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Details of Equation-of-State and Opacity Models
Igor V. Sokolov,
 The radiation is treated within the multi-group model: the radiation energy
density is assumed to be integrated over some frequency range, for each
group
 The multi-group radiation diffusion model is coupled to the hydrodynamic
motion. The following effects are accounted for in this way:
University of Michigan, and CRASH-team
General features.
Opacities.
The input parameters for both inline EOS and inline opacities are
Energies of different states (of atom and ions).
Bound-bound transitions: we use of the Kramers formula for the oscillator strengths. At the present
time we neglect the transitions with no change in the principal quantum number.
The states differ with:
charge number
 the radiation “advection”;
 the effect of the radiation pressure on the plasma motion (momentum and
energy sources in the hydrodynamic equations of motion);
 radiation diffusion;
 absorption and spontaneous emission.
 The models for Equation Of State (EOS) and for opacities are needed to close
the following hydrodynamic equations:
principal quantum number for ground and excited states
for the given state its energy is a total of the ionization energy and
excitation energy. The ionization energies are all taken from databases,
some of the excitation energies are approximated.
Populations and partition function ('ionization equilibrium'), which
determine relative and absolute densities of different atoms/ions.
Ionization equilibrium is calculated by minimizing the Helmholtz free energy.
We do this with account of:
 Effects of Fermi statistics in free electron gas
 The energy of electrostatic interaction ('the continuum lowering') in the
approximation of the Madelung energy.
Hydro equations + multi-group radiation diffusion
Excitation energy and its possible dependence on the plasma density
('pressure ionization')

   ( u)  0,
t
u
   u  u  (P  Prad )I   0,
t
From the properly minimized Helmholtz free energy we also derive the energy
density and pressure (if the equal ion and electron temperatures are assumed),
or electron pressure and electron energy density. They are used as the equation
of state in out hydrodynamic simulations.
u
u
 (   E)
2

t
 u 2

  u(   E  P  Prad ) Prad   u,
2


E g
 (E g )
1
   (uE g )    u 
d 
t
3

g
1
 E g  u 
3
cCg (Tg )
  (
Tg )  c PlanckCg (T)(T  Tg )
3 Ross
P  PEOS (E,  ), T  TEOS (E,  ),
1
Tg (E g ) : E g   B(Tg )d, Prad   E g ,
3
g
g
Cg (T) 

g
dB(T)
d, E g 
dT
3 / 2 

1
eV M a T 
F(T, )  TN a log   2  


Na
N
2

h
 a


 j1

2

  Ii  j E M 
j




0
TN a log  g j ge  exp 
 e ,
j 0


T






 e 
1.8Ry
ge  exp , E M 
,
 T 
(riono /a0 )

2
 E d
g

 me T 
4 3
Na
riono  V, e  2VT 2  Fe3 / 2 (ge ),
2h 
3



1
x dx
Fe (ge ) 
,

x
(  1) 0 ge e  1
3/2
2V  me T 
Z
 2  Fe1/ 2 (ge )
N a 2h 
3/2
Free-free transitions: approximated Gaunt factor, account for Fermi statistics effects in the electron
gas.
Bound-free transitions: account for Fermi statistics effects in the electron gas. Neglect the
absorption by core electrons from inner electron shells.
For multi-group opacities the absorption coefficients are averaged with Planckian weight function
over the frequency group range, to obtain the set of Planck opacities. The inverse of the absorption
coefficient is averaged with the Planckian temperature derivative, giving the inverse of the Rosseland
opacities.
The use of the same partition functions as the inputs for EOS and opacities ensures the
consistency of the overall model.
   bb   bf   ff ,

bb
line_ width
d 
e
2

m e c  ,n,m
f nm
N ,n
,
V
N ,n w n E n 
 bf  7.9 10 [m ] 

2 


V



1


 ,n,m
3
22
2
N c   1
22
2
 ff  7.9 10 [m ] 
9  V
v

2
2
e (N e /V )



2

  
(T,)  1 exp  (T,),
 T 

1 dB
 B(T,)d
 dT d
1
g
g
 Planck 
,

,
dB

B(T,

)d


Ross
d

g
dT
g
Bound-bound absorption in polyimide, T = 10 eV
1/ 3



3