Finite Element Method Validity on Biconic Cusp Magnetic Confinement Jonathan Kelley June 3, 2017 1 Abstract Energy generation through nuclear fusion has been a focus of research for several decades. One such concept employs inertial electrostatic confinement of charged ions, accelerating fusion fuel with sufficient kinetic energy to satisfy the lawson criterion. This concept was analyzed using the finite element method using the EMWorks FEM pacakge for SOLIDWORKS, with a focus on magnetic and electrostatic field interactions in a biconic and quasipsherical cusp system. The SOLIDWORKS model was then used to construct a physical system with according characteristics. An analysis was performed measuring the magnetic field of the physical model with comparison to the finite element model. A single and dual coil biconic cusp system was constructed to match the SOLIDWORKS model specifications and tested with a hall effect magnetic field probe. The results proved that the β field behavior of the FEM model was indeed similar to the experimental devices. However, continual development of the FEM model and coil construction is required to numerically validate the FEM model for use in inertial electrostatic confinement fusion. 1 Final Paper EE442 1 2 Introduction Nuclear fusion with the aim of generating usable electric power has undergone significant research for several decades [20]. According to Ongena and Ogawa, nuclear fusion is appealing as a new source of power because of its desirable en- vironmental considerations and virtually inexhaustible fuel supply [15]. There are currently many different strategies for generating the fusion conditions re- quired for efficient atomic nuclei reactions. Some systems involve containing a heated plasma while others focus on bringing solid fuel to ignition conditions. This paper explores the design of the electromagnetic coils for fusion systems with the unique characteristic of a combination electrostatic and magnetostatic fuel acceleration. The β fields are studied in depth against a new finite ele- ment method that allows for rapid iteration and design in the SOLIDWORKS software package. Modern research in the development of nuclear fusion devices follows two main strategies. The first strategy involves the use of magnetic fields to contain a very energetic dense plasma of light atomic nuclei[14]. This approach is com- monly found in systems like stellerators and tokamaks such as the Wendelstein 7-X project in Germany and the ITER project in France. A second approach fol- lows inertial confinement of a solid fuel pellet that is heated to fusion conditions by a laser or ion beam [4]. This technique has been successfully demonstrated in the National Ignition Facility at the Lawrence Livermore National Laboratory. While these techniques have been moderately successful, there are several other methods of achieving fusion conditions with the intent to generate electrical energy. Of the other various nuclear fusion strategies, the formation of a potential well through the confinement of electrons has shown promising results. EMC2 has reported neutron emission of 1E9 fus/sec [19] with a quasi-spherical mag- netic trap and ionized deuterium injection. The potential well created by mag- netic cusp confinement of electrons generates substantial electric potential for the positively charged deuterium gas to reach sufficient kinetic energy for fusion. One such device that utilizes this strategy is the Polywell fusor. The Polywell system is an improvement on traditional electrostatic confinement as there are no electron losses due to conduction with physical cathode. The electrical po- tential as generated by the confined electrons serves as a "virtual cathode," accelerating the fuel ions. Despite being an adequate solution for conduction losses, the official Polywell design [8] continues to struggle from losses due to bremsstrahlung radiation [3], conduction losses to unshielded construction [16], and cusp losses [11] by elec- tron leakage [2]. In the 1991 paper on electron leakage, Bussard describes that cusp losses constitute the majority of losses in the simulated Polywell system. Conduction losses to joints and unshielded surfaces contributed the majority of loss for the early Polywell design, but this was identified and rounder solenoids and shielding was implemented to mitigate conduction effects [8]. It is estimated that no more than 3E-5 fractional surface area of the machine can be unshielded to fully mitigate conduction losses [19]. Electron leakage through point cusps Final Paper EE442 2 generated by the magnetic field serves as the current major loss mechanism for Polywell theory[19]. As electron leakage from the containment field is the major contribution to cusp loss, it is important to identify the factors that lead to poor confinement. There are several aspects to quasi-spherical magnetic trapping that impact effi- ciency and potential well depth. Efficiency when concerning this system relates the amount of electrons injected into the well and the depth of the well itself. Systems that do not trap the electrons for a substantial amount of time will be much more inefficient as more electrons are required to maintain the well. The electrons lost in this system are predicted and verified to escape through point cusps generated by parallel magnetic fields located in near the vertexes and face centers of the quasi-spherical system. In a complete cubic Polywell system, this corresponds to 27 point cusps [5] and 10 linear cusps when spaced far apart. The classic biconic cusp system has only two point cusps and a very large ring cusp along the mid plane of the two electromagnets. It has been found that linear and ring cusps contribute to the majority of loss for a Polywell system [17], however attempts to reduce such cusps in favor of more point cusps has only demonstrated a very small improvement. This is because electron loss is proportional to the total amount of point cusps in the system. Despite the similar loss rates, confinement time of electrons was increased by 2.5 times that of the standard 27 point cusp system [18]. While the losses and transit times associated with the point cusps are easier to calculate, the cusps created by coils spaced less than their critical distance are much more complicated and thusly named "funny cusps." It is key when designing a magnetic trapping system to minimize spacing between coils as the complex cusps are eliminated. In addi- tion to minimzing loss, recent simulations have also demonstrated an increased transit time for electrons entering the null point [13]. This is valuable as per the Lawson Criteria [12] where greater confinement time increases the likelihood of fusion. Despite the realization that high beta confinement increases the potential well in a Polywell system, the plasma must first enter low beta and experience similar loss mechanisms. There have been several studies that successfully model electron trajectory in a low beta systems because electrostatic effects between the electrons is ignored; the plasma pressure is low enough to assume indepen- dent particle motion[13, 5, 6]. Ultimately, a Polywell system with reduced loss will consist of high beta plasma confinement and the reduction of point cusps. This will increase electron density and minimize loss due to poor cusp confine- ment [8]. A major point of interest is reducing losses due to point and line cusps generated by coil arrangement. There are several ideas that involve electrostatic repulsion as a virtual barrier to electrons escaping through point and line cusps. In a traditional magnetic mirror system, plasma is confined axially symmetric through two sets of electromagnetic coils. However, the charged particles of the plasma follow the magnetic field lines and expand in the space between the coils [1]. Early researchers implemented Joeffe bars as a way to stabilize plasma in this region, but it was determined that Joeffe bars do not completely close ring cusps, but rather form sets of linear cusps [17]. In addition, the geometry of the Final Paper EE442 3 Joeffe bars does not integrate with the quasi-spherical polyhedron arrangement of the confinement coils. The second theory of eliminating point cusp loss is through the use of elec- trostatic repeller plates that generate an electric field barrier. In a low beta environment, point cusp loss can be modeled by the Child-Langmuir law and the amount of power lost can be estimated. While repeller plates demonstrate loss in theory, they have not yet been proven to adequately reduce electron loss [5]. Dolan explored the concept of penning traps and electrostatic plugging in a magnetic confinement system with traditional plasma behavior. However, it is noted that the Polywell and penning trap systems would benefit from elec- trostatic plugging despite inadequate performance on spindle cusp systems[7]. However, it is noted that repeller plates would suffer from conduction losses if placed within an effective range for the point cusps. Despite this, the concept of electrostatic plugging remains a valid solution to reducing point cusp losses. A simulation by an Australian research team includes both magnetic and electro- static fields contained within the device with the electric field source centered about the coil geometry [10]. The resulting structure is a combination of classic insulating magnetic coils and an external electric field that generates a mirror electric point cusp within the coil face to prevent electron loss. In a hybrid system that combines electric and magnetic fields as described in the Sydney experiment, the confinement time of electrons is increased. Due to the addition of the electric field, the strength of the virtual anode is significantly increased and more energy is transferred to the ion fuel, raising the rate of fusion. The virtual cathode generated by electrons in the magnetic confinement system works in tandem with a new virtual anode generated by the electric field. Currently, this is the only study that investigates the combination of electric fields and magnetic fields in a biconic cusp configuration with high fidelity.