carsten fortmann- bremsstrahlung in dense plasmas: a many-body theoretical approach

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Bremsstrahlung in Dense Plasmas:

A Many-Body Theoretical Approach

Dissertation

zurErlangung des akademischen Grades

doctor rerum naturalium (Dr. rer. nat.)der Mathematisch-Naturwissenschaftlichen Fakultat

der Universitat Rostock

vorgelegt vonCarsten Fortmann, geb. am 24.7.1978 in Eutin

aus Rostock

Rostock, 29. August 2008



Liste der Gutachter

• Prof. Dr. Gerd RopkeInstitut fur PhysikUniversitat Rostock

• Prof. Dr. Wolf-Dietrich KraeftInstitut fur PhysikUniversitat Greifswald

• Prof. Dr. Wojciech RozmusTheoretical Physics InstituteUniversity of AlbertaEdmonton, Kanada

Datum der Verteidigung: 28. Oktober 2008



Wenn aber gleich alle un-

sere Erkenntnis mit der Er-

fahrung anhebt, so entspringt

sie darum doch nicht eben

alle aus der Erfahrung.

Immanuel Kant

Kritik der reinen Vernunft



Kurzfassung

Bremsstrahlung ist ein fundamentaler Prozess der Licht-Materie Wechselwirkung. Sie be-

zeichnet die Strahlungsemission wahrend der Streuung von geladenen Teilchen. Bremsstrah-lungsspektroskopie an Plasmen ermoglicht die genaue Bestimmung wichtiger Plasmapa-rameter. In dichten Plasmen kommt es infolge von Vielteilcheneffekten zu einer Modi-fizierung des Emissionsspektrums. Beispiele hierfur sind die Abschirmung des Streupoten-tials und wiederholte Teilchenstoße wahrend des Emissionsvorgangs. Diese fuhren zu einerAbsenkung des Bremsstrahlungsspektrums bei kleinen Photonenenergien. Die in dieserArbeit vorgestellte Vielteilchentheoretische Beschreibung von Bremsstrahlung erlaubt diesystematische Darstellung des Einflusses von statischen und dynamischen Korrelationen auf das Emissions und Absorptionsspektrum. In diesem Formalismus sind die Korrelationenin der Einteilchen-Selbstenergie enthalten, die die endliche Lebensdauer von Einteilchen-

zustanden im Medium beschreibt. Die Berechnung der Selbstenergie geschieht mithilfe derGW (0)-Methode, wodurch uber die Mean-Field Naherung hinausgegangen wird. Vorherunbekannte Resultate fur die GW (0)-Selbstenergie in einem nichtentarteten Plasma wer-den prasentiert. Ergebnisse zur Modifikation der Bremsstrahlung im Medium werden inForm des Gauntfaktors angegeben.



Contents

Abstract/Kurzfassung i

Table of content iv

I Correlated plasmas in radiation fields v

1 Introduction 11.1 Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Characterization of plasmas and plasma parameters . . . . . . . . . . . . . 11.3 Interaction of Radiation with Matter . . . . . . . . . . . . . . . . . . . . . 41.4 Many-particle Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 51.5 Bremsstrahlung in dense media . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Aim and structure of this work . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Many-particle theory and application to plasmas 132.1 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Many-particle field theory and the GW -approximation . . . . . . . . . . . 332.3 Matsubara Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Application: The self-energy in GW (0) approximation . . . . . . . . . . . . 412.5 Optical properties of dense plasmas . . . . . . . . . . . . . . . . . . . . . . 42

3 Plasmas and radiation 453.1 Basic processes and plasma diagnostics . . . . . . . . . . . . . . . . . . . . 45

3.2 Line emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Recombination radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Bremsstrahlung radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.6 Radiation transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Selected problems 534.1 Bremsstrahlung vs. Thomson Scattering . . . . . . . . . . . . . . . . . . . 534.2 Bremsstrahlung and Line Spectroscopy of Warm Dense Al Plasma . . . . . 574.3 Landau-Pomeranchuk-Migdal Effect in Dense Plasmas . . . . . . . . . . . . 59



iv CONTENTS

4.4 Optical Properties and Spectral Function . . . . . . . . . . . . . . . . . . . 634.5 Self-Consistent Spectral Function and Analytic Scaling Behaviour . . . . . 65

Summary 76

Outlook 78

Bibliography 96

II Published articles 97

5 Bremsstrahlung and the LPM effect 99

6 Bremsstrahlung vs. Thomson Scattering 117

7 Optical Properties and Spectral Function 133

8 Spectroscopy of Warm Dense Al Plasma 147

9 Self-Consistent Spectral Function and Scaling Behaviour 155

10 Single Particle Spectral Function for the Classical Plasma 185

Curriculum vitae 200

List of publications/Liste der Veroffentlichungen 201

Declaration of authorship/Selbststandigkeitserklarung 205



Part I

Correlated plasmas in radiation fields



Chapter 1

Introduction

1.1 Plasmas

As to our latest knowledge, only 5% of the matter present in our universe is visible matter,i.e. sensitive to electromagnetic radiation [Kra01]. The remaining portion consists of darkmatter (23%) and dark energy (72%). Out of the 5% visible matter, more than 99%[GB05] are in the so-called plasma state. The term “plasma” was originally introducedby Langmuir [Lan28] to describe an ionized gas in an arc discharge. Today, plasmas areunderstood as matter in a state characterized by a significant (e.g. > 10%) degree of ionization. Typical every-day examples are tube discharges, flames or plasma lamps. Mostplasmas occurring in nature are found in astrophysical objects, such as the interior of

stars [Ich93], giant planets [Bag92] or in the interstellar medium [Alf60]. In this work, theterm plasma is used mainly in the context of Coulomb systems, i.e. matter consisting of electrons and atomic nuclei, that interact via the Coulomb force. In this sense, also systemslike excited semi-conductors and metals, where electrons in the conduction band behavesimilar to free electrons in a plasma, make part of the considerations. More fundamentalparticles (in the sense of elementary particle physics), e.g. hadrons and quarks, will onlybe mentioned in marginal remarks. It should be noted at this point, that also the state of hadrons dissolved into quarks and gluons is referred to as a plasma, the so-called quark-gluon plasma [HM96].

1.2 Characterization of plasmas and plasma parame-

ters

In general, plasmas are in a non-equilibrium state that needs a detailed characterization.The concept of the time dependent distribution function is used to describe the plasmakinetics [Ich92]. However, quite often, conditions are met where the plasma state can beassumed to be in a global or local thermodynamic equilibrium, characterized by a fewparameters such as temperature and density, possibly depending on the position. Near



2 Introduction

equilibrium states can be described by the degree of ionization as a function of time andposition and assuming different temperatures for the various subspecies, such as atoms,unbound electrons, and ions [Chi00]. Fluctuations near equilibrium are considered in linearresponse theory, see e.g. [ZMR96a].

A very useful illustration of plasmas and Coulomb systems in general is their repre-sentation in the parameter plane spanned by the density of unbound electrons ne andthe temperature T , as shown in Fig. 1.1. On the left plane we see the typical parameter

Figure 1.1: Density-temperature plane with important examples for plasmas and isolinesfor the plasma coupling parameter Γ and the electron degeneracy parameter θ. Plasmasoccurring in nature (astrophysical plasmas) are given on the left, the right plane showsregions where laboratory plasmas are found or are expected to be found. This graphcourtesy of Dr. A. Holl.

ranges for plasmas occurring in astrophysical objects. The right plane shows the parame-

ters of plasmas that are produced in a laboratory, i.e. fusion plasmas, such as magneticallyconfined plasmas (“Tokamak”) and plasmas generated during inertial confinement fusion(“ICF”). Furthermore, the parameter ranges of typical condensed matter objects (metalsand semiconductors, the electron density is the concentration of carriers in the conductionband) are shown as well as the parameters for ion traps. Besides the examples, Fig. 1.1also shows isolines of the electron coupling parameter Γe (dark blue lines) and the electrondegeneracy parameter θe (light blue lines). The coupling parameter

Γc =Z 2c e2

4π0dkBT (1.1)



1.2 Characterization of plasmas and plasma parameters 3

is the ratio of the Coulomb interaction among two particles of species c at the mean inter-particle separation d = (3/4πnc)

1/3 and the thermal energy kBT . Plasmas characterizedby Γc < 1 are called weakly coupled. Their behaviour is largely governed by the tem-perature of the system. The individual particles can be considered as free particles, eachmoving independently of the others. Plasma observables are easily calculated by aver-aging the corresponding single-particle quantity over the Maxwell-Boltzmann distributionfunction.

In the opposite case, Γc > 1, the term “strongly coupled plasma” is used. The mutualCoulomb interaction is larger than the mean kinetic energy of each particle. Hence, thecalculation of observables is much more complicated, since the interaction of the particleshas to be accounted for. Examples for strong coupling effects are the formation of boundstates (at low temperatures, e.g. in traps), the dissolution of bound states due to the Motteffect (at high densities), the structuring of the plasma and formation of a crystal lattice

in the limit of low temperatures and high densities (liquids, semiconductors, metals).Note, that the coupling parameter looses its meaning as the ratio of potential energy

to kinetic energy, when the temperature becomes smaller than the Fermi energy. In thiscase, the mean kinetic energy of the system scales with the Fermi energy rather thanthe temperature. As the relevant parameter characterizing dense and cold systems, oneintroduces the degeneracy parameter θc as the ratio of the thermal energy to the Fermienergy,

θc =kBT

E F,c, E F,c =

2k2F,c2mc

, kF,c =

6π2nc

2sc + 1

1/3, (1.2)

with kF,c being the Fermi momentum and mc being the mass of particles of species c, scis the species’ spin quantum number. The limit θc 1 characterizes plasmas that can betreated as ideal Fermi gases, here, the quantum nature of the particles becomes importantand determines the plasmas comportment. This happens usually at high densities, wherethe mean particle separation becomes so small, that the wavefunctions overlap and quan-tum interference sets in. Observable effects are the Pauli blocking for Fermions or quantumcondensation like Bose-Einstein condensation (BEC) or Bardeen-Cooper-Schrieffer-pairing(BSC).

In degenerate systems, the Fermi-Dirac distribution function

nF,c( p) = [1 + exp(2 p2/2mc − µc]/kBT )

−1, (1.3)

approaches a step function with a discontinuity at the Fermi energy E F,c = µc(T = 0); µcis the chemical potential of species c.

In the limit θ 1 we speak of a classical plasma, the particles can be treated as classicalobjects, i.e. their equilibrium momentum distribution function can be approximated by aMaxwell-Boltzmann distribution,

nM,c( p) =ncΛ

3c

2sc + 1exp(−

2 p2/2mc) , (1.4)

Λc = (2π2/mckBT )1/2 is the thermal de-Broglie wavelength for species c.



4 Introduction

Most real-world plasmas are neither ideal quantum gases nor classical, weakly coupledplasmas. The theoretical description of these plasmas is difficult. The Coulomb interactionamong the particles and their quantum nature leads to an enormous degree of complex-ity. Whereas in ideal systems, the observables can be derived from the single-particlebehaviour, spatial and temporal correlations among the constituents govern the physicsof real plasmas and lead to the emergence of new phenomena. Collective modes appear,leading to a modification of the response of the plasma to external fields. To deal withthese complexities, it is helpful to introduce the notion of quasi-particles interacting witheach other via effective forces [KKER86]. Also bound states can be included in this pictureas a distinct particle species. E.g. the many-electron problem of an electron scattering onbound electrons in an atom is reduced to the one-particle problem of an electron scatteringin the effective potential of the atom [RDO05]. Similarly, plasmon modes are described asquasi-particles giving rise to the dynamical screening of the Coulomb potential.

The weakening of the interaction due to screening can lead to the disintegration of bound states above a certain critical density, the so-called Mott-effect [KSK05, FIK06]. Inthis way, temperature and density directly influence the thermodynamic properties of theplasma, i.e. its equation of state.

Another important category of plasma properties is the response of the plasma toexternal perturbations. Examples are the plasmas DC conductivity (response to electricfields), and magnetic susceptibility (response to magnetic fields), optical properties likeopacity, reflectivity and emission (response to electromagnetic fields) and the response toparticle beams, like the stopping power.

1.3 Interaction of Radiation with Matter

The interaction of plasmas with electromagnetic radiation is a long standing issue in boththeoretical and experimental physics. Since light propagation in a medium is limited towavelengths below the critical wavelength λc = 2πc/ωpl, with the electron plasma frequency

ωpl =

nee2/0me, investigations of condensed matter and plasmas near solid density orabove require light sources in the far UV or x-ray regime. Light sources of the so-calledfourth generation, such as the free electron laser at DESY-Hamburg (FLASH) [A+06],provide unprecedented parameters in brilliance, temporal and spatial coherence and at the

same time short wavelengths [AAA

+

07]. Notably in the domain of plasma diagnostics,FELs offer intriguing possibilities, e.g. via collective Thomson scattering [HBC+07]. Fur-ther challenging projects aim at time resolved investigations, e.g. of chemical and biologicalprocesses [Tsc04, TT05, HRCK+07].

On the other hand, the development in short pulse and high power laser technologyopened up a whole catalogue of interesting new phenomena to investigate [GF96]. Intenselasers provide an excellent tool to generate high energy density plasmas, e.g. in inertialconfinement fusion research [Lin95]. Effects related to high intensities are multi-photonionization, generation of higher laser harmonics [TPS+03], attosecond pulse generation[CMK97, WCL+08], and the laser self-focussing effect [SS99]. At relativistic laser in-



1.4 Many-particle Quantum Electrodynamics 5

tensities (> 1018 W/cm2), when the quiver energy of electrons becomes comparable orlarger than the electrons rest mass [Gib06], bubble-acceleration [Puk03] sets in, openingnew perspectives for small-scale particle accelerators [GTT+04, FGP+04]. From a theo-retical point of view, ultrarelativistic laser-plasma effects expected at highest intensities(> 1021 W/cm2), e.g. the birefringence of the vacuum [HLA+06], pair creation by vacuumtunneling [BPR+06] and the Schwinger mechanism [Sch51a, DW97, SW92], i.e. the “breakdown” of the quantum vacuum, are of special interest. These effects relate to fundamentalquestions of quantum field theory. A quantum field theoretical formulation of laser-plasmainteraction at such conditions is indispensable.

The most general approach to the interaction between charged particle systems and elec-tromagnetic fields is quantum electrodynamics (QED). Its basic concepts will be outlinedbriefly in section 2.1, with special emphasis on the response of plasmas to electromagneticfields. For more detailed presentations, I refer to textbooks e.g. by Itzykson and Zuber

[IZ80], Gross [Gro93], or Bjorken and Drell [BD65]. In the end, expressions for the absorp-tion, emission, and the scattering of electromagnetic waves or photons will be presented,that are obtained from QED via a perturbation expansion of the QED Hamiltonian. Thequestion of non-perturbative treatments will be covered in section 2.2.

1.4 Many-particle Quantum Electrodynamics

A theoretical approach to the interaction of electromagnetic fields with correlated matter,that can be extended to such extreme conditions as outlined above, needs to attend three

main issues: First, the correlations in the system have to be considered as well as (second)the interaction between the light and the matter constituents. Both these questions shouldbe addressed within a common framework. Only in this way, modifications of the basicprocesses of absorption, emission, and scattering due to correlations and the creation ordestruction of correlations in the plasma through radiation fields, notably coherent radi-ation fields, can be accounted for systematically and without double counting of one orthe other. Thirdly, taking on the example of a plasma created by interaction of intenselaser light with condensed matter, it is clear that also non-equilibrium aspects need tobe incorporated, such as density and temperature gradients, instabilities and relaxationprocesses. A field-theoretical generalization of linear response theory is straightforward

[MRH02a, MRH02b].In plasma physics, many-body theoretical methods [Mah81, FW71], have a long tra-dition of application to thermodynamic properties [KKER86], transport cross sections[RRRW00], and collective excitations. Correlations due to the Coulomb interaction amongthe constituent particles and their Bose- or Fermi statistics are calculated from the systemsHamiltonian after a perturbative expansion in terms of the interaction. Often, summationschemes are applied in order to include also inherently non-perturbative phenomena, suchas the occurrence of bound states or the dielectric screening of the interaction potential.Standardly, the Hamiltonian contains the free movement of the particles, i.e. the kineticenergy, and the electrostatic interaction (Coulomb interaction).



6 Introduction

On the other hand, Quantum Electrodynamics is successfully applied to the calcula-tion of those processes, were electrically charged particles, that can be regarded as freeparticles in good approximation, interact with electromagnetic fields, i.e. photons. Here,the Hamiltonian, besides the free movement, contains the coupling between Fermions andgauge Bosons (photons), which is treated as a perturbation. Quantum electrodynamics isrelativistically covariant and both massive particles and gauge bosons are considered asquantum objects. However, the Coulomb interaction and the resulting correlations andcollective phenomena, which are accounted for in many-body theory, is either not dealtwith at all or it is regarded as a part of the unperturbed Hamiltonian. The latter allowsfor the treatment of interaction of atoms, molecules, or ions with the radiation field.

Besides established analytic techniques, based on perturbation theory, also numericalapproaches have matured over the last years. By use of Dirac equation solvers [BSG99,MK04], various effects of interaction between ultra-intense (> 1020 W/cm2) laser pulsesand atoms have been investigated, such as bremsstrahlung emission by laser acceleratedelectrons [SLJK07], high harmonic generation [FLK06] or induction of nuclear reactions[BEK06] have been accessed.

However, complex plasma dynamics, involving dynamical screening, ionization and re-combination, or occurrence of collective modes are often not accounted for in QED. Many-particle quantum electrodynamics [Rei05], i.e. the synthesis of both theoretical concepts,provides a systematic approach to the description of strongly correlated systems exposed

to external electromagnetic fields. The interplay between statical and dynamical corre-lations on the one side and the coupling of matter to the radiation field can be studiedwithin a common, consistent theoretical framework. The starting point is the Lagrangianof QED. A central task is to find consistent non-perturbative approximations. A system-atic approach to this problem is the Φ-derivable technique [Bay62] and Dyson-Schwingerequations [RW94]. Further details are provided in chapter 2.

This work is contained in the frame of an ambiguous and progressive research centersituated at the Institute of Physics of the University of Rostock, the “Collaborative Re-search Center 652: Strong Correlations and Collective Phenomena in Radiation Fields:

Coulomb systems, Clusters, and Particles”. In this center, correlated systems submittedto electromagnetic radiation are studied under various aspects, both experimentally andtheoretically. The dynamical response of metal clusters in strong, short-pulse laser fields,plasmas interacting with free electron laser radiation, quantum dynamics of ions and atomsin traps, optical properties of highly excited semiconductors, and quantum condensationin strongly correlated electron-hole systems are the main topics, that are investigated in13 projects. The subject of this thesis work, emanates from the project A4 (“Many-bodyQuantum Electrodynamics and Dielectric Response”). Besides bremsstrahlung in denseplasmas, another PhD thesis in this project deals with optical properties of metal clusters,investigated by means of molecular dynamic simulations [RRR07].



1.5 Bremsstrahlung in dense media 7

1.5 Bremsstrahlung in dense media

Bremsstrahlung denotes the emission of a photon by an accelerated charge, e.g. during the

scattering of an electron in the Coulomb potential of an ion. A relativistic and quantummechanical description of bremsstrahlung is provided by QED within second order pertur-bation theory. On the other hand, bremsstrahlung is the dominant process of emission of radiation from a hot and highly ionized plasma. The inverse process, absorption of photonsduring particle collisions, so-called inverse bremsstrahlung, is used e.g. for plasma heating[SRB06, CR85, BRS+03].

The precise prediction of the bremsstrahlung spectrum is of primary importance forplasma diagnostic purposes [Hut87]. Bremsstrahlung spectroscopy allows for the deter-mination of many important plasma parameters, such as the density of the plasma, itscomposition, and its temperature. In principle, the electron velocity distribution func-

tion can be reconstructed from the bremsstrahlung spectrum [BEH+06], and thus alsonon-equilibrium features, such as non-thermal components, can be characterized.

But also from a more theoretical point of view, bremsstrahlung is an interesting sci-entific issue: The emission (or absorption) of radiation by free or quasi-free particles (e.g.electrons) requires the presence of a third particle or collision partner (the ion) to carryaway the recoil momentum, otherwise, conservation of momentum would be violated. Thismeans that in order to describe bremsstrahlung in a many-particle environment, one hasto adopt not only the formalism of Quantum Electrodynamics to describe the couplingof emitted or absorbed radiation and the particles, but one also has to account for thecorrelation and interaction among charged particles. Thus, bremsstrahlung is particularly

well suited to study the interplay between correlations in the plasma and the coupling tothe radiation field.

An example for the modification of the bremsstrahlung emission spectrum due tocorrelations in the plasma is the so-called Landau-Pomeranchuk-Migdal (LPM) effect[LP53, Mig56, Kle99]. At small photon energies, when the wavelength of the emittedphoton is of the same order as the mean distance between nuclei, the scattering ampli-tudes of successive scatterings interfere. In a disordered system, this interference is mainlydestructive, which leads to a suppression of the bremsstrahlung emission at small photonenergies. The LPM effect was observed in high energy electron scattering experiments bymany different groups, see e.g. [Kas85, ABSB+97, HUB+03].

Besides successive scattering, also the screening of the scattering potential is knownto effect the bremsstrahlung emission. Ter-Mikaelyan [TM53] showed, that screening of the interaction potential leads to a further decrease of the bremsstrahlung emission andat very low frequencies (in the vicinity of the medium’s plasma frequency) dominates theLPM-suppression.

Being approved unambiguously in the case of high energy electron scattering, the ques-tion remains, if the same effects are also present in the bremsstrahlung emission from a hotand dense plasma. In a consistent, many-body field theoretical approach to bremsstrahlungthey should naturally be included.

The approach is straightforward: In general, optical properties of charged particle



8 Introduction

systems, e.g. absorption and emission, are described by the dielectric tensor of the medium[Jac75]. The latter is a functional of the single-particle Green function and the electron-photon vertex function [KKER86]. These functions are well defined objects of quantumstatistics, describing the propagation of particles in the medium and the coupling to theradiation field, respectively. Thus, we have a relation between the macroscopic properties,such as emission and absorption spectra, to the microscopic determinants of the system.The account for many-particle effects in the calculation of the Green function and thevertex opens the possibility to describe medium effects on optical properties within aself-contained formalism. Here, the focus lies on the calculation of the influence of many-particle effects on the bremsstrahlung emission and inverse bremsstrahlung absorptionfrom a plasma in thermal equilibrium. However, further macroscopic quantities, such astransport coefficients (electrical and thermal conductivity, stopping power) or the equationof state of the plasma can be accessed within the same approach [KKER86].

1.6 Aim and structure of this work

This work aims at the description of bremsstrahlung from dense, correlated media, in theframework of many-particle QED. This general approach enables us to account for thevarious effects of correlations on the bremsstrahlung emission in a systematic way. Tradi-tionally, corrections of the bremsstrahlung spectrum due to medium and quantum effects,are known as the Gaunt factor [Gau30, BH62]. It gives the deviation of the bremsstrahlungspectrum with respect to an expression that goes back to Kramers [Kra23], who neglected

correlations and quantum effects altogether. In this sense, the work aims at a many-bodyfield theory of the Gaunt factor.

Being a prototypical process of QED, the study of bremsstrahlung in a plasma directlyleads to a number of questions that derive from the more general problems discussed inthe previous sections. The electron-ion scattering becomes screened, which also affectsthe bremsstrahlung emission. Furthermore, the picture of the emitting electron as a freeparticle before and after the scattering breaks down. The account for the finite life time of single-particle states, i.e. momentum states, in the calculation of bremsstrahlung is a mayortopic of this work. Other important issues are e.g. bremsstrahlung in non-equilibriumsystems [May07], coherence effects, such as bremsstrahlung of laser-accelerated electrons

[SLJK07], coherent superposition of bremsstrahlung photons from electron scattering inion lattices [BK05] or the question of bremsstrahlung in magnetized plasmas [NV83].

Bremsstrahlung is mainly emitted during electron-ion collisions. Electron-electronbremsstrahlung is negligible at least at non-relativistic energies. Therefore, a first cen-tral point in this work is the analysis of the electron-ion scattering process beyond Bornapproximation, i.e. beyond the perturbative result provided by QED. Born approximationis appropriate for small transfer momenta. At large transfer momenta, it drastically over-estimates the scattering cross section[RRRW00]. Here, the concept of the t-matrix has tobe used, i.e. the summation of single photon exchange diagrams that make up the Bornseries. This leads to the Sommerfeld approximation for the Gaunt factor [WMR+01].



1.6 Aim and structure of this work 9

The Sommerfeld Gaunt factor is applied to answer a question that is related to thecomparison of bremsstrahlung to another plasma diagnostic technique, i.e. Thomson scat-tering. Thomson scattering provides a well suited tool to infer the plasma temperature anddensity directly from the scattering spectrum [LGE+01, GGL+03, GLN+07]. In the caseof high energy density plasmas, the problem comes up, if the scattered intensity overcomesthe bremsstrahlung background level. A detailed analysis of the ratio of Thomson scatter-ing to bremsstrahlung emission over a broad range of plasma densities and temperaturesis performed. Results are presented in Ref. [FRR+06]. In particular, threshold conditionsfor the Thomson probe intensity and the spectral resolution of the detector are determinedassuming the scenario of a Thomson scattering experiment at the free electron laser atDESY-Hamburg (FLASH) as proposed in Ref. [HBC+07]. As a result, bremsstrahlungdoes not present a major obstacle for such an experiment if the minimum requirementsregarding the FEL intensity (I > 1012 W/cm−2) and detector resolution (∆λ/λ < 10−2)

are met.Another application of the Sommerfeld Gaunt factor is presented in section 4.2. Here,

XUV emission spectra from aluminum plasmas are analysed. The plasma was generatedby interaction of a solid Al target with intense free electron laser radiation at the FLASHfacility. By comparison of the experimental data to the theoretical prediction, the plasmatemperature (kBT = 40 eV) and the free electron density (ne = 4 × 1022 cm−3) is inferred.These values are in excellent agreement with data from the analysis of the spectral lineswhich were measured in the same experiment and also with radiation hydrodynamics sim-ulations. These results prove the excellent applicability of bremsstrahlung spectroscopy fordiagnostics of high energy density plasmas.

In a dense plasma, the successive scattering of the emitting electron on different ions be-comes important as known from the LPM effect, which was discussed above, see section 1.5.Formally, the account for successive scattering is achieved by resummation of certain sub-classes of self-energy diagrams, describing the single scattering process. In Ref. [ FRRW05]calculations for the absorption coefficient using the full self-energy are compared to theBorn-approximation. In order to preserve gauge invariance and conservation of charge,also vertex corrections are performed on the level of the polarization function as dictatedby the Ward-Takahashi identities [War50, Tak57]. It is found, that the account for suc-cessive collisions in fact leads to a decrease of the absorption spectrum at low photonenergies, while at high photon energies, the results converge with the Born approximation.

For thermal bremsstrahlung from a dense plasma, this has not been shown before. As afurther novelty, an increase of the absorption coefficient at intermediate photon energies,is discovered. Both, decrease at low energies and increase at intermediate energies, arein the range of a few percent to a few ten percent compared to the Born approximation,depending on the plasma parameters. Details are discussed in section 4.3 and in the pub-lications [FRRW05] and [FRW07]. Also, partial compensation between self-energy andvertex corrections is observed.

The role of vertex corrections is investigated also on the level of the self-energy itself.This is done by comparing the two self-energy diagrams that appear in the second order of the interaction. Also here, partial compensation between the pure self-energy contributions



10 Introduction

and the vertex corrections is obtained. The self-energy decreases by 20% due to the vertexterm.

A second central point is the consideration of electron-electron correlations. Electron-electron correlations are of a more dynamical nature than electron-ion correlations, due tothe large mass of the atomic nucleus compared to the electron mass. Well known effectsof electron-electron correlations are for example the dynamical screening of the Coulombpotential and the appearance of collective plasma modes, so-called plasmons.

As before, also the electron-electron correlations are included via the electron self-energy. This is achieved in the framework of the so-called GW approximation, whichwas introduced for the homogeneous electron gas at zero temperature by Hedin [Hed65],but which can also be applied at finite temperature and also for multicomponent systems[FSK+06].

The GW approximation allows the non-perturbative treatment of dynamic electron-

electron correlations. The self-energy is calculated beyond the mean-field level (Hartree-Fock approximation). Its foundations and various applications will be discussed in detailin section 2.2.

The GW self-energy (i.e. the GW (0)-variant, see section 2.4) was calculated for abroad range of plasma densities and temperatures to address a number of key questions:How does the electron spectral function, as the determinant for the optical response andother properties, behave under various plasma conditions? How is the transition from aweakly coupled, dilute system to a degenerate quantum liquid, reflected in the spectralfunction? Under which conditions do collective excitations, visible as peaks in the spectralfunction, appear? Which are the dominant processes of damping of collective states and of

single-particle states? A computer code was developped, that gives the GW (0)

self-energyfor arbitrary density and temperature. The numerical results for the self-energy and thespectral function will be discussed in sections 4.4 and 4.5.

Complementary to the pure numerical treatment, much effort was made to get ananalytic solution for the GW (0) self-energy. In this way, deeper insight into the relevantphysics behind the mathematical formalism can be gained than from the numerical data.Of course, analytic solutions can only be obtained after performing certain approximations,which are valid only in limited cases of the plasma parameters.

Here, I focus on the case of weakly coupled, low density and classical plasmas. This caseoffers the opportunity to precisely elaborate on the effects of increasing density. Thereby,

the influences of decreasing degeneracy parameter θ = kBT /E F (see equation (1.2)) and in-creasing plasma coupling parameter Γ (equation (1.1)) become accessible. At the center of these considerations is the imaginary part of the single-particle self-energy. The imaginarypart of the self-energy describes the damping of single-particle states due to the interactionwith other plasma particles. Evaluated at the quasi-particle dispersion (see section 4.5 andRefs [For08, For09] for details), it is equal to the inverse of the life time of single-particlestates. For free particles, the self-energy tends to zero, and the life time is infinite.

Naively, one could expect that in weakly coupled plasmas, a series expansion of the self-energy in terms of the coupling parameter gives a sufficient approximation to the exactsolution. This is not the case. The perturbative quasi-particle self-energy is far away from



1.6 Aim and structure of this work 11

the precise numerical solution. Furthermore, it is independent of density [FW74, KKER86],see also Ref. [Le 96]. This means, that the single-particle life time stays finite even in thevacuum, which is an unphysical result. This paradox remained unresolved for a long time.

In Refs. [For08] and [For09], it is shown, that a physically sound description of thesingle-particle life time in a many-particle environment can only be obtained from a non-perturbative ansatz. In the first of these papers, the quasi-particle self-energy at smallmomenta is investigated. It is shown, that the Born approximation for the self-consistentself-energy can be solved analytically and the result complies very well with the full nu-merical solution. The derived expression for the quasi-particle damping or inverse life timeis valid in the regime of plasma densities and temperatures where the Debye screeningparameter κ =

nee2/0kBT is smaller than the inverse Bohr radius. In this regime, it

behaves proportional to n1/4. Thus, the particle life time becomes infinite in the limit of vanishing density, as one would expect from simple physical arguments. At larger densi-

ties, and also at larger momenta, the influence of collective excitations becomes dominant,which can no longer be accounted for using the Born approximation.

The case of large momenta is studied in a second paper [For09]. Here, the relevantdamping mechanism is the coupling to longitudinal electron plasma oscillations, the plas-mons. Using an appropriate approximation to the screened interaction, i.e. the plasmon-pole approximation [KKER86], an analytic solution for the imaginary part of the self-consistent quasi-particle self-energy could be derived also in this case. At large momentum p, the quasi-particle damping behaves as T /p ln( p2), i.e. it decreases very slowly.

The low momentum limit and the large momentum limit are finally combined via a two-point Pade approximation [Mag82], to form an interpolation formula for the quasi-particle

damping width that covers the complete range of momenta from p = 0 to p → ∞.The spectral function, computed via the interpolation formula complies with the full

numerical solution within an error bar of less than 10%. The validity is limited to densitiesand temperatures where the Debye screening parameter κ < 1 a−1B .

Having the single-particle spectral function disposable in an analytic form, greatlyreduces the numerical cost for calculations of higher order correlation functions that needthe spectral function as an input quantity. An examples is the already mentioned dielectricfunction which is given by a convolution of two spectral functions and the vertex function.

The discussion of bremsstrahlung in dense plasmas and associated problems, as outlinedabove, will be limited to non-relativistic plasmas, i.e. the plasma temperature is small

compared to the electron rest mass, kBT mec

2

. A generalized linear response approachto relativistic plasmas is given in Refs [MRH02a, MRH02b]. Secondly, the intensity of theexternal radiation field is always assumed to be so small that no non-linear effects appearand that no relativistic electrons are generated during the absorption of the fields. To givean order of magnitude, non-linear effects (e.g. birefringence of the vacuum, generationof higher order harmonics, self-focussing of laser beams, etc.) appear at intensities of roughly 1018 W/cm2 [Jac75]. Inverse bremsstrahlung in strong laser fields was recentlytreated by Brantov et al. [BRS+03], see also Bornath et al. [BSHK01]. Furthermore, theplasma is assumed to be of infinite extension in each direction; no boundary effects, suchas reflection, refraction, transition radiation, etc. will be considered. For calculations of



12 Introduction

inverse bremsstrahlung in clusters, see the work by Mulser et al. [MKH05].Out of the various problems discussed so far, two will be analyzed in more detail in

this work: The question is raised, how the finite life time of the scattering particles canbe calculated in a consistent way. This leads to the GW (0) approximation for the self-energy. Using the self-energy, it is subsequently investigated how the account for the finitelife time affect the bremsstrahlung emission, i.e. the emission spectrum. Furthermore,the applicability of bremsstrahlung spectroscopy for plasma diagnostic purposes will bedemonstrated.

The work is organized as follows: After this introductory chapter, the next chapterwill give a brief outline of the theoretical framework. After a short discussion of quantumelectrodynamics as the basic theory for the interaction of charged particles with electro-magnetic radiation, the fundamental aspects of many-particle field theory will be outlined.Central objects within this approach are the Dyson-Schwinger equations for the particle

and field propagators and their self-energies. As a special approximation and truncationscheme of Dyson-Schwinger equations, the rainbow-ladder approximation will be discussedand the relation to the GW -approximation, known from solid-state theory will be illumi-nated. Also, a short discussion of the GW approximation in the context of the so-calledΦ-derivable theory by Kadanoff and Baym [BK61, Bay62] will be presented. After a shortsummary of the Matsubara Green function technique, the GW (0) approximation will beintroduced, which is the working horse in this work. The chapter closes with a discus-sion of the theory of optical properties of charged particle systems in the framework of many-particle field theory.

In chapter 3 the basic processes of interaction between photons and charged particles

will be discussed, i.e. emission, absorption, and scattering via various interaction channels.In particular, the role of the various processes for the purpose of plasma diagnostics willbe delineated. Also, the problem of radiation transport is presented shortly.

In chapter 4, selected problems connected with the many-body field theoretical descrip-tion of bremsstrahlung in dense plasmas will be discussed in more detail. The main resultsof the present thesis are shortly summarized. The detailed presentation of the results canbe found in Part II, which contains a compilation of three published, one accepted, andtwo submitted original articles.



Chapter 2

Many-particle theory and applicationto plasmas

2.1 Quantum Electrodynamics

Quantum Electrodynamics (QED) is todays most precise theory for the behaviour of elec-trically charged particles (electrons, nucleons, i.e. fermions), electromagnetic fields, i.e.photons, and their mutual interaction. The quantum nature of both massive particles andfields is contained, and the theory is relativistically invariant. Precision measurements e.g.of the 1s − 2s transition in atomic hydrogen comply with theoretical predictions obtainedfrom QED up to relative errors of 10−17 [H06].

QED is the most general starting point for a theoretical description of plasmas and theirinteraction with external electromagnetic fields. The role of the fields is twofold: On theone hand, the interaction between the charged particles is mediated by the electromagneticfield through the exchange of virtual photons. On the other hand, one wishes to study thebehaviour of a plasma submitted to an external field, e.g static electric or magnetic fieldsor time-dependant fields, such as laser pulses. The emission, absorption, and scattering of radiation by plasmas is of fundamental interest, since these processes and the respectivespectra heavily depend on the macroscopic and microscopic state of the plasma, e.g. itsdensity- and temperature distribution, its composition and so forth. Precise measurementsof emission- absorption- and scattering spectra therefore are an important tool for plasma

diagnostics. The objective of this chapter is to outline the fundamental aspects of QEDand show how expressions for emission, absorption and scattering of radiation in a plasmacan be obtained.

2.1.1 Lagrangian formalism

A convenient starting point is the Lagrangian density of Quantum Electrodynamics [IZ80,PS95]. It can be written as the sum

L(x) = LD(x) + LM(x) + Lint(x) , (2.1)



14 Many-particle theory and application to plasmas

over the Dirac part LD(x) describing the free movement of massive fermions, the Maxwellpart LM describing free electromagnetic waves, while Lint is the interaction part, i.e. thecoupling between fermions and the electromagnetic fields. Here and in the following, xdenotes a four vector, i.e. x = (ct, r), with time t, coordinate vector r = (x,y,z) andthe velocity of light in vacuum c = 299792458 ms−1. The Einstein convention for thescalar product of 4-vectors is used throughout this chapter, i.e. xµyµ :=

3µ,ν =0 xµgµν xν =

x0y0 − x1y1 − x2y2 − x3y3, with the standard metric tensor gµν = gµν [IZ80].The different parts of the Lagrangian density read

LD(x) =c

ψc(x)

icγ µ∂ µ − mcc2

ψc(x) , (2.2)

LM(x) = −0c2

4F µν (x)F µν (x), (2.3)

Lint(x) =c

Z cejµc(x)Aµ(x) . (2.4)

ψc(x) and ψc(x) are the Dirac spinor and its hermitean adjoint for fermions of species c, F µν

is the field tensor of the electromagnetic field. It is derived from the four potential A(x) =(φ(x)/c, A(x)) (φ(x) is the scalar potential, A(x) is the vector potential) via F µν (x) =∂ µAν (x) − ∂ ν Aµ(x). The four-current jc(x) = (cρc(x), jc(x)) is given by the particle fields(Dirac spinors), jµc (x) = cψc(x)γ µψc(x), the γ µ’s are the Dirac gamma matrices. Z c isthe charge-number of species c, e = 1.602176487 × 10−19 As is the elementary charge and0 = 8.854187817 · 10−12 As/Vm is the vacuum electrical permittivity.

From the Lagrangian density, the equations of motion of the particle and radiation fieldsfollow from the principle of extremal action, i.e. the variation of the action S =

d4xL(x)with respect to a dynamical field ϕ vanishes, δS = 0. Evaluation of the (functional)variation yields the Euler–Lagrange equations

∂ L∂ϕ

− ∂

∂xµ

∂ L∂ (∂ϕ/∂xµ)

= 0 . (2.5)

For example, variation with respect to ψ leads to the Dirac equation,

(icγ µ∂ µ

−mc2 + Z ceγ µAµ(x))ψc(x) = 0 , (2.6)

and variation with respect to ψ(x) leads to the hermitean adjoint Dirac equation. Thevariation with respect to Aµ(x) gives

(∂ ν ∂ ν )Aµ(x) = jµ(x) + ∂ µ(∂ ν Aν (x)) . (2.7)

The last equation, written for every component Aµ(x) leads to the more familiar Maxwellequations ×B(r, t) = µ0 j(r, t) + 0∂ E(r, t)/∂t and ×E(r, t) = −∂ B(r, t)/∂t(r, t). Theremaining Maxwell equations · E(r, t) = ρ(r, t)/0 and · B(r, t) = 0 are equivalent tothe definition of the potentials φ(r, t), and A(r, t): B(r, t) = × A(r, t) and E(r, t) =



2.1 Quantum Electrodynamics 15

−φ(r, t) − ∂ A(r, t)/∂t. These equations are not explicitly time-dependent and thereforethey are not obtained through the variational principle.

The Lagrange density (2.1) is gauge invariant. Local phase transformations of the Diracfields ψ(x) → ψ(x) exp(−ief (x)/c) and ψ(x) → ψ(x)exp(+ief (x)/c) do not alter theLagrangian density, if the four potential Aµ(x) is simultaneously transformed as Aµ(x) +∂ µf (x). The proof is straightforward by inserting the transformed fields into the Lagrangiandensity (2.1). The term −eγ µ∂ µf (x) from the differentiation of ψ(x)exp(−ief (x)/c) iscompensated by the term eγ µ∂ µf (x) from the gauge-transformed four-vector potential.This is a characteristic feature of any gauge field theory: The synchronous transformationof the particle fields and the interaction fields, or gauge fields, leave the Lagrangian densityand thereby the equations of motion unchanged. This argument can also be turned aroundin the following sense: Suppose the scattering of a Dirac fermion on some scattering center.The scattering induces a phase shift in the fermion’s wave function. Due to the gauge

invariance, this phase shift entails a perturbation of the electromagnetic field, i.e. theemission or absorption of a photon whilst the scattering. This is precisely bremsstrahlung,or inverse bremsstrahlung in the case of the photon being absorbed. In a similar way, therequest for gauge invariance leads to the W and Z bosons in the case of weak interactionand to the gluons as the gauge bosons in quantum chromodynamics (QCD).

The gauge function f (x) has to satisfy certain general mathematical requirements whichare not to be discussed here. For a detailed presentation of this issue, see Ref. [Wei96].

In a given physical problem, one has to fix the gauge in order to obtain meaningful quan-titative results. The two most frequently used gauges are the Lorentz gauge ∂ µAµ(x) = 0and the Coulomb gauge · A(x) = 0. While the former has the advantage of being rela-

tivistically invariant, the latter is more suitable when it comes to formulate a quantizationscheme for the Maxwell field. Furthermore, the Coulomb gauge is convenient, since it sep-arates the electromagnetic field into a transverse part, the photon field and a longitudinalpart, which gives the Coulomb interaction among charged particles.

The Lagrange density (2.1) is Lorentz invariant. It is valid in any special frame of reference related via xµ = Λµν xν with the Lorentz boost tensor Λ [LL97a]. In order toperform canonical quantization, a special frame of reference will be introduced. However,the results can be transformed into any frame of reference by means of Lorentz transform.

2.1.2 Hamiltonian

Besides using the principle of extremal action, the equations of motion for the dynamicalfields can be derived via the Hamiltonian. The Hamiltonian formalism is more suited forquantization of the theory via introduction of commutation relations among the generalizedcoordinates and their conjugate momenta, whose equations of motion are given by theHamiltonian of the system.

The Hamiltonian is obtained from the Lagrangian L =

d3rL(x) via the Legendre

transformation H =

i P iQi − L, with the generalized coordinates Q and the conjugatemomenta P i = δL/δQi. Note, that in order to define spatial integration and differentiationwith respect to time, a special frame of reference has to be chosen. Since the only time




derivatives in the Lagrange density (2.1) are those of the Dirac fields ψ(x) and the vectorpart A(x) of the four potential1, only ψ(x) and A(x) are generalized dynamical coordinates.The corresponding generalized momenta are

P ψ(x) = δL/δψ(x) = iψγ 0 = iψ†(x) , (2.8)

and PA(x) = δL/δA(x) = 0

φ(x) +

∂

∂tA(x)

= −0E(x) . (2.9)

Applying the Legendre transform with respect to these two dynamical coordinates, theHamiltonian

H (ψ, A; P ψ, PA) =

d3r

iψ†ψ + 0

φ(x) +

∂

∂tA(x)

· A

− L

=

d3r

P ψγ 0

i

−icγ ·+ mc2 − ecγ · A

ψ + 0c2

2(×A)2 + 1

20P2

A

=

d3r

P ψγ 0

i


ψ +02

c2B + E2

,

(2.10)

is obtained. The first term is recognized as the Hamiltonian for a Dirac fermion in anelectromagnetic field, whereas the second term is the energy density of the electromagneticfield 0(c2B2 + E2)/2.

Two remarks should be made at this point: In the derivation of the above equation,Gauss’ law · E = ρ/0, which is one of Maxwell’s equations, has been used. However,

this does not affect our scheme of deriving the dynamical equations of motion from theHamiltonian, since Gauss’ law does not represent such a dynamical equation, i.e. it is timeindependent and simply reflects the conservation of charge. Secondly, it was used, thatthe term · (PAφ) vanishes when the volume integral

d3r is performed, which is again

a consequence of charge conservation.Equation (2.10) is the Hamiltonian of relativistic quantum mechanics. The fields ψ, A

and their conjugate momenta are not quantized. The equations of motion for the gener-alized coordinates Q lead to the Dirac equation and the Maxwell equation. The solutionof the Dirac equation gives the wave function for a Dirac Fermion in a classical electro-magnetic field, which is itself governed by Maxwell’s equations. The next step consists inthe quantization of the Dirac fields and the Maxwell field. In this way, effects like particleproduction, the spontaneous decay of excited atoms, or the vacuum polarization becomeaccessible[Gro93].

In the following section the quantization of both the Dirac and the Maxwell field willbe delineated. As already mentioned, this task requires to fix a system of reference for thedefinition of commutation relations among the field operators. Secondly, the Maxwell fieldhas to be gauged. Here, the Coulomb gauge will be applied. This choice is advantageous

1In the Lagrange density (2.1) only the term ∂ψ/∂t appears, not ∂ ψ/∂t. The time derivative of thescalar potential ϕ does not appear as can be seen by expanding the Maxwell part of the Lagrangian usingthe definition of the Maxwell field tensor.




in a twofold sense: First, canonical quantization is more simple to perform, since already aspecial frame of reference had to be chosen in order to fix the Coulomb gauge itself, sincethe latter involves spatial derivatives. Second, in the theory of electron-ion plasmas, the ionsubsystem defines a frame of reference, since ions barely move due to their comparativelylarge mass. Thus, the laboratory frame can be chosen as the one in which the Coulombgauge is fixed and where the quantization is defined.

The Maxwell field A(r, t) is transverse in the Coulomb gauge, i.e. the longitudinal com-ponent does not represent a dynamical degree of freedom. Consequently, the longitudinalpart of the generalized momentum PA(r, t) has to be eliminated as a dynamical degree of freedom. This is done by decomposing PA(r, t) = PA⊥(r, t) + PA(r, t) and identifyingE(r, t) = −PA/0 with the Coulomb field, satisfying Gauss’ law · E(r, t) = ρ/0.

The Hamiltonian (2.10) in Coulomb gauge is rewritten as

H =

d3r

P ψγ 0i


ψ + 02

c2B + E2

⊥

+ e

2ρφ

, (2.11)

with the scalar potential φ(x). The scalar potential does not represent a dynamical degreeof freedom, since it is related in an instantaneous manner to the charge density eρ(x). It canbe given completely in terms of the Dirac fields, i.e. the charge density eρ(x) = eψ†(x)ψ(x),

φ(x) =e

2

d3x

ρ(x)

4π0 |r − r| . (2.12)

E⊥ is the transverse part of the electric field, i.e.

·E⊥ = 0.

2.1.3 Quantization of the fields

In order to quantize the Hamiltonian (2.11), commutation relations have to be imposed onthe field operators ψ(x), P ψ(x), A(x), and PA(x), which replace the generalized coordinatesand conjugate momenta. The field operators are usually represented as a Fourier series overcreation and annihilation operators of field-modes. The creation and annihilation operatorsare particularly suited for the formulation of perturbation theory and for the calculationof cross-sections, e.g. the reaction of particles among themselves and their coupling toexternal electromagnetic fields.

2.1.4 Quantization of the Maxwell field in Coulomb gauge2

In Coulomb gauge, · A = 0, the transversality of the Maxwell field has to be reflectedalso in the commutation relation. Therefore, the commutation relation between A and PA

reads Ai(x), P jA(x)

= Ai(x)P jA(x) − P jA(x)Ai(x) = −iδ⊥ij(r − r)δ(t − t) , (2.13)

2For other quantization schemes, e.g. using Lorentz gauge, see Ref. [Reb05].




with the transverse delta function

δ

⊥

ij(r − r

) = d3k

(2π)3 e

ik·(r−r)δij −

kik j

k2

. (2.14)

The transverse delta function is solenoidal, i.e. δ⊥ij = 0. The Fourier decomposition of the vector potential reads

A(x) = Ω00

d3k

2ωk(2π)3

2λ=1

ελk

bλ(k) e−ikx + b†λ(k) eikx

, (2.15)

where the annihilation and creation operators b†λ(k) and bλ(k) for a photon of wavevectork = (ωk/c, k), kµkµ = 0 and polarization ελk were introduced. The transversality of A

imposes the conditions

ελk · k = 0 ,

ελk · ελ

k = δλλ

,(2.16)

on the polarization vectors ελk. Inversion of Eq. (2.15) leads to

bλ(k) =i

(2π)3√

2ωkΩ00

d3x eikx

↔

∂ 0 ελk · A(x) ,

b†λ(k) =−i

(2π)3√2k0Ω00 d3x e−ikx

↔

∂ 0 ελk

·A(x) ,

(2.17)

where we used the short-hand notation f (x)↔

∂ g(x) = f (x)∂g(x)−(∂f (x))g(x). By using thedefinition (2.15), the commutator between creation and annihilation operators is evaluatedto

bλ(k), b†λ(k)

= δλλδ(k − k) ,

[bλ(k), bλ(k)] =

b†λ(k), b†λ(k)

= 0 .(2.18)

In order to prepare the next section, were the perturbation expansion for QED will beoutlined, I also give the propagator for the transverse photon field. The transverse photonpropagator is defined as

D⊥ij(x, x) =

1

ic0| T B

Ai(x)A j(x)

|0 , (2.19)

where the field modes satisfy the Coulomb gauge condition. The propagator is the ex-pectation value in the ground state |0 of the time-ordered product of two field oper-ators at different coordinates. The time-ordered product for Bose particles is definedas T B A(x)B(x) = θ(t − t)A(x)B(x) + θ(t − t)B(x)A(x). The propagator D⊥

ij(x, x)




can most easily be evaluated in momentum space, using the plane-wave decompositionEq. (2.15). One obtains

D⊥ij(x − x) =

d4k

(2π)4D⊥ij(k) exp(−ik(x − x)) , (2.20)

D⊥ij(k) =

2λ=1

ελi · ελ j1

k2 + iη, η → 0 . (2.21)

Note, that the propagator depends only on the difference x − x as a consequence of space-time homogeneity.

2.1.5 Quantization of the Dirac field

In order to quantize the Dirac fields ψ(x) and P ψ(x) = iψ†(x), anticommutators insteadof commutators have to be used to reflect the Fermi-Dirac statistics of spin 1/2 particles.In this way, the total energy of the Dirac field is positive definite, using commutators wouldlead to unphysical negative energies. The anticommutator relations imposed on the Diracfield operators are

ψα(x), P ψα

(x)

= ψα(x)P ψα

(x) + P ψα

(x)ψα(x) = δααδ(r − r) ,ψα(x), ψα(x)

=

P ψα(x), P ψα

(x)

= 0 .(2.22)

The index α denotes the spinor component. The plane-wave decomposition for the Diracfields reads

ψα(x) =

√mc2 (2π)3

4α=1

d3 p

E paα(p)uα(p)e−ξα p·x ,

P ψα(x) = i

√mc2 (2π)3

4α=1

d3 p

E pγ 0a†α(p)uα(p)eξα p·x ,

(2.23)

with ξα = 1 for α = 1, 2 and ξα = −1 for α = 3, 4. aα(p) is an annihilation operatorfor fermions (α = 1, 2) in momentum state p and a creation operator for antifermions

(α = 3, 4), while a†α(p) acts as a creation operator for fermions and annihilation operator

for antifermions. Antifermions are solutions of Dirac’s equation with opposite charge asfermions and travelling backwards in time. The restmass m > 0 is the same for fermionsand for antifermions. uα(p) is the amplitude of the Fourier component for momentum p,i.e. the solution of the free Dirac equation (icγ µ pµ − ξαmc2)uα(p) = 0. For the creationand annihilation operators the anticommutation relations

aα(p), a†α(p)

= δααδ(p − p) ,

aα(p), aα(p) =

a†α(p), a†α(p)

= 0 ,(2.24)




hold.The fermion propagator is defined as

S (x − x) = −i0| T F

ψ(x) ψ(x) |0 . (2.25)

The time-ordered product for fermions T F A(x)B(x) = θ(t−t)ψ(x)ψ(x)−θ(t−t)ψ(x)ψ(x)correctly reflects the Fermi-Dirac statistics i.e. the anticommutation relations instead of the commutation relations as in the case of the Bose time-ordered product. Using Eq. (2.23)the free fermion propagator is obtained as

S (x − x) =

d4 p

(2π)4S ( p)exp(−ip(x − x) (2.26)

S ( p) =γ µ∂ µ + mc

p2 − m2c2 + iη, η

→0 . (2.27)

Note, that S (x − x) is a 4x4 matrix acting in the Dirac spinor space.

2.1.6 The QED Hamiltonian

Using the quantized expressions for the Maxwell field Eq. (2.15) and the Dirac fieldsEq. (2.23), the Hamiltonian of QED is obtained as

H = Ω0

4

α=1 d3 p

(2π)3

E pa†α(p)aα(p) + Ω0

2

λ=1 d3k

(2π)3ωkb†λ(k)bλ(k) +

1

2

− emc34

α,α=1

2λ=1

Ω5/20

d3 p

(2π)3

E p

d3k

(2π)3√

2ωk

(γ · εkλ) uα( p)uα( p)

×

exp(i t

ξαE p − ωk − ξαE ξαp−ξαk

)

E ξαp−ξαk

a†α(p) bλ(k) aα(ξαp − ξαk)

+exp(i t

ξαE p + ωk − ξαE ξαp+ξαk

)

E ξαp+ξαk

a†α(p) b†λ(k) aα(ξαp + ξαk)

+

1

2

4α,α=1

Ω0 d3 p

(2π)3d3 p

(2π)3e2

0k2a†

α(p − k) a

†

α(p

+ k) aα

(p

) aα(p) .

(2.28)

Ω0 is the normalization volume, E p is the fermion energy E p = +

m2c4 + p2c2. ThisHamiltonian features an infinite contribution due to the zero-point energy

d3kωk/2(2π)3

of the photon field. However, this contribution is treated as a constant (albeit infinite),additive term to the total energy operator, which is not observable, since only energy dif-ferences can be measured. However, it should be noted that under special experimentalarrangements, the zero-point energy of the electromagnetic field has an observable effect.One such example is the Casimir effect [Cas48, PMG86], which describes the attractive




force between two uncharged metal plates due to small difference in their respective “vac-uum” energies. On the other hand, the well-known and “every-day” observation of thespontaneous radiative decay of excited atoms or molecules can only be understood as aresult of the interaction of the atom with the vacuum fluctuations of the electromagneticfield [Mes99].

Also the free Fermion part (first term) contains an infinite contribution. This can beseen, by expanding the sum over Dirac components as

Ω0

4α=1

d3 p

(2π)3E pa†α(p)aα(p) =

Ω0

2α=1

d3 p

(2π)3E pa†α(p)aα(p) + Ω0

4α=3

d3 p

(2π)3E paα(p)a†α(p) − 2Ω0 d3 p

(2π)3E p . (2.29)

The first two terms are the kinetic energies of the fermion and the antifermion subsystem,respectively (note that a3/4 is an antifermion creation operator), the third term stemsfrom the antifermion commutation relation. As in the case of the zero-point energy of thephoton field, this constant but infinite contribution is subtracted, since not contributingto observable effects.

To avoid the appearance of infinite energy contributions, it is common use in quantum

field theory, to write the Hamiltonian in normal order . Operators with the phase factorexp(−iωt) alway appear left from those whose time dependence is given by exp(+iωt). Inthis way, the infinite terms in the Hamiltonian are omitted right from the beginning, whichmeans that only the physical masses appear. The normal ordering of two operators A andB is denoted by : AB :. The following identity holds

AB =: AB : ±0|T

AB

|0 , (2.30)

where the negative sign appears only if the operators are fermionic creation- or annihilation

operators.

The third and fourth term of the Hamiltonian describe the coupling of the Diracfermions to the transverse photon field (third term) and their mutual Coulomb interaction(fourth term). The third term exhibits the typical structure of QED processes, where theinteraction of fermions and photons is described as the creation and annihilation of fermionsin different energy- and momentum states and the creation or annihilation of photons withmatching energy and momentum, such that the total four-momentum is conserved. Thecoupling term is the starting point for the formulation of perturbation theory, which leadsto the cross-sections for the various interaction channels in QED.




2.1.7 Non-relativistic limit

Since typical energies considered in this work are non-relativistic, I should give the non-

relativistic limit of the QED Hamiltonian, as well,

H = Ω0

s

d3 p

(2π)3E s(p)a†s(p)as(p) + Ω0

2λ=1

ωkb†λ(k)bλ(−k)

− Ω0

2λ=1

j(k) · εkλ

Ω0

20ωk

1/2 b†λ(−k) + bλ(k)

+1

2Ω30

ss

d3 p d3 p d3k

(2π)9V ss(k)a†s(p − k)a†s(p + k)as(p)as(p) . (2.31)

Here, the non-relativistic single-particle energy of particles of species s, E s(p) =

2

p2

/2mswas introduced. Note, that the index s labels different species and further quantum num-bers such as the particles spin. Only particles, no antiparticles appear in the non-relativisticHamiltonian, since the typical energy scales (given e.g. by the plasma temperature) aresmall compared to twice the rest mass of the particles, kBT 2mcc

2. Only at this energyscale, pair production sets in. The current density operator j(k) is defined as

j(k) =s

d3p

(2π)3Z se

ms

p a†s(p − k/2)as(p + k/2) . (2.32)

2.1.8 Perturbation theory

As already mentioned, the objective of QED is the calculation of cross sections for vari-ous processes involving the emission, absorption, and scattering of photons by electricallycharged particles. To this purpose, a perturbation series known as the S-matrix expansionis developed in terms of the interaction part of the Hamiltonian (2.28), i.e. the couplingof electrons to transverse photons and the Coulomb interaction. The S-matrix expansionis most conveniently treated in the interaction picture or Dirac picture. In the Dirac pic-ture, the time-evolution of an observable O due to the interaction part of the HamiltonianH int = H − H 0 is formally eliminated via the unitary transform

OD(t) = eiH 0t/OSe−iH 0t/ . (2.33)

OS is the observable in the Schrodinger picture. One can easily show that the equation of motion for the observable in the Dirac picture involves only the non-interacting Hamilto-nian, i.e.

∂

∂tOD(t) =

i

H 0, OD(t)

. (2.34)

On the other hand, the state vectors in the Dirac pictureψD(t) = eiH 0t/ |ψS(t) obey a

wave equation which involves only the interaction part of the Hamiltonian,

i∂

∂t ψD(t) = H Dint(t)

ψD(t) . (2.35)




This equation can formally be integrated as

ψ

D

(t) =ψ

D

(t0) +

1

i tt0 dt1H

D

int(t1)ψ

D

(t1) . (2.36)ψD(t0) is the quantum state at a given initial time t0, which is characterized by a vanishinginteraction part of the Hamiltonian. To eliminate any ambiguity in the choice of t0, thelimit t0 → −∞ is considered and the state

ψD(t → −∞) is denoted the initial state|i. Since the interaction part of the Hamiltonian is assumed to vanish at t → −∞, theinitial state in the Dirac picture is equal to the initial state in the Schrodinger picture. Bysuccessive iteration of Eq. (2.36) on arrives at the Dyson series for the wave function,

ψD(t) = |i +

1

i t

−∞

dt1 H Dint(t1) |i +1

(i)2 t

−∞ t1

−∞

dt1 dt2H Dint(t1) H Dint(t2) |i + . . . .

(2.37)The S-matrix is defined as the transition amplitude ψD(t → ∞)|i. Its matrix elements

Sfi = f | S| i measure the probability amplitude that in the limit t → ∞ the quantumsystem is in the final state |f . As for the initial state, the interaction is assumed to vanishfor t → ∞ and the Dirac- and Schrodinger picture are equivalent. By projecting the Dysonseries (2.37) on the final state f |, the S-matrix elements can be calculated iteratively. Incompact form, using the time-ordered product T , which sorts its arguments such that latetimes stand to the left of early times, one obtains

Sfi =

f

|ψD(t

→ ∞)

=

f

|

∞

n=0 1

in 1

n!

∞

−∞

dt1 ∞

−∞

dt2 . . . ∞

−∞

dtn−1 ∞

−∞

dtn

T H Dint(t1)H Dint(t2) . . . H Dint(tn−1)H Dint(tn) |i . (2.38)

The factor 1/n! is the correct normalization to the number of possible permutations of theoperators in the time-ordered product. Note, that this equation is only valid, if the numberof fermion creation and annihilation operators is even. In QED, this is indeed the case asbecomes clear from the Hamiltonian (2.28).

The evaluation of the perturbation expansion of the S-matrix directly gives the tran-sition amplitudes for various QED processes, where a transition from the initial state |ito the final state |f appears. The evaluation of the first order is simple. Since only one

time argument appears, the time-ordering T has no effect. The interaction term withtransverse photons gives eight terms, which correspond to photon emission and absorptionby fermions and by antifermions (4 terms) and fermion-antifermion creation (annihilation)together with photon creation (annihilation) (4 terms). However, non of these processes isa real physical process, since energy- and momentum cannot be conserved simultaneously.

In higher orders, the emerging terms contain a large number of products of creation- andannihilation operators for photons and fermions. Out of these many products only thosegive a finite contribution to the amplitude, which contain the right number of annihilationoperators to destroy all particles in the initial state and the right number of creationoperators to create all particles in the final state.




Wick [Wic50] showed that the time-ordered product of N normal ordered field opera-tors, as is the case in the interaction Hamiltonian, can be written as a sum over productsof M propagators and N

−2M external operators and the sum extends from M = 0 to

M = N/2. The evaluation of the second order of the perturbation series will be given as anexample. In second order of the S-matrix expansion, terms appear which involve the inter-action of charged particles with free, transverse photons. Due to Wick, the time-orderedproduct in the second-order contribution to the S-matrix

S (2) =e2

2(i)2

d4x1 d4x2 T

ˆψ(x1)γ µψ(x1)Aµ(x1) ˆψ(x2)γ ν ψ(x2)Aν (x2)

(2.39)

can be decomposed into the sum

S (2) =e2

2(i)2 d4x

1d4x

2

: ˆψ(x1)γ µψ(x1)Aµ(x1) ˆψ(x2)γ ν ψ(x2)Aν (x2) :

+ 2 : ˆψ(x1)γ µiS (x2 − x1)γ ν ψ(x2)Aµ(x1)Aν (x2) :

+ : ˆψ(x1)γ µψ(x1) ˆψ(x2)γ ν ψ(x2)iDµν ⊥ (x2 − x1) :

+ 2 : ˆψ(x1)γ µiS (x2 − x1)γ ν ψ(x2)iDµν ⊥ (x2 − x1) :

+ : iS (x2 − x1)γ µiS (x1 − x2)γ ν Aµ(x1)Aν (x2) :

+ : iS (x2 − x1)γ µiS (x1 − x2)γ ν iDµν ⊥ (x2 − x1) : .

(2.40)

These terms can most conveniently be depicted by Feynman diagrams [Vel94], which allowfor a very intuitive interpretation of the different terms and to sort out those terms whichdescribe a special process. The first term is again vanishing since it describes two parallelprocesses of first order. In the second term, one finds the Compton scattering of photonson electrons (positrons) and the process of pair production by two-photon decay and pairannihilation by two-photon production. These processes are depicted by the Feynmangraphs in Fig. 2.1

Whereas the second term contains the interaction with an external transverse field,the third term contains the fermion-fermion scattering (Møller scattering) and fermion-

antifermion scattering (Bhabha scattering). It is depicted by the Feynman graphs inFig. 2.2In the fourth term in (2.40) appears what is called the one-loop fermion self-energy,

i.e. the product of one fermion propagator and one photon propagator. The one-loopself-energy gives the first order correction to the bare fermion propagator. The fifth termcontains the photon self-energy or polarization function, i.e. the product of two fermionpropagators. The self-energy corrections to the bare propagators are depicted by theFeynman diagrams in Fig. 2.3

The evaluation of the one-loop fermion self-energy leads to a divergent integral. Similarto the vacuum fluctuations of the Maxwell field, the electron self-energy is treated as a




(a) Fermion Compton scattering (b) Antifermion Compton scat-tering

(c) Pair annihilation

(d) Fermion Compton scattering(exchange term)

(e) Antifermion Compton scat-tering (exchange term)

(f) Pair creation

Figure 2.1: Feynman diagrams of fermion(antifermion)-photon interaction

constant, infinite contribution to the electron’s rest mass. To obtain the correct physicalmass of the electron, the so-called bare mass, i.e. the mass without self-energy corrections,is itself fixed a infinity, such that the sum of the bare mass and the self-energy gives thephysical mass. The mathematical rendition of this concept is known as renormalization[Kak93].

Figure 2.3(c) gives the one-loop photon self-energy or polarization function [Gro93].Finally, the last term is the so-called vacuum contribution ground state energy, see Fig. 2.4It can be shown, that this graph belongs to a series of terms that constitutes a phase factorof the total S-matrix and therefore has no observable effect. On the other hand, thesecompletely closed vacuum graphs are the starting point for the path-integral formulationof quantum field theory [GR96, FH65].

In a many-particle system, not only the transverse photons contribute to the inter-action part of the Hamiltonian, but also the Coulomb interaction has to be considered.In principle, all the graphs in Fig’s. 2.1, 2.2, 2.3, and 2.4 also exist with the transversephotons replaced by the longitudinal Coulomb potential. For example, the diagram in




(a) Fermion Møller scattering (b) Antifermion Møller scatter-ing

(c) Bhabha scattering

Figure 2.2: Feynman diagrams of fermion-fermion scattering

(a) Fermion self-energy correc-tion

(b) Antifermion self-energy (c) Photon self-energy correction

Figure 2.3: Fermion and photon one-loop self-energy corrections

figure 2.2(a), with the transverse photon replaced by the longitudinal Coulomb potential,describes the elastic scattering among two fermions. Let’s consider the case when one of the scattering partners is an ion and the other one is an electron. Due to it’s heavy mass,the ion can be treated as fixed in the laboratory frame of reference. Evaluation of thetransition amplitude, averaging over all initial spin polarizations and summation over allfinal spin polarizations leads to the elastic scattering cross section

dσ

dΩ

elastic

=Z 2α4

1 − β 2 sin2 θ/2

4β 2|pi|2

1

sin4 θ/2. (2.41)

Here, the finestructure constant α = e2/4π0c 1/137 was introduced. In the non-relativistic limit, this expression reduces to the well-known Rutherford cross-section. Onegets the same result for elastic positron scattering [GR84].

Further terms in second order of the Coulomb interaction are the self-energies due tothe Coulomb self-interaction, and the screening of the Coulomb interaction as well as thevacuum contribution of the Coulomb field. The mixed terms, where one Coulomb inter-action and one transverse photon appear in the Feynman diagram are shown in Fig. 2.5.They describe the emission of a photon whilst the scattering of a fermion in the Coulombpotential of an ion, so-called bremsstrahlung emission.




Figure 2.4: Vacuum contribution in 2nd order of the S-matrix expansion

(a) Electron bremsstrahlung (b) Electron bremsstrahlung (ex-change term)

Figure 2.5: Feynman diagrams for electron bremsstrahlung

Out of the many processes that are obtained in second order of the S-matrix expansion,four are of special importance also for plasma physics. For the interaction with externalfields these are Compton scattering and bremsstrahlung and for the internal fields theelectron and photon self-energy. Compton scattering describes the scattering of electro-magnetic waves on electrons. As will be discussed later, the scattering spectrum is sensitiveto the macroscopic properties of the plasma, i.e. temperature, density, and the equation of state. Therefore, scattering of electromagnetic waves is a suitable tool for plasma diagnos-

tics. On the other hand, bremsstrahlung is the most dominant mechanism of absorptionand emission in a hot, highly ionized plasma. Also in this case, details in the emission orabsorption spectrum can give valuable information about the plasma’s state.

In a many-particle environment, the electron and the photon self-energy give the mod-ifications of single-particle properties due to correlations. An example is the change of the energy-momentum relation for massive particles due to the effective mass. The ef-fective mass is given by the fermion self-energy. Similarly, the propagation of photons inthe medium, commonly described via the dielectric function, is determined by the photonself-energy, i.e. the polarization function.




2.1.9 Bremsstrahlung: Bethe-Heitler formula

Bremsstrahlung is understood as the emission of a photon by a charged particle which is

submitted to an external field. In the case of the external field being a magnetic field, theemitted radiation is usually called synchrotron radiation, while the term bremsstrahlungis mostly associated with emission of radiation by electrons in the Coulomb field of othercharged particles.

In the most simple case, an electron scatters on an ion, which is assumed not to moveduring the process and whose finite extension is neglected, such that it’s field can betaken as a pure Coulomb field. Evaluation of the Feynman diagrams in Fig. 2.5 givesthe well known Bethe-Heitler formula for bremsstrahlung [BH34]. This approximation willbe referred to as Born approximation in the following. Note, that the presence of thescattering center (the ion) breaks the translational invariance, and therefore conservation

of three-momentum is no longer required. The Bethe-Heitler cross section reads [IZ80]

d3σ

dk0 dΩγ dΩe=

Z 2α3

(2π)2|pf |

|pi| |k|4 k0

× |pf |2 sin2 θf

( p0f − |pf | cos θf )2(4( p0i )2 − |k|2) +

|pi|2 sin2 θi( p0i − |pi| cos θi)2

(4( p0f )2 − |k|2)

+ 2k20|pi|2 sin2 θi + |pf |2 sin2 θf

( p0f − |pf | cos θf )( p0i − |pi| sin θi)− 2

|pi| |pf | sin θi sin θf cos ϕif

( p0f − |pf | cos θf )( p0i − |pi| cos θi)

×(4 p0i p

0f

− |k

|2 + 2ω2) . (2.42)

θf and θi are the angles enclosed by the photon’s momentum k and the electron’s initial andfinal momentum pi and pf , respectively. ϕif is the angle enclosed by the planes spannedby k and pi, and pf , respectively.

Two important limiting cases should be discussed at this point. In the limit of vanishingphoton energy k0 = ω → 0 one obtains

limk0→0

d3σ

dk0dΩe dΩγ

=

dσ

dΩe

elastic

× e2 d3k

2k0(2π)3 |

pf

|2 sin2 θf

(k0 p0f − |k| |pf | cos θf )2 + |pi

|2 sin2 θi

(k0 p0i − |k| |pi| cos θi)2

− 2|pf | |pi| sin θf sin θi cos ϕif

(k0 p0i − |k| |pi| cos θi)(k0 p0f − |k| |pf | cos θf )

. (2.43)

The prefactor is the elastic scattering cross section Eq. (2.41). Eq. (2.43) shows thatthe Bethe-Heitler cross section diverges as 1/k0 = 1/ω when the photon energy goesto 0. This is known as the infrared divergence of bremsstrahlung. However, the crosssection is not a directly measurable quantity. In a scattering experiment, one measuresthe radiation intensity, i.e. the radiated power per unit area of the detection device. It




is proportional to the product of the cross section and the photon’s energy and thereforetends towards a finite value at small photon energy. Another way of discussing the infrareddivergence of the bremsstrahlung cross section takes into account terms in the third orderof the S-matrix expansion: Assuming that in an experiment we want to measure the

Figure 2.6: Radiative corrections of the elastic scattering amplitude

bremsstrahlung spectrum with a spectrometer of spectral bandwidth ∆ω. For photonfrequencies ω < ∆ω, we cannot distinguish between elastically scattered electrons andthose which have emitted a bremsstrahlung photon. Therefore, in the theory, both elasticand inelastic scattering amplitudes have to be added coherently up to the same order inαQED. Since the bremsstrahlung cross section is one order of αQED higher than the elasticscattering cross section, also higher order contributions to the elastic scattering amplitudehave to be considered, so-called radiative corrections. The corresponding diagrams areshown in Fig. 2.6. It can be shown, that these diagrams are also infrared divergent andthat the divergent terms exactly cancel with those terms appearing in the bremsstrahlung

amplitude.The second limiting case discussed here is the non-relativistic case, p0i mec. Then,

the bremsstrahlung cross section turns intodσ

dω

nr

=16

3

Z 2e2

4π0c

e2

4π0mec2c2

v2i2 ln

√

E i +√

E i − ω√ω

, (2.44)

vi is the initial velocity. Again, this expression is infrared divergent.

2.1.10 Non-relativistic bremsstrahlung and Gaunt factors

Quantum field theory led us to an expression for the bremsstrahlung cross-section, basedon second order perturbation theory, the Bethe-Heitler formula (2.42).

Historically, expressions for the emission of electromagnetic radiation from a pointcharge, accelerated in the Coulomb field of an ion, where first given by Kramers [ Kra23].He used Maxwell’s theory of electromagnetism and assumed classical trajectories of theemitting particles: The radiated energy per frequency interval dW (ω)/dω of emission fromaccelerated point charges is given by Larmor’s law,

dW (ω)

dω=

2

3πc3e2

4π0|a(ω)|2 , (2.45)




with the Fourier components of the acceleration a(ω) = ∞−∞

r(t) exp(iωt)dt. Thus, theknowledge of the particles trajectory r(t) allows the calculation of the power spectrum.Unbound electrons in a Coulomb field follow hyperbolic trajectories, i.e. the distance r(t)between the ion and the electron as a function of time is given by

r(t) =b2

b⊥(1 +

1 + (b/b⊥)21/2

cos θ(t)), (2.46)

with the impact parameter b, θ(t) being the angle between the initial velocity vector vi =v(t → −∞) and the velocity at time t, v(t); and the impact parameter for 90 scatteringb⊥ = Ze2/4π0mev20 .

In general, one is interested in the power spectrum, i.e. in the radiated energy perunit time. This is obtained by integrating the radiated energy spectrum over all scattering

angles and impact parameters and multiplying by the density of ions ni and the velocityof the incoming electrons, i.e.

dP (ω; v0)

dω= niv02π

∞0

db bdW (ω; b)

dω. (2.47)

The integrations can be performed analytically [LL97b], which leads to the expression

dP (ω; v0)

dω=

16πω

3√

3

Z 2ni

m2ec3v0

e2

4π0

3G(ω) . (2.48)

Here, the frequency dependent Gaunt factor G(ω) was introduced,

G(ω) =π√

3

4uH (1)u (u) H (1)u

(u) , u =

iωZe2

4π0mev30, (2.49)

with the Hankel function H (1)n (z) of first kind and of order n. The derivative H

(1)n

(z) of

the Hankel function is given by

H (1)n

(z) =

d

dzH (1)n (z) =

nH (1)n (z)

z− H (1)n+1(z) (2.50)

The Gaunt factor has the special property, that for large dimensionless frequencies u,u 1, it converges towards unity, G(ω) = G(u(ω)) → 1 [Miy80]. Thus, Kramers’ formulafor the bremsstrahlung power spectrum predicts a frequency independent behaviour, if either the frequency becomes large, or if the initial velocity becomes small. Immediately,this behaviour leads to an ultraviolet catastrophe, since the total radiated energy, W = ∞0

dωdP (ω/dω) diverges at the upper integration limit. This is due to the fact that thequantum nature of electromagnetic waves has been neglected. A natural upper cutoff for the frequency integration is introduced, if the photon energy is accounted for in theconservation of energy during the emission, i.e. mev20/2 = mev2(t → ∞)/2 +ω. Since the




energy of the emitted photons cannot overcome the initial kinetic energy of the scatteringelectrons, the spectrum terminates at ω = mv20/2.

When the theory of bremsstrahlung, as sketched here, was developed, the calculationof Hankel functions was considerably complicated. Therefore, various approximations forthe Gaunt factor have been developed that lead to more simpler expressions and whichare suitable in different regions of the parameter space spanned by the initial velocity, thefrequency and the charge of the central ion. A summary can be found in the review articleby Karzas and Latter [KL61].

Nowadays, the term Gaunt factor is used in a more general context. It is common use tonormalize calculations of the bremsstrahlung power spectrum to the frequency independentpart of Kramers’ formula (2.48) and refer to the remaining factor as the Gaunt factor,that describes corrections beyond the high frequency limit of Kramers’ classical formula

[BH62, KM59]. Such corrections take into account e.g. the quantum nature of the emittingparticles. In principle, the classical trajectories are replaced by Coulomb wavefunctions[Som49]. The resulting expression, named Sommerfeld Gaunt factor, reads

G(ω) = π√

3−ξ d

dξ|F (iν f , iν )i, 1, ξ)|2

(1 − exp(−2πν f ))(exp(2πν i) − 1), ν i/f = Z

2Ryme/pi/f (2.51)

with the complete hypergeometric function F (α , β, γ ; z). The derivative is given bydF (α , β, γ ; z)/dz =

−αβF (α + 1, β + 1, γ + 1, z). Initial and final momentum are related

by conservation of momentum, |pi−pf | = ω/c. In the situation when both incoming andoutgoing electron energies are large compared to the potential energy at the point of closestapproach, i.e. ν f , ν i 1, the derivative of the complete hypergeometric function can beapproximated by F (1, 1, 1, ξ) −ν f ν iF (1, 1, 2, ξ) = ν f ν i ln(1 − ξ)/ξ and the completehypergeometric function itself tends to F (0, 0, 1, ξ) = 1. Then, we find again the Bornapproximation for the non-relativistic case, see Eq. (2.44) .

The Bethe Heitler formula (2.42) represents the Gaunt factor for relativistic electron-ion bremsstrahlung. The Born-Elwert approximation [Elw39] represents an improved non-relativistic Born approximation, that reproduces the full Sommerfeld result up to photonenergies only slightly below the endpoint of the spectrum.

Also in the case of bremsstrahlung emission from a plasma, the term Gaunt factoris used. Here, it describes corrections due to many-particle effects, such as screening of the Coulomb potential [WMR04], multiple scattering [WMR+01], correlations among thescattering centers [FRRW05], and correlations among the scattering electrons [FRW07],which go beyond the classical Kramers result, averaged over the distribution function of the emitting particles in the medium. Expressions for the bremsstrahlung emission froma plasma will be given in section 2.5. In principle, these are obtained by averaging thebremsstrahlung cross section for the individual scattering event with the electron momen-tum distribution function.




2.1.11 Electron-photon Scattering: Klein-Nishina formula

The cross section for scattering of polarized photons on a single electron (Compton scat-

tering) is obtained by evaluation of the Feynman diagrams 2.1(a) and 2.1(d). One obtainsthe Klein-Nishina formula [KN29],

dσ

dΩ

KN

=r2e2

ωf

ωi

2 ωf

ωi

+ωi

ωf

+ 4(εf · εi)2 − 2

, ωf = ωi

1 +

ωi

mec2(1 − cos θ)

−1.

(2.52)The second equation reflects conservation of 4-momentum, ωi/f is the frequency of thephoton before and after the scattering, respectively, εi/f are the initial and final polarizationvectors. Ω is the solid angle the photon scatters into. Furthermore, re denotes the so-calledclassical electron radius, re = e2/4π0mec2.

Summing over all initial and final polarizations yields the unpolarized cross-section,

dσ

dΩ

KN

=r2e2

ωf

ωi

2 ωf

ωi+

ωi

ωf

− sin2 θ

, (2.53)

θ is the scattering angle.

In the limit of small photon energies, ωi mec2, the classical Thomson cross section

dσ

dΩ=

r2e

2(1

−cos2 ϕ sin2 θ) . (2.54)

is found. As already noted, Compton or Thomson scattering play an important role inplasma diagnostics. First, scattering on uncorrelated electrons results in a Doppler broad-ened scattering spectrum due to the thermal movement of the electrons. The Dopplerwidth is related to the temperature of the system or the Fermi energy in the case of Fermidegenerate systems. Secondly, in correlated electron systems, such as dense plasmas, thescattering takes place on collective excitations (plasmons, phonons), which show up aspeaks in the scattering spectrum. Their position and amplitude are also directly related tothe system’s temperature and density [Thi07]. The role of Compton scattering or Thom-son scattering as a tool for plasma diagnostic purposes will be discussed in more detail insection 3.5.

As a final remark, it should be noted, that both bremsstrahlung and Compton scatteringcan be understood as the scattering of photons on electrons. In the case of bremsstrahlung,one of the transverse photons, present in the Compton scattering diagrams, is replaced bya virtual photon, the longitudinal Coulomb interaction. Both processes are contained inthe 4x4 Klein-Nishina matrix, which gives the scattering cross section of polarized photonson electrons. The 0-components of this matrix give the bremsstrahlung cross-section, whilethe other components give the “true” Compton scattering cross-sections, accounting forthe polarization of the incoming and the outgoing photon.



2.2 Many-particle field theory and the GW -approximation 33

2.2 Many-particle field theory and the GW -approximation

2.2.1 Dyson-Schwinger equationsPerturbative QED, as outlined section 2.1, provides a systematic and powerful tool tocalculate cross-sections for the various interaction processes between the radiation field(gauge bosons) and electrically charged particles (fermions). Examples where given in sec-tions 2.1.9 and 2.1.11 for bremsstrahlung and Compton scattering, respectively. However,there also exist phenomena that cannot be described by perturbation theory. Examplesare the occurrence of bound states, e.g. atoms, or the appearance of collective modesin a correlated many-particle system, such as plasmons (collective plasma oscillations),phonons (collective modes of crystal lattices) or giant-resonances in atomic nuclei. To ac-count for these kind of phenomena, non-perturbative methods have to be applied. These

methods can be regarded as an extension of perturbation theory in the sense that certainsub-classes of diagrams are summed up to infinite order, and thereby give a solution tothe underlying equations of motion, e.g. of the particle propagator, that also contains thenon-perturbative effects. Examples will be discussed below.

Formally, the many-particle problem can be cast into a hierarchy of non-linear coupledintegro-differential equations, known as the Dyson-Schwinger equations [Dys49, Sch51b,Sch51c]. This approach was originally formulated for the vacuum, i.e. at zero temperatureand zero density. Its generalization to finite temperature and finite density was accom-plished more recently, see e.g. the review by Roberts and Schmidt [RS00] and referencestherein. An excellent introduction to Dyson-Schwinger equations can also be found in[HRW06].

We will discuss the lowest order Dyson-Schwinger equations. These are the equationsfor the two-point functions, i.e. the single-particle Green function G and the photonpropagator D and their respective self-energies, the single-particle self-energy Σ and thepolarization function Π. A detailed derivation can be found in Ref. [HRW06]. Theseequations are represented by the following Feynman diagrams

G=

G(0)

+G(0)

ΣG

, (2.55)

Σ = G

D

Γ(0) Γ , (2.56)

D=

D(0)

+D

(0)

ΠD

, (2.57)

Π =

G

G

Γ(0) Γ . (2.58)




Note, that the wiggly lines denote full photon propagators. In section 2.1, the wigglylines where used to denote free photons, which are now represented by the dashed lines.Furthermore, the full vertex Γ is introduced as the solid dot. It will be discussed furtherbelow.

Equation (2.55) is the Dyson-Schwinger equation for the single-particle Green func-tion, which involves the self-energy as the driving term or inhomogeneity. The self-energydescribes the influence of many-particle correlations on the behaviour of the fermions. Ob-servable effects are the finite life time of single-particle states in the medium (containedin the imaginary part of the self-energy) and modifications of the dispersion relation, ex-pressed via the real part of Σ. Equation (2.56) is the Dyson-Schwinger equation for thefermion self-energy itself. It describes that correlations enter the self-energy via the fullphoton propagator D, see equation (2.56). The fermion, being electrically charged, polar-izes the surrounding medium (or vacuum) via emission of a virtual photon and the polarized

medium acts back on the fermion. The virtual photon, as well as real photons, obey theDyson-Schwinger equation (2.57) (rigorously, equation (2.57) contains a summation overthe polarization states of the photon, which is omitted here to keep the notation short. Seealso section 2.1.3 for a complete notation and remarks concerning the gauge of the photonfield). Here, the polarization function Π plays the same role as the fermion self-energy Σfor the fermion Green function G. Hence, each equation is directly or indirectly connectedto each of the other three equations, yet the system is not closed. The vertex functionΓ, which describes the effective fermion-photon coupling constant, obeys another integralequation, which involves the two-particle effective interaction K , i.e. a 4-point function.The vertex equation can be depicted as

Γ = Γ(0) + K Γ (2.59)

The label G at the full fermion Green functions has been omitted for convenience.The Dyson-Schwinger equations represent an exact, non-perturbative quantum-field

theoretical approach to many-particle systems. In fact, the quantum nature of both themassive “matter” particles (fermions) as well as of the gauge bosons and the externalfields are accounted for. On the other hand, many-particle correlations are consistentlyintegrated.

Albeit having been developed originally for QED, also other quantum field theories,such as the electroweak theory and quantum chromodynamics are formulated within thesame framework [RS00].

However, in order to perform practical calculations e.g. of experimentally accessibleobservables, Dyson-Schwinger equations themselves are only of limited use. One has closethe Dyson-Schwinger hierarchy. This means that in order to determine Dyson-Schwingerfunctions up to an order n by solving the corresponding Dyson-Schwinger equations, then + 1-point function either has to be parametrized by using an appropriate ansatz or




neglected altogether. This procedure is similar to the virial expansion in statistical ther-modynamics [AP92] and the introduction of closure relations in the BBGKY hierarchy of kinetic equations [ZMR96b].

2.2.2 Closure of Dyson-Schwinger equations and Φ-derivable ap-proximations

The first non-trivial closure of the Dyson-Schwinger hierarchy consists in neglecting higherorder term in the vertex function Γ, i.e. setting Γ(12, 3) = δ(1, 2)δ(2, 3). In this way,

a closed system of equations for the two-point functions G and D and their respectiveself-energies is obtained. In quantum field theory, notably in QCD, this approximation isknown as the rainbow-ladder approximation [RW94].

The rainbow-ladder approximation belongs to the class of so-called Φ-derivable ap-proximations introduced by Baym and Kadanoff [BK61, Bay62]. The theory of Baym andKadanoff states, that an approximation to the self-energy Σ is energy, momentum, andcharge conserving, if and only if there exists a functional Φ, such that the self-energy canbe written as the functional derivative Σ = δΦ[G]/δG.

The Φ-functional is obtained as the sum of all closed two-particle irreducible diagrams,i.e. those diagrams, that cannot be separated into two disconnected graphs by removingtwo fermion lines. For the rainbow-ladder approximation, the Φ functional is given by thefollowing sum

ΦRL = Φrainbow + Φladder

= + + + + + . . .

+ + + . . .

(2.60)

Differentiation with respect to a Green function G corresponds to the removal of any one




fermion line in each graph. One obtains for the self-energy

Σ = + + +

+ + . . .

+ + + . . .

= +

T

− .

(2.61)

In the last line, we made use of the Dyson-Schwinger equation (2.57) for the photon toresum all fermion loop diagrams. Furthermore, the T -matrix was introduced [KKER86],i.e. the resummation of all ladder -diagrams. Since the Born approximation (third term inthe first line) is contained in both the rainbow and the ladder term, it has to be subtracted

once, to avoid double counting of this contribution.In the context of Coulomb systems, equation (2.61) is also known to as the Gould-DeWitt approximation [GD67] for the self-energy. The first terms describes the influence of the dynamical screening, while the second term describes strong collisions, i.e. collisionsthat involve large transfer momenta. Weak scatterings are contained in the Born approxi-mation, i.e. the third term. The first term, known as the GW approximation, can be seenas a modification of the Fock term (second term in equation ( 2.61)), with the Coulombinteraction replaced by the dynamically screened interaction.

Being a Φ-derivable approximation, the rainbow-ladder approximation, as well as theGW approximation lead to expressions for the Green function, which allows energy, mo-




mentum, and charge conserving calculations of higher order correlation functions, e.g.two-particle correlation functions or collision integrals.

The drawback of Φ-derivable approximations is, that the Ward-Takahashi identities areviolated. Ward-Takahashi identities provide an exact relation between the vertex functionΓ, i.e. the effective electron-photon coupling in the medium, and the self-energy, which inrelativistic notation read

qµΓµ( p + p, q) = qµqµ + Σ( p) − Σ( p) . (2.62)

Ward-Takahashi identities follow directly from the exact Dyson-Schwinger equations. Theyreflect the gauge invariance of the theory. In the GW or rainbow-ladder approximation,they are violated simply because corrections to the vertex beyond zero order are neglectedaltogether.

As an example consider the self-energy in second order of the screened interaction,given by

Σ(2) = + . (2.63)

While the first term is contained in the GW approximation as the second iteration, thesecond term is missing. It represents a vertex correction, similar to the radiative correctiondiscussed in the context of bremsstrahlung in section 2.1.9.

The problem of unfulfilled Ward-Takahashi identities in Φ-derivable approximationstouches on a fundamental problem in many-body theory and field theory, namely the

question how to preserve gauge invariance in an effective, i.e. approximative theory,without violating basic conservation laws. A detailed analysis of this question with ap-plication to nuclear physics is presented in a series of papers by van Hees and Knoll[HK01, HK02b, HK02a]. In particular, they show how to implement vertex correctionson top of the self-consistent solution of Φ-derivable approximations, in a way that thedesirable features of the Φ-derivable approximations are retained and gauge invariance isrestored.

Approximations for the self-energy, that also contain the vertex are often referred to asGW Γ approximations. An application to solid state physics can be found in Ref. [Tak01],where the spectral function of electrons in aluminum is calculated using a parametrized

vertex function. An interesting result obtained in that work is that vertex correctionsand self-energy corrections entering the polarization function, largely cancel. This is as aconsequence of Ward-Takahashi identities. Thus, and in order to reduce the numerical cost,it is a sensible choice to neglect vertex corrections altogether, and to keep the polarizationfunction on the lowest level, i.e. the random phase approximation (RPA) which is theconvolution product of two non-interacting Green functions in frequency-momentum space.The corresponding self-energy is named the GW (0) self-energy and has been introduced byHolm and von Barth [BH96], who were also the first to study the fully self-consistent GW approximation [HB98].

For a detailed presentation of various implementations of the GW approximations and




applications in the field of solid state physics, I refer to the review articles [AG98, ORR02,Mah94, Hed99].

Before coming to the evaluation of the GW (0) approximation for the self-energy, themethod of thermodynamic Green functions will be introduced in the next chapter. Withits help, many-particle Feynman diagrams can be evaluated at finite temperatures.

2.3 Matsubara Green functions

In this section the method of thermodynamic Green functions will be outlined briefly. It isalso referred to as the Matsubara Green function technique. For a detailed presentation of this subject I refer to the textbooks by Mahan [Mah81] and Baym and Kadanoff [KB62].

The Matsubara Green function for fermions of species c in the momentum state p is

defined as Gc(p, τ , τ ) = −Tτ ac,p(τ )a†c,p(τ ) . (2.64)

τ and τ are imaginary times, being defined on the interval −iβ ≤ τ ≤ iβ . The dependenceof the creation and annihilation operators a†p(τ ) and ap(τ ) on the imaginary time is givenin the modified Heisenberg picture,

ac,p(τ ) = e−iHτ ac,peiHτ , (2.65)

with the effective Hamiltonian H = H −c,p µca†c,pac,p, µc is the chemical potential of

particles of species c, and H is the Hamiltonian of the many-particle system (see equa-tion (2.11)), i.e containing the mutual interaction among the particles and external fields,if present. The bracket . . . in Eq. (2.64) denotes the average in the grand canonicalensemble with the equilibrium statistical operator

O = Tr ρeqO = eβ ΩTr

e−β HO

, (2.66)

Ω = − 1β

ln Tr

e−β H

is the normalization of the grand-canonical statistical operator. Trdenotes the trace of an operator. Finally, the time-ordering operator Tτ sorts the operatorsto its right in descending order, i.e. the smallest time is standing at the rightmost position.

The Matsubara Green functions are very convenient, because they are directly con-nected to physical properties. For example, the single-particle Green function Gc(p, τ , τ )

or its Fourier transform, gives the spectrum of single particle excitations, i.e. the prob-ability to find a particle with a certain energy and momentum. Such probability can bemeasured in angular-resolved photoemission spectroscopy (ARPES) [Din98]. (PES).

I shall now discuss two important properties of thermodynamic Green functions. Thefirst property is, that the Green function depends only on the difference of its time-arguments, Gc(p, τ , τ ) = Gc(p, τ −τ ). This can easily be seen by setting τ = 0, τ = τ −τ

in Eq. (2.64) and using the cyclic properties of the trace. The second property is the so-called Kubo-Martin-Schwinger condition (KMS),

Gc(p, τ ) = −Gc(p, τ + iβ ), −iβ < τ < 0 , (2.67)



2.3 Matsubara Green functions 39

i.e. the Green function is periodic on the imaginary times axis with period 2iβ . From thisinformation one immediately knows the Fourier spectrum of the Green function,

Gc(p, τ ) = iβ

∞ν =−∞

eizντ Gc(p, zν ) , Gc(p, zν ) = −i

iβ

0

dτ e−izντ Gc(p, τ ) , (2.68)

with the fermionic Matsubara frequencies

zν =(2ν + 1)πi

2β , ν = 0, ±1, ±2 . . . . (2.69)

The Green function for free, i.e. non-interacting fermions can easily be calculated, using

e−iH0τ ac,peiH0τ = eiεc(p)τ ac,p , (2.70)

with the unperturbed effective Hamiltonian H0 =

c,p(2 p2/2mc−µc)a†c,pac,p. One arrivesat the expression

G(0)c (p, zν ) =

1

zν − εc(p), (2.71)

where εc(p) = 2 p2/2mc − µc is the single-particle energy of species c, relative to thechemical potential µc. Summation over all Matsubara frequencies leads to the momentumdistribution function,

1

2− 1

β

zν

1

zν − εc(p)=

1

exp(βεc(p)) + 1= nF,c( p). (2.72)

The Green function for interacting fermions is the solution of Dyson equation

Gc(p, zν ) = G(0)c (p, zν ) + G(0)

c (p, zν ) Σc(p, zν ) Gc(p, zν ) =1

zν − εc(p) − Σc(p, zν ), (2.73)

where the self-energy Σc(p, zν ) has been introduced. It contains the interaction among thefermions and the external fields.

The Green function can be expressed by a Cauchy integral over its imaginary part,

Gc(p, z) = − ∞−∞

dω

π

Im Gc(p, ω)

z − ω. (2.74)

The imaginary part of the Green function on the real frequency axis is also called thespectral function, i.e.

Ac(p, ω) = −2Im Gc(p, ω + iδ) =−2ImΣc(p, ω + iδ)

[ω − εc(p) − Re Σc(p, ω)]2 + [Im Σc(p, ω + iδ)]2.

(2.75)In a similar way, the photon Green function (propagator) in the Matsubara framework

can be derived, starting from the time-ordered product

Dµν (q, τ , τ = − T Aµ(q, τ )Aν (−q, τ ) . (2.76)




Due to the permutation symmetries of the bosonic creation- and annihilation operatorswhich appear in the field modes A(q, τ ), the KMS condition for the bosonic propagatorreads

Dµν (q, τ ) = Dµν (q, τ + β ) . (2.77)

The Fourier coefficients for D(q, τ ) are

Dµν (q, ωλ) =

β 0

dτ exp(iωλτ ) Dµν (q, ωλ) , ωλ =2πλ

β , λ = 0, ±1, ±2 · · · . (2.78)

For the free photon propagator, one obtains

D(0)µν (q, ωλ) = −4π(δµν − qµqν /q2)

ω2λ + ω2

q

. (2.79)

The full photon Green function obeys the photon Dyson equation

Dµν (q, ωλ) = D(0)µν (q, ωλ) + D(0)

µρ (q, ωλ)Πρσ(q, ωλ)Dσν (q, ωλ) . (2.80)

Remind, that throughout this chapter, the Coulomb gauge · A(q, ωλ) = 0 is applied,which separates the electromagnetic field into photons, and the longitudinal Coulomb in-teraction.

The Coulomb potential V (r) = Z 1Z 2e2/4π0r2 is the propagator of the Poisson equationfor an isolated point charge, i.e. for the non-interacting case.

− ∆V (r)/Z 2e = Z 1e δ(r) . (2.81)

This equation is easily solved in momentum space, one obtains the well-known relationV (q) = Z 1Z 2e2/0q2.

For the case of an interacting many-particle system, the Green function or potentialW (q, ωλ) obeys the Dyson equation,

W (q, ωλ) = V (q) + V (q) Π(q, ωλ) W (q, ωλ) =V (q)

1 − V (q)Π(q, ωλ), (2.82)

c.f. equation (2.57).

Comparing equation (2.82) to the definition of the internal interaction potential asknown from electrodynamics, W (q, ω) = V (q)/(q, ω), the following relation between thelongitudinal polarization function and the longitudinal component of the dielectric tensorcan be established:

(q, ω) = 1 − V (q)Π(q, ω) . (2.83)

Finally, the spectral representation for the effective interaction potential reads

W (q, z) = V (q)

1 +

∞−∞

dω

π

Im −1 (q, ω + iδ)

z − ω

. (2.84)



2.4 Application: The self-energy in GW (0) approximation 41

2.4 Application: The self-energy in GW (0) approxima-

tion

The GW -approximation for the self-energy is given by the diagram (c.f. section 2.2)

Σ(p, zν ) =G

W

Γ(0) Γ(0) = −T ωµ

q

G(p − q, zν − ωµ) W (q, ωµ) .(2.85)

The index c will be ignored in the following and electronic quantities will be assumed.Equation (2.85) is evaluated as follows: First, the Green function G and the screenedinteraction W are replaced by their spectral representations (2.74) and (2.84). One arrivesat

Σ(p, zν ) = −T q,ωµ

V (q)

∞−∞

dω

2π

A(p − q, ω)

zν − ωµ − ω

1 +

∞−∞

dω

π

Im −1RPA(q, ω)

ωµ − ω

. (2.86)

Note that the dielectric function is taken in the random phase approximation (RPA)[AB84]; this defines the GW (0) approximation. In GW , also the screened interactionhas to be calculated as a functional of the Green functions, c.f. equation (2.57). Aftersummation of the bosonic Matsubara frequencies,

Σ(p, zν ) =q

V (q) ∞−∞

dω

2πA(p − q, ω)

×

1 − nF(ω) +

∞−∞

dω

π

Im −1RPA(q, ω) [nB(ω) + 1 − nF(ω)]

zν − ω − ω

,

is obtained. This expression contains the Hartree-Fock self-energy of the interacting sys-tem,

ΣHFint (p) = −

q

∞−∞

dω

2πA(p − q, ω)nF(ω)V (q) , (2.87)

and the correlated self-energy

Σcorr(p, zν ) =q

V (q)

∞−∞

dω

2πA(p − q, ω)

× ∞−∞

dω

π

Im −1RPA(q, ω) [nB(ω) + 1 − nF(ω)]

zν − ω − ω. (2.88)

For convenience, the upper index “corr” is skipped in the following and it is only distin-guished between the frequency dependent self-energy Σ(p, ω + iδ) and the Hartree-Fockterm ΣHF

int (p), in the following.




After analytic continuation zν → z = ω +iδ, δ → 0, the imaginary part of the correlatedself-energy is evaluated using Dirac’s identity limδ→0 1/(x ± iδ) = P 1/x iπ δ(x),

ImΣ(p, ω + i0+) = 1nF(ω)

q

∞

−∞

dω

2πV ee(q)A(p − q, ω − ω)

×Im −1RPA(q, ω) nB(ω) nF(ω − ω) , (2.89)

where the exact relation nB(ω) + 1 − nF(ω − ω) = −nB(ω) nF(ω − ω)/nF(ω) was used.This equation represents a non-linear integral equation for the imaginary part of the self-energy ImΣ(p, ω + i0+), since the latter also enters the spectral function via the Dysonequation 2.75.

The real part of the self-energy can be obtained from the Kramers-Kronig relation

ReΣ(p, ω) = ΣHFint (p) +

∞−∞

dω

πImΣ(p, ω + iδ)

ω − ω. (2.90)

2.5 Optical properties of dense plasmas

The knowledge of the polarization function or, equivalently, the dielectric function allowsfor the determination of various optical properties of the system under consideration. Thetransverse polarization function is related to the index of refraction n(q, ω) and the ab-sorption coefficient α(q, ω) via

⊥(q, ω) =

n(q, ω) + ic

2ω2

. (2.91)

The absorption coefficient is defined as the relative attenuation per unit length of theintensity of an electromagnetic wave.

The above relations can be solved for the index of refraction and the absorption coeffi-cient, themselves,

n(q, ω) =1√

2[Re ⊥(q, ω) + |⊥(q, ω)|]1/2 (2.92)

α(q, ω) =ω

c n(q, ω)Im ⊥(q, ω) . (2.93)

Since in the limit q → 0, the transverse dielectric function coincides with the lon-gitudinal one, we may neglect the distinction between both cases and write (q, ω) =⊥(q, ω) ≡ (q, ω), instead, as long as the wavelengths involved are large compared toatomic dimensions, i.e. λ aB = 0.53 × 10−10 m.

As will be explained further below in section (3.6), the knowledge of the absorptioncoefficient also allows for the determination of the emission spectrum from the plasma.Furthermore, the reflectivity

r(q, ω) =1 − n(q, ω)

1 + n(q, ω), (2.94)



2.5 Optical properties of dense plasmas 43

is known via the dielectric function. Thus, in order to determine the optical properties of a plasma, a theory of the dielectric function is needed.

A quantum statistical approach to the dielectric function, which allows for the system-atic inclusion of many-particle correlations is described in detail in [Rei05]. Central to thisformulation is the so-called generalized Drude ansatz, which relates the dielectric functionin the limit of long wavelengths to the so-called complex collision frequency ν (ω),

(q → 0, ω) = 1 − ω2pl

ω(ω + iν (ω)). (2.95)

The collision frequency, on the other hand, can be expressed via a force-force correlationfunction,

ν (ω) =β Ω0

0ω2

pl

limk→0 J

z

k, J z

kω+iη , (2.96)

with J zk the longitudinal component of the force operator

J k =1

Ω0

c,p

ecmc

p a†c,p−k/2ac,p+k/2 , (2.97)

ρ0 is the equilibrium statistical operator.The evaluation of the force-force correlation function is performed by making use of its

representation in terms of a force-force Green function GJ J [ZMR96a]

J z0 , J z0

ω+iη =

i

β

∞

−∞

dω

π

1

ω + iη − ω

1

ω

Im GJ J (ω + iη) . (2.98)

By exploiting Dirac’s identity

limη→0

1

x ± iη= P 1

x iπδ(x) , (2.99)

the real part of the force-force correlation function is evaluated to

Re J z0 , J z0 ω+iη =1

βωIm GJ J (ω + iη) . (2.100)

The time derivative of the electron current density operator is calculated as

J z0,e =i

H, J z0,e

=

ie

meΩ0

pkq

V (q)eiqza†e,p a†i,k ai,p−q ae,p+q (2.101)

using the non-relativistic many-particle Hamiltonian (2.31).With Eq.(2.101), we identify the Green function as a four particle Green function. Its

diagrammatic representation is shown in figure 2.7.Further evaluation requires the replacement of the four-particle Green function G4, that

in principle contains all correlations, by a suitable approximation. Evaluation of the force-force correlations using various approximations is demonstrated e.g. in Refs [RRRW00,




G4

Figure 2.7: Diagrammatic representation of the current-current Green function GJ J (ω) asa four-particle Green function.

WMR+01]. In [FRRW05], it is investigated, how the use of full Green functions instead of

their non-interacting counterparts modifies the result for the force-force correlation functionand hence for the absorption coefficient, compared the Born approximation, where all Greenfunctions are non-interacting Green functions, and no interaction “inside” the correlationfunction is taken into account.

From the force-force correlation function, the absorption coefficient for inverse bremsstrahlungcan be calculated via equations (2.96), (2.95), and (2.93). The results can be cast into theform

α(ω) =16π2Z 2neni

3mecω3

e2

4π0

32π

3mekBT

1/2[1 − exp(−ω/kBT )] gff (ω) , (2.102)

with the averaged free-free Gaunt factor gff (ω). The averaged Gaunt factor contains themedium and quantum corrections. A central point in this thesis is the calculation of theaveraged Gaunt factor. To this end, the in-medium electron propagator and the in-mediumvertex is calculated and plugged in the force-force correlation function to determine theabsorption coefficient and by comparison to equation (2.102) the Gaunt factor.

The Born approximation, discussed above, results in a simple expression for the Gauntfactor, namely

gBff (ω) =

√3

πexp(ω/2kBT ) K 0(ω/2kBT ) . (2.103)

K 0(x) is the modified Bessel function of order 0 [AS70]. In most cases, instead of comparing

to the Kramers approximation gff = 1, the results for the absorption coefficient will becompared to the Born approximation αB(ω) = α(ω)gBff (ω). In this way, the influencesbeyond the Born approximation, i.e. beyond the perturbative treatment of the Gauntfactor are studied.



Chapter 3

Plasmas and radiation

3.1 Basic processes and plasma diagnostics

Plasma spectroscopy is a versatile too to infer macroscopic properties from the plasma,like the plasma temperature and the density of its various components (electrons, ions,neutrals). Also non-equilibrium properties, like gradients in temperature and density,instabilities, and non-linear effects can be detected and characterized by spectroscopicanalysis. In particular, absorption and emission spectroscopy should be mentioned as im-portant methods of plasma diagnostics [Gri64], besides other techniques e.g. refractometryand magnetic diagnostics [Hut87]. From the details of the spectrum, detailed information

about the plasma’s state can be extracted. The intensity of spectral lines, resulting fromradiative transitions in ions, atoms, or molecules gives access to the plasma temperatureand the degree of ionization. The form (width and position) of the lines also allows for thedetermination of the plasma density [Hut87]. The continuum part of the spectrum, due tofree-bound and free-free transitions, is very sensible to the plasma temperature, density,and composition, as well. Fig. 3.1 (solid line) shows an exemplary emission spectrum froma laser produced plasma at an electron temperature of kBT e = 170 eV compared to theblackbody continuum at the same temperature (dashed line). The figure is taken fromRef. [Sig91]. Clearly, the three typical features of an emission spectrum can be identi-fied, namely the spectral lines, the bound-free continuum, limited by sharp edges at theseries limits, and the free-free continuum, which is the only contribution at long wave-length, i.e. small photon energies. The blackbody and the bremsstrahlung continuummerge at long wavelengths (λ > 300 nm). This behaviour is due to the re-absorption of thebremsstrahlung (inverse bremsstrahlung) and subsequent thermalisation of the radiation,which therefore becomes blackbody radiation. This point will be discussed in more detailin section 3.6.

The characteristics of the emission or absorption spectrum are sensitive to the density,temperature, and composition of the plasma. In order to interpret the experimental dataand to determine these plasma parameters, an accurate theory of the optical properties of a realistic plasma is required. The complex dynamics of the interaction of electromagnetic



46 Plasmas and radiation

Figure 3.1: Exemplary emission spectrum from a laser produced plasma compared to theblack-body spectrum at the same temperature. Taken from [Sig91]

radiation with a highly correlated system, such as a partially or fully ionized plasma, hasto be investigated. In this spirit, the present work is devoted to the theory of opticalproperties of dense plasmas. The focus will lie especially on the question how the free-freecontinuum (bremsstrahlung) behaves in a dense plasma.

Other authors, using an approach similar to the one presented in this work (c.f. sec-tion 2.3), have concentrated on the theory of line spectra, such as G unter [Gun95], Sorge

et al.[SWR+

00], Koennies et al. [KG94] for hydrogen spectra, Omar et al. for helium likespectra [OGWR06], and Lorentzen [Lor08] for hydrogen like lithium spectra.

3.2 Line emission

The transition of a bound electron from an excited state |n to a final state |m by emissionof a photon of energy ω = E n−E m results in a sharp spectral line in the detection device.For the purpose of plasma diagnostics, the determination of the spectral line’s width andthe shift from its position as measured in a dilute system (gas) or known from first principle



3.2 Line emission 47

calculations, is of interest. To this end, the resolution of the spectrometer needs to be fineenough (typically below 1 nm).

To understand the underlying principle of plasma diagnostics using spectral lines, onehas to look at the several mechanisms which are responsible for the finite spectral line width.First of all, the line has a natural line width, since the life time of an excited atomic stateeven in vacuum is finite. This is due to the coupling of the atom to the vacuum fluctuationsof the electromagnetic field as discussed in section 2.1.1. The natural line width ∆ν nat isconnected to τ via via the Heisenberg uncertainty relation, i.e. ∆ν nat = 1/τ . Typicalvalues for the natural line width for dipole transitions range from 107 Hz to 109 Hz, whichcorresponds to life times in the order of 1 ns to 100 ns. E.g. for the hydrogen Balmer-αline (3 p − 2s), one finds for the natural line width ∆ν = 6.5 × 107 Hz [WSG66]. Dipoleforbidden transitions, e.g. the 2s − 1s transition in atomic hydrogen, have a much smallernatural line width of about 1.3 Hz.1

In a hydrogen gas or a plasma, the thermal movement of atoms or molecules leads tothe so-called Doppler broadening. If the radiator moves away from the detector in themoment of the radiative decay, the wavelength of the emitted photon is shifted to largerwavelengths (redshift), while in the opposite case, when the radiator moves towards thedetector, the Doppler effect leads to a blueshift. The distribution of wavelength shiftsaround the unperturbed position reflects the velocity distribution function of the radiatorsin the plasma. Since line spectroscopy is usually carried out on dilute, hot plasmas, themomentum distribution function can well be approximated by the Maxwell-Boltzmanndistribution. Its width (FWHM) is given by the temperature of the system,

∆ν = ν 0

2 ln(2) kBT mAtc2

, (3.1)

with the atom’s mass mAt and the unperturbed line position ν 0.Therefore, Doppler spectroscopy allows for the determination of the plasma tempera-

ture. The Doppler width of the hydrogen Balmer-α line at a plasma temperature of 1 eV is∆ν 2× 1010 Hz, e.g. about two orders of magnitude larger than the natural line width of the same transition. This means, that the natural line width can be neglected in most caseswhere temperature measurements using line spectroscopy of hot plasmas are performed.

Finally, in a dense plasma, the spectral line width and position is strongly influenced bythe interaction of the radiator with surrounding particles. The width of the spectral line due

to the interaction with third particles is called pressure broadening. The electromagneticfield exerted on the radiator by the surrounding particles leads to a shift (Stark shift) inthe case of an electric field and possibly Zeeman splitting (in the case of strong currentsand external magnetic fields) of the levels involved in the particular transition. Since themagnitude of the shift of the upper and lower level involved in the particular transitiondepend on the respective quantum numbers, these processes result in a shift in wavelengthof the emitted photon. The field strength is distributed according to the spatial distribution

1This feature makes the 2s− 1s transition suitable for high precision measurements. An example is theinvestigation of the variation of fundamental physical constants with time [H06].




of the perturbing particles, which is reflected in a broadening and a shift of the spectralline. As opposed to the Doppler broadening, pressure broadening heavily depends on thedensity of the medium. Therefore, the measurement of the spectral line width and shiftalso allows for the determination of the plasma density. As an example, the hydrogenBalmer-α line width due to pressure broadening is reported in [Hut87] as ∆ν 3×1010 Hzat a plasma temperature of kBT = 1 eV and ion density of ni = 1021 m−3. This value isof the same order of magnitude as the Doppler broadening of the same line at the sametemperature. Thus, to accurately determine the temperature and the density in denseplasmas from the spectral line broadening, both effects have to be taken into account. Thepressure shift (Stark shift) is of the same order of magnitude as the Stark broadening.

For further details on spectral line broadening, I refer to the extensive literature onthis subject, e.g. the monograph by Griem [Gri64], the collection [LH95], and the originalarticles [OGRW06, OGWR06], plus further references therein.

Another method to determine the plasma temperature from the line spectrum uses astatistical argument. Assuming that the population of the levels involved in a particu-lar radiative transition follows a Maxwell-Boltzmann distribution, the temperature of theplasma can be obtained by comparison of the integrated line intensities of two distinctspectral lines [LH95]. This method is particularly useful, if the spectrometer does notallow for a detailed analysis of the spectral line shape, i.e. if the spectrometer’s spectralresolution is not sufficient to determine the Doppler broadening and/or the pressure broad-ening. The statistical distribution of the levels involved in the radiative transition is alsoused in the temperature measurement via spectroscopy of continuum radiation, as showne.g. in Ref. [ZFF+08], see also section 4.2.

3.3 Recombination radiation

When the plasma is partially ionized, two additional processes contribute to to the emissionspectrum of the plasma. Besides transitions between continuum states (bremsstrahlung),free-bound transitions occur. The case of a transition from a continuum state |k at en-ergy E k to a bound state |n at a discrete energy E n and the accompanying emission of aphoton of energy ω = E k − E n is called recombination radiation, whereas the opposite

process is referred to as photoionization. The corresponding emission (absorption) spec-trum has a sharp edge at the line series limit as the most remarkable feature. Besides theseedges, which allow for the identification of ion species and ionization stages, the intensityof bound-free transitions in a plasma in thermal equilibrium behaves roughly proportionalto exp(−ω/kBT ), as follows from averaging the individual transition probability with theMaxwell-Boltzmann distribution function of the initial states. This allows for the determi-nation of the temperature from the slope of the frequency (energy) spectrum. Furthermore,by measuring the height of the absorption edge, the penetration depth of the radiation intothe medium can be determined, provided, the absorption coefficient at both sides of theedge is known.



3.4 Bremsstrahlung radiation 49

3.4 Bremsstrahlung radiation

Bremsstrahlung is the radiative transition between two continuum states. In section 2.1.9,various expressions for the bremsstrahlung cross-section have been reviewed, i.e. the Bethe-Heitler formula (2.42) (full relativistic Born approximation), the Sommerfeld expression(2.51) (quantum-mechanical, non-relativistic), and the Kramers formula (2.48) (classical,non-relativistic). By averaging the cross-section, which depends on the initial velocity(momentum) of the emitting particle, with the momentum distribution function, an ex-pression for the bremsstrahlung emission from a plasma can be derived. Thus, the accuratemeasurement of the bremsstrahlung spectrum emitted from a plasma allows for the deter-mination of many plasma parameters [Hut87]. In general, bremsstrahlung spectra can beused to reconstruct the momentum distribution of electrons in the plasma [BEH+06] bydeconvolution of the bremsstrahlung scattering cross-section from the measured spectrum.

In the case of a classical, completely ionized plasma in thermal equilibrium, where theelectron distribution function is a Maxwell-Boltzmann distribution, i.e. completely char-acterized by the electron temperature, the bremsstrahlung photon energy spectrum has anexponential decay behaviour at large photon energies, I bs(ω) ∝ exp(−ω/kBT ), c.f. sec-tion 2.5. In this case, it is sufficient to determine the slope of the bremsstrahlung continuumto obtain the electron temperature [Ric95]. For partially ionized plasmas, the situation ismore complicated, since especially in the high frequency part of the spectrum, bound-freetransitions become important. However, the latter have a similar frequency dependenceand can be used for temperature measurements in the same way. An example of applicationcan be found in Ref. [SWR+00]. In non-equilibrium plasmas, where we have additional

electron modes (instabilities), the bremsstrahlung spectrum is no longer exponential butrather displays a power law behaviour [HSSE03, BEH+06]. The bremsstrahlung emissioncarries a prefactor Zn2

e/√

T e. Thus, if the emission spectrum is absolutely calibrated, alsothe electron density can be determined.

Bremsstrahlung spectroscopy is widely used in astronomy and astrophysics to determineimportant parameters of the surface of stars, e.g. it’s temperature, density, and electrondistributions. An example can be found in [KEM+07].

3.5 Scattering

Besides emission and absorption, the scattering of photons is the third class of processesbelonging to the response of the medium to external fields. In general, two kinds of scat-tering have to be distinguished, namely elastic and inelastic scattering. Elastic scattering,also called Rayleigh scattering, denotes those processes, where the energy of the photon isconserved, only the momentum changes, i.e. the photon is scattered into some angle withrespect to its initial trajectory. In contrast, inelastic scattering involves also a change inthe photons energy, i.e. energy is transferred from the scattering particles to the scatteredparticles or vice versa. Several phrases are used in connection with scattering, denotingvarious channels of energy and momentum transfer. For a detailed discussion, see e.g. the




monograph by Sheffield [She75].Thomson scattering refers to the elastic scattering of photons on free or weakly bound

electrons. For long wavelengths, i.e. in the optical and the infrared regime, the classicalpicture of an oscillatory movement of the electron in the external field applies, which leadsto dipole radiation from the electron [LGE+01]. Due to the thermal movement of theelectrons, there appear red- and blue shifted photons in a detector that is set up at aparticular position. Thus, the scattering spectrum is symmetrically broadened and thewidth gives the plasma temperature, assuming, the velocity distribution is given by theMaxwell-Boltzmann distribution. In cold systems, e.g. metals, when the temperature iscomparable or below the Fermi energy, the width of the scattering spectrum is given bythe Fermi energy, which allows for the determination of the electron density [ LNC+ew].Note that scattering of electromagnetic waves in cold targets requires the use of x-rays,since optical probes cannot penetrate the target when the probe frequency is below the

plasma frequency of the target.Scattering of photons involving atomic or molecular transitions is called Raman scat-

tering. During the interaction, a bound electron is transferred from its initial state to someintermediate, unoccupied state and from there to a final state. Depending on weather thefinal state is energetically higher or lower than the initial state, the wavelength of the scat-tered photon is shifted towards longer wavelengths (“Stokes line”) or shorter wavelengths(“Anti-Stokes line”), respectively. The term Raman scattering is also used for the scat-tering of infrared photons involving transitions between vibrational or rotational states of molecules.

Finally, Brillouin scattering denotes the inelastic scattering of photons on collective

excitations of the medium. Typically, the term stands for those processes that involve theabsorption or creation of phonons by light scattering. But also scattering on other excita-tions, such as plasmons or magnons (spinwaves), can be referred to as Brillouin scattering.However, the scattering on plasmons as the longitudinal excitations in a plasma is mostlycalled “collective Thomson scattering”, since it involves scattering on free electrons whichperform collective motion.

Optical Thomson scattering has been applied since many years in diagnostics of dilutegases and plasmas, such as Tokamak plasmas [Hug75] or arc discharges [SLR93]. Alsoelectron-hole plasmas in semi-conductors have been investigated using Thomson scattering[UW77]. Nowadays, researchers are looking for reliable and robust techniques to diagnose

also dense systems. To a large extent this is due to the increasing research activities ininertial confinement fusion research [Lin95] and laboratory astrophysics [TLS+06]. In thesefields, the structure and the dynamics of plasmas which are at the same time hot (withtemperature between several eV and several hundred eV) and dense, i.e. whose densityranges in the vicinity of solids, near 1023 cm−3, are investigated. At these parameters,the plasma is no longer penetrable by optical probes, since the plasma frequency is largerthan the probe frequency, i.e. the radiation is reflected and penetrates the plasma only inthe skin-layer of several nm in depth. However, latest achievements in the developmentof narrow bandwidth, high intensity UV and x-ray sources, such as backlighter sources[GGL+03, SGK+07] and free electron lasers [AAA+07], made plasma diagnostics using



3.6 Radiation transport 51

Thomson scattering also applicable in this so-called warm dense matter regime [GLN+07,GND+08].

3.6 Radiation transport

Dealing with radiation from and inside a plasma of finite spatial extensions, emission,absorption as well as scattering have to be treated on equal footing. External fields getabsorbed and scattered in the medium, at the same time radiation is produced by radiativetransitions inside the matter and possibly reabsorbed. The question of radiation transportbecomes important.

Radiation transport accounting for both absorption and emission and scattering is acomplicated issue, that cannot be covered in these introductory remarks. Therefore, I focus

on radiation transport that includes only absorption and emission. Since all processes of emission and absorption are assumed to be isotropic (we neglect e.g. the case of magnetizedplasmas), the problem of radiation transport can be formulated in one dimension. Note,that when including scattering, this simplification is no longer valid, since the momentumtransfer during the scattering involves a change of the photon’s direction while propagatingthrough the medium.

The central quantity is the absorption coefficient α, defined as the relative attenuationper unit length of the intensity of electromagnetic waves propagating through a medium.In general, the absorption coefficient depends on both time and place. Also, it dependsheavily on the frequency of the electromagnetic wave, i.e. its Fourier spectrum. The basic

equation that governs the propagation of radiation through a medium is the macroscopicradiative transfer equation,

dI (ω; s)

ds= −α(ω; s)I (ω; s) + j(ω; s) , (3.2)

which is a balance equation for the loss and the gain of the radiation intensity I (ω; s),i.e. the radiation power emitted into the solid angle dΩ and frequency interval dω, mea-sured at a detector of cross-section dA at the distance s from the incident surface of themedium, I (ω; s) = dP rad(ω; s)/dAdΩ dω. Here, the discussion is restricted to the one-dimensional transfer equation, for the much more complicated three-dimensional case, see

the monograph by Chandrasekhar [Cha60]. The loss of intensity is described by the firstterm in equation (3.2), the second term describes the gain, i.e. the emitted power per unitfrequency interval, unit solid angle and unit volume at s.

It is convenient to define the dimensionless optical depth τ as the integral of the ab-sorption coefficient along the path taken by the radiation through the medium of length las

τ (ω; l) =

l0

ds α(ω; s) . (3.3)

j(ω; s) is the emittance, i.e. the radiated power per unit volume, l is the length of the




medium. Then, the radiative transfer equation is rewritten as

dI (ω; τ (ω))

dτ (ω) = −I (ω; τ (ω)) + S (ω, τ (ω)) , (3.4)

with the source function S (ω), defined as the ratio of the emissivity and the absorptioncoefficient, S (ω) ≡ j(ω; s)/α(ω; s).

Eq. (3.4) has the formal solution

I (ω; τ (ω)) = I (ω, 0)e−τ (ω) +

τ (ω)0

e−(τ (ω)−τ (ω))S (ω, τ (ω)) dτ (ω) . (3.5)

I (ω, 0) is the intensity of the wave at the incident surface. In the case of a homogeneousmedium, the optical depth is given by the product τ (ω) = l

·α(ω) and the solution of the

radiation transport equation becomes

I (ω; l) = I (ω, 0)e−α(ω)l + S (ω)(1 − e−α(ω)l) . (3.6)

In general, the source function can have arbitrary complexity. In the case of thermalradiation, i.e. radiation in thermodynamic equilibrium, the source function is the Planckblackbody spectrum, given by

S T (ω) =ω3/4π3c2

eω/kBT − 1, (3.7)

or, as a function of the wavelength,

S T (λ) =

dω

λ

S T (ω) =4πc2/λ5

e2πc/λkBT − 1. (3.8)

Two limiting cases of Eq. (3.6) should be discussed more closely. In the limit τ (ω) 1,we have a transparent medium, only a small amount of radiation is absorbed. In thiscase, Eq. (3.6) turns into Kirchhoff’s law [RL75], i.e. the emitted intensity is I (ω; l) −I (ω; 0) = α(ω)S T (ω) · l ≡ jT (ω) · l. When we have a strongly absorbing medium, externalfields and self-emission are repeatedly absorbed and re-emitted, such that thermodynamicequilibrium is established between the matter and the radiation. The resulting self-emittedintensity I (ω; l) − I (ω; 0) = S T (ω) is just Planck’s law of a black-body radiator.

We have now defined the quantities that describe the propagation of electromagneticwaves through a given medium, i.e. the absorption coefficient and the source function,and we have outlined the relations between them. Next, we need to establish a link tothe microscopic level, i.e. to the physical behaviour of the elementary constituents of themedium. Since we will consider the electromagnetic spectrum only for photon energiessmaller than γ -ray energies (ω < 1 MeV), we are allowed to treat electrons and ions as“elementary particles” and we do not have to consider hadronic or partonic degrees of freedom.



54 Selected problems

In order to resolve the plasmon resonances and their separation from elastic Rayleighscattering, small bandwidth sources are required. However, the applicability of opticallasers is limited to rather low density plasmas, due to the critical density of about 1020 cm−3

for optical wavelengths.For solid density plasmas, narrow bandwidth x-ray sources are the method of choice.

Pioneering work in this direction was performed by Glenzer et al. showing the possibilityof Thomson scattering in both the non-collective scattering regime (α < 1) [GGL+03] aswell as in the collective scattering regime [GLN+07]. Recently, the same technique wasalso demonstrated for shock compressed matter, see Refs. [GND+08, LNC+ew].

The regime of near solid density (ne = 1021 cm−3 − 1023 cm−3) at temperatures of several eV is of particular interest for Thomson scattering. In this so-called warm densematter (WDM) regime, a complex interplay between strong coupling effects (collisions) andquantum effects can be observed, since both the plasma coupling parameter Γ as well as the

degeneracy parameter θ are close to 1. The theory of Thomson scattering has to accountfor these effects to allow for precise prediction of the scattering spectrum. A many-bodytheoretical approach is presented in Ref. [TRRR06].

A collective Thomson scattering experiment at the free electron laser facility FLASHat DESY-Hamburg, operating at XUV wavelengths (currently 6-100 nm), was proposedin Ref. [HBC+07]. An important issue related to the feasibility of this experiment is thequestion of the signal to noise ratio. At the sought plasma conditions, bremsstrahlungrepresents a considerable source of background photons, which has to be overcome by thescattering signal.

A detailed comparison of the bremsstrahlung level and the intensity of scattered FEL

photons over a broad range of plasma parameters was performed in Ref. [FRR+

06]. Dif-ferent from previous studies of the same subject, various Gaunt factors were compared.Besides the standard expression for bremsstrahlung, i.e. Kramers’ formula, the Born ap-proximation and the Sommerfeld expression were applied. In particular the latter approx-imation allowed to study the influence of strong collisions on the bremsstrahlung emis-sion. The Thomson scattering spectrum was calculated in random phase approximation[HRR+04].

Two key questions were of primary interest: What is the minimum intensity of theFEL required to overcome the bremsstrahlung level at various densities and temperatures?What is the required detector resolution to achieve this? The detector resolution is taken

into account by performing convolution of the simulated scattering spectra [HBC

+

07] witha Gaussian of the corresponding width.

4.1.2 Results

We consider Thomson scattering of photons with a wavelength of λ0 = 32nm on Z = 2fold ionized aluminum under an angle of θ = 120. Figure 4.1 (black dashed curve) showsthe raw scattering spectrum, the solid curve is obtained by convolution of the raw spec-trum with a Gaussian function of relative width ∆λ/λ = 10−2 to model the detectorresolution. The laser intensity is I L = 5 × 1010 W/cm2. The electron temperature is



4.1 Bremsstrahlung vs. Thomson Scattering 55

30 31 32 33 34

λ [nm]

0

1×105

2×105

3×105

4×105

d P / d V d λ d Ω [ W

/ c m

3 n

m s r ]

Bremsstr. (Kramers)

Bremsstr. (Born)

Bremsstr. (Sommerfeld)

Thomson ( I L=5x1010 W/cm2)

Thomson, not convoluted

Figure 4.1: (a) Thomson scattering spectrum for laser wavelength λ0 = 32 nm at scatteringangle θ = 120 for aluminum at ionization degree Z = 2. The laser intensity is I L =5 × 1010 W/cm2. The thin dashed curve gives the raw scattering spectrum, the solidcurve accounts for the detector resolution (∆λ/λ = 10−2) via Gaussian convolution. Thegrey curves give the bremsstrahlung emission level assuming different expressions for the

Gaunt factor, i.e. Kramers’ approximation (dashed), Born approximation (dotted), andSommerfeld approximation (solid). The plasma parameters are ne = 1020 cm−3 and kBT e =10 eV. The scattering parameter is α = 1.25.

fixed at T e = 10eV/kB, the electron density is ne = 1020 cm−3. At these conditions, thescattering parameter is α = 1.25, i.e. we are in the collective scattering regime. Corre-spondingly, plasmon peaks appear in the spectrum, separated from the central Rayleighpeak by roughly 0.5 nm. The grey curves mark calculations of the bremsstrahlung emis-sion for the same conditions, applying different expressions for the averaged Gaunt factor,c.f. equation (2.102), i.e. Kramers’ approximation (gff = 1, dashed curve), Born ap-proximation (equation (2.103), dotted curve), and the temperature averaged Sommerfeldexpression (equation (2.51), solid curve). All bremsstrahlung curves are below the peaksof the Thomson scattering signal, i.e. at the present condition, the Thomson scatteringexperiment is feasible. Comparing the different bremsstrahlung curves, the Sommerfeldapproximation gives the most conservative estimation, i.e. the highest bremsstrahlungintensity. Born approximation, gives a much lower bremsstrahlung level. The Kramersresult, i.e. neglecting quantum and medium effects is close to the Sommerfeld result at thepresent conditions.

In the same manner, the maximum Thomson scattering intensity is compared to the




10 100 1000

k BT e [eV]

1020

1021

1022

1023

1024

n e

[ c m - 3 ]

1014 W/cm2

1013 W/cm2

1011 W/cm2

1012 W/cm2

1010 W/cm2

BornKramersSommerfeld

109

W/cm2

Figure 4.2: Threshold intensity as a function of electron temperature T e and free electrondensity ne. For each curve, (ne − T e)-points above and to the right are not accessible byThomson scattering since the bremsstrahlung level is above the Thomson signal. Belowand left of each curve, the Thomson signal is stronger than the bremsstrahlung background.Various expressions for the Gaunt factor have been used, Kramers (dashed), Born (dot-

ted) and Sommerfeld (solid). The detector resolution is fixed at ∆λ/λ = 10−4

, the FELwavelength is 32 nm and the scattering angle is 120. The target material is Al, Z=2.

bremsstrahlung intensity over a broad range of electron density and temperature. In thisway, for each point in the (ne − T e) plane, a threshold intensity can be determined, fromwhich on the Thomson scattering intensity overcomes the bremsstrahlung background.The result is shown in figure 4.2 for the same scattering parameters as above. This time,the detector resolution is set to 10−4. The curves are isolines of the threshold intensity asa function of electron density and temperature. Having a certain laser intensity available,

all points in the (ne − T e) diagram below and to the left of the corresponding isolineallow for Thomson scattering, whereas bremsstrahlung is dominant for points above andto the right of the isoline. Here, higher laser intensities are needed. As before, differentGaunt factors are compared. While the Born approximation (dotted curves) shows arather large deviation from the Kramers result (dashed), especially at increased densities,the Sommerfeld result (solid) is closer again to Kramers. The Sommerfeld expressionfor the Gaunt factor is the most accurate expression which contains the exact scatteringwavefunctions. Hence, it should be used to estimate the bremsstrahlung intensity. Itgives slightly higher thresholds as compared to the Kramers result, which was used inRef. [BDFR02].



4.2 Bremsstrahlung and Line Spectroscopy of Warm Dense Al Plasma 57

Summarizing, we determined threshold intensities for a Thomson scattering experimentusing free electron laser radiation at warm dense matter conditions. It was shown thatmoderate intensities of 1012 W/cm2 are sufficient to perform Thomson scattering in theWDM regime. Furthermore, a comparatively low detector resolution of 10−2 is sufficient,10−4 would be largely enough.

4.2 Bremsstrahlung and Line Spectroscopy of Warm

Dense Al Plasma

4.2.1 Introduction

The production of a hot plasma (i.e. at temperatures of several eV to several hundreds

of eV) at solid density is a challenging task to researchers. The main interest in sucha plasma stems from its relevance for inertial confinement fusion research [ Lin95] andlaboratory astrophysics [TLS+06]. Among the various strategies, shock wave experiments[DSCC+97], heavy ion beams [HBN+05, HFL+02], and high power lasers are the mostsuccessful.

A promising further option to produce high energy density plasmas are x-ray lightsources of the fourth generation, free electron lasers [A+06]. Due to the shorter wavelengthas compared to optical lasers, the radiation can penetrate deep into the target and canproduce free electrons via direct photoionization [FZG04]. A homogeneous, volumetricplasma can be produced. In contrast, optical laser radiation is usually absorbed in a thin

skin-layer of several nm producing a plasma with high density and temperature gradients.

4.2.2 Experimental setup and results

A pioneering proof-of-principle experiment [ZFF+08] has recently been conducted at thefree electron laser at DESY-Hamburg (FLASH) [A+06, AAA+07]. FEL radiation of 13.5 nmwavelength was focused on a sample of aluminum under 45 degree incident angle. The FELwas run in multibunch mode at 5 Hz repetition rate and with 20 pulses in each bunch. Thepulse length of each individual pulse was 15 fs and the pulse energy was 33 µJ. The focalspot size was 30 µm in diameter which gives an intensity of 2 × 1014 W/cm2. The plasma

emission was measured with a high throughput transmission grating XUV spectrometer[JTT+94] under 90 angle with respect to the FEL. The setup is shown in figure 2 inRef. [ZFF+08].

The resulting spectra are shown in figure 4.3. The summed photon yield of five separatemeasurements (total integration time was 13.5 minutes) is shown as a function of the photonwavelength. Various features are discernible: The main peak at 13.5 nm is the elasticallyscattered FEL light. In its right wing, a spectral line at λ = 12.7 nm can be identifiedoriginating from Al4+ (Al V). Further spectral lines mainly from Al IV ions are locatedat 11.6nm and at 16.2 nm. These lines have been identified with the help of NIST datatables [KM91]. At 17.0 nm, the usual L-shell absorption edge shows up, accompanied by




6 8 10 12 14 16 18wavelength λ [nm]

103

104

105

106

10

7

e m i t t e d p h o t o n s · ∆ Ω - 1

· ∆ λ

- 1

experiment

k BT e = 46.0 eVk

BT

e= 40.5 eV

k BT

e= 35.0 eV

11.6 nm (Al IV)

12.7 nm (Al V)

13.5 nm

16.2 nm (Al IV)

17.2 nm (LII/III

M)

17.0 nm (L-edge)

Rayleigh scattering

Figure 4.3: Plasma emission spectrum featuring spectral lines from Al IV, Al V, theL-shell absorption edge and Rayleigh scattering. The solid curves show bremsstrahlungcalculations at three different temperatures. Best fit is obtained for kBT e = 40.5eV.

an LII/III transition line at 17.2 nm. These features sit on a background that is mainlycaused by bremsstrahlung. At short wavelengths (λ < 10 nm), bound-free transitions showup as step-like structures.

4.2.3 Data analysis

From the spectroscopic data, valuable information about the plasma parameters can beinferred. Notably the plasma temperature is of interest, since it gives information aboutthe effectiveness of plasma heating using FEL radiation. The plasma temperature can be

determined in two complementary ways: First, the background radiation was compared tocalculations for the bremsstrahlung radiation at various temperatures. These calculationsare given in figure 4.3 as coloured lines (red: kBT = 46 eV, green: kBT e = 40.5 eV, blue:kBT e = 35 eV). The best fit to the experimental data is given by the 40.5 eV bremsstrahlungcurve. In this analysis, the Gaunt factor in Sommerfeld approximation was used, seesection 2.1.10.

Alternatively, the plasma temperature is given by the ratio of integrated line intensities[LH95]. We compared the Al IV lines at 11.6nm and 16.2nm, and obtained kBT e =(34.6 ± 6) eV for the plasma temperature. Within the error bars, this value agrees wellwith the temperature that was obtained from the bremsstrahlung background.



4.3 Landau-Pomeranchuk-Migdal Effect in Dense Plasmas 59

Since the spectrum gives absolute photon numbers, also the electron density can bedetermined via the bremsstrahlung background. The value of ne = 4.2×1022 cm−3 indicatesthat in fact a hot plasma at solid density conditions was produced. Furthermore, thesevalues comply well with radiation hydrodynamics simulations using the code HELIOS[MGW06].

In conclusion, the reported experiment has shown that FEL lasers provide a wellsuited tool to produce warm dense matter under defined laboratory conditions by ho-mogeneous plasma heating. The emitted radiation in the XUV spectral range, notablythe bremsstrahlung continuum, allows for an accurate characterization of the main plasmaparameters, i.e. density and temperature. In this case, temperatures around 40 eV anddensities of 4.2 × 1022 cm−3 have been found.

4.3 Landau-Pomeranchuk-Migdal Effect in Dense Plas-mas

4.3.1 Introduction

The modification of the bremsstrahlung emission spectrum in a dense medium is a longstanding issue. Since the pioneering work by Bethe and Heitler [BH34, Hei54], it wasclear, that perturbative Quantum Electrodynamics provides a rather incomplete pictureof the bremsstrahlung process. Notably the infrared-divergence dσ/dω ∝ 1/ω of thebremsstrahlung cross-section was discussed. Two main lines of arguments can be identi-

fied: On the one hand, the more academic problem of the isolated bremsstrahlung event,i.e. the scattering of a single electron on a single ion is considered going beyond the Bornapproximation. Compensation between divergent higher order terms in the elastic scatter-ing cross section and the infrared divergence of the Bethe-Heitler formula is found, as wasalready discussed in section 2.1.9.

A different approach takes its arguments from the fact that bremsstrahlung never oc-curs completely isolated but rather in a more or less correlated environment. The basicidea is the following: If the oscillation period of the emitted photon is of the same order asthe average time between collisions with ions, i.e. the life time of the single-particle state,the scattering amplitudes of successive scatterings have to be added coherently in order

to find the total scattering probability. In disordered media, the interference of successivescattering amplitudes is destructive, the bremsstrahlung cross-section is reduced as com-pared to the isolated bremsstrahlung process. For the case of two scattering centers, thiswas first shown by Landau and Pomeranchuk [LP53] using semiclassical arguments. Later,a quantum mechanical theory was developed by Migdal [Mig56]. He described the prop-agation of the electron through the scattering medium as a diffusion process. The ratioof the resulting bremsstrahlung cross-section in the medium to the Bethe-Heitler result isproportional to

√ω. Hence, LPM suppression does not remove the infrared divergence

but the order of divergence is lowered from (ω)−1 to (ω)−1/2.The influence of screening of the ion potential and dispersion of the emitted photons




in the medium was first considered by Ter-Mikaelyan [TM53] using the dielectric function.The resulting bremsstrahlung cross-section behaves proportional to ω at low photon ener-gies, i.e. the infrared divergence is removed. This result is often referred to as the dielectricsuppression of bremsstrahlung.

Both LPM suppression and dielectric suppression have been investigated by variousexperimental groups. The first experiments made use of highly energetic cosmic ray parti-cles. Particular features in the energy distribution of secondary particles could be tracedback to the LPM effect [Kas85]. First accelerator based studies of the LPM effect wereperformed by Varfolomeev et al. [V+75]. However, the results suffered from poor statis-tics. More clear evidence of LPM suppression was obtained by Anthony et al. [ABSB+97],although other effects, such as dielectric suppression and also transition radiation [Kle99]could not always be separated. The experiments by Hansen et al. [HUB+03, HUB+04]

are today regarded as the first unambiguous confirmation of LPM suppression. Due to thelarge electron energies of several hundred GeV, all energy scales could be separated, i.e.the various effects were identifiable. A review of the theory of the LPM effect and exper-imental studies can be found in Ref. [Kle99]. In that work, the author also mentions thepossibility of LPM suppression at non-relativistic energies in the thermal bremsstrahlungemission from hot and dense media, e.g. high energy density plasmas.

4.3.2 Many-body theoretical approach to the LPM effect in denseplasmas

In Ref. [FRRW05], the effect of successive scattering on the thermal bremsstrahlung emis-sion and absorption in a dense plasma is analyzed. A non-relativistic, fully-ionized hydro-gen plasma is assumed. Within the framework of linear response theory [ZMR96b], theabsorption coefficient is related to equilibrium force-force correlation functions, which arecalculated with the help of thermodynamic Green functions.

Successive scatterings are accounted for by resummation of the self-energy that de-scribes the single scattering event, i.e. solving the corresponding Dyson equation. A sim-

ilar approach was also suggested by Knoll and Voskresensky [KV96]. They showed, thatthe account for the finite life time of single-particle states, i.e. a finite width of the spectralfunction, leads to a suppression of the bremsstrahlung spectrum at small photon energies.The emissivity is obtained from the polarization tensor which, in one-loop approximation,is given as the convolution of two spectral functions. However, the imaginary part of theself-energy, was not explicitly calculated but rather set as a pure parameter. Thus, onlyqualitative results could be obtained. A microscopic approach to the self-energy and thusto the LPM effect in plasmas was still missing.

In this work, a self-consistent equation for the self-energy is derived, that describes the



4.3 Landau-Pomeranchuk-Migdal Effect in Dense Plasmas 61

scattering of electrons on fixed ions. It is given by the following diagrammatic expression:

Σ(p, ω) =

G

V D i V D

i

. (4.1)

The approach is similar to the GW approximation [Hed65], with the exception that astatically screened Debye potential is used instead of a dynamically screened one. In thisway, the Born approximation is automatically included as the lowest order contribution tothe self-energy. Effects due to dynamical screening, such as the dielectric suppression, areexcluded. Hence, the LPM effect can be studied independently.

Having the full electron propagator at our disposal, we proceed to calculate the brems-

strahlung spectrum via the force-force correlation function. In order to fulfill Ward-Takahashi identities [War50, Tak57], also vertex corrections to the force-force correlationfunction are performed. For a broader discussion of this issue, see section 2.2.1. Only thelowest order vertex correction that stems from electron-ion scattering is considered here.The self-consistent calculation of the vertex function in the same manner as for the self-energy, i.e. the solution of the corresponding Bethe-Salpeter equation was not performed.

4.3.3 Results

The central result of this study is shown in figure 4.4. The absorption coefficient α(ω) for

inverse bremsstrahlung, calculated from the force-force correlation, is given as a functionof the photon energy ω. The absorption coefficient is normalized to the result obtained inBorn approximation, see equation (2.103). The plasma parameters are ne = 10−6 a−3 =6.7 × 1018 cm−3 for the electron density and kBT = 2 Ry = 27.2 eV for the plasma temper-ature, i.e. a classical, weakly coupled plasma is studied.

Two results are compared: The solid line represents the complete calculation includingthe self-energy corrections as well as the lowest order vertex correction. The dashed linebelongs to the absorption coefficient with only the self-energy corrections taken into accountwhile neglecting vertex corrections.

In both cases, a significant deviation of the absorption coefficient from the Born ap-

proximation is obtained. Notably at small photon energies (

ω < 0.08 Ry), the absorptionis largely suppressed. At high photon energies (ω > 0.5 Ry) the effect vanishes, the curvesconverge with the Born approximation, i.e. α(ω)/αB(ω) → 1. The suppression was foundto increase as a function of the imaginary part of the self-energy. This coincides with theresults by Knoll and Voskresensky [KV96], using an energy and momentum independentself-energy.

At intermediate photon energies, an increase of the absorption coefficient with respect tothe Born approximation is found with a maximum of α(ω)/αB(ω) reached at ω = 0.2Ry.This resonant behaviour is a novel feature, that has not been observed before and must beattributed to the dynamical character of the self-energy. Note, that the plasma frequency




Figure 4.4: Inverse bremsstrahlung absorption coefficient as a function of the photon energyfor a classical plasma. (ne = 10−6 a−3B 6.7 × 1018 cm−3 and kBT = 2Ry = 27.2eV).The results are normalized to the Born approximation result αB(ω), see equation (2.103).

Solid line: Sum of self-energy and vertex contribution. Dashed line: Only self-energycontribution.

at the present density is ωpl = 7.1 × 10−3 Ry, thus the increase in the absorption cancertainly not be attributed to plasmon resonance. This enhancement feature certainlydeserves further attention but goes beyond the scope of this work.

By comparison between the solid curve (vertex corrections + self-energy) and the dashedcurve (only self-energy), the effect of the vertex corrections can be studied. The increaseof absorption at intermediate frequencies is lowered from α(ω)/αB(ω) = 1.06 to a value of 1.04. This decrease also seen in the calculation of spectral line shapes [Gun95]. Also there,

vertex corrections largely cancel with self-energy effects.At low photon energies, the suppression of inverse bremsstrahlung absorption is ampli-fied, here the vertex corrections and the self-energy add up to a large effect. It remains tobe scrutinized if this large suppression is an artefact of the perturbative treatment of thevertex correction, i.e. if this behaviour changes if also the vertex function is determinedself-consistently by solving the corresponding Bethe-Salpeter equation (2.59).

Summarizing, it was shown, that the successive scattering of electrons on randomlydistributed ions leads to a decrease in the inverse bremsstrahlung absorption efficiency.This is comparable to the LPM effect that is well known from high energy scatteringexperiments. As a further interesting feature it was found, that the account for the self-



4.4 Optical Properties and Spectral Function 63

energy also leads to an increase of the absorption at intermediate photon energies which,however is reduced, if vertex corrections are considered. The latter is required to fulfillWard-Takahashi identities, i.e. to preserve gauge invariance of the theory.

4.4 Optical Properties and the One-Particle Spectral

Function in Hot Dense Plasmas

4.4.1 Introduction

In the previous section, the Green function method was applied to determine the effectof successive scattering on the inverse bremsstrahlung absorption coefficient in a classical,

weakly coupled plasma, see also Ref. [FRRW05]. In this section, some further reachingquestions shall be addressed: First, the effect of successive scattering is investigated atsolar core conditions. The solar core plasma is a prototypical example for high energydensity matter. Values for the mass density of ρ = 1.5 × 102 g/cm3 and a temperature of T = 1.6 × 107 K 1360 eV are extracted from observations [BPW95]. By comparing tothe results obtained in section 4.3, we can study the effect of increased density.

Second, we analyze the importance of vertex corrections in the self-energy itself, whereasbefore, the vertex was only considered inside the polarization loop. To this end, GW Γcalculations (see section 2.2.2) are performed.

4.4.2 Results

First, we consider the impact of successive electron-ion scattering on the absorption coeffi-cient following the same lines as in the previous section. As our model system, a completelyionized hydrogen plasma at solar core conditions is chosen. The density of ions and elec-trons is ni = ne = 7 × 1025 cm−3 and the temperature is kBT = 100 Ry = 1360 eV.

The results are shown in figure 4.5. Using only self-energy corrections (solid curve) ,a similar behaviour as in the former case was found, i.e. suppression of bremsstrahlungat low photon energies, an enhancement at photon energies above the plasmon energyωpl and convergence to the Born approximation in the high energy limit. Again, vertex

corrections are applied to the force-force correlation function. At solar core parameters, thevertex corrections largely dominate the self-energy effects (dashed curve). In particular,the enhancement at intermediate photon energies is compensated. Similar results have alsobeen obtained by [BTADS97].

The question of vertex corrections in the self-energy itself is investigated by evaluatingthe self-energy in second order of the electron-ion scattering,

Σ(2)(p, ω) = + . (4.2)




1 10 100 1000

ω [Ry]

0.85

0.9

0.95

1

1.05

1.1

α /

α B

αΣ

αΣ+α

V

Figure 4.5: Inverse bremsstrahlung absorption coefficient as a function of the photon energyfor the solar core plasma. Results including self-energy (solid curve) and self-energy plusvertex corrections (dashed curve) are shown, normalized to the Born approximation αB(ω).

Whereas the first diagram is included in the second iteration of equation 4.1, the sec-ond diagram represents a vertex correction, which is not included. Note, that the upperpropagator lines in equation (4.2) represent the ions, while the base lines represent theelectrons.

It was found, that the vertex term gives a correction of at most 20%, see figure 4.6.Here, the first and second iteration of the self-consistent equation for the self-energy (4.1)(dashed and dotted curves, respectively) are shown, as well as the converged result (solidcurve). The inset shows the comparison between the iterations of the self-consistent self-

energy and the vertex correction in second order, see equation (4.2). The vertex term givesa correction of at most 20%. For the plasma parameters ne = 1021 cm−3 and kBT = 1 Ryare chosen. A detailed discussion can be found in Ref. [FRW07].

In conclusion, it was shown, that vertex corrections are important for the calculationof optical properties because they partially compensate pure self-energy effects. This wasseen in the case of the force-force correlation function, where the vertex correction compen-sates the enhancement feature at intermediate energies that is observed when taking onlyself-energy effects into account. On the other hand, also the self-energy itself is reducedsignificantly (20% at the considered parameters) by the vertex counterterm.



4.5 Self-Consistent Spectral Function and Analytic Scaling Behaviour 65

0 0.5 1 1.5ω [Ry]

-0.15

-0.1

-0.05

0

I m Σ

( 0 . 1 , ω

) [ R y ]

GW0, 1. iteration

GW0, 2. iteration

GW0, stable

0.9 1-0.01

-0.005

0

0.005

0.01

Vertex correction

Figure 4.6: Imaginary part of the self-energy at p = 0.1 a−1B for electron-ion scattering,see equation (4.1). The first and second iteration (dashed and dotted curves, respectively)are shown as well as the converged result (solid curve). The inset shows the comparison

between the iterations of the self-consistent self-energy and the vertex correction in secondorder, see equation (4.2). The vertex term gives a correction of at most 20%. Plasmaparameters: ne = 1021 cm−3, kBT = 1 Ry.

4.5 Self-Consistent Spectral Function and Analytic

Scaling Behaviour

4.5.1 Introduction

The single-particle spectral function A(p, ω) gives the probability density to find a particleat energy ω for a given momentum p. Starting from the spectral function, a number of interesting plasma observables can be determined, such as its equation of state [ VSK04],transport properties like thermal and electric conductivity [RRRW00], optical properties[FRW07], see also the preceding sections, or the stopping power of fast particles traversing amedium [GSK96, MR96]. This is achieved by calculating higher order correlation functions,i.e. convolutions of single-particle spectral functions and vertex functions.

The analytic structure of the spectral function, i.e. the location of resonances and theirwidth, is determined by the real part and the imaginary part of the self-energy Σ( p, ω),respectively. In the case that the spectral function contains only a single peak, one speaks




of a broadened quasi-particle spectral function. The position of the quasi-particle resonanceis given by the solution of the quasi-particle dispersion

E (p) =2 p2

2m− µ + Re Σ(p, ω) |ω=E (p) . (4.3)

The imaginary part of the self-energy at the quasi-particle dispersion Im Σ(p, ε(p)) is ameasure for the width of the quasi-particle resonance. Often, this quantity is referred toas the quasi-particle damping width or inverse of the quasi-particle life time. The quasi-particle life time is the average time in which the particle is in a defined momentum statebefore its momentum is changed due to collisions with other particles in the system.

The GW (0) approximation for the self-energy is a well established method, which wasalready extensively discussed in section 2.4. It allows the self-consistent calculation of

the self-energy beyond the Hartree-Fock level, i.e beyond the mean-field approximation,where particle correlations are neglected. The GW (0) approximation yields a set of coupledintegral equations for the electron self-energy Σ and the spectral function A(p, ω). Thescreened interaction potential is described by the random phase approximation for the po-larization function. Furthermore, the chemical potential has to be adjusted to the spectralfunction to ensure conservation of the number of particles. Rigorously, this set of equationsis solved only numerically, using an iteration algorithm that starts from a suitable initial-ization of the spectral function or the self-energy. Usually, the algorithm converges afterabout 10 iterations, i.e. the functions do not change any more from iteration to iteration.

4.5.2 Numerical Results

As a representative example, the spectral function for a one-component electron plasmais studied at conditions similar to the ones found at the solar core, i.e. the temperatureis kBT = 1000 eV, the density is 7 × 1025 cm−3. A three-dimensional representation isgiven in figure 4.7. The spectral function is shown as a function of the particle energy(frequency) shifted by the chemical potential, ω + µe for five different momenta. At lowmomenta, plasmaron resonances show up as shoulders in the spectral function, shiftedfrom the central quasi-particle peak by roughly the plasma frequency, which at the presentdensity is ωpl = 23 Ry. At increased momenta a single, broadened quasi-particle peak is

obtained.The spectral function at various densities is studied in figure 4.8. The temperature isthe same for all calculations, kBT = 1360 eV, the momentum is fixed at p = 0, where thecollective effects become most visible. Again, at the solar core density n = 7 × 1025 cm−3,the characteristic central peak (quasi-particle peak) is accompanied by plasmaron satellites(indicated by arrows), shifted by the plasma frequency ωpl = 23 Ry from the quasi-particlemaximum. The quasi-particle peak itself is shifted from the single-particle dispersionE (0) = −µ by roughly 7 Ry and considerably broadened. Both, shift and width, decreaseas the density is lowered. This can be understood a consequence of the decreased couplingconstant Γ ∝ n1/3. In the limit of vanishing density, the spectral function converges towards




] - 1

B

m o m e n t u m p [ a

012 3

45678910 [ R y ]

e µ +ω

f r e q u e n c y -40 -20 0 20 40 60 80 100120

140

) [ 1 / R y ]

ω

A ( p ,

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4.7: Electron spectral function at solar core conditions (kBT = 1000eV, ne =7×1025 cm−3). The spectral function is shown as a function of the particle energy shifted bythe chemical potential µe for five different momenta. At low momenta, plasmaron satellitesshow up in the spectral function, whereas at increased momenta a single, broadened quasi-

particle peak is obtained. The solid black line at the bottom denotes the free single-particledispersion ω = 2 p2/2m − µ. The chemical potential is µ = −1880 eV (138 Ry).

an on-shell delta distribution A(p, ω) → 2πδ(ε(p) − ω)) with the free particle dispersionε(p) = 2 p2/2m − µ.

4.5.3 Analytical expression for the quasi-particle damping widthat small momenta

It is worth to look for an analytic expression for the quasi-particle damping width σ p =ImΣ(p, E (p)) for the following reasons: On the one hand, analytic results can be used asbenchmarks for the numerical algorithm and give accurate results in limiting cases thatare hard to access numerically. In particular, this applies to the case of large momenta,and low densities or low temperatures. In each case, the spectral function features verynarrow resonances that are difficult to resolve on the numerical grid. Second, analyticalexpressions permit a deeper insight into the relevant physical processes that lead e.g. tothe finite spectral width and shift of the spectral function.

Naturally, analytical results can only be obtained through approximations to the exactexpressions and equations that otherwise can only be solved numerically. These approxi-




0

0.2

0.4

0.6

0.8

1

1.2

-40 -20 0 20 40

s

p e c t r a l f u n c t i o n A ( p = 0 , ω

) [ R y ]

energy −hω+µ [Ry]

n=7.0x1025

cm-3

n=7.0x1024

cm-3

n=7.0x1023

cm-3

n=7.0x1022

cm-3

n=7.0x1021

cm-3

Figure 4.8: Spectral function at vanishing momentum p = 0 as a function of energyω + µ. The different curves correspond to different densities between 7 × 1021 cm−3 and7 × 1025 cm−3. The temperature is kept constant at kBT = 1360 eV. At low density,the spectral function becomes a narrow, unstructured quasi-particle resonance, while at

increased density, the plasmaron satellites appear in the wings of the central peak (arrows).

mations are only valid for certain plasma parameters. Here, I focus on the case of weaklycoupled, classical plasmas, i.e. plasmas at low density and high temperature. As a furtherapproximation, only temperatures that are large against the binding energy of hydrogenare considered, kBT > 13.6 eV = 1 Ry. Thus, bound states do not need to be taken intoaccount in the calculation of the self-energy. At lower temperatures, bound states becomeimportant and must be included in the self-energy. This can be achieved by use of thet-matrix [KKER86].

The limit of weak coupling is of special interest, since here a vanishing self-energy isexpected, such that the spectral function converges to a delta-distribution. This behaviourwas also seen in the numerical results, see figure 4.8. At small coupling parameters, onemight expect that the self-energy is already given by the first iteration of the correspondingintegral equation (2.89) starting from the quasi-particle limit, i.e. replacing the spectralfunction on the r.h.s. by an on-shell delta-distribution. This so-called quasi-particle damp-ing was investigated e.g. by Fennel and Wilfer in Ref. [FW74]. However their result is notsatisfying. The quasi-particle damping is far from the self-consistent numerical solution.Secondly, an expression is obtained, which is independent of density. This means, that afinite quasi-particle life time is obtained regardless of the density and even in the vacuum




limit n = 0. This is unphysical, since free particles are always in a defined momentumstate.

This paradox can only be resolved if the self-consistency implied in the equation forthe self-energy is consequently elaborated also in the analytical treatment. In Refs [For08]and [For09], the analytic solution for the self-consistent quasi-particle damping is derived.Expressions are found, that give the quasi-particle damping for a weakly coupled one-component plasma in the limit of vanishing momentum p = 0 and in the asymptoticcase p → ∞. By means of a two-point Pade approximation, an interpolation formula isconstructed that covers the complete p range.

The difficulty in solving the integrals in equation (2.89) is mainly caused by the termIm −1(q, ω). Here, suitable approximations have to be used. It was found, that at smallmomenta p, the one-loop approximation (Born approximation) for the screened potentialleads to the same form of the self-energy than the full RPA expression. Replacing the

inverse dielectric function by the one-loop approximation and applying certain approxi-mations to the remaining integrands leads to the analytic form of the self-energy valid atsmall momenta,

Σ(p, ω) =ω − 2 p2/2m + µ

2

− sign(ω − 2 p2/2m + µ)

ω + iδ − 2 p2/2m + µ

2

2− κe2

4π0kBT

1/2. (4.4)

For the derivation, see Ref. [For08].

Equation (4.4) reproduces the numerical data for the self-energy and the spectral func-tion at vanishing momentum with an accuracy of better than 7%. This is true as long asno plasmaron resonances appear in the spectral function. At the single-particle dispersionω = E (p), equation (4.4) simplifies to

σ0 := Im Σ(p, E (p) = −

κe2kBT /4π0 , (4.5)

the so-called non-collective damping width. It can also be expressed by the plasma couplingparameter Γ as

σ0 = −(3Γ3)1/4kBT . (4.6)

The range of plasma parameters, where equation (4.4) is valid, can be estimated from

the ratio of the plasma frequency (i.e. the energy scale where plasmaron peaks appear)and the non-collective damping width. The plasmaron peaks are hidden in the broadenedquasi-particle peak, as long as the parameter η = −ωpl/σ0 < 1. The parameter η isidentical to the square-root of the Debye screening parameter, measured in inverse Bohrradii,

η =√

κ aB . (4.7)

The non-collective damping width σ0 behaves proportional to n1/4. It vanishes in thelimit n → 0 as expected from first principle arguments. The high accuracy by which thissimple expression reproduces the numerical results is illustrated in figure 4.9.




10-3

10-2

10-1

Γ

10-2

10-1

- I m Σ

( 0 , E 0

/ h_ ) / k

B T

η=1

η=1

η=1

η=0.5

η=0.5

η=0.5

θ = 1 θ = 1 θ = 1

k B

T = 10 Ry

k BT = 100 Ry

k BT = 1000 Ry

~Γ 3/4

~Γ 3/4

Figure 4.9: Numerical results for the quasi-particle damping width at vanishing momentum,ImΣ(0, E (0))/kBT , normalized to the temperature as a function of the plasma couplingparameter Γ. Results are shown for three different temperatures, kBT = 10 Ry (solid),kBT = 100 Ry (dashed), and kBT = 1000 Ry (dash-dotted). At small coupling parametersand for η = −ωpl/σ0 = √κ aB < 1, the numerical data comply with the derived scaling lawfor the non-collective damping σ0 ∝ Γ3/4, see equation 4.6. When η becomes larger than 1,the plasmaron satellites appear in the spectral function and the width of the quasi-particlepeak decreases accordingly.

The numerical data for the quasi-particle damping at vanishing momentum, σ0, normal-ized to the plasma temperature is shown as a function of the plasma coupling parameterΓ for three different temperatures. At small values of η (see equation (4.7)), the numer-ical data nicely follow the analytic expression (4.6), i.e σ0/kBT ∝ −Γ3/4. Only when η

becomes larger than 1, the numerical data deviate from the scaling law. This is due to theappearance of collective modes in the spectral function. Correspondingly, spectral weightis transferred from the central peak into the resonances and the quasi-particle resonancenarrows.

The regime of validity for the quasi-particle damping width σ p as given by (4.4) isillustrated in figure 4.10. The grey colored area is limited by the green line, here theparameter η =

√κ aB is equal to 1, i.e. ωpl = −σ0. At higher densities, the plasmaron

resonances separate from the central peak and appear as distinct features in the spectralfunction. Furthermore, the grey region is limited by the horizontal line at kBT = 1 Ry =13.6 eV, i.e. the binding energy of the ground state hydrogen atom. Below this line, bound




1020

1021

1022

1023

1024

1025

1026

1027

ne

[cm-3

]

100

101

102

103

104

k B

T

[ e V ]

k BT = 1 Ry

θ = 1

Γ = 1

κ = 1 aB-1

Γ = 1

θ = 1 κ = 1 aB

-1

σ0

= −(e2κ k

BT /4πε

0)1/2

Figure 4.10: Density-temperature plane and isolines of various plasma parameters. Thegrey colored region marks the validity region of the analytic scaling law (4.5) for the quasi-particle damping width.

following form is obtained,

σPade p =

a0 + a1 p

1 + b1 p + b2 p2ϕ( p) ,

a0 = −π

2

√κT , a1 = −κ

π

2

3/2, b1 =

πκ

2T , b2 =

πκ

2T ,

ϕ( p) = ξ( p) − ln ξ( p) +ln ξ( p)

ξ3( p)− ln ξ( p)

ξ2( p)+

ln ξ( p)

ξ( p)− 3 ln2 ξ( p)

2ξ3( p)+

ln2 ξ( p)

2ξ2( p)+

ln3 ξ( p)

3ξ3( p),

ξ( p) = ln(e + 2

π κp2 exp(A/T )/T ) ,

A = −1.3357ωpl

2[nB(ωpl) exp(ωpl/T ) − nB(−ωpl)exp(−ωpl/T )] .

(4.8)

Here, Rydberg units have been used, i.e. e2/4π0 = 2, = 1, me = 1/2, and kB = 1.

The quasi-particle damping width is plotted in figure 4.11 as a function of p (dashedcurve). Density and temperature of the plasma are fixed at n = 7 × 1020 cm−3 and kBT =100 eV, respectively. The solid curve is obtained by fitting the numerical data for the




0 20 40 60 80 100

momentum p [aB

-1]

-0.5

-0.4

-0.3

-0.2

q u a s i - p a r t i c l e d a m p i n g σ p

[ R y ]

GW (0)

-fit

Interpolation formula

Figure 4.11: Effective quasi-particle damping σ p as a function of momentum p for plasmadensity n = 7 × 1020 cm−3 and temperature T = 100 eV. The fit-parameters for theGaussian fit to the full GW (0)-calculations are given as solid line, the dashed line denotes

the analytic interpolation formula (4.8).

spectral function using a Gaussian of the form

AG(p, ω) =

2π

σ2 p

exp

−(ω − ε(p))2

2σ2 p

. (4.9)

The Pade formula agrees with the fit data within an error bar of less than 10%. At p = 0,the quasi-particle damping starts at the finite value σ0 = −π

e2κkBT/8π0/21 and slowly

decreases towards higher p, such that in the limit p

→ ∞, a narrow, on-shell quasi-particle

spectral function is obtained. This is also illustrated in figure 4.12.Here, the full numerical solution for the spectral function in GW (0) approximation (solid

curve) is compared to the Gaussian ansatz (4.9) (dashed curve) using the quasi-particledamping width σ p from equation (4.8). Results are shown for three different momenta, p = 0 (a), p = 50 a−1B (b), and p = 100 a−1B (c). The plasma density is ne = 7 × 1020 cm−3,the temperature is kBT = 100 eV. The Debye screening parameter is κ = 1.9 × 10−2 a−1B ,

1This value is obtained from equation 4.5 by multiplication with the factor π/2√

2. In this way, themaximum of the Gaussian function matches the maximum of the Lorentzian form of the spectral function(2.75).




i.e. we are in the range of validity of the non-collective damping width, as discussed in theprevious subsection.

At small momenta, the Gaussian does not completely reproduce the numerical solution,although the height of the maximum and the overall width coincide at least approximately.The steep rise in the wings of the numerical solution and the plateau around ω + µe = 0are not reproduced by the Gaussian ansatz. These structures result from the plasmaronresonances that merge with the central peak at the considered plasma parameters.

At higher momenta an almost perfect agreement between the numerical data and theGaussian spectral function is observed. Here, the plasmaron resonances are damped outand only the broadened quasi-particle peak remains.

In conclusion, an analytic expression for the quasi-particle damping width was pre-sented, that accurately reproduces the numerical data for the spectral function at arbitrarymomentum p by use of a Gaussian ansatz. In the regime of validity (see figure 4.10), this

result greatly facilitates the calculation of higher order correlation functions via the spectralfunction. An example is the polarization function which yields the optical properties of thesystem. Further interesting quantities, e.g. transport coefficients [RRRW00], the stoppingpower[GSK96] or the equation of state [VSK04] can be accessed as well. The spectral func-tion does not need to be recalculated by solving the GW (0) equations for each density andtemperature but can be approximated by the Gaussian ansatz (4.9) in combination withthe analytic expression for the quasi-particle damping σ p, equation (4.8). Furthermore, inthose cases, that the spectral function needs to be known with high accuracy (i.e. betterthan 10%), the Gaussian ansatz should be used as initialization for the iterative algorithm.Convergence is then reached after only two or three iterations.




0

2

4

6

8

10

-3 -2 -1 0 1 2 3

s p e c t r a l f u n c t i o n A ( p , ω

) [

1 / R y ]

frequencyω+µe [Ry]

p=0 aB-1

GW(0)

Gauss fit

(a) p = 0

0

2

4

6

8

10

2497 2498 2499 2500 2501 2502 2503


) [ 1 / R y ]


p=50 aB-1

GW(0)Gauss fit

(b) p = 50 a−1B

0

2

4

6

8

10

9997 9998 9999 10000 10001 10002 10003


) [ 1 / R y ]

frequency ω+µe [Ry]

p=100 aB-1

GW(0)

Gauss fit

(c) p = 100 a−1B

Figure 4.12: Spectral function in GW (0)-approximation (solid lines) and Gaussian ansatz(dashed lines) with quasi-particle damping width σ p taken from equation (4.8) for threedifferent momenta. Plasma parameters: n = 7 × 1020 cm−3, T = 100 eV. The plasmacoupling parameter is Γ = 2.1 × 10−2, the degeneracy parameter is θ = 3.5 × 102, the

Debye screening parameter is κ = 1.9 × 10

−2

a

−1

B .



76 Summary

Summary

The present work is devoted to the process of bremsstrahlung occurring in a correlatedmedium, such as a dense plasma. A many-body quantum field theoretical approach ispresented. In this frame, the bremsstrahlung emissivity and the absorption due to inversebremsstrahlung are expressed via the force-force correlation function or, equivalently, thepolarization function. This allows for a microscopical description based on the Greenfunction technique. In this way, many-particle effects are included in a systematic andintuitive manner.

The key quantity within the many-particle theoretical description is the single-particlespectral function. With its help, single-particle properties can be calculated as moments of the spectral function. Higher order correlation functions, such as the polarization functionor the two-particle Green function are given by convolution products of the single-particle

spectral function, and the vertex function, which is itself related to the spectral functionvia the Bethe-Salpeter equation, see section 2.2.

The influence of correlations is described by the complex single-particle self-energy. Theexact determination of the self-energy requires the rigorous solution of the Dyson-Schwingerequations which amounts to the exact diagonalization of the many-particle Hamiltonian.Appropriately chosen approximations have to be performed, depending on the special prob-lem that is regarded.

Here, special attention was payed to the question, how the bremsstrahlung emissionand inverse bremsstrahlung absorption are modified due to correlations among plasmaconstituents. Both electron-ion and electron-electron correlations were analysed. The

results are given as a Gaunt factor, i.e. a correction factor that Kramers’ formula forbremsstrahlung has to be multiplied with.

As a well known example for the influence of strong electron-ion correlations on thebremsstrahlung emission, the Landau-Pomeranchuk-Migdal effect was considered, i.e. thesuppression of bremsstrahlung at small photon energies due to multiple scattering on dif-ferent ions whilst the emission of the bremsstrahlung photon (c.f. section 4.3). Beingtheoretically predicted and experimentally confirmed in the case of ultrarelativistic elec-tron scattering in dense targets, a quantitative analysis of this question for the case of thermal bremsstrahlung from a plasma was still missing.

In our calculations, where we took into account the successive scattering via resum-

mation of the self-energy in Born approximation, we obtained a similar effect than theLPM theory. At small photon energies, the emission spectrum is suppressed comparedto the Bethe-Heitler formula, i.e. the Born approximation. The suppression can becomesignificantly large, in the case of a plasma near solar core conditions, 15% suppressionhave been observed. The suppression increases systematically as a function of the widthof the spectral function, i.e. the imaginary part of the self-energy. As the self-energy, thesuppression reaches a maximum in the regime of moderately coupled (Γ 1) and moder-ately degenerate (θ 1) plasmas, where also the imaginary part of the self-energy peaksas a function of density and temperature. At large photon energies, the results convergetowards the Bethe-Heitler result for bremsstrahlung. At an intermediate energy, slightly



Summary 77

above the plasma frequency, an enhancement of bremsstrahlung emission was obtained.However, it remains unclear, if this enhancement is an artefact which could result from aninconsistent treatment of the vertex function, i.e. counterparts to the pure self-energy cor-rections. This assumption is enforced by the observation that the enhancement is reducedif vertex corrections are taken into account on a perturbative level.

The question of vertex corrections was furthermore investigated on the level of the self-energy itself in section 4.4. To this end, the self-energy arising from electron-ion scatteringwas evaluated in second order. As predicted by first principle relations (Ward-Takahashiidentities), partial compensation of the direct self-energy term through the exchange termwas observed. In the considered case, a reduction of 20% was found.

In order to study electron-electron correlations, special care has to be taken for theirdynamical character. Dynamical screening of the potential and the emergence of collec-tive excitations in the spectral function and the self-energy are well known effects due to

electron-electron correlations.A straightforward method to compute the electron self-energy beyond the mean-field

level is the GW approximation. Here, we used the GW (0) variant, i.e. the screenedinteraction is fixed on the level of the random phase approximation. A systematic behaviourof the single-particle spectral function was obtained. In the low density limit, the spectralfunction consists of a single peak, the quasi-particle peak. Its width decreases slowlywith increasing momentum. At large momenta, the spectral function converges towards anarrow on-shell delta function.

With increasing density, the spectral function becomes more complex. In this case,coupled electron-plasmon modes (plasmarons) appear as shoulders in the wings of the

quasi-particle peak at small momenta. At increased momenta, the plasmarons decay andonly the quasi-particle remains, becoming again a narrow on-shell resonance.

The simple, single-peak behaviour at low densities and high temperatures was analyzedin more detail. Using a Gaussian ansatz for the spectral function, the quasi-particle damp-ing width was derived by analytic solution of the GW (0) equation. The obtained formulaereproduce the numerical data with less than 10% relative deviation. The quasi-particledamping width scales with density as n1/4. Thus, in the vacuum limit (n = 0), free par-ticles are obtained consistently. This is in contrast to the quasi-particle approximation tothe self-energy which yields a density independent damping width.

Using the Sommerfeld expression for the Gaunt factor, bremsstrahlung as a competing

process of emission in a Thomson scattering experiment was analyzed in section 4.1. Bycomparing the bremsstrahlung photon yield and the Thomson scattering cross-section as afunction of the plasma temperature and density, threshold conditions for the laser intensityand the detector resolution were determined. In particular, the case of a Thomson scat-tering experiment at the free electron laser facility Hamburg (FLASH), operating at XUVwavelengths, was considered. It was found, that the typically available FEL intensities of about 1012 W/cm2 and moderate detector resolutions of ∆λ/λ 10−2 are sufficient for theThomson scattering signal to exceed the thermal bremsstrahlung background.

The paramount importance for bremsstrahlung as a plasma diagnostic tool was demon-strated in the context of an experiment aiming at the production of a solid density, homoge-



78 Outlook

neously and volumetrically heated plasma. Ultrashort FEL pulses were focussed on a solidAl target and the resulting plasma radiation was measured in an XUV spectrometer. Bothbremsstrahlung and line components of the spectrum concordantly yield a plasma temper-ature of about 40 eV at an electron density of 4.2 × 1022 cm−3. Thus, warm dense matterconditions were accessed. This experiment established a new method for the productionof warm dense matter under defined laboratory conditions using FEL radiation.

Outlook

Further investigations of the single-particle self-energy for dense plasmas are currently per-formed. The focus is on the microscopical description of plasmas at arbitrary degeneracy.This is of large interest e.g. to trace the excitation of a hot dense plasma by interaction of

intense laser radiation with cold solid state material, such as the one outlined above. Hav-ing the spectral function disposable over the complete range of density and temperature,a unified picture of the absorption in the cold target, the dynamic evolution of the plasmaand also recombination processes could be given.

Other open questions in this field are related to the consistent treatment of the ver-tex. Here, it is worth to consider strategies that are successfully applied in quantumfield theory, notably in hadron physics. The basic idea is to postulate a vertex func-tion first, constrained by sum-rules and identities such as the Ward-Takahashi identities[BC80a, BC80b, EAC+08]. Subsequently, the propagators and self-energies are obtainedby functional differentiation of the vertex function. In this way, the Ward-Takahashi identi-ties are automatically fulfilled. Furthermore, this ansatz being Φ-derivable, conservation of energy, momentum, and particle number is assured on the level of the expectation values.An application to the problem of particle beam-plasma interaction in a hot QED plasma,following these lines, was recently discussed by Morozov et al. [MR06].

As has been demonstrated, bremsstrahlung plays an important role in the field of plasma diagnostics of warm dense matter. Thus, it is obligatory to advance also the theoryof bremsstrahlung. As an important result of this work, it should be kept in mind thatmany-particle effects can significantly modify the bremsstrahlung emission spectrum orthe inverse bremsstrahlung absorption spectrum. The detection of features such as thesuppression at low photon energies or the enhancement above the plasma frequency offernew possibilities also for plasma diagnostic techniques.



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Part II

Published articles



Chapter 5

Bremsstrahlung from Dense Plasmasand the

Landau-Pomeranchuk-Migdal Effect

Authors: Carsten Fortmann, Heidi Reinholz, August Wierling, and Gerd Ropke

Appeared in volume 20 of the series Condensed Matter Theories, Nova Science, New York,2006, pages 317-332.

Listing of contributions by authors:

•C.F.: Preparation of the manuscript (sections 1, 3A, 3B, 3C, 4), derivation of the

algebraic equation for the self-energy, numerical calculations.

• H.R.: Preparation of manuscript (section 2)

• G.R.: Preparation of manuscript (section 1)

• A.W.: Preparation of manuscript (section 3D), numerical calculation of vertex con-tributions




Bremsstrahlung from dense plasmasand the Landau-Pomeranchuk-Migdal effect

C. Fortmann, H. Reinholz, G. R¨ opke, and A. Wierling

Institute of Physics, Rostock UniversityD-18051 Rostock, Germany

1. INTRODUCTION

In a fully ionized plasma, bremsstrahlung and inverse bremsstrahlung are theonly emission and absorption processes, respectively. For partially ionized plasmasthese processes contribute to some extent to the continuous emission/absorption spec-trum. We introduce the emission coefficient j(ω) as the rate of radiated energy perunit volume, frequency and solid angle, and the absorption coefficient α(ω) as the rel-ative attenuation of the intensity of electromagnetic waves propagating in the mediumper unit length. For a thermally equilibrated plasma, these quantities are linked byKirchhoff’s law [1]

j(ω) = L(ω)α(ω) , (1)

with the Planck distribution L(ω) = hω3/

4π3c2(exp(hω/kBT ) − 1)

. Thus, it issufficient to study one or the other, i.e. j(ω) or α(ω) . Here, we choose the absorptioncoefficient α(ω).

The absorption spectrum can be determined according to quantum electrody-

namics (QED) [2] from the interaction part of the QED Lagrangian

Lint(x) =c

Z cejcµ(x)Aµ(x) , (2)

x being a four-dimensional space-time variable. This Lagrangian describes a minimalcoupling between the particle current jcµ and the vector potential Aµ. The index cdenotes the species of the particles involved, carrying the charge Z ce. Introducinga Hamiltonian H , the transition rate between asymptotically free states of electrons|pin , |pout with the energies E ein, E

eout respectively, follows from Fermi’s Golden

rule aswin,out =

2π

h

pout|H int|pin2δ(E ein + hω − E eout) , (3)

101



with H int = − d3xLint(x). Assuming single, uncorrelated scattering from differentions only, the absorption coefficient is given by

α(ω) = niΩ0

Ω0d3 pin

(2π)3Ω0d3 pout

(2π)3f (E ein) 1

cwin,out . (4)

Here, niΩ0 is the total number of ions in the volume Ω0 and win,out is the transitionprobability that a photon of momentum k = pout−pin and polarization λ is absorbedby a single electron of momentum pin in the Coulomb potential of an ion, leaving theprocess with momentum pout. The momentum distribution function of the incoming

electrons is denoted as f (E ein) =

exp

(h2 p2in/2me − µe)/kBT

+ 1)−1

. The factor1/c arises due to the current of incoming photons.

Evaluation of the transition rate in Born approximation with

pout|H int|pin =Z ie

2h2

ǫ0meΩ0

e2

2hǫ0Ω0hω

(pout − pin)z|pout − pin|2 (5)

gives the well known Bethe-Heitler formula [3] for the emission coefficient Eq. (1).

In the low frequency limit, the Bethe-Heitler cross section behaves roughly like1/ω. Within QED, this infrared-divergence is discussed as a consequence of theneglect of vertex corrections [4]. The infrared-divergent terms can be shown to cancelwith corresponding contributions to the form factor of the source particle which is amanifestation of the Ward-Takahashi identities [5].

In the non-relativistic limit and for soft photons, the absorption coefficient for ahydrogen plasma (Z i = 1) is given by

αB(ω) =C

hω3ne ni sinh

hω

2kBT

K 0

hω

2kBT

, (6)

where C −1 = 3√

2ǫ30π3/2m

3/2e c (kBT )

1/2 /e6 and K 0(x) = ∞0

cos(x sinh t)dt is themodified Bessel function of zeroth order. Here, the electron density ne has beenintroduced, which is equal to the ion density ne = ni in charge neutral systems.

This Born approximation can be improved by taking Coulomb wave functions

for the initial and final states. In this case, an analytic result for the absorptioncoefficient was given by Sommerfeld [6]. Due to the occurrence of hyper-geometricfunctions, simpler approximations such as the Born-Elwert approximation [7] havebeen developed. In the classical limit, the Sommerfeld expression reduces to a resultobtained earlier by Kramers [8].

In a dense plasma, the influence of the collective behavior of the system and themodification of single-particle properties of the emitting and absorbing particles aswell as the bremsstrahlung photons are important. Retaining the single-scatteringpicture of Eq. (4), medium effects can be taken into account e.g. by a modificationof the potential. Instead of a Coulomb potential, a static or dynamically screenedpotential should be used [9]. A quantum-statistical approach based on a systematicperturbative treatment of the force-force correlation function has been developed andapplied to inverse bremsstrahlung in Ref. [10].

102 Bremsstrahlung and the LPM effect



How does multiple scattering of the emitting electron change the cross section?This question was first treated by Landau and Pomeranchuk in a semi-classical wayand soon afterwards by Migdal [11] using kinematic considerations. They showed

that the account of successive collisions leads to a suppression of the bremsstrahlungcross section compared to the Bethe-Heitler result at photon energies low againstthe energy of the scattering electron. Both Bethe-Heitler and Landau-Pomeranchuk-Migdal (LPM) theory give the same result in the limiting cases of high photon energiesand/or low densities, i.e., in those cases where the Born approximation is applica-ble. Migdal’s result has been rederived more recently using different methods, e.g.,path-integral calculations [12] and quantum kinetic equations [13]. A comprehensiveoverview of theoretical approaches is given in [14]. There, it is also pointed out thatthe LPM effect might play an important role for the emission/absorption spectrumof a plasma even in the non-relativistic regime due to the large number of free chargecarriers.

Due to the large energies required to observe notable effects, it took quite a longtime until experiments could unambiguously approve LPM theory. Experimentalinvestigations have been performed since the late 1950’s using high energy electrons(some MeV to GeV) from cosmic rays and accelerators [14]. These early experimentssuffered from poor statistics and were unable to confirm LPM theory. More recentexperiments at SLAC and CERN [15,16] have indeed shown the LPM effect.

Migdal’s theory is not completely microscopic but relies on the definition of amacroscopic parameter, namely the coherence length l, initially introduced by Ter-Mikaelyan [14] and first applied to the theory of bremsstrahlung by Landau andPomeranchuk [11]. The coherence length gives the scale of photon energy, below

which the suppression of bremsstrahlung becomes important. It is basically deter-mined by the density of the medium. Knoll and Voskresensky [17] were the first totreat the question of the LPM effect in the context of a many particle system, whereevery constituent has to be regarded as an emitter of bremsstrahlung. They wereable to show a suppression of the emission/absorption spectrum at low frequenciesby using medium modified single particle propagators. The particles are assigned afinite lifetime τ , given by the width of their spectral function Γ = h/τ . However,this quantity is not calculated from a microscopic approach, but simply set as a pa-rameter. It is related to the aforementioned coherence length by the simple relationτ ≃ l/c [14].

In this work, we present a completely microscopic calculation of the absorptionspectrum, where the spectral function of the electron is obtained in a self consis-tent manner. Thus, we present for the first time a fully microscopic treatment of the LPM effect for a many-particle system. The absorption coefficient is defined inlinear response theory and is expressed through thermodynamical correlation func-tions using a diagram technique equivalent to the well known Feynman diagrams[18]. Medium effects are accounted for systematically in terms of self-energy andvertex corrections, which are evaluated in certain approximations. We show thatthe Bethe-Heitler bremsstrahlung spectrum follows from this approach in lowest or-der perturbation theory using free particle propagators. The account of coherentsuccessive scattering, as in LPM theory, can be achieved by a partial summation of

self-energy diagrams. This procedure leads to a finite width of the single-particlespectral function which reflects the modification of the energy-momentum dispersion

103



relation due to the medium. It is found that the absorption coefficient calculated onthe basis of these medium modified propagators is significantly altered in the low fre-quency range in comparison to the Born approximation. In the high frequency limit

as well as in the low density limit, the Born approximation is reproduced. Thus, themain features of LPM theory can be rederived within our microscopic approach.

2. LINEAR RESPONSE THEORY

We consider the interaction of soft photons with a non-relativistic, homogeneousplasma. A key quantity to describe the propagation of electro-magnetic waves in amedium is the dielectric tensor ǫij(k, ω) [19]. In the isotropic case, the tensor can bedecomposed into a transverse ǫt(k, ω) and a longitudinal ǫl(k, ω) part with respectto the wave vector k. Here and in the following, we take k to point along the z-axis,k = kez. In the long-wavelength limit k

→0, the longitudinal and the transverse

part coincide ǫ(ω) = limk→0 ǫl(k, ω) = limk→0 ǫt(k, ω). The absorption coefficientcan be obtained from the dielectric function according to

α(ω) =ω

c

Im ǫ(ω)

n(ω), (7)

where the index of refraction n(ω) is also linked to the dielectric function by

n(ω) =1√

2(Re ǫ(ω) + |ǫ(ω)|)1/2 . (8)

The relation between ǫl(k, ω) and the longitudinal response function χl(k, ω)

ǫl(k, ω) =

1 − e2

ǫ0k2χl(k, ω)

−1, (9)

allows for a microscopic approach to the dielectric function. Within linear responsetheory, the Kubo formula relates the response function to the current-current corre-lation function [20]

χl(k, ω) = iβ Ω

0

k2

ω J zk

; J zk ω+iη

, (10)

where the correlation functions for two observables A,B are defined according to

(A;B) =1

β

β 0

dτ Tr[A(−ihτ )B†ρ0] , A;Bω+iη =

∞ 0

dt ei(ω+iη)t (A(t);B) . (11)

The time dependency of the operators is taken in the Heisenberg picture. The currentdensity operator is given as

Jk = 1Ω0

c,p

ecmc

hp a†c,p−k/2ac,p+k/2 . (12)




a†c,p and ac,p are creation and annihilation operators for momentum states, respec-tively. The species and further quantum numbers such as spin are labeled by c , Ω0

is a normalization volume, ρ0 is the equilibrium statistical operator, β = 1/(kBT ) is

the inverse temperature. Note, that in Eq.(10), a small but finite imaginary part ηhas been added. In the final results, the limit η → 0 is taken. The inverse responsefunction can also be expressed as [10]

χ−1l (k, ω) =−iω

β Ω0k2(J zk ;J zk )2

iω(J zk ;J zk ) − J zk ; J zk ω+iη +

J zk ;J zk ω+iηJ zk ; J zk ω+iηJ zk ;J zk ω+iη

.

(13)

This transformation of the current-current correlation function into a force-force cor-relation function

J zk ; J zk

ω+iη with J = i

h [H,J] , which has the meaning of a force as

the time derivative of momentum, is more suited for a perturbative treatment [10].Also, it is convenient to introduce a generalized collision frequency ν (ω) in analogy tothe Drude relation [10] ǫ(ω) = 1 − ω2

pl/ [ω (ω + iν (ω))] where ω2pl =

c nce

2c/(ǫ0mc)

is the squared plasma frequency.

By comparison with Eq. (9), we establish an expression for the collision frequencyin terms of correlation functions

ν (ω) =β Ω0

ǫ0ω2pl

limk→0

J z

k, J z

k

ω+iη

− J zk ; J zk ω+iη J zk ; J zk ω+iηJ zk ; J zk ω+iη

, (14)

taking into account that (J zk ; J zk ) = ǫ0ω2pl/β Ω0 is an exactly known quantity. Furtherdetails can be found in Ref. [10]. Making use of Eq. (7), the absorption coefficientcan be expressed as

α(ω) =ω2pl

c

Re ν (ω)

(ω2 − 2ω Im ν (ω) + |ν (ω)|2)n(ω). (15)

In the high frequency limit ω ≫ ωpl, the index of refraction is unity and thecollision frequency is small compared to the frequency ω. Then, we can consider theapproximation

α(ω) =ω2pl

c ω2Re ν (ω) =

β Ω0

cǫ0ω2ReJ z

0, J z

0

ω+iη

, (16)

where the collision frequency is given in the form of a force-force correlation function,cf. Ref. [10]. Thus, the absorption coefficient is directly proportional to the real part of the force-force correlation function, which itself can be determined using perturbationtheory.

As well known, the deviation of the diffraction index n(ω) from unity at fre-quencies near the plasma frequency ωpl is responsible for the so-called dielectricsuppression of the bremsstrahlung spectrum. We refer to the pioneering work of Ter-Mikaelyan [21]. In our approach, making use of Eqs. (8) and (14), the index of refraction can be determined from the force-force correlation function. However, due

105



·

µ

·

¥

¥

µ

·

µ

·

µ

·

µ

·

Figure 1. Diagrammatic representation of GJ J (ωµ). (a) full account of all mediumeffects by a four particle Green function, (b) factorization into two polarization bub-bles, (c) Born approximation, (d) account for successive scattering of electrons onions via electronic spectral function (full electron propagator), (e) vertex correction.

to the choice of the frequency range in consideration (ω ≫ ωpl), this effect will not beconsidered in the present work. We will focus only on the medium effects obtaineddirectly from the evaluation of the force-force correlation function.

3. GREEN FUNCTION APPROACH AND DENSITY EFFECTS

A convenient starting point for a perturbative treatment of the force-force cor-relation function is the representation in terms of a Green function GJ J in the limitk → 0, see Ref. [22]

J z0 , J z0 ω+iη =i

β

∞−∞

dω

π

1

ω + iη − ω

1

ωImGJ J (ω + iη) . (17)

By exploiting Dirac’s identity limη→01

x±iη = P 1x ∓ iπδ(x) , the absorption coefficient

reads in the high-frequency limit

α(ω) =πΩ0

cǫ0ω3ImGJ J (ω + iη) . (18)

The time derivative of the electron current density operator is calculated as

J z0,e =i

h

H, J z0,e

=

ie

meΩ0

pkq

veiq qza†e,p a

†i,k ai,k−q ae,p+q (19)

with the Hamiltonian

H =c,k

E cka†c,kac,k +

1

2

c,d

kpq

vcdq a†c,k+qa†d,p−qad,pac,k (20)

and E ck = h2k2/2mc. The spin is not given explicitly but is included into the single-particle quantum number c. Due to conservation of total momentum of electrons,only electron-ion collisions contribute to Eq.(19). With Eq.(19), we identify theGreen function as a four-particle Green function.




Its diagrammatic representation is shown in Fig. 1 (a). Here, G4 denotes afour-particle Green function that contains all interactions between the consideredparticles [10]. We perform a sequence of approximations by selecting certain diagrams

contributing to G4. Considering the electron-ion interaction determining the forceJ z0,e only in lowest order, i.e. in Born approximation, but the full correlations withinthe electron and ion subsystem, respectively, we are led to Fig. 1 (b) showing onlythose diagrams which can be factorized into two polarization bubbles. Πe denotesthe electronic polarization function, Πi is the corresponding ionic quantity. In thisway, we keep the description of the single scattering event on the level of the Bornapproximation. However, higher order interactions between the full electron and ionsubsystem such as a ladder approximation (t-matrix) are ignored, cf. also Ref. [10].The t-matrix corrections and in particular the reproduction of the Sommerfeld result[6] have been studied in Ref. [10]. Diagrams (d) and (e) account for successivescattering of electrons on ions via the electronic spectral function (d) and the vertexcorrection (e). They will be discussed in subsections 3.B, 3.C and 3.D, while in thefollowing subsection 3.A we discuss the Born approximation.

A. Born approximation

As a simple example and prerequisite for further improvements, we consider theBorn approximation. In this case, G4 is a product of four single particle propagatorsand all single particle propagators are replaced by free propagators. We obtain for theGreen function the diagram given in Fig. 1 (c). For a Maxwellian plasma (f (E e p) =

neΛ3e/2 exp(−βE e p)), we obtain

ImGBornJ J

(ω) =nineΛ3

ee6(1 − e−βhω)

24π3ǫ20h4ω2

dE e p e

−βEe

p×

×−2 E e p

hω + E e p

h2κ2

me

(hω + h2κ2

2me

)2 + 2E e ph2κ2

me

+1

2ln

E e p + hω + E e p2

+ h2κ2

2me E e p + hω − E e p2 + h2κ2

2me

.

(21)

Note, that the spin-degeneracy factor 1/2 for fermions is compensated by a factor 2from the summation over spin variables in the calculation of the correlation functions.Λe = (2πh2/mekBT )1/2 is the thermal de-Broglie wavelength. The inverse screeninglength κ occurs due to the use of a statically screened potential of Debye-Huckel type

veiq = − Zie2

ǫ0Ω0(q2+κ2), with κ2 =

c ncZ

2c e

2/(ǫ0kBT ), to ensure convergence at ω = 0.

For ω = 0, we can consider the Coulomb limit κ = 0. Performing the integration wearrive at Eq. (6). Note, that this result is sometimes also written as

αBorn(ω) =nineΛ3

ee6

24h4ω3ǫ30cπ3

(1 − e−βhω)

∞ 0

dE e p e−βEe

p ln

E e p + hω +

E e p

E e p + hω − E e p

, (22)

see Ref. [23]. It is instructive to study the different terms in Eq.(22) in more de-tail: The integrand contains the distribution function (Maxwell distribution). Fur-thermore, a logarithm that depends on both electron and photon energy appears.

107



¥

·

·

·

·

·

·

·

Figure 2. Lowest order correction to Born approximation. Self-energy (second andthird diagram) and vertex correction from scattering of electrons on ions.

Taken with appropriate prefactors, this logarithm is equal to the differential crosssection for inverse bremsstrahlung in the non-relativistic limit (Bethe-Heitler for-

mula) [19]. Thus, we have a reasonable relation between the absorption coefficientas a macroscopic property and the underlying microscopic process, namely inversebremsstrahlung. The absorption spectrum is obtained through integration of thecross section of the microscopic process weighted with the distribution function of the absorbing particles.

We will now show how the Born approximation can be improved. As alreadymentioned above, improvements based on a more sophisticated description of thesingle scattering process via a t-matrix approach have been obtained recently [10].Effects due to dynamical screening have also been considered. Both effects removethe infrared divergence mentioned before. In this work, we want to include mediumeffects such as the successive scattering of the absorbing particles on ions during

the absorption of the photon, a process becoming more important with increasingdensity. This is also the basic idea of the LPM effect.

The correction in lowest order to Born which takes account of medium effectsin the propagator can be achieved by performing either one self-energy insertion orone vertex correction in the sense of Ward-Takahashi identities. The correspondingFeynman diagrams are given in Fig. 2. The first loop on the r.h.s. of Fig. 2 yieldsthe Born result as shown previously. In the following contributions we shown theself-energy and the vertex correction, respectively. We use i+ to indicate loops withionic propagators.

However, one does not obtain a finite result from this ansatz in the case of

the self-energy correction. Instead, a partial summation of all self-energy termsleading to a spectral function is necessary. The results are presented in subsection3.C. In constrast, the last diagram of Fig. 2 describing the vertex correction gives afinite contribution as shown in subsection 3.D. A full self-consistent treatment of thevertex, i.e. solving the corresponding Bethe-Salpeter equation [18], has not yet beenperformed.

B. Spectral function

We now discuss the influence of multiple scattering of the source particles (elec-trons) on ions. This can be accounted for by using dressed propagators in the calcu-lation of the force-force correlation function, i.e. by replacing the free electron Green




function by the expression

Ge(p, zν) =

∞ −∞

dhω

2π

Ae(p, hω)

zν − hω , (23)

where Ae(p, hω) is the electronic spectral function. According to Dyson’s equa-tion [18], the Green function can be represented by a complex electron self-energyΣe(p, zν). We perform the analytic continuation of the discrete Matsubara frequen-cies into the upper half of the complex energy plane via zν = hω + iη and decom-pose the self-energy into the real and imaginary part, Σe(p, hω + iη) = ∆e(p, hω) −iΓe(p, hω)/2. Then, the spectral function is related to the self-energy:

Ae(p, hω) =Γe(p, hω)

(hω −E e p − ∆e(p, hω))2 + Γ2e(p, hω)/4 . (24)

For the self-energy, we describe the scattering of the electron on an ion by a stat-ically screened ion potential of the Debye type as discussed before. The Hartreeterm vanishes due to charge neutrality. The Fock term of the electron self-energy isnot relevant, since we assume a non-degenerate system. We consider here the self-consistent first loop correction to the Hartree-Fock self-energy due to the electron-ioninteraction. This approximation can be improved by taking into account a partialsummation of further loops, leading to the so-called GW approximation for the self-energy [20]. After analytic continuation of the Matsubara Green function we have

Σe(p, hω + iη) = ni

d3q

(2π)3veiq 2 1

hω + iη −E ep+q − Σe(p + q, hω + iη). (25)

Eq. (25) can be solved numerically by iteration starting from a suitable initialization.From the form of the Debye potential, we note that the main contribution to theintegral in Eq. (25) arises from terms with small momentum q. Therefore, we willdiscuss an approximation where the argument p + q in the self-energy is replaced byp, i.e. Σe(p + q, hω + iη) ≈ Σe(p, hω + iη) on the r.h.s. of Eq. (25). After droppingthe shift of the momentum variable in the self-energy, the q integral can be performedanalytically. The result

Σe(p, hω + iη) = −nimee4

4πǫ20h2

1

κ×

κ2 + p2 − 2

me

h2(hω + iη − Σe(p, hω + iη)) − 2iκ

2me

h2(hω + iη − Σe(p, hω + iη))

−1

(26)

is solved numerically for the real and imaginary part.

In Fig. 3 (a), we show the self-energy, the dispersion relation, and the resultingspectral function obtained from a self-consistent solution of Eq. (26). The spectralfunction in Fig. 3 (a) shows a broadened quasi-particle resonance at the energy hω =

109



0

1

2

(b)

-0.5

0

0.5

-0.5

0

0.5

0.7 0.8 0.9 1 1.1 1.2h_ω [units of Ry]

0

500

1000

0

0.1

0.2

Γ e

( p , ω

)

(a)

-0.05

0

0.05

∆ e

( p , ω

)

-0.5

0

0.5

h_ ω -

E e p - ∆

e ( p , ω

)

0.7 0.8 0.9 1 1.1 1.2h_ω [units of Ry]

0

50

100

A e

( p , ω

)

p = 1 aB

-1

Figure 3. Imaginary and real part of the electrons self-energy Σe( p, hω+iη), disper-sion relation, and spectral function Ae( p, hω) for p = 1 a−1B (a) from self-consistentcalculation cf. Eq. (26) and (b) using free propagators, cf. Eq. (27). The parametersare: Electron density ne = 10−6a−3B , temperature kBT = 27.2 eV.

h2( p2 + κ2)/2me. As expected, its shape is primarily determined by the imaginarypart of the self-energy. This can be seen by a comparison of Ae( p,ω) with Γe( p,ω). Wemention, that analytic constraints on the self-energy function such as Kramers-Kronigrelations as well as the first sum-rule for the spectral function

∞−∞

dhω2π Ae(p, hω) = 1 ,

are fulfilled within the numerically achievable precision.

For the sake of comparison, we discuss a simplified calculation in which weneglect the self-consistent propagator and replace it by a free propagator. Thus, theself-energy on the r.h.s. of Eq. (25) disappears. Then we find

Σ0e(p, hω+iη) = − nime4

4πǫ20h2

1

κ

κ2+ p2−2

me

h2(hω+iη)−2iκ

2me

h2(hω + iη)

−1, (27)

which can be separated into real and imaginary part in the limit η → 0:

∆0e(p, hω) = − nime4

2(2π)2ǫ20h2

π

κ

p2/2 + κ2/2 −meω/h

( p2/2 + κ2/2 −meω/h)2 + 2κ2meω/h,

Γ0e(p, hω) =

πnime4

(2π)2ǫ20h2

2meω/h

( p2/2 + κ2/2 −meω/h)2 + 2κ2meω/h. (28)

From the imaginary part Γ0e(p, hω) we see that the contribution to the spectral

function near the free-particle energy hω = E e p is damped out to a large extent.

These functions as well as the corresponding dispersion relation hω−E e

p−∆0

e

(p, hω)and spectral function are plotted on the r.h.s. of Fig. 4. The spectral functionexhibits two separate peaks corresponding to the roots of the dispersion relation and




no peak at the quasi-particle energy E e p. This contribution from the central root at

the free-particle energy is damped out due to the large value of Γ0e(p, hω = E e p) at

the free particle dispersion. Note also, the order of magnitude change in the damping

Γe between the first iteration shown on the r.h.s. of Fig. 4 and the self-consistentresult shown on the l.h.s. Thus, these structures can clearly be identified as artifactssince they disappear in the self-consistent calculation, which is necessary to obtainrelevant results.

C. Effects of the electron self-energy

The Born approximation can be improved accounting for self-energy and vertexcorrections, see Fig. 2. However, the contribution of the two diagrams containing the

self-energy is diverging so that we performed partial summations of higher orders,leading to the Dyson equation and the spectral function discussed above. In thisway, the free electron propagator is replaced by the full electron propagator whencalculating the force-force correlation function. This approximation exactly reflectsthe point made by Migdal and Landau/Pomeranchuk to account for a finite life-time(damping rate) of electron states propagating in a dense medium.

The force-force Green function is obtained from the evaluation of the diagramshown in Fig. 1 (d):

GΣJ J

(ωµ) =h2nie

6

m2eǫ

20 zν

d3 p

(2π)3 d3q

(2π)3q2z

(q2 + κ2)2×

×∞

−∞

dω′

2π

Ae(p − q, ω′)

zν + ωµ − hω′

∞ −∞

dhω′′

2π

Ae(p, hω′′)

zν − hω′′. (29)

After summation over the fermionic Matsubara frequencies zν and shifting variables,we obtain the imaginary part of GΣ

J J (ω).

ImGΣJ J

(ω) =

=

nie6

6m2eǫ20 d3 p

(2π)3 d3q

(2π)3

(p

−q)2

((p − q)2 + κ2)2×

×∞

−∞

dhω′

2πAe(q, hω′)Ae( p, hω + hω′) (f (hω′ + hω) − f (hω′))

=nie

6h

3m2eǫ

20(2π)5

∞ 0

d p

∞ 0

dqpq

2

ln

( p + q)2 + κ2

( p− q)2 + κ2

− 4κ2 pq

(( p + q)2 + κ2)(( p− q)2 + κ2)

×

×∞

−∞

dω′Ae(q, hω′ + hω)Ae( p, hω′) (f (hω′ + hω) − f (hω′)) , (30)

where the integrals over the angular parts have been performed.

111



0.1h_ω [units of Ry]

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

α Σ ( ω ) / α

B ( ω )

Free propagator

Self-consistent self-energyLorentzian

Figure 4. Correction factor αΣ(ω)/αB(ω) as a function of the photon energy hω withfree propagators in self-energy diagram, the self-consistent spectral function, and aLorentzian ansatz. Parameter values: Electron density ne = 10−6a−3B , temperaturekBT = 27.2 eV .

The further evaluation requires an expression for the spectral function. Note,that in the limit of free particles, where the spectral function is given by a δ-function,the Born approximation, Eq. (21), is recovered. We will use the result obtained above

within our approximation for the self-energy.

The result is shown in Fig. 4. The correction factor αΣ(ω)/αB(ω) (cf. Eq.(18))is plotted as a function of the frequency for three different approximations with theparameters ne = 10−6 a−3B and kBT = 27.2 eV. The full line presents the resultof the self-consistent treatment, Eq. (26), and is compared to a calculation withfree propagators in the self-energy diagrams, see Eq. (27), and a calculation using aLorentzian ansatz of the spectral function with a width taken at the on-shell energyΓ(p, hω = E e p). This corresponds to the introduction of a finite life-time in theapproach of Knoll and Voskresensky [17].

For the lowest frequencies considered here, all approximations show a suppression

of the absorption coefficient in comparison to the Born result. At high frequencies,all curves tend towards unity, i.e., the Born result is recovered. For intermediatefrequencies, an enhancement of up to 35 % is found for the calculation using freepropagators. Making use of the self consistency, the enhancement is reduced to 6 %at most. For the Lorentzian ansatz, no enhancement at all appears. Thus, the widthof the imaginary part of the self-energy as a function of frequency ω (Fig. 3) plays acrucial role for the size of the enhancement. We expect that any increase in the widthof the self-energy will further decrease the enhancement or even lead to a suppressionfor all frequencies as in the case of the Lorentzian ansatz, which corresponds to aninfinite width of the imaginary part of the self-energy. A further broadening of theimaginary part of the self-energy could result from an extension to higher orders inthe set of diagrams for the self-consistent calculation of the spectral function, e.g. byinclusion of vertex terms.




D. Vertex corrections

As known from the Ward-Takahashi identities, self-energy and vertex corrections are

intrinsically related. In particular, if medium corrections arise in a certain order of a small parameter like the density from the self-energy, also contributions from thevertex corrections are expected in the same order. This well-known fact is used toconstruct so-called conserved approximations [24]. Thus, it is necessary to studythe vertex corrections corresponding to the self-energy considered before. However,the solution of the vertex-equation is a technically very challenging task and hasbeen solved so far only in certain approximations [25]. In lowest order, the vertexcorrection is obtained through insertion of one ion-loop inside the electron-loop: Wefind for the imaginary part of the force-force Green function

ImGVJ J

(ω + iη) = nineΛ3ee624π3ǫ20h

3meω

∞ 0

d ppe−βh2 p2/2meΓV( p) ln

E e p +

E e p + hω

E e p − E e p + hω

+

+

∞ √

2meω/h

d p p e−βh2 p2/2meΓV( p) ln

E e p +

E e p − hω

E e p − E e p − hω

,

(31)

with the vertex part

ΓV

( p) =

π2nie4m

(2π)3h2ǫ20κp2 , (32)

taken in lowest order in κ. This expression can be evaluated numerically. In the limitof high frequencies, where the second integral becomes negligible compared to the firston, the absorption coefficient is proportional to K 1(ω)/ω4. Since the K 1 function hasthe same asymptotic behavior as the K 0 function, which appeares in the Born resultfor the absorption coefficient Eq. (6), the ratio αV/αB (cf. Eq. (18)) behaves like1/ω in the high frequency limit. For all frequencies considered, a suppression withrespect to the Born approximation is found. For the considered energy range, thecorrections are small and decrease with increasing energy. Using the set of parametersne =

·10−6 cm−3 and kBT = 27.2 eV, the expected high frequency behavior

∝ω−1

arises for energies larger than 1 Ry. The corrections are small for low densitiescompared to higher densities.

We consider the absorption coefficient α(ω) including all of the improvements.Since the Born approximation is already included in the self-energy contributionαΣ(ω) we have

α(ω) = αV(ω) + αΣ(ω). (33)

The relative change α(ω)/αB(ω) is presented in Fig. 5. For the sake of comparison,the self-energy correction is shown as well. For small frequencies, the self-energy con-tribution and the vertex contribution add to a net suppression. For higher energies,the self-energy term shows an enhancement, which is partially compensated by thevertex. However, the net result is still an enhancement. In the high frequency limit,the Born result is reproduced.

113



0.01 0.1 1h_ω [units of Ry]

0.8

0.9

1

1.1

α ( ω ) / α

B ( ω )

Self-energy

Self-energy + vertex

Figure 5. Total absorption coefficient taken relative to the Born result αV(ω)/αB(ω)as a function of the photon energy hω. Parameter values: Electron density ne =10−6a−3B , temperature kBT = 27.2 eV.

4. CONCLUSIONS

In this paper, we have studied the influence of the surrounding medium on thebremsstrahlung spectrum in non-ideal plasmas. The interaction with the mediumleads to a finite life-time of the electron states. Instead of free quasi-particles, thespectral function has to be used to describe the electron properties in the medium.Thus, the use of the single-particle spectral function is a quantum-statistically soundimplementation of the original idea of successive scatterings by Landau/Pomeranchukand Migdal.

Our approach, namely the microscopic treatment of the dynamical self-energy,extends a recent work of Knoll and Voskresensky [17], where a Lorentzian ansatzfor the spectral function with a frequency-independent quasi-particle lifetime was

considered. The Lorentzian ansatz for the spectral function was discussed above insubsection 3.C., taking the imaginary part of the self-energy at the quasi-particleenergy. Then, a suppression of the bremsstrahlung spectrum was observed. In gen-eral, the microscopical treatment leads a frequency-dependent imaginary part of theself-energy, and, according to the Kramers-Kronig relation, to a non-vanishing realpart. In particular, the inclusion of the real part of the self-energy in the spectralfunction influences the medium modification of the bremsstrahlung spectrum.

In the present paper, the one-loop approximation was taken for the self-energy.It has been shown that a self-consistent treatment has to be used in order to avoidunphysical artifacts which arise, if instead of the full propagator the free propagatoris taken to evaluate the self-energy. This is already known from the treatment of thespectral functions in plasma physics in the so-called GW approximation [20] wherethe interaction with the medium is implemented by a screened potential. It should




be mentioned that in this case a self-consistent treatment of the spectral function onthe level of the GW approximation has been performed [26]. Any iterative solutionstarting from the free propagator leads to non-physical structures in the spectral

function.Within our approach, we found a switch from a suppression at low frequencies

to an enhancement at high frequencies for the bremsstrahlung spectrum. The ap-proximation for the self-energy can be improved by considering further diagrams. Inparticular, the vertex correction would be of interest which modifies the couplingto the interaction potential. The inclusion of vertex corrections is also necessary toobtain conserved approximations and has been shown in Subsection 3.D., where afurther suppression of the bremsstrahlung spectrum was observed. The modificationof the bremsstrahlung spectrum by the surrounding medium is sensitively dependenton the approximation used.

We cannot elaborate further on the switch from suppression to enhancementseen in the self-energy correction. In order to verify the existence of such a switch,higher order calculations are necessary. A consistent procedure would consist in a)using the full propagator in the calculation of the vertex correction, b) solving thefull vertex equation with full propagators and finally c) solve the Dyson equation forthe single particle propagator with the solution of the vertex equation.

The importance of vertex corrections in the self consistency relations has alsobeen shown in the description of the spectral function of the homogeneous electrongas. There, notable differences between a so-called GWΓ approximation [25] includ-ing vertex terms and a GW approximation [27] arise. It should be mentioned thatself-consistent Schwinger-Dyson equations for the self-energy have been consideredin field theory [28] to find solutions for the QCD running coupling problem. A cor-responding treatment would lead to a better description of the modification of thebremsstrahlung spectrum in a dense medium, but would exceed the frame of thepresent work. Also, at low frequencies, further effects such as the dielectric suppres-sion are of importance. It has not been considered in this approach, but can easilybe obtained from the force-force correlation function as well.

ACKNOWLEDGEMENTS

We would like to thank J. Knoll, D. Voskresensky and V. Morozov for stimulatingdiscussions. C.F. would also like to thank the Gesellschaft fur Schwerionenforschung(GSI) for its hospitality and the Studienstiftung des Deutschen Volkes for a scholar-ship.

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115



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[10] H. Reinholz, R. Redmer, G. Ropke, and A. Wierling, Phys. Rev. E 62, 5648(2000), A. Wierling, Th. Millat, G. Ropke, and R. Redmer, Phys. Plasma 8,3810 (2001).

[11] L. D. Landau and I. J. Pomeranchuk, Dokl. Akad. Nauk SSSR 92, 535 (1953), A.B. Migdal, Phys. Rev. 103, 1811 (1956).

[12] B. G. Zakharov, JETP Lett. 64, 781 (1996).

[13] A. V. Koshelkin, J. Phys. A 35, 8763 (2002).

[14] S. Klein, Rev. Mod. Phys.f 71, 1501 (1999).

[15] J. F. Bak, Nucl. Phys. B 302, 525 (1988).

[16] H. D. Hansen et al., Phys. Rev. D 69, 032001 (2004).

[17] J. Knoll and D. Voskresensky, Ann. Phys. 249, 532 (1996).

[18] W. D. Kraeft, D. Kremp, W. Ebeling, and G. Ropke, Quantum Statistics of

Charged Particle Systems (Akademie-Verlag, Berlin, 1986).[19] J. D. Jackson, Classical Electrodynamics (Wiley & Sons, New York, 1975).

[20] G. D. Mahan, Many-Particle Physics (Plenum Press, New York and London,1981).

[21] M. L. Ter-Mikaelyan, Dokl. Akad. Nauk SSSR 94, 1033 (1953).

[22] D. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium Processes (Akademie-Verlag, Berlin, 1996), Vol. 2.

[23] I. H. Hutchinson, Principles of Plasma Diagnostics (Cambridge University Press,Cambridge, 1987).

[24] G. Baym and L. Kadanoff, Phys. Rev. 124, 287 (1961).[25] Y. Takada, Phys. Rev. Lett. 87, 226402 (2001).

[26] A. Wierling and G. Ropke, Contrib. Plasma Phys. 38, 513 (1998).

[27] B. Holm, Phys. Rev. Lett. 83, 788 (1999).

[28] C. D. Roberts and S. M. Schmidt, Prog. Part. Nucl. Phys. 45, S1 (2000).




Chapter 6

Bremsstrahlung vs. Thomsonscattering in VUV-FEL Plasma

Physics Experiments

Authors:Carsten Fortmann, Ronald Redmer, Heidi Reinholz, Gerd Ropke, August Wier-ling, and Wojciech Rozmus

Appeared as regular article in Journal of High Energy Density Physics, Volume 2, pages57-69, May 2006.


• C.F.: Preparation of manuscript, all numerical calculations

• R.R.: Preparation of manuscript

• H.R.: Preparation of manuscript

• G.R.: Preparation of manuscript

• A.W.: Preparation of manuscript

• W.R.: Code for Thomson scattering calculations



118 Bremsstrahlung vs. Thomson Scattering



A u t h o r

' s p e r s

o n a l c

o p y

Bremsstrahlung vs. Thomson scattering in VUV-FEL plasma experiments

C. Fortmann a,*, R. Redmer a, H. Reinholz a, G. Ropke a, A. Wierling a, W. Rozmus b

a Institute for Physics, Rostock University, 18051 Rostock, Germanyb Theoretical Physics Institute, University of Alberta, Edmonton, Canada T6G 2J1

Received 21 February 2006; received in revised form 13 April 2006; accepted 14 April 2006

Available online 15 May 2006

Abstract

We determine the spectral photon yield from a hot dense plasma irradiated by VUV-FEL light in a Thomson scattering experiment. The

Thomson signal is compared to the emission background mainly caused by bremsstrahlung photons. We determine experimental conditions

that allow for a signal-to-background ratio larger than unity. By derivation of the Thomson and the bremsstrahlung spectrum from linear

response theory we present a consistent quantum statistical approach to both processes. This allows for a systematic treatment of medium

and quantum effects such as dynamical screening and strong collisions. Results are presented for the threshold FEL-intensity as a function

of density and temperature. We show that the account for quantum effects leads to larger thresholds as compared to previous work.

Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Thomson scattering; Bremsstrahlung; Free electron laser; Plasma diagnostics; Threshold intensity; Dielectric function; Dynamic structure factor

1. Introduction

Thomson scattering is a well-established technique for ex-

perimental investigation of plasma parameters. Examples can

be found in Refs. [1e6]. Observables like particle density, tem-

perature, composition, and degree of ionization can be spatially

and temporally resolved by analysis of the scattering spectrum

[7]. Until recently, coherent light sources have been availableonly for the visible and near UV part of the electromagnetic

spectrum. Due to small critical density ncrit ¼u2e0me /

e2z 1020 cmÀ3 of free charge carriers for optical probes,the ap-

plicability of Thomson scattering using coherent sources has

been limited to targets of relatively low density.

Glenzer et al. [8,9] have shown andexploredthe possibility of

X-ray Thomson scattering in solid density targets using the Ti

He-a line at 4.75 keV as probe light [10]. A new alternative

emerged with the development of VUV-free electron lasers

(VUV-FEL), providing pulses of coherent radiation in the far

(vacuum-) ultraviolet. At the moment, the VUV-FEL at DESY

Hamburg operates at 32 nm wavelength [11], corresponding to

38 eV photons. With this coherent light source, dense matter

up to solid densities of 1023 cmÀ3 can be penetrated, see Refs.

[12,13]. Under these conditions, the Thomson spectrum permits

the determination of electron temperature and density directly

from the position and height of collective resonances, i.e. plas-

mons, showing up in the scattering signal [9]. First experimentswill be performed in the near future at the VUVeFEL facility at

DESYat l¼ 32 nm FEL wavelength,while in later stages of the

project, wavelengths from 13 nm (VUVeFEL) down to 0.1 nm

(X-FEL) will be available.

Due to the large number of free charge carriers at the tem-

peratures and densities considered, thermal bremsstrahlung

emission, resulting from inelastic freeefree scattering, con-

tributes significantly to the emission background. Therefore,

experimental conditions such as scattering angles, spectral

properties of the probe and the detector have to be chosen as

to obtain a maximum signal-to-background ratio. Background

is to be understood as bremsstrahlung radiation, whereassignal corresponds to the photons having undergone Thomson

scattering.* Corresponding author.

E-mail address: [email protected] (C. Fortmann).

1574-1818/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.hedp.2006.04.001

High Energy Density Physics 2 (2006) 57e69www.elsevier.com/locate/hedp

119



A u t h o r

' s p e r s

o n a l c

o p y

So far, classical formulas for the bremsstrahlung emission

level going back to Kramers [14] have been used to determine

threshold intensities of the external source to overcome thebackground due to bremsstrahlung [15,16]. The Kramers re-

sult, given below in Eq. (19), is derived from the assumption

of Keplerian trajectories of the emitting electron in the Cou-

lomb field of an ion and integration over all initial velocities

weighted with the Maxwellian velocity distribution function.

Quantum features are only accounted for in a semiclassical

way: the velocity integral extends over velocities y, fulfilling

the condition my2 /2 ! hu, i.e. the kinetic energy has to be

larger than the photon energy. Further quantum properties,

such as the finite photon momentum as well as the quantum

mechanical nature of the scattering process are not accounted

for. By comparing the Thomson signal strength at the laserwavelength l¼ 14.7 nm to the bremsstrahlung photon yield

calculated from Kramers formula, Baldis et al. found threshold

intensities of 1013 W/cm2 for typical values of free electron

density ne ¼ 1022 cmÀ3 and temperature k BT ¼ 100 eV [16].

In this work, we evaluate threshold conditions (intensities)

using improved expressions for the bremsstrahlung spectrum.

As usual, corrections to Kramers formula for bremsstrahlung

are described by the so-called Gaunt factor [17]. In the sim-

plest approach it is obtained by taking into account collisions

between electrons and fixed ions in Born approximation. In

dense plasmas, many-particle effects as dynamical screening

become important. Moreover, strong collisions have to be ac-

counted for. We show in this paper how the Gaunt factor canbe derived from linear response theory [18] in a general way.

Within this framework, modifications of the emission spec-

trum beyond Born approximation can be included in a system-

atic manner [19] as will be discussed later.

We then apply our formulas to determine the threshold in-

tensities for a broad range of experimental parameters (wave-

length, spectral properties of detectors, and different

materials), relevant for future experiments at DESY. Further-

more we compare Thomson and bremsstrahlung photon yield

over a finite spectral range. Thereby, and by taking improved

expressions for the bremsstrahlung cross section, we show that

even higher thresholds have to be reached in order to obtaina Thomson signal above the bremsstrahlung level at-least

near the plasmon resonances. These peaks are much lower

than the central peak, being essentially an ion feature.

The present work is organized as follows: in the first sec-

tion we review the basic physics of Thomson scattering and

bremsstrahlung and how they can be expressed in terms of

the dynamic structure factor and the dielectric function, re-

spectively. Since these two quantities are related to each other

via the fluctuationedissipation theorem [20], we are able to

describe both processes on a common and consistent basis.

We then compare the emission level due to bremsstrahlung to

the Thomson signal whose strength is proportional to the flux of

incoming photons, i.e. the power density of the external source.

Thereby, we find expressions for the threshold power density as

a function of particle density and temperature. In the last section

we discuss our results for various sets of experimental parame-

ters relevant for future experiments at the VUV-FEL.

2. Thomson scattering and bremsstrahlung

The central quantity of interest is the spectral power densityd P /dV dl dU, i.e. the rate of energy radiated per unit scattering

volume dV , wavelength dl, and solid angle dU. The total spec-

tral power density is the sum of the corresponding quantity for

every radiative process in the plasma. In this work we focus on

Thomson scattering and bremsstrahlung, i.e.

d3 Ptot

dV dl dU¼ d3 PTh

dV dl dUþ d3 Pbr

dV dl dU: ð1Þ

To unambiguously identify the Thomson signal, we require

that the Thomson power density is at least equal to the brems-

strahlung level,

d3 PTh

dV dl dU! d3 Pbr

dV dl dU: ð2Þ

The Thomson spectrum is given by the intensity of the probe

laser I L and the Thomson scattering cross section d2sTh /dudU.

To account for the finite spectral bandwidth of the detector,

one has to convolute each power spectrum with a detector

function G(l). In practice, this is only relevant for the Thomson

signal since the bremsstrahlung spectrum is slowly varying in

the relevant frequency region. The Thomson power spectrum

reads

d3 PTh

ðl

ÞdV dl dU ¼I

LZ dlG

ðlÀlÞ

d2sTh

ðul

ÞdU du ¼I

L R

ðlÞ: ð

3Þ

We have introduced the response function RðlÞ, where the

bar denotes the convolution with the detector function. Note

that we assume an optically thin plasma, thus radiation trans-

port is neglected. Also, due to the short pulselength of the

VUV-FEL (20e120 fs), we neglect the heating of the plasma

due to the probe beam.

The bremsstrahlung spectrum does solely depend on the

plasma parameters density and temperature, so that, with

a suitable formula for the bremsstrahlung spectrum, which

we will abbreviate by the notation d3 Pbr /dV dl dUh j (l) in

the following, Eq. (2) defines a threshold intensity

I threshðlÞ ¼ j ðlÞ RðlÞ: ð4Þ

We will now briefly describe how expressions for the

Thomson scattering and the bremsstrahlung spectrum can be

obtained from a common starting point, i.e. the dielectric func-

tion of the plasma.

2.1. Thomson scattering

The cross section for Thomson scattering in a plasma can

be given in terms of the dynamic structure factor (DSF),Sð k;uÞ:

d2sThð k;uÞdUdu

¼

dsðUÞdU

Th

k 1

k 0Sð k;uÞ; ð5Þ

58 C. Fortmann et al. / High Energy Density Physics 2 (2006) 57 e69




A u t h o r

' s p e r s

o n a l c

o p y

see Refs. [21,22] for details. The variables ð k;uÞ are related to

the transferred linear momentum and energy, respectively,

while k 0 ¼u0 / c and k 1 ¼ j k0 þ kj are incoming and outgoinglinear momenta of the laser field, respectively. (ds(U)/dU)Th is

the Thomson scattering cross section for the isolated scattering

event. It is given by the KleineNishina formula, its derivation

can be found in standard textbooks of quantum electrodynam-

ics, e.g. Ref. [23].

In the nonrelativistic limit, the Thomson cross section for

linearly polarized light is given byds

dU

Th

¼ r 2e2

À1 À cos2 4 sin2 q

Á: ð6Þ

Here, r e ¼ e

2

/4p

e0mec

2

is the classical electron radius, q is thescattering angle between k1 and k0, and 4 is the angle enclosed

between the plane of polarization and the scattered wave-

vector k1. For unpolarized light, averaging over all possible

polarizations yields ðds=dUÞTh ¼ r 2e ð1 þ cos2 qÞ2. For the

VUV-FEL, the light is linearly polarized and Eq. (6) should

be applied. However, details of the experimental setup with re-

spect to the direction of the polarization plane are still open.

Henceforth, we use the unpolarized Thomson cross section,

thereby eliminating the dependence on the parameter 4. Fur-

thermore, with regard to the comparison of Thomson scatter-

ing to bremsstrahlung, we keep the estimation of the

Thomson signal as restrained as possible.

Eq. (5) shows that plasma collective properties can be mea-sured in a Thomson scattering experiment. However, this re-

quires an accurate theory of the dynamic structure factor.

The dynamic structure factor Sð k;uÞ is closely related to the

longitudinal dielectric function by the fluctuationedissipation

theorem [24]

Sð k;uÞ ¼ À e0hk 2

pe2ne

Im eÀ1l ð k;uÞ

1 À expð À hu=k BT Þ : ð7Þ

This relation can be utilized by applying an appropriate

approximation to the dielectric function. Due to the mass ratio

me=mi ( 1, Thomson scattering on ions can be neglected.Therefore, we will approximate Sð k;uÞ by Seeð k;uÞ, the elec-

tronic DSF.

2.2. Bremsstrahlung

Radiative freeefree transitions of electrons, known as

bremsstrahlung, represent the main source of emission in

a hot, fully ionized plasma. However, the bremsstrahlung pro-

cess requires the presence of a scattering partner that carries

the recoil momentum. Kinematically, emission by electrons

which scatter on ions is dominant.

For a thermally equilibrated plasma, emission, character-

ized by the spectral power density j (u), and the absorption

coefficient a(u) as the relative attenuation of the intensity

of electromagnetic waves propagating in the medium per

unit length, are linked by Kirchhoff’s law [25]

j (u) ¼ L(u)a(u), with the Planck distribution LðuÞ ¼ hu3=

½4p3c2ðexpðhu=k BT Þ À 1Þ. The wavelengths which are of in-

terest here are in the 10e100 nm-range and large against

atomic length scales of a few A, so that the long wavelengthlimit k / 0 is applied throughout this paper. Therefore, the

emissivity j (u) and absorption a(u) are only functions of fre-

quency u in this limit. For effects of nonlocality on the brems-

strahlung spectrum, we refer to Ref. [26]. The absorption

coefficient can be obtained from the imaginary part of the

transverse dielectric function according to

aðuÞ ¼ u

c

Im et ðuÞnðuÞ ; ð8Þ

where the index of refraction n(u) is also linked to the trans-

verse dielectric function by

nðuÞ ¼ 1 ffiffiffi2

p ðRe et ðuÞ þ jet ðuÞjÞ1=2: ð9Þ

In the long wavelength limit, the longitudinal dielectric

function, which appears in the fluctuationedissipation theo-

rem (7) and the transverse dielectric function coincide [27].

Thus, relations (7) and (8) enable us to treat both Thomson

scattering and bremsstrahlung on a common basis, namely

by using an appropriate theory for the longitudinal dielectric

function.

2.3. Consistent approximations

The aim of this work is to compare both radiation pro-

cesses, Thomson scattering and bremsstrahlung, calculated

in a consistent approximation. The comparison has to be car-

ried out between the contributions of either process in leading

order of density.

Bremsstrahlung occurs in second order of the coupling con-

stant aQED ¼ 1/137 as can be seen from the transition ampli-

tude wbrfi expressed by Feynman diagrams,

giving the BetheeHeitler cross section, see Ref. [23]. The

transition amplitude involves a longitudinal field (Coulomb

field) i.e. a scattering partner, say an ion of effective charge

Zeff . Thus, bremsstrahlung is naturally of second order in

density. A free electron does not emit bremsstrahlung, un-

less collisions take place. The challenge is then to accu-

rately describe the scattering process itself.

Born approximation, as given by Eq. (10) does not describe

the correct behaviour of the bremsstrahlung spectrum in the

case of collisions involving high transfer momenta, so-called

strong scatterings. These can be included by ladder-summa-

tion of all one-photon exchange processes, which leads to

the t -matrix. Thereby, the electroneion interaction is treated

accurately in all orders. In the Coulomb limit, one finds

59C. Fortmann et al. / High Energy Density Physics 2 (2006) 57 e69

121



A u t h o r

' s p e r s

o n a l c

o p y

4.1. Born approximation

As the lowest order contribution with respect to the interac-tion potential, Fig. 1(b) shows the Born approximation. This

approach reproduces the well-known BetheeHeitler formula

in the nonrelativistic limit:

Ecp is the free particle energy, for nonrelativistic particles

Ecp ¼ h2 p2=2mc holds. Details of the calculation are dis-

cussed in Ref. [29]. In Eq. (17) the effective ion charge

Zeff is used to account for the screening of the charge Ze

of the nucleus due to inner shell electrons. In Ref. [16],

Zeff is calculated as a function of temperature and density

in the framework of ThomaseFermi theory, see Ref. [30]

for details. Alternatively, Zeff can be obtained in the so-

called chemical picture by solving Saha’s equation, seeRef. [31]. Especially for high temperatures, the chemical

picture gives more reliable results, as the ionization equilib-

ria between different ionization stages are satisfied. We will

therefore use the chemical picture in this work and calcu-

late the mean ionization level with COMPTRA04 [31].

The ThomaseFermi model will only be applied for the pur-

pose of comparison of results obtained by Baldis et al. [16].

The logarithm in Eq. (17) is the nonrelativistic limit of the

BetheeHeitler cross section [32,33]. The drawback of this re-

sult is the divergence of the bremsstrahlung spectrum in the

limit u/ 0. However, physically this is of no importance,

since for low frequencies the index of refraction is different

from unity and modifies the spectrum significantly. This is

known as dielectric suppression [34].

It is in common use to write formulas for the emission and

absorption due to (inverse) bremsstrahlung in terms of

Kramers classical result [14]

j K

ðu

Þ ¼

8 Z 2eff neni

3mec3

e2

4pe0

3

b

6p

me

1=2

exp

ðÀbhu

Þ;

ð19

Þmultiplied with a correction factor gff ðuÞ, called Gaunt factor

[17], which takes into account medium effects as well as quan-

tum corrections, i.e.

j ðuÞ ¼ j K ðuÞ$gff ðuÞ: ð20Þ

with Eq. (18), the Gaunt factor in Born approximation reads

gBff ðuÞ ¼

ffiffiffi3

p

pexpðbhu=2Þ K 0ðbhu=2Þ: ð21Þ

4.2. Strong collisions

The Born approximation assumes free particles as the in

and out states in the scattering amplitude. Taking the true

scattering states, i.e. Coulomb wavefunctions, leads to the

so-called Sommerfeld formula [28],

gSff ðu;T Þ ¼ 1

k BT

Z N0

d Ei eÀ Ei=k BT gSff ðu; EiÞ; ð22Þ

with

gSff ðu; EiÞ ¼ 4

ffiffiffi3

p

p

h2i þ h2

f þ 2h2i h

2f

2hihf

I 0

À À1 þ h2

i

Á1=2

1 þ h2f

1=2

I 1

! I 0; ð23Þ

and

I l ¼1

4

"4 ffiffiffiffiffiffiffiffiffi

Ei Ef

p

À ffiffiffiffiffi Ei

p À ffiffiffiffiffi Ef

p Á

2

#l þ1 ffiffiffiffiffi

Ei


p

ffiffiffiffiffi Ei

p þ ffiffiffiffiffi

Ef

p

iðhiþhf ÞepjhiÀhf j=2

ÂGðl þ 1 þ ihiÞGÀl þ 1þ ihf

ÁGð2l þ 2Þ

Â 2 F1

l þ 1À ihf ; l þ 1 À ihi;2l þ 2;

À 4 ffiffiffiffiffiffiffiffiffi

Ei Ef

p À ffiffiffiffiffi

Ei


p Á2

; Ef ¼ Ei À hu: ð24Þ

h2i=f ¼ Z 2eff Ry= Ei=f is the Sommerfeld parameter with the Ryd-

berg energy Ry ¼ mee4=2ð4pe0hÞ2x13:6 eV; 2 F1ða; b; c; d Þ, is

the hypergeometric function [35] and G( x ) is the Gamma func-

tion. As shown in Ref. [19], Sommerfeld’s expression is also

obtained by a t -matrix ladder-summation with a statically

screened Debye potential (Fig. 1(c)) if the limit of vanishing

inverse screening length (Coulomb limit) is taken. It gives

the correct Gaunt factor in the low density limit, which is con-

sidered here.

Fig. 1. Diagrammatic representation of G_ J _ J

Àum

Á. (a) full account of all me-

dium effects by a four-particle Green function, (b) Born approximation, (c)

t -matrix approximation.


123



A u t h o r

' s p e r s

o n a l c

o p y

Fig. 2 shows the dependence of the Gaunt factor on the

photon energy for electron temperatures k BT ¼ 10 eV,

100 eV and 1000 eV. Over the large energy interval shown,

the Gaunt factor in either calculation for a fixed temperature

does not vary much and is of order unity, thus the widely

used approximation to set gff ðuÞz1 is justified for estimates

of the emission level as done in Refs. [15,16]. More specific,

gff ðuÞ ¼ 1 is a good approximation for photon energies com-

parable to the temperature -u=k BT x1 and for temperatures in

the vicinity of the ionization energy (k BT x Z 2Ry for hydro-genic systems). If one of these conditions is not met, one

should use the full Sommerfeld expression (22) or appropriate

approximations, see the detailed discussion in Ref. [36]. For

the VUV-FEL experiments, both requirements are satisfied

only roughly: the laser provides photon energies of

hux40 eV and the optical pump laser will excite the plasma

to temperatures of k BT x10.50 eV, which is of almost the

same order of magnitude as the first ionization energies of alu-

minum (6 eV, 19 eV, 28 eV), or hydrogen (13.6 eV) [37], ma-

terials presumably used as target in the experiments. Thus we

take into account the Gaunt factor in t -matrix approximation

(Eq. (22)) in our calculations.In Ref. [19], a consistent treatment of the impact of dynam-

ical screening and strong collisions on bremsstrahlung based

on the GouldeDeWitt scheme [38] is given. It is shown that

the high frequency behaviour of the Gaunt factor is dominated

by the t -matrix contribution, whereas dynamical screening can

be neglected as long as the considered frequencies are large

compared to the plasma frequency.

5. Dynamic structure factor

The total dynamic structure factor is defined as [39]

Sccð k;uÞ ¼ 12p N

Z þN

ÀN

dt eiut hrcð k; t ÞrcðÀ k;0Þi; ð25Þ

h.i denotes the ensembleequilibrium average. Here, our discus-

sion is focused on the electron structure factor Seeð k;uÞ. The

fluctuationedissipation theorem (7) in connection with Eq. (12)

allows to express the electronic part of the dynamical structure

factor via the electronic dielectric susceptibility ceeð k;uÞ,

Seeð k;uÞ ¼h

pne

Im ceeð k;uÞ1 À expð À hu=k BT eÞ

: ð26Þ

We will consider the classical limit, where the denominator

of Eq. (26) can be approximated by hu / k BT e. In RPA, the fol-

lowing equation for the susceptibility holds [40]:

cRPAcc0 ð k;uÞ ¼ c0

cð k;uÞdcc0 þ c0cð k;uÞV sc

cc0ð k;uÞc0c0ð k;uÞ: ð27Þ

V scð k;uÞ is the screened interaction potential which acts

between particles of species c and c0. It satisfies the equation

V sccc0ð k;uÞ ¼ V cc0ð kÞ þ

Xd

V cd ð kÞc0d ð k;uÞV sc

dc0ð k;uÞ: ð28Þ

The free susceptibility c0cð k;uÞ is obtained from Eq. (14),

taking the currentecurrent correlation function only in zeroth

order with respect to the interaction, i.e.

c0cð k;uÞ ¼ ibU0

k 2

uqcqc0

D J zk ;c; J zk ;c0

E0

uþih

¼ dcc0X

p

f c pþ k=2 À f c

pÀ k=2

Ec pþ k=2 À Ec

pÀ k=2 À hðuþ ihÞ: ð29Þ

f c p is the momentum distribution function of species c. Again,

a small but finite imaginary frequency ih, h> 0 has to be intro-

duced, in order to fix the sign of the imaginary part and to obtain

convergent results. In the case of a classical two-component

plasma with different temperatures T c, we take the Maxwell dis-

tribution. In the limit -/ 0 the resulting susceptibility reads

c0;clc ðk ;uÞ ¼ ÀU0ncW ð x cÞ=k BT c; ð30Þ

with the plasma dispersion function

W ð x cÞ ¼ 1 À 2 x c eÀ x 2cZ

x c

0

et 2

dt À i ffiffiffiffipp x c eÀ

x 2c

; ð31Þ

and the dimensionless variable x c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2mc=2k 2k BT c

p . Eqs.

(27) and (28) can be solved algebraically for the RPA-suscep-

tibilities cRPAcc0 ð k;uÞ as shown in Appendix B, see also

Ref. [40]. Evaluation of Eq. (26) with cRPAee ð k;uÞ, see Eqs.

(29) and (B.5), yields the electronic DSF in RPA, i.e.

Seeð k;uÞ ¼ 1 þ Z eff a

2ðT e=T iÞW ð x iÞ1 þ a2W ð x eÞ þa2 Z eff ðT e=T iÞW ð x iÞ

2 x e exp

ÀÀ x 2eÁ

u ffiffiffiffip

p

þ Z eff Àa2W ð x eÞ

1 þ a2W ð x eÞ þ a2 Z eff ðT e=T iÞW ð x iÞ2

Â x i expÀÀ x 2i

Áu ffiffiffiffip

p : ð32Þ

Here, the scattering parameter

1

2

3

g f f ( ω

)

_

10 100 1000

hω [eV]

10 eV

10eV

100 eV

1000 eV

100 eV

1000 eV

BornSommerfeld

Fig. 2. Gaunt factor gff ðuÞ in Born approximation Eq. (21) and using Sommer-

feld’s formula Eq. (22), respectively. Results are presented as a function of the

photon energy -u for various temperatures.





A u t h o r

' s p e r s

o n a l c

o p y

a ¼ ke=k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nee2=e0k BT ep =k ð33Þ

has been introduced. It separates the regime of collective

Thomson scattering ða[1Þ, where collective resonances

(plasmons) appear in the spectrum, and the regime of noncol-

lective Thomson scattering ða ( 1Þ, where the spectrum

reflects the single-particle distribution function of electrons

without further structures. Expression (32) was also used by

Baldis et al. [16].

Eq. (32) for the DSF contains two terms: the first term is

basically the DSF of a free electron gas. However, it also con-

tains the ionic dispersion function, which accounts for the

screening effect of ions. The second term gives the scattering

signal from electrons which are close to ions and are thereforedetermined by the dynamics of these heavy particles. This

term is dominant at small u where W ( x e) can be approximated

by its static limit, i.e. W ð x e ( 1Þ/1. For larger values of u,

the first term dominates the spectrum, because the ion part is

damped out more rapidly than the electron part. In particular,

plasmon resonances, which appear in the spectrum due to

a vanishing real part of the denominator, only survive in the

first term, while they are damped out in the second. In so-

called Salpeter approximation [41] the DSF (32) is separated

into two terms depending only on electronic and ionic vari-

ables, respectively, electroneion correlations are neglected

to a large extent. Therefore, the full expression (32) should

be used, as done throughout this work.

Chihara [22] gives the DSF in terms of the local field cor-

rection factor, thereby including electroneion collisions. In

appropriate limits, his expression coincides with Eq. (32),

see Ref. [21]. On the other hand, due to the equivalence of

the local field correction and the collision frequency in the

high frequency limit, Chihara’s expression leads to the same

formula for the absorption coefficient a(u) as given in Eq.

(8) [42]. Thus, Chihara presents an alternative approach to

the question of photoabsorption and emission and Thomson

scattering in plasmas starting from the DSF, whereas in this

work the dielectric function is the central quantity.

Having the thermal emission spectrum (20) and thedynamic structure factor (32) at our disposal, we are now in

the position to compare the signal from Thomson scattering

to the bremsstrahlung background thereby treating both pro-

cesses on a common basis.

6. Results

We now compare power spectra for Thomson scattering

and bremsstrahlung emission. Figs. 3e6 show results for dif-

ferent combinations of electron density and temperature for

a hydrogenic plasma. We consider densities ne

¼1020 cmÀ3

and 1022 cmÀ

3 and temperatures of 10 eV and 50 eV as exam-

ples for possible experimental conditions with cryogenic

targets. Furthermore, backscattering geometry (scattering

angle 120) is chosen and a VUV laser wavelength of

l¼ 32 nm is assumed.

To model the detector, we assume a detector function G(l)

of Gaussian shape, its width given by the relative spectral

bandwidth Dl / l¼Du / u [16]

GðlÞ ¼ 1 ffiffiffiffiffiffiffiffiffiffiffi2ps2

p eÀl2=2s2

; s¼ 0:425Dl: ð34Þ

For the case of bremsstrahlung, we neglect the effect of the

finite bandwidth of the detector, since the corresponding spec-

trum is only slowly varying with frequency. The effective ion

charge Z eff , calculated with COMPTRA04 [31] is close to

Z eff ¼ 1 in the case of hydrogen for the present values of elec-

tron density and temperature.

In Fig. 3, the black dashed curve represents the pure Thom-

son signal, i.e. no convolution with the detector function G(l)

has been performed. For the present parameters

ne ¼ 1020 cmÀ3 and k BT e ¼ 10 eV, the Thomson spectrum con-

tains a very narrow ion peak, situated at the laser wavelength,

and two satellites, which can be identified as electronic plas-

mon resonances. The central peak dominates these electronic

resonances by a factor of 10, approximately. The solid black

30 31 32 33 34

λ [nm]

0

4×105

3×105

2×105

1×105


/ c m

3 n

m s r ]

Bremsstr. (Kramers)

Bremsstr. (Born)


Thomson, not convoluted

Thomson ( I L=5x1010

W/cm2)

Fig. 3. Emission power spectra for Thomson scattering and bremsstrahlung.

The black dashed curve represents the unconvoluted Thomson spectrum. Pa-

rameters: k BT e ¼ 10 eV, ne ¼ 1020 cmÀ3, laser wavelength l¼ 32 nm, scatter-

ing parameter a¼ 1.25, Z eff ¼ 1.

30 31 32 33 34

λ [nm]

0

4×109

2×109

6×109

8×109


/ c m

3 n

m s r ]

Bremsstr. (Kramers)

Bremsstr. (Born)


Thomson ( I L=1013

W/cm2)


Parameters: k BT e ¼ 10 eV, ne ¼ 1022 cmÀ3, laser wavelength l¼ 32 nm, scat-

tering parameter a¼ 12.5, Z eff ¼ 1.


125



A u t h o r

' s p e r s

o n a l c

o p y

curve is obtained by convolution of the pure Thomson signal

(dotted black curve) with the detector function. The detector

resolution is set to Dl¼ 0.32 nm, which corresponds to 1%

of the central wavelength. Due to the broad detector function,

the central peak is lowered and broadened such that the plas-

mon resonances do not show up as separate structures any-

more. Instead, they only provide sidewings in the spectrum.

In Fig. 4 (ne ¼ 1022 cmÀ3 and k BT e ¼ 10 eV), the electronic

peaks are totally absorbed in the central peak. In this case,

the ion peak in the unconvoluted signal (not shown) is severalorders of magnitude larger than the electronic resonances,

which therefore do not contribute to the convolution integral

(3). Fig. 5 shows spectra for ne ¼ 1020 cmÀ3 and k BT e ¼ 50 eV.

At these parameters, one obtains a¼ 0.56 < 1 for the scatter-

ing parameter, we are in the noncollective regime. Already the

unconvoluted Thomson signal is free of sharp electronic reso-

nances, only two shoulders appear in the broad, noncollective

signal. Finally, in Fig. 6 (ne ¼ 1022 cmÀ3 and k BT e ¼ 50 eV),

we have the same structure as in Fig. 4, no plasmon peaks

are visible due to the dominance of the ion feature.

In all four calculations of the Thomson spectrum, collisions

have been neglected, since the electroneion collision frequency

nei [43] is small as compared to the plasma frequency. Even for

the most unfavorable set of parameters (Fig. 4), the ratio

nei / upl ¼ 0.16 is still allowing for collisionless approximation.In Figs. 3e6, the intensity of the laser is chosen such that

the central peak is situated clearly above the bremsstrahlung

level. Three approximations for bremsstrahlung are shown,

namely Kramers formula (19), Born approximation (18) and

Sommerfeld’s formula (22). Born approximation gives larger

deviations from Kramers result than the Sommerfeld expres-

sion. This corresponds to the behaviour of the Gaunt factor

shown in Fig. 2: the Sommerfeld result is always closer to

unity than Born approximation.

In the following, we evaluate the threshold intensity

I thresh(u) defined in Eq. (4) using Eqs. (3), (5), and (32) for

the Thomson power spectrum and Eq. (20) with the Gauntfactor in either Born approximation (21) or t -matrix approxi-

mation (22) for the bremsstrahlung power spectrum.

Figs. 7e10 show contour plots of the threshold intensity in

part of the neeT e-plane. In Fig. 7 we compare Kramers result

(dashed curves) to Born approximation (dotted curves) assum-

ing aluminum as target material, scattering angle q¼ 20 ,

Dl / l¼ 10À4. In this case, the effective ion charge Z eff is cal-

culated using ThomaseFermi theory [30] as was also done

by Baldis et al. [16], who used the same set of parameters.

Their results are reproduced by using Kramers formula for

bremsstrahlung (dashed curve).

For low temperatures, where the Gaunt factor in Born ap-

proximation is smaller than unity at the considered photon en-ergy, cf. Fig. 2, higher densities are accessible as compared to

Kramers result. For high temperatures, the opposite becomes

true, gBff ðuÞ > 1 leads to lower accessible densities.

In Figs. 8e10, three approximations have been calculated

for the bremsstrahlung level. Besides Kramers formula

(dashed curves) we show results for Born approximation (dot-

ted curves) and Sommerfeld’s formula (solid curves). Compar-

ing the Sommerfeld result to Born approximation, it can be

noted that Born approximation gives larger corrections, while

Sommerfeld’s theory leads to smaller deviations from Kramers

result. This is one important result of this work: taking into

account quantum effects in a rigorous way via Sommerfeld’sexpression for the Gaunt factor leads only to small corrections

of threshold intensities, while Born approximation tends to

overestimate these effects. This underlines the importance to

go beyond Born approximation. For moderate and high tem-

peratures, the Sommerfeld result lies systematically below

the Kramers result, due to the increasing Gaunt factor at

high temperatures.

Furthermore, we investigated the influence of other experi-

mental parameters, namely the laser wavelength, the material,

and the spectral bandwidth of the spectrometer. The latter

parameter turns out to be of great importance: by comparison

of Fig. 8 with Fig. 9, one observes that notably higher densities

are accessible in the case of small spectral bandwidth (Dl /

l¼ 10À4 in Fig. 9) than for the relatively large bandwidth

(Dl / l¼ 10À2 in Fig. 8).

Comparing Fig. 7 to Fig. 9, the influence of the Z -number

of the material becomes apparent: low Z -materials produce

30 31 32 33 34

λ [nm]

0

8×1010

6×1010

4×1010

2×1010


/ c m

3 n

m s r ]

Bremsstr. (Kramers)

Bremsstr. (Born)


Thomson ( I L=1014

W/cm2)




30 31 32 33 34

λ [nm]

0

5×106

4×106

3×106

2×106

1×106


/ c m

3 n

m s r ]

Bremsstr. (Kramers)

Bremsstr. (Born)

Bremsstr. (Sommerfeld)Thomson ( I L=10

12W/cm

2)








A u t h o r

' s p e r s

o n a l c

o p y

less bremsstrahlung than high Z -elements due to the Z 2-

proportionality of the bremsstrahlung cross section, cf.

Eq. (19). On the other hand, the Z -dependence of the dynam-

ical structure factor largely cancels out for the free electron

part of the structure factor while it is roughly Z /(1 þ Z )2 for

the ionic part, cf. Eq. (32).Finally, important differences are noted upon changing the

laser wavelength from 32 nm (Fig. 9) to 13 nm (Fig. 10) espe-

cially at low temperatures. Both data sets have been calculated

using the same spectral resolution (Dl / l¼ 10À4) and material

(H). Looking at Fig. 10 (l¼ 13 nm) at low temperatures the

threshold contours are almost parallel to the density axis.

For higher temperatures, both wavelengths 32 nm and 13 nm

give nearly equal threshold intensities, which do not depend

on temperature. This can be understood from the bremsstrah-

lung spectrum, which becomes nearly independent of fre-

quency and temperature at low hu / k BT .

Since the information about temperature and density is

stored in the position and height of the plasmon resonances,

we will now focus on experimental conditions to be met inorder to separate the plasmon peak from bremsstrahlung back-

ground. Results for the threshold intensity are given in Table 1.

The electron density is set to ne ¼ 1021 cmÀ3, for

ne ¼ 1020 cmÀ3 and 1022 cmÀ3 see Figs. 3e6. Two wave-

lengths are considered, l¼ 32 nm, the momentary VUVeFEL

wavelength at DESY, and l¼ 13 nm, envisaged wavelength in

the near future. Since the latter wavelength allows for very ef-

ficient X-ray optics to be applied, the spectral resolution has

10 100 1000

k BT e [eV]

1024

1023

1022

1021

1020

n e

[ c m - 3 ]

108

W/cm2

Born

Kramers

1010

W/cm2

1011

W/cm2

1012

W/cm2

1013

W/cm2

1014

W/cm2

1015

W/cm2

Fig. 7. Threshold curves in the densityetemperature plane for Al. Parameters: laser wavelength 14.7 nm, q¼ 20, Dl / l¼ 10À4, T e / T i ¼ 2. For bremsstrahlung,

three approximations are applied: Kramers (dashed curves), Born (dotted curves), and Sommerfeld (solid curves).

Born

Kramers

Sommerfeld

10 100 1000

k BT e [eV]

1024

1023

1022

1021

1020

n e

[ c m - 3 ]

1010

W/cm2

1011

W/cm2

1012

W/cm2

1013

W/cm2

1014

W/cm2

1015

W/cm2

Fig. 8. Threshold curves in the densityetemperature plane for H. Parameters: laser wavelength 32 nm, q¼ 120, Dl / l¼ 10À2, T e / T i ¼ 2. For bremsstrahlung, three

approximations are applied: Kramers (dashed curves), Born (dotted curves), and Sommerfeld (solid curves).


127



A u t h o r

' s p e r s

o n a l c

o p y

been reduced to Dl / l¼ 10À4, while 32 nm wavelength allows

only for Dl / l¼ 10À2. Finally the scattering angle has been

chosen such that the plasmon peak is on the one hand well pro-

nounced but on the other hand shifted far enough from the

central peak as to be resolved by the spectrometer. The wave-

length shift of the plasmon resonance from the FEL wave-length is given in the fifth column labelled by Dlres. For

detailed discussion of the Thomson scattering spectrum at

various plasma parameters, we refer to Refs. [12,13]. The

rightmost column of Table 1 gives the Gaunt factor in Som-

merfeld approximation (Eq. (22)). Note that in the case of

the 32 nm wavelength, the threshold intensity is increased by

14e22% due to the Gaunt factor.

7. Conclusion

In this work we have studied Thomson scattering and brems-

strahlung emission in warm dense matter in the context of

plasma diagnostic experiments to be performed in near future.

Out of these two competing, not only Thomson scatteringis sup-posed to serve as a probe for plasma parameters, but also brems-

strahlung gives an important contribution to the photon yield

from highly ionized plasmas. Thus, experimental conditions

have to be determined that allow for a maximum signal-to-back-

ground ratio. Here, we focused on the laser intensity. To this pur-

pose, expressions for the Thomson signal, which is given by the

dynamical structure factor of the plasma as well as for the

Born

Kramers

Sommerfeld

10 100 1000

k BT e [eV]

1024

1023

1022

1021

1020

n e

[ c m - 3 ]

1010

W/cm2

108

W/cm2

1011

W/cm2

1012

W/cm2

1013

W/cm2

1014

W/cm2

1015

W/cm2

Fig. 10. Threshold curves in the densityetemperature plane for H. Parameters: laser wavelength 13.0 nm, q¼ 120, Dl / l¼ 10À4, T e / T i ¼ 2. For bremsstrahlung,

three approximations are applied: Kramers (dashed curves), Born (dotted curves), and Sommerfeld (solid curves).

Born

Kramers

Sommerfeld

10 100 1000

k BT e [eV]

1024

1023

1022

1021

1020

n e

[ c m - 3 ]

10

10

W/cm

210

9W/cm

2

1011

W/cm2

1012

W/cm2

1013

W/cm2

10

14

W/cm

2

Fig. 9. Threshold curves in the densityetemperature plane for H. Parameters: laser wavelength 32 nm, q¼ 120, Dl / l¼ 10À4, T e / T i ¼ 2. For bremsstrahlung, three

approximations are applied: Kramers (dashed curves), Born (dotted curves), and Sommerfeld (solid curves).





A u t h o r

' s p e r s

o n a l c

o p y

bremsstrahlung spectrum have been derived from a common

starting point, namely linear response theory. This approach al-

lows for a systematic treatment of medium and quantum effects,

such as dynamical screening and strong collisions.

We have applied our formulas to determine threshold inten-

sities for the external photon source (FEL) as function of den-

sity and temperature as well as their dependence on further

parameters, namely the laser wavelength, the spectral resolu-

tion of detectors, and the material. In the discussion we fo-

cused on the bremsstrahlung spectrum. It was shown that

Born approximation overestimates the effect of collisions,

i.e. it leads to relatively high Gaunt factors if compared to

Sommerfeld’s expression. Sommerfeld’s formula (22) gives

the correct Gaunt factor in a weakly coupled plasma. At tem-peratures k BT e ! 20 eV threshold intensities calculated with

the Gaunt factor in Sommerfeld approximation are larger

than those obtained by Baldis et al. [16], who used Kramers

expression, i.e. gff ¼ 1.

The Thomson signal was analysed within RPA which gives

the contribution in lowest order of density. For densities inves-

tigated here, collisions are also relevant for the Thomson

signal and lead to a pronounced change of the electronic

part in Sð k; lÞ. This was investigated in Ref. [44]. However,

the account for collisions does not alter the results for the

threshold intensities performed in this work, since the Thom-

son scattering signal was evaluated at the laser wavelength,where the ionic feature of the DSF dominates.

Finally, we have demonstrated that Thomson scattering can

easily overcome the bremsstrahlung background, if laser in-

tensities of 108e1013 W/cm2 are provided. From this point

of view, Thomson scattering experiments for plasma diagnos-

tics using VUVeFEL radiation seem feasible.

Acknowledgements

This work has been supported by the Virtual Institute

VH-VI-104 Plasma Physics Research using FEL Radiation

of the Helmholtz Society and the DFG-Sonderforschungsber-

eich 652 Starke Korrelationen und kollektive Phanomene im

Strahlungsfeld: Coulombsysteme, Cluster und Partikel . We

gratefully acknowledge A. Holl and J. Chihara for stimulating

discussion. C.F. would like to thank DESY for hospitality and

support during a two week visit.

Appendix A. Correlation functions and green functions

Correlation functions for two observables A, B are definedaccording to

ð A; BÞ ¼ 1

b

Z b0

dt TrÂ AðÀihtÞ Byr0

Ã; h A; Biuþih

¼Z N

0

dt eiðuþihÞt ð Aðt Þ; BÞ; ðA:1Þ

r0 is the equilibrium statistical operator. The time dependence

of observables is taken in the Heisenberg picture, using the

system Hamiltonian

H ¼X p;c

h2 p2

2mc

ay p;ca p;c þ 1

2

X pkq;cd

V cd ðqÞay pþq;ca

y kÀq;d a k;d a p;c: ðA:2Þ

Performing integration by parts, the currentecurrent corre-

lation function (16) can be expressed through a forceeforce

correlation function as

cÀ1ð k;uÞ ¼ i

bU0

uq2

k 21À

J zk ; J zk

Á2

"À iu

À J zk ; J zk

Áþ _ J zk ;

_ J zk

uþih

À _ J zk ; J zk uþih J zk ;_ J zk uþih

J zk ; J zk uþih

#; ð A:3

Þ

with

_ J z0;e ¼ ih H ; J z0;e

i.h:

Also, it is convenient to introduce a generalized collision

frequency n(u) in analogy to the Drude relation [20]

eðuÞ ¼ ð1 À u2plÞ=ðu2 þ iunðuÞÞwhere u2

pl ¼ Pc nce2

c=ðe0mcÞ is the squared plasma frequency.

By comparison with Eq. (12) usingÀ J zk ; J zk

Á ¼ e0u2pl=bU0;

we establish an expression for the collision frequency in terms

of correlation functions

nðuÞ¼ bU0

e0u2pl

limk /0

"_ J zk ;

_ J zk

uþih

À_ J zk ; J zk

uþih

J zk ;

_ J zk

uþih

J zk ; J zk

uþih

#: ðA:4Þ

Further details can be found in Ref. [20]. Making use of

Eq. (8), the absorption coefficient can be expressed as

aðuÞ ¼ u2pl

c

Re nðuÞÀu2 À 2u Im nðuÞ þ jnðuÞj2

ÁnðuÞ : ðA:5Þ

In the high frequency limit u[upl, the index of refraction

is unity, the imaginary part of the collision frequency tends to

zero and the collision frequency is small compared to the fre-

quency u. Then, we can consider the approximation aðuÞ ¼u2

plRe nðuÞ=u2c ¼ bU0Re_ J z0; _ J z0

uþih

=ce0u2, where the col-

lision frequency is given in the form of a forceeforce correla-

tion function, cf. Ref. [20]. Thus, the absorption coefficient is

directly proportional to the real part of the forceeforce

Table 1

Experimental parameters and resulting threshold intensities for different elec-

tron temperatures and laser wavelengthsk BT e[eV]

l

[nm]

q Dl / l Dlres[nm]

I thresh[W/cm2]

gSff

10 32 120 10À2 1.1 1.37 Â 1012 1.14

20 32 120 10À2 1.2 4.92 Â 1012 1.17

50 32 120 10À2 1.4 9.29 Â 1012 1.22

10 13 120 10À4 0.3 2.95 Â 109 1.03

20 13 45 10À4 0.2 1.55 Â 1011 1.04

50 13 45 10À4 0.2 3.49 Â 1012 1.06

Dlres is the position of the plasmon resonance in the Thomson spectrum rela-

tive to the laser wavelength. Fixed parameters: Z ¼ 1, ne ¼ 1021 cmÀ3.


129



A u t h o r

' s p e r s

o n a l c

o p y

correlation function, which itself can be determined using per-

turbation theory.

The relation to the thermodynamic Green function of twoobservables is given by

h A; Biuþih¼i

pb

Z NÀN

du

u

1

uþ ihÀuIm G ABðuþ ihÞ: ðA:6Þ

Using Dirac’s identity one obtains Reh A; Biuþih ¼Im G ABðuþ ihÞ=ub; which directly leads to Eq. (16).

Appendix B. The RPA-susceptibility

Eq. (27) is obtained from the exact relation for the response

function ccc0ð k;uÞ,

ccc0ð k;uÞ ¼ c0cð k;uÞdcc0 þ

Xd

c0cð k;uÞ V sc

cd ð k;uÞcdc0ð k;uÞ;

ðB:1Þ

by truncation after the first iteration, i.e. cdc0 ¼ c0dc0 in the sec-

ond term of Eq. (B.1). Insertion of the screened potential Eq.

(28) yields the closed equation

cRPAcc0 ð k;uÞ ¼ c0

cð k;uÞdcc0 þ c0cð k;uÞ

"V cc0ð kÞ

þX

d

V cd ð kÞc0d ð k;uÞV sc

dc0ð k;uÞ#c0

c0ð k;uÞ

¼ c0cð k;uÞdcc0 þ

Xd

c0cð k;uÞ V cd ð kÞcRPA

dc0 ð k;uÞ:

ðB:2ÞFor a two-component plasma, i.e. an electroneion plasma

c ¼ e, i, we obtain by matrix inversion

cRPAee ¼ c0

e

À1 Àc0

i V iiÁ

1 À c0eV ee À c0

i V ii þ c0i c

0eðV iiV ee À V eiV ieÞ; ðB:3Þ

cRPAie ¼ c0e V eic0i

1 À c0eV ee À c0

i V ii þ c0i c

0eðV iiV ee À V eiV ieÞ: ðB:4Þ

cRPAei and cRPA

ii are obtained by interchanging indices i and e in

Eq. (B.4) and Eq. (B.3), respectively.

The imaginary part of the electronic susceptibility in RPA,

cRPAee ð k;uÞ (Eq. (B.3)) for a hydrogen plasma (V ee ¼ V ii ¼

ÀV ei ¼ À V ieh V ) now evaluates to

Im cRPAee ð k;uÞ ¼

1 À V ð kÞc0i ð k;uÞ

1 À V ð kÞ½c0eð k;uÞ þc0

i ð k;uÞ

2

Im c0eð k;uÞ

þ V

ð k

Þc0

e

ð k;u

Þ1 À V ð kÞ½c0eð k;uÞ þ c0

i ð k;uÞ2

Im c

0

i ð k;uÞ:ðB:5Þ

This expression can also be derived from kinetic theory, i.e.

the perturbative expansion of Vlasov’s equation for a two-

component plasma [15]. It was used in Ref. [16] to compare

the Thomson signal to the emission background caused by

thermal bremsstrahlung photons. Generalizing for differentelectron and ion temperatures T e and T i and inserting the ex-

plicit expression for the response function in RPA (30), one

obtains Eq. (32).

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8 (2001) 3810.

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New York, 1980.

[24] S. Ichimaru, Statistical Plasma Physics, vol. 1, Addison-Wesley,

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[25] H.R. Griem, Principles of Plasma Spectroscopy, Cambridge University

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[27] G.D. Mahan, Many-Particle Physics, second ed. Plenum Press, New York

and London, 1981.

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weig, 1949.

[29] C. Fortmann, H. Reinholz, G. Ropke, A. Wierling, Condensed Matter

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[34] M. Ter-Mikaelyan, Dokl. Akad. Nauk SSSR 94 (1953) 1033.[35] M. Abramowitz, A. Stegun (Eds.), Handbook of Mathematical Functions

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4365.


131



Chapter 7

Optical Properties and One-ParticleSpectral Function in Non-Ideal

Plasmas

Authors: Carsten Fortmann, August Wierling, and Gerd Ropke

Appeared as regular article in Contributions to Plasma Physics, Vol. 47, Issues 4-5, pages297-308, June 2007.


•C.F.: Preparation of manuscript (mainly sections 3-6), all numerical calculations

• A.W.: Preparation of manuscript (mainly sections 1-2,7)




134 Optical Properties and Spectral Function



Contrib. Plasma Phys. 47, No. 4-5, 297 – 308 (2007) / DOI 10.1002/ctpp.200710040

Optical Properties and One-Particle Spectral Function in Non-Ideal Plasmas

C. Fortmann∗, G. R opke, and A. Wierling

Institut fur Physik, Universitat Rostock, 18051 Rostock, Germany

Received 19 December 2006, accepted 19 December 2006

Published online 6 June 2007

Key words Non-Ideal plasmas, bremsstrahlung, spectral function, vertex corrections.

PACS 52.25.Mq,52.25.Os,52.27.Gr

A basic concept to calculate physical features of non-ideal plasmas, such as optical properties, is the spectralfunction which is linked to the self-energy. We calculate the spectral function for a non-relativistic hydrogenplasma in GW -approximation. In order to go beyond GW approximation, we include self-energy and vertexcorrection to the polarization function in lowest order. Partial compensation is observed. The relation of ourapproach to GW and GW Γ calculations in other fields, such as the band-structure calculations in semicon-ductor physics, is discussed. From the spectral function we derive the absorption coefficient due to inversebremsstrahlung via the polarization function. As a result, a significant reduction of the absorption as comparedto the Bethe-Heitler formula for bremsstrahlung is obtained.

c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Spectroscopy can serve as a versatile tool to infer properties of dense and hot plasmas from the emitted radiation[1,2]. In non-ideal plasmas, where the coupling parameter can exceed unity, properties often deviate significantly

from their form in ideal plasmas due to the importance of interaction effects [3]. With the advent of femtosecond

laser pulses [4], it has become possible to produce non-ideal plasmas with table-top systems. In particular,

it is nowadays possible to create conditions similar to those in the center of astrophysical objects such as the

sun or giant planets. Also, the FAIR-project at GSI Darmstadt combined with the high energy laser systems

PHELIX and NHELIX available at GSI will provide ideal tools to produce and diagnose plasmas where the

effects under discussion here are important [5]. Effects such as dynamical screening, dissolution of bound states,

Pauli-blocking, and the importance of collisions have been observed [3]. Taking proper account of interaction

effects is a challenge to any theoretical description of non-ideal plasmas.

Many-body perturbation theory presents a toolbox to determine various properties of non-ideal plasmas [6–

8]. The use of Green’s function techniques allows for a systematic and intuitive consideration of many-body

effects. In particular, one-particle properties can be obtained from the one-particleGreen’s function or its spectralrepresentation, the one-particle spectral function. A number of important mechanisms in dense plasmas such as

dynamical screening of the Coulomb interaction can be described by partial summation of certain diagrams.

A particularly successful concept is the quasi-particle picture [6–8]. However, with increasing coupling the

quasi-particle pictures breaks down. This shows up in a broadening of the spectral function. To go beyond

the quasi-particle picture in a consistent way poses serious problems of self-consistency. Vertex and self-energy

corrections have to be taken into account on the same footing to obey sum rules and other exact knownproperties.

Using Ward identities [9] or the Kadanoff-Baym scheme of the conserving vertex [10] enables one to construct

consistent sets of diagrams, alas the resulting integral-equations are complicated to solve.

The GW approximation is a particularly scheme for the self-energy approximating it by a product of the

Green’s function G and an effective interaction W [11]. Being introduced by Hedin in 1965 [12], it has a long

history of applications, which is reviewed in Ref. [13] and [14]. However, the full self-consistency implied in

∗Corresponding author: e-mail: [email protected], Phone: +49 381498 6947, Fax: +49381 4986942

c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

135



298 C. Fortmann, G. Ropke, and A. Wierling: Optical properties and one-particle spectral function

Hedin’s original proposal was not carried out so far. Instead, the ideal Green’s function or a quasi-particle picture

was used. In this manner, the GW approximation was successfully applied to determine shifts in the ionization

potential, approximations for the equation of state, and an effective interaction to describe bound states in a

medium [8]. Also, the band structure of different types of materials, like semiconductors [15], alkali [16], and

transition metals [17] was determined. Due to the efficiency of modern computer technology, it has recently

become feasible to address the self-consistencyimplied in Hedin’s scheme to some extent, seee.g. [18–21]. Some

of these calculations improve solely on self-energy corrections,while others take into account vertexcontributions

as well. It is customary to call the later GW Γ approximations.

In this paper, we perform calculations for the optical properties of a hydrogen plasma at solar core conditions,

as an important example for astrophysical plasmas. Typical parameters of the solar core plasma are temperatures

of about T = 100Ry/kB 1360 eV/kB and particle densities reaching n 7 · 1024 cm−3. These parameters

justify the model of a classical (non-degenerate) and weakly coupled plasma, being characterized by values of

θ = kBT /E F 10 for the degeneracyparameter and Γ = (Ze2/4π0)/(3/4πn)1/3kBT 0.03 for the coupling

parameter.

Central to the description of optical properties of plasmas is the dielectric function (k, ω) [1]. Within theframe of the approach presented here, it can be obtained from the polarization function, which is a member

of Hedin’s set of equations. In an earlier paper [22], the suppression of the bremsstrahlung cross section due

to successive scattering has been treated by a taking into account self-energy and vertex corrections. However,

these corrections were only consideredwithin a one-loopapproximation. In this way, dynamical screening effects

were neglected. It is the objective of this communication to present results with an improved one-particle spectral

function as obtained with the GW Γ approximation.

In Sec. 2, we will review the GW approximation making use of Hedin’s equations. Sec. 4 presents an

illustrative example for the GW 0 approximation. Implications for the absorption coefficient are discussed in Sec.

6. A discussion and conclusions will be given in Sec. 7.

If not otherwise indicated, we apply the Rydberg system of units where = kB = 1 , e2 = 2 , 0 =1/4π , and me = 1/2.

2 The GW Γ approximation

A convenient starting point of our approach are Hedin’s equations [23]. It is a closed set of equations relating the

full Green’s function G, the non-interacting Green’s function G0, the self-energy Σ, the dynamically screened

interaction W ab [11], the polarization function Π, and the vertex function Γ. In detail, the full Green’s function

is given by

Ga(12) = Ga,0(12) +

d(34) Ga,0(13)Σa(34) Ga(42) . (1)

Here and in the following, the shorthand notation (1) ≡ (r1, τ 1,σ1 . . .) for spatial variables r, imaginary times

τ as well as quantum numbers such as spin is used. The self-energy Σa(12) is obtained from the dynamically

screened interaction and the vertex function according to

Σa(12) = i

d(34) Ga(13) W aa(41)Γa(32, 4) . (2)

The dynamical screened interaction is given via

W ab(12) = V ab(12) +c

d(34)V ac(13)Πcc(34)W cb(42) (3)

by the polarization function

Πaa(12) =

d(34)Ga(13)Ga(41)Γa(34, 2) . (4)

Finally, the vertex function obeys a Bethe-Salpeter like equation

Γa(12, 3) = δ(12) δ(13) +b

d(4567)

δΣa(12)

δGb(45) Gb(46) Gb(75)Γb(67, 3) . (5)

www.cpp-journal.org c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim




Contrib. Plasma Phys. 47, No. 4-5 (2007) 299

The set of equations represents a perturbation expansion in terms of the screened interaction W ab and is expected

to show better convergence properties compared to an expansion in the inter-particle interaction V ab.By truncating the set of equations (1)-(5) on a certain level, various approximations for these quantities can

be defined. A particular simple approximation is the G0W 0 approximation [12,23]. It is obtained by taking the

bare vertex Γ(12, 3) = δ(12)δ(13) and inserting non-interacting Green’s functions into the expression for the

polarization function. This leads to a self-energy given by ΣGW a (12) = iGa(12)W aa(21). Next, the Green’s

functions in this expression as well as in the screened interaction W are taken as the free Green’s function G0.

This expression turns out to be quite successful. However, in dense plasmas as well as in a number of materials

in solid state physics, there is need to improve beyond this simple approximation.

Such an improvement is the GW 0 approximation which was studied in Ref. [24] for the electron gas at T = 0and in Ref. [25] for the solar core plasma. In the latter case, a considerably broadened quasiparticle and a

featureless behaviour at the plasma frequency was found. No plasmon-likesatellite structures survived the partial

self-consistency treatment.

Another straightforward extension is a self-consistent solution of Dyson’s equation and the screened inter-

action while keeping the vertex function in lowest order [19]. This will be termed GW approximation in thispaper. In Ref. [19], an increase in the quasiparticle bandwidth and a featureless satellite structure is found in

contradiction to experimental evidence. Also, such an approximation leads to a drastic violation of the f-sum for

the inverse dielectric function. The results indicate the importance of vertex corrections [26]. Calculations for

a one-dimensional semiconductor [27] lead to similar conclusions. Large cancellations between self-energy and

vertex corrections have also been found in Ref. [28–32].

Recently, a numberof approximation schemes to take into account vertex correctionshave been proposed [33].

In this work we will apply a sequence of approximations described in the following:

Taking the self-energy in the GW -approximation, the functional derivative occurring in the vertexequation (5)

yields in lowest order of W aa

δΣa(12)

δGb(45)= W 0aa(12)δ(14)δ(25)δab . (6)

Note that the screened interaction is taken in the one-loop approximation for Π, the so-called random-phase

approximation (RPA). This term leads to a ladder approximation for the vertex in terms of the dynamically

screened interaction

Γa(12, 3) = δ(12)δ(13) +

d(67)W 0aa(12)Ga(16)Ga(72)Γa(67, 3) . (7)

Already this equation is challenging to solve [34], even in Shindo approximation [35]. As a result, the improved

self-energy in this approximation is given in second order of the dynamically screened interaction by the term

studied in Ref. [18],

ΣGW 0Γa (12) = Ga(12)W 0(21) +

d(34) Ga(13) W 0aa(23) Ga(34) W 0aa(41)Ga(42) . (8)

Also, the improved polarization function is given besides the loop diagram by an exchange diagram with respectto W 0

ΠGGΓaa (12) = Ga(12)Ga(21) +

d(34)Ga(13)Ga(41)W 0aa(43)Ga(24)Ga(32) . (9)

At this stage the first iteration of Hedin’s equations (1)-(5) is completed. We will not go beyond this first iteration,

in particular the screened interaction potential W aa is kept on the level of RPA.

To address the deviations from the quasi-particle picture, the introduction of the one-particle spectral function

Aa( p,ω) is convenient. With its help, the spectral representation of the full Green’s function is given by

Ga( p,z) =

∞

−∞

dω

2π

Aa( p,ω)

z − ω, (10)

p and z denote momentum and energy (frequency) of the particle. The Green’s function Ga(12) is obtainedfrom the Green’s function in momentum-frequencyrepresentation Ga( p,z) by means of Laplace transform. The


137




quasi-particle approximation itself can be stated as a δ-like form of the spectral function

Aa( p,ω) = 2πδ (ω −E a( p)) , (11)

where the energy E a( p) is obtained as

E a( p) =p2

2ma+ Re Σa( p,E a( p)) . (12)

Note, that the spectral function itself obeys a normalization relation ∞

−∞

dω

2πAa( p,ω) = 1 . (13)

Furthermore, sum rules for the first and second moment of the spectral function are known [24],

∞

−∞

dω ω Aa( p,ω) = E HFa , (14)

∞

−∞

dω ω2 Aa( p,ω) =

∞

−∞

dω ImΣca( p,ω) +

E a( p)HF

2, (15)

where the index HF refers to the energy in Hartree-Fock approximation and the index c indicates the use of the

correlated self-energy. From the spectral function, a number of thermodynamic quantities can be obtained, e.g.

the one-particle density [8,36],

na(µa, β ) =

d3 p

(2π)3

∞

−∞

dω

2πf a(ω) Aa( p,ω) , (16)

with the distribution function f a(ω) of particles of species a.

Optical properties, which are under consideration in this work, can be obtained from the well-known relation

between the dielectric function (q, ω) and the polarization function Eq. (4), i.e.

(q, ω) = 1 −a

V aa(q)Πaa(q, ω) , (17)

with the interaction potential V aa(q). In particular, the absorption coefficient, which gives the attenuation of

electromagnetic radiation traversing the plasma is given by the imaginary part of the long wavelength limit of the

dielectric function as

α(ω) =ω

cIm (q → 0, ω), (18)

a relation that holds for wavelengths long against interatomic distances and frequencies high compared to the

plasma frequency ωpl = 4(πn)1/2. For details, we refer to Refs. [1,25].

3 Applications of GW approximation for the self-energy

The GW approximationhas been very successfully used for a broad variety of problems in many-particle physics

for a long time. Here we would like to mention only a few examples.

One of the first applications in solid state physics was the calculation of band structures combining density

functional theory (DFT) and the GW -method. Northrup et al. [16] showed how this approach improves the band-

gap problem of the local density approximation (LDA), which predicts too small band-gaps for most materials,

whereas the use of GW helps to decrease the deviation of the theoretical from the experimental value below

0.1 eV in the case of silicon, compared to about 1 eV as obtained from pure LDA-DFT. Similar results were

obtained for many different semi-conductors and insulators [14, 38]. Optical properties of highly excited semi-

conductors using a combined GW and T-Matrix approach have been studied extensively by Schmielau et al., see

Ref’s [39–41]. Also in the case of finite systems, the GW approximation has been applied successfully. For the

case of the Na4 tetramer, good agreement between the theoretically predicted photoabsorption and experimentaldata was obtained [42].






Nuclear and quark matter have been investigated using techniques very similar to the approach presented here.

The problem of dynamical chiral symmetry breaking, which cannot be described in a perturbative calculation of

quark propagators, was demonstrated to emerge in a nonperturbative approach known in quantum field theory as

rainbow-ladder approximation [43,44].

Fehr [45] and Wierling et al. [25] investigated spectral properties of electrons in nonideal plasmas making

use of the GW approximation. Whereas Fehr applied the perturbative G0W 0 approach (c.f. section 2) to both

equation of state and optical properties, Wierling already used a self-consistent GW 0 approximation which is

also used here in the following section.

4 GW 0 approximation for the solar core plasma

In the GW 0 approximation, the set of Hedin’s equation reduces to a self-consistent solution of Dyson’s equation

with the correlated part of the self-energy,

Ae( p,ω + iη) =−

2ImΣe( p,ω + iη)[ω − p2 −Re Σe( p,ω + iη)]2 + [Im Σe( p,ω + iη)]2

, (19)

Σe( p,ω + iη) = −

∞ −∞

dω dω

(2π)2

∞ 0

d3q

(2π)38π

q2Im −1RPA(q, ω + i0) [1 + nB(ω)]

Ae( p− q, ω)

ω + iη − ω − ω.

(20)

As indicated by W 0, the dielectric function (q, ω) is taken in random phase approximation (RPA) and the Bose

function is given as nB(ω) = (exp(−ω/T ) − 1)−1

. Here, as in the following, electrons labeled by a = e are

considered. Note that the Hartree-Fock self-energy, which appears as an additional term in Eq. (20) is small for

the considered parameters and is henceforth neglected.

We give results for a hydrogen plasma using the conditions at the solar core center. We ignore other ions such

as helium etc. in these exploratory calculations. In Fig. 1, the self-consistent spectral function calculated in GW 0

approximation, i.e. the solution of the set of Eqs. (19) and (20), is shown for a fixed momentum of p = 0.21a−1B .

The grey curve is the initial ansatz for the spectral function, i.e. the input for the r.h.s. of Eq. (20). It has been

chosen of Lorentzian form with a width of γ = 10 Ry. The first iteration is given by the dashed-dotted curve.

The peak of the spectral function is shifted to smaller frequencies and the function is asymmetrically broadened.

The second and third iteration give only minor modifications to the first iteration, the forth iteration (not shown)

does not vary significantly from the third iteration.

-50 0 50

ω [Ry]

0

0.1

0.2

0.3

0.4

A e

( p , ω )

[ 1 / R y ]

initial (Lorentzian, γ =10 Ry)

QPA1. iteration2. iteration3. iteration

Fig. 1 Electron spectral function in GW

0

for solar coreconditions (ne = 7 ·1024 cm−3, T e = 100 Ry 1360 eV)

at momentum p = 0.21a−1B . Dashed-dotted curve: first iter-

ation of GW 0, (quasiparticle approximation [QPA]). Grey

solid curve: Ansatz for the spectral function in the iterative

solution of GW 0. Dotted curve: 1st iteration, dashed: 2nd

iteration, solid: 3rd iteration. The 4th iteration is not distin-

guishable from the 3rd iteration.

For comparison, we also show the G0W 0 approximation, i.e. the first iteration of Eq. (20) starting from

a spectral function of vanishing width (delta-distribution). In this case, there is no quasiparticle peak at the

quasiparticle resonance ωQP = p2, but four satellites and minima appear. The latter are due to sharp peaks in the

response function Im −1RPA(k, ω) and have been observed earlier [46,47].

As a result of this calculation, we observe, that the self-consistent calculation leads to a spectral function thatis physical easily understandable, i.e. it contains a broadened and shifted quasiparticle resonance. However, the


139




signatures of collective effects, such as the dynamical screening, which are present in the G0W 0 approximation

(dash-dottedcurve in Fig. 1) vanish completely in the self-consistent result. In contrast, the G0W 0 result does not

contain a broadened quasiparticle peak but is completely determined by the behaviour of the response function,

which contains the collective excitations of the plasma (plasmons), as also shown by Fehr et al., see Ref. [48].

0-6

-4

-2

0

I m Σ

0-30

-20

-10

0

0-4

-2

0

2

4

R e Σ

0

-20

-10

0

10

20

0-40-20

020

40

ω - p

2 - R e Σ

0-40-20

020

40

-40 -20 0 20 40ω [Ry]

0

0.1

0.2

0.3

0.4

A e

[ 1 / R y ]

-40 -20 0 20 40ω [Ry]

0

1

2

3

Fig. 2 Self-energy (real and imaginary part), dispersion relation and spectral function calculated self-consistently by solving

Eq. (21) (left) and quasiparticle approximation (right), i.e. the r.h.s of Eq. (21) evaluated with Σ( p, ω) = 0. Parameters:

ne = 7 · 1024 cm−3, T e = 100 Ry 1360 eV. The momentum is fixed at p = 0.21 a−1B .

-40 -20 0 20 40

ω [Ry]

0

0.1

0.2

0.3

0.4

A e

( p = 0 . 2

1 , ω

) [ 1 / R y ]

Alg. solution for Born approx. (Lorentz plasma)

full RPA solution (TCP)

Fig. 3 Self-consistent spectral function computed in full

RPA (two component plasma) (dashed curve) and in static

Born approximation including only electron-ion collisions

(solid curve). The neglectance of e − e collisions leads to

a smaller width of the spectral function. Solar core parame-

ters: ne = 7 · 1024 cm−3, T e = 100 Ry.

Although convergence is already achieved after 3-4 iterations of Eqs. (19) and (20), the calculation is too

time-consuming to be used in the computation of physical observables such as equation of state or the polar-

ization function, the latter involving convolution integrals over two spectral functions in both momentum and

frequency domain. Therefore, we make use of some further approximations. First, we replace the full RPA-like

screened interaction by a one-loop approximation, which takes into account the scattering among particles in Born

approximation. In this case we only consider electron-ion scatterings. Ions are treated in adiabatic approximation

and the interaction is mediated by a statically screened potential of Debye-type, i.e. V ei(q) = 8π/(q2 + κ2),

where κ = (8πn/T )1/2 is the inverse Debye screening length. This approximation is justified by the fact that the

corresponding absorption cross-section leads to the nonrelativistic limit of the Bethe-Heitle formula for inversebremsstrahlung in the limit of vanishing κ, as shown in [22].






For small κ, the main contributions to the integral in Eq. (20) come from small q. Therefore, we can neglect

the shift in the momentum variable in the self-energy on the r.h.s. As shown in Ref. [22], one then obtains a

particularly simple equation for the self-energy in this approximation, which we will denote by the suffix GW 0s,

ΣGW0s( p, ω+iη) = −8πniκ

κ2 + p2 − ω − iη + ΣGW0s( p, ω + iη)− 2iκ

ω + iη − ΣGW0s( p, ω + iη)

−1

,

(21)

which can be solved by standard root-finding algorithms. Fig. 2 shows the solution of Eq. (21) for the same

set of parameters as used in the RPA calculation (Fig. 1). To the left, the self-energy (imaginary and real part),

as well as the dispersion relation, and the spectral function as obtained from the numerical solution of Eq. (21)

are shown. The spectral function is centered around the quasiparticle energy. Its overall shape is given by the

imaginary part of the self-energy, while the real part determines the dispersion. In the self-consistent solution,

the dispersion relation contains only a single root near the quasiparticle energy, while there are three roots inthe quasiparticle approximation (r.h.s of Fig. 2). The upper and lower root give rise to the two satellites in the

corresponding spectral function, while the central root does not yield a resonance due to the large imaginary part

of ΣG0W0s around ω = 1 Ry. The separation of the satellites from the quasiparticle energy is approximately

given by the plasma frequency. This is a general feature, which has been observed in earlier calculations carried

out at lower particle densities and temperatures [22].

0 2 4 6 8 10

20

0

20

40

60

80

100

120

p [aB -1

]

ω[Ry]

0

0.1

0.2

0.3

0.4

0.5

A e (p, [ω) Ry]

Fig. 4 Contour plot of the self-consistent

spectral function with simplified self-energy

(Eq. (21) at solar core conditions as function

of frequency ω and momentum p. The spectral

function is asymetrically broadened and shifted

from the quasi-particle energy ωQP = p2.

At high momentum the spectral function con-

verges into a sharp quasi-particle resonance at

the free-particle energy. Parameters: ne =

7 · 1024

cm−3

, T e = 100 Ry 1360 eV.

In Fig. 3 we compare the spectral functions obtained from the two-component RPA and the simplified Born

approximationcontaining only the scattering of electrons on fixed ions (Lorentz-Plasma). Since electron-electron

collisions are neglected in the latter case, the corresponding spectral function is narrower than in the RPA cal-

culation. Furthermore, in the one-loop approximation there are no plasmon degrees of freedom which lead to

further damping in the RPA calculation. On the other hand, the Lorentz plasma calculation already contains

all characteristic features of the RPA spectral function. Namely the asymmetrical broadening and the shift of

the peak towards lower energies show up. These features become less pronounced at higher momenta, where

many-particle effects are expected to be less important. With increasing momentum, the width of the spectral

function decreases, while at the same time the height of the quasiparticle peak increases, as can be seen in Fig. 4.

This follows naturally from the normalization condition for the spectral function Eq. (13). In conclusion, we

have shown that our simplified Lorentz plasma model for the self-energy leads to physically intuitive one-particlespectral function which includes the electron-ion collisions in a consistent way.


141




5 Applications of GW Γ-approximation for the self-energy

In the last years, substantial effort has been made to go beyond the GW -approximation and to include the vertex

to some extent. In a pioneering paper, Takada [21] demonstrated the feasibility of self-consistent GW Γ approx-

imation. He used an ansatz for the vertex-function which fulfils certain sum-rules and conservation laws. As

a result, he showed the subtle cancellation of contributions from self-energy and vertex-corrections to spectral

properties of charged particles, exemplified for the damping of plasmons in Al. Very recently, Ziesche [49] has

reviewed the calculation of direct and exchange contributions (vertex correction) to the on-shell self-energy of

the homogenous electron-gas.

For plasmas, Vorberger et al. [36] systematically studied contributions to theequation of state beyondMontroll-

Ward, including all exchange terms (vertex corrections). In the same spirit, we will now investigate the lowest

order vertex correction to the self-energy in second order of the interaction, given by the following diagrammatic

expression.

Σ(2)( p, ω) = + . (22)

The second diagram is the lowest order diagram which does not appear in the GW approximation, whereas the

first diagram is automatically included. The base lines are free electron propagators, while the upper loops are

composed of ionic propagators taken in the adiabatic approximation. To ensure convergence,a statically screened

potential V ei(q) is used. The numerical evaluation of Eq. (22) is shown in Fig. 5. The dashed and dotted curves

give the first and second iteration respectively of the GW 0s approximation, the solid curve is the self-consistent

result. For comparison, the first order vertex correction as obtained from the exchange diagram in Eq. (22) is

shown in the inset. The vertex term gives at most a 20% reduction of the self-energy.

0 0.5 1 1.5ω [Ry]

-0.15

-0.1

-0.05

0

I m Σ

( 0 . 1 , ω

) [ R y ]

GW0, 1. iteration

GW0, 2. iteration

GW0, stable

0.9 1-0.01

-0.005

0

0.005

0.01

Vertex correction Fig. 5 Imaginary part of the self-energy as calculated from

Eq. (21). The first (dashed) and second (dotted) iteration

as well as the convergent result (solid) are shown. Inset:

Comparison of GW approximation and lowest order ver-

tex correction (dashed-dotted), cf. Eq. (22). The exchangediagram gives a correction of at most 20%. Parameters:

ne = 1021 cm−3, T e = 1Ry.

6 Modification of the absorption coefficient

As an illustrative example, we calculate the impact of the broadened electron spectral function on the absorption

coefficient due to inverse bremsstrahlung. The absorption coefficient for radiation in a plasma can be obtained

from the dielectric function (q, ω), Eq. (18). According to Eq. (4), nonideality effects enter via self-energy

corrections of the full Green’s functions and by vertex corrections beyond the bare vertex. Within the GW

approximation, only self-energy corrections are considered, while the GW Γ also accounts for vertex terms in thepolarization function.






6.1 Self-energy corrections

Using the self-consistent spectral function obtained in simplified GW 0

(Eq. (21)) to calculate the polarizationfunction, leads us to the result plotted as solid curve in Fig. 6. We normalize our result to Kramers’ formula

corrected by a Gaunt-factor in Born approximation [50,51], which corresponds to the non-relativistic limit of the

Bethe-Heitler cross section for inverse Bremsstrahlung [52]. In the infrared limit (ω → 0) theBorn approximation

shows a logarithmic divergence.

1 10 100 1000

ω [Ry]

0.85

0.9

0.95

1

1.05

1.1

α Σ

/ α

B

Gaussian SF, σ = 6.5 Ry

Gaussian SF, σ =13.0 Ry

Self-consistent SF

Fig. 6 Free-free absorption coefficient calculated from

broadened electron propagators. The absorption coefficient

is normalized to the Born approximation. At low frequen-

cies, suppression of inverse bremsstrahlung is obtained. For

high photon energies, the improved result converges to the

Born approximation, while around ω = 10Ry enhance-

ment sets in. Also shown: α(ω) calculated with Gaus-

sian spectral functions (dotted and dashed line) of different

widths. Here, the convergence into the Born result is much

slower, since the Gaussian spectral function does not yield

the correct quasiparticle limit at high frequencies. Parame-

ters: ne = 7 · 1024 cm−3, T e = 100 Ry (solar core).

The result for solar core parameters is shown in Fig. 6. For small frequencies, a reduction of the free-free

absorption of about 15% as compared to Born approximation is observed, while at high frequencies our approach

converges to the Born approximation. At intermediate frequencies, we obtain a slight enhancement of our result

relative to the Born approximation. These characteristics have already been observed in earlier calculations for

lower densities and temperatures [22]. We compare our results to calculations which use parametrized spectral

functions of Gaussian shape in the polarization loop (dashed and dotted curve). The width σ is given. Also in

this case, an enhancement is observed which decreases with decreasing width σ as can be seen by comparing the

dotted (σ = 6.5 Ry) and the dashed curve (σ = 13Ry). The value σ = 6.5 Ry gives a Gaussian spectral function

of similar shape as the self-consistent calculation at small momenta. As shown in Fig. 4, the self-consistently

calculated spectral function converges to a quasi-particle resonance at large momenta, i.e. large frequencies.

This behaviour leads to the faster convergence of the absorption coefficient to the Born result as compared to the

calculation using Gaussians with frequency and momentum independent widths. Remember that the Born result

for the absorption coefficient is obtained by inserting delta-like spectral functions in the polarization function

Eq. (4).

6.2 Vertex corrections

We have shown that the account of a broadenedone-particle spectral function leads to a suppressionof the infrared

behaviour of the inverse bremsstrahlung spectrum. However, the calculations have been carried out on the level

of GW 0-approximation. Vertex corrections contribute to the polarization function in the same order with respect

to the statically screened potential. Moreover, these vertex corrections tend to cancel the self-energy correction

to some extent. In order to account for both types of corrections on the same footing, we focus on contributions

from the vertex equation, Eq. (5), up to first order in the screened interaction in the polarization, Eq. (4),

Π(q, ω) = + . (23)

In Fig. 7 we compare the suppression of the absorption coefficient obtained from the inclusion of self-energy

only (solid curve) and with included vertex correction in the polarization loop (dashed curve), c.f. Eq. (23). At

the present parameters, the vertex correction is the most dominant term. Over the whole range of frequenciesconsidered here, we obtain a suppression of the absorption coefficient as compared to the Born approximation.


143




1 10 100 1000

ω [Ry]

0.85

0.9

0.95

1

1.05

1.1

α

/ α

B

αΣ

αΣ+α

V

Fig. 7 Absorption coefficient α(ω) for solar core condi-

tions (ne = 7 · 1024 cm−3, T = 100Ry) as function of

the photon energy ω. α(ω) is calculated from Eq. (4) with

broadened electron propagators using the self-energy ob-

tained from Eq. (21) (solid curve). Additionally, the first

vertex correction is calculated, cf. Eq. (22). The sum of

both terms is given by the dashed curve. The vertex correc-

tion dominates the behaviour of the absorption coefficient.

In particular, the enhancement at ω 10Ry present in the

pure self-energy calculation vanishes completely.

Also, the enhancement of absorption at frequencies around ω = 10Ry, which was observed in the calculation

using only self-energy corrections (solid curve), vanishes if the vertex is taken into account (dashed curve). This

shows the importance of vertex corrections.

Here, the vertex correction is only taken in lowest order of the interaction and density, while the self-energy

based result contains a summation to all orders of density. In order to compare self-energy and vertex contri-

butions to the modification of the absorption spectrum in a fully consistent way, one would have to go beyond

the perturbative calculation presented here and solve for the vertex equation Eq. (5) using at least free-particle

propagators. This task goes beyond the scope of this paper, where only the lowest order corrections are to be

studied. Finally, we remark, that in the mentioned earlier calculations presented in Ref. [22], the vertex correc-

tion modifies the self-energy result only to a minor extent, i.e. the enhancement at intermediate frequencies is

reduced by 40%. This is due to the lower density (1019 cm−3) used in that work. In all calculations, the high

frequency behaviour converges nicely to the Born result.

For the infrared part of the spectrum, the behaviour of the absorption coefficient needs further considerations,going beyond the present work. As shown in Fig. 7, the self-energycorrection in the polarizationfunction reduces

the absorption coefficientby a constant factor of about 15% in the low frequency limit, but does not regularize the

infrared divergence of the Born approximation. The vertex term, on the other hand, induces a large suppression

so that in the low frequency limit higher orders of the vertex correction have to be considered. Furthermore, it

is well-known, that below the plasma frequency (7 Ry for the solar core parameters) dynamic screening plays an

important role. This needs a further summation of diagrams going beyond the present scheme. The impact of

dynamical screening on bremsstrahlung emission has been discussed by Ter-Mikaelyan in Ref. [53].

7 Conclusions

A systematic treatment of optical properties in non-idealplasmas is possible in the framework of Green’s functionmethods. Corrections beyond the quasi-particle pictures can be generated using Hedin’s equations. However, a

consistent solution of Hedin’s equations is a formidable task. Here, we considered a simplified set of equa-

tions. Notably the GW 0-approximation, mainly used in solid state physics, has been shown to lead to sensible

results also in the field of plasma physics. Results for the self-energy and the spectral function are presented.

Plasmon-like structures, present in perturbative calculations as performed in Ref. [48] vanish completely. An

asymmetrically broadened and shifted spectral function is obtained. Furthermore, it is shown, that a simplified

model, where the dynamically screened interaction is approximated by static screening leads to similar results

for both self-energy and spectral function as compared to the full RPA results. Deviations can be understood as a

consequence of neglecting collisions among particles of equal species.

The simplified GW 0 approximation for plasmas is now available for a broad range of parameters, i.e. density

and temperature. Also, the correct asymptotic behaviour, i.e. convergence to a delta-like quasiparticle resonance

at largemomenta, is obtained. In this paper we used parameters corresponding to the solar core, whereas Ref. [22]contains similar results for lower density and temperature.






Vertex corrections to the self-energy have been studied on a perturbative level. So far, only the high frequency

behaviour of the exchange diagram second order in the screened interaction was evaluated. It leads to 20%

reduction of the self-energy with respect to the second order of GW 0.

The availability of the one-particle spectral function for any momentum and frequency makes it possible to

use it in calculations of further physical observables as the equation of state and the optical properties. Here

we focused on the absorption of electromagnetic radiation due to free-free transitions, which is obtained from

the polarization function. Insertion of the broadened particle propagators in the one-loop approximation for Πleads to a significant modification of the absorption spectrum at low frequencies. Besides the suppression in the

infrared, enhancement of absorption is observed at intermediate frequencies.

Since the one-loop approximation using full propagators is an inconsistent approximation with regards to con-

servation laws, suchas Ward identities, we investigated the vertex-correctioninside the polarization loop in lowest

order. It turned out, that the vertex correction is by far the most important correction to Born approximation, i.e.

the self-energy effects contained in the full spectral functions, are dominated by the vertex correction. However,

this is not a general feature as becomes clear if comparing to calculations for other parameters in Ref. [22]. On

the other hand, a consistent comparison of both self-energy and vertex corrections on the same level of approxi-mation necessitates the summation of vertex-correctionsby solution of the Bethe-Salpeter equation (5). This task

will be accomplished in the future.

Acknowledgements This article was supported by the DFG within the Sonderforschungsbereich 652 ’Starke Korrelationen

und kollektive Phanomene im Strahlungsfeld: Coulombsysteme, Cluster und Partikel.’ A.W. would like to thank the Center

of Atomic and Molecular Technologies of Osaka University for its hospitality. C.F. acknowledges stimulating discussion with

J. Vorberger and W.D. Kraeft and thanks R. Zimmermann for many helpful advice.

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Chapter 8

Bremsstrahlung and LineSpectroscopy of Warm Dense

Aluminum Plasma generated byXUV Free Electron Laser

Authors: Ulf Zastrau, Carsten Fortmann, Roland Rainer Faustlin, Lei Feng Cao, TiloDoppner, Stefan Dusterer, Siegfried H. Glenzer, Gianluca Gregori, Tim Laarmann, HaejaLee, Andreas Prsystawik, Paul Radcliffe, Heidi Reinholz, Gerd Ropke, Josef Tiggesbaumker,Robert Thiele, Xuan Truong Ngyuen, Ingo Uschmann, Sven Toleikis, August Wierling,Thomas Tschentscher, Eckhart Forster, and Ronald Redmer

Appeared as regular article in Physical Review E, Vol. 78, Issue 6, pages 066406 1–5, Dec.2008.


• U.Z.: Execution of experiment, data processing, writing of manuscript

• C.F.: Bremsstrahlung calculations, writing of manuscript

• R.R.F.: Execution of experiment, HELIOS simulations, writing of manuscript

• A.W.: Calculation of relative line intensities, writing of manuscript

• L.F.C.: Execution of experiment, writing of manuscript

• R.R.: Writing of manuscript

• G.R.: Writing of manuscript

• H.R.: Writing of manuscript

• S.H.G.: Execution of experiment, writing of manuscript

• all others: Execution of experiment



148 Spectroscopy of Warm Dense Al Plasma



Bremsstrahlung and line spectroscopy of warm dense aluminum plasma heated by xuv

free-electron-laser radiation

U. Zastrau,1,* C. Fortmann,

2R. R. Fäustlin,

3L. F. Cao,

1T. Döppner,

4S. Düsterer,

3S. H. Glenzer,

4G. Gregori,

5

T. Laarmann,3

H. J. Lee,6

A. Przystawik,2

P. Radcliffe,3

H. Reinholz,2

G. Röpke,2

R. Thiele,2

J. Tiggesbäumker,2

N. X. Truong,2

S. Toleikis,3

I. Uschmann,1

A. Wierling,2

T. Tschentscher,3

E. Förster,1

and R. Redmer2

1 Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, Max-Wien Platz 1, 07743 Jena, Germany

2 Institut für Physik, Universität Rostock, 18051 Rostock, Germany

3 Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg, Germany

4 Lawrence Livermore National Laboratory, University of California, P.O. Box 808, Livermore, California 94551, USA

5Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

6 Department of Physics, University of California, Berkeley, California 94720, USA

Received 25 April 2008; revised manuscript received 20 October 2008; published 30 December 2008

We report the creation of solid-density aluminum plasma using free-electron laser FEL radiation at

13.5 nm wavelength. Ultrashort pulses were focused on a bulk Al target, yielding an intensity of 2

1014 W /cm2. The radiation emitted from the plasma was measured using an xuv spectrometer. Bremsstrah-

lung and line intensity ratios yield consistent electron temperatures of about 38 eV, supported by radiation

hydrodynamics simulations. This shows that xuv FELs heat up plasmas volumetrically and homogeneously at

warm-dense-matter conditions, which are accurately characterized by xuv spectroscopy.

DOI: 10.1103/PhysRevE.78.066406 PACS numbers: 52.50.Jm, 52.25.Os, 52.27.Gr, 52.70.La

I. INTRODUCTION

The physics of warm dense matter WDM 1 has gained

increasing interest because of its location in the transition

region from cold condensed materials to hot dense plasmas.

WDM states are of paramount importance to model astro-

physical objects such as giant planets 2 or brown dwarfs3. Furthermore, WDM occurs as a transient state in novel

experiments to generate high-energy densities in materials,

most notably the realization of inertial confinement fusion4,5. The first experimental investigations of WDM have

been performed, e.g., with shock-wave experiments 6 and

with laser-excited plasmas 7–11.

WDM describes materials at temperatures of several eV at

solidlike densities. Its creation and investigation under con-

trolled conditions in the laboratory is a difficult task. Using

common optical short-pulse lasers, nonlinear absorption

leads to rapid temporal variations, steep spatial gradients,

and a broad spectrum of plasma physical processes. Pioneer-

ing techniques such as laser-driven shock heating, x-ray

heating, ion-heating techniques 6,12–15, and x-ray Thom-son scattering 16 have been developed in order to improve

the plasma heating mechanism.

In WDM the electron temperature is comparable to the

Fermi energy, i.e., the degeneracy parameter = k BT e / E F is

close to unity. Furthermore, the ion coupling parameter

= Z 2 /4 0k BT i4 ni /31/3 is greater than or equal to unity,

i.e., the interparticle Coulomb correlation energy is equal or

exceeds the thermal energy; Z is the ion charge, and ni is the

ion density. Thus, electrons as well as ions exhibit strong

temporal and spatial correlations which depend strongly on

the plasma parameters, temperature, and density.

A proper description of WDM is also a tremendous chal-

lenge to many-particle physics. Both the theory for ideal

plasmas and for condensed matter fail in this regime. Clas-

sical plasma theory based on expansions of the correlation

contributions in powers of the coupling parameter breaks

down since 1, and strong-coupling effects among the

various species have to be taken into account. On the other

hand, the plasma is too hot to be considered as condensed

matter, i.e., expansions in terms of the degeneracy parameter also fail. Thus, precise knowledge of physical properties as

a function of the plasma parameters temperature and density

is of primary importance.

In this article, we demonstrate that xuv free-electron la-

sers FELs open a promising possibility to heat matter volu-

metrically and homogeneously. Furthermore, we show that

xuv bremsstrahlung and line spectroscopy allow the determi-

nation of the plasma temperature and free-electron density.

II. EXPERIMENTAL SETUP

A. Free-electron laser characteristics

The fourth-generation light source FLASH free-electron

laser in Hamburg uses the self-amplified spontaneous emis-

sion SASE principle to generate brilliant xuv pulses17,18. In the experiment, pulses of 91.8 eV photon energywavelength = 13.5 nm, 155 fs duration were focused

to a 30-m spot by a carbon-coated ellipsoidal mirror with

grazing incidence angle of 3° and 2 m focal length, as shown

in Fig. 1.

Since the FEL process starts from spontaneous radiation,

it shows an intrinsic shot-to-shot pulse energy spread. The

relevant distribution of pulse energies for this experiment is

shown in Fig. 2. The total number of pulses in our measure-

ment was 81 000, including 3% of zero-energy events and a

significant fraction of high-energy pulses up to 130 J. Themean value is 48 J, measured at the end of the undulators.

Since the beamline transmission is known to be T =0.68, the

average pulse energy at the experiment is 33 J.*[email protected]

PHYSICAL REVIEW E 78, 066406 2008

1539-3755/2008/786 /0664065 ©2008 The American Physical Society066406-1

149



B. xuv plasma spectroscopy

Each pulse irradiates a bulk aluminum target under 45°

incident angle, resulting in an average intensity of 21014 W /cm2; see Fig. 1. The polarization is linear in the

horizontal plane. At the chosen wavelength, the critical den-

sity for penetration into the bulk, ncrit = 2 c2

0me/

e22

=6.11024 cm−3, is about 40 times higher than the valence

electron density in cold solid aluminum, ne =1.6

1023 cm−3; therefore, the initial absorption length is 40 nm19. The pulse energy is deposited in a target volume of

15 m240 nm, generating WDM. The number of 1012

atoms in this volume is in the same order of magnitude as the

incident photon number.

At fixed target position, the emission spectrum of about

104 exposures was recorded before moving to a fresh site.

Since the very first FEL pulse ablates a few-nanometer-thin

surface layer, a further surface cleaning technique was not

necessary. The target was at ambient temperature, and

vacuum was kept constant at 10−7 mbar.

The FEL was run in multibunch operation mode at a 5-Hz

repetition rate, with 20 bunches per train giving 100 FEL

pulses per second. Five separate measurements of different

durations adding up to a total interaction time of 13.5 min

were performed. Since the individual spectra look identical,

we assume that plasma formation and emission processes

vary very slowly during the measurement.

xuv emission spectra in the region of 6–18 nm were mea-

sured with a high-throughput spectrometer, featuring a toroi-

dal Ni-coated focusing mirror and a free-standing transmis-

sion grating. The spectrometer is described in detail in Ref.20. The spectral resolution was limited to 0.2 nm due to a

slightly fluctuating plasma position. A back-thinned xuv

charge-coupled device CCD camera with 1313 m2

pixel size and a quantum efficiency of =0.45 served as the

detector. From the measured spectra, absolute photon num-

bers per wavelength interval and solid angle are calculatedusing the tabulated efficiency of all components.

III. DATA ANALYSIS

Figure 3 shows the sum of all spectra in logarithmic scale

after correction for the spectrometer throughput and detec-

tion efficiency. The error bars arise essentially from statisti-

cal signal-to-noise ratios beside uncertainties of the spec-

trometer components. The main peak at 13.5 nm stems from

Rayleigh scattering of FEL photons by bound aluminum

electrons. It is broadened symmetrically by 0.4 nm due to

artifacts originating from the support grid of the transmission

grating. Spectral lines from Al IV and Al V are identifiedusing the NIST tables 21 as listed in Table I. The con-

tinuum emission is formed by free-free transition radiationbremsstrahlung and free-bound recombination radiation.

FIG. 1. Color online Experimental setup. a The xuv pulses

from the FEL undulators are focused by an elliptical beamline mir-

ror on the bulk aluminum target, creating a 30-m focus. b The

target is hit under 45° and plasma is created. A high-resolution xuv

spectrometer observes the plasma emission under 45° to the surface

normal.

( )

FIG. 2. Color online Histogram of the energy spread of

13.5-nm xuv pulses generated by self-amplified spontaneous emis-

sion SASE. Only pulses contributing to the measured spectrum of

the performed experiment are shown. The top ordinate shows the

corresponding irradiation intensities on target.

6 8 10 12 14 16 18wavelength λ [nm]

103

104

105

106

107

e m i t t e d p h o t o n s ∆ Ω - 1 ∆ λ -

1

experimentk

B

T

e

= 46.0 eV

k BT

e= 40.5 eV

k BT

e= 35.0 eV

11.6 nm (Al IV)

12.7 nm (Al V)

13.5 nm

16.2 nm (Al IV)

17.2 nm (LII/III

M)

17.0 nm (L-edge)

Rayleigh scattering

FIG. 3. Color Experimental xuv photon spectrum per solid

angle = 410−4 sr and wavelength interval =0.025 nm

symbols with error bars and bremsstrahlung calculations for dif-

ferent electron temperatures. The spectrum is corrected for the spec-

trometer throughput and detection efficiency.

ZASTRAU et al. PHYSICAL REVIEW E 78, 066406 2008

066406-2




Since the target is heated volumetrically, the continuumemission is partly reabsorbed and we observe a steplike fea-

ture at 17.0 nm that is consistent with the absorption LII/III

edge 19. This indicates deep deposition of energy into the

target, as expected for xuv photon-matter interactions. The

corresponding absorption L edge at a similar excitation flux

was also analyzed in laser-excited silicon 22. Finally, the LII/III M fluorescence line 23 is observed at 17.2 nm.

A. Bremsstrahlung

The experimental spectra allow the determination of the

plasma parameters using fundamental relations 24. The

electron temperature and density are inferred from the con-

tinuum background radiation due to bremsstrahlung. We

compare the experimental data to Kramers’ law 25

jff = e2

4 03

ne2 16 Ze−2 c/k BT e

3mec2

26 k BT eme

gT 1

for the free-free emissivity jff . Here, me is the electron

mass and gT is the wavelength-dependent Gaunt factor26, accounting for medium and quantum effects. It is cal-

culated in a Sommerfeld approximation 27. We assume an

average ion charge of Z =4, which is supported by calcula-

tions of the relative ion abundances using the codeCOMPTRA04 28; see below. In the wavelength range from

7.1 nm to 8.0 nm marked by the dotted box in Fig. 3, we

expect no essential contributions from bound-bound and

bound-free transitions. Statistical analysis of the data in this

range yields 40.5 eV for the temperature with a rms error of

5.5 eV. Bremsstrahlung spectra for 35 eV, 40.5 eV, and

46 eV are shown in Fig. 3. Reabsorption was considered

using tabulated opacity data 19; in this way, the L edge at

17.0 nm is reproduced. The height of the L edge corresponds

to a transmission through 40 nm of cold aluminum.

Kramers’ law Eq. 1 depends on the square of ne. From

the absolute photon number at =17.0 nm, N photon

=51 424−1 −1, we calculate the free-electron densityusing Eq. 1 as ne =4.01022 cm−3, taking the inferred

plasma temperature of 40.5 eV. This value is consistent with

radiation hydrodynamics simulations; see below.

B. Transition line ratio

Independently, the electron temperature can be obtained

from the ratio of integrated line intensities I for the identi-fied transition lines from the Boltzmann distribution 29 as

follows:

I 1

I 2= 1

3

23

f 1

f 2e− 1− 2/k BT e, 2

with the corresponding photon frequencies and oscillator

strengths f , having in mind that the plasma is optically thin.

Here, the Al IV lines doublets at 1 = 16.169 nm and 2

=11.646 nm, with oscillator strengths given in Table I, are

used. Integration was performed from 15.9 to 16.3 nm and

from 10.8 to 11.9 nm after subtraction of the bremsstrahlung

continuum, respectively. The resulting temperature is346 eV. Within the error bounds, this is consistent with

the temperature inferred by analysis of the bremsstrahlung

continuum, so that we can state the plasma temperature with

about 38 eV.

A full compliance of line and continuum temperature can-

not be expected, since the plasma dynamics affects both

emission processes in different ways. At early times after the

laser-target interaction, the system is still very dense and the

excited levels of the transitions under consideration are pos-

sibly dissolved into the continuum. Only after expansion are

the excited levels well defined and radiative transitions take

place. Bremsstrahlung, on the other hand, is most relevant at

early times due to the ne2 dependence of its emissivity; see

Eq. 1. Thus, in the bremsstrahlung emission we expect a

higher temperature than in the line spectrum.

C. Relative ion abundance

The relative abundance of aluminum ions was calculated

with the code COMPTRA04 28. Results for electron tempera-

tures from 10 to 50 eV are shown in Fig. 4, assuming a solid

density sol=2.7 g /cm3 and 0.5 sol—i.e., slightly expanded.

The concentration of ion species for the relevant tempera-

tures between 34 and 40 eV complies with the observed line

emission spectrum. At an averaged temperature of 38 eV, the

ion fractions of Al IV, Al V, and Al VI take the values of

about 7%, 45%, and 46%, respectively. Amounts of Al I – III

as well as Al VII and higher are negligible.

All expected spectral lines from Al IV in the observed

spectral range are either observed at 16.2 nm or covered by

the strong FEL signal at 12.9 nm. For transition details see

Table I. Weak lines from Al V between 11.9 and 13.2 nm

overlap with the Rayleigh peak and only the 12.607-nm line

can be identified. Transition lines from Al VI, which are lo-

cated between 8.6 and 10.9 nm, are not significantly present

compared to the detector noise. This indicates that the corre-

sponding high-lying excited levels are dissolved and do not

contribute.

Emission lines from Al VII and higher are expected to

play a significant role only at temperatures exceeding 40 eV,as shown in Fig. 4. These lines have also not been found, but

have previously been observed in optical laser-matter inter-

action experiments 30. This contrast is well understood by

TABLE I. List of the spectral lines emitted from the plasma,

identified with the NIST database 21,23.

Experimental

data

nm

Reference

data

nm

Relative

intensity

NIST

Oscillator

strength f

NIST Transition

11.60.2 11.646 250 0.332 Al IV: 2s22 p6

−2s22 p52 p1/2

0 4d

12.70.2 12.607 800 Al V: 2s22 p5

−2s22 p41 D3s

16.20.2 16.169 700 0.247 Al IV: 2s22 p6

−2s22 p52 p1/2

0 3s

17.20.2 17.14 LII/III M

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151



scrutinizing the different mechanisms of absorption and ion-

ization in the case of optical light as opposed to the case of

xuv photons.

IV. DISCUSSION

A. Different photon absorption mechanisms

In optical laser-matter interactions, dominant absorption

mechanisms are multiphoton ionization, nonlinear processes,

inverse bremsstrahlung, and resonance absorption 31.When the critical free-electron density of the optical laser is

exceeded, most light is reflected and absorption is limited to

the skin layer, leaving behind a plasma with steep density

and temperature gradients.

For xuv photons, nonlinear absorption is negligible at the

considered intensity and resonance absorption is not impor-

tant since the plasma is undercritical. Thus, “hot”-electron

production in the keV–MeV range is unlikely. Photoexcita-

tion and inverse bremsstrahlung are the only possible mecha-

nisms. Since there are no prepulses in the FEL case, the

deposition of photons starts in a cold target and the energy is

distributed volumetrically and homogeneously throughout

the interaction zone.

A change in the polarization of the FEL e.g., using cir-

cularly polarized light could lead to less heating and cer-

tainly to a reduction in the scattered intensity Rayleigh

peak . For atomic systems, a systematic decrease in the

photoionization yield has been reported upon changing the

laser’s polarization 32, while such studies for solid-state

targets at xuv conditions remain to be done.

The plasma production mechanism presented here takes

advantage of photons exceeding a 2 p-level binding energy of

72.8 eV. In particular, a 2 p bound electron is photoionized

with a cross section of PI =7 Mbarn 19. This electron is

transferred into the conduction band, leaving behind a hole

in the 2 p shell. For partially ionized aluminum the photoab-

sorption cross section increases below the LII/III edge slightlywith temperature 33. Due to their high excess energy of

about 20 eV, further electrons at lower energies are excited

via impact ionization and Auger processes 1. The electrons

equilibrate at a temperature of several eV up to hundred fem-

toseconds 34,35, forming typical WDM. During this time,

electrons can also recombine with the 2 p holes by emitting

fluorescence radiation at 17.2 nm wavelength. This state of

matter cools down on a picosecond time scale by energy

transfer to the lattice via electron-phonon scattering 24.

B. Hydrodynamics simulation

To illustrate the hydrodynamic processes and to estimate

the electron temperature, one-dimensional radiation hydro-

dynamics simulations using HELIOS 36 have been per-

formed. HELIOS features a Lagrangian reference frame—i.e., the grid moves with fluid, separate ion and electron

temperatures, and flux-limited Spitzer thermal conductivity.

It allows the deposition of laser energy via inverse brems-

strahlung as well as bound-bound and bound-free transitions,

using a SESAME-like equation of state. Per atom, 2.6 con-

duction band electrons were assumed to contribute to the

laser absorption 37. The results are shown in Fig. 5. On the

time scale of the FEL pulse, both electron density and tem-

perature rise up to values of ne 1023 cm−3 and k BT e 26 eV, respectively, without any steep gradient. These val-

ues are in good qualitative agreement with the results for ne

and T e obtained by the spectral analysis.

This hydrodynamics simulation uses an average pulse en-

ergy of 33 J on the target as an input parameter. As dis-

cussed above, Fig. 2 illustrates that a significant fraction of

pulses have much higher pulse energies up to 130 J. Since

the free-free emissivity Eq. 1 depends strongly nonlin-

early on both electron temperature and density, we expect the

observed radiation to be rather dominated by the high-energy

fraction of the xuv pulses than by its average value. This

finally explains the slight underestimation of the electron

temperature in the hydrodynamics simulation compared to

the experimental results.

Additionally, the simulation shows that these WDM con-

ditions exist for about 200 fs, at almost constant plasma den-

sity and temperature, and hydrodynamic motion is negligible

38. Plasma expansion as well as electron diffusion to thecold matter of the bulk target and heat conduction becomes

important during the first several picoseconds 39, while the

density decreases about a factor of 2, influencing the relative

20 30 40 50electron temperature k

BT

e[eV]

0

0.2

0.4

0.6

0.8

1

r e

a t v e

o n

a

u n

a n c e

Al III, Z=2Al IV, Z=3

Al V, Z=4Al VI, Z=5Al VII, Z=6

FIG. 4. Color Calculation of the relative Al ion species abun-

dance from COMPTRA04 as a function of the electron temperature.

The solid line is for the Al density sol =2.7 g /cm3, dashed line

0.5 sol =1.35 g /cm3.

t i m e

[ p s

]

0.0

0.1

0.2

depth [µm]0 -0.10.1

(a) Electron density

0[1/cm ]

3

2.5x1023

(b) Electron temperature

0.025 25.7[eV]

laser

0 -0.1

0.3

FIG. 5. Color HELIOS simulation results for the electron den-

sity left and electron temperature right as a function of time and

radius.

ZASTRAU et al. PHYSICAL REVIEW E 78, 066406 2008

066406-4




abundance of ion species only slightly, as demonstrated in

Fig. 4.

V. SUMMARY AND CONCLUDING REMARKS

The capability of xuv FEL radiation to create WDM

by interaction with a solid aluminum target was demon-

strated. The analysis of the xuv line and continuum emission

spectra yields an electron temperature of 346 eV and40.55.5 eV, respectively. The observed line spectrum is

compatible with predicted ion abundances. Together with ra-

diation hydrodynamics modeling, we get a sound picture of

complex xuv laser-plasma interaction dynamics. The simula-

tions confirm the volumetric heating of the target without

strong gradients. Our results provide complementary infor-

mation to results that were reported for optical laser-matter

interactions 30.

Further and detailed studies of WDM will include spa-

tially and temporally resolved experiments to determine the

electron temperature and density. For this regime, novel di-

agnostic techniques such as x-ray interferometry 40,41 and

x-ray Thomson scattering 34,42,43 have been developed.

In combination with these techniques, the xuv FEL will be a

unique platform for WDM investigations. This will be im-

portant for shock-wave physics, applied-material studies,

planetary physics and inertial confinement fusion, and other

forms of high-energy density-matter generation.

ACKNOWLEDGMENTS

We thankfully acknowledge financial support by the Ger-

man Helmholtzgemeinschaft via the Virtual Institute VH-VI-

104, the German Federal Ministry for Education and Re-

search via Project No. FSP 301-FLASH, and the Deutsche

Forschungsgemeinschaft DFG via the Sonderforschungs-

bereich SFB 652. T.L. acknowledges financial support from

the DFG under Grant No. LA 1431/2-1. R.R.F. received DFG

funds via Grant No. GRK 1355. The work of S.H.G. was

performed under the auspices of the U.S. Department of

Energy by Lawrence Livermore National Laboratory under

Contract No. DE-AC52-07NA27344. S.H.G. was also sup-

ported by LDRDs 08-ERI-002, 08-LW-004, and the Alex-

ander von Humboldt foundation. The work of G.G. was par-

tially supported by the Science and Technology Facilities

Council of the United Kingdom. Finally, the authors are

greatly indebted to the machine operators, run coordinators,

and scientific and technical teams of the FLASH facility for

enabling an outstanding performance.

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153



Chapter 9

Self-Consistent Spectral Function forNon-Degenerate Coulomb Systems

and Analytic Scaling Behaviour

Author: Carsten Fortmann

Appeared as regular article in Journal of Physics A: Mathematical and Theoretical, Vol.41, Issue 44, pages 445501 1–27, Sept. 2008



156 Self-Consistent Spectral Function and Scaling Behaviour



IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL

J. Phys. A: Math. Theor. 41 (2008) 445501 (27pp) doi:10.1088/1751-8113/41/44/445501

Self-consistent spectral function for non-degenerateCoulomb systems and analytic scaling behaviour

Carsten Fortmann

Institute of Physics, Rostock University, 18051 Rostock, Germany

E-mail: [email protected]

Received 29 May 2008, in final form 11 September 2008Published 7 October 2008

Online at stacks.iop.org/JPhysA/41/445501

Abstract

Novel results for the self-consistent single-particle spectral function and self-

energy are presented for non-degenerate one-component Coulomb systems at

various densities and temperatures. The GW (0)-method for the dynamical self-

energy is used to include many-particle correlations beyond the quasi-particle

approximation. The self-energy is analysed over a broad range of densities

and temperatures (n = 1017 cm−3 –1027 cm−3, T = 102 eV/kB –104 eV/kB).

The spectral function shows a systematic behaviour, which is determined

by collective plasma modes at small wavenumbers and converges towards a

quasi-particle resonance at higher wavenumbers. In the low density limit, thenumerical results comply with an analytic scaling law that is presented for the

first time. It predicts a power-law behaviour of the imaginary part of the self-

energy, Im ∝ −n1/4. This resolves a long time problem of the quasi-particle

approximation which yields a finite self-energy at vanishing density.

PACS numbers: 52.27.Aj, 52.65.Vv, 71.10.Ca, 71.15.−m

1. Introduction

Strongly correlated Coulomb plasmas, found e.g. in planetary interiors [1, 2], fusion plasmas

[3], and plasmas excited by lasers or ion beams [4], are characterized by a high degree of

spatial and temporal correlations, which lead to the emergence of phenomena like collective

plasma modes, dynamical screening of the interparticle interaction potential, and dissolution

of bound states. In particular, laser-excited plasmas cover a broad range of densities and

plasma temperatures. Values range from typical condensed matter conditions to hot, weakly

coupled plasmas.

Theoretical approaches to the physical properties of such systems have to deal with a

great complexity. A particular challenge is the formulation of a coherent theory, which is valid

over a wide range of densities (n) and temperatures (T ), thereby allowing to describe matter

1751-8113/08/445501+27$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1

157



J. Phys. A: Math. Theor. 41 (2008) 445501 C Fortmann

in various states, e.g. a solid-state target, being transferred into a plasma by interaction with

high-power lasers and its subsequent relaxation [5]. Many-particle perturbation theory [6]

presents a general approach to many-body systems like condensed matter [7], partially and

fully ionized plasmas [8], and nuclear matter, to mention only a few. Also for non-abelian

systems, such as the quark–gluon plasma [9], there exist similar approaches to that described

here for Coulomb systems, e.g. the concept of Schwinger–Dyson equations, see the review

article [10]. The thermodynamic properties as well as the response to external perturbations

of these systems in various situations can be studied systematically [11].

The central quantity within the many-body theoretical approach is the single-particle

spectral function A(p, ω). It represents a physical observable which can be measured via

angular resolved photoemission spectroscopy (ARPES) [12 – 14]. Starting from the spectral

function, a number of interesting questions related to the physics of many-particle systems

can be addressed. The equation of state [15], transport cross-sections [16] (e.g. electrical

conductivity, thermal conductivity and stopping power [17]) and optical properties [18]

(emission and absorption of electromagnetic radiation) become accessible.In this work, the focus is on the spectral function of plasmas. As an example, a one-

component electron plasma is considered which is charge compensated by a homogeneously

distributed background of positively charged ions (jellium model). The plasma is characterized

by the degeneracy parameters θ and the plasma coupling parameter which are defined as

θ = 2mkBT

h2(3π 2n)2/3, = e2

4π 0kBT

4π n

3

1/3

. (1)

Here, the electron mass m was introduced, kB is the Boltzmann constant. In this work, we

consider only non-degenerate systems, θ 1, i.e. the thermal energy kBT is large compared

to the Fermi energy EF = h2(3π 2n)2/3/2m.

The calculation of the spectral function becomes challenging in the regime of strong

coupling, i.e. when the plasma coupling parameter becomes comparable or larger than unity.The coupling parameter measures the ratio of the Coulomb interaction energy of two particles

at a mean distance to their thermal energy kBT . At 1, particle collisions become frequent,

involving transfer of both momentum and energy. The interparticle potential is screened due

to the presence of nearby third particles. These correlations significantly modify the plasma

observables and have to be accounted for in the calculation of the spectral function. This

is accomplished via the single-particle self-energy (p, ω), which is a complex function of

both wavevector p and frequency ω, leading to a structured spectral function. Though, the

main task of many-particle theory, applied to strongly coupled systems, is to calculate the

self-energy in a suitable approximation.

The simplest approximation, often found in the literature on Coulomb systems, is the

mean-field or Hartree–Fock approximation [8]. One obtains a frequency-independent self-

energy which induces a shift in the spectral function’s pole, the so-called Hartree–Fock or

quasi-particle shift. For dilute plasmas, this correctly describes the lowering of the chemical

potential due to the averaged field of the plasma particles. Also, the shift of the ionization

energy for bound states is obtained [19, 20]. However, in dense systems, the mean-field

approximation breaks down since the dynamical screening and collective excitations cannot

be accounted for. One has to go beyond the quasi-particle picture.

A particularly successful approximation for the self-energy, including these dynamical

effects, is the so-called GW -approximation [21, 22]. Correlations are accounted for via

the dynamically screened interaction potential W (q, ω), rather than via the bare Coulomb

interaction. The GW approximation knows a long history of applications in the field of

condensed matter theory. Examples are the calculation of single-particle spectra in the

2





homogeneous electron gas [23 – 25], bandgaps in semiconductors [26], effective masses of

metal electrons [27], optical and electronic properties of insulators [28], electronic structure

of superconductors [29], but also atomic and molecular systems [30 – 32]. In particular, GW

self-energy corrections systematically improve band-gap calculations performed by means of

density functional theory [33 – 35].

Recently, the GW approximation has been applied also to dense plasmas. Whereas

Fehr et al [36] performed lowest order (one-loop) self-energy corrections to the equation of

state, Wierling et al [37] carried out pioneering self-consistent calculations of the electron

self-energy in the solar-core plasma. An asymmetrically broadened, otherwise featureless

spectral function was obtained. In this work, the GW self-energy and the corresponding

spectral function is investigated for non-degenerate, one-component electron plasmas. Only

unbound electrons are considered, bound state contributions can be accounted for via T -matrix

calculations, as done in [38]. The self-energy is evaluated for a broad range of densities and

temperatures, going from ideal, weakly coupled plasmas (

1) to the strong coupling

regime 1. As a novel contribution to the field, an analytic scaling law for the GW self-energy at low densities is derived which accurately describes the numerical data in this

limit. This expression can be combined with corresponding formulae that are valid in the

degenerate case, when kBT EF, to construct a fit formula for the self-energy which then

covers a large portion of the density–temperature plane.

Formerly, analytic expressions for the self-energy have been derived that base on the

quasi-particle approximation [39]. In particular, the completely degenerate electron gas at

T = 0 was considered, using the plasmon-pole approximation [40], and also weakly coupled

( 1), classical plasmas (EF kBT ), using the Born approximation for the self-energy

[8]. The latter result exhibits several problems: the imaginary part of the quasi-particle self-

energy is independent of density and carries a prefactor ∝1/h. Thus, there is an unphysical

finite damping of single-particle states even in the vacuum and the classical limit h → 0 is

not defined. On the other hand, from physical arguments, one expects that the self-energyvanishes at zero density and that it is a purely classical expression (h = 0), when θ 1. This

problem has remained unresolved up to now. The real part of the quasi-particle self-energy is

well behaved, i.e. it vanishes at zero density and is purely classical.

The new analytic expression for the self-energy presented in this paper is derived without

the quasi-particle approximation, i.e. it is a non-perturbative result. It is shown that only this

non-perturbative treatment leads to an expression that is classical for both the real and the

imaginary part and vanishes exactly in the vacuum limit n → 0.

The work is organized as follows: after a brief recapitulation of the single-particle spectral

function and the GW -method in section 2, numerical results for the self-consistent spectral

function and self-energy will be discussed in section 3. Section 4 contains the derivation of the

non-perturbative scaling law and comparison to the numerical results. In section 5, it will be

analysed why the quasi-particle picture is incapable to give a physically consistent result for

the imaginary part of the self-energy. Conclusions will be drawn in section 6. The appendixcontains detailed calculations that are only summarized in the main part of the paper.

2. Spectral function and self-energy

The derivation of the GW -approximation involves some lengthy manipulations. In this section,

only the most relevant formulae are given, while appendix A contains the detailed steps.

Central to the description of electronic properties in a many-body system, which

is in thermodynamic equilibrium, is the thermodynamic electron single particle Green

function G(p, zν ), defined at the discrete Matsubara frequencies zν = (2ν + 1)π ikBT /h,

3

159




ν = 0, ±1, ±2, . . .. It is related to the single-particle self-energy (p, zν) via Dyson’s

equation

G(p, zν ) = G(0)(p, zν ) + G(0)(p, zν )(p, zν )G(p, zν )

=

G(0)−1(p, zν ) − (p, zν )

−1, (2)

with the free Green function G(0)(p, zν ) = [hzν − εp]−1. Also, the single-particle energy

εp = h2p2/2m − μ is introduced, μ is the electron chemical potential. G(p, zν ) contains the

thermodynamic properties of a single particle coupled to a thermal bath at a given temperature

T . For example, the momentum distribution function is easily obtained by summation of the

Green function over all Matsubara frequencies,

n(p) = kBT

zν

G(p, zν ). (3)

Instead of the complex Matsubara Green function, it is more convenient to operate on the

real valued spectral function A(p, ω), defined on the real frequency axis. It carries the sameinformation as the Green function and is defined via the spectral representation of the latter,

G(p, zν ) = ∞

−∞

dω

2π

A(p, ω)

zν − ω. (4)

Here, ω is a real valued frequency. This relation can be resolved for A(p, ω),

A(p, ω) = − limδ→0+

2 Im G(p, ω + iδ) (5)

= limδ→0+

−2 Im (p, ω + iδ)

[hω − εp − Re (p, ω)]2 + [Im (p, ω + iδ)]2, (6)

i.e. the spectral function is obtained after analytic continuation of the Green function from the

Matsubara frequencies to arbitrary complex frequencies as the imaginary part of G(p, ω + iδ),

when δ approaches zero from positive values. In this way, the sign of the imaginary part of the

self-energy is fixed, i.e. Im (p, ω) < 0 for δ > 0. The real part of the self-energy behaves

unambiguous for δ = 0.

The spectral function usually exhibits several resonances, including a central peak, located

at the quasi-particle energy Ep, i.e. the solution of the quasi-particle dispersion

Ep = εp + Re (p, Ep/h), (7)

accompanied by symmetrically distributed satellites which are attributed to collective modes

in the many-particle system [40]. The width of the resonances in the frequency domain is

commonly identified with the inverse life-time of these excitations.

Let us first look at the lowest order approximation to the self-energy, the Hartree–Fock

term HF

(p). It is given by the convolution of a non-interacting Green function with theunscreened Coulomb potential V(q) = e2/0q20 (0 is a normalization volume),

HF(p, zν ) = −kBT ωμ,q

G(0)(p − q, zν − ωμ)V(q) (8)

=

q

[1 − nF(εp−q)]V(q) ≡ HF(p), (9)

with the Fermi distribution function nF(hω) = [exp(hω/kBT ) + 1]−1. In the first line,

summation takes place over the Bosonic Matsubara frequencies ωμ = 2π iμkBT /h, μ =4





0, ±1, ±2, . . . . The first term q V(q) (Hartree term) diverges, but it is exactly compensated

by the same term from the positive charge background. The second term (Fock term or

exchange term) gives a finite contribution. Closed expressions can be given in the case of non-

degenerate plasmas [8, 39] and completely degenerate Fermi gases [7]. One finds HF(p) ∝ n

in the high temperature limit (kBT EF) and HF(p) ∝ n1/3 in the quantum degenerate

case kBT EF, see [8] for details. Thus, the Hartree–Fock self-energy fulfils the physical

constraint to vanish at zero density.

The Hartree–Fock term is a real function of momentum, only. The corresponding spectral

function is shifted from the free particle dispersion,

AHF(p, ω) = 2πδ(εp + HF(p) − hω). (10)

No imaginary part of the self-energy appears in this approximation, i.e. the life-time of the

Hartree–Fock quasi-particles is infinite. This is consequence of the mean-field approximation,

where no fluctuations of the electric field, i.e. no dynamics of the surrounding plasma particles

are taken into account. Recently, also the second-order exchange contribution to the self-energy has been obtained in closed form [41, 42], see also [43]. However, this term and all

higher order terms, involving only the bare Coulomb potential, do not lead to a finite particle

life-time, only a shift of the dispersion relation is obtained.

To describe the situation in a dense and strongly correlated system, where the single-

particle states are spectrally broadened, i.e. they acquire a finite life-time, one has to go beyond

the quasi-particle approximation, and take into account the screening of the interaction. The

GW -approximation, can be regarded as the generalization of the Hartree–Fock theory to

dynamically screened interactions. It was introduced by Hedin [23] for the homogeneous

electron gas, and is defined as

(p, zν ) = −kBT q,ωμ

G(p − q, zν − ωμ)W (q, ωμ). (11)

W (q, z) is the dynamically screened interaction. Note that the GW approximation is a self-consistent ansatz, since the self-energy appears on the lhs as well as in the Green function on

the rhs of (11). Also, the screened interaction W (q, ωμ) is a functional of the Green function

via the dielectric function (q, ωμ), i.e. the polarization function (q, ωμ):

W (q, ωμ) = V(q)

(q, ωμ)= V(q)

1 − V (q)(q, ωμ). (12)

In GW -approximation, (q, ωμ) is given by the inner product of two Green functions,

(q, ωμ) = −kBT

p,zνG(q + p, zν + ωμ)G(p, zν ). The ‘double’ self-consistency implied in

this ansatz makes the GW -approximation complicated and a numerically demanding problem.

On the other hand, the full GW -approximation suffers from deficiencies due to the neglect

of vertex-corrections [44], such as violation of the f -sum rule [45]. This problem can be

avoided by keeping the dynamically screened interaction on the level of the random phaseapproximation (RPA) [46], defined by the RPA polarization function,

RPA(q, ωμ) = −kBT p,zν

G(0)(p − q, zν − ωμ)G(0)(p, zν ), (13)

RPA(q, ω + iδ) = −

k

nF(εk+q/2) − nF(εk−q/2)

h(ω + iδ) + εk−q/2 − εk+q/2

. (14)

The use of the RPA polarization function leads to the so-called GW (0) approximation for

the self-energy. It has been shown to give more accurate quasi-particle energies [25]

5

161




than the full GW -approximation. Additionally, it is known that higher order corrections

beyond GW (0), such as vertex-corrections and corrections in the polarization function beyond

RPA, partially compensate. Therefore, ignoring them altogether is expected to give better

results than accounting for one or the other [22]. The f -sum rule is fulfilled. Further sum

rules, e.g. for the moments of the spectral function can be derived [24] which are useful to

control the numerical treatment of the integral equations to solve.

The inverse dielectric function −1RPA(q, ω) describes the propagation of electromagnetic

waves in the plasma. As a main feature, it contains the longitudinal plasma oscillations or

plasmons. These resonances show up as peaks in the inverse dielectric function, located

at the roots of the plasmon dispersion Re RPA(q, ω) = 0. For non-degenerate systems, as

considered here, the plasmon dispersion can be expanded in powers of the wavenumber q,

and one finds the Gross–Bohm relation [47] ω2res(q) = ω2

pl(1 + q2/κ2) + (hq2/2m)2 for the

plasmon resonance frequency ωres(q). Here, the plasma frequency ωpl and the inverse Debye

screening length κ,

ωpl =

ne2

0m

1/2

, κ =

ne2

0kBT

1/2

, (15)

have been introduced. A detailed discussion of the plasmon resonance in dense plasmas can

be found in [48]. For the present discussion, it is important to keep in mind that the collective

plasma excitations are accounted for via the inverse dielectric function in RPA. This is the

main advantage of the GW (0)-approximation compared to the mean-field or Hartree–Fock

approximation. Depending on the choice of parameters like density and temperature, these

plasmon resonances determine the shape of the self-energy as a function of the frequency and

thereby also the spectral function, where satellites besides the quasi-particle peak indicate

coupled electron–plasmon modes, often referred to as plasmarons [40].

It should be noted at this point that contributions from bound states to the self-energy are

not accounted for in this work. The description is limited to fully ionized plasmas. Bound

state contributions can be included using the concept of the T -matrix, see e.g. the work bySchmielau et al [38].

Using the spectral representation (4) and the screened interaction (12), the following

equation for the imaginary part of the self-energy in GW (0)-approximation is obtained after

summation over the Bosonic Matsubara frequencies ωμ,

Im (p, ω + iδ) = h

nF(hω)

q

∞

−∞

dω

2πV (q)A(p − q, ω − ω)

× Im −1RPA(q, ω)nB(hω)nF(hω − hω), (16)

with the Bose–Einstein distribution function nB(hω) = [exp(hω/kBT ) − 1]−1. The real part

of the self-energy is obtained by means of Hilbert transform as

Re (p, ω) =

HF

int (p) +P ∞

−∞

dω

π

Im (p, ω)

ω − ω . (17)

P denotes the Cauchy principal value integration, HFint (p) is the Hartree–Fock self-energy of

the interacting system,

HFint (p) = −h

q

∞

−∞

dω

2πA(p − q,ω)nF(hω)V (q). (18)

Finally, to close the set of equations, the chemical potential has to be fixed by inversion of the

density relation

n(μ,T) = 2h

0

p

∞

−∞

dω

2πA(p,ω)nF(hω). (19)

6





The factor 2 in front of the rhs stems from the summation over the spin components. Together

with Dyson’s equation (6), (16)–(19) constitute a system of nonlinear integral equations for

the self-energy.

Besides the normalization of the spectral function

h

∞

−∞

dω

2πA(p, ω) = 1, (20)

similar sum-rules can be derived also for higher moments of the spectral function [24]. These

are independent of the concrete approximation used for the self-energy. In second order,

one obtains an equation relating the first moment of the spectral function to the interacting

Hartree–Fock self-energy (9),

h2

∞

−∞

dω

2πωA(p, ω) = εp + HF

int (p). (21)

Similarly, the second moment is related to the Hartree–Fock energy and the frequencyintegrated imaginary part of the self-energy, which is itself a conserved quantity, at least

within the GW (0) approximation, see (23) below,

h3

∞

−∞

dω

2πω2A(p, ω) = h

∞

−∞

dω

πIm (p, ω + iδ) +

εp + HF

int (p)2

. (22)

For the GW (0) self-energy, Holm and von-Barth have found the following identity, relating the

integrals over the imaginary part of the self-energy to the totally integrated response function,

h

∞

∞

dω

πIm (p, ω + iδ) = h

q

∞

−∞

dω

2πV(q)Im −1

RPA(q,ω). (23)

In the next section, results for the self-energy will be presented that are obtained via

numerical solution of (16). The sum rules given above are used to check the accuracy of thenumerical results.

3. Numerical results

The GW (0)-approximation is evaluated numerically for various sets of plasma parameters in

the following. A typical example of a weakly coupled ( = 0.07), moderately degenerate

(θ = 2.2) plasma is the plasma at the solar core, with temperatures of T 100 Ry/kB 1360 eV/ kB and electron densities of n 7×1025 cm−3 [49]. The solar core plasma has been

investigated using the GW (0)-method in a number of previous publications, see [18, 37, 50].

Here, most attention is paid to a systematic analysis of the single-particle spectral function and

the self-energy over a broad range of densities and temperatures, however, sticking to non-

degenerate plasmas and neglecting bound states. We therefore start with a plasma temperature

that equals the solar core temperature and a density that is 10% of the solar core electron

density. Later, higher and lower temperatures will be considered as well, i.e. kBT = 10 Ry

and kBT = 1000 Ry. Note that kBT is always chosen large against typical binding

energies of atoms which are usually of the order of several Ry. Thus, bound states can be

neglected.

The numerical solution of equation (16) is performed by means of an iterative algorithm,

starting from a suitable initialization of the spectral function. Typically, the algorithm

converges after 5–10 iterations. The threefold integral (16) is evaluated on a two-dimensional

grid with roughly 100 nodes in the frequency coordinate and 10–20 nodes in the momentum

coordinate. The angular integral is performed first, followed by the frequency integration and

7

163




] - 1

B

m o m e n t u m p [ a

0

5

10

15

20 [ R y ]μ+ωhe ne rg y-50 0 50 100 150 200 250 300 350 400 450

) [ 1 / R y ]

ω

A ( p ,

0

0.1

0.2

0.3

0.4

0.50.6

Figure 2. Spectralfunctionfor plasma density n = 7×1024 cm−3 and temperature kBT = 1360 eV(solar core temperature) as a function of momentum and density. The black line on the bottomrepresents the free dispersion relation hω = εp = h2p2/2m − μ. At the present parameters thechemical potential is μ = −377 Ry.

real part of the self-energy (second graph from top) is a rather smooth function, leading to

only small variations in the dispersion (third graph from top).

Next, the dependence of the spectral function on the wavenumber p is analysed.

In figure 2, the spectral function A(p, ω) is shown for five different wavenumbers, i.e.

p = 0, 5a−1B , 10a−1

B , 15a−1B and 20a−1

B , aB = 4π 0h2/me2 is the Bohr radius. The density

and temperature are the same as before, n = 7 × 1024

cm−3

, kBT = 100 Ry. At increasedwavenumber

p 5a−1

B

, enhanced complexity of the spectral function is observed. The

plasmaron peaks, which at hp = 0 appear as small shoulders in the otherwise broad central

resonance, are better defined. The central quasi-particle peak itself becomes narrower and the

plasmaron peaks separate. At the highest momenta consideredhp > 15a−1

B

, the plasmarons

themselves are damped out, and a single, narrow resonance forms, located near the single-

particle energy hω = εp = h2p2/2m − μ, i.e. the quasi-particle picture is restored. Some of

these features, especially the plasmaron satellites are already known from literature [40].

Now that the general characteristics of the spectral function have been discussed, the

central concern of this paper can be worked out, i.e. the analysis of the dependence of the

self-energy and the spectral function on the plasma parameters density and temperature.

In figure 3, the spectral function at p = 0 is shown for five different densities between

n=

7×

1025 cm−3 (solar core conditions) and0.01% of thesolar core density. The temperature

is kept constant at T = 100 Ry/kB = 1360 eV/kB. The spectral function drastically changes

with varied density. In the case of the highest density considered, a narrow quasi-particle peak

accompanied by two separate plasmaron satellites (indicated by arrows) is observed. The

quasi-particle peak is notably shifted from the free dispersion ε0 = μ, due to the real part of

the self-energy. Going to lower densities, the plasmaron satellites merge into the central peak,

as can be seen in the case of the spectral function for n = 7 × 1024 cm−3 and also the quasi-

particle shift is reduced. Finally, at the lowest densities considered, n = 7 × 1022 cm−3 and

7 × 1021 cm−3, a single, narrow quasi-particle resonance is obtained which is centred around

the free dispersion. The width decreases with the density which is the expected behaviour in

the low density limit.

9

165




0

0.2

0.4

0.6

0.8

1

1.2

-40 -20 0 20 40

s p e c t r a l f u n c t i o n A ( p = 0 , ω

) [ R y ]

energy −hω+μ [Ry]

n=7.0x1025

cm-3

n=7.0x1024

cm-3

n=7.0x1023

cm-3

n=7.0x1022

cm-3

n=7.0x1021

cm-3

Figure 3. Spectral functions at p = 0 for different plasma densities, ranging from the density atthe solar core, n = 7 × 1025 cm−3 (solid curve) to 0.01% of the solar core (dash-dot-dotted curve).The plasma temperature is T = 100 Ry/ kB = 1360 eV/ kB for all five curves.

1 0 1 7

1 0 1 8

1 0 1 9

1 0 2 0

1 0 2 1

1 0 2 2

1 0 2 3

1 0 2 4

1 0 2 5

1 0 2 6

1 0 2 7

1 0 2 8

density n [cm-3

]

10-2

10-1

- I m

Σ ( 0 , E 0

/ h_ ) / k

B T

θ = 1

θ = 1θ = 10

θ = 10

θ = 10 θ = 1

~n1/4

~n1/4

~n

1/4

k BT = 10 Ry

k BT = 100 Ry

k BT = 1000 Ry

Figure 4. Effective quasi-particle damping width at p = 0, −Im (0, E0/h) normalized to thethermal energy as a function of the plasma density. The arrows indicate for each temperature the

density at which the degeneracy parameter θ takes the values θ = 10 and θ = 1.

In order to study the dependence of the self-energy on density and temperature in more

detail, the effective quasi-particle self-energy (p, Ep/h) as a function of the density at

various temperatures is considered. This quantity gives the shift and width of the central peak

in the spectral function A(p, ω), i.e. when ω is close to the quasi-particle frequency Ep/h,

see (7). The results for the imaginary part of the effective quasi-particle self-energy at p = 0

as a function of the plasma density are shown in figure 4. Three different temperatures have

been assumed, T = 10, 100 and 1000 Ry/ kB. Towards low densities, a systematic decrease

10





of −Im (0, E0/h) with the density is observed which is also known from the literature

[37]. The asymptotes to the low density behaviour, shown as thin dotted lines, indicate that

Im (p, Ep/h) scales proportional to −n1/4. This behaviour will be analysed in more detail

in section 4, where an analytic solution for the GW self-energy is derived that exhibits the

same n1/4 proportionality.

At higher densities, the power law behaviour terminates and the self-energy starts to

decrease. This can be understood by looking again at figure 3. Here, it was shown that at

increased density, the plasmaron satellites separate from the central quasi-particle peak, i.e.

spectral weight is shifted to the satellites and the central peak narrows. The calculations have

only been performed for non-degenerate systems, i.e. for densities, where the degeneracy

parameter θ = kBT /EF is still large compared to unity. The extension to degenerate systems

is straightforward and is covered in another paper [51], but will not be treated in this work.

The real part of the self-energy (effective quasi-particle shift), at the densities and

temperatures considered here, was found to follow exactly the Hartree–Fock behaviour, i.e.

Re (0, E0/h) = −h2κ2/2m ∝ −n [8]; κ is the inverse Debye screening length, see (15).

4. Analytic solution for the GW (0) self-energy in Born approximation: classical limit

4.1. Derivation of the analytic solution

As discussed, thespectral function in thelow density limit is lacking any plasmaron resonances,

only a broadened quasi-particle peak appears, see figure 3. In order to understand this

behaviour, the GW (0)-equation (16) is reconsidered applying a sequence of approximations

as described in the following. In this way, an analytic solution is found that is valid at low

coupling parameters.

It will be shown that the observed scaling is obtained correctly, if the imaginary part of the

self-energy is kept finite also on the rhs of the self-energy integral equation (16). It thereforerepresents a generically non-perturbative result. Details of the calculations can be found in

appendix B.

Since collective excitations do not show up in the self-energy and the spectral function

at low densities, it is obvious to neglect these features already in the screened interaction.

Formally, this is achieved by replacing the complete inverse dielectric function by the Born

approximation,

Im −1(q, ω) − Im (q, ω)

|(q, 0)|2. (24)

For the static dielectric function appearing in the denominator, we use the Debye expression

D(q, 0) = 1 + κ 2/q2, with the inverse Debye screening length. In other words, instead of the

interaction via a dynamically screened potential, electron–electron collisions via a statically

screened potential are considered using the Born approximation. Then, (16) turns into

Im (p, ω + iδ) =

2mkBT

π 3

e2κ2

4π0

1

−1

dcos θ

∞

−∞dω

× ∞

0

dq q

(q2 + κ 2)2exp

− mω2

2q2kBT

exp

hω

2kBT

× Im (p − q, ω + iδ − ω)

[hω − hω − εp−q − Re (p − q, ω − ω)]2 + [Im (p − q, ω + iδ − ω)]2.

(25)

11

167




-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-4 -2 0 2 4

I m

Σ ( p = 0 , ω

) [ R y ]

energy−

hω+μ [Ry]

q=0q finite

Figure 5. Imaginary part of the self-energy at momentum hp = 0 for plasma parametersn = 7 × 1021 cm−3 and T = 100 Ry/ kB. The self-consistent Born approximation (finite q,dashed curve) is compared to the calculation where the momentum shift hq is neglected in theself-energy on the rhs of the self-energy equation (solid curve).

Note that the dielectric function is taken in the classical limit, i.e. the Fermi–Dirac distribution

is replaced by the Maxwell distribution, leading to the exponentials in the first line of ( 25).

Due to the statically screened Coulomb potential, important contributions to the q-integral

stem from values q κ. Therefore, we neglect the shift of momentum in the self-energy

on the rhs of equation (25), i.e. we write (p

−q, ω

−ω)

(p, ω

−ω). To justify this

approximation, we show the numerical solution for the imaginary part of the self-energy (25)in figure 5 (dashed curve). The solid curve corresponds to the solution that is obtained by

neglecting the momentum shift in the argument of the self-energy on the rhs of (25). As can be

seen, this approximation does not modify the result significantly. In fact, the small deviations,

which are only observable around hω + μ 0, are already in the order of the numerical

accuracy.

Subsequently, the remaining terms in (25) are expanded in powers of q/κ , as described

in detail in the appendix. Finally, the threefold integral can be performed and the equation

[Im (p, ω + iδ)]2 + [h2p2/2m − μ − hω + Re (p, ω)]2 = kBT κe 2

4π 0

(26)

is obtained. The lhs is just the denominator of the spectral function, cf ( 6). Together with the

spectral representation of the Green function (4), we then find the equation

[hz − h2p2/2m + μ − (p, z)]−1 = 4π 0

κe 2kBT (p,z), (27)

which, in the limit z = ω + iδ, δ → 0+ has the solution

(p, ω + iδ) = hω − h2p2/2m + μ

2− sign(hω − h2p2/2m + μ)

×

hω + iδ − h2p2/2m + μ

2

2

− κe 2

4π 0

kBT

1/2

. (28)

12





The signum function,

sign(ω) =

1 ⇔ ω 0

−1 ⇔ ω < 0,(29)

ensures the correct sign of the imaginary part of the self-energy, i.e. Im (p, ω + iδ) < 0 for

δ > 0.

4.2. Comparison to the numerical solution

The imaginary part of (28) is plotted in figure 6 for T = 100 Ry/kB and n = 7 × 1021 cm−3,

i.e. for the smallest density considered in figure (3). The analytic formula is compared to the

full numerical solution for two different wavenumbers, p = 0 (a) and p = 1/aB (b). Thedotted vertical line indicates the position of the quasi-particle dispersion Ep. In the first case,

both numerical and analytic calculation agree reasonably well, albeit the analytic solution lies

systematically above the numerical data. However, the overall deviation is smaller than 7%.

In the second case (p = 1/aB), the upshifted plasmon peak, present in the numerical result,

is not reproduced by the analytic formula. Thus, the analytic formula is applicable only for

small momenta, while at higher momenta, the dynamical features of the interaction become

important.

On the other hand, the analytic formula is very useful to initialize the numerical algorithm.

This is analysed in figure 7. Here, the spectral function, that is obtained in the first iteration

of the algorithm, was computed in two different ways for the same parameters as above,

n = 7 × 1021 cm−3 and kBT = 100 Ry. The dashed curve gives the first iteration starting

from the analytic formula (28) for the self-energy, the dotted curve is the same calculation butstarting from a narrow Gaussian spectral function with a width of 0.3 Ry (FWHM). In plot (a)

the wavenumber is p = 0, while in (b), p = 1/aB was chosen. For p = 0, the analytic ansatz

leads to a good resemblance with the converged result (solid curve). The converged result is

taken here as the 20 iteration starting from the Gaussian ansatz. The Gaussian ansatz, iterated

once, results in a two-peak structure which is far from the converged spectral function. Also

at p = 1/aB, starting from the analytic ansatz gives a much better overall correspondence

than the calculation starting from a Gaussian spectral function, although subtle details like the

plasmaron peak at hω + μ 4 Ry is not reproduced in the first iteration.

In order to perform a quantifiable comparison between both initializations and their impact

on the convergence of the algorithm, we determine the mean squared deviation of the spectral

function in a given iteration ν from the converged result S ν = N −1

N i=1(A(ν) (0, ωi ) −

A(20)(0, ωi ))2 with N the number of points on the ω-grid of the spectral function, ωi the

grid points. The result is shown in figure 8 for the Gaussian ansatz (marked +) and

the initialization using the analytic self-energy (marked ×). During the first four iterations,

the mean squared deviation of the second method is by two orders of magnitude smaller than

if using the Gaussian spectral function. While the mean squared deviation using the analytic

self-energy becomes smaller than 10−2 already after five iterations, it takes eight iterations for

the Gaussian ansatz to get to this point. Also, it was found that a Gaussian with the width

fixed at the imaginary part of the effective quasi-particle self-energy, does not improve the

convergence, since the special form of the self-energy and the spectral function with a broad

plateau and steep edges cannot be reproduced by such an ansatz and the analytic self-energy

given in (28) should be used instead.

13

169




-2.5

-2

-1.5

-1

-0.5

0

-10 -5 0 5 10

I m

Σ ( p = 0 , ω

) [ R y ]

energy

−

hω+μ[Ry]

GW(0)

analytic formula

(a) p = 0

-2.5

-2

-1.5

-1

-0.5

0

-10 -5 0 5 10

I m

Σ ( p = 1 / a

B , ω

) [ R y ]

energy −hω+μ[Ry]

GW(0)

analytic formula

(b) p = 1/aB

Figure 6. Imaginarypart of the self-energy for plasma density n = 7×1021 cm−3 and temperatureT = 100 Ry/ kB. Results for p = 0 (a) and for p = 1/aB (b) are shown. The self-consistentGW (0)-calculation (solid curve) is compared to the analytic formula (28) given as dashed curve.The dotted vertical line indicates the quasi-particle dispersion hω = Ep.

4.3. Analytic solution at the quasi-particle dispersion

In the following, the analytic solution (28) with the frequency fixed at the quasi-particle

dispersion ω = Ep/h shall be considered in more detail. The numerical results for (p, Ep/h)

at p = 0 have already been discussed in section 3, see figure 4. Since the only dependence on

frequency and wavenumber is given by the trivial term hω − h2p2/2m + μ = hω − εp, the

discussion may be restricted to the case p = 0 and hω = ε0 = −μ. Note that due to (28)

Re (p, εp) = 0, therefore Ep = εp. Then, the imaginary part of (28) reads

Im (0, −μ/h) = −

κe 2kBT

4π 0

= −

e2

4π0

3

4π nkBT

1/4

, (30)

14





the damping is mainly non-collective, and a single, broadened resonance appears in the spectral

function, cf figure 3.

The parameter η can also be expressed through the plasma coupling parameter and the

degeneracy parameter θ , or through the inverse Debye screening length κ,

η =

27

35/2π 2

1/6

−1/4θ−1/2 0.9698 −1/4θ−1/2, (33)

η = √ κaB. (34)

Thus, the derived scaling law is only valid for large θ , i.e. classical systems. This was already

shown in figure 4. When θ approaches 1, quantum effects set in. For example, collisions

become less probable due to Pauli blocking which leads also to a decrease of the self-energy.

Obviously, the Bohr radius aB sets the relevant length-scale that is to be compared to

the inverse screening length κ in order to estimate the importance of non-collective damping.

Non-collective damping is the dominant mechanism, as long as the screening length is large

compared to the Bohr radius. When the screening length becomes smaller than the Bohr

radius, i.e. η > 1, which, due to (32), is equivalent to having the energy of plasma oscillations

larger than the non-collective damping width, the plasmaron satellites begin to separate from

the broadened quasi-particle peak. Spectral weight is transferred from the wings of the central

peak into the plasmaron satellites leading to a more defined quasi-particle resonance, i.e. a

decreased damping of the central peak, see figure 3. Concluding, the analytic result is only a

good approximation at low densities, when η 1. At higher densities, dynamical screening

becomes important, leading to satellites in the spectral function.

5. Deficiencies of the quasi-particle approximation

The non-collective damping width was introduced above in (30) as the value of the imaginarypart of the self-energy at vanishing momentum and frequency, p = 0, ω + μ/h = 0. Of

course, the same result is also obtained if this choice of variables was already made at the very

beginning of the calculations leading to (28). However, in the latter case, the manipulations

can be performed in a different manner. At an intermediate step of the calculation, one

can identify the reason why the quasi-particle damping Im (p, Ep/h) as given in [8, 39],

behaves unphysical in the low density and classical limits. Detailed calculations are given in

appendix C, while here only the most important steps are summarized.

Setting p = 0 and hω + μ = 0 in (16), neglecting the momentum shift in the argument of

the self-energy on the rhs and replacing the dynamically screened potential by the statically

screened Born approximation as before, we obtain

Im (0, −μ/h) = −e2κ2

π 0hmkBT

2π ∞

0

dq q

(q2 + κ 2)2 Re[exp(−z2

) erfc(iz)], (35)

with

z = h2q2 + i2m Im (0, −μ/h)

2hq√

2mkBT . (36)

Most contributions to the integral stem from small values of the wavevector, q κ .

Therefore, we may neglect the real part of z and write z = i√

2m Im (0, −μ/h)/2hq√

kBT .

Using the expansion

limx→+∞

exp(x2) erfc(x) = 1√ π x

− 1

2√

π x3+ O(x−5) (37)

17

173




in lowest order only, the q-integral can be performed, resulting in

Im (0, −μ/h) = −κe 2kBT /4π 0, (38)

which coincides with (28) at p = 0 and hω = −μ.

From (35), one can also derive the quasi-particle approximation for the imaginary part of

the self-energy: if the imaginary part of z, i.e. the self-energy, is neglected on the rhs (this

is just the quasi-particle approximation), and furthermore the limit q → 0 is considered, the

expression

Im QP(0, −μ/h) = − e2κ 2

π 0h

mkBT

2π

∞

0

dq q

(q2 + κ2)2

= − e2

4π0h

2mkBT

π, (39)

is obtained.This coincides with the formula for the imaginary part of the quasi-particle self-energy

as given in [8, page 114, equation (4.164)]. There, the spectral function on the rhs of

the integral equation for (p, ω) is replaced by an on-shell delta distribution (free particle

spectral function), i.e. the self-energy is set to 0 on the rhs. The resulting integral is evaluated

at the free particle dispersion hω = εp.

As a result, one obtains the expression

Im (p, εp/h) = − e2

4π 0h

2mkBT

π1F 1(1, 3/2;−εp/2kBT ), (40)

with 1F 1(α, β; z) being the confluent hypergeometric function [52]. Note that in the given

reference, instead of the imaginary part of the self-energy, the quasi-particle damping(p, εp/h) = −2Im (p, εp/h) is given. Also, the original formula differs from (40) by a

factor of 1/4. However, the formula given here was approved through private communication

by W-D Kraeft.

Obviously, (40) is independent of density. The neglect of Im (0, −μ/h) in the complex

variable z leads to a different analytical structure of the equation. Therefore, the quasi-particle

approximation has no chance to ever obtain the correct behaviour at low densities. Low

densities, and therefore small inverse screening lengths κ shift the supporter of the q-integral

to small q, where contributions from Im z are important, whereas the real part of z vanishes at

q = 0 and leads to a result which is independent of κ .

In the same way, one can understand why the quasi-particle limit diverges when

considering the classical limit h → 0. The imaginary part of z has h in the denominator

which after the integration cancels the h in the prefactor in equation (35). No cancellation

takes place, if the imaginary part is neglected, i.e. in the quasi-particle approximation. Thisleads to the divergence of the final result.

6. Conclusion

In this work, the single-particle self-energy of the one-component electron plasma was

investigated. The spectral function was calculated self-consistently using the GW (0)-

approximation which allows the systematic treatment of dynamical correlations in the

plasma. The spectral function contains at small momenta a broadened quasi-particle

18





peak and two plasmaron satellites which, at low densities, merge into the central quasi-

particle resonance. At increased momenta, for a given density and temperature, the

spectral function converges to a single, sharp quasi-particle resonance. Special attention

was paid to a systematic investigation of the self-energy and the spectral at different

densities and temperatures. Here, only non-degenerate plasmas were considered, i.e.

the temperature is large compared to the Fermi temperature. Also, bound states were

neglected.

It was found that at low densities, the imaginary part of the on-shell self-energy, i.e.

the inverse single particle lifetime, follows a universal scaling law Im (p, Ep/h) ∝ −n1/4.

For the first time, an analytic result for the on-shell single-particle self-energy was found

that contains the correct low-density limit, i.e. a vanishing self-energy at n = 0. This

is a major progress compared to the well-known quasi-particle approximation that yields

a finite damping width even at zero density. The new on-shell single-particle damping

width is

−Im (p, Ep/h)

=(33)1/4kBT . Since it is derived in Born approximation, i.e.

no collective excitations contribute to the damping mechanisms, this quantity is called thenon-collective damping width. By comparison of the numerical results to the new analytic

formula, the parameter η = √ κaB was identified to separate the regime of non-collective

damping (η 1) from the regime, where the coupling between single-particle states and

collective excitations dominantly determine the single-particle damping (η 1) at small

momenta. This analysis complements earlier work on the electron spectral function based on

the plasmon-pole approximation.

For η 1, the analytic formula (28) is a good approximation for the self-energy.

Furthermore, the use of the analytic formula for the self-energy as an initialization of the

iterative algorithm leads to significantly faster convergence as compared to other methods,

where a Gaussian ansatz is used as the initial spectral function.

The non-collective damping is a purely classical result, no powers of h appear. This is

fundamentally different from the quasi-particle approximation to the imaginary part of the self-energy which has no classical limit, i.e. the self-energy diverges in the limit h → 0. It could

be shown that this problem, as well as the paradox of being density independent, stem from

the inherently inconsistent treatment of the self-energy in the quasi-particle approximation.

The long-time open question of the classical limit of the single-particle self-energy can now

be regarded as settled.

The results reported in this work are of paramount importance for many-particle theory and

applications to dense plasmas. In particular, simple analytic expressions for the single-particle

spectral function and self-energy in the classical and in the degenerate limit are needed to

construct Pade-like interpolation formulae that cover the complete density–temperature plane.

Such expressions would greatly simplify the calculation of equation of state, transport and

optical properties of dense, high energy plasmas, solid state devices but also nuclear, hadronic

and partonic matter, and provide benchmarks for numerical approaches, i.e. simulation

techniques. One part of this task, the analytic formula for non-degenerate dilute plasmashas been accomplished in this work.

Acknowledgments

This work was performed with financial support from the German Research Society (DFG)

under grant SFB 652 (Collaborative Research Center ‘Strong Correlations and Collective

Phenomena in Radiation Fields: Coulomb Systems, Clusters, and Particles’). Special thanks

go to G Ropke and A Wierling for many suggestions and comments on the paper and to W-D

Kraeft for stimulating discussions.

19

175




Appendix A. Details on the GW (0)-approximation

Throughout the appendix, the Rydberg system of units will be applied to keep the formulae

short and readable. In these units h = kB = 1, e2 = 2, 0 = 1/4π and m = 1/2.

We start from the representation of the self-energy in terms of the full Green

function G(p, zν ), the dynamically screened potential W (q, ωμ) and the vertex function

(p, p + q; zν , zν + ωμ), given by the diagram

Σ(p, zν ) =

G

W

Γ(0) Γ

.

(A.1)

In the GW -approximation, the vertex is replaced by the bare vertex (0)

=e, i.e. the charge

of the considered particles, electrons in this case,

Σ(p, zν ) =

G

W

Γ(0)

Γ(0)

(A.2)

= −T q,ωμ

G(p − q, zν − ωμ)W (q, ωμ), (A.3)

which is equation (11). The dynamically screened interaction is taken in the random phase

approximation [46],

W (0)(q, ωμ) = V(q)

RPA(q, ωμ), (A.4)

RPA(q, ω + iδ) = 1 − V (q)RPA(q, ω + iδ) (A.5)

RPA(q, ω + iδ) = −

k

nF(εk+q/2) − nF(εk−q/2)

ω + iδ + εk−q/2 − εk+q/2

. (A.6)

Using the spectral representations of both the Green function (4) and the screened interaction

in RPA,

W (0)(q, z) = V(q)

1 +

∞

−∞

dω

π

Im −1RPA(q, ω + iδ)

z − ω

, (A.7)

leads to

(p, zν ) = −T q,ωμ

V(q)

∞

−∞

dω

2π

A(p − q, ω)

zν − ωμ − ω

1 +

∞

−∞

dω

π

Im −1RPA(q, ω)

ωμ − ω

, (A.8)

and after summation of the Bosonic Matsubara frequencies,

(p, zν ) =

q

V(q)

∞

−∞

dω

2πA(p − q, ω)

×

1 − nF(ω) +

∞

−∞

dω

π

Im −1RPA(q, ω)

nB(ω) + 1 − nF(ω)

zν − ω − ω

, (A.9)

20





is obtained. This expression contains the Hartree–Fock self-energy of the interacting system,

HFint (p) = −

q

∞−∞

dω2π

A(p − q,ω)nF(ω)V (q), (A.10)

and the correlated self-energy

corr (p, zν ) =

q

V(q)

∞

−∞

dω

2πA(p − q, ω)

× ∞

−∞

dω

π

Im −1RPA(q, ω)[nB(ω) + 1 − nF(ω)]

zν − ω − ω . (A.11)

For convenience, we skip the upper index ‘corr’ in the following and only distinguish between

the frequency-dependent self-energy (p, ω + iδ) and the Hartree–Fock term HFint (p), in the

following.

After analytic continuation zν → z = ω + iδ, δ → 0, the imaginary part of the correlatedself-energy is evaluated using Dirac’s identity limδ→0 1/(x ± iδ) = P 1/x ∓ iπδ(x),

Im (p, ω + i0+) = 1

nF(ω)

q

∞

−∞

dω

2πV ee(q)A(p − q, ω − ω)

× Im −1RPA(q, ω)nB(ω)nF(ω − ω), (A.12)

where the exact relation nB(ω) + 1 − nF(ω − ω) = −nB(ω)nF(ω − ω)/nF(ω) was used.

This equation is given as (16) in the main text.

Appendix B. Analytic self-energy for the classical one-component plasma

In the high temperature limit kBT

EF, we replace the Fermi–Dirac distributions in the

self-energy equation (16) as well as in the dielectric function (14) by the Maxwell–Boltzmann

distribution, nF(εk) → f(k) = n3

2exp(−εk/T ) with the thermal de-Broglie wavelength

= (4π / T )1/2. In this approximation, the polarization function takes the form [39],

Re RPA(q, ω) = n

2qT

ω

q− q

1F 1

1, 3/2, −

ω

2q√

T − q

2√

T

2

−

ω

q+ q

1F 1

1, 3/2, −

ω

2q√

T +

q

2√

T

2

(B.1)

Im RPA(q, ω) = T n3

8π q

exp

−

ω

2q√

T +

q

2√

T 2

− exp

−

ω

2q√

T − q

2√

T 2

(B.2)

= −T n3

8π qexp

− ω

2T

exp

−

ω2

4q2T +

q2

4T

1

nB(ω). (B.3)

Then, the imaginary part of the self-energy writes

Im (p, ω + iδ) = 2κ2

π 3/2

√ T

1

−1

dcos θ

∞

0

dq

∞

−∞dω 1

q3

exp− ω2

4q2T + q2

4T

|(q,ω)|2

exp

ω

2T

× Im (p − q, ω + iδ − ω)

[ω − ω − εp−q − Re (p − q, ω − ω)]2 + [Im (p − q, ω − ω)]2. (B.4)

21

177




Furthermore, the Born approximation is applied, i.e. the dielectric function in the denominator

is replaced by the Debye expression, D

(q, 0)=

1 + κ2/q2. Diagrammatically, the self-energy

in this approximation is written as

Σ(p, zν ) =

G

V D

ΠRPA

V D .(B.5)

V D denotes the Debye potential V D(q) = e2/0(q2 + κ 2)0.

Since the main contribution to the q-integral stems from momenta q < κ, we neglect the

transfer wavenumber q in the argument of the self-energy on the rhs and write

Im (p, ω + iδ) = 2κ2

π 3/2

√ T

1

−1

dcos θ

∞

0

dq

∞

−∞dω

× 1

q3

exp−

ω24q2T

+ q2

4T

1 + κ 2

q2

2exp

ω

2T

× Im (p, ω + iδ − ω)

[ω − ω − εp−q − Re (p, ω − ω)]2 + [Im (p, ω − ω)]2. (B.6)

Furthermore, we neglect the term q2/4T in theexponential which is small forhigh temperatures

and for q < κ,

Im (p, ω + iδ) =2κ2

π 3/2

√ T 1

−1dcos θ ∞

0 dq ∞

−∞dω

× q

[q2 + κ2]2exp

− ω2

4q2T

exp

ω

2T

× Im (p, ω + iδ − ω)

[ω − ω − εp−q − Re (p, ω − ω)]2 + [Im (p, ω − ω)]2. (B.7)

This equation was given in section 4 as (25).

Now, the integration over the angle θ can be performed as

Im (p,ω + iδ) = 2κ 2

π 3/2

√ T

1

−1

dcos θ

∞

0

dq

∞

−∞dω q

[q2 + κ2]2

× exp− ω

2

4q2T

exp

ω2T

Im (p, ω + iδ − ω)

4p2q2

×

ω − ω − p2 − q2 + μ − Re (p,ω − ω)2pq

+ cos θ

2

+

Im (p, ω + iδ − ω)

2pq

2 −1

(B.8)

= κ2

π 3/2p

√ T

∞

0

dq

∞

−∞dω 1

[q2 + κ2]2exp

− ω2

4q2T

exp

ω

2T

22





×arctan

(p + q)2 − ω − μ + Re (p, ω − ω)

Im (p, ω + iδ − ω)

× arctan

(p − q)2 − ω − μ + Re (p, ω − ω)

Im (p, ω + iδ − ω)

, (B.9)

where the integral

dx/[(a + x)2 + b2] = b−1 arctan((a + x)/b) was used.

In the limit q → 0 the identity

limq→0

1

2q√

T e−ω2/4q2T = √

πδ(ω), (B.10)

allows us to perform the frequency integration,

Im (p,ω + iδ) = κ2

π 3/2p

√ T

∞

0

dq2q

√ T

[q2 + κ2]2

∞

−∞dω exp

ω

2T

exp

− ω2

4q2T

2q

√ T

× arctan

(p + q)2

−ω

−μ + Re (p, ω

−ω)

Im (p, ω + iδ − ω)

− arctan

(p − q)2 − ω − μ + Re (p, ω − ω)

Im (p, ω + iδ − ω)

= 2κ2T

π 3/2p

∞

0

dqq

[q2 + κ 2]2

∞

−∞dω exp

ω

2T

√ πδ(ω)

×

arctan

(p + q)2 − ω − μ + Re (p, ω − ω)

Im (p, ω + iδ − ω)

− arctan

(p − q)2 − ω − μ + Re (p, ω − ω)

Im (p, ω + iδ − ω)

=2κ2T

πp

∞

0

dqq

[q2 + κ 2]2

×

arctan

(p + q)2 − ω − μ + Re (p, ω)

Im (p, ω + iδ)

− arctan

(p − q)2 − ω − μ + Re (p, ω)

Im (p, ω + iδ)

. (B.11)

Using the following power expansion of the arctan-function

arctan(1 + x) = π

4+

x

2+ O(x3), (B.12)

i.e.

arctan

(p + q)2 − ω − μ + Re (p, ω)

Im (p, ω + iδ)

− arctan

(p − q)2 − ω − μ + Re (p, ω)

Im (p, ω + iδ)

= 4pq

Im (p, ω + iδ)

1 +p2

−ω

−μ + Re (p, ω)

Im (p, ω + iδ)2−1

+ O(q3),

(B.13)

we obtain

Im (p,ω + iδ) = 2κ2T

πp

∞

0

dqq

[q2 + κ 2]2

× 4pq

Im (p, ω + iδ)

1 +

p2 − ω − μ + Re (p, ω)

Im (p, ω + iδ)

2−1

,

(B.14)

23

179




which can be turned into

[Im (p, ω + iδ)]2 + [p2 − ω − μ + Re (p, ω)]2 = 8κ2

T π

∞0

dq q2

(q2 + κ2)2

= 2κT . (B.15)

To solve this single equation for the two unknown Re (p, ω) and Im (p, ω + iδ), we make

use of the spectral representation of the Green function

G(p, z) = [z − p2 − (p, z)]−1 = ∞

−∞

dω

2π

A(p, ω)

z − ω

= ∞

−∞

dω

2π

Im (p, ω + iδ)

z − ω

(Im (p, ω + iδ))2 + (ω + μ − p2 − Re (p,ω))2

−1

(B.16)

= ∞

−∞

dω

2π

Im (p, ω + iδ)

z − ω

1

2κT = (p, z)

2κT . (B.17)

In the last step we also used the spectral representation of the correlated self-energy. The last

equation has the solution

(p, z) = z − p2 + μ

2±

z − p2 + μ

2

2

− 2κT

1/2

. (B.18)

With z = ω + iδ and having in mind that Im (p, ω + iδ) < 0 for δ > 0, we finally find

(p, ω + iδ) = ω + μ − p2

2− sign(ω + μ − p2)

ω + μ + iδ − p2

2

2

− 2κT

1/2

, (B.19)

i.e. equation (28).

Appendix C. Details for the quasi-particle self-energy

We start from (25) for the imaginary part of the self-energy using the Born approximation for

the screened interaction potential:

Im (p, ω + iδ) = −√

T κ2

π 3/2

1

−1

dcos θ

∞

0

dq

∞

−∞dω q

(q2 + κ2)2

× A(p − q, ω − ω) exp

ω

2T

exp

− ω2

4q2T

. (C.1)

By assuming a frequency and momentum independent self-energy (p, ω) ≡(0, E0/h), this becomes

Im (0, E0/h) = 2√

T κ 2

π 3/2

1

−1

dcos θ

∞

0

dq

∞

−∞dω q

q2 + κ 22

× Im (0, E0/h)

[ω + q2]2 + [Im (0, E0/h)]2exp

ω

2T

exp

− ω2

4q2T

. (C.2)

Since the self-energy is assumed to be independent of the frequency, the real part of the

correlated self-energy vanishes exactly. For the Hartree–Fock part of the self-energy is

proportional to n3 in the classical limit [8], we also neglect this term, since it gives

contributions of higher order in n, whereas we are only interested in the lowest order.

24





After eliminating Im (0, E0/h) on both sides, performing the trivial integration over the

angle θ , which yields a factor 2, the frequency integration is performed by the help of [ 52] ∞

0

exp(−t 2)dt

z − t = π

2izexp(−z2) erfc(−iz), (C.3)

leading to

1 = 4

T

π 3κ 2

∞

0

dq q

(q2 + κ2)2

∞

−∞

exp(−ω2/4q2T ) dω

[q2 + ω]2 + [Im (0, E0/h)]2(C.4)

= −4

T

π 3κ 2

∞

0

dq q

(q2 + κ2)2

π Re[exp(−z2) erfc(iz)]

Im (0, E0/h), (C.5)

which is rewritten as

Im (0, E0/h) = −4

T

πκ2

∞

0

dq q

(q2 + κ 2)2Re[exp(−z2) erfc(iz)], (C.6)

with

z = q2 + iIm (0, E0/h)

2q√

T . (C.7)

Equation (C.6) is given as (35) in section 5. It should be noted at this point that the integral

converges only for finite κ, i.e. the Coulomb limit κ → 0 does not yield a finite result.

Most contributions to the integral stem from small values of the wavevector, q κ .

Therefore, we may neglect the real part of z and write z = i Im (0, E0/h)/2q√

T . Using the

expansion

limx→+∞

exp(x2) erfc(x) = 1√ π x

− 1

2√

π x2+ O(x−3) (C.8)

in lowest order only, the q-integral can be performed, resulting in

Im (0, E0/h) = −4

T

πκ2

∞

0

dq q

(q2 + κ 2)2(C.9)

× Re[exp(−(i Im (0, E0/h)/2q√

T )2) erfc(i · i Im (0, E0/h)/2q√

T )]

= −4

T

πκ2

∞

0

dq q

(q2

+κ2)2

(C.10)

× Re[exp((−Im (0, E0/h)/2q√

T )2) erfc(−Im (0, E0/h)/2q√

T )]

−2

T

πκ2

∞

0

dq q

(q2 + κ2)2

1√

π (−Im (0, E0/h)/2q√

T )(C.11)

= 8T κ2

π Im (0, E0/h)

∞

0

dq q 2

(q2 + κ 2)2= 8T κ 2

π Im (0, E0/h)

π

4κ. (C.12)

25

181




[47] Bohm D and Gross E P 1949 Phys. Rev. 75 1851–64

[48] Thiele R, Bornath T, Fortmann C, Holl A, Redmer R, Reinholz H, Ropke G, Wierling A, Glenzer S H and

Gregori G 2008 Phys. Rev. E 78 026411

[49] Bahcall J N, Pinsonneault M H and Wasserburg G J 1995 Rev. Mod. Phys. 67 781–808

[50] Fortmann C, Ropke G and Wierling A 2007 Pulsed Power Conference (PPPS-2007) Digest of Technical Papers

(IEEE) ed E Schamiloglu and F Peterkin p 194

[51] Fortmann C, Ropke G and Wierling A (in preparation)

[52] Abramowitz M and Stegun A (ed) 1970 Handbook of Mathematical Functions with Formulas, Graphs and

Mathematical Tables 9th edn (New York: Dover)

27

183



Chapter 10

Single-Particle Spectral Function forthe Classical One-Component Plasma

Author: Carsten Fortmann

Appeared as regular article in Physical Review E, Vol. 79, Issue 1, pages 016404 1-11, Jan.2009.



186 Single Particle Spectral Function for the Classical Plasma



Single-particle spectral function for the classical one-component plasma

C. Fortmann* Institut für Physik, Universität Rostock, 18051 Rostock, Germany

Received 15 August 2008; published 8 January 2009

The spectral function for an electron one-component plasma is calculated self-consistently using the GW 0

approximation for the single-particle self-energy. In this way, correlation effects that go beyond the mean-field

description of the plasma are contained, i.e., the collisional damping of single-particle states, the dynamical

screening of the interaction, and the appearance of collective plasma modes. Second, a nonperturbative analytic

solution for the on-shell GW 0 self-energy as a function of momentum is presented. It reproduces the numeri-

cal data for the spectral function with a relative error of less than 10% in the regime where the Debye screening

parameter is smaller than the inverse Bohr radius, 1aB−1. In the limit of low density, the nonperturbative

self-energy behaves as n1/4, whereas a perturbation expansion leads to the unphysical result of a density-

independent self-energy Fennel and Wilfer, Ann. Phys. Leipzig 32, 265 1974. The derived expression will

greatly facilitate the calculation of observables in correlated plasmas transport properties, equation of statethat need the spectral function as an input quantity. This is demonstrated for the shift of the chemical potential,

which is computed from the analytical formulas and compared to the GW 0 result. At a plasma temperature of

100 eV and densities below 1021 cm−3, the two approaches deviate by less than 10% from each other.

DOI: 10.1103/PhysRevE.79.016404 PACS numbers: 52.27.Aj, 52.65.Vv, 71.15.m

I. INTRODUCTION

The many-particle Green function approach 1 allows for

a systematic study of macroscopic properties of correlated

systems. Green functions have a long history of applications

in solid state theory 2, nuclear 3, and hadron physics 4,

and also in the theory of strongly coupled plasmas 5. In the

last case, optical and dielectric properties 6,7 have beenstudied using the Green function approach, as well as trans-

port properties like conductivity 8 and stopping power9,10, and the equation of state 11. Modifications of these

quantities due to the interaction among the constituents can

be accessed, starting from a common starting point, namely,

the Hamiltonian of the system.

The key quantity for electronic properties in a correlated

many-body environment is the electron spectral function Ap , , i.e., the probability density to find an electron at

energy frequency for a given momentum p. It is related

to the retarded electron self-energy p , + i , 0, via

Dyson’s equation

Ap, =− 2 Im p, + i

− p − Re p, 2 + Im p, + i 2 .

1

Here, the single-particle energy

p = p2 − e 2

has been introduced, where e is the electron chemical po-

tential. Note that here and throughout the paper the Rydberg

system of units is used, where = 1, me = 1 /2, and e2/4 0

=2. Furthermore, the Boltzmann constant k B is set equal to 1,

i.e., temperatures are measured in units of energy.

The self-energy describes the influence of correlations onthe behavior of the electrons. A finite, frequency-dependentself-energy leads to a finite lifetime of single-particle statesand a modification of the single-particle dispersion relation.Hence, the calculation of the electron self-energy is the cen-tral task if one wants to determine electronic properties, e.g.,those mentioned above.

The Hartree-Fock approximation 12 represents the low-est order in a perturbative expansion of the self-energy interms of the interaction potential 13. Because it is a mean-field approximation, effects due to correlations in the systemcannot be described. Examples are the appearance of collec-tive modes, the energy transfer during particle collisions, andthe quasiparticle damping. The next-order term is the Bornapproximation, where binary collisions are taken into ac-count via a bare Coulomb potential. However, the Born ap-proximation leads to a divergent integral, due to the long-range Coulomb interaction. Therefore, the perturbationexpansion of the self-energy has to be replaced by a nonper-turbative approach, accounting for the dynamical screening

of the interparticle potential.A nonperturbative approach to the many-particle problemis given by the theory of Dyson 14 and Schwinger 15,16generalized to finite temperature and finite density 17. An

excellent introduction to the Dyson-Schwinger equations can

also be found in 4. The Dyson-Schwinger equation for the

self-energy contains the full Green function G, the

screened interaction W , and the proper vertex . Since each

of these functions obeys a different Dyson-Schwinger equa-

tion itself, involving higher-order correlation functions, the

Dyson-Schwinger approach leads to a hierarchy of coupled

integro-differential equations. In order to provide soluble

equations, this hierarchy has to be closed at some level, i.e.,

correlation functions of a certain order have to be either pa-

rametrized or neglected.One such closure of the Dyson-Schwinger hierarchy con-

sists in neglecting the vertex, i.e., the three-point function,

and considering only the particle propagators and their re-

*[email protected];

http://everest.mpg.uni-rostock.de/ carsten

PHYSICAL REVIEW E 79, 016404 2009

1539-3755/2009/791 /01640411 ©2009 The American Physical Society016404-1

187



spective self-energies, i.e., two-point functions. One arrivesat the so-called GW approximation, introduced in solid-state

physics by Hedin 18. Hedin was led by the idea of includ-ing correlations in the self-energy by replacing the Coulombpotential in the Hartree-Fock self-energy by the dynamicallyscreened interaction W . In this way, one obtains a self-consistent, closed set of equations for the self-energy, thepolarization function , the Green function, and the screenedinteraction.

It can be shown 19 that the GW approximation is one of the so-called -derivable approximations 20,21. As such, itleads to energy-, momentum-, and particle-number-conserving expressions for higher-order correlation func-tions. It has been successfully applied in virtually allbranches of solid state physics. An overview of theoreticalfoundations and applications of the GW approximation can

be found in the review articles 22–24.The drawback of the GW approximation is that the Ward-

Takahashi identities are violated. The Ward-Takahashi iden-

tities provide an exact relation between the vertex function

, i.e., the effective electron-photon coupling in the medium,

and the self-energy, and follow from the Dyson-Schwinger

equations. They reflect the gauge invariance of the theory. In

GW theory, they are violated simply because corrections to

the vertex beyond zero order are neglected altogether. This

issue touches on a fundamental problem in many-body

theory and field theory, namely, the question of how to pre-

serve gauge invariance in an effective, i.e., approximate,

theory, without violating basic conservation laws. A detailed

analysis of this question with application to nuclear physicsis presented in a series of papers by van Hees and Knoll25–27.

Approximations for the self-energy that also contain the

vertex are often referred to as GW approximations. An ex-

ample can be found in Ref. 28, where the spectral function

of electrons in aluminum is calculated using a parametrized

vertex function. An interesting result obtained in that work is

that vertex corrections and self-energy corrections entering

the polarization function largely cancel. This can be under-

stood as a consequence of the Ward-Takahashi identities.

Thus, and in order to reduce the numerical cost, it is a sen-

sible choice to neglect vertex corrections altogether, and to

keep the polarization function on the lowest level, i.e. the

random phase approximation RPA, which is the convolu-

tion product of two noninteracting Green functions in

frequency-momentum space. The corresponding self-energy

is named the GW 0 self-energy and has been introduced by

von Barth and Holm 29, who were also the first to study the

fully self-consistent GW approximation 30. Throughout

this work, the GW 0 self-energy will be analyzed.

Having been used in solid state physics traditionally, theGW 0 method was recently also applied to study correlations

in hot and dense plasmas. The equation of state 31,32, as

well as optical properties of electron-hole plasmas in highly

excited semiconductors 33 and dense hydrogen plasmas 7were investigated.

In general, the calculation of such macroscopic observ-ables of many-particle system involves numerical operations

that need the spectral function as an input. Since the self-

consistent calculation of the self-energy, even in the GW 0

approximation, is itself already a numerically demanding

task, it is worth looking for an analytic solution of the GW 0

equations which reproduces the numerical solution at least ina certain range of plasma parameters. Such an analytic ex-

pression then also allows study of the self-energy in various

limiting cases, such as the low-density limit or the limit of

high momenta, which are difficult to access in the numerical

treatment. Furthermore, an analytic expression that is already

close to the numerical solution permits the calculation of the

full GW 0 self-energy using only few iterations.

Analytical expressions for the single-particle self-energy

have already been given by other authors, e.g., Fennel and

Wilfer 34 and Kraeft et al. 12. They calculated the self-

energy in first order of the perturbation expansion with re-

spect to the dynamically screened potential. Besides being

far from the converged GW

0

self-energy, their result is in-dependent of density, i.e., the single-particle lifetime is finite

even in vacuum. As shown in 35, this unphysical behavior

is a direct consequence of the perturbative treatment. By us-

ing a nonperturbative ansatz, an expression for the self-

consistent self-energy in a classical one-component plasma

was presented that reproduces the full GW 0 self-energy at

small momenta, i.e., for slow particles. The behavior of the

quasiparticle damping at larger momenta remained open and

will be investigated in the present work. Second, based on

the information gathered about the low- and high-momentum

behavior, an interpolation formula will be derived that gives

the quasiparticle damping at arbitrary momenta.

The work is organized in the following way. After a brief

outline of the GW 0 approximation in the next section, nu-

merical results will be given in Sec. III for the single-particle

spectral function for various sets of parameters, electron den-

sity n, and electron temperature T . In Sec. IV the analytic

expression for the quasiparticle damping width is presented

and comparison to the numerical results is given. Section V

deals with the application of the derived formulas to the

calculation of the chemical potential as a function of density

and temperature. An Appendix contains the detailed deriva-

tion of the analytic self-energy. As a model system, we focus

on the electron one-component plasma; ions are treated as a

homogeneously distributed background of positive charges jellium model.

II. SPECTRAL FUNCTION AND SELF-ENERGY

We start our discussion with the integral equation for the

imaginary part of the single-particle self-energy in the GW 0

approximation:

Im p, + i =1

nF

q

−

d

2 V q Ap − q, −

Im RPA−1 q, nB nF − . 3

V q = 8 /q20 is the Fourier transform of the Coulomb po-

tential with the normalization volume 0. It is multiplied by

the inverse dielectric function in the RPA,

C. FORTMANN PHYSICAL REVIEW E 79, 016404 2009

016404-2




marons appear; only a featureless, single resonance is ob-tained. At higher momenta, the plasmarons merge into the

central peak. As in the low-density case, the position of the

maximum approaches the single-particle dispersion, due to

the decreasing Hartree-Fock shift at high p. Again, the width

of the spectral function decreases with increasing momen-

tum.

This is visible more clearly in Figs. 2–7. Here, the solid

curves represent the GW 0 spectral function as a function of

frequency. Results are shown for three different momenta,

p = 0aB−1 a, 50aB

−1 b, and 100aB−1 c. Two different tem-

peratures are considered, T =100 Figs. 2–4 and 1000 eVFigs. 5–7, and for each temperature three different densities

are studied. With increasing momentum p, the spectral func-tion becomes more and more narrow, converging eventually

into a narrow on-shell resonance, located at the unperturbed

single-particle dispersion +e = p2.

As a general feature, one can observe an increase of the

spectral function’s width with increasing density and with

increasing temperature. The increase with density is due to

the increased coupling, while the increase with temperature

reflects the thermal broadening of the momentum distribu-

tion function nF that enters the self-energy and thereby

also the spectral function. From these results, we see that the

spectral function has a quite simple form in the limit of low

coupling, i.e., at low densities and high temperatures.

The numerical results are compared to a Gaussian ansatz

for the spectral function, shown as the dashed curve in Figs.

2–7. The explicit form of the Gaussian is given as Eq. 11,

below. Its sole free parameter is the width, denoted by p. Ananalytic expression for p will be derived in Sec. IV. The

coincidence is in general good at high momenta, whereas at

low momenta the spectral function deviates from the Gauss-

ian. In particular, the steep wings and the smoother plateau

that form at low momenta are not reproduced by the Gauss-

ian. Also, the plasmaron peaks appearing in the spectral

function at high density see Fig. 7 cannot be described by

the single Gaussian.

Determination of p via least-squares fitting of the Gauss-

ian ansatz to the numerical data at each p leads to the solid

curve in Fig. 8, obtained in the case of n = 71020 cm−3 andT =100 eV. Starting at some finite value 0 at p =0, the width

falls off slowly toward larger p. The dashed curve shows

pas obtained from the analytic formula that will be derived in

the following section.

IV. ANALYTICAL EXPRESSION FOR THEQUASI-PARTICLE SELF-ENERGY

The solution of the GW 0 equation 3 requires a consid-

erable numerical effort. So far see, e.g. the work by Fennel

and Wilfer in 34, attempts to solve the integral 3 analyti-

cally were led by the idea of replacing the spectral function

on the right-hand side RHS by its noninteracting counter-

part, A0p , = 2 p − , i.e., going back to the pertur-

bation expansion of the self-energy and neglecting the im-

plied self-consistency. At the same time, the inverse

dielectric function is usually replaced by a simplified expres-

0

2

4

6

8

10

12

14

-3 -2 -1 0 1 2 3


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

12

14

9997 9998 9999 10000 10001 10002 10003


) [ 1

/ R y ]


p=100 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

12

14

2497 2498 2499 2500 2501 2502 2503


) [

1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

(a) (b) (c)

FIG. 2. Spectral function in GW 0-approximation solid lines and Gaussian ansatz dashed lines with quasiparticle damping width ptaken from Eq. 17 for three different momenta p = a 0, b 50aB

−1, and c 100aB−1. Plasma parameters: n = 71019 cm−3, T =100 eV. The

plasma coupling parameter is =1.010−2, the degeneracy parameter is =1.6103, and the Debye screening parameter is =6.0

10−3aB−1.

0

2

4

6

8

10

-3 -2 -1 0 1 2 3


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

2497 2498 2499 2500 2501 2502 2503


) [ 1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

9997 9998 9999 10000 10001 10002 10003


) [ 1 / R y ]


p=100 aB-1

GW(0)

Gauss fit

(a) (b) (c)

FIG. 3. Spectral function in GW 0 approximation solid lines and Gaussian ansatz dashed lines with quasiparticle damping width ptaken from Eq. 17 for three different momenta p = a 0, b 50aB



10−2aB−1.


016404-4




sion, e.g., the Born approximation or the plasmon-pole ap-proximation 12. Whereas the second simplification is indis-

pensable due to the complicated structure of the inverse

dielectric function, the first one, i.e., the recursion to the

quasiparticle picture, is not necessary, as was shown by the

author in Ref. 35. In fact, the result that one obtains in the

quasiparticle approximation is far from the converged result,

at least in the high-temperature case. Second, if the quasipar-

ticle approximation is used, the imaginary part of the self-

energy is not density dependent, i.e., a finite lifetime of the

particle states is obtained even in vacuum. This unphysical

result can be overcome only if one sticks to the self-

consistency of the self-energy, i.e., if one leaves the imagi-

nary part of the self-energy entering the RHS of Eq. 3finite.

Using the statically screened Born approximation, which

describes the binary collisions among electrons via a stati-

cally screened potential, a scaling law Im(p , QPp)3/4 was found 35. Hence, the spectral function width

vanishes when the plasma coupling parameter see Eq. 7tends to 0. An expression for the self-energy was found that

reproduces the converged GW 0 calculations at small mo-

menta, p . At higher momenta, the derived expression

ceases to be valid.

In this work, a different approximation to the dielectric

function is studied, namely, the plasmon-pole approximation12. This means that the inverse dielectric function is re-

placed by a sum of two functions that describe the location

of the plasmon resonances,

Im RPA−1

q, → Im PPA−1

q,

= −

2

pl2

q − q + + q . 8

For classical plasmas, the plasmon dispersion q can be ap-

proximated by the Bohm-Gross dispersion relation 38

q2

= pl2 1 +

q2

2 + q4. 9

Many-particle and quantum effects on the plasmon disper-

sion have recently been studied in 39.

The plasmon-pole approximation PPA allows one toperform the frequency integration in Eq. 3, resulting in the

expression

Im p, + i = pl

2

4

qV q

1

q Ap − q, − q

nB qexp q/T − Ap − q, + q

nB− qexp− q/T . 10

We will first study the case of high momenta, i.e., momenta

that are large against any other momentum scale or inverse

length scale, such as the mean momentum with respect to the

Boltzmann distribution, p¯

=3T /2, or the inverse screening

length =8 n /T .

0

2

4

6

8

10

-3 -2 -1 0 1 2 3


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

2497 2498 2499 2500 2501 2502 2503


) [

1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

0

2

4

6

8

10

9997 9998 9999 10000 10001 10002 10003


) [

1 / R y ]


p=100 aB-1

GW(0)

Gauss fit

(a) (b) (c)

FIG. 4. Spectral function in GW 0 approximation solid lines and Gaussian ansatz dashed lines with quasiparticle damping width ptaken from Eq. 17 for three different momenta p = a 0, b 50aB



10−2aB−1.

0

0.5

1

1.5

2

-10 -5 0 5 10


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

0.5

1

1.5

2

2490 2495 2500 2505 2510


) [ 1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

0

0.5

1

1.5

2

9990 9995 10000 10005 10010


) [ 1 / R y ]


p=100 aB-1

GW(0)

Gauss fit

(a) (b) (c)

FIG. 5. Spectral function in GW 0 approximation solid lines and Gaussian ansatz dashed lines with quasiparticle damping width ptaken from Eq. 17 for three different momenta p =a 0, b 50aB



10−2aB−1.

SINGLE-PARTICLE SPECTRAL FUNCTION FOR THE… PHYSICAL REVIEW E 79, 016404 2009

016404-5

191



As discussed in the previous section, the numerical results

for the spectral function at high momenta can well be repro-

duced by a Gaussian. Thus, we make the following ansatz forthe spectral function:

AGaussp, = −2

pexp−

− p − HFp2

2 p2 . 11

Note that only the Hartree-Fock contribution to the real part

of the self-energy appears. The frequency-dependent part is

usually small near the quasiparticle dispersion,

QPp = p + Re p, = QPp, 12

which therefore can be approximated as QPp =p+ Re HFp. In the following, we make use of the knowl-

edge about the width parameter p that we gathered already

through simple least-squares fitting of the Gaussian ansatz to

the spectral function in order to solve the integrals in Eq.

10.

First, we replace the spectral function on the RHS by the

Gaussian ansatz 11 and evaluate the emerging equation at

the single-particle dispersion QPp. By claiming that the

Gaussian and the spectral function have the same value at the

quasiparticle energy, we identify p = /2Im (p , QPp).

Figure 8 shows that the quasiparticle damping p is a smooth

function of p that varies only little on the scale of the screen-

ing parameter . Since the latter defines the scale on whichcontributions to the q integral are most important, we can

neglect the momentum shift in the self-energy on the RHS,

i.e., we can replace the spectral function on the RHS of Eq.10 by

Ap − q,p + HFp q

→ −2

pexp−

p q − p−q2

2 p2 , 13

and can now perform the integral over the angle between

the momenta p and q,

2

p

−1

1

d cos

exp−p q − p2 − q2 + 2 pq cos + e2

2 p2

=

2 pqErf p + q2 − p2 q

2 p

− Erf p − q2 − p2 q

2 p . 14

The remaining integration over the modulus of the trans-fer momentum q can be performed after some further ap-

proximations, explained in detail in Appendix A. For large p,

one finally obtains the transcendental equation

p = − 1.3357

2

pl

2 pnB plexp pl/T

− nB− plexp− pl/T −

2

T

2 pln 2 p2

/ p2.

15

The solution of this equation can be expanded for large ar-

guments of the logarithm, yielding

p = −

2

T

p p,

p = p − ln p +ln p

p−

ln p

2 p

+ln p

p−

3 ln2 p

2 3 p+

ln2 p

2 2 p+

ln3 p

3 3 p

+ O p−3,

p = ln 2

p2 exp A/T /T , 16

A = − 1.3357 pl

2nB plexp pl/T

− nB− plexp− pl/T .

Equation 16 is a solution of Eq. 15 provided the argumentof the inner logarithm is larger than Euler’s constant e, i.e.,2 / p2 exp A /T /T e, i.e., at large p. The case of small

p, where the previous inequality does not hold, has to be

treated separately; see Appendix B.

Together with an expression for the quasiparticle damping

at vanishing momentum taken from 35 and scaled such that

the maximum of the spectral function at p =0 is reproduced,

0 = − T /2, an interpolation formula Padé formula was

derived that covers the complete p range:

pPadé

=a0 + a1 p

1 + b1 p + b2 p2 ˜ p ,

˜ p = ˜ p − ln ˜ p +ln ˜ p

˜ p−

ln ˜ p

˜ 2 p+

ln ˜ p

˜ p

−3 ln2 ˜ p

2 ˜ 3 p+

ln2 ˜ p

2 ˜ 2 p+

ln3 ˜ p

3 ˜ 3 p ,

˜ p = lne + 2

p2 exp A/T /T ,

a0 = −

2 T , a1 = −

23/2

, b1 =

2T , b2 =

2T .

17

The function ˜ p in the last equation differs from p in Eq.

16 in that Euler’s constant e 2.7183 has been added to the


016404-6




argument of the logarithm. In this way, the function ˜

p isregularized at small p and tends to 1 at p =0, i.e., the quasi-

particle damping goes to the correct low- p limit. At large p,

this modification is insignificant, since the original argument

rises as p2. For the detailed derivation, see Appendix B.

Expression 17, used in the Gaussian ansatz 11, leads to

a spectral function that well reproduces the numerical data

from full GW 0 calculations: Figure 8 dashed curve shows

the effective quasiparticle damping width p as a function of

momentum p for the case n = 71020 cm−3 and T =100 eV.

The solid curve gives the best-fit value for p obtained via

least-squares fitting of the full GW 0 calculations assuming

the Gaussian form 11 see Sec. III. The two curves coin-

cide to a large extent. The largest deviations are observed in

the range of p 20aB−1. At this point, the validity of expres-

sion 16 as the solution of Eq. 15 ceases, since the argu-

ment of the logarithm becomes smaller than e. As already

mentioned, we circumvented this problem by regularizing

the logarithm, adding e to its argument. The deviation at p

20aB−1 of up to 15% is a residue of this procedure. At

higher momenta, the deviation is generally smaller than 10%

and the analytic formula evolves parallel to the fit param-

eters.

At smaller densities, the correspondence is even better as

can be seen by comparing the spectral functions shown in

Figs. 2–7. The dashed curves give the Gaussian ansatz for

the spectral function with the quasiparticle width taken from

the interpolation formula 17. As a general result, the ana-

lytic expression for the quasiparticle damping p leads to aspectral function that nicely fits the numerical solution for

the spectral function at least at finite p. At very small values

of p, the overall correspondence is still fair, i.e., the position

of the maximum and the overall width match, but the de-

tailed behavior does not coincide. In particular, the steep

wings and the central plateau that form in the GW 0 calcu-

lation are not reproduced by the one-parameter Gaussian. For

this situation, the analytic formula for self-energy given in35 should be used instead.

By comparing the numerical data for the spectral function

to the Gaussian ansatz at different densities, it is found that

the Gaussian spectral function is a good approximation as

long as the Debye screening parameter is smaller than the

inverse Bohr radius, 1aB−1. This becomes obvious by

comparing Figs. 6 and 7. In the first case n = 71023 cm−3,

T = 1000 eV, we have = 0.19, while in the second casen = 71025 cm−3, T = 1000 eV, =1.9 is found. As already

noted in the discussion of the numerical results in Sec. III, in

the case of increased density, the plasmaron satellites appear

as separate structures in the wings of the central quasiparticle

peak, whereas they are hidden in the central peak at smaller

densities. Therefore, a single Gaussian is not sufficient to fit

the spectral function at increased densities. Since the position

of the plasmaron peak is given approximately by the plasma

frequency pl, whereas the width of the central peak at small

p is just the quasiparticle width 0, we can identify the ratio

of these two quantities, − pl/ 0 , as the parameter that

0

0.2

0.4

0.6

0.8

1

-20 -15 -10 -5 0 5 10 15 20


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

0.2

0.4

0.6

0.8

1

2 48 0 24 85 24 90 24 95 25 00 2 50 5 2 51 0 2 515 2 520


) [

1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

0

0.2

0.4

0.6

0.8

1

9 98 0 9 98 5 9 99 0 9 99 5 1 00 00 1 00 05 1 00 10 1 00 15 1 00 20


) [

1 / R y ]


p=100 aB-1

GW(0)

Gauss fit

(a) (b) (c)




10−1aB−1.

0

0.2

0.4

0.6

0.8

1

-40 -20 0 20 40


) [ 1 / R y ]


p=0 aB-1

GW(0)

Gauss fit

0

0.2

0.4

0.6

0.8

1

2460 2480 2500 2520 2540


) [ 1 / R y ]


p=50 aB-1

GW(0)

Gauss fit

0

0.2

0.4

0.6

0.8

1

9960 9980 10000 10020 10040


) [ 1 / R y ]


p=100 aB-1

GW(0)

Gauss fit

(a) (b) (c)



plasma coupling parameter is =9.610−2, the degeneracy parameter is =1.6, and the Debye screening parameter is =1.9aB−1. Here, the

Gaussian fit is no longer sufficient due to the appearance of plasmaron resonances in the spectral function shoulders at −30 and

20 Ry.


016404-7

193



tells us if plasmaron peaks appear separately pl− 0 or

not pl− 0. Since the plasma frequency increases as a

function of n1/2, whereas the quasiparticle width scales as

n1/4 see Eq. 17, the transition from the single-peak behav-

ior to the more complex behavior including plasmaron reso-

nances appears at increased density. Neglecting numerical

constants of order 1 in the ratio of plasma frequency to

damping width, we see that − pl/ 01 is equivalent to

1, which was our observation from the numerical results.Therefore, we can identify the range of validity of the pre-

sented expressions for the spectral function and the quasipar-

ticle damping. It is valid for those plasmas where we have

densities and temperatures such that 1.

The physical origin of the requirement 1 can be un-

derstood in the following way 35. At length scales smaller

than the Bohr radius, one typically expects quantum effects,

e.g., Pauli blocking. These effects are not accounted for in

the derivation of the quasiparticle damping. Therefore, it ap-

pears to be a logical consequence that the validity of the

results is limited by the length scale at which typical quan-

tum phenomena become important.

The regime of validity of the analytic formula can also be

expressed via the plasma coupling parameter and the tem-

perature as T −2/3. Since we restrict ourselves to plasma

temperatures where bound states can be excluded, i.e., T

1 Ry, this is equivalent to saying that 1.

Although the correspondence between the accurate GW 0

calculations and the parametrized spectral function at small

momenta is not as good as in the case of large momenta, the

parametrized spectral function can be applied in the regime

of validity to the calculation of plasma observables without

introducing too large errors. As an example, this will be

shown for the case of the chemical potential in the next

section.

V. APPLICATION: SHIFT OF THE CHEMICALPOTENTIAL

To demonstrate the applicability of the presented formulas

for quick and reliable calculations of plasma properties, we

calculate the shift of the electron’s chemical potential

=−free, i.e., the deviation of the chemical potential of the

interacting plasma from the value of the noninteracting

system free. The chemical potential of the interacting system

is obtained by inversion of the density as a function of T

and , Eq. 5. The free chemical potential free is obtained

in a similar way by inversion of the free density,

nfreeT ,free = 2p

nFp − free. 18

Figure 9 shows the shift of the chemical potential as a

function of the plasma density n for a fixed plasma tempera-

ture T =100 eV. Results obtained by inversion of Eq. 5 us-

ing the parametrized spectral function 11 with the quasipar-

ticle damping width taken from Eq. 17 solid curve are

compared to those results taking the numerical GW 0 spec-

tral function dashed curve.

The GW 0 result gives slightly smaller shifts than the pa-

rametrized spectral function, i.e., the usage of the analytical

damping width leads to an overestimation of the shift of the

chemical potential. However, the deviation remains smaller

than 20% over the range of densities considered here, i.e., for

1. At small densities, i.e., for n1020 cm−3, the param-etrized spectral function yields the same result as the full

GW 0 calculation.

The deviation at increased density can be reduced by im-

proving the parametrization of the spectral function at small

momenta. To this end, the behavior of the quasiparticle

damping width at high momenta, Eq. 16 should be com-

bined with the frequency-dependent solution for p at van-

ishing momentum, as presented in Ref. 35. However, this

task goes beyond the scope of this paper, where we wish to

present comparatively simple analytic expressions for the

damping width that yield the correct low-density behavior of

plasma properties.

VI. CONCLUSIONS AND OUTLOOK

In this paper, the GW 0 approximation for the single-

particle self-energy was evaluated for the case of a classical

0 20 40 60 80 100

momentum p [aB

-1]

-0.5

-0.4

-0.3

-0.2

q u a s i - p a r t i c l e d a m p i n g σ

p [ R y ]

GW (0)

-fit

Interpolation formula

FIG. 8. Effective quasiparticle damping p as a function of mo-

mentum p for plasma density n = 71020

cm−3

and temperature T =100 eV. The fit parameters for the Gaussian fit to the full GW 0

calculations are given as the solid line; the dashed line denotes the

analytic interpolation formula 17.

1e+19 1e+20 1e+21 1e+22

density n [cm-3

]

1e-02

1e-01

- ∆ µ

[ R y ]

Gaussian spectral function

GW (0)

T = 100 eV

FIG. 9. Shift of the chemical potential as a function of the

plasma density for a plasma temperature T =100 eV. Results for

using the parametrized spectral function solid line are compared

to full numerical calculations, using the GW 0 approximation

dashed line.


016404-8




one-component electron plasma, with ions treated as a ho-mogeneous charge background. A systematic behavior of the

spectral function was found, i.e., a symmetrically broadenedstructure at low momenta and convergence to a sharp quasi-particle resonance at high p. At increased densities, plasma-ron satellites show up in the spectral function as satellitesbesides the main peak.

In the second part, an analytic formula for the imaginarypart of the self-energy at the quasiparticle dispersion QPp=p +HFp was derived as a two-point Padé for-mula that interpolates between the exactly known behavior at p =0 and p→. The former case was studied in 35, whilean expression for the asymptotic case p→ was derivedhere. The result is summarized in Eq. 17. In contrast topreviously known expressions for the quasiparticle damping,based on a perturbative approach to the self-energy 34, the

result presented here shows a physically intuitive behavior inthe limit of low densities, i.e., it vanishes when the system

becomes dilute. Use of the Gaussian ansatz 11 for the spec-

tral function in combination with the quasiparticle width

leads to a very good agreement with the numerical data for

the spectral function in the range of plasma parameters

where 1aB−1; the relative deviation is smaller than 10%

under this constraint.

Thus, a simple expression for the damping width of elec-

trons in a classical plasma has been found, which can be used

to approximate the full spectral function to high accuracy.

This achievement greatly facilitates the calculation of ob-

servables that take the spectral function or the self-energy as

an input, such as optical properties inverse bremsstrahlungabsorption, conductivity, or the stopping power.

Furthermore, it was demonstrated that the derived expres-

sions allow for quick and reliable calculations of plasma

properties without having to resort to the full self-consistent

solution of the GW 0 approximation. As an example, the

shift of the chemical potential was calculated using the pa-

rametrized spectral function, and compared to GW 0 results.

For densities of n1021 cm−3, the two approaches coincide

with a relative deviation of less than 10%, going eventually

up to 20% as the density approaches 1022 cm−3. At low den-

sities both approaches give identical results. This shows the

extreme usefulness of the presented approach for the calcu-

lation of observables via the parametrized spectral function.

As a further important application of the results presented

in this paper, we would like to mention the calculation of

radiative energy loss of particles traversing a dense medium,

i.e., bremsstrahlung. A many-body theoretical approach to

this scenario is given by Knoll and Voskresensky 40, using

nonequilibrium Green functions. They showed that a finite

spectral width of the emitting particles leads to a decrease in

the bremsstrahlung emission. This effect is known as the

Landau-Pomeranchuk-Migdal effect 41,42. It has been ex-

perimentally confirmed in relativistic electron scattering ex-

periments using dense targets, e.g., lead 43,44. In 45, it is

shown that thermal bremsstrahlung from a plasma is also

reduced due to the finite spectral width of the electrons in the

plasma. In the cited papers, the quasiparticle damping widthwas either set as a momentum- and energy-independent pa-

rameter in 40, or calculated self-consistently using sim-

plified approximations of the GW 0 theory in 45, which

itself is a very time-consuming task and prohibited investi-

gations over a broad range of plasmas parameters. Now,

based on this work’s results, calculations on the level of thefull GW 0 approximation become feasible, since analytic for-

mulas have been found that reproduce the GW 0 self-energy.

Effects of dynamical correlations on the bremsstrahlung

spectrum can be studied starting from a consistent single-

particle description via the GW 0 self-energy.

ACKNOWLEDGMENTS

The author acknowledges much helpful advice from Gerd

Röpke and fruitful discussion with W.-D. Kraeft as well as

with C. D. Roberts. Financial support was obtained from the

German Research Society DFG via the Collaborative Re-

search Center “Strong Correlations and Collective Effects in

Radiation Fields: Coulomb Systems, Clusters, and Particles”SFB 652.

APPENDIX A: ANALYTIC SOLUTION FOR THE GW (0)

SELF-ENERGY USING THE PLASMON-POLEAPPROXIMATION

After the angular integration which was performed in Eq.14, the imaginary part of the self-energy at the quasiparticle

dispersion =p reads

Im p, p2 = pl

2

4 p

0

dq

q qErf q2 + 2 pq + q

2 p

− Erf q2

− 2 pq + q2 p nB qexp q/T

− Erf q2 + 2 pq − q

2 p

− Erf q2 − 2 pq − q

2 pnB− qexp− q/T .

A1

This equation represents a self-consistent equation for

Im p , = p2=2 / p.

Our aim is to derive an analytic expression that approxi-

mates the numerical solution of Eq. A1 for arbitrary p. Tothis end, we first look at the case of large momenta, p ,

and later combine that result with known expressions for the

limit of vanishing momentum p→0, to produce an interpo-

lation “Padé” formula that covers the complete p range.

We perform a sequence of approximations to the integral

in A1. First, we observe, that at large p, the term 2 pq

dominates in the argument of the error function. We rewrite

Eq. A1 as

p =

2Im p, p2 =

2

pl2

4 p

0

dq

q qErf 2 pq

2 p

− Erf − 2 pq

2 p nB qexp q/T − Erf 2 pq

2 p− Erf − 2 pq

2 pnB− qexp− q/T A2


016404-9

195



=

2

pl2

4 p

0

dq

q q2 Erf 2 pq

2 p

nB qexp q/T

− nB− qexp− q/T . A3

The integrand in Eq. A3 contains a steeply rising part at

q− p / p and a smoothly decaying part for at large q, i.e.,

when q− p / p. Therefore, we separate the integral in the

equation into two parts, one going from q =0 t o q = q =

− p / p and the other from q to infinity. In the first part of the

integral, the values for q are so small that we can replace the

plasmon dispersion by the plasma frequency pl. In the sec-

ond term, the argument of the error function is large and the

error function can be replaced by its asymptotic value at

infinity, lim x→ Erf x =1. This leads to

p = 2 pl2

4 p 0

q¯

dqq pl

2 Erf 2 pq2 pnB plexp pl/T

− nB− plexp− pl/T + 2q

dq

q qnB qexp q/T

− nB− qexp− q/T . A4

Finally, we expand the last term in powers of q /T , which is

justified at low densities q pl, and keep only the first

order,

nB qexp q/T − nB− qexp− q/T =2T

q+ O q−3.

A5

We obtain

p =

2

pl2

4 p

0

q dq

q pl

2 Erf 2 pq

2 pnB plexp pl/T

− nB− plexp− pl/T + 4T q

dq

q q2 . A6

Both integrals can be performed analytically:

0

q

dqq

Erf 2 pq2 p = − 2

2

pq¯

p2F 21/2,1/2; 3/2,3/2;

− 2 p2q 2/ p2

= − 2 2

2F 21/2,1/2; 3/2,3/2;− 2

= − 1.3357,

q

dq

q pl2 1 + q2

/ 2=

1

2ln1 + 2/q 2 =

1

2ln1 + 2 p2

/ p2,

A7

where q¯

= − p / p was used. Note that, in the second integral,the q4 term in the plasmon dispersion 9 is omitted.

2F 2a1 , a2 ; b1 , b2 ; z is the generalized hypergeometric func-

tion 46.

We arrive at the equation

p = − 1.3357

2

pl

2

2 pnB plexp pl/T − nB− pl

exp− pl/T −

2

T

2 pln1 + 2 p2

/ p2. A8

At large p, the term 2 p2/ p dominates the argument of the

logarithm, i.e., we can write ln1 + 2 p2/ p

2 ln 2 p2/ p

2.

Then, we arrive at Eq. A1, given in Sec. IV.

APPENDIX B: PADÉ APPROXIMATION

From the knowledge of the behavior of p in the limits

p→

0 and p→

, a two-point Padé interpolation formula canbe constructed. For the value of the quasiparticle damping

width at p =0 we take the expression

0 = −

2 T , B1

which is the exact solution of the self-consistent Born ap-

proximation 35.

The Padé interpolation formula is constructed in the fol-

lowing way. We make the ansatz

pPadé

=a0 + a1 p

1 + b1 p + b2 p2 ˜ p , B2

where the function ˜

p contains the logarithmic terms

present in the behavior of p at large p cf. Eq. 16:

˜ p = ˜ − ln ˜ +ln ˜

˜ −

ln ˜

˜ 2+

ln ˜

˜

−3 ln2 ˜

2 ˜ 3+

ln2 ˜

2 ˜ 2+

ln3 ˜

3 ˜ 3 , B3

˜ = lne + 2

p2 exp A/T /T . B4

The coefficients a0 , a1 , b1 , b2 are determined by power ex-pansion at p =0 and p→,

lim p→0

pPadé

= a0 + a1 − a0b1 p + O p2 , B5

lim p→

pPadé

= a1

b2 p+

a0b2 − a1b1

b22 p2

+ O p−3 ˜ p , B6

and comparison to the behavior of p in these limiting cases,

e.g., Eq. 16 for large p and Eq. B1 for p→0. Setting the

slope of p at p =0 to zero, as well as the coefficient in front

of the p−2 term of the asymptotic expansion, we arrive at the

following equations for the coefficients of the interpolation

formula:

a0 = −

2 T , a1 − a0bq = 0 , B7


016404-10




a1 = − T

2b2, a0b2 − a1b1 = 0 . B8

The solution reads

a0 = −

2 T , a1 = −

23/2

, B9

b1 =

2T , b2 =

2T , B10

which is given as Eq. 17 in the main text, Sec. II.

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016404-11

197



200 Curriculum vitae

Curriculum vitae

24.07.1978 Geboren in Eutin

07/1989 – 06/1998 Besuch des Gymnasiums, Johann-Heinrich Voss Schule, Eutin20.06.1998 Abitur

10/1998 – 10/1999 Zivildienst, Universitatsklinikum Freiburg. i. Br.10/1999 – 09/2001 Grundstudium im Studiengang Physik (Diplom), Universitat Rostock

07/2001 Vordiplomprufungen10/2001 – 09/2002 Studienaufenthalt an der Unversite de Nantes, Frankreich10/2002 – 09/2003 Hauptstudium im Studiengang Physik (Diplom), Universitat Rostock

07/2003 Diplomprufungen10/2003 – 08/2004 Diplomarbeit zum Thema

“Bremsstrahlung in Dichter Materie und

Landau-Pomeranchuk-Migdal Effekt”Wahrend des Studiums gefordert durch dieStudienstiftung des Deutschen Volkes

09/2004 – 12/2004 Stipendiat des Gradiuiertenkollegs 567“Stark korrelierte Vielteilchensysteme”

01/2005 – 06/2005 Wissenschaftlicher Mitarbeiter im Virtuellen Institutder Helmholtzgemeinschaft“Plasma Physics Research using FEL Radiation”

07/2005 – heute Wissenschaftlicher Mitarbeiter im Sonderforschungsbereich 652 der DFG“Starke Korrelationen und Kollektive Phanomene im Strahlungsfeld:Coulombsysteme, Cluster und Partikel”



List of publications 201

Liste der Veroffentlichungen und Fachvortrage

Eingeladene Vortrage1. Diagnostics of Dense Plasmas using Thomson Scattering

C. Fortmann, R. Redmer, H. Reinholz, G. Ropke, R. Thiele, S. Glenzer, H. LeeGSI Plasmaphysik-SeminarJuni 2008

2. Bremsstrahlung and Thomson Scattering in Dense Plasmas,C. Fortmann, G. Ropke, and A. Wierling,Lawrence Livermore National Laboratory, Livermore, California, USA,Juni 2007.

Konferenzvortrage

1. Spectral Function for Dense Plasmas,C. Fortmann, G. Ropke, and A. Wierling,Fruhjahrstagung der Deutschen Physikalischen Gesellschaft,Darmstadt, Marz 2008.

2. Thomson Scattering at FLASH ,C. Fortmann,Physics of High Energy Density Matter Conference,Hirschegg, Osterreich, Januar 2008.

3. Theory of Thomson scattering in Warm Dense Matter ,C. Fortmann,Abschluss-Meeting des Virtuellen Instituts “Plasma Physics Research Using FELRadiation”,Hamburg, Dezember 2007.

4. Thomson Scattering in Warm Dense Matter ,C. Fortmann,2. Laserlab Europe Workshop,Darmstadt, Oktober 2007.

5. FLASH scattering and absorption in liquid H2 ,C. Fortmann,Peak-Brightness Collaboration Meeting,Hamburg, September 2007.

6. Bremsstrahlung from Dense Plasmas: A Many-Body Theoretical Approach ,C. Fortmann, G. Ropke, and A. Wierling,IEEE Pulsed Power and Plasma Sciences Conference,Albuquerque, New Mexico, USA, Juni 2007.



202 List of publications

7. Einteilchen-Spektralfunktion und optische Eigenschaften dichter Plasmen ,C. Fortmann, G. Ropke, A. Wierling,Fruhjahrstagung der Deutschen Physikalischen Gesellschaft,Dusseldorf, Marz 2007.

8. Bremsstrahlung und optische Spektren aus dichten Plasmen ,C. Fortmann, G. Ropke, J. Tiggesbaumker, T. Doppner,Workshop des SFB 652 “Starke Korrelationen und kollektive Phanomene im Strahlungs-feld: “Coulombsysteme, Cluster und Partikel”,Warnemunde, Februar 2007.

9. Optical Properties and the One-particle Spectral Function in Non-ideal Plasmas,C. Fortmann, G. Ropke, and A. Wierling,

12th International Workshop on Physics of Non-ideal Plasmas (PNP12),Darmstadt, September 2006.

10. Spectral function for charged particles: GW and beyond ,C. Fortmann, G. Ropke,Workshop des SFB 652 “Starke Korrelationen und kollektive Phanomene im Strahlungs-feld: “Coulombsysteme, Cluster und Partikel”,Dorf Mecklenburg, April 2006.

11. Bremsstrahlung vs. Thomson scattering in VUV-FEL plasma experiments,C. Fortmann, R. Redmer, H. Reinholz, G. Ropke, W. Rozmus, and A. Wierling,

Physics of High Energy Density Matter Conference,Hirschegg, Osterreich, Januar 2006.

12. Landau-Pomeranchuk-Migdal Effect in Dense Plasmas,C. Fortmann, H. Reinholz, G. Ropke, and A. Wierling,Fruhjahrstagung der Deutschen Physikalischen Gesellschaft,Berlin, Marz 2005.

13. Bremsstrahlung in Dense Matter and LPM Effect ,C. Fortmann, G. Ropke,Workshop of the Virtual Institute “Dense Hadronic Matter and QCD Phase Transi-tion”,Bad Honnef, Juli 2004.

Veroffentlichungen in Peer-Review Journalen

1. Carsten Fortmann,Self-consistent Spectral Function for Non-Degenerate Coulomb Systems and AnalyticScaling Behaviour ,J. Phys. A: Math. Gen. (accepted).



List of publications 203

preprint: http://arxiv.org/abs/0805.4674.

2. R. Thiele, T. Bornath, C. Fortmann, A. Holl, R. Redmer, H. Reinholz, G. Ropke, A.Wierling, S. H. Glenzer, G. Gregori,Plasmon resonance in warm dense matter ,Phys. Rev. E 78, 026411 (2008).

3. S. H. Glenzer, P. Neumayer, T. Doppner, O. L. Landen, R. W. Lee, R. J. Wallace,S. Weber, H. J. Lee, A. L. Kritcher, R. Falcone, S. P. Regan, H. Sawada, D. D.Meyerhofer, G. Gregori, C. Fortmann, V. Schwarz, and R. Redmer,Compton scattering measurements from dense plasmas,J. Phys.: Conf. Series 112, 032071 (2008).

4. A. Holl, Th. Bornath, L. Cao, T. Doppner, S. Dusterer, E. Forster, C. Fortmann, S.H.Glenzer, G. Gregori, T. Laarmann, K.-H. Meiwes-Broer, A. Przystawik, P. Radcliffe,R. Redmer, H. Reinholz, G. Ropke, R. Thiele, J. Tiggesbaumker, S. Toleikis, N.X.Truong, T. Tschentscher, I. Uschmann, and U. Zastrau,Thomson Scattering from Near-solid Density Plasmas Using Soft X-Ray Free Electron Lasers,Journal of High Energy Density Physics 3,120 (2007).

5. C Fortmann, G. Ropke, and A. Wierling,

Optical Properties and the One-particle Spectral Function in Non-ideal Plasmas,Contrib. Plasma Phys. 47, 297 (2007).

6. C. Fortmann, R. Redmer, H. Reinholz, G. Ropke, W. Rozmus, and A. Wierling,Bremsstrahlung vs. Thomson Scattering in VUV-FEL Plasma Physics Experiments,Journal of High Energy Density Physics 2, 57 (2006).

Bei Peer-Review Journalen eingereichte Artikel

1. C. Fortmann,Single Particle Spectral Function for the Classical One-Component Plasma ,eingereicht bei Phys. Rev. E.preprint: http://arxiv.org/physics/0808.2151.

2. U. Zastrau, C. Fortmann, R. R. Faustlin et al.Bremsstrahlung and Line Spectroscopy of Warm Dense Aluminum Plasma generated by XUV Free-Electron Laser Radiation ,eingereicht bei Phys. Rev. Lett.



204 List of publications

Konferenzproceedings und Institutsberichte

1. Carsten Fortmann, R. Redmer, H. Reinholz, G. Ropke, and R. Thiele Thomson

Scattering on Solid Density and Compressed Be,High Energy Density Physics with Intense Ion and Laser Beams: GSI Annual Report2007.

2. Bremsstrahlung from dense matter and the Landau-Pomeranchuk-Migdal effect,C. Fortmann, H. Reinholz, G. Ropke, and A. Wierling,Condensed Matter Theories, Vol. 20, p. 317 (Nova Science, New York, 2006).

3. Carsten Fortmann, R. Redmer, H. Reinholz, G. Ropke, W. Rozmus, and A. Wierling,Bremsstrahlung vs. Thomson scattering in VUV-FEL plasma experiments,High Energy Density Physics with Intense Ion and Laser Beams: GSI Annual Report

2005.

4. C. Fortmann, H. Reinholz, G. Ropke, and A. Wierling,Bremsstrahlung in dense matter and Landau-Pomeranchuk-Migdal effect ,GSI Plasma Physics Annual Report 2004.



Declaration of authorship 205

Erklarung

Hiermit erklare ich an Eides statt, dass ich die vorliegende Arbeit selbstandig angefertigt

und ohne fremde Hilfe verfasst habe, keine außer den von mir angegebenen Hilfsmittelnund Quellen dazu verwendet habe und die den benutzten Werken inhaltlich und wortlichentnommenen Stellen als solche kenntlich gemacht habe.

Rostock, den 28.8.2008

carsten fortmann- bremsstrahlung in dense plasmas: a many-body theoretical approach

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