Dense-plasmas opacity calculations using average-atom (AA) … · 2017-07-11 · Dense-plasmas...

Dense-plasmas opacity calculations using

average-atom (AA) models

(1)CEA, DAM, DIF, F-91297 Arpajon, France.(2)Laboratoire ''Interactions, Dynamiques et Lasers'', UMR 9222, CEA-CNRS-Université Paris-Saclay, Centre d'Etudes de Saclay, F-91191 Gif-sur-Yvette Cedex, France.

Current challenges in the physics of white dwarf stars

Santa Fe, June 15th, 2017 R. PIRON(1) | T. BLENSKI(2)

2/38R. PIRON, T. BLENSKI | Santa Fe | June 15th, 2017 |

INTRODUCTIONAre actual AA models relevant to the physics of WD ?

0.6M⊙ DA-type white dwarf data after [Fontaine et al., EPJ Web of Conf. 43, 05001 (2013)]Solar data after [Bahcall, Pinsonneault Rev. Mod. Phys. 67, 781-808 (1995)]



Carbon (~99% of mass)(estimates using theThomas-Fermi-Dirac model)

0.6M⊙ DA-type white dwarf data after[Fontaine et al., EPJ Web of Conf. 43, 05001 (2013)]

WD-relevant domain



Helium (~1% of mass)(estimates using theThomas-Fermi-Dirac model)


WD-relevantdomain



Hydrogen (~0.01% of mass)(estimates using theThomas-Fermi-Dirac model)


WD-relevant domain


INTRODUCTIONThe opacity issue in the physics of WD

The AA models we will deal with address partially ionized, moderately coupled plasma, at any degeneracy. → Part of the WD-relevant domain does fall into this category.

Opacities are a long-standing issue in the modeling of WD cooling [Marshak, 1940] and WD accretion phenomenon (polar) [Frank, King, Raine, Accretion power in astrophysics (2002)]

Carbon/Oxygen core + Helium shell▶ Competition between conduction and radiative transfer in highly ionized, degenerate matter

→ quite simple models could be relevant (OCP, cold plasma)

Thin but opaque atmosphere regulating the cooling through radiation transfer▶ Hydrogen (DA spectral type) or Helium (DB/DO), with possible traces of higher-Z elements (DZ?)

→ Silicium, Iron... [Holberg et al., ApJ 416, 806 (1993) ; Barstow et al., MNRAS 279, 1120 (1996)]▶ Carbon (DQ) [Pelletier, Fontaine et al. ApJ 307, 242 (1986) ; Dufour et al. Nature 450, 522 (2007)]▶ Cooling problem → relevance of the high-density/low-temperature/low-frequency regime

– quasi-neutral elements, negative ions, molecular effects in the coolest, outermost shells of the atmosphere [Iglesias, Rogers, Saumon, ApJ 569, L111 (2002) ; Ferguson et al., ApJ 623, 585 (2005)]

– Strong Stark broadening of lines, Zeeman in some cases (Magnetic WD)▶ But higher-frequency radiative properties are also relevant

– hard-X-ray irradiation by an accretion shock (~10-50keV)– re-emission of soft-X-rays by the atmosphere [Ramsay, Cropper MNRAS 347, 497 (2004) ; König, Beuermann,

Gansicke A&A 449, 1129 (2006)]– Radiation transfer from accretion impact zones (blob model) [King ApJ 541, 306 (2000)]

Disclaimer: I am not astrophysicist! Please, do not hesitate to correct if I made any mistakes.


OUTLINE OF THIS TALK

In this talk we will focus on screening of ions in dense plasmas through the average-atom (AA) model, and its use to calculate opacities.

Models of atoms in a plasma: a short, non-exhaustive survey

Basics of opacity calculations: independent-particle linear response

→improvement 1: accounting for fluctuactions

→improvement 2: accounting for collective effects

CEA | 10 AVRIL 2012

MODELS OF ATOMS IN A PLASMA :A SHORT, NON-EXHAUSTIVE SURVEY


THOMAS-FERMI « ION-IN-CELL » MODEL

Electrondensity

radius

Central ion charge

Neutral Wigner-Seitz sphere

The Thomas-Fermi model was the first statistical model of an atom in a plasma

[Feynman, Metropolis, Teller, PR 75, 1561 (1949)]The TF model can be formulated as the self-consistent field model of hydrostatic equilibrium of an ideal electron gas “seeing” the potential of a central ion in a neutral cell.

▶ This relies on a local density approximation to the kinetic-entropic term of the free energy

The model is variational: it can be obtained through a minimization of the model electron free-energy per ion. The model fulfills the virial theorem.

▶ One can rigorously write the thermodynamic quantities → useful for the equation of state

The restriction of the equations to the WS sphere yields a versatile model: one can either interpret the ion cell as periodic [Slater, Krutter, 1935] or as embedded in a jellium [Liberman, 1979].

The model was also used for the calculation of radiative properties, either from rigorous approaches to the dynamic semiclassical model [Ball, Wheeler, Firemen, 1973], [Ishikawa, Felderhof, Blenski, Cichocki, 1998], [Caizergues, Blenski, Piron, 2014] or from a heuristic use of the TF potential [Latter, 1955]

▶ Need for quantum extension of the TF model



Electrondensity

radius

Central ion charge



[Feynman, Metropolis, Teller, 1949]The TF model can be formulated as the self-consistent field model of hydrostatic equilibrium of an ideal electron gas “seeing” the potential of a central ion in a neutral cell.









Electrondensity

radius

Central ion charge

Spread-out non-central ions

+ uniform electron density

cavity



[Feynman, Metropolis, Teller, 1949]The TF model can be formulated as the self-consistent field model of hydrostatic equilibrium of an ideal electron gas “seeing” the potential of a central ion in a neutral cell.








LIBERMAN'S « INFERNO » MODEL

Quantum electrondensity

radius

Central ion charge


+ uniform electron density

cavity

step

The electron and ion charge densities cancel outside the

WS sphere


The INFERNO model: a first attempt to extend the TF model to quantum calculations

[Liberman, PRB 20, 4981 (1979)]:

The model was built in a heuristical way, substituting the semiclassical electron density by a DFT-like quantum density inside the ion cell.

▶ Indeed, the “step” in the density is related to the non-variational character of this model, which does not fulfill the virial theorem.

As a consequence of the heuristic character of this model, several definitions of the thermodynamic quantities can be set, leading to different variants of the model (model A, T [Liberman, PRB 20, 4981 (1979)], use of the virial pressure [Bar-Shalom, Oreg, HEDP 3, 12 (2007)])

However, this model yields equations that are relatively simple to implement, notably due to their restriction to the finite-size WS sphere.


VARIATIONAL AVERAGE-ATOM MODEL (« VAAQP »)

Quantum electrondensity

Non-neutral WS sphere

radius

Central ion charge


+perturbed electron density

cavity

The VAAQP model solves the problem of thermodynamic consistency [Blenski, Cichocki, PRE 75, 056402 (2007)], [Piron, Blenski, Cichocki, PRE 83, 026403 (2011)]

In this model, the condition of overall neutrality replaces the neutrality of the WS cell. This was already at the origin of F. Perrot's unpublished model AJCI.

The VAAQP model is variational and fulfills the virial theorem. This was not the case of AJCI.

In a large temperature/density domain, the VAAQP model confirms (and therefore justifies) INFERNO results

▶ Discrepancies appear in the high-density/low-temperature region.

This model opened the way to attempts to go beyond the cavity model for ion-ion correlations [Starrett, Saumon, PRE 85:026403 (2012)], [Chihara, HEDP 2016]


DIFFERENCES BETWEEN THE MODELS

Mean ionization of aSi plasma at 5eV temperature(significant differences appear above 1/10 solid density)

Self-consistent density and potential for Be at 2eV temperature and solid density

Al principal Hugoniot(with an OCP ion contribution added)

CEA | 10 AVRIL 2012

BASICS OF OPACITY CALCULATIONS : INDEPENDENT-PARTICLE LINEAR RESPONSE


INDEPENDENT-PARTICLE LINEAR RESPONSE

The photoabsorption cross-section is directly related to the density susceptibility (or, in other words, to the dynamic atomic polarizability):

The crudest approach is to use the susceptibility obtained from the independent-particle approximation

▶Independent-particle approximation:

▶This leads to the well-known AA Kubo-Greenwood formula for the photoabsorption cross-section:

▶Within this approximation, the photoabsorption can be decomposed into bound-bound, bound-free,

and free-free contributions

▶This decomposition is used in nearly ALL opacity calculations, from AA to detailed line accounting !

(in the dipole approximation)

the being the Kohn-Sham AA orbitals (i.e. one-electron wavefunctions)

(makes the link with the DFT)


EXAMPLES OF OPACITY

Role of bound-bound, bound-free, and free-free contributions

(solid density)

(4x solid density)

(1/100 solid density)

opacity of a Si plasma at 5eV temperature(simplified-DCA calculation from the VAAQP code,

without line broadening and collective effects)


EXAMPLES OF OPACITY

Infrared region: essentially free-free contributes

UV / soft X-rays: Δn=0 transitions when some shells are open X-rays: Δn≠0 transitions

and bound-free

Pressure-ionized plasma: shells tend to be either full or empty, bound-bound transitions disappear

Free-free can also matter here

Role of bound-bound, bound-free, and free-free contributions

opacity of a Si plasma at 5eV temperature(simplified-DCA calculation from the VAAQP code,

without line broadening and collective effects)


BOUND-BOUND / BOUND-FREE CONTRIBUTIONDirect use of the independent-particles linear response

Si plasma at 5eV, 0.236 g.cm-3 (AA calculation from the VAAQP code, with arbitrary 0.1 eV broadening)

What you can get from the linear-response calculation

▶ Self-consistent static screening effect on the 1-electron

energies, i.e. « continuum lowering », Inglis-Teller limit,

and pressure ionization

Among features that are not included:

▶ Energy positions of the bb transitions and bf thresholds

are approximate:

– LDA energies→ this can be improved by using the

HF expression [Wilson, Liberman, Springer, JQSRT 54, 857 (1995)]

– no orbital relaxation effects (1-electron energies are

the same in the initial and final state)

→ important for spectroscopy application

▶ Dirac-δ for each bb transition (In the figure, they are

broadened by a 0.1eV arbitrary width)

→ neither physical broadening nor statistical

broadening, i.e. effects of the fluctuations


FREE-FREE CONTRIBUTIONDirect use of the independent-particles linear response

What you can get from the simple linear-response calculation

▶ Realistic calculation of free-free opacities

(static gaunt factor only)

Among features that are not included :

▶ Dynamic self-consistent effects.

In theories based on an effective system of

independent particles, the collective behavior

stems from the self-consistency. Here the

equilibrium system is computed self-consistently,

but the linear response is not.

i.e. particle-hole modes only, no collective modes

Typically, collective effects are expected to play a

role at photon frequencies near the plasma

frequency

(Ehrenfest theorem [Bethe, Salpeter, Quantum Mechanics of one- and two-electron atoms (1957)])

The free-free contribution is calculated using the acceleration form of the dipole matrix element, which leads to absolutely convergent integrals.

Si plasma at 5eV, 2.36 g.cm-3

(free-free opacity calculation from the VAAQP code)


BOUND-BOUND / BOUND-FREE / FREE-FREERemark on the continuity of observables

3s–3p2s–3p2p–3d

Opacity of a Si plasma at 5eV temperature: delocalization of the 3p subshell

(VAAQP calculation, uncorrected LDA)

3p–continuum

A careful treatment of resonances in the continuum allows one to preserve the continuity of the opacity and equation of state.

In a screened potential, the number of bound state is finite. When the density changes, a bound state can disappear. It is however compensated by a resonance in the continuum.

Example: at 5eV, the 3p orbital of Si gets delocalized between 0.96 and 0.98 g.cm-3 in the VAAQP model. It has a mean population of 1.8 e–

▶ The 2s–3p and 3s–3p transitions are replaced by

contributions of the 3p resonance to the 2s and 3s

bound-free spectra

▶ The 3p bound-free contribution is replaced by the

contribution of the 3p resonance to the free-free

spectrum

Any improvement in which a distinction is made between bound and free electrons, will lead to slight discontinuities.


BOUND-BOUND / BOUND-FREE CONTRIBUTIONCorrection of the LDA energies using the HF formula

Si plasma at 5eV, 0.236 g.cm-3 (AA calculation from the VAAQP code, with arbitrary 0.1 eV broadening)

Transition or threshold energy (eV)

transition FAC* HF corrected** LDA**

3s – 3p 8.9 9.8 [+10%] 8.6 [-3%]

3s – cont. 44.8 43.7 [+3%] 37.3 [-17%]

2p – 3d 121.8 125.6 [+3%] 112.5 [-8%]

2p – cont. 148.8 151.3 [+2%] 130.2 [-13%]

2s – 3p 163.1 163.8 [+0.5%] 143.8 [-12%]

2s – cont. 202.6 202.4 [-0.1%] 172.5 [-15%]

1s – 3p 1852.5 1869.2 [+0.9%] 1780.9 [-4%]

1s – cont. 1893.7 1909.8 [+0.8%] 1809.6 [-4%]

* data from FAC [Gu, Can J Phys 86, 675 (2008)] for the configuration 1s2 2s2 2p6 3s1 of Si

**data from VAAQP for Si at T=5eV, ρ=2.36 10-4 g.cm-3

(average configuration: 1s2 2s2 2p6 3s1)

Using the Hartree-Fock expression of the configuration-averaged energy in order to calculate the transition energies and photoionization thresholds allows one to obtain more realistic spectra

▶ At low density, an agreement within a few percent

with the FAC code (isolated ion model) is found.

▶ Most of the energy differences that remain are due to

the effect of orbital relaxation

CEA | 10 AVRIL 2012

IMPROVEMENT :ACCOUNTING FOR THE FLUCTUATIONS


ACCOUNTING FOR THE FLUCTUATIONSSimplified DCA models

In principle, the Average-Atom model only gives the average equilibrium state of the atom.However, absorption and emission of light by atoms occurs on short timescales, which makes them sensitive to fluctuations around the average.

A way to account for the fluctuations is to construct a detailed configuration (DCA) model, starting from the quantities calculated from the average-atom

▶ Terms of the configuration energies related to bound electrons are taken from the HF formula

▶ Terms involving free electrons play a non-negligible role in the configuration energies.

Several simplified configuration-energy models can be proposed

▶ Starting from the INFERNO model, one can scale the free-electron density according to the

ion charge,see for instance [Perrot PRA 35, 1235 (1987)], probably some LANL, LLNL codes are also

based on a similar model

▶ Starting from the variational model of configurations [Blenski, Cichocki, PRE 75, 056402 (2007)], a

simplified model using only average-atom quantities was proposed [Piron, Blenski, HEDP 9:702 (2013)]

The simplified DCA model can still involve a large number of configurations, statistical treatments can

therefore be useful

▶ Gaussian approximation [Perrot, Physica A 150:357 (1988)], [Piron, Blenski, HEDP 9:702 (2013)]

▶ Super Transition Arrays [Bar-Shalom PRA 40, 3183 (1989)], [Blenski, Grimaldi, Perrot PRE 55, R4889 (1997)]


ACCOUNTING FOR THE FLUCTUATIONSExamples of charge states distributions

Charge state distributions of Iron at 40eV temperature, for various densities(simplified-DCA model from the VAAQP code)


ACCOUNTING FOR THE FLUCTUATIONSConsequences on bound-bound transition arrays

2p–3d transition array with an arbitrary additional broadening of 0.5 eV

Si plasma at 5 eV temperature, 0.00236 g.cm-3 matter density(simplified-DCA model from the VAAQP code)

Charge state distribution

CEA | 10 AVRIL 2012

IMPROVEMENT :ACCOUNTING FOR COLLECTIVE EFFECTS(EMISSION OF PLASMONS, CHANNEL MIXING, etc.)


ACCOUNTING FOR COLLECTIVE EFFECTSWhere could collective effects matter ?

Carbon (~99% of mass)(estimates using theThomas-Fermi-Dirac model)


WD-relevant domain



Helium (~1% of mass)(estimates using theThomas-Fermi-Dirac model)


WD-relevantdomain



Hydrogen (~0.01% of mass)(estimates using theThomas-Fermi-Dirac model)


WD-relevant domain


ACCOUNTING FOR COLLECTIVE EFFECTSA heuristic approach to recover the Drude behavior

In [Perrot, LPB 14, 731 (1996)], a very simple, heuristic approach is proposed in order to recover the Drude-like collective behavior at low frequency (very similar approaches are also described in [Johnson, Guet, Bertsch JQSRT 99, 327 (2006)] and [Johnson, HEDP 5, 61 (2009)])

One identifies the collision frequency from the theory of weak scattering in the expression of the f-f conductivity or photoabsorption cross-section :

By comparison with Ziman's formula for static conductivity [Ziman, Phil Mag 6,1013 (1961)], one deduces a correcting factor, which allows one to recover Ziman's result at low frequency:

This factor forces a Drude-like behavior

Here, for the sake of consistency of the presentation, we present the effect on the opacity, but below the plasma frequency, we should rather speak in terms of dynamic conductivity.

Si plasma at 5eV, 2.36 g.cm-3

(free-free opacity calculation from the VAAQP code)


SELF-CONSISTENT LINEAR RESPONSEIntroduction, RPA

Using a Time-Dependent Density Functional formalism to calculatethe linear response leads to a Random Phase Approximation Self-Consistent scheme [Stott, Zaremba, PRA 21:12 (1980), Zangwill, Soven PRA 21:1561 (1980)]:

Using this kind of approach, one can in principle account for the channel mixing (mixing between bound-bound, bound-free and free-free channels) and collective phenomena.

These equations are solved in [Zangwill, Soven PRA 21:1561 (1980)] to obtain photoabsorption cross-sections of neutral rare gases, which do not involve free-free channels.

In the case of a plasma, difficulties arise due to the contribution of free-free transitions, which involve non-localized wavefunctions.

The cluster expansion, on which the VAAQP model is based, is an efficient method to substract the non-integrable contribution of the homogeneous medium.

(in the Adiabatic LDA)


SELF-CONSISTENT LINEAR RESPONSESum rule

(induced dipole)

(SC induced density) (SC perturbing potential)

(equilibrium potential) (equilibrium density)

(“Bremsstrahlung” term) (“collective” term)(cold plasma dielectric function)

From this self-consistent linear-response formalism, a sum rule for the atomic dipole was derived in [Blenski, JQSRT 99:84 (2006)]:

In the independent particle approximation, this sum rule is equivalent to the well-known relation between length and acceleration form of the oscillator strength (Ehrenfest theorem)

This sum rule provides a way of calculating the atomic dipole considering only localized quantities.

This sum rule can be used to define two contributions to the photoabsorption cross-section▶ A contribution that we can associate to Bremsstrahlung, which is the only one

considered in the independent particle approximation

▶ A contribution that we can associate to the effect of collective phenomena


SELF-CONSISTENT LINEAR RESPONSESC linear response of the Thomas-Fermi model

In the TF approximation, the VAAQP model is equivalent to the usual TF ion-in-cell model.

The TF dynamic properties can be derived in the framework of Bloch's hydrodynamics [F. Bloch Z. Phys. 81, 363 (1933)]

Using the method of [Ishikawa, Felderhof Blenski, Cichocki, JPP 60:787 (1998)], the self-consistent linear response of the finite-temperature TF model was studied in [Blenski, Piron, Caizergues, Cichocki, HEDP 9:687 (2013), Caizergues, Blenski, Piron, HEDP, 12:12 (2014)]. It was also shown that the sum rule remains valid in the TF approximation

The dynamic TF model is a good theoretical test-bed, but leads to unphysical results, especially at high frequencies [Ishikawa Physica A 253, 541 (1998)]

opacity of an Al plasma at 10eV temperature, and 2.7 g.cm-3 (solid density),calculated from the SC linear response of the finite-temperature TF model

~200 eV(beware: TF calculation !)

negative Bremsstrahlung contribution here


SELF-CONSISTENT LINEAR RESPONSESC linear response of the Thomas-Fermi model

opacity of an Al plasma at 10eV temperature, and 2.7 g.cm-3 (solid density),calculated from the SC linear response of the finite-temperature TF model

In the TF approximation, the VAAQP model is equivalent to the usual TF ion-in-cell model.

The TF dynamic properties can be derived in the framework of Bloch's hydrodynamics [F. Bloch Z. Phys. 81, 363 (1933)]

Using the method of [Ishikawa, Felderhof Blenski, Cichocki, JPP 60:787 (1998)], the self-consistent linear response of the finite-temperature TF model was studied in [Blenski, Piron, Caizergues, Cichocki, HEDP 9:687 (2013), Caizergues, Blenski, Piron, HEDP, 12:12 (2014)]. It was also shown that the sum rule remains valid in the TF approximation

The dynamic TF model is a good theoretical test-bed, but leads to unphysical results, especially at high frequencies [Ishikawa Physica A 253, 541 (1998)]


CONCLUDING REMARKS

Opacity calculations from AA models are useful to the calculation of broad-band radiative properties such as Rosseland mean.

▶ Progress in dense-plasma modeling through AA models can lead to progress in opacity calculations (not only in equation of state calculations)

▶ However AA opacities are not well-suited to precise spectrosopy of low density plasmas

Simplified configuration models can be used to account for fluctuations

▶ Treatment of continuum electrons can be made in different ways

First-principle accounting for the dynamic screening remains difficult to address

▶ The semiclassical (TF) dynamic response model was studied

▶ Although progress was achieved in the understanding of the problem, the study of the self-consistent linear response in the quantum case still lead to unconclusive results [Caizergues, Blenski, Piron, HEDP 18, 7 (2016)]


ACKOWLEDGEMENTS

Bogdan Cichocki (collegue from Warsaw university), Clément Caizergues (former PhD student)

Didier Saumon, Charles Starrett, Jerome Daligault, for giving us this opportunity to meet

Emeric Falize, Gilles Fontaine for their opinion on some slides, as well as for useful discussions on the physics of white dwarfs and accretion phenomena.

All of the participants, for the many things I’m learning during this workshop

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