H. Reinholz et al- Optical and Transport Properties in Dense Plasmas Collision Frequency from Bulk to Cluster

8/3/2019 H. Reinholz et al- Optical and Transport Properties in Dense Plasmas Collision Frequency from Bulk to Cluster

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International Journal of Modern Physics BVol. 22, Nos. 25 & 26 (2008) 46274641c World Scientific Publishing Company

OPTICAL AND TRANSPORT PROPERTIES IN DENSE PLASMAS

COLLISION FREQUENCY FROM BULK TO CLUSTER

H. REINHOLZ, T. RAITZA and G. ROPKE

Universitat Rostock, Institut fur Physik, D-18051 Rostock, [email protected]

I. V. MOROZOV

Joint Institute for High Temperatures of RAS, Izhorskaya 13/2, Moscow 125412, [email protected]

Received 31 July 2008

The dielectric function of dense plasmas is treated within a many-particle linear response

theory beyond the RPA. In the long-wavelength limit, the dynamical collision frequencycan be introduced which is expressed in terms of momentum and force auto-correlationfunctions (ACF). Analytical expressions for the collision frequency are considered for bulkplasmas, and reasonable agreement with MD simulations is found. Different applicationssuch as Thomson scattering, reflectivity, electric and magnetic transport properties arediscussed. In particular, experimental results for the static conductivity of inert gasplasmas are now well described.

The transition from bulk properties to finite cluster properties is of particular inter-est. Within semiclassical MD simulations, single-time characteristics as well as two-timecorrelation functions are evaluated and analyzed. In particular, the Laplace transformof current and force ACFs show typical structures which are interpreted as collectivemodes of the microplasma. The damping rates of these modes are size dependent. They

increase for the transition from small clusters to bulk plasmas.

Keywords: Laser excited clusters; dielectric function; molecular dynamics simulations;absorption; dynamical conductivity.

1. Introduction

The plasma state is the most abundant state of matter in the universe, and the

properties of plasma are subject of many investigations. The approach to the plasma

state considered in the present paper is based on a quantum statistical treatment

of a charged particle system, where interactions are governed by the Coulomb law.In contrast to high-temperature, low-density ideal plasmas where correlations can

be neglected, we are interested in moderately to strongly coupled plasmas where

the interaction energy e2/(40d) of particles with charge e, at average distance d

given by the electron density ne, is comparable or larger than the thermal energy

4627


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4628 H. Reinholz et al.

kBT = 1. The plasma parameter

= e2

40

4ne

3

1/3

(1)

characterizes the nonideality of the plasma and is in the order of 1 or larger for

non-ideal plasmas.

Strongly coupled plasmas occur in astrophysical objects such as compact stars

and planets. In laboratory experiments with schock compressed or laser excited con-

densed matter, plasmas within the warm dense matter regime are produced. They

are characterized by a density of 1020 1026 particles/cm3 and temperatures in

the region of 103 108 K. Such plasmas are of relevance for inertial fusion exper-

iments which is one of the emerging fields in recent research. Another application

is the production of nanoplasmas by irradiation of matter with high intense, short

pulse laser in the visible or VUV/X-ray region. Currently, the FLASH facility at

DESY, Hamburg, is available with wavelengths ranging from 750 nm in the VUV

region.1 A XFEL at Hamburg is proposed2 as well as a free electron laser at the

Stanford Linear Acceleration Center (SLAC).3 In particular, we will consider laser

excited small clusters. A fascinating issue is the transition from condensed matter

as described in solid state physics to the so-called warm dense matter.

For the sake of simplicity, we consider here only singly charged ions and restrict

ourselves to the nonrelativistic case. We are interested in the dielectric properties

ruling the interaction of the strongly coupled Coulomb system with the radiation

field.4 Optical properties are of interest for plasma diagnostics but also for excitation

and absorption processes. Thus the approach given here allows to consider the

static and dynamic conductivity, scattering processes like Thomson scattering,57

reflectivity,8,9 bremsstrahlung radiation10,11 and optical transitions.12,13 Further

effects which arise in magnetic fields such as Hall effect or spectral line shapes in

strong magnetic fields can also be considered,14 but will not be presented here.

A more delicate question is the relation between classical and quantum descrip-

tion. Quantum effects have two origins. On the one hand, we have the statistical

treatment where the thermal wavelength occurs, describing the degeneracy of the

constituents of the plasma, in particular electrons. Here we can introduce the de-

generacy parameter

=2me2

32ne

2/3

, (2)

describing the ratio of thermal energy to the Fermi energy. On the other hand,

the quantum effect is related to the interaction where we have bound states and

scattering phase shifts to be calculated from a quantum approach. A correspondingparameter is the ratio of the thermal energy to the binding energy.

We can include all quantum effects in a systematic way using a quantum statisti-

cal approach as shown in Sec. 2. Within linear response theory, transport coefficients

are expressed in terms of equilibrium correlation functions which can be evaluated


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Collision Frequency from Bulk to Cluster 4629

using the Green function formalism. However, this is a perturbative approach with

respect to the interaction, and can be applied only for 1. Alternatively, MD

simulations are applicable for the evaluation of equilibrium correlation functions at

any coupling strengths, but are based on a classical treatment. To understand the

optical properties such as absorption, excitation, reflection, scattering and emission

of light we investigate different ACFs such as dipole, current and force, see Sec. 3.

Of particular interest are microplasmas produced in small clusters if they are

irradiated by intense short-pulse laser beams. Recently, MD simulations codes have

been developed to describe the properties and the dynamical evolution of such ex-

cited clusters.15,16 Vlasov and VUU simulations17 were also applied to study the

interaction of short laser pulses with clusters. Calculations for Na icosahedrons are

shown in Sec. 4. Compared with bulk plasmas, the ACFs show a more complex

structure which is related to different, single-particle properties, Sec. 5, as well as

collective, excitation modes, Sec. 6. The position and the damping rates of the ex-

citations are discussed in Sec. 7. Varying the number of atoms bound in a cluster,

we will investigate size effects for the cluster properties and compare with bulk

properties.

2. Dielectric and Optical Response

The response of homogeneous, two-component plasma to external fields is given bythe dielectric function

(k, ) = 1 +i

0(k, ) = 1

2pl( i(k, ))

, (3)

where the plasma frequency is given by pl =

nee2/0me. Assuming local thermal

equilibrium, and according to the fluctuation dissipation theorem,4,8,18 the calcu-

lation of dynamical conductivity (k, ) and dynamical collision frequency (k, )

can be performed within linear response theory evaluating the current-current cor-

relation function19 or, more generally, force-force and dipole-dipole ACFs.

In particular, we will consider the long-wavelength limit k 0 where only thefrequency dependence remains. The frequency-dependent conductivity is given by

equilibrium ACFs of the current j or the velocity R of the charge carriers (because

of the large ion to electron mass fraction, we can restrict ourselves to electrons only

when considering electronic properties)

() = j;j+i =e2

R ; R+i ,

where the Laplace transform of the correlation functions are defined by equilibrium

averages according to

A; Bz =

0

dt eizt0

dA(t i)B . (4)

The equilibrium correlation functions can be evaluated using quantum statisti-

cal methods such as thermodynamic Green functions. Within perturbation theory,


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a diagram representation can be given. Performing appropriate partial summations,

different effects such as dynamical screening, strong collisions or higher correlations

are included. This approach treats quantum effects in a rigorous way. Analytical ex-

pressions in the weak-coupling limit can be found, but are questionable for coupling

parameters 1. On the other hand, MD simulations can be performed based on

a classical description, calculating the trajectory of the system in the phase space

of N particles and replacing the ensemble average by the time average, details are

given below in Sec. 4. Using an appropriate pseudopotential, quantum effects can

be included, and good agreement of results between analytical quantum statistical

calculations and MD simulations has been found.20

Numerically as well as analytically, it is of advantage to use the force ACFs

instead of the current ACFs since they are directly related to the dynamical collision

frequency22 for the different relavant scattering mechanisms between electrons and

electrons (ee), ions (ei) and atoms (ea), respectively,

() =

ne

P; P

+i

, P = F = Fei + Fee + Fea . (5)

This formalism has been applied to different phenomena such as the conductivity

in fully ionized plasmas, or to the evaluation of optical properties such as refraction

index n() and absorption coefficient () for bulk according to

limk0

(k, ) =

n() +

ic

2()

2(6)

which allows to calculate bremsstrahlung10 and spectral line profiles.12

The dynamical structure factor has been evaluated according to

S(k, ) =0k2

nee21

e 1Im 1(k, ) , (7)

and comparison with MD simulations has been performed.2022 An important issue

is the account of correlations in determining Thomson scattering cross section

d2

d d=

8 r2e3

k1k0

S(k, ) (8)

where the transfer wave number k = k0 k1 is given by the difference between

incident and scattered wave number, = 0 1, and re is the classical electron

radius. Decomposing the dynamical structure factor into different contributions,

the free electron contribution can be analyzed to infer the temperature and densityof the plasma.5,7 An adequate treatment of interaction and collisions is needed to

provide relations valid also in the non-ideal plasma region. In particular, at very high

densities, Thomson scattering is one of the favoured methods for plasma diagnostics

since X-rays can penetrate the plasma at condensed matter densities.6


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3. Electrical Conductivity of Partially Ionized Argon Plasmas

As an example for recent progress in evaluating transport coefficients in non-idealplasmas, we consider the electrical conductivity of partially ionized inert gas plas-

mas.23,24 We report in particularly on argon. The conductivity of a fully ionized,

non-ideal plasma has been investigated extensively, and reasonable agreement be-

tween analytical calculations, MD simulations and experiments with shock-induced

dense plasmas has been obtained.4 However, experiments are generally performed

involving partially ionized plasmas. An important issue is the inclusion of the in-

teraction with bound states when calculating the electrical conductivity.

The dc electrical conductivity is related to the static collision frequency accord-

ing to

dc = lim0

() =0 2pl

dc=

e2nemedc

. (9)

As shown in Eq. (5), the collision frequency can be decomposed into different parts

determined by the different interactions present. In particular, the interaction be-

tween electrons and atoms will give a contribution to the collision frequency and

lead to a considerable reduction of the electrical conductivity. Using empirical cross

sections for the electron-atom scattering, it is possible to incorporate special effects

such as the Ramsauer minimum which is characteristic for the electron scattering by

inert gases.24 Employing these empirical cross sections instead of the polarization

potential approximation,24 it was possible to develop a consistent description of the

collision frequency and find excellent agreement with the corresponding conductiv-

ity. Results for argon are shown in Fig. 1.

Similar excellent agreement with experimental results has been obtained also

for other inert gases.24 Whereas the inclusion of bound states in evaluating the

Fig. 1. Left: Experimental results of the ea momentum transfer cross section as a function ofenergy. Right: Electrical conductivity of argon as a function of temperature at different densities.Experimental results (Shilkin) are compared with calculations within linear response theory (LRT)with the ea interaction from polarization potential (PP) or experimental data (Ex), see left sideof figure. For details and references see Adams et al.24


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dynamical collision frequency is an open problem at present, this special example of

static problems gives an idea how the transition from the fully ionized to a partially

ionized non-ideal plasma can be performed.

4. Laser Excited Clusters

Recently, clusters at nearly solid densities are more accessible to experimental in-

vestigations25 using laser intensities of 1013 1016 Wcm2. We are interested in

the cluster size dependence of the collisional damping rates in comparison to bulk

systems. Laser excited Na55, Na147 and Na309 nano-clusters irradiated by a short

pulse laser, are simulated using a MD code.16 Initially, electrons are positioned on

top of singly charged ions, see Fig. 2, for which isocahedral geometry is assumed.MD simulations were performed, using an error function pseudo-potential

Vei(r) = Verf(r) = Zie2

40rerf r

(10)

where = 6.02 aB in order to reproduce the correct ionisation energy IP = Verf(r

0) = 5.1 eV for sodium.

-20 -10 0 10 20

r in a0

-5

-4

-3

-2

-1

0

Vei(r)in

eV

Fig. 2. Error function potential (full line) used for MD calculations of clusters consisting ofsodium atoms in comparison to Coulomb potential (dashed line).

Fig. 3. Evolution of a Na55 cluster irradiated by a 100 fs laser pulse of I = 0.5 1012 Wcm2

intensity. Electrons become delocalized and form a nanoplasma. The ion geometry (straight lines)

expands.

Absorbing energy from the electromagnetic field, electrons become delocalized

within the cluster, forming a nano-plasma. We consider a cosine square laser pulse

with intensities of I = 0.5; 1; 5 1012 Wcm2. From the ion geometry, we define a


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cluster radius rc =

5/3

ir2i which marks the spatial extension of the cluster.

The scale of electron motion is in the fs region. On the time scale of 100 fs, some

electrons escape this cluster extension rc thus charging the cluster. Ion motion

becomes essential on the timescale of ps, leading to the expansion of the cluster.

Figure 3 illustrates the expansion of a Na55 cluster due to a 100 fs laser pulse of

I = 0.5 1012 Wcm2 intensity.

0 500 1000800t in fs

0

100200

300

400

500

rrm

sinaB

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Na55

tfreeze

Na55

tfreeze

I = 5 1012

W cm-2

I = 1 1012

W cm-2

I = 0.5 1012

W cm-2

Fig. 4. Evolution of the Na55 cluster radius after irradiation with a 100 fs laser pulse of differentintensities I. After 800 fs, the time of freezing the ions tfreeze , full MD simulations are continuedby restricted MD simulations where the cluster radius does not expand any further.

During the expansion of the excited cluster, we assume local thermal equilibrium

because of the small electron mass compared with the Na ions. In the following, weapplied a restricted MD simulations scheme. For this, ion positions were frozen after

an expansion time tfreeze, in order to analyze nano-plasmas at different thermody-

namical parameters. Restricted MD simulations at those given ion positions allow

to consider only electron dynamics. The evolution of cluster radius rc is shown in

Fig. 4 for three parameter sets. Different intensities at otherwise same conditions of

pulse duration and starting geometry, leads to different densities and temperatures.

5. Single-Time Properties

After the freezing time, the electrons are allowed to equilibrate until a constant

temperature Te and cluster charge Z is reached. Here, we consider these single-time

properties of the nanoplasma in quasi-equilibrium.

Equilibration of electron temperature, calculated as the mean kinetic energy of

the electrons remaining within the cluster, is shown in Fig. 5 over a time period

of 100 ns for a Na+1155 cluster. The freezing time was tfreeze = 100 fs and a radius

of rc = 16 aB was obtained. The insert of Fig. 5 illustrates that the momentum

distribution

fe(p) = 4p2

K

3/2exp

Kp2

, (11)

with K = 1/(2mekBTe) taken from all times after 20 ns, has a Boltzmann like be-

haviour and the temperature can be deducted accordingly. An electron temperature

ofTe = 1.23 eV was calculated as a time average of N = 2 108 steps of t = 0.1 fs.


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0 20 40 60 80 100

t in ns

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Te

ineV

0 0.1 0.2 0.3 0.4p in eV fs / a

B

0

1

2

3

4

fe(p)

Fig. 5. Electron temperature at frozen ion geometry. After 20 ns, stable thermodynamical equi-llibrium is reached. From this moment, the time average fe(p) of the momentum distributionfunction, seen in the insert, was calculated.

0 0.5 1 1.5 2

r in nm

0

5

10

15

20

25

30

ne

in10

21c

m-3

N = 55

N = 147

N = 309

Fig. 6. Radial dependence of the electron density for different cluster sizes after irradiation withlaser pulse of intensity I = 0.5 1012 W cm2, and equilibration after freezing time tfreeze = 100 fs.

The density profile of the electrons remaining within the cluster can be de-

ducted from the MD simulations. Figure 6 shows the result for a varying number

of ions. A plateau is observed in the core which extends with increasing cluster

size. Alternatively, the electron density can be calculated via a mean field potential

(ne exp(U(r)/kBTe), where U(r) is extracted from MD simulations), see Fig. 7

for a Na55 cluster.

6. Two-Time Properties

Once local equilibrium is reached, we have also determined two-time properties. We

calculated the momentum auto-correlation function (ACF) for the total momentumPe(t) =

Nei=1 pi(t) of electrons in the equilibrated cluster within the restricted MD


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0 0.5 1 1.5 2 2.5 3-20

-15

-10

-5

U(r)ineV

mean field potential

0 0.5 1 1.5 2 2.5 3

r in nm

0

10

20

30

40

ne

in10

21

cm

-3

simulated ne

mean field ne

Fig. 7. Left axis: The electron density profile for a Na+1155

cluster from same conditions as in Fig.5 deducted directly from the simulation data (solid line) and calculated using an external mean

field potential (dashed line). Right axis: Corresponding mean field potential.

simulations

K(t) = Pe(t), Pe(0) 1

N

Ni=1

Pe(t + i) Pe( i) (12)

with N = 1 106

the number of averages. The normalized Laplace transformationof the momentum ACF Eq. (12) is

K() =1

P2e

0

dteitPe(t), Pe(0) . (13)

The Laplace transforms of the momentum ACFs, Eq. (13), for three different

cluster sizes with the same parameters as used in Fig. 6, are shown in Fig. 8. A dou-

ble resonance structure is observed. A shift of the resonances is seen which depends

not only on the cluster size but also on other cluster parameters as temperature,

ionization degree and electron density.

0 2 4 6 8

in fs-1

0.1

1

10

ReK() N = 55

N = 147

N = 309

Fig. 8. The momentum ACF spectrum calculated from Laplace transform of the time dependentmomentum ACF for different cluster sizes which correspond to the density profiles of Fig. 6.


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In Fig. 9, the time evolution of the momentum ACF spectrum for the Na55cluster is shown for different freezing times, thus following the expansion of the

cluster and investigating different instants of the evolution. Considering a local

equilibrium at each time step, the momentum ACF was calculated with restricted

MD simulations. The freezing times considered were the time directly after the end

of the laser interaction and four times later, taking a time step of t = 50 fs, till 200

fs afterwards. The electron density decreases, because the ion geometry expansion

evolves with each time step, which also dilutes the electron gas. Collective modes

can be seen for all electron densities, with changing positions and widths.

0 2 4 6 8 10

in fs-1

0.1

1

10

ReK()

ne

= 17.1981021

cm-3

ne

= 11.1811021

cm-3

ne

= 6.6821021

cm-3

ne

= 4.0181021

cm-3

ne

= 2.4041021

cm-3

Fig. 9. The momentum ACF spectrum calculated from Laplace transform of the time dependentmomentum ACF taking different freezing times. The electron density decreases with respect tothe expansion of the ion geometry.

In a two-component bulk plasma, the dielectric function is given by the frequency

dependent ACF K() via

1

()= 1 i

pl

K() (14)

with the plasma frequency 2pl = e2ne/(0me). Considering a generalized Drude

approximation for the dielectric function,4,8 Eq. (3), the momentum ACF can be

directly related to the collision frequency ()

K() =

() i

2 2pl

(15)

which has a Lorentzian form. Thus, information about the collisions in the systemscan be gathered. We would like to apply this approach to finite systems using

correlation functions to determine optical properties.

As can be seen from the structure of the ACFs in Fig. 8 and Fig. 9, a single

Lorentzian fit as in bulk according to Eq. (15) is not possible in the case of clusters.


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7. Collective Excitations

One can calculate the collective modes of a spherically symmetric hydrodynamicelectron system. Assuming harmonic oscillations a(r, t) = a(r)eitez in z-direction

with a radially dependent amplitude a(r), the eigenfrequencies can be calculated.

The following equation of motion has to be solved:

mea(r, t) = me2a(r)eitez = FU,z(r, t)ez (16)

where the change of the force FU(r, t) is due to the mean field produced by the

change of the surrounding electron cloud ne(r) which can be expressed as

ne(r) =

z [a(r)neq

e (r)] . (17)

using the hydrodynamic equation of continuity. An additional term due to the

changing pressure has been neglected so far. Then we find the following differential

equation

me2a(r) =

2

z2

d3r1 a(r1)n

eqe (r1)Vee(r r1). (18)

In order to calculate the eigenfrequencies, this equation was symmetrized, using

(r) = a(r)neqe (r),

me2(r) =

2

z2

d3r1(r1)

neqe (r1)Vee(r r1)

neqe (r). (19)

Introducing a normalized set of basic functions

i(r)j(r)d3r = ij, the eigen-

values of the resonance frequency can be calculated via diagonalization, similar to

the Ritz variational approach in quantum mechanics. After transformation from

variable z to the radial distance r, we find

me2 =

4

3

0

dr r2(r)neqe (r)

2

r2

d3r1(r1)n

eqe (r1)Vee(r r1)

8

3

0

dr r(r)

neqe (r)

r

d3r1(r1)

neqe (r1)Vee(r r1). (20)

In the following, we use Hermitean basis functions with a width parameter b as the

normalized basis set,

0,b(r) =exp

r

2

b2

2

b3/2 , (21)

1,b(r) =exp r2

b2

2

b3/2 83

r2

b2 3

4

, (22)

2,b(r) =exp

r

2

b2

2

b3/2

32

15

r4

b4

5

2

r2

b2+

15

16

. (23)


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For the case shown in Fig. 5, calculations of the resonance frequencies were done.

The density profile from the restricted MD simulations was taken as the equilib-

rium density neqe (r). Analyzing the Laplace transfom of the electron momentum

ACF, three collective modes were found at 0 = 6.16 fs1, 1 = 3.80 fs

1 and

2 = 1.98 fs1. Using now the hydrodynamical approach, the eigenvalues for the

resonance frequencies in dependence on the width parameter b have been calculated

and illustrated in Fig. 11 . For b = 6 aB, the resonance frequencies are maximal.

0 5 10 15 20

b in aB

0

1

2

3

4

5

6

7

Ri

nfs-1

collectiv modes

0

1

2

Fig. 11. The calculated eigenvalues of the resonance frequencies are shown in dependence on thewidth parameter b. The thick horizontal solid lines display the resonance frequencies as obtainedfrom the Laplace transform of the ACF. The first (solid line), second (dashed line) and third(dotted line) eigenvalue were calculated taking the mean field interaction into account.

They were calculated as 0 = 4.59 fs1, 1 = 2.23 fs

1 and 2 = 0.82 fs1. The

ratio between the resonance frequencies is of the same order as for the results taken

directly from the simulations. However, the position is too low. Further investiga-

tions are necessary to improve this result, e.g. using a more involved hydrodynamical

approach taking into account corrections due to pressure gradients. The spatially

dependent amplitude a(r) was calculated via (r) = a(r)ne(r). The results areshown in Fig. 12.

8. Discussion and Conclusion

We have shown that the evaluation of equilibrium ACF for the electron momentum

or the force allows to calculate the dynamical collision frequency which determines

the dielectric function. Having the dielectric function to our disposal, different op-

tical and transport properties of dense plasmas can be determined. Applications

are dc conductivity, absorption, emission, reflection, bremsstrahlung, and Thomson

scattering.In particular, the methods developed to investigate homogenous non-ideal plas-

mas can also be applied to finite nano-plasmas, produced by laser irradiation of

clusters. Using restricted MD simulations, we determined single-time properties

(density and momentum distribution) as well as two-time properties, in particular


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10 20 30r in a

B

0

0.5

1

a

(r)

first mode (from 0

)

second mode (from 1)

third mode (from 2)

Fig. 12. Spatially dependent amplitudes a(r) for the width parameter b = 6 aB, derived from themaximal resonance frequencies using the mean field contribution.

the momentum ACF spectrum. The single-time properties are well reproduced from

equilibrium statistics. The momentum ACF shows a peak structure which is ex-

plained via collective excitations of the nano-plasma. Within exploratory calcula-

tions, collective modes are obtained, solving the self consistent mean field equation.

The exact position of the peaks, however, needs further investigations.

Of interest are the damping rates which are related to the collision frequency.

We found that damping rates obtained by Lorentzian fits to the excitation spectrumcan be identified, which show a dependence on the size of the cluster. The collision

frequency is increasing with increasing cluster size due to more frequent collisions

with other particles in the average. A smooth transition to the bulk behavior is

expected, see Ref. 26. More systematic investigations are needed to derive the size

effects in the collision frequency, which will be the subject of future work.

Acknowledgments

We would like to thank E. Suraud for many fruitful discussions. The authors grate-fully acknowledge financial support by the Deutsche Forschungsgesellschaft within

the Sonderforschungsbereich SFB 652. I.V. Morozov acknowledges CRDF and Dy-

nasty foundation.

References

1. R. Bonifacio, C. Pellegrini and L. Narducci, Opt. Comm. 50, 373 (1984).2. R. Brinkmann, K. F. B. Faatz, J. Rossbach, J. R. Schneider, H. Schulte-Schrepping, D.

Trines, T. Tschentscher and H. Weise (eds.), TESLA XFEL Technical Design Report

(Supplement) (DESY, Hamburg, Germany, DESY Rep. 2002-167, 2002).3. LCLS Design Study Report (The LCLS Design Study Group, SLAC, Stanford, CA,

1998), SLAC-R-0521.4. H. Reinholz, Les Annales de Physique 30, 45 (2006).5. R. Thiele, R. Redmer, H. Reinholz and G. Ropke, J. Phys. A 39, 4365 (2006).6. A. Holl, Th. Bornath, L. Cao, T. Doppner et al., HEDP 3, 120 (2007).


15/15


7. S. H. Glenzer, O. L. Landen, P. Neumayer, R. W. Lee, K. Widmann, S. W. Pollaine,R. J. Wallace, G. Gregori, A. Holl, T. Bornath, R. Thiele, V. Schwarz, W.-D. Kraeft

and R. Redmer, Phys. Rev. Lett. 98 (2007).8. H. Reinholz, Yu. Zaparoghets, V. Mintsev, V. Fortov, I. Morozov and G. Ropke, Phys.

Rev. E 68, 036403 (2003).9. T. Raitza, H. Reinholz, G. Ropke, V. Mintsev and A. Wierling, J. Phys. A 39, 4393

(2006).10. A. Wierling, T. Millat, G. Ropke, R. Redmer and H. Reinholz, Phys. Plasma 8, 3810

(2001).11. C. Fortmann, R. Redmer, H. Reinholz, G. Ropke, A. Wierling and W. Rozmus, HEDP

2, 57 (2006).12. B. Omar, S. Gunter, A. Wierling and G. Ropke, Phys. Rev. E 73, 056405 (2006).13. A. Sengebusch, S. H. Glenzer, H. Reinholz and G. Ropke, Contrib. Plasma Phys. 47,

309 (2007).14. J. R. Adams, H. Reinholz, R. Redmer and M. French, J. Phys. A 39, 4329 (2006).15. M. Belkacem, F. Megi, P.-G. Reinhard, E. Suraud and G. Zwicknagel, Eur. Phys. J.

D 40, 247 (2006).16. H. Reinholz, T. Raitza and G. Ropke, Int. J. Mod. Phys. B 21, 2460 (2007).17. J. Kohn, R. Redmer, K.-H. Meiwes-Broer and T. Fennel, Phys. Rev. A 77, 033202

(2008).18. H. Reinholz, R. Redmer, G. Ropke and A. Wierling, Phys. Rev. E 62, 5648 (2000).19. R. Kubo, J. Phys. Soc. Jpn. 120, 570 (1957).20. I. Morozov, H. Reinholz, G. Ropke, A. Wierling and G. Zwicknagel, Phys. Rev. E 71,

066408 (2005).21. A. Selchow, G. Ropke, A. Wierling, H. Reinholz, T. Pschiwul and G. Zwicknagel,Phys. Rev. E 64, 056410 (2001).

22. H. Reinholz, I. Morozov, G. Ropke and Th. Millat, Phys. Rev. E 69, 066412 (2004).23. J. R. Adams, N. S. Shilkin, V. E. Fortov, V. K. Gryaznov, V. B. Mintsev, R. Redmer,

H. Reinholz and G. Ropke, Phys. of Plasmas 14, 062303 (2007).24. J. Adams, H. Reinholz, R. Redmer, V. B. Mintsev, N. S. Shilkin and V. K. Gryaznov,

Phys. Rev. E 76, 036405 (2007).25. T. Doppner, Th. Diederich, A. Przystawik, N. X. Truong, Th. Fennel, J.

Tiggesbaumker and K. -H. Meiwes-Broer, Phys. Chem. 9, 4639 (2007).26. L. Ramunno, C. Jungreuthmayer, H. Reinholz and T. Brabec, J. Phys. B 39, 4923

(2006).

H. Reinholz et al- Optical and Transport Properties in Dense Plasmas Collision Frequency from Bulk to Cluster

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