Thermodynamic and Transport Properties of Strongly-Coupled Degenerate Electron-Ion Plasma by First-Principle Approaches
Levashov P.R.Joint Institute for High Temperatures, Moscow, Russia
Moscow Institute of Physics and Technology, Dolgoprudny, Russia
JIHTRAS
In collaboration with: Minakov D.V.
Knyazev D.V.
Chentsov A.V.
Khishchenko K.V.
Outline
• 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.)
– – –
• Hypervelocity impact, laser, electronic, ionic beams,powerful electric current pulse etc.
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Coupling and Degeneracy in Plasma
• Coupling parameter:
• Degeneracy parameter
(for electronic subsystem)
- Strong coupling
- Strong degeneracy
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Strongly coupled degenerate non-relativistic plasma(warm dense matter)-
EOS: Traditional Form
Traditional form of semiempirical EOS. Free energy
Cold curve Thermal contributionof atoms and ions
Thermal contributionof electrons
Semiempiricalexpressions
DFT, mean atom models
Adiabatic (Born-Oppenheimer) approximation (me << mi)
Free energy ofelectrons in thefield of fixed ions
Free energy of ionsinteracting with potential depending on V and T
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Mean atom models or DFT calculations might help to decrease the number of fitting parameters
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
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Finite-Temperature Thomas-Fermi Model
00 rr
ZrrV r 0 00 rV 0
0rrdr
rdV
θ
μrVIθπ
rδZπVΔ2
123224
r0
Z
Feynman R., Metropolis N., Teller E. // Phys. Rev. 1949. V.75. P.1561.
Poisson equation
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)
Thomas-Fermi model is realistic but crude at relatively low temperatures and pressures.Thermal contributions to thermodynamic functions is a good approximation.
HARTREE-FOCK-SLATER MODEL AT T>0
Atom with mean populations
nl
nllN
exp
rVrVrV exc
r r
rc drrrdrrrrr
ZrV ''''''
Potential:
rrrrrVex .
Exchange potential:
Poisson equation solution:
εnl – energy levels in V(r)
Iterative procedure for determination of ρ(r), εnl and V(r)
Nikiforov A.F., Novikov V.G., Uvarov V.B. Quantum-statistical models of high-temperature plasma. M.: Fizmatlit, 2000.
From the radial Schrödinger equation
RADIAL ELECTRON DENSITY r2 (r) BY HARTREE-FOCK-SLATER MODEL
0.00 0.05 0.10 0.150
20
40
60
80
100
120
140
160
4
r2 (r
)
(r/r0)1/2
Auρ = 10-3
g/cm3
4·102 K4·105 K
Thomas-Fermi
Hartree-Fock-Slater
Density Functional Theory
• Thomas-Fermi theory is the density functional theory:
kinetic energy external potential
exchange energy Hartree energy
• Is it a general property?
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Density Functional TheoryFor systems with Hamiltonian
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).
the following theorems are valid:
Hohenberg, Kohn, 1964
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Kohn-Sham FunctionalThe system of interacting particles is replaced by the system of non-interacting particles:
kinetic energy ion-ioninteraction
exchange-correlationfunctional
All many-body effects of exchange and correlation are included into EXC[n]
true systemnon-interacting system
Kohn, Sham, 1965
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The minimization of EKS leads to the system of Kohn-Sham equations
Minimization in HFS and DFT
• In Hartree-Fock(-Slater) method we find
• In DFT we find
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Density Functional Theory: All-Electron and Pseudopotential Approaches
(S. Yu. Savrasov, PRB 54 16470 (1996),G. V. Sin'ko, N. A. Smirnov, PRB 74 134113 (2006)
Full-potential approach: all electrons are taken into account (FP-LMTO)
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).
Calculations were made in the unit cell at fixed ions and heated electrons
EF EF
T = 0 T > 0Coreelectrons
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At T > 0: occupancies are ( , , ) 1/ 1 exp ( ( , )) /f T T T
Aluminum. Cold Curve
0 2 4 6 8 10 120
50
100
150
200 Al
Pre
ssur
e (M
bar)
/0
FP-LMTO
VASP
Al is more compressiblein VASP calculationsthan in FP-LMTO
Compression ratio
Pres
sure
(Mba
r)
Levashov P.R. et al., JPCM 22 (2010) 505501
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Potassium. Cold Curve
0 5 10 150
5
10
15
20 K
Pre
ssur
e (M
bar)
/0
VASP
FP-LMTO3p, 4s
3s, 3p, 4s K is less compressiblein VASP calculationsthan in FP-LMTO
Less number of valentelectrons leads to bigger disagreementwith FP-LMTO
Levashov P.R. et al., JPCM 22 (2010) 505501
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DFT-calculations of Fe(Te, V)
0 5 10 150
5
10
15
20W
Cve
Te (eV)
1
2
• 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
Heat capacity of W, V0
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Thomas-Fermi with corrections
Why can we use DFT for thermodynamic properties of electrons?
bcc
fcc
( , )ee T eV
e V
E TCT
24x24x24 mesh of k-pointsNbands = 94Ecut = 1020 eV
Tungsten, Ti = 0, V = V0. Electron Heat Capacity
0 2 4 6 8 10
4
8
12
0.0
0.4
0.8
Isoc
horic
hea
t cap
acity
Temperature, eV
W, bcc
Num
ber o
f ele
ctro
ns
IFG, Z = 1
IFG, Z = 6
Thomas-Fermi
VASP
FP-LMTO
Levashov P.R. et al., JPCM 22 (2010) 505501
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Core electrons becomeexcited at ~3 eV
0 2 4 6 8 10 120
1
2
3
4
5
Ther
mal
pre
ssur
e, M
bar
Temperature, eV
W, bcc
Tungsten, Ti = 0, V = V0. Thermal Pressure
IFG, Z = 1
Thomas-Fermi
IFG, Z = 6
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Levashov P.R. et al., JPCM 22 (2010) 505501
• Pressure is determined by free electrons only
• Interaction of electronsshould be taken intoaccount
VASP
FP-LMTO
Thermal contribution of ionsAb-initio molecular dynamics (AIMD) simulations
From AIMD From AIMD From DFT Analyticexpression
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
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
Full energy during the simulations
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Isotherm T = 293 K for Al
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.000
1
2
3
4
5
6
7
8
- VASP - Wang et al. (2000) - Greene et al. (1994) - Akahama et al. (2006), fcc - Akahama et al. (2006), hcp
Pres
sure
, Mba
r
/
Al
QMD results show good agreement with experiments and calculations for fcc.
Shock Hugoniots of AlPressure – compression ratio
Shock Hugoniot of Al. Melting
Higher melting temperatures by QMD calculations might be caused by a small number of particlesNB: we can trace phase transitions in 1-phase simulation
600 800 1000 1200 1400 1600 1800 2000 22003000
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
- VASP Hugoniot - MPTEOS Hugoniot - MPTEOS melting curve - Exp. melting Boehler & Ross 1997
Pressure, kbar
Tem
pera
ture
, K
Al
Melting criterion for aluminumEquilibrium configurations of Al ions at 5950 and 6000 K
Steinhardt P.J. et al, PRL 47, 1297 (1981)
Rotational invariants of 2nd (q6) and 3rd (w6) orders
Cumulative distribution functions C(q6) and C(w6) are extremely sensitive measure of the local orientation order
Klumov B.A., Phys. Usp. 53, 1045 (2010)
This criterion may be applied to other structural phase transitions
Melting curve of aluminum
Boehler R., Ross M. Earth Plan. Sci. Lett. 153, 223 (1997)
QMD, 108 particles, equilibrium configurations analysisSize effect should be checked
QMD
Shock-waveShaner et al., 1984
DAC
Release Isentropes of Al
0 1 2 3 4 5 6 7 80.01
0.1
1
10
100
1000
- Zhernokletov et al. (1995) - VASP - MPTEOS
Mass velocity, km/s
Pres
sure
, kba
rAl
correspond to the experiments from M. V. Zhernokletov et al. // Teplofiz. Vys. Temp. 33(1), 40-43 (1995) [in Russian]
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Release Isentropes of Al
6 8 10 12 14 16 18 20 22 24 26
10
100
1000
10000
- Knudson et al. (2005) - VASP - MPTEOS
Mass velocity, km/s
Pres
sure
, kba
rAl
correspond to the experiments fromKnudson M.D., Asay J.R., Deeney C. // J. Appl. Phys. V. 97. P. 073514. (2005)
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Sound Velocity in Shocked Al
1 2 3 4 5 6 7
7
8
9
10
11
12
13Al
- Al'tshuler et al. (1960) - McQueen et al. (1984) - Neal et al. (1975) - MPTEOS - VASP
Soun
d ve
locity
, km
/s
Mass velocity, km/sOscillations are caused by errors of interpolation
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T0 = 7.6 kK (AIMD)T0 = 3.1 kK (EOS)T0 = 2.1 kK (SAHA-D)T0 = 1 kK (ReMC)
Deuterium
Compression Isentrope of Deuterium
Ab initio complex electrical conductivity
Complex electrical conductivity:
The real part of conductivity is responsible for energy absorption by electrons and is calculated by Kubo-Greenwood formula:
Kubo-Greenwood formula includes matrix elements of the velocity operator, energy levels and occupancies calculated by DFT
L.L. Moseley, T. Lukes, Am.J.Phys, 46, 676 (1978)
The imaginary part of conductivity is calculated by the Kramers-Kronig relation:
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occupancies for the energy level ε
broadening is required
electrical field strengthcurrent
density
Transport properties: Onsager coefficients
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Definition of Onsager coefficients Lmn:
current density
heat flow density
electric fieldstrength
temperaturegradient
L11 – electrical conductivity
Using the Onsager coefficients the thermal conductivity coefficient becomes:
2 2
2
( )( ) 3K T kLT T e
Wiedemann-Frantz law:
Lorentz number
at
thermoelectric term
Onsager coefficients are symmetric:
L12 = L21
Optical properties
2 11 2
0 0
( ) ( )( ) 1 ; ( ) ;
12 2 2
( )2( )( )
P d
1 11 1( ) | ( ) | ( ); ( ) | ( ) | ( );2 2
n k
Imaginary part of electrical conductivity is calculated from the real one by the Kramers-Kronig relation:
Real and imaginary parts of complex dielectric function:
Complex index of refraction:
Reflectivity:2 2
2 2
[1 ( )] ( )( )[1 ( )] ( )n krn k
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
Thermoelectric correction is not more than 2% at T < 10 kK
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
Recoules and Crocombette, Phys. Rev. B, 72, 104202 (2005)Rhim and Ishikawa, Rev. Sci. Instrum., 69, 10, 3628-3633 (1998)
108 particles
256 particles
Thermal conductivity coefficient
108 particles
256 particles
Al, 1000 K – 20000 K, 2.35-2.70 g/cm3
Dependence of Lorentz number on temperature
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)
Al ρ=2.35 г/см3
Thermal conductivity and Onsager L22 coefficient, temperature dependence
Al ρ=2.70 г/см3
ThermoelectricCorrection, ~10%
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Beyond DFT and Mean Atom: Path Integral Monte Carlo and Wigner Dynamics
PATH INTEGRAL MONTE-CARLO METHODfor degenerate hydrogen plasma
(V. M. Zamalin, G. E. Norman, V. S. Filinov, 1973-1977)
• Binary mixture of Ne quantum electrons, Ni classical protons
!!,,,,, ieieie NNNNQVNNZ
• Partition function:
• Density matrix:
HHH expexpexp n+1
1 nkT1
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PATH INTEGRAL MONTE-CARLO METHOD
P V
nΝN
idrdrrq P
ei1
33 11
;,,
,;,,;,, 11 PSrPrqrrq nn
permutationoperator
spinmatrix
thermalwavelengths of
electron and ion
parity ofpermutation
Pa
qaproton
ee m 2eb
electron
rb
λe
r
λΔem
2r(1) = r + λΔξ(1)
r(2)
r(n)r(n+1) = rσ’ = σ
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PATH INTEGRAL MONTE-CARLO METHODPath integral representation of density matrix:
V P
nknn PPSdRdRrq P 111 ˆ'ˆ,1;,,
iii rqR ,spin matrix
permutationoperator
e
e
el
N
ss
nab
sN
N
p
lpp
n
l
U Cerq0
1,
11
det;,,
iHii ReR ˆ1
cUKH ˆˆˆ epc
ec
pcc UUUU ˆˆˆˆ ,
exchangeeffects
kinetic partof densitymatrix
epl
el
pl UUU
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KELBG PSEUDOPOTENTIAL
abababx r
First order perturbation theory solution of two-particle Bloch equation for density matrix in the limit of weak coupling
12erf,,
1
0
'
ab
ab
abbaabab
ab dd
deerr
'1 abababd rr
ababx
abab
baab
ab xxexee
ab erf11, 2
r
Diagonal Kelbg potential:
ab
baee
0abr
ab
ba
ree
abab r
abab 22111 baab mm
Ebeling et al., Contr. Plasma Phys., 1967
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ACCURACY OF DIRECT PIMC
HHH eee ˆˆˆˆ
n+1UKH ˆˆˆ TkB1 1 n
UKUKH eeeeˆ,ˆ
2ˆˆˆ2
ˆ
1 at n -> ∞Error ~ (β / n)2 for every multiplier
Total error Δρ ~ β2 / n -> 0 at n->∞
potfreeHigh-temperaturepseudopotential
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TREATMENT OF EXCHANGE EFFECTS
Inside main cell – exchange matrix
Outside main cell – by perturbation theory
Accuracy control – comparison with idealdegenerate gas
300~3een
MainMC cell
ee
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)
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HYDROGEN, PIMC-SIMULATION, n = 1021 cm-3
Ne = Ni = 56, n = 20
T = 10 kK
Γ = 2.7nλ3 = 0.41
T = 50 kKρ = 1.67x10-3 g/cm3
Γ = 0.54nλ3 = 3.7 x10-2
- proton
- electron
- electron
104 105 1060.01
0.1
1
10 a)
- 1020 cm-3, DPIMC - 1021, DPIMC - 1020, SC - 1021, SC
P, GP
a
T, K
104 105 106
1013
1014
- 1020 cm-3, DPIMC - 1021, DPIMC - 1020, SC - 1021, SC
relati
ve nu
mber
conc
entra
tion
E, erg
/ g
T, K0.0
0.2
0.4
0.6
0.8
1.0b)
- 1020
- 1021
HYDROGEN, PIMC-SIMULATION AND CHEMICAL PICTURE, ne = 1020, 1021 cm-3
Pressure Energy
D. Saumon, G. Chabrier, H.M. Van Horn, Astrophys. J. Suppl.Ser. 1995. V.99. P.713
DEUTERIUM SHOCK HUGONIOTS
0,4 0,6 0,8 1,0346810
20
406080100
200
4006008001000
2000
- NOVA - NOVA - gas gun - Z-pinch - Mochalov - Mochalov - Trunin - REMC - DPIMCP ,
GPa
, g/cm30.1335 g/cm3
1550 bar
0.153 g/cm3
2000 bar
0.171 g/cm3
2720 bar
Grishechkin et al. Pis’ma v ZhETF. 2004. V.80. P.452(hemishpere)
HYDROGEN, PIMC-SIMULATION, T = 10000 KNe = Ni = 56, n = 20
n = 1023 cm-3
ρ = 0.167 g/cm3
Γ = 12.5nλ3 = 41
n = 3x1022 cm-3
ρ = 0.05 g/cm3
Γ = 8. 4nλ3 = 12.4
- proton
- electron
- electron
T = 10000 K, n = 3x1025 cm-3, ρ = 50.2 g/cm3
Γ = 84nλ3 = 12400
PIMC SIMULATION Ne = Ni = 56, n = 20
protons ordering
1
2
3
4
5
0.1 0.20 r/aB
gee
gii
gei
30r2gee
JETP Letters 72, 245 (2000)
- proton
- electron
- electron
50
ELECTRON DENSITY DISTRIBUTION IN TWO-COMPONENT DEGENERATE SYSTEM
mh= 800me= 2.1<r>/aB = 3
top view side view
ρ = 25T / Eb = 0.002
- hole
- electron
- electron
- holeIN COULOMB CRYSTAL
51
QUANTUM DYNAMICS IN WIGNER REPRESENTATION
,Ht
i
deqqpqW ipNd
L ,,
dpepqqWqq pqqiL ,,
dsqstqspdsW
qW
mp
tW L
LL
,,,
'sin, sqqqUqdqs Nd
pW
qU
qW
mp
tW LLL
mpq
qUp
Density matrix:
Wigner function:
Evolution equation:
Classical limit ħ -> 0: Characteristics (Hamilton equations):
Quasi-distribution function in phase space for the quantum case qqqq *, C
RW L
SOLUTION OF WIGNER EQUATION
''''''' ,',,',,;,,'
,,,;,,,,
qsqspdsWqptqpdqdpd
dqdpqpWqptqptqpW
Wt
W
'''' ',,;', qqpqtqFdtpd
'''' ',,;', pqppmtpdtqd
Dynamical trajectories:
',, '' qp
pq
',,;()',,;()',,;,,( '''''' qptqqqptppqptqp ttW
Propagator:
0 50 1000,1
1
10
100T = 2 104 K0 1rs = 43.2 2
3 4 5
10
5 Re{
p }
p
Quantum dynamics in Wigner representation
λ
mtp
dtqd
tqFdtpd
)(
))((
random+ momentum jumps
))(),(( qpWinitial quantum
distribution
)()(~ ptpFT
Dynamic conductivity
Veysman et al.Tkachenko et al.Bornath et al
T = 2·104 Krs = 43.2
QDh = Ry
0 5 10 15 20
0
5
10
15
20
25
30
35
T=2 104 K0 rs=5
1 2 3
10
5 Re{
p }
p ω / ωp
ω / ωp
T = 2·104 Krs = 5
hν = Ry
Bornath et al.Ternovoi et al.
QD
QD5000-20000 configurations
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
1' '' 2 5 3 ' 2 '
3 2 1 2 1 22 3 20
3 25 3 2a
T TT T TS F T u I u T T I u T I duT
3 2' '' '
1 2 3 2 1 22 ,
2 52 3T VT T N V
P F I I IT T T T
3 2' '' '
1 22 ,
22V VV V N T
P F IT
Second derivatives of free energy
Thermodynamic Functions of Thomas-Fermi Model
5 2 1 15 5
3 2 3 2 1 220 0
2 2( , ) 8 3 ( )aTF V T I u I du u I duT
- dimensionless atomic potential, = / (u2T), a – cell volume, u = (r / r0)1/2
Free energy:
Expressions for 1st derivatives of F (P and S) are known.
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:
From the expressions for (N’T)N,, (’T)N, и (N’)T,N one can obtain:
TVNTTVNTVN ,,,,,,,,
TVNTTVNTVNNN ,,,,,,,,
T
T
TN NN
V ,
,
,
TNV N
TNT ,
,
,
TTN N
TNTT ,,
,
,,
We need6 derivatives in the grand canonical ensemble
T,'
,'T
,'
T
,
'T
N ,'
TN ,'TN
Shemyakin O.P. et al., JPA 43, 335003 (2010)
TF Potential and its Derivatives on , and T
2
'
2' 3 3 2
1 2 2
'0 0 1 1
;
2 ;
2 ;
, 0.
u
u
u u u u
W uW uV
W uV au T ITu
W Z r W W
Poisson equation
'
,
'
3 3 2' 1 2
1 2 1 2
'1 1
;
2 ;
4 ;3
0.
T
u
u
u u u
L
L uM
au TM I auT I L
L L
Derivative of the Poisson equation on :
' 2
,
' 1 2 21 2
' 1 2 21 2
'1 1
;
;
;
0.
T
u
u
u u u
u
auT u I
auT u I
F
Derivative on :
'
,
'
' 3 1 2 1 21 2 1 2 1 2
'1 1
;
2 ;
3 ;
0.
T
u
u
u u u
Q
Q uR
R au T I I auT QI
Q Q
Derivative on T:
'
,TN
'
,TN
'
,TN
Adiabatic Sound Velocity by Thomas-Fermi Model
V
V, atomic units
Ideal Fermi-gas
Isotherms
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)
105 106 1070
2
4
6
8
10
12
<Z>
T, K
= const
Al = 10-3 g/cm3
Hartree-Fock-Slater
Chemical plasmamodel
Thomas-Fermi
TYPICAL CONFIGURATION OF PARTICLESH + He mixture, T = 105 K, ne = 1023 cm-3
proton
988.0HHeHe mmm
α-particle
electron,
electron,
He+
He
40 α-particles, 2 protons, 82 electrons
ENERGY IN H + He MIXTURE ON ISOTHERMS
200 kK
100 kK
50 kK
40 kK
Ideal, 100 kK
1020 1022 1024
0.0
0.5
1.0
1.5
2.0 E
/ NR
y
ne, cm-3
234.0 HeH
He
mmm
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
- hole
- electron
- electron
- hole
<r>/aB = 0.63
T = 0.064Eb
64
ELECTRON-HOLE PLASMA. PIMC SNAPSHOTstill crystal, mh(eff) = 100, me(eff) = 1
- hole
- electron
- electron
- hole
<r>/aB = 0.63
T = 0.064Eb
65
liquid, mh(eff) = 25, me(eff) = 1
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
- hole
- electron
- electron
- hole
<r>/aB = 0.63
T = 0.064Eb
66
unordered plasma, mh(eff) = 1, me(eff) = 1
ELECTRON-HOLE PLASMA. PIMC SNAPSHOT
- hole
- electron
- electron
- hole
<r>/aB = 0.63
T = 0.064Eb
67
PAIR DISTRIBUTION FUNCTIONS. QUANTUM MELTING
<r>/aB = 0.63
T = 0.064Eb
M = mh / me
h-he-eh-h
e-e
h-h
e-e
h-he-e
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
Coreelectrons(pseudopotential)
Valentelectrons(plane waves)
Full-potential approachmuffin-tin orbitals
for all electrons
APPROXIMATIONS IN PSEUDOPOTENTIAL APPROACHPseudopotential 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
Electron Heat Capacity for W at T = 11eV. Return of Free Electrons into 4f-state under Compression
W, bcc
JIHTRAS
Electrons return to 4f state under compression
1.0 1.2 1.4 1.6 1.80
2
4
6
8
92
94
96
98
100W
(%)
Cv e(
0 )
N4f
ion (
0) - N
4fio
n ()
/0
Cv e()
N4f
ion (
0)(%
)
W, bcc, T = 11 eV
CV / CV(ρ0)Number ofreturned electrons (%)Fr
actio
n of
f-el
ectr
ons
Compression ratio
1.0 1.2 1.4 1.6 1.80
4
8
12
16W
12 eV
Cve
/0
1 eV
Elec
tron
hea
t cap
acity
Compression ratio