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Thermodynamic and Transport Properties
of Strongly-Coupled Degenerate ElectronIon Plasma by First-Principle Approaches
Levashov P.R.
Joint Institute for High Temperatures, Moscow, Russia
Moscow Institute of Physics and Technology, Dolgoprudny, Russia
*pasha@ihed.ras.ru
In collaboration with:
Minakov D.V.
Knyazev D.V.
Chentsov A.V.
Khishchenko K.V.
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Outline
•
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Strongly coupled degenerate plasma
Ab-initio calculations
Quantum-statistical models
Density functional theory
Quantum molecular dynamics
Path-Integral Monte Carlo
Wigner dynamics
Conclusions
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Ab-initio calculations
• Thermodynamic, transport and optical
properties
• Use the following information:
– fundamental physical constants
– charge and mass of nuclei
– thermodynamic state
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Extreme States of Matter
• Kirzhnits D.A., Phys. Usp., 1971
• Atomic system of units:
–
me = ħ = a0 = 1
• Extreme States of Matter (Kalitkin N.N.)
– P = e2 a04 = 294.2 Mbar
– E = e2 a0 = 27.2 eV
– V = a03 = 0.1482 A
• Hypervelocity impact, laser, electronic, ionic beams,
powerful electric current pulse etc.
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Coupling and Degeneracy in Plasma
• Coupling parameter:
G=
U pot
Ekin
e2
=
k BT r
1
- Strong coupling
G
(for electronic subsystem)
• Degeneracy parameter
2
2
p
nele3 , le2 =
me kBT
nele3
G
nele3
1
1
1
-
- Strong degeneracy
Strongly coupled degenerate non-relativistic plasma
(warm dense matter)
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EOS: Traditional Form
Adiabatic (Born-Oppenheimer) approximation (me << mi)
( { }) + F (V,T,{R })
( )
F V,T = Fe V,T, R
0
t
Free energy of
electrons in the
field of fixed ions
n
Free energy of ions
interacting with potential
depending on V and T
Traditional form of semiempirical EOS. Free energy
( )
( )
0
t
( )
( )
F V,T = Fc V + Fi V,T + Fe V,T
Cold curve
Thermal contribution Thermal contribution
of electrons
of atoms and ions
Semiempirical
expressions
DFT, mean atom models
Mean atom models or DFT calculations might help
to decrease the number of fitting parameters
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Electronic Contribution to
Thermodynamic Functions:
Hierarchy of Models
• Exact solution of the 3D multi-particle
Schrödinger equation
• Atom in a spherical cell
• Hartree-Fock method (1-electron
wave functions)
• Hartree-Fock-Slater method
• Hartree method (no exchange)
• Thomas-Fermi method
• Ideal Fermi-gas
r0
Z
4 3 1
p r0 =
3
n
Finite-Temperature Thomas-Fermi Model
 The simplest mean atom model
 The simplest (and fully-determined) DFT model
 Correct asymptotic behavior at low T and V (ideal Fermi-gas)
and at high T and V (ideal Boltzmann gas)
r0
Z
Poisson equation
Δ V   4 π Z δ r  
0  r
rV r  r  0  Z
 r0 
2
π
2 θ 
V r0   0
3 2
 V r   μ 
I1 

2
θ


dV r 
dr
r  r0
0
Thomas-Fermi model is realistic but crude at relatively low temperatures and pressures.
Thermal contributions to thermodynamic functions is a good approximation.
Feynman R., Metropolis N., Teller E. // Phys. Rev. 1949. V.75. P.1561.
HARTREE-FOCK-SLATER MODEL AT T>0
Nikiforov A.F., Novikov V.G., Uvarov V.B. Quantum-statistical models of hightemperature plasma. M.: Fizmatlit, 2000.
Atom with mean populations
N nl 
  l  
  exp  nl    
εnl – energy levels in V(r)
Potential:
From the radial Schrödinger equation
V r   Vc r   Vex r 
Poisson equation solution:
r
Z
 r 
Vc r  
 
r '  r ' dr '   r ' r ' dr



r
r
r
Exchange potential:
Vex r  
 r r  
 r 




r  
 
   .    


 

 

Iterative procedure for determination of ρ(r), εnl and V(r)
'



RADIAL ELECTRON DENSITY r2 (r)
BY HARTREE-FOCK-SLATER MODEL
160
140
Hartree-Fock-Slater
Au
ρ = 10-3 g/cm3
120
2
4r (r)
100
Thomas-Fermi
80
60
40
4·105 K
20
0
0.00
4·102 K
0.05
0.10
(r/r0)
1/2
0.15
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Density Functional Theory
• Thomas-Fermi theory is the density functional
theory:
kinetic energy
ETF [ n ] = C1 ò d rn ( r )
3
C2 ò d rn ( r )
3
43
53
external potential
+ ò d 3rVext ( r) n ( r) +
1 3 3 n ( r) n ( r¢)
+ d rd r¢
2
r - r¢
exchange energy
• Is it a general property?
Hartree energy
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Density Functional Theory
For systems with Hamiltonian
2
1
1
e
Hˆ = - å Ñi2 + åVext ( ri ) + å
2 i
2 i¹ j ri - rj
i
the following theorems are valid:
Theorem 1. For any system of interacting particles in an external potential Vext(r) the
potential Vext(r) is determined uniquely, except for a constant, by the ground state
particle density of electrons n0(r).
Therefore, all properties are completely determined given only the ground state
electronic density n0(r).
Theorem 2. A universal functional for the energy E[n] in terms of the density n(r) can
be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact
ground state energy of the system is the global minimum value of this functional, and
the density n(r) that minimizes the functional is the exact ground state density n0(r).
Hohenberg, Kohn, 1964
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Kohn-Sham Functional
The system of interacting particles is replaced by the system of non-interacting
particles:
EKS [n] = Ts [n]+ ò drVext ( r) n ( r) + EHartree [ n] + EII + EXC [n]
exchangeion-ion
interaction correlation
functional
kinetic energy
All many-body effects of exchange and correlation are included into EXC[n]
EXC [n] = Tˆ - Ts [n]+ Vˆint - EHartree [n]
true system
The minimization of EKS leads to the system of
Kohn-Sham equations
non-interacting system
Kohn, Sham, 1965
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Minimization in HFS and DFT
• In Hartree-Fock(-Slater) method we find
min WéëY i ( r)ùû
Y i (r)
• In DFT we find
min Wéën ( r)ùû
n(r)
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Density Functional Theory: All-Electron
and Pseudopotential Approaches
Full-potential approach: all electrons are taken into account (FP-LMTO)
(S. Yu. Savrasov, PRB 54 16470 (1996),
G. V. Sin'ko, N. A. Smirnov, PRB 74 134113 (2006)
At T > 0: occupancies are f (  ,  , T )  1 / 1  ex p  (    (  , T )) / T 
Pseudopotential approach: the core is replaced by a pseudopotential, the Kohn-Sham
equations are solved only for valent electrons (VASP)
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); 49, 14251 (1994).
G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).
EF
Core
electrons
T=0
EF
T>0
Calculations were made in the unit cell at fixed ions and heated electrons
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Aluminum. Cold Curve
Pressure
(Mbar)
Pressure (Mbar)
200
Al
Al is more compressible
in VASP calculations
than in FP-LMTO
FP-LMTO
150
VASP
100
50
0
0
2
4
6
8
10
12
Compression
/ratio
0
Levashov P.R. et al., JPCM 22 (2010) 505501
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Potassium. Cold Curve
K
Pressure (Mbar)
20
3s, 3p, 4s
VASP
15
10
FP-LMTO
3p, 4s
5
0
0
5
10
K is less compressible
in VASP calculations
than in FP-LMTO
Less number of valent
electrons leads to
bigger disagreement
with FP-LMTO
15
/0
Levashov P.R. et al., JPCM 22 (2010) 505501
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DFT-calculations of Fe(Te, V)
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Why can we use DFT for thermodynamic properties of electrons?
20
ThomasFermi with
corrections
e
15
Cv
e
¶F
( r ,Te )
e
2
T
PT = -r
- Pc
¶r
T
Heat capacity of W, V0
W
e
  E Te (  , Te ) 
C 


T
e

V
10
e
V
2
bcc
5
1
fcc
0
0
5
10
15
24x24x24 mesh of k-points
Nbands = 94
Ecut = 1020 eV
Te (eV)
• Cold ions, hot electrons
• Heat capacity is very close for fcc and bcc structures of W
• It should be close to unordered phase at the same density
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Tungsten, Ti = 0, V = V0.
Electron Heat Capacity
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0.8
Thomas-Fermi
8
0.4
VASP
IFG, Z = 6
4
Number of electrons
12
Isochoric heat capacity
Core electrons become
excited at ~3 eV
W, bcc
IFG, Z = 1
0
2
4
6
8
0.0
10
Temperature, eV
Levashov P.R. et al., JPCM 22 (2010) 505501
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Tungsten, Ti = 0, V = V0.
Thermal Pressure
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5
Thermal pressure, Mbar
W, bcc
• Pressure is determined
by free electrons only
• Interaction of electrons
should be taken into
account
Thomas-Fermi
4
3
IFG, Z = 6
2
1
0
IFG, Z = 1
0
2
4
6
8
Temperature, eV
10
12
Levashov P.R. et al., JPCM 22 (2010) 505501
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Thermal contribution of ions
Ab-initio molecular dynamics (AIMD) simulations
( )
( )
( )
( )
F V,T = Fc V + Fi V,T + Fe V,T
FAIMD (V,T ) - Fe (V,T ) - Fc (V ) + Fions,kin (V,T )
From AIMD
From AIMD
From DFT
Analytic
expression
But it’s better to use AIMD to calibrate the EOS by changing fitting parameters
Desjarlais M., Mattson T.R., Bonev S.A., Galli G., Militzer B., Holst B., Redmer R.,
Renaudin P., Clerouin J. and many others use AIMD to compute EOS for many substances
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Details of the AIMD simulations
• We use VASP (Vienna Ab Initio Simulation Package) - package for
performing ab-initio quantum-mechanical molecular dynamics
(MD) using pseudopotentials and a plane wave basis set.
• Generalized Gradient Approximation (GGA) for Exchange and
Correlation functional
• Ultrasoft Vanderbilt pseudopotentials (US-PP)
• One point (Γ-point) in the Brillouin zone
• The QMD simulations were performed for
108 atoms of Al
• The dynamics of Al atoms was simulated
within 1 ps with 2 fs time step
• The electron temperature was equal to the
temperature of ions through the
Fermi–Dirac distribution
• 0.1 < ρ / ρ0 < 3, T < 75000 K
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); 49, 14251 (1994).
G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).
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Al, ionic configurations
Full energy during the simulations
Normal conditions, solid phase
T=1550 K; ρ=2.23 g/cm3; liquid phase
108 atoms US pseudopotential; 1 k-point in the Brillouin zone; cut-off energy 100 eV; 1 step – 2 fs
Configurations for electrical conductivity calculations are taken after equilibration
Isotherm T = 293 K for Al
8
7
6
Pressure, Mbar
Al
- VASP
- Wang et al. (2000)
- Greene et al. (1994)
- Akahama et al. (2006), fcc
- Akahama et al. (2006), hcp
5
4
3
2
1
0
1.00
1.25
1.50
1.75
2.00
/
2.25
2.50
2.75
3.00
QMD results show good agreement with experiments and
calculations for fcc.
Shock Hugoniots of Al
Pressure – compression ratio
Shock Hugoniot of Al. Melting
8000
- VASP Hugoniot
- MPTEOS Hugoniot
- MPTEOS melting curve
- Exp. melting Boehler & Ross 1997
7500
7000
Temperature, K
6500
6000
5500
5000
4500
4000
Al
3500
3000
600
800
1000
1200
1400
1600
1800
2000
2200
Pressure, kbar
Higher melting temperatures by QMD calculations might be caused by
a small number of particles
NB: we can trace phase transitions in 1-phase simulation
Melting criterion for aluminum
Equilibrium configurations of Al ions at 5950 and 6000 K
Cumulative distribution
functions C(q6) and
C(w6) are extremely
sensitive measure of
the local orientation
order
This criterion may be
applied to other
structural phase
transitions
Rotational invariants of 2nd (q6) and 3rd (w6) orders
Klumov B.A., Phys. Usp. 53, 1045 (2010)
Steinhardt P.J. et al, PRL 47, 1297 (1981)
Melting curve of aluminum
Shock-wave
Shaner et al., 1984
DAC
QMD
QMD, 108 particles, equilibrium configurations analysis
Size effect should be checked
Boehler R., Ross M. Earth Plan. Sci. Lett. 153, 223 (1997)
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Release Isentropes of Al
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correspond to the experiments from
M. V. Zhernokletov et al. // Teplofiz. Vys. Temp. 33(1), 40-43 (1995) [in Russian]
- Zhernokletov et al. (1995)
- VASP
- MPTEOS
Al
1000
Pressure, kbar
100
10
1
0.1
0.01
0
1
2
3
4
5
Mass velocity, km/s
6
7
8
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Release Isentropes of Al
RAS
correspond to the experiments from
Knudson M.D., Asay J.R., Deeney C. // J. Appl. Phys. V. 97. P. 073514. (2005)
10000
Al
- Knudson et al. (2005)
- VASP
- MPTEOS
Pressure, kbar
1000
100
10
6
8
10
12
14
16
18
Mass velocity, km/s
20
22
24
26
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Sound Velocity in Shocked Al
13
12
Sound velocity, km/s
Al
- Al'tshuler et al. (1960)
- McQueen et al. (1984)
- Neal et al. (1975)
- MPTEOS
- VASP
11
10
9
8
7
1
2
3
4
5
6
Mass velocity, km/s
Oscillations are caused by errors of interpolation
7
Compression Isentrope
of Deuterium
Deuterium
T0 = 7.6 kK (AIMD)
T0 = 3.1 kK (EOS)
T0 = 2.1 kK (SAHA-D)
T0 = 1 kK (ReMC)
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Ab initio complex
electrical conductivity
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Complex electrical conductivity:
current
density
(
)
jw = s 1 (w ) + is 2 (w ) Ew
electrical field
strength
The real part of conductivity is responsible for energy absorption by electrons
and is calculated by Kubo-Greenwood formula:
2p e 2 2 N N 3 é
s 1 (w ) = 2 åååë f ei - f e j ùû Y j Ña Y i
3m wW i=1 j=1 a =1
( ) ( )
occupancies for the
energy level ε
2
d (e j - ei - w )
broadening is required
Kubo-Greenwood formula includes matrix elements of the velocity operator,
energy levels and occupancies calculated by DFT
The imaginary part of conductivity is calculated by the Kramers-Kronig relation:
s 1 (n )w
s 2 (w ) = - P ò 2
dn
2
p 0 (n - w )
2
¥
L.L. Moseley, T. Lukes, Am.J.Phys, 46, 676 (1978)
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Transport properties:
Onsager coefficients
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Definition of Onsager coefficients Lmn:
current
density
heat flow
density
electric field
strength
L12 ÑT ö
1æ
j = ç L11E ÷
eè
T ø
L22ÑT ö
1æ
jq = 2 ç eL21E ÷
T
e è
ø
jq = -KÑT
temperature
gradient
L11 – electrical conductivity
Onsager coefficients are symmetric:
j =0
at
L12 = L21
Using the Onsager coefficients the thermal conductivity coefficient becomes:
L12 L21 ö
1 æ
K = 2 çç L22 ÷÷
L11 ø
eTè
Wiedemann-Frantz law:
K (T )
 (T )  T
 L 

2
k
2
3 e
2
Lorentz number
thermoelectric term
Optical properties
Imaginary part of electrical conductivity is calculated from the real one by the KramersKronig relation:
 2 ( )  
2

P
 1 ( )
(   )
2
2
d
Real and imaginary parts of complex dielectric function:
 1 ( )  1 
 2 ( )
 0
; 2 ( ) 
 1 ( )
 0
;
Complex index of refraction:
n ( ) 
1
2
|  ( ) |   1 ( ) ;k ( ) 
1
2
|  ( ) |   1 ( ) ;
[1  n ( )]  k ( )
2
[1  n ( )]  k ( )
2
2
Reflectivity:
r ( ) 
2
Static electric conductivity is calculated by linear extrapolation (by 2 points) of
dynamic electrical conductivity to zero frequency
Al, T=1550 K, ρ=2.23 g/cm3
Real part of electrical conductivity
Real part of dielectric function
256 atoms; US pseudopotential; 1 k-point in the Brillouine zone; cut-off energy 200 eV;
δ-function broadening 0.1 eV; 15 configurations; 1500 steps; 1 step – 2 fs
In liquid phase the dependence of electrical conductivity is Drude-like.
The agreement with reference data is good
Krishnan et al., Phys. Rev. B, 47, 11 780 (1993)
Al, 1000 K – 10000 K, ρ = 2.35 g/cm3
Static electrical conductivity
Thermal conductivity coefficient
108 particles
108 particles
256 particles
256 particles
256 atmos; US pseudopotential; 1 k-point in the Brillouin zone; energy cut-off 200 eV; δ-function broadening 0.07 eV - 0.1
eV ; 15 configurations; 1500 steps; 1 step – 2 fs
Thermoelectric correction is not more than 2% at T < 10 kK
Recoules and Crocombette, Phys. Rev. B, 72, 104202 (2005)
Rhim and Ishikawa, Rev. Sci. Instrum., 69, 10, 3628-3633 (1998)
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Al, 1000 K – 20000 K, 2.35-2.70 g/cm3
Dependence of Lorentz number
on temperature
Thermal conductivity and Onsager L22
coefficient, temperature dependence
Thermoelectric
Correction, ~10%
Al ρ=2.35 г/см3
Al ρ=2.70 г/см3
256 atoms; US pseudopotential; 1 k-point in the Brillouin zone; energy cut-off 200 eV; δ-function broadening
0.07 eV - 0.1 eV; 15 configurations; 1500 steps; 1 step – 2 fs
Distinction of Lorentz number from the ideal value is about 20%.
Thermoelectric correction is substantial only at 20 kK (10%).
Recoules and Crocombette, Phys. Rev. B, 72, 104202 (2005)
Beyond DFT and Mean Atom: Path
Integral Monte Carlo
and Wigner Dynamics
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PATH INTEGRAL MONTE-CARLO METHOD
for degenerate hydrogen plasma
(V. M. Zamalin, G. E. Norman, V. S. Filinov, 1973-1977)
• Binary mixture of Ne quantum electrons,
Ni classical protons
• Partition function:
Z  N e , N i ,V ,    Q  N e , N i ,   N e ! N i !
Q ( N e , Ni , b ) = å ò dq dr r ( q,r,s ; b )
s V
• Density matrix:


  exp   H  exp    H     exp    H
n+1
  1 kT
    n  1

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PATH INTEGRAL MONTE-CARLO METHOD
electron
eb
r(n+1) = r
σ’ = σ
 e  2    m e
r(2)
 
Pa

 2    m e
λΔ
r(1) = r + λΔξ(1)
λe
r(n)
proton
qa
r
rb
parity of
permutation
 q , r , ;   
1
 3i N  3Ν
i
thermal
wavelengths of
electron and ion
P
e
   1
P
1   dr  n  
dr

V
 q , r , r ;     q , r
1 
n 
, Pr
 n 1 
permutation
operator
;   S   P   ,
spin
matrix
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PATH INTEGRAL MONTE-CARLO METHOD
RAS
Path integral representation of density matrix:
 q , r , ;     dR
1 
 dR
n 
1 
 
R
 q
i 
,r
i 

ˆ  ' Pˆ   n  1




1
S

,
P

kP
P
V
i 
n 
  i   R  i 1 e    H R  i 
ˆ
spin
matrix
permutation
operator
Hˆ  Kˆ  Uˆ c , Uˆ c  Uˆ cp  Uˆ ce  Uˆ cep
 n  U l    N e l
  q ,r , ;      e
  pp

p 1
 l 1
U      U le      U
p
l
ep
l
  
 Ne s
n ,1
  C N e det  ab
 s 0
kinetic part
of density
matrix
s
exchange
effects
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KELBG PSEUDOPOTENTIAL
RAS
First order perturbation theory solution of two-particle Bloch equation
for density matrix in the limit of weak coupling
d ab  


rab ,r ,     e a e b 
erf 

d ab    2  ab  1    
0
1

ab
'
ab
d
'
d ab     rab  1   rab
1
 ab
 m a 1  m b 1
Diagonal Kelbg potential:

rab  0
 eaeb
 ab
ab
rab ,    
 ab   2  2  ab
eaeb
 ab x ab
1 e
rab   ab
2
 x ab
  x ab 1  erf  x ab  
x ab  rab  ab
eaeb
r ab
Ebeling et al., Contr. Plasma Phys., 1967
JIHT
ACCURACY OF DIRECT PIMC
RAS
ˆ  e
  Hˆ
  1 k BT
ˆ    e
   Hˆ
e
   Hˆ
  e
n+1
Hˆ  Kˆ  Uˆ
e
   Kˆ
e
   Uˆ
   Hˆ
  

e
   2
2
Error ~ (β / n)2 for every multiplier
n  1 
Kˆ ,Uˆ 
1 at n -> ∞
Total error Δρ ~ β2 / n -> 0 at n->∞
     free  pot    
High-temperature
pseudopotential
JIHT
RAS
TREATMENT OF EXCHANGE EFFECTS
Inside main cell –
exchange matrix
e Maine
MC cell
e
Outside main cell –
by perturbation theory
Accuracy control –
comparison with ideal
degenerate gas
n e  ~ 300
3
e
Filinov V.S. // J. Phys. A: Math. Gen. 34, 1665 (2001)
Filinov V.S. et al. // J. Phys. A: Math. Gen. 36, 6069 (2003)
HYDROGEN, PIMC-SIMULATION, n = 1021 cm-3
Ne = Ni = 56, n = 20
T = 50 kK
ρ = 1.67x10-3 g/cm3
Γ = 0.54
nλ3 = 3.7 x10-2
T = 10 kK
Γ = 2.7
nλ3 = 0.41
- proton
- electron
- electron
HYDROGEN, PIMC-SIMULATION AND CHEMICAL
PICTURE, ne = 1020, 1021 cm-3
Pressure
Energy
1 .0
10
14
b)
E , e rg / g
P, GPa
0 .8
1
- 10
20
0 .6
-3
cm , D P IM C
21
- 1 0 , D P IM C
20
0 .4
- 10 , SC
-3
cm , D P IM C
21
- 1 0 , D P IM C
0 .1
- 10
20
10
20
21
- 10 , SC
13
- 10
- 10 , SC
21
- 10
- 10 , SC
20
0 .2
21
T, K
0 .0 1
10
4
10
5
0 .0
10
6
10
4
10
5
T, K
D. Saumon, G. Chabrier, H.M. Van Horn, Astrophys. J. Suppl.Ser. 1995. V.99. P.713
10
6
re la tive n u m b e r co n ce n tra tio n
a)
10
DEUTERIUM SHOCK HUGONIOTS
2000
- NOVA
- NOVA
1000
800
600
400
P, GPa
Grishechkin et al. Pis’ma
v ZhETF. 2004. V.80. P.452
(hemishpere)
- gas gun
- Z-pinch
- Mochalov
200
- Mochalov
- Trunin
100
80
60
40
- REMC
- DPIMC
20
10
8
6
4
3
0.171 g/cm3
2720 bar
0.153 g/cm3
2000 bar
 , g/cm
3
g/cm3
0.1335
1550 bar
0,4
0,6
0,8
1,0
HYDROGEN, PIMC-SIMULATION, T = 10000 K
Ne = Ni = 56, n = 20
n = 3x1022 cm-3
ρ = 0.05 g/cm3
Γ = 8. 4
nλ3 = 12.4
n = 1023 cm-3
ρ = 0.167 g/cm3
Γ = 12.5
nλ3 = 41
- proton
- electron
- electron
PIMC SIMULATION
- proton
- electron
Ne = Ni = 56, n = 20
protons ordering
- electron
5
30r2gee
4
gii
3
gee
2
T = 10000 K,
n = 3x1025 cm-3,
ρ = 50.2 g/cm3
Γ = 84
nλ3 = 12400
1
gei
0
JETP Letters 72, 245 (2000)
0.1
0.2
r/aB
50
ELECTRON DENSITY DISTRIBUTION IN TWOCOMPONENT DEGENERATE SYSTEM
IN COULOMB CRYSTAL
mh= 800
me= 2.1
<r>/aB = 3
top view
- hole
ρ = 25
T / Eb = 0.002
- hole
- electron
side view
- electron
51
QUANTUM DYNAMICS IN WIGNER
REPRESENTATION
Quasi-distribution function in phase space for the quantum case
 q , q    
Density matrix:
Wigner function:
W
L
q , p  
*
q   q  

 
Nd
Evolution equation:
t
p W
L

m
W
t
L

p W
m
q
L




,q 

i
 
e

 i  q   q   p
, p e
dp

L
q
 s , q  
Classical limit ħ -> 0:

 q   q 

W

 q 
 q , q     W 
L
 C
 dsW

  Nd
L
 ip 

t
 Hˆ ,  
d
W
L
R
 p  s , q , t  s , q ds
  sq ' 
U q  q   sin
d
q

  


Characteristics (Hamilton equations):
U W
q
p
L
 
q 
p
m
p  
U
q
SOLUTION OF WIGNER
EQUATION
W p, q ,t  

W
 p , q , t ; p  , q  ,   W   p  , q  dp  dq 


t
 d  '   dp  'dq  ' 

W
 p , q , t ; p ' , q  ' , '    dsW  p '  s , q  ' , '  s , q  ' 

Dynamical trajectories:
dp
dt
dq
dt
 F q t , q  '  ' ; p ' , q  ' , '   q  '
q
p
 p t  m , p '  ' ; p ' , q  ' , '   p '
Propagator:
p ' , q ' ,  '
 ( p , q , t ; p ' , q ' ,  ' )    p  p t ( t ; p ' , q ' ,  ' )  ( q  q t t ; p ' , q ' ,  ' 
W
Quantum dynamics
in Wigner representation
Dynamic conductivity
100
4
W ( p (  ), q (  ))
initial quantum
distribution
5
10 Re{p}
λ
T = 2 10
rs = 43.2
Veysman et al.3
Tkachenko et 4al.
Bornath et al 5
10
1
0,1
dq
dt

p (t )
5000-20000 configurations
random
+ momentum
jumps
m
 ~ FT  p ( t ) p (  )  
35
50
QD
30
5
10 Re{p}
dt
 F ( q ( t ))
hν = Ry
QD
0
dp
T0 = 2·104 K
K
1
rs = 43.22
25
100
ω/ω

pp
4
T=2 T
10= K2·10
rs=5 K
4
0
h =1 Ry
r
=
5
s
QD
20
15
Bornath et al.
Ternovoi et al.
10
2
3
5
0
0
5
10
15
20
ω/ ωp
Conclusions
• Ab initio methods are useful for calculation of
different properties of matter at high energy
density
• The main goal of ab initio methods is to replace
experiment; in some cases it’s already possible
• Growth of computational possibilities will allow
to apply more general approaches for calculation
of plasma properties (quantum filed theory)
• Currently, however, semiempirical approaches
are main workhorses; ab initio methods are used
for calibration of such models
Thermodynamic Functions
of Thomas-Fermi Model
Free energy:
F (V ,T ) 
2 2 aT

5 2
2
1
1


 
5
5
 I 3 2    8  u I 3 2    d u  3  u  I 1 2 (  )d u 
0
0
T 


 - dimensionless atomic potential,  =  / (u2T), a – cell volume, u = (r / r0)1/2
Expressions for 1st derivatives of F (P and S) are known.
Second derivatives of free energy
PV   FV V 
'
''
PT   FV T 
'
S
'
T
''
 F
''
TT

 2 
2
2
 2 
2
3 2
3 2
2
3 2 a
 T
2
3 2
 
'
I1 2    V
T 
 

 '
I
 1 2   T
T 

N ,T
 
1
 5T 2 u 5 I
3

2

N ,V
 

I 3 2    I1 2 
3
T  T
T
5
    3 u 3   T' T 2






 2 T I1 2     u   T T   I  1 2    d u

'
0
Shemyakin O.P. et al., JPA 43, 335003 (2010)
Second Derivatives
of the Thomas-Fermi Model
The number of particles and potential are the functions of the grand canonical ensemble
variables, which are in turn depend on the variables of the canonical ensemble:
N  N    N ,V ,T ,  N ,V ,T ,T  N ,V ,T 
      N ,V ,T ,  N ,V ,T ,T  N ,V ,T 
From the expressions for (N’T)N,, (’T)N, и (N’)T,N one can obtain:
  N  T  ,
  
 


  N    ,T

T

V ,N
We need
6 derivatives in the grand
canonical ensemble
  N     ,T
  
 


  N    ,T

V

 N ,T
 '  ,T
 T'  ,
 ' T ,
  N  T   ,    
  
  
 




 



T

T

N





 N ,

  ,
 ,T
 ,T 
NT'  ,
N ' T ,
N' T ,
Shemyakin O.P. et al., JPA 43, 335003 (2010)
TF Potential and its Derivatives on ,  and T
Derivative of the Poisson equation on :
Poisson equation
W    u  ;
 '
W u  2 u V ;

V '  2 a u 3T 3 2 I
12
 u

W
 Z r0 ,W
u 0

 
2
 W  u 2

2
T
u

 Wu

;

'
u 1
u 1
 0.
L   '
;

 ,T

 L'  2 u M ;
 u
3
3 2

4au T
'
12
I1 2     a u T I  1 2    L ;
M u 
3

'
L
 Lu u 1  0 .
u

1

N 
'


Derivative on :
 
2
   '
u ;

 ,T

 '  auT 1 2   u 2 I
 ;
1 2  
 u

'
12
2
  u I 1 2    ;
 u  auT

'
  u  1  Fu u  1  0 .






'
N

 ,T
 
 ,T
Derivative on T:
 Q   T'
;
 ,

Q '  2 u R ;
 u
 '
3
12
 3 I 1 2      I  1 2      a u T 1 2Q I  1 2    ;
Ru  au T



'
 Q u  1  Q u u  1  0 .
N 
'
T
 ,
Adiabatic Sound Velocity by ThomasFermi Model
Isotherms
V
Ideal Fermi-gas
V, atomic units
PROBLEMS OF TF MODEL
• Thomas-Fermi method is quasiclassical; if one
calculates energy levels in VTF(r) using the
Shrödinger equation and then electron density
quant(r), it will differ from the original TF electron
density (r)
• Mean ion charge is roughly determined
• The solution is to make  (r) self-consistent and
use the corrected potential and electron density
MEAN ION CHARGE (HFS)
 = const
12
Al
 = 10-3 g/cm3
<Z>
10
Thomas-Fermi
8
Chemical plasma
model
6
4
Hartree-Fock-Slater
2
0
T, K
5
10
6
10
7
10
TYPICAL CONFIGURATION OF PARTICLES
H + He mixture, T = 105 K, ne = 1023 cm-3
m He
m He
 m H   0 . 988
40 α-particles, 2 protons, 82 electrons
α-particle
proton
electron, 
He+
electron, 
He
ENERGY IN H + He MIXTURE ON ISOTHERMS
2 .0
1 .5
m He
E / NRy
m H  m He
200 kK
 0 . 234
1 .0
Ideal, 100 kK
0 .5
100 kK
50 kK
0 .0
40 kK
10
20
10
22
n e , cm
-3
10
24
D. Saumon, G. Chabrier, H.M. Van Horn, Astrophys. J. Suppl.Ser. 1995. V.99. P.713
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
crystal, mh(eff) = 800, me(eff) = 1
<r>/aB = 0.63
T = 0.064Eb
- hole
- hole
- electron
- electron
64
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
still crystal, mh(eff) = 100, me(eff) = 1
<r>/aB = 0.63
T = 0.064Eb
- hole
- hole
- electron
- electron
65
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
liquid, mh(eff) = 25, me(eff) = 1
<r>/aB = 0.63
T = 0.064Eb
- hole
- hole
- electron
- electron
66
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
unordered plasma, mh(eff) = 1, me(eff) = 1
<r>/aB = 0.63
T = 0.064Eb
- hole
- hole
- electron
- electron
67
PAIR DISTRIBUTION FUNCTIONS. QUANTUM MELTING
<r>/aB = 0.63
e-e
T = 0.064Eb
e-e
h-h
h-h
M = mh / me
e-e
e-e
h-h
h-h
68
PSEUDOPOTENTIALS IN DFT
• Diminish the number of plane waves necessary for the good
representation of inner electrons wave functions
• Part of electrons are considered as a core, part as valent
• Pseudopotential is constructed at T = 0 and doesn’t depend on
pressure and temperature
Pseudopotential approach
Valent
electrons
(plane waves)
Core
electrons
(pseudopotential)
Full-potential approach
muffin-tin orbitals
for all electrons
APPROXIMATIONS IN
PSEUDOPOTENTIAL APPROACH
Pseudopotential describes electrons with energies less than the
Fermi energy – errors at relatively high temperatures
EF
EF
T=0
T>0
Spatial distribution of core electrons in a pseudopotential is unchanged
– errors at relatively high pressures
P=0
P>0
HOLE-HOLE DISTANCE FLUCTUATIONS
Lindemann criterion
71
JIHT
Electron Heat Capacity for W at T = 11eV.
Return of Free Electrons into 4f-state under
Compression
RAS
W, bcc
8
12 eV
(%)
ion
ion
8
4
W
6
98
4
96
0
N4f (0) - N4f ()
12
100
CV / CV(ρ0)
2
Number of
returned electrons (%)
0
1.0
1.2
94
1 eV
0
1.0
1.2
1.4
/
1.6
0
Compression
ratio
1.8
v
e
capacity
Electron heat
Cv
16
ion
N4f (of
0) f-electrons
Fraction
W
W, bcc, T = 11 eV
1.4
1.6
/0
Compression
ratio
Electrons return to 4f state under compression
1.8
92