The study of the so-called Warm Dense Matter (WDM) is an emerging and challenging field that is at the crossroads of condensed matter and plasma physics. Here the density goes from solid density up to 10 times its value ; the temperature varies from 0.1 to 100 eV. In this regime, matter is mostly degenerate, strongly coupled and non-ideal. That gives rise to a physical complexity that ought to be a fertile ground for exciting scientific discoveries, as illustrated by the important number of recent international workshops and schools related to WDM. Warm Dense Matter science covers a wide range of physical phenomena from astrophysics and geophysics (stellar and planetary interiors), inertial confinement fusion (early stage initial conditions) up to industrial applications such as laser ablation, laser machining and laser damaging.
Great uncertainties remain regarding the physics of these states of matter. From the theoretical point of view, one has to deal with ab initio calculations (molecular dynamics), which work well at low temperatures (T < 1 eV), or variational methods (Density Functional Theory – DFT) which can be very efficient in a plasma phase. However for most of the WDM “phase diagrams”, these methods are pushed to their intrinsic limits and experiments are the only way to discriminate between the theories. Therefore, the key issue is to create those conditions in the laboratory, which is still a challenge.
Figure 1. Principle of isochoric heating inducing the ultra-fast phase transition from cold solid up to Warm Dense state. This state of matter is very transient, since it is followed by a rapid expansion whitin a few tens of ps.
We work on the production of homogeneous Warm Dense Plasmas (WDP) and on the study of the ultrafast dynamics of their structural (short-range order expected) and electronic properties during the ultra-fast phase transition from solid to WDM. We propose to generate WDM by isochoric heating (Cf. Fig. 1) with an ultra-short laser pulse (femtosecond scale) and with an ultra-short laser-produced proton burst (picosecond scale). Two complementary laser installations are used : high repetition rate and moderate energy at CELIA for study of WDM in the 0.1 – 1 eV range (laser heating) ; single shot high energy laser at LULI (30 J) to extend this study up to the 1 – 10 eV range (proton heating). Mainly, two time-resolved diagnostics are used. The first one is the ultra-fast near-edge X-ray absorption spectroscopy to probe the short-range order of ions, and more generally the frontier between bound and free electronic states. The second one consists in the measurement of the phase shift and reflectivity of an optical beam to probe the free electron dynamic (Fourier Domain Interferometry).
This work is a collaborative project with colleagues from CEA – DIF – DPTA and LULI, taking advantage of complementary experimental, numerical and theoretical skills. It is the result of a discussion within the frame of the working group MEHDOC (Matière à Haute Densité d’Energie et Ondes de Choc, resp. M. Koenig) gathering the principal investigators in the field of WDM in the French community. The development of laser-based ultra-fast near-edge X-ray absorption spectroscopy to probe correlated matter has been also identified as a major topic in the GDR AppliX (Applications des nouvelles sources X, resp. A. Klisnick).
Ultra-fast X-ray Absorption Near-Edge Spectroscopy (XANES)
When irradiating solid targets of high Z material, an ultra-short (ps) high repetition rate (1 kHz) X-ray source has been optimized, offering intense broadband emission over a wide spectral range (1.5 – 1.75 keV). The objective was to realize experiments of X-ray Absorption Fine Spectroscopy, around the K-edge of Al (1.559 keV). Spectral structures that can be observed near the K-edge (1.55 – 1.65 keV) are called XANES (for X-ray Absorption Near-Edge Spectroscopy). Further (1.6 – 1.75 keV) is the EXAFS domain (for Extended X-ray Absorption Fine Structure). The analysis of these structures leads to a rich and direct information about the local atomic arrangement in the probed sample.
Figure 2. Left : X-ray absorption spectra (XANES) measured near the K-edge of solid cold Al, with different sample thicknesses (1000, 2000, 5000 Å, 1 µm and 2 µm from bottom to top). Measurements are artificially vertically shifted for clarity reasons. They are obtained when cumulating 30 s laser shots (4 mJ, 3 ps, 1 kHz) focused on an erbium target. Right : XANES spectra of cold solid Al calculated with a multiple scattering method, considering 11 and 43 atoms in the vicinity of the absorbing Al atom.
A first experiment has been performed in 2007 with the “Aurore” laser of CELIA. The X-ray source has been optimized by irradiating an erbium target. X-ray absorption spectra have been registered near the K-edge of a cold aluminum sample. The XANES structures are clearly observed and resolved, as can be seen in Fig. 2, even with very thin Al sample (1000 Å). EXAFS structures are still weaker and need additional work on the X-ray source. Now, we plan to extend these measurements to the study of laser heated Al samples (up to a few eV), using the “pump (optical) – probe (X-ray)” technique. This work is part of the PhD of Marion Harmand.
In order to associate these results with dense matter models, we have developed a XANES code based on the multiple scattering method. This code calculates precisely the X-ray absorption cross-section near the K-edge of an isolated atom surrounded by a “cluster” of others atoms. It has been validated in the case of a crystalline solid, where the atomic positions are well defined. The objective being now to study the Warm Dense Matter, a procedure has been achieved to generate atomic configurations in space, from the Radial Distribution Function (“g(r)” function). These configurations are built to sample the g® et are used as an input in the XANES code mentioned above. This g® function is derived from a HNC/DFT approach : ions are treated classically with the HNC approximation, eventually improved by the use of a Bridge function, while electrons are treated in the frame of the Density Functional Theory (DFT). Under these assumptions, complete Al K-edge XANES profiles have been calculated.
Fourier Domain Interferometry
The control of the sample temperature and of its confinement at solid density will be achieved using the Fourier Domain Interferometry technique (FDI). This powerful diagnostic lies on the interference in the spectral domain of two ultra-short delayed laser pulses. First pulse is the reference and is reflected by the sample before to be heated. Second pulse is reflected after the heating. The diagnostic measures its reflectivity and its relative phase compared to the reference (Cf. Fig. 3).
Figure 3. Example of registered spectrum with the FDI technique. Vertical axis is the space (one-dimensional spatial resolution). The horizontal one is the spectral coordinate that eventually codes the temporal response in case of chirped pulses. Fringes come from the interference between the two laser pulses in the Fourier domain. The reflectivity of the second pulse probing the heated matter can be observed through the more luminous signal from the plasma (close to the image central axis). The phase shift is derived through the fringe shift.
This device has been designed by a team of collaborators from LULI. It measures the heating zone expansion with femtosecond resolution and has been already used with success to get the temporal behavior of plasmas temperature and density. This improved technique will also permit to get unprecedented time-resolved measurements of the free electron dynamic at the sample surface, during the transition from the solid state to the warm dense matter regime.