# Evaluation of 4-point probe and eddy current methods as plasma facing components erosion measurement methods

A. Deleau, C. Hernandez, A. Skarlatos, Zarel Valdez Nava, P. Castelan, D. Malec

► To cite this version:

A. Deleau, C. Hernandez, A. Skarlatos, Zarel Valdez Nava, P. Castelan, et al.. Evaluation of 4-point probe and eddy current methods as plasma facing components erosion measurement methods. Nuclear Fusion, 2025, 66, pp.016005. 10.1088/1741-4326/ae03d7. hal-05346936

HAL Id: hal-05346936

https://hal.science/hal-05346936v1

Submitted on 4 Nov 2025

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

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

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

Distributed under a Creative Commons Attribution 4.0 International License
# Evaluation of 4-point probe and eddy current methods as plasma facing components erosion measurement methods

A. Deleau<fnref n="1" />,<fnref n="2" />*, C. Hernandez<fnref n="1" />, A. Skarlatos<fnref n="3" />, Z. Valdez-Nava<fnref n="2" />, P. Castelan<fnref n="2" /> and D. Malec<fnref n="2" />

E-mail: amelia.deleau@laplace.univ-tlse.fr

Received 20 March 2025, revised 20 August 2025

Accepted for publication 5 September 2025

Published 31 October 2025

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

Impurities resulting from the erosion of plasma-facing components (PFCs) in the plasma chamber disrupt the normal operation of tokamaks. Additionally, continuous erosion leads to the premature aging of these components, affecting their long-term performance and reliability. To quantify erosion and redeposition directly within the machine between operational phases, non-destructive testing methods are crucial. This study evaluates the 4-point and eddy current (EC) methods for measuring the erosion of PFCs and the design of dedicated erosion markers composed of tungsten and alumina multilayers. The 4-point method, sensitive to thickness loss, necessitates electrically insulating layer. Conversely, the EC method requires a conductivity difference between materials but eliminates the need for interlayers. These non-destructive techniques can be effectively integrated into the tokamak to monitor erosion without requiring component removal. Both methods demonstrate potential for accurate thickness measurements. Our results demonstrate that tungsten erosion of less than 2 µm can be assessed using both the four-point probe and EC methods.

Keywords: tokamak, erosion, eddy current, 4-point probe method

(Some figures may appear in colour only in the online journal)

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹CEA, IRFM, F-13108 Saint-Paul-Lez-Durance, France</footnote>

<footnote n="2">²Univ Toulouse, Toulouse INP, CNRS, LAPLACE, Toulouse, France</footnote>

<footnote n="3">³Université Paris-Saclay, CEA, List, Palaiseau F-91120, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
# 1. Introduction

The WEST tokamak (W Environment in Steady-State Tokamak) aims to evaluate the performance and reliability of tungsten (W) plasma facing units (PFUs) under ITER-relevant conditions. In particular, it focuses on critical operational challenges such as surface erosion, impurity generation, and thermal fatigue in the lower divertor region [1]. Each PFU in WEST consists of 35 tungsten monoblocks (MBs) brazed onto a CuCrZr cooling pipe, and is subjected to surface heat fluxes that can reach up to 10 MW m⁻², leading to erosion and crack formation [2–4] Understanding and quantifying tungsten erosion is crucial for assessing the lifespan of PFUs and for optimizing future reactor operations. Existing diagnostic techniques such as ion beam analysis (IBA), Rutherford backscattering (RBS), nuclear reaction analysis (NRA), proton-induced x-ray emission (PIXE), and laser induced breakdown spectroscopy (LIBS) are known for their high accuracy, and have been used on devices like WEST [3], JET [5] and W7X [6]. However, these techniques are typically destructive or require component removal, which limits their use for routine in situ monitoring. Both methods were applied to a series of calibrated multilayer samples, simulating the future erosion marker design to be integrated into WEST PFUs. These markers consist of an alumina insulating layer deposited on the PFU surface, followed by a thin tungsten coating. This architecture ensures electrical isolation while maintaining a fully tungsten-facing plasma surface. The thicknesses of both layers must be carefully optimized: thick enough for reliable measurements, but thin enough to avoid mechanical delamination or performance degradation under thermal cycling. To address this limitation, the present study investigates two non-destructive methods that are compatible with in situ deployment, including with robotic systems such as WEST’s Articulated Inspection Arm (AIA) [7]: the 4-point probe method and the eddy current (EC) method. The 4-point probe method relies on injecting current and measuring voltage at separate contacts to determine the electrical resistivity of a surface layer. When combined with a known material law or resistivity model, this allows for estimating the thickness of a conductive coating based on its electrical resistance. The EC method uses the interaction between induced magnetic fields and the test sample to measure changes in impedance, which also depend on layer thickness, conductivity, and substrate properties. Frequency tuning and impedance modeling make it possible to extract erosion levels from impedance measurements. This study combines experimental measurements, analytical modeling, and numerical simulations to evaluate the accuracy, sensitivity, and limits of both techniques across a wide range of tungsten thicknesses (1–30 µm), alumina layer configurations, and substrate types (W, CuCrZr, and porous W). Special attention is given to the effects of substrate conductivity, coating microstructure, and measurement dispersion, with the goal of guiding the future in situ integration of these diagnostics into fusion devices.

# 2. Material and methods

## 21. 4-point probe method

The 4-point probe method is widely employed for measuring electrical resistivity in applications such as microelectronics and semiconductor manufacturing [7–10]. It operates by injecting current through two probes while measuring voltage between two additional probes (figure 1(a)). This configuration minimizes contact resistance effects. For thin deposited films with finite dimensions, the resistivity (ρ) of deposited material is determined using the formula [7]:

$$ \rho_{2D}^{\text{line}} = F(t, s) \frac{\pi t V}{\ln(2)} I \quad (2.1.1) $$

with t the thickness of the thin film (m), s the distance between the points (m), F(s, t) a correction factor linked to the distance of the probes from the edges of the sample, V the voltage (V) and I the current (A).

Correction factors adjust for measurement errors caused by edge proximity and film geometry [11]. Assuming the resistivity of the deposited material remains constant despite the erosion process, the thickness of the deposited layer can be determined by measuring its resistance this way:

$$ t = \frac{\rho_{2D}^{\text{line}} \ln(2) I}{F(t, s) \pi V} \quad (2.1.2) $$

However, as shown in figure 1(b), this approximation breaks down beyond a critical thickness (~2.22 × 10⁻³ m in our configuration), where the resistance becomes almost independent of thickness. This corresponds to the thick-film regime, where current paths are no longer laterally confined and become vertical. As a consequence, the resistance becomes constant and the method loses sensitivity to small thickness variations. This has important implications for erosion measurements: if the initial tungsten coating is too thick, even a 1 µm erosion will cause no measurable change in resistance. Therefore, careful selection of the initial thickness is essential to ensure that the system operates within the sensitive range.

To further improve detection sensitivity, a thin alumina interlayer can be introduced below the tungsten coating (figure 2(b)). This insulates the conductive substrate, preventing current leakage and enhancing the confinement of current in the tungsten layer, especially in multilayer structures. Because the expected resistance variations are extremely small (μΩ), a nanovoltmeter is used for precise voltage measurements. The injected current is limited to 0.1 A to avoid Joule heating at the probe tips and preserve measurement integrity.

## 22. EC method

EC testing is extensively used for non-destructive inspection in metallurgy [13]. Alternating current applied to a coil
<figure><img src="image_4.png" /><figcaption>Figure 1. (a) 4-point method with equal space between points on an infinite plate. (b) Theoretical variation in resistance in the case of an infinite plate (2.1.1) as a function of the thickness of a tungsten plate. Tungsten electrical resistivity is 5.48 × 10⁻⁶ Ω·m (0 °C) [12].</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2. (a) Monoblocks are components of plasma facing unit in WEST lower divertor. (b) A type of sample used for 4-point and eddy current testing. The alumina layer acts as an electrical insulator for the 4-point probe method.</figcaption></figure>

generates magnetic fields that induce currents (ECs) in conductive materials (figure 3). These currents interact with the coil's magnetic field, altering its impedance. Impedance variation is given by:

$$ \Delta Z = Z_0 - Z_{\text{Sample}} = (R_0 - R_{\text{sample}}) + j(X_0 - X_{\text{sample}}) $$

where $R_0$, $X_0$ are coil resistance and reactance in the free space and $R_{\text{sample}}$, $X_{\text{sample}}$ are the corresponding coil resistance and reactance values in the presence of the plate.

The change in coil impedance is directly related to the thickness and conductivity of the measured material, making possible to accurately estimate the thickness of a deposited layer [14–17]. The coil frequency affects the sensitivity of the method due to the skin effect, which limits the penetration depth of the induced currents. As the frequency increases, the penetration depth decreases, making the method more sensitive to surface variations but less effective for deeper layers. Using a calibrated reference sample or an inverse method [18], the erosion thickness of the plate can be determined. Reference samples consisting of layers with known thicknesses are used to calibrate the measurement process, ensuring accurate estimations of the deposited layer's erosion.

## 3. Samples

### 31 Elaboration of the erosion markers

In order to assess accuracy of the methods with view to implementing erosion markers in WEST tokamak, a number samples was fabricated with multilayer coatings of tungsten and alumina. Tungsten serves as the primary plasma-facing material, while alumina acts as an electrical barrier. Due to its high resistivity (10¹⁴ Ω·cm), a very thin alumina layer can provide excellent electrical insulation minimizing mechanical constraints within the layers, and prevent delamination. Samples were manufactured using physical vapor deposition (PVD) method [19, 20] to achieve controlled alumina and tungsten thicknesses. To evaluate the sensitivity of both measurement methods, a wide range of coating thicknesses was tested ranging from 0.2 µm to 5 µm for the alumina layer and from 1 to 30 µm for the tungsten layer respectively (figure 2(b)). The substrates used included solid tungsten, porous tungsten, and CuCrZr alloys. Porous substrates introduced controlled roughness for sensitivity testing under realistic conditions, while CuCrZr was used to verify the effectiveness of alumina as a barrier. The fabricated samples had different substrate materials and dimensions: bulk tungsten substrates (rectangular, 28 mm × 12 mm × 1.5 mm), porous tungsten substrates (rectangular, 40 mm × 12 mm × 2.5 mm), and
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) Schematic of the primary and secondary magnetic fields generated by the presence of a tungsten plate with alumina interlayer.</figcaption></figure>

<table><caption>**Table 1.** Alumina and tungsten coatings on the three types of substrate have been produced by PVD (physical vapor deposition). On the three types of substrate, there are alumina deposits of between 0.2 and 5 µm and tungsten deposits of between 1 and 30 µm.</caption><thead><tr><th rowspan="3">Number of samples</th><th rowspan="3">Alumina coating thickness (µm)</th><th rowspan="3">Tungsten coating thickness (µm)</th><th colspan="4">Roughness Ra (µm)</th></tr><tr><th colspan="3">Substrate</th><th rowspan="2">Wp CuCrZr</th></tr><tr><th>W</th><th>Wp</th><th>Wp CuCrZr</th></tr></thead><tbody><tr><td>3</td><td>0.205</td><td>4.955</td><td>0.23</td><td>4.0</td><td>0.40</td></tr><tr><td>3</td><td>0.516</td><td>4.955</td><td>0.32</td><td>4.1</td><td>0.50</td></tr><tr><td>3</td><td>1.011</td><td>4.955</td><td>0.24</td><td>2.5</td><td>0.44</td></tr><tr><td>4</td><td>5.210</td><td>0.910</td><td>0.29</td><td>2.3</td><td>0.44</td></tr><tr><td>4</td><td>5.210</td><td>2.984</td><td>0.29</td><td>3.5</td><td>0.49</td></tr><tr><td>4</td><td>5.210</td><td>5.260</td><td>0.24</td><td>2.9</td><td>0.41</td></tr><tr><td>4</td><td>5.210</td><td>10.524</td><td>0.31</td><td>3.8</td><td>0.42</td></tr><tr><td>5</td><td>5.210</td><td>29.273</td><td>0.32</td><td>3.7</td><td>0.44</td></tr></tbody></table>

CuCrZr substrates (circular, 30 mm in diameter and 5 mm thick). Table 1 summarizes the characteristics of the fabricated samples, including coating thicknesses, and roughness parameters.

# 4. Experimental

## 4.1. 4-point setup

For the 4-point method, resistance measurements were taken using an 4-point probe (SP4, Microworld, Grenoble, France) under controlled conditions (figure 4). The Probe characteristics are 1.6 mm spacing between tips, 45 g spring pressure, and tips of tungsten carbide material with a tip radius of 254 µm. The resistance measurement was carried out at the center of each sample. The probe tips were aligned along the longest axis of the sample. A current source (SourceMeter 2410, Keithley Instruments, Cleveland, OH, USA) supplied a current of 0.1 A, while a precision voltmeter (Nanovoltmeter 2182A, Keithley Instruments, Cleveland, OH, USA) recorded the voltage values. To mitigate thermal drift, measurements were completed within one minute for each sample.

## 4.2. EC setup

For EC testing, impedance variations were measured over a frequency range of 100 Hz–200 kHz using a impedance analyzer (E4990A, Keysight Technologies, Santa Rosa, CA, USA). The coil used had 421 turns, with an inner radius of

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Experimental setup for 4 points method, consisting of a nanovoltmeter, a sourcemeter and an the used 4-point probe. Picture shows a sample with a bulk tungsten substrate on which the probe is placed.</figcaption></figure>

0.9 mm and an outer radius of 5 mm (figure 5). Each measurement was taken at the center of the samples to avoid edge-effects. This coil will be referred to as coil N5 in the following sections.

# 5. Results

## 5.1. ANSYS model of 4-point probe method and experimental data

Simulations performed using the ANSYS finite element modeling software [21] (DC Conduction) enabled the estimation of resistance variations as a function of tungsten
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Picture of a sample with CuCrZr substrate on which a coil connected to an impedance meter is placed.</figcaption></figure>

coating thickness. Additionally, the influence of alumina thickness on resistance measurements was analyzed to determine the minimum required thickness for effective electrical insulation. This optimization ensures that the alumina layer remains sufficiently thin to mitigate the risk of tungsten coating delamination while maintaining adequate insulation, particularly under the thermal stresses encountered in the tokamak environment. Several assumptions were made: the materials were considered homogeneous, the electrical contacts were assumed to be perfect, and Joule heating effects were neglected. In the simulation, the tungsten and alumina layers have a fixed resistivity of $3.7 \times 10^{-7} \Omega \cdot m$ and $1.1 \times 10^{14} \Omega \cdot m$ regardless of thickness.

Figures 6(a) and (b) show respectively the mean resistance as a function of alumina thickness (for a fixed tungsten thickness of 5 µm) and of tungsten thickness (for a fixed alumina thickness of 5 µm). It is worthy noting that roughness has no impact on the resistance measurement. In fact, there is no difference between the resistance measurements for tungsten substrates and porous tungsten substrates, which have different surface roughness (table 1). The experimental results validate the numerical model, confirming that the alumina layer acts as an effective electrical barrier, regardless of its thickness. This property is essential for the integration of erosion markers into PFUs, as it allows for electrical isolation while potentially limiting delamination risks by using thinner insulating layers. A specific anomaly was observed, however, for samples composed of 5 µm W on alumina on CuCrZr substrates. As shown in figure 6(a), these samples exhibit resistance values that deviate significantly from the general trend. Interestingly, they also show nearly twice the resistance of equivalent samples on other substrates. To investigate this discrepancy, we repeated the resistivity measurements on additional samples from the same batch, which confirmed the results. Complementary LIBS analyses were also performed to explore possible interfacial or compositional anomalies, but no conclusive irregularities were found. While the exact cause remains unclear, this behavior could be linked to local imperfections in the alumina barrier. This data point has been retained for completeness but is interpreted with caution. Aside from this outlier, the experimental resistance measurements are generally consistent with the analytical model for tungsten thicknesses ranging from 3 to 30 µm, across all three substrate types (W, Wp, and CuCrZr). However, for very thin tungsten layers (~1 µm), the measured resistance is

consistently higher than predicted by the model assuming constant resistivity. The hypothesis of an error in the thicknesses is ruled out by scanning electronic microscope (SEM) observation of the coating. The only possibility is therefore a variation in resistivity for the 1 µm thickness of tungsten (figure 7(a)). One hypothesis for the variation in resistivity at small thicknesses is that it is influenced by the structure of the coating [22], particularly the size and distribution of the nucleation grains formed during the PVD deposition process (figure 7(b)).

## 52 Resistivity model to determine tungsten thickness

As we saw in equation (2.1.2), the thickness of the deposit can be determined by knowing the resistivity and a measure of resistance. Since the resistivity is not constant (particularly for very thin layers) it is necessary to determine an appropriate mathematical model to minimize the error induced by its variation. To do so, we first selected 30 samples representative of various W/Al₂O₃/W configurations to establish the calibration model. These samples span a wide range of tungsten thicknesses and alumina interlayers. Based on their experimental resistance measurements (figure 7(a)), we tested three models for predicting resistivity as a function of tungsten thickness (figure 8): an affine hyperbolic model $\rho(t) = (a/t + b)$, an affine inverse square function $\rho(t) = (c/t^2) + d$ and an inverse cubic model with offset $\rho(t) = e/t^3 + f$, where the coefficients were optimized using the Levenberg-Marquardt algorithm (see table 2).

To evaluate the generalization capacity of these models and avoid overfitting, the remaining 60 samples (not used in the model construction) were treated as an independent test set. For each sample, the W thickness was estimated using the three candidate models, and the results were compared to the known manufacturer values (see table 3). This partitioning ensures that several thickness values in the test set were not explicitly used during the training phase. Among the three models, the inverse square function $c/t^2 + d$ and the cubic model $e/t^3 + f$ provided the best compromise between accuracy and stability, with relative errors between 0.2% and 4.1%. It is also important to critically reflect on the limitations of the resistivity-thickness models used. In particular, the number of experimental points in the low-thickness range (below 3 µm), where resistivity variation is most significant, remains limited. This inevitably affects the robustness of the fitted models in that specific regime. However, this does not compromise the primary goal of our approach. In real integration scenarios, tungsten layers will typically be deposited with initial thicknesses of about 30–40 µm, and erosion between campaigns is expected to remain well above the critical threshold where resistivity deviations become problematic. Consequently, in practice, measurements will be conducted in a range where resistivity remains approximately constant, and where the analytical model is most reliable. Our objective in presenting this resistivity variation is not to propose a highly accurate model for very thin films, but to inform the reader of the potential risk of misinterpretation if this variation is ignored.
<figure><img src="image_9.png" /><figcaption>**Figure 6.** (a) Average electrical resistance as a function of alumina deposit thickness. The tungsten thickness is 5 µm. (b) Average electrical resistance as a function of tungsten deposit thickness. The alumina thickness is 5 µm. The error bars represent the 95% confidence interval, calculated as the standard error multiplied by the critical value of the Student distribution (2.7, 3.1 or 4.3, depending on the sample number).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** (a) Average resistivity of the various samples as a function of the thickness of the tungsten deposit with 5 µm alumina coating. (b) SEM picture of one sample with 30 µm of tungsten and 5 µm alumina PVD coating.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** (a) Experimental mean resistivity for the W/Al₂O₃/W samples and the different models. (b) Relative error in tungsten thickness as a function of the model used.</figcaption></figure>

<table><caption>**Table 2.** Constants of the various resistivity models.</caption><thead><tr><th>Constants</th><th>a</th><th>b</th><th>c</th><th>d</th><th>e</th><th>f</th></tr></thead><tbody><tr><td>Values</td><td>3.33 × 10<sup>-13</sup></td><td>3.40 × 10<sup>-07</sup></td><td>2.82 × 10<sup>-19</sup></td><td>3.80 × 10<sup>-07</sup></td><td>2.51 × 10<sup>-25</sup></td><td>3.88 × 10<sup>-07</sup></td></tr></tbody></table>
<table><caption>**Table 3.** Comparison of tungsten thickness estimation using different resistivity models on tungsten substrate.</caption><thead><tr><th>Tungsten coating<br>constructor thickness<br>(μm)</th><th>Tungsten coating<br>thickness (μm)<br>estimate with<br>model a/t + b</th><th>Tungsten coating<br>thickness (μm)<br>estimate with<br>model c/t<sup>2</sup> + d</th><th>Tungsten coating<br>thickness (μm)<br>estimate with<br>model e/t<sup>3</sup> + f</th><th>Tungsten coating<br>thickness (μm)<br>estimated with<br>constant resistivity<br>ρ = 38 μΩ·cm</th></tr></thead><tbody><tr><td>0.91</td><td>0.93</td><td>0.94</td><td>0.95</td><td>0.5</td></tr><tr><td>2.984</td><td>3.4</td><td>3.1</td><td>2.9</td><td>2.9</td></tr><tr><td>5.26</td><td>5.3</td><td>5.2</td><td>5.2</td><td>5.0</td></tr><tr><td>10.524</td><td>9.8</td><td>10.1</td><td>10.2</td><td>10</td></tr><tr><td>29.273</td><td>27.4</td><td>29.6</td><td>30.2</td><td>29.4</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Comparison of experimental measurements and simulations: resistance and reactance as a function of frequency for different substrates and tungsten thicknesses.</figcaption></figure>

### 53 CIVA model for EC method and experimental data

CIVA [23] is an advanced simulation software platform for non-destructive testing which offers dedicated simulation modules for the basic techniques, namely ultrasonic, EC and radiography testing. The eddy-current module simulates electromagnetic interactions between probes, materials, and defects by solving Maxwell's equations to generate the resulting probe signals. To enhance the computational performance of the EC testing module, a semi-analytical modelling approach based on Green's dyads is implemented [24]. Using the CIVA software [24], the variation in reactance and resistance was calculated as function of the thickness of alumina and tungsten layers. The alumina layer has no noticeable impact on the coil's impedance variation due to its small thickness and its orientation with respect to the ECs.

All experimental impedance measurements and model fittings shown in this section were obtained using coil N5

described in section 4.2. The EC results are now presented through two complementary figures to better illustrate the gap between experimental measurements and numerical simulations, and to justify the use of a multivariate modeling approach for thickness estimation. Figure 9 shows the variation of resistance and reactance as a function of frequency, for various tungsten thicknesses, and for two different substrates (W and CuCrZr). Three curves are shown for each thickness: raw experimental data (dotted), experimental data corrected for non-ideal coil behavior using the Harrison correction method [25] (solid), and simulation using CIVA (dashed). These comparisons reveal a clear discrepancy between simulated and experimental values in absolute terms. Despite correction, the experimental impedance remains systematically different from the simulated one, both in resistance and reactance. This discrepancy prevents the use of direct inverse modelling based on simulations for quantitative thickness extraction.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Normalization of experimental measurements and comparison with CIVA simulations: evolution of resistance and reactance at 100 kHz as a function of tungsten thickness.</figcaption></figure>

However, when we analyze the normalized results in figure 10, which presents the relative variations in impedance (resistance and reactance values normalized as a function of tungsten thickness at a fixed frequency of 100 kHz), a consistent trend emerges. The experimental data show the same overall dependence on thickness as predicted by simulation, particularly in terms of slope direction and monotonicity. This confirms that, while absolute matching is poor, the relative variation with thickness is captured and can be used as a basis for empirical modelling.

Because of the observed limitations of simulation-based inversion, we adopted a multivariate polynomial regression model to estimate W thickness from experimental impedance data. This model expresses the measured quantity $Z(f, t)$ (real or imaginary part of impedance) as a second-order polynomial function of both frequency $f$ and tungsten thickness $t$:

The coefficients $a_i$ are determined using least-squares fitting from a training dataset of measured impedance values acquired over a set of known thicknesses and frequencies. The fitted coefficients for both the real and imaginary parts of the impedance, using coil N5 and two substrates (W and CuCrZr), are presented in table 4.

The model allows estimating the tungsten thickness from a set of impedance measurements taken at different frequencies. For each frequency, an independent estimate of thickness is computed. The final thickness value is obtained by

averaging over a selected frequency band (typically at higher frequencies), where the sensitivity to thickness is stronger and the noise level is lower (table 5). While the model performs well overall, prediction accuracy is reduced for the CuCrZr substrate, due to higher measurement noise and larger signal variability, as also observed in the normalized plots. This is attributed to stronger electromagnetic interactions between the highly conductive CuCrZr and the coil, which amplify instability in impedance measurements at high frequencies.

54 Comparison of the accuracy of the two methods

Figure 11 compares the thicknesses of tungsten deposits measured using the two methods. The 4-point probe method exhibits lower measurement dispersion compared to the EC method (average error of 1.2 µm for the 4-point probe method versus 1.4 µm for the EC method). In both cases, the results demonstrate that these methods are capable of detecting micrometer-scale variations in tungsten thickness, confirming their potential for erosion monitoring when alumina-based markers are used.

Substrate effects must also be taken into account when evaluating the accuracy of both methods:

* For the 4-point probe method, the substrate has no significant influence on the measurement, provided the insulating layer is uniform and defect-free.
<table><caption>**Table 4.** Coefficients of the second-order polynomial model for approximating the impedance Z(f, t) for coil N5.</caption><thead><tr><th>Substrate</th><th></th><th>a<sub>0</sub></th><th>a<sub>1</sub></th><th>a<sub>2</sub></th><th>a<sub>3</sub></th><th>a<sub>4</sub></th><th>a<sub>5</sub></th></tr></thead><tbody><tr><td rowspan="2">W/Wp</td><td>Z<sub>re</sub></td><td>0.945 94</td><td>2.5965 × 10<sup>-5</sup></td><td>0.010 1774</td><td>-9.429 53 × 10<sup>-12</sup></td><td>-8.586 47 × 10<sup>-11</sup></td><td>-4.354 55 × 10<sup>-4</sup></td></tr><tr><td>Z<sub>im</sub></td><td>2.365 25</td><td>1.023 89 × 10<sup>-3</sup></td><td>-0.040 6256</td><td>-1.7328 × 10<sup>-11</sup></td><td>1.862 01 × 10<sup>-7</sup></td><td>1.100 78 × 10<sup>-3</sup></td></tr><tr><td rowspan="2">CuCrZr</td><td>Z<sub>re</sub></td><td>1.303 36</td><td>1.069 89 × 10<sup>-4</sup></td><td>-0.005 9624</td><td>8.753 28 × 10<sup>-11</sup></td><td>-7.201 89 × 10<sup>-8</sup></td><td>1.688 45 × 10<sup>-4</sup></td></tr><tr><td>Z<sub>im</sub></td><td>2.230 65</td><td>2.209 67 × 10<sup>-3</sup></td><td>0.027 6877</td><td>-2.709 58 × 10<sup>-10</sup></td><td>5.0362 × 10<sup>-7</sup></td><td>-8.827 49 × 10<sup>-4</sup></td></tr></tbody></table>

<table><caption>**Table 5.** Thickness of the tungsten layer and the associated relative error.</caption><thead><tr><th rowspan="2">Tungsten coating constructor thickness (μm)</th><th colspan="2">W/Al<sub>2</sub>O<sub>3</sub>/W</th><th colspan="2">W/Al<sub>2</sub>O<sub>3</sub>/Wp</th><th colspan="2">W/Al<sub>2</sub>O<sub>3</sub>/CuCrZr</th></tr><tr><th>Tungsten thickness estimation (μm)</th><th>% relative error</th><th>Tungsten thickness estimation (μm)</th><th>% relative error</th><th>Tungsten thickness estimation (μm)</th><th>% relative error</th></tr></thead><tbody><tr><td>0.91</td><td>0.44</td><td>52</td><td>2.07</td><td>127</td><td>-0.05</td><td>105</td></tr><tr><td>2.984</td><td>2.88</td><td>4</td><td>2.16</td><td>28</td><td>-1.92</td><td>164</td></tr><tr><td>5.26</td><td>5.1</td><td>3</td><td>3.13</td><td>41</td><td>7.82</td><td>49</td></tr><tr><td>10.524</td><td>11.36</td><td>8</td><td>12.46</td><td>18</td><td>13.85</td><td>32</td></tr><tr><td>29.273</td><td>29.07</td><td>1</td><td>29.12</td><td>1</td><td>28.27</td><td>3</td></tr></tbody></table>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Comparison of the thickness of tungsten deposited using the two methods mentioned above: 4-point method and eddy current method on tungsten substrate.</figcaption></figure>

For the EC method, the situation is more complex. In theory, a strong contrast in electrical conductivity between the coating and the substrate (e.g. tungsten on CuCrZr) should enhance sensitivity to thickness variations. However, our experiments show that while CuCrZr substrates indeed generate stronger signal variations, they also introduce greater measurement dispersion.

This highlights that substrate choice impacts not only sensitivity but also measurement reliability, particularly for the EC method. In our study, tungsten substrates were chosen to reflect conditions relevant to the lower divertor region of WEST, which consists of bulk tungsten MBs. However, if future applications target the upper divertor, where CuCrZr substrates with tungsten coatings are used, the increased

variability observed on CuCrZr must be carefully considered when selecting and calibrating diagnostic strategies.

## 6. Conclusions

This study evaluated the 4-point and EC methods as non-destructive techniques for measuring the erosion of plasma-facing components in tokamaks. The results demonstrated that both methods are suitable for detecting micrometer-scale thickness variations in tungsten coatings, though each has unique strengths and limitations. The 4-point method exhibited higher accuracy for thinner coatings (<5 µm) due to its enhanced sensitivity when using an alumina insulating layer. Experimental results showed an average error of 1.2 µm across all measurements. However, its dependency on an insulating interlayer introduces complexity in practical applications, particularly for *in situ* integration. In contrast, the EC method demonstrated better accuracy in predicting thickness variations when they ranged between 10 and 30 µm, with an average error of 1.4 µm, compared to its performance for thinner coatings (1–10 µm). In addition, this method requires no insulating layer, simplifying its implementation. Simulations and experiments highlighted its sensitivity to coil design and operational frequency, with higher frequencies (200 kHz) providing better accuracy for thin layers. These findings emphasize the importance of optimizing coil parameters for enhanced precision. The choice between the two methods depends on operational needs. The complementary nature of these methods allows for robust erosion diagnostics when used together. Future work should aim to integrate these methods into tokamak diagnostic systems, particularly in the divertor region, where erosion is most severe. Additional studies are needed to evaluate long-term performance under high heat fluxes and
plasma exposure. Understanding the effects of thermal cycling and structural changes on alumina and tungsten layers will further enhance measurement accuracy and reliability. It is important to acknowledge that the erosion rates of bulk tungsten and PVD-deposited tungsten coatings differ significantly, which may affect the direct transposition of erosion measurements. Further investigations are required to quantify these differences and assess how they influence the interpretation of diagnostic results. In addition, real plasma-facing surfaces are often modified by processes such as boronization, fuel implantation, or material migration, which lead to the formation of mixed or contaminated layers with unpredictable electrical and electromagnetic properties. These conditions can introduce significant uncertainty in thickness measurements. Despite these limitations, both methods remain promising for relative erosion monitoring between campaigns, especially when the initial state of the surface is well characterized.

## ORCID iDs

A. Deleau 0000-0002-7610-3036

C. Hernandez 0000-0002-3291-7548

A. Skarlatos 0000-0001-5589-5145

Z. Valdez-Nava 0000-0002-9353-8250

D. Malec 0000-0001-7821-6299

## References

[1] Bucalossi J. et al 2014 The WEST project: testing ITER divertor high heat flux component technology in a steady state tokamak environment *Fusion Eng. Des.* **89** 907

[2] García-Rosales C. 1994 Erosion processes in plasma-wall interactions *J. Nucl. Mater.* **211** 202

[3] Balden M. et al 2021 Erosion and redeposition patterns on entire erosion marker tiles after exposure in the first operation phase of WEST *Phys. Scr.* **96** 124020

[4] Diez M. et al 2024 Experimental characterization of leading edge cracking on bulk tungsten divertor components during 2017–2019 WEST operation *Nucl. Mater. Energy* **41** 101746

[5] Catarino N., Widdowson A., Baron-Wiechec A., Coad J.P., Heinola K., Rubel M., Barradas N.P. and Alves E. 2020 Deposition in the tungsten divertor during the 2011–2016 campaigns in JET with ITER-like wall *Phys. Scr. T* **171** 014044

[6] Mayer M. et al 2021 Erosion of tungsten marker layers in W7-X *Phys. Scr.* **96** 124070

[7] Miccoli I., Edler F., Pfnür H. and Tegenkamp C. 2015 The 100th anniversary of the four-point probe technique: the role of probe geometries in isotropic and anisotropic systems *J. Phys.: Condens. Matter* **27** 223201

[8] Valdes L.B. 1954 Resistivity measurements on Germanium for transistors *Proc. IRE* **42** 420

[9] Pfennigstorf O., Petkova A., Guenter H.L. and Henzler M. 2002 Conduction mechanism in ultrathin metallic films *Phys. Rev. B* **65** 045412

[10] Singh Y. 2013 Electrical resistivity measurements: a review *Int. J. Mod. Phys.: Conf. Ser.* **22** 745

[11] Topsøe H. 1968 *Geometric factors in four point resistivity measurement* Bulletin 472-13, 2nd revised edition
Haldor Topsøe Semiconductor Division, Vedbæk
pp 1–63

[12] Elwell W.T. 1971 *Analytical Chemistry of Molybdenum and Tungsten* (Including the Analysis of the Metals and their Alloys) 1971 edn, edWood D.F. (Pergamon Press)

[13] García-Martín J., Gómez-Gil J. and Vázquez-Sánchez E. 2011 Non-destructive techniques based on eddy current testing *Sensors* **11** 2525

[14] Wang H., Li W. and Feng Z. 2015 Noncontact thickness measurement of metal films using eddy-current sensors immune to distance variation *IEEE Trans. Instrum. Meas.* **64** 2557

[15] Moulder J.C., Uzal E. and Rose J.H. 1992 Thickness and conductivity of metallic layers from eddy current measurements *Rev. Sci. Instrum.* **63** 3455

[16] Tai C., Rose J.H. and Moulder J.C. 1996 Thickness and conductivity of metallic layers from pulsed eddy-current measurements *Rev. Sci. Instrum.* **67** 3965

[17] Huang R. et al 2020 Thickness measurement of metallic plates with finite planar dimension using eddy current method *IEEE Trans. Instrum. Meas.* **69** 8424

[18] Norton S.J. and Bowler J.R. 1993 Theory of eddy current inversion *J. Appl. Phys.* **73** 501

[19] Moshfegh A.Z. 2004 PVD growth method: physics and technology *Physics and Technology of Thin Films* 2004 edn (World Scientific) pp 28–53

[20] Tracton A.A. 2005 *Coatings Technology Handbook* 3rd edn (CRC Press) p 936

[21] Stolarski T., Nakasone Y. and Yoshimoto S. 2018 *Engineering Analysis with ANSYS Software* 2018 edn (Butterworth-Heinemann) p 562

[22] Mayadas A.F. and Shatzkes M. 1970 Electrical-resistivity model for polycrystalline films: the case of arbitrary reflection at external surfaces *Phys. Rev. B* **1** 1382

[23] Calmon P., Mahaut S., Chatillon S. and Raillon R. 2006 CIVA: An expertise platform for simulation and processing NDT data *Ultrasonics* **44** e975

[24] Gilles-Pascaud C. et al 2008 Modelling of eddy current inspections with CIVA 17th World Conf. on Nondestructive Testing (Shanghai, 2008) ed C. Pichenot

[25] Harrison D.J., Jones L.D. and Burke S.K. 1996 Benchmark problems for defect size and shape determination in eddy-current nondestructive evaluation *J. Nondestruct. Eval.* **15** 21