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

*pasha@ihed.ras.ru

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