Spectral Intensity of Electron Cyclotron Radiation Emerging from the Plasma to the First Wall in ITER

Abstract

It is predicted that in ITER, due to high values of electron temperature and magnetic field strength, electron cyclotron (EC) radiation emitted by plasma will be a significant source (together with external EC radiation injected for auxiliary plasma heating and non-inductive current drive) of additional thermal and electromagnetic loads for microwave and optical diagnostics. The spectral distribution of plasma EC radiation is particularly important to consider in millimeter-wave diagnostics, namely for high- and low-magnetic-field side reflectometry, plasma position reflectometry, and collective Thomson scattering diagnostic, because the transmission lines of these diagnostics yield the transport of EC waves emitted by the plasma. The development of semi-analytical methods used to describe the spectral distribution of plasma-generated EC radiation in tokamaks, starting from the work of S. Tamor, is based on the dominance of multiple reflections of this radiation from the first wall in a toroidal axially symmetric vacuum chamber. Here, we present calculations using the CYNEQ code of the spectral intensity of the EC radiation emerging from the plasma to the first wall and port plugs for five scenarios of ITER operation. This code uses the symmetry-based effect of approximate isotropy and homogeneity of radiation intensity in a substantial part of the phase space and has been successfully tested by comparison with first-principles codes. The energy flux density in the range of 30–200 kW/m² is predicted for wall reflectance in the range of 0.6–0.95. The possible effect of this radiation on in-vessel components and diagnostics is assessed by calculating the surface density of the energy absorbed by various materials of the ITER first wall.

1. Introduction

Electron cyclotron (EC) radiation in ITER is expected to play an important role in electron-power-loss balance due to the high electron temperature and strong magnetic field [1,2]. This radiation is also a source of additional thermal and electromagnetic loads for microwave and optical diagnostics [3,4]. The EC radiation generated in the plasma dominates over the stray radiation from electron cyclotron resonance heating (ECRH) and current drive (ECCD) microwave power sources in high-performance discharges at the flat-top stage of discharge, so its effect on diagnostics should be investigated [3]. This is especially important for millimeter-wave diagnostics in ITER, such as microwave reflectometers and the collective Thomson scattering system, whose transmission lines yield the transport of EC waves emitted by the plasma and, in principle, additional measurements of EC radiation spectra [5]. Transmission lines for high field side (HFS) reflectometry are planned to be used as waveguides to observe the radiation of extraordinary waves (X-mode) in the frequency range of 12–90 GHz and of ordinary waves (O-mode) in the 18–140 GHz frequency range. These diagnostics have an operating frequency range that lies significantly below the main cyclotron frequency on the toroidal axis of the vacuum chamber (148 GHz) and the operating frequency of the additional heating and non-inductive current generation (170 GHz) in ITER. However, the antennas and waveguides of the discussed diagnostics are susceptible to the entire spectrum of electromagnetic radiation at frequencies above 10 GHz. Because the absorption of the energy of electromagnetic waves by the metal walls of waveguides and other elements of diagnostic systems increases noticeably with an increase in the frequency of the waves, the EC radiation from the plasma can be the cause of their strong heating.

This paper is structured as follows. In Section 2, “Materials and Methods”, we describe the CYNEQ code [2,6,7], which is used to simulate the transport of the EC radiation in the plasma at moderate and high harmonics of the EC fundamental frequency, and briefly describe the algorithm [8,9] used to calculate the EC radiation spectrum from low to high harmonics of the EC frequency. Section 3, “Results”, presents detailed calculations of the spectral intensity of the EC radiation and its properties for five reference scenarios of ITER operation, obtained by the simulation of the discharges using the free-boundary transport modelling code CORSICA [10]. Compared to previous works [8,9], which describe the calculation algorithm and provide calculations of the plasma-generated EC radiation intensity in ITER and DEMO (in a baseline scenario predicted using the ASTRA [11] and CRONOS [12] codes, respectively), the present calculations are made for the reference ITER scenarios, which are modeled with CORSICA to cover a wide range of ITER operation parameters. Additionally, the present work includes a comparison of the EC radiation energy flux density and the absorbed energy density for two different sources, namely the plasma-generated EC radiation and the stray radiation produced by the reflection of the radiation injected by ECH/ECCD systems. Section 4 contains a discussion and the conclusions on the results obtained for the absorption of the EC radiation by various materials of the ITER first wall in the view of its possible effect on in-vessel components and diagnostics.

2. Materials and Methods

2.1. CYNEQ Code for EC Radiation Transport in Tokamak Reactors at High and Moderate Harmonics

The current state of the art in modeling the transport of plasma-generated EC radiation in tokamaks with high-temperature plasma and highly reflecting vacuum chamber walls is described in detail in a recent review [13]. Under tokamak reactor conditions (high-temperature plasma with a volume average electron temperature < T_e >_V≥ 10 keV; non-circular poloidal cross-section of the torus; moderate values of aspect ratio A~3; and multiple reflections of radiation from the vacuum chamber wall), the transport of EC radiation appears to be non-local (non-diffusive), i.e., most of the EC radiation energy is carried by electromagnetic waves, for which plasma is optically thin for one pass between the reflections from the wall. The CYNEQ code [2,6,7] provides a semi-analytical solution to the EC-radiation-transport problem in tokamak reactors, using the assumption of the angular isotropy of the intensity of EC radiation, proposed by the advanced first-principles simulations of EC radiation transport using the SNECTR code [14,15].

In [16,17], it was shown that under tokamak reactor conditions, the results of the CYNEQ code are in good agreement with other EC radiation transport codes: (1) the SNECTR code [14,15], which is based on the Monte Carlo simulation of the EC wave emission and absorption in an axisymmetric toroidal plasma with specular or diffuse reflection of waves from the walls of a vacuum chamber; and (2) the RAYTEC code [18], which integrates the radiative transfer equation along the trajectories of EC waves, taking into account the reflection of radiation from the wall.

The approach of the CYNEQ code to solve the problem of the EC-radiation-transfer equation in tokamak reactors is close to the CYTRAN code [19] but, at the same time, improves it by using the escape probability method developed for radiation transfer in spectral lines of atoms and ions in the case of radiation transfer by free electrons (the relations between CYTRAN and CYNEQ are discussed in detail in [13] in Section 2.2).

The conclusion of the benchmarking [2,16,17] was that the modified codes CYNEQ [7] and CYTRAN [19,20] (in the version that participated in the benchmarking [16]) are suitable for use in global transport codes (e.g., ASTRA [11]) for self-consistent 1.5D transport simulations of plasma evolution in tokamak reactors because they provide good approximation and computational efficiency. A comparison between the CYNEQ and CYTRAN results and the latest results [21] from RAYTEC simulations of ECR power loss density for DEMO-like high-temperature plasmas in [13] confirmed this conclusion. Therefore, the use of the CYNEQ code is acceptable in estimating the thermal loads on the first wall in ITER.

Within the CYNEQ code approach, the plasma is divided into an optically thick inner region and an optically thin outer region. In the optically thin region, the intensity of the outgoing EC radiation, J, assuming its angular isotropy, is a function of the wave frequency and wave mode and is given by a simple equation, which in the case of neglecting the conversion of the EC wave modes in wall reflections takes the following form:

$$J(\nu ,\mathsf{\zeta })=\frac{{\displaystyle \int d{\Omega }_n}{\displaystyle \underset{V_{thin}}{\int }dV\hspace{0.17em}q(\varphi ,r)}}{{\displaystyle \int d{\Omega }_n}{\displaystyle \underset{(ndS)\hspace{0.33em}>\hspace{0.33em}0}{\int }(ndS)(1-R_{\mathrm{w}}(\varphi ,r))}+{\displaystyle \int d{\Omega }_n}{\displaystyle \underset{V_{thin}}{\int }dV\hspace{0.17em}\kappa (\varphi ,r)}}$$

(1)

where ϕ = [ν, n, ζ] is the set of the wave parameters in which ν is the wave frequency, n is the wave direction, ζ is the wave mode (polarization) (O: ordinary wave; X: extraordinary wave), r is the spatial coordinate (which, due to the toroidal symmetry of magnetic confinement systems, is reduced to the dimensionless radial coordinate, ρ, which is the square root of the normalized toroidal magnetic flux and, in fact, a label of the magnetic flux surface), q and κ are the coefficients of emission and absorption of the EC waves, and R_w is the reflection coefficient of the EC radiation from the wall of the vacuum chamber. Note that the absorption and emission coefficients in Equation (1) are averaged over the angles of wave propagation, Ω_n, and the volume, V_thin, of the optically thin plasma region. In the CYNEQ code, the coefficients of absorption and emission of the EC waves are calculated using the Schott–Trubnikov formulas as described in [22,23] (Equations 3.13 and 6.15, respectively) and [24] (Equation 2.2.24). A simplification of the calculations using Equation (1) is made: instead of the exact value of the integral with an average of the reflection coefficient over the area of the chamber surface, S, a constant value of surface-averaged R_w is used, which is a free-input parameter in our simulations. We do not take into account the polarization scrambling of EC waves in reflections from the wall because, for the considered high values of R_w and the domination of the frequency range of moderate and high harmonics in EC radiation intensity, this effect is not important [25].

The non-local nature of the EC radiation transfer in the optically transparent region manifests itself in Formula (1) in the fact that the intensity of this radiation is a function of the integral characteristics of the plasma (namely, volume-averaged coefficients of emission and absorptions of the EC waves). Energy losses are due to the escape of photons from the device with a free path of EC waves in relation to their absorption by the plasma, significantly exceeding the characteristic width of the toroidal plasma column in its poloidal cross-section. The latter is due to the strong reflection of EC radiation from the wall of the vacuum chamber.

The optically thin outer plasma layer used in Equation (1) is defined by the following equation:

$${\displaystyle \underset{{\rho }_{thin}}{\overset{1}{\int }}\hspace{0.17em}a\hspace{0.17em}\kappa (\rho ,\hspace{0.17em}\nu ,\hspace{0.33em}\mathsf{\zeta })d\rho \hspace{0.17em}\hspace{0.33em}\le {\tau }_{\mathrm{crit}}}$$

(2)

where a is the effective minor radius of toroidal plasma (considering the ellipticity of the last closed magnetic-flux surface) and τ_crit is an effective value of the optical thickness (which was chosen as equal to 1.5, so that the semi-analytical approach of the CYNEQ code better matches the first-principles modelling of the SNECTR code). For tokamak reactor conditions, this region dominates moderate and high EC harmonics with harmonic number n ≥ 3 of the fundamental EC wave frequency, ν_c0 = e B₀/(2π m_e), where e is the electron charge, B₀ is the vacuum magnetic field on the torus axis, and m_e is the electron mass (see, e.g., Figure 3 in [13] for an illustration of the size of the optically thick region of plasma for an ITER-like scenario of operation). The optically thick inner region of the plasma is the region where diffusive transport dominates. For this part of the phase space, EC radiation intensity is assumed to be equal to the blackbody intensity. To improve accuracy, the boundary between intensity (1) and blackbody intensity can be smoothed by interpolation between these two regions (see Equation (10) in [2]).

2.2. Estimation of EC Radiation Outgoing from Plasma at Low Harmonics

While for the modeling of the EC radiation spectrum at the moderate and high harmonics of the fundamental EC wave frequency, n ≥ 3, we use the approach of the CYNEQ code, to estimate the outgoing radiation at low harmonics, n = 1 and n = 2, we use the following simplified model. For EC waves at harmonics n = 1 and n = 2, the plasma in ITER will be optically thick. In this case, the local values of the EC radiation intensity are close to those of blackbody radiation with a temperature equal to that of the local electron temperature. This yields a relationship between (in a reasonable approximation) the spectral intensity of the outgoing EC radiation and the spatial profile of the electron temperature T_e along the chord of observation. We assume that the radiation temperature is defined in the standard way:

$$T_{\mathrm{rad}}(\nu )=\frac{c^2}{{\nu }^2}J(\nu )$$

(3)

For harmonics n = 1 and n = 2, the radiation temperature is equal to the electron temperature in the resonance region of the wave–particle interaction:

$$\begin{array}{l}{T}_{\mathrm{rad}}(\nu )=T_e(r_{res}(\nu )),\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\nu =\hspace{0.17em}n\hspace{0.17em}\hspace{0.33em}\frac{e\hspace{0.17em}B(r_{res})}{2\pi \hspace{0.17em}{m}_e},\end{array}$$

(4)

where r_res is the radial coordinate (in the torus poloidal cross-section) of the wave–particle resonance region for the propagation of the EC wave perpendicular to the magnetic field and B(r) is the local magnetic field (in which the tokamak is proportional to 1/R, where R is the torus radial coordinate in the poloidal cross-section). The spectral intensity of the EC radiation, differential in solid angle, can be calculated by Equations (3) and (4). The accuracy of this approach can be estimated from the comparison in Figure 3 in [26] of two angles of the observation/collection of EC radiation, which shows the simulation of EC emission spectra at low harmonics in ITER using the SPECE code [27] and calculations using Equations similar to (3) and (4). From the results of [26] and the present work, it is clear that the accuracy of Equations (3) and (4) in estimating the frequency-integrated thermal loads is satisfactory.

Note that the EC radiation intensity at low harmonics coming from the optically thick (for a given frequency) region of plasma is isotropic. When the direction of the observation chord deviates from the direction perpendicular to the magnetic field, and when the chord crosses the toroidal plasma column in a short section near the plasma surface, Equation (4) is no longer applicable, and a detailed calculation is required for the spectral–angular distribution of the outgoing radiation intensity in the corresponding range of angles and frequencies. In general, for the simulation of the EC radiation spectra at low harmonics, it is necessary to consider all the features of the propagation of EC waves in plasma, including the reflection of waves from cut-off zones, polarization scrambling at wall reflections, etc. Accurate modelling of the EC radiation spectra from plasma at low harmonics is performed using the following methods: single-pass ray-tracing calculations using the SPECE code [26,27]; and the approach described in [28], which relies on the ensemble-averaging of rays traced through many randomized wall reflections. However, the details of the spectrum of the outgoing EC radiation at low harmonics, as we will see below, are not critical for the assessment of the possible contribution of plasma-generated EC radiation to the thermal and electromagnetic loads on various diagnostics, on the first wall, and in regions behind the blankets and in the port plugs.

To analyze the effect of EC radiation on in-vessel components, it is convenient to use the total (i.e., integrated over frequencies and angles) EC radiation energy flux density, emerging from the plasma to the first wall, which is calculated by the following formula:

$$F=\pi {\displaystyle \sum _{\zeta =X,O}{\displaystyle \int J(}}\nu ,\zeta )d\nu$$

(5)

2.3. Calculation of Stray Radiation Produced by External Injected Radiation

The energy density of the flux of the plasma-generated EC radiation on the ITER first wall can be compared with the stray microwave radiation produced by multiple reflections of the unabsorbed external radiation, injected into the plasma from the gyrotrons for ECRH/ECCD and diagnostics purposes. The energy flux density of the stray radiation from the gyrotron radiation of the collective Thomson scattering diagnostic (average power of 500 kW at 60 GHz) is in the range of 11–45 kW/m² (see Table 4 in [4]). In the case of an isotropic and homogeneous stray radiation field from gyrotrons of the ECH system, the calculation of the respective energy flux density, F_inj, from the steady-state power balance equation, without taking into account the absorption of the injected EC radiation in plasma, yields the following relation, which can also be derived from Equations (1) and (5) for negligible absorption:

$$F_{inj}=\frac{P_{inj}}{(1-R_{\mathrm{w}})S},$$

(6)

where P_inj is the total injected power. At the start-up phase of the discharge in ITER with an input power of P_inj = 6.7 MW, and R_w in range of 0.6–0.95, the energy flux density will be in the range 19–153 kW/m², respectively.

The way to obtain Equation (6) can be considered as a simplification (to a single resonator) of the multi-resonator model proposed in [29] and applied in [3,4] to estimate the energy flux density of stray radiation from the ECRH/ECCD system in ITER. Calculations of the stray radiation energy fluxes in 55 sectors of the ITER vacuum vessel are presented in [4]. The simplification, which uses a single resonator with an effective surface-averaged reflection coefficient, is appropriate because this coefficient can be found by solving an inverse problem using calculations with a multi-resonator model (see Section 7 in [4]), and then it can be used to estimate the energy flux density for plasma-generated EC radiation with high accuracy, which is more isotropic and homogeneous than the ECRH/ECCD-produced stray radiation.

3. Results

3.1. Scenarios of Tokamak Reactor ITER Operation

Tokamak reactor ITER has the following characteristic parameters: torus major and minor radii R₀ = 6.2 m and a = 2.0 m, respectively; elongation k_elong = 1.9; triangularity δ = 0.6 of the plasma poloidal cross-section; and magnetic field strength on the torus axis B₀ = 5.3 T. Calculations of the spectral intensity of EC radiation were performed for five scenarios of ITER operation:

(1)

baseline inductive scenario, operating in high confinement mode (H-mode) with plasma current I_p = 15 MA (we use label HMODE13 for this scenario);

(2)

hybrid operation scenario with electron cyclotron heating (ECH), I_p = 12.5 MA (HYBRID_EC11);

(3)

hybrid operation scenario with ECH and lower hybrid heating, I_p = 12.5 MA (HYBRID_LH03);

(4)

steady-state operation scenario with injection of 13.3 MW of ECH power, I_p = 8.5 MA (STEADY_EC42);

(5)

steady-state scenario with injection of 20 MW ECH power, I_p = 9.0 MA (STEADY_EC1F).

These scenarios, developed in 2016–2021, were obtained by a simulation of the ITER discharges with the free-boundary transport modelling code CORSICA [10]. The ITER baseline scenario is published in [30], hybrid scenarios are presented in [31], and steady-state scenarios are given in [32]. The CORSICA-simulated scenarios used in this study have been developed and provided before the recent decision to remove the LHCD system from ITER upgrade options [33]. However, the analysis conducted in this work mainly depends on the global 0D plasma parameters and major plasma profiles, such as the temperature and density. Considering the large number of uncertainties on the assumptions applied to the ITER DT scenarios, the range of global 0D parameters and plasma profiles would still be in the foreseen ranges, and the interpretation of EC power loss analysis and the validity of the developed tool would be unchanged.

Table 1. Parameters of ITER scenarios for the flat-top stage of discharge.

N	1	2	3	4	5
Scenarios	HMODE13	HYBRID EC11	HYBRID_LH03	STEADY_EC42	STEADY_EC1F
Te(0), keV	22.1	29.2	31.4	35.9	37.4
Te(1), keV	0.2	0.1	0.1	0.1	0.1
<Te>V, keV	10.6	13.2	14.3	14.2	15.3
ne(0), 1019 m−3	9.7	8.5	8.5	6.5	6.0
ne(1), 1019 m−3	3.9	3.0	3.0	2.3	2.1
<ne>V, 1019 m−3	8.7	7.4	7.4	5.7	5.2
Ip, MA	15.0	12.5	12.5	8.5	9.0
Pnbi, MW	32.4	32.8	32.8	22.6	17.0
Pech, MW	6.5	20.0	20.0	13.3	20.0
Pich, MW	9.8	0	0	0	0
Plh, MW	0	0	20.0	20.0	20.0
Paux, MW	48.7	52.8	72.8	56.0	57.0

T_e(ρ) is the electron temperature profile as a function of the magnetic surface label, ρ; the label of the magnetic surface ρ is defined as the square root of the normalized toroidal magnetic flux, where ρ = 0 corresponds to the axis of nested magnetic flux surfaces, and ρ = 1 is the label of the last closed magnetic surface; n_e(ρ) is the electron density profile; < >_V is the result averaged over the plasma volume; I_p is the plasma (toroidal) current; P_src is auxiliary plasma heating power from various external sources; src = nbi is the heating by neutral beam injection; src = ech is electron cyclotron resonant heating; src = ich is ion cyclotron resonant heating; src = lh is lower hybrid heating; P_aux is total power of auxiliary plasma heating by all external sources.

shows the comparison of the main parameters of these ITER scenarios for the flat-top stage of discharge.

3.2. Spectral Intensity and Energy Flux Density of EC Radiation from Plasma at Low EC Harmonics

Figure 1 shows the results obtained using Formulas (3) and (4) for the spectral intensity emerging from the plasma to the first wall for low EC harmonics n = 1 and n = 2 and the corresponding EC energy flux density for all considered scenarios of ITER operation. Note that, in our calculations, the EC radiation spectrum at low EC radiation harmonics does not depend on the EC radiation reflection from the wall.

3.3. Spectral Intensity and Energy Flux Density of EC Radiation from Plasma for Moderate and High EC Harmonics

The evolution of the frequency-integrated EC radiation energy flux density for moderate and high harmonics, n ≥ 3, F, emerging from the plasma to the first wall calculated using the CYNEQ code for R_w = 0.9 is shown in Figure 2. We assume that the total surface of the wall of the vacuum chamber, excluding the internal surface of the port plugs and all other internal cavities of the chamber, is S = 876 m². It can be seen that the energy flux density, F, strongly depends on the electron temperature profile. This behavior is in qualitative agreement with the Trubnikov scaling law for the total (i.e., integrated over the entire frequency range and over the plasma volume) EC radiation power losses from a homogeneous fusion plasma [18,34], which yields the following scaling law for the EC radiation energy flux density (also shown in Figure 2 for comparison with the CYNEQ results):

$$F\propto \frac{\sqrt{n_e}\hspace{0.17em}{T}_e{}^{5/2}\hspace{0.17em}{B}_0{}^{5/2}\hspace{0.17em}\sqrt{a}}{\sqrt{1-R_{\mathrm{w}}}\hspace{0.17em}},$$

(7)

where the volume-averaged values of electron temperature and density are used.

Figure 2 shows also the duration of the steady-state (“flat-top”) phase of discharge in ITER, during which the loads from the plasma-generated EC radiation prevail (by orders of magnitude), as was estimated earlier in [3] and will be shown below, over the loads from the scattered radiation from the ECRH/ECCD systems and the duration of the initial stage of discharge, during which the above-mentioned radiation (ECRH/ECCD-produced stray radiation) dominates in radiation loads.

Because the reflection coefficient of EC radiation from the wall of the vacuum chamber in ITER is not specified in the ITER database, it is reasonable to conduct an analysis with a variation in the reflection coefficient in a certain range of values. The predicted value of the effective reflection coefficient of the ITER vacuum chamber is very high R_w ≈ 0.95 (even taking into account the port plugs, the space under the divertor and the cryopumps) and is a rather weak function of the frequency of the EC wave [4] (see Figure 7 therein) but in reality, this value could be lower due to surface roughness.

Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 show the calculations of the spectral intensity emerging from the plasma to the first wall, J, for all considered scenarios at fixed values of the reflection coefficient of the EC wave from the vacuum chamber wall, R_w = [0.6, 0.8, 0.9, 0.95]. Figure 8 shows J as a function of EC cyclotron harmonic number, n, and wall reflection coefficient, R_w, in the scenario of ITER operation with improved confinement (scenario No. 1 in Table 1, HMODE13).

The neglect of the weak spectral dependence of the reflection coefficient on radiation from the wall is justified by the fact that, taking into account such a dependence in the most sensitive case, namely the largest reflection coefficient R_w = 0.95, yields a difference in the results of no more than 1% (Figure 9).

For practical purposes, it is worth obtaining the following approximate formulas for the normalized EC radiation energy flux density and normalized frequency, ν*, at which the EC radiation intensity (1), emerging from the plasma, is at a maximum. These formulas are obtained from a parametric analysis of the results of our calculations (with RMSE equal to 8% and 4%, respectively):

$$\widehat{F}=\frac{F(R_{\mathrm{w}})}{F(R_{\mathrm{w}}=0.8)}\approx \frac{0.51}{{(1-R_{\mathrm{w}})}^{0.43}}$$

(8)

$$\widehat{\nu }=\frac{{\nu }^{*}(R_{\mathrm{w}})}{{\nu }^{*}(R_{\mathrm{w}}=0.8)}\approx \frac{0.83}{{(1-R_{\mathrm{w}})}^{0.13}}$$

(9)

The EC radiation energy flux density emerging from the plasma to the wall as a function of R_w and the scaling law for its normalized value for all considered scenarios are shown in Figure 10. Frequencies corresponding to the maximum EC radiation intensity as a function of R_w and the scaling law for the normalized frequency are shown in Figure 11.

3.4. Comparison of Energy Flux Density and Thermal Loads for Plasma-Generated EC Radiation and Stray Radiation Produced by ECRH/ECCD

The characteristics of the plasma-generated EC radiation intensity and energy flux density, which were calculated with the algorithm described in Section 2, in all considered scenarios and the ECRH/ECCD-produced stray radiation energy flux density, which were calculated using Equation (6), are summarized in

Table 2. Characteristics of the plasma-generated EC radiation emerging from the plasma in various ITER scenarios from Table 1.

	Scenario	HMODE13			HYBRID_EC11			HYBRID_LH03			STEADY_EC42			STEADY_EC1F
Frequency range and reflection		ν* [GHz], Jmax [1 × 10−15 MW m−2 sr−1 Hz−1], Fn [kW/m2]
n = 1		170.0	0.1	0.2	170.0	0.1	0.3	155.3	0.1	0.3	155.3	0.2	0.3	155.3	0.2	0.3
n = 2		301.8	0.4	1.7	316.5	0.6	2.1	301.8	0.6	2.2	301.8	0.7	2.2	316.5	0.7	2.4
n ≥ 3, Rw = 0.60		653.5	1.7	26.7	741.4	2.4	46.8	741.4	2.7	55.7	726.8	3.0	62.4	741.4	3.2	69.0
n ≥ 3, Rw = 0.80		668.1	2.0	37.0	756.1	3.0	65.8	785.4	3.4	78.7	770.7	3.7	89.3	814.7	4.0	99.0
n ≥ 3, Rw = 0.90		770.7	2.5	50.1	858.6	3.8	90.6	858.6	4.3	108.7	844.0	4.7	124.8	873.3	5.1	138.8
n ≥ 3, Rw = 0.95		873.3	3.0	66.7	902.6	4.6	122.4	946.5	5.3	147.4	961.2	5.9	171.3	975.8	6.3	191.2
F [kW/m2]
n ≥ 1, Rw = 0.60		28.6			49.1			58.2			64.9			71.8
n ≥ 1, Rw = 0.80		38.9			68.2			81.2			91.8			101.7
n ≥ 1, Rw = 0.90		52.1			92.9			111.2			127.3			141.6
n ≥ 1, Rw = 0.95		68.6			124.7			149.9			173.8			194.0

ν* is the frequency at which the EC radiation intensity (1), emerging from the plasma, is at its maximum; J_max is the maximum value of the EC radiation intensity for the specified frequency range; F_n is the energy flux density of the EC radiation emerging from the plasma to the first wall, for the specified frequency range, including the EC harmonics n = 1, n = 2 and n ≥ 3; F is the total (integrated over all frequencies) value of energy flux density for EC harmonics n ≥ 1. Total surface of the wall of the vacuum chamber, S = 876 m².

and

Table 3. Energy flux density for the stray radiation produced by ECRH/ECCD.

Reflection Coefficient	Finj [kW/m2], Pinj = 6.7 MW
Rw = 0.60	19.1
Rw = 0.80	38.2
Rw = 0.90	76.5
Rw = 0.95	153.0

F_inj is the energy flux density of the ECRH/ECCD-produced stray radiation incident on the first wall at the start-up stage of discharge in ITER when absorption of the EC radiation in plasma is negligible (total input heating power P_inj = 6.7 MW). Total surface of the first wall of the vacuum chamber, S = 876 m².

, respectively.

From the data in Table 2, it can be understood that the contribution to the energy flux density from moderate and high harmonics of the fundamental EC frequency significantly exceeds the contribution from low harmonics.

The calculations presented in Table 3 show the results for the worst case, when the conversion of the injected (and not absorbed by plasma) radiation into stray radiation is not suppressed with the help of special absorbers in the first wall. It should also be noted that, for the start-up phase of discharge in ITER, an injection of 5.87 MW of ECRH from the upper launcher is currently considered; for later phases, 6.7 MW will be injected from the equatorial launcher. In our estimations using Equation (6) for stray radiation, this difference in the total injected power of the ECRH yields a 12% difference in the estimate of the energy flux density shown in Table 3.

It is important to note that, as shown in [4], the lower the quality factor of the resonators (i.e., the lower the value of the reflection coefficient of the walls), the greater the variation in the fluxes of stray radiation in the vessel (see Figure 3 in [4]). Thus, for the real design of diagnostics, taking into account the possible deterioration of the reflectivity of the walls, it is necessary to use the results of modeling [4] of the ECRH/ECCD-produced stray radiation.

For the estimation of the thermal loads on various internal chamber elements caused by the absorption of plasma-generated EC radiation and of ECRH/ECCD-produced stray radiation, we calculated the absorption coefficient of EM waves for metal surface (using Fresnel equations), averaged over polarizations and incidence angles, for three materials (stainless steel, tungsten, beryllium) at different wall temperatures (for detailed description of this approach, see Section 5.8.5 in [35] and Equations (2)–(10) in [4]). Data on the electrical resistivity of tungsten and beryllium are taken from the table in Section 12–42 in [36], and data on the resistivity of stainless steel (type 316L(N)-IG) are taken from Table 2 in [37]. The material absorption coefficients as a function of frequency for isotropic stray radiation are shown in Figure 12. For the considered wall temperatures (100–300 °C for the start-up stage of discharge and 600 °C for the flat-top stage of discharge), the values of the absorption coefficient, η_s, for beryllium and tungsten are very close, while the absorption coefficient for stainless steel is approximately twice as high. Calculations of the absorbed energy density of plasma-generated EC radiation, F_abs, in the considered scenarios of ITER operation and of the stray radiation from ECRH/ECCD systems, F_abs,inj, at the start-up stage of the discharge in ITER are presented in

Table 4. Absorbed energy density for plasma-generated EC radiation emerging from the plasma in various ITER scenarios from Table 1 for different materials at flat-top state of discharge.

	Scenario	HMODE13		HYBRID_EC11		HYBRID_LH03		STEADY_EC42		STEADY_EC1F
Reflection		Absorbed energy density, Fabs [kW/m2], for stainless steel and tungsten for temperature 600 °C
Rw = 0.60		1.3	0.6	2.3	1.2	2.7	1.4	3.1	1.6	3.4	1.7
Rw = 0.80		1.8	0.9	3.3	1.7	3.9	2.0	4.5	2.3	5.0	2.5
Rw = 0.90		2.4	1.2	4.6	2.3	5.6	2.8	6.4	3.3	7.2	3.7
Rw = 0.95		3.3	1.7	6.3	3.2	7.7	3.9	9.0	4.6	10.1	5.2

and

Table 5. Absorbed energy density for ECRH/ECCD-produced stray radiation in ITER for different materials at start-up stage of discharge.

Reflection	Absorbed Energy Density for ECRH/ECCD-Produced Stray Radiation, Fabs,inj [kW/m2], for Stainless Steel and Tungsten for Temperature 300 °C (Pinj = 6.7 MW)
Rw = 0.60	0.4	0.2
Rw = 0.80	0.9	0.4
Rw = 0.90	1.8	0.8
Rw = 0.95	3.6	1.5

for stainless steel and tungsten materials. F_abs and F_abs,inj values of stainless steel are approximately twice as large as those of tungsten. The ratio of F_abs and F_abs,inj for stainless steel material is shown in Figure 13 for five scenarios of ITER operation.

4. Discussion and Conclusions

Here, we present calculations using the CYNEQ code of the spectral intensity of the EC radiation emerging from the plasma to the first wall and port plugs for five scenarios of ITER operation, which cover a wide range of basic plasma parameters. This code uses the symmetry-based effect of approximate isotropy and the homogeneity of radiation intensity in a substantial part of the phase space and has been successfully tested by comparison with first-principles codes. A detailed analysis of the spectral and frequency-integrated characteristics of the outgoing EC radiation made it possible to find the spectral range of the maximum power of this radiation and to estimate the role of radiation at low harmonics of EC frequency. It is shown that, for all considered ITER operation scenarios, the main contribution to the energy flux density of EC radiation emerging from the plasma to the first wall is made by moderate and higher harmonics of fundamental EC frequency with harmonic number n > 3. The energy flux density in the range of 30–200 kW/m² for the wall reflection coefficient in the range of 0.6–0.95 is predicted.

An approximate formula is proposed which, with an accuracy of 10%, describes the dependence of normalized energy flux density of the plasma-generated EC radiation on the surface-averaged reflection coefficient, R_w, of EC waves from the walls of the vacuum chamber (averaging of R_w over the surface of the first wall assumes that the radiation exits only through the port plugs).

The possible effect of this radiation on in-vessel components and diagnostics is assessed by calculating the surface density of the energy absorbed by various materials of the ITER first wall. The calculations performed for the maximum possible values of the reflection coefficient show the relevance of a detailed study of the possible effect of the EC radiation of the plasma in ITER on the in-chamber elements of various diagnostics and its effect in the areas behind the blankets and in the port plugs. The presented results can be used as a boundary condition for the problem of the propagation and absorption of EM radiation in port plugs and various diagnostic elements, including, first of all, all microwave diagnostics and all other diagnostics, which, unlike the first wall, are subject to outgoing EC radiation and do not have intensive cooling.