Matter under extreme conditions experiments at the Linac Coherent Light Source

MEC is equipped with two frequency doubled Nd:Glass laser systems [23, 71] that each deliver green (527 nm) light on target with energies up to 18 J for pulse durations larger than 15 ns. For typical applications with pulse lengths of several nanoseconds, the peak laser power is 1–2 GW per beam with variable pulse shaping and rise times of 0.5–1 ns see figure 8. The arrival time of the optical laser beams with respect to the x-ray beam can be changed during the experiment, and is accurate within 25 ps rms. Typical repetition rates for the Nd:glass laser are 1 shot every 7 min when using both lasers simultaneously.

To access a broad range of interesting regimes by dynamic compression it is desirable to further improve the existing systems. At highest priority is increasing the total laser energy on target, to provide precision laser pulse shaping, and shot-by-shot pulse monitoring with the goal to drive larger samples with spatially flat topped intensity profiles for planar shocks. In addition, cross timing and equivalent plane focal spot monitoring will enable measurements of the laser-beam uniformity at the plane of the target. For this purpose, the monopotassium phosphate frequency doubling crystals may be replaced by lithium triborate crystals for a 30% increase of laser energy on target. In addition, for tailoring the focal spot profile on target a suite of phase plates are available to deliver spot diameters in the range of 50 μm < d < 400 μm. To further reduce the effects of laser-beam non-uniformities experiments on solid targets have employed low-Z ablators of varying thickness from 2 μm < d < 50 μm. Future improvements in laser technology and additional amplifier stages for delivering increased laser energy on target will be important to continue accessing important new regimes of the HED phase space, see figure 2.

The existing MEC short pulse laser system is shown schematically in figure 9. The system is capable of producing 1 J pulses with a temporal width of 40 fs, and thus delivers a peak power of 25 TW. When focused to a ∼ 5 μm spot size on a target, this provides peak intensities in the relativistic regime where the electron quiver velocity approaches relativistic velocities, i.e. at intensities exceeding 10¹⁹ W cm⁻². The repetition rate of this system is determined by the final amplifier GAIA pump laser and is currently 5 Hz.

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At relativistic laser intensities, the contrast ratio between the main pulse and any possible pre-pulses or pedestal pulses is important. To reduce the pre-pulse intensity a cross-polarized wave generator (XPW) has been implemented. In this case, the pulse is compressed after the regenerative amplifier (Legend). The resulting short pulse can then be cleaned in a saturable absorber or XPW device. These pulse 'cleaning' techniques have demonstrated contrast ratios in the 10¹⁰ range, albeit at the loss of pulse energy. To compensate the loss an additional multi-pass amplifier (MPA) has been added. Because these MPA are pumped by CW diode-pumped Nd:YLF lasers, they provide an exceptionally stable, typically better than 0.2% rms energy stability, and a near perfect Gaussian spatial profile. Because these lasers run at 120 Hz, this will provide the option to have lower energy 30 mJ short pulses at 120 Hz to match the pulse rate of LCLS. The addition of a MPA that can be pumped by a 7 J, GAIA laser provides higher-energy high-power experiments at 25 TW albeit at reduced repetition rates of 5 Hz.

The system can be further upgraded to 200 TW peak laser power by adding a third MPA to the laser to be pumped by the MEC nanosecond laser. At this stage, larger diffraction gratings may be needed in the pulse vacuum compressor in order to accommodate the increased energy without damage. Also, a deformable mirror and wavefront sensor system will be fielded to optimize the wavefront for the tightest possible focal spot. This should provide 6–8 J of output energy in a ∼45 fs pulse width with good spatial wave front and temporal contrast. This high-power laser will produce focused peak intensities in the 10²⁰ W cm⁻² range at a repetition rate that is defined by the nanosecond laser at 1 shot every 3–10 min. It may further be possible to develop a burst mode which will deliver a 0.01 Hz shot rate followed by an extended cool down period that may be used for other experimental activities.

The size of the MEC target chamber with a radius of 1 m provides sufficient space to place in-vacuum optical and x-ray diagnostics. An extensive suite of x-ray scattering diagnostics is available including highly efficient graphite crystal spectrometers for single shot measurements of scattering spectra, high-resolution spectrometers for x-ray emission and absorption measurements, and areal detectors for angularly resolved scattering measurements, and phase contrast imaging. The suite of x-ray diagnostics is complemented by optical measurement capabilities, FDI and VISAR. In addition, high magnification optical backlighting has been applied using the optical short pulse laser. For alignment purposes, questar long distance microscopes are available providing a field of view of 0.5–5 mm with a spatial resolution of 7 μm. Below, we discuss examples of x-ray and optical measurements applied during the first shock-compressed matter experiments at MEC.

2.5.2.1. Highly annealed pyrolytic graphite (HAPG) spectrometer

To maximize the collection fraction, x-ray scattering experiments require spectrometers with high efficiency, but with high resolving power of ΔE/E < 10⁻³. In the multi-keV x-ray regime, artificial crystals from low-Z materials such as carbon with a mosaic spread of ∼0.1° approach these requirements. To further improve collection efficiency the distances from the crystal to both source and detector are typically chosen to be the same, resulting in mosaic focusing and high spectral resolution [73]. In addition, a cylindrically bent crystal in von-Hamos geometry will further optimize collection by about a factor of 2 [74].

At MEC, HAPG crystals have been fielded [75, 76]. These crystals have been shown to provide advantages over previously used HOPG [74, 77]. The point-spread function is dominated by so-called depth broadening. Due to negligible photo-absorption (the attenuation length of 8 keV x-rays in C is about 1 mm), x-rays penetrate deep into the crystal until they encounter a crystallite that is properly oriented to reflect the ray. In von-Hamos geometry, a x-ray point source will therefore be broadened in the detector plane and in the dispersion plane by 2D/tan(θ_B) with the crystal thickness D and the Bragg angle θ_B. Ray tracing simulations for a point source and monochromatic 8 keV x-rays have been performed for HAPG crystals of 30 × 30 mm² size and 51.7 mm radius of curvature (θ_B = 13°). The calculations show that the point spread function of the 40 μm thick crystal is about 6 eV wide with E/ΔE = 1300 suitable for resolving plasmon scattering spectra in dense plasmas. This crystal has been used for the forward scattering measurements. In backscatter, a 100 μm thick crystal has been employed with lower spectral resolution of 14 eV and E/ΔE = 550. This crystal results in a higher reflectivity, as determined by the integral over the reflection curve. Some decrease of reflectivity in the high-photon energy tail is observed due to photo-absorption in the crystal volume.

Figure 10 compares the instrument function for a single shot with the one for experiments that average over 700 shots with a seeded x-ray beam. Here, we observe that statistical fluctuation in the x-ray laser energy result in additional broadening of the effective profile. Even when taking these fluctuations into account the diagnostic is well suited for resolving the plasmon scattering spectrum and the plasmon frequency shift.

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2.5.2.2. Plasmon measurements

Plasmon measurements were first validated at MEC utilizing high-resolution measurements from isochorically heated targets at 120 Hz repetition rates. Subsequently, the technique was applied for single shot measurements on shock-compressed aluminum. Figure 11 shows the experimental scattering spectra from solid-density aluminum from Fletcher et al [57]. Results are shown from the forward scattering and backward scattering spectrometers with the x-ray beam in seeded and SASE mode of operation. The seeded x-ray beam provides a bandwidth of 1 eV and combined with 8 eV spectrometer resolution resolves the plasmon that is down-shifted from the elastic 8 keV scattering feature by 19 eV.

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Further, figure 11 shows the results of the backward scattering spectrometer which observes elastic scattering at 8 keV reflecting the instrument function of the spectrometer convolved with a Gaussian profile that accounts for the statistical fluctuations of the x-ray energy of a seeded beam over 700 shots. These measurements do not indicate an inelastic scattering feature in the vicinity of the plasmon peak validating that bound-free scattering is not affecting the results of the forward scattering measurements; bound-free scattering features are predicted to be negligible in this energy range and no feature has been observed with the backscattering spectrometer. For these measurements, the crystal in the forward scattering spectrometer is 40 μm thick while in backscattering we employed a 100 μm thick crystal giving rise to slightly different instrument functions and consequently slight differences in the spectral shape of the elastic scattering feature.

Because the x-ray seed power is generated from the random SASE process that fluctuates in energy and amplitude, less than 1/3 of the shots provide sufficient photons with adequate spectrum for single shot x-ray scattering experiments. When summing over 700 shots the spectrum shows a broad foot, but the bandwidth is mostly preserved greatly improving the resolution when compared to the SASE spectrum. In forward direction, the x-ray scattering spectra with SASE operation show a slight broadening of the red wing of the scattering spectrum, see figure 11, but the SASE scattering spectrum is not suitable for inferring the dense plasma conditions.

Figure 12 shows the theoretical fit of the dynamic structure factor, see section 3, to the experimental forward scattering spectrum. The shift of the plasmon peak yields the electron density of n_e = 1.8 × 10²³ cm⁻³ ± 5%. The electron temperature of the fit is 2 eV accounting for the isochoric heating of the aluminum by the x-ray laser pulse [22]. The ions are cold with temperatures of order 0.1 eV consistent with the observation of Debye–Scherrer rings from solid aluminum. Here, the plasmon frequency is extremely sensitive to the electron density, but temperature effects are too small to affect the plasmon energy shift. The accuracy of the electron density measurement is very high due to the sensitivity of the plasmon resonance to the plasma frequency. With n_e = Z/A 6.03 × 10²³ ρ (the electron density n_e in cm⁻³ and the mass density ρ in g cm⁻³), assuming the ionization state of Z = 3 and using the mass A = 26.98 u, the measured electron density results in ρ  = 2.7 g cm⁻³ as expected for solid aluminum.

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The MEC instrument is equipped with a line imaging VISAR system for velocity measurements of rapid surface motions that are typically in the range of 1–20 km s⁻¹. This diagnostic tool obtains accurate continuous velocity-time histories of surface motions. The VISAR measures velocities by using the Doppler shift of a green laser beam diffusely reflected from the moving surface. The reflected light is collected and collimated into a Michelson-type interferometer. The imaging field of view is 0.5–1 mm with a spatial resolution of 10 μm. The interferometer splits the light into equal parts so that the apparent optical path lengths of the two arms are equal to allow interference of spatially incoherent light, but at the same time with light from one arm delayed in time so that fringe shifts occur with wavelength changes. These requirements are satisfied by the use of an etalon in one arm. At MEC, several etalons of 25 mm diameter and with varying thickness between 5.072 mm–15.01 mm and of 50 mm diameter and with varying thickness in the range of 25.036 mm–75.04 mm are available. Two Hamamatsu streak cameras measure the fringes in two arms with different etalons providing a temporal window of 1 ns–1 ms and temporal resolution of approximately 20 ps. The system can operate at a repetition rate of about 0.1 Hz.

Figure 13 shows an example of a VISAR streak camera record when used in combination with the LCLS x-ray beam. This experiment uses a graphite target irradiated by a 1 TW frequency-doubled drive beam launching a shock wave that propagates through the target. The probe beam reflection from VISAR shows steady interference fringes from the backside of target that is coated with a reflective aluminum surface. The fringes disappear at the time of shock breakout when the optical reflectivity of the surface drops significantly. With exact cross timing of the VISAR probe laser, streak camera and drive laser these data yield the averaged shock velocity. In addition, step targets or wedge targets will provide more accurate measurements of the shock velocity. However, a unique feature can be observed at the time when the LCLS x-ray beam is fired (in this case about ¾ through the optical laser drive). At that time, the reflectivity of the back coating is reduced providing an accurate measurement of both the x-ray probe time with respect to shock breakout and the exact location of the LCLS beam in the target.

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Figure 14 shows examples of radiation-hydrodynamic design calculations of shock-compressed solid aluminum experiments that utilize experimental laser powers, pulse shapes and phase-plate focal spots [78] available at MEC. A 2 μm thick CH coating was applied on the laser drive sides. The simulations use the radiation-hydrodynamic code HELIOS [79]. The nanosecond laser beams are frequency converted to green delivering 527 nm laser light on target with the pulse shape shown in blue in figure 8 in a focal spot of approximately 50 μm.

The simulations indicate that the shocks propagate through the solid with an averaged shock velocity of 14 km s⁻¹, traversing 25 μm thick aluminum plus 2 μm CH and break out or coalesce in the center at about 1.9 ns after the start of the laser irradiation. These shock waves initially compress the solid by a factor of 2–3 and during coalescence they reach 4 times solid density and multi mega bar pressures. These conditions can be compared with the Rankine–Hugoniot relations with ρ₁/ρ₀ = D/(D − u), where D is the shock velocity, u is the particle velocity. For two-fold compression, i.e. ρ₁/ρ₀ = 2, we infer ½ D = u, and with a shock velocity of 14 km s⁻¹ (20 km s⁻¹) we obtain pressures of ΔP = ρ₀ Du ∼ 2.6 Mbar (5.4 Mbar). Here, the density and pressure define the Hugoniot curve of the shocked material. Experiments that will simultaneously measure the temperature will determine the thermodynamic state of the shock wave and will be important to estimate heat capacity and energy partition of the shocked material.

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To reach high compression states of solid targets by intense laser radiation in the laboratory, the technique of coalescing shocks is often applied [39, 44]. It is well known that a large number of shocks with a fixed pressure change results in adiabatic compression also described with negligible entropy changes. At MEC, the counterpropagating shock technique has been applied in experiments that reach several times solid density, temperatures of several 10 000 K, and pressures in excess of 1 Mbar. Here, the entropy largely determined by the conditions in the first shock. These extreme states of matter serve as model systems to investigate the thermodynamic state, the material structure, and the physical (microscopic) properties.

For these studies, the adiabatic exponent $\gamma =(\partial \;\mathrm{ln}\;p/\partial \;\mathrm{ln}\;\rho )$_S=const is a central quantity. For perfect gases, γ = c_p/c_v is the ratio of heat capacities at constant pressure and volume, respectively. Furthermore, γ determines the sound speed, and the EoS [1, 47, 48, 80–82]. For the examples below we choose ρ₀ = 2.7 g cc⁻¹. Here and in the following, we denote conditions in the uncompressed material by the index 0, the single shock compressed matter by index 1, and the region of shock overlap by index 2.

We apply the Rankine–Hugoniot equations [47] to determine the polytropic index γ. We describe each shock by the Rankine–Hugoniot equation which relates the particle velocities u₀, u₁, pressures p₀, p₁ and mass densities ρ₀, ρ₁ in the uncompressed and in the compressed region, respectively and which follow from the conservation of total mass, energy, and momentum. One finds for the energy jump across the shock front

$$\begin{eqnarray}{\mathcal{E}}_{0}-{\mathcal{E}}_{1} & = & {u}_{0}^{2}-{u}_{1}^{2}+\displaystyle \frac{{p}_{0}}{{\rho }_{0}}-\;\displaystyle \frac{{p}_{1}}{{\rho }_{1}}\\ & = & \displaystyle \frac{1}{2}({p}_{0}+{p}_{1})({\rho }_{1}^{-1}-{\rho }_{0}^{-1}).\end{eqnarray} \tag{ 1 }$$

These relations hold for each individual shock as well as for the two shock fronts that evolve after the shock collision between the single and the doubly shocked material and which travel in opposite directions away from each other. The adiabatic exponent γ is introduced to eliminate the internal energy ε = p/(γ−1) ρ. Accounting for different values of γ before and after compression, one derives for the compression ratio

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{1}}{{\rho }_{0}}=\;\displaystyle \frac{1+({p}_{1}/{p}_{0})\displaystyle \frac{{\gamma }_{1}+1}{{\gamma }_{1}-1}}{{p}_{1}/{p}_{0}+\displaystyle \frac{{\gamma }_{0}+1}{{\gamma }_{0}-1}}.\end{eqnarray} \tag{ 2 }$$

In the strong shock limit p₁/p₀ → ∞, one recovers the familiar result

$$\begin{eqnarray}&&\mathrm{lim}\\ &&{p}_{1}/{p}_{0}\to \infty \displaystyle \frac{{\rho }_{1}}{{\rho }_{0}}=\displaystyle \frac{{\gamma }_{1}+1}{{\gamma }_{1}-1}.\end{eqnarray} \tag{ 3 }$$

Note that the strong shock compression depends only on γ₁, i.e. the conditions in the compressed region. For a perfect monoatomic gas with γ = 5/3, one obtains ${\mathrm{lim}}_{{p}_{1}/{p}_{0}\to \infty }{\rho }_{1}/{\rho }_{0}=4.$ The pressure ratio is derived from equation (3) as

$$\begin{eqnarray}&&\displaystyle \frac{{p}_{j}}{{p}_{i}}=1+\displaystyle \frac{2{\gamma }_{j}-2{\gamma }_{i}\displaystyle \frac{{\rho }_{j}}{{\rho }_{i}}\;\displaystyle \frac{\;{\gamma }_{j}-1}{\;{\gamma }_{i}-1}\;\;}{\displaystyle \frac{{\rho }_{j}}{{\rho }_{i}}({\gamma }_{j}-1)-{\gamma }_{j}-1}.\end{eqnarray} \tag{ 4 }$$

The jump in particle velocities can be expressed as

$$\begin{eqnarray}&&{({u}_{i}-{u}_{j})}^{2}=\left(1-\displaystyle \frac{{\rho }_{i}}{{\rho }_{j}}\right)\displaystyle \frac{{p}_{i}}{{\rho }_{i}}\left(\displaystyle \frac{{p}_{j}}{{p}_{i}}-1\right).\end{eqnarray} \tag{ 5 }$$

In the case of two colliding strong shocks, symmetry of the problem with respect to the contact surface between the two shock waves at the moment of collision competely determines the final compression [47]. In particular, the material at the contact surface is at rest, u₂ = 0. Furthermore, the material velocity in the unshocked material u₀ is negligible compared to u₁, the velocity in the first, strongly shocked material region. For a system with constant heat capacity ratio γ₂ = γ₁ = γ₀, one recovers the following result for the second compression:

$$\begin{eqnarray}{\rho }_{2}/{\rho }_{1} & = & [2+(\gamma +1)({\rho }_{1}/{\rho }_{0}-1)]/\\ & & [2+(\gamma -1)({\rho }_{1}/{\rho }_{0}-1)].\end{eqnarray} \tag{ 6 }$$

For the ideal gas with γ = 5/3 one finds ρ₂/ρ₁ = 2.5. For a system where both density jumps are accurately measured one easily inverts the equations to obtain γ₁ and γ₂. For the example above with ρ₁/ρ₀ = 2 and γ₂ = γ₁ = 2 (see [44]), we find ρ₂/ρ₁ = 5/3 or ρ₂/ρ₀ = 10/3. Further, by varying the laser intensity on both sides of the foil and measuring the density jumps with, e.g, XRTS, a range of conditions can be accessed, and γ can be directly determined.

Spectrally resolved x-ray scattering has emerged as a powerful tool to measure the plasma conditions of dense compressed states of matter and as a core capability to determine the physical properties of MEC. XRTS provides the dynamic structure factor S(k, ω), which is central for calculating e–i collisions for wave damping, electrical conductivity, thermal conductivity, free–free opacity, line-broadening, and plasma oscillations. Experiments on laser-produced dense plasmas utilize fast detection of the angular and frequency resolved x-ray scattering spectra. Previous studies at large kilojoule type laser facilities [20, 35–46] have demonstrated spectrally resolved scattering techniques measuring the x-ray Compton backscattering and the plasmon forward scattering spectra providing the temperature and density of the plasma. The temporal resolution achieved in previous studies is about 100 ps using fast framing camera detectors, or about 10 ps using a laser-produced K-α source.

The collimated high-energy LCLS x-ray beam enables novel investigations of HED states using the high photon flux in 50 fs pulses with photon energies of 4–8 keV that are unique to the MEC end station. Experiments with ultrafast 50 fs x-ray scattering techniques and high spectral resolution can now be performed at orders of magnitude higher repetition rates than previous studies so that high signal-to-noise ratios and precision spectral measurements will be achieved. Furthermore, using the LCLS x-ray beam as opposed to laser-produced plasma sources will provide a well-defined incident scattering vector; these facts combined with a seeded x-ray beam with 1 eV spectral bandwidth will provide precision spectra and highly accurate measurements of the scattering features. Combining this unique x-ray capability with powerful lasers allows investigating the microscopic and hydrodynamic physics properties of HED plasmas.

The x-ray scattering process is significantly different from optical scattering [20, 82] because the energy of the incident x-ray photon with frequency ω₀ is large enough to give a significant Compton shift to the frequency of the scattered radiation. During the scattering process, the incident photons transfer momentum ħk and energy ħω = (ħk)²/2m_e to the electrons. The magnitude of the k-vector is given by

$$\begin{eqnarray}&&k=|{\boldsymbol{k}}|=\displaystyle \frac{4\pi {{\rm E}}_{0}}{hc}\;\mathrm{sin}\left(\displaystyle \frac{\theta }{2}\right)\end{eqnarray} \tag{ 7 }$$

with E₀ = ħω₀ being the energy of the probe x-rays and θ the scattering angle. Momentum and energy are primarily transferred to the free and weakly bound electrons whose binding energy is less than the Compton energy ħω. In the non-collective (Compton scattering) regime, the contribution of the free electrons that carry the information on the Fermi energy (i.e., electron density) or the electron temperature of the dense plasma is measured in the inelastic scattering feature. Moreover, bound electrons with ionization energies larger than ħω (states deep in the Fermi sphere) cannot be excited, and no Compton energy transfer occurs during the scattering process.

Thus, the total XRTS spectrum results in elastic and inelastic spectral features that are described by the dynamic form factor, S (k, ω) [83]. The ion structure is measured from the un-shifted elastic scattering component at E₀ that is commonly referred to as Rayleigh peak. For the analysis of x-ray scattering experiments, we developed a comprehensive analysis tool [84] that is based on the Chihara formula [85]. The formula describes the contributions to elastic scattering and inelastic scattering from free (delocalized) and weakly bound electrons by the dynamic form factor

$$\begin{eqnarray}S({\bf{k}},\omega ) & = & | {f}_{{\rm{I}}}({\bf{k}})+q({\bf{k}})| {}^{2}{S}_{{\rm{ii}}}({\bf{k}})+{Z}_{{\rm{f}}}{S}_{{\rm{ee}}}(k,\omega )\\ & & +{Z}_{{\rm{C}}}{\int }^{}{S}_{{\rm{CE}}}({\bf{k}},\omega -{{\rm{\omega }}}^{\text{'}}){S}_{{\rm{S}}}({\bf{k}},{\rm{\omega }}){\rm{d}}\omega ^{\prime} \end{eqnarray} \tag{ 8 }$$

with Z_f and Z_C denote the number of free and bound electrons, respectively. It is primarily the intensity of the elastic (Rayleigh) peak described by the first term that provides the ionic structure while the intensity and spectral dispersion of the inelastic free-electron scattering described by the second term provides information on the electronic system. The last term of equation (8) includes inelastic scattering by weakly bound electrons, which arises from bound–bound and bound-free transitions to the continuum of core electrons within an ion, S_CE(k, ω), modulated by the self-motion of the ions, represented by S_S(k, ω). The corresponding spectrum of the scattered radiation is that of a Raman-type band. Recently, there has been much activity to improve the theoretical description of this term, e.g., [86, 87]. For plasmon scattering applied in this study, we find that this term is not important when compared to the free electron dynamic structure. Principally, the relative contributions of the bound-free feature (third term in equation (8)) and the free–free feature (second term in equation (8)) can be utilized to infer the number of bound electrons and hence the ionization degree and the density of the dense plasma [87].

We have developed a theoretical fitting code that calculates S(k, ω). Where physics uncertainties exist, several theoretical models have been developed for comparisons and to establish sensitivities. They are available for comparisons with experimental data. There are three areas, (1) for term 1, four different models have been implemented for calculations of the ion–ion structure factor S_ii(k) to describe the elastic (Rayleigh) scattering peak [88]. They include a Debye model, a one component plasma model and a screened one component plasma (SOCP) approximation with and without negative screening; these have been used previously with some success; (2) for term 2, calculations of the free–free contribution include e–i collisions in the Born–Mermin approximation (BMA) [89–91] together with local field corrections (LFC) [92, 93] which has been applied with good success to describe the dispersion [42] and width of plasmons [36]. These models show little effect for calculations of the Compton profile which can be tested by comparison with the random phase approximation (RPA) [94]; (3) for term 3, calculations of the bound-free contribution include the form factor approximation and the impulse approximation [84] that show differences for low-temperature HED plasmas with little ionization and which are presently under scrutiny for testing of continuum lowering models. Importantly, the present models include different continuum lowering physics and finite temperature effects.

LCLS provides new opportunities to develop novel physics areas using x-ray scattering. Among those are first-principal electron temperature measurements using detailed balance, observations of plasmon damping by collisions, LFC effects on the plasmon dispersion and effects of viscosity and damping on phonon modes on the ion feature. For high-pressure conditions, elastic and inelastic bound-free scattering can become important and will be utilized in a novel way to characterize ultrafast phase transitions, continuum lowering or ion–ion correlations.

Initially, XRTS was developed in isochorically-heated matter. This path was chosen because the advantage of a priori known mass density allows the electron density to be directly inferred from the degree of ionization. The first experiments have tested calculations of the Compton scattering feature that shows significant broadening due to the thermal motion of the electrons resulting in factor of >2 larger width than scattering spectra from cold beryllium [35]. In many dynamic compression experiments, the conditions are close to the Fermi degenerate state. Here, the total width of the Compton peak depends on $\surd {E}_{{\rm{Fermi}}}\sim {{n}_{e}}^{\mathrm{1/3}}$ in the partly Fermi degenerate regime with E_Fermi = ħ²(3π²n_e)^2/3/2m_e. Thus, the width of the Compton scattering profile is fairly insentitive to the electron density increase during shock collision. Figure 15 shows an example for 8.5 keV x-ray probing. The challenge will be observing the whole scattering spectrum with sufficient energy range and sufficient resolution to measure this effect.

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Finally, with experiments aimed towards producing high-temperature states, Compton scattering is the technique of choice to measure the transition from a Fermi parabolic distribution towards a Boltzmann Gaussian profile whose width provides the electron temperature with high accuracy.

To access collective scattering requires that the scattering parameter must be larger than one:

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{1}{k{\lambda }_{{\rm{S}}}}=\displaystyle \frac{\mathrm{Scattering}\;\mathrm{length}}{\mathrm{Screening}\;\mathrm{length}}\gt 1.\end{eqnarray} \tag{ 9 }$$

The scattering length is defined by the scattering vector k (equation (7)) and is given by the probe energy and scattering angle: λ = λ₀/[4π sin(θ/2)]. In the limits of degenerate or classical plasmas the screening length assumes the Thomas–Fermi length or the Debye length, respectively. In case of α < 1, the probe radiation with wavelength λ₀ is scattered from electron density fluctuations on a spatial scale λ smaller than the screening length λ_s. In this case, no collective effects like electron plasma oscillations can be observed and the scattered radiation shows the Compton scattering feature providing the electron velocity distribution function. In case of α > 1, the scattering wavelength λ is larger than the screening length λ_s. In a weakly degenerate solid-density plasma with an electron temperature of the order of the Fermi temperature, T_e ∼ T_F = 15 eV, collective scattering will occur in forward scattering geometry with θ < 40° and x-ray probe energies of E₀ > 3 keV. In this regime, the scattered spectrum shows collective effects corresponding to scattering resonances from electron plasma (Langmuir) oscillation, i.e., plasmons, and from ion-acoustic modes.

In this study, our measurements observe a plasmon shift of ΔE = 25 eV that is approximated by the dispersion relation for plasmons in dense matter, often expressed in the familiar expression [95, 96]:

$$\begin{eqnarray}&&{\omega }^{2}\sim {\omega }_{{\rm{pe}}}^{2}+3{(k{\nu }_{{\rm{th}}})}^{2}+{\left[\displaystyle \frac{\hbar{k}^{2}}{2{m}_{{\rm{e}}}}\right]}^{2}.\end{eqnarray} \tag{ 10 }$$

For HED states, the first term of equation (10) dominates the frequency shift. With ${{\omega }_{{\rm{pe}}}}^{2}\;=\;{n}_{{\rm{e}}}{e}^{2}/{\varepsilon }_{0}{m}_{{\rm{e}}}$ the first term being sensitive to the electron density, n_e. The second term is known as the thermal correction from the Bohm–Gross dispersion relation and is of order <5% of the total shift. However, T_e will influence the damping of the plasmon and a rough estimate can be inferred from the fit to the width of the experimental data. The third term is the well-known Compton shift and exactly calculated with equation (1). Higher order terms have been estimated analytically; they are included in numerical results of e.g., [96].

Under the assumption that inter particle interactions are weak, so that the nonlinear interactions between different density fluctuations are negligible, the plasmon shift has been first calculated in the RPA. In the classical limit, it reduces to the usual Vlasov equation. In the limit of the RPA, strong coupling effects are not accounted for, thus limiting the model validity. To expand the theoretical modeling into the WDM regime, the BMA has been developed. The theory was subsequently improved to allow for different models for the dynamic collision frequency and for LFC. Its application to laser experiments on compressed boron has shown that the description of the plasmon dispersion is greatly improved with this theory [42]. However, at the time of this writing, the theoretical description of many-body effects in WDM is not complete.

Figure 16 demonstrates that for small forward scattering angles the calculated plasmon energy shift is not sensitive to the choice of the model for S_ee(k, ω). For a scattering angle of 13° the plasmon shift calculated with BMA and BMA–LFC agree over a range of densities. The shift reflects the relative increase in plasma frequency with density and is well understood compared to the absolute plasmon energy at arbitrary wavenumber. This is apparent in the calculations with increasing scattering angle that show discrepancies between these models. It becomes clear that MEC provides a great opportunity for future investigations of these and related phenomena when using larger scattering angles. For the purpose of the present analysis we conclude that it is possible to achieve an error in density of 5% when choosing conditions where differences in the theoretical approximations are minimized.

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For HED matter at significant temperature, the collective x-ray forward scattering spectrum will show two plasmon features. In addition to the downshifted feature, an up-shifted plasmon will occur on the high-energy wing with nearly the same frequency shift. Compared to the intensity of the downshifted plasmon, the up-shifted intensity is reduced by the Bose–Einstein function dictated by the principle of detailed balance. The intensity ratio of these plasmon features thus offers a first-principal method to measure the ultrafast electron temperature evolution of HED plasmas.

The relation that demonstrates this method is derived from the fluctuation–dissipation theorem. The density fluctuations in the plasma, described by the dynamic structure factor, are related to the dissipation of energy, described by the dielectric function ε(k, ω)

$$\begin{eqnarray}&&S({\bf{k}},{\rm{\omega }})=\displaystyle \frac{{\varepsilon }_{0}{\hbar}{k}^{2}}{\pi {e}^{2}{n}_{{\rm{e}}}}\;\displaystyle \frac{{\rm{Im}}({\varepsilon }^{-1}({\bf{k}},\omega ))}{1-\mathrm{exp}({\hbar}\omega /{k}_{{\rm{B}}}{T}_{{\rm{e}}})}.\end{eqnarray} \tag{ 11 }$$

Independent of evaluating the dielectric function, the shape of the structure factor, and modeling collisional effects, the dielectric function fulfills the requirement

$$\begin{eqnarray}&&\displaystyle \frac{S({\bf{k}},\omega )}{S(-{\bf{k}},-\omega )}={{\rm{e}}}^{-\hbar{\rm{\omega }}/{k}_{{\rm{B}}}{T}_{{\rm{e}}}}.\end{eqnarray} \tag{ 12 }$$

This relation is referred to as the detailed balance relation. As a general consequence, the structure factor shows an asymmetry with respect to k and ω. When applied to plasmons, it provides the electron temperature independent of a detailed formulation of dissipative processes in the plasma. In past laser experiments, due to small signal-to-noise ratios limited information was obtained on T_e [36, 97], making a future systematic studies with small noise amplitudes highly interesting.

The width of the plasmon feature is sensitive to collisions, thus providing a measurement of the e–i collision frequency and consequently a novel method for measuring conductivity and equilibration. Figure 17 shows the comparison between measured and calculated collision frequency ν_ei and sensitivity of the plasmon width to ν_ei. From the e–i collision rate the conductivity can be inferred through the Ziman formalism [98, 99]. This can readily be seen from the dependence of the x-ray scattering spectrum on the dielectric function ε(k, ω). The generalized expression for the dielectric function in the Mermin approximation employs the standard collision-less value obtained in the RPA, ε^RPA(k, ω), and the collision frequency ν(ω)

$$\begin{eqnarray}{\varepsilon }^{{\rm{M}}}({\bf{k}},\omega ) & = & f({\varepsilon }^{{\rm{RPA}}}\;({\bf{k}},\omega +{\rm{i}}v(\omega )),\\ & & {\varepsilon }^{{\rm{RPA}}}({\bf{k}},\omega ),v(\omega )).\end{eqnarray} \tag{ 13 }$$

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The collision frequency obtained in this way is related to the conductivity through the plasma coupling parameter Γ [100]

$$\begin{eqnarray}&&\sigma (\omega )=\lambda ({\rm{\Gamma }})\displaystyle \frac{{e}^{2}{n}_{{\rm{e}}}}{{m}_{{\rm{e}}}v(\omega )}{\rm{and}}\;{\rm{\Gamma }}=\displaystyle \frac{{e}^{2}}{k{T}_{{\rm{e}}}}{(4\pi {n}_{{\rm{i}}}/3)}^{\mathrm{1/3}}.\end{eqnarray} \tag{ 14 }$$

Thus, measurements of the plasmon spectrum have the potential to provide critically important information on equilibration and hydrodynamics but must use high spectral resolution and high signal-to-noise detection unique to LCLS.

LCLS high photon flux of narrowband and coherent x-rays are now available to study the dynamics of the low-frequency correlations (the ion-acoustic waves). The dispersion curve of and the details of ion correlations have long remained elusive in the HED physics regime [82, 88], with almost no comparison with experimental data except for the case of liquid metals at room temperature [102]. New studies at LCLS will be able to investigate contributions of LFC and the relevance of hydrodynamics and generalized hydrodynamics formulations on plasma transport properties [103, 104]. Further, predictions for additional dispersion at large wave numbers can be tested [63, 105].

Of particular interest are pump–probe experiments where unequal electron and ion temperatures will be produced. Thus, the measurement of the ion acoustic resonances will provide a direct estimate of the ion temperature, while the measurement of plasmon resonances will give the electron temperature. This platform can provide new insights in the e–i coupling in HED experiments at MEC.

Unique to LCLS MEC is the fact that one can further develop x-ray scattering diagnostic to simultaneously resolve both the spectral and angular (k-vector) components of the XRTS signal during a dynamic laser and x-ray heating experiment. In particular, when significant compression occurs, for example due to strong shock waves, accurate measurements can be achieved. The momentum-resolution of such methods has not previously been achieved in WDM research and is critically needed to measure the material structure in conditions with simultaneous spectrally-resolved scattering measurements. Such experiments will take advantage of the strong correlation peaks in the scattering intensity of the elastic scattering component (1st term of equation (8)).

The ionic structure factor peak has been observed recently for the first time in shock-compressed aluminum. In these experiments, aluminum was shocked from one side [46] and data were accumulated over 3 separate shot days at the Omega laser [45]. The absolute strength of the scattering amplitude and overall predicted shape have shown that HNC calculations with repulsive potentials and SOCP models provide an improved description while Debye–Hückl screening models show weaker correlations and smaller scattering amplitudes than the experimental data.

For matter with strong ion–ion correlations the frequency integral of equation (8), S^total(k), is given by the Fourier transform of the pair–correlation function g(r):

$$\begin{eqnarray}&&{S}^{{\rm{total}}}({\bf{k}})=1-n{\displaystyle \int {{\rm{d}}}^{3}{\bf{r}}\mathrm{exp}(-{\rm{i}}{\bf{k}}{\bf{r}})[g({\bf{r}})-1]}^{}\end{eqnarray} \tag{ 15 }$$

with

$$\begin{eqnarray}&&{\rm{g}}({\bf{r}})=\displaystyle \frac{1}{Nn}\displaystyle \int {{\rm{d}}}^{3}{\bf{r}}^{\prime} \;n({\bf{r}}+{\bf{r}}^{\prime} )n({\bf{r}}^{\prime} ),\end{eqnarray} \tag{ 16 }$$

where n is the average density, N is the number of particles in the scattering volume. Hence, S^total(k) is a direct measure of pair correlations in dense matter. Equations (15) and (16) predict a shift of the calculated maximum elastic x-ray scattering amplitudes to higher wave number with increased density. For higher densities, the mean separation between ions decreases providing scattering peaks at larger scattering angles and thus larger wave number k. Here, the small k-vector spread of the incoming LCLS beam is required to resolve the shift of the structure peak. Importantly, different theoretical models of the ion–ion correlation predict differences in the widths, which have been resolved in experiments at MEC and which are discussed further below.

Figure 19 shows the schematic of dynamically compressed matter experiments at MEC. Also shown are data records from the x-ray scattering detectors. The spectrometers have been fielded for density and temperature measurements from plasmons and angular resolved scattering provides diffraction and structure factor data. Here, two 4.5 J laser beams irradiate 50 μm thick Al foils (initial density of ρ₀ = 2.7 g cm⁻³) that are coated with 2 μm thick parylene. The laser beams are absorbed in the parylene, heat the material and launch two counter-propagating multi-megabar shock waves into the solid aluminium by ablation pressure. For the conditions achieved in this experiment, we find that the shocks propagate inward at 14 km s⁻¹, each shock wave compressing the aluminium target to about 5 g cm⁻³.

The shock speed and planarity over the central 10 μm diameter region were verified by shock breakout measurements from 25 μm thick foils irradiated by a single beam. Thus, when adding the second beam on the opposite side of the target, we doubled the foil thickness to 50 μm to mirror the shock propagation. Figure 20 shows that the averaged shock velocity varies by <2% over the region probed by the x-ray beam. Also, the shock velocity is seen to scale with laser intensity for two different focal spot sizes. When the shocks coalesce in the center of the foil, the density is expected to significantly increase to about 7.5 cc⁻¹ reaching peak pressures of 5 Mbar.

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In this study, accurate measurements of compression have been achieved by measurements of the electron plasma (Langmuir) oscillations. These measurements test the predictions of compression based on equations (1)–(6) by applying the theory of equations (8) and (10). Figure 21 shows the plasmon scattering spectra and theoretical fits of the dynamic structure factor S(k, ω) near peak-compression at t = 1.9 ns. These data are compared with measurements from uncompressed aluminium. The plasmon feature is downshifted from the incident 8 keV x rays as determined by the generalized Bohm–Gross dispersion relation, see equation (10), and its resonance frequency is used as a sensitive marker of the electron density, see figure 16. While the data from uncompressed aluminium yield a total plasmon shift of 19 eV providing a free electron density of 1.8 × 10²³ cm⁻³ the total shift increases to 29 eV at shock coalescence. For the example shown in figure 21, a density of 4.1 × 10²³ cm⁻³ has been measured and corresponding to 2.3× compression. The error in electron density is ±5% determined by noise in the data and the fit of the theoretical dynamic structure factor.

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The maximum density measured at shock coalescence is 7 g cc⁻¹ slightly less than expected from estimates discussed above. Specifically, we find that compression in the first shock is slightly lower than predicted due to varying total peak powers and probably also due to the slow rise time of the laser pulse. Typical values are 1.2 < ρ₁/ρ₀ < 1.9, which when combined with the compression due to coalescence of ρ₂/ρ₁ ∼ 5/3 results in final compression values of ρ₂/ρ₀ > 2.

The scattering spectra further provide a measure of the temperature by analyzing the intensity of the elastic scattering features. For this purpose, structure factors are determined from wavenumber resolved scattering while the absolute intensity is determined by the intensity ratio with the plasmon feature; the plasmon intensity is determined by the f-sum rule (particle conservation)

$$\begin{eqnarray}&&\displaystyle {\int }_{-\infty }^{\infty }{\left(\displaystyle \frac{{{\rm{d}}}^{2}\sigma }{{\rm{d}}{\rm{\Omega }}{\rm{d}}\omega }\right)}_{{\rm{Plasmon}}}\omega {\rm{d}}\omega =\displaystyle \frac{Z\hslash {k}^{2}}{2{m}_{{\rm{e}}}},\end{eqnarray} \tag{ 17 }$$

where (d²σ/dΩ dω)_Plasmon is the measured plasmon intensity from the free electron Langmuir oscillations. Here, Z = 3 is the ion charge state, $\hslash$ is Planck's constant, k is the wavenumber and m_e the electron mass. At peak compression, the intensity of the elastic scattering feature increases by a factor of 2.8 over the cold scattering amplitude. We use a model [84] to fit the measured intensities; the potential and structure factors are shown in section 4.1.2; they are in excellent agreement with the experimental data and DFT-MD simulations, see figure 18. For the example shown in figure 21 the temperature is T = 20 000 K, i.e., 1.75 eV ± 0.5 eV. For our conditions, the ultrafast x-ray pulse deposits 3500 K or 0.3 eV per electron into the target. A fraction of the energy can be expected to heat the electron fluid, but the total energy is small compared to shock heating and will not affect the plasmon shift. In addition, within the duration of the x-ray pulse, the ionic correlations have no time to respond to changes of the screening properties as demonstrated by the observation of solid and solid compressed states of aluminium.

Figure 22 shows the wavenumber resolved scattering data from which we obtain the high-pressure ion–ion structure factors. At t = 0, before the rise of the optical laser beam power, the data show peaks from Debye–Scherrer rings indicating the ionic lattice in the solid. The peaks at 38°, 45°, and 65° correspond to (111), (200), and (220), respectively. When the shocks are launched and compress the solid, the lattice spacing d is reduced and we observe that the peaks shift to larger scattering angles as determined by the Bragg scattering equation nλ_x-ray = 2d sinθ, here λ_x-ray = 1.55 Å the wavelength of the incident x-ray laser, and 2θ the scattering angle.

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With increasing pressures, the aluminium melts and transitions into the WDM state. We first observe a comparatively broad ion–ion correlation peak at an angle of 45° at 1.38× compression together with shifted Bragg scattering of solid compressed aluminium. For these conditions the comparison with simulations shows that the wave number of both the ion–ion correlation peak and the shifted Bragg peak provide the same density indicating a co-existence regime. With higher pressure accessed later in time and with higher laser intensity, we compress aluminium up to 1.9× the initial density and the strong ion–ion correlation peak shifts to 2θ = 51°. At shock coalescence, the peak shifts further to 2θ = 55° and the data indicate 2.3× compressed aluminium consistent with densities inferred from the simultaneous measurements of the plasmon shift.

The measured ion–ion correlation feature is sensitive to the ion–ion structure factor, see equation (8). For nearly elastic scattering the scattering angle 2θ determines the wavenumber of the correlations. With k = 2k_x-ray sinθ and k = 2π/λ determining the scale length of the density fluctuations measured in this scattering experiment, we find 1.38 Å <  λ_SL < 5.7 Å. On these atomic scale lengths, the scattering amplitude is probing important details of the ion–ion interaction potential V_ii.

For the analysis of the measured scattering data and the ion–ion correlation feature various theoretical approximations have been considered. The data depend on both material compression, temperature, and on the choice of the many particle theory to describe the interaction physics. Here, densities and temperatures are inferred from the plasmon scattering spectra providing an excellent experimental test of theory and simulations by comparing predictions with the measured wavenumber-resolved scattering data. In previous studies, the measurements of the elastic scattering amplitude alone was often not sufficient to provide information about the state of the dense plasma. This is due to the fact that calculations of S_ii(k), e.g., using the HNC approximation, must assume an effective interaction potential. In this study, S_ii(k) is directly obtained from the measured wavenumber resolved scattering data using W(k) = S_ii(k)(f(k) + q(k))² with the atomic form factor f(k) and the screening function q(k) calculated from the number of bound core electrons, Z_C = 10 for aluminium.

Figure 23 shows various screening functions q(k) and atomic form factors f(k). Combining these results with our measurements of the wavenumber resolved scattering data, W(k), demonstrates the sensitivity of the choice of the function q(k). HNC calculations with a screened Coulomb potential using a Yukawa screening term (Y) together with a SRR term provide excellent agreement with the measured data. This is illustrated as the full curves in figure 22 that agree with both experiment and the results from the DFT-MD simulations. Similar results can be obtained with a hard sphere model. However, it is apparent that a Coulomb potential or a screened Coulomb potential cannot account for the experimental observations, see dashed curves in figure 22. At small wavenumbers, we note that these models show different sensitivity to the temperature. This is due to the fact that the repulsive core part of the potential is temperature independent; the remaining sensitivity is due to the temperature dependence of the screening cloud. Here, DFT-MD simulations shown in figure 18 provide the temperature dependence that is applied for the interpretation of the elastic scattreing amplitude in the plasmon spectra, see figure 21. The error in temperature can be estimated to be in the range of 0.5 eV–0.7 eV including contributions from both uncertainties in the models and noise of the data.

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The analysis presented in figures 22 and 23 provide the following physical picture for the ion–ion structure factor in WDM: by matching the pair correlation functions g(r), the effective ion–ion pair potentials can be directly extracted from the DFT-MD simulations that show the importance of including both SSR and screening. The interaction potential can be described as

$$\begin{eqnarray}&&\beta {V}_{{\rm{ii}}}(r)=\displaystyle \frac{{Z}^{2}{e}^{2}}{r{k}_{{\rm{B}}}T}{{\rm{e}}}^{-\kappa r}+\displaystyle \frac{{\sigma }^{4}}{{r}^{4}},\end{eqnarray} \tag{ 18 }$$

with r being the distance between ions, k_B is Boltzmann's constant, e the electron charge number. The exponent κ = κ (n_e,T_e) is the Yukawa screening function. For our conditions, κ approaches the Thomas–Fermi screening length with accuracy of better than 1%. The last term in equation (19) accounts for the additional repulsion from overlapping bound electron wave functions, with the parameter σ fitted to match DFT-MD simulations [59]. Figure 22 (top three panels) shows that the DFT-MD simulations, results based on equation (18), and the wavenumber resolved scattering data agree well with each other for densities obtained independently from the plasmon or Bragg scattering data. Importantly, linearly screened Coulomb potentials, denoted here by Yukawa in figure 22, show no agreement with the experimental data.

The data and simulations can further be used to describe the angle of the maximum scattering amplitude; the latter is sensitive to the density. Figure 24 compares the angle of maximum scattering amplitude with the density from the plasmon x-ray scattering data. Also shown are various theoretical approximations. The result show that using equation (18) for the calculation of the dynamic structure factor S(k, ω) provides a model to determine density from wavenumber resolved scattering.

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The therory and simulations can now be applied to analyze the scattering data for potential gradients in the x-ray probing region. Potential non-uniform shock conditions could give rise to different conditions either transversly over the 10 μm x-ray probe cross section or along the shock propagation direction. Clearly, the latter will always include the low density blow-off plasma and if probed early in time the unshocked material ahead of the shock. However, the x-ray scattering amplitude is weighted towards the highest density and the broad ion–ion correlation feature can only be observed after shock melting. To quantitativly estimate the effect of gradients the wavenumber resolved scattering can be compared with calculations for the ion–ion structure factor for a range of density conditions.

Figure 25 shows the x-ray scattering data together with calculations for a density of ρ = 2.32 ρ₀ ± Δρ. The density range is assumed to be linear over the ranges indicated in the figure. The results indicate that density gradients are negligibly small for the present experimental conditions and are less than 5%.

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Finally, the experimental and theoretical structure factors further provide a new method for determining the material pressures. In particular, the measured wavenumber resolved scattering data W(k) yield the structure factor S(k) using W(k) = S(k)[f(k) + q(k)]² with the atomic form factor f(k) and the screening function q(k) that can be integrated and which provides a novel method for determining the pressure [106]. The total pressure consists of ionic and electronic contributions with P_tot = P_i + P_e. In nearly Fermi degenerate conditions, P_e includes well-known density-sensitive contributions accounting for Fermi pressure, degeneracy, Coulomb, and exchange terms. On the other hand, the ion pressure, when divided into an ideal gas part P_G = n_ik_BT and the correlation part P_i = p^x + P_G, is dominated by the (negative) ion excess pressure

$$\begin{eqnarray}&&{p}^{x}=\displaystyle \frac{{n}_{{\rm{i}}}U}{3N}-\displaystyle \frac{{n}_{{\rm{i}}}{(Ze)}^{2}}{12{\pi }^{2}}\displaystyle {\int }_{0}^{\infty }S({\bf{k}})\displaystyle \frac{{\kappa }^{4}}{{({k}^{2}+{\kappa }^{2})}^{2}}{\rm{d}}{\bf{k}}\end{eqnarray} \tag{ 19 }$$

with U/N being the internal energy. In equation (19), the structure factor directly obtained from the measured wavenumber resolved scattering data can be used.

Figure 26 compares the experimental pressure data [22] with available empirical data [107] and state-of-the-art simulations. In particular, we show previous DFT-MD simulations for temperatures calculated for the shock Hugoniot [62] together with our DFT-MD simulations performed for temperatures and densities of the present experiment [61]; the simulations provide excellent agreement with the data from the experiment. The data further show no indication of Bragg peaks for compressed aluminium above P = 1.2 Mbar consistent with previous melt line measurements on the Hugoniot [108]. The present data set extends further showing the disappearance of Bragg scattering data above 1.2 Mbar and up to densities of 4.5 g cc⁻¹. At higher pressures, the shock coalescence data are on the isentrope [109] slightly above the isotherm [107] validating our understanding of dense aluminium utilizing pressure, temperatures and densities solely based on measurements.

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These new experimental and theoretical findings demonstrate that spectrally and wavenumber resolved x-ray scattering is applicable for thorough testing of radiation-hydrodynamic calculations and EoS models. The method presents unique highly resolved data for dynamic high-pressure material science studies that require accurate knowledge of material properties and the thermodynamic state at high densities and are applicable for future studies aimed at observing effects of ionization on the EoS at high compression.