The impact of low-mode symmetry on inertial fusion energy output in the burning plasma state

Indirect Drive Inertial Confinement Fusion Experiments on the National Ignition Facility (NIF) have achieved a burning plasma state with neutron yields exceeding 170 kJ, roughly 3 times the prior record and a necessary stage for igniting plasmas. The results are achieved despite multiple sources of degradations that lead to high variability in performance. Results shown here, for the first time, include an empirical correction factor for mode-2 asymmetry in the burning plasma regime in addition to previously determined corrections for radiative mix and mode-1. Analysis shows that including these three corrections alone accounts for the measured fusion performance variability in the two highest performing experimental campaigns on the NIF to within error. Here we quantify the performance sensitivity to mode-2 symmetry in the burning plasma regime and apply the results, in the form of an empirical correction to a 1D performance model. Furthermore, we find the sensitivity to mode-2 determined through a series of integrated 2D radiation hydrodynamic simulations to be consistent with the experimentally determined sensitivity only when including alpha-heating.

Achievement of Target Gain Larger than Unity in an Inertial Fusion Experiment

On December 5, 2022, an indirect drive fusion implosion on the National Ignition Facility (NIF) achieved a target gain G_{target} of 1.5. This is the first laboratory demonstration of exceeding "scientific breakeven" (or G_{target}>1) where 2.05 MJ of 351 nm laser light produced 3.1 MJ of total fusion yield, a result which significantly exceeds the Lawson criterion for fusion ignition as reported in a previous NIF implosion [H. Abu-Shawareb et al. (Indirect Drive ICF Collaboration), Phys. Rev. Lett. 129, 075001 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.075001]. This achievement is the culmination of more than five decades of research and gives proof that laboratory fusion, based on fundamental physics principles, is possible. This Letter reports on the target, laser, design, and experimental advancements that led to this result.

Dynamics and Power Balance of Near Unity Target Gain Inertial Confinement Fusion Implosions

The change in the power balance, temporal dynamics, emission weighted size, temperature, mass, and areal density of inertially confined fusion plasmas have been quantified for experiments that reach target gains up to 0.72. It is observed that as the target gain rises, increased rates of self-heating initially overcome expansion power losses. This leads to reacting plasmas that reach peak fusion production at later times with increased size, temperature, mass and with lower emission weighted areal densities. Analytic models are consistent with the observations and inferences for how these quantities evolve as the rate of fusion self-heating, fusion yield, and target gain increase. At peak fusion production, it is found that as temperatures and target gains rise, the expansion power loss increases to a near constant ratio of the fusion self-heating power. This is consistent with models that indicate that the expansion losses dominate the dynamics in this regime.

Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment

For more than half a century, researchers around the world have been engaged in attempts to achieve fusion ignition as a proof of principle of various fusion concepts. Following the Lawson criterion, an ignited plasma is one where the fusion heating power is high enough to overcome all the physical processes that cool the fusion plasma, creating a positive thermodynamic feedback loop with rapidly increasing temperature. In inertially confined fusion, ignition is a state where the fusion plasma can begin "burn propagation" into surrounding cold fuel, enabling the possibility of high energy gain. While "scientific breakeven" (i.e., unity target gain) has not yet been achieved (here target gain is 0.72, 1.37 MJ of fusion for 1.92 MJ of laser energy), this Letter reports the first controlled fusion experiment, using laser indirect drive, on the National Ignition Facility to produce capsule gain (here 5.8) and reach ignition by nine different formulations of the Lawson criterion.

Record Energetics for an Inertial Fusion Implosion at NIF

Inertial confinement fusion seeks to create burning plasma conditions in a spherical capsule implosion, which requires efficiently absorbing the driver energy in the capsule, transferring that energy into kinetic energy of the imploding DT fuel and then into internal energy of the fuel at stagnation. We report new implosions conducted on the National Ignition Facility (NIF) with several improvements on recent work [Phys. Rev. Lett. 120, 245003 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.245003; Phys. Rev. E 102, 023210 (2020)PRESCM2470-004510.1103/PhysRevE.102.023210]: larger capsules, thicker fuel layers to mitigate fuel-ablator mix, and new symmetry control via cross-beam energy transfer; at modest velocities, these experiments achieve record values for the implosion energetics figures of merit as well as fusion yield for a NIF experiment.

Observation of hohlraum-wall motion with spectrally selective x-ray imaging at the National Ignition Facility

The high fuel capsule compression required for indirect drive inertial confinement fusion requires careful control of the X-ray drive symmetry throughout the laser pulse. When the outer cone beams strike the hohlraum wall, the plasma ablated off the hohlraum wall expands into the hohlraum and can alter both the outer and inner cone beam propagations and hence the X-ray drive symmetry especially at the final stage of the drive pulse. To quantitatively understand the wall motion, we developed a new experimental technique which visualizes the expansion and stagnation of the hohlraum wall plasma. Details of the experiment and the technique of spectrally selective x-ray imaging are discussed.

Automated analysis of hot spot X-ray images at the National Ignition Facility

At the National Ignition Facility, the symmetry of the hot spot of imploding capsules is diagnosed by imaging the emitted x-rays using gated cameras and image plates. The symmetry of an implosion is an important factor in the yield generated from the resulting fusion process. The x-ray images are analyzed by decomposing the image intensity contours into Fourier and Legendre modes. This paper focuses on the additional protocols for the time-integrated shape analysis from image plates. For implosions with temperatures above ∼4 keV, the hard x-ray background can be utilized to infer the temperature of the hot spot.

First high-convergence cryogenic implosion in a near-vacuum hohlraum

Recent experiments on the National Ignition Facility [M. J. Edwards et al., Phys. Plasmas 20, 070501 (2013)] demonstrate that utilizing a near-vacuum hohlraum (low pressure gas-filled) is a viable option for high convergence cryogenic deuterium-tritium (DT) layered capsule implosions. This is made possible by using a dense ablator (high-density carbon), which shortens the drive duration needed to achieve high convergence: a measured 40% higher hohlraum efficiency than typical gas-filled hohlraums, which requires less laser energy going into the hohlraum, and an observed better symmetry control than anticipated by standard hydrodynamics simulations. The first series of near-vacuum hohlraum experiments culminated in a 6.8 ns, 1.2 MJ laser pulse driving a 2-shock, high adiabat (α∼3.5) cryogenic DT layered high density carbon capsule. This resulted in one of the best performances so far on the NIF relative to laser energy, with a measured primary neutron yield of 1.8×10(15) neutrons, with 20% calculated alpha heating at convergence ∼27×.

Thin shell, high velocity inertial confinement fusion implosions on the national ignition facility

Experiments have recently been conducted at the National Ignition Facility utilizing inertial confinement fusion capsule ablators that are 175 and 165  μm in thickness, 10% and 15% thinner, respectively, than the nominal thickness capsule used throughout the high foot and most of the National Ignition Campaign. These three-shock, high-adiabat, high-foot implosions have demonstrated good performance, with higher velocity and better symmetry control at lower laser powers and energies than their nominal thickness ablator counterparts. Little to no hydrodynamic mix into the DT hot spot has been observed despite the higher velocities and reduced depth for possible instability feedthrough. Early results have shown good repeatability, with up to 1/2 the neutron yield coming from α-particle self-heating.

Diagnosing residual motion via the x-ray self emission from indirectly driven inertial confinement implosions

In an indirectly driven implosion, non-radial translational motion of the compressed fusion capsule is a signature of residual kinetic energy not coupled into the compressional heating of the target. A reduction in compression reduces the peak pressure and nuclear performance of the implosion. Measuring and reducing the residual motion of the implosion is therefore necessary to improve performance and isolate other effects that degrade performance. Using the gated x-ray diagnostic, the x-ray Bremsstrahlung emission from the compressed capsule is spatially and temporally resolved at x-ray energies of >8.7 keV, allowing for measurements of the residual velocity. Here details of the x-ray velocity measurement and fitting routine will be discussed and measurements will be compared to the velocities inferred from the neutron time of flight detectors.

Reconstruction of 2D x-ray radiographs at the National Ignition Facility using pinhole tomography (invited)

Two-dimensional radiographs of imploding fusion capsules are obtained at the National Ignition Facility by projection through a pinhole array onto a time-gated framing camera. Parallax among images in the image array makes it possible to distinguish contributions from the capsule and from the backlighter, permitting correction of backlighter non-uniformities within the capsule radiograph. Furthermore, precise determination of the imaging system geometry and implosion velocity enables combination of multiple images to reduce signal-to-noise and discover new capsule features.

2D X-ray radiography of imploding capsules at the national ignition facility

First measurements of the in-flight shape of imploding inertial confinement fusion (ICF) capsules at the National Ignition Facility (NIF) were obtained by using two-dimensional x-ray radiography. The sequence of area-backlit, time-gated pinhole images is analyzed for implosion velocity, low-mode shape and density asymmetries, and the absolute offset and center-of-mass velocity of the capsule shell. The in-flight shell is often observed to be asymmetric even when the concomitant core self-emission is round. A ∼ 15 μm shell asymmetry amplitude of the Y(40) spherical harmonic mode was observed for standard NIF ICF hohlraums at a shell radius of ∼ 200 μm (capsule at ∼ 5× radial compression). This asymmetry is mitigated by a ∼ 10% increase in the hohlraum length.

Early-time symmetry tuning in the presence of cross-beam energy transfer in ICF experiments on the National Ignition Facility

On the National Ignition Facility, the hohlraum-driven implosion symmetry is tuned using cross-beam energy transfer (CBET) during peak power, which is controlled by applying a wavelength separation between cones of laser beams. In this Letter, we present early-time measurements of the instantaneous soft x-ray drive at the capsule using reemission spheres, which show that this wavelength separation also leads to significant CBET during the first shock, even though the laser intensities are 30× smaller than during the peak. We demonstrate that the resulting early drive P2/P0 asymmetry can be minimized and tuned to <1% accuracy (well within the ±7.5% requirement for ignition) by varying the relative input powers between different cones of beams. These experiments also provide time-resolved measurements of CBET during the first 2 ns of the laser drive, which are in good agreement with radiation-hydrodynamics calculations including a linear CBET model.

Hot-spot mix in ignition-scale inertial confinement fusion targets

Mixing of plastic ablator material, doped with Cu and Ge dopants, deep into the hot spot of ignition-scale inertial confinement fusion implosions by hydrodynamic instabilities is diagnosed with x-ray spectroscopy on the National Ignition Facility. The amount of hot-spot mix mass is determined from the absolute brightness of the emergent Cu and Ge K-shell emission. The Cu and Ge dopants placed at different radial locations in the plastic ablator show the ablation-front hydrodynamic instability is primarily responsible for hot-spot mix. Low neutron yields and hot-spot mix mass between 34(-13,+50)  ng and 4000(-2970,+17 160)  ng are observed.

Numerical modeling of the sensitivity of x-ray driven implosions to low-mode flux asymmetries

The sensitivity of inertial confinement fusion implosions, of the type performed on the National Ignition Facility (NIF) [1], to low-mode flux asymmetries is investigated numerically. It is shown that large-amplitude, low-order mode shapes (Legendre polynomial P(4), resulting from low-order flux asymmetries, cause spatial variations in capsule and fuel momentum that prevent the deuterium and tritium (DT) "ice" layer from being decelerated uniformly by the hot spot pressure. This reduces the transfer of implosion kinetic energy to internal energy of the central hot spot, thus reducing the neutron yield. Furthermore, synthetic gated x-ray images of the hot spot self-emission indicate that P(4) shapes may be unquantifiable for DT layered capsules. Instead the positive P(4) asymmetry "aliases" itself as an oblate P(2) in the x-ray images. Correction of this apparent P(2) distortion can further distort the implosion while creating a round x-ray image. Long wavelength asymmetries may be playing a significant role in the observed yield reduction of NIF DT implosions relative to detailed postshot two-dimensional simulations.

Stochastic ion heating from many overlapping laser beams in fusion plasmas

In this Letter, we show through numerical simulations and analytical results that overlapping multiple (N) laser beams in plasmas can lead to strong stochastic ion heating from many (~N(2)) electrostatic perturbations driven by beat waves between pairs of laser beams. For conditions typical of inertial-confinement-fusion experiment conditions, hundreds of such beat waves are driven in mm(3)-scale plasmas, leading to ion heating rates of several keV/ns. This mechanism saturates cross-beam energy transfer, with a reduction of linear gains by a factor ~4-5 and can strongly modify the overall hydrodynamics evolution of such laser-plasma systems.

Characterization of single and colliding laser-produced plasma bubbles using Thomson scattering and proton radiography

Time-resolved measurements of electron and ion temperatures using Thomson scattering have been combined with proton radiography data for comprehensive characterization of individual laser-produced plasma bubbles or the interaction of bubble pairs, where reconnection of azimuthal magnetic fields occurs. Measurements of ion and electron temperatures agree with lasnex simulations of single plasma bubbles, which include the physics of magnetic fields. There is negligible difference in temperatures between a single plasma bubble and the interaction region of bubble pairs, although the ion temperature may be slightly higher due to the collision of expanding plasmas. These results are consistent with reconnection in a β∼8 plasma, where the release of magnetic energy (<5% of the electron thermal energy) does not appreciably affect the hydrodynamics.

Imaging of high-energy x-ray emission from cryogenic thermonuclear fuel implosions on the NIF

Accurately assessing and optimizing the implosion performance of inertial confinement fusion capsules is a crucial step to achieving ignition on the NIF. We have applied differential filtering (matched Ross filter pairs) to provide broadband time-integrated absolute x-ray self-emission images of the imploded core of cryogenic layered implosions. This diagnostic measures the temperature- and density-sensitive bremsstrahlung emission and provides estimates of hot spot mass, mix mass, and pressure.

Precision shock tuning on the national ignition facility

Ignition implosions on the National Ignition Facility [J. D. Lindl et al., Phys. Plasmas 11, 339 (2004)] are underway with the goal of compressing deuterium-tritium fuel to a sufficiently high areal density (ρR) to sustain a self-propagating burn wave required for fusion power gain greater than unity. These implosions are driven with a very carefully tailored sequence of four shock waves that must be timed to very high precision to keep the fuel entropy and adiabat low and ρR high. The first series of precision tuning experiments on the National Ignition Facility, which use optical diagnostics to directly measure the strength and timing of all four shocks inside a hohlraum-driven, cryogenic liquid-deuterium-filled capsule interior have now been performed. The results of these experiments are presented demonstrating a significant decrease in adiabat over previously untuned implosions. The impact of the improved shock timing is confirmed in related deuterium-tritium layered capsule implosions, which show the highest fuel compression (ρR~1.0 g/cm(2)) measured to date, exceeding the previous record [V. Goncharov et al., Phys. Rev. Lett. 104, 165001 (2010)] by more than a factor of 3. The experiments also clearly reveal an issue with the 4th shock velocity, which is observed to be 20% slower than predictions from numerical simulation.

Assembly of high-areal-density deuterium-tritium fuel from indirectly driven cryogenic implosions

The National Ignition Facility has been used to compress deuterium-tritium to an average areal density of ~1.0±0.1 g cm(-2), which is 67% of the ignition requirement. These conditions were obtained using 192 laser beams with total energy of 1-1.6 MJ and peak power up to 420 TW to create a hohlraum drive with a shaped power profile, peaking at a soft x-ray radiation temperature of 275-300 eV. This pulse delivered a series of shocks that compressed a capsule containing cryogenic deuterium-tritium to a radius of 25-35 μm. Neutron images of the implosion were used to estimate a fuel density of 500-800 g cm(-3).

Impeding hohlraum plasma stagnation in inertial-confinement fusion

This Letter reports the first time-gated proton radiography of the spatial structure and temporal evolution of how the fill gas compresses the wall blowoff, inhibits plasma jet formation, and impedes plasma stagnation in the hohlraum interior. The potential roles of spontaneously generated electric and magnetic fields in the hohlraum dynamics and capsule implosion are discussed. It is shown that interpenetration of the two materials could result from the classical Rayleigh-Taylor instability occurring as the lighter, decelerating ionized fill gas pushes against the heavier, expanding gold wall blowoff. This experiment showed new observations of the effects of the fill gas on x-ray driven implosions, and an improved understanding of these results could impact the ongoing ignition experiments at the National Ignition Facility.

Three-wavelength scheme to optimize hohlraum coupling on the National Ignition Facility

By using three tunable wavelengths on different cones of laser beams on the National Ignition Facility, numerical simulations show that the energy transfer between beams can be tuned to redistribute the energy within the cones of beams most prone to backscatter instabilities. These radiative hydrodynamics and laser-plasma interaction simulations have been tested against large-scale hohlraum experiments with two tunable wavelengths and reproduce the hohlraum energetics and symmetry. Using a third wavelength provides a greater level of control of the laser energy distribution and coupling in the hohlraum, and could significantly reduce stimulated Raman scattering losses and increase the hohlraum radiation drive while maintaining a good implosion symmetry.

Demonstration of ignition radiation temperatures in indirect-drive inertial confinement fusion hohlraums

We demonstrate the hohlraum radiation temperature and symmetry required for ignition-scale inertial confinement fusion capsule implosions. Cryogenic gas-filled hohlraums with 2.2 mm-diameter capsules are heated with unprecedented laser energies of 1.2 MJ delivered by 192 ultraviolet laser beams on the National Ignition Facility. Laser backscatter measurements show that these hohlraums absorb 87% to 91% of the incident laser power resulting in peak radiation temperatures of T(RAD)=300 eV and a symmetric implosion to a 100 μm diameter hot core.

Lorentz mapping of magnetic fields in hot dense plasmas

Unique detection of electromagnetic fields and identification of field type and strength as a function of position were used to determine the nature of self-generated fields in a novel experiment with laser-generated plasma bubbles on two sides of a plastic foil. Field-induced deflections of monoenergetic 15-MeV probe protons passing through the two bubbles, measured quantitatively with proton radiography, were combined with Lorentz mapping to provide separate measurements of magnetic and electric fields. The result was absolute identification and measurement of a toroidal magnetic field around each bubble and determination that any electric field component parallel to the foil was below measurement uncertainties.

Pressure-driven, resistive magnetohydrodynamic interchange instabilities in laser-produced high-energy-density plasmas

Recent experiments using proton backlighting of laser-foil interactions provide unique opportunities for studying magnetized plasma instabilities in laser-produced high-energy-density plasmas. Time-gated proton radiograph images indicate that the outer structure of a magnetic field entrained in a hemispherical plasma bubble becomes distinctly asymmetric after the laser turns off. It is shown that this asymmetry is a consequence of pressure-driven, resistive magnetohydrodynamic (MHD) interchange instabilities. In contrast to the predictions made by ideal MHD theory, the increasing plasma resistivity after laser turn-off allows for greater low-mode destabilization (m>1) from reduced stabilization by field-line bending. For laser-generated plasmas presented herein, a mode-number cutoff for stabilization of perturbations with m> approximately [8pibeta(1+D_{m}k_{ perpendicular};{2}gamma_{max};{-1})];{1/2} is found in the linear growth regime. The growth is measured and is found to be in reasonable agreement with model predictions.

Observations of electromagnetic fields and plasma flow in hohlraums with proton radiography

We report on the first proton radiography of laser-irradiated hohlraums. This experiment, with vacuum gold (Au) hohlraums, resulted in observations of self-generated magnetic fields with peak values approximately 10;{6} G. Time-gated radiographs of monoenergetic protons with discrete energies (15.0 and 3.3 MeV) reveal dynamic pictures of field structures and plasma flow. Near the end of the 1-ns laser drive, a stagnating Au plasma (approximately 10 mg cm;{-3}) forms at the center of the hohlraum. This is a consequence of supersonic, radially directed Au jets (approximately 1000 microm ns;{-1}, approximately Mach 4) that arise from the interaction of laser-driven plasma bubbles expanding into one another.

Monoenergetic-proton-radiography measurements of implosion dynamics in direct-drive inertial-confinement fusion

Time-gated, monoenergetic radiography with 15-MeV protons provides unique measurements of implosion dynamics in direct-drive inertial-confinement fusion. Images obtained during acceleration, coasting, deceleration, and stagnation display a comprehensive picture of spherical implosions. Critical information inferred from such images, hitherto unavailable, characterizes the spatial structure and temporal evolution of self-generated fields and plasma areal density. Results include the first observation of a radial electric field inside the imploding capsule. It is initially directed inward (at approximately 10(9) V/m), eventually reverses direction ( approximately 10(8) V/m), and is the probable consequence of the evolution of the electron pressure gradient.

Observation of megagauss-field topology changes due to magnetic reconnection in laser-produced plasmas

The spatial structure and temporal evolution of megagauss magnetic fields generated by interactions of up to 4 laser beams with matter were studied with an innovative, time-gated proton radiography method that produces images of unprecedented clarity because it uses an isotropic, truly monoenergetic back-lighter (14.7-MeV protons from D3He nuclear fusion reactions). Quantitative field maps reveal precisely and directly, for the first time, changes in the magnetic topology due to reconnection in a high-energy-density plasma (n(e) approximately 10(20)-10(22) cm(-3), T(e) approximately 1 keV).

Observation of the decay dynamics and instabilities of megagauss field structures in laser-produced plasmas

Monoenergetic proton radiography was used to make the first measurements of the long-time-scale dynamics and evolution of megagauss laser-plasma-generated magnetic field structures. While a 1-ns 10(14) W/cm2 laser beam is on, the field structure expands in tandem with a hemispherical plasma bubble, maintaining a rigorous 2D cylindrical symmetry. With the laser off, the bubble continues to expand as the field decays; however, the outer field structure becomes distinctly asymmetric, indicating instability. Similarly, localized asymmetry growth in the bubble interior indicates another kind of instability. 2D LASNEX hydrosimulations qualitatively match the cylindrically averaged post-laser plasma evolution but even then it underpredicts the field dissipation rate and of course completely misses the 3D asymmetry growth.

Quenching of the nonlocal electron heat transport by large external magnetic fields in a laser-produced plasma measured with imaging thomson scattering

We present a direct measurement of the quenching of nonlocal heat transport in a laser-produced plasma by applying large external magnetic fields (>10 T). The temporally resolved Thomson-scattering measurements of the electron temperature profile show that the heat front propagation transverse to a high-power laser beam is slowed resulting in extremely strong local heating. We find agreement with hydrodynamic modeling when including a magnetic field model that self-consistently evolves the fields in the plasma.

Measuring E and B fields in laser-produced plasmas with monoenergetic proton radiography

Electromagnetic (E/B) fields generated by the interaction with plasmas of long-pulse, low-intensity laser beams relevant to inertial confinement fusion have been measured for the first time using novel monoenergetic proton radiography methods. High-resolution, time-gated radiography images of a plastic foil driven by a 10(14) W/cm(2) laser implied B fields of approximately 0.5 MG and E fields of approximately 1.5 x 10(8) V/m. Simulations of these experiments with LASNEX+LSP have been performed and are in overall (though not exact) agreement with the data both for field strengths and for spatial distributions; this is the first direct experimental test of the laser-generated B-field package in LASNEX. The experiments also demonstrated that laser phase plates substantially reduce medium-scale chaotic field structure.

Proton radiography of a laser-driven implosion

Protons accelerated by a picosecond laser pulse have been used to radiograph a 500 microm diameter capsule, imploded with 300 J of laser light in 6 symmetrically incident beams of wavelength 1.054 microm and pulse length 1 ns. Point projection proton backlighting was used to characterize the density gradients at discrete times through the implosion. Asymmetries were diagnosed both during the early and stagnation stages of the implosion. Comparison with analytic scattering theory and simple Monte Carlo simulations were consistent with a 3+/-1 g/cm3 core with diameter 85+/-10 microm. Scaling simulations show that protons>50 MeV are required to diagnose asymmetry in ignition scale conditions.

Effect of nonlocal transport on heat-wave propagation

We present the first direct measurements of spatially and temporally resolved temperature and density profiles produced by nonlocal transport in a laser plasma. Absolutely calibrated measurements have been performed by Rayleigh scattering and by resolving the ion-acoustic wave spectra across the plasma volume with Thomson scattering. We find that the electron temperature and density profiles disagree with flux-limited models, but are consistent with nonlocal transport modeling.

Nonlinear theory of the ablative Rayleigh-Taylor instability

A fully nonlinear sharp-boundary model of the ablative Rayleigh-Taylor instability is derived and closed in a similar way to the self-consistent closure of the linear theory. It contains the stabilizing effect of ablation and accurately reproduces the results of 2D DRACO simulations. The single-mode saturation amplitude, bubble and spike evolutions in the nonlinear regimes, and the seeding of long-wavelength modes via mode coupling are determined and compared with the classical theory without ablation. Nonlinear stability above the linear cutoff is also predicted.

Shell mix in the compressed core of spherical implosions

The Rayleigh-Taylor instability in its highly nonlinear, turbulent stage causes atomic-scale mixing of the shell material with the fuel in the compressed core of inertial-confinement fusion targets. The density of shell material mixed into the outer core of direct-drive plastic-shell spherical-target implosions on the 60-beam, OMEGA laser system is estimated to be 3.4(+/-1.2) g/cm(3) from time-resolved x-ray spectroscopy, charged-particle spectroscopy, and core x-ray images. The estimated fuel density, 3.6(+/-1) g/cm(3), accounts for only approximately 50% of the neutron-burn-averaged electron density, n(e)=2.2(+/-0.4)x10(24) cm(-3).