Thermonuclear Fusion in an Equilibrium Z Pinch* Uri Shumlak University of Washington Zap Energy, Inc. Lawrence Livermore National Laboratory HEDS Seminar Series 23 September 2021 *This work has been funded by grants from ARPA-E, AFOSR, DOE/FES, & NNSA. Support from the Erna and Jakob Michael Visiting Professorship at the Weizmann Institute of Science and as a Faculty Scholar at the Lawrence Livermore National Laboratory is gratefully acknowledged. University of Washington | Zap Energy Flow Z-Pinch Experiment & Plasma Modeling Personnel UW Faculty & Staff: Ray Golingo, Eric Meier‡, Brian Nelson (Co-PI)‡, Uri Shumlak (PI), Anton Stepanov‡, Tobin Weber‡, Yue Zhang LLNL Staff: Christopher Cooper, Drew Higginson (Co-PI), Harry McLean (PI), James Mitrani, Andréa Schmidt, Kurt Tummel UW Graduate Students: Elliot Claveau*, Daniel Crews, Iman Datta*, Bennett Diamond, Eleanor Forbes*, Brian Henderson, Andrew Ho, Aria Johansen, Aqil Khairi, Tim Lloyd, Hannah Meek‡*, Collin Parsons*, Chris Provencher*, Sina Taheri*, Yu Takagaki, Whitney Thomas UW Undergraduate Students: David Arnold*, Zack Draper‡*, James Ferrese*, Ben Fetters, Harry Furey-Soper, Spencer Goldstein, Michelle Graebner*, Sara Hines, Daniela Nankova, André Nicolov*, Daniel Roberts, Hannah Schnelz*, Katrina Teo*, Grace Zoppi * recently graduated; ‡ currently at Zap Energy Presentation Outline - Z-pinch equilibrium scales to plasma parameters reaching fusion conditions in a compact device - Sheared-flow stabilization (SFS) applied to the Z pinch – theoretical findings and experimental investigation - Experimental data from ZaP (Z Pinch) device indicating stable behavior coincident with sheared plasma flow - Stabilizing effect works for higher performing plasmas, as demonstrated with the ZaP-HD device, supported by computational investigations - FuZE (Fusion Z-pinch Experiment) explores the SFS Z pinch for fusion – extended experimental capability and numerical simulations - Detailed experimental profile measurements → a tightly compressed plasma with high density, magnetic field, and temperature - Sustained neutron production from thermonuclear fusion - Looking forward: Scaling the SFS Z pinch to a compact core for fusion energy and to a fusion space thruster The Z-pinch equilibrium has no applied axial fields as described by: (B_theta / (mu_0 * r)) * d/dr(r * B_theta) + d/dr(n * (T_i + T_e)) = 0 Increasing the current and the resulting azimuthal magnetic field compresses the plasma to fusion conditions in a compact device – no magnetic field coils. Assuming adiabatic compression and radial force balance gives scaling relations for higher current: a_2 / a_1 = sqrt((n_1 / n_2) * (N_2 / N_1)) = (I_1 / I_2)^(1/(gamma - 1)) * (N_2 / N_1)^(gamma / (2 * (gamma - 1))) n_2 / n_1 = (T_2 / T_1)^(1/(gamma - 1)) = (I_2 / I_1)^(2/(gamma - 1)) * (N_1 / N_2)^(1 / (gamma - 1)) These results are predicated on having a stable plasma. Z-pinch research predates nuclear fusion understanding - 1790: Earliest 'Z pinch' research by Martinus van Marum - 1905: Observation of crushed lightning rod by Pollock & Barraclough - 1907: 'Pinch phenomenon' in liquid conductor by Northrup - 1934: Theoretical model of plasma Z pinch by Bennett - 1950: Z pinch was Project Sherwood Jim Tuck's preferred approach to achieve controlled fusion - 1957: Theory and experiments demonstrated virulent instabilities, m = 0, 1 - 1998: Performance of Z pinches using frozen deuterium fibers was severely limited by these instabilities (Lebedev et al., PoP 1998) Flow Z pinch observations and concept - 1969: Long-lived plasma column observed during coaxial plasma gun research at LANL (Newton, Marshall, & Morse) - 1973: High-density plasma beam formed with continuous-flow Z pinch as a NASA project (Cheng & Wang) - 1977: Conceptual fusion reactor based on the flow Z pinch (Hartman et al.) - 1990: Development of coaxial quasi-steady-state plasma accelerators in the USSR produces 4 m long stable plasma stream (Ananin et al., Morozov) Stabilizing the Z pinch has proven challenging While the equilibrium is attractive, the Z pinch is classically unstable to MHD modes: m = 0 sausage and m = 1 kink. Stability can be provided by limiting the pressure gradient, introducing an axial magnetic field, or installing a close-fitting conducting wall. These approaches are incompatible with magnetically confining a high-temperature, high-density plasma. Work from a collaboration with Charles Hartman at LLNL suggests the m = 1 mode can be stabilized with a sheared axial flow: dv_z / dr >= 0.1 * k * V_A though the value is sensitive to axis treatment. Flow shear is observed to stabilize m = 0 modes as well. Schematic illustrates flow Z-pinch formation: - t_1: Neutral gas is injected through puff valves into the annulus of a coaxial plasma accelerator. - t_2: Neutral gas expands before a capacitor bank is discharged across the electrodes. - t_3: The plasma accelerates down the coaxial accelerator due to generated currents and magnetic fields. Schematic illustrates flow Z-pinch formation: - t_4: The plasma continues down the accelerator in a snow-plow manner. - t_5: At the end of the accelerator the plasma assembles into a Z-pinch configuration. - t_6: Inertia and gun currents maintain the plasma flow and supply until the accelerator plasma empties or the capacitor current vanishes. ZaP Flow Z-Pinch Experiment The ZaP Flow Z-Pinch project investigated the concept of using flows to stabilize an otherwise unstable plasma configuration. Components diagrammed include: Acceleration Region, Assembly Region, Inner Electrode, Outer Electrode, Vacuum Vessel, Neutral Gas Injection Plane, z=0 line, and Electrode End Wall (1 m assembly region). Diagnostics measure plasma flow and stability ZaP diagnostics measure equilibrium plasma parameters, plasma flow, and magnetic mode activity (stability). - Surface-mounted magnetic field probes -> Determine magnetic topology, magnetic fluctuations, and plasma stability - Fast framing camera with optical filters -> Qualitative measure of plasma structure - Four-chord HeNe interferometry -> Measure plasma density profile - 0.5 m imaging spectroscopy with 20 input chords and an intensified CCD detector -> Doppler shift for plasma flow profile, Doppler broadening for ion temperature, Zeeman splitting for magnetic fields, Stark broadening for density - Thomson scattering system using a ruby laser and a Hibshman spectrometer -> Measure electron temperature - Digital holographic interferometer -> Measure two-dimensional plasma density structure Magnetic fluctuations diminish after pinch forms Fluctuations of the magnetic modes are significantly reduced for ≈ 37 μs after pinch forms. Mode activity increases again after this quiescent period. Experimental data suggests the quiescent length is limited by plasma supplied from the accelerator. Plot: B_m / B_0 for m = 1, 2, 3 and I_p (kA) versus Time (μs), showing the quiescent period from ~25 to 62 μs. Optical images show a stationary plasma pinch Visible emission images are obtained of the pinch, every 200 ns, through a 5 cm hole with an Imacon fast-framing camera. - Images show a stationary plasma pinch during the quiescent period. Note hollow structure (Pulse 40115035). - Gross kink & sausage instabilities appear at the end of the quiescent period (Pulse 40127041). Multichord spectrometer provides velocity profiles A 20-chord imaging spectrometer is connected to an intensified CCD detector to measure the Doppler shifts of impurity emission lines. The chords are spaced out of plane across the assembly region. Large flow velocity exists during the quiescent period Large Doppler blue-shift of the C-III impurity line (229.7 nm) during the quiescent period (Pulse 726025, Shumlak et al., PRL 2001). Flow velocity decreases at end of the quiescent period Small Doppler shift of the C-III impurity line (229.7 nm) at the end of the quiescent period (Pulse 726027). Flow profile is correlated to plasma stability - Plasma assembly (tau < 0), the axial plasma velocity is high and uniform: dv_z / dr ≈ 0 - 4 x 10^6 s^-1 - Start of quiescent period (tau = 0), the velocity profile is high at the plasma edge and lower at the axis: dv_z / dr ≈ 7 - 12 x 10^6 s^-1 - End of quiescent period (tau = 1), the plasma velocity profile is low and uniform: dv_z / dr ≈ 0 - 6 x 10^6 s^-1 For these plasma parameters, the theoretical growth time ≈ 20 ns. Linear theory specifies a shear threshold of ≈ 5 x 10^6 s^-1. ZaP & ZaP-HD investigated sheared-flow stabilization ZaP (1999 – 2009) / ZaP-HD (2009 – 2015) Flow Z-Pinch projects investigated using sheared flows to stabilize an unstable plasma: - generated sheared-flow-stabilized (SFS) Z-pinch plasmas 50 – 126 cm long - observed the coincidence of plasma stability and a sheared-flow state - characterized the plasma equilibrium and demonstrated radial force balance Simulations show formation of high-pressure plasma - Time-dependent simulations of ZaP-HD indicate the formation of a high-pressure plasma that extends through the Z-pinch assembly region and has a high degree of axial uniformity. - Whole-device simulations are performed with the Mach2 resistive MHD code using realistic circuit solvers from two capacitor banks (t = 30 μs, t = 60 μs, t = 90 μs). - Multi-fluid plasma model using WARPX are also applied to study the onset of drift instabilities and turbulence. Magnetic fluctuations diminish after pinch forms Fluctuations of the magnetic modes along the entire length of the Z-pinch plasma are significantly reduced for 40 – 80 μs after pinch forms. Mode activity increases again after this quiescent period. The quiescent period is coincident with a sheared axial flow, v'_z != 0. The velocity profiles agree with calculations of Braginskii ion viscosity for spatially dependent magnetization. Viscous damping time is consistent with the stable Z-pinch lengths observed. Holographic interferometry measures density structure Digital holographic interferometer uses a Nd:YAG laser and digital SLR to provide high-resolution measurements of the plasma density structures, which can be used to characterize the equilibrium. The phase change gives the line-integrated density: Delta phi = (lambda * e^2) / (4 * pi * c^2 * epsilon_0 * m_e) * integral(n_e(x) dx) Holographic interferometry measures density structure The digital hologram is recorded on the camera and then numerically reconstructed to give the complex wave field, which yields the phase change. After unwrapping, a 2D map of phase change results. The line-integrated density is Abel inverted to yield electron number density profiles n_e(r) along the axial extent of the hologram. The data plot retains the displacement of the pinch axis relative to the geometric axis. Notice a peak density of ≈ 2 x 10^17 cm^-3 and a pinch radius of ≈ 3 mm along the axial extent of the image. Holographic interferometry measures density structure Profiles at 3 axial locations (z = 7.7 cm, 8.0 cm, 8.3 cm) are extracted and analyzed. Holographic interferometry measures density structure Radial electron density profiles n_e(r) at z = 7.7 cm, z = 8.0 cm, and z = 8.3 cm showing peak density > 2 x 10^17 cm^-3 and radial falloff out to 0.8 cm (Pulse 160524021). B_theta(r) and T_e(r) are computed from the density profiles Assuming a uniform drift speed, v_d = v_i - v_e, and matching the experimentally measured plasma current, magnetic field profiles B_theta(r) are computed as: B_theta(r) = (mu_0 * e * v_d / r) * integral_0^r (n_e * r' dr') Notice a peak magnetic field of 8.5 T at r ≈ 0.3 cm (z = 8.0 cm). B_theta(r) and T_e(r) are computed from the density profiles Integrating the radial force balance equation gives the electron temperature profile T_e(r): T_e(r) = (1 / (2 * mu_0 * n_e)) * integral_0^r ( (B_theta / r') * d/dr'(r' * B_theta) dr' ) assuming T_e = T_i, consistent with the expected collisionality. Notice a peak electron temperature of ≈ 1 keV. Independent spectroscopic measurements indicate T_i ≈ 800 eV. FuZE Project advances the SFS Z pinch for fusion Fusion Z-pinch Experiment, FuZE, expands on the success of ZaP and ZaP-HD: - more robust device that achieves fusion - concerted effort on kinetic and multi-fluid modeling - additional personnel/capabilities by LLNL & UW FuZE benefits from detailed numerical simulations Nonlinear fluid & kinetic simulations using Mach2 (MHD), WARPXM (2-fluid), and LSP (PIC) to: (a) study sheared-flow stabilization (b) design experimental details (c) model whole device (d) predict neutron yield Results show plasma stabilization with sufficient flow shear. Simulations provide insight to gas injection dynamics. Multi-fluid models are solved in balance law form Multi-fluid plasma models capture physical phenomena beyond single fluid models. The 5N-moment multi-fluid model is solved using the WARPXM code using a high-order discontinuous-Galerkin finite-element method: 1) Mass conservation: ∂rho_s / ∂t + grad · (rho_s * u_s) = 0 2) Momentum conservation: ∂(rho_s * u_s) / ∂t + grad · (rho_s * u_s * u_s + p_s * I + Pi_s) = (rho_s * q_s / m_s) * (E + u_s x B) - sum_{s*} R_{s,s*} 3) Energy conservation: ∂epsilon_s / ∂t + grad · ((epsilon_s + p_s) * I + Pi_s) · u_s + h_s = (rho_s * q_s / m_s) * u_s · E + sum_{s*} Q_{s,s*} where the total energy is epsilon_s = (1 / (gamma - 1)) * p_s + (1/2) * rho_s * u_s^2, for each species s. The fluids are coupled to each other and to the electromagnetic fields through Maxwell's equations and collisional transport terms. Multi-fluid (5N-moment) plasma model simulations using the WARPXM code demonstrate that sheared flows (v_z^a = 0.5 V_A vs v_z^a = 0.0 V_A) effectively mitigate macro (fluid) instabilities over time steps t = 0, 5, 10, 15, 20, 25, 30, and 35 tau_A. While the static case rapidly disrupts, the sheared case remains confined. Multi-fluid models study sheared-flow stabilization Defining pinch mass and thermal energy as integral_0^a n_i dx^3 and integral_0^a (p_i + p_e) dx^3 provides a measure of confinement. - Confinement rapidly degrades in the static Z pinch as a result of instabilities. - Flow shear enhances confinement. Even when instabilities appear in the sheared-flow Z pinch, confinement is better maintained. - Flow direction matters, with increased stabilization for Omega · B < 0, similar to recent Kelvin-Helmholtz Instability (KHI) studies. Kinetic models also study sheared-flow stabilization - Fully kinetic simulations using the LSP PIC code suggest that sheared flows are effective in mitigating micro (kinetic) instabilities, in addition to macro (fluid) instabilities (comparison of v_z^a = 0.00 V_A, 0.25 V_A, 0.50 V_A across 2 to 8 tau_A). - Gyrokinetic simulations of the SFS Z pinch with COGENT show a stabilizing effect with increased flow shear, but not complete linear stabilization. Whole-device modeling provides insight to experiment Time-dependent simulations of the FuZE device indicate the dynamics of the Z-pinch plasma formation, heating, and fusion neutron production from an extended line source for approximately 10 μs (shown at t = 20, 30, and 40 μs for ion density n_D, ion temperature T_D, and neutron production rate dn_n/dt). FuZE produces stable, high-performance plasmas - Stable behavior observed during quiescent period. Pinch current rises above 200 kA until it equals total plasma current. - Doppler broadened spectra during quiescent period (t = 25 μs) show T_i ≈ 1.0 – 1.5 keV from He-like C V emission; presence suggests high T_e and well confined electrons. - Digital holographic interferograms indicate pinched plasma density during quiescent period (t = 25 μs): a ≈ 3 mm, B_theta ≈ 10 T, T_e + T_i ≈ 1.8 keV. Fusion neutrons from FuZE deuterium plasmas - When gas mixtures containing deuterium, D2 – H2, are used to make FuZE plasmas, sustained fusion neutron production (≈ 8 μs) is detected coincident with quiescent period and large pinch current. - Measurements indicate a steady neutron emission to within statistical expectations consistent with a thermonuclear process (Pulse 180222035, neutron yield ~ 1.3 x 10^5). Fusion neutrons scale with deuterium concentration - Neutron counts disappear for plasmas with no deuterium (0% D2 / 100% H2). - Neutron count dependence scales quadratically with D2 concentration (tested at 0%, 10%, 20% D2), agreeing with expected scaling with n_D^2. Fusion neutron yield agrees with theoretical expectation Expected neutron yield can be calculated from experimental measurements: - ion temperature (gives thermonuclear reaction rate parameter) - radial profile of the ion density - neutron emission time - plasma volume Y_n = integral ( 1/2 * n_D^2 * <sigma v> * tau dx^3 ) Measured neutron yield is 10^5, agrees with theoretical thermonuclear process with T_i ≈ 1.2 keV. Spectroscopy indicated T_i ≈ 1.0 – 1.5 keV. Neutron energy isotropy excludes D+ beams >8 keV - Neutron energy spectra are characterized from proton recoil signals of upstream and downstream plastic scintillator detectors. - Maximum neutron energy relation: E_n,max = (1/8) * [ sqrt(E_b) + sqrt(3*(E_b + 2*E_f)) ]^2 - FuZE neutron energy spectra are statistically identical on both detectors. - Measurement uncertainty of 140 keV limits the maximum deuteron beam energy to below 7.5 keV, indicating fusion originates from a thermonuclear process. Spatially-resolved measurements indicate line source - Determining the neutron emission volume is accomplished by making measurements from multiple detectors at varying axial positions. - The data are fit to an analytical calculation for a uniform source of arbitrary length and center. - Least squares fit to the data gives a neutron emission geometry that is an extended line source: 32.8 cm length, 16.4 cm centroid. Neutron yield increases with pinch current - Adiabatic scaling relations predict a neutron yield that strongly depends on Z-pinch current: Y_n ∝ I^11. - Two-temperature MHD simulations of the SFS Z pinch support a similar scaling. - Experimental campaign on the FuZE SFS Z pinch to increase pinch current shows a yield consistent with this expectation. SFS Z pinch achieves fusion breakeven at 650 kA Starting with experimentally achieved plasma parameters at 50 kA, increasing the current with a fixed linear density rapidly reaches breakeven conditions. Fusion core remains compact even at high Q. Large instantaneous power avails modest duty cycle operation. Sample instantaneous conditions at 2 MA: - I_p = 2 MA - T = 32 keV - L = 75 cm - a = 120 μm - Q = 29 - P_f = 3.1 TW SFS Z-pinch reactor conceptual design is underway Bechtel, WSI, and Dec. Sys. SFS Z-Pinch Study w/ 3 Cores: - Liquid LiPb serves multiple functions: outer electrode, heat transfer fluid, biological shield, tritium-breeding blanket. - SFS Z-pinch reactor conceptual design: pulsed at 10 Hz, 190 MWth each core; multiple cores share tritium-handling facility. SFS Z pinch achieves D-3He fusion Q=1 at 1.9 MA The D-3He reaction results in ionized fusion products, which are directed axially for thrust. Lower reactivity requires higher currents, but can still achieve high Q for a low-alpha (kg/kW) fusion space thruster with high specific impulse ≈ 10^6 s and high thrust ≈ 10^6 N. Sample instantaneous conditions: - I_p = 4 MA - Q = 10 - L = 30 cm - P_f = 13 TW Fuel mixtures diluted with non-fusing gases provide throttle-ability. The device operates as a fusion-augmented MPD thruster. SFS Z-Pinch Fusion Space Propulsion System Operating a 4 MA SFS Z-pinch fusion thruster at 1 Hz with a 0.001% duty cycle could deliver a 2,100 kg payload to a 125 AU orbit (interstellar space) in 9 years with a 70,000 kg spacecraft. Advancing the propulsion system requires: - Maintaining plasma stability and control at higher pinch currents - Developing electrode designs that withstand plasma contact - Integrating systems (spacecraft modeling and mission architecture) Open Questions and Future Research - Is compression in the sheared-flow-stabilized Z pinch an adiabatic process? Do shocks occur during the Z-pinch assembly process? - Under what conditions will shear-flow stabilization breakdown? - Will compressing the plasma to even smaller radii increase the drift speed faster than the thermal speed – potentially leading to drift or kinetic instabilities? - What barriers may prevent the Z pinch from being scaled to even higher plasma parameters? - Can flow shear replace or augment magnetic shear in other plasma confinement configurations? Is this already happening? - Does flow shear play a role in astrophysical objects? Summary & Conclusions - The sheared-flow-stabilized Z pinch produces equilibrium plasmas that exhibit gross stability during an extended quiescent period. - The quiescent period is coincident with a sheared plasma flow that is consistent with sheared-flow-stabilization theory. - Combining fluid & kinetic numerical simulations with well-diagnosed experiments – ZaP, ZaP-HD, & FuZE – has demonstrated scaling of the SFS Z pinch to high energy density plasmas. - Experimental measurements show an axially uniform, compressed plasma with high parameters: a ≈ 3 mm, n_e ≈ 10^17 cm^-3, B_theta ≈ 10 T, and T_e ≈ T_i ≈ 1 keV. - FuZE demonstrates sustained neutron production during the quiescent period through a thermonuclear fusion process. - SFS Z pinch has no magnetic field coils resulting in a compact device for terrestrial fusion energy and would translate to a low-alpha fusion space thruster.