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Local Thermodynamic Equilibrium
In thermodynamic equilibrium the state of the plasma is specified uniquely by two thermodynamic variables (like the absolute temperature and the density) via the well known equations of statistical mechanics. However, this situation is not valid in astrophysics and laboratory plasmas when electron–ion collisions or photon–ion interactions have a small probability. The problem lies in the deviations from the thermal distribution of the plasma photons. This mainly results from the ability of the photons to escape from the plasma without being reabsorbed; this occurs, for example, in the corona stellar regions (and also in many laboratory plasmas), where the photon mean free path is larger than the plasma dimensions. As a result, in a plasma some particles may obey thermal distributions (possibly ions and much more probably electrons), something that is ensured by their relatively high collisional rates [2].
In this situation the plasma statistics can be described with the model of local thermodynamic equilibrium (LTE). In LTE model we assume that locally the plasma particles are in thermodynamic equilibrium, even though the photons deviate from it, and thus they can be characterized by a temperature and density determining their distributions. LTE is a strictly local theory and does not permit to correlate the conditions of separate plasma regions, let say by radiation exchange, and thus, we should understand that the plasma parameters of temperature and density have a purely local scope [3].
For LTE to be valid, two conditions should be fulfilled [4] : (1) The collisions mean free paths of the particles, and mainly of the electrons, should be smaller than the dimensions of the gradients exhibiting the thermodynamic parameters, and (2) the photons mean free path should not exceed strongly the gradients dimensions. The former ensures that within the volume of the locally characterized plasma a large number of particles collisions take place which tends to thermalize its behavior. The latter means that the photons interacting locally with the particles come from a region with not very different thermodynamic properties. The non local radiation tends to drive the particles distributions away from their equilibrium state, competing the particles collisions. Concerning the global characteristics of the plasma these conditions are equivalently fulfilled when it has a high density, and it is optically thick.
At thermodynamic equilibrium, the populations of the plasma particles are ruled respectively by the Maxwell, Boltzmann, and Saha distributions for the free electrons, the ionic levels and the ionization stages.
Non–Local Thermodynamic Equilibrium–Collisional Radiative Model
Non–local thermodynamic equilibrium (NLTE) characterizes a plasma when at least one of the particles distribution functions deviates from the corresponding thermal distribution; explicitly, when the occupation number of the bound atomic levels do not follow the Boltzmann distribution, or an ionization state density deviates from the Saha value, or a velocity distribution is non–Maxwellian [2]. Even if the ions in the plasma obey LTE conditions as described above, the photons do not usually follow Planck’s law.
The main cause of the existence of NLTE regimes is the ability of the photons to escape from the plasma (optically thin) and the fact that collision rates are not much higher than radiative rates. Due to this loss of energy the plasma particles participating in the photons emission (ions in bound–bound and bound–free, or electrons in free–free radiative transitions) deviate from their thermal distribution, and, when the plasma density is relatively low, the collisional processes can not compensate this deviation. Other reasons can be the rapid temporal variation in a plasma not being in a steady state, or transport processes due to the steep gradients of the plasma parameters. It is important to notice that because the NLTE situations are mainly due to the non–Planckian behavior of the radiation field, the frontier between LTE and NTLE states is never very clear.
The distributions functions of the plasma particles under NLTE conditions cannot be found in an analytical form from basic principles, as with in LTE. Here, we have to write down explicitly the kinetic equations ruling the statistics of the plasma particles. The state density of a specific particle (ion, electron, or photon) can be obtained by solving rate equations, taking into account all the processes that increase or decrease its population.
In the standard NLTE modeling approach we account in detail for the different atomic states, while the free electrons are still assumed to have a Maxwellian distribution due to their high collision rates in hot plasmas [5]. So, the ionization distribution and the bound level populations are determined by solving a system of rate equations taking into account all the collisional and radiative processes. The general form of the rate equation is − dNi = Ni ∑Ri→ j − ∑N j R j→i dt j j.
The code TRANSPEC/AVERROES
The code TRANSPEC/AVERROES developed by O. Peyrusse [5] implements a NLTE collisional radiative model (CRM) supported by the superconfiguration (SC) formalism [6] for the description of the ions states. This model permits to simulate the complex emission and absorption spectra of medium and high–Z elements plasmas under NLTE. The code is divided in two parts. The first part, the code AVERROES, calculates the collisional and radiative rates of the different atomic processes, as well as the mean energies and variances of the super–transition arrays. Then, these quantities are used by a standard time–dependent CRM such as TRANSPEC [18–19] that calculates the plasma kinetic populations to obtain the synthetic emission spectrum.
Concerning the calculations of the atomic orbitals, AVERROES uses the following approximations : (1) The calculations are performed within the “isolated atom” model. In particular, no static screening by the free electrons is introduced in the atomic potential. Furthermore, the orbitals are not confined into a finite ion–cell.
(2) The orbitals are non relativistic, and the corrections to the radial Schrodinger equation are introduced with the Pauli approximation described in [20].
(3) Exchange contribution to the atomic potential is calculated with the local density approximation. To calculate the synthetic emission spectrum, the code can be used either in a single–cell mode, assuming an homogeneous plasma, or as a post–processor of a hydrodynamic code as MULTI [12]. In the second case the hydro–code gives in a tabulated format the spatial and temporal profiles of the plasma parameters, and thus provides the evolution of the plasma cells used by TRANSPEC.
The TRANSPEC/AVERROES package is used in the analysis of the XUV emission of the xenon and krypton NLTE plasmas (Chapter 4). The time–resolved emission spectrum of each element is measured in the range of 20–200 Å, and the code is used to simulate the transitions structures n = 5–4 and 4–4 of the xenon, as well as the structures n = 4–3 and 3–3 of the krypton. In our work the kinetic code TRANSPEC is used in single–cell mode, and the plasma parameters are measured with the Thomson scattering technique providing their average values. Simultaneously, the x–ray spectrum of the elements is measured in the keV range. In particular, the xenon spectrum is studied in the range of 12 – 15 Å where the 3d–4f transitions appear, and that of krypton in the range 6 – 8 Å corresponding to the 2–3 and 2–4 transitions. In this range, and under the given experimental conditions, the rich emission spectra of the elements are of particular interest by themselves, and the synthetic spectra calculated by the code permit to analyze them, identifying the transitions of the different ions. Moreover, this identification of the ionic species can provide an independent means of inferring the heating conditions of the laser irradiated plasmas.
MULTI one dimensional radiation hydrodynamic code
The MULTI (MULTIgroup MULTIlayer) code developed by R. Ramis et al. [1], simulates the hydrodynamics of a one dimensional plasma coupled with a radiation field. The code is based on a finite set of difference equations, which result from the physical model describing the coupling of the plasma with the radiation, as follows :
• the spatial variation of the plasma fluid motion, and of the radiation transport through it, are described in discrete cells, while the temporal variation is calculated in discrete time steps sampling the evolution of the plasma–radiation coupling.
• the frequency domain is optimally divided in discrete groups.
This is achieved by the multigroup approach of the plasma emissivity and opacity, which is a powerful extension of the simplified single mean opacity approximation used to solve the non grey radiation transfer problem [6].
Physical model of the radiation field–plasma coupling
As we saw in paragraph 2.2.1, for a “static” (u/c << 1) and one dimensional planar plasma, the Lagrange representation of the equations describing its coupling with an intense radiation field are given by (2.3a, b, c’) and (2.10). The equations of state for the pressure and the internal energy, required to close the hydrodynamic system of equations, are provided to the code in a tabulated format using the data of the SESAME library [7]. The term q in Eq. (2.3.c ΄) represents the heat flux carried by electrons, and in the quasi–equilibrium limit is given by the Fourier law q = − KT 5 / 2∂ xT (2.17).
Radiation confinement in the interior of an open spherical gold cavity
The confinement of the radiation field in the interior of a cavity can be viewed as the composite result following from the heating of its wall and the radiation reemission accompanying this heating. Though these phenomena are strongly correlated, we approach them separately in the following paragraphs. First, we analyze the energy exchange inside the cavity, which determines the heating of each wall element. Then, we study the behavior of a single wall element due to the radiation incident on it, which determines its radiation reemission. Finally, we calculate the total radiation heating the absorption foil. Radiation transfer through a plasma–Cavities thermal radiation and solid foil heating Chapter 2
Energy exchange in the cavity interior
The energy exchange inside the cavity is described in Figure 2.2. This figure shows a spherical cavity with open sections having a direct correspondence with the cavities used in the experiment. In particular, the cavity contains two holes with area AH (i.e. the diagnostic holes, one of which is covered by the absorption foil, AH = AF), and an external source occupying an area AS, and which irradiates the interior of the cavity with a flux S0 (r, t) . For the cavity used in the experiment this radiation flux corresponds to the x–ray emission at the rear side of the conversion foil heated by the main beam. If the wall material of the cavity participating in the radiation confinement has an area AW, the simple geometrical formula follows AC = AW +AS +2AH (2.24).
where AC is the equivalent area of a closed spherical cavity with the same radius. r Let us consider a differential wall element at an arbitrary point P(r ) . The radiation energy incident on this element is the sum of the radiation emitted from the source Ss and the radiation emitted from all the other wall elements Si. This energy is balanced by the energy absorbed by the wall element Sw, and the reemitted energy from it Sr. The energy balance in term of radiation fluxes is written r r r r (2.25) Si (r, t) + Ss (r, t) = Sr (r, t) + Sw (r, t).
Table of contents :
Chapter 1 Elements of plasma atomic physics
1.1 Introduction
1.2 Plasma atomic processes
1.2.1 Detailed balance principle
1.2.2 Bound–bound processes
1.2.3 Bound–free processes
1.2.4 Free–free processes (Bremsstrahlung radiation)
1.3 Plasma statistical physics
1.3.1 Local Thermodynamic Equilibrium
1.3.2 Non–Local Thermodynamic Equilibrium–Collisional Radiative Model
1.4 Calculation of complex plasma spectrum–Superconfigurations approach
1.5 Atomic codes of plasma physics
1.5.1 The code HULLAC
1.5.2 The code SCO
1.5.3 The code TRANSPEC/AVERROES
Chapter 2 Radiation transfer through a plasma–Cavities thermal radiation and solid foil heating
2.1 Introduction
2.2 Radiation transfer problem
2.2.1 Hydrodynamics–radiation equations coupling
2.2.2 MULTI one dimensional radiation hydrodynamic code
2.3 Radiation confinement in the interior of an open spherical gold cavity
2.3.1 Energy exchange in the cavity interior
2.3.2 Radiation reemission of the cavity wall–Basko scaling law
2.3.3 Total radiation flux coupled with the absorption foil
Chapter 3 Laser and Instrumentation
3.1 Introduction
3.2 LULI2000 laser facility
3.2.1 Laser beam chains
3.2.2 Focalization of the laser beam–KDP crystal and random phase plate
3.3 Targets description
3.3.1 Gas–jet for spectral emission characterization
3.3.2 Spherical gold cavity for absorption spectra measurements
3.4. Diagnostic instruments
3.4.1 Bragg crystal spectrograph
3.4.2 Transmission grating XUV spectrograph
3.4.3 Thomson scattering diagnostic
3.5 Auxiliary devices and other diagnostics
3.5.1 Streak Camera
3.5.2 CCD Cameras
3.5.3 Kodak DEF film
3.5.4 Pinhole cameras
3.5.5 Streak camera differential pumping system
Chapter 4 X–ray emission spectroscopy of xenon and krypton plasmas in NLTE conditions
4.1 Introduction
4.2 Experimental setup
4.2.1 Laser beam and gas–jet targets
4.2.2 Diagnostic instruments
4.3 X–ray spectra data processing
4.3.1 Emission spectra in the keV range
4.3.2 XUV emission spectra
4.4 Determination of the plasma parameters–Thomson scattering spectra analysis
4.4.1 Helium plasma parameters
4.4.2 Xenon and krypton plasmas parameters
4.5 Characterization of the x–ray emission spectra in the keV range
4.5.1 Xenon plasma emission spectra
4.5.2 Krypton plasma emission spectra
4.6 Analysis of the x-ray spectra with TRANSPEC/AVERROES
4.6.1 Calculation of the synthetic emission x–ray spectra in the keV range
4.6.2 Calculation of the XUV emission spectra
4.7 Conclusions
Chapter 5 XUV absorption spectroscopy of radiatively heated ZnS and Al plasmas.
5.1 Introduction
5.2 Experimental setup and data processing
5.2.1 Laser beams and targets description
5.2.2 Diagnostic Instruments
5.2.3 Experimental data processing
5.3 Experimental methods
5.3.1 Deduction of the plasma transmission
5.3.2 Al and ZnS plasmas absorption spectra
5.4 Hydrodynamics of the radiatively heated Al absorption foil
5.4.1 Calculation of the radiation heating the Al foil
5.4.2 Hydrodynamic simulation of the Al foil expansion
5.5 Analysis of the Al plasma absorption spectrum
5.5.1 Characterization of the Al absorption spectrum
5.5.2 Comparison of the HULLAC synthetic Al spectrum with the measured absorption
5.6 Analysis of the ZnS plasma absorption spectrum
5.6.1 Characterization of the ZnS absorption spectrum
5.6.2 Hydrodynamic simulation of the ZnS foil
5.6.3 Comparison of the SCO synthetic ZnS spectrum with the measured absorption.
5.7 Conclusions
Conclusions and perspectives
Appendix A Elements of the Thomson scattering theory
Appendix B Mach–Zehnder interferometry experiment
Appendix C Publications.
