Transcript EOS and Opacity Models in CRASH

EOS and Opacity Models in CRASH
Igor Sokolov
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
Page 2
Why do we need the EOS and opacity DATA?

   ( u)  0,
t
u
   u  u  (P  Prad )I   0,
t
u2
(
 E)


u2
2
   u( 
 E  P  Prad ) Prad   u,
t
2


E g
 (E g )
1
1
   (uE g )    u 
d   E g   u 
t
3

3

g
  (
cCg (Tg )
Tg )  c PlanckCg (T)(T  Tg )
3 Ross
P  PEOS (E, ), T  TEOS (E, ),

Eg 
Tg ( E g ) :

B(Tg )d,
Prad 
g

Cg (T) 

g
dB(T )
d,
dT
Eg 

g
1
 Eg,
3 g
E d
Page 3

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:
 
 
 2
P
P
 P
T P
Cs    
,     
 

P  T CV P T 
 S
• Therefore we also need:
- all thermodynamic derivatives…
- …along the ionization equilibrium curve.
• We need multi-group opacities now and frequencydependent opacities in the future.
Page 4
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.
Page 5
Why not use black-box external model?
Bremsstrahlung
Photoionization
from excited “H”
n=3 n=2
Broadened
“H-alpha”
100 group opacity for Be.
Density 0.1 g/cm3,
temperature 0.1 keV.
CRASH vs SESAME.
Photoionization
from “H”
Page 6
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
INPUTS:
Atomic
Concentration, Na
Trial Ne
Database
Ionization/
excitation
energies
Te
Pressure/
Energy density
Trial Te
Partition over ion charge number and principal
quantum number, for all mixture components.
Iterate
Averaging
Ne=(total) - (bound)
Crosssections,
lines
Electron
temperature
Averaging
Iterate
Pressure, energy density
Derivatives: specific heat...
Absorption coefficient
Electron heat conduction
Averaging
Multi-group opacities
OUTPUT
Page 8
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:
F
F F


0
N i1 N e N i
• 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.
Page 9
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.
Page 10
Comparison of the multi-group opacity model: Al
100 group opacities in Al
Density 0.1 g/cm3.
Temperature 0.1 keV.
CRASH vs SESAME
Bremsstrahlung
Lines from “Li”
Photoionization Photoionization
from “He”
from “Li”
Line – Planck Opacity/SESAME
Symbols – Planck Opacity/CRASH
Page 11
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.
Page 12