jiht ras% thermodynamicandtransportproperes of …theor.jinr.ru/~klopot/slides/levashov.pdf ·...
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Thermodynamic and Transport Proper2es of Strongly-‐Coupled Degenerate Electron-‐Ion Plasma by First-‐Principle Approaches
Levashov P.R.
Joint Ins1tute for High Temperatures, Moscow, Russia Moscow Ins1tute of Physics and Technology, Dolgoprudny, Russia
JIHT RAS
In collabora1on with: Minakov D.V. Knyazev D.V. Chentsov A.V. Khishchenko K.V.
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Outline
• Strongly coupled degenerate plasma • Ab-‐ini1o calcula1ons • Quantum-‐sta1s1cal models • Density func1onal theory • Quantum molecular dynamics • Path-‐Integral Monte Carlo • Wigner dynamics • Conclusions
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Ab-‐ini2o calcula2ons
• Thermodynamic, transport and op1cal proper1es
• Use the following informa1on: – fundamental physical constants – charge and mass of nuclei – thermodynamic state
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Extreme States of MaCer
• Kirzhnits D.A., Phys. Usp., 1971 • Atomic system of units:
– me = ħ = a0 = 1 • Extreme States of MaXer (Kalitkin N.N.)
– – –
• Hypervelocity impact, laser, electronic, ionic beams, powerful electric current pulse etc.
P = e2 a04 = 294.2 Mbar
E = e2 a0 = 27.2 eVV = a0
3 = 0.1482 A
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Coupling and Degeneracy in Plasma
• Coupling parameter:
• Degeneracy parameter
Γ =Upot
Ekin
= e2
kBT r(for electronic subsystem)
neλe
3, λe2 = 2π2
mekBT
Γ1 -‐ Strong coupling
neλe3 1 -‐ Strong degeneracy
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Γ1 neλe
3 1Strongly coupled degenerate non-‐rela1vis1c plasma (warm dense maXer) -‐
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EOS: Tradi2onal Form
F V,T( ) = Fc V( )+Fi V,T( )+Fe V,T( )Traditional form of semiempirical EOS. Free energy
Cold curve Thermal contribution of atoms and ions
Thermal contribution of electrons
Semiempirical expressions
DFT, mean atom models
Adiaba1c (Born-‐Oppenheimer) approxima1on (me << mi)
F V,T( ) = Fe V,T, Rt0{ }( )+Fn V,T, Rt0{ }( )Free energy of electrons in the field of fixed ions
Free energy of ions interac1ng with poten1al depending on V and T
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Mean atom models or DFT calcula1ons might help to decrease the number of fiang parameters
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Electronic Contribu2on to Thermodynamic Func2ons:
Hierarchy of Models • Exact solu1on of the 3D mul1-‐par1cle Schrödinger equa1on
• Atom in a spherical cell • Hartree-‐Fock method (1-‐electron wave func1ons)
• Hartree-‐Fock-‐Slater method • Hartree method (no exchange) • Thomas-‐Fermi method • Ideal Fermi-‐gas
r0
Z
43πr0
3 =1n
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Finite-‐Temperature Thomas-‐Fermi Model
( )00 rr <≤
( ) ZrrV r ==0 ( ) 00 =rV( ) 0
0==rrdr
rdV
( ) ( ) ( )⎟⎠⎞⎜
⎝⎛ ++−=
θµrVIθ
πrδZπVΔ
21
23224
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 asympto1c behavior at low T and V (ideal Fermi-‐gas) and at high T and V (ideal Boltzmann gas)
Thomas-‐Fermi model is realis1c but crude at rela1vely low temperatures and pressures. Thermal contribu1ons to thermodynamic func1ons is a good approxima1on.
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HARTREE-‐FOCK-‐SLATER MODEL AT T>0
Atom with mean popula1ons
( )( )θµε −+1
1+22=nl
nllN
exp
( ) ( ) ( )rVrVrV exc +=
( ) ( ) ( ) ⎥⎦⎤
⎢⎣⎡ +14−= ∫ ∫0
2 0r r
rc drrrdrrrrr
ZrV '''''' ρρπ
Poten1al:
( ) ( ) ( ) ( ) 31−
3
24
23 ⎥⎦
⎤⎢⎣
⎡3
+75+1=θρπ
θρ
θπ rrrrrVex .
Exchange poten1al:
Poisson equa1on solu1on:
εnl – energy levels in V(r)
Itera1ve procedure for determina1on of ρ(r), εnl and V(r)
Nikiforov A.F., Novikov V.G., Uvarov V.B. Quantum-‐sta1s1cal models of high-‐temperature plasma. M.: Fizmatlit, 2000.
From the radial Schrödinger equa1on
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RADIAL ELECTRON DENSITY r2ρ (r) BY HARTREE-FOCK-SLATER MODEL
0.00 0.05 0.10 0.150
20
40
60
80
100
120
140
160
4π
r2 ρ(r)
(r/r0)1/2
Au ρ = 10-‐3 g/cm3
4·∙102 K 4·∙105 K
Thomas-‐Fermi
Hartree-‐Fock-‐Slater
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Density Func2onal Theory
• Thomas-‐Fermi theory is the density func1onal theory:
ETF n[ ] =C1 d3rn r( )5 3∫ + d3rVext r( )n r( )∫ +
C2 d3r∫ n r( )4 3 + 12d3rd3 "r
n r( )n "r( )r− "r
kine1c energy external poten1al
exchange energy Hartree energy
• Is it a general property?
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Density Func2onal Theory
H = −12
∇i2 + Vext ri( )
i∑
i∑ +
12
e2
ri − rji≠ j∑
For systems with Hamiltonian
Theorem 1. For any system of interac1ng par1cles in an external poten1al Vext(r) the poten1al Vext(r) is determined uniquely, except for a constant, by the ground state par1cle density of electrons n0(r). Therefore, all proper4es are completely determined given only the ground state electronic density n0(r). Theorem 2. A universal func1onal for the energy E[n] in terms of the density n(r) can be defined, valid for any external poten1al Vext(r). For any par1cular Vext(r), the exact ground state energy of the system is the global minimum value of this func1onal, and the density n(r) that minimizes the func1onal is the exact ground state density n0(r).
the following theorems are valid:
Hohenberg, Kohn, 1964
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Kohn-‐Sham Func2onal The system of interac1ng par1cles is replaced by the system of non-‐interac1ng par1cles:
EKS[n]= Ts[n]+ drVext r( )n r( )+EHartree n[ ]∫ +EII +EXC[n]
kine1c energy ion-‐ion interac1on
exchange-‐ correla1on func1onal
All many-‐body effects of exchange and correla1on are included into EXC[n]
EXC[n]= T −Ts[n]+ Vint −EHartree[n]
true system non-‐interac1ng system
Kohn, Sham, 1965
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The minimiza1on of EKS leads to the system of Kohn-‐Sham equa1ons
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Minimiza2on in HFS and DFT
• In Hartree-‐Fock(-‐Slater) method we find
• In DFT we find
minΨi (r)
Ω Ψ i r( )#$ %&
minn(r)
Ω n r( )"# $%
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Density Func2onal Theory: All-‐Electron and Pseudopoten2al Approaches
(S. Yu. Savrasov, PRB 54 16470 (1996), G. V. Sin'ko, N. A. Smirnov, PRB 74 134113 (2006)
Full-‐poten1al approach: all electrons are taken into account (FP-‐LMTO)
Pseudopoten1al approach: the core is replaced by a pseudopoten1al, the Kohn-‐Sham equa1ons 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).
Calcula1ons were made in the unit cell at fixed ions and heated electrons
EF EF
T = 0 T > 0 Core electrons
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At T > 0: occupancies are [ ]{ }( , , ) 1/ 1 exp ( ( , )) /f T T Tε ρ ε µ ρ= + −
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Aluminum. Cold Curve
0 2 4 6 8 10 120
50
100
150
200 Al
Pre
ssur
e (M
bar)
ρ/ρ0
FP-LMTO
VASP
Al is more compressible in VASP calculations than in FP-LMTO
Compression ra2o
Pressure (M
bar)
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)
ρ/ρ0
VASP
FP-LMTO
3p, 4s
3s, 3p, 4s K is less compressible in VASP calculations than in FP-LMTO Less number of valent electrons leads to bigger disagreement with FP-LMTO
Levashov P.R. et al., JPCM 22 (2010) 505501
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DFT-‐calcula2ons 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 correc1ons
Why can we use DFT for thermodynamic proper1es of electrons?
bcc
fcc
PTe = −ρ2
∂FTe (ρ,Te )∂ρ
Te
− Pc
( , )ee T eV
e V
E TCTρ⎡ ⎤∂= ⎢ ⎥∂⎣ ⎦
24x24x24 mesh of k-‐points Nbands = 94 Ecut = 1020 eV
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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
Levashov P.R. et al., JPCM 22 (2010) 505501
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Core electrons become excited at ~3 eV
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0 2 4 6 8 10 120
1
2
3
4
5
Th
erm
al p
ress
ure,
Mba
r
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
• Interac1on of electrons should be taken into account
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Thermal contribu2on of ions
F V,T( ) = Fc V( )+Fi V,T( )+Fe V,T( )Ab-‐ini1o molecular dynamics (AIMD) simula1ons
FAIMD V,T( )−Fe V,T( )−Fc V( )+Fions,kin V,T( )From AIMD From AIMD From DFT Analy1c
expression
But it’s beXer to use AIMD to calibrate the EOS by changing fiang parameters
Desjarlais M., MaXson 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 simula2ons • We use VASP (Vienna Ab Ini1o Simula1on Package) -‐ package for
performing ab-‐ini1o quantum-‐mechanical molecular dynamics (MD) using pseudopoten1als and a plane wave basis set.
• Generalized Gradient Approxima1on (GGA) for Exchange and Correla1on func1onal
• Ultrasox Vanderbilt pseudopoten1als (US-‐PP) • One point (Γ-‐point) in the Brillouin zone • The QMD simula1ons were performed for
108 atoms of Al • The dynamics of Al atoms was simulated
within 1 ps with 2 fs 2me step • The electron temperature was equal to the
temperature of ions through the Fermi–Dirac distribu1on
• 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
Pressure, M
bar
ρ/ρ0
A l
QMD results show good agreement with experiments and calcula1ons for fcc.
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Shock Hugoniots of Al Pressure – compression ra1o
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Shock Hugoniot of Al. Mel2ng
Higher mel1ng temperatures by QMD calcula1ons might be caused by a small number of par1cles NB: we can trace phase transi1ons in 1-‐phase simula1on
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
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Mel2ng criterion for aluminum Equilibrium 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
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Mel2ng curve of aluminum
Boehler R., Ross M. Earth Plan. Sci. Lett. 153, 223 (1997)
QMD, 108 particles, equilibrium configurations analysis Size effect should be checked
QMD
Shock-wave Shaner et al., 1984
DAC
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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
Mas s ve loc ity, km/s
Press
ure, kba
rA l
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
Mas s ve loc ity, km/s
Press
ure, kba
rA l
correspond to the experiments from Knudson 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
13
A l
-‐ A l'ts huler et a l. (1960) -‐ McQ ueen et a l. (1984) -‐ Nea l e t a l. (1975) -‐ MP T E O S -‐ V A S P
Sou
nd veloc
ity, k
m/s
Mas s ve loc ity, km/s
Oscilla1ons are caused by errors of interpola1on
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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
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Ab initio complex electrical conductivity
jω = σ1(ω)+ iσ 2 (ω)( )
Eω
σ1(ω) =2πe22
3m2ωΩf εi( )− f ε j( )#$
%&
α=1
3
∑j=1
N
∑i=1
N
∑ Ψ j ∇α Ψ i
2δ ε j −εi − ω( )
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:
σ 2 (ω) = −2πP σ1(ν )ω
(ν 2 −ω 2 )dν
0
∞
∫
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occupancies for the energy level ε
broadening is required
electrical field strength current
density
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j = 1eL11E − L12∇T
T
#
$%
&
'(
jq =
1e2eL21E − L22∇T
T
#
$%
&
'(
Transport properties: Onsager coefficients
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Defini1on of Onsager coefficients Lmn:
current density
heat flow density
electric field strength
temperature gradient
L11 – electrical conduc1vity
jq = −K∇T
K =1e2T
L22 −L12L21L11
"
#$$
%
&''
Using the Onsager coefficients the thermal conduc1vity coefficient becomes:
2 2
2
( )( ) 3K T kLT T e
πσ
= =⋅
Wiedemann-‐Frantz law:
Lorentz number
j = 0at
thermoelectric term
Onsager coefficients are symmetric:
L12 = L21
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Optical properties
2 11 2
0 0
( ) ( )( ) 1 ; ( ) ;σ ω σ ωε ω ε ωωε ωε
= − =
12 2 2
( )2( )( )
P dσ ν ωσ ω νπ ν ω
= −−∫
1 11 1( ) | ( ) | ( ); ( ) | ( ) | ( );2 2
n kω ε ω ε ω ω ε ω ε ω= + = −
Imaginary part of electrical conduc1vity is calculated from the real one by the Kramers-‐Kronig rela1on:
Real and imaginary parts of complex dielectric func1on:
Complex index of refrac1on:
Reflec1vity: 2 2
2 2
[1 ( )] ( )( )[1 ( )] ( )n krn kω ωωω ω
− +=+ +
Static electric conductivity is calculated by linear extrapolation (by 2 points) of dynamic electrical conductivity to zero frequency
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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 conduc1vity is Drude-‐like.
The agreement with reference data is good
Krishnan et al., Phys. Rev. B, 47, 11 780 (1993)
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Al, 1000 K – 10000 K, ρ = 2.35 g/cm3
Sta1c electrical conduc1vity
Thermoelectric correc1on is not more than 2% at T < 10 kK
256 atmos; US pseudopoten1al; 1 k-‐point in the Brillouin zone; energy cut-‐off 200 eV; δ-‐func1on broadening 0.07 eV -‐ 0.1 eV ; 15 configura1ons; 1500 steps; 1 step – 2 fs
Recoules and CrocombeXe, Phys. Rev. B, 72, 104202 (2005) Rhim and Ishikawa, Rev. Sci. Instrum., 69, 10, 3628-‐3633 (1998)
108 par1cles
256 par1cles
Thermal conduc1vity coefficient
108 par1cles
256 par1cles
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Al, 1000 K – 20000 K, 2.35-2.70 g/cm3
Dependence of Lorentz number on temperature
256 atoms; US pseudopoten1al; 1 k-‐point in the Brillouin zone; energy cut-‐off 200 eV; δ-‐func1on broadening 0.07 eV -‐ 0.1 eV; 15 configura1ons; 1500 steps; 1 step – 2 fs
Dis1nc1on of Lorentz number from the ideal value is about 20%. Thermoelectric correc1on is substan1al only at 20 kK (10%).
Recoules and CrocombeXe, Phys. Rev. B, 72, 104202 (2005)
Al ρ=2.35 г/см3
Thermal conduc1vity and Onsager L22 coefficient, temperature dependence
Al ρ=2.70 г/см3
Thermoelectric Correc1on, ~10%
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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
( ) ( ) !!,,,,, ieieie NNNNQVNNZ ββ =
• Par11on func1on:
Q Ne,Ni ,β( ) = dqdrρ q,r,σ ;β( )V∫
σ∑
• Density matrix:
( ) ( ) ( )HHH expexpexp βββρ Δ−××Δ−=−= …n+1
( )1+=Δ nββkT1=β
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PATH INTEGRAL MONTE-‐CARLO METHOD
( ) ( ) ( ) ( )×−= ∑ ∫Δ P V
nΝN
idrdrrq P
ei…133 11 κ
λλβσρ ;,,
( )( ) ( ) ( )( ) ( ),;,,;,, 11 σσβρβρ ′,ΔΔ + PSrPrqrrq nn…
permuta1on operator
spin matrix
thermal wavelengths of electron and ion
parity of permuta1on
Pa
qa proton
ee mβπλ 22 = 2eb electron
rb
λe
r
λΔ emβπλ Δ= 22Δ 2
r(1) = r + λΔξ(1)
r(2)
r(n) r(n+1) = r σ’ = σ
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PATH INTEGRAL MONTE-‐CARLO METHOD Path integral representa1on of density matrix:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∑ +−=V P
nknn PPSdRdRrq P 111 ˆ'ˆ,1;,, ρσσρρβσρ ……
( ) ( ) ( )( )iii rqR ,=spin matrix
permuta1on operator
( ) ( ) ∑∏∏∑===
−⎟⎟⎠
⎞⎜⎜⎝
⎛=e
e
el
N
ss
nab
sN
N
p
lpp
n
l
U Cerq0
1,
11det;,, ψφβσρ βΔβΔ
σ
( ) ( ) ( )iHii ReR ˆ1 βΔρ −−=
cUKH ˆˆˆ += epc
ec
pcc UUUU ˆˆˆˆ ++=,
exchange effects
kine1c part of density matrix
( ) ( ) ( )βΔβΔβΔ epl
el
pl UUU ++
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KELBG PSEUDOPOTENTIAL
abababx λr=
First order perturba1on theory solu1on of two-‐par1cle Bloch equa1on for density matrix in the limit of weak coupling
( )( )
( )( )⎟⎠
⎞⎜⎝⎛
−= ∫ ααλ
αααβΔΦ
12erf,,
1
0
'
ab
ab
abbaabab
ab dddeerr
( ) ( ) '1 abababd rr ααα −+=
( ) ( )[ ]{ }ababx
abab
baab
ab xxexee
ab erf11, 2 −+−= − πλ
βΔΦ r
Diagonal Kelbg poten1al:
ab
baeeλπ
0→abr
ab
ba
ree
abab λ>>r
abab µβλ 22=111 −−− += baab mmµ
Ebeling et al., Contr. Plasma Phys., 1967
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ACCURACY OF DIRECT PIMC
HHH eee ˆˆˆˆ βββρ Δ−Δ−− ××== …n+1 UKH ˆˆˆ +=TkB1=β
( )1+=Δ nββ( ) [ ]UKUKH eeee
ˆ,ˆ2ˆˆˆ2
ˆβ
ββββρ
Δ−Δ−Δ−Δ−Δ ==
1 at n -‐> ∞ Error ~ (β / n)2 for every mul1plier
Total error Δρ ~ β2 / n -‐> 0 at n-‐>∞
( )βρρρ β Δ=Δ potfreeHigh-‐temperature pseudopoten1al
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TREATMENT OF EXCHANGE EFFECTS
Inside main cell – exchange matrix
Outside main cell – by perturba1on theory
Accuracy control – comparison with ideal degenerate gas
300~3een λ
Main MC cell
e e
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 = 2.7 nλ3 = 0.41
T = 50 kK ρ = 1.67x10-‐3 g/cm3
Γ = 0.54 nλ3 = 3.7 x10-‐2
-‐ proton
-‐ electron
-‐ electron
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104 105 1060.01
0.1
1
10 a)
- 1020 cm-3, DPIMC - 1021, DPIMC - 1020, SC - 1021, SC
P, G
Pa
T, K
104 105 106
1013
1014
- 1020 cm-3, DPIMC - 1021, DPIMC - 1020, SC - 1021, SC
rela
tive
num
ber c
once
ntra
tion
E, e
rg /
g
T, K0.0
0.2
0.4
0.6
0.8
1.0
b)
- 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
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DEUTERIUM SHOCK HUGONIOTS
0,4 0,6 0,8 1,0 3 4 6 8 10
20 40 60 80 100
200 400 600 800 1000
2000
- NOVA - NOVA - gas gun - Z-pinch - Mochalov - Mochalov - Trunin - REMC - DPIMC P , G
Pa
ρ , g/cm 3 0.1335 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)
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HYDROGEN, PIMC-‐SIMULATION, T = 10000 K Ne = Ni = 56, n = 20
n = 1023 cm-‐3 ρ = 0.167 g/cm3
Γ = 12.5 nλ3 = 41
n = 3x1022 cm-‐3 ρ = 0.05 g/cm3
Γ = 8. 4 nλ3 = 12.4
-‐ proton
-‐ electron
-‐ electron
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T = 10000 K, n = 3x1025 cm-‐3, ρ = 50.2 g/cm3
Γ = 84 nλ3 = 12400
PIMC SIMULATION Ne = Ni = 56, n = 20 protons ordering
1
2
3
4
5
0.1 0.2 0 r/aB
gee gii
gei
30r2gee
JETP LeXers 72, 245 (2000)
-‐ proton
-‐ electron
-‐ electron
50
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ELECTRON DENSITY DISTRIBUTION IN TWO-‐COMPONENT DEGENERATE SYSTEM
mh= 800 me= 2.1 <r>/aB = 3
top view side view
ρ = 25 T / Eb = 0.002
-‐ hole
-‐ electron
-‐ electron
-‐ hole IN COULOMB CRYSTAL
51
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QUANTUM DYNAMICS IN WIGNER REPRESENTATION
[ ]ρρ ,Ht
i =∂∂
( ) ( ) ∫ −⎟⎠⎞⎜
⎝⎛
2−
2+
21= ξξξρπ
ξdeqqpqW ipNd
L ,,
( ) ( )∫ ′′−′⎟⎠⎞⎜
⎝⎛
2′′+′=′′′ dpepqqWqq pqqiL ,,ρ
( ) ( )∫ −=∂∂+
∂∂ dsqstqspdsW
qW
mp
tW L
LL
,,, ω
( ) ( ) ( )∫ ⎥⎦⎤
⎢⎣⎡2′−′
22=
'sin, sqqqUqdqs Ndπ
ω
0=∂∂
∂∂−
∂∂+
∂∂
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 ′′′=′′′ ψψρ *, C∈ψ
RWL ∈
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SOLUTION OF WIGNER EQUATION
( ) ( ) ( )
( ) ( ) ( )∫ ∫ ∫∫
∫∞
∞−0
0000000
−×Π
+×0Π=
''''''' ,',,',,;,,'
,,,;,,,,
τττττττ ωτττ qsqspdsWqptqpdqdpd
dqdpqpWqptqptqpW
Wt
W
( )( ) ( ) '''' ',,;', ττττ ττ qqpqtqFdtpd ==
( ) ( ) '''' ',,;', ττττ ττ pqppmtpdtqd ==
Dynamical trajectories:
',, '' τττ qp
pq
( ) ( )',,;()',,;()',,;,,( '''''' τδτδτ ττττττ qptqqqptppqptqp ttW −−=ΠPropagator:
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0 50 1000,1
1
10
100T = 2 104 K0 1rs = 43.2 2 3 4 5
10
5 Re{
σ / ω
p }
ω/ωp
Quantum dynamics in Wigner representation
λ
mtp
dtqd
tqFdtpd
)(
))((
=
= random + momentum jumps
))(),(( 00 qpWinitial quantum
distribution
( )>0< )()(~ ptpFTσ
Dynamic conductivity
Veysman et al. Tkachenko et al. Bornath et al
T = 2·104 K rs = 43.2
QD h = 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{
σ / ω
p }
ω/ωp ω / ωp
ω / ωp
T = 2·104 K rs = 5
hν = Ry
Bornath et al. Ternovoi et al.
QD
QD 5000-20000 configurations
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Conclusions • Ab ini1o methods are useful for calcula1on of different proper1es of maXer at high energy density
• The main goal of ab ini1o methods is to replace experiment; in some cases it’s already possible
• Growth of computa1onal possibili1es will allow to apply more general approaches for calcula1on of plasma proper1es (quantum filed theory)
• Currently, however, semiempirical approaches are main workhorses; ab ini1o methods are used for calibra1on of such models
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( ) ( ) ( ) ( ) ( )υ φ ϕ ϕ φ ϕ ϕ ϕ φπ −
⎡ ⎤= − = + − − −⎣ ⎦∫1
' '' 2 5 3 ' 2 '3 2 1 2 1 22 3 2
0
3 2 5 3 2aT TT T TS F T u I u T T I u T I du
T
( ) ( )θ µ µ µ µµπ
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
3 2' '' '
1 2 3 2 1 22 ,
2 52 3T VT T N V
P F I I IT T T T
( ) ( )θ µ µπ
⎛ ⎞= − = ⎜ ⎟⎝ ⎠
3 2' '' '
1 22 ,
22V VV V N T
P F IT
Second deriva2ves of free energy
Thermodynamic Func1ons of Thomas-‐Fermi Model
( )υ µ φ φ φπ
⎡ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∫ ∫
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 poten1al, = / (u2T), a – cell volume, u = (r / r0)1/2
Free energy:
Expressions for 1st deriva1ves of F (P and S) are known.
Shemyakin O.P. et al., JPA 43, 335003 (2010)
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Second Deriva2ves of the Thomas-‐Fermi Model
The number of par1cles and poten1al are the func1ons 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 ,,,,,,,, υµϕϕ =
( ) ( ) ( )[ ]TVNTTVNTVNNN ,,,,,,,, υµ=
( )( ) T
T
TN NN
V ,
,
, υ
µ
µυµ∂∂∂∂
−=⎟⎠⎞⎜
⎝⎛∂∂
( )( ) TNV N
TNT ,
,
, υ
µυ
µµ
∂∂∂∂
−=⎟⎠⎞⎜
⎝⎛∂∂
( )( ) TTN N
TNTT ,,
,
,, υυ
υµ
υµυ µϕ
µϕϕ
⎟⎠⎞⎜
⎝⎛∂∂
∂∂∂∂
−⎟⎠⎞⎜
⎝⎛∂∂=⎟
⎠⎞⎜
⎝⎛∂∂
We need 6 deriva1ves in the grand canonical ensemble
( ) T,'
µυϕ ( ) υµϕ ,'T ( )
υµϕ ,'T
( )υµ ,
'T
N ( ) µυ ,'TN( ) µυ ,
'TN
Shemyakin O.P. et al., JPA 43, 335003 (2010)
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TF Poten2al and its Deriva2ves on , and T
ϕ µ
µ
= = =
⎧ = −⎪
=⎪⎪ ⎛ ⎞+⎨ = ⎜ ⎟⎪ ⎝ ⎠⎪⎪ = = =⎩
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 equa1on
( )
( ) ( )
υ µϕ
φ φυ −
= =
⎧ =⎪⎪ =⎪⎨
= +⎪⎪⎪ = =⎩
'
,
'
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
Deriva1ve of the Poisson equa1on on :
( )( ) ( )( ) ( )
µ υϕ
φ
φ−
−
= =
⎧Φ = −⎪⎪Φ = Φ +⎪⎨Ψ = Φ +⎪
⎪Φ = =⎪⎩
' 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
Deriva1ve on :
( )
( ) ( ) ( )
µ υϕ
φ φ φ φ− −
= =
⎧ =⎪⎪ =⎪⎨
⎡ ⎤= − +⎪ ⎣ ⎦⎪
= =⎪⎩
'
,
'
' 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
Deriva1ve on T:
ϕ ( )υ µ'
,TN
( )µ υ'
,TN( )µ υ'
,TN
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Adiaba2c Sound Velocity by Thomas-‐Fermi Model
V
V, atomic units
Ideal Fermi-‐gas
Isotherms
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PROBLEMS OF TF MODEL
• Thomas-‐Fermi method is quasiclassical; if one calculates energy levels in VTF(r) using the Shrödinger equa1on and then electron density quant(r), it will differ from the original TF electron density (r)
• Mean ion charge is roughly determined • The solu1on is to make (r) self-‐consistent and use the corrected poten1al and electron density
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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 plasma model
Thomas-‐Fermi
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TYPICAL CONFIGURATION OF PARTICLES H + He mixture, T = 105 K, ne = 1023 cm-‐3
proton
( ) 988.0HHeHe =+mmm
α-‐par1cle
electron, é
electron, ê
He+
He
40 α-‐par1cles, 2 protons, 82 electrons
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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 / NRy
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
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ELECTRON-‐HOLE PLASMA. PIMC SNAPSHOT crystal, mh(eff) = 800, me(eff) = 1
-‐ hole
-‐ electron
-‐ electron
-‐ hole
<r>/aB = 0.63
T = 0.064Eb
64
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ELECTRON-‐HOLE PLASMA. PIMC SNAPSHOT s1ll crystal, mh(eff) = 100, me(eff) = 1
-‐ hole
-‐ electron
-‐ electron
-‐ hole
<r>/aB = 0.63
T = 0.064Eb
65
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liquid, mh(eff) = 25, me(eff) = 1
ELECTRON-‐HOLE PLASMA. PIMC SNAPSHOT
-‐ hole
-‐ electron
-‐ electron
-‐ hole
<r>/aB = 0.63
T = 0.064Eb
66
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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
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PAIR DISTRIBUTION FUNCTIONS. QUANTUM MELTING
<r>/aB = 0.63
T = 0.064Eb
M = mh / me
h-‐h e-‐e h-‐h
e-‐e
h-‐h
e-‐e
h-‐h e-‐e
68
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PSEUDOPOTENTIALS IN DFT
• Diminish the number of plane waves necessary for the good representa1on of inner electrons wave func1ons
• Part of electrons are considered as a core, part as valent • Pseudopoten1al is constructed at T = 0 and doesn’t depend on
pressure and temperature
Pseudopoten1al approach
Core electrons (pseudopoten1al)
Valent electrons (plane waves)
Full-‐poten1al approach muffin-‐1n orbitals for all electrons
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APPROXIMATIONS IN PSEUDOPOTENTIAL APPROACH Pseudopoten1al describes electrons with energies less than the Fermi energy – errors at rela1vely high temperatures
EF EF
T = 0 T > 0
Spa1al distribu1on of core electrons in a pseudopoten1al is unchanged – errors at rela1vely high pressures
P = 0 P > 0
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HOLE-‐HOLE DISTANCE FLUCTUATIONS
Lindemann criterion
71
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Electron Heat Capacity for W at T = 11eV. Return of Free Electrons into 4f-state under
Compression W, bcc
JIHT RAS
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(ρ
0 )
N4f
ion (ρ
0) - N
4fio
n (ρ)
ρ/ρ0
Cv e(ρ)
N4f
ion (ρ
0)(%
)
W, bcc, T = 11 eV
CV / CV(ρ0) Number of returned electrons (%) Fr
ac2o
n of f-‐electron
s Compression ra2o
1.0 1.2 1.4 1.6 1.80
4
8
12
16W
12 eV
Cve
ρ/ρ0
1 eV
Electron
heat cap
acity
Compression ra2o