Z-pinch fusion

Nuclear fusion holds the promise of essentially limitless energy with more manageable wastes than some existing energy sources. Fusion also offers a means to enable interstellar space travel,^1,2 which is not presently realizable. Controlled fusion, with reactions sustained over long durations, has been stymied by fast growing plasma instabilities. A viable approach to controlled fusion continues to be pursued through the study of different confinement approaches—each with unique advantages and different levels of scientific maturity. Recent breakthroughs have renewed interest in the Z pinch as a viable candidate for fusion because it has achieved performance comparable to more mature concepts that have benefited from decades of dedicated research and development. A fusion device based on the Z pinch^3–6 would be particularly attractive since it would be geometrically simple, inherently compact, and of relatively low-cost.

Devising a configuration capable of confining a high-temperature high-density plasma for sufficiently long durations has proven difficult. Successful approaches include large toroidal configurations,^8,9 such as the tokamak and stellarator, which use magnetic coil sets to produce intense external fields that confine and stabilize the plasma. Associated with the large size of these toroidal configurations are high cost, complexity, and long development time. These considerations motivate a search for more compact approaches.

The absence of magnetic field coils greatly simplifies the design of a fusion device by eliminating the need for shielding the coils from the energetic fusion neutrons. As a consequence, a Z-pinch fusion device, if realizable, would be more cost-effective and compact than fusion devices that require coils. Furthermore, since the radius of the plasma-facing surface is not constrained (as it would be if coils and shielding were required), the inner radius of the blanket surrounding the cylindrical fusion volume (effectively $r w$⁠) can be set to optimize the incident heat and neutron flux densities.

Based on the Z-pinch equilibrium of Eq. (3), increasing the axial plasma current compresses the plasma, yielding higher densities and temperatures. The temperature increase is described by the Bennett relation,^3, $μ 0 I 2=8π ( 1 + Z ) N i k B ⟨ T ⟩$⁠, where $I$ is the pinch current, $μ 0$ is the permeability of free space, $Z$ is the ionization state, $k B$ is the Boltzmann constant, $⟨ T ⟩$ is the average temperature, assuming $T i= T e$⁠, and $N i$ is the linear ion number density, defined as $N i= ∫ a n i(r)2πrdr$⁠. Adiabatic scaling relations^6,10 for a “sharp pinch”¹¹ indicate the dependence of plasma temperature $T$⁠, density $n$⁠, and radius $a$ on current. Figure 2 illustrates the application of the scaling relations to the plasma parameters measured from the ZaP Z-pinch device.¹² The curves provide the expected plasma parameters as the pinch current is varied and $N i$ is held constant. A “sharp pinch” provides a convenient model in that current is assumed to be localized at the pinch radius $a$⁠, such that it confines a plasma with uniform density and temperature. Based on the adiabatic scaling relations, $T∝ I 2$⁠, $n∝ I 2 / ( γ − 1 )$⁠, and $a∝ I − 1 / ( γ − 1 )$⁠, which for an adiabatic index $γ=5/3$ gives $n∝ I 3$ and $a∝ I − 3 / 2$⁠. From these relations, the fusion reaction rate, given by the integral in Eq. (2), is seen to increase rapidly with current, $E f u s i o n∝ I 3⟨σv⟩$⁠, where the reactivity can be a strong function of temperature, e.g., proportional to $T 4$ for D-T fusion in the range of 1–10 keV. Using this approximation for reactivity, the relations indicate fusion reaction rate scales strongly with current, namely, $E f u s i o n∝ I 11$⁠, and suggest a path to reach fusion conditions in the Z pinch.

Z-pinch history predates understanding of thermonuclear fusion. Though a description of the historical development of the Z pinch will not be presented in this brief Perspective article, a few highlights help contextualize the pursuit of Z-pinch fusion. For additional information, Refs. 14–17 provide excellent reviews of Z-pinch research.

While the Z pinch is one of the first concepts to be investigated as a potential fusion device, the various embodiments were found to be susceptible to instabilities that severely limited the plasma lifetime. For example, a dense plasma focus^18–22 facilitates fusion reactions through a Z-pinch effect, but the reactions primarily rely on instabilities²³ to produce large axial electric fields that accelerate ions and produce short-lived beam-target fusion.

Achieving thermonuclear fusion requires taming the instabilities^11,24–26 that manifest in the Z pinch. The most virulent unstable modes are the MHD $m=0$ “sausage” and $m=1$ “kink,”where $m$ is the azimuthal mode number. Circumventing these instabilities can be accomplished through several means.

Embedding an axial magnetic field^24,27–29 in the plasma provides an approach to stabilize the Z pinch. However, this approach has drawbacks. The axial magnetic field modifies the equilibrium force balance given by Eq. (3), reduces the achievable beta and confinement efficiency, and increases thermal conduction losses to the electrodes that supply the axial current. Plasma contacting the electrodes is depicted in Fig. 1.

Another approach to Z-pinch instabilities relies on slowing their growth and rapidly forming and compressing the plasma before instabilities can develop to the point of disruption. Annular cylindrical solid liners,^30–32 wire arrays,^33–35 and gas-puffs^36–39 have been imploded using pulsed-power drivers to assemble transient Z-pinch plasmas on axis. After stagnating on axis, the plasma disassembles due to instabilities or expansion with timescales of the order of the acoustic propagation time across the pinch radius. Advances in pulsed-power technology have increased the plasma parameters that can be achieved from annular implosions. Without complete stabilization, however, these approaches produce short-lived dynamic Z-pinch plasmas.

A conducting wall placed close to the surface of the plasma can stabilize the Z pinch,^40,41 but the wall must be within 20% of the pinch radius, $r w/a<1.2$⁠. Maintaining a high-temperature plasma in such close proximity to a solid wall would likely be challenging and makes this approach unlikely to be viable as a fusion Z-pinch concept.

The combination of an embedded axial magnetic field in a plasma that is then compressed by an imploding cylindrical liner^42–44 or plasma⁴⁵ has also been pursued. This configuration of magneto-inertial fusion (MIF)⁴⁶ uses the axial magnetic field to reduce radial thermal conduction to the liner while relying on the imploding liner or plasma to provide stability and to add inertia. Inertia delays disassembly after peak compression. The magnetized liner inertial fusion concept (MagLIF)^42–44,47 at Sandia National Laboratories is an example of an MIF concept that forms a dynamic Z pinch. Neutron measurements⁴⁴ from neutron time of flight (NTOF) detectors support a thermonuclear origin of the fusion reactions. By combining an axial magnetic field embedded in a preheated plasma with a cylindrical liner implosion, the MagLIF concept is able to lower the requirements on implosion velocity and compression to achieve ignition. Indeed, MagLIF has reported impressive fusion performance.

While able to generate fusion reactions that increase with current, any radial equilibrium that may develop in dynamic Z-pinch plasmas is transitory.^48,49 The strong scaling of fusion performance with current described in Sec. I A relies on an assumption of radial equilibrium. The dynamic or “fast” Z pinch is the subject of continuing research and will not be the focus of this article; a review of dynamic Z pinches is provided in Ref. 15.

The equilibrium Z pinch maintains a radial force balance for durations much longer than other plasma timescales, e.g., acoustic propagation time across the pinch radius. Plasma conditions evolve slowly through a series of quasi-equilibrium states, for example, in response to an increasing plasma current. If the Z pinch can be stabilized such that stationary equilibria can be produced and maintained, the demands on pulsed-power and rapid plasma assembly would be greatly relaxed, and the strong scaling of fusion performance with current may be realized.

Recent theoretical and experimental research has shown that MHD instabilities in the Z pinch can be eliminated using sheared flows. Specifically, sheared axial flows have been investigated with theory and simulations^50–52 as a means to stabilize the Z pinch in a way that does not alter the equilibrium described by Eq. (3) and can be indefinitely sustained, in theory. The flow shear required to provide stability can be achieved with subsonic flows.

Observations from early experiments that formed plasmas with coaxial accelerators^53–56 reported constricted plasma columns with axial flows that exhibited unexplained enhanced stability properties. While measurements of plasma flow profiles have not been published, the observed stability may be explained by the presence of sheared flows.

Sheared-flow stabilization of the Z pinch was studied in the ZaP device^57,58 at the University of Washington. Closely coupled experimental and computational investigations characterized the plasma equilibrium and stability properties.^12,57–60 Z-pinch plasmas were produced that persisted much longer than expected for a static plasma, i.e., thousands of radial acoustic or Alfvén wave propagation times. Sheared-flow stabilization was effective for Z-pinch lengths of 50, 100, and 126 cm with no evidence of wall stabilization.⁴¹ Experimental results¹² indicated a stationary equilibrium with the pressure gradient balanced by magnetic forces, as in Eq. (3), and measurements of $T e≈ T i$ supporting ion-electron equilibration, as expected from theoretical calculations of equilibrium conditions.

Advances in computational capabilities and diagnostic techniques have improved our scientific understanding and have facilitated the progress and renaissance that dynamic and equilibrium Z pinches are experiencing. Enabled by high-order-accurate mathematical algorithms and by high performance computers, numerical simulations based on high-fidelity plasma models^61–66 have provided new insights. Simulations capture the interactions among the constituent plasma species and electromagnetic fields, shed new light on the complex dynamics, and accelerate understanding of Z pinches, including the effects of drift modes,⁶⁰ neutron production,^20,21 effects of end losses,⁴³ plasma formation dynamics,⁶ and predictions of plasma behavior at fusion reactor conditions.⁶⁷ Given the difficulty of measuring spatiotemporal properties of plasmas with high energy densities, numerical simulations have provided a holistic view of plasma evolution through validation with experimental measurements.^6,21,43,67 Validated simulations facilitate informed extrapolations and enable detailed exploration of experimental design modifications.

Characterizing experimental plasmas requires accurate measurements of their properties, which can change rapidly, e.g., over the course of 10 ns, and have short gradient lengths, e.g., 100  $μ$m. Advances in nonperturbative measurements have enabled finer spatiotemporal characterization of plasmas. Spatially and temporally resolved emission spectroscopy has revealed temperature and velocity profiles from diagnostics of modest costs.⁶⁸ Collection optics⁶⁹ have been designed to improve the accuracy of spatial deconvolution analysis.⁷⁰ Spectroscopic analysis has also provided internal measurements of density⁷¹ and magnetic field^72,73 by exploiting the Stark and Zeeman effects. Advances in Thomson scattering techniques^74,75 have allowed detailed local measurements of the electron and ion properties. The availability of low-cost, high-quality scientific lasers and high resolution digital cameras has led to the development of digital holographic interferometry^76,77 to resolve two-dimensional structures in the electron density. These diagnostic developments have improved our ability to characterize Z-pinch plasmas through measurements that can be independently corroborated. This in turn has provided metrics for validation with complementary high-fidelity numerical simulations.

New efforts funded by the U.S. Advanced Research Projects Agency-Energy (ARPA-E) are currently being initiated that will develop simulation and diagnostic capabilities for Z pinch, MIF, and other intermediate density fusion approaches. These capabilities are expected to improve scientific understanding and to accelerate progress toward a viable compact fusion concept.

The sheared-flow-stabilized (SFS) Z-pinch concept seeks to produce and sustain an equilibrium Z pinch. Supported by the simulations and diagnostics described in Sec. II, foundational experimental and computational investigations of the SFS Z pinch have led to an improved understanding of the plasma behavior and stability and have facilitated increasing the plasma’s density, temperature, and fusion reaction rate. Recent results from SFS Z pinches have demonstrated the potential of the concept to scale to fusion conditions in a quasi-steady-state configuration.

The ZaP-HD device, which is a derivative of ZaP,¹² was able to maintain long-lived stable Z-pinch equilibria⁶ even as plasma current was increased, compressing the plasma radius, $a=0.3$ cm, with corresponding large values of magnetic field, $B θ=8.5$ T, and plasma temperature, $T e≈ T i=1$ keV. Measurements of the axial and azimuthal topology of the magnetic field during the approximately 50  $μ$s stable quiescent period indicated an axisymmetric and axially uniform Z-pinch plasma. The ZaP-HD device used a triaxial electrode design to drive a larger pinch current than was possible in the coaxial ZaP device. The larger pinch current compressed the plasma without degrading stability and led to the increased plasma parameters reported in Ref. 6, as expected from the adiabatic scaling relations.

Motivated by the encouraging results from the ZaP and ZaP-HD devices, a collaborative project between the University of Washington and Lawrence Livermore National Laboratory was initiated to explore the fusion performance of the SFS Z pinch. The project has benefited from a methodical exploration that has coupled fluid and kinetic computational investigations with experimental investigations that include detailed diagnostic measurements to characterize the plasma equilibrium and stability properties. The Fusion Z-pinch Experiment (FuZE) was designed using coaxial electrodes in a geometry similar to the ZaP device and represents the next progression in the series of SFS Z-pinch experiments. A machine drawing of the FuZE device is shown in Fig. 3 and identifies the major components. Compared to the ZaP-HD power supply, which produced approximately 200 kA, the FuZE power supply is capable of driving larger plasma currents, over 600 kA into a matched load.

A primary objective of the FuZE device is to explore fusion reactions in the SFS Z pinch. Similar to results from ZaP-HD, the FuZE device has demonstrated elevated plasma parameters, $a=0.3$ cm, $n e≈ 10 17$ cm $− 3$⁠, and $T e≈ T i≈1$ keV, that result from increasing the pinch current. Operating with deuterium mixtures produces fusion reactions, which are quantified using calibrated plastic scintillator neutron detectors⁷⁸ operating in pulse counting mode. Steady neutron emissions¹³ are sustained for approximately 10  $μ$s during the quiescent period when the pinch current is sufficient to compress the plasma to fusion conditions.

Figure 4 shows the general behavior of the SFS Z pinch, as quantified by currents, magnetic field fluctuation levels, and neutron signals. The plasma pulse initiates when the plasma current $I p l a s m a$ begins to rise at 2  $μ$s. The plasma forms in the acceleration region and moves toward the pinch assembly region that is depicted in Fig. 3. Magnetic field topology and fluctuation levels are measured using 60 probes, which are mounted on the inner surface of the outer electrode and are distributed axially and azimuthally throughout the assembly region. The field probe locations can be seen in Fig. 3. The plasma assembles on axis beginning at 15  $μ$s and exhibits large magnetic fluctuations along the length of the Z pinch. Magnetic fluctuations are measured from azimuthally distributed magnetic probes in the outer electrode and are Fourier analyzed to obtain the normalized $m=1$ fluctuation level, $B 1/ B 0$⁠. The red trace in Fig. 4 averages the magnetic fluctuation levels at three axial locations in the assembly region and the gray shading represents the standard deviation among the signals from the three locations. The fluctuation levels diminish after 20  $μ$s and a quiescent period ensues. During this period, a sheared-flow state is observed, and the Z-pinch plasma is stable and well-centered along the geometric axis. Currents in the accelerator, indicated in the figure by the difference between the plasma current $I p l a s m a$ and pinch current $I p i n c h$⁠, supply plasma to the pinch and maintain the sheared-flow state. When the accelerator current vanishes, the sheared-flow state degenerates, leading to pinch instabilities. Neutrons emanate from the Z pinch during the quiescent period when the pinch current is sufficiently large to compress the plasma to fusion conditions. The neutron signal disappears when the pinch current decreases below approximately 150 kA.

Experimental measurements indicate that plasma instabilities are absent during the sustained neutron signal. The neutron emission volume was determined by analyzing counts from multiple neutron detectors located at different axial locations along the 50 cm Z-pinch plasma column. The measurements revealed a uniform line source of neutrons with an axial extent comparable to the length of the Z pinch.⁷⁸ Furthermore, varying the deuterium concentration of the injected working gas resulted in neutron yields that were proportional to the square of the concentration, as shown in Fig. 5. The measured neutron yields agree with calculations based on independently measured plasma properties and temperature-dependent fusion reactivities⁷ for a thermonuclear process.

In addition to the linear SFS theory,⁵⁰ nonlinear fluid simulations¹² have demonstrated the effectiveness of sheared flows to stabilize the Z pinch and have provided insight into the plasma acceleration, assembly, and fusion performance of SFS Z-pinch experiments. A two-temperature MHD model is solved with Mach2⁷⁹ with a plasma current computed from a SPICE calculation of the FuZE power supply providing a peak plasma current of 530 kA. Simulation results of the FuZE device operating with 100% deuterium are shown in Fig. 6. Fusion neutron yield estimates are computed by incorporating reaction models⁷ into the fluid simulations, with the burn duration and resulting yield computed self-consistently from the simulations based on the time-dependent plasma dynamics. Using the same current waveform but scaled by a constant multiplier, fluid simulations are used to explore the fusion neutron yield at higher plasma currents. As shown in Fig. 7, the neutron yield calculated from these simulations is observed to scale with the pinch current approximately as $I 11$ beyond a minimum current value. The agreement with the simple adiabatic scaling relations described in Sec. I A is remarkable since the simulations incorporate many experimental details that are not included in the scaling relations, such as the plasma acceleration and the pinch assembly.

Numerical simulations of the SFS Z pinch using the LSP PIC code suggest that sheared flows are effective in mitigating micro (kinetic) instabilities,⁶⁷ in addition to macro (fluid) instabilities. The kinetic simulations were performed using typical experimental parameters from the FuZE device to predict that sheared subsonic flows should stabilize the Z-pinch plasma. Simulations were also performed to extrapolate the SFS Z pinch to plasma parameters that might be expected in a fusion reactor core, and sheared flows were shown to effectively stabilize even these high performing plasmas. The general agreement between kinetic and fluid simulations supports the prospect of scaling SFS Z-pinch experiments to higher currents and plasma parameters that reach reactor conditions.

The encouraging experimental results from multiple devices and the improved understanding gained through validated numerical simulations suggest that sheared-flow stabilization may provide an effective means of mitigating instabilities. In fact, the SFS Z pinch may offer the best path to achieving sustained thermonuclear fusion in the Z pinch, as perhaps envisioned by Tuck during Project Sherwood.⁸⁰ 

The SFS Z pinch has produced intriguing scientific results, which motivates continued development of the concept. As shown in Fig. 2, Z-pinch plasma parameters elevate rapidly with increasing pinch current. At a fundamental level, achieving sustained fusion reactions is the overarching goal, which, based on existing understanding, can be accomplished by increasing the pinch current. Aligned with this goal are several research topics and open questions that define the future directions for the SFS Z pinch.

Maintaining stability is critical for the equilibrium Z pinch. Sheared flows have mitigated instabilities for existing SFS Z-pinch experiments,^{6,12,13,57,59} but the question remains if sheared flows will continue to be effective to stabilize Z pinches with higher fusion performance, e.g., $Q>1$⁠, where $Q≡ P f u s i o n/ P i n p u t$ is the fusion power gain. Fusion $Q$ scales with Z-pinch plasma parameters and is plotted in Fig. 2 for D-T fusion using a temperature-dependent reactivity^81,82 valid up to 25 keV. The input power for an SFS Z pinch includes the power required to compress the plasma, to generate the axial flow, and to compensate for radiative losses.¹⁰ While numerical simulations suggest that sheared-flow stabilization is robust, close coupling between numerical modeling and experimental studies is needed to successfully increase pinch current. There are many topics that need further investigation, such as extending the plasma quiescent period to increase the fusion burn duration. Questions related to plasma stability at higher plasma parameters need to be addressed since stability will likely set the upper bound on fusion performance of the SFS Z pinch. Determining the upper bound on fusion $Q$ will dictate the utility of the concept for fusion.

Plasma parameters have been experimentally observed to increase with pinch current, approximately in accordance with the adiabatic scaling relations. Compared to the scaling of Fig. 2, which extrapolates from measurements on the 50 kA ZaP device, the FuZE parameters at 200 kA, which are also plotted in the figure, trend toward higher temperatures and lower densities. The discrepancy may be explained by differences in the machine details, such as power supply or neutral-gas injection, or it may be explained by intrinsic physical effects such as shock formation in the Z-pinch assembly process, as suggested by numerical simulations. The validity of assuming adiabaticity during plasma compression can be tested using flexible power supplies that can adjust the current amplitude and waveform in the same experimental device. Appropriate diagnostics with the required spatiotemporal resolution can then identify and characterize shock structures internal to the Z-pinch plasma. The experimental measurements can be combined with numerical simulations using physical models with sufficient fidelity to capture the multi-fluid and kinetic physics of high-temperature plasmas. High-fidelity plasma models are needed, for example, to describe plasma-neutral interactions at the pinch periphery and finite gyroradius effects close to the magnetic field null on axis.

Sustained neutron production from an extended emission volume measured on the FuZE device seems to indicate a thermonuclear fusion process, as described in Sec. III A. However, a direct measure of the neutron energy spectrum would provide a definitive determination. The sustained neutron signals complicate the application of conventional NTOF approaches to measure neutron energy, but using calibrated plastic scintillator neutron detectors may prove to be a successful technique. Pulse heights from plastic scintillator detectors are a function of recoil proton energies, which in turn are a function of incident neutron energies.⁸³ Improving the counting statistics by operating at higher fusion reaction rates would allow time-resolved measurements of the neutron emission volume and the neutron energy spectrum, which could better identify any beam-target contribution to the fusion reactions.

Research into viable electrode designs will be an important future topic. The SFS Z-pinch concept brings a fusion-grade Z-pinch plasma into direct contact with electrodes, which must supply the pinch current. Plasma–electrode interactions are likely important in present experiments and increasingly important as the plasma energy density increases. The impact of plasma–electrode interactions has not been sufficiently investigated at high plasma temperatures and current densities. Electrode erosion can degrade plasma temperature by introducing impurities⁸⁴ that enhance radiation losses. Metallic electrodes that were used on the ZaP and ZaP-HD devices experienced substantial erosion at higher plasma parameters and have been replaced with graphite electrodes on FuZE. Other electrode solutions may be required as the pinch current is increased. Possible solutions include liquid metal electrodes⁸⁵ or nanostructured surfaces.⁸⁶ Including plasma–surface interactions into numerical simulations remains challenging. Nevertheless, understanding how boundaries affect plasma behavior is important and would benefit from numerically tractable models that capture the essence of plasma–electrode interactions.

The Z pinch was envisioned from the beginning of magnetic confinement research as a simple approach to fusion. Recent scientific progress in the development of methods to overcome instabilities has been facilitated by advances in computational capabilities and diagnostic techniques combined with dedicated experimental investigations. One of the promising concepts to emerge is based on the equilibrium Z pinch. By addressing the critical stability issue using sheared flows, the SFS Z pinch has demonstrated that an equilibrium can be formed and maintained for extended durations. Furthermore, sustained fusion reactions have been produced by maintaining a stable Z-pinch plasma and compressing it to high densities and temperatures by increasing the pinch current. The experimental results generally agree with expectations from numerical simulations and sheared-flow-stabilization theory, though many questions remain unanswered about the behavior of the SFS Z-pinch plasma as its parameters are further increased.

New simulation and experimental capabilities are being developed and are likely to bring additional insights as the performance limits of Z-pinch fusion are explored. The strong scaling of plasma parameters with current offers the potential to rapidly increase fusion performance. If the plasma remains stable as current is increased, the intrinsic advantages of the Z pinch will provide a simplified path to a compact fusion device.

The author would like to thank H. S. McLean, B. A. Nelson, and G. V. Vogman for stimulating discussions. The author gratefully acknowledges support of the Erna and Jakob Michael Visiting Professorship at the Weizmann Institute of Science, support as a Faculty Scholar at the Lawrence Livermore National Laboratory, and support from Zap Energy Inc. The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy (DOE) under Award No. DE-AR-0000571, the National Nuclear Security Administration under Grant No. DE-NA0001860, the U.S. Department of Energy (DOE) under Grant No. DE-FG02-04ER54756, and the Air Force Office of Scientific Research under Grant No. FA9550-15-1-0271.

The data that support the findings of this study are available within the article.