The canonical high pressure equation of state measurement is to induce a shock wave in the sample material and measure two mechanical properties of the shocked material or shock wave. For accurate measurements, the experiment is normally designed to generate a planar shock which is as steady as possible in space and time, and a single state is measured. A converging shock strengthens as it propagates, so a range of shock pressures is induced in a single experiment. However, equation of state measurements must then account for spatial and temporal gradients. We have used x-ray radiography of spherically converging shocks to determine states along the shock Hugoniot. The radius-time history of the shock, and thus its speed, was measured by radiographing the position of the shock front as a function of time using an x-ray streak camera. The density profile of the shock was then inferred from the x-ray transmission at each instant of time. Simultaneous measurement of the density at the shock front and the shock speed determines an absolute mechanical Hugoniot state. The density profile was reconstructed using the known, unshocked density which strongly constrains the density jump at the shock front. The radiographic configuration and streak camera behavior were treated in detail to reduce systematic errors. Measurements were performed on the Omega and National Ignition Facility lasers, using a hohlraum to induce a spatially uniform drive over the outside of a solid, spherical sample and a laser-heated thermal plasma as an x-ray source for radiography. Absolute shock Hugoniot measurements were demonstrated for carbon-containing samples of different composition and initial density, up to temperatures at which K-shell ionization reduced the opacity behind the shock. Here we present the experimental method using measurements of polystyrene as an example.

## I. INTRODUCTION

Measurements of states of matter at elevated pressures are often made using shock compression,^{1} in which the induced high-pressure material is confined inertially and so is not limited by the strength of surrounding components, as is the case with static presses such as diamond anvil cells.^{2} Normally, experiments to measure shock states seek accuracy by designing the shock-loading system to be as uniform as possible in space and time. Spatial and temporal averaging can then increase the precision, and different aspects of the state can be measured at different locations.

The objective of the work reported here was to measure states at pressures higher than those that could be achieved with a uniform planar drive. Geometrical convergence was used to increase the shock pressure. Converging shocks are inherently non-uniform in space and time, which reduces the options for measuring the shocked state. The approach taken here was to use radiography to simultaneously measure the speed of the shock wave and the mass density immediately behind, as discussed below. This direct, *in situ* measurement of the shock also gives absolute state data, in contrast to other techniques such as shock transit times in comparison with a reference material, which then depend on the equation of state (EOS) of Ref. 1. Absolute state data are particularly important for experiments above the range where reference measurements have been made.

Detailed Hugoniot results and discussion for each material studied will be published separately.^{3–5} Here we describe the radiographic analysis process in detail, focusing particularly on poly(*α*-methyl styrene) (PaMS) driven at relatively low energy such that the shock heating was not high enough to significantly alter the opacity of the sample.

## II. SPHERICALLY CONVERGING SHOCK

Experiments were performed at the National Ignition Facility (NIF), with supporting experiments at the Laboratory for Laser Energetics (Omega laser, University of Rochester) to test radiation-hydrodynamic predictions of drive symmetry. High energy pulsed lasers were used to induce a uniform field of soft x-rays within a hohlraum,^{6} which induced high pressures by ablating the outer surface of a spherical sample assembly within the hohlraum. This configuration took advantage of the large body of work devoted to the symmetric implosion of hollow capsules for inertial confinement fusion (ICF),^{7,8} but for our experiments a solid sphere of the sample material was used. In a solid sphere, a converging shock wave grows stronger as it propagates inward, because radial flow behind the shock causes isentropic compression, increasing the pressure. The shock is subsonic with respect to states behind it, so the increased pressure propagates forward and strengthens the shock itself. As the shock approaches the center, its pressure grows rapidly.

Hugoniot measurements can be obtained most readily from a continuous measurement of the speed and compression of the shock, using radiography. The most straightforward configuration for radiography is a streak record: a continuous source of x-rays (for the few nanosecond duration of the experiment), recorded on a streak camera so that one dimension of the data is diameter and the other is time. At any instant of time, the radial variation of mass density can be obtained from the variation of x-ray attenuation with radius, if the radial symmetry is known. One indication of asymmetry is side-to-side variation in the streak image. A more direct measurement is from conventional spatial radiographs. Framing camera radiographs were obtained on complementary Omega experiments,^{9} demonstrating the symmetry of the shock, though simultaneous framing camera measurements were not possible on the NIF experiments.

The symmetry of the radiation drive in a hohlraum depends on the detailed conditions in the plasma. Laser-plasma interactions cause phenomena such as the transfer of energy between laser pulses propagating in the plasma.^{10} Simulations of the detailed loading conditions are computationally challenging, so the experiments were designed to be close to configurations developed for inertial confinement fusion (ICF), for which extensive simulations and supporting experiments have been performed. In particular, the hohlraum, gas fill, and laser pulse shape followed ICF designs as much as possible, and the sample was a sphere ∼1 mm in radius, with an ignition-derived ablator deposited outside.^{30}

When the shock speed increases with pressure, converging shocks are stable with respect to perturbations. A region of the shock that is lagging induces a greater curvature, so the effect of isentropic compression behind the shock increases and the shock accelerates locally. Conversely, a region of the shock that runs ahead becomes less curved, so the effect of convergence reduces and the shock decelerates locally. The effect of asymmetry was assessed using two-dimensional hydrocode simulations with perturbations to the shape of the ablator or to the spatial variation of the driving pressure. Simulations were performed with Eulerian (Cartesian) and Lagrangian (conformal) methods, and purely spherical configurations were included to verify neither that asymmetries occurred when they should not (a concern with Eulerian simulations) nor that asymmetric flow incorrectly became symmetric (a possible concern with radial-azimuthal zoning).^{11} Realistic EOS for component materials was used, from the Sesame^{12} and Leos^{13} libraries. Unlike an imploding shell, which is formally unstable, a converging shock is stable for normally behaved equations of state, for which sound speed increases monotonically with pressure. The effect of asymmetries in a drive or shape typically causes a damped oscillation about a spherically symmetric shape. The simulations indicated that the shock in the solid sample was ∼50 times less sensitive to perturbations than a nominal ICF hollow-shell implosion, by the time the shock had propagated to ∼1/20 times the outer radius, where convergence effects increased the pressure by an order of magnitude. In the NIF experiments, radiography became challenging around this point because of spatial and temporal resolution in the detector system.

Laser and radiation drives can potentially induce preheating ahead of the shock, by radiation or electron transport. Because the radiations causing preheating do not focus, the degree of preheating depends most on the drive pressure. Thus the increase in shock pressure in converging geometry makes shock measurements inherently less sensitive to preheat, until shock temperatures are high enough that transport from the shocked material itself becomes important. This dependence is supported by radiation hydrodynamics simulations of hohlraum-driven converging-shock experiments.^{14}

## III. ABSOLUTE HUGONIOT DATA FROM RADIOGRAPHY

The simplest type of shock measurement is the symmetric impact: a flat-nosed projectile of the material of interest is accelerated to some speed *u*_{f} and made to collide with a flat-faced target of the same material. The speed *u*_{s} of the shock in the sample is measured by its transit time through a known thickness, and the speed of material behind the shock—the particle speed, *u*_{p}—is, by symmetry, *u*_{f}/2. For a steady shock wave, conservation of mass, momentum, and energy across the shock lead to the Rankine-Hugoniot relations,^{1}

where $v$ is the reciprocal of the mass density *ρ*, *e* is the specific internal energy, *p* is the pressure,^{31} and the subscript “0” denotes material ahead of the shock (with *u*_{p} = 0). The state ahead of the shock is known, leaving five quantities to be determined ($v$, *p*, *e*, *u*_{p}, *u*_{s}). If any two of these quantities are measured, the Rankine-Hugoniot equations determine the rest.

At higher pressures, where other techniques than projectile impact are used to induce the shock wave, it is often simpler in practice to determine Hugoniot data relative to a standard material.^{1} The sample material of interest is mounted on a layer of the standard. The shock is induced in the standard, and its pressure is deduced, usually from the speed of the shock. The speed of the shock in the sample is measured, e.g., by its transit time over a known distance. The shock state in the sample is the intersection of the release or reshock state from the initial shock state in the reference material with the Rayleigh line in the sample material: from the Rankine-Hugoniot relations, *p* = *p*_{0} + *ρ*_{0}*u*_{s}*u*_{p}. Thus the Hugoniot state in the sample can be found, but its uncertainty depends on the uncertainty in principal Hugoniot and secondary Hugoniot or isentrope in the reference material.

In the present work, x-ray radiography was used to measure the shock speed *u*_{s} and the mass density *ρ* at the shock front, as described in more detail below. The Rankine-Hugoniot relations were then used to deduce the pressure,

and hence *e* and *u*_{p}: an absolute measurement of the mechanical state on the shock Hugoniot. Absolute Hugoniot measurements have been performed previously at lower pressures,^{15} but the shock was approximately planar (and thus potentially susceptible to edge effects in a way that does not occur in radiography of one-dimensional spherical or cylindrical systems) and the two mechanical quantities inferred were *ρ* and *u*_{p}, taken to be the speed of the material pushing the sample. Spherically converging shocks, driven by the detonation of chemical explosive, have been used in previous EOS studies, generally employing diagnostics of shock arrival or velocity history at a surface.^{16,17} Some of these results, such as those for water,^{16} have subsequently proven inaccurate, likely because the surface measurement of transit time through a finite thickness of the sample did not allow for adequate correction for varying shock speed.^{18} Radiographic measurements of isentropic compression have been made similarly, in converging geometry, by following the trajectory of components surrounding a low-*Z* sample material.^{19} By contrast, the measurements of *ρ* and *u*_{s} obtained here were at the location of the shock front, making this configuration less susceptible to any temporal variation in shock pressure, which is of course essential here as the converging shock is inherently unsteady. In principle, the approach adopted here could be used at lower pressures to measure multiple-wave structures in the sample, such as elastic and plastic waves or phase transition waves.

The discussion above relates to the inference of mechanical states involving the quantities appearing in the Rankine-Hugoniot relations. These quantities are not sufficient to define a thermodynamically complete EOS, as temperature or entropy is not included. The mechanical EOS is sufficient for analysis of any dynamically deforming system except where temperature is needed to capture some relevant material behavior, such as the kinetics of phase transitions, chemical reactions, or plastic flow.

## IV. EXPERIMENTAL CONFIGURATION

The experimental configuration was based on indirect-drive ICF designs, where a soft x-ray flux inside a laser-heated gas-filled hohlraum implodes a spherical shell containing fusion fuel. In this work, the shell was replaced with a solid spherical target, comprising the sample material of interest within a plastic ablator, collectively referred to as the bead. The hohlraum wall was Au, 30 *μ*m thick. On NIF, the ablator included a graded Ge dopant, following a standard ICF prescription, which increased the conversion of hohlraum radiation to material pressure and shielded the sample from preheat, in particular, by Au *M*-band x-rays around 1.5 keV. The bead was held in position by thin plastic membranes (Fig. 1).

The density distribution in the bead was measured during the experiment with x-ray radiography from an area backligher, a thermal plasma source heated by eight of the NIF laser beams. Slits were cut in the hohlraum walls to allow the probe x-rays to pass through; a diamond flat was glued in each slit to contain the fill gas and to delay closure by ablated Au from the hohlraum wall. The x-ray pulse was 5 ns long, allowing the shock in the bead to be followed for a significant proportion of its transit time to the center. An x-ray streak camera was used to acquire the radiograph, imaging a line across the diameter of the bead with temporal resolution for the duration of the x-ray source. X-rays were imaged onto the photocathode of the streak camera through perpendicular slits, at different axial locations, i.e., giving different magnifications and spatial blurring in the axial and radial directions (Fig. 2).

Design simulations were performed using several hydrocodes, but particularly Hydra,^{20} LLNL’s principal ICF design program. The simulations closely followed the configuration used for hohlraum-driven ICF targets at NIF, particularly for the treatment of radiation transport and ablation of the hohlraum and bead. The simulations were important for adjusting the laser pulse shape and energy to induce the desired range of shock pressures and to predict the radiograph signal level and shock timing.^{14}

Radiation hydrodynamics simulations were used to estimate acceptable deviations from the nominal drive energies and pulse shapes. The NIF lasers delivered the hohlraum drive and backlighter pulses to within the requested tolerances. The hohlraum was heated rapidly to around 150 eV and then rose gradually to a peak close to 200 eV at the end of the drive pulse (Fig. 3).

## V. RADIOGRAPH AND ANALYSIS

The x-ray streak camera record showed on visual inspection the compression of the sample (with doped layer visible throughout) and the shock running ahead and accelerating (Fig. 4).

Data along the shock Hugoniot can be deduced by reconstructing the density distribution *ρ*(*r*, *t*), locating the locus of the shock front *r*_{s}(*t*) and hence the shock compression *ρ*(*r*_{s}(*t*), *t*)/*ρ*_{0}. The shock speed *u*_{s}(*t*) is obtained by differentiating *r*_{s}(*t*), giving a continuum of states along the Hugoniot: *ρ*(*u*_{s}).

### A. Iterative profile-matching

In principle, given a streak radiograph over the diameter of the sample, the density distribution could be reconstructed for each instant of time by Abel inversion:^{21} starting at the outside of the image, calculate the attenuation of the shell of material at that radius and hence its projection over each smaller radius, allowing the density at the next radius inward to be deduced. For the radiographs obtained in the NIF experiments, the signal intensity was relatively low, and the noise and uncertainty accumulating during this unfold led to uncertainties of tens of percent in shock compression if performed for a single row of pixels. The shock speed was high enough that simple spatial averaging introduced too much blurring of the shock position. Furthermore, the mass density is known ahead of the shock, which constrains the reconstructed density distribution in a way that is not natural to implement through the Abel unfold.

Instead, the density distribution was deduced by iterative optimization of a parameterized model of the density distribution *ρ*(*r*, *t*; *c*), where *c* = {*c*_{i}} is a set of parameters. Given a guess *ρ*(*r*, *t*; *c*), the resulting radiograph *I*_{s}(*x*, *t*; *c*) was simulated by a forward model of the radiographic system. A total deviation *D* was calculated from a pixel-by-pixel comparison with the experimental radiograph *I*_{x} (*x*, *t*),

where $w$ is a weighting array used, for example, to mask out irrelevant features in the radiograph, and *l* is the norm, equal to 2 for least-squares optimization. The parameters were then adjusted and *I*_{s} and *D* recalculated until the minimum of *D* was found with respect to variations in the {*c*_{i}}. The fitting algorithm used was a modified form of Bayesian profile-matching such as has been applied previously to medical imaging and x-ray radiographs of assemblies driven by chemical explosives.^{22}

The density distribution was split into regions between which there could be a discontinuity (in particular, the shock), and the locus of the boundary of each region was represented by a function varying smoothly with time. The temporal and spatial variation within each region was represented either by functions of time and radius or by functions of time and the fractional radius between the inside and outside of the region. The latter was found to describe the measured radiograph to the same residual but with fewer parameters, thus giving a smaller uncertainty. In the simplest case, profile-matching can be considered as a way to allow averaging over space and time for a radiograph in which averaging over adjacent times at the same radius would blur out a moving feature, such as the shock wave.

To investigate the reduction in uncertainty from profile-matching compared with Abel inversion, we can consider radiography of the density distribution at an instant of time. Taking a density profile *ρ*(*r*) from a hydrocode simulation of a spherically converging shock, the areal mass distribution *m*_{a} measured by radiography as a function of distance *x* across the object is

where *z* is the distance through the object from the source toward the detector and *R* is the outer radius of the object (Fig. 5). In the treatment of the actual experiment, as described below, *y* is the direction across the imaging slit, an additional integral. Equation (6) holds without loss of generality for parallel or point-projection with a source at any position outside the object (i.e., at a distance greater than *R*) by changing the mapping of *x* to the detector plane. In the absence of noise, *ρ*(*r*) can be recovered from *m*_{a}(*x*) by Abel inversion, which in essence uses the radial derivative of Eq. (6),

where $r=x2+z2$, starting at *r* = *R*, to infer *ρ*(*r*) and thus calculate the contribution to *m*_{a} (*x* < *r*). The Abel unfold over discrete intervals of *r* is thus the solution of an upper triangular matrix equation.

In the presence of noise, inaccuracy in *ρ* deduced by Abel inversion increases as *r* decreases. This makes Abel inversion inappropriate for use in the analysis of converging shocks because the unshocked density *ρ*_{0} ahead of the shock is known. The deduced solution for *ρ*(*r*) propagates inward, and *ρ*_{0} is reconstructed approximately, and with noise, from the radiograph. One key advantage in profile-matching is that the known value of *ρ*_{0} can be imposed. Instead of unfolding *m*_{a}(*x*) to obtain *ρ*(*r*), the current set of profile parameters ${c\u0303i}$ defines a trial $\rho \u0303(r;c)$, and a trial $m\u0303a(x;c\u0303)$ is calculated using Eq. (6). The sensitivity of $ma(x)\u2212m\u0303a(x;c\u0303)$ with respect to the ${c\u0303i}$ is determined and used to find the optimum solution {*c*_{i}}. The {*c*_{i}} can include parameters for the location of the shock and the density behind the shock. Trial calculations analyzing a single time slice from a hydrocode simulation demonstrated that, in the presence of noise, the accuracy with which the density at the shock front was recovered was orders of magnitude more accurate than with Abel inversion, even when the profile further behind the shock was not represented with enough degrees of freedom in the {*c*_{i}} to reproduce it accurately everywhere (Figs. 6–8).

For the x-ray streak camera data, the profile model *ρ*(*r*, *t*; *c*) provided additional constraints by assuming implicitly that the flow behind the shock varied smoothly in *r* and *t*. The effect is to further reduce the contribution of radiographic noise to the uncertainty in the density field.

In the streak radiograph, the image of a line across the object is swept in the time direction. The line image has a finite thickness, in the *y*-direction in the coordinate system defined above, contributing to the modulation transfer function (MTF) in the temporal direction. In the low-drive PaMS experiment on NIF, *y* was parallel with the axis of the hohlraum, and the thickness of the line was defined by the width of the hohlraum slit, initially 100 *μ*m across. In constructing simulated radiographs from model density distributions, the effect of the axial slit was incorporated by applying a radially dependent MTF integrating the radiographic intensity over the axial direction,

If the centerline of the slit is aligned with the axis, *y*_{1} = −*y*_{0} = $w$/2, where $w$ is the slit width. If the camera slit is tilted with respect to the case slit, the axial limits are linear functions of *r* (Fig. 9).

The imaging slit was cut into a Ta plate, 200 *μ*m thick and nominally 16 *μ*m wide, although the width varied slightly along its length, and the edges were slightly irregular. Because the width of the slit was much less than the depth, its effective width decreased with radius over the image of the sample. Unlike a constant slit width (e.g., as produced by a slit with an arc cross section), we did not find a way to represent a varying effective slit width as a modulation transfer function that could be applied to the image simulated with an infinitesimal slit. The variation can be accounted for directly in the simulated radiograph by sampling the transmission across the width of the slit along different rays, which is computationally intensive. Although the effective slit width varied from 16 *μ*m at the center of the image to 14 *μ*m at the edges, the Hugoniot deduced by treating the slit as a constant width of these limiting values was not significantly different.

### B. X-ray source and background

The backlighter was driven by two groups of laser beams and by inspection was not symmetric in brightness side-to-side. The spatial form was estimated from a previous, backlighter-only shot and found to be represented by a pair of supergaussians. The amplitude, width, and center of each supergaussian were treated as adjustable parameters when optimizing the density distribution. The temporal variation of the x-ray intensity from each group of beams was estimated from the brightness of the region visible outside the sphere in the radiograph. This was found to be almost identical to the variation calculated from an average over the complete half of the radiograph illuminated by that group. The deduced x-ray intensity followed the main features of the corresponding laser pulse shape, though the pulse shapes themselves did not reproduce the intensity history accurately enough to be used to describe the temporal variation of the x-ray source, which is as one would expect given that the laser-matter interaction involved in a thermal plasma backlighter removes any inherent linearity in x-ray production. The x-ray intensity histories deduced from the radiograph were filtered to remove high-frequency noise and used without further adjustment in the optimization (Figs. 10 and 11).

The background level was estimated from the image of the fiducial wires in the radiograph. The temporal variation was consistent with a high-energy x-ray component able to pass through the wires. This was represented as a background proportional to the radius-time variation of the backlighter, not significantly attenuated by the bead.

### C. Calibration of imaging and detection configuration

The accuracy of the Hugoniot measurement depends critically on the calibration of the radiographic system. During analysis, several unanticipated corrections were found necessary. The target was positioned at the center of the experimental chamber to a few microns precision, using a target insertion mechanism; its position was adjusted using telescope observations. The imaging slit and streak camera were mounted and positioned on a diagnostic insertion mechanism. The nominal depth of insertion was defined by the component dimensions, but the actual depth of insertion was typically less and was not measured directly, reducing the magnification of the target at the DISC [DIM (diagnostic instrument manipulator) Imaging Streak Camera] streak camera. The insertion depth could be inferred to an accuracy ∼1% by the alignment imagers, which would make it a significant contribution to the uncertainty of the Hugoniot measurement. However, the ARIANE x-ray camera^{23} imaged the x-ray backlighter using a pinhole array mounted on the snout of the DISC; the alignment of these images allowed the DISC insertion depth to be determined more accurately, giving the magnification to ∼0.25%.

Data from the DISC x-ray streak camera were corrected automatically for spatial and temporal distortions as determined from calibrations by the manufacturer (National Security Technologies, Livermore) and during acceptance tests.^{24} However, the camera exhibited shot-to-shot variations that were not corrected in this way. These variations were at the several-percent level in inferred Hugoniot, and we developed ways to account for them.

One such source of uncertainty was the temporal sweep rate of the DISC camera: the time scale determines the shock speed and hence the pressure. The sweep rate and its deviation from linearity were slightly different each time the camera was operated, depending, for example, on the environmental temperature and humidity. In principle, the sweep parameters and absolute timing could be measured on each experiment using a series of fiducial pulses from a reference laser, fed to the edge of the photocathode, but the fiducial system was not available during these experiments. However, the voltage history applied to the plates deflecting the electrons inside the camera was measured and recorded on each experiment. The acceleration voltage of the camera was ∼10 kV, and the deflection voltage was ∼1 kV applied over the sweep duration of nominally 10 ns. For cameras of the DISC design operated in this regime, the lateral deflection (i.e., temporal in the streak record) of electrons reaching the detector was proportional to the instantaneous deflection voltage, so the voltage record could be used to determine a deflection-time relation for each shot. Examination of the voltage history from a set of trial triggerings of the camera showed a variation of up to 3% in sweep rate between experiments (Figs. 12 and 13). There was some uncertainty in relating a given voltage to the corresponding point in the streak record because of jitter in the triggering voltage and residual high-frequency noise in the voltage measurement, but the uncertainty in instantaneous sweep rate was reduced to around 0.4% by using the as-shot voltage history.

Fiducial wires were mounted in the DISC snout between the imaging slit and the photocathode of the streak camera. These did not move during the experiment. However, the location of the shadow cast by the wires (in x-rays) was found to vary during the camera sweep, by ∼20 *μ*m. This was an unexpected time-dependent camera distortion and indicated a change in both scale and centering as the sweep progressed. The cause was not investigated in detail, but this type of distortion can be the result of the particular three-dimensional configuration of the camera, where inductances and capacitances can affect the fields accelerating the electrons differently as they are swept in the time direction. As with the nonlinear sweep rate, the location of the wires was used to correct the mapping of spatial points.

### D. Density reconstruction

Since the doped layer in the ablator remained visible in the radiograph throughout the duration of the x-ray pulse, its radius-time variation was well defined.^{32} For a given density distribution, the mass enclosed within the marker layer was compared with the known mass of the sample. This comparison provided an independent measure of the accuracy of a given density distribution. If the apparent mass enclosed was found to decrease monotonically with time, it could also indicate that the opacity of the sample to the backlighter x-rays was decreasing as a result of heating and ionization. No statistically significant decrease in opacity was found in the low-drive experiment. If the opacity is assumed to remain constant, the apparent mass enclosed can be used as an additional constraint on the density distribution. When this was done, the uncertainty in the density distribution, and hence the Hugoniot, was reduced. The precise degree of reduction depends on the relative weighting given to deviations in the radiograph and deviations in the mass. As the mass weighting was increased, the Hugoniot uncertainty initially decreased and then remained close to constant.

As the actual density distribution contained the radially converging shock, which was a density discontinuity at the spatial resolution of the radiographic configuration, the parameterized distribution *ρ*(*r*, *t*; *c*) was split into regions separated by a discontinuity, as shown in the 1D example in Sec. V A. The interface between the regions was defined by the radius-time locus of the shock, *r*_{s}(*t*). The outside of the shocked region was defined by the radius-time locus of the marker layer, *r*_{m}(*t*). The shocked region was defined by functions describing the compression history at the shock front and the marker layer, *ρ*_{s}(*t*) and *ρ*_{m}(*t*), and a radius-time function capturing the variation in between, with the radius represented by the fractional distance between *r*_{s}(*t*) and *r*_{m}(*t*). The density distribution from the marker layer to the ablation surface was represented by a lower-order radius-time function, as the 1D study demonstrated that the detailed profile away from the shock did not need to be captured to reconstruct the density accurately at the shock.

Initial guesses for the loci *r*_{s}(*t*) and *r*_{m}(*t*) were made by estimating a set of points along each locus by inspection of the radiograph and fitting a function. The parameters in each function were included in the {*c*_{i}}, i.e., the smoothed by-eye loci were not used directly in deducing the Hugoniot. However, the final fit was found to be close to the initial guess. The manually identified points along the shock locus were fit well with the function

where *α*, *β*, and *t*_{0} were fitting parameters. The corresponding points along the marker layer and the outside of the ablator were fit well with the function

where the *α*, *β*, and *r*_{0} were fitting parameters, independent for each locus fitted. It was found possible to fit functions separately to the outside and inside of the marker layer, although the thickness of the layer was close to the calculated spatial resolution of the radiograph.

A variety of functions were used to represent the radius-time variation of the density in the shocked region, as we experimented to find functional forms which could efficiently reproduce the radiograph and investigated the sensitivity of the reconstructed density to parameter perturbations and to the choice of fitting functions. The density at the shock front, *ρ*_{s}(*t*), was represented variously by tabulated densities and by polynomials. Tabulations were optimized by varying the ordinates {*t*_{i}} or the values {*ρ*_{i}}. The density at the marker layer and in the ablation region was represented by lower-order polynomials. The interior density was represented as a linear variation between the value at the shock and that at the marker layer, plus a term describing the nonlinear internal variation,

Preference was given to functions with analytic integrals for the areal mass [Eq. (6)] over a finite region in radius. The areal mass for power law variations (including the constant unshocked region and the linear term above in the shocked region) can be expressed analytically, as can tabulations with polynomial interpolation. A particular family of interest was a Gaussian multiplied by an even power of radius,

as this gives shapes similar to those from hydrocode simulations of a converging shock. The distribution around the peak of the exponential is asymmetric in radius, the asymmetry depending on the integer parameter *n*. A strategy followed for representing the complete radius-time density distribution was to add multiple terms like this, making the continuous parameters {*c*_{i}} functions of time,

The time variation chosen depended on the parameter: density-like variations were typically represented as a tabulation or low-order polynomial and radius-like variations by a tabulation or by a nonlinear function like those used for *r*_{s} and *r*_{m}.

The approach above attempts to find an analytic function representing the density distribution over the full range of the streak radiograph. We also followed the complementary approach of dividing the radiograph into slices of time and fitting lower-order functions independently to each slice. In order to include enough pixels to give a reasonable fitting uncertainty, the slices had to be wide enough that the shock speed changed appreciably over the width. Thus *r*_{s}(*t*) and *r*_{m}(*t*) were represented by quadratics or functions of higher order. Similarly, the time-variation of parameters in the density functions was represented by tabulations or quadratics. The fitting functions deduced from the full radius-time loci had the same number of parameters as a quadratic, so they could be used directly (re-adjusting the parameters to optimize the fit to the slice) without appreciably changing the significance of the fit. Alternatively, the global fit to these loci was used to initialize a quadratic fit. The slice-wise fits gave Hugoniot points that were consistent with the global fit, but with larger uncertainties.

In this way, the most likely distribution of density was reconstructed as a function of radius and time, over the duration of the x-ray streak record, along with the uncertainty in density (Figs. 14–16). As the parameterization of the density distribution was chosen to include the radius-time history of the shock and the density at the shock front explicitly, the corresponding Hugoniot states could be deduced directly from the fitting functions (Fig. 17). The uncertainty in Hugoniot states can be estimated from the uncertainty in the fitting parameters, but the sensitivity of the radiograph fit to variations in the parameters was far from orthogonal—i.e., the variations were correlated—so the parameter uncertainties cannot be combined simply. Instead, the region in parameter space around the best fit was explored, and the uncertainty in the Hugoniot expressed as a probability distribution accumulated from trial Hugoniots with perturbed parameters, giving a less optimal fit to the radiograph. Rather than a set of discrete Hugoniot points, each with an uncertainty in each EOS parameter, the most natural representation of the measured Hugoniot is this probability distribution, which can be plotted, for instance, as the optimal fit (the ridge of the probability distribution) together with 1 − *σ* contours.

Contributions to the overall uncertainty include the statistical uncertainty in profile-matching parameters from the fit to the radiograph and also systematic contributions from uncertainties in the configuration and performance of the x-ray streak measurement. These systematic contributions include the uncertainty in magnification and modulation transfer functions of the imaging system and in the shock speed from the uncertainty in the sweep speed of the x-ray streak camera and its linearity. These contributions can be combined to estimate the overall uncertainty, but, as with the profile-matching parameters, a simple combination discards correlations between these uncertainties. Although the systematic contributions make the precise location of the Hugoniot less certain, they do not change its shape. Thus we prefer to plot uncertainty contours derived from the statistical uncertainty in reconstructing the radiograph from the density distribution and show the systematic uncertainty as a separate error bar which to a good approximation applies to the location of the reconstructed Hugoniot without changing its shape.

Unlike the well-established planar case, the converging shock wave is intrinsically unsteady. In principle, the Rankine-Hugoniot equations should be modified to include a term describing the effect of curvature. The thickness of a shock in a homogeneous material is on the order of the mean free path of the constituent molecules,^{29} which is at least four orders of magnitude smaller than the minimum radius of the shock that could be resolved in these experiments, so the effect of acceleration should be negligible. An equivalent argument is that the strength of the shock changes slowly with respect to the radius (measured relative to the shock thickness) so the local behavior of the shock is close to steady. Our hydrocode simulations used artificial viscosity to prevent unphysical oscillations caused by the shock,^{11} giving the shock an artificially large thickness. The Hugoniot inferred from simulated data reproduced the Hugoniot of the EOS used in the hydrocode, giving additional confidence that the steady shock analysis was justified.

## VI. CONCLUSIONS

Time-resolved x-ray radiography of a laser-driven spherically converging shock provided a way to measure shock Hugoniot states absolutely—without reference to an EOS standard—over a range of states in a single experiment. Convergence increased the shock pressure significantly over the externally applied drive pressure without the drawbacks of increasing a driving radiation intensity, such as preheat. With a hohlraum-based platform, the pressure drive was spatially uniform, and Hugoniot states were measured at higher pressure than previously obtained in laboratory experiments.

Profile-matching, using the known state of the unshocked material, provided a strong constraint on the shock compression and provided a framework for averaging out noise in the temporal as well as the spatial direction. The resulting uncertainty was orders of magnitude less than would be the case using density reconstruction by independent Abel inversion at each instant of time.

The accuracy of the measurement depends very heavily on the precise assembly of the target, the accurate knowledge of the magnification and modulation transfer function of the radiographic configuration, and the behavior of the x-ray streak camera. It was necessary to make unanticipated corrections for linearity of sweep speed and time variations in spatial imaging to obtain data of the desired accuracy.

## ACKNOWLEDGMENTS

This work was performed in support of Laboratory Directed Research and Development Project No. 13-ERD-073 (Principal Investigator: Andrea Kritcher), under the auspices of the U.S. Department of Energy under Contract No. DE-AC52-07NA27344 and Fusion Energy Sciences FWP No. 1001082.

## REFERENCES

The converging shock experiments departed from ICF reference designs in one potentially important respect, that the outside of the solid sphere was compressed inward significantly more slowly than a hollow inertial fusion capsule, so the plasma conditions in the hohlraum would diverge as time progressed.

For materials in which material strength is significant, *p* is the normal stress rather than the mean pressure.

The outer surface of the ablator was not well defined, as the material was ablated into the hohlraum and became mixed with the gas fill and wall material. In an experiment at higher drive energy, the doped layer was ablated by ∼kilovolt x-rays, and its location could not subsequently be determined to any useful accuracy.