X-Ray Thomson Scattering in High Energy Density Plasmas

REVIEWS OF MODERN PHYSICS, VOLUME 81, OCTOBER–DECEMBER 2009

X-ray Thomson scattering in high energy density plasmas

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 ͑Published 1 December 2009͒

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.

DOI: 10.1103/RevModPhys.81.1625 PACS number͑s͒: 52.25.Os, 52.35.Fp, 52.50.Jm

CONTENTS D. Detectors 1650 E. Photometrics 1650 V. X-ray Thomson Scattering Experiments 1651 I. Introduction 1625 A. Isochorically heated matter 1651 A. Overview 1625 B. Matter heated by supersonic heat waves 1654 B. Historical development 1627 C. Shock-compressed matter 1655 C. Physical properties of dense plasmas 1628 D. Coalescing shocks 1657 D. X-ray scattering regimes 1629 VI. Summary and Outlook 1658 E. Sensitivity of scattering spectra to plasma List of Acronyms and Deﬁnitions 1659 parameters 1629 Acknowledgments 1659 II. Facilities for X-Ray Thomson Scattering Experiments 1632 References 1660 III. Theory of the Dynamic Structure Factor 1634 A. General expressions 1634 B. Application to Thomson scattering 1635 I. INTRODUCTION C. Elastic scattering feature 1637 A. Overview D. Free-electron feature 1637 1. Two-component screening 1637 Accurate measurements of the plasma conditions in- 2. Ionic correlations and electronic screening 1638 cluding temperature, density, and ionization state in E. Bound-free transitions 1638 dense plasmas are important for understanding and F. Numerical results 1639 modeling high energy density physics experiments. For 1. Collision frequency 1639 applications in this regime, new scattering techniques 2. Electron feature of the dynamic structure have been developed by Glenzer, Gregori, Lee, et al. factor 1639 ͑2003͒ and Glenzer, Landen, et al. ͑2007͒ that make use 3. Bound-free spectrum 1641 of x rays to penetrate dense or compressed matter. This 4. Ion feature 1641 capability is particularly important for laboratory astro- G. Theoretical spectra 1642 physics and inertial conﬁnement fusion ͑ICF͒ experi- IV. Experimental Techniques 1643 ments ͑Lindl et al., 2004; Remington et al., 2006͒, and is A. Design of scattering experiments 1643 broadly applicable in the dense plasma community. Ex- B. X-ray probe requirements 1645 amples of important applications include the measure- C. Spectrometers 1648 ment of compression and heating of shock-compressed

0034-6861/2009/81͑4͒/1625͑39͒ 1625 ©2009 The American Physical Society 1626 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … matter and the fundamental characterization of thermo- tional to the number of tightly bound electrons as well dynamic properties including phase transitions ͑Beule et as the static structure factor. The latter takes into ac- al., 1999, 2001; Fortov et al., 2007͒ and new states of count the dependence on the ion-ion correlations. For matter ͑Bonev, Schwegler, et al., 2004͒. Specifically, x-ray the noncollective scattering conditions where the static scattering is applicable as a tool to resolve fundamental structure factor approaches 1, the intensity of the elastic physics questions such as the equation of state of dense scattering feature provides an accurate measure of the matter ͑Collins et al., 1988; Young and Corey, 1995͒, ionization state. Absolute calibration is provided by structure factors in two-component plasmas ͑Hansen comparing with the inelastic scattering component and McDonald, 2006͒, limits of the validity of the ran- whose intensity is well known from the sum rules ͑Kohn dom phase approximation ͑RPA͒͑Pines and Bohm, and Sham, 1965͒. For isochorically heated matter where 1952; Bohm and Pines, 1953͒, and the role of collisions the ion density is known a priori, the ionization state ͑Reinholz et al., 2000; Redmer et al., 2005͒. inferred from the ratio of the inelastic Compton to the The spectrally resolving x-ray scattering technique is elastic Rayleigh scattering components yields the elec- now widely applied in many large ͑Gregori et al., 2004; tron density. Gregori, Glenzer, Chung, et al., 2006; Sawada et al., The collective properties of the dense plasmas can be 2007͒ and medium-size laboratories ͑Ravasio et al., 2007; accessed by the proper choice of the scattering angle and Kritcher, Neumayer, Castor, et al., 2008; Garcia Saiz, x-ray probe energy. In the forward scattering regime, Gregori, Gericke, et al., 2008͒ to study dense plasma collective plasmon ͑Langmuir͒ oscillations ͑Tonks and properties. Accurate and precise data are obtained simi- Langmuir, 1929͒ have been observed by Glenzer, lar to data from low density plasmas when optical Th- Landen, et al. ͑2007͒. The plasmon frequency shift from omson scattering was first introduced to study plasmas the incident x-ray probe energy directly provides the lo- in the 1960s. Powerful x-ray sources ͑Landen, Farley, et cal electron density via the Bohm-Gross ͑Bohm and al., 2001; Lee et al., 2003͒ take the place of optical lasers Gross, 1949͒ dispersion relation. Besides well-known to penetrate through dense or compressed matter and to thermal corrections, the dispersion relation in dense access the dense plasma physics regime with electron plasmas includes additional terms for degeneracy and densities of solid and above. These x-ray sources must quantum diffraction. Experiments in isochorically fulfill the stringent requirements ͑Urry et al., 2006͒ on heated solid-density plasmas show that the density from photon numbers and bandwidth for spectrally resolved downshifted plasmons agrees with the values inferred x-ray Thomson scattering measurements in single-shot from noncollective scattering. In addition, forward scat- experiments ͑Landen, Glenzer, et al., 2001͒. tering on plasmons is not dependent on knowledge of X-ray Thomson scattering was first demonstrated em- the ion density and is thus directly applicable to charac- ploying laser-produced He-␣ x-ray sources to perform terize compressed matter and to determine the optical scattering measurements in isochorically heated solid- properties. density matter ͑Glenzer, Gregori, Lee, et al., 2003; Glen- Plasmons are affected by Landau damping as well as zer, Gregori, Rogers, 2003͒. Measurements in backscat- electron-ion collisions. Thus, a consistent description of ter geometry have accessed the Compton scattering the plasmon spectrum describing both the frequency regime where the scattering process is noncollective and shift and the spectral shape requires a theory beyond the the spectrum shows the Compton downshifted line that RPA. Experiments have been successfully described is broadened by the thermal motion of the electrons. with the Mermin approach ͑Mermin, 1970; Höll et al., Thus, the inelastic scattering spectrum reflects the elec- 2004; and Thiele et al., 2006͒ that takes into account the tron velocity distribution function, and for a Maxwell- dynamic collision frequency for calculating damping. In Boltzmann distribution function, yields the electron cases where collisional damping is not the dominant temperature with high accuracy. For Fermi-degenerate mechanism, the Born approximation has been shown to states of matter, on the other hand, the Compton scat- be sufficient. In future studies of cold plasmas with small tering spectrum provides the Fermi energy and thus the Landau damping, plasmon measurements will provide electron density. Experiments observing these distribu- an experimental test on the theory of collisions yielding tions have been interpreted employing first-principles conductivity ͑Röpke, 1998; Redmer et al., 2003; Kuhl- methods ͑Pines and Bohm, 1952; Gregori, Glenzer, Roz- brodt et al., 2005; Höll et al., 2007͒, which is an important mus, et al., 2003; Redmer et al., 2005͒ that includes de- dense plasma property for radiation-hydrodynamic generacy effects and provide the plasma conditions with modeling of high energy density physics experiments. high accuracy. Besides observing plasmons, collective scattering pro- In addition to the Compton scattering feature from vides information on the structure of dense plasmas. In inelastic scattering by free and weakly bound electrons, this regime, the elastic scattering intensity includes addi- the noncollective scattering spectrum shows the un- tional dependency on the density fluctuations associated shifted Rayleigh scattering component from elastic scat- with electrons in the screening cloud around the ion. tering by tightly bound electrons. The latter occupy The latter is sensitive to both the ion-ion and electron- quantum states with ionization energy larger than the ion structure factors, and no simple approximations are Compton energy deep in the Fermi sea that cannot be applicable. In addition, the interaction is dependent on excited, e.g., due to the Pauli exclusion principle. Thus, electron and ion temperatures. Hypernetted chain the intensity of the elastic scattering feature is propor- ͑HNC͒ equations with quantum potentials or density-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1627 functional theory ͑DFT͒ have been successfully applied the same year by Chandrasekhar. At that time, it was to describe experiments ͑Schwarz et al., 2007; Wünsch et thought impossible to observe the broadening of the al., 2008͒, but only few well-characterized conditions ex- Compton scattering feature for the ranges of tempera- ist where the plasma conditions have been determined tures available in the laboratory ͑cf. Chandrasekhar, by noncollective scattering. Ultimately, the study of 1929͒. Scattering experiments on cold solids, however, structure factors in the collective scattering regime will have continued throughout the last century with im- provide the means to determine the thermodynamic proved x-ray sources including synchrotrons. For ex- properties and the atomic structure in dense plasmas. ample, these experiments have investigated the wave The outline of this review is as follows. After a histori- functions of bound electrons, x-ray Raman scattering, cal review, Sec. I provides an introduction to the prop- and collective long-range structure effects ͑Schülke, erties of dense plasmas and to the scattering regimes and 1991͒. plasma parameters accessible by spectrally resolved In the late 1940s, low density laboratory plasma phys- x-ray Thomson scattering experiments. Section II de- ics experiments began to study hot matter. Magnetically scribes the facilities with the x-ray and plasma driver confined plasmas with the goal to perform thermo- capabilities to produce and probe dense plasmas in the nuclear fusion experiments have accessed matter at un- high energy density physics regime. A comprehensive precedented high temperatures. However, accurate tem- theory of the dynamic structure factor of the x-ray Th- perature measurements using scattering required the omson scattering spectrum is given in Sec. III. The ex- development of powerful light sources due to the small perimental techniques and requirements to perform cross section for Thomson scattering, x-ray Thomson scattering measurements are discussed 8␲ ␴ = r2 = 0.665 ϫ 10−24 cm2, ͑1͒ in Sec. IV. Section V describes the proof-of-principle ex- Th 3 0 periments and highlights present state of the art experimental work. A summary is given in Sec. VI that in- Ϸ ϫ −15 where r0 2.8 10 m is the classical electron radius. cludes an outlook to future experiments and physics For the photon energies and applications of interest in issues where the x-ray Thomson scattering measure- this review, the cross section is not dependent on energy. ments will be required to enhance our knowledge of For high energy gamma-ray photons, very strong mag- high-energy density plasmas. A list of acronyms and netic fields, or extremely intense radiation, the cross sec- symbols used in this review is also provided. tion is modified ͑Bethe and Guth, 1994͒. After the introduction of the ruby laser by Maiman ͑1960͒ and the first B. Historical development scattering experiments in hot plasmas by Funfer et al. ͑1963͒ and Kunze et al. ͑1964͒, Thomson scattering has Scattering experiments have played a central role in become a powerful plasma diagnostic technique that is all areas of modern physics to study the microscopic widely employed to measure noncollective and collec- structure and the state of matter. The discovery of x rays tive features of laboratory plasmas. A large body of the- by Wilhelm Röntgen in 1895 and their first use in spec- oretical ͑Dougherty and Farley, 1960; Fejer, 1960; trally resolved scattering experiments are closely con- Renau, 1960; Salpeter, 1960͒ and experimental studies nected to the emergence of quantum mechanics and sta- ͑Kunze, 1968; Evans and Katzenstein, 1969; Sheffield, tistical physics in the beginning of the last century. The 1975͒ was performed, principally oriented to low density scattering of optical light from free electrons was first plasmas ranging from magnetically confined fusion re- observed in 1906 and is referred to Thomson scattering search plasmas ͑Peacock et al., 1969͒ to ICF hohlraums ͑Thomson, 1906͒. For geometries, where no component ͑Glenzer, Alley, et al., 1999; Glenzer, Rozmus, et al., of the electron velocity is probed, the frequency of the 1999͒. These studies are limited to electron densities of scattered optical radiation is unchanged and observa- 1021 cm−3 and below where ultraviolet probe lasers tions have been explained by the classical theory of elec- propagate through the plasma. tromagnetic waves scattered by charged particles. It was With the advent of the field of high energy density Arthur Holly Compton ͑1923͒ who applied x rays for physics ͑Drake, 2006͒ in this millennium, the need for scattering experiments on solid-density materials ob- accurate measurements of the conditions, physical prop- serving the redshifted Compton line referred to as the erties, and structure of dense plasmas has emerged. Compton effect or Compton scattering. The frequency Typically conditions with pressures above 1 Mbar shift has been explained by the momentum transfer of =102 GPa=105 Jcm−3 are of interest with densities ap- light quanta to the electrons of the material and has proaching solid density or higher and with temperatures provided direct evidence for the quantum nature of above several eV. Powerful x-ray sources enable these light. measurements with spectrally resolved x-ray Thomson In 1928, spectrally resolved x-ray scattering measure- scattering. They require a large photon flux in a short ments of the Compton redshifted line provided a direct x-ray burst with high spectral purity to resolve the measurement of a parabolic distribution function re- Compton and plasmon features in short-lived states of flecting the Fermi energy distribution of electrons in sol- matter. Experiments have employed He-␣ and Ly-␣ ids ͑Dumond, 1929͒. These measurements were per- sources to demonstrate Compton and plasmon measure- formed with week-long exposures of x-ray film and have ments in isochorically heated beryllium. These proof-of- verified the statistical model for electrons developed in principle experimental studies have been extended to

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1628 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

are comparable to conditions of the interior of stars, brown dwarfs, and Jovian planets. In addition to accessing dense plasmas, x-ray Thom- son scattering is also important for the characterization of warm dense states of matter at lower densities. It has been a long-standing goal to study the microscopic properties in the regime where the transition from an ideal plasma to a degenerate strongly coupled plasma occurs. In this regime, conventional diagnostic techniques and standard plasma theory, which treat the interactions between particles as a small correction, are often not applicable. In particular, transient plasma behavior at these conditions is observed in dynamical experiments, like laser and particle-solid interactions or Z-pinch experi- FIG. 1. Density-temperature phase space indicating areas of ments, and are of particular importance for ICF experi- Fermi-degenerate matter ͑dark gray area͒, strongly coupled ments. This regime is defined by the degeneracy and plasmas ͑light gray area͒, and the ideal plasma regime ͑white coupling parameters. area͒. The solid lines indicate the scattering parameter ␣ from The degeneracy parameter ⍜ defines the Fermi- ͑ ͒ ␪ Eq. 5 for backscattering at =180° and ultraviolet and x-ray degenerate regime by estimating the role of quantum probes. Also shown is the ionization balance for solid density ͑ ͒ statistical effects in the system. The parameter is defined beryllium dashed curve . From Landen, Glenzer, et al., 2001. by the ratio of the thermal energy and the Fermi energy ⑀ F, shock-compressed matter and the characterization of k T ប2 shocks with picosecond temporal resolution employing ⍜ B e ⑀ ͑ ␲2 ͒2/3 ͑ ͒ = ⑀ , F = 3 ne , 2 K-␣ x rays. In parallel with the experimental demonstra- F 2me tion, the theory of x-ray scattering has been developed with kB the Boltzmann constant, Te the electron tem- including an appropriate description of the inelastic scat- perature, n the electron density, h=2␲ប the Planck’s tering component beyond the random phase approxima- e constant, and me the mass of the electron. In a degener- tion with the Mermin approach and close coupling cor- ate plasma, the Fermi energy is larger than the thermal rections. Also, density-functional theory has been energy, i.e., ⍜Ͻ1, and most electrons populate states applied for calculating the elastic scattering component. inside the Fermi sea where quantum effects are of im- The experimental and theoretical investigations in this portance. field are reviewed in the following sections. They include The coupling parameter ⌫ is defined as the ratio of the observations of the scattering spectrum in both forward Coulomb energy between two charged particles at a and backscatter directions covering a large range of en- ¯ ergy shifts. These conditions result in scattering spectra mean particle distance d and the thermal energy ͑Ichi- where the Compton effect dominates the frequency of maru, 1973͒. Using the Wigner-Seitz radius for d¯ ,wefind the inelastic scattering component to plasmons scatter- an expression for the electron coupling parameter, ing where the Compton effect is small compared to the 2 4␲n −1/3 plasma frequency. Hence the term x-ray Thomson scat- ⌫ e ¯ ͩ e ͪ ͑ ͒ ee = , d = . 3 tering is used throughout this review to describe the ␲␧ ¯ 3 4 0dkBTe wide range of plasma conditions and scattering features ␧ accessed in the high energy density physics regime. Here e is the electric charge and 0 is the permittivity of free space. For ⌫Ͻ1 the plasma is weakly coupled, and ⌫Ӷ1 denotes the ideal plasma regime. Correlations be- C. Physical properties of dense plasmas come more important in the coupled plasma regime, where ⌫տ1. Figure 1 shows the density-temperature phase space In Fig. 1 the regions of strongly coupled plasma and ͑⌫ Ͼ Ͻ⑀ ͒ indicating the range of parameters accessed in high en- Fermi-degenerate regimes ee 1 and Te F are sepa- ⌫ ⍜ ergy density physics experiments. X-ray Thomson scat- rated by the ee=1 and =1 curves in electron density– tering has been developed to probe conditions ap- electron temperature phase space. For a given density at proaching solid densities and higher and with the lowest temperatures, the plasmas are either Fermi temperatures of several eV. These systems span the full degenerate or only partially ionized, and hence in a range of plasmas from Fermi degenerate to strongly sense only weakly coupled. At higher temperatures, they coupled and to high temperature ideal gas plasmas. Scat- behave as ideal gases with insignificant inter-particle tering experiments have been demonstrated for plasmas coupling. In between, the ideal gas approximation for 24 −3 with electron densities of up to ne =10 cm , but ex- plasma behavior breaks down. periments at even higher densities will become possible In the strongly coupled plasma regime, various statis- in future ICF capsule implosion experiments. Electron tical mechanics models predict electron-ion collisionali- 26 −3 densities approaching ne =10 cm are expected, which ties that differ by factors of 2–5. Material properties

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k /2 noncollective scattering regime, the density ﬂuctuations of individual electrons are resolved. k S k /2 ͑ ͒ ␬ /2/2 In Eq. 6 , the inverse electron screening length e is v written in terms of Fermi integrals k0 Detector + + ϱ ␯ + 1 dxx F␯͑y͒ = ͵ , ͑7͒ S ⌫͑␯ ͒ ͑ ͒ +1 0 exp x − y +1 X Rays 1/1/k where ⌫͑␯͒ is the gamma function ͑Zimmerman, 1987͒. ͑ ͒ ͑ ͒ Equation 6 is valid for any degeneracy via the param- FIG. 2. Color online Scattering geometry indicating the ␩ ␮ ␮ length of the scattering vector k and the relation to the screen- eter e = e /kBTe with e the chemical potential. The ␭ usual Debye screening length in the nondegenerate case ing length S of the plasma. ͑⍜ӷ1͒ gives n e2 1/2 such as electrical and thermal conductivity, opacity, and ␬ ͑␩ ͒ ␭−1 ͩ e ͪ ͑ ͒ lim e e = D = , 8 ␩ → ϱ ␧ equation of state have been studied in this regime to e − 0kBTe attempt to resolve theoretical and calculational uncer- and the Thomas-Fermi screening length for degenerate tainties. However, the usefulness of such measurements systems ͑⍜ഛ1͒, has been impaired because of the lack of an independent measurement of temperature and density. m e2 3n 1/3 ␬ ͑␩ ͒ ␭−1 ͱ e ͩ e ͪ ͑ ͒ lim e e = TF = . 9 ␩ ӷ ␲␧ ប2 ␲ e 1 0

D. X-ray scattering regimes Analytically the screening length with appropriate limits can also be calculated using the effective tempera- ͑ 2 2͒1/2 ͑ ͒ ⑀ ͑ Figure 2 shows the geometry and k vectors for a back- ture Tcf= Te +Tq in Eq. 8 . Here Tq = F / 1.3251 scattering experiment. The plasma is irradiated either by ͱ ͒ ¯ ͑ −0.1779 rs , rs =d/aB, and aB is the Bohr radius Gre- nonpolarized x rays from a laser-plasma source or by a gori, Glenzer, Rozmus, et al., 2003͒. This corrected tem- linearly polarized free electron x-ray laser beam along perature is chosen such that the temperature of an elec- the direction of the incident wave vector k0 with k0 tron liquid obeying classical statistics gives exactly the ␲ ␭ =2 / 0. The detector observes the scattered radiation at same correlation energy of a degenerate quantum fluid the scattering angle ␪ in the direction of the scattered at Te =0 obtained from quantum Monte Carlo calcula- wave vector kS at a distance much larger than the tions ͑Dharma-wardana and Perrot, 2000͒. This ap- plasma extension. proach was shown to reproduce finite-temperature static During the scattering process, the incident photon response of an electron fluid, valid for arbitrary degen- transfers on average momentum បk and the Compton eracy. ប␻ ប2 2 ប␻ ប␻ ͑ ͒ ͑ ͒ ␭ energy = k /2me = 0 − 1 to the electron, where As seen from Eqs. 8 and 9 , S is determined by the ␻ Ã 1 is the frequency of the scattered radiation. In the plasma conditions, density, and temperature, while ␭ ͑ប␻Ӷប␻ ͒ nonrelativistic limit 0 and for small momentum =2␲/k is primarily determined by the x-ray probe en- ͑␪ ͒ transfers we have k=2k0 sin /2 ; cf. Fig. 2. Thus, the ergy and the scattering angle ͓i.e., by the setup of the scattering geometry and the probe energy determine the scattering experiment, Eq. ͑4͔͒. The latter statement is scattering vector k through only approximately fulfilled. When appropriate, well- known corrections to the length of the scattering vector E k = ͉k͉ =4␲ 0 sin͑␪/2͒. ͑4͒ for the finite frequency shift of the scattered radiation hc and due to the dispersion of the radiation in the dense Equation ͑4͒ determines the scale length ␭Ã Ϸ2␲/k of plasma need to be included. In particular the latter is the electron density fluctuations measured in the scatter- important in case of soft x-ray or vacuum-ultraviolet ͑ ing experiment. Comparison with the screening length probe radiation employed by free-electron lasers Höll ͒ ␭ determines the scattering regime. The dimensionless et al., 2007 . Finally, for extremely dense states of matter S encountered in ICF research, high x-ray probe energies scattering parameter ␣, of Eജ10 keV are required to penetrate through the 1 26 −3 ␣ ͑ ͒ dense plasma of up to 10 cm . This case results in = ␭ , 5 large k vectors limiting scattering experiments at the k S highest densities to the noncollective regime. ␭−2 with S , n e2 F ͑␩ ͒ E. Sensitivity of scattering spectra to plasma parameters ␭−2 = ␬2 = e −1/2 e , ͑6͒ S e ␧ k T F ͑␩ ͒ 0 B 1/2 e With the scattering vector k defined by Eq. ͑4͒, the ␣Ͼ is used. Here 1 defines the collective scattering re- scattered power PS from N electrons into a frequency gime. Density fluctuations at a scale larger than the interval d␻ and solid angle d⍀, i.e., the scattering spec- screening length are observed, while at ␣Ͻ1, i.e., in the trum ͑Sheffield, 1975͒, is determined by

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1630 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

v P r2d⍀ z P ͑R,␻͒d⍀d␻ = 0 0 NS͑k,␻͒d␻ S 2␲A ϫ͉ ˆ ϫ ͑ ˆ ϫ ˆ ͉͒2 ͑ ͒ kS kS E0 . 10 v ͑ ͒ In Eq. 10 , P0 denotes the incident x-ray power, A is the plasma area irradiated by the probe x rays, and S͑k,␻͒ is the total electron dynamic structure factor. The polariza- v tion term reflects the dependence of the scattered power x vx on the polarization of the incident radiation; for linearly polarized light, e.g., from a free-electron laser ͑ ͒ ͉ ˆ ϫ͑ ˆ ϫ ˆ ͉͒2 ͑ 2 ␪ 2 ␸͒ FEL , kS kS E0 = 1−sin cos . The electric v dvx field direction of the incident radiation is given by E0 y with the hat denoting unit vectors, and ␸ determining the angle between the plane of the scattering vectors and E0. On the other hand, for unpolarized light, FIG. 3. Volume element in three-dimensional velocity space ͉ ˆ ϫ͑ ˆ ϫ ˆ ͉͒2 that determines the number of electrons contributing to the e.g., from a laser-plasma source, kS kS E0 ͑ 1 2 ␪͒ 1 ͑ 2 ␪͒ velocity component vx. = 1−2 sin = 2 1+cos . Clearly, for linearly polarized laser sources, a detector at ␸=␲/2 will measure the maximum scattered light signal. Generally, this term To calculate the number of electrons that contribute must be included for measurements with absolutely cali- to the Doppler broadening via Eq. ͑11͒, the three- brated detectors. dimensional velocity space is considered in Fig. 3. The ͑ ͒ By multiplying Eq. 10 with the length of the scatter- number of electrons within the velocity interval vx and ing volume ᐉ it can be seen that the scattered power vx +dvx is proportional to the volume described by an depends on the setup of the scattering experiment ͑ini- annulus around vx that must be integrated to include all tial probe power, scattering angle, probe wavelength, possible projections of velocity vectors v onto vx. probe polarization, and detector solid angle͒, the elec- ␤ From Fig. 3 and with the angle between v and the vx tron density, the plasma length, and the total dynamic axis the number of electrons is given by structure factor S͑k,␻͒. The structure factor is defined ϱ as the Fourier transform of the electron-electron density v= dv f͑v ͒dv = ͵ n͑v͒2␲ͱv2 − v2 dv . ͑12͒ fluctuations, containing the details of the correlated x x x ␤ x v sin many-particle system; cf. Ichimaru ͑1994͒. Section III is x devoted to the theory of S͑k,␻͒ including the elastic and The first term in the integral is the occupation number, inelastic scattering components in the collective and described by either Fermi-Dirac or Maxwell-Boltzmann noncollective regimes. In this section, some approxima- statistics, the second term is the circumference, the third tions are discussed, demonstrating the fundamental prin- is the annulus width, and the last term is the thickness. ␪ ciples probed with spectrally resolved x-ray scattering. With vx =v cos ,wefind These relations illustrate the dependence of the mea- ␲ 2 /2 v v sured scattering spectra on the dense plasma conditions. f͑v ͒dv = ͵ nͩ x ͪ x tan ␤d␤. ͑13͒ x x ␤ 2 ␤ The frequency shift of a scattered photon by a free 0 cos cos electron is determined by the Compton and Doppler ef- Thus, x-ray scattering in the noncollective regime will fects, probe the distribution function. For a Fermi degenerate ␻ =−បk2/2m ± k · v. ͑11͒ plasma the width of the downshifted peak is propor- e tional to the Fermi energy. With the Fermi distribution The former is due to the transfer of momentum from the function applied for weakly degenerate plasmas, kBTe Ͻ⑀ ͑ ⑀ ͒1/2 incident photon to the electron during the scattering F, where vF = 2 F /me is the Fermi velocity, process resulting in the Compton downshift of the scat- Landen, Glenzer, et al. ͑2001͒ showed tered radiation to lower x-ray energies. This term de- ␲ v /2 ͑v /v cos ␤͒2 tan ␤d␤ fines the inelastic scattering component in fully noncol- f ͩ x ͪ ϰ ͵ x F , ͑14͒ 0 ͓͑v /v cos ␤͒2−␩͔/͑T /⑀ ͒ lective x-ray Thomson scattering experiments. The vF 0 e x F e F +1 Compton shift contains no diagnostic information about ␩ the dense plasma. The intensity of this feature, however, with obtained according to the Sommerfeld expansion, indicates the number of electrons and states available ␲2 2 Te for inelastic scattering processes. On the other hand, the ␩ =1−ͩ ͪͩ ͪ . ͑15͒ 12 ⑀ spectrum of the Compton-downshifted line reflects the F velocity component of the electron in the direction of The second term accounts for the fact that the chemical the scattering vector k. This fact has been explored in potential in the expression for the occupation of states ͓ ͑ ␮͒ ͔−1 the late 1920s when the spectrum from solids have for fermions, exp E− /kBTe +1 , has some tempera- shown the Fermi velocity distribution function. ture dependence at finite temperature.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1631

23 -3 ne =6x10 cm ) eV ( 30 E

23 -3 ne =3x10 cm

10 FIG. 4. Calculated Thomson backscattered spectra for various 1 2 ⑀ ⑀ k (10-10 m-1) ratios of kBTe / F and for F =15 eV. Solid, long dashed, and ⑀ short dashed correspond to kBTe / F =0.1, 0.2, and 0.4, respectively. The spectral shift corresponding to an electron velocity FIG. 5. ͑Color online͒ The plasmon dispersion with the plas- component equal to the Fermi velocity is denoted by a vertical mon energy shift from the incident x-ray energy E0 vs k. The line. Note that only one side of spectrum is shown, and that calculations are for beryllium plasmas with Te =12 eV, Z=2.5, ϫ 23 −3 ϫ 23 −3 ⌬␯=0 corresponds to the Compton shifted frequency. From E0 =4.75 keV, ne =3 10 cm , and ne =6 10 cm . Solid, Landen, Glenzer, et al., 2001. long dashed, and short dashed correspond to calculations with Born-Mermin theory, the random phase approximation, and with Eq. ͑16͒, respectively. From Thiele et al., 2008. Figure 4 shows the calculated profiles, indicating that the shape and width of the velocity distribution are probed. In a degenerate system, applying Eqs. ͑14͒ and second term represents the effect on propagation of the ͑15͒ will provide the spectrum of the Compton line with oscillation from thermal pressure. The third term in- ͑⑀ ͒1/2 ͑ ͒1/3 the width proportional to F and hence to ne , cludes degeneracy effects from Fermi pressure, and the yielding the density of the system. In a nondegenerate last term is the quantum shift ͑Haas et al., 2000͒. The plasma, on the other hand, the Compton scattering spec- quantum shift and the electron oscillation terms are trum will reflect a Maxwell-Boltzmann distribution, pro- temperature independent, and, for the partially degener- viding a measure of the electron temperature. In addi- ate plasmas of interest here, the thermal pressure is tion to the Compton scattering component, elastic weakly dependent on T . Therefore, for small tempera- scattering at the incident x-ray energy as well as possible e contributions of bound electrons to the inelastic scatter- tures, no accurate knowledge of the temperature is re- ing component will also need to be taken into account. quired, and the plasmon energy provides a sensitive The general structure factor of Sec. III includes these measure of the plasma electron density. effects for the full range of temperatures accessible in Figure 5 shows the energy shift of the plasmon versus experiments and further accounts for finite values of ␣ the length of the scattering vector k for a dense beryl- and effects of collisions. lium plasma. The conditions are chosen such that the With decreasing scattering angle and with moderate scattering is collective with ␣Ͼ1 spanning the range of x-ray probe energies, the scattering length increases and scattering angles between 30° and 90°. These calcula- ␣Ͼ 1 results in collective scattering. In this case, collec- tions show that the random phase approximation agrees tive plasmon oscillations have been observed. The plas- closely with the Born-Mermin theory for a wide range of mon frequency shift from E0 is determined by the plas- conditions; discrepancies observed for kϾ10−10 m−1 and mon dispersion relation and the width is determined by n =3ϫ1023 cm−3 are smaller than 1 eV, which is difficult Landau damping and collisional damping processes. e While these damping processes are treated in Sec. III, to resolve with present experimental capabilities. This we note here that the plasmon frequency shift can be study shows that error bars in the measurement of the approximated for small values of k using an inversion of density from the plasmon dispersion can be expected to Fermi integrals given by Zimmerman ͑1987͒ that results be of order 5% or less when applying these theories. On in a modified Bohm-Gross dispersion relation ͑Höll et the other hand, Eq. ͑16͒ shows deviations from the al., 2007; Thiele et al., 2008͒ Born-Mermin theory of order 2 eV, indicating that the analytical approximation needs to be applied with cau- បk2 2 ␻2 ␻2 2 2 ͑ ⌳3͒ ͩ ͪ ͑ ͒ pl = p +3k vth 1 + 0.088ne e + , 16 tion. 2me The relation that demonstrates the fundamental prin- ␻ ͱ 2 ␧ where p = nee / 0me is the plasma frequency, vth ciples probed with spectrally resolved x-ray scattering is ͱ ⌳ = kBTe /me is the thermal velocity, and e derived from the fluctuation-dissipation theorem. The ͱ ␲ =h/ 2 mekBTe is the thermal wavelength. density fluctuations in the plasma, described by the dy- In Eq. ͑16͒ the first term is a result of electron oscil- namic structure factor, are related to the dissipation of lations in the plasma ͑Tonks and Langmuir, 1929͒, the energy, described by the dielectric function ␧͑k,␻͒,

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1632 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

FIG. 6. ͑Color͒ Schematic view of the National Ignition Facility ͑NIF͒ showing the main components of the laser system ͑Moses and Wuest, 2005͒.

␧ បk2 1 more National Laboratory in the United States and the ͑ ␻͒ 0 ␧−1͑ ␻͒ ͑ ͒ S k, = ប␻ Im k, . 17 Laser Megajoule ͑Cavailler, 2005͒ in France, two ␲e2n 1−e /kBTe e premier laser facilities for accessing the high energy Independent of evaluating the dielectric function, the density physics regime are becoming available. The NIF shape of the structure factor, and modeling collisional will include both the capability for high energy and effects, the dielectric function fulfills the requirement ultrashort-pulse laser-produced x-ray probings of un- precedented x-ray fluence and energy. A total of 192 S͑k,␻͒ ប␻ − /kBTe ͑ ͒ beamlets arranged in 48 quads of beams will deliver a ͑ ␻͒ = e . 18 S − k,− total energy of 1.8 MJ in 500 TW for up to 30 ns long This relation is referred to as the detailed balance rela- laser pulses, cf. Fig. 6. A quad of four beams, entering tion. As a general consequence, the structure factor the target chamber through one port, will exceed the shows an asymmetry with respect to k and ␻ that has laser energy used in present experiments at the Omega been observed in many experiments. In principle, this laser facility ͑Soures et al., 1993; Boehly et al., 1997͒, relation could also be used to infer the temperature providing sufficient energy for x-ray probing with scat- since, when applied to plasmons, it provides the electron tering or radiography. A quad of four beamlets operat- temperature independent of a detailed formulation of ing at 351 nm can provide up to 40 kJ at 351 nm, or dissipative processes in the plasma. However, conditions 84 kJ at 526 nm in several nanosecond long pulses. The with sufficiently high temperatures and small noise am- laser intensity on target is tailored using continuous plitudes are required to reliably measure the upshifted phase plates ͑Dixit et al., 1996͒, spanning the range of plasmon. This measurement has recently been demon- intensities of 1014–1016 Wcm−2 per quad of beams strated by Döppner et al. ͑2009͒. where optimum conversion efficiency for thermal laser- produced He-␣ and Ly-␣ x-ray sources have been dem- II. FACILITIES FOR X-RAY THOMSON SCATTERING onstrated; cf. Sec. IV.B. EXPERIMENTS Experiments on NIF with the first quad have begun in 2003 and results on laser-plasma interaction instabilities, With the completion of the National Ignition Facility hydrodynamics instabilities and radiation hydrodynam- ͑NIF͒͑Moses and Wuest, 2005͒ at the Lawrence Liver- ics have been reported by Glenzer et al. ͑2004͒, Blue et

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1633 al. ͑2005͒, Dewald et al. ͑2005͒, Landen et al. ͑2006͒, and plasma conditions are produced with energetic laser Glenzer, Froula, et al. ͑2007͒. The goal of NIF experi- beams. This is accomplished by individually pointing the ments is to show the feasibility of inertial confinement beams to various targets and by timing and pulse shap- fusion, perform high energy density physics experi- ing the laser pulses to effectively produce the x-ray ments, and study astrophysics processes in the labora- probe and the dense plasma. An alternative approach is tory. These studies share the need to prepare materials to employ pulsed power to create the latter. For ex- at extreme conditions accessing the high energy density ample, the Z machine at Sandia National Laboratory physics regime for which accurate probing with x-ray can generate a current of 20 MA and has been refur- scattering will be an important diagnostic for measuring bished to increase the current to 30 MA ͑Matzen et al., the conditions and properties. 2005͒. The discharge through a wire array allows pulse In addition to the long-pulse laser beams, the NIF will shaping control and efficient multi-keV x-ray produc- also provide an ultrashort-pulse laser capability ͑ARC͒ tion. Plasmas have been produced with x-ray energies ͑Barty et al., 2004͒. In one NIF beamlet aperture, it is approaching 2 MJ at powers of Ͼ200 TW for inertial planned to deliver a total energy between 1 and 3 kJ in confinement fusion and high energy density physics ϳ1 to 50 ps long pulses at the fundamental laser wave- ͑HEDP͒ experiments. In addition, shock strength above length of 1053 nm. Up to eight short-pulse beams in four 1 Mbar and plasma jets have been produced. NIF apertures with 10 kJ in 10 ps will provide a capabil- These conditions can be probed with a large-aperture ity greatly surpassing present ultrashort-pulse laser pa- ͑30-cm͒ kilojoule-class Nd:glass laser system, the Z bam- rameters where laser-produced K-␣ x rays are applied let ͑Rambo et al., 2005͒. The laser, operating with typical for x-ray probing ͑Kritcher, Neumayer, Castor, et al., pulse durations from 0.3 to 1.5 ns, employs a sequence 2008͒. The intensity on target per short-pulse beam will of successively larger multipass amplifiers to achieve up reach ϳ1018 Wcm−2 for 10 ps long pulses that can be to 3 kJ energy at a wavelength of 1054 nm. Large- independently timed with up to 20 ns delay with respect aperture frequency conversion and long-distance beam to the long-pulse beams. As discussed in Sec. IV.B,at transport can provide on-target energies of up to 1.5 kJ these intensities K-␣ probe radiation is efficiently pro- at 527 nm. Although less energy than employed at duced. Omega, this capability significantly exceeds the energy The Omega laser facility at the Laboratory for Laser for x-ray probing that is available at medium-sized laser Energetics ͑Soures et al., 1993; Boehly et al., 1997͒ has facilities. been the first facility where x-ray Thomson scattering On several medium-sized lasers, namely the LULI measurements have been successfully performed. This is 2000 laser at the Ecole Polytechnique, the Vulcan laser a 60-beam facility delivering routinely 500 J energy per at the Rutherford Appleton Laboratory, and the Titan beam on target at 351 nm wavelength and in a 1 ns long laser at the Lawrence Livermore National Laboratory, flat-top pulse. The single beam laser pulse duration can x-ray Thomson scattering experiments have been suc- be varied over 0.1−4 ns, and can be flat-topped, Gauss- cessfully fielded. While the scattering experiments on ian, or custom-designed shapes. Typically 3.5 to 12 kJ of the former two facilities by Riley et al. ͑2000͒ and Rava- laser energy has been employed for producing chlorine sio et al. ͑2007͒ have primarily investigated diffraction on Ly-␣ at 2.96 keV, titanium He-␣ at 4.75 keV, and zinc warm dense matter, experiments with the Titan laser He-␣ at 9 keV. These x-ray scattering probes have been have demonstrated inelastic x-ray Thomson scattering successfully fielded for single-shot experiments. These on plasmons with 1012 x-ray probe photons at the dense sources have provided 1014 photons at the dense plasma plasma. The Titan laser beams, which originate from the sample with appropriate bandwidth for spectrally resolv- Janus laser, consist of one 300-J petawatt-class beam to- ing Compton and plasmon spectral features. gether with a second nanosecond laser that can operate The Omega enhanced performance upgrade described independently from each other. While the short pulse by Stoeckl et al. ͑2006͒ has added energetic ultrashort- beam provides 0.7–50 ps long pulses with intensities up pulselaser capability to nanosecond laser beams. This fa- to 1018 Wcm−2, the long-pulse beam employs NIF-like cility uses a similar architecture of chirped pulse ampli- pulse shaping capable of producing multiple shock fication ͑Strickland and Mourou, 1985͒ as that on NIF. waves with pressures of 3–4 Mbar. The x-ray Thomson When completed, two short-pulse beams can be com- scattering experiments by Kritcher, Neumayer, Castor, et bined collinearly and routed into the 60-beam target al. ͑2008͒ probed multiple coalescing shock waves with chamber delivering up to 2.6 kJ at 10 ps pulse length or 10 ps temporal resolution via short-pulse laser-produced 1 kJ at 1 ps pulse length in one beam. The other beam Ti K-␣ radiation. can deliver up to 2.6 kJ at pulse lengths of ജ80 ps. Al- In addition, there are several medium-size laser facili- ternatively, the short-pulse laser beams can also be com- ties with laser energy and power capabilities in the ap- bined with up to four 10 ns, 6.5 kJ beams of NIF-type propriate range for producing powerful laser-produced architecture into a separate target chamber. In this case, x-ray probes. Among those are the Trident laser ͑Mont- the short-pulse beams can be operated with up to 2.6 kJ gomery et al., 2001; Froula et al., 2002; Niemann et al., at ജ10 ps or 1 kJ at 1 ps. 2004͒ at the Los Alamos National Laboratory, the LIL These kilojoule laser facilities allow experiments laser in France ͑Eyharts et al., 2006͒, the Xingguang II where both the x-ray probe and high energy density laser facility ͑Bai et al., 2001͒, and the Gekko laser

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1634 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

͑Kodama et al., 2001; Snavely et al., 2007͒. In addition, development of a consistent quantum statistical theory new laser facilities will come online in the near future, for strongly correlated systems ͑Ichimaru, 1973; Kremp e.g., the Fast Ignition Realization Experiment ͑Azechi et al., 2005͒. X-ray Thomson scattering offers the possi- and Project, 2006; Mima et al., 2007͒ or the Phelix laser bility to test various concepts of many-particle theory at the Gesellschaft für Schwerionenforschung, Germany for strongly coupled plasmas and the validity of standard ͑Neumayer et al., 2005͒. In addition, soft x-ray laser fa- approximations in a conclusive way. cilities ͑Klisnick et al., 2006; Rus et al., 2007͒ with ener- The dynamic structure factor S͑k,␻͒ is related to the gies of 10 ␮J–1 mJ at 13.9 nm have also been consid- imaginary part of the inverse dielectric function ␧−1͑k,␻͒ ered for scattering experiments in the extreme via the fluctuation-dissipation theorem ͑Kubo, 1966͓͒see ultraviolet spectral range by Baldis et al. ͑2002͒. Eq. ͑17͔͒. The ͑longitudinal͒ dielectric function ␧͑k,␻͒ is A powerful alternative to long-pulse laser-produced connected with the polarization function ⌸͑k,␻͒ and the He-␣ and Ly-␣ or short-pulse laser-produced K-␣ probe dynamic conductivity ␴͑k,␻͒ via x rays are FELs ͓see, e.g., Lee et al. ͑2003͔͒. They pro- 1 i vide new x-ray capabilities suited to fulfill the stringent ␧͑k,␻͒ =1− ⌸͑k,␻͒ =1+ ␴͑k,␻͒. ͑19͒ ␧ 2 ␧ ␻ simultaneous requirements on photon flux, energy, and 0k 0 bandwidth for x-ray scattering experiments in the high The dielectric or polarization function have to be calcu- energy density physics regime ͑Höll et al., 2007͒. The lated for the many-particle system under consideration Free-Electron Laser at Hamburg ͑Ackermann et al., in order to derive the dynamic structure factor or other 2007͒ delivers pulses with on-target energies of approxi- physical quantities. For example, the long-wavelength mately 50 ␮J with 1012–1013 photons in the extreme ul- limit of the transverse dielectric function yields optical traviolet spectral range between about 6.5 and 40 nm. properties such as the index of refraction n͑␻͒ and the The temporal pulse width of ϳ20 fs and the bandwidth absorption coefficient ␣͑␻͒ according to of ⌬E/E=0.5% is suitable for x-ray probing, capable of resolving rapidly changing hydrodynamic conditions. 2 ␧ ͑ ␻͒ ͩ ͑␻͒ ic ␣͑␻͒ͪ ͑ ͒ The complete transverse coherence results in diffraction lim tr k, = n + , 20 k→0 2␻ limited spots with a focal spot size as low as 2 ␮m diameter and intensities approaching 1017 Wcm−2. These while the longitudinal dielectric function is related to the properties together with the high repetition rate of 104 stopping power and the transport properties ͓see Zwick- pulses per second allows scattering experiments with nagel et al. ͑1999͒ and Reinholz ͑2005͔͒. femtosecond temporal and micrometer spatial resolu- Quantum effects and strong correlations in solid- tions on dense plasmas. For example, cryogenic hydro- density plasmas require the application of consistent gen or helium targets that can be heated by an optical quantum statistical methods which is still an issue of on- ultra short-pulse laser beam have been proposed by Höll going research. The various sum rules ͑ ͒ ϱ et al. 2007 to study matter at electron densities of + d␻ ജ1022 cm−3, cf. Tschentscher and Toleikis ͑2005͒. ͵ ␻n ␧±1͑ ␻͒ ±͑ ͒͑͒ ␲ Im k, = Cn k 21 With the construction of the Linac Coherent Light −ϱ Source in the USA and the X-ray Free-Electron Laser have to be fulfilled when constructing approximate solu- projects in Germany and Japan, x-ray energies of tions for the dielectric function ͓see Röpke and Wierling 10 keV and an order of magnitude larger photon flux ͑1998͒ and Reinholz et al. ͑2000͔͒. For example, the will become available to probe dense plasmas produced − ␻2 f-sum rule C1 =− pl follows from particle conservation in compressed matter experiments. First results on the and describes the sum of oscillator strengths for transi- former injector have recently been reported by Akre tions from the ground to excited states. The conductivity ͑2008͒. In particular, higher photon flux in short pulses sum rule C+ =␻2 results from the integral over the dis- will allow resolving smaller plasma regions than current 1 pl sipative part of the dielectric function, the dynamic con- experiments, eliminating the need to produce large ho- ductivity ␴͑k,␻͓͒see Eq. ͑19͔͒. mogeneous plasma samples. This capability will allow Approximate expressions for the dielectric function of for the first time the measurement of the dynamics of strongly correlated plasmas are usually expressed by dense matter thermal equilibration rates and the forma- means of the known result for an ideal collisionless tion of thermal equilibrium. plasma, the RPA ͓see Eq. ͑38͒ below͔. Extensions for strongly correlated plasmas have been developed within III. THEORY OF THE DYNAMIC STRUCTURE FACTOR the concept of dynamical local field corrections ͑LFC͒ ͑Ichimaru et al., 1982͒, A. General expressions ⌸RPA͑k,␻͒/␧ k2 ͑ ␻͒ ␧͑k,␻͒ =1− 0 . ͑22͒ The dynamic structure factor S k, is directly related 1+G͑k,␻͒⌸RPA͑k,␻͒/␧ k2 to the spectrally resolved Thomson scattering signal via 0 Eq. ͑10͒ and contains, as the spectral function of the A first simple expression for the static LFC G͑k͒ was electron-electron density fluctuations, important infor- derived by Hubbard ͑1958͒, accounting for the mation on the correlated many-particle system. This exchange-correlation hole in a dense electron gas. quantity is, therefore, of paramount importance for the Singwi et al. ͑1968͒ developed a self-consistent scheme

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1635 for calculating the dynamic LFC by accounting for the the system. Note that the original Drude ansatz con- sum rules mentioned above. Ichimaru et al. ͑1984͒ ex- tained only a constant collision frequency ␯. The dy- tended this approach by treating the ionic correlations namic collision frequency directly determines the dy- within classical HNC calculations and treating screening namic conductivity via Eq. ͑19͒, effects within linear response theory. They were able to construct interpolation formulas for the static LFC G͑k͒ ͑ ͒ Ichimaru et al., 1982 which have found wide applica- ␧ ␻2 tion since then. ␴͑ ␻͒ 0 pl ͑ ͒ k, = ␻ ␯͑ ␻͒ , 24 Alternatively, collisions between the electrons and i + k, ions represent important correlations and the dominat- ing dissipation mechanism in dense plasmas which can be treated systematically within an effective linear re- and is, therefore, a key quantity in this context. It can sponse theory ͑Röpke, 1998͒. As a result, a generalized directly be related to the electrical conductivity in the expression for the dielectric function accounting for col- plasma, e.g., the Ziman or Spitzer formula ͑Röpke, lisions in terms of force-force correlation functions can 1998͒. Mermin ͑1970͒ constructed a generalized dielec- be derived which is applicable in the entire ͑k,␻͒ space. tric function by adopting the original Drude ansatz for The expression can also be related to the classical Drude the whole dynamics of the RPA dielectric function and ͑ ͒ 1890 formula, by exploiting local particle conservation. This has been ␻2 extended further by accounting for local energy conser- pl ␧͑k,␻͒ =1− , ͑23͒ ͑ ͒ ␻͓␻ + i␯͑k,␻͔͒ vation Röpke et al., 1999; Selchow et al., 2002 so that the following expression using the dynamic collision fre- ␯͑ ␻͒ where the dynamic collision frequency k, represents quency ␯͑␻͒ was derived: the damping of plasma oscillations due to collisions in

͓1+i␯͑␻͒/␻͔͓␧RPA k,␻ + i␯͑␻͒ −1͔ ␧M͑k,␻͒ −1= „ … , ͑25͒ ͓␯͑␻͒ ␻͔͓␧RPA ␻ ␯͑␻͒ ͔ ͓␧RPA͑ ͒ ͔ 1+i / „k, + i … −1 / k,0 −1

which is exact in the long-wavelength limit k→0 and interactions ͓see Selchow et al. ͑2001͒ and Morozov et al. also serves as a good approximation for finite transfer ͑2005͔͒. momenta k. It is still an open question whether or not the concept of dynamic LFC, Eq. ͑22͒, or the extended Mermin ansatz, Eq. ͑25͒, is able to describe strongly B. Application to Thomson scattering coupled plasmas with ⌫ജ1 and ⍜ഛ1 correctly ͑see Fig. 1͒. For the analysis of the respective Thomson scattering While short-range correlations such as exchange inter- signal, we need the total electronic dynamic structure actions are well described within the concept of LFC, factor or the inverse total electronic dielectric function long-range correlations such as screening and collisions ␧−1͑k,␻͒ via the fluctuation-dissipation theorem ͓see Eq. are effectively accounted for by means of a collision fre- ͑17͔͒. We introduce the density-density response func- ␹ ͑ ␻͒ quency. Both concepts are closely related as can be seen tion cd k, between particles of species c and d ac- from Eqs. ͑22͒ and ͑23͒, cording to Ichimaru ͑1973͒, ␻2 1 1 ␯͑␻͒ pl ͑ ␻͒ ͑ ͒ =1+ ͚ ␹ ͑k,␻͒, ͑27͒ =−i G 0, . 26 ␧͑ ␻͒ ␧ 2 cd ␻ k, 0k cd An alternative promising approach to determine the full which describes the induced density fluctuations of spe- wave vector and frequency-dependent correlation func- cies c due to the influence of an external field on par- tions in strongly correlated plasmas is to calculate the ticles of species d. It measures the polarization of the density-density autocorrelation function via time- system ͑i.e., the induced charge density͒ in terms of the averaged ab initio molecular dynamics simulations and external field to perform an ensemble average over a sufficient num- ⌸͑k,␻͒ ber of different runs with varying initial microscopic ͚ ␹ ͑ ␻͒ ͑ ͒ configurations. However, this scheme requires long cd k, = ␧͑ ␻͒ . 28 cd k, simulation times ␶ and a large number of particles N so that it has been implemented up to now only within clas- The total electronic dynamic structure factor is then sical molecular dynamics simulations with effective pair given by

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1636 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

ប 1 mus, et al. ͑2003͒ showed that the following approxima- S ͑k,␻͒ =− Im ␹tot͑k,␻͒, ͑29͒ ͑ ␻͒ ͑ ͒␦͑␻͒ ee ␲ tot 1−e−␤ប␻ ee tion is sufficient, Sii k, =Sii k , thus calculating ne ͑ ͒ the static structure factor for ion-ion correlations, Sii k tot ͑ ͒ with ne being the total electron density defined below. see Sec. III.C . In case of a mixture of several ionic The total electron density-density response function species, the scattering contributions from all species as ␹tot can be given as ͑Salpeter, 1960; Selchow et al., 2001͒ well as their mutual correlations have to be included: cf. ee ͑ ͒ ϱ Gregori, Glenzer, and Landen 2006 . i + ͑ ͒ ␹tot͑ ␻͒ ⍀ ͵ i͑␻+i␦͒t The second term in Eq. 32 gives the scattering con- ee k, = 0 ប dte 0 tribution from the free electrons that do not follow the ion motion. Here S0 ͑k,␻͒ is the high frequency part of ϫ͓͗␦ tot͑ ͒ ␦ tot͑ ͔͒͘ ͑ ͒ ee ne k,t , ne − k,0 , 30 the electron-electron correlation function ͑Ichimaru, ͒ ͑ with ␦ntot=ntot−͗ntot͘ the density fluctuations of all elec- 1973 and it reduces to the usual electron feature Sal- e e e ͒ trons relative to the average density and the brackets peter, 1960 in the case of an optical probe. Assuming denoting the ensemble average; the normalization vol- that interparticle interactions are weak, so that the non- ⍀ ␦→ linear interaction between different density fluctuations ume is 0. The limit 0 has to be taken after the thermodynamic limit. is negligible, the dielectric function can be derived in ͑ ͒ The solution of Eq. ͑30͒ for a two-component RPA Pines and Bohm, 1952; Pines and Nozieres, 1990 . electron-ion plasma accounting for bound states is a In the classical limit, it reduces to the usual Vlasov equa- nontrivial problem ͓see Röpke and Der ͑1979͔͒. Here we tion. In the limit of the RPA, strong coupling effects are follow the idea of Chihara ͑1987, 2000͒ who applied a not accounted for, thus limiting the model validity to chemical picture and decomposed the electrons into free plasma conditions in the range of the nonideality param- ⌫ϳ −2/3Ͻ ͑ ͒ b eter of Ns 2, where Ns is the number of electrons and bound i.e., core electrons of density ne and ne , respectively. With Z free and Z bound electrons per inside the Debye sphere. f b → b In the limit Te 0, which corresponds to an electron nucleus, ne and ne are related to the ion density ni ac- b gas in the ground state, the dielectric function takes the cording to ne =Zfni and ne =Zbni. Writing the total elec- ͓ tot b Lindhard-Sommerfeld form see Pines and Nozieres tron density ne =ne +ne , we can decompose the total ͑ ͔͒ ͑ ͒ 1990 . In the case of scattering from uncorrelated elec- density response function in Eq. 30 according to trons, the form of the dynamic structure factor follows ␹tot ␹ ␹ ␹ ␹ ␹ ␹ ␹ ͑ ͒ the electron velocity distribution function ͑Dumond, ee = ff + bb + bf + fb = ff + bb +2 bf, 31 1929͒ showing a parabolic Fermi function ͑Sec. I.E͒. with the electron density response functions of the free- The last term of Eq. ͑32͒ includes inelastic scattering free, bound-bound, bound-free, and free-bound systems by strongly bound core electrons, which arises from Ra- ␹ ␹ ␹ ␹ ff, bb, bf, and fb, respectively. The response functions man transitions to the continuum of core electrons are defined as in Eq. ͑30͒ with the corresponding elec- within an ion, S˜ ͑k,␻͒, modulated by the self-motion of tron densities. We also used ␹ =␹ . Note that Z in- ce bf fb f the ions, represented by S ͑k,␻͒. For the experimental cludes both the truly free ͑removed from the atom by s spectra described in Sec. V, we find that this contribution ionization͒ and the valence ͑delocalized͒ electrons. is quite small compared to the free-electron dynamic Assuming the system is isotropic, as in the case of structure factor. In addition, the Raman band has a interest here ͑liquid metals or plasmas͒, the dynamic width comparable or larger than the Compton band ͑Is- structure factor depends only on the magnitude of k ͒ ͉ ͉ solah et al., 1991 , so we can regard this type of contri- = k , not on its direction. The next step consists in sepa- bution as yielding a contribution to the wing of the free- ͑ ͒ rating the total density fluctuations between the free Zf electron feature. ͑ ͒ and bound Zb electron contributions, and separating Treating bound electrons in a simplifying, frozen core the motion of the electrons from the motion of the ions. approximation, we find from Eq. ͑31͒ for a two- ͑ The details of this procedure are given by Chihara 1987, component electron-ion plasma 2000͒, thus obtaining for the total electron dynamic structure factor ␹ ␹RPA ␹ 2␹RPA ␹ ␹RPA ͑ ͒ ff = ee , bb = Zb ii , bf = Zb ei , 33 ͑ ␻͒ ͉ ͑ ͒ ͑ ͉͒2 ͑ ␻͒ 0 ͑ ␻͒ See k, = fI k + q k Sii k, + ZfSee k, where the partial response functions are calculated, e.g., ͵ ␻Ј˜ ͑ ␻ ␻Ј͒ ͑ ␻Ј͒ ͑ ͒ + Zb d Sce k, − SS k, . 32 in RPA or improved approaches. In order to evaluate the Thomson scattering signal it is helpful to extract the The first term in Eq. ͑32͒ accounts for the density cor- ionic terms from the electron contribution which de- relations of electrons that dynamically follow the ion scribe the screening of the ions by quasifree electrons so motion. This includes both the bound electrons, repre- that Höll et al. ͑2007͒ derived the following expression ͑ ͒ sented by the ion form factor fI k , and the screening from Eq. ͑31͒: cloud of free ͑and valence͒ electrons that surround the ͑ ͒ ion, represented by q k . The ion-ion density correlation ␹RPA 2 function S ͑k,␻͒ reflects the thermal motion of the ions ␹tot = ␹RPA + ͩZ + ei ͪ ␹RPA. ͑34͒ ii ee ee b ␹RPA ii and/or the ion plasma frequency. Gregori, Glenzer, Roz- ii

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C. Elastic scattering feature D. Free-electron feature

In the frozen core approximation discussed above, see The free-electron contribution to the dynamic struc- ͑ ␻͒ 0 ͑ ␻͒ Eq. ͑34͒, we neglect bound-free transitions and treat the ture factor Se k, :=ZfSee k, is computed via the ͑ ͒ ions adiabatically, i.e., the frequency dependence is re- fluctuation-dissipation theorem, Eq. 29 , i.e., the free- ␹ ͑ ␻͒ duced to a delta function, electron density response function ee k, that was introduced in RPA as first term in Eq. ͑34͒. However, for strongly correlated plasmas we have to account for col- S ͑k,␻͒ = ͉f ͑k͒ + q͑k͉͒2S ͑k,␻͒ i I ii lisions. They are treated within the Mermin ansatz, Eq. S ͑k͒ 2 ͑25͒, so that for the free-electron density response func- ͯ ͱ ei ͯ ͑ ͒␦͑␻͒ ͑ ͒ = Zb + Zf ͑ ͒ Sii k . 35 tion follows Sii k i␻ ␹M͑ ␻͒ ͩ ͪ ͑ ͒ ͑ ͒ ee k, = 1− We have Zb =limk→0fI k and the screening function q k ␯͑␻͒ is extracted from the ionic terms of the RPA response RPA RPA ͑ ͒ ͑ ͒ ␹ ͑k,z͒␹ ͑k,0͒ function. Sei k and Sii k are the static electron-ion and ϫͩ e e ͪ, ͑37͒ ␹RPA͑ ͒ ͓ ␻ ␯͑␻͔͒␹RPA͑ ͒ ion-ion structure factors, respectively. Numerical results e k,z − i / e k,0 for S ͑k͒ and S ͑k͒ as well as for the total elastic contri- ei ii with z=␻−Im␯͑␻͒+i Re␯͑␻͒. The collisionless response bution S ͑k͒ are given in Sec. III.F.4. i function ␹RPA͑k,␻͒ is evaluated in RPA numerically Note that Eq. ͑35͒ can be improved by calculating the e without any further approximation, i.e., for any degen- ionic form factor f ͑k͒ via the wave functions of the I eracy parameter ⍜ defined in Eq. ͑2͓͒see Arista and bound core electrons using of appropriate methods, e.g., Brandt ͑1984͔͒, the Hartree-Fock self-consistent field method of James ͑ ͒ 1 f ͑p + k/2͒ − f ͑p − k/2͒ 1962 . For low-Z elements such as hydrogen or beryl- ␹RPA͑ ␻͒ ͚ e e ͑ ͒ lium, one can use hydrogenic wave functions for the e k, = ⍀ ⌬ ͑ ͒ ប͑␻ ␩͒ , 38 0 p Ee p,k − − i ͑ ͒ ͑ 2 2 ͒2 K-shell 1s electrons which gives fI,1s k =1/ 1+k as /4 , with ⌬E ͑p,k͒=E ͑p+k/2͒−E ͑p−k/2͒=បk·p/m . The with as an effective Bohr radius; its size can be found e e e e elsewhere ͑Pauling and Sherman, 1932͒. The static elas- Fermi distribution functions are given by ͑ ͒ tic scattering feature is then given instead of Eq. 35 by 1 ͑ ͒ ͑ ͒ ͑ ͒ Gregori, Glenzer, Rozmus, et al. 2003 , fe p = ͕͓ ͑ ͒ ␮ ͔ ͖ . 39 exp Ee p − e /kBTe +1 S ͑k͒ 2 The collision frequency is taken in Born approxima- ͑ ͒ ͯ ͑ ͒ ͱ ei ͯ ͑ ͒ ͑ ͒ ͑ ͒ Si k = ZbfI,1s k + Zf ͑ ͒ Sii k . 36 tion from Reinholz 2005 , Sii k

ϱ ␧ ⍀2n 1 ␯B͑␻͒ =−i 0 0 i ͵ dkk6͓VS ͑k͔͒2S ͑k͒ ͓␧RPA͑k,␻͒ − ␧RPA͑k,0͔͒. ͑40͒ ␲2 2 ei ii ␻ e e 6 e mene 0

S ͑ ͒ 2 ␧ ͑ 2 ␬2 ͒ Vei k =Zfe / 0 k + sc is the statically screened poten- conductivity but is rather involved for the dynamic col- ␬ ͑ ͒ ␯͑␻͒ tial with the inverse screening length sc. Sii k is the lision frequency . Therefore, we restrict all further aforementioned static ion-ion structure factor. Going be- calculations to the Born-Mermin approximation ͑BMA͒ yond the Born approximation is, in principle, possible which is valid for strongly coupled plasmas with ⌫ജ1. but evaluating multifrequency contributions in a Further simplifying assumptions can be made at this screened ladder T-matrix ͑TM͒ equation is rather in- level in order to study the inﬂuence of different effects. volved and not solved yet ͓see Reinholz ͑2005͒ for de- tails͔. Furthermore, the so-called Gould-DeWitt ͑GDW͒ 1. Two-component screening approximation, ␯͑␻͒GDWϵ␯͑␻͒TM+␯͑␻͒LB−␯͑␻͒B which treats strong collisions on TM level and dynamical We ﬁrst neglect ionic correlations in Eq. ͑40͒, i.e., ͑ ͒ screening effects on the level of the Lenard-Balescu Sii k =1, and consider electrons and ions as contributing ␬ ͑LB͒ collision term, already requires a self-consistent so- to the k-independent inverse screening length sc, lution for the dielectric function since dynamic screening ␬2 = ␬2 + ␬2. ͑41͒ effects are considered in the LB collision integral explic- sc e i ␬ itly. The electronic screening length e is calculated via Eq. As shown by Redmer et al. ͑1990͒ the GDW approxi- ͑6͒, i.e., accounting for the full quantum degeneracy of mation works well for static quantities such as the dc electrons. Ions can be treated classically in all cases, i.e.,

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1638 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

TABLE I. Possible approximations for electronic and ionic Stewart and Pyatt ͑1966͒ calculated the continuum correlations in collision integral ͑40͒. The ion-ion structure fac- lowering using a finite-temperature Thomas-Fermi ͑ ͒ tor is given by Sii k =1, taken in DH approximation, or calcu- model which reproduces both the classical Debye lated by solving the HNC equations numerically. Screening in screening length ␭ for low density plasmas and the ion- ͑ ͒ D the potential is considered by both components TCS or only ¯ by electrons in the other cases; see Eq. ͑41͒ sphere correlation length d for high density strongly coupled systems. In their model, the lowering of the ion- ͑ ͒ ␬2 ization potential is given by Approx. Sii k sc TCS 1 ␬2 +␬2 Z e2 e i ⌬ f ͑ ͒ EB = , 44 k2 ␬2 4␲␧ ␭ DH e 0 s k2 +␬2 i with the effective screening length ␬2 HNC via HNC eqs. e 2 Z1/3d¯ ͑Z1/3d¯ /␭ ͒2 ␭ f f D ͑ ͒ s = . 45 3 ͓͑Z1/3d¯ /␭ ͒3 +1͔2/3 −1 ␬2 2 ␧ f D the Debye expression i =Zfnie / 0kBT is valid and used ͓see also Eq. ͑8͒ for electrons͔. A perturbative treatment of many-particle effects ͑dy- namic screening and self-energy, Pauli blocking, and statistical correlations͒ on the eigenvalue spectrum of 2. Ionic correlations and electronic screening bound states in a plasma applying the Green’s function In this case we consider ionic correlations directly by technique, on the other hand, revealed that the con- ͑ ͒ calculating the ion-ion structure factor Sii k either tinuum is lowered and that the binding energies remain within the simple Debye-Hückel ͑DH͒ approximation almost constant ͑Zimmermann et al., 1978͒. This leads to a respective lowering of the ionization energies with the k2 SDH͑k͒ = , ͑42͒ density and, whenever the bound state energy merges ii 2 ␬2 k + i into the continuum, to pressure ionization ͑Mott effect͒ ͓ ͑ ͒ ͑ ͔͒ or by solving the integral equations for the two-particle see Rogers et al. 1970 and Seidel et al. 1995 . correlation function numerically within the HNC ap- The relativistic correction for a bound-free term for proximation. In this way one also takes into account an electron, initially in a state with principal quantum higher-order ionic correlations as in a classical fluid. Be- number n and orbital quantum number l, can be calcu- cause the ionic correlations are treated explicitly, only lated analytically assuming that the final state is repre- electronic screening is considered in the potential in or- sented by a plane wave and that the bound states are ␬ ␬ given by simple Coulomb wave functions. In this form der to avoid double counting, i.e., sc= e. It is possible to include the k dependence of electronic factor approximation, the bound-free term arising from screening effects in the inverse screening length via the the sth electron in the n,l configuration can then be RPA dielectric function as follows: written as ␬ ͑k͒ = kͱRe ␧RPA͑k,0͒ −1. ͑43͒ 24͑l+1͒n2͑n − l −1͒!͑l!͒2 ͱ1+p2m e e S˜ nl,s͑k,␻͒ = e ce ␲2͑ ͒ ␣ ប In this way, the collision frequency in Born approxima- 2 n + l ! f kZnl,s tion, Eq. ͑40͒, converges to the same value as the LB ϫ͓ ͑ ␰2͒ ͑ ␰2͔͒ ͑ ͒ Jnl 1+ − Jnl 1+1/ , 46 approximation ͑Reinholz et al., 2000͒ at vanishing fre- − + ␣ quency. However, we found that the k dependence is of where f is the fine structure constant, Znl,s is the effec- minor importance for the results and, therefore, ne- tive nuclear charge felt by the sth electron in the n,l glected in what follows. shell, The different approximations which are possible within the framework described above are summarized ប␻ Enl,s 2 ͱͩ iz ͪ ͑ ͒ in Table I. p = 2 −1+ 2 −1, 47 mec mec

nl,s E. Bound-free transitions with Eiz the ionization energy for the sth electron, and n͑p − kប/m c͒ n͑p + kប/m c͒ In the case of very dense plasmas, the potential distri- ␰ e ␰ e ͑ ͒ − = ␣ , + = ␣ . 48 bution of a given ion is inﬂuenced not only by its own fZnl fZnl bound electrons but also by the charges from neighbor- ing atoms. The net effect is a lowering of the ionization The shell function Jnl has been explicitly calculated by potential ͑continuum lowering͒. Such lowering depends Schumacher et al. ͑1975͒, and for K-shell electrons ͑n on the total number of ions that participate in the modi- =1, l=0͒ can be written as ﬁcation of the potential around a test ion, which in turn 1 is a function of the screening distance of the Coulomb J ͑y͒ = . ͑49͒ forces. 10 3y3

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1639

HNC RPA DH TCS TCS DH HNC

FIG. 8. ͑Color͒ Electron feature of the dynamic structure fac- FIG. 7. ͑Color͒ Real part of the collision frequency in Born tor for compressed ͑blue͒ and uncompressed ͑red͒ beryllium. approximation using various approximations for electronic and The RPA ͑collisionless plasma͒ result is compared to the BMA ionic correlations in the collision integral; see Table I. Results result by taking into account the ionic structure factor and the are shown for three different electron densities. The mean ion inverse screening length according to the models speciﬁed in charge is Zf =2 and the plasma temperature is Ti =Te =12 eV. Table I. Further parameters: Zf =2, Te =Ti =12 eV, photon en- ប␻ ͑ ␣͒ ␪ ergy 0 =2960 eV Cl-Lyman- , scattering angle =40° for ប␻ ͑ ␣͒ ␪ The full bound-free term that appears in Eq. ͑32͒ is red curve and 0 =6180 eV Mn-He- , scattering angle then constructed by adding all bound electron contribu- =25° for blue curve. tions, ϫ 23 −3 ˜ ͑ ␻͒ ˜ nl,s͑ ␻͒ ͑ ͒ sity of ne =7.5 10 cm that represent conditions Sce k, = rk͚ ͚ Sce k, . 50 nl s reached in experiments with compressed beryllium, a reduction in the collision frequency is observed which is The normalization constant rk accounts for the possibil- due to Pauli blocking in the expression for the collision ity of coherent scattering and is given by frequency, Eq. ͑40͒. This is more than balanced by con- ͉f ͑k͉͒ 2 sidering ionic correlations in DH and HNC approxima- ϳ ͩ I ͪ ͑ ͒ rk 1− . 51 tions. At frequencies larger than the electron plasma fre- Zc ␻ quency pl, collisions become less effective. Schumacher et al. ͑1975͒ showed that in the limit of In the static limit ␻→0, the collision frequency ␯͑␻͒ is ͑ nl→ ͒ small binding energies Eiz 0 , the form factor ap- connected with the dc electrical conductivity via Eq. proximation reduces to the well-known impulse approxi- ͑24͒, ͑ ͒ mation Eisenberger and Platzman, 1970 which has ␧ ␻2 been used to describe Compton scattering from static ␴ 0 pl ͑ ͒ dc = . 52 cold matter ͑as in synchrotron experiments͒. ␯͑␻ =0͒ ប␻Ͼ Transitions into the continuum require EB so This quantity can be evaluated beyond the BMA, e.g., in that the position of the Raman band is independent of k, the GDW approximation ͑see Sec. III.D͒, so that besides and the respective threshold measures the ionization en- dynamic screening effects in ﬁrst Born approximation ergy of K-shell electrons in a dense plasma. Therefore, strong collisions are also accounted for with respect to a ⌬ different models for the continuum lowering EB in statically screened potential ͑Röpke, 1998͒. However, dense plasmas as given by Stewart and Pyatt ͑1966͒ or the consideration of the full dynamics of the collision the improved approaches of Rogers et al. ͑1970͒ and frequency is an important issue when calculating the dy- Zimmermann et al. ͑1978͒ can, in principle, be probed by namic structure factor ͓see Redmer et al. ͑2005͔͒. x-ray Thomson scattering.

2. Electron feature of the dynamic structure factor F. Numerical results Figure 8 shows the electronic contribution to the dy- 1. Collision frequency ͑ ␻͒ 0 ͑ ␻͒ namic structure factor Se k, =ZfSee k, for uncom- ͑ ϫ 23 −3͒ ͑ Figure 7 shows the real part of the dynamical collision pressed ne =3 10 cm and compressed ne =7.5 frequency in Born approximation, Eq. ͑40͒, treating ϫ1023 cm−3͒ beryllium, probed with a photon energy of ប␻ ͑ ␣͒ ␪ electronic and ionic correlations in the collision fre- 0 =2960 eV Cl-Lyman − at a scattering angle of quency in the approximations summarized in Table I =40° as a function of the energy shift from the incident ͓see Thiele et al. ͑2006͒ and Höll et al. ͑2007͔͒. Three photon energy. Results for compressed beryllium are densities have been investigated. Up to electron densi- calculated for higher energy x rays and at a scattering ties reaching solid conditions and Fermi degeneracy, the angle of 25°. The collisionless ͑RPA͒ calculation is collision frequency increases with the density which is shown as a thin dotted line, while the thick full curves due to increased coupling. However, for the highest den- show the results obtained using the BMA and TCS as

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1640 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

30 including ion-ion correlations in the DH and HNC mod- a) n =1019 cm-3 e els, scattering is increased and the damping is strongest. ] -1 20 The variation in the electron contribution to the dynamic structure factor is shown in Fig. 9 for a wide range

) [Ryd ͓ of plasma densities and temperatures see also Gregori, (k, 0 10 a3 Glenzer, Rozmus, et al. ͑2003͒ and Redmer et al. ͑2005͔͒. ee S a2 We consider weakly nonideal ͑⌫Ӷ1͒ and nondegenerate a1 ͑⍜ӷ ͒ 1 plasmas in sets ai, the warm dense matter region 0 with strong coupling ͑⌫ϳ1͒ and partial degeneracy ͑⍜ b) n =1021 cm-3 e ജ ͒ 1 in sets ci, and the strongly correlated quantum liq- ]

-1 ͑⌫ജ ⍜ഛ ͒ ͑ ͒ 2 uid 1, 1 in sets ei see Table II . The parameters ai are accessible with optical lasers, ) [Ryd

͑ ͒ the parameters ci with extreme ultraviolet EUV light, (k,

0 1 ee and the parameters e with x-ray sources. Figure 9 shows S c3 i c2 that for the cases ai collisions are less important than for c1 0 higher densities, i.e., the cases ci and ei. For the param- c) n =1023 cm-3 eter sets c , the inﬂuence of collisions is largest. Pauli e i

] 0.3 blocking prevents the collisions to have a major inﬂu-

-1 RPA BM ence in the case of highest densities, i.e., sets ei.

) [Ryd 0.2 The well-known form for the dynamic structure factor

(k, in RPA is recovered by the calculations within the 0

ee ͑ S 0.1 BMA. For small recoil energy compared to the thermal ͒ e3 energy as in case ai two symmetric peaks are found e1 e2 0 which are located slightly above the plasma frequency. -8-6-4-202468 / With increasing recoil energy, i.e., shorter wavelengths, pl these peaks become asymmetric and the redshifted peak FIG. 9. ͑Color online͒ Calculations of the electron feature for is most pronounced. This can be used to infer the tem- the dynamic structure factor according to Gregori, Glenzer, perature since the underlying detailed balance relation Rozmes, et al. ͑2003͒ and Redmer et al. ͑2005͒ shown for vari- ͓see Eq. ͑18͔͒ is independent of the theoretical approxi- ous plasma conditions, see Table II. RPA results are compared mations made in evaluating the dielectric function with the BMA that includes collisions according to TCS. and/or the dynamic collision frequency. The influence of collisions is most effective in conditions that can be outlined in Sec. III.D. The dashed and dash-dotted probed with extreme ultraviolet radiation at densities 21 −3 curves correspond to the electronic and ionic correlation around ne =10 cm , i.e., for sets ci. For these param- models in the collision integral specified in Table I. The eters it can be seen that collisions broaden the structure energy shift increases with the density independent of factor and the position of the peak is redshifted. This the approximation. Inclusion of collisions in BMA leads domain of nearly solid densities can be probed with FEL to significant damping of the plasmon resonance. When facilities ͓see Höll et al. ͑2007͔͒.

TABLE II. Parameters for the calculation of the electron feature of the dynamic structure factor: ␻ ␭ ͑⌫͒ Free-electron density ne, plasma frequency pl, probe wave length 0, temperature Te, coupling , ͑⍜͒ ␣ ␪ and degeneracy parameter as well as scattering parameter ; the scattering angle is always S =60°.

␻ ␭ ne pl 0 Te ͑cm−3͒ ͑eV͒ ͑nm͒ ͑eV͒ ⍜⌫␣

a1 200 11800 0.0025 2.5 a2 1019 0.117 532 600 35500 0.0008 1.5 a3 3000 177000 0.0002 0.7 c1 0.5 1.371 4.642 4.0 c2 1021 1.174 4.13 2.0 5.485 1.161 2.0 c3 8.0 21.943 0.290 1.0 e1 0.8 0.10 13.468 2.0 e2 1023 11.742 0.26 3.0 0.38 3.591 1.0 e3 13.0 1.65 0.829 0.5

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1641

(a) (c)

1 T=1eVe 1 Te =40 eV 23 -3 23 -3 ne =2.5x10 cm ne =2.5x10 cm Z=2 Z=2 0.8 f 0.8 f λ=0.26 nm free λ=0.26 nm

0.6 0.6

free 0.4 core (x5) 0.4

0.2 0.2 core (x2.5)

0 0

4500 4550 4600 4650 4700 4750 4800 4850 4500 4550 4600 4650 4700 4750 4800 4850 Energy (eV) Energy (eV) (b) (d)

1 1 T=1eVe Te =40 eV 23 -3 23 -3 ne =2.5x10 cm ne =2.5x10 cm 0.8 Z=2f 0.8 Z=2f λ=0.78 nm λ=0.78 nm 0.6 0.6 free core (x100) 0.4 free 0.4 core (x50) 0.2 0.2

0 0

1350 1400 1450 1500 1550 1600 1650 1700 1350 1400 1450 1500 1550 1600 1650 1700 Energy (eV) Energy (eV)

FIG. 10. ͑Color online͒ Comparisons between the RPA dynamic structures arising from free and core electrons are shown ͑ ͒ according to Sec. III.E and Gregori, Glenzer, Rozmus, et al. 2003 for the case of a beryllium plasma. At Te =1 eV, beryllium is assumed in its normal state, Zf consists only of conduction electrons with a K-shell ionization potential EB =111.5 eV. At Te ␪ =40 eV, beryllium is assumed doubly ionized Zf includes only free electrons with EB =159.5 eV. The scattering angle is =160°. The scattering parameter is ͑a͒ ␣=0.46, ͑b͒ ␣=1.36, ͑c͒ ␣=0.23, and ͑d͒ ␣=0.67.

3. Bound-free spectrum 5 Figure 10 displays the contributions of free and core 4 electrons to the dynamic structure factor for a solid- 3 n =3.0 x 1023 cm-3 density beryllium plasma as derived by Gregori, Glen- e 2 zer, Rozmus, et al. ͑2003͒. For both x-ray wavelengths ͑␭ ͒ 1 0 =0.26 and 0.78 nm as well as for both temperatures actor ͑ ͒ f Te =1 and 40 eV considered, the contribution from 0 transitions of core electrons into the continuum is small 4 SDH ii structure compared to the scattering signal of free electrons. Fur- structure HNC 3 S thermore, the width of the respective Raman band is of ii n =7.5 x 1023 cm-3 SHNC e the order or even larger than the Compton band con- 2 ei S tributing to its low energy wing. 1 i

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 k[a -1] 4. Ion feature B

Figure 11 shows the static ion-ion and electron-ion ͑ ͒ ͑ ͒ ͑ ͒ FIG. 11. Color online Static ion-ion and electron-ion struc- structure factors Sii k and Sei k for uncompressed and ͑ ϫ 23 −3͒ ture factors for uncompressed ne =3.0 10 cm and com- compressed beryllium as function of the wave number k ͑ ϫ 23 −3͒ ͑ ͒ pressed beryllium ne =7.5 10 cm calculated in DH Sii calculated in the HNC scheme as outlined by Schwarz et and HNC approximation ͑S and S ͒ for Z =2 and T =T ͑ ͒ ͑ ͒ ii ei f e i al. 2007 . In addition, the simple DH result for Sii k is =12 eV. Curves: Total elastic contribution to the dynamic ͓ ͱ ͑ ͒ ͑ ͔͒2 ͑ ͓͒ ͑ ͔͒ shown as well as the total elastic contribution Si to the structure factor Si = Zb + ZfSei k /Sii k Sii k see Eq. 35 .

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1642 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

1 1 the other hand, strong interaction potentials enhance 0.8 0.8 correlations over large scales leading to high elastic scat-

) 0.6 0.6 tering amplitudes. k (k) ( ii

ee The influence of the ion correlations on measured S 0.4 S 0.4 Thomson scattering intensities is further outlined in Fig. 0.2 0.2 26 for beryllium and Fig. 31 for lithium. Ionic and elec- 0 0 0246 810 0246 810 tronic correlations are connected in the approach of k/k k/k ͑ ͒ ͑ ͒ De De Chihara 1987, 2000 via the screening function q k and 1 2 rely on defining bound and free electrons with a chemi- 0.8 cal picture model. Future studies will allow a first- 1.5 principles approach to the ion-ion static structure factor ) 0.6 k ( 1 by ab initio quantum molecular dynamics simulations ei q(k) S 0.4 which treat electronic and ionic correlations in a strict 0.5 0.2 physical picture by combining finite-temperature DFT 0 0 for the electrons and classical molecular dynamics for 0246 810 0246 810 ͑ ͒ k/k k/k the ions, cf. Desjarlais et al. 2002 , Bonev, Militzer, and De De Galli ͑2004͒, Garcia Saiz, Gregori, Gericke, et al. ͑2008͒, ͑ ͒ ͑ ͒ Garcia Saiz, Gregori, Khattak, et al. 2008 , and Wünsch, FIG. 12. Color online The calculated static structure factors ͑ ͒ ͑ ͒ ͑ ͒ Vorberger, and Gericke 2009 . Scd k and screening charges q k for a beryllium plasma with ϫ 23 −3 ͑ ͒ ZA =4 at ne =2.5 10 cm , Zf =2, and Ti /Te =1 solid line , ͑ ͒ ͑ ͒ Ti /Te =0.5 dashed line , Ti /Te =0.1 dash-dotted line . The transfer momentum k is normalized with respect to the inverse G. Theoretical spectra ␭ ͓ ͑ ͔͒ electron Debye screening length kDe=1/ D see Eq. 8 . From Gregori, Glenzer, and Landen, 2006. In order to generate theoretical scattering spectra comparable to the data measured with Thomson scattering experiments, we proceed as follows. First, we calcu- dynamic structure factor according to Eq. ͑35͒. Further late the collision frequency in the range ͓ ␻ ␻ ͔ ͑ ͒ parameters are Zf =2 and Ti =Te =12 eV. −100 pl,100 pl on a logarithmic grid; see Eq. 40 . Sec- The HNC approximation also takes into account ond, we generate the free electron contribution ͑ ␻͒ 0 ͑ ␻͒ ͓ ␻ ␻ ͔ ͓ higher-order ion-ion correlations as in a classical liquid, Se k, =ZfSee k, in the range −10 pl,3 pl ; see whereas the DH result is only valid for low densities. Eqs. ͑17͒ and ͑25͔͒. Electron-ion correlations become more pronounced Third, we take into account the finite resolution of the with increasing density as expected. The total elastic detector as well as the finite bandwidth of the probe contribution to the dynamic structure factor is sensitive beam. The calculated spectra have to be convoluted with with respect to an analysis of measured Thomson scat- an appropriate weighting function g͑␻͒. Here we use a tering spectra ͑see Sec. V. C ͒. Gaussian of the form The dependence of the dynamic structure factor and 1 ␻2 screening function on k is displayed in Fig. 12 from Gre- g͑␻͒ = expͩ− ͪ. ͑53͒ gori, Glenzer, and Landen ͑2003͒. The ion-ion and ͱ2␲␴ 2␴2 electron-electron structure factors approach unity for ␴2 ␴ ϫ large k while electron-ion correlations decrease corre- The variance is fixed by =0.425 FWHMtot, with spondingly. In this limit, the resulting screening function the total full width at half maximum FWHMtot given by 2 2 2 q͑k͓͒see Eq. ͑35͔͒ is rapidly approaching zero and the FWHMtot=FWHMdetector+FWHMprobe. Finally, the elas- ͑ ͒ ͑ ͒ ͑ ͒ϳ 2 tic feature Si k via Eq. 35 and if necessary bound-free ion feature Si k Zb. Also shown are the results for nonequilibrium plasmas relevant for ultrashort excita- transitions are added. tion processes and relaxation phenomena. In summary, theoretical spectra result by computing ͑ ͓͒ ͑ ͒ ͑ ͔͒2 the following expression using Eq. ͑32͒: The value of Sii k fI k +q k at small values of k is ϱ small in the limit of a neutralizing electron background 1 such as described by one component plasma ͑OCP͒ S͑k,␻͒ = ͵ S ͑k,␻Ј͒ ͱ ␲␴ ee models or weak electron-ion interaction potentials. In 2 −ϱ this case, the weak electron-ion interaction is described ͑␻ ␻ ␻ ͒2 − 0 − Ј by small values of q͑k͒. In addition, cold ions in the ϫexpͩ− ͪd␻Ј. ͑54͒ 2␴2 presence of uniform electron background arrange to minimum electrostatic potential lattice ͑for ion Debye Theoretical x-ray Thomson scattering spectra have length smaller than the interparticle spacing͒ and corre- been calculated by Gregori, Glenzer, Rozmus, et al. lations on spatial scales longer than the interparticle ͑2003͒ with the dynamic structure factor S͑k,␻͒ for be- spacing are reduced ͑see Fig. 12 for small values of k or ryllium targets with ZA =4 at various temperatures, den- ͒ Ti /Te . Thus, for latticelike structures the scattering is sities, and ionization stages. The results are shown in coherent and spatial uniformity results in weak ͑elastic͒ Fig. 13. The probe radiation is E=4.75 keV and the scat- scattering amplitudes with exception of Bragg peaks. On tering angle is ␪=160°.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1643

FIG. 13. Theoretical x-ray Thomson scattering spectra calculated with the dynamic structure factor S͑k,␻͒ for beryllium targets with ZA =4 at various temperatures, densities, and ionization stages. The probe radiation is E=4.75 keV and the scattering angle is ␪=160°. From Gregori, Glenzer, Rozmus, et al., 2003.

The temperature and density dependent Thomson from laser-illuminated solid target plasmas ͑Glenzer, scattering spectrum is shown for beryllium in Figs. Gregori, Lee, et al., 2003; Gregori et al., 2004; Glenzer, ͑ ͒ ͑ ͒ ͒ 13 a –13 c for Zf =2. The inelastic Compton scattering Landen, et al., 2007 or by direct-drive compression feature is strongly broadened with increasing tempera- ͑Sawada et al., 2007; Lee, 2008͒. A separate multi-keV ture. Figures 13͑d͒–13͑f͒ show that the elastic scattering line radiator, with adequate energy to penetrate through feature is strongly reduced relative to the inelastic scat- the dense sample, has been produced from a second de- tering feature with increasing Zf. In isochorically heated layed laser plasma providing the spectrally narrow x-ray matter of known mass density, the electron density can probe for x-ray Thomson scattering before the plasma be inferred from the intensity ratio. For liquid hydrogen sample cools or disassembles. Scattered photons in near targets a similar analysis has been performed by Höll et forward scattering and near backscattering geometries al. ͑2007͒ for x-ray Thomson scattering experiments us- have been collected and spectrally dispersed by high re- ing FELs. flectance crystal spectrometers coupled to a gated fram- ing camera, measuring spectra with a temporal resolution of 80 ps. Approximately 1014 x-ray probe photons IV. EXPERIMENTAL TECHNIQUES have been delivered to the sample and Ͼ105 photons A. Design of scattering experiments have been detected for high-quality single shot experiments. Subsequently, larger curved crystal spectrometers In the first x-ray Thomson scattering experiments at and higher efficiency detectors have been employed at the Omega laser, submillimeter scale samples of light the Titan ͑Kritcher, Neumayer, Castor, et al., 2008͒, elements, e.g., beryllium or carbon, have been heated LULI 2000 ͑Ravasio et al., 2007͒, or Vulcan ͑Garcia Saiz, volumetrically using either multi-keV L-shell x rays Gregori, Gericke, et al., 2008͒ lasers allowing measure-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1644 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

250 µmBe ments with significantly lower numbers of x-ray photons; Ta shield with solid target 12 400 µm at 10 photons at the sample. 11 Heater Beams pinhole [smoothed with phase The use of unfocussed laser-produced x-ray probe ra- 500 µm plates] 12 Probe diation requires that extended fairly homogeneous Beams plasma samples are produced by energetic drivers. Fur- thermore, the quasimonochromatic x-ray probe must be Mn foil produced efficiently and placed in close proximity to the sample. At a distance of 0.3–3 mm these two indepen- Target µ dent laser plasmas will avoid interactions among them- positioner 250 m aperture 8m selves. At this distance, the sample subtends a solid m −2 Au/Fe angle of 10 ഛd⍀ഛ2 sr. Thus, it is necessary to heat or Shield compress a dense plasma sample homogeneously over a [CH coated] Crystal spectrometer surface diameter of d=0.35 mm to match the solid angle FIG. 14. ͑Color online͒ Schematic of x-ray scattering target and to use the maximum amount of x-ray probe photons after Sawada et al. ͑2007͒ and Lee et al. ͑2009͒. A pinhole is provided by the laser plasma. Moreover, to maximize used for collimating the x-ray probe radiation. Shields and a the probability for a scattering process and to achieve a 250 ␮m aperture limit the field of view of the x-ray spectrom- high scattering fraction it is advantageous to heat or eter to the compressed material. compress the dense plasma over a length of order of the diameter ᐉ=0.4 mm, resulting into a total volume of the dense plasma of V=5ϫ10−5 cm3. from the isochorically heated beryllium. In recently de- Heating such a volume of solid density matter with a signed experiments, these techniques have been com- mass density on the order of 1 g cm−3 to temperatures bined. of several eV requires a driver that delivers about Figure 14 shows the scattering target developed by 10 eV/atom. For example, the total number of atoms in Sawada et al. ͑2007͒ for x-ray Thomson scattering mea- −3 a beryllium sample with mass density of 1.85 g cm , Zf surements on shock-compressed matter. This target plat- ͑ ͒␳ ϫ 18 =1, and A=9.012 amu is N=neV=NA Z/A V=6 10 . form has been subsequently employed by Lee et al. ϫ 23 −1 ͑ ͒ Here NA =6.022 10 mol is the Avogadro constant 2009 for use with different sample materials and scat- ͑or Loschmidt’s number͒. To heat this sample requires tering angles. A Ta shield with a pinhole restricts the ⍀ Ӎ that 10 J of energy deposited resulting in a total energy solid angle to d x ray 2 sr so that the x rays interact density of 2ϫ105 Jcm−3, thus requiring a high energy with a sample surface area of 0.005 cm−2 and a volume density physics facility. of VӍ10−4 cm3. The 8 mm-long Au/Fe shields restrict In experiments at the Omega or Titan laser, a uniform the field of view of the x-ray spectrometer to the beryl- ␮ plasma of 50 m scale length to about 1 mm has been lium sample and prevent the Mn plasma from moving produced. Shields have been applied to define the scat- into the field of view at the time when the scattering tering volume and to avoid scattering from the nonuni- spectrum is measured with a gated detector. form part of the target. In addition, they also prevent a Because of the relatively small scattering signals, it is direct view of the x-ray probe plasma by the detector. At important to avoid the x-ray fluorescence of filters and Omega, up to 1 J of x-ray energy has been applied for shields ͑Tertian and Claisse, 1982͒ induced by direct probing, which itself heat the sample; experiments have bremsstrahlung emission or high energy electrons from observed heating up to about 2 eV. With the use of fo- laser-plasma interactions. Fluorescence by inner-shell cused x-ray beams ͑e.g., from FELs͒, these volume and ␣ ␤ heating restrictions for probing the dense plasma sample K- and K- transitions has been investigated by Kyrala ͑ ͒ will be mitigated by achieving a scattering volume as and Workman 2001 . The incident spectral photon flux small as 2 ␮m in diameter, while the length of the scat- density I␭͑0͒ is measured in units of photons per unit tering volume can be chosen to achieve the required bandwidth interval per unit area and are attenuated at a −␮␭␳x scattering fraction. Thus, these x-ray sources will allow a depth x according to Beer’s law I␭͑0͒d␭e . Here ␮␭ s p ␲ lower energy driver for producing a smaller scale-length =␮␭ +␮␭ +␮␭ is the total absorption coefficient at the plasma sample. wavelength ␭ due to scattering, photoelectric effect, and The first angularly resolved x-ray diffraction experi- pair production. At the depth x the photons in the beam ments on dense plasmas by Riley et al. ͑2000͒ and Riley, −␮␳x are further depleted in a sheet dx, I␭͑0͒ d␭ e ␯␭␳ dx. Weaver, et al. ͑2002͒ employed two apertures to colli- Only photons absorbed by the photoelectric effect de- mate the incident x rays and to observe scattering from a ␮p free-standing shocked foil at different angles on multiple scribed by ␭ lead to fluorescence, and the fraction of shots. The x-ray Thomson scattering experiments of vacancies in the K shell that subsequently lead to K shell Glenzer, Gregori, Lee, et al. ͑2003͒, Glenzer, Gregori, fluorescence is determined by the jump factor in the Rogers, et al. ͑2003͒, Gregori et al. ͑2004͒, and Glenzer, photoelectric absorption coefficient at the K edge, rk. Landen, et al. ͑2007͒ used shields to restrict the field of This leads to the fluorescence of a filter of thickness h in view of the spectrometer to measure scattering spectra direction of the detector of

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1645

␮p rk −1 ␭ Time-integrated ͑ ͒ ␭ ͫ ͬ ␻ ͫ ͬ emission BK h d = c k 1 ␮ ␮ )) rk ␭ − k ␳ ͑␮ ␮ ͒ ␮ ␳ − h ␭− k − k h ϫ͓1−e ͔I␭͑0͒d␭ e . ͑55͒ sr keV /( J ( 100 ps gate The fluorescence at the wavelength of the K lines is de- at peak velocity ␮ -10 noted by k. With absorption coefficients from Henke et 10 al. ͑1982, 1993͒, estimates by Kyrala and Workman ͑2001͒ showed that the flux in high-Z filters can be significant reaching up to the same order of magnitude as X-ray emission the incident radiation. In scattering experiments where -20 the photon flux is of order 10−6 of the primary x-ray 10 probe radiation, this signal would be a significant source 100 101 102 of high background and noise levels. Target shields with X-ray energy (keV) coatings of low-Z plastic as well as alternating materials with appropriate K edges have been shown to be effec- FIG. 15. ͑Color online͒ X-ray bremsstrahlung emission as tive for avoiding fluorescence radiation. In addition, function of radiation energy from dense capsule implosion free-standing crystals have been applied by Kritcher, conditions. The time-integrated emission is compared with the Neumayer, Castor, et al. ͑2008͒ to avoid fluorescence in emission during a 100 ps time interval at maximum implosion velocity. the spectrometer housing material. Collimating the probe x rays and restricting the scattering volume is also important for accurately defining plasmas of interest. Thus, for plasma samples with in- the scattering angle, to avoid sampling a large spread of creasingly higher densities and temperatures, x-ray k vectors and to achieve partial spatial coherence of the sources with higher flux or energy are required to pen- source ͑Wolf, 1987͒. In noncollective scattering experi- etrate through the dense matter and to obtain larger ments, broadening of spectral features has been kept scattering signals compared to the bremsstrahlung emis- small compared to the thermal broadening of the Comp- sion. ton line by choosing ⌬k/kϽ0.2; cf. Eqs. ͑11͒ and ͑14͒. Equation ͑57͒ further indicates that bremsstrahlung For collective scattering experiments, Gregori, Tom- emission in the x-ray region has to be accounted for in masini, et al. ͑2006͒ estimated the effect of an extended conditions with high electron temperatures. For dense source. hydrogen plasmas, Fortmann et al. ͑2006͒ showed that a The coherence length in the plasma defined by Eq. free-electron laser source at 80 eV provides a factor of ͑4͒, ␭Ã, will be decreased due to additional path length ten larger scattering signal above the bremsstrahlung from light rays on opposite ends of the source. Thus, the emission. On the other hand, for high temperature gas- scattering angle in Eq. ͑4͒ must be replaced by ␪+⌬␪/2 bag plasmas, Gregori, Glenzer, Chung, et al. ͑2006͒ with ⌬␪ the angular extent of the source as seen by each chose a high energy x-ray probe, the Zn He-␣ probe point in the scattering volume. The coherence length at radiation at 9 keV, to obtain a larger signal from x-ray scattering than from bremsstrahlung emission. the sample from an extended source of length LS can ⌬ ⌬ ϳ Figure 15 shows the calculated bremsstrahlung emis- then be estimated from the condition LS k 1 with ⌬ ⌬␪ sion from an inertial confinement fusion capsule implo- k0 =k0 /2. These geometric considerations lead to sion experiment by Hatchett ͑2008͒. This calculation in- ⌬␪ cludes radiation transport and assumes a field of view of ␣ ജ 1+ ͑56͒ 2␪ 300 ␮m in diameter as well as a low concentration of deuterium ͑H:D:T=1:0.0025:1͒ to limit the neutron to observe collective scattering features. yield. These experiments will allow probing extremely Finally, the size of the scattering volume determines dense conditions. The time-integrated emission is shown the amount of bremsstrahlung emission from the dense together with the emission gated over 100 ps at about plasma measured by the x-ray scattering detector. The peak implosion velocity. Quantitative estimates pro- emitted power by ͑free-free͒ bremsstrahlung emission ͑ប␻͒ ប␻ vided in Sec. IV.E indicate that a gated detector enables Pff , at radiation energy into the energy band- ͑ប␻͒ ⍀ measurements in this regime using a powerful x-ray width d , solid angle d , and volume V has been source as described in Sec. II. discussed by Griem ͑1997͒, and assuming radiation transport effects not to be important B. X-ray probe requirements 32 ␲ 1/2 1/2 ͑ប␻͒ ͑ប␻͒ ͩ ͪ ͩ EH ͪ Pff d = r0c 3 3 kBTe For x-ray Thomson scattering experiments with laser- ប␻ produced x-ray sources, it is important to develop con- ϫZ2n n ge− /kBTed͑ប␻͒͑57͒ i e ditions with high x-ray conversion efficiency. In this sec- with EH the ionization energy of hydrogen. High density tion, the x-ray yield Y of a laser source is defined as the corrections by Fortmann et al. ͑2006͒ showed that the number of photons that are produced per 1 J of incident gaunt factor g varies by factors of 2–3 for the dense laser energy and in the full sphere. With EL the total

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1646 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

laser energy on target the total number of photons pro- Girard, 3 1ns duced is YE . The x-ray conversion efficiency is then Fournier, 3 1ns -2 L sr)) 10 y ␩ ␯ Yaakobi, defined as x =Yh , and is a dimensionless parameter. 10 3 500ps The important quantity is the number of x-ray photons in a single x-ray spectral line produced in a single laser Glenzer, 3 1ns Riley, 2 300ps shot. In most cases, the source size is sufficiently small to

photons/(J 4 Riley, 2 80ps fulfill the requirements outlined in Sec. IV.A. These 12 Matthews, 2 700ps 10-3 sources must further provide a narrow bandwidth of 1 Matthews, 3 700ps ⌬E/EӍ0.01 for noncollective scattering ͑Landen, Glen- Matthews, 1 600ps Conversion efficienc zer, et al., 2001͒ and ⌬E/EӍ0.002 for collective scatter- Yield (10 ing ͑Urry et al., 2006͒, respectively. While FEL x-ray sources are designed ab initio to provide optimum spec- Matthews, 1 100 ps 10-4 tral and temporal properties with fixed photon numbers 0.1 0.1 110 100 for x-ray Thomson scattering applications, laser- 15 -2 produced plasmas require development to obtain the de- Intensity(10 Wcm ) sired properties. FIG. 16. ͑Color͒ X-ray yield and x-ray conversion efficiency For example, probing solid-density beryllium plasmas for the titanium He-␣ x-ray emission at 4.75 keV as function of at temperatures of 50 eV and electron density of ne =3 laser intensity from various studies ͑see text͒. A typical error ϫ 23 −3 ␪ 10 cm in backscatter geometry, =125°, and with a bar of 50% in conversion efficiency due to uncertainties of the moderate x-ray energy source such as titanium He-␣ at absolute detector calibration, crystal reflectivity, and shot-to- ͉ ͉ E0 =4.75 keV will access a scattering vector of k =4 shot reproducibility is shown. ϫ10−10 m−1; cf. Eq. ͑4͒. In this case, with ␣=0.3 the experiment results in noncollective scattering. The Comp- strahlung absorption. These studies have been reviewed ⌬ ប2 2 ton feature will be shifted from E0 to EC = k /2me by Kauffman ͑1991͒. They are aimed primarily at devel- =70 eV, and the spectral broadening by thermal motion oping x-ray backlighter sources for radiography ͑Molito- will result in a width of 200 eV. To resolve this feature, ris et al., 1992; Hammel et al., 1993͒ where the require- an x-ray probe bandwidth of 40 eV such as provided by ments on x-ray bandwidth and spectral purity is less ␣ the titanium He- spectrum is sufficient. Specifically, the stringent than for scattering applications. Yaakobi et al. He-␣ probe spectrum consists of the resonance, inter- ͑1981͒, Matthews et al. ͑1983͒, Kodama et al. ͑1986͒, Phil- combination, and dielectronic satellite lines. On the lion and Hailey ͑1986͒, and Urry et al. ͑2006͒ studied the other hand, for forward scattering measurements on dependence of the number of x-ray photons on laser plasmons, a spectral resolution of 6 eV is necessary and color, pulse length, intensity, and energy. In addition, ef- has been provided, e.g., by the chlorine Ly-␣ doublet or fects of polarization and angle of incidence have been titanium K-␣ transitions. studied by Tallents et al. ͑1990͒ and Kyrala et al. ͑1992͒. Plasmon measurements suffer from the fact that these For 4–10 keV He-␣ and Ly-␣ x rays, a conversion effi- ␣ ␣ Ͻ␩ Ͻ features are weak and, when using He- or Ly- probes, ciency of 0.0005 x 0.01 has been measured. elastically scattered dielectronic satellite lines can blend Figure 16 shows the dependence of the conversion ef- with the plasmon signal. Temporal gating is required to ficiency and x-ray yield on laser intensity for the tita- minimize the satellite signal which is typically emitted nium He-␣ transition. The experiments on solid-density late in time during the plasma evolution after the laser disk targets by Matthews et al. ͑1983͒ showed ␩Ϸ10−3, beams that produce the x-ray probe are turned off. roughly corresponding to 1012 titanium He-␣ ͑4.75 keV͒ When applying a gated detector, the contributions from x-ray photons per 1 J of incident laser energy and into these unwanted spectral features have been shown to be 4␲ sr. These data have been reproduced by Yaakobi et ഛ5% of the main chlorine Ly-␣ transitions; cf. Fig. 22. al. ͑1981͒, Riley, Woolsey, et al. ͑2002͒, and Glenzer, Gre- In addition, measurements at high density, such as per- gori, Rogers, et al. ͑2003͒ to within about a factor of 2–5. formed by Lee et al. ͑2009͒, are advantageous providing Furthermore, a linear scaling with laser energy and weak plasmon features separated sufficiently from the He-␣ dependence on pulse length has been measured. The lat- or Ly-␣ probe energy where no satellite features exist. ter has also been observed by Riley, Woolsy, et al. ͑2002͒; Thus, for collective scattering, short-pulse laser- they reported a factor of 2 variation in conversion effi- produced K-␣ radiation has the advantage over long- ciency when increasing the laser pulse width from pulse laser-produced He-␣ or Ly-␣ transitions because 80 to 300 ps. While some discrepancies exists among the no spectral features exist on the red wing of the K-␣ various studies, a strong dependence on laser frequency spectral line. for low energy x rays ͑Ͻ4 keV͒ has been observed with A large body of experimental and theoretical studies up to an order of magnitude higher conversion efficiency has investigated the x-ray conversion efficiency from la- measured with 351 nm ͑3␻͒ compared to 1053 nm ͑1␻͒ ser to x-ray photons by laser-irradiating solid targets laser irradiation. with nanosecond pulses. He-␣ and Ly-␣ x rays are emit- Girard et al. ͑2005͒ showed further improvements in ted by highly ionized heliumlike and hydrogenlike atoms conversion efficiency, reaching values in excess of 10−2 in ablation plasmas that are heated by inverse brems- by employing a laser prepulse to generate an extended

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1647

ablation plasma before heating with a high intensity Khattak, 400nm

pulse. Extended plasma sources such as under dense sr)) ͑ ͒ ͑ aerogels Fournier et al., 2004, 2006 , gas targets Back et Riley, 800nm 45fs 10-4 al., 2001, 2003͒, and gas jets ͑Kugland et al., 2008͒ have 0.1 Kritcher, 1054nm 0.5 ps also been shown to improve the x-ray conversion efﬁ- Kritcher, photons/(J 4 1054nm 5 ps ciency. For titanium He-␣ emission, Fournier et al. 12 ͑ ͒

2004 reported a conversion efﬁciency of about 1%. Reich, Conversion efficiency 800nm 90fs

Further, recent studies on under dense targets indicate Yield (10 agreement with radiation-hydrodynamic modeling. To Jiang, 1053nm 0.4 ps date, extended under dense sources have not been 10-5 0.01 tested for scattering experiments; x-ray scattering probes 0.01 1 100 have been solely produced by laser-foil interactions. Intensity(1017 Wcm-2) A threshold for the onset of x-ray yield has been measured depending on the x-ray transition and energy. For FIG. 17. ͑Color online͒ The x-ray conversion efficiency, along example, the threshold intensity for 4.75 keV titanium with a typical error bar of 50%, for the titanium K-␣ x-ray He-␣ is I=2ϫ1014 Wcm−2, increasing to I emission at 4.5 keV as function of the short-pulse laser inten- Ϸ1016 Wcm−2 for 9 keV Zn He-␣ radiation. As shown sity. in Fig. 16, further increasing the laser intensity results in decreasing x-ray yield due to decreased coupling with the K-␣ yield varied by a factor of 1.8 for different laser the onset of laser-plasma interaction instabilities. This intensities, while the variation in the density scale length trend has been reproduced for other elements emitting ␣ ͑ has been observed to provide a gain factor up to 4.6. below 10 keV up to the Zn He- line Workman and Kritcher et al. ͑2007͒ showed moderate enhancement ͒ ␣ Kyrala, 2001 . Generally, all studies of thermal He- or with perylene coatings. Khattak et al. ͑2006͒ showed a ␣ Ly- sources with laser-irradiated foils show a strong factor of 4 enhancement using frequency-doubled radia- reduction with increasing x-ray energy. For example, tion compared to Riley et al. ͑2005͒. However, because ͑ ͒ ͑ ͒ Workman and Kyrala 2001 and Ruggles et al. 2003 of the lack of efficient frequency conversion, this im- ͑ ͒ showed a conversion efficiency of 0.01–0.2 % for 9 keV provement in conversion efficiency is often canceled by Zn He-␣ radiation compared to a range of ͑0.1–1͒% for the use of smaller laser energy on target. Similarly, high Ti radiation at 4.75 keV. conversion efficiencies with shorter pulse length ob- In contrast to these He-␣ and Ly-␣ sources, irradiat- served by Kritcher, Neumayer, Lee, et al. ͑2008͒ do not ing targets with ZϾ20 with short-pulse lasers results in lead to more powerful x-ray probes because of restric- K-␣ conversion efficiencies that are almost constant or tions of laser energy at high power operations. slightly increasing with increasing K-␣ x-ray energy. Other studies have developed improved target con- With increasing Z, radiation processes dominate over figurations such as nanostructured velvet targets nonradiative Auger processes and fluorescence x rays ͑Kulcsar et al., 2000͒, or cones ͑van Woerkom et al., are emitted by inner-shell transitions in relatively cold 2008͒ enhancing laser absorption. Improved conversion atoms where the electron holes are created by interac- efficiencies by more than one order of magnitude have tions with fast electrons from short-pulse laser interac- been reported, suggesting that ongoing developments in tions with the solid target. These studies have been re- this area will provide very efficient and versatile x-ray viewed by Park et al. ͑2006͒. In general, after sources. optimization of laser energy, power, polarization, Future x-ray FELs are expected to provide a peak prepulse, angle of incidence, and spot size on target, brilliance of 1032–1034 photons per second mrad2,mm2, −5Ͻ␩ Ͻ −4 2 4 these sources yield 10 x 10 . and 0.1% bandwidth for photon energies of 10 –10 eV Figure 17 shows the dependence of the conversion ef- ͓see Ackermann et al. ͑2007͔͒. Although the laser energy ficiency and x-ray yield on laser intensity for titanium can be tuned and highly focussed, the number of x-ray K-␣ radiation with data from Jiang et al. ͑1995͒, Reich et photons available for probing is smaller compared to al. ͑2003͒, Riley et al. ͑2005͒, Khattak et al. ͑2006͒, and laser sources. For present single-shot x-ray scattering ap- Kritcher, Neumayer, Castor, et al. ͑2008͒. The results are plications ͑Ackermann et al., 2007͒, a total of 1012 probe reproducible to within a factor of 2–5 with laser energy x rays can be provided at the dense plasma for x-ray and power varying from Ͻ1 to 300 J and 30 to 200 TW. energies up to 92 eV within a bandwidth of ⌬E/E However, comparison with the theory developed by ഛ0.5% and in a 20 fs long pulse. At lower x-ray ener- Salzmann et al. ͑2002͒ has shown that these data are a gies, the number of photons available is slightly higher factor of 5–10 smaller than calculations indicating that with total energy deliverable on target being constant. full optimization of laser and target parameters are chal- At the future FEL ͑Akre, 2008͒, a total of 1012 probe x lenging. rays are projected at an energy of up to 8.5 keV. Improvements of conversion efficiency values have Figure 18 compares the x-ray probe energy from been reported. Rousse et al. ͑1994͒ and Bastiani et al. plasma and FEL sources as function of x-ray probe en- ͑1999͒ showed high x-ray yield with production of a pre- ergy. Data for thermal He-␣ or Ly-␣ sources are taken plasma. Ziener et al. ͑2002͒ independently showed that from Workman and Kyrala ͑2001͒, the chlorine data are

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1648 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

1015 ently being considered to probe the ﬁnal stages of im-

He-alpha ploding capsule conditions. On the other hand, 10 keV sources, 100 ps x-ray probes are largely sufﬁcient to probe the implod- ing shell for studies of the assembly of thermonuclear fuel. Free-electron lasers provide 1012 photons on target for energies of about 92 eV in a 20 fs long pulse. Al- otons on target 13 FEL h 10 sources, 20 fs though the solid angle can be made very small by tightly focussing the beam, the total number of x-ray photons -ray p K-alpha X sources, 10 ps available for scattering does not change. At future free electron laser facilities, x-ray probes with an order of He-alpha photons/sr/1kJ/100ps pulse K-alpha photons/sr/300J/10ps pulse magnitude more photons are expected. FEL photons/sr/20 fs pulse 1011 103 104 105 X-ray energy (keV) C. Spectrometers

FIG. 18. ͑Color online͒ X-ray photons available for x-ray scat- Measuring the x-ray Thomson scattering signal re- tering experiments from laser-produced plasmas sources, along quires highly efficient spectrometers as well as moderate with a typical error bar of 50%, as function of x-ray energy. spectral resolution to discriminate between the elastic ␣ ͑ ␻͒ For He- sources data produced with frequency-tripled 3 and inelastic scattering components. For free-electron or ␣ lasers were chosen while K- data are taken from experiments soft x-ray laser experiments in the vacuum-ultraviolet without frequency conversion. The dashed line indicates the spectral range, 6.7Ͻ␭Ͻ32 nm, reflection grating spec- photon numbers expected from future x-ray free-electron la- trometers fielded by Nakano et al. ͑1984͒, Beiersdorfer et sers. al. ͑1999͒, and Harada et al. ͑1999͒ are available. They are advantageous compared to transmission grating ͑ ͒ from Urry et al. ͑2006͒, and the titanium data from Riley, spectrometers Jasny et al., 1994 because they are highly Woolsey, et al. ͑2002͒ and Glenzer, Georgi, Rogers, et al. reflective and further do not show ghosts that occur due ͑2003͒. High energy ͑EϾ50 keV͒ K-␣ conversion effi- to the support structure of transmission gratings. ciencies are from Schnurer et al. ͑1996͒ and Tillman et al. For x-ray wavelength below 6.5 nm, crystals with high ͑ ͒ ͑1997͒, medium energy K-␣ conversion efficiencies reflectivity are available Beiersdorfer et al., 2004 .In ͑ Ͻ Ͻ ͒ ͑ ͒ x-ray scattering experiments in the range 3ϽEϽ9 keV, 10 keV E 50 keV are from Beg et al. 1997 , Ya- ͑ suike et al. ͑2001͒, and Park et al. ͑2004͒, and low energy Bragg crystals in the mosaic focusing mode Yaakobi and Burek, 1983͒ have been employed to spectrally dis- conversion efficiencies are from Rousse et al. ͑1994͒ and perse the scattered photons onto an efficient detector, cf. Wharton et al. ͑1998͒. The chlorine data are from Urry et Sec. IV.D. The crystals intersect a solid angle of al. ͑2006͒ and the titanium data from Kritcher, Neu- 0.1 radϫ3 mrad/4␲ to measure the spectrum for a de- mayer, Castor, et al. ͑2008͒. fined k vector ͑Pak et al., 2004͒. Alternatively, a wide Thermal He-␣ foil sources provide a conversion effi- ␩ Ӎ angle spectrometer with a highly oriented pyrolytic ciency of x 0.004 at 4.75 keV into a bandwidth of ͑ ͒ ⌬ graphite HOPG crystal has been developed by Garcia E/E=0.5%, providing a total photon number of 5 Saiz et al. ͑2007͒ to measure the energy-resolved spec- ϫ 12 10 photons per1Jofincident laser energy and into a trum of scattered x rays from a dense plasma over a ␲ solid angle of 4 sr. Assuming laser pulses with total wide range of angles in a single shot. energy of 1 kJ per 100 ps and laser intensities of The ideal mosaic crystal consists of mosaic blocks with 15 −2 2 10 Wcm giving a surface area of 1 mm . As outlined slightly different orientation of the reflecting planes above the probe x rays will interact with the dense characterized by the mosaic spread ␥. These blocks must ⍀ plasma subtending a solid angle of d =1 sr. This will individually fulfill the Bragg condition n␭=2d sin ␪ , ͑ ͒ B yield a peak x-ray brilliance of B 4.75 keV =5 where n is the diffraction order, ␭ is the x-ray wave- ϫ1012 photonsϫ103 ϫ͑4␲͒−1=4ϫ1014 photons/100 ps ␪ length, and B is the Bragg angle. This property leads to ͑0.5% bw/sr͒ at the dense plasma in a single shot. a broad rocking curve as shown in Fig. 19. In addition, At 4.5 keV, K-␣ sources provide two orders of mag- the mosaic crystal should be thick enough so that it is nitude smaller photon numbers in approximately 10 ps almost completely reflective in the middle of the mosaic x-ray pulses; this pulse length has been measured by distribution; a reflectivity of 2–9 mrad has been re- Chen et al. ͑2007͒ for picosecond laser irradiation. In the ported by Marshall and Oertel ͑1997͒ and Pak et al. present experiments, 300 J of short-pulse laser energy ͑2004͒ for HOPG crystals with a ␥=0.4° and ␥=3.5°. has been routinely delivered on target with These measurements have been performed in first and B͑4.75 keV͒=1.7ϫ1012 photons/10 ps ͑0.5% bw/sr͒,to second order and with x-ray energies in the range 4.5 be substantially improved with more energetic short- ϽEϽ9 keV. For the 12.6 keV Kr K-␣ line, Döppner et pulse laser systems. Therefore, with increasing x-ray en- al. ͑2008͒ have shown that the reflectivity of HOPG ergy, K-␣ sources become more favorable providing drops by nearly 2 orders of magnitude from 4 mrad to roughly the same x-ray probe energy as thermal He-␣ or 0.07 mrad when going from first to fourth diffraction Ly-␣ sources at about 22 keV, the probe energy pres- order. The reflectivity for the 22 keV Ag K-␣ line is

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1649

According to Ice and Sparks ͑1990͒, the resolution in the dispersive direction is determined by geometric ab- ⌬ Ӎ⌬␪2 ␪ ͑ ͒ errations with xa F/2 tan B angles in radians . These aberrations result in negligible broadening. Other broadening mechanisms which affect the focus and resolution of the emission lines have been considered by Pak et al. ͑2004͒. Estimates for 4.51 keV ͑Ti K-␣͒ up to 8.64 keV ͑Zn K-␣͒ indicate that finite source size and natural linewidth result in ⌬E/EϽ5ϫ10−4 while intrin- sic defocusing accounts for ⌬E/EӍ2ϫ10−4. Volume ͑depth͒ broadening, however, appears to be significant resulting in ⌬E/EӍ1ϫ10−3. The latter also induces an asymmetry enhancing the high energy wing of the spec- FIG. 19. Rocking curves for HOPG and LiF crystals at tral feature. On the other hand, in the sagittal direction ⌬ ␥ ␪ 4.51 keV. From Marshall and Oertel, 1997. the mosaic broadening of ym =2 F sin B is the dominant mechanism severely limiting the spatial resolution. about a factor of 3 smaller than at 12.6 keV with a In case of a curved HOPG crystal, the aberrations in the smaller reduction in reflectivity for higher diffraction or- sagittal and dispersive direction mix, thus limiting the ⌬ Ӎ ders spectral resolution to E 10 eV. The use of a mosaic By differentiating the Bragg law, we obtain a simple crystal further results in a nonuniform intensity distribu- estimate of the crystal dispersion ⌬␭/␭=⌬E/E tion in the focal plane that need to be corrected by flat- ⌬␪ ␪ ⌬␪ fielding the spectrometers, e.g., with in situ measure- = /tan B, where E is the x-ray energy and is the angular spread of the incident x rays. On the other hand, ments of bremsstrahlung emission. the spatial distribution of energies ͑i.e., the dispersion͒ Figure 20 shows the results of the spectral resolution at a normal image plane is measurement of the graphite crystal using a titanium K-␣ x-ray source at 4.5 keV. When the distance between ⌬E E ͑ ͒ the source and the crystal equals the distance between ⌬ = ␪ , 58 x 2F tan B the crystal and the detector plane, a focussed image with ⌬x=2F⌬␪ and F the focal length ͑source-to-crystal ͑mosaic focusing͒ with a spectral resolution of ⌬␭/␭ distance͒. To obtain optimum signal collection, crystals =0.003 has been observed. If this ratio deviates signifi- curved along the sagittal ͑i.e., nondispersive͒ plane with cantly from 1, the mosaic structure of the crystal is ob- a radius of curvature of R=115 mm have been fielded by served, resulting in a rather poor spectral resolution. Urry et al. ͑2006͒. Best focusing in both the dispersive These tests have further confirmed that the graphite and sagittal directions is thus achieved when the source- crystal has a high reflectivity, a factor of 3 larger then to-crystal and crystal-to-image distances are given by F LiF, and more than 1 order of magnitude larger than ␪ =R/sin B. PET or KAP crystals.

ab a 10 b

Ti K source 3 mm; Mosaic 4.5 keV cell d/ =0.02

a “Focus” 1 ab b 0.5 mm; Ti K d/ =0.003 source 4.5 keV

FIG. 20. ͑Color online͒ 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 ⌬␭/␭=0.003 for spectrally resolved x-ray Thomson scattering measurements. From Glenzer, Gregori, Rogers, et al., 2003.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1650 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

1 CCD DQE up to x-ray energies of about 100 keV, and less sensitive Image Plate DQE than CCD cameras for EϽ4 keV. The DQE of IP detectors has been estimated by Amemiya et al. ͑1988͒ to be 80% for x-rays with energies

ency 8–20 keV. The high values are explained by high ab- i c CCD IP sorption efﬁciency of the phosphor and 1/300 lower Effi background levels than ﬁlm. The fading time constants 0.1 have been measured by Meadowcroft et al. ͑2008͒ to be

uantum sufficiently large for applications with high power lasers. Q The CCD and IP data shown in Fig. 21 have been ob- MCP tained in laser-produced K-␣ experiments by Neumayer ͑2008͒, where the DQE of IP is less than ideal due to additional noise from the short-pulse laser experiment. 0.01 According to Amemiya and Miyahara ͑1988͒ a higher 1 10 100 DQE is possible in more ideal conditions. The dynamic Energy (keV) range of 1–105 with a high linearity and the spatial res- ␮ ͑ ͒ FIG. 21. ͑Color͒ Quantum efficiency of CCD ͑back illumi- olution of 150 m full width at half maximum match nated͒ and MCP detectors ͑Bateman, 1977͒ as function of well for measuring the scattering features described in x-ray energy. Measured detection quantum efficiency data of Sec. III and from plasma samples considered in Sec. CCD and IP detectors from Neumayer ͑2008͒ and Izumi ͑2008͒ IV.A. shown together with the IP calculated absorption scaled to the Significantly higher efficiency of gated detectors than measured data at 3 and 4.5 keV. the values indicated in Fig. 21 with data by Bateman ͑1977͒ appears possible. A larger DQE of CsI photo- cathodes by at least one order of magnitude has been D. Detectors reported by Lowney et al. ͑2004͒ for use with grazing incidence x rays with energies Eഛ1 keV. For pulsed mi- In the x-ray energy range of Ͻ10 keV, charged crochannelplate detectors, the dependence on the angle coupled device ͑CCD͒ cameras are routinely used for of incidence has been investigated by Landen, Lobban, detection, primarily because of their high detection effi- et al. ͑2001͒. A factor of about 3 improvement can be ciency and versatility. The CCD linearity is ±0.25% for achieved when applying grazing incidence detection at ϳ signals between Ͻ0.5% and Ͼ80% of the full dynamic 3° as opposed to the configuration in present experi- range. Also important for quantitative measurements, ments at the Omega laser that have been performed at ϳ the CCD readout noise of Ͻ5 electrons is advantageous 8°. when compared to charge injection device ͑CID͒ cam- In the future, complementary metal oxide semicon- ͑ ͒ eras that show a linearity of ±3% and a readout noise of ductor CMDS image sensors may be also applied. The Ͻ240 electrons. Multiple readouts can partly improve quantum efficiency is comparable to front illuminated readout noise, but the number of readouts is limited by CCD detectors but with a fill factor of 65%. Their read- dark current. To measure scattering spectra temporally out noise as well as gain nonuniformity is significantly larger than for CCD cameras; the noise is 200 electrons resolved with a resolution of Ӎ100 ps, microchannel- per pixel with 3.6 eV per electron hole pair. The CMOS plate ͑MCP͒ detectors combined with CCD cameras development is directed toward smaller features and have been successfully fielded. lower voltage both affecting imaging performance, but Figure 21 shows the quantum efficiency ͑QE͒ for these factors also imply less sensitivity to neutron- CCD, MCP, and image plate ͑IP͒ detectors ͑Bentley, induced noise. For neutron energies of 3–10 MeV, 1977; Meadowcroft et al., 2008͒. Generally, the signal- CMOS detectors have Ͻ2ϫ10−10 damage sites per pixel to-noise ratio of the detected signal relates to the signal- and per incident neutron. CCDs have 4ϫ10−8 damage to-noise ratio of the input signal via the detective ͑ ͒ ͱ sites per pixel and per incident neutron, two orders of quantum efficiency DQE , SNRout= DQESNRin magnitude more than for CMOS. These features indi- ͑ ͒1/2 1/2 = DQENphotons , with Nphotons the statistical quantum cate that strong shielding is required in a neutron envi- ͑ ͒ noise Zanella 2002 . In case that other additional noise ronment and will likely limit routine scattering experi- ͑ sources are negligible e.g., readout noise or experimen- ments with these detectors to neutron flux environments ͒ tal fluorescence noise , we have DQE=QE, otherwise of Ͻ1014/4␲ neutrons. DQEϽQE. CCD cameras show a high QE approaching 1 for x rays with energies EϽ4 keV and becoming insensitive E. Photometrics for energies EϾ20 keV. For higher x-ray energies and also in high emp, neutron, and high gamma flux environ- In x-ray Thomson scattering experiments, the intense ments, film has traditionally been the alternative to x-ray source, the closely coupled geometry, and the high CCD cameras, and is now being replaced by IP ͓see detection efficiency result in a substantial fraction of ͑ ͔͒ ϫ 23 −3 Izumi et al. 2006 . These x-ray detectors are sensitive scattered photons. For ne =3 10 cm , the Thomson

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1651

scattering cross section, and a path length of ᐉ=0.1 cm, Photometrics sets the practical limits on the temporal ␴ᐉ the scattering fraction is ne =0.02, close to the maxi- and spatial resolution of x-ray Thomson scattering mea- mum desirable for avoiding multiple scattering. Coupled surements. Scattering experiments with future x-ray with a source solid angle of 1 sr, the scattered efficiency FELs will provide single-shot detection signals compa- is 10−3, which is substantially larger than that available rable to Kritcher, Neumayer, Castor, et al. ͑2008, 2009͒ for optical Thomson scattering experiments, allowing with the added benefit of 20 fs temporal resolution and single-shot experiments with 1012 photons at the plasma possible spatial resolution of order 1 ␮m. In addition, sample. high repetition rate capability will provide spectra with Ͼ Ϫ The total number of the detected photons Nph,d,is signal-to-noise ratios of 100 compared to 10 20 in estimated from Eq. ͑10͒, single-shot experiments. For future experiments on NIF, a suitable x-ray probe ⍀ ␴ ᐉ EL plasma ne Th at energies of 22 keV where x rays penetrate through N = ͩ ␩ ͪͩ ␩ ͪͩ ͪ ph,d x att 2 ␩ Ϸ h␯ 4␲ ͑1+␣͒ the dense capsule with att 1/e, must overcome brems- −7 ͑ ͒−1 ⍀ strahlung emission of 10 J eV sr corresponding to ϫͩ det ␩ ͪ ͑ ͒ 104 detected photons per eV; cf. Fig. 15. In this case, ␲ Rcrystal d , 59 4 more than 2 orders of magnitude higher electron densi- ␩ ties than the experiment by Kritcher, Neumayer, Castor, where EL is the laser energy, x is the conversion effi- ␩ et al. ͑2008, 2009͒ will result in acceptable signal-to-noise ciency from the laser energy into the probe x rays, att is the attenuation of the probe xrays through the dense ratio using the short-pulse laser capability on the NIF. ⍀ With these K−␣ x-ray sources, a temporal resolution of plasma, plasma is the solid angle that the plasma subtends with respect to the x-ray source, ⍀ is the solid 10 ps and a scattering volume with scale length of det ␮ angle subtended by the detector as determined by the 100 m single have been achieved in single-shot experi- crystal and detector size and distance from the plasma, ments. ␩ Rcrystal is the integrated reflectivity of the crystal, and d is the efficiency of the detector including the MCP efficiency and filter transmission. The factor ͑1+␣͒−2 ap- V. X-RAY THOMSON SCATTERING EXPERIMENTS plies for estimates of the inelastic scattering signal. ␩ Ϸ ␣ A. Isochorically heated matter With att 1/e, a He- source at 4.75 keV effectively ͑ ␯͒␩ ͑⍀ ␲͒␩ Ϸ ϫ 14 scatters EL /h x plasma/4 att 2 10 photons at Figure 22 shows a schematic of backward and forward ͑ ͒ the target Sec. IV.B . The collection efficiency of scattering experiments on isochorically heated solid- ϫ ␲Ӎ ϫ −5 ͑ ͒ 0.1rad 3mrad/4 3 10 Sec. IV.C and a detection density beryllium. The beryllium is homogeneously and ␩ Ӎ efficiency for a MCP detector plus filter of d 0.01 isochorically heated by L-shell x rays from a mid-Z foil ͑ ͒ ϫ −7 Sec. IV.D result in a collection fraction of 3 10 . ͑1–2 ␮m thick Mg, Rh, or Ag foils wrapped around the Combined with a scattering fraction of 0.02, the total Be͒. The dense plasma is probed after t=0.5 ns with the 6 number of collected photons is 10 for the experiments narrowband x-ray probe from titanium or chlorine He-␣ ͑ ͒ of Glenzer, Gregori, Lee, et al. 2003 , Glenzer, Gregori, or Ly-␣ radiation. The high laser energy for producing ͑ ͒ ͑ ͒ Rogers, et al. 2003 , Gregori et al. 2004 , Gregori, Glen- the narrow band x-ray probe and the broadband heating ͑ ͒ ͑ ͒ zer, Chung, et al. 2006 , Glenzer, Landen, et al. 2007 , radiation provides sufficient photons for producing ho- ͑ ͒ and Lee et al. 2009 . mogeneous HEDP states of matter and probing in The detection efficiency has been substantially im- single-shot experiments. The same target platform has ␣ proved for experiments with K- sources. A curved been subsequently employed by Gregori et al. ͑2004͒ and crystal improved the collection efficiency by a factor of 2 Gregori, Glenzer, Chung, et al. ͑2006͒ to probe isochori- ͑ ͒ Urry et al., 2006 and use of a IP detector improved the cally heated carbon. Results for carbon are presented in detection efficiency by a factor of about 50. At 4.5 keV, Sec. V. B . Isochoric heating of experiments has the ad- 12 the K-␣ source provides 2ϫ10 photons into 1 sr. For vantage that the mass density is known a priori so that the experiments of Kritcher, Neumayer, Castor, et al. the electron density can be directly inferred if the ion- ͑2008͒, the target subtends 0.25 sr and ᐉ=0.01 cm yield- ization state is known or vice versa. Ӎ ϫ 3 ing Nphotons 3 10 photons collected in a single shot. Figure 23 shows an example of a backscattering spec- For use with a CCD camera with a gain of gCCD=3.5 trum from heated beryllium. Elastic scattering from ͑ ͒ eh/count electron holes per count and geh=3.6eV/eh is both the titanium He-␣ radiation at 4.75 keV and Ly-␣ the energy required to liberate an electron hole ͑eh͒ pair radiation at 4.96 keV is observed together with the per incident photon of energy E, a total number of downshifted Compton scattering feature. The Compton ϫ counts of Nphotons E/gCCDgeh is expected. Experiments scattering feature shows significant broadening due to by Kritcher, Neumayer, Castor, et al. ͑2008, 2009͒ and the thermal motion of the electrons resulting in factor of Kritcher, Neumayer, Lee, et al. ͑2008͒ have employed Ͼ2 larger width than scattering spectra from cold beryl- curved HOPG crystal spectrometers illuminating about lium, i.e., no heater laser beams ͑see Glenzer, Gregori, 50ϫ20 pixels. Thus, the collected photons are distrib- Lee, et al., 2003͒. The spectrum is fitted by theoretical uted over 103 pixels resulting in a scattering signal of spectra, employing the measured spectrum of the tita- about 1000 counts. nium probe x rays, spectrometer resolution, and the dy-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1652 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

6 Best fit: Best fit ) Te =53eV X-ray scattering data 4 kS T =53eVfrom 4.75 keV e arb. Units the Compton =125 ( k0 Backward scattering wing k Scatter 2 70 eV Intensity (arb. Units) Intensity 30 eV

0 =0.3 = 125˚ 4.4 4.6 4.8 5.0 4.4 4.6 4.8 Energy (keV) Energy (keV) kS 2.96 keV ͑ ͒ ͑ ͒ =40 k Forward FIG. 23. Color Scattering spectrum blue dots from isochori- Scatter ͑ ͒ k0 cally heated beryllium. Elastic and inelastic Compton scattering components are observed from both the titanium He-␣ and Ly-␣ probe x rays ͑left͒. The spectrum is well fitted with theoretical scattering spectra employing the dynamic structure factor of Sec. III ͑red line͒. The fit to the red wing of the Compton scattering spectrum is sensitive to the temperature FIG. 22. ͑Color͒ Schematic of scattering experiments employ- ͑right͒. These data provide temperatures of T =53 eV and ing the titanium He-␣ spectral line in backscattering at ␪ e densities of n =3.3ϫ1023 cm−3 characterizing the solid density =125° or the chlorine Ly-␣ line in forward scattering at ␪ e plasma regime with an error bar of about 10%. From Glenzer, =40°. The inset shows a Ly-␣ source spectrum with an inten- Gregori, Lee, et al., 2003. sity contrast of dielectronic satellites to Ly-␣ of ഛ5% for up to 7 kJ of laser energy to produce the narrowband x-ray probe radiation. The k-vector diagrams show the averaged scattering To access collective scattering, in a weakly degenerate vectors in both geometries yielding noncollective or collective solid-density beryllium plasma with electron tempera- Ӎ scattering. Up to 15 kJ of laser energy has been used for pro- ture of the order of the Fermi temperature, Te TF ducing the broadband L-shell heating radiation for isochori- =15 eV, requires forward scattering with ␪=40° and cally heating the target. A density contour plot from radiation- ͑␭Ϸ ͒ x-ray probe energies of order E0 =3 keV 0.4 nm . hydrodynamic modeling using the code LASNEX has been With these conditions, the scattering is predominantly overlayed on the Be target, indicating that only homoge- probing k vectors with k=1 Å−1. Calculating the screen- neously and isochorically heated beryllium is in the field of ␣ view of the spectrometer at the time of the measurements. ing length at the effective temperature results in =1.6. Note that the color map of the density contour is displayed. In this regime, the scattered light spectrum shows collec- From Glenzer, Gregori, Lee, et al., 2003 and Glenzer, Landen, tive effects corresponding to scattering resonances off et al., 2007. ion acoustic waves and off electron plasma waves, i.e., plasmons. Figure 24 shows experimental scattering spectra from namic structure factor of Sec. III, providing temperatures and densities from the broadening of the Compton Ion feature downshifted line and from the intensity ratio of elastic to 3 Best fit of Plasmon [elastic scattering] 23 -3 2 ne =3x10 cm inelastic scattering components, respectively. The figure Best fit n =3x1023 cm-3 e n = also shows the sensitivity of the shape of the red Comp- 2 Plasmon T =12eV e e 23 -3 ton scattering wing to the electron temperature. The lat- scattering with collisions 4.5x10 cm Intensity

Intensity Detailed ter is inferred from these data with an accuracy of about 1 10%. balance A similar analysis has been performed when inferring 0 0 the ionization state and the electron density from the 23 -3 ne =1.5x10 cm E intensity ratio of elastically to inelastically scattered ra- 2.9 3.0 2.90 2.92 2.94 2.96 diation. With these conditions with large values of k and Energy (keV) Energy (keV) where the contribution of weakly bound electrons to the ͑ ͒ ͑ ͒ intensity of the inelastic scattering component is small, FIG. 24. Color Scattering spectrum gray line from isochori- the intensity of the elastic scattering feature approaches cally heated beryllium. Plasmon scattering downshifted and upshifted in energy is observed, where the intensity of the up- f2 ͑Peyrusse, 1990͒. For the data in Fig. 23 the ratio shifted plasmon is reduced according to detailed balance, cf. yields the density with accuracy of about 15%. For lower Eq. ͑18͒. The spectrum is best ﬁt with theoretical scattering temperature conditions, however, where bound elec- ϫ 23 −3 ͑ ͒ spectra of Sec. III for Te =12 eV and ne =3 10 cm left . trons scatter nonelastically, the contributions from free Also shown is the comparison of the plasmon spectrum with and weakly bound electrons blend. Nonetheless, the ϫ 23 calculations for two different densities, ne =4.5 10 and 1.5 ϫ 23 −3 bound-free spectrum is well known and can be distin- 10 cm , indicating that the spectrum is best ﬁt for ne =3 guished from the Compton scattering spectrum of the ϫ1023 cm−3. The dashed curve represents a calculation for a ϫ 23 −3 ͑ ͒ free electrons yielding temperature and density informa- density of ne =3 10 cm that is neglecting collisions right . tion with slightly larger error bars. From Glenzer, Landen, et al., 2007.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1653 isochorically heated beryllium measured in forward scat- 4 ␣ ACTEX tering of the chlorine Ly- line at 2.96 keV. To resolve Non-collective x-ray the plasmon frequency shift and damping in forward scattering data LASNEX Comptra )

scattering, the x-ray bandwidth has to be smaller com- 3

- SCAALP

pared to Compton scattering measurements. In Fig. 24, cm 3 the plasmon shift of E=28 eV was resolved by the chlo- 23 rine Ly-␣ probe radiation with an effective bandwidth of 7.7 eV and no signiﬁcant dielectronic satellite radiation

on the red wing of the Ly-␣ doublet ͑see inset in Fig. 22͒. Density (10 Also shown are theoretical scattering profiles that rep- Collective x-ray 2 scattering data resent a convolution of the theoretical form factor S͑k,␻͒, calculated for the range of k vectors of the experiment, with the spectral resolution of 7.7 eV. 0 20 40 60 The ion feature is observed as an elastic scattering Temperature (eV) peak at E0 =2.96 keV that is not resolved in this experiment. On the lower energy wing of the ion feature, we FIG. 25. ͑Color͒ The measured densities from noncollective observe a strong plasmon resonance at 2.93 keV. On the and collective x-ray Thomson scattering experiments shown as higher energy wing with nearly the same frequency shift, functions of the measured electron temperature together with the data show a weak up shifted plasmon signal. Com- calculations from the codes ACTEX, LASNEX, COMPTRA, and pared to the intensity of the down-shifted plasmon, the SCAALP. The error bars in electron density are larger for the intensity is reduced by the Bose function ͓cf. Eq. ͑18͔͒ collective scattering data because of noise. reflecting the principle of detailed balance. The intensity ratio of these plasmon features is thus sensitive to the In experiments, the temperature of the beryllium temperature. In this experiment, the signal-to-noise ra- plasma has been varied using different mid-Z foils, i.e., tio only allows deducing an upper limit from the inten- Mo, Rh, and Ag, to convert the laser energy to L-shell sity ratio of T Ͻ25 eV. e heating x rays. In addition, on some shots only half of Collision effects on plasmons in S͑k,␻͒ are accounted the laser energy per beam has been applied, probing a for in the BMA which also incorporates degeneracy ef- temperature range of up to T =53 eV. Figure 25 shows fects. While the frequency shift is not affected, colli- e the comparison between experimental data and calcula- sional damping cannot be neglected. The fit provides a tions of densities and temperatures using various chemi- temperature of Te =12 eV and density of ne =3 cal and physical picture models. ϫ 23 −3 ͑ ͒ 10 cm . Figure 24 right indicates that the density is At the highest temperatures, the measured electron accurately determined by the shift of the plasmon. For density agrees with various ionization balance models, the small k vectors probed in the present experiment, i.e., the codes ACTEX from Rogers and Young ͑1997͒ and significant deviations from the random phase approxi- Rogers ͑2000͒ using the activity expansion method, the ͑ ͒ mation are not expected see also Fig. 5 yielding an code COMPTRA from Kuhlbrodt et al. ͑2005͒ using the error bar in density that is solely determined by the partially ionized plasma model, LASNEX from Zimmer- signal-to-noise ratio and the quality of the fit. The calcu- man and Kruer ͑1975͒, and SCAALP from Renaudin et al. ϫ 23 ϫ 23 −3 lations for densities of ne =4.5 10 and 1.5 10 cm ͑2002͒ using a density-functional plasma model. On the indicate that the error bar in density is of order 20%; a other hand, at lower electron temperatures, the role of value that has been improved with better signal-to-noise delocalized electrons requires the calculation of all pos- ratio; cf. Sec. V. C . sible interactions between the plasma constituents in- Inferring the temperature from the collective scatter- cluding the screening of the bound states. For large den- ing spectrum requires accurate measurements of the up- sities, the classical Debye-Hückel ͑Yukawa͒ potential shifted plasmon and the application of detailed balance, needs to be replaced by quantum interaction potentials or alternatively the width of the plasmon may be used. that approach the thermal de Broglie wavelength. This The latter is determined by Landau damping and colli- allows the calculation of the number of electrons that sional damping. For the conditions in isochorically are no longer bound to a single ion. These electrons are heated beryllium, collisions account for an additional free or weakly bound like the conduction electrons in a broadening of approximately 5 eV. The difference can metal and are accounted for in ACTEX. On the other be as large as a factor of 1.5 for scattering spectra when hand, LASNEX calculations that use interpolation func- k-vector blurring is negligible. Scattering experiments in tions between the zero and high temperature conductiv- a regime where damping by collisions dominates the ity limits from Desjarlais ͑2000͒; Desjarlais et al. ͑2002͒ broadening of plasmons are of interest to test models of showed deviations from the measured data at small tem- the collisionality and conductivity in dense plasmas. In- peratures. cluding the corrections due to collisions provides a good For plasmon measurements, temperatures up to fit of the plasmon spectrum for a temperature of 12 eV 15 eV have been accessed compared to the noncollec- and with the dynamic collision frequency calculated in tive scattering data because ͑1͒ the detector was gated the Born approximation. earlier during the heating of the plasma, ͑2͒ fewer laser

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1654 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

Ͼ Quantum potential dence. On the other hand, for k 2/aB, the HNC calcu- S (k)(f + q)2 3 ii lation overestimates the measured intensity of the elastic scattering signal. Calculations by Gregori, Glenzer, Roz- RPA Quantum potential mus, et al. ͑2003͒ using the analytical structure factor by 2 S (k)(f)2 Sii(k)(f + q) ii

actor ͑ ͒

f Arkhipov and Davletov 1998 agree with the data at 2 this k vector, but the k dependence is not reproduced. The apparent agreement appears to be due to the fact tructure S that the simple long k-vector limit is correctly obtained. 1 Future calculations employing physical picture modeling of the electron-ion interaction will be needed to obtain a more complete theoretical description of the observa- =40˚ = 125˚ 0 tions that include the known limits for small k. 0 1 2 3

-1 k(aB ) B. Matter heated by supersonic heat waves FIG. 26. ͑Color online͒ Elastic scattering amplitude data are shown that have been measured as function of the scattering Back et al. ͑2000͒, Edwards et al. ͑2000͒, and Constan- angle from isochorically heated beryllium for a density of ne tin et al. ͑2005͒ studied matter heated by supersonic heat ϫ 23 −3 =3 10 cm and temperature of Te =12 eV. Also shown are waves for testing different opacity models and to de- calculations using HNC with quantum potential models of ͓ ͑ ͒ velop efficient x-ray sources see Sec. IV.B and works by Schwarz et al. 2007 and the analytical RPA model of Back et al. ͑2001͒ and Fournier et al. ͑2004͔͒. In particu- Arkhipov and Davletov ͑1998͒. lar, the hydrodynamic efficiency of CH capsule ablators and the growth of the Rayleigh-Taylor hydrodynamic in- beams have been used to illuminate the L-shell con- stability for ICF studies depend on the soft x-ray opacity verter, and ͑3͒ a higher-Z element was used, i.e., Ag and the number of remaining bound carbon K-shell compared to the higher temperature experiments. Cal- electrons at temperatures of 200–300 eV. In this area, culations for Ag show 20–30 % lower conversion effi- x-ray Thomson scattering measurements are important ciency into L-shell radiation than for Rh or Mo. For since they provide data at the required accuracy to criti- these studies, the experimental temperature data agree cally test kinetics calculations. Moreover, target geom- well with radiation hydrodynamic simulations ͑Glenzer, etries that make use of hohlraum radiation to drive a Landen, et al., 2007͒, but the densities are underesti- heat wave are advantageous because radiation heating mated because the role of delocalized electrons is not can be accurately characterized. accounted for in integrated radiation-hydrodynamic The experiments of Gregori et al. ͑2008͒ have em- modeling with the code LASNEX. ployed 1.2 mm long, 1.2 mm diameter gold hohlraums The data in Fig. 25 show that the plasma parameters heated through a laser entrance hole with 6.2 or 8.5 kJ inferred from the plasmon spectrum measured in for- of laser energy at 351 nm in a 1 ns square pulse. The ward scattering are consistent with the data from the hohlraum produces a relatively uniform soft x-ray field Compton scattering spectrum measured in backscatter- that is emitted through the laser entrance hole and mea- ing. In this regime, the plasmon spectrum provides a sured with DANTE, an absolutely calibrated diode array robust density diagnostic. Nonetheless, future measure- broadband x-ray flux detector, providing a time depen- ments of the plasmon dispersion at large k will be of dent radiation temperature with maximum drive of ͑ interest to test the theory of local field corrections Gre- TRAD=270 eV. On the other end of the hohlraum, the gori, Glenzer, and Landen, 2003; Tauschwitz et al., 2007͒ radiation field drives a supersonic ionization wave into a and may directly lead to precision measurements of de- 300 ␮m thick carbon foam. The experiments employed tailed balance, collisions, and quantum diffraction. the titanium He-␣ line scattered off the heated foam Figure 26 shows the experimental elastic scattering providing temporally resolved temperature and ioniza- amplitude for various scattering angles from isochori- tion state data. cally heated beryllium. The absolute values for the am- These measurements have provided electron tempera- plitude have been calculated from the inelastic scatter- tures inside the heated foam of Te =130 eV and indicating feature, namely, the plasmon intensity for ing rapid heating and cooling of 100 eV/ns. The dynam- measurements at ␪=40° and the intensity of the Comp- ics of the supersonic heat waves driven by the hohlraum ␪ ton scattering feature at =125°. The HNC calculations radiation field at radiation temperature TR has been suc- with the quantum potential of Klimontovich and Kraeft cessfully modeled with a radiation flux model, where the ͑1974͒ suggest that the increase in elastic scattering is incident radiation flux is balanced by absorption, re- primarily dominated by the k-vector dependence of the emission, and convection. In particular, applying the structure factor, with the q factor a small correction. measured temperatures results in a correct description Ͻ The experimental data of Fig. 26 for k 1/aB are con- of heat wave propagation and burn through time. sistent with HNC calculations by Schwarz et al. ͑2007͒ The complete ionization balance has been measured with weak quantum potentials ͓see, e.g., Klimontovich in isochorically heated carbon by Gregori, Glenzer, and Kraeft ͑1974͔͒, which is leading to OCP-like depen- Chung, et al. ͑2006͒ employing a variety of target plat-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1655

2 Ion feature Ion feature Scattering data Scattering data n =7.5x1023 cm-3,T =13eV n = 7.5 x 1023 cm-3,T =13eV e e e e n =3x1023 cm-3,T =13eV 6 2 e e n =12x1023 cm-3,T =13eV e e Laser driven CH gas (gasbag) 1 Soft x-ray driven foams Plasmon Plasmon scattering scattering 4 (hohlraum) 1 Intensity (arb. Units) Intensity (arb. Units)

Ionization state Hard x-ray driven foams (Rh L-shell) 0 0 2 5.8 6 6.2 6 6.1 6.2 Shock driven Energy (keV) Energy (keV) CH foils FIG. 28. ͑Color͒ Experimental and theoretical scattering spec- 0 0 100 200 300 tra are shown from shock-compressed Be at 25° scattering angle ͑Lee et al., 2009͒. The best ﬁt to the data yields n =7.5 Electron temperature (eV) e ϫ 23 −3 ͑ ͒ 10 cm , Zf =2, and Te =13 eV left . The plasmon energy is ͑ ͒ FIG. 27. ͑Color online͒ The measured ionization state of car- a sensitive measure of the electron density right . bon is shown as function of the measured electron temperature ͑Gregori, Glenzer, Chung, et al. 2006͒. These experiments em- the transition region between heliumlike to hydrogen- ploy noncollective x-ray Thomson scattering experiments on like carbon, the ionization balance calculations agree various target platforms. The experimental data points corre- well with the experiments. These results warrant future spond to laser driven gas bag experiments, soft x-ray driven work on kinetics collisional radiative modeling to im- carbon foams, hard x-ray driven carbon foams, and cold plastic prove the accuracy of the calculations in the aforemen- foils. Also shown are calculations from kinetic calculations us- tioned transition region. ing the code FLYCHK for densities of 1021 cm−3 ͑dotted line͒, 1022 cm−3 ͑dashed line͒, and 1023 cm−3 ͑solid line͒ and COMPTRA for carbon at 0.2 g cm−3 ͑solid dark line͒. C. Shock-compressed matter

Fermi-degenerate dense plasmas are accessed in laser forms that span electron temperatures from cold to fully shock-compressed foils using the target platform shown ionized carbon at 280 eV. For this purpose, Gregori, in Fig. 14. The laser beams directly illuminate the foil ͑ ͒ Glenzer, Chung, et al. 2006 extended the expressions in with intensities in the range 1014ϽIϽ1015 Wcm−2 com- Sec. III to calculate the x-ray scattering form factor in pressing the foil by ablation pressure in the range multicomponent plasmas. Figure 27 shows experimental 20–60 Mbar. In the experiments by Lee et al. ͑2009͒, data from various experiments. The mid-to-high Te data beryllium was employed and x-ray Thomson scattering −3 points were obtained on low density ͑0.2 g cm ͒ carbon measurements at various times during the drive resulted ͑ foam targets driven by hohlraum soft x rays Gregori et in electron densities of n =8ϫ1023 cm−3 and tempera- ͒ e al., 2008 , while the mid-to-low Te points correspond to tures of 13 eV. The experiments by Lee et al. ͑2009͒ es- hard x rays isochorically heated higher density tablished x-ray Thomson scattering as a direct diagnostic −3 ͑0.7 g cm ͒ carbon foams ͑Gregori et al., 2004͒. A low for characterizing compressed states of matter. temperature data point on cold CH plastic foils Figures 28 and 29 show the experimental scattering ͑1gcm−3͒ from Sawada et al. ͑2007͒ and a high tempera- data for 25° and 90° scattering angle accessing the col- ture data point from a fully ionized gas bag plasma lective and noncollective regimes, respectively. Also ͑MacGowan et al., 1996; Glenzer et al., 1997͒ are also shown in the ﬁgure. The complete data set presents the 2 2 Ion feature Ion feature full ionization balance curve for carbon over a large Scattering Data Scattering Data n =7.5x1023 cm-3 T =13eV,T =13eV Ͻ e e i 23 -3 T =8eV,T =8eV range of electron temperatures in the range 2 T ) n =3x10 cm e i e e ts

i T =30eV,T =30eV 23 -3 e i 21 n n =12x10 cm Ͻ Ͻ e 300 eV and densities in the range 10 ne U . Ͻ 23 −3 b ar

10 cm . ( 1 1 ty Comparison with ionization balance calculations is i ntens given in Fig. 27 for the COMPTRA ͑Kuhlbrodt et al., 2005͒ I Compton scattering Intensity (arb. Units) Compton scattering and FLYCHK ͑Chung et al., 2003͒ codes. The calculation reported in the ﬁgure for COMPTRA corresponds to the −3 0 0 case of solid density carbon at 0.2 g cm . The model 5.8 6 6.2 5.8 6 6.2 shows good agreement with the data except in the Energy (keV) Energy (keV) 120–170 eV range, which is near the transition from he- FIG. 29. ͑Color͒ Experimental and theoretical scattering spec- liumlike to hydrogenlike carbon. Similar behavior is also tra are shown from shock-compressed Be at 90° scattering shown by FLYCHK for which three calculations are ͑ ͒ angle Lee et al., 2009 . The best ﬁt to the data yields ne =7.5 shown with densities of 1021 cm−3 ͑dotted line͒, ϫ 23 −3 10 cm , Zf =2, and Ti =13 eV. The width of the Compton 1022 cm−3 ͑dashed line͒, and 1023 cm−3 ͑full line͒. From energy is a sensitive measure of the electron density ͑left͒ and Saha equilibrium, the calculations for higher densities the intensity ratio of elastic to inelastic scattering is sensitive to result in a smaller ionization states. With exception of the ion temperature ͑right͒.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1656 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … shown are corresponding theoretical spectra. In these experiments, the 6.2 keV He-␣ x-ray spectrum has been 14 used for probing the compressed beryllium with the advantage that these x rays are of sufﬁciently high energy 12 to penetrate through the high density plasma, and with Te(eV) the disadvantage that the probe consists of two spectral features of comparable intensity, namely, the 6.181 keV 10

Mn He-␣ and 6.153 keV intercombination x-ray probe Parameter n (1023 cm-3) lines. Gaussian convolution has been performed with 8 e FWHM=11 eV in the case of the intercombination line and FWHM=14 eV for the He-␣ spectral feature, re- 6 spectively. Three weak satellite lines whose intensities relative to the main spectral features have also been de- 15 20 35 termined with direct measurements of the x-ray probe Pressure (Mbar) spectrum have been included in the fit. For forward scattering at an angle of 25°, the collec- FIG. 30. ͑Color͒ Comparison between modeled and measured tive scattering regime has been accessed with ␣Ӎ1.6. densities ͑in units of 1023 cm−3͒ and temperatures ͑in units of Two plasmons have been observed, one corresponding eV͒ from x-ray Thomson scattering measurements shown as to the He-␣ and one to the intercombination line. The functions of pressure. Black square and black circle symbols plasmons are downshifted in energy by 50 eV, indicating represent densities and electron temperatures from plasmon scattering. Red squares and red circles represent the densities an electron density of n =7.5ϫ1023 cm−3. Assuming an e and ion temperatures inferred from Compton scattering. Also ionization state of Zf =2 consistent with simulations and shown as dashed lines are the results from radiation- temperature measurements in the noncollective regime hydrodynamic modeling. ͑cf. Fig. 29͒ results in a compression of 3.3 for a laser drive intensity of 3ϫ1014 Wcm−2 and a calculated pressure of 28 Mbar. matter these contributions become important due to the Also shown in Fig. 28 are theoretical spectra for two lowering of the ionization energy. electron densities, i.e., noncompressed and approxi- Figure 30 shows the measured densities and tempera- mately five times compressed beryllium indicating that tures as function of the pressure. The Compton scatter- the electron density is accurately measured from the fre- ing data are consistent with the plasmon data with the quency shift of the plasmons. Calculating the standard inclusion of bound-free transitions using the model of deviation between the data and theoretical spectra for Schumacher et al. ͑1975͒. Without bound-free transi- variety of conditions shows that the error bar is approxi- tions, the density inferred from the tail of the Compton mately 10%. line is larger than obtained from the plasmon frequency ⑀ These densities result in a Fermi energy of F =30 eV. shift. For these compressed states of matter, Eq. ͑44͒ With these conditions, the Compton scattering spectrum indicates about 20 eV additional lowering of the ioniza- of the He-␣ and intercombination x-ray probe lines tion energy when compared to isochorically heated mat- measured in non-collective scattering with ␣Ӎ0.5 at a ter. This observation indicates that future measurements scattering angle of 90° on a separate shot and with the of the ionization energy lowering in compressed matter same laser drive shows a paraboliclike spectrum. As ex- at densities beyond the Inglis-Teller limit will be acces- pected for a Fermi-degenerate plasma, the width of the sible. The experimental data are also consistent with Compton spectrum provides the Fermi energy. In addi- radiation-hydrodynamic simulations of this experiment tion, the intensity ratio of the elastic to inelastic scatter- with the code HELIOS ͑MacFarlane et al., 2006͒ indicating feature from Fermi-degenerate plasmas is sensitive ing that density and temperature are modeled to accu- to the ion temperature because elastic scattering is de- racy of order 10%. These findings demonstrate that con- pendent on the ion-ion structure factor. ditions reached in ICF capsule implosion experiments For the theoretical scattering spectra the electron den- can be directly measured with x-ray Thomson scattering. sity and the temperature have been varied. For the Generally shock-compressed matter experiments have analysis, Te =Ti and Z=2 have been assumed consistent begun delivering data, indicating that critical tests of col- with calculations and with observations from isochori- lisions, bound-free contributions, ionization energy low- cally heated Be. Density and temperature obtained in ering, and ion-ion structure factor calculations are acces- ϫ 23 −3 this way are ne =7.5 10 cm and T=13 eV. The error sible. For example, the plasmon spectra of Fig. 28 are bar due to noise for these measurements is of order well fitted with the theory presented in Sec. III. How- 5–10 %. The agreement between the density inferred ever, data with higher spectral resolution and less noise from the plasmon and Compton data indicate good shot- will be needed to test the different calculations for the to-shot reproducibility. Moreover, the results are consis- collision integral and the plasmon width as presented in tent with the structure factors used at large k and with Fig. 8; while none of these calculations can be ruled out, the bound-free contributions to the Compton line shape the calculation using TCS provides a consistent fit to the that result in low energy wings. For compressed states of data.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1657

Burkel ͑1996͒. The scattering experiments on isochorically heated beryllium and carbon, on the other hand, showed strong elastic scattering peaks as described by Peyrusse ͑1990͒. Scattering from shocked CH-foils by Sawada et al. ͑2007͒ also showed elastic scattering consistent with Eq. ͑32͒, and when adding bromide impurities to the CH-foil, the elastic scattering amplitude has been observed to increase accordingly. To test the structure of correlated systems, future work employing angularly resolved scattering from states of matter spanning high coupling and high densities will be of great interest.

FIG. 31. ͑Color͒ Elastic scattering amplitude data shown as function of the scattering angle from shock-compressed Li by D. Coalescing shocks Garcia Saiz, Gregori, Gericke, et al., 2008. Also shown are calculations using various HNC models together with DFT In pump-probe experiments on the medium-size Titan Monte Carlo simulations. laser, Kritcher, Neumayer, Castor, et al. ͑2008͒ investigated lithium-hydride plasmas employing a nanosecond Measurements in laser shocked aluminum by Riley et laser beam to shock compress the solid target. Density al. ͑2007͒ attempted modeling of bound-free contribu- and temperature conditions have been measured with tions to the scattering spectra. Initial results indicate that x-ray Thomson scattering from a short-pulse laser- the data are bound by the impulse approximation of produced 10 ps K-␣ source. The power of the nanosec- Holm and Ribberfors ͑1989͒ and the form factor ap- ond laser beam was shaped with a 4 ns long foot at proximation of Schumacher et al. ͑1975͒. However, more 1013 Wcm−2 and a 2 ns long peak at 3ϫ1013 Wcm−2, accurate data and characterization of plasma tempera- launching two shock waves into the target. Radiation- ture, density, and ionization state such as obtained in hydrodynamic simulations with LASNEX showed that the beryllium will be required to obtain a critical compari- shock waves coalesce and compress the target by a fac- son with modeling. tor of three over the 600 ␮m focal spot area at 7 ns after Experiments by Garcia Saiz, Gregori, Gericke, et al. the beginning of the laser drive. ͑2008͒ on shocked lithium allowed inferring the ion-ion The conditions before and during coalescence are di- static structure factor when calibrating the measured rectly measured by varying the delay between the nano- elastic scattering amplitude with the measured signal second laser beam and the short-pulse laser produced from free electrons; the latter is determined by the f-sum K-␣ x rays. The x rays penetrate the dense material and rule ͓see Eq. ͑21͔͒ measurements of the elastic and inelastic scattering components at different times during the shock evolution ϱ 2␴ ប 2 are observed with a HOPG crystal spectrometer at a ͵ ͩ d ͪ ␻ ␻ Z k ͑ ͒ ⍀ ␻ d = . 60 scattering angle of about 40°. −ϱ d d free 2me Figure 32 shows the scattering spectra at t=4 and 7 ns Figure 31 shows the ion-ion structure factor data to- along with the K-␣ source spectrum measured with the gether with calculations using DFT for all electrons same spectrometer. Also shown are theoretical ﬁts to the coupled with molecular dynamics for the ions. Also measured scattering data using the theoretical form fac- shown are HNC calculations with various interaction po- tor, Eq. ͑32͒, convoluted with a Gaussian proﬁle that tentials. Within the error bars, these models describe the accounts for both the measured source width and crystal experimentally observed relative reduction in scattering resolution. for small k. However, in contrast to isochorically heated At t=7 ns, a plasmon is observed that is downshifted beryllium, unscreened HNC calculations that essentially in energy by about 25 eV. The shift is dominated by the use OCP-like behavior underestimate the total structure plasma frequency, i.e., by the electron density, while factor, indicating that screening effects are more impor- quantum diffraction results in a correction of order tant in lithium. The HNC calculations that use the po- 7 eV. For the low temperatures and high densities in the tential inferred from the DFT simulations show good shock wave, the system is degenerate with ␣Ӎ1 and in agreement with the data while calculations using the this case the width and shift of the plasmon yields the ϫ 23 −3 quantum potential of Deutsch ͑1977͒ are marginal. A electron density of ne =1.7 10 cm . Due to noise, the similar conclusion has recently been reached for error bar for the density is about 20%. shocked plastic by Barbrel et al. ͑2009͒. A temperature history of shock progression through A reduction in the elastic scattering features has also the compressed material has been extracted from the been reported from shocked aluminum by Ravasio et al. relative intensities of the elastic scattering components; ͑2007͒, and interpreted as a Debye-Waller-like correc- see Fig. 33. The intensity is calibrated by the inelastic tion due to high coupling parameters. These effects may scattering component for data at t=7 ns. This fact allows be expected when transitioning from a solid to a liquid inferring ion temperatures from the elastic scattering in- state; observations for lithium are reported by Sinn and tensity earlier in the experiment when the inelastic scat-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1658 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

FIG. 32. ͑Color͒ Experimental and theoretical scattering spectra from shock-compressed LiH shown at various times. At t FIG. 33. ͑Color͒ The temperature of shocked LiH shown as =7 ns, the scattering parameter is ␣Ӎ1 and the inelastic scat- function of time from x-ray Thomson scattering measurements tering component shows a plasmon downshifted by 25 eV from and from radiation-hydrodynamic modeling using different the K-␣ energy. At t=4 ns, the lack of inelastic scattering indi- EOS models ͑Kritcher, Neumayer, Castor, et al., 2008͒. The cates small temperatures and negligible ionization. Also shown range of temperatures for each model is accounting for LiOH is the K-␣ x-ray probe spectrum. From Kritcher, Neumayer, surface impurities ͑lower bounds͒ to no impurities ͑upper Castor, et al., 2008. bounds͒. 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-␣ x-ray pulse. tering component is weak. In particular, the dynamic structure factor provides the electron temperature as- on solutions of the single-particle quantum levels in the suming Te =Ti. In addition, the intensity of the elastic self-consistent field of an atom. The peak temperature scattering is insensitive to density. At early times, t and the experimentally observed coalescence time are in Ͻ 6 ns, the lack of inelastic scattering is a strong indica- good agreement with modeling that uses the SESAME tion of low ionization states, ZϽ0.1, and simultaneously EOS. These results demonstrate the capability to mea- Ͻ low temperatures, Te 0.5 eV. Although this procedure sure the temperature and density in dense matter during relies on knowledge of the structure factor in the range shock compression with 10 ps temporal resolution and of small k vectors, weak interaction potentials consistent to resolve small differences in coalescence time pre- with Fig. 26 have been employed, and the error bar is dicted by modeling. expected to be dominated by noise and has been esti- The experiments have shown the transition to a me- mated to be 20%. tallic plasma state in the solid phase resulting in the ob- At shock coalescence, rapid heating to temperatures servation of plasmons. This feature has further allowed of 25 000 K is observed when the scattering spectra testing of radiation-hydrodynamic calculations with dif- show the collective plasmon oscillations that indicate the ferent EOS models for shock-compressed matter. This transition to the dense metallic plasma state. At peak technique is opportune for HEDP experiments that will temperature, the calculations indicate pressures in the achieve extreme densities, e.g., on the National Ignition range of P=300–420 GPa. With Z=1 as determined Facility. In addition, with future x-ray FELs, high repeti- from the shape of the plasmon with an error bar of 10% tion experiments will provide the same photons for scat- and the electron density from the plasmon shift yields a tering experiments as employed in these studies, and mass density of ␳=2.25 g/cm3, corresponding to three data with high accuracy will be possible due to reduced times compression in excellent agreement with the cal- noise levels from multiple shots. culations. This value is consistent with the equation of ͑ ͒ state EOS data obtained from density functional per- VI. SUMMARY AND OUTLOOK turbation theory by Yu et al. ͑2007͒, and approach conditions at which Wang et al. ͑2003͒ predicted the X-ray Thomson scattering has been developed to pro- insulator-metal phase transition. vide accurate characterization of high energy density The range of temperatures shown in Fig. 33 are from plasmas. The interpretation of the inelastic scattering re- calculations that include various amounts of impurities sults from low to moderately coupled systems are based and oxide layers consistent with target characterization. on first principles. In particular, x-ray Thomson scatter- The quotidian equation of state model is consistent with ing allows noninvasive probing of the plasma conditions the wide range of temperatures accessed in this experi- and properties of dense and optically opaque matter. ment. The SESAME EOS includes atomic structure based Results from forward and backward scattering measure-

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy … 1659 ments in isochorically heated as well as shock- LIST OF ACRONYMS AND DEFINITIONS compressed matter have been shown to yield mutually ARC Advanced radiographic capability consistent results. BMA Born-Mermin approximation In particular, the electron density inferred from the CCD Charge coupled device frequency shift of plasmons agrees with the ionization CID Charge injection device balance measurements from the intensity ratio of the DFT Density-functional theory Compton scattering to elastic scattering component. DH Debye-Hückel Moreover, the electron temperature is inferred from the DQE Detective quantum efficiency spectrum of the Compton downshifted line that is di- EOS Equation of state rectly reflecting the electron velocity distribution provid- FEL Free-electron laser ing temperatures. GDW Gould-DeWitt These results show that the x-ray scattering technique HEDP High energy density physics can be applied to measure the compressibility of dense HNC Hypernetted chain matter from the plasmon frequency shift, or in case of a HOPG Highly oriented pyrolytic graphite degenerate system, from the Compton scattering feature ICF Inertial confinement fusion that will reflect the Fermi temperature. Future applica- IP Image plate tions will include measurements of the conductivity from LB Lenard-Balesen approximation the collisional damping of plasmons and investigations LFC Local field corrections of the validity of the random phase approximations at MCP Microchannelplate large values of the scattering vector k. NIF National Ignition Facility In addition, it is expected that the x-ray Thomson OCP One component plasma scattering technique will find new applications with the QE Quantum efficiency development of target platforms for testing important QEOS Quotidian equation of state physics processes in dense or strongly coupled plasmas. RPA Random phase approximation While first results on transitions to dense metallic TCS Two component screening plasma states have been obtained, future studies are ex- TM T matrix pected that will range from stopping power measurements to fast ignition studies and laboratory astrophysics experiments. An important application for measurements of the temperature and density include inertial ACKNOWLEDGMENTS confinement fusion capsule implosions. Accurate data This work was performed under the auspices of the will allow testing the fuel adiabat, i.e., the ratio of elec- U.S. Department of Energy by Lawrence Livermore Na- tron to Fermi temperature, and assess assembly of the tional Laboratory under Contract No. DE-AC52- fuel. First assessments have indicated that the scattering 07NA27344. The work of S.H.G. was also supported by will be dominated by the dense capsule fuel, thus, char- Laboratory Directed Research and Development acterizing the conditions most wanted for measuring the Grants No. 08-ERI-002 and No. 08-LW-004, and the assembly of thermonuclear fuel. Alexander-von-Humboldt Foundation. The work of Experiments to determine the physical properties of R.R. was supported by the Helmholtz association VH- these dense states of matter have begun to measure the VI-104 and by the Deutsche Forschungsgemeinschaft effects of collisions to yield the conductivity in high den- SFB 652. We thank O. L. Landen and G. Gregori for sity and strongly coupled plasmas. Since plasmons will their important contributions to the physics of x-ray Th- be observed simultaneously with the elastically scattered omson scattering. We further thank P. Beiersdorfer, Th. ion feature, these studies will also determine the struc- Bornath, D. Bradley, L. Cao, J. Castor, H.-K. Chung, C. ture factor in two component plasmas. Constantin, J. K. Crane, M. P. Desjarlais, L. Divol, T. Measurements of structure factors in addition to a di- Döppner, R. W. Falcone, G. Faussurier, R. Fäustlin, E. rect measure of density and ionization state in shocked Förster, C. Fortmann, D. H. Froula, D. O. Gericke, F. matter further indicate that scattering experiments can Girard, S. Haan, B. A. Hammel, J. Hastings, S. Hatchett, provide new insights in the ongoing controversy about A. Höll, B. Holst, N. Izumi, M. Koenig, W. D. Kraeft, A. the equation of state of deuterium. The studies will go Kritcher, N. Kugland, H.-J. Kunze, H. J. Lee, R. W. Lee, beyond testing the equation of state models in inte- B. J. MacGowan, A. J. Mackinnon, K.-H. Meiwes-Broer, grated radiation-hydrodynamic calculations by yielding D. D. Meyerhofer, E. Morse, D. H. Munro, P. Neu- directly compression or Hugoniot data when combined mayer, A. Ng, C. Niemann, A. Pak, A. Pelka, S. W. Pol- with a shock velocity measurement. laine, A. Przystawik, S. Regan, H. Reinholz, D. Riley, H. X-ray FEL facilities will allow experiments with 20 fs F. Robey, F. J. Rogers, G. Röpke, M. Roth, W. Rozmus, resolution and with much higher repetition rate. Future M. Schollmeier, V. Schwarz, B. Spears, R. Thiele, J. Tig- x-ray Thomson scattering experiments will be aimed to gesbäumker, S. Toleikis, R. Tommasini, T. Tschentscher, perform high precision measurements and, in particular, M. Urry, I. Uschmann, J. Vorberger, S. Weber, K. Wid- to determine thermodynamic properties and equilibra- mann, and U. Zastrau for many valuable discussions re- tion in high energy density plasmas. lated to this work.

Rev. Mod. Phys., Vol. 81, No. 4, October–December 2009 1660 Siegfried H. Glenzer and Ronald Redmer: X-ray Thomson scattering in high energy …

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