# FRIENDS3 project - Preliminary simulation report

Wenling Dong, Vladimir Manea

► To cite this version:

Wenling Dong, Vladimir Manea. FRIENDS3 project - Preliminary simulation report. IJCLab. 2022. [hal-05234546](https://hal.science/hal-05234546v1)

HAL Id: hal-05234546

https://hal.science/hal-05234546v1

Submitted on 2 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
FRIENDS³ project

Preliminary simulation report

Wenling Dong¹ and Vladimir Manea¹

¹Université Paris-Saclay, IN2P3/CNRS, IJCLab, 91405 Orsay, France

19 April 2022

Contents

<table><tr><td>1 Introduction</td><td>1</td></tr><tr><td>2 Phenomena involved in gas-cell simulations</td><td>2</td></tr><tr><td>2.1 Simulation of the gas flow</td><td>3</td></tr><tr><td>2.2 Electric and magnetic fields</td><td>4</td></tr><tr><td>2.3 Ion motion in the gas environment</td><td>4</td></tr><tr><td>2.3.1 Individual ion flying</td><td>5</td></tr><tr><td>2.3.2 Density-based approach</td><td>6</td></tr><tr><td>2.4 Densities of diluted species</td><td>7</td></tr><tr><td>2.5 Interaction of ions with the diluted species</td><td>7</td></tr><tr><td>3 Simulation of the gas cell</td><td>7</td></tr><tr><td>3.1 General considerations</td><td>7</td></tr><tr><td>3.2 COMSOL model</td><td>9</td></tr><tr><td>3.2.1 Geometry and electric field optimization</td><td>9</td></tr><tr><td>3.2.2 Simulation of the gas flow</td><td>10</td></tr><tr><td>3.2.3 Simulation of the ion motion</td><td>12</td></tr><tr><td>3.2.4 Simulation of electron motion</td><td>14</td></tr><tr><td>3.3 SIMION simulation studies</td><td>16</td></tr><tr><td>3.3.1 COMSOL benchmark and effect of diffusion</td><td>16</td></tr><tr><td>3.3.2 Effect of the exit hole diameter</td><td>19</td></tr><tr><td>3.3.3 Effect of the tube radius</td><td>19</td></tr><tr><td>3.3.4 Efficiency and extraction-time maps</td><td>22</td></tr><tr><td>4 Conclusions and outlook</td><td>24</td></tr></table>

1 Introduction

The gas cell represents an essential component of the S³-LEB installation, being the interface between the high-energy S³ beam line and the low-energy ion-transport system [1]. The fusion-evaporation recoils from S³ enter the gas cell through a thin window that separates the argon-filled volume from the spectrometer vacuum and lose energy by collisions with the buffer gas. The top view of the first-generation S³-LEB gas cell and the S-shape RFQ is shown in fig. 1. A detailed description of the setup is given in [2].

As discussed in the scientific document of the FRIENDS³ project [3], the first version of the S³-LEB gas cell shown in fig. 1 is limited in extraction time to about 500 ms for
a 1 mm exit hole diameter (and about half for a 2 mm diameter) [4, 5, 6], and relies for neutralization on electron-ion recombination in the plasma created by the stopped ion beam, which is a process dependent on the beam intensity. The interest in a faster and neutralization-capable gas cell for studying the shorter lived nuclei at S³ constitutes the motivation of the FRIENDS³ project. The first approach of the project is to boost the ion extraction from the stopping region (where the gas flow is very slow) by the use of an electrical field and to perform the neutralization by an artificially created electron cloud close to the gas-cell exit (where the gas flow is very fast). A similar approach is being pursued at GSI, where the neutralization mechanism is a heated filament onto which the ions are guided, then neutralized, before being re-evaporated [7].

<figure><img src="image_2.png" /><figcaption>Figure 1: Top view of the first version of the S³-LEB gas cell.</figcaption></figure>

An important work package of the FRIENDS³ project is dedicated to the simulation of the gas-cell phenomena relevant to the project objectives in a geometry compatible with the additional constraints imposed by the S³-LEB setup (S³ interface, space and layout, pumping capacity etc.). The simulated geometry would eventually translate into a gas-cell prototype which will be built and tested off-line. This prototype will later be converted into a design which can be implemented at S³-LEB. This two-stage approach ensures that in the initial development phase some of the S³-LEB constraints can be relaxed for a gain in simplicity and cost, while focusing on the more fundamental aspects of the studied phenomena. It also means that the first simulations can use a simplified geometry too.

The first stage of the simulation work package is a preliminary study of the software tools (programs, physics interfaces, modules and approaches) required for describing the gas-cell phenomena, as well as a validation, using these tools, of the chosen simplified geometry in terms of key performance parameters such as the gas cell extraction efficiency, extraction time and time available for the neutralization process. The present report is the deliverable of this project stage, consisting of two main sections. The first section is an overview of the phenomena which need to be accounted for in gas-cell simulations and of the different approaches which can be used for describing them, while the second section presents some simulation results using these approaches, including benchmarks and first optimization studies.

## 2 Phenomena involved in gas-cell simulations

A comprehensive simulation of a gas cell for the stopping, extraction and neutralization of radioactive ion species needs to address five important classes of phenomena:
1. The gas flow through the cell until its evacuation through the exit hole and the resulting gas pressure, temperature and velocity fields. For the phenomena taking place in the gas-cell volume, the properties of the supersonic jet formed after the exit from the gas cell are not relevant and only the outflow rate plays a role.

2. The distribution of electric and magnetic fields in the gas cell, generated by a given voltage or current distribution.

3. The motion of the ions of interest under the combined action of the gas flow and of the electric (and/or magnetic) fields.

4. The density distributions of diluted species, electrons and impurities, either charged or neutral, present in the buffer gas or generated artificially (for example electrons induced by a discharge or energy deposition by a beta source or the heavy-ion beam itself).

5. The interaction of the ions of interest with the diluted species, leading to different reactions, among which the neutralization.

In the following, we will discuss the available methods for simulating these phenomena.

## 2.1 Simulation of the gas flow

The gas flow in the cell can be simulated using COMSOL [8], which can solve the equations for compressible flow, derived from the Navier-Stokes equations:

$$ 
\begin{aligned}
\rho(\mathbf{u} \cdot \nabla)\mathbf{u} &= \nabla \cdot [-p\mathbf{I} + \mathbf{K}] + \mathbf{F} \\
\nabla \cdot (\rho\mathbf{u}) &= 0 \\
\mathbf{K} &= \mu (\nabla\mathbf{u} + (\nabla\mathbf{u})^T) - \frac{2}{3}\mu(\nabla \cdot \mathbf{u})\mathbf{I},
\end{aligned}
\quad (1) $$

where $\rho$ is the gas density, $\mathbf{u}$ is the gas velocity, $p$ is the pressure, $\mathbf{I}$ is the identity matrix, $\mu$ is the dynamic viscosity of the gas and $\mathbf{F}$ is the volume force vector, which in our case is zero. In this framework, eq. (1) establishes a relation between the gas density/pressure and velocity, which can be solved by also taking into account the boundary conditions and the gas properties. The equations are solved assuming laminar flow and a low Mach number ($M < 0.3$). In the present simulations, the effect of turbulence from the gas injection is not taken into account. The gas is injected by an inlet and the exit hole of the gas cell is defined as an outlet. The inlet condition is given by a certain mass-flow value. The outlet can be defined as a constant pressure boundary ($p = 0$), but close to the outlet this would lead to velocities comparable to the speed of sound, hence a breaking of the condition of $M < 0.3$. The solution is to end the gas-cell volume early, at an axial coordinate where the $M < 0.3$ condition is always satisfied. The outlet would thus be slightly before the actual position of the nozzle throat. Instead of the pressure condition, one could define the outlet using a volume flow condition. The value would be taken equal to the volume flow through the nozzle, which is given by [4, 6]:

$$ Q_V = \sqrt{\frac{\gamma RT}{M}} \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma+1}{2(\gamma-1)}} \frac{\pi}{2} d^2, \quad (2) $$

where $\gamma$ is the isoentropic expansion coefficient, which for argon and helium is 5/3, $R$ is the universal gas constant, $T$ and $M$ are the gas absolute temperature and molar mass and $d$ is the nozzle throat diameter. A quick formula for computing the volume flow rate from eq. (2) is [6]:

$$ Q_V[\text{L/s}] = 0.052(d[\text{mm}]^2)\sqrt{\frac{T}{A}}, \quad (3) $$
where A is the atomic mass of the gas. By modifying the inlet mass flow, one can obtain stationary solutions corresponding to different pressures in the gas cell.

The temperature is only an input parameter of the simulation, but does not enter the equations, its role being limited to defining the values of the material properties. Coupling the Navier-Stokes problem to a heat-transfer problem is however possible, allowing to include the temperature values from a dedicated simulation. The effect of temperature gradients is however a higher-order refinement of the simulations of the gas cell performance, therefore simply giving the temperature as a constant input parameter is sufficient for this work. The physics interface required to solve these equations in the laminar, compressible-flow regime with small Mach number is included in the basic COMSOL license and does not require the purchase of any special module.

The simulation results can be re-injected as input parameters to other COMSOL simulations, or they can be saved to a file and used with other simulation programs, such as SIMION [9] with the Statistical Diffusion Simulation model [10], which requires the gas velocities, pressures and temperatures to be given as input arrays. A conversion of the data from one software to the other will be necessary in this case.

## 2.2 Electric and magnetic fields

Computing electric and magnetic fields requires solving Maxwell's equations using certain boundary conditions given by the gas-cell and electrode geometry together with the applied voltages and currents. Both COMSOL and SIMION have the capability to solve these equations for given configurations of electrodes and coils. In the case of COMSOL, the basic license is sufficient to perform electrostatic calculations, which are the goal of the preliminary simulations. The system of equations which is solved is:

$$ \begin{gathered} \nabla \cdot (\epsilon_0 \epsilon_r \mathbf{E}) = \rho_c \\ \mathbf{E} = -\nabla V, \end{gathered} \quad (4) $$

where **E** is the electrical field intensity, **V** is the electrical potential, $\rho_c$ is the charge density and $\epsilon_{0,r}$ are the vacuum and relative electrical permeability of the medium.

The fact that each software can perform its own calculations means that no data exporting will be necessary.

## 2.3 Ion motion in the gas environment

The pressure in the gas cells to be simulated in this work varies between 10 – 500 mbar. In such a high-pressure gas environment, ions evolve under the combined action of the acceleration by the electrical field and of collisions with the background buffer gas. Normally, all relevant dynamics can be accounted for if the collisions between the ions and the buffer gas are modelled explicitly in addition to the electrical forces, however this approach can only work in a primary vacuum regime (pressures below or maximum around 1 mbar). For the pressures of interest in this work, accounting explicitly for the ion-neutral collisions would lead to unreasonably long calculation times.

The collisions of the ions with the buffer gas give rise to two types of phenomena. The first one is the so-called drag force, which is a continuum-type effect (also present in fluid mechanics) manifesting whenever there is a difference between the ion velocity and the average gas velocity. The second one is the diffusion phenomenon, through which the ion cloud (now looking from a density perspective) is expected to gradually disperse in the direction of the concentration gradient. From a single-ion perspective, diffusion would manifest as a random walk of the ion with respect to the trajectories resulting from the deterministic forces at play (electrical and drag).

In the following we will discuss two possible approaches for studying the ion motion in the gas environment. The first approach aims to integrate the equation of motion for each
ion individually, taking into account the forces acting upon it. The problem can be solved for one ion or many and provides the trajectories and velocities of individual ions, however many ions must be flown in order to also perform a statistical analysis of their properties. It is also a natural framework to study the transport efficiency of the gas cell and to determine propagation/extraction times. The second approach treats ions as a diluted, charged fluid propagating in the buffer gas and integrates the equations governing the diffusion, convection and migration of the fluid density (or concentration). One thus loses the information on the individual ions, but naturally obtains statistical properties governing the ion ensemble.

### 2.3.1 Individual ion flying

The standard approach to computing ion trajectories in the gas cell would be to integrate the equation of motion of the individual ions under the combined action of the given forces. Ignoring in a first instance the diffusion phenomenon, a first approximation of the ion trajectories can be obtained by only solving for the electrical and drag force. The former has the standard expression as a function of the electrical field $F_e = qE$, while the latter can be expressed as $F_{vis} = -m\delta(v - u)$, where $v$ is the ion velocity and $u$ is the macroscopic gas flow velocity. This drag force is also called Stokes drag or viscous-damping force, or sometimes Stokes damping, due to its similarity to the Stokes drag force of fluid mechanics, which is the friction force between a laminar fluid and a perfect sphere. In the case of a stationary gas, the resulting motion will lead to an equilibrium (or drift) velocity along the electrical fields lines:

$$F_e + F_{vis} = 0 \rightarrow v_{drift} = \frac{q}{m\delta}E = KE, \qquad (5)$$

where K is by definition the ion mobility, often expressed as:

$$K = K_0 \frac{T/T_N}{P/P_N}, \qquad (6)$$

with $K_0$ the so-called reduced mobility, $T/P$ the absolute gas temperature and pressure and the $N$ subscript denoting the “normal” conditions ($T_N = 273.15$ K and $P_N = 1$ atm).

Both COMSOL and SIMION can model the electrical forces (which is native to the two programs) and can include the drag force as a user-defined one. It is therefore possible to model the force in the same way with both programs. In case of COMSOL, the relevant physics interface is the Charged Particle Tracing, while the pressure, temperature and gas-velocity fields required for determining the drag force can be computed in a prior calculation (see the subsection on the simulation of the gas flow). In the case of SIMION, the Statistical Diffusion Simulation model can be employed [10], which includes a drag force, however the gas velocities, pressures and temperatures must be provided as input files and therefore will rely on a prior COMSOL calculation.

In addition to the motion under the action of the electrical and drag forces, the diffusion phenomenon would manifest as a random walk of the ions away from the deterministic trajectories resulting from the former two. This phenomenon however cannot be modeled by a deterministic force and requires a statistical treatment which can mimic its effect without having to deal with all the individual collisions. Currently, COMSOL does not include diffusion in the Charged Particle Tracing physics interface, which is only equipped with the possibility to simulate individual collisions and hence is limited in the pressure range of interest.

An alternative in COMSOL to Charged Particle Tracing is the physics interface called Particle Tracing for Fluid Flow. This interface is dedicated primarily to fluid mechanics and particles in the sense of small spherical solids or droplets in a fluid. The particles are defined using density and diameter. Concerning the particle-fluid interaction, the defining fluid parameter is the viscosity. Mobility does not enter explicitly. Nevertheless, electrical forces can be included. The advantage of this physics interface is that it can also include by built-in domain definitions both a drag force (which can take slightly different formulations)
and a so-called “Brownian force”, the latter allowing to model the diffusion phenomenon. The disadvantage is that the fluid-mechanics formulation requires establishing effective links between the parameters of ions in the gas phase (masses of the ion and of the buffer-gas atoms, collision cross section, temperature and mobility) and the native parameters of the Particle Tracing for Fluid Flow interface. This approach is more difficult to define and later validate against experiment, therefore the Charged Particle Tracing physics interface is favored.

For simulating the effect of diffusion, the only current solution is the SDS model of SIMION [10]. The model is defined through a user file which implements both the drag force and the diffusion behavior. The modeling of diffusion starts with the observation that, after a sufficiently high number of collisions with the buffer gas atoms, the average distance travelled by an ion from the starting position is proportional to the square root of the number of collisions, which is true also when the ion mass differs significantly from the mass of the buffer gas atoms. In the pressure regime of interest, the collision rate is sufficiently large that the condition of a large number of collisions is satisfied for one integration step. Furthermore, the SDS model is supplied with the results of a large Monte Carlo simulation performed in advance, giving the distribution of unit-normalized scattering radii (assuming unit scattering per collision) after 100000 collisions, for different mass ratios between the ion of interest and the buffer gas (one distribution per mass ratio). For a given simulation, one can after each integration step sample the predetermined distributions and then linearly interpolate the extracted values to the real mass ratio, thus obtaining a value of the unit-normalized scattering radius for the given integration step and for 100000 collisions. By knowing the duration of the integration step, the local pressure and temperature of the gas, it is possible to compute the expected number of collision in the given step. This allows to scale by the square root law the scattering radius obtained as described above from 100000 to the expected number of collision. Finally, the unit-normalized scattering radius is converted to a radius in units of length by multiplying it with the mean free path of the ion at the given pressure and temperature conditions. For each integration step, after computing the trajectory using the deterministic forces, the SDS model induces a jump of the ion in a random direction by the scattering radius determined for the given step as described above. More information about the numerical procedure can be found in the manual of the model which is copied by default in the local SIMION installation folder (in the *Examples* subfolder).

### 2.3.2 Density-based approach

Alternatively to the approach detailed in the previous subsection, one can describe the ions as a diluted fluid, using as degrees of freedom, instead of the particle positions, the concentration of the diluted fluid in the background gas at every point in space. The evolution of the ions would then be studied by solving the equations which describe the migration (evolution in an electrical field), convection (evolution with the fluid flow) and diffusion of their corresponding concentration. This can be achieved with COMSOL by using the Transport of Diluted Species physics interface. In this interface, the global equation for the concentration c of an ion species is written as:

$$ \frac{\partial c}{\partial t} + \nabla \cdot (-D\nabla c + zKFc\mathbf{E} + cu) = 0, \quad (7) $$

where the first term between parentheses describes the diffusion phenomenon (D is the diffusion constant), the second term describes migration (z is the ion charge state, K is the mobility, F is the Faraday constant and E is the electrical field) and the third term describes convection (u is the fluid velocity). The right-hand side of the equation can be 0 or a production rate R (allowing to also include the effect of reactions).

This approach has been used in describing the extraction of neutralized species from the stopping region of the S³-LEB gas cell [4, 5, 11]. Because no electrical charges or fields were taken into account, the migration term of eq. (7) was not included and the simulation
only addressed the convection and diffusion phenomena. The gas velocity involved in the convection term was obtained by a preceding laminar flow calculation. In order to use this physics interface for simulating transport efficiencies, the starting region of the neutralized species was modeled as a source domain with a non-zero production rate *R* and subsequently a stationary study was performed. The total source intensity was calculated as the integral flux through the boundary of the source domain, while the surviving intensity was calculated as the corresponding flux through the gas-cell output boundary (corresponding to the nozzle throat). The ratio of the two quantities gave the extraction efficiency. The extraction time was computed by defining the source domain as a volume of some known concentration and no production rate and performing a time-dependent calculation, while recording the concentration at the level of the exit hole. The extraction efficiency and time could also be calculated as a function of the starting position in the beam stopping volume of the gas cell, by defining the source as a small sphere and performing the stationary and the time-dependent calculations for different sphere positions. This allowed obtaining efficiency and extraction-time maps (see for example the results of [6]).

## 2.4 Densities of diluted species

Low-concentration species in the gas environment are either neutral impurities ($H_2O$, $N_2$, $O_2$, $CO_2$ and others), or electrons and ions of various species. If originating as impurities of the gas bottle, these species can be considered uniformly distributed in the gas environment and have the same properties as the predominant buffer gas. Therefore, no additional calculation is required. Certain neutral species can however also be evaporated from a filament or oven, or injected in the gas cell through dedicated inlet, case in which a calculation would be required, using eq. (7). Ion/electron pairs produced by ionization mechanisms such as a discharge or radioactive source can be simulated by solving the plasma equations with migration in electrical field for both electrons and the heavy ion species. For simpler simulations, the densities can also be modeled statically, based on independent calculations or estimates. For realistic calculations, the plasma module of COMSOL is necessary.

## 2.5 Interaction of ions with the diluted species

The interaction of the charged particles with the diluted species, in particular the free electrons in the gas, can be simulated using collisions, giving for each process of interest the corresponding reaction cross-section. As mentioned before, collisions with the buffer gas cannot be modeled individually due to their very high rate, so processes involving buffer gas atoms need to use an alternative approach, for example, the definition of a "decay" constant which is linked to the buffer-gas density. Three-body processes involving the buffer gas need to be converted into two-body processes with a reaction rate depending on the buffer-gas density. For the diluted species, either neutrals, ions or electrons, collisions can be modeled individually because their concentration is typically lower than $10^{-6}$, therefore the collision rate will not be unreasonably high. One possibility to consider is the simulation of reactions with high-density components (including the buffer gas) by re-normalizing the reaction cross section.

# 3 Simulation of the gas cell

## 3.1 General considerations

The simulations of the FRIENDS³ gas-cell prototype are constrained by the project objectives and by the requirement of compatibility with S³-LEB. In addition, the prototype design is bound (at least at this stage) by the requirement of simplicity even at a moderate expense in performance (such as efficiency). This explains for example the choice for a pure DC guiding field, despite its limitations in efficiency. The question of efficiency and the possible inclusion of RF fields will be considered at a later stage of the project. For the
same reason, all studies have been performed so far in cylindrical symmetry to significantly reduce the required computation time.

The length of the gas cell is constrained by the available space in the S³-LEB vacuum chamber, taking into account that the S-RFQ would also be redesigned. Furthermore, the S³ beam enters the gas cell through a dedicated inlet window with a diameter of 74 mm, adapted to the S³ beam size in convergent mode. Ideally, we would maintain the window holder design and by this also maintain a broad acceptance. The standard diameter of the gas cell exit hole is 1 mm and can be increased up to a maximum of 2 mm, the latter being the limit from the available pumping speed.

The typical S³ beam in convergent mode has a radius of about 20 - 30 mm and the longitudinal straggling in the argon gas spreads the stopped ions longitudinally over 10 mm. In the present simulations the initial particle release distribution in the stopping volume take the realistic beam shape and depth into account. For the existing Ar-filled gas cell, the ion distribution in the stopping volume has been calculated for several S³ beams with different entrance window thickness and under different operating pressures (100 to 500 mbar). Thus, it is assumed in our simulations that ions are released in a 3D Gaussian distribution with depth dispersion of σz = 5 mm and radius dispersion of σr = 10 mm.

In general terms, the gas cell is designed to have two functional sections:

1. An electric field region for the fast extraction, focusing and guiding of the ions.

2. A no-field region before extraction where the ion transport is ensured solely by the gas flow and where an efficient and fast neutralization technique will be applied, in particular by the inclusion of electrons in equilibrium with the buffer gas. The residence time of ions in this volume should be sufficient for the neutralization process to occur and the free electrons generated should not be affected by the electric field.

There are several optimization goals for the performance of the gas cell :

1. To extend the laser spectroscopy study to short-lived S³ products, the extraction time of the ions is expected to be approximately, or better still, less than 100 ms.

2. The neutralization process prior to the extraction in a supersonic jet is estimated to take a few tens of ms. Adequate neutralization time should be ensured before the extraction.

3. To maximize the extraction efficiency of the radioactive atoms.

4. To generate an electron cloud of sufficient density within the gas flow near the exit hole and to control its dynamics, for good neutralization efficiency. The risk of electrons leaving the neutralization zone rapidly due to the electric field should be avoided.

In addition, it is of great significance to describe the multiphysics of the gas cell in the simulation. The ions in the gas cell are subjected to the combined action of the electric force and collisions with the buffer gas atoms. The collisions give rise to two types of phenomena, the deterministic drag force manifesting whenever there is a velocity difference between the ion and the gas flow and the diffusion phenomenon characterized by the random thermal motion of particles. Since it is not possible to simulate collisions at high pressure on a case-by-case basis during the optimization process, effective estimates of drag forces and of diffusion effects are required.

COMSOL is chosen for the first stage of optimization, taking into account the simplicity of changing the geometry and the accuracy of the multiphysics implementation. The gas-cell operating parameters (i.e. pressure, voltage configuration) and geometry are optimized in COMSOL without diffusion effects. The diffusion is then re-coupled via SIMION with the design and gas flow from COMSOL as input parameters.
## 3.2 COMSOL model

The volume and geometry of the gas cell was initially constructed and optimized in COMSOL. Electrical fields were applied in order to extract the radioactive ions from the gas cell in much shorter time than the gas flow would allow. In addition, as stated in section 2.3, rather than explicitly modeling the individual collisions between the ions and the buffer gas, a Stokes type viscous damping force was defined as an effective drag force. It should be noted that the simulations for optimization are conducted in fine meshing [8] for high optimization efficiency without seriously sacrificing the accuracy of the physics interpretation. In the ultimate stage of the simulation work, the mesh size is adjusted to extremely fine for more accurate results.

The optimization is performed with $^{133}$Cs$^{+}$ due to its well-known reduced ion mobility, $k_0(^{133}\text{Cs}^{+}) = 2.1 \text{ cm}^2\text{V}^{-1}\text{s}^{-1}$[10] and due to the fact that it is a likely species to be used for experimental verifications. The Argon gas is added as a user-defined material in COMSOL by specifying two material properties:

Dynamic viscosity $\mu = 2.26 \times 10^{-5} \text{Pa}\cdot\text{s}$ [12],

Pressure dependent density $\rho(p) = 1.664 \text{ kg/m}^3 \cdot p/(1 \text{ atm})$.

The COMSOL simulation is performed by the following procedure:

1. Build the geometry of the gas cell.

2. Perform a stationary study to simulate the gas flow by using the Laminar Flow and Electrostatics physics interfaces, by which the gas velocity, density, pressure, temperature and the electric field in the gas cell were calculated.

3. Perform a time dependent study by using the Charged Particle Tracing interface to calculate the motion of the particles with the stationary study results used as input environment parameters.

4. Optimize the geometry and physical-parameter settings. Then repeat the above steps to obtain satisfactory extraction time and efficiency, taking into account the previously mentioned simulation objectives and constraints.

### 3.2.1 Geometry and electric field optimization

The preliminary design of a cylindrical gas cell is shown in fig. 2(a). The electric field is shaped by applying a gradually decreasing DC voltage on two sets of electrodes [13]: the cage, comprising five similar cylindrical electrodes, and the funnel, which consists of several (so far 8) electrodes with decreasing diameters. The exit of the funnel is connected to a long no-electrical-field tube. The ions are focused and quickly extracted in the cage and the funnel regions and neutralized in the tube.

In the electric field region the electrical guiding is much more significant than the gas-flow guiding, therefore, after the geometry optimization, the next priority is to optimize the electric field. The number, dimensions and voltage settings of the electrodes are optimized to align the electric field with the desired ion trajectory which close to the neutralization area should follow as much as possible the gas flow. The voltage setting on the window and window ring is optimized to first converge the beam particles toward the central axis of the gas cell, before moving them in the axial direction. The cage and funnel electrodes are optimized to focus and extract the ions. In addition, the entrance part of the neutralization tube is segmented into three sections and the voltage is gradually reduced to zero, so as to avoid a large voltage drop between the tube and the exit of the funnel, which will cause the ions to be guided to the grounded wall of the tube, reducing the extraction efficiency. The performance of the preliminary design is presented in Section 2.2.3.

The optimized DC voltage configuration is shown in section 3.2.1. The components with voltage settings are indicated and numbered in fig. 2(a). The optimized electric field
described by potential contours is presented in fig. 2(b), the electrostatic force acting on the ions will be normal (at 90 degrees) to the potential contour.

<figure><img src="image_3.png" /><figcaption>Figure 2: Optimized gas-cell geometry with resulting voltages and electric field (shown as potential contours).</figcaption></figure>

<table><caption>Table 1: Optimized voltage settings.</caption><thead><tr><th>No.</th><th>Component</th><th>DC voltage (V)</th><th>No.</th><th>Component</th><th>DC voltage (V)</th></tr></thead><tbody><tr><td>1</td><td>Window + window ring</td><td>246</td><td>10</td><td>Funnel 3</td><td>150</td></tr><tr><td>2</td><td>Cage disk</td><td>256</td><td>11</td><td>Funnel 4</td><td>120</td></tr><tr><td>3</td><td>Cage 1</td><td>260</td><td>12</td><td>Funnel 5</td><td>90</td></tr><tr><td>4</td><td>Cage 2</td><td>252</td><td>13</td><td>Funnel 6</td><td>60</td></tr><tr><td>5</td><td>Cage 3</td><td>248</td><td>14</td><td>Funnel 7</td><td>30</td></tr><tr><td>6</td><td>Cage 4</td><td>244</td><td>15</td><td>Funnel 8</td><td>16</td></tr><tr><td>7</td><td>Cage 5</td><td>240</td><td>16</td><td>Segmented tube 1</td><td>8</td></tr><tr><td>8</td><td>Funnel 1</td><td>212</td><td>17</td><td>Segmented tube 2</td><td>2</td></tr><tr><td>9</td><td>Funnel 2</td><td>180</td><td>18</td><td>Segmented tube 3</td><td>0</td></tr></tbody></table>

### 3.2.2 Simulation of the gas flow

The properties of the gas flow are simulated by a stationary study through the Laminar Flow and Electrostatics interfaces. In the first stage of the simulation, the Argon gas is set to enter the gas cell homogeneously through the 74 mm diameter inlet window and exit through the 1 mm hole. In the real gas cell, a lateral gas inlet would be used, but the cylindrical symmetry does not allow such an inlet to be represented. Nevertheless, for the scope of the simulation (and not considering turbulence) using the window as the inlet is sufficient, because it provides an accurate representation of the gas flow close to the neutralization tube. The pressure of the gas cell can be controlled by changing the inlet mass flow rate. The performance of the designed gas cell was simulated for 50 - 500 mbar pressure in steps of 50 mbar, while 100 mbar is expected to be a good working condition for the gas cell with a 1 mm diameter throat nozzle. The simulated gas velocity is shown in fig. 3.
Velocity magnitude (m/s)

max: 308 m/s

▲ 309

<figure><img src="image_4.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b)</figcaption></figure>

Figure 3: (a) The gas velocity resulting from the Laminar Flow study. The maximum and minimum values of the presented gas properties are indicated on the top and bottom of the legend. (b) A new outlet slightly lower than the exit hole is set to redefine the simulation volume.

Velocity magnitude (m/s)

<figure><img src="image_6.png" /><figcaption>(a)</figcaption></figure>

Pressure (mbar)

<figure><img src="image_7.png" /><figcaption>(b)</figcaption></figure>

Figure 4: The velocity and pressure values of the gas resulting from the Laminar Flow study with lower outlet.

As stated before, all the equations describing the gas properties are solved assuming a laminar, compressible flow with low Mach number ($M < 0.3$), using a constant-pressure boundary condition at the outlet ($p = 0$ mbar). One notices however that the resulting velocity near the exit hole is upwards of 300 m/s. In order to address the problem of the velocity exceeding the scope of the laminar flow, the simulation volume is restricted by a 2 mm lower outlet, as shown in fig. 3(b). The boundary condition at the new outlet can not be maintained as a constant pressure value of 0 mbar, because it would act as an effective reduction of the gas-cell length. Instead, a constant volume flow is used as boundary condition, calculated by eq. (3). This is possible because, the gas-cell pressure being constant across the extraction tube, the total volume flow through any section of the tube is the same and given by the nozzle flow rate of eq. (3). The geometry and all other configurations of the gas cell remain unchanged. The gas velocity and pressure in the
redefined simulation volume are shown in fig. 4. By a fine comparison one notices that the redefinition of the simulation volume does not change the results in a significant way.

### 3.2.3 Simulation of the ion motion

The Charged Particle Tracing physics interface of COMSOL was used to simulate the ion motion by a time-dependent study. Ions can be flown within the domain simulated by the previous stationary study under the action of the combined forces acting during their flight. In the current simulation, $^{133}$Cs$^+$ ions without initial velocity (already thermal) are created at the expected stopping position in the gas cell with a typical Gaussian distribution to imitate the S$^3$ beam. The trajectories, extraction time and extraction efficiency were initially calculated through a time-dependent study for the optimized geometry and settings but for different pressures. The gas pressure, temperature and velocity fields obtained from the stationary study were utilised as input parameters. The results for 100 mbar are presented in fig. 5.

The COMSOL Particle Tracing Module allows defining a Particle Counter, which is a domain or boundary feature that provides information about particles arriving on a set of selected domains or surfaces from a particle release. Two Particle Counters were created to on the one hand count the particles entering the no-field region of the tube and exiting it through the outlet hole and on the other hand to determine their arrival time. The positions of the counters are shown on the right panel of fig. 5 by the red and blue arrows. The panel on the left shows the extraction efficiency as a function of the elapsed time: the blue curve for the entrance into the no-field region, the red curve for crossing the exit boundary. The "efficiency" defined in this sense is a function of the elapsed time at the two boundaries. The efficiencies in the common sense are given by the maxima of the two curves. The residence time of the ions in the volume between the two particle counters is the time available for neutralization.

<figure><img src="image_8.png" /></figure>

Figure 5: Extraction efficiency and time of the ions as a function of elapsed time from their release, for 100 mbar pressure. The red and blue curves present the transmission time and efficiency recorded by the two particle counters. The maximum value of the curves shows the final transmission efficiency and the dashed lines at the half maximum indicate the median arrival times of the particles the two counters. The red curve shows that 91% of the particles arrive the no-field tube region with the median arrival time of 62 ms, and the blue curve presents that 77% of the particles are extracted from the gas cell with the median time of 98 ms.

The designed gas cell was simulated by COMSOL at different pressures from 50 mbar to 500 mbar, in steps of 50 mbar. The calculated extraction efficiency, median extraction time and the time available for neutralization, are represented as a function of pressure in fig. 6.
One must note that in the absence of diffusion, the efficiency values are not yet representative of the actual performance of the system. The efficiency taking into account also diffusion will be simulated using SIMION in section 3.3 In the absence of diffusion, the trends of extraction efficiency and time with pressure depends mainly on the impact of pressure on the drag phenomenon, where the ion mobility is inversely proportional to pressure. Higher pressure leads to lower mobility. As illustrated by eq. (5) and eq. (6), this leads to a smaller ion drift velocity and therefore longer time for the ions to converge towards the center in the cage area and be extracted. In addition, it leads to a stronger drag force at the entrance to the no-field area, where the gas velocity becomes important, meaning a more significant "resistance" to the electrical-force components guiding the ions towards the wall.

The residence time of the ions in the no-field region, which is indicated as the blue volume in fig. 5, is the time available for neutralization, $T_N$. Since a sufficient residence time of the ions in the tube is crucial to improve the neutralization efficiency, the length and radius of the tube can be optimized to obtain a longer transport time. Figure 7 presents the influence of the tube length and radius on $T_N$ and the extraction efficiency. All the simulation results were carried out at 100 mbar pressure and the same geometry of the gas cell as before.

<figure><img src="image_9.png" /><figcaption>Figure 6: Simulated COMSOL results for the designed gas cell as a function of pressure (a) The extraction efficiency (b) The extraction time (c) The residence time of the ions in the no-field tube, which is the time difference between the blue and red points in (b). The preliminary design guarantees around 35 ms for neutralization under all simulated pressures.</figcaption></figure>
From fig. 7 it becomes evident that the longer and wider the tube, the more time is available for neutralization, without affecting the efficiency (so far not considering diffusion). Nevertheless the ion extraction is also slowed down by an increase of the tube dimensions. Because the length of the tube is constrained by the available space at S³-LEB, in the preliminary design a 70 mm length and 7 mm radius tube is adopted, which is a compromise for having a sufficient T<sub>N</sub>, short extraction time(<100 ms) and compatibility with the S³-LEB layout.

<figure><img src="image_10.png" /><figcaption>Figure 7: The effect of modifying the dimensions of the neutralization tube on the corresponding transport time: (a) increasing the length of the tube from 60 mm to 80 mm in 5 mm increments; (b) increasing the radius of the tube from 6 mm to 10 mm in 5 mm increments.</figcaption></figure>

### 3.2.4 Simulation of electron motion

The gas-cell design aims to perform the neutralization in a high-density electron cloud created in the long no-field tube before leaving the gas cell. Therefore it is important to obtain an appropriate dynamics of the electrons in the tube for high-efficient neutralization. In this respect, the attraction by the electrical field in the gas cell might lead to an instantaneous loss of the electrons generated in the tube or unwanted avalanche effects. Therefore, a starting point is to study the electron motion by COMSOL.

Similarly to ions, the electron motion is simulated by the Charged Particle Tracing interface in COMSOL, the drag force exerted by the Ar gas on the electrons being added as a Stokes type user-defined force. The reduced mobility of electrons in argon is $\mu_0 = 5.22 \cdot 10^3$ cm²V⁻¹s⁻¹[14]. In all the simulations of the electron motion, the pressure of the gas cell was kept at 100 mbar.

Three hundred electrons were released without initial velocity in a Gaussian distribution at different positions in the neutralization tube. Figure 8 shows the trend of electron movement, highlighting the fact that all the electrons released below a certain coordinate (the red line) are pulled down by the electric field, while electrons released above the red line are dragged out by the gas flow, unaffected by the electric field.
<figure><img src="image_11.png" /></figure>

Figure 8: The trajectories of electrons after being released at an arbitrary position in the no-field tube.

For electrons released in Gaussian distribution centered at the coordinate $z = 195$ mm, the simulation results are shown in fig. 9. The electrons pulled down by the electric force almost completely follow the inverse potential gradients due to their particularly light mass and large mobility.

<figure><img src="image_12.png" /></figure>

Figure 9: Dynamics of the electrons after being released at coordinate $z = 195$ mm, in the neutralization tube: (a) trajectories of the electrons; (b) integrated number of electrons crossing the two particle counters used for the ion simulations as a function of time. About 30% of electrons are rapidly pulled down by the electric field within a few milliseconds, and about 60% of the electrons follow the gas flow slowly towards the exit hole.
By releasing the electrons at coordinate $z = 204$ mm or above, almost all the electrons can follow the gas flow without being affected by the electric field, the corresponding electron motion simulation is shown in fig. 10. It is worth noting that the residence time of the electrons in the tube in the absence of an electrical field is similar to the one of the ions, which is both expected and an important ingredient for efficient neutralization.

<figure><img src="image_13.png" /><figcaption>Figure 10: Same as fig. 9 but for the release distribution centered at $z= 204$ mm.</figcaption></figure>

## 3.3 SIMION simulation studies

### 3.3.1 COMSOL benchmark and effect of diffusion

The standard geometry optimized with COMSOL was implemented in a SIMION gem file with a resolution of 1 mm/grid unit and used to perform simulations using the SDS module, as discussed in section 2.3.1. The SDS module requires as input an array of gas velocities, which was provided based on a COMSOL stationary study with the finest meshing possible. Because COMSOL uses an irregular mesh compared to SIMION which uses an equidistant one, a Python script implementing the "griddata" class of the "scipy.interpolate" module was used to linearly interpolate the equidistant grid values from the COMSOL mesh-based ones. The same voltages were used in SIMION as in COMSOL, but the electrostatic fields were computed by SIMION.

The $^{133}$Cs⁺ ions were released with the same initial conditions. In addition to the mobility of $^{133}$Cs⁺ which is an input parameter of the SDS model, the radii of $^{133}$Cs⁺ and of the Ar buffer-gas atoms are also required. The radii are necessary in order to estimate the collision cross section and the corresponding frequency and thus the so-called "kinetic radii" are required. For Ar, a recent value from the literature was used, $r_{Ar} = 0.17$ nm [15]. For $^{133}$Cs⁺, the radius was automatically calculated by the SDS module based on the mobility.

Figure 11 shows the ion trajectories calculated by SIMION with and without the diffusion effect. One notices how the drag force is sufficient to describe the average ion trajectories and how diffusion adds a random component and an enlargement of the ion-beam cross-section.
<figure><img src="image_14.png" /><figcaption>Figure 11: The trajectories of ions extracted from the gas cell calculated by SIMION without (a) and with (b) diffusion</figcaption></figure>

An important first cross-check between COMSOL and SIMION is the comparison of the ion transport when only the electrical and drag forces are taken into account and the diffusion phenomenon is neglected. If the same voltages, gas pressure and ion starting conditions are used, the comparison would verify the quality of the transfer of gas velocities from COMSOL to SIMION plus the relative accuracy of the geometry implementation and of the solving of the electrical-field and ion-transport problems. In a second step, the effect of diffusion can be quantified using SIMION. Finally, in a third step other questions can be solved directly with SIMION, taking diffusion into account.

Figure 12 shows the ion extraction efficiency and time as a function of pressure for the gas cell design simulated in the previous section, with exit-hole diameter $D_{hole} = 1$ mm and tube radius $R_{tube} = 7$ mm.

From the comparison it can be seen that the calculations without the diffusion effect performed with COMSOL and SIMION give similar efficiencies and a comparable trend with pressure. The difference of 10% at high pressure up to 20% at low pressure can be due to the poorer resolution of the SIMION geometry which leads to a less accurate representation of near-wall trajectories. The simulated extraction times are almost the same for the two programs. The diffusion effect significantly reduces the efficiency but has almost no effect on the average extraction time. This means that when transport times are the main focus of a simulation, the diffusion effect is not a relevant ingredient and simulations can be performed with COMSOL only. The simulations show that the diffusion effect is very important for pressures around and below 100 mbar and leads to an efficiency below 20%. For these low pressures and the obtained voltages, total extraction times (including the neutralization time) around or below 100 ms are possible. The efficiency increases towards 40% if the pressure is around 200 mbar, with a total extraction time around 160 ms.

Following the good quantitative comparison between the SIMION and COMSOL results
<figure><img src="image_15.png" /><figcaption>Figure 12: Comparison of COMSOL (blue points) and SIMION simulations without (green points) and with diffusion (red points) for D<sub>hole</sub> = 1 mm, R<sub>tube</sub> = 7 mm. (a) extraction efficiency; (b) average time of flight; panel (c) shows the time of flight differences between the two programs for the calculations without diffusion.</figcaption></figure>
on the standard configuration, it is possible to further use SIMION for studying the effect
of certain design choices, such as the diameter of the exit hole and the diameter of the
neutralization tube ($D_{hole}$ and $R_{tube}$, respectively). Because in both cases the efficiency is
a crucial output, SIMION is preferred because of its description of diffusion. This will be
presented in the following two sections.

### 3.3.2 Effect of the exit hole diameter

Once the diameter of the exit hole is increased, a larger inlet gas flow rate is required to maintain the same pressure in the gas cell. In general, if the hole diameter is increased by a factor of two, an approximately 4 times larger inlet flow rate is required to maintain the same pressure, which can also be verified by Eq. 2 (scaling with the square of the diameter). The simulation results for $D_{hole}$ = 2 mm and $R_{tube}$ = 7 mm are presented in fig. 13.

It can be seen from fig. 13(b) that increasing the diameter of the exit hole can significantly reduce the extraction time due to the significant increase in gas velocity, in which case smaller particle-gas relative velocity (u - v) leads to lower drag force along the trajectories of the ions. In addition, because the ions spend much less time in the gas flow, the diffusion losses are greatly reduced, therefore the extraction efficiency is significantly improved, as shown in fig. 13(a).

However, the extraction efficiency in the case of $D_{hole}$ = 2 mm does not show the same trend with pressure as in the case of $D_{hole}$=1 mm, i.e., for high pressure values the efficiency does not increase but decreases with pressure. This is due to the shorter axial transport time, leading to the fact that the ions have less time to converge towards the center of the gas cell before reaching the entrance of the neutralization tube. Therefore the losses at the entrance of the tube become more important. This problem of insufficient convergence is more pronounced at high pressures, where the lower mobility leads to a lower drift velocity, therefore the ions move towards the center more slowly. The trajectories of the ions at 100 mbar and 500 mbar pressure in the gas cell with $D_{hole}$ = 2 mm and $R_{tube}$ = 7 mm are shown in fig. 14, illustrating this centering issue.

<figure><img src="image_16.png" /><figcaption>Figure 14: Comparison of ions trajectories calculated by SIMION for an exit hole diameter $D_{hole}$ = 2 mm and a tube radius $R_{tube}$ = 7 mm at two pressures. (a) 100 mbar and (b) 500 mbar. The ion losses at the entrance of the tube become greater for higher pressure due to insufficient convergence.</figcaption></figure>

### 3.3.3 Effect of the tube radius

The ion losses mostly occur at the entrance of the no-field tube due to insufficient centering and in the tube region due to the diffusion effect, therefore, by increasing the tube radius, the extraction losses should be reduced and the extraction efficiency is expected to be improved. The simulations were performed for four combinations of the tube radius and exit hole diameters: $D_{hole}$ = 1 mm, $R_{tube}$ = 7 mm; $D_{hole}$ = 2 mm, $R_{tube}$ = 7 mm; $D_{hole}$ = 1 mm, $R_{tube}$ = 9 mm; $D_{hole}$ = 2 mm, $R_{tube}$ = 9 mm. Figure 15 shows the calculations at different pressures.
<figure><img src="image_17.png" /><figcaption>Figure 13: Comparison of COMSOL and SIMION simulations for different exit hole diameters $D_{hole}$ = 1 mm or 2 mm: (a) extraction efficiency. (b) extraction time. (c) time difference between COMSOL and SIMION for simulations not considering the diffusion effect and $D_{hole}$ = 2 mm.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>(a) Extraction efficiency</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(b) Extraction time</figcaption></figure>

Figure 15: Comparison of simulation results for different tube radii $R_{tube} = 7$ mm or 9 mm and exit hole diameters $D_{hole} = 1$ mm or 2 mm: (a) extraction efficiency. (b) average extraction time of the ions from the gas cell.

As shown in fig. 15, comparing to the tube radius, the exit hole diameter plays a much more important role in improving the extraction efficiency and reducing the extraction time. It can be concluded from fig. 15(b) that a larger exit hole significantly reduces the extraction time and a narrower tube also leads to slightly faster extraction. From fig. 15(a) it can be seen that a larger exit hole improves the extraction efficiency over the full range of simulated pressures by reducing the time available for diffusion losses to occur. However, as described in section 3.3.2, this improvement is limited at high pressures by the increase of the losses at the tube entrance. Nevertheless, at high pressures, a wider tube helps improve the extraction efficiency by reducing losses at the tube entrance and diffusion losses in the tube. The simulation results with $D_{hole} = 2$ mm, $R_{tube} = 9$ mm give the highest extraction efficiency and shortest extraction time.

However, a drawback is that the reduction of the extraction time in all the cases mentioned above corresponds also to a reduction of the time available for neutralization $T_N$. Therefore, the $T_N$ simulated by SIMION for the full range of pressures with the four sets of
geometry configurations, by setting the end of the simulation at the two limiting coordinates of the neutralization region (z = 171 mm and z = 215 mm) and calculating the corresponding average times of flight and their difference. The calculated $T_N$ values are presented in fig. 16. The calculations by COMSOL for the case $D_{hole} = 1$ mm, $R_{tube} = 7$ mm are also plotted for comparison with the calculations by SIMION.

<figure><img src="image_20.png" /><figcaption>Figure 16: Gas-cell extraction time for tube radii $R_{tube} = 7$ mm or 9 mm and exit hole diameters $D_{hole} = 1$ mm or 2 mm.</figcaption></figure>

As shown in fig. 16, the calculations by COMSOL and SIMION for the same geometry (the green open circle points and green full circle points) are comparable. COMSOL gives a slightly larger result because the outlet in COMSOL is set to z = 229 mm, while the outlet in SIMION is set to z = 215 mm. Therefore it can be concluded that the $T_N$ simulations by SIMION is quite reliable. It is obvious from fig. 16 that the time available for neutralization is too short when $D_{hole} = 2$ mm. In addition, in order to determine the optimal diameter of the exit hole, several other constraints need to be considered:

* Pumping speed limit: the larger the exit hole, the higher gas flow needs to be handled by the pumping system.

* Jet properties: a larger exit hole will produce a wider gas jet and therefore a wider laser beam will be required. For twice the diameter, four times the laser power will be required to saturate atomic transitions.

* Radiation problem: a lower flow rate allows more time for the radioactive species to decay in the S³ chimney before being released into the atmosphere.

The conclusions of this study are that the optimized values of the exit hole diameter $D_{hole}$, tube length $L_{tube}$ and tube radius $R_{tube}$ should be chosen in a compromise between extraction efficiency, extraction time and time available for neutralization, taking into account the above-mentioned limitations. Based on the current design, the tube length can be increased as long as the S³-LEB layout allows, the exit hole diameter can be slightly increased with larger tube radius. The geometry with an exit hole diameter $D_{hole} = 1$ mm, tube length $L_{tube} = 70$ mm and tube radius $R_{tube} = 9$ mm is preliminarily adopted.

### 3.3.4 Efficiency and extraction-time maps

The beams of radioactive ions, discussed in the context of the designed gas cell, are typically fusion evaporation residues. The ions transmitted through S³ are envisaged to be stopped, then extracted and neutralized in the fast gas cell. The stopping position of the ions, i.e.
the particle release position in the current simulations, depends on the energies and mass of the beam species, the window thickness and the pressure of the gas cell. Therefore the performance of the designed gas cell may also vary from case to case due to the variation of the stopping position. In all the previous simulations, the initial distribution of the ions was assumed taken with a size comparable to the expected S³ beam in convergent mode and to be centered at coordinate z = 50 mm. For a comprehensive study, the calculations were also performed by SIMION for different center coordinates at 100 mbar and 200 mbar, for gas cell with D<sub>hole</sub> = 1 mm, L<sub>tube</sub> = 70 mm and R<sub>tube</sub> = 9 mm, respectively. Figure 17 shows the simulation results for the possible ranges.

<figure><img src="image_21.png" /><figcaption>Figure 17: SIMION simulation of extraction efficiency, extraction time and time available for neutralization for different particle release positions.</figcaption></figure>

It can be concluded from fig. 17 that there is an optimized position for the particles to be released for maximum extraction efficiency at both pressures, which is due to the special shape of the voltage gradients in the cage area. This position is at around coordinate z = 47 ± 1 mm for 100 mbar and z = 44 ± 2 mm for 200 mbar, respectively. The shift and widening of the efficiency curve with pressure is due to the drag force being proportional to pressure. In the designed gas cell, the electric field near the entrance window points towards
the window, which guides some of the ions towards it, leading to losses. At higher pressure, the greater drag force counteracts more the electric force and therefore reduces the ion losses on the window side. In addition, the extraction time is reduced for the ions released at a deeper position, while the average time available for neutralization is almost independent of the initial release position.

Another useful information would be how the extraction efficiency and time depends on the exact particle stopping position. In this case, instead of a wide ion distribution corresponding to the full S³ beam, a set of much narrower ion distributions needs to be flown from different starting positions, allowing to determine an efficiency and an extraction time for each of them. The simulation was performed for a pressure of 100 mbar. In each of the runs, 10000 ¹³³Cs⁺ ions at rest were released from a 3D gaussian distribution with σz = 1 mm and σr = 2 mm centered at different (z, r) coordinates. The resulting contour maps are shown in fig. 18, from which the optimal stopping volume for high efficiency can be estimated.

<figure><img src="image_22.png" /><figcaption>Figure 18: Extraction efficiency and time map for 100 mbar pressure simulated by SIMION.</figcaption></figure>

# 4 Conclusions and outlook

In this report, the phenomena involved in the gas-cell were described and it was shown that several simulation approaches are possible. In order to have a accurate implementation of the multiphysics in the gas cell, as well as to be able to easily adjust the geometry during the optimization, the gas-cell operating parameters and geometry were firstly optimized in COMSOL considering only the electric force and gas drag force on the ions, then the diffusion effect was re-coupled via SIMION with the design and gas flow from COMSOL as input parameters. The preliminary design of the fast gas cell and the simulation results of its performance under different operating pressures were presented. These results verified the tentative conception of the fast gas cell composed of two functional sections: an electric field extraction region and no-field neutralization region. In addition, the comparable results between COMSOL and SIMION software without diffusion effects gave confidence in the implemented simulation approach.

The optimization work was always a compromise between extraction efficiency, extraction time and time available for neutralization, taking into account the limitation (maximum 2 mm) of the exit hole diameter and the available space in the S³-LEB vacuum chamber. After the first phase optimization by COMSOL without considering diffusion effects, the preliminary design of the electric field region of the gas cell was determined. Further optimization of the neutralization tube where the diffusion losses manifest, was then carried out based on calculations of both COMSOL and SIMION, by which the design with exit hole diameter $D_{hole} = 1$ mm, tube length $L_{tube} = 70$ mm and tube radius $R_{tube} = 9$ mm were adopted.

The performance of the designed gas cell was calculated by SIMION for 50 - 500 mbar
pressure with the complete physics interpretation. As shown in fig. 15, at 100 mbar pressure the efficiency is around 23% and the total extraction time is around 120 ms, with about 50 ms available for neutralization. The efficiency can be as high as 65% for 500 mbar, however the extraction time will be much longer, around 280 ms.

The efficiency and time maps covering the full range of the stopping volume in the designed gas cell were also explored by SIMION, from which the optimal stopping volume for high efficiency and fast extraction are estimated.

Two promising conclusions concerning the electrons creation can be drawn from the simulation:

* A electron source can be placed at a certain coordinate of the neutralization tube, where the generated secondary electrons will not be affected by the electric field. Therefore, an electron cloud with high density and good volume can be generated in the tube.

* One possibility to increase the volume of the electron cloud in the tube is to increase the tube length as much as the S³-LEB layout allows.

Further simulations for the neutralization process will be performed with the COMSOL Plasma module. The electron generation and dynamics will be studied and compared with the results obtained with the Charged Particle Tracing module.

## References

[1] F. Déchery, et al., Nucl. Instrum. Meth. B 376, 125 (2016).

[2] J. Romans, et al., Atoms 10, 21 (2022).

[3] V. Manea, et al., 'Fast radioactive ion extraction and neutralization device for s³', project ANR-21-CE31-0001 (2021).

[4] E. Mogelevskiy, 'Diffusion inside the gas cell for laminar flow', Tech. rep., KU Leuven (2013).

[5] E. Mogelevskiy, 'Losses in the horn cell', Tech. rep., KU Leuven (2013).

[6] Y. Kudryavtsev, et al., Nucl. Instrum. Meth. B 376, 345 (2016).

[7] S. Raeder, et al., Nucl. Instrum. Meth. B 463, 272 (2020).

[8] COMSOL Multiphysics v. 5.6., www.comsol.com.

[9] D. A. Dahl, Int. Jour. Mass Spectrom. 200, 3 (2000).

[10] A. D. Appelhans, D. A. Dahl, Int. Jour. Mass Spectrom. 244, 1 (2005).

[11] A. Zadvornaya, Characterization of the spectral resolution of the in-gas-jet laser ionization spectroscopy method, Ph.D. thesis, KU Leuven (2018).

[12] M. Huber, A. Harvey, Viscosity of Gases. CRC-Press, Boca Raton, FL, 92nd edn. (2011).

[13] M. Wada, Nucl. Instrum. Meth. B 317, 450 (2013).

[14] M. Huyse, M. Facina, Y. Kudryavtsev, P. V. Duppen, ISOLDE collaboration, Nucl. Instr. Methods B 187, 535 (2002).

[15] S. Kunze, R. Groll, B. Besser, J. Thöming, Scientific Reports 12, 2057 (2022).