Claire Bauche-Arnoult

Opacity Studies of Iron in the 15-30eV Temperature Range

Absorption of the 2p-3d transitions of iron has been measured using point projection spectroscopy. Thin C tamped Fe foils were heated around 20 eV by X-rays generated in gold spherical hohlraums irradiated by the high-power laser ASTERIX IV. Absorption of Fe V to Fe X has been observed in the spectral vicinity of 730 eV (17 A). The Ag backlighter source and absorbed spectra were recorded on the same shot by a TlAP crystal spectrograph. The experimental spectra have been reproduced by the two superconfiguration local thermodynamic equilibrium codes SCO and STA. Detailed statistical calculations of the different ionic structures have also been performed with the Spin Orbit Split Arrays method, allowing the determination of ion populations. The electron temperature and average ionization obtained by fitting the experiment with the different calculations were compared with radiative hydrodynamic simulations.

Absorption measurements of radiatively heated multi-layered Al/Ni foils

Abstract Mixtures of light and mid- Z elements have been used to measure the absorption of the mid- Z element Ni, the temperature is inferred from the K-shell absorption of the light element Al. Here we test this method by comparing the temperatures deduced from Al K α transitions and nickel L-shell absorption spectra in Al/Ni multilayers and bilayers. The ionisation state is obtained by comparison of the Al and Ni spectra with calculations assuming local thermodynamic equilibrium. The temperatures obtained from the experiment are compared with hydrodynamic simulations predictions. Simulation code results show that the density differs by a factor of 2 in the two elements. This has to be taken into account in the determination of the temperature.

Statistics of electric-quadrupole lines in atomic spectra

In hot plasmas, a temperature of a few tens of eV is sufficient for producing highly stripped ions where multipole transitions become important. At low density, the transitions from tightly bound inner shells lead to electric-quadrupole (E2) lines which are comparable in strength with electric-dipole ones. In this work, we propose analytical formulas for the estimation of the number of E2 lines in a transition array. Such expressions rely on statistical descriptions of electron states and J-levels. A generalized ‘J-file’ sum rule for E2 lines and the strength-weighted shift and variance of the line energies of a transition array nlN + 1 → nlNn′l′ of inter-configuration E2 lines are also presented.

Superconfigurations and Super Transition Arrays

This chapter is a continuation of Chap. 6 in the sense that, for very complex ions, not only the number of levels is overwhelmingly large, but also the number of configurations. The concept of superconfiguration has been introduced for a proper gathering of configurations through the definition of supershells (collections of ordinary subshells). This concept relies on the notion of partial Local Thermodynamical Equilibrium, i.e., all the configurations belonging to a given superconfiguration are distributed according to a Boltzmann law at some temperature.

Static and dynamical equilibrium in plasmas

The plasma properties such as ionization, internal energy, emissivity or opacity depend on the populations of the energy levels, i.e., on the thermodynamical state of the plasma. The Local Thermodynamical Equilibrium (LTE) is the simplest state, but a non-LTE description is often necessary. The LTE laws are reviewed. The Saha-Boltzmann law is thoroughly derived in the framework of statistical mechanics. The most general description, however, requires considering all the microscopic processes (excitation, de-excitation, ionization, recombination, etc.) and building a rate equation for each atomic level. The system of coupled equations for the populations is called the collisional-radiative system. A major difficulty of this detailed level accounting (DLA) approach is the need to compute a large set of rates for all the processes between the energy levels. In each case, the link between the detailed balance principle and the microreversibility of the processes is discussed.

Distribution functions. Energy levels

The frequency distribution functions are universal quantities for describing the statistics of large ensembles. These functions are generally represented by their distribution moments, of various orders. In atomic physics, such moments are computed by means of the tensor-operator formalism, as sums of products of Wigner n-j coefficients. When the summation problem appears to be untractable, two methods may bring a decisive help: the second-quantization formalism developed in atomic physics by Judd, and the graphical methods elaborated by Jucys and his team. After a (limited) number of moment values have been obtained, one enters them into the distribution function of the “best” statistical model, which is a matter of choice (due to the limitation in the number of moments).

Global approach to plasmas in LTE equilibrium

The computation of a quantity such that the LTE opacity of a plasma by using a full detailed level accounting approach is, even in a limited spectral range, very demanding in terms of computational resources. For moderate- to high-Z plasmas, global concepts such as Unresolved Transition Arrays (between configurations) and Super Transition Arrays (between superconfigurations) can be very useful tools to handle large sets of radiative transitions. The relevance and the efficiency of these global methods are illustrated through the example of an absorption spectrum calculated in a line-by-line calculation and compared with the result of a superconfigurationaccounting approach. Also, the comparisons of these global calculations with experimental measurements of the plasma opacity in some selected energy ranges corresponding to specific nl - n´l´ transitions assess the usefulness of these methods. In particular, for moderate-Z plasmas, a UTA approach gives satisfactory results, notwithstanding the approximate accounting of the Boltzmann factors which can slightly alter the results at low temperatures. For higher-Z plasmas, the superconfiguration-accounting approach gives good results and makes tractable the treatment of very complex ions.

Applications to hot-plasma radiation

Some results of calculations using the methods proposed in the book are presented as examples. The plasma characteristics are needed either for studying its structure, or for computing its time evolution. Among the former, there are the free-electron temperature and density, and the chargestate distribution; among the latter, there are the radiative power losses, the cooling coefficients, and the Rosseland mean opacities. In the field of applications, several examples of fine experiments are addressed.

The central-field configurational model

In the central-field configurational model, the basis functions of the atomic states are products of a radial part, an angular part, and a spin part. They must obey the antisymmetry principle. They can be gathered into electronic configurations. The atomic states (wavefunctions) are the eigenvectors of the H Hamiltonian, which is essentially the sum of the electronic kinetic energies, of the electrostatic interactions between the nucleus and the electrons, and of those between the electrons, plus the magnetic interactions between the orbital and spin magnetic moments of the electrons. They are obtained by diagonalizing H over the basis functions of a single configuration (in intermediate coupling), or of several configurations (in configuration interaction). The (3p 2 + 3d 2) mixing is taken as an example.

Modeling of ionic spectra

In the case of coalescence of the lines, an unresolved transition array (UTA) can be simply represented, as a Gaussian (or more complicated) curve, using the computed distribution moments. Here, there appears a conspicuous difference between the emission and absorption spectra. The linewidths, and the overlapping of the lines (and the gaps between them) have a critical effect on the value of the total absorption, but not on that of the total emission. The emission spectrum can be represented accurately as a collection of Gaussian curves, one per UTA, but not the transmission spectrum.