Metallic Neutral Vapours Diffusion in Electron Cyclotron Resonance Ion Sources: Fluid Dynamics and Particle Tracing Simulations

Resistive oven technique is used to inject vapours of metallic species in electron cyclotron resonance (ECR) plasma traps, where plasma provides step-wise ionization of neutral metals, producing charged ion beams for accelerators. We present a numerical survey of metallic species suitable for oven injection in ECR ion sources, studying neutrals diffusion and deposition under molecular flow regime. These aspects depend on geometry of the evaporation inlet, thermodynamics, and plasma parameters, which strongly impact on ionization and charge-exchange rate, thus on the fraction of reacting neutrals. We considered diffusion of metals with and without plasma. The plasma and its parameters have been modelled considering an established self-consistent particle-in-cell model. Numerical predictions might be relevant to reduce the metal consumption, to increase the overall efficiency, and to improve the plasma ion source performances. As test case, we studied the 134Cs isotope, as one of the alkali metals of interest for the modern nuclear physics.


Introduction
Electron cyclotron resonance ion sources (ECRIS) are largely employed to deliver ion beams at several accelerator facilities [1].Their usage is particularly considered for metal ion beams , produced from materials in the solid state, which must be transformed into the gaseous state to feed the plasma [2].Several metal injection techniques can be used depending on many aspects -materials, long time stability, long operation periods without maintenance, constant beam parameters, and low material consumption when using rare isotope materials [3,4].The thermal evaporation of the solid compound via resistively heated ovens is one of the most employed.The development of metal ion beams is one of the most important areas in the accelerator technology, and continuous efforts to optimize and predict the atom-to-ion conversion yield of ovens in ECRIS are required.In this work, we study the metal diffusion dynamics via evaporation in an upcoming ECR facility at the INFN-LNS, the PANDORA plasma trap [5], which will make use of rare radioisotopes for multidisciplinary experimental study in ECR plasmas.This work has been motivated both by the aim to improve the global conversion efficiency of rare and expensive isotope atom to ions (e.g., 48 Ca, 176 Lu), and to explore viable ways to reduce the neutral metals deposition on the plasma chamber wall (e.g,.by 134 Cs), poisoning the signal-to-noise ratios of certain designed measurements in the PANDORA project [5,6] .We performed numerical simulations of 134 Cs atom diffusion, as test case, both as particle fluid and single particle motion, under non-collisional gas regime, for its relevance in the PANDORA research.Calculations have been performed by using COMSOL-Multiphysics © Molecular Flow and the Mathematical Particle Tracing Modules, as well as a Monte Carlo model developed in MATLAB ® to include the impact of plasma ionization and charge-exchange in the atom-to-ion conversion/tracking dynamics.Simulations might help in predicting metal atom conversion efficiency depending on the metal species, plasma target and parameters, and to improve the metal evaporation technique.Results of this study underlines the plasma role on a space-dependent conversion yield, which reflects the strongly anisotropic and in-homogeneous ECR plasma features.Moreover, the non-axisymmetric injection scenario, considered to support the presence of biased disk, led to peculiar deposition and ion transport led by the confinement magnetic field.

Simulations: vapours diffusion in molecular flow regime
The simulation's geometry includes a cylindrical vacuum/plasma chamber reproducing the PANDORA ECR plasma trap geometry, and the oven inlet, reduced to an emitting surface at given pressure, p, and temperature, T. No details on the atom flux from the inner parts of the crucible, or on the impact of the crucible surface's thermodynamics on the evaporation-rate enhancement [7] have been considered.Possible temperature-dependent crossing from noncollisional to collisional regimes have been neglected at the moment.
The caesium gas non-collisional behaviour has been assessed calculating the Knudsen number, K = λ/L, where λ = k B T/(πd 2 P) is the atom mean free path (k B the Boltzmann constant, T the oven temperature, d the atomic diameter, and P the chamber pressure), and L is the characteristic length of the system studied.Parameters assumed for calculations are reported in Table 1.For the vacuum chamber K ∼ 7, which confirms the goodness of the considered regime.

Monte Carlo model for Particle Tracing Study
To provide deeper insights on the ECR plasma dynamics effects to the atom-to-ion conversion, a Monte Carlo (MC) routine was developed in MATLAB, based on the data retrieved from self-consistent (SC) plasma simulations of electron dynamics in the PANDORA plasma trap [11,8,12].Plasma trap parameters for the SC simulations are in Table 1, and further details can be found in [8].SC simulations provided 3D electron density and energy density maps, later used to compute dominant collisional processes reaction rates in the metal atom dynamics: electron-impact ionization (EI) and charge-exchange (CEX).Because of typical in-homogeneous and anisotropic plasma parameters, this resulted in space-dependent reaction rate maps [12], which are inversely related to the EI and CEX cross-sections, σ EI and σ CEX , respectively.The cross-section σ EI was computed according to the general formalism of Lotz [13], while σ CEX was computed according to the standard Müller-Salzborn formula [14].An example of the space dependency for the ionization rate is shown in Fig. 2(c).Data on the cross-sections and then on the reaction rate were used to calculate a space-dependent probability map that thermally diffusing metal atoms go into reactions, considering a plasma support gas of oxygen.In addition, reacted atoms (ions) are transported through the magneto-static field lines till reaching the chamber wall, where neutralization condition was assumed.Figures 2(a-b) show the deposited neutrals on the chamber wall as from simulations.In particular, in Fig. 2(a) the plasma is OFF, deposition is uniform, with some anisotropy originating from the non-axisymmetric oven inlet, and maximum of deposit close to the inlet region -accordingly to FMF/MPT results.This choice was not random, but tailored to the spatial constraints imposed by the biased disk of the trap [15].On the other side, Fig. 2(b) shows a peculiar deposition map strongly depending on the reaction dynamics (plasma ON) , and on the ion transport.The deposited is strongly notuniform and highly anisotropic, with maxima close to the inlet and along the closest magnetic field branch.In this case, the resulting percentage of the deposited material over the amount fluxed for this plasma configuration was around 78%, less than the MPT results, reasonable in the regard of an extended reacting plasma volume and of the more accurate plasma and reactions modelling.This also indicates that about 22% of initial metal atoms reacted and get trapped into the plasma.The number is quite close to the generally experienced overall efficiency of the vaporization technique around the 30%.Finally, the combined plasma dynamics and geometrical injection aspects in this study, have provided relevant data to investigate on the short-lived isotopes chamber-wall poisoning, underlining that part of the trap wall is completely not affected by the material deposition.

Conclusion
The numerical tool and results have been presented in the framework of the PANDORA plasma trap.They can be easily extended to other ECR plasmas and traps, helping in predicting the metal vapors diffusion and dynamics, and hence in giving hints on viable ways to improve the technique efficiency.The vapors dynamics depends both on species and the plasma target.Thus, having a reliable target model would support theoretical predictions of the overall atom-to-ion conversion yield.In this view, and in collaboration with the ion source department of the GSI, we considered to apply the tool to study more cases, e.g., the evaporation of rare earths and rare isotopes, such as 48 Ca, highly required from the nuclear physics community.This will provide experimental benchmark of the numerical study, e.g., based on the deposition tracks at the extraction electrodes, plasma chamber, and injection plunge, as well as an R&D activity on this metal ion source technique, towards radioactive metal ion beam production improvements.It was possible to provide deposition data for further simulations regarding the nuclear β-decay rate of neutral radioisotope, e.g., 134 Cs, which is one of the physics case of PANDORA.The γdecay rate of neutral-Cs's daughter, 134 Ba nuclei, can poison the signal-to-noise ratio, enriching the plasma background, in γ-tagging operations designed to study the β-decay rate change in IOP Publishing doi:10.1088/1742-6596/2687/5/0520275 plasma [16].The content of these calculations, and further improvements of the diffusion tool, including thermal re-emission at the chamber and multiple charge-state ion dynamics impact on deposited metal, will be part of another ongoing work.

Figure 1 .
Figure 1.(a) Model geometry: cylindrical chamber, resistive oven inlet tilted towards the center of the plasmoid boundary reaction surface.Vacuum conditions applied to outer boundaries reproduce vacuum pump operations.Freezing conditions are applied to the chamber wall, while sticking (else diffusing) ones with probability γ = 1 − λ ion /L p are applied to the plasmoid.(b) Total particle number density Log ρ [m −3 ] reconstruction plot: results from FMF module (top) and MPT (bottom) are contrasted, evincing the most deposited region close to the oven inlet.

2. 1 .
Fluid Dynamics StudyA fictitious volume corresponding to the plasmoid core volume within the ECR isomagnetic surface was assumed in the model inside the chamber, as shown in Fig.1(a).This to keep account of an approximated lastreacting boundary surface on which the atomto-ion conversion mostly takes place with a certain average probability, γ = 1 − λ ion /L p .It also allows to simulate the depletion of neutrals from the oven flux, so as from the deposited.The mean free path λ ion for neutral going ionized was evaluated at the ECR surface plasma conditions.The Free Molecular Flow (FMF) stationary study in COMSOL uses the angular coefficient method[9], which computes the molecular flow by summing the flux arriving at a surface from all other surfaces in its line of sight.The emitted molecular flux, J in = pN A / √ 2πRMT [1/m 2 s], follows from the known Hertz-Knudsen model[10].Free from kinetic effects in the non-collisional regime, a time-dependent mathematical particle tracing (MPT) has been also performed tracking N p particles, with thermal average velocity, v th = k B T/M, for a simulation time, t span .Results of reconstructed particle number density [1/m 3 ] from both methods are contrasted in Fig.1(b), showing a good agreement on the anisotropic deposition region.The percentage of the deposited material over the amount fluxed, from MPT calculations, for this configuration is around 85%.