X-ray Thomson scattering in high energy density plasmas

Siegfried H. Glenzer and Ronald Redmer

Rev. Mod. Phys. 81, 1625 – Published 1 December 2009

Abstract

Accurate x-ray scattering techniques to measure the physical properties of dense plasmas have been developed for applications in high energy density physics. This class of experiments produces short-lived hot dense states of matter with electron densities in the range of solid density and higher where powerful penetrating x-ray sources have become available for probing. Experiments have employed laser-based x-ray sources that provide sufficient photon numbers in narrow bandwidth spectral lines, allowing spectrally resolved x-ray scattering measurements from these plasmas. The backscattering spectrum accesses the noncollective Compton scattering regime which provides accurate diagnostic information on the temperature, density, and ionization state. The forward scattering spectrum has been shown to measure the collective plasmon oscillations. Besides extracting the standard plasma parameters, density and temperature, forward scattering yields new observables such as a direct measure of collisions and quantum effects. Dense matter theory relates scattering spectra with the dielectric function and structure factors that determine the physical properties of matter. Applications to radiation-heated and shock-compressed matter have demonstrated accurate measurements of compression and heating with up to picosecond temporal resolution. The ongoing development of suitable x-ray sources and facilities will enable experiments in a wide range of research areas including inertial confinement fusion, radiation hydrodynamics, material science, or laboratory astrophysics.

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DOI:https://doi.org/10.1103/RevModPhys.81.1625

Authors & Affiliations

Siegfried H. Glenzer

L-399, Lawrence Livermore National Laboratory, University of California, P.O. Box 808, Livermore, California 94551, USA

Ronald Redmer

Institut für Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany

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Vol. 81, Iss. 4 — October - December 2009

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Figure 1
Density-temperature phase space indicating areas of Fermi-degenerate matter (dark gray area), strongly coupled plasmas (light gray area), and the ideal plasma regime (white area). The solid lines indicate the scattering parameter $α$ from Eq. (5) for backscattering at $θ = 180 °$ and ultraviolet and x-ray probes. Also shown is the ionization balance for solid density beryllium (dashed curve). From 124.Reuse & Permissions
Figure 2
(Color online) Scattering geometry indicating the length of the scattering vector $k$ and the relation to the screening length $λ_{S}$ of the plasma.Reuse & Permissions
Figure 3
Volume element in three-dimensional velocity space that determines the number of electrons contributing to the velocity component $v_{x}$ .Reuse & Permissions
Figure 4
Calculated Thomson backscattered spectra for various ratios of $k_{B} T_{e} ∕ ϵ_{F}$ and for $ϵ_{F} = 15 eV$ . Solid, long dashed, and short dashed correspond to $k_{B} T_{e} ∕ ϵ_{F} = 0.1$ , 0.2, and 0.4, respectively. The spectral shift corresponding to an electron velocity component equal to the Fermi velocity is denoted by a vertical line. Note that only one side of spectrum is shown, and that $Δ ν = 0$ corresponds to the Compton shifted frequency. From 124.Reuse & Permissions
Figure 5
(Color online) The plasmon dispersion with the plasmon energy shift from the incident x-ray energy $E_{0}$ vs $k$ . The calculations are for beryllium plasmas with $T_{e} = 12 eV$ , $Z = 2.5$ , $E_{0} = 4.75 keV$ , $n_{e} = 3 \times 10^{23} {cm}^{- 3}$ , and $n_{e} = 6 \times 10^{23} {cm}^{- 3}$ . Solid, long dashed, and short dashed correspond to calculations with Born-Mermin theory, the random phase approximation, and with Eq. (16), respectively. From 204.Reuse & Permissions
Figure 6
(Color) Schematic view of the National Ignition Facility (NIF) showing the main components of the laser system (143).Reuse & Permissions
Figure 7
(Color) Real part of the collision frequency in Born approximation using various approximations for electronic and ionic correlations in the collision integral; see Table . Results are shown for three different electron densities. The mean ion charge is $Z_{f} = 2$ and the plasma temperature is $T_{i} = T_{e} = 12 eV$ .Reuse & Permissions
Figure 8
(Color) Electron feature of the dynamic structure factor for compressed (blue) and uncompressed (red) beryllium. The RPA (collisionless plasma) result is compared to the BMA result by taking into account the ionic structure factor and the inverse screening length according to the models specified in Table . Further parameters: $Z_{f} = 2$ , $T_{e} = T_{i} = 12 eV$ , photon energy $ℏ ω_{0} = 2960 eV$ (Cl-Lyman- $α$ ), scattering angle $θ = 40 °$ for red curve and $ℏ ω_{0} = 6180 eV$ (Mn- $He − α$ ), scattering angle $θ = 25 °$ for blue curve.Reuse & Permissions
Figure 9
(Color online) Calculations of the electron feature for the dynamic structure factor according to 78 and 160 shown for various plasma conditions, see Table . RPA results are compared with the BMA that includes collisions according to TCS.Reuse & Permissions
Figure 10
(Color online) Comparisons between the RPA dynamic structures arising from free and core electrons are shown according to Sec. 3E and 78 for the case of a beryllium plasma. At $T_{e} = 1 eV$ , beryllium is assumed in its normal state, $Z_{f}$ consists only of conduction electrons with a $K$ -shell ionization potential $E_{B} = 111.5 eV$ . At $T_{e} = 40 eV$ , beryllium is assumed doubly ionized $Z_{f}$ includes only free electrons with $E_{B} = 159.5 eV$ . The scattering angle is $θ = 160 °$ . The scattering parameter is (a) $α = 0.46$ , (b) $α = 1.36$ , (c) $α = 0.23$ , and (d) $α = 0.67$ .Reuse & Permissions
Figure 11
(Color online) Static ion-ion and electron-ion structure factors for uncompressed $(n_{e} = 3.0 \times 10^{23} {cm}^{- 3})$ and compressed beryllium $(n_{e} = 7.5 \times 10^{23} {cm}^{- 3})$ calculated in DH $(S_{i i})$ and HNC approximation ( $S_{i i}$ and $S_{e i}$ ) for $Z_{f} = 2$ and $T_{e} = T_{i} = 12 eV$ . Curves: Total elastic contribution to the dynamic structure factor $S_{i} = {[Z_{b} + \sqrt{Z_{f}} S_{e i} (k) ∕ S_{i i} (k)]}^{2} S_{i i} (k)$ [see Eq. (35)].Reuse & Permissions
Figure 12
(Color online) The calculated static structure factors $S_{c d} (k)$ and screening charges $q (k)$ for a beryllium plasma with $Z_{A} = 4$ at $n_{e} = 2.5 \times 10^{23} {cm}^{- 3}$ , $Z_{f} = 2$ , and $T_{i} ∕ T_{e} = 1$ (solid line), $T_{i} ∕ T_{e} = 0.5$ (dashed line), $T_{i} ∕ T_{e} = 0.1$ (dash-dotted line). The transfer momentum $k$ is normalized with respect to the inverse electron Debye screening length $k_{De} = 1 ∕ λ_{D}$ [see Eq. (8)]. From 76.Reuse & Permissions
Figure 13
Theoretical x-ray Thomson scattering spectra calculated with the dynamic structure factor $S (k, ω)$ for beryllium targets with $Z_{A} = 4$ at various temperatures, densities, and ionization stages. The probe radiation is $E = 4.75 keV$ and the scattering angle is $θ = 160 °$ . From 78.Reuse & Permissions
Figure 14
(Color online) Schematic of x-ray scattering target after 205 and 127. A pinhole is used for collimating the x-ray probe radiation. Shields and a $250 μ m$ aperture limit the field of view of the x-ray spectrometer to the compressed material.Reuse & Permissions
Figure 15
(Color online) X-ray bremsstrahlung emission as function of radiation energy from dense capsule implosion conditions. The time-integrated emission is compared with the emission during a $100 ps$ time interval at maximum implosion velocity.Reuse & Permissions
Figure 16
(Color) X-ray yield and x-ray conversion efficiency for the titanium $He − α$ x-ray emission at $4.75 keV$ as function of laser intensity from various studies (see text). A typical error bar of 50% in conversion efficiency due to uncertainties of the absolute detector calibration, crystal reflectivity, and shot-to-shot reproducibility is shown.Reuse & Permissions
Figure 17
(Color online) The x-ray conversion efficiency, along with a typical error bar of 50%, for the titanium $K − α$ x-ray emission at $4.5 keV$ as function of the short-pulse laser intensity.Reuse & Permissions
Figure 18
(Color online) X-ray photons available for x-ray scattering experiments from laser-produced plasmas sources, along with a typical error bar of 50%, as function of x-ray energy. For $He − α$ sources data produced with frequency-tripled $(3 ω)$ lasers were chosen while $K − α$ data are taken from experiments without frequency conversion. The dashed line indicates the photon numbers expected from future x-ray free-electron lasers.Reuse & Permissions
Figure 19
Rocking curves for HOPG and LiF crystals at $4.51 keV$ . From 134.Reuse & Permissions
Figure 20
(Color online) Demonstration of mosaic focusing mode using a HOPG Bragg crystal. This configuration provides simultaneously a high reflectivity from the carbon crystal and a sufficient wavelength resolution of $Δ λ ∕ λ = 0.003$ for spectrally resolved x-ray Thomson scattering measurements. From 69.Reuse & Permissions
Figure 21
(Color) Quantum efficiency of CCD (back illuminated) and MCP detectors (16) as function of x-ray energy. Measured detection quantum efficiency data of CCD and IP detectors from 183 and 98 shown together with the IP calculated absorption scaled to the measured data at 3 and $4.5 keV$ .Reuse & Permissions
Figure 22
(Color) Schematic of scattering experiments employing the titanium $He − α$ spectral line in backscattering at $θ = 125 °$ or the chlorine $Ly − α$ line in forward scattering at $θ = 40 °$ . The inset shows a $Ly − α$ source spectrum with an intensity contrast of dielectronic satellites to $Ly − α$ of $⩽ 5 %$ for up to $7 kJ$ of laser energy to produce the narrowband x-ray probe radiation. The k-vector diagrams show the averaged scattering vectors in both geometries yielding noncollective or collective scattering. Up to $15 kJ$ of laser energy has been used for producing the broadband $L$ -shell heating radiation for isochorically heating the target. A density contour plot from radiation-hydrodynamic modeling using the code LASNEX has been overlayed on the Be target, indicating that only homogeneously and isochorically heated beryllium is in the field of view of the spectrometer at the time of the measurements. Note that the color map of the density contour is displayed. From 68 and 70.Reuse & Permissions
Figure 23
(Color) Scattering spectrum (blue dots) from isochorically heated beryllium. Elastic and inelastic (Compton) scattering components are observed from both the titanium $He − α$ and $Ly − α$ probe x rays (left). The spectrum is well fitted with theoretical scattering spectra employing the dynamic structure factor of Sec. 3 (red line). The fit to the red wing of the Compton scattering spectrum is sensitive to the temperature (right). These data provide temperatures of $T_{e} = 53 eV$ and densities of $n_{e} = 3.3 \times 10^{23} {cm}^{- 3}$ characterizing the solid density plasma regime with an error bar of about 10%. From 68.Reuse & Permissions
Figure 24
(Color) Scattering spectrum (gray line) from isochorically heated beryllium. Plasmon scattering downshifted and upshifted in energy is observed, where the intensity of the upshifted plasmon is reduced according to detailed balance, cf. Eq. (18). The spectrum is best fit with theoretical scattering spectra of Sec. 3 for $T_{e} = 12 eV$ and $n_{e} = 3 \times 10^{23} {cm}^{- 3}$ (left). Also shown is the comparison of the plasmon spectrum with calculations for two different densities, $n_{e} = 4.5 \times 10^{23}$ and $1.5 \times 10^{23} {cm}^{- 3}$ , indicating that the spectrum is best fit for $n_{e} = 3 \times 10^{23} {cm}^{- 3}$ . The dashed curve represents a calculation for a density of $n_{e} = 3 \times 10^{23} {cm}^{- 3}$ that is neglecting collisions (right). From 70.Reuse & Permissions
Figure 25
(Color) The measured densities from noncollective and collective x-ray Thomson scattering experiments shown as functions of the measured electron temperature together with calculations from the codes ACTEX, LASNEX, COMPTRA, and SCAALP. The error bars in electron density are larger for the collective scattering data because of noise.Reuse & Permissions
Figure 26
(Color online) Elastic scattering amplitude data are shown that have been measured as function of the scattering angle from isochorically heated beryllium for a density of $n_{e} = 3 \times 10^{23} {cm}^{- 3}$ and temperature of $T_{e} = 12 eV$ . Also shown are calculations using HNC with quantum potential models of 189 and the analytical RPA model of 6.Reuse & Permissions
Figure 27
(Color online) The measured ionization state of carbon is shown as function of the measured electron temperature (73). These experiments employ noncollective x-ray Thomson scattering experiments on various target platforms. The experimental data points correspond to laser driven gas bag experiments, soft x-ray driven carbon foams, hard x-ray driven carbon foams, and cold plastic foils. Also shown are calculations from kinetic calculations using the code FLYCHK for densities of $10^{21} {cm}^{- 3}$ (dotted line), $10^{22} {cm}^{- 3}$ (dashed line), and $10^{23} {cm}^{- 3}$ (solid line) and COMPTRA for carbon at $0.2 g {cm}^{- 3}$ (solid dark line).Reuse & Permissions
Figure 28
(Color) Experimental and theoretical scattering spectra are shown from shock-compressed Be at 25° scattering angle (127). The best fit to the data yields $n_{e} = 7.5 \times 10^{23} {cm}^{- 3}$ , $Z_{f} = 2$ , and $T_{e} = 13 eV$ (left). The plasmon energy is a sensitive measure of the electron density (right).Reuse & Permissions
Figure 29
(Color) Experimental and theoretical scattering spectra are shown from shock-compressed Be at 90° scattering angle (127). The best fit to the data yields $n_{e} = 7.5 \times 10^{23} {cm}^{- 3}$ , $Z_{f} = 2$ , and $T_{i} = 13 eV$ . The width of the Compton energy is a sensitive measure of the electron density (left) and the intensity ratio of elastic to inelastic scattering is sensitive to the ion temperature (right).Reuse & Permissions
Figure 30
(Color) Comparison between modeled and measured densities (in units of $10^{23} {cm}^{- 3}$ ) and temperatures (in units of eV) from x-ray Thomson scattering measurements shown as functions of pressure. Black square and black circle symbols represent densities and electron temperatures from plasmon scattering. Red squares and red circles represent the densities and ion temperatures inferred from Compton scattering. Also shown as dashed lines are the results from radiation-hydrodynamic modeling.Reuse & Permissions
Figure 31
(Color) Elastic scattering amplitude data shown as function of the scattering angle from shock-compressed Li by 61. Also shown are calculations using various HNC models together with DFT Monte Carlo simulations.Reuse & Permissions
Figure 32
(Color) Experimental and theoretical scattering spectra from shock-compressed LiH shown at various times. At $t = 7 ns$ , the scattering parameter is $α ≃ 1$ and the inelastic scattering component shows a plasmon downshifted by $25 eV$ from the $K − α$ energy. At $t = 4 ns$ , the lack of inelastic scattering indicates small temperatures and negligible ionization. Also shown is the $K − α$ x-ray probe spectrum. From 112.Reuse & Permissions
Figure 33
(Color) The temperature of shocked LiH shown as function of time from x-ray Thomson scattering measurements and from radiation-hydrodynamic modeling using different EOS models (112). The range of temperatures for each model is accounting for LiOH surface impurities (lower bounds) to no impurities (upper bounds). The experiments and calculations demonstrate efficient heating to $2.2 eV$ by shock coalescence with small differences in shock timing resolved by the short $K − α$ x-ray pulse.Reuse & Permissions

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