# Numerical simulation of a helium Plasma-Material Interaction experiment in
GyM linear device through SOLPS-ITER and ERO2.0 codes

F. Mombelli1, G. Alberti1, E. Tonello2, C. Tuccari1, A. Uccello3, C. Baumann4,
X. Bonnin5, J. Romazanov4, M. Passoni1,3 and the GyM team6 1 Politecnico di
Milano, Department of Energy, Milan, 20133, Italy  2 Ecole Polytechnique
Fédérale de Lausanne, Swiss Plasma Center, Lausanne, 1015, Switzerland  3
Istituto per la Scienza e Tecnologia dei Plasmi, Consiglio Nazionale delle
Ricerche, Milan, Italy  4 Forschungszentrum Jülich GmbH, Institut für Energie-
und Klimaforschung – Plasmaphysik, Partner of the Trilateral Euregio Cluster
(TEC), 52425 Jülich, Germany  5 ITER Organization, 13067 St Paul Lez Durance
Cedex, France  6 See the author list of A. Uccello et al 2023 Front. Phys.
11:1108175<EMAIL_ADDRESS>
###### Abstract

Learning how to safely handle Plasma-Material Interaction (PMI) is a key
challenge towards the commercialisation of energy from nuclear fusion. In this
respect, linear plasma devices are ideal experimental testbeds, and numerical
codes play a crucial complementary role. In this paper, a numerical
investigation of PMI-relevant helium plasma experimental discharges in GyM
linear device is presented, in which SOLPS-ITER and ERO2.0 codes are coupled
for plasma background generation and material erosion investigation
respectively, with the aim to support the interpretation and complement the
available experimental dataset. On the plasma side, simulated profiles are
validated against experimental data to provide a realistic plasma background,
and the role of He metastable states is assessed for the first time in SOLPS
simulations. On the material side, the erosion and deposition effects due to
the introduction of the sample-holder in the simulation volume are
investigated, now considering also the real stainless steel composition as
wall material.

Keywords— Plasma-Material Interaction, linear plasma device, helium plasma,
SOLPS-ITER, ERO2.0, GyM

##  1 Introduction

The path towards the profitable implementation of a magnetic confinement
nuclear fusion reactor, which sees ITER as a crucial milestone [1], raises the
compelling need to address the topic of Plasma-Material Interaction (PMI) [2].
On one hand, the build-up of plasma particle and heat fluxes towards solid
plasma-facing components (PFC) results in their erosion and surface
modification, threatening their properties and integrity [3]. On the other,
the transport of eroded impurities into the plasma may result in excessive
fuel dilution and enhanced radiation losses, while their deposition may
contribute to tritium retention and promote dust formation [2].

Coherently, the investigation of PMI has gained high priority within the
European fusion research programme coordinated by EUROFusion Consortium [2]:
the Work Package Plasma-Wall Interaction and Exhaust (WP PWIE) is meant to
pursue this objective exploiting both ad-hoc experimental and numerical
methods.

Although several PMI-relevant experimental campaigns have been carried out in
tokamaks [4, 5], particle and heat fluences to PFCs in present-day devices are
orders of magnitude lower than those expected in future reactors due to
generally low pulse duration [6]. For steady-state ITER operation, ion fluxes
$\leq 10^{25}$ ions$\cdot$ m-2s-1 and heat fluxes of 10 MW$\cdot$ m-2 are
expected onto the divertor tiles [7], to be integrated over the long duration
of the plasma discharges ($\sim$ 1000 s) [8]. Linear plasma devices (LPDs),
thanks to their simple, cost-effective and flexible design, have thus been
widely exploited to fill such experimental gaps, appearing as ideal testbeds
for PMI phenomena [9, 10].

Among the several LPDs which operate worldwide, the present paper considers
GyM linear device, hosted at Istituto per la Scienza e Tecnologia dei Plasmi -
Consiglio Nazionale delle Ricerche (ISTP - CNR), Milan, whose schematic is
visible in figure 1 and whose characteristics are comprehensively detailed in
reference [11]. GyM LPD generates both hydrogenic and non-hydrogenic plasma
[11]: namely, the relevance of helium (He) plasma to fusion research arises
from the fact that He ashes will always be present in the thermonuclear plasma
as reaction products, assessing its role in PMI is therefore crucial.

Figure 1: 2D schematic of GyM LPD: the position of magnetic coils, gas puff,
turbo-molecular pumps (TP, in the plane), sample-holder (SH) insertion and
Langmuir probes (LP, within blue dashed box) are visible. The approximate
position of the ECRH resonant surface (ECRH RS), where the external power is
favourably absorbed by electrons, is pointed out on the same side of the gas
puff valve. The reference radial axis is shown on the right, the origin lying
on the axis of the cylinder.

Besides the experimental investigation, numerical codes offer the opportunity
to interpret the experimental outcomes, gaining insight into the underlying
physical mechanisms. Once validated, codes also acquire the potential to
predict relevant scenarios which are not experimentally accessible. Moreover,
the flexibility of LPD configures them as ideal testbeds for the development
and validation of numerical tools [11, 12]. It is in the groove of the
numerical analysis that this contribution places itself.

Two state-of-the-art codes are exploited for the present modelling activity,
i.e. SOLPS-ITER [13, 14] and ERO2.0 [15]. The first one couples a 2D multi-
fluid edge plasma solver (B2.5) [16, 17] with a 3D kinetic Monte-Carlo code
simulating the transport of neutral species (EIRENE) [18]. Conversely, ERO2.0
is a 3D Monte-Carlo code simulating the material erosion by a plasma, the
transport and deposition of the released impurities, both at a global and at a
local microscopic scale [15].

Although the SOLPS package is optimised for the simulation of the Scrape-Off
Layer of toroidal devices, it has already been applied to several LPDs [19,
20, 21, 22]. Concerning the specific case of non-hydrogenic plasma in GyM,
SOLPS was successfully applied for both argon (Ar) [23] and He plasmas [24].
ERO2.0 has also been applied to the study of PMI in LPDs [25], while a
morphology analysis at the microscale of samples exposed to a He plasma in GyM
was presented in [26].

More recently, a novel comprehensive numerical analysis of PMI-relevant He
plasma discharges was presented, in which a coupling procedure between SOLPS-
ITER and ERO2.0 codes was implemented [12]. In such work, the He plasma was
firstly generated with SOLPS, yielding a 2D distribution of electron and ion
quantities. Such distributions were eventually exploited as inputs, or
background, for the ERO2.0 simulation of the global erosion of GyM internal
walls, impurity transport and deposition as a function of the wall material
and bias voltage applied, setting a one-way coupling between the two codes
[12]. This work represented a first step towards the integrated plasma-
material modelling of a realistic PWI experiment in GyM, yet it left several
points open for further studies. Indeed, neither the plasma profiles nor the
erosion outcomes were validated against experimental data. Also, neither the
plasma nor the material simulations accounted for the presence of a central
sample-holder and its manipulator within the geometry of the device. Finally,
the erosion simulations considered the walls of the device to be made of
either iron (Fe) or copper (Cu), rather than of realistic stainless steel
(SS), since the ERO2.0 database did not include the sputtering yield for a He
plasma on SS steel at that time.

The present work intends to extend the analysis initiated in [12], i.e. to
present a SOLPS-ITER and ERO2.0 coupled numerical investigation of plasma and
material global erosion in GyM, yet partially overcoming the limitations
associated to the previous study. To this purpose, realistic PMI experimental
scenarios featuring He plasma discharges have been considered as a reference
and exploited for the validation of SOLPS numerical results on the plasma
side. For the first time, the effect of including long-lived metastable states
among the neutral population of a pure He plasma in SOLPS runs is assessed.
Hence, an optimised He plasma background has been used as an input for ERO2.0
simulations. In the step of defining GyM geometry for ERO2.0, a sample-holder
in which samples may be arranged for exposure to plasma has been included.
Erosion, transport and deposition of impurities originating both from the
walls of the device and from the sample-holder have been evaluated as a
function of the bias voltage applied and the wall material, now including the
SS composition.

This work is articulated in four main sections as follows: in section 2, the
reference experimental framework on which the paper is based is described. In
section 3, the setup and major results of the SOLPS-ITER numerical
investigation of the He plasma are discussed. Similarly does section 4 as
concerns the ERO2.0 material erosion simulation. In section 5, the main
outcomes of the work are recapped, and possible continuations are discussed.

##  2 Reference experimental framework

A series of six PMI-relevant GyM discharges are considered as a reference
experimental framework for the validation of plasma simulations. They all
feature He as the main plasma species, with a He puffing strength of 42 sccm
provided through the inlet valve at one of the bases of the SS cylindrical
vessel (radius and length of $R_{0}=0.125$ m and $L_{0}=2.11$ m). Two turbo-
molecular pumps guarantee a pumping speed of 500 L/s each, and external power
is supplied by one of the two available ECRH gyrotron sources at 2.45 GHz and
3 kW nominal power (”SM” source with reference to [11], connected to port 2U
in figure 1).

The axial magnetic field is generated by the surrounding coils, and the six
experiments are categorised and labelled into three couples according to the
intensity of the current flowing in such coils ($I_{coil}$), of 560 A, 600 A
and 640 A respectively. Experimentally, the coil current configuration of 600
A is the one usually exploited for conducting PMI experiments in GyM [27, 28,
29]. Conversely, the values of 560 A and 640 A coil current are the minimum
and the maximum ones allowing to have the resonant surfaces located inside the
vacuum vessel: experiments at these $I_{coil}$ were conducted for plasma
characterisation through spectroscopic techniques and Langmuir probes rather
than for PMI.

For each value of $I_{coil}$, two experiments were executed, in the presence
and absence of a sample-holder (SH) in which samples for PMI studies could be
arranged (see figure 10(b) below). The electron density $n_{e}$ and
temperature $T_{e}$ radial profiles were acquired by three Langmuir probes
(LP) located at axial positions 3U, 4U and 5U respectively (see figure 1).
When present, the SH was inserted on the axis of the cylinder by means of a
manipulator in correspondence of axial position 4U, thus replacing the
corresponding LP.

At present, the only available experimental data concern plasma parameters: no
measurements of the global erosion and deposition within the GyM vessel are
available at this stage. Figure 2 shows the experimental $n_{e}$ and $T_{e}$
radial profiles for the three couples of discharges at the positions of LPs.
In all cases, the electron density is in the order of $\sim 10^{16}$ m-3, yet
higher in the 600 A case, with relative maxima at R = 0 (centre of the
machine) and R $\simeq$ 6 cm. The electron temperature is in the order of
$\sim 5-10$ eV.

Figure 2: Experimental profiles of electron density ($n_{e}$, first row) and
temperature ($T_{e}$, second row) acquired by Langmuir probes (LP), as a
function of the radial coordinate (spanning from the cylinder axis to the
lateral wall as depicted in figure 1). Each column is associated with one of
the three values of $I_{coil}$. Each colour is associated with the axial
position of one of the three LP (see figure 1), while the line style refers to
the presence (dotted) or absence (solid) of the SH respectively. The thin grey
vertical line represents the boundary of the SH (where present), which extends
from the axis of the device (R = 0) to R $\sim$ 2.5 cm.

Comparing homologous discharges in presence and absence of SH (dotted and
solid lines of the same colour respectively, in figure 2), the only relevant
difference observed concerns the $n_{e}$ profiles measured by LP 5U, which
display a remarkable decrease at values of the radial coordinate $\lesssim 3$
cm, i.e. at the centre of the device, when the SH is inserted. Indeed, LP 5U
is situated behind the SH, i.e. on the opposite side of the SH with respect to
the positions of the gas puff valve and ECRH resonant surface (see figure 1):
the suppression of $n_{e}$ profiles observed is thus reasonably due to the
shadowing effect associated to the presence of the SH itself. For all plasma
experiments, the values of neutral pressure measured by the two hot-cathode
gauges at sections 3 and 5 of the device lie in the range of 0.09-0.1 Pa: this
is essentially independent of the magnetic field configuration but is rather
fixed by the volume of the vessel once the gas puff and pumping speed are set.

As concerns the setup of plasma simulations described in section 3, the
possible presence of the central SH is neglected and more details about this
methodological choice are presented in subsection 3.1. On the contrary, the
presence of the SH is included in the global erosion simulation, as it will be
analysed in section 4.

##  3 SOLPS-ITER plasma simulation

The present section focuses on the numerical investigation of GyM plasma. In
subsection 3.1 the simulation setup and main modelling hypotheses are
discussed, while the outcomes of the analysis are summarised in 3.2. The
latter is further articulated into 3.2.1 about a parametric sensitivity scan,
3.2.2 in which the model is validated against experimental data and 3.2.3, in
which the effect of including He metastable states in SOLPS runs is assessed
and compared with respect to a point model for GyM plasma [24].

###  3.1 Simulation setup

The first step towards the SOLPS-ITER plasma simulation is the construction of
the computational mesh, which is done for linear machines starting from the
reconstruction of magnetic equilibria for each discharge, according to the
procedure described in detail in reference [23]. The resulting B2.5 and EIRENE
meshes, structured quadrangular and unstructured triangular respectively, are
plotted in figure 3 for the plasma at $I_{coil}$ = 600 A. Since the magnetic
configuration is almost purely axial for all cases, no significant differences
in the appearance of the computational meshes are observed for the three
different $I_{coil}$.

The computational meshes displayed cover only one half of the physical domain
and cylindrical symmetry around the axis of the machine is assumed. Also note
that, while the EIRENE neutral mesh covers the entire volume of the device
upon symmetrization, the B2.5 field-aligned plasma mesh extends between the
two bases (targets) of the cylinder axially, and to the first point of
tangency radially, i.e. not to the physical lateral wall of the device.

All plasma simulations include He+ and He++ as ion species and neutral ground
state He, the standard SOLPS reaction database for a pure He case is
implemented. However, for the sole simulations discussed in subsection 3.2.3,
long-lived singlet and triplet He metastable states are also included among
neutral species in addition to the ground state. For such runs, a wider
reaction database is coherently implemented as described in more detail in
3.2.3.

A gas puff strength of $1.88\times 10^{19}$ neutral He atoms$/$s is set to
match the experimental puffing conditions. All the wall surfaces but those
corresponding to the turbomolecular pumps are considered as saturated with He
atoms, their particle absorption probability is thus set as $p_{a}=0$. On the
contrary, a particle absorption probability at EIRENE pumping surfaces of
$p_{a}=0.013$ is set to mimic the experimental pumping speed111The particle
absorption probability, or albedo, at the pumping surfaces $p_{a}$ is set to
model the effect of turbomolecolar pumps. Its value is related to the turbo-
pump speed $S$ (Ls-1) by: $S=A\times p_{a}\times 3.638\times\sqrt{T/m}$, where
$T$(K) is temperature and $m$(amu) is the mass of the neutral species. In the
formula, $A$(cm2) is the effective area of the pumping surface: since axial
symmetry is assumed in EIRENE, the area of each pumping surface to be
considered for the computation of $p_{a}$ is $A=2\pi r_{p}L_{p}$, where
$r_{p}$ and $L_{p}$ are the distance from the axis and the width of the
pumping surfaces [30].. Drift effects are totally neglected. The cross-field
transport is modelled as diffusive, which requires the specification of ion
diffusivity $D_{n}$ and ion/electron heat diffusivities $\chi_{i,e}$, treated
as free input parameters.

Figure 3: B2.5 and EIRENE computational mesh at $I_{coil}$ = 600 A plotted on
the GyM vessel. Gas puff location and pumping surfaces are also displayed.

Note that, depending on $I_{coil}$, the axial position of the resonant surface
- i.e. the locus where the electron cyclotron frequency matches the gyrotron
frequency of 2.45 GHz and energy is favourably absorbed by electrons, which
happens for a magnetic field intensity of 87.5 mT - is displaced (see figure
4(a)). Therefore, the external power delivered to the plasma is modelled as an
energy source for electrons distributed around the axial position $z_{res}$ of
the ECRH resonance.

In all simulations, the power density axial profile is modelled as a Gaussian
centered at $z_{res}$ with fixed width $\sigma=0.1$ m. Although the nominal
power of the ECRH source is set, the efficiency in the absorption of energy by
the electron population is unknown. Therefore, the actual amount of power
$P_{ext}$ absorbed by the plasma (i.e. the 3D integral of the power density
distribution) is treated as a free input parameter.

In the lack of experimental data or first-principle simulations of plasma-
microwave interaction, the radial distribution of external power density
source is also treated as a further degree of freedom. The modelling approach
devised in this work includes either constant or Gaussian radial
distributions, or a linear combination of the two (figure 4(b)). Again, any
radial distribution of the transport coefficients or power source is specified
only on one half of the computational domain (i.e. for $R>0$) and cylindrical
symmetry is assumed.

(a) The solid coloured curves plotted on the GyM vessel represent the loci of
points where the electron cyclotron resonance condition (i.e. B = 87.6 mT) is
met for the three values of $I_{coil}$ considered. The crosses identify the
points at which the curves intersect the cylinder axis, whose axial
coordinates are named $z_{res}$ and listed in the legend. Note that in the
$I_{coil}$ = 640 A case, two distinct points on the axis satisfy the resonance
condition.

(b) Contour plots of possible external power density source distributions
implemented in this work to model the ECRH power absorption by the magnetized
plasma. Although the resonance surface is not flat (i.e. the axial position of
the resonance varies with the radial coordinate, see figure 4(a)), the axial
profile of the external power source is modeled as a Gaussian curve centred at
$z_{res}$ (taken as radially constant) with a fixed width $\sigma=0.1$ m. In
turn, the amplitude of the Gaussian is modulated by a radial function, either
constant or a with a variable number of Gaussian peaks.

Figure 4: ECRH external power source modelling in SOLPS-ITER simulations of
GyM.

As concerns the setup of boundary conditions, the sheath boundary conditions
are enforced at the bases of the cylinder. On the symmetry axis of the
cylinder, zero particle and energy fluxes, as well as zero parallel velocity
gradient for the parallel momentum equation and zero current for the potential
equation are imposed. On the lateral boundary, zero parallel velocity gradient
and zero current are fixed, together with a leakage condition prescribing
particle and energy outflows suppressed by a factor $\alpha_{leak}=1\times
10^{-3}$ with respect to the corresponding flows at the plasma sheath
entrance.

Although some experimental setups described in section 2 include the presence
of a sample-holder along the plasma stream for the exposure of selected
samples for PMI experiments, the plasma simulations performed, described and
validated against experimental data in the following sections do not account
for its presence.

This choice is dictated by the limitations of the B2.5 mesh generator adopted
and technical restrictions in SOLPS-ITER when modelling linear geometries,
which require the specification of no more than two targets where the contact
between the plasma and the solid boundary occurs. This constraint prevents the
straightforward construction of a regular plasma computational mesh covering
the entire volume of the device and including the sample holder. This is
possible in principle, as demonstrated by the work of Rapp et al. [20], where
modifications to the SOLPS5.0 code were made to accommodate such a geometry,
but is beyond the scope of the present paper.

A possible straight-forward alternative to the code modification would be to
restrict the B2.5 computational mesh to the sole portion of the volume of the
device between one of the two bases and one of the two faces of the SH
including the gas puff valve and the ECRH resonant surface. However, this
approach would exclude the portion behind the SH, where - according to the
experimental data discussed in 2 \- the only significant differences with
respect to the plain scenario are expected, and therefore it is disregarded.

In summary, as long as an overall plasma analysis is concerned, the presence
of the SH is neglected. However, it will be taken into account properly in the
following investigation of material erosion (section 4).

###  3.2 Results

####  3.2.1 Parametric scan

As a first outcome of the numerical investigation of the He plasma in GyM, a
sensitivity scan of the code’s free parameters (enumerated in 3.1, i.e.
particle and heat anomalous cross-field transport coefficients, external power
source absorbed by the plasma) is presented. In each column of the left
portion of figure 5, the impact of tuning one input parameter at the time is
assessed for the $I_{coil}$ = 600 A scenario on the electron density and
temperature radial profiles evaluated at the axial position of LP 4U (similar
results hold for 560 A and 640 A scenarios as well). The values assumed by the
parameter being scanned are reported in legend, while the values kept fixed
for the other input parameters are listed at the bottom of each column.

  1. (a)

Particle diffusivity $D_{n}$. An increase in the anomalous cross-field
particle diffusion coefficient (figure 5a) reflects in a smoothing of the
electron density gradient from the periphery of the device inward, and in a
corresponding peaking of the electron temperature profile for $D_{n}\lesssim$
5 m${}^{2}/$s. For $D_{n}\gtrsim$ 5 m${}^{2}/$s, an increase in $D_{n}$
translates the profiles to lower densities and higher temperatures
respectively at constant shape. In figure, the outcome of a simulation with
$D_{n}\propto r^{2}$ ranging between 1 m${}^{2}/$s and 10 m${}^{2}/$s - which
will be used for achieving a better agreement with experimental data in
section 3.2.2 \- is also shown.

  2. (b)

Heat diffusivities $\chi$. A change in the ion and electron heat diffusivities
(figure 5b) has negligible impact over both electron and temperature profiles,
which is consistent with the fact that GyM plasma conditions - characterised
by low densities and small parallel temperature gradients - fall into the
sheath-limited regime, dominated by convection rather than conduction [11].

  3. (c)

External power source $P_{ext}$. An increase in the strength of the external
power source (figure 5c) results in an overall increase of the electron
density profile at constant electron temperature. This is coherent with the
GyM plasma being a low density one: the extra power supplied is spent to
increase the plasma ionization fraction rather than to heat up electrons.
Moreover, at fixed total amount of external power supplied, the modulation of
the radial distribution of the power density source - according to a linear
combination of constant and Gaussian functions - results in a local increase
of electron density and temperature in correspondence of the peaks of the
distribution.

According to figure 4(a), for the sole 640 A coil current scenario, two
resonant surfaces along the cylinder axis - and two corresponding values of
$z_{res}$ \- may be identified in sectors 2 and 6 of the device (with
reference to figure 1): in principle, the absorption of external energy occurs
at both axial positions. The right portion of figure 5d compares the electron
density and temperature profiles yielded by three simulations for which the
same amount of input total power (i.e. $P_{ext}$ = 200 W) is split in
different ratios between the two resonant surfaces (0-200 W, 100-100 W, 200-0
W, where the first number is the amount of external power deposited at sector
2 and the second at sector 6).

The temperature profiles are coincident, and only limited difference in the
magnitude of electron density profiles at constant shape is observed: $n_{e}$
evaluated at the axial position of LP 4U increases as the amount of external
power deposited at the second resonance in sector 6 increases from 0 to 200 W.
This is true not only at the axial position of LP 4U, but at all axial
positions.

For simplicity and similarity with the other coil current scenarios, in the
continuation of the analysis, the total amount of power - treated as a totally
free parameter - will be modelled as absorbed only at the resonance in sector
2, i.e. on the gas puff side of the device.

Figure 5: Sensitivity scan of one SOLPS-ITER free parameter at the time for
GyM. For the $I_{coil}$ = 600 A case: a) particle diffusivity $D_{n}$, b)
Electron/ion heat diffusivity $\chi_{i,e}$, c) total external power source
$P_{ext}$. For the $I_{coil}$ = 640 A case: d) scan in the relative
distribution of the same external total power source between the two resonance
surfaces.

####  3.2.2 Validation

In the present subsection, a validation of the numerical results against
experimental data is proposed. This is meant (i) to provide a realistic plasma
background for the eventual ERO2.0 erosion analysis, (ii) to gain physical
insight on the possible mechanisms responsible for the observed experimental
profiles, mostly in relation to the different magnetic field configurations
taken into account, (iii) to complement the dataset and enrich the
characterisation of GyM He plasma.

The parametric scan presented in subsection 3.2.1 pointed out that, among the
plasma parameters for which experimental data are available, electron density
profiles are the most sensitive to a change in the input parameters. On the
contrary, the values of electron temperature - especially at the centre of the
device - are scarcely dependent on the free parameters and are rather fixed by
the throughput strength, i.e. the overall balance of gas puffing and pumping.

Operatively, an optimum set of input parameters (cross-field transport
coefficients, external power absorbed by the plasma and radial distribution of
the external power density) is identified for each $I_{coil}$ scenario to
provide a satisfactory qualitative and quantitative reproduction of the
experimental density profiles, and the temperature ones are obtained
accordingly.

In all three cases, a parabolic profile is imposed for particle diffusivity
(i.e. $D_{n}\propto r^{2}$), such that $D_{n}(R=0)\sim 1$ m${}^{2}/$s and
$D_{n}(R=R_{0})\sim 10$ m${}^{2}/$s, at fixed and radially uniform heat
diffusivity $\chi_{i,e}=1$ m${}^{2}/$ s. The magnitude of the external power
input required to match the experimental profiles for the three coil current
intensity cases needs to be different ($P_{ext}=$ 200 W, 300 W and 250 W
respectively). Also, seeking for a satisfactory agreement with the density
profiles - which all show relative maxima at R = 0 and R $\simeq 6$ cm (see 2)
- a radial distribution for the external power density source is introduced
(namely a linear combination of a constant and two Gaussian functions in the
form:
$\rho(r)=N\times\left\\{c_{0}+\sum_{i=1}^{2}c_{i}\exp\left[-\left(r-r_{i}\right)^{2}/\sigma_{i}^{2}\right]\right\\}$,
with $c_{0}$, $c_{i}$, $r_{i}$, $\sigma_{i}$ free parameters and $N$
normalization factor to ensure that the volume integral of the power density
distribution matches the desired external power $P_{ext}$). Physically, this
suggests that the efficiency of ECRH-plasma coupling and energy transfer is
largely dependent on the specific magnetic field configuration, which in turn
can promote the absorption of energy at selected radial spots, where higher
electron density is found. In the absence of specific experimental data, a
first-principle numerical simulation of wave propagation in the GyM magnetized
plasma by means of $ad-hoc$ multiphysics codes could provide further physical
insight, but this is beyond the scope of this paper.

The results and a summary of the adopted parameters are visible in figure 6.
The numerical temperature profiles are coincident at the three LP axial
positions; however, while they are in good agreement with the corresponding
experimental ones for the 640 A scenario, they tend to underestimate them in
the 600 A and even more in the 560 A scenario.

Figure 6: Validation of SOLPS-ITER simulation results against electron density
and temperature experimental profiles discussed in section 2. Blue and red
lines represent the experimental and numerical radial profiles respectively.
Each tone of blue indicates one of the three LP and is associated to a
corresponding tone of red, which indicates the axial position of the LP itself
at which the numerical profiles are taken. At the bottom of the figure, the
first horizontal band summarises the free parameters (anomalous cross-field
transport coefficients and external power source) adopted for the numerical
simulations presented, while the second horizontal band provides the
parameters used to model the radial distribution of the power source $\rho(r)$
according to the formula
$\rho(r)=N\times\left\\{c_{0}+\sum_{i=1}^{2}c_{i}\exp\left[-\left(r-r_{i}\right)^{2}/\sigma_{i}^{2}\right]\right\\}$.
For the 600 A case, the results of simulations at constant radial power
deposition - used for the erosion analysis presented in section 4 \- are
plotted in dotted line style.

For the 600 A case, the dotted line in figure 6 shows the reference plasma
background used for the erosion analysis presented in section 4. This
simulation uses the same anomalous transport coefficients enforced in the
solid-line simulations ($D_{n}\propto r^{2}$ in the range of 1-10 m${}^{2}/$s,
$\chi_{i,e}=1.0$ m${}^{2}/$s), with a radially uniform external electron
energy source of 400 W. The numerical results well reproduce the observed
radial $n_{e}$ and $T_{e}$ overall trends, although the density peak at
$R\simeq 6$ cm is not reproduced. However, note that the radial portion of
plasma relevant to the ERO2.0 erosion study is the one striking the SH, which
extends well below the radial position of the $n_{e}$ peak. Therefore, these
simplified numerical profiles will constitute the plasma background for the
ERO2.0 erosion simulation discussed in section 4.

By considering the latter simplified 600 A model with some detail from now on
(similar considerations hold for the other scenarios as well), the 2D
distributions of neutral He, He+, He++ densities and of neutral He pressure
are plotted on a longitudinal section of the device. The density of neutral He
is about three orders of magnitude greater than the density of He+, which is
in turn five orders of magnitude greater than the one of He++, which determine
an ionization fraction $\alpha\simeq 0.002$. The presence of He++ may be thus
reasonably neglected in the low-density and low-temperature GyM plasma,
contrary to the situation typical of tokamaks, where the higher ionization
state is generally dominant [31].

The neutral pressure distribution is approximately uniform throughout the
central volume of the device and is peaked at the bases of the cylinder, where
the ions striking the solid surfaces are recycled as neutrals and puffing is
performed. Also note that the He pressure provided by the model in
correspondence of the pressure gauges is about 0.09 Pa: this value is totally
compatible with the experimental one (see 2).

Figure 7: 2D distribution of He, He+, He++ density and He pressure plotted on
a longitudinal section of GyM for the $I_{coil}$ = 600 A scenario. Note that
$n_{\ce{He}}\gg n_{\ce{He+}}\gg n_{\ce{He++}}$.

####  3.2.3 Role of metastable states

The He atom possesses two long-lived metastable (MS) states: both feature the
$1s^{1}2s^{1}$ electronic configuration, and are distinguished by the spin
orientation of the two electrons into singlet and triplet states, which will
be labeled as He^*(1) (with lifetime $\tau\simeq$ 20 ms) and He^*(3)
($\tau\simeq$ 7800 s) respectively [32].

In low-temperature, low-density plasma conditions typical of LPDs, MS states
and their transport within the chamber could affect the plasma properties. In
this work, the possible relevance of including such He MS states among the
plasma neutral populations is preliminarily investigated for the first time in
a purely He plasma by means of ad-hoc SOLPS-ITER simulations. Operationally,
this goal is achieved by splitting the neutral population treated by EIRENE
into three separate species (the ground state He and the two MS states) and
enlarging the reaction database to include the MS-resolved ones extracted from
the AMJUEL database and reported in table 1 [33].

| Reactions | Type | Database
---|---|---|---
1 |  e- + He -¿ 2e- + He+ | EI | AMJUEL 2.3.9b
2 |  e- + He -¿ e- + He^*(1) | EI | AMJUEL 2.3.9c
3 |  e- + He -¿ e- + He^*(3) | EI | AMJUEL 2.3.9d
4 |  e- + He^*(1) -¿ e- + He | EI | AMJUEL 2.3.9e
5 |  e- + He^*(1) -¿ 2e- + He+ | EI | AMJUEL 2.3.9f
6 |  e- + He^*(1) -¿ e- + He^*(3) | EI | AMJUEL 2.3.9g
7 |  e- + He^*(3) -¿ e- + He | EI | AMJUEL 2.3.9h
8 |  e- + He^*(3) -¿ e- + He^*(1) | EI | AMJUEL 2.3.9i
9 |  e- + He^*(3) -¿ 2e- + He+ | EI | AMJUEL 2.3.9j
10 |  e- + He+ -¿ He | RC | AMJUEL 2.3.13b
11 |  e- + He+ -¿ He^*(1) | RC | AMJUEL 2.3.13c
12 |  e- + He+ -¿ He^*(3) | RC | AMJUEL 2.3.13d
Table 1: List of He MS-resolved atomic reactions included in SOLPS-ITER model,
extracted from AMJUEL database. EI and RC stand for electron impact collision
and recombination respectively [33].

Figure 8 compares the electron density and temperature radial profiles at a
fixed axial position yielded by two simulations, MS-unresolved and resolved
respectively. The input parameters set in the two cases are those of the
simplified model for the $I_{coil}=$ 600 A case presented in 3.2.2 (dotted red
lines in figure 6: $D_{n}\propto r^{2}$ in the range of 1-10 m${}^{2}/$s,
$\chi_{i,e}=1.0$ m${}^{2}/$s, $P_{ext}=400$ W uniformly distributed radially).
The inclusion of MS neutral states yields a slightly higher electron density
profile, acting (check the power scan in 3.2.1) as if a larger amount of power
was supplied to the plasma: the presence of He MS species thus appears to
promote ionization processes and increase the overall ionization fraction.
Conversely, the two electron temperature profiles are practically
indistinguishable. Also, the distribution of both MS species provided by
Eirene is approximately uniform throughout the chamber.

Figure 8: Electron density and temperature radial profiles yielded by
homologous MS-resolved and unresolved simulations based on the simplified
$I_{coil}$ = 600 A case described in 3.2.2 and evaluated at the axial position
of the LP 4U. In background, experimental data acquired by LP 4U (600 A) are
showed for reference.

Nevertheless, in the absence of a proper spectroscopic diagnostics, the actual
relevance of He MS states in GyM plasma goes undetected under the experimental
point of view, and the resolved model may not be validated. As a partial
compensation for such limitation, a comparison between the outcomes of the
SOLPS-ITER runs of figure 8 and the results of a simplified MS-resolved 0D
plasma model is proposed. Such 0D model was developed in [31] extending the
MS-unresolved counterpart firstly presented in [24], and its main features are
retraced in Appendix A.

In the 0D model, the experimental values of gas puff and pumping speed (42
sccm and 500 Ls-1 respectively) are set, a decay length for density of
$\lambda_{n}=5$ cm is set coherently with the choice made in reference [24].
Concerning the cross-field diffusion coefficient, the SOLPS simulations
validated in section 3.2.2 all assumed a radially variable profile. In a 0D
treatment, any spatial dependence is suppressed, and the specification of a
single value of diffusivity $D_{\perp}$ is required. Therefore, a scan in the
value of the diffusion coefficient is made investigating the effect of its
choice on the outcomes.

Clearly, the strongly different hypotheses on which SOLPS and the 0D model are
based make it impossible to perform a direct quantitative benchmark between
the two. Most importantly, the 0D model includes one MS species only and is
based on the ADAS database, while SOLPS treats both MS states and is based on
the AMJUEL database. However, the global 0D model is a useful simplified tool
to compare relative trends between MS-resolved and unresolved cases.

The results of the comparison are summarised in figure 9. The bars report the
values of the mean metastable and electron density and temperature for the two
SOLPS simulations - MS-unresolved and resolved respectively - whose outcomes
are plotted in figure 8 according to the legend. Conversely, the line plot in
the centre shows the same quantities as foreseen by the 0D point model as a
function of the diffusion coefficient $D_{\perp}$ input parameter.

As a first observation, both the SOLPS simulation and the 0D model -
independently on the $D_{\perp}$ set - predict a negligible increase in the
electron temperature $T_{e}$ when turning from the unresolved to the resolved
case. Also, the ratio between the mean MS density and the mean electron
density foreseen by the 0D model increases from $\sim 20\%$ to $\sim 50\%$ as
$D_{\perp}$ increases from 0.5 to 10 m2/s: the ratio at $D_{\perp}$ = 5 m2/s
is $\sim 30\%$ and is consistent with the one predicted by the SOLPS
simulation.

Turning from the unresolved to the resolved model in SOLPS foresees an
increase in the mean $n_{e}$ of $\sim 20\%$. As far as the point model is
concerned, the magnitude and the sign of the variation in $n_{e}$ following
the introduction of the MS population appears to depend on $D_{\perp}$. For a
choice of $D_{\perp}$ = 10 m2/s, an increase of $\sim 20\%$ in $n_{e}$ is
observed, which is compatible with SOLPS results -. However, lower values of
$D_{\perp}$ are associated to a decrease of $n_{e}$ upon introduction of MS
species.

Figure 9: Comparison between the results of SOLPS-ITER and 0D point model as
concerns MS-unresolved and resolved He plasma simulations. The bar charts are
related to SOLPS results: on the left, the mean electron density in the MS-
unresolved and resolved case (solid and dashed respectively) and the mean
density for MS species (dotted) are reported. On the right, the mean electron
temperature is reported for the unresolved (solid) and resolved (dotted)
simulations. In the centre, the same quantities provided by the 0D model
described in Appendix A are shown with the same line styles as a function of
the diffusion coefficient.

Overall, although no rigorous validation may be provided at this stage, one
may argue that the current implementation of He MS populations in SOLPS-ITER
is at least partially supported by the compatibility of its results with the
ones of a much simpler 0D model.

Although assessing the role of MS species in a pure He plasma is crucial for
the characterisation of the plasma itself and a validation of the MS-resolved
model is urgent, in the following erosion analysis we will continue to refer
to the unresolved simulations presented in subsection 3.2.2, which were
validated against experimental data. In the first instance, it appears
reasonable to assume that the actual role of MS states in GyM plasma is
scarcely relevant as long as the numerical investigation of PMI is concerned.
In fact, since erosion results from the impact of energetic ions on solid
surfaces, and since the external power source is also treated as a free
parameter, once the numerical electron density profiles match the experimental
ones, it makes little difference that this is achieved without MS states among
the neutral population and slightly higher input power or with MS states and
slightly lower input power respectively.

##  4 ERO2.0 global erosion simulation

The present section focuses on the material side, i.e. it presents the ERO2.0
simulations performed to assess the erosion of plasma-facing structures in
GyM, of PMI-relevant tungsten (W)-based samples exposed to the plasma stream
and the deriving deposition. The main goals, in this respect, are (i) the
comparison between the outcome of simulations based on simplified assumptions
for the wall material with respect to the full steel composition, (ii) the
assessment of the effect related to SH presence on wall erosion and,
conversely, (iii) the role of wall impurities on sample erosion. Note that,
although no experimental data concerning material erosion are available for
model validation at this stage, the results of this work will guide the design
of an experiment devoted to the collection of erosion-deposition data, to be
performed by placing collectors at selected spots of the vessel.

###  4.1 Simulation setup

As far as the simulation setup is concerned, the 3D GyM wall geometry is
reproduced (figure 10) and the 2D distribution of relevant plasma parameters
(electron and ion density and temperature, plasma flow) yielded by the
simplified SOLPS-ITER simulation at $I_{coil}$ = 600 A - taken as a reference
and discussed in detail in the second part of 3.2.2 \- is imported as a plasma
background, assuming cylindrical symmetry. Note that, according to the
observations made in 3.2.2, He+ may be safely considered as the only relevant
ion species, thus excluding the He++ contribution from the input plasma
background. Moreover, the aforementioned sample-holder is included in the
definition of the simulation domain at fixed plasma background, figure 10
showing its geometry and materials. In all simulations, sputtering yields are
retrieved from the SDTrimSP database available in ERO2.0. The energy
distribution of sputtered species is modelled as a Thompson one, while their
angular distribution is assumed to be a function of the incidence angle of the
impinging ion, with the forward direction with respect to it being favoured
[34].

(a) All 3D components of GyM drawn and imported in ERO2.0, with a zoom on the
sample-holder. Only half of the lateral wall with relative bushings on it is
represented for better visibility. Colours represent the materials adopted in
ERO2.0 simulation, as reported in legend.

(b) Photograph of GyM sample-holder.

Figure 10: 3D GyM geometry considered in this work.

A first parameter which has been varied in the simulations is the wall
material. Although the GyM vessel is made of AISI 304L SS, the previous study
treated it as made of either pure Fe or pure Cu [12]. Such a choice was
dictated by a lack of sputtering yield data for He ion species on chromium
(Cr) and nickel (Ni), i.e. the main alloying elements of SS, and their
similarity - in terms of physical properties - to Cu, for which a much wider
database was available and thus taken as proxy [12]. Since then, the dataset
has been enlarged by means of SDTrimSP simulations, and this work considers
and compares pure Fe, pure Cu and realistic AISI 304L SS (73% Fe, 18% Cr, 9%
Ni) as possible wall materials: the results of this investigation are
presented in 4.2.1.

Secondly, the possibility of setting a bias voltage ($V_{bias}$) to the SH is
foreseen. Applying a tunable bias voltage to the SH enables to modulate the
kinetic energy of the plasma ions impinging onto it. The application of a bias
voltage in experiments influences the conditions in the sheath. However, since
SOLPS-ITER background plasma only extends up to the sheath entrance, no
influence of the bias voltage on the plasma parameters imported in ERO has
been considered. For $V_{bias}=0$, the energy of impinging ions depends solely
on the plasma potential, which is about 20 V. Although the bias voltage
applied is always negative (thus accelerating impinging ions), we will always
refer to its absolute value for simplicity.

The $V_{bias}$ values applied to SH in this simulations reflect the ones
usually set experimentally, in the range 0 - 320 V. As in experiments, the
bias voltage is applied only to samples and molybdenum (Mo) mask. Also note
that the tantalum (Ta) mask, being unbiased during experiments, is expected
not to be eroded, and it has been assumed as made of W due to a lack of
sputtering yield data for all material combinations with Ta and to reduce the
number of different species in the simulations. The choice of W is justified
by the high sputtering threshold for He on Ta ($\sim$ 90 eV), which is
comparable with the one on W (105-110 eV) [35].

###  4.2 ERO2.0 results

####  4.2.1 Wall material analysis

In this section, the effect of wall material choice on GyM wall erosion is
investigated. Three different wall material compositions have been considered:
(i) pure Cu, (ii) pure Fe and (iii) the real AISI 304L SS. For each one of
these, five different values of $V_{bias}$ have been set, namely 0, 50, 120,
200 and 320 V, in agreement with the ones commonly adopted in experiments. The
dependence of the SDTrimSP sputtering yields used in this work on projectile
energy and incidence angle is reported in figure 11. Due to the numerous
combinations of projectile - target materials (36), only the most relevant
ones are shown, namely He, as main plasma species, and Mo, as main eroded
species, projectiles. For the SS case, according to the local incoming flux of
He+ ions and eroded particles, the code evaluates the sputtering yield for
each element separately, and the eroded flux is then weighted on the material
composition. No composition evolution is considered in this first attempt.

Figure 11: Sputtering yields for He (left) and Mo (right) projectile on
different target materials as a function of energy at orthogonal incidence
(first row) and of impinging angle at 200 eV (second row, normalized to 0°
value).

The estimated gross erosion of GyM walls, i.e. the number of sputtered atoms
per unit area and time without considering their possible deposition, as a
function of wall material with no bias voltage applied to the SH and expressed
in $[$atoms $\cdot$ m-2s${}^{-1}]$ is: (i) 7.92$\times 10^{17}$ for pure Cu,
(ii) 5.28$\times 10^{16}$ for pure Fe, and (iii) 3.86$\times 10^{16}$ for SS.

For the case of SS wall, the overall gross erosion arises from the sum of the
gross erosion individual contributions associated to each element in the SS
composition. Pure Cu wall presents a severely higher gross erosion with
respect to both pure Fe and SS, whose behaviour is comparable. This hierarchy
is in accordance with the relative magnitude of the sputtering yields reported
in figure 11, where Cu presents the highest erosion for both He and Mo
projectiles. Note that the observed trend is unchanged also at a larger
$V_{bias}$, for which a few percent increase in the global gross erosion
amount is obtained for all wall materials investigated: this is consistent
with the higher kinetic energy acquired by plasma ions impinging on the SH,
which reflects in a higher amount of energy retained by Mo and W atoms eroded
from the SH itself, eventually impinging on surrounding walls and fostering
their erosion.

The overall 30% increase in gross erosion for Fe with respect to SS walls may
be ascribed to the presence of Cr in SS, which is characterized by the lowest
sputtering yield. Indeed, about 80% of SS gross erosion is due to Fe, with a
10% contribution due to Cr and Ni each: the presence of Fe will thus reduce as
the fluence increases and erosion proceeds, in favour of a Cr-enriched
surface. This could further reduce the overall SS gross erosion. Considering
GyM vessel as entirely made of of Fe is thus expected to yield a reasonable
upper limit for the wall erosion, while limiting the complication in the
interpretation of the simulation results deriving from the large number of
projectile-target combinations for SS. Therefore, this modelling approach will
be adopted in the continuation of this work.

####  4.2.2 Effect of sample-holder on GyM wall erosion

In the previous work [12], a proof of concept of the coupling between SOLPS-
ITER and ERO2.0 in a linear device (GyM) was presented. In this work, the
objective is moving forward in the direction of modelling a real PMI
experiment, thus including also the SH in ERO2.0 simulation domain.

Due to its position in the middle of the device, impurities eroded from the SH
are likely to impinge onto the lateral wall, which in turn may undergo
erosion. Note that the magnetic field lines are parallel to the lateral wall
itself: according to the leakage boundary conditions enforced in the setup of
plasma simulations, the ion radial flux at the lateral boundary of the plasma
computational mesh is 1/1000 of the corresponding one at the entrance of the
sheath which builds up in front of a solid surface orthogonal to the magnetic
field lines. Therefore, the lateral wall erosion is mainly due to impurities
eroded elsewhere, with the SH possibly playing a major role as it is the only
biased component during GyM experiments. All the other parts of GyM wall are
not substantially influenced by the presence of the SH, thus will not be
considered in this section: as a reference, at maximum $V_{bias}$, impurities
eroded from the samples and SH are responsible for just $\sim 1\%$ of the
gross erosion of the cylinder bases, which may be practically fully ascribed
to the striking of plasma ions.

Figure 12 shows the distribution of the gross erosion of GyM lateral wall in
different conditions, namely without SH and with SH inserted, at lowest (0 V)
and highest (320 V) $V_{bias}$. Complementarily, figure 13 presents the
relative contributions of the different species (i.e. eroded impurities) to
the erosion of the lateral wall as a function of the bias voltage applied to
the SH. Again, He plasma ions are not included since they are not striking the
lateral wall and causing its direct erosion.

For $V_{bias}$ = 0, the influence of the SH on lateral wall erosion is limited
and concentrated in the proximity of the SH itself. In this condition, only Fe
from the SH support and cylinder bases is eroded and may in turn erode the
lateral wall, while Mo and W are not: the contribution from the Fe SH support
(see figure 10(a)) is then what differentiates the case with and without SH in
figure 12. As $V_{bias}$ is increased to 50 V, the kinetic energy of plasma
ions overcomes the sputtering threshold for Mo. Eroded Mo impurities in turn
retain sufficient energy to largely dominate the erosion of the lateral wall
and increase its absolute value by one order of magnitude (figure 13). At
higher and higher $V_{bias}$, the magnitude of the lateral wall erosion
increases more and more and is firmly dominated by Mo. W eroded from samples
plays a marginal role only for V${}_{\text{bias}}>$ 120 V, when the sputtering
threshold for He on W is overcome. According to the colorbars in figure 12 and
the table in figure 13, the difference in both local and global gross erosion
between the lowest and highest $V_{bias}$ reaches a few orders of magnitude.

Figure 12: Gross erosion distribution of GyM lateral wall in different
conditions, namely without SH inserted and including it with lowest (0 V) and
highest (320 V) $V_{bias}$. Note the different boundaries for the colorbars in
the two cases.

(a)

$V_{bias}$ (V) | Gross erosion (m-2 s-1)
---|---
0 | $1.68\times 10^{11}$
50 | $1.20\times 10^{12}$
120 | $8.46\times 10^{12}$
200 | $4.47\times 10^{13}$
320 | $2.04\times 10^{14}$
(b)

Figure 13: Contribution of the different eroded impurities to the erosion of
lateral wall as a function of the bias voltage applied to the SH. He plasma is
not included since it is not causing erosion of lateral wall. The overall
gross erosion of the lateral wall is reported in table.

####  4.2.3 Effect of eroded materials on sample erosion

Once investigated the effect of SH inclusion on the erosion of GyM walls, the
influence of wall impurities on the sample erosion is assessed. This is
particularly relevant for the possible interpretation of experimental data,
since both the increased erosion caused by wall impurities and their
deposition on samples could influence experimental outcomes of mass loss
measurements.

Figure 14 shows the contribution of the various impinging particles to the
gross erosion of samples as a function of the bias voltage applied to SH. In
this case, He plasma is included, since samples are directly eroded by it. As
for lateral wall, only Fe components are eroded by He plasma with no bias,
thus only Fe neutrals and ions possess enough energy to slightly contribute to
sample erosion. For $V_{bias}$ = 50 V, Mo mask starts to be eroded by the
plasma, and it immediately becomes the major contributor to sample erosion.
Indeed, both Mo eroded from the mask and W eroded from samples may be ionized
in plasma, be captured by axial magnetic field and come back to sample-holder
due to plasma flow.

As soon as He plasma ions, whose flux onto the samples is $\Gamma\simeq
3.5\times 10^{20}$ m-2s-1, are able to directly erode W samples (for
$V_{bias}\geq$ 120 V), they dominate sample erosion: this is consistent with
the fact that the eroded impurities account for just $\sim 0.3\%$ of the total
particle flux striking the samples, which is thus largely dominated by the
plasma itself. However, it should be noted that around the sputtering
threshold for He on W, i.e. for $V_{bias}$ = 120 V, a 15-20% of W sample
erosion is still caused by impurities eroded from Mo mask. The contribution
due to eroded impurities then decreases to few percent at higher bias in
favour of He. Also note that, at every step in the $V_{bias}$, the sample
gross erosion increases by orders of magnitude.

(a)

$V_{bias}$ (V) | Gross erosion (m-2 s-1)
---|---
0 | $1.94\times 10^{10}$
50 | $1.19\times 10^{13}$
120 | $5.04\times 10^{16}$
200 | $1.76\times 10^{18}$
320 | $6.01\times 10^{18}$
(b)

Figure 14: Contribution of the different impinging projectiles to the erosion
of samples as a function of bias voltage applied to the SH. The overall gross
erosion of the samples is reported in table.

Finally, the fractional deposition of impurities on samples is evaluated in
figure 15. For each $V_{bias}$ investigated, the origin (i.e. the erosion
location within the vessel) of the deposited species is displayed. For
$V_{bias}$ = 0, the main source of deposition is Fe eroded from the SH
support, with a non-negligible contribution due to Fe eroded from bushings
[12] around the SH (see figure 10(a)). As it is evident from the first row of
figure 15, the bases of the GyM wall are too far to significantly influence
sample erosion/deposition. As $V_{bias}$ increases, Mo coming from the mask
plays a dominant role even at the highest $V_{bias}$, when its contribution
remains higher than redeposition from samples themselves. Looking at table in
figure 15, it should be noted that deposition on samples is well below the
gross erosion for $V_{bias}\geq$ 120 V, thus only marginally influencing net
erosion measurements when bias is high enough for He ions to cause W
sputtering. On the contrary, for $V_{bias}<$ 120 V, samples are in net
deposition conditions.

(a)

$V_{bias}$ (V) | Deposition (m-2 s-1)
---|---
0 | $5.60\times 10^{11}$
50 | $3.96\times 10^{13}$
120 | $1.61\times 10^{14}$
200 | $3.38\times 10^{14}$
320 | $5.62\times 10^{14}$
(b)

Figure 15: Deposition fraction of impurities on samples as a function of their
erosion location, for all the $V_{bias}$ investigated. The total deposition on
samples is reported in table.

##  5 Conclusions and future perspectives

The study discussed in this paper presents a comprehensive numerical
investigation of Plasma-Material Interaction (PMI) in GyM linear plasma
device, performed by coupling plasma solver SOLPS-ITER and erosion-deposition
code ERO2.0, according to the procedure firstly described in reference [12].
To this purpose, few PMI-relevant pure He discharges are taken as a reference
experimental framework for plasma background production.

Based on a preliminary sensitivity scan, which points out that the tuning of
SOLPS-ITER free parameters mostly affects the electron density rather than
temperature radial profiles, a validation of the numerical plasma profiles
against available experimental data is presented for three magnetic field
configurations. Optimal input parameters allowing to reproduce experimental
Langmuir probe profiles are identified, allowing (i) to recognise the strength
and radial shape of the external power source as the major responsible for the
difference in the profiles at different coil current intensity, (ii) to
generate a realistic plasma background (at $I_{coil}$ = 600 A) taken as a
reference for eventual erosion investigation.

Secondly, a novel investigation of the role of neutral long-lived He
metastable states is carried out through proper SOLPS simulations. Primarily,
the inclusion of MS populations in GyM conditions results in an overall
increase of the electron density at constant external power source, which is
coherent with the results yielded by the analytical global point model
presented in Appendix A of this work for a suitable choice of diffusion
coefficient.

The outcomes of the reference MS-unresolved SOLPS run are taken as input for
ERO2.0 studies of impurity erosion and deposition concerning both the walls of
the device and the samples exposed to the plasma stream and arranged in a
central sample-holder (SH), as a function of the wall material and of the bias
voltage applied to the SH itself. With respect to the preliminary work
presented in [12], the present investigation is extended to account for the
realistic SS composition of GyM vessel on one hand, and for the presence of
the SH in the geometry on the other.

The estimation of the gross erosion as a function of the wall material points
out that SS behaviour is well approximated by that of Fe only, which is then
taken as proxy for the sake of simplicity. Moreover, the SH - mainly through
the impurities eroded from the Mo mask - turns out to dominate the erosion of
the lateral wall of the device.

As far as W-based samples exposed to the plasma stream are concerned, their
erosion by He+ species is the dominant mechanism for sufficient bias voltage,
although the Mo impurities eroded from the SH also contribute to the erosion
of the samples themselves. Finally, the analysis of the impurities deposited
on the samples suggests that they mainly originate from the Mo mask as the
bias voltage exceeds the threshold for Mo sputtering by impinging plasma ions.

Several continuations and future perspectives are envisaged for the present
work. On the plasma side, this includes the application of the new Wide Grid
SOLPS-ITER release to GyM geometry, possibly enabling to build a computational
mesh accounting for the presence of the SH. Also, the investigation of the
role of He metastable states is only at the beginning: for instance, further
numerical campaigns and more refined theoretical models are required to assess
their contributions in tokamaks, possibly supported by experimental data for
validation and comparison purposes.

Still on the plasma side, the first-principle modelling of wave transport in
plasma through proper multiphysics code may shed light on the absorption of
external power by electron cyclotron resonance at specific spots of the
resonant surface, which in this work is taken as a purely free parameter and
adapted to match the experimental profiles.

On the material side, an experimental campaign is foreseen to fill the current
lack of data concerning the erosion and deposition of both GyM vessel and
samples. The experiment, whose design is supported by the ERO2.0 numerical
simulations, will be carried out by placing deposition monitors at selected
spots of the device - where higher erosion and deposition rates are expected -
and collected data will in turn allow to validate the outcomes of the
numerical simulations presented in this paper.

A further possible continuation of the ERO2.0 numerical investigation is about
assessing the effect of residual carbon or oxygen present in the GyM vacuum
chamber on the erosion and deposition processes. Finally, the implementation
of a back-coupling scheme between the two codes is of interest, such that the
ERO2.0 impurity sources are provided as input to second-order SOLPS-ITER
simulations.

##  Appendix A MS-resolved 0D model for GyM plasma

In this appendix, some details about the MS-resolved 0D model used in section
3.2.3 are reported. Such analytical model is developed in [31] extending the
MS-unresolved one presented in reference [24], which is based on the volume
averaging of SOLPS-ITER equations.

GyM plasma is described by a set of space-independent variables, i.e. density
$n_{p}$ and temperature $T_{p}$ of each one of the considered populations $p$.
The MS-unresolved model of reference [24] included ground state (GS) He as the
only neutral species and He+ as the only ion species, thus neglecting the
presence of He++ ions and allowing to identify the ion and electron density
$n_{\ce{He+}}=n_{e}$ to ensure neutrality. Moreover, the unresolved reaction
rates were obtained by solving the ADAS radiative collisional model neglecting
the lifetime of MS states.

Conversely, a resolved treatment explicitly introduces a density balance
equation for every MS species considered. For simplicity, the 0D model
developed in [31] and hereby retraced still considers He+ as the only ion
species and a single MS state He∗ accounting for both the singlet and triplet
states, which thus requires the introduction of a single additional density
balance equation. Ion and neutral temperatures are assumed to be constant and
equal to the room temperature of 0.025 eV. The reaction rates are extracted
from the corresponding ADAS MS-resolved database.

The equations of the resolved model are reported below, where the unknowns
$n_{\ce{He+}}=n_{e}$ are the densities of He+ ions and electrons respectively,
$n_{\ce{He}}$ and $n^{*}$ are the densities of the ground state neutral He and
of the single MS species respectively, and $T_{e}$ is the electron
temperature.

$\frac{dn_{\ce{He+}}}{dt}=R_{iz}n_{e}n_{\ce{He}}+R^{*}_{iz}n_{e}n^{*}-R_{rc}n_{e}n_{\ce{He+}}-R^{*}_{rc}n_{e}n_{\ce{He+}}-\Gamma_{i,\ce{wall}}n_{\ce{He+}}$
(1)

$\begin{split}\frac{dn_{\ce{He}}}{dt}&=-R_{iz}n_{e}n_{\ce{He}}+R_{rc}n_{e}n_{\ce{He+}}+X_{\ce{(MS)\rightarrow(GS)}}n_{e}n^{*}\\\
&-X_{\ce{(GS)\rightarrow(MS)}}n_{e}n_{\ce{He}}+\Gamma_{n,\ce{recyc}}n_{\ce{He+}}-\Gamma_{n,\ce{pump}}n_{\ce{He}}+\frac{\Gamma_{n,\ce{puff}}}{V}\end{split}$
(2)

$\begin{split}\frac{dn^{*}}{dt}&=-R^{*}_{iz}n_{e}n^{*}+R^{*}_{rc}n_{e}n_{\ce{He+}}+X_{\ce{(GS)\rightarrow(MS)}}n_{e}n_{\ce{He}}\\\
&-X_{\ce{(MS)\rightarrow(GS)}}n_{e}n^{*}-\Gamma_{n,\ce{pump}}n^{*}\end{split}$
(3)

$\begin{split}\frac{3}{2}n_{e}\frac{dT_{e}}{dt}&=\frac{P_{ext}}{V_{e}}-E_{iz}R_{iz}n_{e}n_{\ce{He}}-\left(E_{iz}-E^{*}_{iz}\right)R^{*}_{iz}n_{e}n^{*}\\\
&-E_{n,\ce{rad}}R_{n,\ce{rad}}n_{e}n_{\ce{He}}-E^{*}_{n,\ce{rad}}R^{*}_{n,\ce{rad}}n_{e}n^{*}\\\
&-E^{*}_{iz}X_{\ce{(GS)\rightarrow(MS)}}n_{e}n_{\ce{He}}-E_{i,\ce{rad}}R_{i,\ce{rad}}n_{e}n_{\ce{He+}}-\Gamma_{e,\ce{wall}}T_{e}n_{e}\\\
&-\frac{3}{2}\frac{2m_{e}}{m_{i}}R_{i,\ce{el}}n_{\ce{He+}}n_{e}\left(T_{e}-T_{i}\right)\\\
&-\frac{3}{2}\frac{2m_{e}}{m_{n}}R_{n,\ce{el}}\left(n_{\ce{He}}+n^{*}\right)n_{e}\left(T_{e}-T_{n}\right)-\frac{3}{2}\frac{dn_{\ce{He+}}}{dt}T_{e}\end{split}$
(4)

The meaning of the coefficients which appear in the balance equations above is
summarized in table 2. Note that it is assumed that puffing and plasma
recycling only produce GS He atoms, while turbo-pumps act as sinks also for
the MS population.

The input parameters of the 0D model are the external power $P_{ext}$, the gas
puff strength and pumping speed, which determine $\Gamma_{n,\ce{puff}}$ and
$\Gamma_{n,\ce{pump}}$ according to the expressions valid for the unresolved
point model contained in [24]. Moreover, since the expression for the ion-to-
wall sink is retrieved from the integration of SOLPS-ITER equations, it is
required to specify a radial decay length for the density $\lambda_{n}$ \-
mimicking the decay boundary condition scheme - and a cross-field diffusion
coefficient $D_{\perp}$ for its computation according to the formula derived
in reference [24].

Symbol | Physical meaning | Units
---|---|---
$R_{iz}$ |  Ionization rate: e- + He -¿ 2e- + He+ |  (cm-3 s-1)
$R^{*}_{iz}$ |  Ionization rate: e- + He^* -¿ 2e- + He+ |  (cm-3 s-1)
$R_{rc}$ |  Recombination rate: e- + He+ -¿ He |  (cm-3 s-1)
$R^{*}_{rc}$ |  Recombination rate: e- + He+ -¿ He^* |  (cm-3 s-1)
$R_{n,\ce{rad}}$ | Excitation rate for GS |  (cm-3 s-1)
$R^{*}_{n,\ce{rad}}$ | Excitation rate for MS |  (cm-3 s-1)
$R_{i,\ce{rad}}$ |  Excitation rate for He+ |  (cm-3 s-1)
$R_{i,\ce{el}}$ | Ion - electron elastic collision rate |  (cm-3 s-1)
$R_{n,\ce{el}}$ | Electron - neutral elastic collision rate |  (cm-3 s-1)
$X_{\ce{(GS)\rightarrow(MS)}}$ |  Metastable cross-coupling rate: e- + He -¿ e- + He^* |  (cm-3 s-1)
$X_{\ce{(MS)\rightarrow(GS)}}$ |  Metastable cross-coupling rate: e- + He^* -¿ e- + He |  (cm-3 s-1)
$\Gamma_{i,\ce{wall}}$ | Ion-to-wall sink |  (s-1)
$\Gamma_{e,\ce{wall}}$ | Electron-to-wall sink |  (s-1)
$\Gamma_{n,\ce{recyc}}$ | Recycling as GS neutrals at the walls |  (s-1)
$\Gamma_{n,\ce{puff}}$ | Neutral GS gas puff |  (s-1)
$\Gamma_{n,\ce{pump}}$ | Pumping (both GS and MS) |  (s-1)
$E_{iz}$ | GS ionization energy: 24.5 eV | (eV)
$E^{*}_{iz}$ | MS ionization energy (triplet): 19.8 eV | (eV)
$E_{n,\ce{rad}}$ | Excitation energy for GS | (eV)
$E^{*}_{n,\ce{rad}}$ | Excitation energy for MS | (eV)
$E_{i,\ce{rad}}$ |  Excitation energy for He+ | (eV)
$m_{e}$ | Electron mass | (A.U.)
$m_{n}=m_{i}$ |  Neutral He and He+ mass (taken as equal) | (A.U.)
$V=V_{e}$ | Volume occupied by neutrals and plasma (taken as equal) |  (cm3)
$P_{ext}$ | External power absorbed by the plasma |  (eV s-1)
Table 2: Summary of the different terms contributing to particle and energy
balance equations 1, 2, 3, 4.

## Acknowledgements

The authors would like to acknowledge Detlev Reiter, who provided an initial
H+He MS-resolved Eirene model. G. Alberti and M. Passoni acknowledge funding
from Eni SpA for the PhD program. This work has been carried out within the
framework of the EUROfusion Consortium (WP-PWIE), partially funded by the
European Union via the Euratom Research and Training Programme (Grant
Agreement No 101052200 — EUROfusion). The Swiss contribution to this work has
been funded by the Swiss State Secretariat for Education, Research and
Innovation (SERI). Views and opinions expressed are however those of the
author(s) only and do not necessarily reflect those of the European Union, the
European Commission, SERI or ITER Organization. Neither the European Union nor
the European Commission nor SERI can be held responsible for them.

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