Helical plasma striations in liners in the presence of an external axial magnetic field

Recently, the MagLIF magneto-inertial confinement fusion concept¹ has attracted interest as an approach to achieving inertial confinement fusion, and preliminary results are encouraging.² In these experiments, a cylindrical current carrying tube, referred to as a liner, containing a preheated plasma is compressed by imploding the liner in a time of the order of 100 ns. An axial magnetic field is applied to provide thermal insulation between the hot plasma and the imploding liner. In experiments without an applied field conducted on the 20 MA Z machine³ at Sandia National Laboratories, an interesting observation in soft X-ray radiographs was the formation of horizontal striations perpendicular to the current flow in the imploding liner. Peterson et al. proposed that these striations were formed early in the current pulse due to an electrothermal instability initiated in the condensed matter states, which later acted as a seed for the Magneto-Rayleigh-Taylor instability.⁴ Similar striations have also been observed on the 1 MA Cornell Beam Research Accelerator (COBRA) with a gated extreme ultraviolet (XUV) imaging diagnostic.⁵ 

In order to mitigate these instabilities, thick dielectric coatings were applied to the surface of the liner in experiments on Z reported by Peterson et al.⁶ Furthermore, applying an external axial magnetic field was shown to tilt these striations, giving them a helical pattern when viewed with soft X-ray radiography.^7,8 Figure 1 shows sketches of how the striations in the liner appear from a side view. A thin-shell model analysis was later applied to give a theoretical picture of the evolution of helical perturbations.⁹ 

In the experiments reported here, we have confirmed the formation of striations with and without an axial magnetic field. The principal diagnostic, a time gated XUV self-emission imaging system, was different from that used on the Z-machine, and the experiments reported here have a factor of 20 less peak current. Our experiments, which used non-imploding liners, were carried out on the 1 MA COBRA machine¹⁰ at Cornell University with 100 ns pulse rise time. A typical COBRA current pulse for all experiments related to our investigation is shown as a solid line in Fig. 2. The figure also shows two $Ḃ$ traces, which will be discussed in Section III A. The two experimental arrangements that were used to generate the axial magnetic fields were significantly different. The first produced a time-varying and spatially non-uniform field by using twisted return current wires. The second produced a steady-state and spatially uniform field by using a Helmholtz coil. These will be described in detail in Section II. While we can confirm the helical morphology qualitatively, we cannot say, anything definitive about the mechanism that generates the striations.

The primary diagnostic system for monitoring the striations produced in the liner was a pair of XUV time gated pinhole cameras positioned side-on and one opposite the other. While Awe's radiographs on Z looked through both sides of the liner in the same image, our images see only one side at a time as the liner is optically thick in the observed wavelength range. Each camera viewed the liner through four pinholes having a diameter of either 50, 100, or 200 μm, resulting in different energy sensitivities as discussed later in Section III B. Each pinhole cast an image on a different quadrant of a microchannel plate. All four quadrant frames had an exposure time of either 5 or 10 ns, with exposures two through four delayed by 10, 20, and 30 ns, respectively, with respect to the first one. We refer to these devices as quad-cams. As there was no observable difference in the data from the 5 and 10 ns exposures, we will only be presenting data that used 10 ns exposures. In all of our experiments, these images were taken significantly after the pulse rise time, as shown in Fig. 2.

Magnetic fields were measured using micro $Ḃ$ probes. Single loop probes, illustrated in Fig. 3(b), were placed outside the liner near its surface, as shown in Fig. 3(a) for the twisted return current wire experiments. Probes could be oriented to measure either axial or azimuthal magnetic fields. A probe was also placed on axis inside the liner, as shown in Fig. 3(a), to measure the axial field that penetrates through the liner.

Two designs were used for the liners. The first was the one-piece Al liner shown in Fig. 4. This liner was machined down to its final profile from an Al rod. The wall thickness of the liner tube shown in Fig. 4(a) was 50 μm. The bottom part of the piece was threaded so that it could be screwed into the brass cathode base, as shown in Fig. 4(a). The length of the tube, measured between anode and cathode, was 1 cm, and the inner diameter of the liners was 4 mm.

The second design consisted of a 4 μm thick Al foil wrapped into a cylinder with one or two turns. This was similar to the design used in the liner instability studies previously done on COBRA.¹¹ The cylindrical foil was placed inside a tube holder by sliding it through a hole in the anode into the bottom holder, which was the cathode, as shown in Fig. 3. Contact was made by the liner on the holder walls. The liners in this design were also 1 cm in length and were 4, 8, or 16 mm in inner diameter.

Different diameters enabled a variation in the azimuthal magnetic field, $Bθ$⁠, on the surface of the liner produced by the 1 MA current pulse flowing through it. The azimuthal field is given by

where r is the radius of the liner, μ₀ is the free space magnetic permeability, and I is the current.

For experiments that did not require an externally applied axial magnetic field, straight return current posts were used, as shown in Fig. 4(b). The experimental results without a field shown below used liners of the first design, though we have also done experiments using the second design, with similar results. In experiments with applied B_z, it was important to reduce the self-generated $Bθ$ from the liner in order to have a detectable striation tilt. Hence, for the twisted return current wires experiments, we used 8 mm diameter liners. However, 50 μm thick liners with that diameter did not emit in the XUV wavelength. They were too massive and heated insufficiently. Thus, the experimental results from the twisted wires shown here were of the second design. They have an inner diameter of 8 mm and are double wrapped.

To produce an axial magnetic field in the configuration shown in Fig. 3, four 2.1 mm diameter copper wires placed inside four equally spaced holes, each a distance 12.7 mm from the center, carried the current between the upper and lower anode plates. The upper anode section was rotated 90° by hand either clockwise or counterclockwise, and was at a height of 2.5 cm with respect to the lower section. Solder was applied at the connections between the copper wires and the brass holders. The part of the anode that touched the liner consisted of an Al tube that could slide freely inside the top brass holder, as shown in Fig. 3. This way, the configuration could be adjusted so that the liner was always 1 cm long before vacuum pump-down (between cathode and anode).

The possible disadvantages of this arrangement were the following. First, the rotation and concentricity of the Cu return current wires were determined only by eye. A more robust method for the rotation could be devised. However, the non-uniformity of the axial magnetic field at the liner due to a possible deviation from concentricity is much less significant than that due to there being only four twisted return current wires. This non-uniformity is visible in our data and will be discussed further in the results section. Second, the $Ḃ$ probes used to measure the B_z field were positioned at varying locations around the liner. As the magnetic field from the return current wires is inherently non-uniform, $BzBθ$ is location specific. However, as we will be looking only at average ratios of the field components and comparing them to average striation tilt angles, the $Ḃ$ measurements are still meaningful for qualitative comparison.

For the second experimental arrangement, a Helmholtz coil having a 150 μs rise time was used, which is considered steady-state on the time scale of the COBRA pulse. The spatial uniformity of the coil, with the load hardware included, was within 5% in the area of interest. This coil produced 0.5 T to 1.5 T axial magnetic fields depending on the driving capacitor bank voltage. A photograph of the coil with a loaded liner, along with a $Ḃ$ probe on axis in the liner, is shown in Fig. 5(a).

Most liners used in the Helmholtz coil experiments were single wrapped of the second design, with an overlap region of 3–5 mm, and are shown in the sketch of Fig. 5(b). Some of these experiments used liners that were single wrapped on one side and double wrapped on the other side. We will refer to such liners as 1.5 turn liners. Experiments with the latter liners investigated what effects doubling the current path area might have on the striations.

The liners were 16 mm in diameter in all of the coil experiments. As mentioned before, the reason for the larger diameter was to reduce the azimuthal magnetic field produced by the machine current running through the liner. For a B_z of 1.5 T, the expected maximum azimuthal magnetic field strength at such a diameter is about 25 T, giving a minimum $BzBθ$ ratio of 0.06. If the helical pitch is determined by this magnetic field ratio, then the minimum pitch angle would be 3.4°.

At this point, let us define the average angle of striations as the projected average, measured via averaging straight lines drawn between the leftmost and rightmost ends of given striations in the images. Note that the measured average angles from our XUV images will be different from the real angles because we are looking at a 2-D representation of a 3-D cylindrical system. For a given striation, the vertical height change from one side to the other in both geometric systems is the same, but the horizontal distance changes from 2r to $π$r between these two systems. This changes the angle by a factor of $2π$⁠, as shown in the following equation:

where θ is the angle in radians, y is the vertical delta, r is the radius of the liner, and a small angle approximation has been used.

Using this, we note that the 3.4° will actually correspond to a 5.3° measured angle, which we should be able to detect in our XUV images. To avoid confusion, from now on all angles given will be the 3D angles that occur in reality as that is what is of interest. We also note that local angles will have their own transformation factors as $2π$ is not appropriate when dealing with short sections of a striation. While most angles presented will be averages, we will explicitly point out when we talk about local angles.

Without an axial magnetic field, horizontal striations with deviation of up to ± 2° were observed in XUV self-emission, as shown in Fig. 6. Note that all deviation values presented in this paper reflect the maximum change in angle and not a standard deviation. Also, all XUV images shown are “negatives,” meaning that darker regions correspond to stronger emission and vice-versa. The no-field images are useful as controls for experiments with axial fields. The wavelength of the striations when comparing no applied field experiments with the alternative did not noticeably change: they all averaged 600–750 μm. While our liners do not implode, we believe that this wavelength is dominated by the Magneto-Rayleigh-Taylor instability in the ablated plasma, as the latter moves slightly in response to the radial forces.

Sample experimental results when the axial field was produced by the twisted return current wires are shown in Fig. 7. The difference between pulse C3317, corresponding to Figs. 7(a) and 7(b), and pulse C3318, corresponding to Figs. 7(c) and 7(d), is the direction of the axial magnetic field determined by the return current wire twist. C3318 is turned clockwise when viewed from above, similar to Fig. 3, producing an axially downward magnetic field at the liner's surface. The opposite is true for experiment C3317.

With four return current wires, the uniformity of the axial magnetic field at the liner radius, both azimuthally and axially, was poor. As a result, $BzBθ$ in the areas of the liner closer to the wires was stronger than elsewhere. However, a meaningful average pitch angle could still be measured from the images for both twist orientations: both cases averaged 8 ± 1° in their respective directions. Considering the sources of error, this matches reasonably well with analysis using a combination of basic calculations of the expected $Bθ$ field and data from $Ḃ$ probes, as will be shown.

Fig. 3 shows the hardware configuration for the data used in the following discussion. According to the probe external to the liner, the maximum axial field strength measured roughly midway between anode and cathode axially, at the midpoint between two return current wires azimuthally, and 2.5 ± 0.5 mm radially outside the liner was 5.6 T (Fig. 2). As the return current wires were at an angle of approximately 45°, it is reasonable to assume that a similar strength azimuthal magnetic field due to a single wire was present. The current at this time is close to 1 MA. Note that the “single wire assumption” will introduce some error as the combination of four wires will reduce the measured $Bθ$ whereas B_z contributions will add. To get the proper fields with the load hardware, a 3-dimensional simulation could be run. However, such a simulation would not be including magnetic field non-uniformities from current flow and plasma ablation dynamics, discussed in Section III. Hence, the following discussion will be semi-quantitative instead as that is useful in illustrating all the factors that need to be taken into account in the twisted wire configuration.

In both twist orientation cases, the azimuthal field created by the wire currents will add to the $Bθ$ generated by the current flowing through the liner. According to Eq. (1), which assumes azimuthal symmetry, the machine current flowing through the liner generates 31 T azimuthal magnetic field at the radius of the probe (4 + 2.5 = 6.5 mm) when the “axial” $Ḃ$ probe reads 5.6 T. Using a superposition of azimuthal fields generated from the current running through the liner and the return current posts, the total azimuthal field at the location of the probe becomes 36.6 T. This $BzBθ$ ratio gives an angle of roughly 7°. However, this ratio is valid only near the probe and not at the liner surface.

We do expect a discrepancy between this $BzBθ$ and that measured from the XUV images. In general, this discrepancy is likely to have four sources: imprecise angle measurement due to spatial and temporal limitation of XUV camera resolution, the “single wire assumption,” probe locations 2.5 mm exterior to the liner, leading to the axial and azimuthal field ratio being different at those locations compared to the ratio at the liner wall, and error in the location and orientation of the probes when placing them. If the striation tilt angle corresponds to $BzBθ$⁠, then the last of these four sources is likely the most important and quite significant. It is reasonable to assume that the orientation of the probe relative to the z-axis may have an offset of up to 10°. This roughly corresponds to measuring 15% of the $Bθ$ in the B_z measurement, giving an error bar of $±ΔBzBz=±60%$⁠.

In addition to the tilted striations, a wide column of plasma can be seen in Fig. 7, stretching from the cathode to the anode at an angle of about 30° from the vertical. The source of that hot plasma is yet to be determined for certain, but it is likely due to the load hardware. It is not due to the presence of a seam in the liner as the quad-cams were positioned 180° from one another and both quad-cams saw such a plasma column.

Another measurement of interest in these experiments was the axial magnetic field inside the liner. An example is shown by the dashed-dotted curve in Fig. 2 for pulse C3321. First, there is a time delay between the axial field outside the liner and that measured inside. This is to be expected as the field has to diffuse through the liner. However, when the field does appear inside the liner, it increases in two bursts rather than a smooth increase, as seen outside the liner. This suggests that the magnetic field does not penetrate through the liner via a simple field diffusion process. One possible explanation for the bursts could be magnetic flux compression via radially converging plasma ablated off the inside surface of the liner. It is possible that a wave reaches the inside surface of the liner and carries plasma from there radially inwards. The presence of converging shock waves inside thick liners has been studied recently by Burdiak et al.¹² In comparison with those experiments, the main differences in our experimental arrangement were that our liners were thin (8 μm, which was significantly less than the pulse penetration depth), made out of double wrapped foil, and had no gas fill.

We also observed a tilt in striations in single wrapped foil experiments using the Helmholtz coil. Referring to Fig. 8, the orientation of the average tilt angle matches in orientation with the twisted wires experiments, i.e., left-handed twist for an upwards axial field and right-handed twist for a downwards applied field. The average angle shows some correlation with the applied axial field strength. In pulse C3626, in which a 0.5 T downward field was applied, the average is 3° whereas in pulse C3624, in which a 1.5 T field was applied, the average is 8°. Multiple measurements of the average angle in an image gave a maximum variation between 0.75° and 1.25° from the averages given above. Note however that local variations in tilt are not reflected in the average value and can be significant. This variation is present both azimuthally and axially along the liner, as shown in Fig. 8. For example, the angles of the highlighted striations in C3624 shown in Fig. 8(b) and determined using local transformation factors change from 6° to 18°.

C3623 stands out as an exception. That experiment had a 1.5 T applied field upwards producing the striations shown in Fig. 8(c). The average angle is 4.7 ± 1°. However, a small section of the striations are tilted in the opposite direction from the average angle orientation. Since the magnetic field was applied only in one direction, this suggests that either the striations observed are not determined purely by the magnetic field ratio, or that there was azimuthal non-uniformity in the current distribution in the liner such that the twisting of the current channels produced oppositely directed B_z fields. Bott-Suzuki et al. showed that vacuum gaps at contacts can cause non-uniform current flow with an azimuthal magnetic field variation of up to 50% that persists along the liner.¹³ While we did not introduce vacuum gaps on purpose, there were bound to be contact gaps at the electrodes in our wrapped liner hardware. Hence, we think it is possible that there was a non-uniform azimuthal and axial current distribution, at least in that particular liner, that persisted throughout the pulse. Such non-uniformity could also explain why there is a large angle variation in the tilts at a given time and why we see larger angle variations in the Helmholtz experiments, where the field is uniform, than in the twisted return current wire experiments. We know from optical self-emission images that the larger diameter liners used in the former experiments created more non-uniform plasma from contact arcs, which would lead to more pronounced current non-uniformity.

We can determine the photon energy scale of our observed striations using the combined results of our 50, 100, and 200 μm diameter pinholes. Only the 100 μm pinholes gave clear images. It may be that our 200 μm pinholes, which did not show striations, did not have a high enough resolution and our 50 μm pinholes required too high an energy range for the liners to reach easily. The second limitation in pinhole cameras comes about from diffraction, which we can estimate using

where y is the radius of the Airy disk minimum at the detector, λ is the wavelength, D is the pinhole diameter, and d is the distance from the pinhole to the detector. If y exceeds the size of the image, then wave diffraction through the pinhole is assumed to be too great to observe a liner image. Using this estimate, the striations shown in this paper are limited to the energy range of 20 to 40 eV.

In all our experiments, the tilt angle stayed constant in time, within the angle variations described above. This in itself does not confirm that the striation angle is a locked in behaviour starting at a certain time or disprove that the striations continuously follow the changing magnetic field lines as the images were taken at nearly constant current, hence nearly constant $Bθ$⁠. However, the angles measured do not correspond to the angles one would expect from the $BzBθ$ at the times of measurement, as shown in Fig. 9. The measured angles are higher, corresponding to a higher $BzBθ$ and suggesting that these angles were locked in earlier in time, when the $Bθ$ was lower.

Negative XUV images of 1.5 turn liner experiments are shown in Fig. 10. Noting that the current pulse penetration depth was significantly larger than the 4 or 8 μm liner thicknesses, we assume that the current flowed through the whole liner, not just an outside layer of it. The wavelengths of the striations on the single and double wrapped sides of the liner are the same. Moreover, we can see that while there is a hot vertical column of plasma near the seam, the striations can be individually traced from one side of the plasma column to the other. The average angle changes across the seam. The angles highlighted in red in Fig. 10, which were determined using local transformation factors, change approximately from 12° to 8° in C3618 and from 11° to 14° in C3621. These changes are closer to upper limits; for the striations that we could trace through the seam with confidence, we measured changes between 1° and 4°. No matter the value of the change, the striations on the single wrapped side have an average stronger tilt than on the double wrapped side in both pulses. One possible explanation could be the following. First, since the current was flowing throughout the whole thickness of the liner, the effective mass and current area on the double wrap side were double. Initially, when the liner was cold and its resistivity low, the liner would be mainly driven by an inductive voltage, meaning that the total current would be evenly distributed azimuthally. The liner would then ohmically heat up, the single wrapped side heating faster due to its smaller mass. At the melting point, the resistive impedance would become comparable to the inductive impedance, meaning that more current would progressively shift to the double wrapped side. From this point on, the resistive impedance of the liner would be dominant and the total resistance on the double wrapped side would be less (true even when their resistivities are equal), leading to significantly more current flowing on the double wrapped side. This would lead to a higher $Bθ$ around that section compared to the single wrapped section. This in turn would lead to an angle in the double wrapped section that is smaller than in the single wrapped section, which is observed. In reality there would probably also be complications added from the electrode contact effects discussed earlier.

We have observed a helical striation pattern in liners in the presence of an axial magnetic field from XUV self-emission. In the case of the twisted wires experiments, the pitch of the tilt matches within experimental error, admittedly large, with the axial to azimuthal magnetic field ratio inferred from $Ḃ$ probes. Furthermore, we observed magnetic field inside the liner that cannot be explained via simple field diffusion. In the case of the Helmholtz coil experiments, we observed a tilt in striations that stayed constant in time. Even though the variation in tilt angle was significant in a given image, there was correlation between average tilt angle and applied field strength. Also, the combined data from single wrapped liners and 1.5 turn liners suggests current non-uniformity exists in single thickness liners.

This research was supported by the National Nuclear Security Administration Stewardship Sciences Academic Programs under Department of Energy Cooperative Agreement No. DE-NA0001836, as well as by the National Science Foundation Grant No. PHY-1102471. The authors would like to thank Pierre Gourdain, Harry Wilhelm, Todd Blanchard, Daniel Hawkes, and William Potter for their excellent technical support. We would also like to thank the very careful anonymous reviewer of our manuscript whose comments and suggestions substantially improved the final manuscript.