EOS and Opacity Models in CRASH

Igor Sokolov

Page 2

Our EOS and opacity functions support our UQ effort

• Outline– Why do we need EOS functions and opacities?– Why do we need the built-in model for them (not

tables?)– Scheme of calculation:

- Pressure, internal energy density, specific heat and other thermodynamic derivatives.

- Planck and Rosseland multi-group opacities.– Helmholtz free energy (statistical sum method).– Cross-model comparison

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Why do we need the EOS and opacity DATA?

€

Tg (Eg ) : Eg = B(Tg )dεε g

∫ , Prad =1

3Eg

g

∑ ,

Cg (T) =dB(T)

dTdε

ε g

∫ , Eg = Eεdεε g

∫

€

∂ρ∂t

+∇ ⋅(ρu) = 0,

∂ρu

∂t+∇ ⋅ ρu⊗u+ (P + Prad )I[ ] = 0,

∂(ρu2

2+ E)

∂t+∇ ⋅ u(ρ

u2

2+ E + P + Prad )

⎡

⎣ ⎢

⎤

⎦ ⎥= Prad∇ ⋅u,

∂Eg∂t

+∇ ⋅(uEg ) −1

3∇ ⋅u

∂(εEg )

∂εdε

ε g

∫ = −1

3Eg∇ ⋅u+

+∇ ⋅(cCg (Tg )

3κ Ross∇Tg ) + cκ PlanckCg (T)(T −Tg )

€

P = PEOS (E,ρ ), T = TEOS (E,ρ ),

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What do we need?

• Relationships between - mass density, - pressure, - electron pressure, - internal energy density, - electron temperature.

For xenon, beryllium and plastic!• For high-resolution schemes we need the sound speed, that

is:

• Therefore we also need: - all thermodynamic derivatives…- …along the ionization equilibrium curve.

• We need multi-group opacities now and frequency-dependent opacities in the future.

€

Cs =∂P

∂ρ

⎛

⎝ ⎜

⎞

⎠ ⎟S

=γP

ρ, γ =

ρ

P

∂P

∂ρ

⎛

⎝ ⎜

⎞

⎠ ⎟T

+T

CVP

∂P

∂T

⎛

⎝ ⎜

⎞

⎠ ⎟ρ

2

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There are tables, why do we develop models?

• First, it is interesting and attractive for all the involved sides. • For the uncertainty quantification: we use the model, based on:

- first principles; - specified assumptions (LTE); - controllable list of the input parameters

- ionization potentials; - excitation energies, multiplicities;- cross-sections; - oscillator strengths etc.

• Consistency: calculate opacities and EOS under the same assumptions.

• We benefit from a capability to verify our models with the “gold standard” models (such as SESAME). However, the use of black-box models sometimes appears problematic.

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Why not use black-box external model?

BremsstrahlungPhotoionizationfrom excited “H”

100 group opacity for Be.Density 0.1 g/cm3, temperature 0.1 keV.CRASH vs SESAME.

Broadened“H-alpha”

Photoionization from “H”

n=2n=3

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Why not use black-box external model?

• Similarity and good overall agreement of the “black-box” model with the “transparent” model.

• The partition functions in SESAME differ from those we use for EOS in CRASH, raising the issues of:

- consistency of EOS and opacity models; - utility of uncertainty quantification.

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EOS and opacity model: general scheme

Atomic Concentration, Na

INPUTS:Electrontemperature

Pressure/Energy density

Te Trial TeTrial Ne

Database

Ionization/excitationenergies

Cross-sections,lines

Partition over ion charge number and principal quantum number, for all mixture components.

Averaging

Pressure, energy densityNe=(total) - (bound)

AveragingIterate Iterate

Absorption coefficient

Averaging

Multi-group opacities

Electron heat conduction

OUTPUT

Derivatives: specific heat...

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Ionization equilibrium.

• The Helmholtz free energy includes the contributions from-Fermi statistics in the free electron gas;-Coulomb interactions (the Madelung energy);-Excited levels;-Pressure ionization (eliminate weakly-bound states)

• Minimizing the Helmholtz free energy yields:

• The ionization equilibrium includes the following effects:-The ‘continuum lowering’ affects not only the absorption spectrum, but also thermodynamics (via ionization).- The Fermi statistics effect, ‘the exchange interaction’, affects the pressure both directly and via the ionization.

€

∂F∂N i+1

+∂F

∂Ne−∂F

∂N i= 0

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Testing EOS

• Comparison with Hyades and SESAME models for EOS: the deviation in the calculated ionization degree is ~0.2.

• Should compare the partition functions, rather than the averages. More challenging is the comparison of opacities.

• Check a separate contribution from Coulomb interaction.

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Comparison of the multi-group opacity model: Al

Line – Planck Opacity/SESAMESymbols – Planck Opacity/CRASH

100 group opacities in AlDensity 0.1 g/cm3.Temperature 0.1 keV.CRASH vs SESAME

Bremsstrahlung

Photoionization from “He”

Photoionization from “Li”

Lines from “Li”

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Plans for a future work

• Extend the database - more realistic cross-sections for photo-ionization;

- more line information.• Improve the line broadening description.• Quantify the electron heat conduction model

- flux limiting- distribution over the ion charge number.