The evolution of the dipole operator D(t) is then obtained: D(t) = U(t) D(0)U(t). The Fourier transform of the dipole operator D(ω) is further used to calculate the line spectrum: I λ (ω) ω 4 fi e λ Dfi (ω) 2. i f The angle brackets denote an averaging over several runs of the code (which corresponds to the averaging over an ensemble of emitters).

20 Computer simulations :: Features Interactions between plasma particles are included. Degenerate and non-degenerate cases are treated correctly within the same framework. Ion-dynamical effects are taken into account naturally. Addition of external magnetic and electric fields is possible. Very powerful but also very computation-resource intensive. Alternatives?

21 Quasi-contiguous (QC) approximation Static Stark effect of the H 9 line F = 10 kv/cm ev 10 ev 100 ev Rectangle, 100 ev ν, cm -1 What seems to be a rather complex pattern gradually becomes a simple rectangle.

22 Quasi-contiguous (QC) approximation Static Stark effect of Ly 9 π σ Intensity -10 -q QC q q QC Intensities of the π and σ components form two parabolae, which, on average, can be substituted with a simple rectangular shape.

23 Quasi-contiguous (QC) approximation Therefore: I (0) n I n (ω) = 2α n F/ for ω α n F 0 for ω > α n F, where I (0) n is the total line intensity, and α n is the linear-stark-effect coefficient: α n = 3 2 (n2 1) ea 0 Z. Generalization for n > 1: α nn = 3 2 (n2 n 2 ) ea 0 Z.

24 QC approximation: quasistatic shape Convolution with a microfield distribution W(F): I(ω) = I (0) nn ω/α n W(F)dF 2α nn F/ I(0) nn L qs (ω), or, with the reduced field strength β = F/F 0 and detuning ω = ω/ 0, L qs ( ω) = 1 H(β) 2 β dβ, where H(β) = W(F)/F 0, 0 = α nn F 0, and ( ) 2/3 4 F 0 = 2π Z p en 2/3 p (the Holtsmark field). 15 β

25 QC approximation: quasistatic shape Ideal plasma Holtsmark distribution: H(β) = 2 π β x sin(βx) exp( x 3/2 )dx. 0 L qs ( ω) = S( ω), where the S function is defined as S( ω) = 1 π 0 cos( ωx) exp( x 3/2 )dx. [Stambulchik and Maron, 2008]; also corrections due to moderate plasma coupling.

26 QC approximation: dynamical effects Quasi-static width: Typical field frequency: w dyn = v r = w qs = 2 ω 0 1/2 0. Introduce a quasistaticity factor f: f = kt m p R R + R 0, ( 4πNp 3 ) 1/3. where R = w qs w dyn and R 0 is a constant of the order of unity. The full Stark width then w qs for R 1, w = f w qs wqs/w 2 dyn N p / T for R 1.

27 QC approximation: examples We compare QC results with results of a computer simulation modeling (SimU) [Stambulchik and Maron, 2006], applied to H Balmer lines, N e = cm 3, kt = 0.16 ev D Balmer lines, N e = cm 3, kt = 4 ev Ne Lyman lines in a D plasma with N e = cm 3, kt = 1000 ev FWHM (cm -1 ) QC SimU n FWHM (cm -1 ) QC SimU n FWHM (ev) QC SimU n 6 7 8

28 QC-FFM :: When FWHM is not enough Applying frequency-fluctuation model (FFM) [Calisti et al., 2010]: L( ν; ω) = 1 π Re J( ν; ω) 1 νj( ν; ω), where Lqs ( ω )d ω J( ν; ω) = ν + i( ω ω ) ν w dyn / 0 For ideal one-component plasma: J( ν; ω) = [Stambulchik and Maron, 2013]. 0 dτ exp ( τ 3/2 i( ω i ν)τ).

29 QC-FFM :: Realistic plasmas Non-ideal OCP: where J( ν; ω) = 0 dτ e i( ω i ν)τ C qs (τ), C qs (τ) F { L qs ( ω) } (τ) = Im τ 1 F { β 1 W(β) } (τ). W(β) can be provided by computer simulations or a model, e.g., APEX [Iglesias et al., 2000]. Or from an analytical model: W(β) = 2 π β x sin(βx) exp [ f(x)]dx 0 (ideal plasma Holtsmark distribution f(x) x 3/2 ). Then, J( ν; ω) = A simple 1D integral! 0 dτ exp [ f(τ) i( ω i ν)τ].

30 QC + FFM: An example Ne X Lyδ D plasma, N e = cm -3, T = 1 kev Ions Electrons Total SimU E (ev) E (ev) Application: z-pinch diagnostics, talk of Yitzhak Maron tomorrow.

31 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

32 He I/II n = 3, 4 2, 3 transitions Fitting ab initio computer simulation spectrum with CR modeling. 2p - 4d,f (s) 2p - 4d,f (t) 2p - 3d (s) 2p - 3d (t) 2s - 3p (s) He II P-α 2s - 3p (t) SimU 4 ev, cm -3 CR 4 ev, cm -3 Intensity (arb. units) E (ev) N e is determined with a 20 30% accuracy. SimU CPU time: 1 week. Lineshapes in CR: 1 ms.

33 K α radiation + satellites Electron shells of Ti +hν 3d 4s 3p 3s 2p 2s 1s Ti VI inner-shell Grotrian diagram (using the FAC code of M.F. Gu) } 1s 1 2s 2 2p 6 3s 2 3p 5 4s 1 1s 1 2s 2 2p 6 3s 2 3p 5 3d Outer electrons stripped; Ti I Ti V. Inner shell ionization Ti VI. 440 Finite T a 3p electron excited. 420 Energy (ev) J = 1/2 J = 3/2 1s 1 2s 2 2p 6 3s 2 3p 6 1s 2 2s 2 2p 5 3s 2 3p 5 4s 1 1s 2 2s 2 2p 5 3s 2 3p 5 3d 1 1s 2 2s 2 2p 5 3s 2 3p 6

34 Ti VI Kα spectrum Ti VI K α spectrum at different bulk temperatures Intensity (arb.units) T = 5 ev T = 8 ev T = 10 ev T = 15 ev T = 20 ev E ph (ev) [Stambulchik et al., 2009]. Thousands of unresolved satellites. Application to diagnostics of WDM: talk of Ulf Zastrau on Thursday.

35 QC-FFM :: High-n series & continuum lowering The approach (closely following [Griem, 1997]): Calculate bound-bound (BB) n l n u shapes for a series of n u until FWHM exceeds E nu E nu +1 (the Inglis-Teller reasoning). (Optional) continue for a few more n u with the same width. Assume the free-bound (FB) edge at the E nu +1 energy. Convolve the FB continuum with the last (n l n u ) BB lineshape. Sum up. No sharp-fb-edge artifacts. No ionization potential depression assumed.

36 QC-FFM :: High-n series & continuum lowering (cont.) H I Lyman series, n e = cm -3, T = 1 ev (without the Boltzmann factor) 8 n = 4 BB FB Total SimU with all n 10 Intensity (arb. units) Energy (hartree) SimU: Hamiltonian with 385 fully interacting states. SimU CPU time: > 1 month. QC-FFM: 1 sec ( 3, 000, 000). E

37 QC-FFM :: High-n series & continuum lowering (cont.) Intensity (arb.units) 1e-10 1e-11 1e-12 1e-13 1e-14 1e-15 γ H I Balmer series T = 1 ev δ ε ξ η θ cm cm cm cm -3 1e-16 1e Energy (hartree) E

38 QC-FFM :: High-n series & continuum lowering (cont.) [Wiese et al., 1972] Intensity (arb. units) 1e-16 1e-17 1e-18 N e = N e = N e = N e = e λ (Å)

39 Outline of the talk 1 Spectral line broadening by plasmas Line broadening processes Stark effect: Isolated and hydrogenlike lines 2 Isolated lines Baranger formalism... and a bit beyond 3 Hydrogenlike transitions and intermediate cases Standard theory Computer simulations Quasi-contiguous approximation 4 Examples He spectrum Kα in WDM Continuum lowering 5 Conclusions

40 Conclusions Lineshape analysis is a very important tool for plasma diagnostics. Lineshapes affect the radiation transfer and level populations. Accurate detailed lineshape modeling requires substantial computational resources; but this should not be an excuse to abandon it in CR calculations altogether: Reasonably accurate and very fast approximate methods do exist. Use simplified Baranger formalism for isolated lines, quasi-contiguous approximation with frequency-fluctuation model for hydrogenlike transitions, and simple interpolations between the two for intermediate cases.

41 Thank you!

42 Bibliography Baranger, M. (1958). Phys. Rev., 112: Calisti, A., Mossé, C., Ferri, S., Talin, B., Rosmej, F., Bureyeva, L. A., and Lisitsa, V. S. (2010). Phys. Rev. E, 81(1): Griem, H. R. (1997). Principles of Plasma Spectroscopy. Cambridge University Press, Cambridge, England. Grutzmacher, K. and Wende, B. (1977). Phys. Rev. A, 16: Iglesias, C., Rogers, F., Shepherd, R., Bar-Shalom, A., Murillo, M., Kilcrease, D., Calisti, A., and Lee, R. (2000). J. Quant. Spectr. Rad. Transfer, 65(1 3): Kelleher, D. E. and Wiese, W. L. (1973). Phys. Rev. Lett., 31: Stambulchik, E., Bernshtam, V., Weingarten, L., Kroupp, E., Fisher, D., Maron, Y., Zastrau, U., Uschmann, I., Zamponi, F., Förster, E., Sengebusch, A., Reinholz, H., Röpke, G., and Ralchenko, Yu. (2009). J. Phys. A: Math. Theor., 42: Stambulchik, E. and Maron, Y. (2006). J. Quant. Spectr. Rad. Transfer, 99(1 3): Stambulchik, E. and Maron, Y. (2008). J. Phys. B: At. Mol. Opt. Phys., 41(9): Stambulchik, E. and Maron, Y. (2013). Phys. Rev. E, 87(5): Stamm, R. and Voslamber, D. (1979). J. Quant. Spectr. Rad. Transfer, 22: Wiese, W. L., Kelleher, D. E., and Paquette, D. R. (1972). Phys. Rev. A, 6(3):