Wire number dependence of the implosion dynamics ...

Wire number dependence of the implosion dynamics, stagnation, and radiation output of tungsten wire arrays at Z driver Michael G. Mazarakis, Christopher E. Deeney, William A. Stygar, Melissa R. Douglas, Jerry Chittenden, Daniel B. Sinars, Michael E. Cuneo, Thomas J. Nash, Gordon A. Chandler, M. Keith Matzen, John L. Porter, Kenneth W. Struve, and Dillon H. McDaniel Citation: Physics of Plasmas 18, 112706 (2011); doi: 10.1063/1.3657421 View online: http://dx.doi.org/10.1063/1.3657421 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/18/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Inner-shell radiation from wire array implosions on the Zebra generator Phys. Plasmas 21, 031207 (2014); 10.1063/1.4865370 Implosion dynamics and radiation characteristics of wire-array Z pinches on the Cornell Beam Research Accelerator Phys. Plasmas 16, 012706 (2009); 10.1063/1.3054537 Neutron emission generated during wire array Z-pinch implosion onto deuterated fiber Phys. Plasmas 15, 032701 (2008); 10.1063/1.2839352 Implosion and stagnation of wire array Z pinchesa) Phys. Plasmas 14, 056315 (2007); 10.1063/1.2671940 Structure of stagnated plasma in aluminum wire array Z pinches Phys. Plasmas 13, 082701 (2006); 10.1063/1.2234284

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PHYSICS OF PLASMAS 18, 112706 (2011)

Wire number dependence of the implosion dynamics, stagnation, and radiation output of tungsten wire arrays at Z driver Michael G. Mazarakis,1 Christopher E. Deeney,2 William A. Stygar,1 Melissa R. Douglas,3 Jerry Chittenden,4 Daniel B. Sinars,1 Michael E. Cuneo,1 Thomas J. Nash,1 Gordon A. Chandler,1 M. Keith Matzen,1 John L. Porter,1 Kenneth W. Struve,1 and Dillon H. McDaniel1 1

Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87185, USA National Nuclear Security Administration, Washington, D.C. 20585, USA 3 Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4 Imperial College, London, SW& 2BW, United Kingdom 2

(Received 3 August 2011; accepted 9 October 2011; published online 14 November 2011) We report results of the experimental campaign, which studied the initiation, implosion dynamics, and radiation yield of tungsten wire arrays as a function of the wire number. The wire array dimensions and mass were those of interest for the Z-pinch driven Inertial Confinement Fusion (ICF) program. An optimization study of the x-ray emitted peak power, rise time, and full width at half maximum was effectuated by varying the wire number while keeping the total array mass constant and equal to 5.8 mg. The driver utilized was the 20-MA Z accelerator before refurbishment in its usual short pulse mode of 100 ns. We studied single arrays of 20-mm diameter and 1-cm height. The smaller wire number studied was 30 and the largest 600. It appears that 600 is the highest achievable wire number with present day’s technology. Radial and axial diagnostics were utilized including crystal monochromatic x-ray backlighter. An optimum wire number of 375 was observed which was very close to the routinely utilized 300 for the ICF program in C 2011 American Institute of Physics. [doi:10.1063/1.3657421] Sandia. V

I. INTRODUCTION

The 20-mm diameter, 10-mm high, 300-wire, 5.8-mg total mass, W wire array is widely used with the Z-accelerator in Sandia National Laboratory and is considered the radiation source of choice for Inertial Confinement Fusion (ICF) research. It is a design that was developed to optimize radiated power based on array height and mass only. Over the years, a number of experiments have been conducted which indicate that the radiated x-ray wire array power should also be optimized based on both wire number and total array mass. In particular, experimental results show that higher number wire arrays provide higher powers for broadband emission (e.g., W) and k-line radiation (e.g., Al). Similarly lighter than 5.8 mg weight arrays, which give faster 80-ns implosion times, yield higher peak radiated x-ray power and faster rise times.1 These observations suggest that the mass and wire number of the 20 mm diameter tungsten wire array may not be optimized for maximum peak power. A number of effects may together influence these observed trends and focus on the wire initiation=pre-acceleration evolution. In particular, the ability to initiate the wires themselves, the ablation times, the amount of plasma precursor produced, the mass left behind, and the global magnetic field structure are all dependent to some degree on wire number and total array mass. To this end, a systematic, controlled experimental campaign was undertaken to investigate only the effect of wire number on the 20-mm diameter, 10-mm high tungsten wire array, keeping the array mass the same and at the assumed optimum of 5.8 mg. Wire numbers ranging from 30 to 600, 1070-664X/2011/18(11)/112706/15/$30.00

corresponding to wire diameters between 35.85l and 8.01l and interwire gaps of 2.09 mm down to 0.104 mm were studied. A similar systematic mass optimization of the array at constant, optimum wire number is also needed and is advisable to follow the present investigation. Early Al wire array experiments were performed by Sanford et al.2 with the Saturn accelerator.3 The maximum current of Saturn is 7-MA, and it can operate in short (35 ns rise time) and in long (up to 250 ns) modes. Those first experiments had a very short pinch time of the order of 35 ns. The inter-wire gaps scanned a range between 6 and 0.4 mm. Because of array assembly difficulties and wire number availability, two load configurations were utilized: one with 8.6-mm radius and total mass between 615 and 656 lg, and another with 12-mm radius and 820-lg total mass. The height of both arrays was 20 mm, and the wire number varied from 10 to 192. For the first time, it was discovered that by increasing the wire number or equally decreasing the inter-wire gap (IWG), the pinch dynamics changed and the stability improved. A critical inter-wire gap of 1.3 mm was measured below which the radiated x-ray power output increased dramatically while the x-ray rise time remained approximately the same. This phenomenon was attributed at that time to a shelllike behavior and increased symmetry of the imploding array for IWG smaller than the critical gap. At the time of those experiments, because of technological difficulties, no interwire gaps smaller than 0.4 mm were attempted. In more recent wire number scan experiments by Coverdale et al.4 again with the Saturn accelerator but in a

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longer pulse mode (time to pinch 165 ns), similar behavior of the Al array was observed. Two configurations were utilized, one of 20 mm and the other of 16 mm radius with total masses of 620 lg=cm and 960 lg=cm, respectively. The IWG range covered with both arrays was from 3.9 mm to 0.36 mm. The results of these experiments, in addition to exhibiting a critical IWG like the previous work, also revealed an optimum gap of (0.7 mm) below which the output power appeared to decrease while the rise time increased. It is interesting to note that despite the fact that both experiments extended the measurements to almost the same minimum size IWG (0.4 mm versus 0.36 mm), the previous short implosion time (35 ns time) results never reached an optimum gap. The power output kept increasing all the way to the 0.4 mm gap. It is possible that this apparent disagreement may be due to the different pinch dynamics between fast 35-ns and long 165-ns implosion times. Also in both experiments, by necessity, two different load configurations and masses were utilized that may have affected the pinch dynamics and the observed trends. In addition for higher dI=dt generators like Saturn, it is possible that a high enough magnetic pressure is reached earlier on before the wire cores have expanded so much and thus the wire core sizes and the optimal gap could possibly be smaller. Having in mind the above observations, in our W wire number scan experiments, we kept all load parameters strictly the same, that is the total mass, the array diameter, the height, the gap between the wire array edge, and the return current can (Fig. 1) as well as the final coaxial magnetically insulated transmission line (MITL) gap.5 We only varied the wire number. We made a special effort to extend the measurements to as small an IWG as technically possible which turned out to be 0.104 mm, and we scanned an IWG range between 0.104 mm and 2.09 mm. Our results clearly demonstrate in addition to the critical IWG of 0.3 mm, an optimum IWG of 0.175 mm, which is approximately 4 times smaller than that observed in aluminum. In the following sections, we describe the experiment (Sec. II). We present experimental results and analysis (Sec. III), discuss plausible explanation of the multi-wire array behavior, and compare it with current general theoretical understanding and available model simulations (Sec. IV).

FIG. 1. (Color) Side section of the load design and the final coaxial selfmagnetic insulated transmission line (MITL) that transfers the total generator current into the load.

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II. EXPERIMENTAL ARRANGEMENT

The experiments presented in this paper were performed with the Z accelerator before its refurbishment,6–14 which could drive up to 20 MA current within 100 ns through a wire array load. The Z pulsed power design is based on the conventional Sandia pulsed power technology of Marx generators, water-pulse-forming and transmission lines, vacuum MITLs, and post-hole convolutes. The oil and water sections contain 36 modules with identical components. The pulses of the 36 modules are combined together in parallel into four equal number groups, 9 each, and feed four biconical constant impedance vacuum MITLs. The four pulses are then combined again via a double post-hole convolute section into a single 20 MA, 2.5 MV pulse which finally drives the Z-pinch load on axis. We fired 17 shots, all with approximately the same mass of 5.8 mg. This mass was selected since it was widely utilized and believed to be the optimum mass for a 300-wire, 20 mm diameter, 10 mm height tungsten single-wire array. A systematic mass optimization of this load geometry had not as yet been undertaken and was not one of the goals of the present study. However, in recent15–17 experiments, it was discovered that the optimum mass for higher power outputs might not be 5.8 mg. Much lesser masses, of the order of 2.4 mg, radiated higher x-ray powers which are closer to those obtained by nested arrays. Figure 1 presents the load design and the final coaxial insulated transmission line (MITL) that transfer the total generator current into the load. For all our shots, we utilized single arrays with the same diameter and height. The anode-cathode gap between the arrays and the return current cylinder (can) was 4 mm while the gap of the final coaxial MITL was 3 mm. This was evaluated to be the optimum gap in previous experiments.18 It was not too small to cause gap closure at peak load current and not unnecessarily large to increase the load inductance. To our knowledge, the present series are the only controlled high current experiments aiming to investigate the effect of the wire number on the pinch quality and x-ray radiated output power, while all the other array parameters remained strictly the same. Therefore, the variations observed during shots of different wire number can only be attributed to the wire number itself (or equivalently the inter-wire gap). Despite the fact that the total cross sectional area of the arrays remained the same, and consequently the total load initial inductance and resistance, the current per wire unavoidably did not. Therefore, our results may be due either to an IWG effect on the pinch dynamics or to the current per wire or to both. Table I gives the actual wire diameter, wire number, IWG, and total mass for each array studied. The return current cylindrical electrode that surrounded the wire array (Fig. 1) had nine 5.6 mm wide slots around its center circumference corresponding to an equal number of lines of sights (LOS) where the various diagnostics observing the pinch were located. The central top anode electrode of the load had a 5 mm diameter circular aperture to facilitate the observation of the pinch from the top by a suite of axial diagnostics similar to those of the side LOS. The side LOS were oriented 12 above the pinch middle plane and


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TABLE I. Summary of load parameters.

Z-shot number 1072, 1073, 1075 734, 914 868 875 733 732 874 731, 872 873, 912, 913, 1074, 1076

Number of wires (n)

Wire diameter (l)

Total array mass (mg)

IWG (mm)

600 500 450 300 194 120 90 50 30

8.01 8.75 9.19–9.21 11.36 14.05–14.07 17.85 20.69 27.78 35.85

5.87 5.84 5.79–5.82 5.91 5.80–5.82 5.79 5.88 5.85 5.88

0.104 0.126 0.140 0.209 0.324 0.524 0.698 1.26 2.09

contained among others a five-channel x-ray diode (XRD) array,19 five diamond photo-conducting detectors, three nickel bolometers integrating the radiated power during a 40 ns interval,20,21 and four microchannel plate pinhole cameras. Different pinhole sizes and filters were used to selfimage the pinch in selected regions of the emitted x-ray spectrum. The pinch power was determined by normalizing a spectrally equalized linear combination of the five XRD signals to the average of the three bolometer energy measurements.22 The spatially integrated x-ray diagnostics observed only the upper half of the axial extent of the pinch while the framing cameras recorded the entire length. The axial diagnostics included two 12-time frame pinhole cameras. Each time-frame had two pinholes with two different filters providing the radial evolution of the pinch into two spectral regions, one up to 200 eV x-ray energy (soft x-rays) and the other above 200 eV (hard x-rays). XRDs, PCDs, and bolometers measured the axially emitted radiation. Those diagnostics provided a very valuable insight into the behavior of the precursor plasma and its time evolution as well as the degree of azimuthal symmetry of the stagnating pinch.23,24 The load current was measured with two magnetic flux monitors (B-dots) which were located at the anode side of the central biplate MITL,18 6 cm away from the pinch axis and in almost diametrically opposite sites 150 apart.

In a limited number of shots, we follow the evolution of the array edge with one of the radial framing cameras and with a crystal monochromatic x-ray backlighter. III. EXPERIMENTAL RESULTS AND ANALYSIS A. Power, energy, and rise time scaling

During the 17 shots, we collected a wealth of experimental results pertaining to the power, energy, rise time, full width at half maximum (FWHM), peak kinetic energy, and spectra of the x-ray radiated power. In addition, we measured the spatial and temporal evolution of the array, specifically the precursor and pinch plasmas both axially and radially. Special attention was paid to observe the precursor plasma and measure the time of its arrival on axis. Fig. 2(a) presents the x-ray pulse waveforms and load currents for two (30 and 450) representative wire numbers. Although the load currents were approximately the same, the x-ray output pulse and power peak location varied from one wire number array to another. The x-ray pulse for the 30 wires was broader; the peak radiation power was lower and occurred later. In addition, the location where the extrapolation of the rise side of the pulse crosses the time axis (zero crossing) came earlier. Fig. 2(b) shows clearer this behavior of the x-ray radiation pulse. Both wire arrays pinch later than the OD prediction of the SCREAMER (Ref. 25) circuit code simulations (blue line). Fig. 3(a) shows the x-ray power pulses for most of the arrays studied. In the same figure, the expected time location of the pinch is presented as calculated by the OD circuit model SCREAMER and serves as a time reference point. Fig. 3(b) presents two time differences: the time difference between the time to peak power (tPeak) and the ideal OD minimum radius time (tRmin), and the time difference of the zero crossing time (tPo) from the same ideal OD minimum radius time (tRmin). The tRmin is the time in the OD calculations that the array reaches the minimum preset radius (here 1 mm). It is obvious from Fig. 3(b) that the x-ray pulse starts earlier and peaks later for larger IWG or smaller wire numbers. It is also interesting to note the following trend: the difference between the time to x-ray peak power and zero crossing (approximately equal to rise time) decreases as the

FIG. 2. (Color) (a) X-ray pulse waveforms and load currents for two (30 and 450) representative wire numbers. SCREAMER circuit code simulation results are also shown for time comparison. (b) X-ray radiation pulse for 30 and 450 wire arrays shown in an enlarged time base for clarity. Both wire arrays pinch later than the OD prediction of the SCREAMER circuit code simulation.


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FIG. 3. (Color) (a) Overlay of the x-ray power pulses for most of the arrays studied. (b) Time differences of the time to peak power (tPeak) and of zero crossing time (tPo) from the ideal OD minimum radius time (tRmin) which is equal to the time in the OD calculations that the array reaches the minimum (here 1 mm) preset radius.

wire number increases (smaller IWG), and the extrapolated limit value appears to be 3.6 ns for an infinite number of wires. This time interval is equal to the ideal kinetic energy thermalization times. From OD model consideration, since all the above arrays had the same mass and same final kinetic energy, they should have reach minimum radius and pinch at the same time. However, the larger IWG arrays behave kinematically as if they had larger masses despite the fact that the radiated power is much smaller. It is obvious from Fig. 3 that the larger the IWG or equally the smaller the wire number the later appears the x-ray radiated pulse. This is in very good agreement with 2D simulations of Ref. 26. Fig. 4 summarizes all x-ray peak power measurements for the different inter-wire gaps and wire numbers. We covered a range of IWG between 0.104 mm and 2.09 mm or equivalently wire numbers between 600 and 30. Fig. 4(a) presents the dependence of peak power (XRPOWER) on the IWG. Fig. 4(b) zooms in the region of the smaller gap studied. Fig. 5 shows the same as Fig. 4 results but now as a function of the wire number. The peak power increases as the wire number increases or equivalently as the IWG decreases. However, this trend does not continue to infinite number of wires or zero IWG.

There is an optimum peak power output at approximately 375 wires or 0.175 mm IWG. Our results exhibit similar but more pronounced behavior as the aluminum results.4 We distinguish a critical interwire gap at about 0.3 mm where the power starts to increase more rapidly and an optimum gap at about 0.175 mm below which the radiated power plummets to levels equal to or smaller than those of the other extreme of very large 2 mm IWG (30 wires). It appears that this behavior observed first in aluminum and now in tungsten may be a universal phenomenon for all the materials. Just the location of those behavioral changes may vary from material to material. The results are suggestive to the fact that smaller effective core size materials like W have optimum and critical IWG of smaller values unlike the aluminum that exhibits similar behavior at relatively larger IWG. Fig. 6(a) summarizes the x-ray pulse rise time measurements while Fig. 6(b) zooms in the smaller IWG region between 0.1 and 0.6 mm. Again an optimum gap appears at exactly the same location (0.175 mm) where the x-ray radiated power becomes maximum (Fig. 5). Fig. 7 compares our peak x-ray power results (Fig. 7(a)) with the first aluminum measurements2 (Fig. 7(b)). The

FIG. 4. (Color) (a) X-ray peak power (XRPOWER) measurements for all the inter-wire gaps studied. An IWG range between 0.104 mm and 2.09 mm was covered. (b) X-ray power measured for the smaller IWG between 0.104 mm and 0.6 mm.


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FIG. 6. (Color) (a) Measurements of the 10%–90% x-ray-power rise time as a function of the array IWG. (b) Rise time measurements for the smaller IWG studied in order to put into evidence the optimum IWG. FIG. 5. (Color) X-ray (XRPOWER) peak power results as a function of array wire number.

critical inter-wire gap for aluminum is at 1.4 mm. The tungsten critical gap is 4.5 times smaller. In Ref. 2, the critical inter-wire gap was attributed at that time to the change in the imploding array behavior from a discrete wire-like to plasma shell-like. The results of Ref. 2 show continuous increase in the peak power down to the minimum of 0.4 mm gap measured without a power decrease below 0.6 mm IWG as in the results of Coverdale et al.4 Fig. 8 compares the rise time results of Ref. 4 (Fig. 8(a)) for aluminum again with our tungsten results (Fig. 8(b)). Both experiments show an optimum IWG where the rise time becomes minimum. For the aluminum this gap is 0.7 mm. The question still remains unanswered why the early experiments of Sanford et al.2 did not show similar behavior as in Ref. 4 and in our results. A plausible explanation may rely on the different pinch times of Refs. 2 and 4. In Ref. 2, the time to implosion was only 40 ns, while in Ref. 4, it was 165 ns. w?>As we found in our recent current scaling experiments,17 same wire number and same geometry arrays but lower mass generate much higher peak powers and shorter rise times for fast current drives. One therefore can speculate that for faster pinches even smaller IWG may be necessary to observe the decrease of the power output beyond a certain number of wires. The most recent thinking about the wire array behavior suggests a competition between the stabilizing effect of the

precursor plasmas which does not exist in foil pinches and the disrupting effects of the Rayleigh-Taylor (R. T.) instabilities. Faster pinches may require more wire numbers and smaller inter-wire gaps to exhibit plasma shell-like behavior and substantial R.T. contribution. It would be of great interest if technologically feasible to extend the experiments of Ref. 2 to even lower IWG or to repeat our experiment with larger mass and larger implosion times. In the latter case, if the above heuristic argument holds, the optimum IWG should be larger than in the presentinvestigation. The total radiated x-ray energy (Fig. 9) seems to follow the same behavior with the change of the wire number as the peak power and the rise time. There is an optimum again at the same number of wires (375 6 25), and the total energy drastically decreases at large as well as at very small wire numbers. B. Stagnated plasma scaling

The plasma size at stagnation was measured with timeresolved framing cameras which provided images with interframe separation of 2 ns. Fig. 10 contains images of the 500 wires near stagnation obtained with one of the side pinhole framing cameras. The top frames are exposed to the entire x-ray spectrum. Only a very thin filter was placed in front of the camera aperture in order to essentially cut off the visible light from striking the micro-channel plates. The bottom series of frames correspond to the harder x-ray spectrum above 200 eV. The axially averaged radial extent of the

FIG. 7. (Color) (a) Comparison of our peak x-ray power results with the first aluminum measurements2 (b). The critical inter-wire gap for aluminum is at 1.4 mm. The tungsten critical gap is 4.5 times smaller.


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FIG. 8. (Color) Comparison of the rise time results of Ref. 4 (a) for aluminum with our tungsten results (b). Both experiments show an optimum IWG where the rise time becomes minimum.

FIG. 9. (Color) Measurements of the total radiated x-ray energy (XENERGY) as a function of the IWG (a) and the wire number (b). The total radiated x-ray energy seems to follow the same dependence on the wire number as the peak power and the rise time. There is an optimum again at the same number of wires (375 6 25), and the total energy drastically decreases at larger as well as at smaller wire numbers.

FIG. 10. (Color) (a) X-ray images of the entire pinch length around pinch time. The plasma size at stagnation was measured with time-resolved framing cameras that provided images with inter-frame separation of 2 ns. The wire number of the array was 500. The top frames are exposed to the entire x-ray spectrum. Only a very thin filter was placed in front of the camera aperture in order to essentially cut off the visible light from striking the micro-channel plates. The bottom series of frames correspond to the harder x-ray spectrum above 200 eV. (b) The axially averaged radial extent of the pinch plasma is shown as a function of machine time (red line with error bars) superimposed on the x-ray radiation pulse (light blue trace, arbitrary units). The numbered negative small pulses (arbitrary units) indicate the time that each frame of (a) was triggered.


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pinch plasma is also shown to be 2 mm. The diameter of the stagnating plasma decreases as the frames approach pinch time. However, the minimum size appears to be 2 ns later past the peak radiated x-ray power. This phenomenon was also observed on the axial framing camera pictures which will be presented later on in this paper. Fig. 11 summarizes the stagnated plasma diameter measurements for a number of IWG. The radial extent of the stagnated plasma follows the same behavior as the power, energy, and rise time relative to the variation of the IWG or wire number. The pinch becomes tighter for the optimum inter-wire gap region as one should expect for more stable and higher yield pinches. C. Plausible explanation of wire number scaling

Table II compares the observed optimum and critical gap of tungsten (present work) with the corresponding values measured at the previous two aluminum experiments.2,4 The critical inter-wire gap IWGc is defined following the heuristic model of Haines27 from the expression IWGc ¼ 2vc tp ;

(1)

where vc is the critical implosion plasma velocity and tp is the time to pinch. Using similar parametrization, we can calculate an optimum plasma velocity (vopt) now utilizing the optimum instead of the critical inter-wire gap, IWGopt ¼ 2vopt tp :

(2)

If the heuristic model was equally valid for fast and slow aluminum pinches, the critical velocity should have been the same and independent of the pinch times. The calculated vc for aluminum arrays by the heuristic model27 is 1.28 cm=ls which agrees very well with the results of Ref. 2. However, the longer 165 ns results of Ref. 4 provide a different critical velocity of 0.86 cm=ls. In the same manner and utilizing expression (1), we calculate a critical velocity vc of 0.15 cm=ls for W. Utilizing expression (2), we estimate an optimum velocity vop of 0.44 cm=ls for Al and 0.09 cm=ls for W. The corona and core expansion velocities in the single wire measurements of Ref. 28 were as follows; for Al, the

corona and core expansion velocities were measured to be, respectively, 2 cm=ls and 0.7 cm=ls. For the W case, the corona expansion velocity was found to be 1 cm=ls and core 0.15 cm=ls More recent measurements by Shelkovenko and coworkers29 with x-ray backlighting techniques, which are not affected by the close to the core coronas, suggest core expansion velocities for our 8 to 9 lm wires (600 and 500 wire arrays) of the order of 0.06 cm=ls. However, taking into account the fact that our wires were shorter, 1 cm instead of 1.5 to 2 cm of Ref. 29, and the extremely small IWGs (0.1 mm versus 3–5 mm of Ref. 29) of our arrays, it is plausible to assume that the cathode electrode core broadening effect may have increased the core expansion velocities for almost the entire wire length bringing them closer to our estimated vop ¼ 0.09 cm=ls. So for our smallest inter-wire gaps, we may have had core merging, and our estimated vop could indeed be that of the core expansion velocity. Of course in a wire array because of the global magnetic field, the tangential to the array circumference expansion velocities will be reduced due to the preferred motion of the coronal plasma and ablated material towards the array axis. Hence, the optimum wire number could be considered as the number beyond which the core merge and the array pinch as a very thin R.T. unstable foil. In the heuristic Haines’ model, the assumption was made that the wire arrays remain immobile until pinch time. So the calculated critical velocity in Eq. (1) is the velocity with which the corona plasma expands around each wire core until they merge and form an azimuthally continuous plasma shell. Then the wire array implodes like a shell. The increased power beyond the critical interwire gap was attributed to the fact that the coronas of the wires merged before the pinch creating a uniform highly symmetric plasma conducive to a more stable pinch. However, this hypothesis seems to be in contradiction with the fact that the smaller the IWG the faster the corona merger and the longer time available for R.T. instabilities. The assumption of the stabilizing effect that may have a plasma distribution in the interior of the array appears more plausible as an explanation for the power increase and pinch quality improvement both as the inter-wire gaps decrease and later again when the pinch quality decreases as the interwire gaps become too small. As the wire number increases,

FIG. 11. (Color) Stagnated axially integrated plasma diameter for a number of wire arrays studied. The measurements were done with side framing x-ray cameras observing the entire length of the pinch. Both the total (red color) and hard spectrum (blue color) plasma sizes are presented. (a) Pinch plasma diameter versus IWG. (b) Pinch plasma diameter versus wire number.


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TABLE II. Comparison of aluminum and tungsten critical and optimum interwire gaps (IWG).

Experiments

Critical IWGc (mm)

Optimum IWGo (mm)

Time to pinch (ns)

1.4 2.2 0.3

Not seen 0.7 0.175

35 165 90

Aluminum, Sanford et al.2 Aluminum, Coverdale et al.4 Tungsten, present work

the ablated mass distribution and plasmas inside the array becomes more azimuthally uniform and lead to a more stable pinches.30,31 However, when the IWG becomes smaller than the optimum, the global magnetic field dominates the private magnetic field and the flow of ablated mass and plasma inside the array quenches very early in the driving pulse current. This happens when the IWG is approximately equal to p times the wire core size.30,32 The array then pinches like a thin plasma shell which is very susceptible to R.T. instability. It behaves like a thin foil. Assuming this explanation for the observation of an optimum gap and utilizing the expression dcore ¼

IWGop ; p

(3)

where dcore is the wire core diameter, we estimate a core size for the aluminum of 220l and for the tungsten 55l. These estimates are in reasonably good agreement with the independently measured core sizes for both materials.28,31–34 The above discussion suggests that the optimum IWGop is the gap where the wire coronas merge very early during the prepulse stage of the load current pulse, the precursor plasma inside the array becomes negligible or non-existent, and the wire array pinches as a thin (50l for W and 220l for Al) shell. Analytical calculations35 and 2D model simulations26 support this assumption and suggest that the plasma distribution inside the array, prior to the onset of the wire motion, enhances the pinch stability. In addition, an increase of the wire number, up to the optimum, improves symmetry which in turn makes the implosion and stagnation even more stable.

D. Precursor plasma observations

We obtained a time estimate of the precursor plasma arrival on axis utilizing high sensitive XRD channels in the side and axial (top) diagnostic ports. We also utilized very sensitive top (axial) framing cameras to image the selfemission of the incoming plasmas towards the pinch axis. We triggered the cameras up to 50 ns before the 100 ns pinch times. Fig. 12 shows pictures of the precursor plasma, for 50 wire arrays, arriving on axis before the pinch. Both top axial cameras were triggered early 45 ns before pinch time. There are two frames per time interval; the left frames record practically all the x-ray spectrum while the right hand side frames record the hard above 200 eV component. As both cameras clearly show, the plasmas arrive on axis in two waves: as early as 45 ns to 42.4 ns there is a lot of plasma on axis. The x-ray radiation produced by these plasmas first increases as time goes by. Then at about 35 ns reaches a minimum and then starts increasing again until the last captured frame of 22.9 ns (camera 2). This behavior appears to be real since we carefully check the sensitivity response of each frame in both cameras and it was the same for all frames. Camera 2 had more sensitive microchanel plates than camera 1. Fig. 13 presents radial line-outs of the intensity of the soft and hard x-ray frames of camera 2 (Fig. 12) along an horizontal line passing through the center of each frame. The colored numbers are the timing of each frame relative to pinch time which is taken as “0” time. From the XRD measurements, we established a lower limit of the precursor velocity for arrays of small wire numbers (30, 50, and 90 in Fig. 13). To obtain the data of Fig. 14, we used very sensitive XRD channels with the axial diagnostic package. Fig. 14(a) demonstrates the measurement technique and Fig. 14(b) summarizes the results for 30, 50, and 90 wires. For the higher wire number arrays, it was difficult to distinguish the arrival of the precursor on axis from the onset of the main x-ray pulse. Because of uncertainties in establishing the precursor plasmas radial motion starting times, the velocities may be underestimated. On the other hand, substantial concentration of plasmas on axis may have occurred before

FIG. 12. (Color) Pictures of precursor plasma arriving on axis for a 50 wire array. Both top axial cameras (number 1 and number 2) were triggered much earlier than pinch times (47.4 ns and 42.9 ns). There are two frames per time interval; the frames of the first left column and the fourth right column record practically all the x-ray spectrum (soft), while the frames of the two middle columns (hard) record the hard x-rays above 200 eV component.


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FIG. 13. (Color) Radial line-outs of the intensity of the soft and hard x-ray frames of camera 2 (Fig. 12). The colored numbers are the timing of each frame relative to pinch time which is taken as “0” time.

reaching sufficient density and temperature for the plasma to emit x-ray fluxes detectable by the XRD detectors. Nonethe-less, it is reassuring that the estimated velocities of 2 6 0.2 107 cm=s (Fig.14(b)) are in reasonable agreement to those evaluated with optical techniques in Ref. 31. During the present investigation, optical diagnostics with the Z driver were not yet available. Fig. 15 gives time snap shots of two different x-ray radiated power levels (0.1 TW and 1 TW) observed on the center of the pinch. This corroborates our previously presented results and statement that the radiated x-ray power pulse starts later and has shorter rise time for larger wire number arrays. The earliest that our axial camera of Fig. 12 was triggered was 47.4 ns. Even that early in the driving pulse, there was enough plasma on axis to emit x-ray radiation. From the intensity of the 47.4 ns frame, it is obvious that the precursor plasma must have arrived earlier. This suggests that our precursor velocity estimates may be a lower limit (Fig. 14(b)).

E. Study of wire motion

We studied the wire core location and motion as a function of time with two side framing cameras, one looking at

the axis of the pinch and the other tangentially observing the edge of the wire array. The latter technique was successful for the smaller wire number arrays (30 and 50). Fig. 16(a) indicates the orientation of framing camera number 1 aiming the axis of the array and camera number 2 observing the edge of the array. Fig. 16(b) shows the camera orientations as seen from the top. Fig. 17 presents the framing pictures taken with camera number 1. This camera was triggered at 20.4 ns. Even as close as at 80% of the pinch time, the wire cores appear stationary at their original location. Fig. 17(a) shows four frames taken with slightly different filters. The array studied in this shot had 50 wires. The axially averaged radial line-out scan of the pictures (Fig. 17(b)) shows maxima and minima with a period of 1.2 mm: the same as the original IWG of the 50 wire array. Similar periodicity with a slightly smaller period is exhibited in the radially averaged axial line-out of the same frames (Fig. 17(c)). These results are showing that the large inter-wire gap arrays do not follow the OD trajectories almost all the way to the pinch times. We have observed similar behavior with the tangential framing camera number 2, observing the x-ray self-emission of the array edge. Figure 18 present the single “wire” and total frame axial line-out of the first (a) and fourth (b) frames of Fig. 17. It is

FIG. 14. (Color) (a) demonstrates how we estimated the precursor arrival on axis for 30, 50, and 90 wire arrays. (a) is for a 30 wire array. The graph is raw data in Volts as obtained by a very sensitive channel of the XRD monitor scope (XRDOA1KM). We estimate an approximate time of 43 6 4 ns of precursor arrival on axis. Since the array radius is 1 cm, we estimate a precursor velocity of 2 6 0.2 107 cm=s. This of course is the time when enough precursor plasma is gathered on axis to produce detectable x-ray radiation. Optical measurements may give even earlier arrivals. (b) Precursor arrival times for 30, 50, and 90 wire arrays.


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FIG. 15. (Color) Times of observed on axis two x-ray radiation power levels (0.1 TW and 1 TW) as a function of wire number (a) and IWG (b).

apparent an evolution in axial modulation. The approximate wavelength is 1 mm. Fig. 19 presents the results taken with the camera 2. The camera 2 had two filters: The bottom frames (Fig. 19(a)) had the same filter which practically allowed the entire x-ray spectrum and cut only the optical radiation (“soft filter”), while the top frame filter (Fig. 19(b)) allowed x-rays to pass about above 200 eV energy (“hard filter”). There were two pinholes for each time sequence, one with the hard x-ray filter and the other with the soft. The camera was triggered 35 ns before the pinch (35 ns), and the interframe separation was 10 ns with frame time exposure of 2 ns. On the left side of each frame picture is also seen the shadow of the oval shaped viewing slot edge. Those pictures were taken with a 30-wire array. The wire cores in Fig. 19 appear to be broken to several vertical sections which remain immobile as the time goes by and as close to the pinch as 5 ns. However, they increase in radial size and the ablating material seems to move upwards. The 5 keV x-ray crystal back-lighter36 aiming again at the array edge agrees with the framing camera results, showing significant structure at the initial location of the 30 wire array at 39 ns into the implosion (Fig. 20). However, the 600 wire array at 32 ns (Fig. 20) has already substantially

moved and ablated a lot of material that obstructs the x-ray passage through a large extent of the diagnostic port. Fig. 21 compares the framing cameras measured locations of the 30 and 50 wire array cores as a function of time with OD and 2D trajectories.37 It is obvious that the cores, or at least a large number of core segments, of small number wire arrays do not follow OD or 2D trajectories and remain practically stationary all the way down to pinch times. It is also possible that we observe only the remnants or the left behind parts of the wire cores while a small part of the wires get ablated, move on axis, and contribute to the pinch. This is in agreement with the findings of the 3D resistive magneto hydrodynamic simulations performed by Chittenden et al.26 F. Clustering and apparent rotation of the stagnated plasmas

Fig. 22(a) presents a typical axial framing camera picture for the soft and hard x-ray spectra around pinch times. The observed array had 300 W wires. There are 12 time frames of each spectral region with 2 ns inter-frame separation. The sensitivity of the micro-channel plates were greatly reduced here as compared with those of Fig. 12 because they were set to observe the x-radiation emitted at pinch times. Radial

FIG. 16. (Color) Orientation of side framing cameras number 1 and 2. (a) Orientation of the cameras (side view). (b) Orientation of the cameras (top view).


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FIG. 17. (Color) Framing pictures taken with camera number 1. (a) Four frames taken with slightly different filters. The wire array had 50 wires. (b) Axially averaged radial line-out scan of the picture of the first frame. (c) Radially averaged axial line-out of the same frame.

intensity line-outs along an horizontal line passing through the center of each frame are also presented (Fig. 22(b)). Both channels’ early frames from 11.7 ns before the pinch and inwards show an annular plasma shock wave which converges and pinches on axis. It initiates the thermalization of the plasma kinetic energy which “lights up,” emitting maximum x-ray radiation 4 ns later. It is interesting to note that the maximum radiated x-ray flux recorded by the microchannel plates happens 4 ns before the peak x-ray power (pinch time) as observed by the radial XRD monitors. However, the tightest pinch is observed at 5.7 ns before pinch time.

Careful evaluation of the diagnostic trigger cable lengths of both axial and side diagnostics ruled out any systematic synchronization error of axial and radial diagnostics. The run-in pinch plasma is not azimuthally uniform, having a spoke-like structure. At pinch time ( 0.2 ns), the stagnated plasma appears hollow and annular with the annulus broken down to 5 clusters. If they were caused by the influence of the nine slots of the current return, then they should be 9 clusters and not 5. In the later frames (þ2.2 ns and up to þ6.2 ns), the annulus appears to rotate in a clockwise fashion. The cluster structure is more clearly shown in

FIG. 18. (Color) The single “wire” and total axial line up of the first (a) and fourth (b) frames of Fig. 17 show evolution in axial modulation. The approximate wave-length is 1 mm.


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FIG. 19. (Color) Frames obtained with camera number 2 (30 wire array). (a) Frames obtained with the “soft filter.” (b) Frames obtained with the “hard filter.” The camera was triggered 35 ns before pinch and took pictures every 10 ns.

Fig. 23, where the blown up frames þ0.2 and þ2.2 ns of Fig. 22 are presented. This is most probably due to an m1 instability growth fed by the magnetic energy stored around the array during the driving current pulse.26,37 We can speculate that the axial images are projections of a stagnated plasma exhibiting a combination of m0 (clustering) and m1 (rotation) instabilities.26 Although, the annular structure can be attributed to a rotating and expanding plasma column, axially jettisoning cold plasmas cannot be precluded. The apparent rotation could also be attributed to different moments of bright spot appearances along the pinch. Nonetheless, the annularity, clustering and possible rotation of the stagnated plasma merit further systematic investigation. (The two right angle white lines faintly visible in Figure 22 are due to thin plastic wire cross-hair fiducials.) IV. DISCUSSION OF THE RESULTS AND COMPARISON WITH 2D AND 3D SIMULATIONS

We have measured the x-ray power output, energy and rise time of the 20 mm diameter, 10 mm height, W wire arrays of mass 5.8 mg for a wire number range between 30

FIG. 20. (Color) 5 keV back-lighter35 transmission images. 30 wire array edge transmission picture at 39 ns. Similar 600 wire picture at 32 ns.

FIG. 21. (Color) Comparison of the framing camera derived locations of the 30 and 50 wire array cores at a number of times into the driving pulse with OD and 2D trajectories.37

and 600 wires. In addition, we observed the precursor plasma behavior and the array implosion dynamics. The stagnated plasma behavior was also followed with x-ray self emission imaging. The power output, energy, rise time, and stagnated plasma size are strong functions of wire number or interwire gap. A critical and optimum wire number and IWG were observed for W similar to those of Al but at four times smaller IWG. It is suggestive that this behavior is a universal phenomenon and most probably occurs in all materials at different wire numbers or IWG locations. The comparison of the two Al wire number scans and our W studies suggests that the optimum and critical interwire gap may be also a function of the driving current rise times. Namely, Sanford et al. have much faster current rise times than Coverdale et al. and us. Therefore, one could speculate that the cores did not have enough time to fully expend to 200l core size before implosion and smaller IWG would be necessary to observe the optimum gap. The later could possibly explain the discrepancies between the two Al experimental results. Present theoretical considerations, 2D, and 3D MHD simulations can explain at least qualitatively the improvement of pinch quality for larger wire numbers. However, we can offer only speculations concerning the pinch quality deterioration at too large wire numbers. 2D and 3D simulations of the wire array implosion dynamics can explain why the higher wire number arrays start moving earlier than the smaller wire number arrays. In the 2D simulations, this behavior is mainly attributed to earlier change of the global magnetic field due to faster filling of the interwire gap with conducting plasma. In the 3D simulations, the faster convergence of higher wire number is attributed to larger ablation rates. The wire array behavior is caused by the early development of axial mass modulations probably due to m0 instabilities (Fig. 2 of Ref. 26). In Ref. 26, these mass modulations are artificially introduced by imposing uncorrelated temperature variations along each wire axis. With higher wire numbers, the current is more concentrated around the outer edge of the array. This causes an increase in the mass ablation rate, which in turn


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Phys. Plasmas 18, 112706 (2011)

FIG. 22. (Color) (a) Axial framing camera picture for the hard (two middle columns of pictures and the soft (two outside columns) x-ray spectra around pinch times. The wire array had 300 wires. (b) Radial intensity line-ups (hard x-ray spectrum). The colored numbers are the timing of each frame relative to pinch time which is taken as “0” time.

leads to earlier development of fully ablated axial sections triggering the onset of implosion phase. The variation of implosion trajectory as a function of wire number was investigated using the Gorgon resistive magneto-hydrodynamics code.26 A similar setup was used to that described in Ref. 37. In this case, a new variant of the code developed for r,h,z geometry was used run in 2D in the r-h plane. Reflective conditions were used for the azimuthal boundaries, with a transmissive “free-flow” boundary placed at a radius of 1 mm. This removed the problem associated with small azimuthal cell widths close to the axis at the expense of not being able to model the pinch stagnation phase. The same current waveform was applied to all simulations to allow direct comparison. With small wire numbers and large inter-wire gaps, a significant fraction of the current is swept downstream into the interior of the array by the precursor flow. With higher wire numbers, the current is more concentrated at the edge of the array adjacent to the ablating wires. This leads to a higher rate of wire ablation and an earlier start to the implosion for larger numbers of wires.37 Figure 24 illustrates how the implosion trajectory varies with the number of wires in 2D simulations. In Fig. 25 at 90 ns (90 ns before pinch time), the majority of the mass is located in the compact ablating wires with both the 30 and 300 wires. By 40 ns, in the 30 wire case, the residual dense wire cores still persist at their original positions but are rapidly running out material. In the 300 wire case, however, the wire cores have already run out of mass by this time and the implosion has already started, forming a quasi-continuous

FIG. 23. (Color) Cluster structures of stagnated plasmas. Selected frames of Fig. 22.

plasma shell which drives an accreting snow-plow through the precursor flow. However, the implosion trajectories alone cannot explain the pinch improvement with the increase of wire number. The earlier wire motion together with the increased uniformity of the implosion plasma surface are the two main factors contributing to higher performance of the arrays with larger wire numbers. In addition, the higher ablation velocities reduce the spread of the mass arrival on axis, or in other words, the mass left behind becomes negligible at pinch times. This results in tighter pinches and shorter rise times. Figures 9 and 10 in Ref. 26 by Chittenden et al. clearly illustrate the above assertions in the case of three Al arrays with different wire numbers. The cause of pinch quality deterioration at too high wire numbers remains unknown. We can only speculate the following plausible explanation which is also on line with Lebedev et al. thinking.31 When the wire number becomes larger than the optimum number (375 6 25), the wire cores start merging very early during the prepulse stage of the load current pulse, the precursor plasma inside the array becomes negligible or non-existent, and the wire array pinches as a thin shell which is prone to instabilities. Applying this explanation to W and Al experimental results, we estimate wire

FIG. 24. (Color) 2D simulations of orbit trajectories for different wire number arrays. Radii are those of the density maximum near the edge of the plasma as a function of time for the 50, 300, and 600 wires.


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FIG. 25. (Color) Contours of the logarithm of mass density for the 30 and 300 wire simulations at 90 ns, 40 ns, and 30 ns before pinch times as well as profiles of the azimuthally integrated mass density versus radius.

core sizes and core expansion velocities close to those independently measured in Imperial College and Cornell University,28,29,33,34 This finding further reinforces the plausibility of our speculations. In addition, analytical calculations35 and 2D model simulations26 suggest the enhanced Z-pinch stability due to plasma distribution inside the array prior the onset of wire motion. ACKNOWLEDGMENTS

The authors are deeply indebted to our colleagues at Sandia National Laboratories, Ktech Corporation, and Team Specialty Products. They also wish especially to thank the Z operation department headed during the performance of this work by Guy L. Donovan, the supporting technologies department headed by Johann F. Seamen, the load design group headed by Dustin Heinz Romero, the diagnostics team headed by Don O. Jobe, and the wire array laboratory headed by Dolores Graham for their superb work and great dedication. The authors are grateful for helpful discussions with Dr. Sergey Lebedev and Dr. Simon Bland of the Imperial College. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S.

Department of Energy under Contract No. DE-AC04-94AL85000. 1

M. G. Mazarakis, M. E. Cuneo, W. A. Stygar, H. C. Harjes, D. B. Sinard, B. M. Jones, C. Deeney, E. M. Waisman, T. J. Nash, K. W. Struve, and D. H. McDaniel, Phys. Rev. E 79, 016412 (2009). 2 T. W. L. Sanford, G. O. Allshouse, B. M. M. Marder, T. J. Nash, R. C. Mock, R. B. Spielman, J. F. Seamen, J. S. McGurn, D. Jobe, T. L. Gilliland, M. Vargas, K. M. Struve, W. A. Stygar, M. R. Douglas, M. K. Matzen, J. H. Hammer, J. S. De Groot, J. L. Eddleman, D. L. Peterson, D. Mosher, K. G. Whitney, and J. W. Thornhill, Phys. Rev. Lett. 77, 5063 (1996). 3 D. D. Bloomquist, R. W. Stinnett, D. H. McDaniel, J. R. Lee, A. W. Sharpe, J. A. Halbleib, L. G. Schlitt, P. W. Spence, and P. Corcoran, in Proceedings of the 6th IEEE Pulsed Power Conference, Arlington, VA, edited by B. H. Bernstein and P. J. Turchi (IEEE, New York, 1987), IEEE Cat. No. 87CH2522-1, p. 310. 4 C. A. Coverdale, C Deeney, M. R. Douglas, J. P. Apruzese, K. G. Whitney, J. W. Thornhill, and J. Davis, Phys. Rev. Lett. 88(6), 065001 (2002). 5 M. G. Mazarakis, C. E. Deeney, M. R. Douglas, W. A. Stygar, D. B. Sinars, M. E. Cuneo, J. Chittenden, G. A. Chandler, T. J. Nash, K. W. Struve, and D. H. McDaniel, Plasma Devices Oper. 13, 157 (2005). 6 R. B. Spielman, W. A. Stygar, J. F. Seamen, F. Long, H. Ives, R. Garcia, T. Wagoner, K. W. Struve, M. Mostrom, I. Smith, P. Spence, and P. Corcoran, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 709.


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P. A. Corcoran, J. W. Douglas, I. D. Smith, P. W. Spence, W. A. Stygar, K. W. Struve, T. H. Martin, R. B. Spielman, and H. C. Ives, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 466. 8 R. J. Garcia, H. C. Ives, K. W. Struve, R. B. Spielman, T. H. Martin, M. L. Horry, R. Waverick, and T. F. Jaramillo, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 1614. 9 H. C. Ives, D. M. Van De Valde, F. W. Long, J. W. Smith, R. B. Spielman,W. A. Stygar, R. W. Wavrik, and R. W. Shoup, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 1602. 10 M. A. Mostrom, T. P. Hughes, R. E. Clark, W. A. Stygar, and R. B. Spielman, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 460. 11 R. W. Shoup, F. Long, T. H. Martin, R. B. Spielman, W. A. Stygar, M. A. Mostrom, K. W. Struve, H. Ives, P. Corcoran, and I. Smith, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 1608. 12 I. D. Smith, P. A. Corcoran, W. A. Stygar, T. H. Martin, R. B. Spielman, and R. W. Shoup, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 168. 13 K. W. Struve, T. H. Martin, R. B. Spielman, W. A. Stygar, P. A. Corcoran, and J. W. Douglas, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 162. 14 W. A. Stygar, R. B. Spielman, G. O. Allshouse, C. Deeney, D. R. Humphreys, H. C. Ives, F. W. Long, T. H. Martin, M. K. Matzen, D. H. McDaniel, C. W. Mendel, Jr., L. P. Mix, T. J. Nash, J. W. Poukey, J. J. Ramirez, T. W. L. Sanford, J. F. Seamen, D. B. Seidel, J. W. Smith, D. M. Van DeValde, R. W. Wavrik, P. A. Corcoran, J. W. Douglas, I. D. Smith, M. A. Mostrom, K. W. Struve, T. P. Hughes, R. E. Clark, R. W. Shoup, T. C. Wagoner, T. L. Gilliland, and B. P. Peyton, in Proceedings of the 11th IEEE International Pulsed Power Conference, edited by G. Cooperstein and I. Vitkovitsky (IEEE, Piscataway, NJ, 1997), IEEE Cat. No. 97CH36127, p. 591. 15 D. B. Sinars, R. W. Lembe, M. E. Cuneo, S. V. Lebedev, E. M. Waisman, W. A. Stygar, B. Jones, M C. Jones, E. P. Yu, J. L. Porter, and D. F. Wenger, Phys. Rev. Lett. 100, 145002 (2008). 16 M. E. Cuneo, private communication (2008). 17 M. G. Mazarakis, M. E. Cuneo, W. A. Stygar, H. C. Harjes, D. B. Sinars, C. Deeney, E. M. Waisman, T. J. Nassh, K. W. Struve, and D. H. McDaniel, Phys. Rev. E 79, 016412 (2009). 18 W. A. Stygar, H. C. Ives, D. L. Fehl, M. E. Cuneo, M. G. Mazarakis, J. E. Bailey, G. R. Bennett, D. E. Bliss, G. A. Chandler, R. J. Leeper, M. K,

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Prandtl number dependence of the viscous ... - Semantic Scholar

Quantum dynamics is infinitesimal qr-number dynamics