Transport and Optical Properties in Dense Plasmas Heidi Reinholz

Transport and Optical Properties in Dense
Plasmas
Heidi Reinholz
in collaboration with J. Adams, C. Fortmann, S. Glenzer, I. Morozov, T. Raitza, R. Redmer,
G. Röpke, A. Sengebusch, R. Thiele
Dense Matter In Heavy Ion Collisions and Astrophysics
21.8.-1.9.2006, JINR, Dubna/Russia
Reinholz, DM06 – p. 1
Outline
Plasma in electric and magnetic fields, interaction with radiation
⇒ production, excitation, diagnostic tool
Many–particle theory
kinetic equations, linear response, molecular dynamics simulations
⇒ dielectric function ǫ(k, ω), dynamical collision frequency ν(ω)
Applications
⇒ transport properties
(dc–conductivity, thermopower, Hall effect)
⇒ optical properties
(dynamical conductivity, reflectivity, absorption,
Thomson scattering, bremsstrahlung, spectral lines)
Reinholz, DM06 – p. 2
Density-temperature regions
temperature [K]
8
10
7
10
6
10
105
4
10
103
2
10
1
10
solar
corona
supernova
explosion
gaseous
nebula
solar
flames
atmosphere
ionosphere
interstellar gas
Tokamaks
105
104
White
dwarf
3
10
Z-pinchs
shock waves
cap. discharge
Brown
dwarf
Jupiter
core
Γ = 10
Γ=1
Γ = 0.1
ICFplasmas
101
0
10
glow
discharge
semi-cond.
traps
2
10
10-1
10-2
-3
10
0
4
8
12
16
20
24
28
32
4
8
12
16
20
24
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
density [cm -3]
density [cm -3]
A. Höll, PhD thesis (Rostock, 2002)
Reinholz, DM06 – p. 3
temperature [eV]
109
θ = 0.1
θ = 10
Sun
core
XFEL
active
galactic
nucleus
106
metals
1010
Electrical Conductivity σ(ω)
induced charge current density J~ in many-particle system under the influence
~
of electric field E
~
J~ = σ E
Reinholz, DM06 – p. 4
Electrical Conductivity σ(ω)
induced charge current density J~ in many-particle system under the influence
~
of electric field E
~
J~ = σ E
σ in 1/(Ωm)
10
5
Xe
100 kK
30 kK
25 kK
10
4
20 kK
15 kK
10
10
[Shilkin 02, Mintsev 03] (7.5-20 kK)
[Mintsev 00] (5-20 kK)
[Urlin 92] (10-20 kK)
[Mintsev 80] (30-100 kK)
[Mintsev 79] (5-20 kK)
[Ivanov 76] (20-30 kK)
3
10 kK
2
10
-1
10
-3
ρ in g/cm
0
10
1
S. Kuhlbrodt et al., Contr Plasma Phys. 45 (2005) 61
Reinholz, DM06 – p. 4
Electrical Conductivity σ(ω)
induced charge current density J~ in many-particle system under the influence
~
n o
of electric field E
~ = Tr ~ˆj ρ̂
J~ = σ E
σ in 1/(Ωm)
10
5
Xe
100 kK
30 kK
25 kK
10
statistical operator ρ̂
4
20 kK
currrent operator
15 kK
10
10
[Shilkin 02, Mintsev 03] (7.5-20 kK)
[Mintsev 00] (5-20 kK)
[Urlin 92] (10-20 kK)
[Mintsev 80] (30-100 kK)
[Mintsev 79] (5-20 kK)
[Ivanov 76] (20-30 kK)
3
~ˆj =
10 kK
2
10
-1
10
-3
ρ in g/cm
0
10
1 X ec ~ˆ
Pc
Ω c mc
1
S. Kuhlbrodt et al., Contr Plasma Phys. 45 (2005) 61
Reinholz, DM06 – p. 4
Linear response theory
statistical operator for generalized grand canonical ensemble by
introducing set of relevant observables {Bn }
ρrel
=
1 −[Ĥ−µN +Pn Φn Bn ]
e
Zrel
Ĥ = Ĥeq −
X
~ cE
~
ec R
c
self-consistency condition for response parameter Φn
Tr (Bn ρrel ) = Tr (Bn ρ)
with statistical operator
ρ = ρrel + ρirrel
solution in linear response:
response equation containing equilibrium correlation functions/
generalized BOLTZMANN equation
~˙ c iec E
~
hBm ; R
=
X
hBm ; Ḃn iΦn
n
Tr {Bn ρ}
=
X
(Bn ; Bm )Φn
m
Zubarev, Morozov, Röpke, Stat. Mech. of Non-equilibrium Processes, Berlin 1996
Reinholz, DM06 – p. 5
Linear response theory
~˙ c iec E
~
hBm ; R
=
X
hBm ; Ḃn iΦn
n
Tr {Bn ρ}
=
X
(Bn ; Bm )Φn
m
equilibrium correlation functions
hA; Biz
=
Z
0
(A (t) ; B)
=
1
β
∞
dt e
Z
∞
izt
i
(A (t) ; B) = −
β
Z
∞
−∞
dω
1
1
Im GAB + (ω − i0)
π z−ω ω
dτ Tr [A (t − i~τ ) B + ρ0 ]
0
~˙
application to electrical current density using set {Bn } = P~ = R
n o
n
o
n o
D E
e
e
˙
~
~ = Tr ρrel P~ = σ E
J~ = ~j = Tr ρ~j = Tr ρR
Ω
Ω
Röpke, Meister, Ann. Phys. 36 (1979) 377; Röpke, PRA 38 (1988) 3001; Reinholz et al. , PRE 52 (1995) 6368
Reinholz, DM06 – p. 6
~˙ c iec E
~
hBm ; R
=
X
hBm ; Ḃn iΦn
n
Tr {Bn ρ}
=
X
(Bn ; Bm )Φn
m
~ =R
~˙
application to electrical current density using set {Bn } = P
D E
n o
n o
n
o
e
e
˙
~ = Tr ρrel P
~ = σE
~
J~ = ~j = Tr ρ~j = Tr ρR
Ω
Ω
solution for electrical conductivity
βe2 (P ; P )2
σ = βΩhj; ji =
Ω me hṖ ; Ṗ i
Kubo-Greenwood formula
⇐⇒
force force correlation functions
(Ṗ = Fei + Fee + Fea )
Röpke, Meister, Ann. Phys. 36 (1979) 377; Röpke, PRA 38 (1988) 3001; Reinholz et al. , PRE 52 (1995) 6368
Reinholz, DM06 – p. 7
Dynamical conductivity
time dependent external field
2
ǫ0 ωpl
σ(ω) =
−iω + ν(ω)
2
e
ne
2
ωpl =
ǫ0 me
generalized Drude formula
Reinholz, DM06 – p. 8
Dynamical conductivity
time dependent external field
2
ǫ0 ωpl
σ(ω) =
−iω + ν(ω)
2
e
ne
2
ωpl =
ǫ0 me
generalized Drude formula
kinetic theory
collision integrals, moment expansion for distribution function, relaxation time ansatz
Reinholz, DM06 – p. 8
Dynamical conductivity
time dependent external field
2
ǫ0 ωpl
σ(ω) =
−iω + ν(ω)
2
e
ne
2
ωpl =
ǫ0 me
generalized Drude formula
kinetic theory
collision integrals, moment expansion for distribution function, relaxation time ansatz
linear response theory
generalized statistical operator, equilibrium correlation functions
β D ~˙ ~˙ E
P ; P ω+iη
ν(ω) =
ne Ω
Reinholz, Redmer, Röpke, Wierling, PRE 62 (2000) 5648
Reinholz, DM06 – p. 8
Dynamical conductivity
2
ǫ0 ωpl
σ(ω) =
−iω + ν(ω)
time dependent external field
2
e
ne
2
ωpl =
ǫ0 me
generalized Drude formula
kinetic theory
collision integrals, moment expansion for distribution function, relaxation time ansatz
linear response theory
generalized statistical operator, equilibrium correlation functions
β D ~˙ ~˙ E
P ; P ω+iη
ν(ω) =
ne Ω
Reinholz, Redmer, Röpke, Wierling, PRE 62 (2000) 5648
molecular dynamic simulations
normalized current auto–correlation function (ACF)
hJ; Jiω+iη =
2
ǫ0 ωpl
βΩ
lim
ǫ→0
Z∞
0
i(ω+iǫ)t
e
K(t) dt
1 1
K(t) =
hJ 2 i δ
Z
δ
dτ J(t + τ )J(τ )
0
Reinholz, DM06 – p. 8
MD simulations
normalized current auto-correlation function (ACF)
K(t)
L/T
1 1
=
hJ 2 i δ
Z
δ
dτ J z/x (t + τ )J z/x (τ )
0
with
N
XX
1
z
~ =
(t)
ec~vc,α
J(t)
Ω0 c α=1
Reinholz, DM06 – p. 9
MD simulations
normalized current auto-correlation function (ACF)
K(t)
L/T
1 1
=
hJ 2 i δ
Z
δ
dτ J z/x (t + τ )J z/x (τ )
0
with
N
XX
1
z
~ =
(t)
ec~vc,α
J(t)
Ω0 c α=1
equation of motion
with mean fieldE z due to
average current density ~j(t) = jz ~ez
has only z-component due to ~k = k~ez
Reinholz, DM06 – p. 9
MD simulations
normalized current auto-correlation function (ACF)
K(t)L/T =
1 1
hJ 2 i δ
Z
δ
1
K(t)
dτ J z/x (t + τ )J z/x (τ )
0
0.5
with
N
XX
1
z
~ =
(t)
ec~vc,α
J(t)
Ω0 c α=1
0
0
equation of motion
1
2
3
with mean fieldE z due to
average current density ~j(t) = jz ~ez
has only z-component due to ~k = k~ez
4
t/τe 5
-0.5
Kelbg pseudopotential
Figure: normalized current ACF for Γ = 1.28, τe = 2π/ωpl – period of electron plasma
oscillations: MD simulations without (o) and including (△) an additional mean-field term
Reinholz et al., PRE 69 (2004) 066412; Morozov et al. PRE 71 (2005) 066408
Reinholz, DM06 – p. 9
Dynamical collision frequency
fully ionized plasma at n = 3.8 × 1021 cm−3 , T = 33 000 K, Γ = 1.28, Θ = 3.2
1
0.5
ν(ω)/ωpl
R
e
hqecyDpeuaonbhfsirttil
M
gm
a
o
s
mpoetr
mI
pmDeuaonbsrttili
M
eohysr
ν(ω)/ωpl
0.4
0.3
0.2
0.1
0
0.1
-0.1
-0.2
0.1
1
ω/ωpl
10
0.1
1
ω/ωpl
10
Reinholz, DM06 – p. 10
Dynamical collision frequency
fully ionized plasma at n = 3.8 × 1021 cm−3 , T = 33 000 K, Γ = 1.28, Θ = 3.2
1
0.5
ν(ω)/ωpl
R
e
hqecyDpeuaonbhfsirttil
M
gm
a
o
s
mpoetr
mI
pmDeuaonbsrttili
M
eohysr
ν(ω)/ωpl
0.4
0.3
0.2
0.1
0
0.1
-0.1
-0.2
1
0.1
ω/ωpl
perturbation theory:
ν(ω) ≈ r(ω)ν
(P0 )
(ω) = r(ω)ν
10
0.1
1
ω/ωpl
10
Reinholz et al., PRE 62 (2000) 5648; PRE 69 (2004) 066412
GD
(ω) = r(ω)(ν
ladder
h
i
Born
LB
ω) − ν
(ω) + ν (ω)
dynamically screened binary collision approximation using Gould-deWitt ansatz (dynamical
screening and strong collisions); higher moments of single–particle distribution function via
renormalization factor (electron-electron interaction)
Reinholz, DM06 – p. 10
Dynamical collision frequency
fully ionized plasma at n = 3.8 × 1021 cm−3 , T = 33 000 K, Γ = 1.28, Θ = 3.2
1
0.5
ν(ω)/ωpl
R
e
hqecyDpeuaonbhfsirttil
M
gm
a
o
s
mpoetr
mI
pmDeuaonbsrttili
M
eohysr
ν(ω)/ωpl
0.4
0.3
0.2
0.1
0
0.1
-0.1
-0.2
1
0.1
ω/ωpl
perturbation theory:
ν(ω) ≈ r(ω)ν
(P0 )
(ω) = r(ω)ν
10
0.1
1
ω/ωpl
10
Reinholz et al., PRE 62 (2000) 5648; PRE 69 (2004) 066412
GD
(ω) = r(ω)(ν
ladder
h
i
Born
LB
ω) − ν
(ω) + ν (ω)
dynamically screened binary collision approximation using Gould-deWitt ansatz (dynamical
screening and strong collisions); higher moments of single–particle distribution function via
renormalization factor (electron-electron interaction)
high frequency asymptotes in Born approximation
Re ν Kelbg (ω) ∝ ω −7/2
ν Coulomb (ω) ∝ ω −3/2
Reinholz, DM06 – p. 10
DC Conductivity comparison
1
n(0)/wpl
0.1
G
0.1
1
collision frequency
from MD simulations at ωpl
• collisional damping of
Langmuir waves, ν = 2δc
(Morozov, Norman, JETP 100 (2005))
Reinholz, DM06 – p. 11
DC Conductivity comparison
1
n(0)/wpl
0.1
G
0.1
1
collision frequency
from MD simulations at ωpl
• collisional damping of
Langmuir waves, ν = 2δc
(Morozov, Norman, JETP 100 (2005))
σdc = lim σ(ω) =
ω→0
2
ǫ0 ωpl
νdc
e2 ne
=
τdc
me
△ — MD simulations T = 33000K
full line — interpolation formula T = 33000K
filled symbols — experimental data (Mintsev et al.)
Reinholz, DM06 – p. 11
DC Conductivity comparison
1
s/4pe0wpl
n(0)/wpl
1
0.1
G
0.1
1
0.1
0.1
collision frequency
from MD simulations at ωpl
• collisional damping of
Langmuir waves, ν = 2δc
(Morozov, Norman, JETP 100 (2005))
1
G
Static conductivity
σdc = lim σ(ω) =
ω→0
2
ǫ0 ωpl
νdc
e2 ne
=
τdc
me
△ — MD simulations T = 33000K
full line — interpolation formula T = 33000K
filled symbols — experimental data (Mintsev et al.)
Reinholz, DM06 – p. 11
DC conductivity in xenon
electrical conductivity of xenon in comparison to experimental data
σ in 1/(Ωm)
10
5
Xe
100 kK
30 kK
25 kK
10
4
20 kK
15 kK
10
10
[Shilkin 02, Mintsev 03] (7.5-20 kK)
[Mintsev 00] (5-20 kK)
[Urlin 92] (10-20 kK)
[Mintsev 80] (30-100 kK)
[Mintsev 79] (5-20 kK)
[Ivanov 76] (20-30 kK)
3
10 kK
2
10
-1
10
-3
ρ in g/cm
0
10
1
partially ionized plasma ⇒ bound states, depletion of free charge carriers,
additional scattering mechanisms (COMPTRA04)
S. Kuhlbrodt et al., Contr Plasma Phys. 45 (2005) 61
Reinholz, DM06 – p. 12
Conclusions I
linear response theory (Zubarev approach) to derive expressions for static
and dynamical conductivity
Kubo formula or force-force correlation functions
systematic and consistent inclusion of strong collisions, dynamical
screening, e-e interaction and effects in partially ionized systems
results from perturbation theory, molecular dynamics simulation and
experiments are consistent in weakly coupled plasmas
high frequency behaviour of collision frequency
quantum statistical simulations: QMD, WPMD, PIMC
Reinholz, DM06 – p. 13
Applications
Hall effect
optical properties
reflectivity
Thomson scattering
Reinholz, DM06 – p. 14
Hall measurement
Reinholz, DM06 – p. 15
Hall measurement
Argon
Xenon
-4
10
-3
10
Incident wave
Refled wave
RH [Ωm]
RH [Ωm]
-4
10
-5
10
-6
10
-8
-6
10
10
2.5
3
D [km/s]
3.5
1.5
Incident wave
Reflected wave
2
D [km/s]
2.5
Shilkin et al. , JETP 77 486 (2003)
Reinholz, DM06 – p. 15
Hall measurement
Hall voltage Uh
“
”
Uh
~
~
~
Eh = Rh J × B =
d
free electron densities ne calculated
from a thermodynamic model
⇒ Rh = rh /(e ne )
Argon
Xenon
-4
10
-3
10
Incident wave
Refled wave
RH [Ωm]
RH [Ωm]
-4
10
-5
10
-6
10
-8
-6
10
10
2.5
3
D [km/s]
3.5
1.5
Incident wave
Reflected wave
2
D [km/s]
2.5
Shilkin et al. , JETP 77 486 (2003)
Reinholz, DM06 – p. 15
Electromagnetic transport properties
~ and magnetic field B
~
many-particle system under the influence electric field E
D E
~ = − e R̂˙
~ + σRh J~el × B
electric current densities: J~ = σ E
Ω
Reinholz, DM06 – p. 16
Electromagnetic transport properties
~ and magnetic field B
~
many-particle system under the influence electric field E
D E
~ = − e R̂˙
~ + σRh J~el × B
electric current densities: J~ = σ E
Ω
time evolution:
i
dR̂n
˙
= R̂n = [Ĥ, R̂n ],
dt
~
e ~ ~ˆ ~ˆ
ˆ
~
~
Ĥ = Ĥ0 − eE R +
B(R × P )
2me
Reinholz, DM06 – p. 16
Electromagnetic transport properties
~ and magnetic field B
~
many-particle system under the influence electric field E
D E
~ = − e R̂˙
~ + σRh J~el × B
electric current densities: J~ = σ E
Ω
time evolution:
˙
me R̂
¨
me R̂
=
=
i
dR̂n
˙
= R̂n = [Ĥ, R̂n ],
dt
~
”
e “~
B × R̂
P̂ +
2
„
“
”«
1
˙
˙
~+
~
P̂ − e E
R̂ × B
me
e ~ ~ˆ ~ˆ
ˆ
~
~
Ĥ = Ĥ0 − eE R +
B(R × P )
2me
relevant observables
{Bn }
~˙ 0
R
=
=
~˙ n
R
˙
R̂
with
Lorentz force
Reinholz, DM06 – p. 16
Electromagnetic transport properties
~ and magnetic field B
~
many-particle system under the influence electric field E
D E
~ = − e R̂˙
~ + σRh J~el × B
electric current densities: J~ = σ E
Ω
time evolution:
˙
me R̂
¨
me R̂
=
=
i
dR̂n
˙
= R̂n = [Ĥ, R̂n ],
dt
~
”
e “~
B × R̂
P̂ +
2
„
“
”«
1
˙
˙
~+
~
P̂ − e E
R̂ × B
me
e ~ ~ˆ ~ˆ
ˆ
~
~
Ĥ = Ĥ0 − eE R +
B(R × P )
2me
relevant observables
{Bn }
~˙ 0
R
=
=
~˙ n
R
˙
R̂
with
Lorentz force
˙
˙
equilibrium correlation functions hP̂m ; P̂n i + ωe 2 hP̂n ; P̂m i
with electron cyclotron frequency ωe =
eB
me
Reinholz, DM06 – p. 16
fully ionized plasma in magnetic field
LRT within five moment approximation
transport cross section in Born approximation for statically screened e-i and e-e potential
(Debye potential)
note that rH = 1.93 for fully ionized Lorentz plasma
Reinholz, DM06 – p. 17
fully ionized plasma in magnetic field
LRT within five moment approximation
transport cross section in Born approximation for statically screened e-i and e-e potential
(Debye potential)
note that rH = 1.93 for fully ionized Lorentz plasma
conductivity is reduced in low density limit (higher moments are not relevant)
maximum in Hall factor shifts to higher densities with decreasing temperature
Reinholz, DM06 – p. 17
Hall coefficient in partially ionized plasma
scattering mechanisms:
F = Fei + Fee + Fea
Reinholz, DM06 – p. 18
transport cross section
Reinholz, DM06 – p. 19
conductivity in magnetic field
Reinholz, DM06 – p. 20
Conclusions II
extension of linear response theory to include magnetic field effects in order to describe
Hall effect
reproduction of results from relaxation time approach
systematic and consistent inclusion of e-e interaction and effects in partially ionized systems
Reinholz, DM06 – p. 21
Conclusions II
extension of linear response theory to include magnetic field effects in order to describe
Hall effect
reproduction of results from relaxation time approach
systematic and consistent inclusion of e-e interaction and effects in partially ionized systems
enhancement of Hall factor (rH =1.2) in low density semiconductors/plasmas suppressed
with increasing correlations (density)
correlations are not relevant for high magnetic fields (rH =1) except in a small parameter
region of intermediate correlation strength
Reinholz, DM06 – p. 21
Conclusions II
extension of linear response theory to include magnetic field effects in order to describe
Hall effect
reproduction of results from relaxation time approach
systematic and consistent inclusion of e-e interaction and effects in partially ionized systems
enhancement of Hall factor (rH =1.2) in low density semiconductors/plasmas suppressed
with increasing correlations (density)
correlations are not relevant for high magnetic fields (rH =1) except in a small parameter
region of intermediate correlation strength
outlook
more detailed calculations necessary for comparison with experiment
(ionization degree, transport cross section)
Hall factor as diagnostic tool for determination of system parameters
Reinholz, DM06 – p. 21
Dielectric and optical response
2
ωpl
i
ǫ(k, ω) = 1 +
σ(k, ω) = 1 −
ǫ0 ω
ω(ω − iν(ω))
dynamical conductivity - generalized Drude formula
Reinholz, DM06 – p. 22
Dielectric and optical response
2
ωpl
i
ǫ(k, ω) = 1 +
σ(k, ω) = 1 −
ǫ0 ω
ω(ω − iν(ω))
dynamical conductivity - generalized Drude formula
2
ic
optical information: lim ǫ(k, ω) = n(ω) +
α(ω)
k→0
2ω
Reinholz, DM06 – p. 22
Dielectric and optical response
2
ωpl
i
ǫ(k, ω) = 1 +
σ(k, ω) = 1 −
ǫ0 ω
ω(ω − iν(ω))
dynamical conductivity - generalized Drude formula
2
ic
optical information: lim ǫ(k, ω) = n(ω) +
α(ω)
k→0
2ω
dynamical structure factor:
1
1
−1
S(k, ω) =
Imǫ
l (k, ω)
π V (k) e−β~ω − 1
Reinholz, Redmer, Röpke, Wierling, PRE 62 (2000) 5648
Reinholz, DM06 – p. 22
Dielectric and optical response
2
ωpl
i
ǫ(k, ω) = 1 +
σ(k, ω) = 1 −
ǫ0 ω
ω(ω − iν(ω))
dynamical conductivity - generalized Drude formula
2
ic
optical information: lim ǫ(k, ω) = n(ω) +
α(ω)
k→0
2ω
dynamical structure factor:
1
1
−1
S(k, ω) =
Imǫ
l (k, ω)
π V (k) e−β~ω − 1
reflectivity at normal incidence
R(ω) =
Reinholz, Redmer, Röpke, Wierling, PRE 62 (2000) 5648
p
ǫ(ω) − 1 2
p
ǫ(ω) + 1 Reinholz, DM06 – p. 22
Reflectivity of Deuterium
0.8
shock wave experiments along
Hugeniot
Reflectivity
0.6
assuming step like shock wave
front
0.4
R(ω)
=
with
Exp. 1064 nm
Exp. 808 nm
+
FVT id 808 nm
0.2
+
FVT id
0
0
100
200
300
400
P [GPa]
RPA ǫ(ω)
˛
˛p
˛ ǫ(ω) − 1 ˛2
˛
˛
˛
˛p
˛ ǫ(ω) + 1 ˛
=
1−
1064 nm
500
600
Drude
ǫ(ω)
=
1−
2
ωpl
ω2
2
ωpl
ω(ω + iνdc )
insulator-plasma transition
Redmer et al. , AIP Conf. Proc. 845, 127 (2006
Reinholz, DM06 – p. 23
Reflectivity of xenon
shock compressed dense plasma:
⇒ pressure 1.6 - 20 GPa, T≈ 33 000 K, density 0.5 - 4 g cm−3
laser:
λ = 1.06 µm, 0.694 µm, 0.532 µm
Reinholz, DM06 – p. 24
Reflectivity of xenon
shock compressed dense plasma:
⇒ pressure 1.6 - 20 GPa, T≈ 33 000 K, density 0.5 - 4 g cm−3
laser:
λ = 1.06 µm, 0.694 µm, 0.532 µm
distribution of density of free electrons ne (z)
in shock wave front in dependence on electron density ne
H 21 cm-3L
neH10
9
5
2
10
5 naF
0.4
0
-0.4 zHΜmL
-0.8
Reinholz, DM06 – p. 24
Reflectivity of xenon
shock compressed dense plasma:
⇒ pressure 1.6 - 20 GPa, T≈ 33 000 K, density 0.5 - 4 g cm−3
laser:
λ = 1.06 µm, 0.694 µm, 0.532 µm
0.8
distribution of density of free electrons ne (z)
in shock wave front in dependence on electron density ne
R
H 21 cm-3L
neH10
9
5
2
10
0.6
0.4
0.2
5 naF
0.4
0
-0.4 zHΜmL
-0.8
0
0
2
4
8
6
21
10
-3
ne in 10 cm
comparison of experiment with theoretical calculations for 1.06 µm: molecular dynamic simulation (N); ERRdc Padé formula for νei,ee (♦);
ERRdc incl. νea (•); full line - shock front profile
Reinholz, DM06 – p. 24
Reflectivity of xenon
0.5
R
0.4
0.3
0.2
0.1
0
0
2
4
6
21
-3
ne in 10 cm
8
10
Reflectivity coefficient R for Xenon calculated with asymmetric Fermi profile in comparison with
measurements (symbols with error bars) for laser wavelengths 1.06 µm (solid line, ), 0.694 µm
(dashed line, N), and 0.532 µm (dotted line, ), the corresponding critical densities ncr
e are
Raitza et al. , J. Phys. A 39 (2006) 4393
indicted with vertical lines
•
Reinholz, DM06 – p. 25
Thomson Scattering
scattering cross section:
k1
d2 σ
= σT S(k, ω)
dΩ dω
k0
k = k0 − k1 , ω = ω0 − ω1
k0 (k1 ): incident (scattered) wavevector
σT : Thomson cross section
Reinholz, DM06 – p. 26
Thomson Scattering
scattering cross section:
k1
d2 σ
= σT S(k, ω)
dΩ dω
k0
k = k0 − k1 , ω = ω0 − ω1
k0 (k1 ): incident (scattered) wavevector
σT : Thomson cross section
O.L. Landen et al., JQSRT 71 (2001) 465
Reinholz, DM06 – p. 26
Thomson Scattering
scattering cross section:
k1
d2 σ
= σT S(k, ω)
dΩ dω
k0
k = k0 − k1 , ω = ω0 − ω1
k0 (k1 ): incident (scattered) wavevector
σT : Thomson cross section
O.L. Landen et al., JQSRT 71 (2001) 465
S(k, ω) = |fI (k) + q(k)|2 Sii (k, ω) + Zf See (k, ω) + ZC
Z
e ω − ω ′ )Ss (k, ω ′ )
dω ′ S(k,
Reinholz, DM06 – p. 26
Thomson Scattering
scattering cross section:
k1
d2 σ
= σT S(k, ω)
dΩ dω
k0
k = k0 − k1 , ω = ω0 − ω1
k0 (k1 ): incident (scattered) wavevector
σT : Thomson cross section
O.L. Landen et al., JQSRT 71 (2001) 465
S(k, ω) = |fI (k) + q(k)|2 Sii (k, ω) + Zf See (k, ω) + ZC
Z
e ω − ω ′ )Ss (k, ω ′ )
dω ′ S(k,
Experiment on Beryllium at 30 kJ
Omega laser facility in Rochester
heating: Rh X-ray (2.7 keV - 3.4 keV)
scattering source: He-like Ti α-line
(4.75 keV)
scattering angle: ΘS = 125◦
Glenzer et al., PRL 90 (2003) 175002
Reinholz, DM06 – p. 26
Dynamic structure factor
See (k, ω) =
1
1
−1
Imǫ
long (k, ω)
−β~ω
π V (k) e
−1
0
10
0.5
S
ν
Γ
(k,ω)
RPA
S
ω
Simulation
10
ω)
Sk(,
-4
ωpl
S(k,ω
(k,ω)
10
0.2
-6
10
0.1
-8
0
0.5
ω
1
ωpl]
unost[fi
0
(k,ω)
-7.5
-2
ωpl]
un)so[tif
0.3
ν,RPA
S
RPA
S (k,ω)
Simulation
0.4
κ=0.36
unost[fi =10k.,/
~
κ=0.36
=10k./
RPA,
Γ
1.5
2
10
1
ω
ωpl]
10
dynamic structure factor for an electron-proton model plasma with Deutsch-like effective
interaction in RPA and in Mermin-like approximation which utilizes a dynamically screened
collision frequency ν(ω)
A. Selchow et al. PRE 64 (2001) 056410
Reinholz, DM06 – p. 27
Thomson scattering
diagnostic tool for warm dense matter:
determination of plasma parameters using VUV and X-ray
relevance of collision
22
-3
ne=10 cm , λ0=1.0nm, θ=60°
T=0.5 eV Born
T=2.0 eV Born
T=8.0 eV Born
T=0.5 eV LB
T=2.0 eV LB
T=8.0 eV LB
T=0.5 eV RPA
T=2.0 eV RPA
T=8.0 eV RPA
4
S(k,ω)
3
2
1
0
-4
-2
0
ω/ωpl
2
Höll et al. , EPJD 29 (2004) 159, Redmer et al. , IEEE Transact. on Plasma Sc. 33 (2005)
4
Reinholz, DM06 – p. 28
Determination of temperature and density
electron density ne is given by position of
plasmon peak, related to dispersion
relation Re ǫ(k, ω) = 0
in RPA: Gross-Bohm relation [1]:
2
2
2
+ 3k2 vth
≈ ωpl
ωR
21
0.5
with ωpl =
e2 n e
ǫ0 me
BM Te= 1eV
BM Te= 5eV
0.4
-1
Y=0.8
0.3
0.2
0.1
Y=0.3
0
-2
q
-3
ne=10 cm , ωpl=1.17 eV, λ0=32 nm, θS=120º
See(k,ω) [Ryd ]
Electron temperature Te can be
determined for all frequencies via
detailed balance:
See (k, ω)
− k ~ω
= e B Te
Y =
See (−k, −ω)
-1.5
-1
-0.5
0
ω/ωpl
0.5
1
1.5
2
2 = k T /m
and the thermal velocity vth
e
B e
Note: for warm dense matter resonance position ωR can be shifted by collisions
[1] Bohm, Gross, PR 75 (1949) 1851
Reinholz, DM06 – p. 29
Conclusions III
The dielectric function governs different physical properties such as the
dc and optical conductivity.
Central quantity in describing transport and optical properties is the
dynamical collision frequency.
Results obtained from linear response theory and MD simulations are in
good agreement for dynamic structure factor as well as in
long-wavelength limit.
In order to describe experimental results
inclusion of bound states (partial ionization, polarization potential,
spectral function, Mott effect)
density and temperature profiles
Thomson scattering in warm dense matter and bremsstrahlung are of
particular interest due to recent developments of experimental
opportunities.
Reinholz, DM06 – p. 30
Transport properties
many-particle system under the influence of external perturbations Xi :
~ temperature gradient ∇T , magnetic field B
~
electric field E,
consider electric charge and energy current densities:
J~el
=
X
J~q
=
i
~ + σN ∇T × B
~
~i = σ E
~ − S∇T + σRH J~el × B
L̂0i X
i
X
L̂1i
~ − L ∇T × B
~
~ i = T S J~el − K∇T − N T J~el × B
X
Reinholz, DM06 – p. 31
Transport properties
many-particle system under the influence of external perturbations Xi :
~ temperature gradient ∇T , magnetic field B
~
electric field E,
consider electric charge and energy current densities:
J~el
=
X
J~q
=
i
~ + σN ∇T × B
~
~i = σ E
~ − S∇T + σRH J~el × B
L̂0i X
i
X
L̂1i
~ − L ∇T × B
~
~ i = T S J~el − K∇T − N T J~el × B
X
σ
−
electrical conductivity
K
−
thermal conductivity
S
−
thermopower
RH
−
Hall coefficient
N
−
Nernst coefficient
L
−
Leduc − Rhigi coefficient
Reinholz, DM06 – p. 31
Transport properties
many-particle system under the influence of external perturbations Xi :
~ temperature gradient ∇T , magnetic field B
~
electric field E,
consider electric charge and energy current densities:
J~el
=
X
J~q
=
i
~ + σN ∇T × B
~
~i = σ E
~ − S∇T + σRH J~el × B
L̂0i X
i
X
L̂1i
~ − L ∇T × B
~
~ i = T S J~el − K∇T − N T J~el × B
X
σ
−
electrical conductivity
K
−
thermal conductivity
S
−
thermopower
RH
−
Hall coefficient
N
−
Nernst coefficient
L
−
Leduc − Rhigi coefficient
Hall voltage UH
~ = UH
~ H = RH J~el × B
E
d
Reinholz, DM06 – p. 31
Transport and Onsager coefficients
~ ⊥B
~
assume geometrical configuration: ∇T k E
~ and B
~ or B
~ =0
components parallel to E
σk
Sk
~
components perpendicular to B
= e2 L01
1
L11
=
h−
eT
L01
σ⊥
S⊥
RH
h - enthalpy
electron cyclotron frequency ωe =
n o
ˆ
J~ = Tr ~j ρ̂
eB
me
=
=
e2 Le01
η
1
eT
L̃11
h−
Le01
Le02
= −
η
2
eme Le01
η=
1+
1+
e e
L02 L12
ωe2
Le01 Le11
e2
2 L02
ωe
Le201
! !
η
!−1
Reinholz, DM06 – p. 32
Relaxation time approach
kinetic approach: solving linearized Boltzmann equation with relaxation time
approach
L̃ij
=
Lij
=
fi
fl
ǫi τ j
ne
me 1 + ωe2 τ 2
ne ˙ i j ¸
ǫ τ
me
Reinholz, DM06 – p. 33
Relaxation time approach
kinetic approach: solving linearized Boltzmann equation with relaxation time
approach
L̃ij
=
Lij
=
fi
fl
ǫi τ j
ne
me 1 + ωe2 τ 2
ne ˙ i j ¸
ǫ τ
me
τ
=
h. . . i
=
1
ve ni Qei
Z
2 me
3
2 df0
d
p
.
.
.
v
−
3 ne ~2
dǫ
Reinholz, DM06 – p. 33
Relaxation time approach
kinetic approach: solving linearized Boltzmann equation with relaxation time
approach
L̃ij
=
Lij
=
σ⊥
rH
fi
fl
ǫi τ j
ne
me 1 + ωe2 τ 2
ne ˙ i j ¸
ǫ τ
me
=
=
e2 n e
me
fi
τ
1 + ωe2 τ 2
−e ne RH = D
0
τ
=
h. . . i
=
fi
fl B
B
B1 + D
@
D
τ
2τ 2
1+ωe
2τ 2
ωe
2τ 2
1+ωe
τ
2τ 2
1+ωe
τ2
2τ 2
1+ωe
E2 D
+
E2
1
ve ni Qei
Z
2 me
3
2 df0
d
p
.
.
.
v
−
3 ne ~2
dǫ
fl2 1
C
C
E2 C ⇒
A
ω2 τ 2
2τ 2
1+ωe
E2 ⇒
e2 n e
hτ i
σk =
me
rH =
˙ 2¸
τ
hτ i2
Lee, More Phys. Fluids 27 (1984)
Reinholz, DM06 – p. 33
Low density limit
σ ∗ lnΛ
{Pn }
−eS/kB
rH
ei
ei + ee
ei
ei + ee
ei
ei + ee
0
0.2992
0.2992
0
0
1
1
0, 1
0.9724
0.5781
1.1538
0.8040
1.5325
1.2586
0, 1, 2
1.0145
0.5834
1.5207
0.7110
1.9786
1.2068
0, 1, 2, 3
1.0157
0.5875
1.5017
0.7139
1.9343
1.2077
0, 1, 2, 3, 4
1.0158
0.5892
1.5004
0.7039
1.9333
1.2036
0, 1, . . . , 10
1.0159
...
1.5000
...
1.9328
...
Relax app
1.0159
−
1.5000
−
1.9328
−
Spitzer2
−
0.591
−
−
−
−
Arbitrary B
0.2992
0.2992
0
0
1
1
[1] Reinholz, Redmer and Tamme, Contrib. Plasma Phys. 29 (1989)
[2] Spitzer and Härm, Phys. Rev. 89 (1953)
Reinholz, DM06 – p. 34