Recent DIII-D progress toward validating models of tungsten erosion, re-deposition, and migration for application to next-step fusion devices

Fundamental mechanisms governing the erosion and prompt re-deposition of tungsten impurities in tokamak divertors are identified and analyzed to inform the lifetime of tungsten plasma-facing components in ITER and other future devices. Various experiments conducted at DIII-D to benchmark predictive models are presented, leveraging the DiMES removable sample exposure probe capability and the Metal Rings Campaign, in which toroidally symmetric rows of tungsten-coated tiles were installed in the DIII-D divertor. In tokamak divertors, the width of the electric sheath is of the order of the main ion Larmor radius, and a vast majority of sputtered tungsten impurities are typically ionized within the sheath. Therefore, W prompt redeposition is mainly governed by the ratio of the characteristic ionization mean-free path of neutral tungsten to the width of the sheath. In-situ monitoring of the prompt redeposition of tungsten impurities in divertors is demonstrated via the use of WII/WI line ratios and the ionizations/photon (S/XB) method in L-mode discharges. Even with this relatively limited set of emission measurements, net erosion measurements were found to be a consistent upper bound to an analytic scaling based on the ratio of the W ionization length, λiz, and the width of the magnetic sheath rather than the ratio of λiz and the W+ gyro-radius. In the far-scrape-off layer (SOL) of the ITER divertor, however, it is calculated that the measurement of photon emissions associated with the ionization of tungsten impurities up to W5+ may be required. Finally, W deposition patterns on DiMES collector probes, interpreted via DIVIMP-WallDYN modelling, reveal the key roles of progressive W erosion/re-deposition staps and E × B drifts in regulating long-range high-Z material migration.


Introduction
Tungsten remains the leading candidate plasma-facing material (PFM) for next-step fusion devices due to its low erosion rates, low tritium retention via implantation, high thermal conductivity, and the potential for development of radiation-hardened alloys.Despite its known drawbacks such as high core radiation [1], W is also the chosen PFM for the first-generation ITER divertor [2], and likely for the first wall as well.The physics of W sourcing, migration, scrape-off-layer (SOL) transport, and core contamination is complex and significant work remains to develop a sufficiently mature validated physics understanding with which to predict high-Z dynamics in next-step tokamak devices.This paper focuses on recent studies in the DIII-D tokamak on the path towards this capability, particularly on advances in understanding tungsten sputtering, spectroscopy, and redeposition in steady-state conditions.Some studies of W gross erosion during edge localized modes (ELMs), as well long-range transport in the SOL, are also discussed.
In tokamak divertors, the charge separation in the sheath region occurs over a length scale of the order of the main ion Larmor radius because magnetic field lines intersect the divertor target surfaces at grazing incidence, leading to the formation of a magnetic pre-sheath, or Chodura sheath [3][4][5].As a result of the large width of the sheath near divertor targets, a vast majority of tungsten impurities sputtered from divertor PFMs are ionized multiple times within the sheath and accelerated back toward the divertor surfaces by the sheath electric field over a distance comparable to the Larmor radius, a process known as prompt re-deposition [6][7][8].The complexity of predictive models for tungsten prompt re-deposition and net erosion in divertors has been noted [9][10][11][12][13][14][15].Experimental measurements of tungsten net erosion in divertors of various devices, such as JET [16], DIII-D [17][18][19], ASDEX-Upgrade [20] or WEST [21], are generally well-reproduced by numerical models, e.g., ERO [22].The overall agreement reported between interpretative modeling and experimental data for various plasma configurations suggests that current numerical models effectively include fundamental physics mechanisms governing tungsten prompt re-deposition and net erosion in tokamak divertors.Nevertheless, the predictive capability of these models remains to be demonstrated, as they typically rely upon fitting parameters inferred from experiments.
In this paper, results from a refined theoretical analysis of the prompt re-deposition of tungsten impurities in tokamak divertors [23], which expands upon the original works of Brooks and Fussmann [6,7] and the more recent study by Chankin [24], are discussed.It is argued that W prompt re-deposition is governed by the ratio of the neutral tungsten ionization mean-free path over the sheath width, and a new scaling law for tungsten prompt redeposition is presented.The impact of tungsten prompt re-deposition on the monitoring of tungsten erosion and redeposition in tokamak divertors is also discussed.Monitoring emission associated with the ionization of tungsten impurities in multiple charge states, typically W 3+ , W 4+ and W 5+ , is shown to be necessary to monitor the prompt redeposition and net erosion of W impurities for plasma conditions expected in the far-SOL of the ITER divertor [25].Modification of the electron distribution in the sheath region and transient populations of metastable levels of W atoms are also shown to potentially impact the ionization and emission rates of tungsten and the ionizations per photon (S/XB) coefficients used to determine the flux of impurities from divertor PFMs [26].
Following this discussion, several recent experimental studies conducted at DIII-D to examine the validity of physics parameters governing tungsten prompt re-deposition are presented.These studies use several different methods to infer prompt re-deposition, including in situ passive spectroscopy [27,28] and post-mortem surface analysis techniques [29][30][31].The results are shown to be largely consistent with theoretical models using a minimal number of interpretive fitting parameters.Finally, we describe recent developments advancing the science of high-Z material migration DIII-D, defined here as any W impurity transport occuring within length scales much longer than the sheath width or W gyro-radius [32].We present refined analysis and modelling efforts from some experiments that elucidate the key role of the E × B drifts in explaining the W deposition patterns observed on collector probes in the divertor.

Scaling law for W prompt Re-deposition
The analysis of the prompt re-deposition of tungsten was conducted with the Monte-Carlo code ERO [22], which simulates the three-dimensional trajectories of tungsten impurities physically sputtered from a flat divertor target during gyro-orbits assuming uniform plasma conditions across and along the divertor material surface.Only the sheath electric field and collisions of tungsten impurities with plasma electrons inducing the ionization of tungsten impurities are considered [23].Results from this analysis are therefore applicable when the characteristic lengths of plasma density and temperature are wider than the length scales governing tungsten prompt re-deposition: the characteristic ionization mean-free path of tungsten , iz W 0 l + the tungsten Larmor radius W r and the sheath width , sheath l which are typically smaller than 1 mm in tokamak divertors (figure 1(a)).The profile of the sheath electric field was taken from kinetic simulations [33] that show the electric potential sheath f can be well approximated by an exponential function when the angle of incidence of magnetic field lines onto divertor targets is below 5 °.The electron temperature T e is assumed to be constant in the sheath region, and the electron density in the sheath is given by the Boltzmann relationship.
A conceptual picture of tungsten re-deposition is shown in figure 2(a).Incident ions are accelerated through the magnetic sheath and sputter tungsten atoms from the surface, which subsequently ionize in the sheath.The effect of the sheath electric field on the subsequent trajectories can be characterized by the potential energy of tungsten impurities normalized to their kinetic energy [24].This ratio can be considered as an effective sheath electric field, , ´-~), as discussed in [12].As a consequence, the sheath electric field impacts the trajectories of heavy impurities like tungsten more dramatically than lighter impurities (figure 1(b)).Because of the large inertia of tungsten, their successive ionizations during the first gyro-orbit are not sufficient to significantly reduce the effective electric field since /  1 Z, sheath W crit s s ~where crit s is the effective kinetic energy of the impurity necessary to escape the sheath region.Therefore, these ionizations do not prevent the prompt re-deposition of tungsten impurities that are re-entering or already in the sheath region.Only neutral tungsten impurities which are ionized near or beyond the sheath entrance do not promptly re-deposit, such that tungsten prompt re-deposition is mainly governed by the ratio of the characteristic ionization mean-free path of neutral tungsten impurities to the sheath width.Nonetheless, the reduction of the Larmor radius of tungsten impurities due to the multiple ionizations during their first gyro-orbit remains critical to avoid sheath re-entry.
The fraction 1 prompt -c of tungsten impurities that do not promptly redeposit is thus strongly correlated to the fraction sheath c of neutral tungsten impurities that are ionized within the sheath region and exhibits a squareroot dependency on : Using the analytical expression of sheath c derived in [33], a new robust scaling law is obtained to estimate the prompt re-deposition of tungsten impurities on divertor targets as a function of the governing parameter iz W 0 This newly derived scaling is only parametrized by the cut-off energy of the sputtered tungsten impurity distribution, highlighting the weak dependency of tungsten prompt re-deposition on the nature of the profile of  + ~Therefore, tungsten prompt re-deposition may be significantly reduced and tungsten net erosion significantly increases in fusion devices operating at high magnetic field.However, devices with higher magnetic fields also tend to have higher electron densities, resulting in lower W ionization MFPs, which may partially or fully offset this effect depending upon details of the target regime.

Spectroscopic monitoring of W net erosion
When removable samples are exposed to static (L-mode) plasma conditions, the level of net erosion can be reliably interpreted via post-mortem analysis [29,34].This technique cannot be used, however, to infer timedependent net erosion that occurs during transient events, such as the start-up phase, ELMs, L-H transitions, or disruptions.This motivates the development of in situ techniques to diagnose high-Z net erosion in real time.
Here we present a refinement of a line-ratio spectroscopy technique originally developed on TEXTOR [27] and JET-ILW [28] to infer tungsten re-deposition, and thus net erosion.
The simplified picture of W re-deposition utilized in this work is shown in figure 2(b), where we invoke the standard set of assumptions in the ionizations/photon method [35].Of particular relevance are (a) steady state; (b) the plasma is homogenous in the near-surface region where significant W ionization/re-deposition occurs; (c) W impurities of charge state Z travel with velocity v Z in the ẑ direction (normal to the surface); and (d) other source/sink effects (transport, recombination, charge exchange) are negligible.Note that assumption (b) neglects any variation of electron density, n , e within the pre-sheath region.This may impact the calculation of S/ XB coefficients; this is discussed in the next section.Applying these assumptions to the continuity equation, we arrive at a modified form of the S/XB formulation applicable for all W charge states: where Z G is the W impurity flux away from the surface, S Z is the effective ionization rate coefficient from charge state Z to Z +1, ( ) XB Z,l is the excitation rate coefficient times the branching ratio for a particular electron transition of the impurity of charge Z with characteristic photon emission wavelength , l and is the line-integrated photon emission flux along the ẑ direction.For the case where Z 0, = i.e., W neutrals, it is assumed the negative ion population is negligible ( G = F l recognizable as the standard equation of the / S XB method for diagnosing gross erosion, i.e., .

G = G
Note that we have moved the Z -notation outside the parentheses for brevity.For W ions (Z 1  ), it is assumed that the whole population is ionized along the line of sight of the diagnostic, such that Z G corresponds solely to the re-deposition flux of W Z+ ions to the target surface.Summing over all charge states, the cumulative re-deposition flux, , redep G is therefore: where N must be, at minimum, the highest W charge state for which any significant W re-deposition occurs.To good approximation, this can be assumed equal to Z , prompt the expected W ionization stage within one gyroorbit.The fraction of W ions that are not promptly re-deposited can thus be expressed as: Equation (4) relates the net erosion of tungsten to S/XB coefficients obtained from atomic physics databases and spectroscopic quantities that are directly measurable in experiments.ERO calculations of Z prompt are shown in figure 3(a).For divertor conditions expected in the far-SOL of ITER, for example, monitoring of tungsten prompt re-deposition will require measuring photon emission associated with the ionization of tungsten impurities in charge states Z > 2+, typically W 3+ , W 4+ , or W 5+ .It is noticeable that while typical ITER far-SOL and DIII-D divertor conditions do not overlap, the expected values of Z prompt are similar in both cases.It should also be noted that in practice measured emission from different W charge states is limited by the available spectroscopic hardware.It becomes progressively more difficult to measure W emission as Z increases because the strongest emission lines of a given W charge state move progressively deeper into the ultraviolet region, requiring more specialized detection equipment.

Sheath effects on tungsten S/XB coefficients
Whereas the ionization of low-Z impurities, such as C or Be, emitted from divertor targets typically takes place far away from the sheath region, tungsten impurities sputtered from divertor targets are mainly ionized inside the sheath region, as for attached divertor plasma conditions (figure 1(a)).The variation of the electron distribution in the sheath impacts the ionization and emissions of sputtered impurities in this region [7], and thus the interpretation of spectroscopic measurements of tungsten emissions using S/XB coefficients.Considering standard assumptions for the electron population in the electric sheath near divertor targets (constant electron temperature and electron density given by the Boltzmann relationship), it can be shown for instance that the S/XB coefficients for neutral tungsten impurities are reduced in high-density divertor plasma conditions due to the reduction of the electron density in the sheath (figure 3(b)) [33].
Furthermore, high-Z atoms like tungsten exhibit complex electronic configurations and can have multiple metastable levels with a population comparable to the ground state population.Using excitation rates from a new non-perturbative Dirac R-matrix electron-impact calculation for WI [36], it can be shown that the total value of the ionization coefficient for neutral tungsten is relatively insensitive to changes in the metastable fraction, whereas photo-emission coefficients strongly depend on the distribution of the electron population among metastable states [37].Consequently, populations of the various metastable levels of tungsten atoms can be important in determining an appropriate S/XB coefficient for those plasma conditions [38].Using the Python solver ColRadPy [39] to provide time-dependent solutions of a collisional-radiative model for tungsten neutral sputtered from divertor surfaces, it is shown that the evolution in time of the population of metastable levels of neutral tungsten can also impact the values of the S/XB coefficient [38] (figure 3(c)).Furthermore, as spectral lines are each dominantly driven by a single metastable state, population of metastable states can be experimentally inferred from the measurements of spectral line ratios [37].

Experimental setup: the DIII-D tokamak
The DIII-D tokamak is a medium-to-large scale experimental fusion device with a major radius R = 1.6 m, minor radius a = 0.6 m, maximum on-axis toroidal magnetic field B T = 2.17 T, and typical plasma current ranging from 1.0-1.8MA.The 2016 DIII-D Metal Rings Campaign (MRC) represented the first large-scale use of high-Z plasma facing components (PFCs) in DIII-D.The hardware and diagnostic tools developed for the MRC have been described in previous works [40][41][42] and are only briefly re-described here.As shown in figure 4(a), two toroidally symmetric rings of isotopically distinct W-coated tiles were installed in the DIII-D lower, outer divertor.The W coatings were deposited on slabs of TZM molybdenum alloy, which were then directly retrofit into the existing set of graphite tiles such that the W coatings were flush with the tile surfaces.The two tungsten-coated tiles rows are identified as the 'floor ring,' consisting of natural W (26.5% 182 W) and the 'shelf ring,' containing isotopically enriched W (93% 182 W).By resolving the isotopic ratios in W depositions collected on removable witness samples installed for a particular set of repeat plasma discharges, this first-of-akind approach allowed for detailed tracing of the source location of tungsten material [43].
The Metal Rings Campaign leveraged a wide collection of DIII-D diagnostics, including several key capability enhancements, as shown in figures 4(b)-(c).Four new views for the Multichordal Divertor Spectrometer (MDS) and the filterscope PMT array were installed to diagnose the tungsten gross erosion source via the S/XB method [42].Filtered imaging of the WI 400.9 nm spectral line also enabled characterization of the W source with very high spatial resolution.Two sight lines for a new UV survey spectroscopy (200-400 nm) diagnostic were also installed, allowing for real-time monitoring of a number of WI and WII spectral lines [44].Sets of graphite rods were installed close to the outboard midplane in the far SOL as collector probes to measure the fluence of tungsten in the far-SOL [41], similar to the approach on ASDEX-Upgrade [45].Two new Langmuir probes (LPs) were installed at the major radius of the shelf ring, which in addition to the existing LP array and Divertor Thomson Scattering (DTS) system, allowed for detailed characterization of the divertor and SOL background plasmas during MRC experiments.
In addition, several local W re-deposition and sheath physics experiments described in this work were conducted using the long-standing Divertor Materials Evaluation System (DiMES) [46] platform, also shown in figure 4(c).This removable sample exposure probe is approximately 5 cm in diameter and can be changed on a shot-by-shot basis, allowing for tremendous flexibility in experimental design, as will be discussed below.

Validation of Re-deposition and sheath models 4.1. Assessment of W net erosion using DiMES
To experimentally assess prompt redeposition of tungsten impurities in divertors, a technique has previously been developed on DIII-D to expose two circular tungsten spots of radius R small = 1 mm and R large = 10-15 mm located at the same radial location in the divertor to steady, attached, L-mode plasma conditions using the DiMES system [29][30][31].The strike point is located away from those samples, such that those samples are exposed to plasma conditions characterized by large radial scale lengths relative to the size of the samples.Plasma conditions can be thus be assumed to be uniform across the samples, as well as the gross erosion flux W gross G and the fraction of promptly redeposited tungsten . is the re-deposition ratio introduced in section 2.1.The coefficients f large and f small are geometric factors that account for the fact that the tungsten spots are finite in surface area.As the tungsten spots approach an infinitely large radius, the expected net erosion does not approach zero, but instead reduces to ( ) 1 .
prompt W gross c -G Here we introduce the simple assumption that no W is re-deposited on any portion of the spot that lies within one neutral ionization length, , iz W 0 l + of the downstream edge of the W spot. Sputtered W atoms from the This geometric model, pictured in figure 5(a), allows for a determination of f large and f small using only the radius of the spot (R large or R small ) and the background plasma conditions.Re-arranging equation ( 5) we arrive at an expression for the experimentally inferred prompt re-deposition fraction: Therefore when plasma conditions are sufficiently uniform above the material samples in both toroidal and radial direction, the redeposited tungsten impurities can be well approximated by this simple expression parameterized by only the characteristic neutral W ionziation length, the width of the sheath electric field, and geometric factors.This reduced model formed by equations (5) and (6) was applied to several previously published experiments conducted on DiMES during which many tungsten disks of various radii were exposed to single attached plasma conditions.The ratio of the net erosion rates obtained from two samples of different size exposed to the same plasma conditions calculated with the reduced model are generally in good agreement with experimental values (figure 5(b)).For one data point, however, the predicted value of 1 prompt c is much higher than the experimental measurement.In this experiment, the electron temperature was measured by a Langmuir probe to be quite high (35 eV).Langmuir probes are known to often over-estimate the electron temperature in strongly attached DIII-D divertor plasmas.When Divertor Thomson Scattering (DTS) measurements are available, they can be lower by a factor of two [31].This experimental data was re-analyzed assuming the near-target electron temperature was lower by factor of two (17.5 eV) and the electron density was higher by a factor of 2 to maintain the measured value of ion saturation current, ~This results in a somewhat better match between the reduced model and the experimental measurement, although it does not entirely resolve the discrepancy.We also note that the reduced model lies in good agreement with the ERO predictions.This is expected because the reduced model is derived from a scaling law fit to ERO simulations.The strong correlation between measured tungsten net erosion rates, assessed by exposing tungsten samples of various sizes into the DIII-D divertor, and the predictions made by the reduced model is encouraging.It suggests that the critical parameters governing W prompt redeposition, namely the ionization rates for tungsten and the width of the magnetic pre-sheath, have been properly identified, while using a simpler set of assumptions than reduced models that have been introduced previously [47].

Assessment of W net erosion using spectroscopy
As discussed in section 2, equation (4) provides a method to measure the total net erosion rate of tungsten spectroscopically.For the experiments described here, only measurements of tungsten neutral (W 0 ) and singly ionized (W 1+ ) emission lines were available.As shown in figure 3(a), using Z prompt = 1 in equation (4) will under-estimate the total prompt re-deposition fraction and thus these measurements can be considered a lower The accuracy of this assumption will decrease as Z prompt increases.
W re-deposition was studied in 'piggyback' during the MRC using repeat, L-mode discharges with ∼3 MW auxiliary heating power [48].Example spectra obtained from the shelf ring view of the UV survey spectrometer in the vicinity of the WI 400.9 nm line and WII 364.1 nm line are shown in figures 6(a)-(b).These peaks are relatively small compared to other components of the spectrum, which originate mainly from the D, C, and B radiation, but can be clearly distinguished from other peaks.The signal-to-noise ratio of these spectral lines was generally not high enough to reliably fit them to Gaussians or another shape, so the integrated intensity under the curve was simply used.A background subtraction was performed to remove any offset due to dark current in the detector or other sources of noise.The time evolution of the intensity of these W spectral lines throughout one discharge is shown in figure 6(b).Error bars on these measurements are propogated using the standard deviation of pixel noise acquired from dark frames and assuming Poisson statistics on the detector count rates.The measured intensity of the WI and WII lines increases sharply when the outer strike-point (OSP) is swept on to the shelf ring around 1500 ms, stays roughly constant for the flat-top phase of the discharge, then quickly decays back to zero when the OSP sweeps back off the W ring at 5000 ms.
An absolute calibration of this diagnostic was not available so it was assumed that the relative sensitivity of the detector was equivalent at the WI 400.9 nm wavelength and the WII 364.1 nm wavelength.This implies that an absolute calibration is not necessary to correctly infer the intensity ratio between these two spectral lines.Using equation (4), the inferred values of The W net erosion rate remains fairly constant through the discharge.The measurement is somewhat noisy because it represents the ratio of the intensities of two relatively weak emission lines.
It should be noted that recent measurements from a high-resolution UV spectrometer conducted on the Auburn Compact Toroidal Hybrid (CTH) device indicate that the WII emission peak measured in the spectrum above is contaminated by several smaller WI peaks, located at 364.013 nm, 364.185 nm, and 364.281 nm, respectively.The local plasma conditions in CTH are not measured, but if the same contaminants are also present in DIII-D, this will result in over-estimation of the WII emission, i.e., an over-estimation of the ionization flux of W + ions, and thus an under-estimation of the W + re-deposition flux.The contaminant line emission observed in the 364 nm range on CTH was approximately 30% the measured intensity of the WII 364.1 line, but only 2% of the intensity of the WI 400.9 nm line.The exact level of contamination in DIII-D discharges is not known, but this motivates future investigations to more accurately quantify WII emission.These corrections, while potentially significant, are ignored in the analysis at present.The S/XB values for the WII line at 364.1 nm were also not directly present in the atomic datafile, and instead were inferred from the WI 365.759 nm S/XB by taking the ratio of the A values for the two transitions.This represents an additional source of uncertainty for the WII S/XB values used in this analysis.
We analyze several discharges from the MRC where the divertor plasma conditions were varied due to movement of the OSP across the shelf ring.The inferred values of normalized W net erosion, 1− , prompt c are plotted in figure 7 as a function of two different parametrizations of the W ionization length.For the purposes of the S/XB analysis, the n e and T e measurements from the Langmuir probes must be representative of the region of the plasma from which WI and WII emission is occurring, so only measurements where iz W < l r

and /
iz sheath l l < 3, i.e., where W atoms are ionized relatively close to the surface, are included.The lowest observed values of 1prompt c are in the range of 0.1-0.2,corresponding a reduction of the W net erosion of a factor of 5 to 10 relative to the W gross erosion.
These spectroscopically inferred values of W net erosion can be compared to re-deposition models.Also overlaid in figure 7 are the analytic approximations assuming that W prompt re-deposition is dominated the electrostatic forces of the magnetic pre-sheath [49], described by the scaling law in equation (1), or by gyromotion, as postulated in [8].While both models tend to approximately capture the measured W erosion behavior, the W + gyro-orbit model under-predicts prompt c at low values of / , W iz l r while as discussed above, the measurements are expected to provide an upper limit.In contrast, the pre-sheath model captures the experimental data fairly well when / iz sheath l l » 1, which is expected because W + is the primary charge state undergoing re-deposition in this regime.In the region of strong ionization ( /  iz sheath l l 1), the pre-sheath model tends to under-predict the measured level of net W erosion.As discussed above, this is likely due to the fact that only W + re-deposition is included in the measurement (assumption of Z prompt = 1), whereas the scaling law formulation in equation ( 1) is derived from ERO simulations that include higher W charge states.Carbon micro-spheres with a mean diameter of 8 μm were deposited onto smooth W surfaces of roughness R a < 1 μm.Using DiMES, the samples were exposed to attached L-mode plasmas in the DIII-D lower divertor near the outer strike-point location for a duration of 30 s, sufficient to expect a modification of ∼10% of each carbon micro-sphere due to physical sputtering by deuterium ions and carbon deposition.The shape of those carbon micro-spheres before and after plasma exposure was obtained by secondary electron microscopy (SEM).Stereoscopic SEM implemented by tilting the sample surface allowed for measurement of the preferential direction toward which the micro-sphere was altered through the formation of a 'nose' (figures 9(a)-(b)).Investigation of the inner structure of these noses revealed an interface between the carbon in the micro-sphere bulk and the nose, suggesting that the deposition of C impurities onto the micro-spheres contributes to the formation of these features.It is assumed that the nose formation is due to direct material deposition in the direction of the local incidence angle.Average values of the angles of incidence 〈θ〉 = 62°and 〈j〉 = 53°were obtained from the direction of the noses with an accuracy better than 5°.The value of sheath l inferred from those experimentally measurements using ERO simulations is about 1.2 i r ´(figures 8(b)-(c)), which is very good agreement with the value predicted by kinetic simulations of the Chodura sheath [33].
In a separate experiment, micro-trenches 30 μm × 30 μm × 2-4 μm deep were etched with a focused ion beam (FIB) on a Si surface partially coated with Al.The root-mean-square (RMS) roughness of the Si surface was approximately 10 nm.These samples were also exposed to L-mode deuterium discharges using DiMES [50].Erosion/deposition patterns within those trenches result from the shadowing effects of the trench on incoming plasma particles following magnetic field lines at grazing incidence and are therefore strongly correlated to the polar and azimuthal angles of incidence of impinging plasma particles.The areal impurity aluminum and carbon deposition patterns on the trench floors were measured by energy-dispersive x-ray spectroscopy (EDS) (figures 9(c)-(d)).Deposition profiles in the trenches were simulated with the Monte Carlo code MPR using distributions of polar and azimuthal angles of incidence calculated with a semi-analytical model [51] for various values of the sheath width.The main features of the measured deposition profiles of carbon and aluminum are well reproduced when sheath l » 1.2-1.5 i r ´ [52].
This excellent agreement between the values of sheath l inferred independently from the micro-sphere and micro-trench experiments provides a robust validation of the characteristic sheath structure in tokamak divertors predicted by kinetic simulations.

Assessing W Re-erosion and E × B drift effects
Finally, mixed-material, high-Z impurity migration models were also tested against DIII-D W divertor experiments.Simulations conducted using the coupled DIVIMP-WallDYN codes indicate that E × B drifts should be the dominant driver of tungsten material migration in the DIII-D divertor during L-mode discharges with the ion B × ∇B drift away from the strike-points (unfavorable direction for H-mode access) [53].To benchmark these calculations experimentally during the MRC, graphite witness samples were installed on DiMES, which was located about 5 cm outboard of the shelf ring, in order to diagnose short-range tungsten migration within the lower divertor region.Incorporating E × B drifts, calculated via integrating Ohm's law along parallel field lines [54], was essential to match the W migration features observed experimentally using post-mortem analysis [48].A peak in the W deposition profile appeared many ionization lengths radially outboard of the ring of W-coated divertor tiles.The WallDYN model agrees with experimental data within a factor of two over the entire radial extent of the W re-deposition pattern when E × B drifts are adjusted to 60% of their calculated value.This suggests (a) high-Z material migration involves multiple erosion/re-deposition events before reaching equilibrium, and (b) effects beyond Ohm's law, such as SOL currents or edge turbulence, may play an important role setting the electric field in the divertor region.

Summary
Recent tungsten divertor experiments in the DIII-D tokamak have made significant progress elucidating key mechanisms responsible for high-Z erosion, re-deposition, leakage, and scrape-off-layer (SOL) transport in fusion devices.The net erosion of tungsten PFMs is primarily determined by the prompt redeposition of tungsten impurities, which is expected to be very large in the high-density, partially attached divertor plasma conditions projected in ITER operational divertor scenarios.A complex interplay between the successive ionizations of tungsten impurities sputtered from divertor PFCs and the sheath electric field ultimately determines the prompt redeposition of tungsten in the divertor regions.Key insights into W gross and net erosion physics discussed in this work include the following: (i) Tungsten prompt redeposition is governed by the ratio of the characteristic ionization mean-free path of neutral tungsten over the sheath width, leading to a new scaling law quantifying the prompt redeposition of tungsten impurities in divertors.This scaling implies that tungsten net erosion strongly increases with the strength of the magnetic field if plasma conditions remain constant.In practice, electron density tends to increase with field, which will lead to shorter W ionization lengths and may partially or fully offset this effect.
(ii) In-situ monitoring of tungsten prompt redeposition in divertors may be feasible by the measurement of photon emissions associated with the ionization of tungsten, and has been demonstrated using UV survey spectrometers on DIII-D.However, measurement of impurities in charge states up to Z = 5 may be necessary for sufficiently hot and/or dense divertor plasma conditions, such as those expected in the far-SOL of ITER or other future devices.
(iii) S/XB coefficients for tungsten impurities can be modified in high-density divertor plasma conditions because of the decrease of the electron density in the sheath and the transient populations of metastable states.
(iv) The characteristic sheath structure strongly impacts ion impact trajectories, affecting W sputtering yields and prompt re-deposition probabilities.Fortunately, kinetic models of sheath width are consistent with measurements in the DIII-D divertor using novel micro-structured surface techniques.
Investigations of tungsten prompt redeposition and net erosion in DIII-D experiments are expected to be relevant to predict tungsten erosion and redeposition in ITER, since key parameters governing tungsten prompt redeposition are similar in DIII-D divertor experiments conducted to investigate tungsten erosion and redeposition, and in the far-SOL divertor plasma expected in ITER.While the DIII-D divertor environment does contain carbon, which will not be present in ITER, this does not impact the interpretation of these model validation studies.Dilution of the W surfaces by carbon reduce the W gross and net erosion by the same proportional factor [9], such that the inferred prompt re-deposition ratios remain accurate.
Experimentally validated PMI models on DIII-D help advance the field towards a predictive capability for the high-Z divertor erosion, re-deposition, and leakage in the mixed-material environment of ITER and other future tokamaks.Existing tools can qualitatively predict patterns of long-range material migration, when known physical processes, such as impurity drift effects, are taken into account.However, free parameters remain in these models due to unaccounted processes such as edge turbulence.Nonetheless, it is demonstrated that these various DIII-D experiments provide a robust experimental framework to (i) assess the validity of the physics parameters governing tungsten prompt re-deposition, such as tungsten ionization rates or the sheath width, and (ii) benchmark predictive modeling of tungsten net erosion in tokamak divertors.Such models are essential to develop/optimize mitigation strategies for minimizing tungsten core contamination, and thus maximizing fusion gain, in future devices.
sheath sand is proportional to the impurity mass/charge ratio and the square of the inverse sheath width (

Figure 1 .
Figure 1.(a) Value of the characteristic ionization mean-free path of neutral tungsten over the sheath width as a function of the divertor plasma electron density and temperature.(b) Value of the effective sheath electric field (see section 2.1 for definition) as a function of the impurity charge state.(c) Fraction of tungsten impurities that do not promptly re-deposit as a function of the characteristic ionization mean-free path of neutral tungsten over the sheath width calculated with ERO and estimated via equation (1).

Figure 2 .
Figure 2. (a) Conceptual picture of W sputtering, ionization, gyro-motion, sheath acceleration, and prompt re-deposition.(b) Simplified picture of W re-deposition used to enable spectroscopic analysis.

Figure 3 .
Figure 3. (a) Highest charge state into which tungsten impurities are ionized during their first gyro-orbit as a function of divertor electron temperature and density.(b) S/XB coefficients for the WI 400.9 nm line without (solid lines) and with (dashed lines) the variation of the electron density in the sheath.(c) S/XB coefficients for the WI 400.9 nm line at an electron temperature of 25 eV with the populations of metastable states (dashed black line) and time-resolved metastable states (solid red line) of tungsten neutral atoms included.Corresponding calculations with neither effect included (solid black line) are also shown.

promptc
As a result, the ratio R of the average tungsten net erosion fluxes from the tungsten disks W

Figure 4 .
Figure 4. (a) Photograph of the two rows of W-coated tiles installed for the DIII-D Metal Rings Campaign.(b) Overview of the diagnostic layout for the MRC to diagnose W sourcing and migration.(c) Zoomed view of the DIII-D lower divertor, including DiMES and boundary diagnostics.

Figure 5 .
Figure 5. (a) Depiction of the analytic-geometric model used to infer the fraction of the W-coated spots on which prompt redeposition occurs in DIII-D DiMES experiments.The direction of toroidal magnetic field and ion flow is also shown.(b) Comparison of experimentally inferred values of 1 prompt c from previously published DIII-D experiments[9,30,31] to those predicted by the reduced model in section 2.1.Previously published 3D Monte Carlo modelling from ERO or WBC-REDEP simulations, found in the same papers, are also shown.

4. 3 .
Validation of sheath models using DiMES ERO simulations of trajectories of deuterium and carbon ions impinging onto divertor targets after entering the Chodura sheath indicate that the average polar and azimuthal incidence angles 〈θ〉 and 〈j〉 (figure 8(a)) of those ions onto divertor targets are strongly correlated to sheath l (figures 8(b)-(c)).The sheath width can thus be experimentally inferred from the experimental measurement of the incidence angles of particles striking onto divertor targets.Such measurements of 〈θ〉 and 〈j〉 were performed on DiMES by exposing two types of microstructures: C micro-spheres deposited onto W surfaces and micro-trenches etched into Si surface [50].

Figure 6 .
Figure 6.Example emission spectra obtained in the region of (a) the WII 364.1 nm line and (b) the WI 400.9 nm line from the UV spectrometer viewing the shelf W ring during the MRC.(c) Time evolution of these emission peaks throughout a stationary L-mode discharge.(d) Inferred values of 1prompt c using equation (3) for this same discharge.

Figure 7 .
Figure 7. Inferred values of 1− prompt c for 4 L-mode discharges during the MRC plotted as a function of W ionization length normalized to (a) the width of the magnetic pre-sheath and (b) the W + gyro-radius.Approximations obtained from analytic models are overlaid in both cases, with corresponding references provided.

Figure 8 .
Figure 8.(a) Spherical coordinate system convention used in this analysis.(b) Average polar and (c) azimuthal angles of incidence of carbon and deuterium ions onto divertor target as a function of sheath l calculated with ERO.Experimental values deduced from the shape of carbon micro-spheres are overlaid.

Figure 9 .
Figure 9. Views of carbon micro-spheres before (a) and after (b) exposure, with the radial and toroidal directions indicated.(c) Carbon deposition patterns on the trench floors measured by energy-dispersive x-ray spectroscopy.(d) Scanning electron microscopy image of a micro-trench before plasma exposure.