# Particle collection with the ergodic divertor of Tore Supra

A Azéroual, Pégourié B., E. Chareyre, Y. Corre, Alexandre Escarguel, R

Guirlet, James Paul Gunn, T. Loarer, E. Tsitrone, Stéphane Vartanian

► To cite this version:

A Azéroual, Pégourié B., E. Chareyre, Y. Corre, Alexandre Escarguel, et al. Particle collection with the ergodic divertor of Tore Supra. Nuclear Materials and Energy, 2000, 40, pp.1651. 10.1088/0029-5515/40/9/309. cea-04976282

HAL Id: cea-04976282

https://cea.hal.science/cea-04976282v1

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# Particle collection with the ergodic divertor of Tore Supra

A. Azéroual, B. Pégourié*, E. Chareyre, Y. Corre, A. Escarguel, G. Granata,

R. Guirlet, J.P. Gunn, T. Loarer, E. Tsitrone, S. Vartanian

Association Euratom-CEA, CEA Cadarache,
St-Paul-lez-Durance, France

**Abstract.** For controlling the edge particle flux, a proposed method of pumping is to capture backscattered neutrals due to atomic processes with a dedicated vented structure. This solution has been tested with the ergodic divertor (ED) of Tore Supra. The investigated ranges of volume averaged density $\langle n_e \rangle$ and total power $P_{tot}$ are (2.5–5.5) $\times 10^{19}$ m$^{-3}$ and 1–6.5 MW, respectively, covering both the high recycling and semidetached plasma regimes. In this domain of parameters, the neutral pressure in the ED plenum always remains proportional to the incident ion flux. It follows that the high recycling regime is the most efficient for density control. This regime is then studied in more detail and the behaviour of the edge temperature and density with $\langle n_e \rangle$ and $P_{tot}$ characterized, together with the location of the transition towards the semidetached regime in the $[\langle n_e \rangle \times P_{tot}]$ plane. The particle balance around the neutralizers is analysed with the code EDCOLL (Ergodic Divertor COLLECTION), the simulations of which show a satisfactory agreement with both the pressure and $D_α$ emission measurements, demonstrating that both neutral recirculation and particle collection with the vented structure are correctly understood. Finally, possible improvements of particle control with the ED are discussed and the code EDCOLL is used to estimate the collection efficiency of an optimized structure. It is shown that the present performance could be multiplied by ≈3, allowing efficient density control in the semidetached plasma regime at a moderate level of power.

## 1. Introduction

In next step machines, obtaining high performance discharges over long times will require simultaneous control of the heat and particle fluxes at the edge. For this purpose, two systems have been developed to manage the plasma-wall interaction, namely limiters and divertors. Whereas limiters have demonstrated extensively their ability to control the density [1–5] they suffer from the presence of a leading edge, upon which the heat load can reach intolerable values during high power discharges and from an insufficient decoupling between the plasma core and the edge, which can lead to significant impurity penetration. This decoupling is achieved with axisymmetric poloidal divertors and has led to good results in terms of confinement, impurity screening and power deposition [6]. Unfortunately, the deleterious effect of spurious recycling on the baffles during ELMs leads to an effective decoupling lower than what is expected. In addition, the ergodic divertor (ED), in which neutral confinement is ensured by the plasma itself, appears as an alternative solution which has demonstrated its ability to control the core

<footnote>* Corresponding author, e-mail: pegourie@drfc.cad.cea.fr</footnote>
contamination and the heat load on the plasma facing components [7].

The different edge density regimes for the two divertor configurations are very similar [8]. Three regimes have been identified and characterized by simple modelling [9]. At low core density ($n_e$) is the sheath limited or linear regime for which the particle and power fluxes vary linearly with the upstream parameters. The radiated power is less than 50% of the total input power and the edge temperature is high (40–60 eV). At intermediate values of $n_e$, the edge density increases rapidly, proportionally to $n_e^\zeta$, with $3 < \zeta < 4$, and the electron temperature drops towards a value close to 10–15 eV. This is the conduction, or high recycling regime, for which there is a high level of ionization on field lines that connect to plasma facing components. At the high-est densities, the semidetached regime is characterized by an ion flux to the neutralizers which levels off while the edge temperature remains in the 8–12 eV range. The most promising regimes for an efficient density control while limiting the heat load on plasma facing components to an acceptable level are the high recycling and semidetached regimes. The latter is envisaged as an operating regime for next step tokamaks. For ensuring an efficient density control despite the vanishing ion flux in this last

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regime, a proposed method of pumping is to capture backscattered neutrals due to atomic processes, namely Franck-Condon (FC) dissociation and charge exchange (CX), with a dedicated vented structure. This solution has been tested on Tore Supra, first with a vented limiter which demonstrated a satisfactory collection efficiency for particles [4], and then with the ED.

The article is organized as follows: in Section 2, the ED is described with its associated neutralizers, pumps and dedicated edge diagnostics. The experimental procedure and database for the present study are presented in Section 3, together with a characterization of the high recycling regime along the lines of the analysis given in Ref. [10]. A brief description of the computer code EDCOLL (Ergodic Divertor COLLection), which has been developed for interpreting and simulating the experiments, is given in Section 4 and the computations are compared with available measurements in Section 5. The relevance of our results with respect to extrapolation towards efficient density control in a reactor environment is discussed in Section 6, with suggestions for possible improvements of the neutralizer geometry leading to a better exhaust efficiency of the ED. The main results are summarized in Section 7.

## 2. The ergodic divertor and associated diagnostics

The ED of Tore Supra is composed of six identical modules located on the low field side of the torus. The angular extension of a module is $\delta\theta = 2\pi/3$ in the poloidal direction and $\delta\varphi = \pi/14$ in the toroidal direction. As shown in Fig. 1, each module consists of an octopolar coil which generates between the current bars, where neutralizers are located, a radial magnetic field directed outwards or inwards, alternately. As a consequence, depending on their poloidal location, the neutralizers are wetted by particles flowing in the co- or counter-current direction, alternately. Only about half the total particle flux falls on the neutralizers themselves, the remaining part falling on the front face of the coils. The magnetic perturbation generated by the ED is resonant for $q_{edge} = 3 \pm 0.5$, and the divertor bars, in the $[\theta \times \varphi]$ plane, are parallel to the corresponding direction of the field lines. For each module, seven neutralizers are located between the eight bars, five of which are equipped with a set of slots (from one to three slots of 30 cm length and 1 cm width; the

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Schematic view of the front face of an ED module with dedicated diagnostics. Langmuir probes are represented by open circles, pressure measurements by open triangles. The two shadowed rectangles are the titanium getter pumps. The neutralizers are labelled from A (top of the vessel) to G (bottom of the vessel).</figcaption></figure>

corresponding molecular conductance at a wall temperature of $T_W = 450$ K is $C = 5.9 \text{ m}^3 \text{ s}^{-1}$ per module) [11]. These slots are connected to a plenum located behind the module, pumped by two titanium getter pumps located on either side, in the toroidal direction (Fig. 1). The effective pumping speed $S$ for deuterium is about $10 \text{ m}^3 \text{ s}^{-1}$ per module, i.e. $60 \text{ m}^3 \text{ s}^{-1}$ for the whole ED.

Three main elements are implied in the recycling and particle collection, which must be characterized correctly for an accurate description of the particle balance during ED operation. They are:

(a) The part of the incident particle flux collected by the neutralizers;

(b) The spatial distribution of the recycling neutrals;

(c) The plasma in front of the neutralizers, whose characteristics (density and temperature profiles, inclination of field lines) determine the magnitude of incident ion flux and effective ionization length.

### 2.1. Collected flux

The number of collected particles can be determined from the pressure balance in the ED plenum, the backward conductance of the slots being known ($5.9 \text{ m}^3 \text{ s}^{-1}$). For each module, the pressure is mea-
<figure><img src="image_3.png" /></figure>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Deconvolution of pressure measurements, comparison between $\Pi_o$ and $\Pi_M$. The two vertical lines at $t = 0$ s and $t = 13$ s correspond to the beginning and end of the discharge, respectively.</figcaption></figure>

sured in the equatorial plane, behind the neutralizer and in the titanium getter pump as indicated in Fig. 1. The ratio between the two measurements yields the effective pumping speed ($10 \text{ m}^3 \text{ s}^{-1}$ per module). The pressure gauges themselves are located outside the machine, at the end of a tube of typical length 5 m and diameter 1.6 cm. The time constant of this tube is long enough to require the signal to be deconvoluted before analysis (it is more than 1 s for deuterium, i.e. longer than the characteristic evolution time of the plasma edge during a pulse of auxiliary heating or a density rampup). The deconvolution procedure is described in Appendix A. An example is displayed in Fig. 2, where both the raw ($\Pi_M$) and deconvoluted ($\Pi_o$) signals are plotted. The error bars are determined by both the gauge accuracy and the noise introduced by the deconvolution procedure.

## 2.2. Ionization source

The ionization source is characterized by the $D_\alpha$ light distribution, which is measured in front of a neutralizer located in the equatorial plane of the machine. The latter is equipped with a set of dedicated diagnostics for visualizing the recycling pattern. It includes a visible imaging system consisting of an endoscope equipped with a CCD camera and a set of interference filters (including one for $D_\alpha$) and four in-situ optical fibres connected to a visible spectrometer. The latter diagnostics measure the $D_\alpha$ line

emission integrated in the toroidal direction at four different depths in front of the neutralizer plate. Its detailed geometry is shown in Fig. 3. Its accuracy is ±15% and its space resolution 2 cm.

## 2.3. Parameters of the plasma edge

The density and electron temperature profiles in the plasma are measured by reflectometry, Thomson scattering, far infrared interferometry and a set of 14 domed Langmuir probes poloidally and toroidally distributed on the different neutralizers of the ED (their locations on a module are indicated in Fig. 1). In addition, for a few discharges, a fast scanning Mach probe, on the top of the vessel, yields information on the electron temperature and density distributions in the first 5 cm of the plasma edge. Using the field line tracing code MASTOC (MAgnetic STOchastic Configuration) [12], these measurements can be mapped to the neutralizers. An example of the resulting density and temperature profiles is displayed in Figs 4(a) and (b). The different measurements are indicated together with the smooth interpolation used for the simulations presented in Section 5 (the complex distribution of the density and temperature at the edge is not taken into account and only averaged values are considered). The code MASTOC is also used to estimate the inclination of the field lines with respect to the neutralizers, allowing the calculation of the magnitude of the incident particle flux.

# 3. Experimental procedure and database

## 3.1. Experimental scenarios

Two different scenarios have been used, for ohmic discharges as well as for those with additional power (ion cyclotron heating, $P_{ICRH} \le 5$ MW). In the first scenario, the volume averaged density $\langle n_e \rangle$ is kept constant during the current flat-top, which is long enough for the pressure to reach steady state in the ED plenum. In the second scenario, $\langle n_e \rangle$ is slowly increased (up to a factor of 2) for a better coverage of the $\langle n_e \rangle \times P_{tot}$ domain. The relation between the parameters of the core and those of the plasma edge is not significantly perturbed by this procedure since the time constant of the rampup, $\langle n_e \rangle/\langle n_{e'} \rangle$, is much longer than the core particle lifetime $\tau_p \approx 0.3-0.7$ s [13]. The investigated ranges of $\langle n_e \rangle$ and total power $P_{tot}$ were (2.5–5.5) $\times 10^{19} \text{ m}^{-3}$ and
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Schematic view of a neutralizer with the lines of sight of the four optical fibres measuring the $D_\alpha$ emission (labelled $L_1-L_4$). The x co-ordinate is the distance from the neutralizer surface and $L_\phi$ is the distance along a needle.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Plasma profiles at the edge. Both the different measurements and fit used in the simulations are displayed as a function of the distance from the neutralizer: (a) electron density, (b) electron temperature.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Time history of the main plasma parameters during ED operation: (a) toroidal current, volume averaged density and total input power; (b) electron temperature and density at the neutralizer surface; (c) neutral pressure in the ED plenum, probe saturation current on the neutralizers and safety factor at the edge. The shadowed region is the safety factor interval required for ED resonance.</figcaption></figure>
1–6.5 MW, covering both the high recycling and semi-detached plasma regimes. The magnetic field was in the range $B_φ = 2.9–3.7$ T and the plasma current was in the range $I_p = 1.3–1.7$ MA, in such a way that the condition of resonance of the ED, $q_a = 3 \pm 0.5$, was satisfied.

An example of the time history of the main discharge characteristics is shown in Fig. 5. The discharge enters the high recycling regime after the end of the RF heating pulse and is terminated by a semidetached regime. The different regimes can be clearly identified in the trace of probe saturation current $J_{sat}$ (Fig. 5(c)). Before $t \approx 2$ s and after $t \approx 10$ s, the condition for ED resonance is not satisfied. The neutralizers are not wetted by the plasma outflux and $J_{sat} \approx 0$. Between $t \approx 4$ s and $t \approx 6$ s ($\langle n_e \rangle \approx 3.8 \times 10^{19} \text{ m}^{-3}$), the discharge is in the high recycling regime. $J_{sat}$ and the edge density $n_a$ increase rapidly with $\langle n_e \rangle$ and the edge temperature $T_a$ decreases slowly from 40 to 15 eV. For $6 < t < 9$ s, the discharge enters the semidetached regime and $J_{sat}$ decreases abruptly until the loss of the ED resonance, where it vanishes ($\approx 10$ s). During this phase, $n_a$ decreases from $1.1 \times 10^{19} \text{ m}^{-3}$ to $7 \times 10^{18} \text{ m}^{-3}$ while $T_a$ remains between 10 and 15 eV. The volume averaged density at which the transition between the two regimes occurs varies from discharge to discharge (between 2.7 and $3 \times 10^{19} \text{ m}^{-3}$). It depends on $n_e$ and $P_{tot}$, but also on $Z_{eff}$ (between 1.4 and 2 for these experiments). A remarkable point is that the pressure in the ED plenum, $\Pi_o$, always remains proportional to $J_{sat}$, even during the semidetached regime. This similarity is not surprising since the lowest edge temperature obtained when operating the ED is $\approx 8$ eV, which is too high for volume recombination to take place.

## 3.2. The Tore Supra database

The characteristics of the 56 selected discharges will now be presented. In Figure 6(a), total input power $P_{tot}$ is plotted versus volume averaged density $\langle n_e \rangle$. The different symbols are for the linear (L, open circles), high recycling (HR, solid squares) and semidetached regimes (SD, open triangles); the procedure used to distinguish between the different regimes is described at the end of Section 3.3. The associated edge parameters, $n_a$ and $T_a$, are displayed in Fig. 6(b). The transition between the high recycling and semidetached regimes occurs for $T_a \approx 12–15$ eV for the whole density range. Figure 6(c) shows that the pressure in the ED plena $\Pi_o$ remains

<figure><img src="image_8.png" /><figcaption>**Figure 6.** The 56 discharges of the Tore Supra database. The labels L, HR and SD refer to the linear, high recycling and semidetached regimes, respectively. (a) Total input power versus volume averaged density, the chain curve is the limit between the domains of the high recycling and semidetached regimes as deduced from the high recycling scaling laws. (b) Plasma parameters at the neutralizer surface: electron temperature versus electron density, the chain line is at $T_a = 15$ eV. (c) Neutral pressure in the ED plenum versus particle incident flux on the neutralizer surface.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 7. Behaviour of $J_{sat}/\langle n_e \rangle$ as a function of $J_{sat} T_a$ for the three edge density regimes.</figcaption></figure>

proportional to the incident particle flux ($n_a T_a^{1/2} \propto J_{sat}$) for the whole database. The dispersion is weak and the points corresponding to the two regimes do not display a different trend, in agreement with the remark at the end of Section 3.1. This emphasizes the interest in operating the ED in the high recycling regime for efficient density control. Indeed, at least at this level of power, the latter corresponds to the best compromise between a high collection probability per particle and a moderate power load on the neutralizer plates. This is demonstrated in Fig. 7 where $J_{sat}/\langle n_e \rangle (\propto \Pi_o/\langle n_e \rangle)$ is plotted versus $J_{sat} T_a$ for three ohmic discharges covering the three regimes of the plasma edge. The former quantity is proportional to the probability a particle of the plasma core has to be pumped (since the flux exhausted is $S\Pi_o$). It displays a maximum at moderate values of $J_{sat} T_a$, which is proportional to the conducted power on the neutralizers. The highest and lowest values of $J_{sat} T_a$ correspond to the linear and semidetached regimes, those for which $J_{sat} \propto n_e$ is maximum to the high recycling regime.

### 3.3. High recycling regime

The dependences of $n_a$, $T_a$ and $\Pi_o$ on the control parameters of the discharge have been determined for the high recycling regime. In the following scaling laws, the densities are in units of $[10^{19} \text{ m}^{-3}]$, the temperatures in [eV], the powers in [MW] and the pressures in [Pa]. We choose $P_{tot}$ and $\langle n_e \rangle$ for the control parameters, since they can be seen as the experimental counterparts of the input of the classical two point model [10]: the upstream heat par-

allel flux $q_u$ and the upstream density $n_u$ on the field line connected to the divertor plate. In the high recycling regime, it has been established that $n_a$ is varying as the cubic power of $\langle n_e \rangle$ [14]. For the present database, one has:

$$n_a \approx (1.75 \pm 0.35) \times 10^{-2} \langle n_e \rangle^{3.4}. \quad (1)$$

(It is remarkable that this scaling does not exhibit any clear dependence on $P_{tot}$.) Taking advantage of the moderate variation of the radiative fraction in this regime ($f_{rad} \approx 60\% \pm 15\%$), one can write $P_{tot} \propto P_{cond} \approx n_a T_a^{3/2}$, which yields for the edge temperature

$$T_a \approx (1.25 \pm 0.2) \times 10^2 \langle n_e \rangle^{-2.3} P_{tot}^{2/3}. \quad (2)$$

Finally, for given $P_{tot}$ and $\langle n_e \rangle$ one can estimate the pressure in the ED plenum by using the proportionality between $\Pi_o$ and $J_{sat}$, which yields for $S=0$ (pumping switched-off):

$$\Pi_o \approx (2.7 \pm 0.4) \times 10^{-3} \langle n_e \rangle^{2.3} P_{tot}^{1/3}. \quad (3)$$

When the pumps are switched on, the number of particles exhausted per module and per second is

$$\Phi_{\text{pump}} \approx \frac{\Pi_o}{kT_W} \frac{SC}{S+C} \quad (4)$$

where $C = 5.9 \text{ m}^3$ is the molecular conductance of the neutralizer slots, $k$ is Boltzmann's constant and $T_W = 450 \text{ K}$ is the temperature of the wall in the plenum. For limiter discharges, a similar scaling for $\Pi_o$ has been reported in a previous work [4]. The dispersion of the individual points with respect to the different scalings can be appreciated in Figs 8(a)-(c) where $n_a$, $T_a P_{tot}^{-2/3}$ and $\Pi_o P_{tot}^{-1/3}$ are plotted versus $\langle n_e \rangle$. The points corresponding to the linear and semidetached regimes are also plotted, and depart significantly from the scaling laws.

For separating the different regimes, we assume that a discharge exits the high recycling regime as soon as its edge density or temperature departs by more than 25% from the values predicted by Eqs (1) and (2). This criterion is similar to the one used on other machines, the DoD on JET for the high recycling to semidetached regime transition for instance [15]. Moreover, its validity can be tested since it is shown, in Fig. 6(b), that the temperature limit between the high recycling and semidetached regimes is $T_a \approx 15 \text{ eV}$. Entering this value in Eq. (2), one obtains the equation of the transition between the high recycling and semidetached regimes in the $[\langle n_e \rangle \times P_{tot}]$ plane,

$$P_{tot} \approx 4.15 \times 10^{-2} \langle n_e \rangle^{3.4} \quad (5)$$
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Behaviour of the neutral pressure and edge parameters with total input power and volume averaged density. The labels L, HR and SD refer to the linear, high recycling and semidetached regimes, respectively. The chain curves are the scaling laws for the high recycling regime, while the dotted curves give the dispersion of the points. (a) Electron density. (b) Electron temperature. (c) Neutral pressure in the ED plenum.</figcaption></figure>

which is plotted in Fig. 6(a). The fact that the line corresponding to Eq. (5) fits the experimental separation between the two regimes with a reasonable accuracy demonstrates the global consistency of the present analysis.

When comparing the dependences in Eqs (1) and (2) with those obtained by the two point model ($n_a \propto n_u^3 q_u^{-8/7}$, $T_a \propto n_u^{-2} q_u^{10/7}$), it is remarkable that the exponents of the upstream density are in close agreement ($n_a \propto \langle n_e \rangle^{3.4}$, $T_a \propto \eta_e^{2.3}$) when those of the upstream parallel heat flow exhibit an obvious discrepancy ($n_a \propto P_{\text{tot}}$, $T_a \propto P_{\text{tot}}^{2/3}$). This can be explained by considering that the dependences on $n_u$ and $q_u$ correspond to two different hypotheses in the physical model. The former is due to the assumption that the plasma pressure is constant along the field line, which is valid for Tore Supra (the proportionality between $P_{\text{tot}}$ and $J_{\text{sat}}$ demonstrates that the neutral pressure always remains low, even at the highest investigated densities). The latter is obtained when assuming that heat transport is dominated by conduction ($q_u \propto T^{7/2}$), which requires verification of $T/\nabla \lambda_e$, where $\lambda_e$ is the electron mean free path. This inequality is not verified in the case of Tore Supra: one has $(T/\nabla T)/\lambda_e = 0.5-5$, depending on the edge conditions and location along the field line. The heat transport is thus not strictly diffusive and might be described either by a proper formalism [16] or, asymptotically, by the limited free flux ($q_u \propto T^{3/2}$) [17]. To reproduce the experimental dependences on $P_{\text{tot}}$ in Eqs (1) and (2), the heat flux must be of the form $\alpha_c n^\alpha T^{\alpha+3/2}$, for any value of $\alpha$, which allows dependences between that corresponding to pure conduction ($\alpha_c \propto T^{7/2}$, $T/\nabla \lambda_e$) and that corresponding to limited free flux ($q_u \propto nT^{3/2}$, $T/\nabla T \lambda_e$), in agreement with the intermediate values of $(T/\nabla T)/\lambda_e$ which characterize the ED of Tore Supra.

## 4. Description of the numerical model EDCOLL

A numerical model has been developed for interpreting the experiments and quantifying the possible improvements of the ED pumping. The inputs are the neutralizer geometry, the magnetic structure, and the density and temperature profiles in the first 20 cm of the plasma. The outputs are the pressure in the ED plenum and in the titanium getter pumps, and the molecular, neutral (FC, reflected and CX) and $D_\alpha$ light distributions. The five neutralizers
equipped with a set of slots are independently mod-
elled. Each neutralizer is discretized in the toroidal
direction only (6–8 elements). For each element, a 1-
D calculation of recycling is performed. The plasma
profiles are taken from experimental data (Figs 4(a)
and (b)) and the inclination of the field lines with
respect to the neutralizer calculated with the field
line tracing code MASTOC: from 4° at the top of the
neutralizer (right hand side in Fig. 3) to 10° at the
bottom, the average value is 6°. Impurities (assumed
to be C3+ in the plasma, and CD4 in the plenum)
are taken into account by the corresponding dilution
of deuterium: nD/n~ (Z-Zeff)/(Z 1) and per-
turbation of the neutral pressure in the plenum.

4.1. Ballistic collection

The incident ion flux falls on the front face of
the neutralizers as well as on the lateral face within
the slots, in proportion to the surface occupied by
the slots or the needles. Once neutralized, it is re-
emitted with an assumed cos θ distribution. A par-
ticle reflected from the lateral face of a slot can be
re-emitted towards the plasma (with a probability
π<sub>out</sub>), towards the plenum (with a probability π<sub>in</sub>) or
experience another reflection from the opposite face
of the slot (with a probability of 1- π<sub>in</sub>-π<sub>out</sub>) where
the process repeats itself. The values of π<sub>in</sub> and π<sub>out</sub>
depend on the depth at which the particle is reflected
from the lateral face of the slot (if edge effects are
neglected). After a series of successive reflections, a
proportion Σ<sub>n</sub> π<sub>in</sub> (n) of the incident flux neutral-
ized in the slots enters the plenum directly, while
the remaining part, Σ<sub>n</sub> π<sub>out</sub> (n) = 1 - Σ<sub>n</sub> π<sub>in</sub> (n),
returns to the plasma. The calculation of π<sub>in</sub> and
π<sub>out</sub> is given in Appendix B.

4.2. Molecular and neutral distributions

All the particles which return to the plasma are
reflected either as molecules or as fast neutrals. The
branch ratio between these two species, as well as
the change in the fast neutral velocity, is taken from
laboratory measurements [18]. For deuterium, the
relevant reactions of atomic physics are the disso-
ciation of molecules and of molecular ions, ioniza-
tion and charge exchange [19]. Due to their short
lifetime, molecular ions are not considered as an
individual species and effective cross-sections have
been used: ⟨σv⟩<sub>eff</sub> = ⟨σv⟩<sup>o</sup>⟨σv⟩<sup>j</sup>/ ∑<sub>i</sub>⟨σv⟩<sup>j</sup>, where the
superscripts o or i refer to the creation or to the dif-
ferent branches of dissociation of molecular ions, in

such a way that both the branch ratios and reac-
tion rates are conserved. The species for which space
distributions are calculated are therefore:

(a) Molecules desorbed at the temperature of the neutralizer surface (T<sub>w</sub> = 450 K),

(b) Fast reflected neutrals,

(c) Neutrals born by molecular dissociation (with a mean energy of 3 eV),

(d) Charge exchange neutrals (at the temperature T of their place of birth).

Since species (c) and (d) are emitted isotropically
inside the plasma, they can either contribute directly
to the neutral flux at a given location, or be reflected
from the wall as molecules or atoms. The calculation
of the distribution of the different species uses the
formalism of the neutral transport code SPUDNUT
[20]. Details of the calculations are given in Ref. [21].

4.3. Particle balance in the ED plena

From the preceding, one can calculate the space
distribution of the different species in the plasma
edge as well as the associated fluxes on the neutraliz-
ers. For each element i of the mesh, with associated
conductance C(i), the number of particles collected
per second is

$$
\Phi_{\text{coll}}(i) = \sum_{j} C_{j}(i)n_{j} \frac{4\Gamma_{j}}{n_{j}\nu_{j}} \quad (6)
$$

where the subscript *j* refers to the different neutral
species: FC and CX (the conductance *C*<sub>j</sub>(*i*) depends
not only on the slot geometry and size of the ele-
ment but also on the temperature of the particle).
For each considered species, Γ<sub>j</sub> is the incident flux,
and *n*<sub>j</sub> and *ν*<sub>j</sub> are the density and mean velocity,
respectively. The ratio 4Γ<sub>j</sub>/(n<sub>j</sub>ν<sub>j</sub>), which is between
2 and 4, accounts for the anisotropy of the the veloc-
ity distribution due to the absorption along the path
between the place of birth of the neutral and the neu-
tralizer (this correction is exact for an aperture in a
sheet of infinitesimal thickness; in practice it remains
valid as long as (slot surface) (slot depth)<sup>2</sup>, which
is the case for the neutralizers of Tore Supra). The
total number of particles collected per second Φ<sub>coll</sub>
is the sum of all the elementary contributions (FC,
CX and particles collected ballistically) over all the
elements of the mesh. The particle balance in the
plenum then determines the resulting pressure, fol-
lowing:

$$
\Pi_o = \frac{k T_w \Phi_{coll}}{S + C}. \qquad (7)
$$
<figure><img src="image_11.png" /><figcaption>Figure 9. (a) Schematic representation of a divertor module, showing the directions of the x axis for the neutralizers equipped with slots. (b) Particle balance in a divertor module. Only the neutralizers equipped with slots are represented. The fluxes are given per unit of slot surface. The fluxes corresponding to the two methods of collection — ballistic, $\Gamma_{bal}$, or from atomic physics processes, $\Gamma_{at}$ — are separately represented. The backflowing flux, $\Gamma_{out}$, is lower for neutralizers B and F because they are equipped with narrower slots of smaller conductance.</figcaption></figure>

The number of particles exhausted per second by the pumping system is $\Pi_o S/(kT_w)$. The remaining part comes back towards the plasma, where it increases the flux of desorbed particles at the temperature of the neutralizer surface ($T_w = 450$ K).

The particle balance for the five neutralizers equipped with slots is shown in Fig. 9. In the vacuum chamber, the neutralizer B is located at the top

of the module and the neutralizer F at the bottom. For each neutralizer, the location of maximum incident flux is situated on the right of the figure. The different traces show the flux collected by backscattering of neutrals due to atomic processes, the flux collected ballistically and the backflowing flux. The latter is constant because there is only one common plenum. On the other hand, the average value of the collected fluxes reflects the distribution of the incident flux, which depends on the poloidal location. For each neutralizer, the flux entering the plenum by neutral collection increases from left to right: it is proportional to the incident flux and vanishes on the unwetted part of the slots. The flux collected ballistically decreases from left to right because it is more sensitive to the inclination of the field lines with respect to the neutralizer: from ≈ 10° at the beginning of the unwetted region to ≈ 4° at the place of maximum flux.

### 4.4. Global results, ionization source and D distribution

The example given in the following is calculated with the profiles of temperature and density given in Figs 4(a) and (b) and for the central cell of a neutralizer D located in the equatorial plane of the machine (Fig. 1). Since the different neutralizers are modelled independently, the same information is available for each neutralizer. The average inclination of the field lines with respect to the neutralizers is 6°. The radial distributions of molecular deuterium, and of 'slow' (FC) and 'fast' (CX and reflected) atoms are shown in Fig. 10(a). Their densities (averaged over the $1/e$ width of the profile) are respectively 120, 10 and 1% of that of the background plasma. This corresponds to 0.2, 2 and 1% of the total plasma pressure at the edge, which is weak enough to validate the assumption of constant plasma pressure in the analysis of Section 3.3. Accounting for the fact that about half the incident ion flux falls on the front face of the coils, 4.5% of the latter is collected and enters the plenum (1.8% by ballistic pumping and 2.7% by atomic processes). Since the effective pumping speed is about equal to 1.4 times the backflowing conductance of the slots, only 60% of the collected flux is effectively exhausted by the pumping system.

The radial distribution of the ionization source is shown in Fig. 10(b). The different contributions are also displayed: the first contribution is limited to the first 0.5 cm and comes from molecular dissociation, the second contribution comes from ionization
<figure><img src="image_12.png" /></figure>

**Figure 10.** (a) Density profiles of molecules and different atomic populations (FC, fast reflected atoms and CX). (b) Contribution of molecules and different atomic populations to the ionization source. The x co-ordinate is the distance from the neutralizer surface.

of FC atoms for the bulk of the source and the third contribution from ionization of CX atoms for the largest penetrations. The ionization source due to the fast reflected atoms remains low everywhere. Knowledge of the ionization source allows calculation of the space distribution of the $D_\alpha$ line emission, using the collisional-radiative model of Ref. [22]. This calculation assumes that:

(a) The incoming neutrals are atoms in the fundamental state (no molecules),

(b) The plasma is homogeneous (i.e. that $n\lambda/\pi n$ and $T/\nabla T$ are much larger than the mean free path of the different species).

<figure><img src="image_13.png" /></figure>

**Figure 11.** Contour plot of the $D_\alpha$ emission distribution. The x co-ordinate is the distance from the neutralizer surface, while $L_\phi$ is the distance along a needle. The lines of sight of the four optical fibres are labelled $L_1-L_4$ (Fig. 3).

Although more than half the recycling flux is in molecular form, the first hypothesis does not invalidate the calculation since the line emission rates and ratios are not dramatically changed for such a value [23]. On the other hand, the second hypothesis is only marginally verified: $n\lambda/\pi n$ and $T/\nabla T$ are of the order of 10–15 cm when the longest mean free paths are slightly less than 5 cm. The contour plot in the $[r \lambda/\pi]$ plane of the $D_\alpha$ emission in front of a neutralizer is shown in Fig. 11, together with the four lines of sight of the optical fiber system described in Section 2.2.

## 5. Simulation of experiments

The predictions of the model have been compared with the measurements of the database of Section 3. The main effort has been devoted to the particle balance around the ED, i.e. the collected and recycled fluxes, characterized by the pressure in the ED plena and the $D_\alpha$ light distribution.

### 5.1. Pressure in the ED plena

Figure 12 shows the calculated and measured pressures for the discharge whose main parameters are
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Time history of the neutral pressure in the ED plenum. Comparison between experimental and calculated values.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Neutral pressure in the ED plenum for 25 discharges. Comparison between experimental and calculated values.</figcaption></figure>

displayed in Fig. 5. For the whole time interval, the mean relative difference is 10%, which is less than the expected accuracy of pressure measurements (Fig. 2). The collection and pumping efficiencies (accounting for the part of the incident integrated flux $\Phi_{inc}$ falling on the front face of the coils) are $\epsilon_{coll} = \Phi_{coll}/\Phi_{inc} = 4.5 \pm 0.3\%$ and $\epsilon_{pump} = \Phi_{pump}/\Phi_{inc} = 2.5 \pm 2\%$, respectively. Due to the proportionality between $\Pi_o$ and $J_{sat}$ (Fig. 6(c)),

<figure><img src="image_16.png" /><figcaption>**Figure 14.** $D_\alpha$ emission distribution along a neutralizer needle; $L_\phi$ is the distance along a needle.</figcaption></figure>

these values of $\epsilon_{coll}$ and $\epsilon_{pump}$ are almost identical for the whole database.

A global comparison between simulations and experiments is shown in Fig. 13, where $\Pi_c$ is plotted versus $\Pi_o$ for the 25 discharges of the database for which the plasma profiles at the edge are well documented. The error bars on $\Pi_c$ are consecutive to those on $J_{sat}$ (one has $\Delta\Pi_c/\Pi_c \approx \Delta J_{sat}/J_{sat}$). The agreement is satisfactory, demonstrating that the two types of particle collection with the vented structure are correctly understood.

## 5.2. D light distribution

The 98% of the incident flux which is not exhausted by the pumping system is re-emitted towards the plasma where it is ionized. The calculated distribution of the $D_\alpha$ line emission shown in Fig. 11 allows, after proper integration, a comparison with measurements. First, when integration is performed along the x co-ordinate, with the CCD camera image which gives the relative light distribution along a neutralizer located in the equatorial plane. This is shown in Fig. 14, where it can be seen that the calculation reproduces the main structures of experimental data. This longitudinal distribution remains almost unchanged in the different edge regimes.

On the other hand, the radial distribution, which is measured by the four optical fibres, displays a significant modification during the transition from the high recycling to the semidetached plasma regimes, where the peak of $D_\alpha$ emission shifts from $L_1$ to
<figure><img src="image_17.png" /><figcaption>**Figure 15.** History of plasma edge parameters and $D_\alpha$ emission profile during a transition to detachment. The three vertical lines indicate the times at which a comparison between experiments and simulation is displayed in Fig. 16. (a) Electron density and temperature at the neutralizer surface. (b) $D_\alpha$ emission: optical fibre measurements (Fig. 3).</figcaption></figure>

$L_3$ [24]. An example is shown in Fig. 15, where the edge temperature, density and the $D_\alpha$ measurements ($L_1-L_4$) are plotted versus time. For $t < 2$ s, the ED resonance condition is not fulfilled and the neutralizers are not wetted by the plasma outflux ($n \approx 0$). Partial detachment occurs at the beginning of the current plateau ($t \approx 3$ s): $T_\alpha$ drops abruptly from 30 to $\approx 10$ eV when $n_\alpha$ increases slowly and finally saturates at $\approx 4.5 \times 10^{18} \text{ m}^{-3}$. Just after the detachment, the brightness measured on the two lines $L_1$ and $L_2$ drops to stabilize at one third of its initial value. Three radial profiles of the $D_\alpha$ light are displayed in Fig. 16(a) for $t = 3, 3.5$ and $4$ s (these times are indicated by solid lines in Figs 15(a) and (b)), showing

<figure><img src="image_18.png" /><figcaption>**Figure 16.** Behaviour of the $D_\alpha$ emission profile during a transition to detachment. The labels $L_1-L_4$ refer to the four optical fibres whose geometry is shown in Fig. 3: (a) experiment, (b) simulation. The chain curves are the calculated profiles, while the solid curves with markers are displayed for comparison with Fig. 16(a).</figcaption></figure>

the shift of the light emission. The measurements can be fitted by the appearance of a drop in the temperature profile: within 3 cm of the neutralizer surface ($T \approx 30$ eV for the attached plasma condition) the electron temperature decreases from 15 to 9 eV between $t = 3$ and $4$ s, but remains unchanged outside this region. No major changes are required in the shape of the density profile. Results are displayed in Fig. 16(b), where the calculated $D_\alpha$ profiles are plotted with the corresponding calculated signals for comparison with Fig. 16(a). The agreement is satisfactory (within $\pm 5\%$) for the shape of the profiles as well as for their magnitude. This drop of electron temperature in a limited region cannot be
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Contribution of the different atomic populations (FC, fast reflected atoms and CX) to the $D_\alpha$ emission profile: (a) attached plasma ($t=3$ s in Fig. 16), (b) semidetached plasma ($t=4$ s).</figcaption></figure>

confirmed experimentally due to the lack of measurement of the temperature radial distribution in front of the neutralizers. The two point model [10] can nevertheless help in determining if this thickness of ≈3 cm is realistic or not. Following this model, the temperature in the parallel direction varies as $s^{2/7}$ from the neutralizer plate ($s$ is the abscissa along the field line). For typical values at the transition between the high recycling and semidetached regimes — downstream temperature≈ 10 eV, upstream temperature ≈ 100 eV and connection length≈ 45 m — one finds that a temperature of 30 eV is reached at ≈25 cm from the neutralizer. Accounting for the 6° of average inclination of the field lines with respect to the neutralizer surface, this parallel distance of 25 cm corresponds to a perpendicular distance of

2.5 cm, which is close to the 3 cm used in the simulations<fnref n="1" />.

During the transition, the ion source distribution follows the appearance of the step in the temperature profile and the neutral effective penetration ($\lambda_0$) increases accordingly: from $\lambda_0$ ≈ 2 cm for fully attached plasmas (≤ 3 s) to ≈8–10 cm when detachment arises ($t > 7.5$ s). The corresponding screening factor ($e^{-\delta/\lambda_0}$, where $\delta \approx 12$ cm is the thickness of the stochastic region) then increases from $2 \times 10^{-3}$ to 0.2–0.3. Experimentally, the fuelling efficiency of gas puff ($\epsilon_f$, ratio between the injected number of particles and the increase of the plasma content) follows the same trend [25]. In the simulations, this is due to the increase of the source of matter due to CX atoms with respect to that due to FC. This is shown in Fig. 17, where the different contributions to the $D_\alpha$ radiation are plotted versus the distance from the neutralizer for an attached and a semidetached plasma. Between these two cases, the relative contribution of the ‘fast’ atoms (CX + reflected) to the $D_\alpha$ radiation increases by more than 30%. This behaviour is in agreement with that of $\epsilon_f$ and has been confirmed qualitatively by shape analysis of the $D_\alpha$ emission line [26], as shown in Fig. 18, where the ratio of the ‘fast’ and ‘slow’ components of the $D_\alpha$ line are plotted versus time during a transition to detachment.

## 6. Improvement of the ED pumping system

From the results presented above, one sees that in the range of investigated $\langle n_e \rangle$ and $P_{tot}$:

(a) The ED controls efficiently the density in the high recycling regime (up to $4 \text{ Pa m}^3 \text{s}^{-1}$ exhausted);

(b) The code EDCOLL allows a satisfactory modelling of the experiments.

Nevertheless, the present database is limited to moderate values of $P_{tot}$ (≤6 MW) and one can wonder if, due to heat load on the plasma facing

<footnote n="1">¹ This discussion is only qualitative because (a) it is shown in Section 3 that the hypothesis of a parallel transport dominated by conduction is only marginally verified in Tore Supra (even if plasma collisionality increases when temperature decreases) and (b) for temperatures of the order of 10 eV, the contribution of convection in the heat parallel transport becomes significant.</footnote>
<figure><img src="image_20.png" /><figcaption>**Figure 18.** Time behaviour of the measured ratio of the 'fast' and 'slow' components of $D_\alpha$ emission during a transition to detachment.</figcaption></figure>

components, operating the ED in the high recycling regime remains a relevant scenario for much higher levels of power. For this reason, it is important to optimize the neutralizer geometry in order to allow efficient density control during ED operation, even in semidetached plasma regimes. Another argument for optimizing the ED collection capability is the pumping of ash. Indeed, if the collection efficiency is almost identical for helium (He) and deuterium/tritium (D/T) in the case of pure ionic collection (throat limiter, ballistically collected flux for a vented structure), this is no longer the case as soon as the collection of backscattered neutrals becomes dominant, due to the differences in the relevant atomic physics processes between helium and deuterium (no FC dissociation for helium and a much smaller CX reaction rate since $n_{He} n_{D}$): the collection efficiency of neutral helium is then a function of helium concentration and is typically a quarter of that of neutral deuterium for [He]/[D]≈10–15% [27].

To improve the ED collection efficiency, two possibilities are available:

(a) Increasing the part of the plasma outflux interacting with the vented structure,

(b) Optimization of the geometry of the slots.

The improvements given in the following were established on the basis of EDCOLL simulations.

<figure><img src="image_21.png" /><figcaption>**Figure 19.** Particle balance for a divertor module equipped with a set of optimized neutralizers: the ballistic collection is maximized (cf. Fig. 9 for comparison with the present geometry).</figcaption></figure>

It was mentioned in Section 2 that about half the incident flux does not wet the neutralizers but falls on the front face of the coils. To move back the latter to allow the whole plasma outflux to fall on the vented structure leads typically to double the collection efficiency. This is paid for by an increase of the divertor current $I_{ED}$ to maintain the magnetic perturbation at the proper level in the plasma edge [28]: one has $\delta \propto \sqrt{I_{ED}}$ and $I_{ED} \propto (1 + \Delta/a)^m$, where $a$ is the plasma minor radius, $\Delta$ the distance between the coils and the plasma edge and $\bar{m}$ the average poloidal wavenumber. For the geometry of Tore Supra that corresponds to an increase of $I_{ED}$ by typically a factor of 4.

Optimizing the slot geometry also improves the collection efficiency. First, by limiting the length of the slots to the part of the latter for which the collected flux is larger than the backflowing flux. Second, by opening their section towards the plenum, with an angle of ≈15° on both sides: such an opening increases significantly the ballistic collection (and thus helium collection) while maintaining the heat load on secondary leading edges at an acceptable level. The particle balance for a neutralizer equipped with a set of such modified slots is shown in Fig. 19. The ballistically collected flux is typically twice as high as it is for the present geometry of the slots (Fig. 9). These two modifications of slot geometry improve the collection efficiency
by a factor ≈1.5–1.7 for deuterium and ≈2–2.3 for helium.<fnref n="2" />

On the whole, it is therefore possible to reach
collection efficiencies of ≈ 14% for deuterium and
≈ 4.5–5.5% for helium (corresponding, for the same
pumping power, to exhaust efficiencies of ≈ 8 and
≈ 3%, respectively). These values are high enough to
meet the requirement of minimum exhaust efficiency
for maintaining a burning reactor grade plasma in
the semidetached plasma regime [29], i.e. as long as
the edge temperature remains high enough for the
ionic flux on the neutralizers to remain significant
(8–10 eV).

7. Conclusion

Maintaining high performance discharges over
long times requires simultaneous control of the heat
and particle fluxes at the edge. This is done in
Tore Supra with the ED, an alternative solution to
the axisymmetric divertor which has demonstrated
its ability to obtain high radiating regimes, low
core contamination and moderate heat load on the
plasma facing components. We concentrate in this
article on the ability of the ED to control the particle
flux and core density. This capability depends on
the screening of the recycling flux and thus on the
edge density regime. For the experiments performed
on Tore Supra (volume averaged density between
2.5 and 5.5 × 10¹⁹ m⁻³; total power between 1
and 6.5 MW), the high recycling regime is the
most favourable for controlling particle flux: up to
4 Pa m³ s⁻¹ has been exhausted. In this regime,
the scaling of the edge density and temperature
follows that predicted by the two point model for
what concerns the upstream density, but departs
significantly from the latter in what concerns the
upstream heat flux, indicating that heat parallel
transport is not purely conductive. For the whole
database (which covers mainly the high recycling
and semidetached regimes), the neutral pressure in
the plenum of the ED remains proportional to the
incident ion flux. This indicates that at this level of
power the neutral pressure remains low in the diver-
tor volume, even in the semidetached plasma regime.

The experimental data have been simulated and the
particle balance studied with the multi-1-D code
EDCOLL. This latter computes the distribution of
the different neutral species in front of the neutral-
izers and the fluxes collected ballistically (ionic col-
lection) as well as by backscattering of neutrals due
to atomic processes (charge exchange and molecu-
lar dissociation). The results of the simulations show
a satisfactory agreement with the measurements for
the pressure in the plenum of the ED as well as for
the profile of the D<sub>α</sub> line emission in front of the
neutralizers, demonstrating that both neutral recir-
culation and particle collection with the vented struc-
ture are correctly understood. The possible improve-
ments of the geometry of the vented structure are
discussed and the code EDCOLL is used to estimate
what would be the collection efficiency of such an
optimized system. It is shown that the present per-
formances could be multiplied by ≈3 for deuterium
and ≈4 for helium, allowing an efficient density con-
trol in the semidetached plasma regime, even at a
moderate level of power.

**Appendix A**

Deconvolution of the pressure signal

The pressure gauge is located at the end of a tube
of length *L* and section *Σ*, the values of the pressure
at the entrance and end of the tube being Π<sub>o</sub>(*t*) and
Π<sub>M</sub>(*t*), respectively. The flow regime, in the tube of
conductance *C*, is molecular and the pressure field
described by the following diffusion equation:

$$
\dot{\Pi} = \nabla(\alpha \nabla \Pi). \tag{8}
$$

The value of α is determined at steady state, for
which the flux Γ is given by

$$
\Gamma = \frac{C}{\alpha \nabla \Pi} = \Sigma (\Pi_o - \Pi_M). \quad (9)
$$

In this case, the pressure gradient is constant and
expressed as $\nabla\Pi = (\Pi_o - \Pi)/L$, yielding

$$
\alpha = \frac{CL}{\Sigma}. \tag{10}
$$

The boundary and initial conditions are Π(x = L, t) = Π₀(t), ∇Π(x = 0, t) = 0 and Π(x, t = 0) = δ(x − L). For Π₀(t) = δ(t), Eq. (8) can be solved and the solution is written as

$$
\Pi_{\delta}(x, t) = \frac{\alpha \pi}{L^2} \sum_{n=0}^{\infty} (-1)^n (2n + 1) \exp \frac{-\alpha \pi^2 (2n + 1)^2 t}{(2L)^2} \\
\times \cos \frac{\pi(2n + 1)x}{2L} . \qquad (11)
$$

<footnote n="2">² This improvement of the collection efficiency of helium with respect to that of deuterium or tritium is only effective in the high recycling and semidetached plasma regimes, for which the particle flux on the neutralizers remains significant, but not in the fully detached regime, for which the particle flux vanishes.</footnote>
<figure><img src="image_22.png" /><figcaption>**Figure 20.** Transfer function of a tube of diameter 1.6 cm and length L = 5 m for deuterium and nitrogen.</figcaption></figure>

Two examples are displayed in Fig. 20, for deuterium and nitrogen at room temperature, for a tube of length 4.8 m and diameter 1.6 cm. For any value of $\Pi_o(t)$, one has then

$$ \Pi ( ) = \Pi ( ) \Pi ( )' \quad (12) \\ \underset{\text{M}}{\text{o}} t \quad \underset{\text{o}}{t} \quad \underset{\text{L}}{t} \quad t-t \quad \text{dt} $$

and, therefore

$$ \Pi (t) = \text{FT}^{-1} \frac{\text{FT}(\Pi_M(L, t))}{\text{FT}(\Pi_\delta(t))} \quad (13) $$

where FT is the Fourier transform.

# Appendix B

## Ballistic collection

The slot is assumed to be of infinite length with respect to its depth and width. Its transverse section is defined by the triplet $(s, \alpha_1, \alpha_2)$, where $s$ = width/depth is the aspect ratio and $\alpha_{1,2}$ the inclination of the lateral faces (Fig. 21). The angular distribution of the velocity of the reflected particles is assumed to obey Lambert's law. After integration along the slot length, the angular emission probability is

$$ \pi(\varphi) d\varphi = \frac{1}{2} \cos\varphi d\varphi \quad (14) $$

where $\varphi$ is the angle between the particle velocity and the normal to its surface of origin, projected on the plane orthogonal to the length of the slot. For a particle reflected from surface i towards surface j, one defines (Fig. 21): $x_i$ the point from where

<figure><img src="image_23.png" /><figcaption>**Figure 21.** (a) Geometry of the slot section, definition of the triplet $(s, \alpha_1, \alpha_2)$. (b) Emission of a particle from surface i, definition of $x_i$, $\varphi_{ij}$, $\Phi_i$ and $\Psi_i$.</figcaption></figure>

the particle is emitted from surface i, $\varphi_{ij}$ the angle between the particle velocity and the normal to surface i, $x_j$ the point where the particle impacts on surface j, $\Phi_i$ the angular limit beyond which the particle comes back to the plasma (towards $z < 0$), $\Psi_i$ the angular limit beyond which the particle enters the plenum (towards $z > 1$). One has

$$ \varphi_{ij} = -\alpha_i \frac{\tan^{-1} \frac{x_i \cos \alpha_i - x_j \cos \alpha_j}{s + (x_i - l_i) \sin \alpha_i + (l_j - x_j) \sin \alpha_j}}{\alpha_j} \quad (15) $$

$$ \Phi_i = -\alpha_i - \tan^{-1} \frac{x_i \cos \alpha_i - l_j \cos \alpha_j}{s + (x_i - l_i) \sin \alpha_i} \quad (16) $$

$$ \Psi_i = -\alpha_i - \tan^{-1} \frac{x_i \cos \alpha_i}{s + (x_i - l_i) \sin \alpha_i + l_j \sin \alpha_j} \quad (17) $$

For a particle neutralized at point $z_o$ on surface 1, with the angular distribution $\pi(\varphi)$ d$\varphi$, the source is given by

$$ J_0(z, x) = \delta_1 \frac{1 - z_o}{\cos \alpha_1} \quad (18) $$

One can then calculate the probability that this particle has to come back towards the plasma or to enter the plenum,

$$ \pi_{\text{out/in}}(z_o, 0) = \begin{cases} \int_{\pi/2}^{\pi/2'} \psi_1 & \pi(\varphi) d\Omega \\ \frac{\pi(\varphi)}{\Phi_1 - \pi/2} & \pi(\varphi) d\Omega \\ \frac{1}{2} [1 \mp \sin(\Phi_1, \Psi_1)] & \pi(\varphi) d\Omega \end{cases} \quad (19) $$
or to be reflected on the opposite face,

$$ \pi_f(z_o, 0) = 1 - [\pi_{in}(z_o, 0) + \pi_{out}(z_o, 0)]. \quad (20) $$

The source distribution on surface 2 is then given by<fnref n="3" />

$$ J_1(z_o, x_2) = \frac{1}{2} \int_{0}^{1/\cos \alpha_1} \cos \varphi_{12} \times \frac{d\varphi_{12}}{dx_2} J_0(z_o, x_1) dx_1. \quad (21) $$

Each point $x_2$ acts as an emitting point and the process repeats, building the following recurrence:

$$ \begin{gathered} \begin{array}{l} \pi_{out}(z_o, 0) \qquad \pi_{out}(z_o, 1) \\ \mathcal{J}_0(z_o, x_i) \quad \rightarrow \quad \mathcal{J}_1(z_o, x_j) \quad \rightarrow \quad \cdots \cdots \cdots \\ \& \qquad \& \\ \pi_{in}(z_o, 0) \qquad \pi_{in}(z_o, 1) \end{array} \\ \begin{array}{l} \pi_{out}(z_o, n-2) \qquad \pi_{out}(z_o, n-1) \\ \mathbf{3} \qquad \mathbf{3} \\ \cdots \cdots \cdots \rightarrow \mathcal{J}_{n-1}(z_o, x_j) \quad \rightarrow \quad \mathcal{J}_n(z_o, x_j) \quad \cdots \\ \& \qquad \& \\ \pi_{in}(z_o, n-2) \qquad \pi_{in}(z_o, n-1) \end{array} \end{gathered} $$

whose general terms are

$$ \pi_{out/in}(z_o, n) = \frac{1}{2} \int_{0}^{1/\cos \alpha_i} J_n(z_o, x_i) \times [1 \mp \sin(\Phi_i, \Psi_i)] dx_i \quad (22) $$

and

$$ J_{n+1}(z_o, x_j) = \frac{1}{2} \int_{0}^{1/\cos \alpha_i} \cos \varphi_{ij} \times \frac{d\varphi_{ij}}{dx_j} J_n(z_o, x_i) dx_i . \quad (23) $$

One obtains finally

$$ \pi_{out/in}(z_o) = \sum_{n=0}^{\infty} \pi_{out/in}(z_o, n). \quad (24) $$

## Acknowledgements

Thanks are due to P. Ghendrih (Association Euratom-CEA, CE Cadarache) for valuable discussions.

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<footnote n="3">³ One verifies that $R_{0}^{1/\cos \alpha_{2}} J_{1}(z_{o}, x_{2}) dx_{2} = \pi_{f}(z_{o}, 0)$.</footnote>
[26] Escarguel, A., et al., J. Nucl. Mater. (in press).

[27] Tsitrone, E., et al., in Fusion Technology (Proc. 19th Symp. Lisbon, 1996), Vol. 1, Elsevier, Amsterdam and New York (1997) 451.

[28] Az’eroual, A., et al., in Fusion Technology (Proc. 20th Symp. Marseille, 1998), Vol. 2, Assoc. Euratom-CEA/Cadarache (1998) 1705.

[29] Gray, D.S., Boedo, J.A., Conn, R.W., Nucl. Fusion 37 (1997) 1437.

(Manuscript received 17 December 1999)

Final manuscript accepted 14 June 2000)

E-mail address of B. P’egouri’e:
pegourie@drfc.cad.cea.fr

Subject classification: J2, Te