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Regular Charged Particle Flow in Pulsed-Power-Driven Nonuniform Transmission Lines John G. Leopold, Raanan Gad, Chaim Leibovitz, and Itamar Navon
Abstract—In electron-emitting nonuniform transmission lines at currents sufficient to support magnetic insulation, it is possible to maintain regular Brillouin electron flow by matching the vacuum impedances of various parts of the transmission line and by ensuring adiabatic transition between them. This idea, first demonstrated numerically in the design of the Brillouin diode (BD), is extended to the structure of the pair known as knob and dustbin. Rather than shunting current to the walls, a “designer” knob–dustbin allows regular flow of the entire current to the BD. Index Terms—Brillouin flow, magnetic insulation, transmission line.
I. I NTRODUCTION
F
OR THE FLASH radiographic study of full-scale highvelocity projectile penetration experiments into solid metals or ceramics, we have introduced the idea of a new type of diode—the Brillouin diode (BD) [1]. This diode enables regular Brillouin flow of the magnetically insulated sheath current along the central stalk of a coaxial cylindrical transmission line to the apex of a truncated coaxial concentric conical transmission line. The connection between these two structures preserves the regularity of the flow as long as their vacuum impedances best match and the transition is adiabatic, i.e., the impedance change is small and the transition is smooth. This requirement follows the theory of Creedon [3] that shows that for constant energy, a constant vacuum impedance of a magnetically insulated transmission line is equivalent to the existence of a canonical constant of motion of regular Brillouin sheath electron flow along the direction of the flow. The theory of the dynamics of magnetically insulated electron flow has been formulated by Mendel et al. [4], [5], extended to nonuniform transmission lines and cases where voltage is added along the axis [6]–[8]. Nonetheless, the application of adiabatic matching of vacuum impedances (equivalent to the canonical momentum in the direction of the flow) of sections of nonuniform transmission lines, at constant energy, to preserve the regularity of the flow, was first suggested for the BD [1]. The truncated concentric coaxial conical diode, on its own, has been experimentally demonstrated as a high-current-density
Manuscript received September 4, 2008; revised December 18, 2008 and January 16, 2009. Current version published April 10, 2009. The authors are with the Department of Applied Physics, RAFAEL Laboratories, Haifa 31021, Israel (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPS.2009.2014761
high-brightness device [9] without considering its matching to the power transmitting line. Other possible high-current-density high-brightness diodes [10] are also connected by a coaxial cylindrical transmission line to the power-generating machine, but only the power flow is usually discussed and not the quality of the flow. Above a certain current limit [11], because of magnetic insulation, electrons emitted along the negative polarity central stalk of this transmission line are confined to flow as a sheath along the central stalk. The sheath is ordinarily shunted to the walls and not incorporated in the diode through direct flow [12] because it was expected that this flow is turbulent and asymmetric. To shunt the sheath away from the diode, the knob in a dustbin is used [2], [13]. The knob is a spherical object, coaxial to the central stalk and of radius larger than the radius of the outer cylinder of the transmission line. The dustbin is a cylindrical tube of even larger diameter attached to the outer cylinder. The dustbin diameter is required to be large enough so that the electric fields on the knob surface are small enough to assume that the knob is nonemitting. The knob and dustbin separate the diode from the rest of the machine. For long pulses or high voltages, some of the anode material can disintegrate and flying anode debris may damage the machine. The knob, in addition to shunting the sheath to the dustbin walls, prevents this debris from reaching the power-generating machine. It was recently shown that the sheath flow is symmetric and regular even for an asymmetric power-generating machine [13]. Even when azimuthal symmetry is assumed, attempts to direct the current around the knob toward the diode resulted in turbulent flow [14]. It was thought impossible to direct the sheath to flow regularly to the diode [15]. The dustbin introduces a further difficulty. It was observed that during the flow, resonant cavity modes develop in it, which interact with the flow, causing unwanted oscillations [13]. It was suggested that the current shunted to the dustbin wall can be reduced and retrapped to the diode if the diode impedance is reduced [12]. When the diode and cylindrical transmission line impedances mismatch, a retrapping wave [16] returns from the diode, rearranging the sheath current allowing for increased diode current accompanied by voltage decrease. The sheath current flows regularly toward the apex of the BD [1] and is completely incorporated in the diode current. We have shown in [1] that for an open upstream boundary, the BD’s operating voltage cannot surpass the self-limit of the magnetically insulated coaxial cylindrical transmission line. With increasing cone length, matching between the diode and
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the cylindrical transmission line improves, and the operating voltage and current density on the anode increase. The BD’s anode end does not have a direct line of sight with most of the machine but on its own; in practice, this may be insufficient to prevent debris damage. In this paper, we present numerical demonstration of an unconventional knob–dustbin design for which the sheath follows the knob’s outline into the BD while, at the same time, anode debris is blocked. This knob need not be nonemitting, and the shape of this designer knob–dustbin needs only to follow the principle of small and smooth change of the vacuum impedance along the electron flow. The designer knob and dustbin pair, the coaxial cylindrical transmission line, and, finally, the diode constitute a geometrically nonuniform transmission line over which continuous regular sheath electron flow is supported. In all our calculations, we have used the ATK/MRC MAGIC PIC code [17]. We are interested in power-generating machines of ∼2 MV [1], and all the results of this paper are for this value. II. “D ESIGNER ” K NOB AND D USTBIN In Fig. 1, we present a reference calculation where a coaxial cylindrical transmission line of b = 20 cm and a = 10 cm is fed at the z = 0 boundary by a voltage pulse rising in 5 ns to Vin = 2 MV. This transmission line is attached to a coaxial concentric conical transmission line of length C = 50 cm truncated to a parallel-plate diode with an rA = 5 mm anode face radius and an anode–cathode separation of dC−A = 1 cm (parameters are defined in [1, Fig. 5]). This entire electron-emitting structure, defines the BD as discussed in [1]. Above an electric field threshold of 230 kV/cm, electrons are emitted along the cathode surfaces with zero initial energy and momentum. We also allow proton emission on the anode walls stimulated by the electron beam impacting the surface as it sweeps along these walls. For this model, proton emission is initiated only on anode regions impacted by the electron beam and above a certain energy threshold. This way, the proton current persists only along regions where the electron beam deposits energy steadily. We chose this model, rather than the MAGIC space-charge-limited emission model from a fixed ion-emitting anode surface, to avoid exaggerated ion current because for the BD, it is difficult to separate the load (the region where electrons cross the cathode–anode gap) from the rest of the structure. In addition, we limit the emitted proton current by the values obtained from space-charge-limited proton current we calculated separately in test calculations. We found that our results are not very sensitive to this choice. For C = 50 cm, impedance matching between the cylindrical and conical sections is within 4%. In Fig. 1, snapshots of charge density contours are drawn at various points in time from early times in the rising pulse [Fig. 1(a)] to steady state [Fig. 1(d)]. Steady state is defined at times where the voltage and currents and the sheath flow’s width stabilize. Prior to steady state, the retrapping wave starting from the downstream edge of the BD [Fig. 1(c)] propagates upstream toward the open boundary [16]. The retrapping wave is evidence for incomplete vacuum impedance matching between the transmission lines and geometrical imperfections such as the truncated end of the conical
Fig. 1. Snapshots of electron charge density contours (pink to white from −0.025 to 0.0 C/m3 , respectively) for a BD of b = 20 cm, a = 10 cm, C = 50 cm, dC−A = 1 cm, rA = 5 mm, Vin = 2 MV, and trise = 5 at (a) 13.604, (b) 16.658, (c) 29.984, and (d) 50.196 ns.
section at the diode’s apex, the discontinuous connection between the cylindrical and conical sections, and the discontinuous electron emission at the open boundary. A small turbulence, which develops due to this emission discontinuity at the open upstream boundary, propagates downstream but does not seem to considerably affect the flow to the apex of the BD. In [1], we have shown that the quality of the Brillouin flow improves as C increases, evident in the increasing amount of current contained within a small fixed radius area on the
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Fig. 2. Time dependence of the voltage V and the total current Itot at the open upstream boundary of the system in Fig. 1, the voltage calculated at the diode’s apex VD , the electron current collected on the anode along the conical section and the diode’s truncated anode face IeD , the electron current collected on the rA = 5 mm anode face Ie(∅10 mm) , and the electron current enclosed within a 2-mm-diameter area on the anode face Ie(∅2 mm) .
Fig. 3. MK&D—A BD comprising a b = 20 cm, a = 10 cm, and 50-cmlong cylindrical section and a concentric coaxial conical section of C = 50 cm, separated from an upstream 50-cm-long cylindrical section by a knob–dustbin structure made up from a 50-cm-long cylindrical section of b = 50 cm, a = 25 cm, connected to the rest of the structure by two 40-cmlong conical sections, one broadening, the other narrowing (with respect to the flow’s direction). rA = 5 mm, dC−A = 1 cm, Vin = 2 MV, and trise = 5 ns. A snapshot of charge density contours at t = 89.953 ns is drawn (pink to white from −0.025 to 0.0 C/m3 , respectively).
Fig. 5. Snapshots of charge density contours for a rounded edge knob and a large diameter cylindrical dustbin attached to the same BD as in Fig. 3. (a) t = 19.990 ns, (b) t = 29.984 ns, (c) t = 40.257 ns, and (d) t = 98.003 ns (pink to white from −0.02 to 0.0 C/m3 , respectively).
Fig. 4. Same as in Fig. 2 for the system of Fig. 3.
anode face. We have also shown that with improved impedance matching, the steady-state voltage increases and the current decreases approaching the self-limiting values.
For this type of calculation, i.e., with an open upstream boundary, the voltage cannot surpass and the current cannot fall beneath the self-limiting values [16] of the corresponding cylindrical transmission line, which for the present conditions have been separately calculated to be VSL = 1.50 MV and ISL = 60.2 kA. In Fig. 2, the time-dependent voltages and
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LEOPOLD et al.: CHARGED PARTICLE FLOW IN PULSED-POWER-DRIVEN NONUNIFORM TRANSMISSION LINES
currents of the system of Fig. 1 calculated at the open boundary and at the diode’s apex are shown. In Fig. 2, we observe that the voltage and the total current at the open boundary reach first the self-limits of the cylindrical transmission line, and when the retrapping wave arrives at ∼45 ns, the voltages and currents reach the same values which are already at steady state at the diode’s apex. The difference between the steady-state Itot and IeD is the positive ion current. The retrapping wave propagates upstream at less than half the speed of light [13] and rearranges the sheath quite dramatically [Fig. 1(c)]. Fig. 2 shows a temporal structure during the rise of the voltage and current signals at the diode’s apex. The electromagnetic wave from the upstream boundary propagates at the speed of light followed by the sheath current which detaches from the upper wall as it propagates [Fig. 1(a)]. The diode’s edge starts to emit after ∼10 ns before the sheath from the cylindrical part arrives [Fig. 1(a)]. When the sheath in the cylindrical section, which shunts at its propagation edge to the upper wall, detaches and joins the sheath in the conical section, there is a temporal current and voltage drop until it recurs as a single sheath current at the diode’s apex [Fig. 1(b)]. This presteady-state structure can change in shape or be absorbed in the diode’s rise time for a different choice of voltage rise time and the relative lengths of the various sections. For the case of Figs. 1 and 2, the steady-state value of VD is ∼15% lower than VSL and the steady-state Itot is ∼8% above ISL , which can be improved by increasing C . To be able to compare to the calculations which we shall present hereinafter, the aforementioned BD was chosen with a relatively long cylindrical section. This choice of such a large system limits the spatial resolution of the PIC calculation at the diode’s downstream edge. In Fig. 2, we draw the current enclosed on a 2-mm-diameter circular area on the rA = 5 mm perpendicular anode face. The ratios of the mean steady-state values Ie(∅2 mm) /IeD = 0.65 and Ie(∅10 mm) /IeD = 0.84 improve with increasing C [1]. In this paper, we wish to introduce a physical obstruction along the cylindrical transmission line of vacuum impedance matched to the rest of the structure to avoid diode edge debris from returning to the power-generating machine. Matching should ensure that the entire current flows to the apex of the BD. The impedance mismatches are required to be small and transitions to be smooth so that the operating diode voltage and current are not seriously affected. The simplest and most natural way to do this is shown in Fig. 3. The conical sections of the knob and dustbin are chosen with vacuum impedances as close as possible to the vacuum impedance of the central stalk. We shall call this structure the matched knob and dustbin or MK&D. We see that although the entire current flows to the BD’s edge, considerable turbulence is apparent. “Wrapping around the knob” has been observed before [14] but for different circumstances, i.e., for a knob and dustbin structure of arbitrary impedance, the shunt current was diverted from the dustbin wall to the diode at a certain decreased anode–cathode distance. It was then preferred to shunt the sheath to the dustbin wall instead and retrap current through the central stalk because the flow was found to be too turbulent and strongly dependent on the diode impedance.
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Fig. 6. Same as in Fig. 2 for the system of Fig. 5.
In Fig. 4, we draw the same values as those drawn in Fig. 2, but for the MK&D of Fig. 3. We see that the steady-state values of all quantities drawn do not differ considerably from those shown in Fig. 2. Moreover, the observed turbulences do not affect considerably the time dependences at steady state seen both at the system’s upstream and downstream ends. The retrapping wave arrives to the upstream end at the same speed as before but in two steps (seen in the behavior of V and Itot ): First, it rearranges the sheath flow along the central stalk and next around the knob. Steady state in both VD and the diode current is reached only after the sheath along the knob has settled, although the change in the latter during this period is small. In Fig. 5, we rounded the knob edges of Fig. 3 and changed the dustbin to a large cylindrical object for the same BD as before. For the MK&D of Fig. 3, the ratio b/a in the knob–dustbin’s cylindrical section is 2, which is the same as b/a in the other cylindrical sections, whereas in Fig. 5, the ratio between the dustbin’s radius and the knob’s height is 3. For this situation, the vacuum impedance of the knob–dustbin region does not match that of the rest of the structure, and turbulent flow is to be expected. In Fig. 5, a much more turbulent development than that obtained for the MK&D of Fig. 3 is observed. In Fig. 6, we draw the time dependence of the voltages and currents for comparison to Figs. 2 and 4. The turbulent behavior shown in the snapshots of Fig. 5 is observed in the noisier behavior at “steady state” in Fig. 6, although it is not extreme. Moreover, although, at steady state, typically illustrated by Fig. 5(d), the mean values of the quantities drawn are similar to those of Fig. 2; a long “settling down” period of time is required to reach this turbulent steady state. The comparison of Figs. 3 and 4 to Figs. 5 and 6, respectively, shows that matching vacuum impedances of the various sections of a nonuniform magnetically insulated transmission line is very important in preserving regular sheath flow. In Fig. 7, we present a modified design over that of the MK&D of Fig. 3 in an attempt to cancel the observed turbulent behavior. We shall name this design the “Designer” Knob and Dustbin, the DK&D. In the DK&D, we round the connections between the conical and cylindrical sections of the MK&D, attempting to make the transitions between the sections more adiabatic. Fig. 7 shows that except for small turbulences, the flow becomes very regular and follows the outline of the
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Fig. 8.
Same as in Fig. 2 for the system of Fig. 7.
Fig. 9. Snapshots of charge density contours (pink to white from −0.025 to 0.0 C/m3 , respectively) for the designer knob and dustbin (DK&D) of Fig. 7 for (a) dC−A = 5 cm at t = 90.685 ns and (b) dC−A = 7.5 cm at t = 94.567 ns.
Fig. 7. Designer knob and dustbin (DK&D) with the same overall geometry as that shown in Fig. 3 (the MK&D) but with rounded connecting surfaces. Snapshots at (a) t = 14.144 ns, (b) t = 20.245 ns, (c) t = 39.935 ns, and (d) t = 60.179 ns of charge density contours are drawn (pink to white from −0.025 to 0.0 C/m3 , respectively).
structure. Fig. 8, similarly to Figs. 2, 4, and 6, shows the time dependences of a few observed voltages and currents for the DK&D. In Fig. 8, the two steps of the retrapping wave propagation (see V and Itot ) are clearer than that in Fig. 4. The differences between the mean steady-state values of the various observed
quantities in Figs. 2, 4, and 8 are not significant. We suspect that the difference between the regular flow shown in Fig. 7 compared to the turbulences in Fig. 3 will be reflected in the quality of the beam reaching the anode face, but because of the limited spatial resolution at the diode’s edge, we are unable to study the details of the electron beam pinch on the anode face. In most diodes, the distance between the cathode and the anode is an important impedance-determining factor [13], [16]. The following results show that the BD is less sensitive to this parameter and that the DK&D works well for a considerable range of dC−A . In Fig. 9, we present steady-state snapshots of electron charge density at steady state for dC−A = 5 and 7.5 cm. Comparison of the steady-state snapshots shown in
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Fig. 11. Contours of steady-state (t = 99.836 ns) electron charge density (pink to white from −5.0 to 0.0 C/m3 , respectively) at the edge of the BD for the DK&D configuration of Fig. 7.
Fig. 12. Contours of steady-state (t = 98.281 ns) electron charge density (pink to white from −5.0 to 0.0 C/m3 , respectively) at the edge of a BD with dC−A = 1 cm, rA = 1.5 cm, and the same DK&D configuration as in Fig. 7. Fig. 10. Same as in Fig. 2 for the system of (a) Fig. 9(a) and (b) Fig. 9(b).
Figs. 9(a) and (b) to 7(d) shows that for the range of dC−A studied, the DK&D performs well and the flow seems little affected. The results of Fig. 10 should be compared to Fig. 8 (dC−A = 1 cm). The steady-state mean values of V , VD , Itot , and IeD change little as dC−A increases, i.e., the impedance of the BD, including the DK&D, depends little on dC−A . On the other hand, as dC−A increases from 1 to 7.5 cm, the ratio Ie(∅10 mm) /IeD decreases from 0.78 to 0.52 and Ie(∅2 mm) /IeD decreases from 0.55 to 0.32, i.e., the beam becomes more spread on the anode face. Preliminary calculations suggest that this spread decreases for higher voltages. In Fig. 10, we also observe that as dC−A increases, the second retrapping wave becomes slower and weak oscillations of period ∼30 ns and more are discerned. This behavior suggests the development of some instability also evident in increasing turbulence and some detachment at the upstream end of the knob in Fig. 9(b). Nevertheless, the change is gradual and leaves considerable range to the geometrical parameters for the optimization of the details of the diode’s end before the truncated end becomes a serious perturbation. Finally, the knob and dustbin optimization is worthwhile only if the electron flow at the end of the BD [1] is not seriously perturbed. In Fig. 11, the magnified edge of the DK&D of Fig. 7 is shown. The details of the edge shown in Fig. 11 are only partially resolved because the overall structure is too large for com-
putational purposes. Nevertheless, one can estimate that 55% of the total electron current reaching the anode is enclosed within a 2-mm-diameter circle on the 1-cm-diameter perpendicular anode edge (Fig. 8). The choice of the edge parameters in Fig. 11 allows only 79% of the total electron current to impact the perpendicular anode edge. Calculations for the corresponding BD of Fig. 1 without the knob and dustbin result in 61% electron current on the 2-mm-diameter circle and 82% on the 1-cm diameter perpendicular anode edge. The dimensions of the diode’s edge in Fig. 11 could be too small to avoid plasma closure [1]. In Fig. 12, the electron charge flow for a more spacious edge choice is shown. For the parameters of Fig. 12, the entire electron beam is contained on the perpendicular anode edge with 49% of the electron current within a 2-mm-diameter circular area. The results shown in Figs. 11 and 12 show that the inclusion of the DK&D has little effect on the performance at the BD’s edge. III. S UMMARY We have shown here that for constant energy, regular sheath electron flow can be sustained along a nonuniform transmission line as long as the constant of motion in the direction of the Brillouin flow is best preserved. To achieve this goal, vacuum impedances of various parts of the nonuniform transmission line need to change little and the crossing between various parts
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be adiabatic. Such regular flow has been first demonstrated in the BD [1] and in this paper for the designer knob and dustbin. In contrast to the ordinary knob and dustbin, our design allows not only protecting the power generator from debris but also the regular flow of the sheath into the diode. The specific DK&D studied by us can be improved by changing the length of the conical sections and improving adiabaticity. To test such changes, more powerful computational tools are required. The DK&D studied in this paper is not the only possible solution to the debris problem—it is though a very neat demonstration of how regular flow can be dynamically tailored. Preserving the regularity of the Brillouin sheath flow by vacuum impedance matching and adiabatic transitions between various sections of nonuniform transmission lines may have other useful applications in plasma physics. We are studying such systems presently. ACKNOWLEDGMENT The authors would like to thank the referees for their professional and critical reading of this paper. R EFERENCES [1] J. G. Leopold, R. Gad, C. Leibovitz, and I. Navon, “Numerical experiments on matching vacuum transmission lines to loads,” IEEE Trans. Plasma Sci., vol. 37, no. 1, pp. 50–57, Jan. 2009. [2] T. J. Goldsack, T. F. Bryant, P. F. Beech, S. G. Clough, G. M. Cooper, R. Davitt, R. D. Edwards, N. Kenna, J. McLean, A. G. Pearce, M. J. Phillips, K. P. Pullinger, D. J. Short, M. A. Sinclair, K. J. Thomas, J. R. Threadgold, M. C. Williamson, and K. Krushelnick, “Multimegavolt multiaxis flash X-ray source development for a new hydrodynamic research facility at AWE Aldermaston,” IEEE Trans. Plasma Sci., vol. 30, no. 1, pp. 239–253, Feb. 2002. [3] J. M. Creedon, “Relativistic Brillouin flow in the high ν/γ diode,” J. Appl. Phys., vol. 46, no. 7, pp. 2946–2955, Jul. 1975. [4] C. W. Mendel, Jr., D. B. Seidel, and S. A. Slutz, “A general theory of magnetically insulated electron flow,” Phys. Fluids, vol. 26, no. 12, pp. 3628–3635, Dec. 1983. [5] C. W. Mendel, Jr., S. E. Rosenthal, and D. B. Seidel, “Low-pressure relativistic electron flow,” Phys. Rev. A, Gen. Phys., vol. 45, no. 8, pp. 5854– 5865, Apr. 1992. [6] C. W. Mendel, Jr. and S. E. Rosenthal, “Modeling magnetically insulated devices using flow impedance,” Phys. Plasmas, vol. 2, no. 4, pp. 1332– 1342, Apr. 1995. [7] C. W. Mendel, Jr. and S. E. Rosenthal, “Dynamic modeling of magnetically insulated transmission line systems,” Phys. Plasmas, vol. 3, no. 11, pp. 4207–4219, Nov. 1996.
[8] C. W. Mendel, Jr. and D. B. Seidel, “Flow impedance of a uniform magnetically insulated transmission line,” Phys. Plasmas, vol. 6, no. 12, pp. 4791–4793, Dec. 1999. [9] P. Gilad, E. Nardi, and Z. Zinamon, “High-current-density relativistic electron beams in conical diodes,” Appl. Phys. Lett., vol. 34, no. 11, pp. 731–732, Jun. 1979. [10] J. Maenchen, G. Cooperstein, J. O’Malley, and I. Smith, “Advances in pulsed power-driven radiography systems,” Proc. IEEE, vol. 92, no. 7, pp. 1021–1042, Jul. 2007. [11] J. M. Creedon, “Magnetic cutoff in high-current diodes,” J. Appl. Phys., vol. 48, no. 3, pp. 1070–1077, Mar. 1977. [12] I. D. Smith, “Induction voltage adders and the induction accelerator family,” Phys. Rev. Spec. Top., Accel. Beams, vol. 7, no. 6, pp. 064 801– 064 841, Jun. 2004. [13] N. Bruner, T. Genoni, E. Madrid, D. Rose, D. Welch, K. Hahn, J. Leckbee, S. Petillo, B. Oliver, V. Bailey, and D. Johnson, “Modeling particle emission and power flow in pulsed-power driven, nonuniform transmission lines,” Phys. Rev. Spec. Top., Accel. Beams, vol. 11, no. 4, pp. 040 401– 040 410, Apr. 2008. [14] K. Hahn, J. Maenchen, S. Cordova, S. A. Drennan, I. Molina, S. Portillo, D. Rovang, D. Rose, B. Oliver, D. Welch, V. Bailey, and D. L. Johnson, “Retrapping studies on RITS,” in Proc. 14th IEEE Int. Pulsed Power Conf., Dallas, TX, 2003, pp. 871–874. [15] D. D. Hinshelwood, R. J. Allen, R. J. Commisso, G. Cooperstein, B. M. Huhman, D. Mosher, D. P. Murphy, P. F. Ottinger, J. W. Schumer, S. B. Swanekamp, S. J. Stephanakis, B. V. Weber, F. C. Young, I. Crotch, J. O’Malley, and J. R. Threadgold, “High-power self-pinch diode experiments for radiographic applications,” IEEE Trans. Plasma Sci., vol. 35, no. 3, pp. 565–572, Jun. 2007. [16] P. F. Ottinger and J. W. Schumer, “Rescaling of equilibrium magnetically insulated flow theory based on results from particle-in-cell simulations,” Phys. Plasmas, vol. 13, no. 6, pp. 063 109–063 117, Jun. 2006. [17] B. Goplen, L. Ludeking, D. Smith, and D. Warren, “User configurable MAGIC for electromagnetic PIC calculations,” Comput. Phys. Commun., vol. 87, no. 1/2, pp. 54–86, May 1995.
John G. Leopold, photograph and biography not available at the time of publication.
Raanan Gad, photograph and biography not available at the time of publication.
Chaim Leibovitz, photograph and biography not available at the time of publication.
Itamar Navon, photograph and biography not available at the time of publication.
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