September, 2014

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA

This work was supported by the U.S. Department of Energy, Grant No. DE-NA0001857, FSC, Grant No. 5-24431, LLE, Grant No. 415935-G, and LLNL, Grant No. B600100. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted. Submitted for publication to Physics of Plasmas.

PSFC/JA-14-26

Utilization of NIF Diagnostic Development Shots for Investigation of Ion Kinetic Effects in Direct-Drive

Exploding-Pusher Implosions

Rosenberg, M. J., Zylstra, A. B., Séguin, F. H., Rinderknecht, H. G., Frenje, J. A., Gatu Johnson, M., Sio, H., Waugh, C. J., Sinenian, N.,

Li, C. K., Petrasso, R. D., McKenty, P. W.1, Hohenberger, M.1, Radha, P. B.1, Delettrez, J. A.1, Glebov, V. Yu.1, Betti, R.1,

Goncharov, V. N.1, Knauer, J. P.1, Sangster, T. C.1, LePape, S.2, Mackinnon, A. J.2, Pino, J.2, McNaney, J. M.2, Rygg, J. R.2,

Amendt, P. A.2, Bellei, C.2, Benedetti, L. R.2, Berzak Hopkins, L.2, Bionta, R. M.2, Casey, D. T.2, Divol, L.2, Edwards, M. J.2,

Glenn, S.2, Glenzer, S. H.2, Hicks, D. G.2, Kimbrough, J. R.2, Landen, O. L.2, Lindl, J. D.2, Ma, T.2, MacPhee, A.2,

Meezan, N. B.2, Moody, J. D.2, Moran, M. J.2, Park, H.-S.2, Remington, B. A.2, Robey, H.2, Rosen, M. D.2, Wilks, S. C.2,

Zacharias, R. A.2, Herrmann, H. W.3, Hoffman, N. M.3, Kyrala, G. A.3, Leeper, R. J.3, Olson, R. E.3, Kilkenny, J. D.4,

and Nikroo, A.4 1Laboratory for Laser Energetics, University of Rochester, Rochester, NY 2Lawrence Livermore National Laboratory, Livermore, CA 3Los Alamos National Laboratory, Los Alamos, NM 4General Atomics, San Diego, CA

Utilization of NIF Diagnostic Development Shots for Investigation of IonKinetic Effects in Direct-Drive Exploding-Pusher Implosions

M. J. Rosenberg,a) A. B. Zylstra, F. H. Seguin, H. G. Rinderknecht, J. A. Frenje, M. Gatu Johnson, H. Sio, C. J.Waugh, N. Sinenian, C. K. Li, and R. D. PetrassoPlasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139,USA

P. W. McKenty, M. Hohenberger, P. B. Radha, J. A. Delettrez, V. Yu. Glebov, R. Betti, V. N.Goncharov, J. P.Knauer, and T. C. SangsterLaboratory for Laser Energetics, University of Rochester, Rochester, NY 14623,USA

S. LePape, A. J. Mackinnon, J. Pino, J. M. McNaney, J. R. Rygg, P. A. Amendt, C. Bellei, L. R. Benedetti, L.Berzak Hopkins, R. M. Bionta, D. T. Casey, L. Divol, M. J. Edwards, S. Glenn, S. H. Glenzer, D. G. Hicks, J. R.Kimbrough, O. L. Landen, J. D. Lindl, T. Ma, A. MacPhee, N. B. Meezan, J. D. Moody, M. J. Moran, H.-S.Park, B. A. Remington, H. Robey, M. D. Rosen, S. C. Wilks, and R. A. ZachariasLawrence Livermore National Laboratory, Livermore, CA 94550, USA

H. W. Herrmann, N. M. Hoffman, G. A. Kyrala, R. J. Leeper, and R. E. OlsonLos Alamos National Laboratory, Los Alamos, NM 87545, USA

J. D. Kilkenny and A. NikrooGeneral Atomics, San Diego, CA 92186, USA

(Dated: 22 September 2014)

On experiments designed for diagnostic calibration at the National Ignition Facility (NIF), ride-along mea-surements of yield, ion temperature, areal density (ρR), shell convergence, and bang time have been madein shock-driven “exploding-pusher” inertial confinement fusion (ICF) implosions to assess the impact of ionkinetic effects. These measurements, which were obtained on D2 and D3He thin-glass-shell implosions, probethe shock convergence phase of ICF implosions, a critical stage in hot-spot ignition experiments. The datacomplement previous studies of kinetic effects in shock-driven implosions. Ion temperature and fuel ρR in-ferred from fusion-product spectroscopy are used to estimate the ion-ion mean free path in the gas. A trend ofdecreasing yields relative to the predictions of 2D draco hydrodynamics simulations with increasing Knudsennumber (the ratio of ion-ion mean free path to minimum shell radius) suggests that ion kinetic effects areincreasingly impacting the hot fuel region, in general agreement with previous results. The long mean freepath conditions giving rise to ion kinetic effects in the gas are often prevalent during the shock phase of bothexploding pushers and ablatively-driven implosions. In addition, a D3He exploding pusher shot (N121128)demonstrates that a precision, monoenergetic proton source could easily be implemented at the NIF for back-lighting a broad range of high energy density (HED) experiments in which fields and flows are manifest, andalso utilized for studies of stopping power in warm dense matter and in classical plasmas.

PACS numbers: 52.57.-z

I. INTRODUCTION

Thin, spherical glass shells were among the first cap-sules used in laser-driven inertial confinement fusion(ICF) research.1–4 Known as “exploding pushers”, theseimplosions are characterized by rapid heating of the thinshell, which explodes and drives a shock wave into thefuel. This shock wave compresses and heats the gasas it converges at the center of the capsule, and pro-duces fusion reactions as it rebounds back through thefuel. This type of implosion is contrasted against im-plosions designed to achieve ignition and energy gain,which, in the final stage, are driven by the ablation ofouter shell material and subsequent hydrodynamic com-pression of the fuel. These ablatively-driven implosions

a)Electronic mail: mrosenbe@mit.edu

are currently under study at the National Ignition Facil-ity (NIF) in the indirect-drive configuration.5–10 Becauseshock-driven exploding-pusher implosions are mainly 1Din nature and less sensitive to the complex hydrody-namic processes that characterize ablatively-driven im-plosions, such as hydrodynamic instabilities and mix,they are an ideal experiment to isolate and study theshock-convergence phase of ICF implosions.

For this reason, exploding pushers are an excellentplatform to probe kinetic effects, which can be significantduring the shock-convergence phase of ICF implosions.11

Kinetic effects are often important in these moderate-density, high-temperature implosions where the meanfree path for ion-ion collisions approaches the size ofthe burn region. Previous experimental work has in-vestigated kinetic effects in shock-driven implosions anddemonstrated the significance of ion diffusion and otherlong-mean-free-path effects that are not usually mod-

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eled in hydrodynamic codes.12,13 In a systematic studyof exploding pushers on OMEGA with a variety of ini-tial gas densities, the ion-ion mean free path λii andthe Knudsen number NK ≡ λii/Rshell, the ratio ofmean free path to minimum shell radius and a key fig-ure of merit of ion kinetic effects, were varied from aregime that is reasonably hydrodynamic-like (NK<1) toa regime that is strongly kinetic (NK�1).12 The trendof increasing deviation from hydrodynamic models (suchas the 1D radiation-hydrodynamics code dued14) withincreasing Knudsen number indicated the influence ofion kinetic effects. In agreement with this picture, an-other recent study by Le Pape et al. shows that ina high-density, low-temperature, short-mean-free-path,(NK�1) indirect-drive exploding pusher implosion, hy-drodynamic codes are able to reproduce with high fidelitythe experimental results.15 In concert, these studies in-dicate that the ion-ion mean free path and the Knudsennumber is a strong determinant of the applicability ofhydrodynamic models.

To further investigate whether the trends observed inthese prior experiments apply generally, under quite dif-ferent experimental conditions (larger capsules, asym-metric illumination, oblate implosions), data were ob-tained, in a ride-along mode, on polar-direct-drive(PDD)16,17 exploding pusher shots at the NIF that wereconducted for diagnostic development and calibration.In addition, given the scarcity of NIF shots for non-programmatic purposes, it is important to cull as muchinformation and physics insight possible from these diag-nostic development shots. These experiments producedcopious DD and D3He reactions, allowing for character-ization of the implosions through measurement of twoseparate fusion yields, two different burn-averaged iontemperatures, fuel ρR in D2 implosions and total ρR inboth D2 and D3He implosions, x-ray images of the im-ploding shell and core, and x-ray and nuclear bang-times.This study employs 2D draco radiation-hydrodynamicssimulations18,19 for comparison to experimental data. Inthis work, the experimentally-inferred ion-ion mean freepath λii ∝ T 2

i /niZ4 (where Ti is the ion temperature,

ni is the ion density, and Z is the ion charge)20 is com-pared to the minimum shell radius Rshell to describe thedegree of ion kinetic behavior, as in prior work.12 It isobserved that the fusion yield relative to draco predic-tions varies inversely with the experimentally-determinedNK , suggesting that ion kinetic effects are beginning todegrade implosion performance. In Section IV, it is dis-cussed that hydrodynamic mix does not account for thesetrends. As shown in Figure 1, it is evident that these re-sults strongly match the findings of the previous exper-iments, which are presented together for guidance. Theentire range of exploding pusher data shown in Figure 1spans three orders of magnitude, between regimes of verylow (10−2) and very high (10) Knudsen numbers. Whilethe OMEGA direct-drive experiments were conducted ina comprehensive, systematic way,12, the experiments de-scribed in this work, conducted in a ride-along mode, pro-

duced a somewhat more complex set of data. However,in concert, these different experimental campaigns showhow the discrepancy relative to hydrodynamic codes withincreasing Knudsen number begins to be observed.

FIG. 1. (Color online) DD YOC as a function of the Knud-sen number (NK) for an indirect-drive exploding pusher onNIF (red diamond),15 three polar-direct-drive (PDD) explod-ing pushers on NIF described in this work for which opti-mal draco simulations (including non-local electron trans-port and/or cross-beam energy transfer, see Section II) wereperformed, from left to right, shots N121128, N130129, andN120328 (black circles), and direct-drive exploding pusherson OMEGA (green circles).12 Filled markers represent D3He-filled implosions, while open markers denote D2-filled implo-sions. Though the drive conditions are quite different, theseexperiments show a unified picture of the increasing impact ofion kinetic effects as a function of increasing Knudsen numberabove NK

>∼0.1. A band centered around NK = 0.5 shows theapproximate Knudsen number at the center of a NIF ignition-relevant indirect-drive or a NIF polar-direct-drive implosionimmediately after shock convergence, while a band centeredaround NK = 2 shows the approximate Knudsen number af-ter shock convergence at the center of a cryogenic layeredimplosion on OMEGA.21

This paper is organized as follows: the experimentsand models used for comparison to experimental dataare described in Section II; experimental and some mod-eled results are shown in Section III; a discussion of thefindings is presented in Section IV, with evidence of ionkinetic effects illustrated in Figure 8; and concluding re-marks are presented in Section V. In the Appendix, theuse of these implosions for development of monoenergeticproton backlighting at the NIF is discussed.

II. EXPERIMENTS AND MODELING

DD and D3He fusion yield, burn-averaged ion tempera-tures, fuel and total ρR, implosion convergence and sym-metry, and bang time were measured in exploding-pusherimplosions at the National Ignition Facility (NIF).22

Experiments were conducted with ∼192 laser beamspointed in the polar direct drive configuration,16,17 de-livering 40 to 130 kJ onto a capsule in a 1.4- or 2.0-ns

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ramp pulse. The experiments, conducted for diagnosticcalibration,23–25 used 2.2 g/cm3 SiO2 shells with a 1530to 1680-µm diameter and a thickness of 4.1 to 4.6 µm,filled with 10 to 12 atm. of D2, D3He, or HD3He gas.Some experimental parameters are summarized in Ap-pendix A.

The primary nuclear reactions used to diagnose theexploding-pusher implosions are:

D + D→ 3He(0.82 MeV) + n(2.45 MeV), (1)

D + D→ T(1.01 MeV) + p(3.02 MeV), and (2)

D + 3He→ α(3.6 MeV) + p(14.7 MeV). (3)

Neutron time-of-flight (nTOF) detectors26,27 and protonspectroscopy28–30 were used to measure DD and D3Heyields and burn-averaged ion temperatures, and a parti-cle time-of-flight (pTOF)31 detector was used to measurenuclear bang time. The yield of the secondary D3He pro-tons was used to determine fuel ρR in D2 implosions.32

These protons are produced in D2 gas-filled implosionswhen 3He fusion products (see equation 1) react withthe thermal D fuel ions

3He(≤ 0.82 MeV) + D→ α+ p(12.6− 17.5 MeV). (4)

The ratio of the secondary D3He-proton to primary DD-neutron yield is proportional to the fuel ρR for fuel ρRbelow ∼10 mg/cm2.32–34 At higher fuel ρR values, thetechnique is saturated. For D3He (D2) gas-filled implo-sions, the energy downshift of the primary (secondary)D3He-proton spectrum is used to infer the total ρR.30

Sample primary and secondary D3He-proton spectra areshown in Figure 2. From the primary spectrum, theD3He yield, D3He-burn-averaged ion temperature, andtotal ρR are inferred; from the secondary spectrum, thefuel ρR and total ρR are inferred. These measurementsare discussed further in Section III.

For illustrative purposes, a 1D lilac35,36 simulation ofmass-element trajectories and time-dependent tempera-ture and fusion burn rates is shown in Figure 3 for atypical D3He gas-filled exploding-pusher implosion at theNIF (shot N110722). Fusion reactions are initiated pri-marily along the shock rebound trajectory(∼1.8-2.0 ns),well before peak compression (∼2.2 ns).

Primarily, the 2D hydrocode draco18,19 was used tosimulate these polar-direct-drive exploding-pusher exper-iments at the NIF. The use of 2D simulations is especiallypertinent in these experiments, where the polar directdrive (PDD) imposes an illumination asymmetry. 3Dray tracing is used to model inverse bremsstrahlung ab-sorption of laser energy, and material equations of statewere taken from SESAME tables. A flux-limited Spitzerthermal conductivity was used, with a flux limiter of0.06.37 draco simulations of ablatively-driven PDD ex-periments at the OMEGA laser facility38 were in good

agreement with measured x-ray radiographs and ρR.39,40

draco simulations have also been found to reasonablyreproduce experimental yields, shell shape, and ρR insymmetrically-driven cryogenic implosions at OMEGAunder conditions of high adiabat,41–44 as in the presentexploding pusher experiments. To properly simulate thelaser-absorption process and approximately capture theimplosion velocity and shape in these exploding-pusherimplosions, it is sometimes necessary to include modelsof non-local electron transport (NLET)45 and cross-beamenergy transfer (CBET).46,47 Including CBET has beenfound to be necessary to reasonably reproduce implosionshape for higher laser intensities (laser energy >100 kJfor these NIF exploding pushers).47 Observables such asyield, ion temperature, ρR, bang time, and shell conver-gence and symmetry are calculated from both the “nom-inal” draco simulations and draco simulations that in-clude NLET and/or CBET.

III. RESULTS

In the following sections, the measured fusion yields,burn-averaged ion temperatures, fuel and shell ρR, bangtimes, shell convergence, and shape are presented and,where appropriate, compared to draco-simulated val-ues. It will be shown later that deviations from dracoyield predictions are suggestive of ion kinetic effects.

A. Yield

DD-neutron yields were measured using the neu-tron time-of-flight (nTOF)26,27 diagnostic suite and DD-proton yields were measured using the solid-state nu-clear track detector CR-39,2548 with an uncertainty of∼±10%.28 The D3He-proton yields were measured usingwedge-range-filter proton spectrometers (WRFs).28–30 Asample WRF primary D3He-proton spectrum is shown inFigure 2a for shot N121128. The measured D3He yieldsshown in Figure 4b are averages of several measurements.The overall uncertainty in the yield measurements on agiven shot is ∼±10%.49

A comparison of measured and draco-simulated DDyields from D2- and D3He-gas-filled exploding push-ers (Figure 4a) reveals that experimental values clusteraround a yield-over-clean (YOC) – the ratio of measuredyield to yield simulated by a “clean” hydrodynamic sim-ulation that does not include a turbulent mix model orkinetic effects – of ∼0.5-0.6. The NLET-CBET dracosimulations have a YOC closer to 1. The experimen-tal and draco-simulated data show that the D3He yieldis, on average, 0.27 of the nominal 2D draco simulatedvalue (with a standard deviation of 0.20). Overall, theobserved implosion performance relative to simulationsis slightly worse for pure D2 and higher-yield (higher-Ti)implosions, such as shot N120328. As will be discussedbelow and in Section IV, these results are consistent with

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FIG. 2. (Color online) (a) Measured primary D3He-proton spectrum from D3He exploding-pusher shot N121128 and (b)measured secondary D3He-proton spectrum from D2 exploding-pusher shot N110131. From the primary spectrum (a), theD3He yield, D3He-burn-averaged ion temperature (related to the spectral width as Ti∝σ2), and total ρR (proportional toenergy downshift) are inferred. From the secondary spectrum (b), the total ρR is inferred from the energy downshift, while theratio of secondary D3He-proton to primary DD-neutron yield is proportional to the fuel ρR for ρR ≤ 10 mg/cm2 (for Te ∼5keV). Note the difference in energy-axis scale, as the secondary spectrum is much broader than the primary spectrum.

influence of ion kinetic effects, which reduce the yield rel-ative to hydrodynamics simulations in implosions withhigher Ti, lower Z, and, thus, longer ion mean free paths.

B. Ion Temperature

In order to explore trends within the yield data to elu-cidate the possible role of ion kinetic effects, the yieldresults are presented in comparison to measured burn-averaged ion temperatures. The DD-burn-averaged ion-temperature was measured by nTOFs and the D3He-burn-averaged ion temperature was determined from theDoppler width of the measured D3He proton spectrum(see Figure 2a), as measured by WRFs.28,50,51 In NIFexploding-pusher implosions, the measured D3He-protonlinewidth is dominated by thermal broadening.52 Othercapsule-related broadening effects (e.g. ρR evolution andimplosion geometry) account for a broadening of σ∼100keV in NIF exploding pusher implosions,53 and are sub-tracted in quadrature from the measured linewidth. Ad-ditional broadening effects that are difficult to quantifyon a shot-by-shot basis are not accounted for in this anal-ysis, and thus set an upper limit on the Doppler-inferredion temperature. However, the Doppler broadening ofthe D3He-proton spectrum should be considered a re-liable measure of the D3He-burn-averaged temperature.Uncertainties in the different measurements are ∼0.5 keVfor the nTOF DD-burn-averaged temperature and ∼2keV for the D3He-burn-averaged temperature.

The measured DD and D3He burn-averaged temper-atures are compared to the draco YOCs in Figure 5.Over the set of experiments, the ion temperature variesas a consequence of variation in laser power, as the im-plosion with the greatest laser power, shot N120328, hasthe highest DD-burn-averaged ion temperature. Bothsets of data show a trend of decreasing YOC with in-creasing burn-averaged Ti over the range of 4 to 15 keV,

with a more apparent trend among the D3He data. InSection IV this YOC trend will be expressed in termsof the Knudsen number NK ∝ (T 2

i /niZ4)/Rshell in a

way that indicates ion kinetic effects may be responsi-ble. The shot with the highest DD-burn-averaged iontemperature, N120328, will be shown to have the high-est Knudsen number and, thus, to be most susceptibleto ion kinetic effects. The measured temperatures aresummarized in Table VI.

In general, nominal draco-simulated ion tempera-tures are in reasonable agreement with experimentalmeasurements. This is especially the case for theD2-filled shots, where measured and nominal draco-simulated DD-burn-averaged ion temperatures differ byno more than 15% over a large range in ion tempera-ture and in YOC. Thus, these results demonstrate thatthe trend of decreasing YOC with increasing Ti is notmerely a consequence of a hydrodynamics-related tem-perature discrepancy.

As an aside, the measured difference between DD-burn-averaged and D3He-burn-averaged ion tempera-tures (x-axis values shown in Figures 5a and 5b) is nota sign of diagnostic disagreement, but rather is symp-tomatic of profiles and temporal evolution of density andtemperature throughout the burn. Relative to the DD fu-sion reactivity, the D3He reactivity has a stronger tem-perature dependence and is weighted more strongly bythe hotter regions of the implosion. Ion temperaturegradients therefore produce measurable differences in theburn-averaged ion temperatures of different reactions.

C. ρR and Convergence

Measurements of fuel ρR, total ρR, and shell conver-gence provide information about the fuel assembly inthese low-convergence, shock-driven implosions, and canbe used to help identify the approximate ion density ni,

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FIG. 3. (Color online) Illustrative 1D lilac-simluated (a)Lagrangian mass element trajectories (black), (b) volume-averaged fuel ion temperature (blue), and rates of DD burn(green - solid) and D3He burn (green - dashed) as a function oftime in NIF D3He exploding-pusher shot N110722 (4.1-µm-thick shell, 1536-µm diameter, 8.6 atm). A representative1.4-ns-ramp laser pulse, approximately like that used in shotN110722, is also shown.

ion mean free path λii, and minimum shell radius Rshell.The ratio of these lengths, NK ≡ λii/Rshell, is a figure-of-merit that will be used in the assessment of ion kineticeffects, as discussed further in Section IV.

As discussed previously in Section II, the fuel ρR is in-ferred from the ratio of secondary D3He-p to primaryDD-n yields in D2 implosions. For shot N110131, asecondary-proton yield of 2.0±0.5×108 (see Figure 2b)and a primary DD-neutron yield of 3.0±0.3×1011 gives afuel ρR of 4.6±1.1 mg/cm2, using a model of uniformfusion production throughout the fuel.33 Similar mea-surements give a fuel ρR of 3.6±1.1 mg/cm2 for shotN120328 and 4.6±1.1 mg/cm2 for shot N130129. Thefuel ρR does not significantly change from one implo-sion to the next, suggesting that differences in implo-sion performance relative to draco are unlikely to becaused by differences in hydrodynamic processes betweenthe different implosions. These fuel ρR measurementsare used to estimate the implosion convergence ratio, asC = (ρRf/ρRf0)1/2, where ρRf0 is the initial fuel arealdensity. With ρRf0∼0.13 mg/cm2 for these three D2-filled implosions, convergence ratios of ∼5.3-6.0 are in-ferred. These measurements will be used subsequently

FIG. 4. (Color online) (a) Measured DD-neutron yield and(b) D3He-proton yield as a function of draco-simulated yield.Open markers denote D2-gas-filled implosions, while filledmarkers denote D3He-gas-filled implosions. The red trianglesrepresent nominal draco simulations, while the black circlesare draco simulations that have included non-local electrontransport and cross-beam energy transfer. Dotted horizontallines connect multiple simulations of the same shot. Yield-over-clean (YOC) values of 1 (solid line) and 0.2 (dashed line)are indicated. The measured yields are averaged over severalmeasurements on each shot, with an overall error of ∼±10%,approximately the size of the symbols.

in Section IV to estimate the ion density and, ultimately,the ion mean free path in these exploding pushers aroundbang time.

Total ρR data is not directly used to evaluatehydrodynamic-kinetic parameters, but is presented forcompleteness and as a consistency check on the approx-imate convergence ratios inferred from the fuel ρR mea-surements. As first discussed in Section II and shown inFigure 2, the total ρR is inferred from the energy down-shift of the secondary or primary D3He-proton spectrum.Assuming an average secondary-proton birth energy of14.96 MeV54 and measured energies of 14.08 ±0.15 MeVon the equator and an average energy of 14.33±0.15 MeVon the pole, total ρR of ∼26±5 and ∼19±5 mg/cm2,respectively, are inferred for shot N110131 (Figure 2b).

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FIG. 5. (Color online) (a) DD YOC as a function of themeasured DD-burn-averaged ion temperature (Ti,DD) and(b) D3He YOC as a function of the measured D3He-burn-averaged ion temperature (Ti,D3He). The red triangles rep-resent nominal draco simulations, while the black circlesare draco simulations that have included non-local electrontransport and cross-beam energy transfer. Dotted verticallines connect multiple simulations of the same shot.

An average total ρR of 18±5 mg/cm2 is inferred on D2

shot N130129, while no total ρR measurement is possiblefor shot N120328 as the spectrum is net upshifted rela-tive to the birth energy, an effect which is present onlyfor that shot.55 For D3He-filled implosions, the birth en-ergy of the primary D3He-p spectrum is taken to be 14.7MeV. D3He shot N121128 was measured to have a totalρR of 9±4 mg/cm2 (see Figure 2a), while a total ρR of13±4 mg/cm2 was obtained on shot N100823 and 11±4mg/cm2 was measured on shot N110722. These totalρR values, generally 10-20 mg/cm2, are a factor of 10-20greater than the initial ρR in the shell of ∼1 mg/cm2.The shell ρR data are roughly consistent with the con-vergence ratio of ∼6 that was inferred from the fuel ρR,when allowing for some blowoff of outer shell material.56

X-ray emission gives a sense of the core size aroundbang time and may be used to estimate the implosionconvergence and the length scale of the fuel, which canbe compared to λii to evaluate the likely impact of ion ki-netic effects. Gated x-ray imaging diagnostics, the hard-ened gated x-ray imager (hGXI)57 and the gated x-raydetector (GXD),58 were used to capture images of the

FIG. 6. (Color online) hGXI x-ray self-emission of the im-ploded core ∼100 ps before bang time in D3He exploding-pusher shot N121128. The image, with 10% peak emissioncontour (white line) indicated, shows an average P0 =168 µmand a P2/P0 = -0.13 (13% oblate). The P0 at bang time isextrapolated to be ∼150 µm. The measured radius Rshell isused as a scale length of the implosion in calculation of theKnudsen number NK ≡ λii/Rshell.

implosions. Figure 6 shows a measured x-ray emissionimage from the imploded core as captured by hGXI onshot N121128, ∼100 ps before bang time (no image atbang time was obtained). The image shows a core ra-dius, expressed as the Legendre mode P0 of ∼168 µmand an oblateness, characterized by a Legendre P2/P0, of∼-0.13. The shell is still converging at this time, and thecore size at bang time is extrapolated to be ∼10% smallerthan inferred from this image. The estimated shell radiusat bang time for N121128 is therefore taken to be 150µm, though this extrapolation has only a small impacton the calculation of NK shown later. The oblateness,present across this set of implosions as a consequence ofthe polar drive (with the x-ray P2/P0 ranging from -0.09to -0.33 oblate, and also evident in the total ρR asym-metry obtained on shot N110131), shows significant dif-ferences from previous, symmetrically-driven implosionson OMEGA,12 which did not measurably deviate fromspherical symmetry. This oblateness also demonstratesthe importance of using a 2D code such as draco forcomparison to experimental results. draco simulationsthat include NLET47 reasonably capture the implosionshape for shot N121128.

A summary of the measured fuel ρR, total ρR, im-plosion size, and symmetry is presented in Table I. Theimplosion radius around bang time (P0) varies as a con-sequence of the different laser pulses. Longer laser pulses,such as on shot N110131, drive the converging shell fora longer period and can produce a smaller implosion.Higher laser intensities, such as on shot N120328, pro-

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duce a more rapid shock convergence, so that bang timeoccurs when the shell is at a somewhat larger radius.The P0 values across this set of exploding pushers implyconvergence ratios of ∼4-8, roughly in agreement withthose inferred from the fuel ρR and total ρR. Theseconvergence data are used in Section IV to estimate theion density59 and implosion radius in order to calculatekey ion kinetic parameters in a manner similar to thatin prior work.12 Overall, the variation in NK across thedata set is most strongly related to variation in Ti and Zand changes only weakly as a consequence of differencesin convergence or ion density.

TABLE I. Measured fuel ρR, total ρR, x-ray emission ra-dius P0 (contour of 10% of maximum brightness for N121128,17% of maximum brightness for the others; the choice of con-tour imposes only a 5% uncertainty), and relative magnitudeof second Legendre mode (P2/P0), the dominant asymmetryin the implosion. This ratio quantifies the deviation in coreshape from spherical symmetry, with a negative value signi-fying an oblate implosion. These measuremets are used toquantify the ion-ion mean free path and implosion size, toassess the likely impact of ion kinetic effects.

Shot Fuel ρR Total ρR X-Ray

(mg/cm2) (mg/cm2) P0 (µm) P2/P0

N100823 13±4

N110131 4.6±1.1 23±5 89 -0.24

N110722 11±4 115 -0.33

N120328 3.6±1.0 182 -0.10

N121128 9±4 150 -0.13

N130129 4.6±1.1 18±5 94 -0.09

D. Bang Time

In order to assess the possible role of ion kinetic effects,it is important to determine that yield trends relative todraco are not the result of discrepancies in the energycoupling to the implosion. The energy coupling and, con-sequently, the implosion velocity, is assessed by measure-ments of nuclear and x-ray bang times – the times of peakfusion and x-ray production in the core. Bang time mea-surements using D3He-protons and DD-neutrons weremade by the particle time of flight (pTOF) diagnostic.31

X-ray bang times were measured using the south polebang time (SPBT) diagnostic60 and gated x-ray imagingdiagnostics, hGXI57 and GXD.58 These measured bangtimes are summarized and compared to draco-simulatedDD-neutron bang times in Table II. Uncertainty in thepTOF-measured DD-neutron bang time is ∼±120 ps,while the uncertainty in the D3He-proton bang time mea-surements is ∼±100 ps. Uncertainty in the x-ray bangtime is ∼±100 ps. pTOF traces used to infer nuclearbang times on shots N121128 and N130129 are shown inFigure 7. Comparing the nuclear bang times, the mea-

sured D3He bang time on shot N121128, 1880±100 ps, isin reasonable agreement with the draco DD bang timeof 2020 ps, while the measured DD bang time on shotN130129, 2470±120 ps, is in good agreement with thedraco DD bang time of 2530 ps. The overall trend in-dicates that draco captures the basic implosion dynam-ics and energy coupling to these shock-driven implosionsfairly well. Therefore, the trend in the yield data is likelynot related to inaccurate modeling of overall implosiondynamics, and may be attributed to ion kinetic effects,as discussed in the following section.

FIG. 7. (Color online) pTOF signal obtained on (a) D3Heshot N121128 and (b) D2 shot N130129, used to infer nuclearbang times.

TABLE II. Measured nuclear and x-ray bang times anddraco-simulated DD-n bang times. The nuclear measure-ments were made using D3He protons (p) on N121128 andwith DD neutrons (n) on N130129.

Shot Measured Bang Times draco Bang Time

Nuclear X-Ray Nuclear (DD-n)

(ps) (ps) (ps)

N110131 2430 2650

N110722 1910 2150

N120328 1770 1990

N121128 1880 (p) 2000 2020

N130129 2470 (n) 2530

IV. DISCUSSION

Though the experimental bang times compare reason-ably well with draco simulations, indicating that theoverall implosion dynamics and energy coupling in theseshock-driven implosions are modeled fairly accurately,trends within the nuclear yield data point to ion kineticeffects impacting fusion production. To estimate the sig-nificance of ion kinetic effects, the ion-ion mean free patharound bang time, λii, which varies in these implosionson the basis of the fuel composition (Z), Ti, and ni, isevaluated. The ion temperature is taken as the DD-burn-averaged Ti in D2 implosions and the yield-weighted aver-

8

age of the DD- and D3He-burn-averaged ion temperaturein D3He implosions. The ion density is estimated fromthe measured fuel ρR as ni = ni0(ρRf/ρRf0)3/2, assum-ing mass conservation and spherical symmetry, where ni0and ρRf0 are the initial gas ion density and the initial fuelρR, respectively. This calculation is performed directlyfor D2 shots on which fuel ρR data was obtained, anda similar implied convergence ratio (∼6) is also assumedfor the D3He shots. The fuel ρR and, therefore, the iondensity, does not change significantly between the differ-ent implosions. These experimental quantities are usedto estimate λii around bang time, which is compared tothe minimum shell radius Rshell as given by the x-ray P0.

As shown in Figure 8, the Knudsen number NK ≡λii/Rshell varies by a factor of ∼7 over these exploding-pusher implosions, largely as a consequence of differ-ent fuel ion Z and ion temperature. For NK �1, theimplosion behaves more hydrodynamically, whereas forNK

>∼0.3, kinetic effects start to become significant.12

The moderate-temperature, D3He-filled shot N121128 isthe most hydrodynamic-like implosion, where Ti = 8.0keV, ni∼7×1022 cm−3, and Z = 1.47, resulting in a λiiof ∼6 µm.61 Under these conditions, and with Rshell =150 µm, NK is ∼0.04 for N121128. In contrast, the high-temperature, D2-filled shot N120328 is the most kinetic-like implosion, where Ti = 11.4 keV, ni∼7×1022 cm−3,and Z = 1, resulting in a λii of ∼50 µm. In this case,Rshell = 180 µm and NK is ∼0.3. Parameters used to cal-culate the experimentally-inferred Knudsen numbers aresummarized in Table III, and YOC values are shown asa function of the experimentally-inferred Knudsen num-bers in Figure 8. For both DD and D3He yields, eachset of simulations shows a trend of decreasing YOC withincreasing NK . For the NLET-CBET draco simula-tions, DD YOC ∼1 for NK ∼0.04 (N121128, the mosthydrodynamic-like implosion), while DD YOC ∼0.4 atNK ∼0.3 (N120328, the most kinetic-like implosion). No-tably, as shown in Figure 1, this most kinetic-like implo-sion among the NIF PDD data set overlaps in both NK

and YOC with the most hydrodynamic-like implosion inthe OMEGA direct-drive exploding-pusher set. As thedraco hydrodynamics code does not account for kineticeffects, ion mean free path effects such as enhanced iondiffusion and Knudsen reactivity reduction due to modi-fication of the ion distribution function62–64 may accountfor this trend. It has been shown previously that kineticeffects reduce shock yields in exploding pushers, and doso more strongly with increasing Knudsen number.12

For these largely shock-driven implosions, hydrody-namic mix at the fuel-shell interface is very unlikely to ex-plain the trend of decreasing YOC with increasing Knud-sen number. In order for mix to explain this trend, themost “kinetic” implosion, N120328, would have to be themost susceptible to mix, to be driven by compressionmore so than the other implosions. However, by virtueof having the highest measured DD-burn-averaged iontemperature – a strong signature of shock heating – itis likely that this shot is the most predominantly shock-

TABLE III. Estimates of ion density, ion temperature, meanion charge, minimum shell radius, Maxwellian-averaged meanfree path for ion-ion collisions, and Knudsen number. Theion-ion mean free path and, therefore, the Knudsen number,varies largely as a consequence of variations in ion tempera-ture and in gas composition (〈Z〉).

NIF ni Ti 〈Z〉 λii Rshell NK

Shot # (cm−3) (keV) (µm) (µm)

N110131 11×1022 5.4 1 9 89 0.1

N110722 6×1022 10.8 1.44 12 115 0.1

N120328 7×1022 11.4 1 50 182 0.3

N121128 7×1022 8.0 1.47 6 150 0.04

N130129 11×1022 4.0 1 5 94 0.05

FIG. 8. (Color online) (a) DD and (b) D3He YOC as a func-tion of the Knudsen number (NK)– the ratio of ion-ion meanfree path (λii) to minimum shell radius (Rshell). The red tri-angles represent comparison to nominal draco simulations,while the black circles represent comparison to draco simu-lations that include NLET and/or CBET to more accuratelysimulate the implosion shape. Dotted vertical lines connectmultiple simulations of the same shot. The trend of decreasingYOC with increasing λii/Rshell across all simulations suggeststhat kinetic effects are starting to impact the experimentalyields.

9

driven, with fusion reactions generated along the shockrebound trajectory. It is therefore unlikely that this shotis preferentially susceptible to the deleterious effects ofmix, and in fact it may be even less compressively-driven(and less susceptible to mix) than the other implosionsin this study.65 Preliminary 2D ares simulations66 thatinclude a KL mix model67 indicate that mix does nothave a significant impact on simulated yields and thata trend of lower yield-over-simulated (YOS) for higher-NK implosions persists. As mix does not appear able toaccount for the trend of decreasing YOC with increasingKnudsen number, ion kinetic effects, rather than mix, areinferred as the likely explanation.

TABLE IV. Estimates of ion density, ion temperature, meanion charge, gas composition, Maxwellian-averaged mean freepath for ion-ion collisions, and Knudsen number after shockconvergence for: a strongly-kinetic exploding pusher onOMEGA from Rosenberg et al.;12 a NIF direct-drive ex-ploding pusher from this study; a strongly hydrodynamic-like indirect-drive exploding pusher on NIF from Le Pape etal.;15, and an ignition-relevant indirect-drive implosion. Theparameters corresponding to hot-spot conditions during thecompression phase of an indirect-drive ignition implosion arealso presented. The three exploding pusher cases span theregimes of strongly kinetic to strongly hydrodynamic-like. Forthe exploding pushers, the ion density and ion temperatureare estimated from experimental measurements, while for theNIF indirect-drive ignition-relevant shock phase case, they aretaken from hydrodynamic simulations of a surrogate implo-sion, near the center of the implosion immediately after shockconvergence. The NIF indirect-drive ignition compressionphase case is an estimate based on recent NIF experiments.68

In both shot N120328 (the most kinetic-like of the NIF direct-drive exploding pushers presented here) and the shock phaseof the NIF indirect-drive ignition-relevant implosion, λii ∼50µm, which approaches the size of the burn region (∼100 µm)in the ignition-relevant case (NK ∼0.5).

Implosion ni Ti 〈Z〉 Fuel λii NK

(cm−3) (keV) (µm)

Rosenberg et al. 4×1021 28 1.5 D3He 900 10

N120328 7×1022 11.4 1 DD 50 0.3

Le Pape et al. 3×1023 3.5 1 DD 2 0.01

NIF Ignition ∼6×1022 ∼10 1 DT ∼45 ∼0.5

(shock phase)

NIF Ignition 1025 4 1 DT 0.1 0.002

(compr. phase)

The results of these NIF direct-drive exploding-pusherexperiments fit within the context of those previouslyobserved in direct-drive exploding pushers at higher NK

on OMEGA12 and indirect-drive exploding pushers atmuch lower NK on NIF.15 Table IV shows the ion-ionmean free path and Knudsen number in each of theseexploding pusher experiments, as well as shortly (∼100ps) after shock convergence in a surrogate, indirect-driveignition-relevant implosion on NIF and during the peak

compression phase of a NIF ignition-relevant implosion.In the OMEGA experiments (Rosenberg et al.),12 theYOC was found to be a strong inverse function of NK

for 0.3<∼NK<∼10; in the NIF indirect-drive exploding

pushers (Le Pape et al.),15 YOC∼1, near-perfect agree-ment with hydrodynamic models, was obtained on im-plosions with NK∼0.01. The present NIF direct-driveexploding-pusher experiments span the NK space in be-tween those extremes and demonstrate that the deviationfrom hydrodynamic models becomes noticeable between0.1<∼NK

<∼0.3. This trend in YOC as a function of NK

over the different experimental campaigns is illustrated inFigure 1, showing a consistent picture of the onset of ionkinetic effects with increasing Knudsen number, despiteradically different drive conditions. Additionally, in boththe more kinetic-like N120328 exploding-pusher exampleand the NIF indirect-drive ignition case, λii∼50 µm aftershock convergence, an appreciable fraction of the size ofthe hot spot. The Knudsen number of ∼0.5 after shockconvergence in the NIF indirect-drive ignition implosionapplies to NIF polar-direct-drive ignition-relevant implo-sions as well, and underestimates the Knudsen numberafter shock convergence in cryogenic layered implosionson OMEGA, as discussed in Section I. Under those con-ditions, ion diffusion may significantly alter the densityprofiles, and Knudsen layer effects may allow the higher-energy ions to escape the hot-spot region.

In a NIF ignition implosion, the continued shell conver-gence after shock rebound greatly increases the fuel iondensity, reducing λii and NK and producing much morehydrodynamic-like conditions around peak compression.The question that will be addressed in the future is: couldthe kinetic-like conditions during the shock convergencephase, including multiple-ion effects,69,70 have any lin-gering manifestations and effects on the subsequent com-pression phase, which is strongly hydrodynamic-like (asshown in Table IV)? Ongoing and future experimentswill explore this open question. Measurements of therelative and absolute timing of shock and compressionbang time on surrogate implosions the NIF, as have beenrecently obtained and will be obtained routinely in thefuture,71 may shed light on this issue.

Further studies will investigate aspects of the shockconvergence phase of ICF implosions using explodingpushers, where ion kinetic effects are likely to be impor-tant. Yield anomalies in mixed-fuel exploding pusherswill be explored and may elucidate species separationduring shock convergence.70 A comparison of exploding-pusher data to a hybrid kinetic treatment like that ofLarroche11,72 is an important continuation of this study.That theoretical work shows that kinetic simulationsproduce weaker and smoother profiles of temperaturesand density near shock-bang time than hydrodynamicsimulations, as the shock front is broadened to ∼λii.Exploding-pusher experiments conducted in the indirect-drive configuration15 will continue to be studied. Theseexperiments in particular more closely approximate theprocess of shock convergence and shock burn as it occurs

10

in NIF indirect-drive implosions.Additionally, and as will be discussed further in Ap-

pendix B, characterization of these implosions aids devel-opment of exploding pushers as a monoenergetic protonsource at NIF. Exploding pushers have been shown to bea reliable source of fusion products insensitive to capsuleand laser illumination uniformity73 and have been usedextensively at the OMEGA laser facility38 as a fusionproduct source for charged-particle radiography of ICFimplosions,74 hohlraums,75 and laser-foil interactions,76

and for studies of stopping power in laser-generated plas-mas. The use of exploding pushers, such as those dis-cussed in this work, as a proton backlighter on NIF hasbeen proposed. These data, including the characteriza-tion of fusion product fluence and energy isotropy pre-sented in Appendix B, provide a promising first step to-ward development of this technique at the NIF. Studyingexploding-pusher implosions both advances understand-ing of the shock convergence phase of ICF implosions,during which ion kinetic effects can be prevalent, andalso enables development of scientific platforms that uti-lize exploding pushers.

V. CONCLUSIONS

In order to fully exploit diagnostic development shotsat the NIF, ride-along measurements of fusion yield,fuel ion temperature, ρR, convergence, and bang timehave been presented for polar-direct-drive, D2 and D3Heexploding-pusher implosions. These data are used toprobe the physics of the shock convergence phase of im-plosions relevant both to shock-driven, exploding-pusherimplosions and to ablatively-driven implosions, when ionkinetic effects can be important. The data have beencompared to 2D draco hydrodynamic simulations andshow a notable trend of decreasing YOC with increas-ing Knudsen number (NK ≡ λii/Rshell). This trend issuggestive of ion kinetic effects, and is consistent withthe results of previous experiments at much higher andmuch lower Knudsen number, even though these implo-sions used different capsules and polar-direct-drive illu-mination. This work also motivates the continued de-velopment of kinetic models of ICF implosions, whichmay be especially pertinent at the high-temperature,moderate-density conditions present at shock burn inboth exploding pushers and the shock-convergence phaseof ablatively-driven implosions.

ACKNOWLEDGMENTS

The authors thank J. Schaeffer, R. Frankel, E. Doeg,M. Cairel, M. Valadez, and M. McKernan for contribut-ing to the processing of CR-39 data used in this work, aswell as the NIF operations crew for their help in execut-ing these experiments. This work is presented in partialfulfillment of the first author’s PhD thesis and supported

in part by US DoE (Grant No. DE-NA0001857), FSC(No. 5-24431), LLE (No. 415935-G), and LLNL (No.B600100).

Appendix A: Experimental Parameters and Measurements

Experimental parameters are summarized in Table V.Experimental measurements are summarized in Table VI.

TABLE V. Capsule and laser parameters for exploding push-ers used in this study, including: capsule outer diameter d;shell thickness ∆r; total laser energy; approximate laser pulseduration; D2 fill pressure; and 3He fill pressure.

NIF d ∆r Energy Pulse D2 fill 3He fill

Shot # (µm) (µm) (kJ) (ps) (atm.) (atm.)

N100823 1567 4.1 80.0 ∼2100 1.4 10.5a

N110131 1555 4.5 52.0 ∼2100 10.0

N110722 1536 4.1 42.7 ∼1400 3.3 5.3

N120328 1555 4.4 130.6 ∼2100 9.9

N121128 1682 4.3 43.4 ∼1400 3.3 5.8

N130129 1533 4.6 51.4 ∼1400 10.0

a Capsule also contained 5.3 atm. H2

Appendix B: Discussion of Monoenergetic ProtonBacklighting Using Exploding Pushers at the NIF

Direct-drive exploding pushers filled with D3He gashave been used extensively as a source of monoenergeticprotons at OMEGA. The data from exploding pushersdiscussed herein are useful towards the development ofthis technique at the NIF. Of particular importance isthe spatial uniformity of both the emitted proton flu-ence and the emitted proton energy. Measurements ofthe uniformity of the D3He-proton spectrum on explod-ing pusher shot N121128, based on seven WRFs (threenear the equator (90,78) and four near the pole (0,0) rel-ative to the NIF geometry) are shown in Table VII andillustrated visually in Figure 9. The spectrum shown inFigure 2a is taken from the Equator-1 WRF.

These data show that the emitted D3He-proton spec-trum was extremely uniform in both fluence and energyon shot N121128. The average fluence variation is <2%from pole to equator, with a 3% standard deviation over-all. The energy varies by only 80 keV between the polarand equatorial measurements, with an overall standarddeviation of 65 keV. Thus, shot N121128 illustrates thepotential for a highly uniform source of protons in bothfluence and energy. While all 192 beams were used todrive this implosion, in a proton backlighting experimentmany fewer beams would be used, as some beams are nec-essary to drive the subject. Even under such conditions,excellent uniformity in proton fluence and energy can be

11

TABLE VI. Measured observables from these exploding pushers, including: DD yield; D3He yield; DD-burn-averaged iontemperature; D3He-burn-averaged ion temperature; bang time (x: x-ray, p: D3He-proton, n: DD-neutron); fuel ρR; and totalρR.

NIF YDD YD3He TiDD TiD3He Bang Time Fuel ρR Total ρR

Shot # (keV) (keV) (ps) (mg/cm2) (mg/cm2)

N100823 1.38×1010 2.32×1010 10.5 16.2 13

N110131 3.01×1011 5.4 2430 (x) 4.6 23

N110722 2.85×1010 1.30×1010 8.8 15.1 1910 (x) 11

N120328 1.00×1012 11.4 1770 (x) 3.6

N121128 7.27×1010 2.09×1010 7.1 11.0 1880 (p) 9

N130129 2.50×1011 4.0 2470 (n) 4.6 18

TABLE VII. Measured D3He-proton yield and energy at sevendifferent WRF positions (including their distance from theimplosion) on D3He exploding pusher shot N121128. Themeasurements on the equator were conducted at a fairly highfluence of protons (6×105 cm−2),77 and the capability to mea-sure the D3He-p energy using WRFs at significantly higherfluences has been developed.78 The random energy uncer-tainty for each measurement is ∼±60 keV, while the randomyield uncertainty for each measurement is ∼±5%. Excellentuniformity of the emitted proton fluence and energy showthat this implosion has great utility as a monoenergetic pro-ton source. Even with many fewer beams, as required in aproton backlighting experiment, such isotropy in proton flu-ence and energy is achievable.

WRF Dist. D3He-p Yield D3He-p Energy

Position (cm) (1010) (MeV)

Equator-1 50 2.01 14.39

Equator-3 50 2.17 14.34

Equator-4 50 2.14 14.40

Eq. Avg. 2.11 14.38

Pole-1 200 2.00 14.54

Pole-2 200 2.05 14.49

Pole-3 200 2.11 14.36

Pole-4 200 2.15 14.45

Pole Avg. 2.08 14.46

achieved. An implosion with this degree of proton emis-sion uniformity could simultaneously backlight multipleexperiments. Additionally, such excellent proton energyisotropy would enable studies of proton stopping powerin laser-generated warm dense matter and classical plas-mas, which rely on an assumption of protons of identicalenergy emitted along different lines of sight. Importantly,as shown in Figure 7, pTOF can be used to measure theD3He-proton bang time, necessary to identify the timeat which protons sample the subject.

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48On shot N130129, the CR-39-based DD-proton yield measure-ment was used to roughly confirm the DD-neutron yield mea-surement.

49The total uncertainty of a given WRF yield measurement is∼±15%. As the yields are averages of 2-7 measurements, theoverall yield uncertainty is reduced to ∼±10%.

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52Instrumental broadening for aluminum WRFs used at NIF isσ∼130-170 keV, while a typical thermal Doppler width of theD3He-p spectrum is σ∼250 keV.

53This broadening arises from differential proton slowing due to dif-ferent amounts of ρR across the burn duration and different pathlengths for protons traversing the shell and sampling different ef-fective amounts of ρR, and is modeled based on the measured ρRand simulated burn duration, assuming uniform proton emissionthroughout the fuel.

54Because the average energy of the 3He reactant ion in secondaryD3He reactions is much greater than the (thermal) reactant ionenergy for primary D3He reactions, the average birth energy ofsecondary D3He-p is ∼300 keV higher than the average birthenergy of a primary D3He-p.

55The measurement of total ρR from energy downshift in the D3He-p spectrum is sometimes complicated by the presence of radialelectric fields around the capsule that cause an upshift in pro-ton energy upon leaving the capsule,79 such that the downshift-inferred total ρR is a lower limit on the actual ρR in the implo-sion. However, when the laser intensity is below ∼4×1014 W/cm2

or when the nuclear bang time is well after the end of the laserpulse, electric-field upshifts are diminished and the birth energyis well known. This is the case on five of the six experiments usedin this study, with shot N120328 the only exception. On all othershots, bang time is >300 ps after the end of the laser pulse.

56The convergence ratio is related to the shell ρR as C ∼√ρRsf/(1 − f)ρRs0, where ρRsf (ρRs0) is the final (initial)

shell ρR and f is the fraction of shell material that has beenblown off (beyond the initial shell radius). For an approximateblowoff fraction of f = 0.6 and ρRsf/ρRs0 ∼10-20, C∼5-7, con-sistent with C∼6 inferred from the fuel ρR data.

13

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59Though there is some inconsistency between the convergence ra-tios inferred from the x-ray P0 measurements and the conver-gence ratios inferred from fuel ρR measurements, for the purposeof inferring an ion density in the later discussion, only the fuelρR is used. The fuel ρR measurement is given greater weight dueto the fact that it is a more direct probe of density.

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