Inverse hall-effect current drive for field-reversed configuration fusion reactors

Abstract

A method (100) for driving current in a field-reversed configuration (FRC) fusion reactor (200) includes creating a negative plasma potential within the FRC by removing positive fusion products from a core plasma region (202) to an energy and ash removal shell (EARS) (206) and generating an electric field perpendicular to a magnetic field of the FRC. The electric field drives a net current in the FRC due to differential ExB drifts between ions and electrons. The electric field is generated predominantly near and perpendicular to a separatrix (204) of the FRC. The EARS (206) is physically separated from the separatrix (204). Gas or plasma is sourced into the EARS (206) in a divertor region (e.g., a divertor chamber 218, 220) via a gas / plasma source and control system (214) to densify the EARS (206). The net current is driven in a toroidal direction of the FRC.

Description

Princeton - 107976INVERSE HALL-EFFECT CURRENT DRIVE FOR FIELD-REVERSED CONFIGURATION FUSION REACTORSCROSS-REFEENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Application No. 63 / 775,112, titled "Inverse Hall-Effect Current Drive for Field-Reversed Configuration Fusion Reactors", filed March 20, 2025 is hereby incorporated by reference in its entirety.STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

[0002] This invention was made with government support under DOE Grant No. DE-AC02-09CH11466 awarded by the Department of Energy. The government has certain rights in the invention.TECHNICAL FIELD

[0003] The present disclosure relates to cunent drive methods for fusion reactors, and more particularly to an inverse Hall-effect current drive technique for field-reversed configuration (FRC) fusion reactors that utilizes differential ExB drifts between ions and electrons to generate net plasma current.BACKGROUND

[0004] Nuclear fusion has long been pursued as a potential source of clean, abundant energy. Among the various approaches to fusion, magnetic confinement devices aim to confine high-temperature plasma using strong magnetic fields. Field-reversed configuration (FRC) reactors represent a class of compact magnetic confinement devices with several potential advantages over other fusion concepts.

[0005] FRC reactors typically consist of a cylindrical chamber containing plasma in a particular magnetic field configuration. The magnetic field lines in an FRC form closed loops within the plasma, creating a self-contained magnetic structure. This configuration allows for a compact reactor design with a high ratio of plasma pressure to magnetic field pressure, known as beta. The closed field line structure and high beta of FRCs create conditions that differ from other fusion concepts such as tokamaks and stellarators.

[0006] One challenge in FRC and other fusion reactor designs is sustaining the plasma current required to maintain the magnetic configuration and plasma confinement. Various methods have been developed to drive current in fusion plasmas, including ohmic current drive,Princeton - 107976neutral beam injection, and radio-frequency wave current drive. In tokamaks, a self-sustaining bootstrap current arises naturally due to the toroidal magnetic field geometry and trapped particle dynamics. However, FRCs lack an equivalent intrinsic current drive mechanism due to their different magnetic field topology, where the magnetic field reaches zero at null points rather than vary ing inversely with major radius as in tokamaks.

[0007] Existing current drive techniques often have limitations in efficiency or applicability to compact FRC devices. Ohmic heating, which relies on plasma resistivity, becomes less effective as plasma temperature increases because resistivity is inversely proportional to electron temperature raised to the three-halves power. Neutral beam injection requires continuous beam production and external power systems to ionize and accelerate particles. Wave current drive similarly requires external power sources and wave generation equipment. These external systems add complexity, cost, and energy consumption to fusion reactor designs.

[0008] In FRC plasmas, charged particles can follow different orbital trajectories depending on their energy and position within the magnetic field structure. These orbit classes include cyclotron orbits, where particles spiral tightly around magnetic field lines; betatron orbits, where higher-energy particles execute larger orbital excursions; and figure-8 orbits, which occur when particles cross the magnetic null line. The different orbital behaviors of ions and electrons in an FRC may influence plasma dynamics and current generation.

[0009] Fusion reactions in certain plasma fuel mixtures, such as deuterium-helium-3, produce energetic charged particles as fusion products. The behavior of these fusion products and their interactions with the background plasma can influence the overall plasma dynamics. In some FRC concepts, positive fusion products are removed from the core plasma to prevent quenching of the fusion reaction and to avoid undesirable secondary reactions. This removal process can affect the charge balance within the plasma.

[0010] The presence of electric fields perpendicular to magnetic fields in a plasma causes charged particles to experience E B drift, where particles move in a direction perpendicular to both the electric and magnetic field vectors. The drift velocity depends on the ratio of the electric field magnitude to the magnetic field magnitude. Different particle species and orbital t pes may experience different effective drift velocities depending on their interaction with the local electric and magnetic fields.

[0011] As research in fusion energy progresses, there is ongoing interest in developing approaches to plasma cunent drive that can reduce reliance on external power systems and improve overall reactor efficiency. Advances in current drive methods could contribute toPrinceton - 107976enhanced plasma stability, improved energy balance, and reduced system complexity in fusion reactor designs.SUMMARY

[0012] This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary’ is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

[0013] According to an aspect of the present disclosure, a method for driving current in a field-reversed configuration (FRC) fusion reactor is provided. The method includes creating a negative plasma potential within the FRC by removing positive fusion products from a core plasma to an energy and ash removal shell (EARS). The method further includes generating an electric field perpendicular to a magnetic field of the FRC. The electric field drives a net current in the FRC due to differential E*B drifts between ions and electrons. This approach enables current drive using energy derived from the fusion reactions themselves, eliminating the need for external power systems and improving overall reactor efficiency.

[0014] According to other aspects of the present disclosure, the electric field may be generated predominantly near and perpendicular to a separatrix of the FRC. Generating the electric field in this location maximizes the interaction between the electric field and the plasma particles near the boundary of the confined plasma region, enhancing the effectiveness of the current drive mechanism.

[0015] According to other aspects of the present disclosure, the method may further include controlling a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field. This control mechanism provides a means to regulate the magnitude of the driven current and maintain stable plasma operation.

[0016] According to other aspects of the present disclosure, the EARS may be physically separated from a separatrix of the FRC. This physical separation allows for independent control of the EARS properties and prevents direct interaction between the core plasma and the ash removal region.

[0017] According to other aspects of the present disclosure, the method may further include sourcing gas or plasma into the EARS in a divertor region to densify' the EARS. Densifying the EARS through gas or plasma injection enhances the ability of the EARS to receive and neutralize the positive fusion products removed from the core plasma.Princeton - 107976

[0018] According to other aspects of the present disclosure, the net current may be driven in a toroidal direction of the FRC. Driving current in the toroidal direction contributes to maintaining the magnetic field configuration and plasma confinement within the FRC.

[0019] According to other aspects of the present disclosure, the net current may result from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits. This differential drift behavior between ion and electron species creates a net current without requiring external current drive systems.

[0020] According to another aspect of the present disclosure, a field-reversed configuration (FRC) fusion reactor is provided. The FRC fusion reactor includes a core plasma region. The FRC fusion reactor further includes an energy and ash removal shell (EARS) surrounding the core plasma region. The FRC fusion reactor further includes a control system configured to control the removal rate of positive fusion products from the core plasma region to the EARS, thereby creating a negative plasma potential and an electric field perpendicular to a magnetic field of the FRC. This reactor configuration enables self-sustaining current drive using the inherent charge imbalance created by fusion product removal.

[0021] According to other aspects of the present disclosure, the electric field may be generated predominantly near and perpendicular to a separatrix of the FRC. Positioning the electric field generation near the separatrix optimizes the coupling between the electric field and the plasma dynamics at the confinement boundary-.

[0022] According to other aspects of the present disclosure, the control system may be further configured to control a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field. This active control capability allows for real-time adjustment of the current drive to respond to changing plasma conditions.

[0023] According to other aspects of the present disclosure, the EARS may be physically separated from a separatrix of the FRC. The physical separation between the EARS and the separatrix provides thermal and particle isolation between the core plasma and the ash removal region.

[0024] According to other aspects of the present disclosure, the FRC fusion reactor may further include a gas or plasma source configured to inject gas or plasma into the EARS in a divertor region to densify the EARS. The gas or plasma source enables control over the density and composition of the EARS to optimize fusion product removal.

[0025] According to other aspects of the present disclosure, the electric field may drive a net current in a toroidal direction of the FRC due to differential E*B drifts between ions andPrinceton - 107976electrons. The toroidal current driven by this mechanism contributes to the magnetic field structure that confines the plasma.

[0026] According to other aspects of the present disclosure, the net current may result from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits. The different orbital characteristics of ions and electrons in the FRC magnetic field geometry enable the differential drift that produces net current.

[0027] According to another aspect of the present disclosure, a method of improving efficiency in a field-reversed configuration (FRC) fusion reactor is provided. The method includes generating fusion reactions in a core plasma of the FRC. The method further includes selectively removing positive fusion products from the core plasma to create a negative plasma potential. The method further includes utilizing the negative plasma potential to drive a net current in the FRC without external current drive systems, collectively removing the inefficiencies in other methods of driving currents in magnetized fusion plasmas. This approach converts fusion product energy directly into plasma current drive, bypassing the energy conversion losses associated with external current drive systems.

[0028] According to other aspects of the present disclosure, selectively removing positive fusion products may include directing the positive fusion products to an energy' and ash removal shell (EARS) surrounding the core plasma. The EARS provides a designated region for collecting and processing the removed fusion products while maintaining the charge imbalance in the core plasma.

[0029] According to other aspects of the present disclosure, the EARS may be physically separated from a separatrix of the FRC. This separation prevents the EARS from interfering with the plasma confinement at the separatrix boundary.

[0030] According to other aspects of the present disclosure, the method may further include sourcing gas or plasma into the EARS in a divertor region to densify the EARS. The densification of the EARS improves its capacity7to absorb and neutralize the positive fusion products.

[0031] According to other aspects of the present disclosure, utilizing the negative plasma potential may include generating an electric field perpendicular to a magnetic field of the FRC. The perpendicular orientation of the electric field relative to the magnetic field creates the ExB drift conditions that drive the plasma current.

[0032] According to other aspects of the present disclosure, the net current may result from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser driftPrinceton - 107976velocities compared to electrons in cyclotron orbits. The orbital dynamics specific to FRC geometry enable this differential response to the electric field, producing a net current that would not occur in configurations where both species follow similar orbital paths.

[0033] The foregoing general description of the illustrative embodiments and the following detailed description thereof are merely exemplary aspects of the teachings of this disclosure and are not restrictive.BRIEF DESCRIPTION OF FIGURES

[0034] Non-limiting and non-exhaustive examples are described with reference to the following figures.

[0035] Figure 1 is a flowchart illustrating a method for driving current in a field-reversed configuration fusion reactor.

[0036] Figure 2 is a schematic diagram of a field-reversed configuration fusion reactor.

[0037] Figure 3 is a flowchart illustrating a method for improving efficiency in a field-reversed configuration fusion reactor.

[0038] Figure 4 illustrates FRC magnetic field lines.

[0039] Figure 5 illustrates toroidal aspect ratio geometry.

[0040] Figure 6 illustrates gy ration of a charged particle in an E X B field.

[0041] Figure 7 illustrates a cyclotron orbit.

[0042] Figure 8 illustrates a figure-8 orbit.

[0043] Figure 9 illustrates a betatron orbit.

[0044] Figure 10 illustrates a schematic of an IHE CD method in an FRC. External to the plasma separatrix is a boundary, here indicted to be metal which is held at a positive potential relative to the plasma. The electric field, E. generated by this boundary’ decreases as one proceeds deeper into the plasma. The magnetic field, B, also decreases going deeper into the plasma, passing through a null line and then reversing direction. Representative electron and ion orbit shapes are shown.

[0045] Figure 11 show s a graph of The drift velocities of an ion and an electron vs applied potential, for the parameters shown in Figure 10 and a skin depth equal to the fuel-ion gyroradius. The ions drift slower than the electrons because they’ spend less time in the high E-field region at the plasma edge. The net effect is the IHE current.Princeton - 107976DETAILED DESCRIPTION

[0046] The following description sets forth exemplary aspects of the present disclosure. It should be recognized, however, that such description is not intended as a limitation on the scope of the present disclosure. Rather, the description also encompasses combinations and modifications to those exemplary aspects described herein.

[0047] Referring to FIG. 1, a method 100 for driving current in a field-reversed configuration (FRC) fusion reactor is illustrated. The method 100 includes a step 102 for creating a negative plasma potential within the FRC by removing positive fusion products from a core plasma to an energy and ash removal shell (EARS). The method 100 further includes a step 104 for generating an electric field perpendicular to a magnetic field of the FRC. The method 100 also includes a step 106 for controlling a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field. Additionally, the method 100 includes a step 108 for sourcing gas or plasma into the EARS in a divertor region to densify the EARS.

[0048] In the method 100, the FRC fusion reactor may utilize deuterium-helium-3 (D-He3) fuel as the plasma composition. The D-He3fuel produces energetic charged particles during fusion reactions. These energetic charged particles may be leveraged for the inverse Hall effect current drive. The magnetic field in the FRC forms closed loops within the plasma with field lines that reverse direction, creating a self-contained magnetic structure. The FRC exhibits a high beta ratio of plasma pressure to magnetic field pressure. In the FRC. current flows perpendicular to the magnetic field direction, distinguishing the FRC from tokamaks where current is driven parallel to field lines.

[0049] FIG. 4 illustrates FRC magnetic field lines. As shown in FIG. 4, the magnetic field O-point line occurs at a radius of -^-rson the z=0 plane of the FRC, where rsis a separatrix radius. The separatnx radius may be approximately 25 cm in a reactor-scale configuration. The magnetic field points in opposing directions on either side of the O-point line.

[0050] FIG. 5 illustrates toroidal aspect ratio geometry. As shown in FIG. 5, an inverse aspect ratio 8 is defined by 8 = a / R, where a is a radius of a tube and R is a radius from a center of a torus to a center of the tube.

[0051] With continued reference to FIG. 1, at step 102, positive fusion products are removed from the core plasma to the EARS. The removal of positive fusion products creates a negative plasma potential within the FRC. The EARS is physically separated from thePrinceton - 107976separatrix of the FRC. The EARS may be densified by gas or plasma sourced into the EARS in the divertor region at step 108.

[0052] At step 104, an electric field is generated perpendicular to the magnetic field of the FRC. The electric field is generated predominantly near and perpendicular to the separatrix of the FRC. The electric field drives a net current in the FRC due to differential E*B drifts between ions and electrons. The net current is driven in a toroidal direction of the FRC.

[0053] FIG. 6 illustrates gyration of a charged particle in an E*B field. As shown in FIG.6, charged particles in the presence of an E*B field move according to the Lorentz force. An ExBelectric field drift velocity VE is calculated as: vE=

[0054] The electric field drift velocity VE represents a transverse velocity that produces helical motion in three dimensions when combined with Larmor gy ration. The magnetic field Bzat a position 2 cm outside the O-point in the z=0 plane is approximately 1.25 T (12.5 kG).

[0055] FIG. 7 illustrates a cyclotron orbit. As shown in FIG. 7, cyclotron orbits occur at low energy7and small mass, where a velocity of a particle is predominantty perpendicular to the magnetic field. Electrons in the FRC have cyclotron orbits near the separatrix with small gyroradii. For a 50 keV electron at 6 T, the gyroradius is approximately 0.01 cm. Electrons in cyclotron orbits spiral tightly around magnetic field lines and stay on either side of the O-point null line.

[0056] FIG. 8 illustrates a figure-8 orbit. As show n in FIG. 8, when a particle in a cyclotron orbit crosses the O-line and traverses the separatrix, the particle enters into a figure-8 orbit. Figure-8 orbits may move in either direction around the FRC.

[0057] FIG. 9 illustrates a betatron orbit. As shown in FIG. 9, when a particle has sufficient momentum, the particle enters into a betatron orbit. Ions in the FRC may occupy betatron orbits with larger gyroradii at much larger radial excursions than electrons. For a 150 keV hydrogen ion at 6 T, the gyroradius is approximately 1 cm.

[0058] The net current results from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits. A betatron orbit moves in the anti-clockwise direction and thus contributes to a net current.

[0059] A plasma resistivity T] follows the relationship:1* e

[0060] where Teis an electron temperature. The plasma resistivity relationship causes ohmic heating to become less effective as plasma temperature increases.Princeton - 107976

[0061] At step 106, a loss rate of electrons that encircle the major axis of the FRC in a single cyclotron period is controlled to maintain a desired value of the electric field. The desired value of the electric field developed between the plasma core and the EARS may be maintained by controlling the loss of electrons that encircle the major axis in a single cyclotron period or other appropriate means.

[0062] At step 108, gas or plasma is sourced into the EARS in the divertor region to densify the EARS. The EARS is physically separated from the separatrix and is densified by gas or plasma sourced into the divertor region rather than by diffusion out of the FRC core. The power to create the electric field comes from fusion events within the plasma without extraction of fusion power, conversion to electricity, or provision of electricity to external equipment for wave or particle beams.

[0063] Referring to FIG. 2, a field-reversed configuration (FRC) fusion reactor 200 is illustrated. The reactor 200 includes a core plasma region 202 where fusion reactions occur. The core plasma region 202 may utilize deuterium-helium-3 (D-He3) fuel as the plasma composition. The D-He3fuel produces energetic charged particles during fusion reactions, and these energetic charged particles may be leveraged for inverse Hall effect current drive. The aneutronic D-He3fuel results in neutron radiation surface power loads approximately 1,000 times lower per unit of power compared to mainline reactors.

[0064] With continued reference to FIG. 2, the reactor 200 includes a separatrix 204 that defines a boundary of the confined plasma region. The separatrix 204 separates the core plasma region 202 from surrounding regions. An electric field is generated predominantly near and perpendicular to the separatrix 204 of the FRC. The magnetic field in the FRC forms closed loops within the plasma with field lines that reverse direction at the separatrix 204.

[0065] The reactor 200 includes an energy’ and ash removal shell (EARS) 206 surrounding the core plasma region 202. The EARS 206 is physically separated from the separatrix 204 of the FRC. The physical separation of the EARS 206 from the separatrix 204 distinguishes the EARS 206 from a scrape-off-layer (SOL) in tokamaks and stellarators. In tokamaks and stellarators, the SOL receives particles by diffusion out of the core plasma. In contrast, the EARS 206 is densified by gas or plasma sourced into a divertor region rather than by diffusion out of the FRC core.

[0066] The reactor 200 may include a first divertor chamber 218 and a second divertor chamber 220, and nozzle coils 208. The divertor(s) receives fusion products and directs exhaust from the core plasma region 202. The divertor(s) provides a region where gas or plasma may be sourced to densify the EARS 206.Princeton - 107976

[0067] As further shown in FIG. 2, the reactor 200 includes field coils 210 that generate the magnetic field configuration. The reactor 200 also includes shielding 212 inside the field coils 210 and other components. The shielding 212 provides protection from radiation and thermal loads.

[0068] The reactor 200 includes a gas / plasma source and control system 214. The gas / plasma source 216 and control system 214 is configured to inject gas or plasma into the divertor region (e.g., into a first divertor chamber 218) to densify the EARS 206. This injection of gas or plasma may be by any appropriate means. The plasma extruder 222 generates the pipe-like form of the EARS. The control system 214 is configured to remove positive fusion products from the core plasma region 202 to the EARS 206, thereby creating a negative plasma potential and an electric field perpendicular to a magnetic field of the FRC. The control system 214 is further configured to control a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field.

[0069] The electric field drives a net current in a toroidal direction of the FRC due to differential ExB drifts between ions and electrons. The net current results from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits. As described previously with reference to FIGS. 8-10, electrons occupy cyclotron orbits with small gyroradii near the separatrix 204, while ions may occupy figure-8 or betatron orbits with larger gyroradii extending over broader radial positions.

[0070] The reactor 200 operates as a compact device small enough to fit on the bed of a truck, enabling niche applications such as space propulsion. The compact size of the reactor 200 results from the high beta ratio of plasma pressure to magnetic field pressure achievable in the FRC configuration and the aneutronic fuel mixture and a novel T exhaust capability. The power to create the electric field comes from fusion events within the core plasma region 202 without extraction of fusion power, conversion to electricity, or provision of electricity to external equipment for w ave or particle beams.

[0071] Referring to FIG. 3, a method 300 for improving efficiency in a field-reversed configuration (FRC) fusion reactor is illustrated. The method 300 includes a step 302 for generating fusion reactions in a core plasma of the FRC, a step 304 for selectively removing positive fusion products from the core plasma to create a negative plasma potential, a step 306 for sourcing gas or plasma into an energy and ash removal shell (EARS) in a divertor region to densify the EARS, and a step 308 for utilizing the negative plasma potential to drive a net current in the FRC without external current drive systems.Princeton - 107976

[0072] At step 302, fusion reactions are generated in the core plasma of the FRC. The core plasma may utilize deuterium-helium-3 (D-He3) fuel, which produces energetic charged particles during fusion reactions. The energetic charged particles include positive fusion products that carry positive charge away from the core plasma region.

[0073] With continued reference to FIG. 3, at step 304, positive fusion products are selectively removed from the core plasma to create a negative plasma potential. Selectively removing positive fusion products comprises directing the positive fusion products to an energy and ash removal shell (EARS) surrounding the core plasma. The EARS is physically separated from a separatrix of the FRC. The physical separation of the EARS from the separatrix distinguishes the EARS from a scrape-off-layer in tokamaks and stellarators, where particles arrive by diffusion out of the core plasma. The selective removal of positive fusion products while retaining electrons in the core plasma creates a net negative charge within the FRC, establishing the negative plasma potential.

[0074] At step 306, gas or plasma is sourced into the EARS in a divertor region to densify the EARS. The densifi cation of the EARS by gas or plasma sourced into the divertor region provides a medium for receiving and neutralizing the positive fusion products directed from the core plasma. The EARS receives the positive fusion products without relying on diffusion from the FRC core.

[0075] At step 308, the negative plasma potential is utilized to drive a net current in the FRC without external current drive systems. Utilizing the negative plasma potential comprises generating an electric field perpendicular to a magnetic field of the FRC. The electric field arises from the potential difference created between the negatively charged core plasma and the EARS. The electric field is generated predominantly near and perpendicular to the separatrix of the FRC, as described previously with reference to the method 100.

[0076] The net current results from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits. As described previously with reference to FIGS. 8-10, electrons occupy cyclotron orbits with small gyroradii near the separatrix, while ions may occupy figure-8 or betatron orbits with larger gyroradii extended radial positions. The differential ExB drift velocities between ions and electrons produce the net current in a toroidal direction of the FRC.

[0077] The method 300 collectively removes the inefficiencies in other methods of driving currents in magnetized fusion plasmas. Ohmic current drive, as described previously, requires external coils to induce plasma current and becomes less effective as plasma temperature increases due to decreasing plasma resistivity. Neutral beam injection requires continuousPrinceton - 107976beam production and external power systems to ionize and inject neutral particles into the plasma. Wave cunent drive requires external systems to generate and couple waves into the plasma for momentum transfer to electrons.

[0078] In contrast, the method 300 derives power for creating the electric field from fusion events within the plasma. The method 300 does not require extraction of fusion power, conversion to electricity, or provision of electricity to external equipment for wave or particle beams. The self-sustaining nature of the inverse Hall effect current drive in the method 300 reduces system complexity and improves overall energy balance of the FRC fusion reactor.

[0079] Gradients drive flows. Considered by itself, this is not a particularly remarkable statement, yet its consequences are clear across all fields in physics and fundamental to many engineering applications: Temperature gradients drive heat flow, pressure gradients prompt liquid flow, while potential gradients generate the electrical currents essential to modem technology. In the abstract, this relationship may be expressed as follows.V (something) = flow (something)

[0080] In the PFRC, current flows perpendicular to the magnetic field direction. This means that the PFRC presents a unique challenge when it comes to plasma current drive because it is easier to drive current in the direction of magnetic field lines, as done in Tokamaks. Here, we investigate the possibility of a novel PFRC current drive method by conducting simulations to assess the relative importance of the recently proposed Inverse Hall Effect (IHE) in which a negative plasma potential following the departure of fusion products from the plasma.

[0081] § 1 Princeton Field Reverse Configuration

[0082] The Princeton Field Reversed Configuration (PFRC) is an alternative fusion reactor concept that reduces technical complexity by addressing more challenging physics: A tradeoff justified by the unique potential of the PFRC to deliver a compact, clean, and safe energy source. In practical terms, the PFRC could be small enough to fit on the bed of a truck and the proposed fuel, aneutronic D — He3, promises significantly reduced radiation emissions -potentially 1,000 times lower per unit of power compared to mainline reactors. These advantages position the FRC for faster, less capital-intensive development via niche applications such as space propulsion when compared to the slower progress of Tokamakbased designs. Show n below is a schematic of the magnetic field in the PFRC. See FIG. 4.

[0083] § 2 Diamagnetic CurrentPrinceton - 107976

[0084] The inherent pressure gradient of a plasma in a magnetic confinement fusion device results in the formation of a diamagnetic current. The nature of this current is revealed following a straightforward manipulation of the ideal magnetohydrodynamics (MHD) equation of motion: [2]duP~dt ^J X B~Vp

[0085] Where pis the mass density, uis the centre of mass velocity, / is the current density, Bis a constant magnetic field, and Fpis the pressure gradient of the plasma. We assume that our plasma is at equilibrium, thus uis not changing in time and our equation simplifies to: / x B = Vp

[0086] Taking the cross product of B with both sides of the equation and simplifying using vector identities, we are left with:\B\2(J - bJu) = B x Vp

[0087] Where bis the unit magnetic field vector, and / ||is the component of / parallel to B. However, subtracting the parallel component of the vector / from itself simply leaves the perpendicular component, / ±. Thus, all that remains is to rearrange our equation for this / ±, which turns out to be our diamagnetic current densify.B x Vp

[0088] Therefore, we see that a current density acting perpendicular to the magnetic field arises naturally from the ideal MHD equation of motion for charged particles in a magnetic field. This current densify is the diamagnetic current which acts to 'push back' against the external magnetic field applied to the plasma.

[0089] However, alone this naturally arising diamagnetic current is insufficient to confine plasma at appropriate conditions for fusion. Therefore, additional current must be driven through the plasma to bridge this gap. The question is then how best to produce this additional driven current such that it becomes self-sufficient.

[0090] § 3.1 Origin of Electric Field in PFRC

[0091] In all FRCs a radial — more precisely, a flux-surface-normal - electric field is expected. It is called the self-consistent field Additionally, in the PFRCfusion products are removed so that the fusion reaction is not quenched and Tritium is not burnt (which would lead to the emission of neutrons). This means that the plasma in the PFRC is net negative, thus a potential difference and subsequent electric field is created. It is this electric field which wePrinceton - 107976investigate here to determine its profile and whether it might have the auxiliary benefit of driving current in the PFRC.

[0092] § 3.2 FRC Particle Orbits

[0093] There are three particle orbit classes in an FRC: cyclotron, betatron, or figure-8.

[0094] Cyclotron orbits occur at low energy', where the velocity7of the particle is predominantly perpendicular to the magnetic field. These particles exhibit small gyroradii and spiral tightly around magnetic field lines. Cyclotron motion dominates in regions with stronger magnetic fields, where the magnetic force effectively confines low-energy^ particles. In an FRC, this means that cyclotron orbit particles stay on either side of the O =point null line. See FIG.7.

[0095] If a particle in a cyclotron orbit crosses the O-line, traversing the separatrix, it enters into a figure- 8 orbit. See FIG. 8.

[0096] Then, if the particle has sufficient momentum, it will enter into a betatron orbit since it is able to reach the single raised-potential-well region. See FIG. 9.

[0097] E x B drift is well defined for cyclotron orbits, however, the E x B dri ft for betatron and figure- 8 orbits must be investigated further. Furthermore, cyclotron and betatron orbits move in opposing directions, whilst figure- 8 orbits may move in either direction.

[0098] § 3.3 Numerical Bellwethers

[0099] In order to drive current in an FRC there must be a net movement of either ions or electrons which is not cancelled by an equivalent flow of oppositely charged particles. The Larmor radius of a particle in an FRC varies with position as a result of the spatially' non-uniform magnetic field. The field, B. is zero at the Opoint (magnetic null) and reaches a maximum at both the centre ( r = 0) and the separatrix radius ( r = rs). This variation in Binfluences the orbital behaviour of particles based on their energy and location as outlined in the previous section. The consequences of this idea are best demonstrated by plugging some real numbers in for aneutronic reactor-like plasmas.

[0100] Consider an electron with an energy of 50 keV. At a position 2 cm outside the O-point in a 25 cm radius FRC reactor, the magnetic field ( Bz) in the z = 0 plane is approximately 1.25 T (or 12.5 kG ). Using the NRL Plasma Formulary's relation for the electron gyroradius:2.38VETeB(fi^) ’Princeton - 107976

[0101] one obtains a gyroradius at this location of approximately 0.04 cm . This value is rather small compared to the electron's radial position, suggesting that the electron must be in a tightly confined cyclotron orbit.

[0102] On the other hand, consider a 150 keV hydrogen ion at the same location. The ion's gyroradius is calculated using the following NRL Plasma Formulary7relation:102VFr ~ -1

[0103] This gives a radius of approximately 3 cm , suggesting that the ion could be in a betatron orbit due to its higher energy and mass. Therefore ions - especially those at high energies - are able to occupy betatron orbits at much larger radial positions than electrons. This difference in orbital behaviour might have important implications for FRC current drive.

[0104] § 4 FRC Current Drive Simulation

[0105] § 4.1 Methodology

[0106] Current is driven by gradients in electric potential. Therefore, it is important to understand the behaviour of an FRC when under the effect of an electric field. RMF code, which models one plasma particle in an FRC over a long period of time, may be used for this purpose. The following sections will outline and present the results of simulations run to determine how the electric field induced by fusion product removal affects current drive in the PFRC device.

[0107] § 4.1.1 Simulation Variables

[0108] The runs presented here each consider one charged particle (ions and electrons respectively) in a reactor-scale environment. Results are obtained for multiple birth radii, with each radius being simulated at varying electric fields.

[0109] Relevant Orbits

[0110] Ions and electrons are only considered in betatron orbits. This is because in cyclotron orbits IHE current drive would act in the direction opposing the B X VB drift which occurs due to a gradient in the magnetic field of an FRC within the separatrix of radius rsy / 2

[0111] The magnetic field O-point line occurs at a radiusof ^-rson the z = Oplane of the FRC. The magnetic field points in opposing directions on either side of the O -point line. The current required to generate this magnetic field must be in the anti-clockwise direction. However, the B X VB drift of a charged particle in a cyclotron orbit moves in the clockwise direction, opposing the magnetic field generation current. On the other hand, a betatron orbit drifts in the anti-clockwise direction and thus contributes to a net current drive.Princeton - 107976

[0112] § 4.1.2 Data Collected

[0113] The dependent variable for these simulations is ^<pfac, a quantity which measures how many times a particle has travelled around the machine with units of radians. Therefore, if ^.cpfac is affected by a change in the electric field, then it suggests that IHE current drive may be feasible - or at least worthy of further investigation. It is important to note that these simulations only consider particles in the z = 0 plane, meaning that a more sophisticated model in 3 -dimensions will be required in future to allow for more complete conclusions to be drawn.

[0114] Further, it will be understood that uniform electric fields applied perpendicular to uniform, low-(3 magnetized plasmas cause charged particle drifts. For charged particles of both signs, the drifts are in the same direction and of precisely the same magnitude. No net current results. These results are when both types of charged particles are in cyclotron orbits and experience the same electric field and magnetic fields throughout their nearly circular orbits.

[0115] Charged particles orbit shapes in FRCs depend on si e(= 0.3rs / Kpi e), particle energy, and canonical momentum. (rsis the separatrix radius, K is the FRC’s elongation, and Pi e is the particle’s gyro-radius at the FRC’s center.) Electrons and ions will have significantly different s, hence have different classes of orbits, see FIG. 10. Typically, near the plasma edge, the electron gyro-radius is a small fraction of the rshence they experience a uniform B throughout each period of their trajectory. In contrast, ions can be in figure-8 or betatron orbits. Thus electrons and ions near the edge will, in the rest of their trajectories, transit regions in the FRC with far different magnetic (and electric) field strengths.

[0116] There is broad agreement on the spatial variation of the magnetic field in FRCs, while that of the electric field is controversial. Electric fields in magnetized plasmas have been studied in cases where the magnetic field is nearly parallel to a material wall and both electrons and ions are in cyclotron orbits. The “usual’' conclusion is that the sheath extends an ion gyroradius into the plasma. However, as noted above, the ions may be in betatron or figure-8 orbits, hence their gyro-radii vary along their trajectories. Also, fusion products in FRC reactors have gyro-radii about 4 times larger than the fuel ions. In brief, a study of the magnetic sheath on FRC fusion reactors is essential. FIG. 11 shows the predicted ion end electron drift velocities for reactor like parameters.

[0117] A number of implementations have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the disclosure. Accordingly, other implementations are within the scope of the following claims.

Claims

Princeton - 107976CLAIMS1. A method for driving current in a field-reversed configuration (FRC) fusion reactor, comprising:creating a negative plasma potential within the FRC by removing positive fusion products from a core plasma to an energy and ash removal shell (EARS); andgenerating an electric field perpendicular to a magnetic field of the FRC, wherein the electric field drives a net current in the FRC due to differential ExB drifts between ions and electrons.

2. The method of claim 1, wherein the electric field is generated predominantly near and perpendicular to a separatrix of the FRC.

3. The method of claim 2, further comprising controlling a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field.

4. The method of any one of claims 1-3, wherein the EARS is physically separated from a separatrix of the FRC.

5. The method of claim 4, further comprising sourcing gas or plasma into the EARS in a divertor region to densify the EARS.

6. The method of any one of claims 1-5, wherein the net current is driven in a toroidal direction of the FRC.

7. The method of claim 6, wherein the net current results from ions in figure- 8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits.

8. A field-reversed configuration (FRC) fusion reactor, comprising:a core plasma region;an energy and ash removal shell (EARS) surrounding the core plasma region; and a control system configured to remove positive fusion products from the core plasma region to the EARS, thereby creating a negative plasma potential and an electric field perpendicular to a magnetic field of the FRC.

9. The FRC fusion reactor of claim 8, wherein the electric field is generated predominantly near and perpendicular to a separatrix of the FRC.Princeton - 10797610. The FRC fusion reactor of claim 9, wherein the control system is further configured to control a loss rate of electrons that encircle a major axis of the FRC in a single cyclotron period to maintain a desired value of the electric field.

11. The FRC fusion reactor of any one of claims 8-10, wherein the EARS is physically separated from a separatrix of the FRC.

12. The FRC fusion reactor of claim 11, further comprising a gas or plasma source configured to inject gas or plasma into the EARS in a divertor region to densify the EARS.

13. The FRC fusion reactor of any one of claims 8-12, wherein the electric field drives a net current in a toroidal direction of the FRC due to differential ExB drifts between ions and electrons.

14. The FRC fusion reactor of claim 13, wherein the net current results from ions in figure-8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits.

15. A method of improving efficiency in a field-reversed configuration (FRC) fusion reactor, comprising:generating fusion reactions in a core plasma of the FRC;selectively removing positive fusion products from the core plasma to create a negative plasma potential; andutilizing the negative plasma potential to drive a net current in the FRC without external current drive systems, collectively removing the inefficiencies in other methods of driving currents in magnetized fusion plasmas.

16. The method of claim 15, wherein selectively removing positive fusion products comprises directing the positive fusion products to an energy and ash removal shell (EARS) surrounding the core plasma.

17. The method of claim 16, wherein the EARS is physically separated from a separatrix of the FRC.

18. The method of claim 17, further comprising sourcing gas or plasma into the EARS in a divertor region to densify the EARS.

19. The method of any one of claims 15-18, wherein utilizing the negative plasma potential comprises generating an electric field perpendicular to a magnetic field of the FRC.

20. The method of claim 19, wherein the net current results from ions in figure- 8 or betatron orbits experiencing less electric field and developing lesser drift velocities compared to electrons in cyclotron orbits.

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