Detection of an anomalous pressure on a magneto-inertial-fusion load current diagnostic

Detection of an Anomalous Pressure on a Magneto-Inertial-Fusion Load Current Diagnostic

M. H. Hess, B. T. Hutsel, C. A. Jennings, J. P. VanDevender, A. B. Sefkow, M. R. Gomez, P. F. Knapp, G. R. Laity, D. H. Dolan, D. C. Lamppa, K. J. Peterson, W. A. Stygar and D. B. Sinars.

Sandia National Laboratories, Albuquerque, New Mexico 87185. A. B. Sefkow also at the Laboratory for Laser Energetics, University of Rochester, 250 East River Road, Rochester, New York 14623-1299.

Sandia release SAND2017-1201J, dated 9 January 2017. PACS number 84.70.+p.

Abstract

Recent MagLIF (Magnetized Liner Inertial Fusion) experiments at the Sandia National Laboratories Z pulsed power facility have featured a PDV (Photonic Doppler Velocimetry) diagnostic in the final power feed section for measuring load current. In this paper, we report on an anomalous pressure that is detected on this PDV diagnostic very early in time during the current ramp. Early time load currents that are greater than both B-dot upstream current measurements and existing Z machine circuit models by at least 1 MA would be necessary to describe the measured early time velocity of the PDV flyer. This leads us to infer that the pressure producing the early time PDV flyer motion cannot be attributed to the magnetic pressure of the load current, but rather to an anomalous pressure. Using the MHD code ALEGRA, we are able to compute a time-dependent anomalous pressure function, which when added to the magnetic pressure of the load current, yields simulated flyer velocities that are in excellent agreement with the PDV measurement. We also provide plausible explanations for what could be the origin of the anomalous pressure.

I. Introduction

Sandia National Laboratories has employed both the PDV and VISAR (Velocity Interferometer System for Any Reflector) diagnostics for determining the velocities of shocked metal flyer surfaces as a function of time. These shocked flyers have been utilized in experiments to infer the applied kinetic pressure in high-pressure dynamic material experiments, as well as to infer electrical currents and associated magnetic pressures in pulsed power experiments.

In the Sandia magneto-inertial-fusion concept called MagLIF, a preheated fuel is compressed to high density and temperature using the Z pulsed power facility. Previously reported results on successful MagLIF experiments showed measured load currents using B-dot probes. These probes were located at a radial distance of 5.9 cm from the load. Unfortunately, it is well-known that B-dot probes have various failure mechanisms, such as in high inductance loads like MagLIF, which can limit their performance. This has led to the development of additional load current diagnostics for MagLIF experiments, including a PDV based approach in the final power feed section.

Recent MagLIF experiments, which incorporate the PDV load current diagnostic, have detected the presence of an additional pressure on the diagnostic early in the current pulse ramp. As we shall later show, the magnetic pressure due to the load current at these early times is too small to account for the measured pressure. Hence, the additional measured pressure is referred to as ‘anomalous pressure’. Our paper is organized as follows: in Sec. II we discuss the experimental setup for the MagLIF load current diagnostic, in Sec. III we show experimental data from the diagnostic and demonstrate that the measured pressure cannot be due to the load current and is therefore anomalous, and in Sec. IV we discuss plausible explanations for the origin of the anomalous pressure and give concluding remarks.

II. Experimental Configuration

The MagLIF experiment utilizes a PDV diagnostic in the final power feed section of the Z machine to measure the load current. The diagnostic operates in the following way. If one surface of the PDV flyer experiences a magnetic field due to the load current, then the magnetic field will induce a current density within the flyer. The magnetic field and current density will cause a net force on the flyer to occur, along with a net motion of the opposite surface. A reflected laser on the opposite flyer surface can measure the surface velocity through interferometry. In addition to the flyer motion generated by the load current magnetic field, a mechanical pressure may also be present which generates flyer motion on the opposite side. The time-dependent magnetic field, and hence load current, as well as the mechanical pressure, can be obtained through simulations of the flyer.

Figure 1a illustrates how the diagnostic works. As shown in Fig. 1a, the magnetic field and induced current density will diffuse through the flyer. Early in time when the current pulse is rising, one expects that the magnetic field and current density will be larger in magnitude near the side where the magnetic field originates. Figure 1b shows the simulated magnitude of the magnetic field at a time t = 3.030 microseconds as a function of radius using the arbitrary-Lagrangian-Eulerian multimaterial code, ALEGRA, developed at Sandia National Laboratories. Further details of the simulations are presented in the next section. At t = 3.030 microseconds, the two sides of the flyer are located at radii r = 1.302 cm and r = 1.339 cm. The density of lines and points shown in Fig. 1a correspond to the magnitude of the current density and magnetic field at t = 3.030 microseconds.

Figure 1a. Illustration of how the PDV load current diagnostic operates. The left-hand side denotes the surface on which the time-dependent magnetic field from the load current originates. The magnetic field (red dots) and the induced current density (black arrows) diffuse through the PDV flyer, and cause a force on the flyer. A time-dependent mechanical pressure may also be present on the left-hand side. A laser that is reflected off of the opposite surface (right-hand side) can be used to measure the flyer surface velocity due to the flyer forces through interferometry.

Figure 1b. Plot of the simulated magnitude of the magnetic field as a function of radius within the flyer at a time t = 3.030 microseconds.

The inner surface of the PDV flyer is initially at a radial location of 1.3 cm from the load and a few millimeters below the slotted return can. The PDV diagnostic was placed in the final feed section to enable the use of the slotted return can for diagnostic access in MagLIF experiments. The flyer replaces a section of the inner MITL (magnetically insulated transmission line) within the final power feed, and the PDV laser is reflected off of the outer radial surface of the final feed to measure the outer surface velocity. Figure 2 shows a schematic of the MagLIF hardware and the location of the PDV diagnostic.

Figure 2. Schematic of the MagLIF experimental hardware showing the PDV flyer at the top of the final power feed section. The dashed red arrow shows the PDV laser and the location of the PDV velocity measurement, and the green arrows show the direction of current flow.

III. Measurement of the Anomalous Pressure

The PDV diagnostic yields a measurement of the flyer surface velocity, which as shown in Fig. 2, corresponds to the outside of the final power feed section. The purple curve in Fig. 3 shows the experimental velocity of the PDV flyer surface for MagLIF shot z2850. Typical velocity uncertainties of the PDV flyer are in the range of 10 m/s. The relative timing uncertainty of the current PDV signal relative to other signals, such as B-dot measurements, is 3 ns. In order to reduce the noise of the PDV data, we often apply a temporal smoothing filter. A smoothed version of the data is shown in red. For the remainder of this paper, we will utilize the smoothed PDV data. In this experiment, the PDV flyer was 384 micrometres plus or minus 1 micrometre thick and made from aluminum-6061 alloy. In addition, Fig. 3 also shows simulated PDV flyer velocities for different specified load currents shown in Fig. 4.

We simulated the motion of the PDV flyer using ALEGRA. The green curve shows the simulated PDV flyer velocity using current computed from the average of multiple B-dot probes at different azimuthal locations. The blue curve is a load current which is computed from a Bertha circuit model of the Z machine. The timing shift between the experimental PDV velocity data and the simulated PDV velocities using both the B-dot measured current and circuit model are well-outside the uncertainties in timing. The black curve shows an inferred load current which is determined from an unfold process similar to the one described by Hess and colleagues for mechanical pressure driven VISAR unfolds. In this case, however, the magnetic pressure is used instead of the kinetic pressure. The ALEGRA simulations utilize the SESAME 3700 equation of state table, the Steinberg-Guinan-Lund material strength model, and the Lee-More-Desjarlais model for computing conductivities. Simulation runs were done in a one-dimensional cylindrical Lagrangian mode with a 2 micrometre radial resolution.

Figure 3. Plots of the PDV flyer velocities: experimental data (purple), smoothed experimental data (red), simulation using the averaged load current from B-dot measurements (green), simulation using the load current from the circuit model (blue), and simulation using the unfolded guess current (black).

Figure 4. Plots of currents: averaged load current from B-dot measurements (green), averaged insulating stack current from B-dot measurements (orange), load current from circuit model (blue), and unfolded guess current (black).

Although the inferred current does an excellent job of reproducing the early time PDV flyer velocity in Fig. 3 compared to the circuit model or load current B-dot measurements, it is highly unlikely that this current is physically relevant. The reason for this is that it is larger than the B-dot measured load current upstream in the power flow section. For example, at a time t = 2.968 microseconds, the averaged B-dot probes measure a load current of approximately 460 kA, whereas the unfold predicts a load current of approximately 2 MA. In addition to the measured and circuit simulated load currents, Fig. 4 also shows the averaged B-dot measurements for the current measured at the insulating stack at a radial location of 166 cm. At t = 2.968 microseconds, the insulating stack current is roughly 700 kA, which is also well below the unfolded current. Previous work has shown that the B-dot uncertainties measuring the load current and insulator stack current are approximately 1.1 per cent and 2.1 per cent, respectively. Therefore, the unfolded guess current is well-outside the uncertainty bounds of both measured B-dot currents. We note that the local peak in the unfolded current near t = 2.968 microseconds corresponds to the peak in the simulated PDV velocity near t = 3.027 microseconds, which has a value of approximately 50 m/s. The time delay between the applied pressure and the measured PDV signal is due to the finite propagation time of sound waves in the flyer. Since this PDV velocity at this peak is well-outside the uncertainty bounds for the diagnostic, that is 10 m/s, we believe that the PDV measurement is corresponding to a real pressure at an earlier time of approximately t = 2.968 microseconds. Moreover, since the unfolded current is well-outside the uncertainty bounds of our circuit model and the measured stack and load currents, we believe that the unfolded current, and its corresponding magnetic pressure, cannot be the cause of this pressure.

If we assume that the load current from the circuit model and/or B-dots are correct, then we are led to the conclusion that there must be an additional pressure or an additional energy deposition on the PDV flyer to account for the difference in the experimental and simulated PDV velocities. We refer to this additional pressure as the anomalous pressure. The anomalous pressure can be included in our ALEGRA simulations by simply placing an additional time-dependent mechanical pressure boundary condition on the inner surface of the flyer (outer radius of the final feed). We can infer anomalous pressure in the following way. Following a similar method as outlined in our earlier work, we can unfold the anomalous pressure from the measured PDV velocity and the velocity calculated using only the magnetic pressure due to load currents given by the averaged B-dot measurement and the circuit model.

Figure 5 shows plots of the unfolded anomalous pressures for the load current corresponding to both the averaged B-dot measurement and the circuit model. We should note two important features about Fig. 5. First, it is obvious that the anomalous pressures for both load currents are in excellent agreement near the start of the pressure at t = 2.94 microseconds, but begin to diverge at a later time closer to the peak near t = 2.968 microseconds. This behavior manifests itself in the timing shift between the B-dot current measurement and current calculated from the circuit model, which is apparent in Fig. 4, as well as the rapid ramp up of the current, shown in green and blue, near t = 2.968 microseconds. A second important feature of Fig. 5 is that the anomalous pressure continues to grow as a function of time. This behavior could be due in part to continued growth of a real mechanical pressure being applied to the flyer. However, it is also possible that the small timing uncertainties, for example of a few ns, of when the current ramp initiates, can cause a larger inferred net velocity, and hence, a larger inferred anomalous pressure. At the very least, we are confident that when the load current is small, that is for times between 2.94 and 2.968 microseconds when the B-dot measured load current is small, the anomalous pressure is real, and Fig. 5 provides a good representation of this pressure. Figure 6 shows the agreement between the experimental PDV velocity and the simulated PDV velocity for simulations using both the circuit model and B-dot load currents, when their respective anomalous pressures are included.

Figure 5. Plots of the anomalous pressures using the averaged load current from B-dot measurements (green) and the circuit model (blue).

Figure 6. Plots of the PDV flyer velocities: experimental data (red), simulation using the averaged load current from B-dot measurements and the corresponding anomalous pressure (green), simulation using the load current from the circuit model and the corresponding anomalous pressure (blue).

IV. Discussion and Conclusions

The obvious question that arises is, what are plausible explanations for the anomalous pressure shown in Fig. 5? We believe that we can rule out the possibility of radiation as the source of the anomalous pressure. From ALEGRA simulations, we know that the temperature increase due to ohmic heating (assuming room temperature at the start of the simulation) on the inside of the final feed is less than 10 K at t = 2.97 microseconds. However, in order for radiation ablation to produce the anomalous pressure at this time, one would need radiation temperatures of at least 9000 K.

Another explanation is the absorption of negatively charged particles (the outside of the power feed is part of the anode) onto the PDV flyer. The following simple one-dimensional analysis of when magnetic insulation occurs for negatively charged particles within the power feed gap may be important for understanding whether or not this could be the source of anomalous pressure. In the power feed gap, the cylindrically symmetric electric and magnetic fields due to the time-dependent applied feed voltage and the axial current are given by the radial electric field, equal to minus the feed voltage divided by the radius times the natural logarithm of the ratio of the outer to the inner radius, and the azimuthal magnetic field, equal to minus the permeability of free space times the current, divided by two pi times the radius, where we assume that the voltage and current are slowly varying in time relative to the transit time of the charged particle and the inner radius is 1.0 cm and the outer radius is 1.3 cm. Additionally, the MagLIF experiment features a uniform axial magnetic field of 10 T which is also present in the final power feed section. Using the relativistic momentum equation for a particle of charge q and mass m, in which the rate of change of momentum equals the charge times the sum of the electric field and the cross product of the velocity with the magnetic field, as well as the energy equation, in which the relativistic kinetic energy equals minus the charge times the feed voltage times the ratio of the logarithm of the radius over the inner radius to the logarithm of the outer radius over the inner radius, we can readily derive expressions for the momenta in the axial and angular directions as functions of radius: the axial momentum equals the charge times the permeability of free space times the current, divided by two pi, times the logarithm of the radius over the inner radius; and the angular momentum equals minus the charge times the axial magnetic field times the difference between the radius and the inner radius.

Inserting these expressions back into the energy equation, and setting the radial momentum to zero at the outer radius, yields the critical current for magnetic insulation across the power feed gap, which is two pi divided by the speed of light times the permeability of free space times the logarithm of the ratio of the outer to the inner radius, all multiplied by the square root of the feed voltage squared, minus twice the rest energy of the particle times the feed voltage divided by the charge, minus the square of the axial magnetic field times the square of the speed of light times the square of the gap width.

As an aside, we checked that the effect of space-charge, as discussed in the well-known work on MITL current flow by Ottinger and Schumer, would be negligible in the early current ramp time during which we detect the anomalous pressure. Hence, the implicit assumption of zero space-charge in our one-dimensional model is appropriate.

Figure 7 shows a plot of the feed voltage as a function of time which is computed within our circuit model. Figure 8 shows plots of the critical magnetic insulation currents computed for electrons and hydrogen negative ions as a function of time compared to plots of the load current from the averaged B-dot probes and the circuit model. Our one-dimensional model predicts that the critical insulation current for electrons is zero, implying that the electrons are well-insulated in the final feed section. A feed voltage of 8.5 MV would be necessary for the critical insulation current to be non-zero and overcome the insulating effect of the axial magnetic field. Meanwhile, our Bertha circuit model predicts a maximum feed voltage of only 1.16 MV at t = 3.066 microseconds, suggesting that the final feed section should be insulated for all time against electron gap crossing. We should note that even though the electrons are insulated in the final feed according to our simple one-dimensional model, three-dimensional simulations of the Z machine that include the complicated geometries of the MITLs, convolute section, and the final power feed section, indicate that electrons originating very early in time upstream may flow into the final power feed section despite being magnetized. Furthermore, electron emission occurs at the earliest times due to the strong electric fields in the final feed. Since the axial magnetic field exceeds the azimuthal magnetic field at very early times, and the axial field lines connect the cathode surface to the PDV flyer plate, it is quite possible that part of the observed anomalous pressure is due to electrons, as well.

Figure 7. Plot of the feed voltage computed using the Bertha circuit code.

Figure 8. Plots of the critical magnetic insulation current for electrons (red) and hydrogen negative ions (black) along with the load currents from the averaged B-dot measurements (green) and the circuit model (blue).

Final feed gap crossing due to negative ions could also be an explanation for the anomalous pressure. The origin of negative ions, such as hydrogen negative ions, could be due to surface contaminants or excess water vapor within the Z machine. Indeed, the presence of negative ions, as well as their time-of-flight spectra, has been extensively studied in MITLs long after electron insulation has occurred. As shown in Figure 8, the region where the critical current for hydrogen negative ions is greater than the load current indicates that the ions would not be insulated and could cross the final feed gap. The turn-on point for uninsulated hydrogen negative ions is approximately t = 2.962 microseconds and t = 2.963 microseconds for the circuit model and averaged B-dot load current, respectively. The turn-off point for uninsulated hydrogen negative ions is approximately t = 2.988 microseconds and t = 2.981 microseconds for the circuit model and averaged B-dot load current, respectively. Therefore, uninsulated hydrogen negative ions crossing the gap and hitting the PDV flyer could be a plausible explanation for part of the observed anomalous pressure.

If we assume that the anomalous pressure is due to hydrogen negative ion current loss, we can make a rough estimate of the radial current density. When energetic ions cross the feed and deposit their energy onto the aluminum anode surface, they will penetrate a certain radial depth into the aluminum. The penetration depth can be computed using the Bethe stopping power formula with appropriate corrections. For example, when hydrogen negative ions have a kinetic energy of 100 keV, such as near t = 2.97 microseconds, the penetration depth is roughly 1 micrometre into the aluminum. As an aside, the thermal conduction length in aluminum for a time scale of 10 ns, the relevant time scale for the anomalous pressure ramp, is roughly 1 micrometre as well. Hence, we can assume that the energy from 100 keV hydrogen negative ions is uniformly deposited onto the aluminum surface within a radial layer thickness of 1 micrometre. At these early times, we expect that the aluminum will be cold, and will be close to its initial density of 2.7 g per cubic centimetre. In this regime, the energy density in joules per cubic metre is roughly proportional to the total pressure in pascals, and is approximately one half of it. By equating the kinetic energy deposited by the hydrogen negative ions after crossing the feed gap with the energy in the deposition layer, we find an expression for the radial hydrogen negative ion current loss given by one half the layer thickness divided by the feed voltage, multiplied by the anomalous pressure ramp rate. Assuming that the anomalous pressure has a uniform ramp rate of 0.013 GPa/ns, as in Fig. 5 in the regime between 2.94 and 2.97 microseconds, then the inferred hydrogen negative ion current loss at 100 keV is approximately 7 A per square centimetre.

Using the computed hydrogen negative ion current loss, we can also estimate if there are enough such ions within the Z machine feed section to account for the computed current loss, and hence, anomalous pressure. At 100 keV, the ions will have a speed of roughly 4 times ten to the sixth metres per second. This yields a density of about ten to the seventeenth per cubic metre using the expression relating radial current density to density, charge and velocity. When the Z machine vacuum section, which includes the inner MITLs and the final feed, is pumped down prior to each MagLIF shot, the typical vacuum pressure is on the order of ten to the minus five torr. The computed hydrogen negative ion density at 300 K prior to the MagLIF shot would correspond to 3 times ten to the minus six torr, which is less than the background vacuum pressure. This indicates that the background vacuum pressure, which likely contains water vapor, could supply the computed current loss. In addition to the contaminants from the background vacuum pressure, surface contaminants in the feed section, such as water, could also contribute to the current loss.

Although we have only considered the magnetic insulation current for hydrogen negative ions, negative ions with larger mass-to-charge ratios will be uninsulated for even longer periods of time compared to hydrogen negative ions. For example, hydroxide negative ions have a turn-on point at t = 2.953 microseconds during the main current ramp of both models, and a turn-off point at t = 3.034 microseconds and t = 3.028 microseconds for the circuit model and averaged B-dot load current, respectively. In order to fully understand the anomalous pressure and its properties, such as its early measured turn-on time of t = 2.94 microseconds and later time-dependency, future three-dimensional simulations of electrons and ions in the final feed section will be necessary. Anomalous pressure that is due to negative ion contaminants could possibly be mitigated in future experiments using methods such as plasma cleaning of the power feed surfaces.

A similar calculation for the current loss can also be performed assuming that the loss is entirely due to electrons. As we mentioned previously, the three-dimensional geometry of our inner MITL and final feed section can enable the electrons to cross the anode-cathode gap despite the simple one-dimensional model showing that they are magnetically insulated. Since electrons are much lighter than ions, their penetration depth into aluminum for the same kinetic energy will be much greater than ions for normal incidence. In particular, we find that the penetration depth of 100 keV electrons for normal incidence onto aluminum is approximately 68 micrometres. The inferred electron current loss at 100 keV would be roughly 400 A per square centimetre. However, since most electrons will impact the aluminum anode surface at a glancing angle while conducting gyro-orbits due to the axial magnetic field, their average penetration depth will be less than 68 micrometres. Hence, the electron current loss estimate of 400 A per square centimetre at 100 keV can only be considered an upper bound. A three-dimensional kinetic simulation of electrons that includes the self-consistent time-dependent fields in the curved inner MITLs and final feed section, the effect of the axial magnetic field, as well as the motion of the PDV flyer due to the absorbed electrons, is necessary to accurately model the current loss and determine whether or not electrons could be responsible for the observed anomalous pressure.

Acknowledgment

M. Hess would like to thank M. Cuneo, R. Lemke, G. Robertson, E. Hamilton, J. Reneker, A. Maurer, J. Gluth, E. Scoglietti, and S. Payne for their helpful input into this paper. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.