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High Energy Density Physics 9 (2013) 251e257

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High Energy Density Physics

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Mach stem hysteresis: Experiments addressing a novel explanation ofclumpy astrophysical jet emission

K. Yirak a,*, J.M. Foster b, P. Hartigan c, B.H. Wilde a, M.R. Douglas a, R. Paguio d, B.E. Blue d,D. Martinez e, D. Farley e, P.A. Rosen b, A. Frank f

a Los Alamos National Laboratory, Los Alamos, NM 87545, USAbAtomic Weapons Establishment, Aldermaston, Reading RG7 4PR, UKcDepartment of Physics and Astronomy, Rice University, 6100 South Main, Houston, TX 77521-1892, USAdGeneral Atomics, 3550 General Atomics Court, San Diego, CA 92121-1122, USAe Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550, USAfDepartment of Physics and Astronomy, University of Rochester, Rochester, NY 14620, USA

a r t i c l e i n f o

Article history:Received 27 July 2012Received in revised form10 January 2013Accepted 11 January 2013Available online 26 January 2013

Keywords:Laboratory astrophysicsShock wave phenomenaWarm dense matter

* Corresponding author.E-mail address: yirak@lanl.gov (K. Yirak).

1574-1818/$ e see front matter Published by Elsevierhttp://dx.doi.org/10.1016/j.hedp.2013.01.006

a b s t r a c t

Recent time-series observations of shock waves in stellar jets taken with the Hubble Space Telescopereveal localized bright knots that persist over nearly 15 years. While some of these features representshock fronts caused by variable velocities in the flow, others appear at the intersection points betweendistinct bow shocks. Theoretically, when the angle between two intersecting shocks exceeds a certaincritical value, a third shock (Mach stem) should form. Because Mach stems form perpendicular to thedirection of flow, incoming particles encounter a normal shock instead of an oblique one, which results inbrighter emission at this location. To study this phenomenon in a controlled laboratory setting, we havecarried out experiments on the Omega laser aimed at understanding the formation, growth, anddestruction of Mach stems in the warm dense plasma regime. Our experimental results indicate how thegrowth rate depends upon included angle, and numerical simulations indicate that it may be possible tostabilize an already-formed Mach stem below the critical angle when certain conditions are satisfied.

Published by Elsevier B.V.

1. Introduction

We investigate the phenomenon of Mach reflection in the lab-oratory setting. Overviews of Mach reflection are offered by Ben-Dor [1] and Courant and Friedrichs [2], among others. The forma-tion of a Mach-reflection shock, known as aMach stem, is importantin a range of high-energy-density areas of study, including astro-physics and inertial-confinement fusion (ICF). We will discuss theastrophysical case of intersecting bow shocks in more detail. OneICF example concerns multiple, compressing shock waves in anignition capsule. The situation may be unstable to the RichtmyereMeshkov, RayleigheTaylor, and KelvineHelmholtz instabilities,which in turn may permit the formation of Mach stems whichchannel material inwards in a nonuniform fashion [3].

Shock waves in the lab are typically created in shock tubes. Suchexperiments are fairly mature on both pulsed-power and laser-driven platforms [4]. However, much remains to be understood

B.V.

about Mach stem formation and destruction in this regime. Earlierresults, for example, suggest that the criteria for formation ofa Mach stem may differ from theory in the laboratory setting [5]. Ithas not yet been addressedwhether this discrepancy has to dowiththe non-ideal-gas aspect or the non-planar geometry of the shocksinvolved.

We have undertaken simulation and experiments of Mach stemformation in a warm, dense plasmadtypically a few eV tempera-ture and a few g cm�3 density, respectively. Our present work ismotivated by recent astronomical observations. Recent HST time-series results [6] suggest that certain emission features inHerbigeHaro (HH) objects may be explained by Mach stem for-mation. Discovered independently by Herbig [7] and Haro [8], HHobjects were originally thought to be stationary nebulae perhapsassociated with hot, blue stars. HH objects in fact are well-collimated astrophysical jets originating from newly-formingstars whose emission is believed to be dominated by shock heat-ing. Just as star formation is common inmolecular clouds [9], so tooare the HH objects associated with them; a reviewmay be found inRef. [10]. Velocity and density variation in the leading shocks ofthese objects is most often assumed to be the source of the

K. Yirak et al. / High Energy Density Physics 9 (2013) 251e257252

emission: dense clumps penetrate, or are overrun by, the bulk fluidmotion, leading to particle acceleration and radiation by theinduced shocks [10]. By assuming a range of velocities and densitycontrasts, simulations can recreate many of the important featuresof such objects [11].

However, certain localized emission features may be alter-natively explained via Mach stem formation. This explanationpostulates that two (or more) co-moving bow shocks may intersectin such a way which support the formation of Mach stems, leadingto areas of enhanced shock heating, and therefore emission. In thepresent work, we seek to better understand and characterize thegrowth, stability, and destruction of a Mach stem in the warmdense plasma regime. In so doing, we may address the centralquestion of whether or not the timescales associated with Machstem formation and destruction are compatible with the observedemission features in HH objects. We discuss this idea further in thenext section and describe our experimental setup. The followingsection will present our results to date, and the final section willoffer a discussion and our conclusions.

Fig. 1. Left: the formation and growth of a Mach stem (Mach reflection) at three dif-ferent times; the included angle f is indicated, where in this case f � fc. Right: theanalogous case of two co-moving bow shocks in the astrophysical case; note that anaxis of symmetry is assumed. The acronyms are as follows: “IS” incident shock; “RS”reflected shock; “MR” Mach reflection; “SL” slip stream; “BS” bow shock. Note that theastrophysical case will also have a slip stream, which will be located on-axis and istherefore not labeled for clarity.

2. Physics and experimental design

As mentioned in the Introduction, Mach reflection is discussedin Refs. [1] and [2]. A similar treatment may be found in Landau &Lifshitz [12]. Briefly, given an incident and reflecting shock wave ona solid boundary, there exists a critical angle between the shock andboundary, fc, above which a fluid parcel cannot pass through bothincident and reflecting shocks and maintain its original trajectory.Instead, once this angle is reached, the solution of the flow de-mands a new configuration in which a shock normal to theboundary offsets the incident/reflecting shock point (now a shockwave triple point) some distance from the surface; see Fig. 1. Onemay consider the analogous astrophysical case of two co-movingbow shocks which, upon reaching an included angle of 2fc willalso form a Mach stemdthis is shown on the right in Fig. 1.

The result is referred to as Mach reflection so as to differentiateit from regular reflection. The only variable upon which this con-figuration can depend in the ideal fluid mechanics case is the ratioof specific heats, g. In the strong-shock limitdi.e., when the ratio ofpost- to pre-shock pressure is much greater than 1dfor g¼ 5/3 theresulting critical angle is fc ¼ 37�. The formula for fc is givenapproximately by Courant and Friedrichs [2]

fc ¼ arcsin�1g

�: (1)

The approximation breaks down as g decreases [13], differing bynearly 10� at g¼ 1.2 (from a practical perspective, exploring g < 1.2has the potential to be unstable to the Vishniac instability [14] andis thereby complicated). Foster et al. [5] have carried out experi-ments in the warm (5e10 eV) dense (1e2 g cm�3) plasma regime inwhich two counter-propagating bow shocks collide. The resultingMach stem differed from theory in that a) it had measurable cur-vature, and b) its critical angle was roughly 48�, or 10� larger thanthat given by theory assuming g w 5/3. These results were con-firmed by the design codes PETRA [15,16] and RAGE [17], the latterof which is discussed below.

As discussed in the Introduction, the present work seeks tobetter understand and characterize the growth, stability, anddestruction of a Mach stem in the laboratory. We choose to addressthis question in two parts. First, what can we learn about thegrowth rate of the Mach stem? Secondly, once a Mach stem isformed, what criteria are necessary and sufficient to destroy it?

The basic experimental design is similar to that of [5], exceptthat only one of the two shocks are required, which is driven into

a foam; see Fig. 4. In the place of a second shock, we insert a goldcone, whose surface profile we have varied between experiments,in the foam (see the description of the two basic cone designsbelow). In this fashion we may dictate the included angle f be-tween the cone surface and the shock. The design takes advantageof cylindrical symmetry which among other aspects allows lesscostly computer modeling than a fully 3D design. As previouslymentioned, our experimental variable is the cone profile. We haveinvestigated two basic cases. In the first case (hereafter “fixed an-gle”), the cone is machined so that the angle f as a function of r andz is constant, i.e.

vzconeðrÞvr

¼ tanðf0Þ (2)

0zconeðrÞ ¼ r$tanðf0Þ þ z0: (3)

While simple, this profile has the disadvantage that the includedangle between cone and bow shock is not constant but rather in-creases over time owing to the curvature of the bow. In the secondcase (hereafter “constant included angle”), we specified a profilesuch that the included angle is held fixed. We derived this profilebased on experimental data such that given a starting height z0 andtime t0, at a given r and z above that the angle between cone andbow shock at that point is a constant f0:

K. Yirak et al. / High Energy Density Physics 9 (2013) 251e257 253

arctan�vzconeðrÞ

vr

�� arctan

�vzshockðr; tÞ

vr

�¼ f0: (4)

Fig. 3. Included angles between cone and shock for two profiles, a 50� / 18.75� “fixedangle” cone (blue, constantly-varying curve) and a 60� / 35� “constant-includedangle” design (orange, flat curve). The dashed line denotes the critical angle from Ref.[5]. The abscissa is a measurement of the position of the bow shock (without Machstem) on the cone surface, therefore equivalent to a measurement of time since thespeed of the shock is known. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Eq. (4) is not amenable to analytic solution; instead, we assumeda time-dependent bow shock shape and speed based on exper-imental data and iteratively solved the above starting at r ¼ 0,z ¼ z0, t ¼ t0 to obtain the profile. These tabulated values were thensupplied to target fabrication at high resolution (data points every1 mm in r proved to be sufficient).

To demonstrate the relationship between “fixed angle” coneangles and Mach stem growth rate, Fig. 2 offers some experimentalresults from cones without a discontinuity. Based on these resultsand numerical simulation, we chose two profiles to investigate thediscontinuous change. The first was a “fixed angle” cone with aninitial angle of 50� which transitions to 18.75� at r ¼ 500 mm. Thesecond was a “constant-included angle” cone which transitionedfrom 60� to 35� also at r ¼ 500 mm. As previously stated, the “fixedangle” profile is so called because the angle between the conesurface and the horizontal is constant before and after transition. Incontrast, the “constant-included angle” cone profile is curved inorder to maintain a constant included angle between cone andshock, before and after transition.

The range of included angles sampled by these two designs isshown in Fig. 3. Fig. 3 demonstrates that in both cases the initialangle is above the critical angle, implying that the Mach stem willform and grow. Again in both cases, once it reaches the transitionpoint at r ¼ 500 mm, the angle drops below the experimentally-derived critical angle (note that this implies it does not dropbelow the theoretical fc, which is roughly 10� less). Based onanalytical theory, the Mach stem should be unsustainable in thisregion of f < fc. While the 50� / 18.75� cone eventually reachesf > fc, the 60� / 35� cone remains fixed at an angle f < fc. Thusone may expect the 50� / 18.75� cone to go through three phases:growth before transition, reduction while f < fc, and secondarygrowth after f > fc. In contrast, based on analytic theory the60� / 35� cone should experience growth prior to transition, anddestruction after. The results from experiment are presented in thenext section. We have also investigated cone designs which featuresinusoidal perturbations of many wavelengths along the conesurface; these designs and results are deferred to a future publi-cation. One important feature of the transition is that it createsa new Mach stem off the cone surface. This new Mach reflection’striple point will move off the surface and eventually mergewith the

Fig. 2. Measured Mach stem sizes for “fixed angle” cones with different cone profileswhich have no discontinuous changes.

original Mach stem’s triple point. We have seen that if this mergerdoes not occur soon enough after transition, the original triplepoint will run into the cone and the Mach stem will be destroyed.

Specific details of the design follow. The cone is 3000 mm indiameter and roughly 1000 mm in height. It is embedded ina resorcinol formaldehyde (RF) aerogel foam (C15H12O4) at0.3 g cm�3 of diameter 4000 mm and length 6000 mm.1 The tip ofthe cone is located 1000e1500 mm above the “pusher”; we haveseen the results to be robust to this positioning. An example pre-shot radiograph of the assembled target is given in Fig. 5. In theseexperiments, the pusher is comprised of 100 mm of 2% brominatedCH plastic at 1.22 g cm�3, on top of which is 300 mm of pure CHplastic at 1.03 g cm�3. As indicated in Fig. 4, we use indirect drive,heating a hohlraum using 12, 450 J/beam laser beams of the Omegasystem [18] in a 1 ns duration, constant-power laser pulse of0.35 mm wavelength. This produces a peak radiation drive tem-perature of close to 175 eV as determined by previous measuringwith the DANTE diagnostic [19]. The experiment is imaged usingnickel backlighter beams positioned roughly orthogonal to eachother (perpendicular to the axis of symmetry) in order to evaluateaxial symmetry. The backlighter targets are located 18 or 21 mmfrom the package and consist of a 5 mm square, 50 mm-thick tan-talum foil illuminated with 3 or 4 425 J beams. The beams areincident on a 400 mmdiameter, 5 mm-thick Ni dot on a 2mm square,50 mm-thick CH foil. Pinhole diameter is either 10 or 20 mmdiameter.

We have investigated the discontinuous change with both thefixed-angle and the constant-included-angle cone types. A fulldescription of the campaign will be offered in a future publication;here we discuss these two examples we have fielded and offerjustification for a third type to be performed in our next set ofexperiments.

The design work was carried out with the LANL code RAGE [17].RAGE is amultidimensional, multiphysics cell-based adaptive meshrefinement (AMR) code with applications for laboratory experi-ments. RAGE utilizes radiation diffusion and the SESAME equation-

1 As it does not affect the experiment, the RF foam is hollowed out above thecone in order to reduce weight, to an extent that structural integrity is notcompromised.

Fig. 4. Experimental configuration with the 60� / 35� “constant included angle” coneprofile. Please see the text for details on the materials used.

K. Yirak et al. / High Energy Density Physics 9 (2013) 251e257254

of-state (EOS) and (gray) opacity tables, and is a “2-temperature”(radiation and material) code. Owing to excellent target fabricationby General Atomics (GA), wewere able to confidently employ RAGEin its cylindrically-symmetric capacity. Typical finest resolutionswere of order 1 mm. Fig. 6 gives examples of synthetic Ni backlitimages from a run modeling the 50� / 18.75� cone. We note thatRAGE does not have laser deposition physics; instead we utilizea volumetric temperature source based upon DANTE data [19] asthis collaboration has done successfully in the past.

Fig. 6. An example of RAGE synthetic Ni backlighter images. The cone design is50� / 18.75�; images are every 25 ns, starting at 65 ns. The Mach stem has formed by90 ns and has reached the transition; it has shrunk by 115 ns, and is regrowing by140 ns. The time at which f again reaches fc is roughly 120 ns. Shown at bottom isa zoomed in view of the Mach stem at 140 ns, when it is roughly 130 mm in size. Theablation front location’s (dark horizontal band behind bow shock) disagreement withexperiment is understood to be an artifact of the volumetric temperature sourceplacement [19]. Note the absence of a preheat “shoulder” as seen in Fig. 7.

3. Results

Experiments addressing discontinuous change in angle wereperformed at the Omega laser facility. Eight shots were conductedwith the above two profiles, four each. Two additional shotsaddressed the topic of preheat, which will make up some discus-sion in this section. Two final shots prototyped a target designemploying a gas fill, which is discussed in the final section.

Fig. 5. A preshot radiograph of a 50� / 18.75� “fixed angle” cone with a discontinuouschange in angle at r ¼ 500 mm. The three stalks used for in-chamber alignment arevisible, as is the (opaque) gold hohlraum. The line in the upper left represents 100 mm.

The pusher design for these experiments is based on pushersused by this collaboration in the past. It is designed so that itsopacity is low enough to not occlude any of the fluid dynamics inthe shocked system. The inclusion of the brominated CH layer isintended to prevent the transmission of M-band radiation from thegold hohlraum during the 1 ns laser pulse. On experiments whichhad a shock wave overrunning a collection of sapphire spheres, noevidence of preheat was observed [20]. Such evidence in experi-ments would have been distortion (growth) of the balls due to theradiation, which was not seen: the dimensions of the spheres inpreshot radiographs agreed excellently with those taken during theshot.

However, evidence of potential preheat in previous shots in thepresent campaign exists, most notably as a measurable expansionof the cone prior to the impact of the shock. This expansion wasverified by comparison with preshot radiographs which are notincluded here, again deferred to a future publication. However, wedo include an example experimental image. Fig. 7 shows50� / 18.75� shot image at t ¼ 90 ns, with the preheat artifactbeing the horizontal “shoulder” after cone transition. Preheat is,therefore, a possible reason for late-time discrepancy betweenexperiment and simulation (see below). This will be tested bychanges to the ablator/pusher design in the next round of shots.

Another issuewith this round of experiments was that we foundthe 60� / 35� design to be too aggressive. That is, slight tilt of thetargets resulted in occlusion of the Mach stem after transition; seeFig. 8. Thus, while the design was attractive from a theoreticalviewpoint, it was shown to be unfeasible from an experimentalviewpoint.

Fig. 7. 50� / 18.75� cone design imaged at t ¼ 90 ns. Evidence of preheat includes thehorizontal “shoulder” on the cone between the leading edge of the bow shock and thetransition. This is believed to be a result of cone expansion and subsequent com-pression by the shock. This is not seen in the RAGE results, which as we have noteddoes not have laser deposition physics capability. The reflected shock off the cone isbarely visible. The dark horizontal band behind the bow shock is the ablation front.Analysis of this image yields a Mach stem size of roughly 80 mm.

Fig. 9. Mach stem size for two cone designs, based on experimental (solid circles) andsimulation (hollow circles) results. While there is reasonable agreement before tran-sition, experimental data after transition are widely scattered and agree poorly withsimulation. The error bars on the experimental data come from the fidelity of theradiographs, which as a rule of thumb cannot resolve features below the pinhole size.The 50� / 18.75� data point at w(0.625,115) has been extracted from Fig. 7.

K. Yirak et al. / High Energy Density Physics 9 (2013) 251e257 255

Fig. 9 shows how the experimental data compare with RAGEsimulation. While the pre-transition data are in rough agreement,the post-transition data are less so. On the assumption that thecone expanded due to preheat this may explain the smaller Machstems than predicted. Cone expansion also increases the difficultyof Mach stem survival after transition, as it reduces the separationbetween Mach stem triple point and cone surface, narrowing thewindow inwhich the Mach stem needs to stabilize and (if possible)start growing again. We believe these reasons explainwhy the finaldata point, at r ¼ 0.9 mm, indicates that the Mach stem had beendestroyed, whereas the simulation results indicate it should havesurvived and begun to grow again by this time.

Finally, analysis of RAGE temperature data (not shown to avoidconfusion) indicates that the Mach stem temperature is 50% higherthan the bow shock temperature before the transition, and in-creases to 100% more after merger and regrowth. This providesa weak implication that indeed in the astrophysical case theemission would be brighter at these locations.

Fig. 8. 60� / 35� cone design imaged at t ¼ 110 ns. The red line is what the initialprofile was believed to be; the image demonstrates line-of-sight tilt/occlusion effectsdiscussed in the text. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

4. Discussion and future work

We wish to better understand the growth and stability/surviv-ability of Mach stems in the laboratory setting. By doing so, we canbegin to address the question of whether or not Mach stems area viable explanation for certain emission features seen in astro-physical jets which are not easily amendable to typical explanation.

There is compelling evidence that our experiments to date mayhave been affected by preheat, causing the cone to expand. Thisexpansion serves to reduce the size of theMach stem and cause it tobe destroyed more rapidly than predicted by RAGE simulation. Wetherefore are taking additional steps to eliminate preheat in ournext set of experiments, including modifying the pusher design toensure enough brominated CH remains unablated to guard againstM-band radiation. We are including Al in the pusher as well, toserve as a guard against another possible source of preheat, namely“laser glint”: laser light reflected off the hohlraumwall at early timebefore the laser-absorbing gold plasma is established. It is hopedthat the modification of the pusher and the addition of this laser-glint shield will yield improved results in the next round ofexperiments.

Based on further RAGE simulation, we will be fielding a modi-fied discontinuous “fixed angle” cone design. Fig. 10 shows that, inthe absence of preheat, this design permits three distinct stages ofevolution after transition (growth before transition is not shown,but may be inferred from Fig. 3). The first stage shows rapiddestruction of the Mach stem. In this case, the Mach stem is beingdestroyed owing to the trajectory of its triple point which, aftertransition, moves toward the cone surface. This may be contrastedwith the typical destruction scenario of a Mach stem, which isindicated by the angle of its slip surface to the surface [1]. Tradi-tionally, if the angle between the solid body and the slip surface ispositive, zero, or negative, the Mach stem will grow, remain stable,or be destroyed, respectively. In our case, however, this angle re-mains positive after transition, suggesting that slip stream anglemeasurement is not a proper diagnostic technique to use with thisgeometry.

The second stage after transition occurs after the triple pointshave merged, but before f > fc. Simulation indicates that this is

Fig. 10. Mach stem size and growth rate predicted by RAGE for an upcoming conedesign, with angles of 55� / 12� “fixed”. The three stages indicated are discussed inthe text.

K. Yirak et al. / High Energy Density Physics 9 (2013) 251e257256

a stage of quasi-stability, with the Mach stem neither growing norshrinking. The final stage occurs once f > fc, and we again seeMach stem growth. We are working on developing a theory for thisevolution which is slated to appear in a future publication.

The angles chosen for the pre- and post-transition angles were55� and 12�, respectively. This choice permits a Mach stem growthrate high enough that, after transition, it will still be resolvable at itssmallest size (roughly 40 mm predicted). Reducing this angle onlyreduces theMach stem size; increasing this angle serves to increasethe angle between Mach stem and bow shock, which reduces thevisibility of the triple point. The choice of 12� post-transition allowsa large window between the triple point merger and the re-achievement of f > fc, thereby creating the roughly equal threewindows discussed above. The choice of using a “fixed angle” conetypedwhich constantly sweeps through a range of includedanglesdis based on the desire to eliminate any tilt occlusion whichwe observed in our 60� / 35� design.

5. Conclusion

We are fielding a set of experiments aimed to improve our un-derstanding of Mach reflection in the laboratory. Our experimentalpackage consists of an indirectly-driven bow shock propagatingthrough a foam, which impinges on a cylindrically-symmetric goldcone. By varying the cone profile, we can observe the Mach stemgrowth as a function of angle and compare this with theory [13] andprevious experiment [5] concerning the critical angle fc. By intro-ducing a discontinuous change of angle in the cone profile, we cantake the first step toward understanding the criteria of Mach stemdestruction once it has formed.

Simulations suggest that two criteria are necessary for Machstem growth after a discontinuous change of angle. The first is thatthe newMach stem launched off the cone surface must merge withthe original in order to change the original’s triple point trajectory.The second is that the included angle must once again reach thecritical angle. If the former condition is met, simulation indicatesthat the Mach stemwill enter a quasi-stable configuration and willonly start growing again once the second condition is satisfied. Weanticipate addressing this with our next set of experiments.

There remain many unaddressed aspects of Mach reflection inthe lab. A large open question has to do with discrepancy betweenexperimental values of fc and theory. There also is opportunity to

investigate our thesis of Mach stem emission in astrophysical ob-jects. Our collaboration has begun to address the latter point withthe University of Rochester 3D MHD AMR code AstroBEAR [21,22]and anticipate a set of simulations complementary to our exper-imental design discussed here. Perhaps most importantly, it re-mains to be seen whether the growth and destruction timescalesobserved in the lab are compatible with those inferred from ob-servations. One important distinction between the astrophysicalcase and our experiments to date concerns the ratio of specificheats g. Our experiments to date all have an effective g ¼ 5/3,however the astrophysical regime features cooling via radiativelythin emission, reducing the effective g. Since the critical angle isa function of g, it would be very interesting to systematicallysample a range of values. We have fielded a prototype designwhichuses a gas fill in place of the RF foam in order to reduce g, similar tothose seen by Reighard et al. [23]. This work is ongoing.

Acknowledgments

The authors wish to thank the anonymous referees for theirinsightful comments on an earlier draft of this manuscript. Theauthors gratefully acknowledge the target fabricators’ craftsman-ship at GA, the staff and operations team of the Omega laser facility,and the conference organizers at which this research was offered inpresentation form. This work was supported by the Los AlamosNational Laboratory and by the NNSA-sponsored NLUF Program forUniversity Research.

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