The Orbitron: A crossed-field device for co-confinement of high energy ions and electrons

Various classes of ion traps have been studied and characterized with respect to their confinement time τ and space charge limited density. In particular, Penning–Malmberg traps,^1–3 Paul traps,⁴ and orbital ion traps^5–7 have all demonstrated long confinement times with τ ≳ 1 s. However, these traps are typically limited by space charge effects to low ion densities. For a 4.5 T magnetic field, the space charge limited density (Brillouin limit) for Be⁺ of n ≈ 6 × 10⁹ cm⁻³ has been achieved in a Penning trap.⁸ Confinement schemes have been explored with Penning traps to exceed this density limit;^9,10 however, research has been limited.

Here, we describe a new approach for reaching ion densities above the space charge limit by co-confining electrons in an orbital ion trap called an Orbitron.¹¹ Orbital ion traps have long been studied for applications to neutralization of electron space charge¹² and mass spectrometry.¹³ Commercial mass spectrometer orbital ion traps operate with confined ion kinetic energies 1–5 keV, and negligible center-of-mass collisional energies E_com due to the use of circularized orbits.¹⁴ In the Orbitron, E_com of 10–60 keV are achieved by scaling the cathode voltage to values on the order of −100 kV and by inserting ions into elliptical orbits. At these high ion energies, reasonable fusion rates are achievable if the ion density is scaled above the ion space charge limit. Therefore, initial investigations of the Orbitron are focused on reaching densities above the ion space charge limit through the co-confinement of electrons with a relatively weak magnetic field. Key challenges are the impacts of Coulomb collisions and particle transport on τ and E_com, plasma stability, and achieving a sufficient ion loading rate.

The rest of this manuscript is structured as follows. In Sec. II, we discuss the principle of operation of the Orbitron device. Sections III and IV describe the experimental apparatus and diagnostics for the Orbitron deuterium–deuterium fusion experiments. In Sec. V, Particle-in-Cell (PIC) simulations are presented, which show the mitigation of the ion space charge limit through co-confinement of electrons. Finally, Sec. VI enumerates areas of investigation underway for assessing the Orbitron’s ability to achieve high fusion reaction rates.

The Orbitron is a crossed-field (E × B) device. As in an orbital ion trap,⁷ ions with sufficient azimuthal (θ) velocity are confined in orbits and accelerated by an electrostatic potential between an outer anode and inner cathode arranged in an annular configuration, see Fig. 1. While orbiting in θ around the cathode, ions simultaneously oscillate back and forth along the z-axis due to the electrostatic pinch formed by the geometry of the electrodes. The cathode is held at a high magnitude negative potential to confine ions in elliptical orbits and accelerate them to center-of-mass energies having high fusion reactivity. To support long ion confinement times, the pressure of neutral particles is held in an ultra-high vacuum (UHV) regime (⁠$<10−8$ Torr) to reduce particle scattering and charge exchange. Simulations (see Sec. V) predict ion density limitations due to space charge are mitigated by co-confining electrons with a longitudinal magnetic field as in a magnetron.¹⁵ 

Magnetron-like electron confinement in the Orbitron is achieved with non-zero values of G, $Bz0$⁠, and $Br0$⁠, see Sec. II A 2. The effect of the magnetic field on ion and electron orbits is related to the cathode voltage and the spatial dimensions of the trap, which are characterized by the maximum radii R_c and R_a of the cathode and anode, respectively (R_c < R_a). For our investigations, $O(Ra)=O(zmax)∼10$ cm, where z_max is the maximum extent of the trap in the ± z directions. For cathode voltages near −100 kV, we focus on |B⁰| ∼ 0.05–0.1 T so that fuel ions (mass $∼2$ amu) are weakly magnetized (Larmor radius ≫ R_a − R_c), while electrons are strongly magnetized (Larmor radius ≪ R_a − R_c).

An ion orbit is illustrated in Fig. 1. With B⁰ = 0, ion confinement has been intensively investigated for both quadro-logarithmic (G = 0)^7,16 and non-quadro-logarithmic (G ≠ 0)^5,12–14,17 potentials. Ions loaded into this azimuthally symmetric potential with sufficient angular momentum are confined radially through the conservation of angular momentum and energy. Axial confinement is provided by the potential well in z formed by the pinch in the geometry of the electrodes at each end of the device.

Adding the magnetic field described in Eq. (2) will perturb the ion dynamics. We explore how the presence of a uniform axially magnetic field changes the ion confinement region in velocity space in a simplified r–θ approximation of Eqs. (1) and (2) with G = 0. For a given radius, we use the conserved energy E = mv · v/2 + qΦ⁰(r, 0) and canonical angular momentum $pθ=mrvθ+r2qBz0(r,0)/2$ to numerically solve for the minimum and maximum v_θ that are confined in the device without hitting the cathode and anode, respectively. This calculation assumes no initial radial velocity. Figure 2 shows the results of this calculation. The gray-shaded region represents velocities confined in the device in the absence of the magnetic field with the lower and upper bounds representing ions colliding with the cathode and anode, respectively. The red-shaded region shows the small perturbation in the confinement region with the addition of a 0.1 T axial magnetic field. The slight perturbation of ion orbital behavior with B⁰ is a trade-off for enabling electron co-confinement described in Sec. II A 2.

Ions confined to elliptical orbits will cross paths with one another within the Orbitron and collide. With high E_com in a collisional event, fusion fuel ions, such as deuterium, will have a probability of fusing. We examine the range of collisional energies in a reduced r–θ model considering only the difference in radial velocity Δv_r. The ranges of ion confinement in phase space depicted in Fig. 2 are associated with a corresponding range of orbital ellipticities. Using the approach in Ref. 18, the ellipticity of an orbit in a logarithmic potential is quantified with the unitless parameter β_O.

For circular orbits, β_O = 0.844. Larger β_O values indicate more elliptical orbits. Figure 3 illustrates the range of E_com vs β_O for deuterium ions with the same apoapsis in an Orbitron at three different cathode voltages. One of the benefits of higher cathode voltages is that less elliptical orbits are required for collisional energies at the deuterium–tritium and deuterium–He³ fusion cross section peaks. Depending on cathode voltage, the range of E_com covers the high fusion reactivity range for deuterium–deuterium, deuterium–tritium, and deuterium–He³ fusion.¹⁹ When high reactivity is coupled with high ion densities, which is the focus of our research, meaningful fusion reaction rates are achievable.

Electron confinement in the Orbitron is magnetron confinement in the r-θ plane coupled with magnetic mirror confinement in z. The magnetic mirror confinement in z is augmented electrostatically by the protrusions on each end of the cathode in the ±z directions, forming an electrostatically plugged magnetic bottle. The E × B field arrangement that supports these mechanisms is illustrated in Fig. 4. Electrons are pushed away from the cathode by the strong electric field shown by the black arrows (∇Φ⁰ ≡ −E). The magnetic field, which provides both radial and axial confinement of the electrons, is shown with the red field lines and gray dashed magnetic field contours.

Two confined electron trajectories are shown in Fig. 4 (blue and magenta curves). Without the magnetic field, electrons would be pushed radially outward from the cathode toward the anode. With the magnetic field as shown, the electrons are radially confined. Electrons undergo E × B orbits in the r–θ plane around the cathode similar to the electron confinement in a magnetron.¹⁵ For illustration purposes, these trajectories were simulated for the geometry and fields shown in Fig. 4 using IBSimu,²⁰ a computer simulation package for ion optics with capabilities for tracking particles in electric and magnetic fields.

As shown in Fig. 4, electrons in the Orbitron encounter increasing |B⁰| as they take excursions in ±z. This increasing magnetic field creates a magnetic mirror, which helps provide axial confinement of the electrons.^21,22 Figure 5 illustrates this axial confinement for the two test particles shown in Fig. 4. These test electrons experience an acceleration from the increasing magnetic field that pushes them back toward z = 0.

The z component of acceleration due to the electric field, E_z, is also shown in Fig. 5. Some of the z electric force on electrons is a necessary by-product of the shaping of the electric field for the purpose of ion acceleration and is, thus, a compromise in electron behavior in order to support ion confinement. Separately, some of the electric force is due to the protrusions on each end of the cathode. These protrusions shape the electric field in order to augment the z-confinement of the electrons. The presence or lack of these protrusions, and their geometry, is an additional design parameter that allows us to trade off the degree to which the magnetic mirror z-confinement of electrons is augmented by electric force vs the effect of the protrusions on ion orbits.

As the charged particle density increases, significant deviations from these single particle models will occur due to collective effects of the plasma, particle energy losses, and particles leaving the confinement region. For example, with non-zero particle conduction loss to the trap walls, the electric field will become perturbed due to sheath formation effects near the walls of the device.²² Sheath perturbation effects will depend on rates of electron and ion flux to the walls, which are affected by the degree to which the particles are confined. In addition to wall sheath effects, our numerical studies described in Sec. V indicate the presence of collective ion-electron space-charge coupling effects in the trap, which also cause field perturbations.

Figure 6(a) illustrates simulated Orbitron electric field perturbations in a high-density scenario (n ≈ 10¹¹ cm⁻³) with co-confined electrons and ions. The presence of this nearly charge neutral plasma alters the confining potential (red dashed curves) from the ideal vacuum potential (black curves). The −100 and 0 kV contour lines correspond to the walls of the cathode and anode, respectively. Figure 6(b) illustrates the magnetic field perturbations in the same high-density scenario. The magnetized electrons rotate azimuthally at a higher velocity than the co-rotating ions. This induces an azimuthal current that alters the magnetic field. Our simulations (see Sec. V) indicate that ion and electron densities above the ion space charge limit are attained in the presence of these field perturbations; however, the particle trajectories are modified from the single particle model discussed previously.

The core of the Orbitron system for deuterium–deuterium (D–D) fusion is illustrated in Fig. 7. A D⁺ or $D2+$ ion beam accelerated across a voltage drop ∼10% of the magnitude of the cathode voltage is injected through a hole in one of the outer anodes. The cathode voltage is nominally −100 kV but is reduced depending on the experiment, and the beam energy is adjusted accordingly. Ions and electrons are confined as described in Sec. II. This section details the design and subsystems of the prototype −100 kV Orbitron, including the vacuum chamber, our high voltage capabilities, the formation of the magnetic field, and the source of ions and electrons. Preliminary experiments on this −100 kV device are ongoing. We also describe future upgrades to the design that will enable higher cathode voltages and stronger magnetic fields.

High neutral background pressures (⁠$>10−8$ Torr) have been shown to reduce the lifetime of pure ion and pure electron plasmas confined at low-energies (⁠$<10$ eV) in Penning–Malmberg traps¹ and reduce the coherence time of higher energy (≲5 keV) ions in orbital ion traps.²³ These traps rely on azimuthal symmetry for confinement in a similar way as the Orbitron. Collisions of the trapped particles with the neutral background exert a torque on the particles causing transport toward the conducting walls and particle loss. With pressures below 10⁻⁸ Torr, particle confinement times greater than a second have been observed in Penning–Malmberg traps¹ and orbital ion traps.²³ In Penning–Malmberg traps, confinement times are limited by azimuthal asymmetries of the device^24–26 in the absence of externally applied torque.²⁷ For the high-energy particles confined in the Orbitron (⁠$>10$ keV), the collisional dynamics will be significantly different and work is currently underway to understand the influence of background pressure on the confinement time. However, to minimize transport from neutral collisions and the loss of ions from charge exchange [a typical challenge in inertial electrostatic confinement (IEC) devices], the Orbitron is typically operated in the ultra-high vacuum (UHV) regime (⁠$<10−8$ Torr).

Figure 8 shows an overview schematic of the vacuum system. A cryopump with a pumping speed of 2500 l/s for H₂ enables a base vacuum pressure near 10⁻⁹ Torr in the Orbitron. The ion source (see Sec. III D) typically operates above 10⁻³ Torr; thus, strong differential pumping is required between the ion source and vacuum chamber. Differential pumping enables ion loading into the Orbitron with a slightly elevated pressure of near 10⁻⁸ Torr.

Given the compact geometry of the Orbitron and the UHV requirement, the generation, transmission, and maintenance of high voltage on the cathode are major challenges. There have been several attempts to develop HV vacuum bushings, such as those for ITER’s Neutral Beam Injector and the University of Wisconsin–Madison’s inertial electrostatic confinement (IEC) reactor.^28,29 These designs, however, either failed to achieve the desired values or were too large for the Orbitron.

Figure 7 depicts our design of a 300 kV UHV bushing.³⁰ This bushing incorporates a MACOR plate as a spacer between UHV and a potting compound at atmospheric pressure. The potting compound, whether oil, room-temperature vulcanizing (RTV) silicone, or resin, ensures the electrical integrity of the cable–cathode connection. The sawtooth pattern on the insulator surface is designed to reduce the probability of surface flashovers. This bushing has been tested below −200 kV on the Orbitron device, and experiments are underway to achieve the −300 kV design milestone.

At these high voltages, the choice of materials, the machining process, and polishing are pivotal to controlling electron emission rates and prevention of flashovers and arcs.³¹ The cathode and anode are constructed from molybdenum and stainless steel, respectively. MACOR is used as the dielectric due to its machinability, ease of use, and high dielectric strength (129 MV/m). Future research on high-voltage materials will aim to investigate alternative substances for the aforementioned components with the objective of enhancing reliability and reducing current loss.

To achieve these high voltages, the cathode is conditioned to safely quench as many sources of prebreakdown current and “primary” microparticle events as possible so that the total number of potential hazards to the stability is significantly reduced. There are conditioning procedures in which the protrusions and field emitters at the surface of the cathode are removed with the aid of controlled discharges, background gases, electric fields, etc. One of the most common methods is current conditioning in which the protrusions are erupted either by their own field-emission current or by the bombardment of the cathode with the desorbed gases ejected from the anode during conditioning. We have adopted two different conditioning methods, current conditioning and gas conditioning.

In the device capable of −100 kV cathode voltages, a 0.05 T magnetic field at the mid-plane (z = 0, r = 6 cm) is sufficient to confine electrons (see Sec. II A 2). For this low magnetic field, we use neodymium magnets in a Halbach array modified by a variable trim coil mounted on the mid-plane as shown in Fig. 7. For electron confinement at higher cathode voltages down to −300 kV, we rely on superconducting magnets for higher magnetic field strengths. Two specially designed, high-temperature superconducting magnet coils placed on either side of the Orbitron enable the desired magnetic field with a field strength of 0.5 T at the mid-plane (z = 0, r = 6 cm). With the addition of a variable trim coil, investigations of how the magnetic field topology affects the electron confinement time will be explored.

Since ions are confined in the Orbitron by their angular momentum, ions must be loaded into the trap with significant azimuthal energy to form elliptical orbits about the cathode. One of the significant achievements of the orbital ion trap used for mass spectroscopy was developing a loading scheme that preserves to a high degree the azimuthal symmetry of the device.⁷ In these traps, the cathode voltage is increased during the first few axial oscillations to reduce the ion’s apoapsis and form stable orbits away from the anode walls. Here, we assume that with a sufficient loading efficiency and ion beam current, cathode ramping will reach ion densities above the space charge limit. Significant work is underway to explore loading schemes that will be used in experiments to demonstrate the operation of the Orbitron above the space charge limit.

To create the highly elliptical orbits needed for fusion, ions are accelerated across a voltage drop of ∼10% of the magnitude of the cathode voltage. The desired beam current necessarily scales with the loading efficiency and confinement time; however, our estimates predict that 1–10 mA will be sufficient for reaching ion densities above the ion space charge limit (⁠$∼109$ cm⁻³ for our device).

Our experiments use a readily available MARK I End-Hall ion source, which we have modified to suit our needs.³² The source outputs $>1$ mA of beam current while operating with 5–10 mTorr of D₂. Ion energies of up to 20 keV are possible by floating this source. Under these nominal operating conditions, the beam composition was measured to be ∼75% $D2+$⁠, 20% $D3+$⁠, and 5% D⁺. This source is typically operated with an anode voltage of 120–150 V and a discharge current of 0.7 A. The beam is focused and steered into the Orbitron using a Sikler lens.³³ 

For our electron source, we are currently taking advantage of the cold field emission off the cathode.³⁴ This leakage current serves as an ideal electron source since it loads the trap from the center of the device. When −100 kV is applied to the conditioned molybdenum cathode, this leakage current is typically $<50μ$A; however, higher source currents are achieved for optimum loading through the choice of the cathode material and secondary emission of electrons through ion impacts on the cathode surface. We have also designed an electron filament,³⁵ which can be floated at the cathode voltage to provide this high loading current if required.

To measure the plasma density, particle energy, particle confinement time, fusion rate, and fusion spatial distribution, we are currently employing an array of diagnostics. For our initial experiments of proving densification of the ions above the space charge limit of the trap, the plasma density will be relatively low ≲10¹⁰ cm⁻³ compared to more traditional quasi-neutral plasmas. This low density rules out some diagnostics such as laser interferometry and makes Thomson scattering borderline. These density diagnostics will be more useful for future higher density experiments. Our plasma is also ideally non-thermal and highly energetic, which adds a level of complication to the analysis of some of the diagnostics we will be discussing in this section.

Depending on the desired experiment, the Orbitron confines pure ion plasmas, pure electron plasmas, or quasi-neutral plasmas by the mechanisms described in Sec. II. By confining pure electron or pure ion plasmas, we are able to characterize the confinement properties of the Orbitron and benchmark simulations in a simpler system. In this section, we will introduce some of the diagnostics we are currently developing for this device.

Optical emission spectroscopy (OES) is commonly used in laboratory plasmas to diagnose plasma purity, electron temperature, and electron density. This technique requires the presence of optical photon-emitting species. In the case of impurities, common low charge state ions, such as C⁺, O⁺/O²⁺, N⁺/N²⁺, and the vibrations of molecular species, can be identified through the assignment of characteristic lines.³⁸ 

In the case of a pure electron plasma, the intentional introduction of a background gas is used to study the electron properties. Initial experiments in the Orbitron have introduced argon gas into the electron plasma and observed lines from excited neutral argon (Ar) and singly ionized argon (Ar⁺) due to collisions of the confined electrons with the background gas. At low cathode voltages (<20 keV), these data may be used to study electron energies or density by the common line ratio analysis.^39–41 At higher cathode voltages, more complex collisional radiative modeling of line emission is being explored, which will require electron gas collisional cross sections at high energies^42,43 and an intensity-calibrated spectrometer. This diagnostic enables narrower lines of sight than microwave interferometry so that it can aid the understanding of the electron density profile.

For experiments with D⁺ ions, line emission will not be observed. In this case, diagnostics that employ charge exchange of the fast D⁺ ion with a neutral gas or beam will be explored. This will give information on the fast ion properties of the confined deuterium beam.⁴⁴ 

The high-energy electrons confined in the Orbitron will emit x-ray radiation when they undergo acceleration from particle collisions, which is referred to as Bremsstrahlung radiation. This radiation is peaked in the soft x-ray range (⁠$<10$ keV) because the dominant collisions are small angle scattering events. For thermal plasmas, measurements of this radiation spectrum are a diagnostic tool to extract the electron temperature.⁴⁵ For our non-thermal plasma, this radiation is present; however, a more detailed analysis is required to deconvolve the electron energy distribution from the energy spectrum of the Bremsstrahlung radiation.

Experiments with pure electron plasmas are investigating the soft x-ray radiation emitted from electron collisions with a neutral gas backfill. The electron collisional cross sections for x-ray radiation from collisions with neutral gasses, such as argon, have been well explored theoretically.⁴⁶ This will enable measurements of the electron energy distribution along a line-of-sight. Density measurements may also be possible with an intensity-calibrated soft x-ray spectrometer by measuring the intensity of this Bremsstrahlung radiation and possibly through measurements of electron collisional excitation of line emission from the background gas. This diagnostic will also be used in the future to quantify power loss in our system from Bremsstrahlung radiation.

For the purpose of optimizing the ion loading process and measuring ion lifetimes in the low-density limit, we employ the method of image current measurement from orbital ion traps used for mass spectroscopy.⁷ The anode is bisected through the mid-plane as shown in Fig. 7 and electrically reconnected via a high-speed, low-noise current sensor, which measures the transfer of image charge between the two halves as the ions oscillate between them. For the typical operating conditions of the Orbitron, the oscillation frequency is in the MHz range. To measure this image current, a pulsed ion beam must be used such that the pulse width is less than or equal to a half period of this oscillation. A packet of ions injected in this way will yield a decaying sinusoidal image current signal. The amplitude of this signal indicates loading efficiency, and the decay rate is approximately equal to the ion confinement time if the ion loss rate is faster than the decoherence time of the ion pulse.

Image current measurements are also routinely used in non-neutral plasmas at higher densities to diagnose space charge waves and instabilities. Similar to Trivelpiece–Gould waves in Penning–Malmberg traps,^47,48 axial waves in the Orbitron should be detectable by the induced axial image current. By segmenting an anode azimuthally, we will also be able to measure azimuthal waves and bulk instabilities like the diocotron mode⁴⁹ (see Sec. VI).

The Orbitron produces fusion products when deuterium ions are used as fuel through beam–beam, beam–background, and beam–target fusions. At high ion densities, beam–beam fusion will dominate since it scales as the ion density squared. However, initial experiments with lower ion densities will need to discriminate between these fusion processes. To this end, we have added several diagnostics to determine both the total and spatial neutron production and the energy spectra of neutrons.

For total neutron production rates, bubble and Helium-3 (He-3) detectors are useful and simple diagnostics. Bubble detectors contain a polymer gel interspersed with small liquid droplets. When a high-energy neutron strikes the liquid, the droplet vaporizes, leaving behind a bubble. Bubble detectors BD-PND (personal neutron dosimetry) from Bubble Technology Industries (BTI Chalk River, Ontario) have a response range from 0.2 to 15 MeV, isotropic angular response, and zero responsivity to gamma radiation.

Helium-3 neutron proportional counters provide a real-time measurement of the fusion rate. These detectors consist of tubes of He-3 gas with a central anode wire surrounded by a cathode. The tubes are encased in a moderator, such as High-Density Polyethylene (HDPE), which converts the fast fusion neutrons into thermal neutrons. Thermal neutrons interact with the He-3 gas to produce H¹ and H³, which both carry kinetic energy. The high-energy particles ionize the surrounding background gas, and electrons move toward the anode, while cations move toward the cathode. There is an avalanche amplification effect that occurs as the moving charges ionize more of the carrier gas. The charge on the anode is recorded on a preamplifier as a voltage pulse and counted.

In addition to measuring total production, properties of the neutrons, including energy and location of production, help distinguish the fusion process. The Orbitron can operate in both pulsed and steady-state modes. Steady-state operation prevents the use of some typical neutron detection systems that rely on time-of-flight measurements for neutron/gamma-ray discrimination and neutron spectroscopy. Instead, we use pulse-shape discriminating scintillators, which employ a scintillating material that produces a pulse of visible light when hit with either a gamma ray or a high-energy neutron.^50–52 By taking the integrated area of short (Q_S) and long (Q_L) time periods that include the tail of the pulse, a pulse shape discriminating ratio is defined as PSD = (Q_L − Q_S)/Q_L. This ratio is small for gamma interactions and large for neutron interactions allowing discrimination of the two events. Measurements of the neutron energy spectrum may be used to distinguish between beam–target fusion at the cathode and beam–beam fusion.⁵³ 

Determining the spatial location of the fusion event will support the discrimination of beam–beam vs beam–target fusions. To measure the spatial location of neutron production, an array of small PSD detectors is embedded in high-density polyethylene, which acts as a collimator. The collimator thermalizes some neutrons that reach detectors off the desired line-of-sight, therefore, lowering their energy. By counting only neutrons that retain their full energy, spatial resolution on the order of centimeters is achieved. Beam–beam fusion will occur slightly away from the cathode where the relative radial energy of the ions is the largest, which will be resolvable with this neutron camera.

Figure 9(a) shows the density evolution of two separate WarpX⁵⁴ particle-in-cell simulations of a pure electron (black) and a pure ion (red) plasma confined in the Orbitron. For these simulations, the cathode voltage is −100 kV and the magnetic field strength is about 0.05 T at the mid-plane (z = 0, r = 6 cm). These simulations assume azimuthal (θ) symmetry. To build up high densities with less computation time, high injection currents are used. In the first 2 µs, the electron and ion injection currents are ramped up to 0.4 A. This loading current remains on for a total of 25 µs. The D⁺ loaded ions in the pure ion simulation are given an initial azimuthal energy of 10 keV from an initial position inside the Orbitron spanning r = 4–5 cm and z = −3 to −2 cm to place them in elliptical orbits around the cathode. External ion loading is not modeled in these simulations. In the pure electron simulations, the electrons are loaded over a thin (z = −1 to 1 mm) radial plane spanning from cathode to anode with an initial energy of 600 eV. The electron and ion macroparticle weight is 1 × 10⁷ particles, the grid size is 0.25 × 0.25 mm², and the time step is 2.0 × 10⁻¹² s. We use the WarpX electromagnetostatic solver option that includes the calculation of self-magnetic fields induced by the plasma. Here, we are plotting the average density over an annulus spanning from the cathode to the anode with a width of 2 cm centered at z = 0.

These non-neutral plasma simulations show the space charge limited density of this trap. The pure electron plasma (black solid line) reaches a max average density of 7.4 × 10⁹ cm⁻³, which is near the predicted Brillouin limit (black dashed line). After the electron source is turned off at 25 µs, the electron density decreases with a loss rate of 20 mA. With this high cathode voltage and weak magnetic field, the electrons are weakly confined; thus, this loss current is not too surprising. Superconducting magnets will enable stronger magnetic fields, which should reduce this loss current. In the pure ion plasma (red line) simulations, the ions are more strongly confined with a loss current of 0.3 mA but are limited by space charge to a lower density of 1.1 × 10⁹. To reach high ion densities relevant for fusion applications, this ion space charge limit must be mitigated.

Figure 9(b) shows a PIC simulation in which electrons are co-loaded with ions in this device to mitigate this ion space charge limit. The simulation parameters and particle loading are identical to the pure electron and pure ion plasma simulations shown in Fig. 9(a). Here, we see that the electron and ion densities couple enabling loading to higher densities. An average ion density of 5.4 × 10¹⁰ cm⁻³, about 50 times larger than the pure ion plasma density, is reached with the same loading conditions. After the loading current is ceased at 25 µs, the density begins to decrease, with the two loss rates tracking together. It is likely that the loss rate is determined by the transport losses of one species, which the other species tracks in accordance with the associated reduction in the space charge limit. These simulations include Coulomb collisions, using the WarpX implementation of the Direct Simulation Monte Carlo method, but the collision time at these high energies is larger than the duration of the simulations.

As illustrated in Fig. 6, the plasma self-fields weaken the magnetic field in some areas. This may be a factor in reducing confinement performance. These self-fields may also limit the achievable density for a given magnetic field and are currently being explored in more detail. These effects are taken into account in calculating Fig. 9.

The spatial density profiles of deuterons and electrons at t = 35 µs are shown in Fig. 10. The two species are illustrated separately; however, in the simulation, they are co-confined together throughout the trap. Both density profiles are rotated 360° in θ around the cathode. The density profile suggests the presence of a collective space-charge coupling effect. In the high-density regions along z = 0, the two densities were calculated to match within ±10%. The ion density in Fig. 10 is more constricted axially than the electrons since the ions are confined by the potential well created by the anode/cathode geometry, whereas the electrons are confined in z by the electrostatically plugged magnetic mirror that extends to the cathode protrusions.

To exceed the space charge limit of this device with a reasonable computation duration, we have artificially increased the loading currents. Experimentally, the initial electron and ion injection currents will be around 1–10 mA. Therefore, to reach the simulated densities, we will require an experimental loading duration on the order of 1–10 ms assuming the ideal simulated loading. On this timescale, collective effects, instabilities, and collisional effects (see Sec. VI) may arise, which are not captured in these simulations. However, the computation time to replicate the exact experimental conditions is outside of the scope of this work. These initial simulations show promising results for exceeding the space charge limit, and we will attempt to understand these possible limitations through experiments.

A key focus of experiments on this device is mitigating the space charge limit. However, to achieve efficient fusion events, ion densities well above this limit must be achieved with a relatively low loss of energy. Instabilities, particle diffusion to the conducting walls, and radiative losses can all limit the fusion efficiency.

Instabilities are a collective process in which a plasma relaxes from a non-thermal state in a time scale faster than a collision time. Initial simulations of pure electron plasmas on this device have seen the classic diocotron instability.⁴⁹ Nascent theory with the support of simulations suggests that this mode might be stabilized at our higher cathode voltages, and experiments are planned to test this voltage suppression. Similar E × B devices have also observed anomalous transport due to the electron cyclotron drift instability.^55,56 Simulations of quasi-neutral plasmas above the space charge limit have not been dominated by configuration-space instabilities, which might be due to damping from the strong shear flow in our device as predicted in mirror machines.⁵⁷ Velocity–space instabilities, such as beam–beam and beam–plasma instabilities, are a concern as a possible source of energy loss from the colliding beams. These instabilities have not been directly observed in simulations of this device; however, they may be an issue at higher densities and will be explored.

Collisional diffusion of the particles to the conducting walls is another source of energy loss. Ion–ion Coulomb collisions are scattering events, which will alter the ideal elliptical trajectory. A feature of this device is that the frequency of 90° scattering events is small compared to the orbital frequency of the ions. Work is in progress to understand the impact of these small angle scattering events on the trajectory of the ions and the timescale at which they cause diffusion to a conducting surface. Electrons will also diffuse across the magnetic field toward the anode due to Coulomb collisions with electrons and ions. With a moderate magnetic field, this diffusion will be on a timescale of multiple collision times.

Particle collisions will not only cause diffusion but also thermalization of the velocity distribution function. The fusion reaction rate will be highly dependent on the velocity distribution of the ions. With the beam–beam velocity distribution predicted in the absence of thermalization, high fusion rates can be achieved at densities and for device scales significantly lower than traditional reactors. In practice, the ion velocity distribution will most likely be somewhere in-between a pure beam and thermal distribution, which will reduce the neutron flux.

An inherent energy loss mechanism of this device is Bremsstrahlung radiation, which is a commonly cited concern for fusion reactors with non-Maxwellian energy distributions.⁵⁸ A key goal of our research is to characterize the Orbitron particle distribution functions and phase space dynamics in order to substantiate a detailed power balance analysis using the methodology described in Ref. 59.

In summary, we have presented the physics of single-charge particle confinement and detailed the experimental apparatus of a new plasma confinement scheme called an Orbitron. This crossed-field device confines ions in orbits around a high-voltage cathode at fusion-relevant energies with co-rotating electrons confined by a relatively weak magnetic field. Particle-in-cell simulations show that these co-rotating electrons enable ion densities above the ion space charge limit. Demonstrating this space charge mitigation will be the focus of initial experiments. After this fundamental science goal is achieved, this device will be scaled up to higher voltages (−300 kV) and stronger magnetic fields (0.5 T) to achieve higher fusion reaction rates.

The authors thanked A. Makarov, S. Tsurkan, C. Reilly, M. Prato, J. Hummelt, and R. Wirz for helpful discussion and a careful review of our manuscript. This material was based upon work supported by the National Science Foundation under Grant No. 2303759. This research used the open-source particle-in-cell code WarpX (https://github.com/ECP-WarpX/WarpX), primarily funded by the U.S. DOE Exascale Computing Project. Primary WarpX contributors are with LBNL, LLNL, CEA-LIDYL, SLAC, DESY, CERN, and TAE Technologies. They acknowledged all WarpX contributors. This research also used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 using NERSC Award No. FES-ERCAP0029121.

The authors have no conflicts to disclose.

M. Affolter: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). R. Thompson: Writing – original draft (equal); Writing – review & editing (equal). S. Hepner: Writing – original draft (equal). E. C. Hayes: Writing – original draft (equal). V. Podolsky: Writing – original draft (equal). M. Borghei: Writing – original draft (equal). J. Carlsson: Investigation (equal). A. Gargone: Investigation (equal); Writing – original draft (equal). D. Merthe: Writing – original draft (equal). E. McKee: Writing – original draft (equal). R. Langtry: Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.