plasma Article An Ion Source’s View of Its Plasma Peter Spädtke Ingenieurbüro für Naturwissenschaft und Programmentwicklung, Junkernstr. 99, 65205 Wiesbaden, Germany; [email protected] Abstract: Modeling of ion beam extraction from an ECRIS requires special procedures in order to achieve results similar to what is found experimentally. The initial plasma conditions must be included for consistency between experiment and simulation. Space charge forces and their compensation of the extracted ion beam become important with increasing beam intensity. Here we consider the various beam-plasma conditions that occur along any beam line. Keywords: ion beam; extraction; space charge 1. Introduction High quality ion beams are required in a growing number of fundamental and tech- nological areas, and a cold plasma from which the ion beam can be formed is a necessary prerequisite. By “high quality” it is meant that all particles within the assemblage behave similarly to each other, implying in turn that their temperature is low. This is equivalent to the requirement of having a high density in six-dimensional phase space, given by three spatial coordinates and three momentum coordinates. According to Liouville it can be shown that under certain conditions (e.g., only conservative forces act) this 6D phase space volume remains constant. Citation: Spädtke, P. An Ion Source’s Many different types of ion sources have been developed and are available, and good View of Its Plasma. Plasma 2021, 4, overviews have been published [1,2]. Still, each ion source has its own set of specific 345–358. https://doi.org/10.3390 working conditions and therefore plasma properties. Some of these unique peculiarities /plasma4020023 have to be considered when computer simulations are used to model ion beam extraction from the ion source and the subsequent beam transport. Academic Editor: Ian G. Brown Computer codes have been developed to model the behavior of charged particles in electric and magnetic fields. They are used to optimize the source extraction geome- Received: 12 May 2021 try, to preserve the quality of the cold plasma, and to not decrease the beam quality by Accepted: 9 June 2021 aberrations due to nonlinear optics or similar effects. Published: 11 June 2021 In the case of a plasma where collisions (not a conservative force) ensure a homoge- neous plasma density distribution near the extraction aperture, Child’s Law describes the Publisher’s Note: MDPI stays neutral 4 p 3 2 matching condition with j = pe0 q/mDF 2 /d , where j is the current density, e0 is the with regard to jurisdictional claims in 9 permittivity of free space, m the mass and q the charge of the ion, and DF is the potential published maps and institutional affil- difference across the acceleration gap d. This matching condition describes the equilibrium iations. between plasma pressure and extraction field strength to achieve the best possible beam. If the plasma density is too low, the ion beam will be overfocused and vice versa if the plasma density is too high the ion beam becomes divergent due to high space charge forces. In between both extreme densities the ion beam has its highest phase space density. This Copyright: © 2021 by the author. condition is called perveance match. Licensee MDPI, Basel, Switzerland. If magnetic fields are not negligible, the plasma is anisotropic, and Child’s Law is This article is an open access article not necessarily fulfilled everywhere over the extraction aperture region. Local differences distributed under the terms and between emission-limited and space-charge-limited flow can occur. conditions of the Creative Commons Whereas in the magnetic field-free case the six dimensions can be separated to fewer Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ dimensions without loss of generality, this is not true anymore in the magnetic case due 4.0/). to the coupling of different subspaces. Coupling between different dimensions of this 6D Plasma 2021, 4, 345–358. https://doi.org/10.3390/plasma4020023 https://www.mdpi.com/journal/plasma Plasma 2021, 4 346 phase space might also take place, for example, if space charge forces are relevant or if collisions between particles become important. The initial conditions of the plasma particles will determine the beam quality: the initial plasma temperature will be conserved of course, and the absolute phase space density will shrink due to the acceleration. Ions starting within a solenoidal field will execute skewed trajectories according to Busch’s theorem. Some ion sources use additional magnetic fields, which may well introduce even more aberrations. Such dependencies need to be taken into account when simulating a system. Plasma physics plays an important role for a correct description of any of these ion optical systems. 2. The Ion Beam Source In this article, the application of computer simulation will be demonstrated for the special case of an Electron Cyclotron Resonance Ion Source (ECRIS) and a recipe will be given for the simulation procedure for this specific kind of ion source. An ECRIS can best be described as a combination of two different machines: an electron accelerator to provide high energy electrons that are capable of stepwise ionizing heavy ions to higher charge states and storage for those ions inside a plasma chamber (PC) until a certain ion charge state distribution (CSD) has been achieved; this is then followed by extraction of these ions from the ion source for their intended application. One version of such an ECRIS is CAPRICE [3], shown in Figure1. This ion source was developed from a precursor device used in early fusion experiments, with which it was shown that the ion particle density ni necessary for fusion of Deuterium and Tritium (2De + 3T ! 4He + n + 17.6 MeV) could not be achieved efficiently enough with a simple mirror device. Additional radial confinement force seemed to be necessary to increase the plasma density. By increasing the radial confinement by means of an additional hexapole magnetic field colinear with the magnetic mirror field, the particle densities were increased, but the numbers were still not adequate for fusion, and the worldwide controlled fusion research effort experiments went on to other devices such as torus-shaped stellarators and tokamaks. Geller [4] recognized the potential of this improved mirror device for application as an ion source, especially for higher charge states of heavy ions. Technical advances in magnet technology (superconductivity) and high frequency power generation (GHz technology in the kW range and up) made this type of ion source possible and attractive. Particle accelerators could greatly benefit from the availability of higher charge states of heavy ions. The use of higher charge states in an accelerator means higher velocity gain for the same potential drop across the acceleration gap. The lower the m/q ratio (mass-to-charge), the higher the velocity of the ions and the more simply (shorter) an rf accelerator can be built. In most cases, the ion velocity has to be matched to the geometry and frequency of the following rf-accelerator. Since the very beginning of ECRIS development, effort has been directed to improve- ments toward higher intensities for higher charge states and the complex dependencies of this multiparameter field on the plasma properties and their impact on beam quality have been clarified. Some of these are listed in the following. The dependency of the intensity and CSD of the extracted ion beam on frequency and magnetic field strength of the ECRIS device was one of the first important features to be observed and clarified, but there are other parameters that also influence the particle density in the plasma from which the ion beam can be extracted. Even small frequency shifts of the heating power have a strong influence on the extracted beam current. These frequency shifts are so small that their effect cannot be explained via changes in the electron cyclotron resonance condition (wce [GHz] = 28 · B [T]). Mixing different frequencies to heat the plasma shows similar effects. Adding various isotopes to the plasma might influence the extractable ion beam current, as does providing additional electrons to the plasma (for example, by using biased probes). The volume of the PC might play a role, via the power density within this volume. Furthermore, there are more dependencies. Different theories Plasma 2021, 4 347 have been proposed to explain the influence of each parametric variation, but up to now no complete theory has been developed to describe the generation and confinement of the ECRIS plasma. Figure 1. The Caprice ECRIS. Normal conducting coils (shown in green) together with a permanent magnet hexapole (shown in red). Microwave power is fed to the source from a high power generator at 14.5 GHz, connected by waveguide to the ECRIS. Iron parts in yellow. Gas is injected from the side opposite to the extraction side. Besides these unknown dependencies, from a mathematical point of view two different types of differential equations have to be solved when simulation programs are used: boundary values are required for the solution of the equations for the electromagnetic potential (Laplace and Poisson) and solving the trajectories is an initial value problem for ray tracing. For the first case a mesh is used on which the differential equation (DE) can be solved, whereas the 2nd order DE for ray tracing can be transformed and integrated twice, using these calculated fields. 3. Ion Beam Extraction Computer simulation can contribute to the optimization of a technical system as well as the experimental approach. Simulation codes for charged particles in electromagnetic fields were developed as early as the 1970s, initially for electrons emitted from a thermionic cathode [5]. Simulation codes followed shortly for ions emitted from a plasma with an assumed homogeneous density distribution [6].