Plasma-Facing Materials

Plasma-facing materials (PFM) are materials used in components that directly face the high-temperature plasma in fusion reactors. They play a crucial role in maintaining plasma performance while withstanding extremely harsh environments in divertors and first walls. The selection of plasma-facing materials is one of the most critical challenges that determine the feasibility of fusion reactors, and decades of materials research are reflected in current designs.

Required Properties of Plasma-Facing Materials

Plasma-facing materials must simultaneously meet multiple stringent requirements. These requirements often have trade-off relationships, making it difficult for a single material to satisfy all of them.

Heat Resistance

At the divertor strike point, steady-state heat fluxes of 10-20 MW/m$^2$ are received. This corresponds to 15-30% of the solar surface radiation intensity (approximately 63 MW/m$^2$) and represents an extremely harsh heat load condition experienced on Earth. During transient events (ELMs, disruptions), instantaneous heat loads exceeding 1 GW/m$^2$ can occur.

The surface temperature rise $\Delta T$ of a material receiving heat flux $q$ is expressed from Fourier’s law in steady state as:

$$\Delta T = \frac{q \cdot d}{\kappa}$$

where $d$ is the material thickness and $\kappa$ is the thermal conductivity. Under conditions of 10 MW/m$^2$ heat flux and 5 mm material thickness, tungsten ($\kappa$ = 173 W/(m·K)) gives $\Delta T$ ≈ 290 K.

For transient heat loads, the thermal diffusion length $\delta$ becomes an important parameter:

$$\delta = \sqrt{\alpha \cdot \tau}$$

where $\alpha = \kappa / (\rho c_p)$ is the thermal diffusivity ($\rho$: density, $c_p$: specific heat), and $\tau$ is the heat load duration. For ELM-induced heat loads ($\tau$ ~ 0.3 ms), tungsten gives $\delta$ ≈ 40 μm, meaning only a very thin surface layer is heated.

The thermal shock parameter $R$ is used as an indicator of whether a material can withstand thermal shock:

$$R = \frac{\sigma_y (1 - \nu) \kappa}{E \alpha_{\text{th}}}$$

where $\sigma_y$ is the yield stress, $\nu$ is Poisson’s ratio, $E$ is Young’s modulus, and $\alpha_{\text{th}}$ is the coefficient of linear thermal expansion. Materials with larger $R$ have better thermal shock resistance.

Sputtering Resistance

When plasma ions collide with the material surface, surface atoms are ejected through momentum transfer (physical sputtering). Material erosion due to sputtering is one of the main factors limiting reactor lifetime.

The sputtering yield $Y$ (number of atoms ejected per incident ion) depends on the incident ion energy $E$ and angle of incidence $\theta$. According to the Yamamura-Tawara empirical formula:

$$Y(E) = 0.042 \cdot Q(Z_2) \cdot \frac{\alpha^*(M_2/M_1)}{U_s} \cdot \frac{S_n(E)}{1 + \Gamma k_e \epsilon^{0.3}} \cdot \left(1 - \sqrt{\frac{E_{\text{th}}}{E}}\right)^s$$

where $Z_2$, $M_2$ are the atomic number and mass number of the target atom, $M_1$ is the mass number of the incident ion, $U_s$ is the surface binding energy, $S_n$ is the nuclear stopping power, $E_{\text{th}}$ is the threshold energy, and $s$ is the exponent parameter.

The threshold energy $E_{\text{th}}$ is approximated from energy and momentum conservation as:

$$E_{\text{th}} = \frac{(M_1 + M_2)^2}{4 M_1 M_2} \cdot U_s$$

Using the mass ratio $\gamma = 4 M_1 M_2 / (M_1 + M_2)^2$, we get $E_{\text{th}} = U_s / \gamma$. When deuterium ($M_1$ = 2) is incident on tungsten ($M_2$ = 184), $\gamma$ ≈ 0.043 is very small, resulting in a high threshold energy.

The angle of incidence dependence is expressed as:

$$Y(E, \theta) = Y(E, 0) \cdot \cos^{-f}(\theta) \cdot \exp\left[-\Sigma(\cos^{-1}(\theta) - 1)\right]$$

where $f$ and $\Sigma$ are material-dependent parameters. For oblique incidence, the reduction in projected area and increased reflection probability compete, generally resulting in maximum sputtering yield at 60-70°.

Tritium Retention Properties

Excessive accumulation of the fuel tritium (T) in materials poses a safety concern. ITER has set an upper limit of 700 g (approximately 2.5×10$^{17}$ Bq) for tritium accumulation within the vacuum vessel. This constraint is set from both radiation safety and fuel economy perspectives.

The hydrogen isotope concentration $c$ in a material follows Sieverts’ law and is proportional to the square root of the equilibrium pressure $p$:

$$c = K_s \sqrt{p} \cdot \exp\left(-\frac{E_s}{k_B T}\right)$$

where $K_s$ is the Sieverts constant, $E_s$ is the heat of solution, $k_B$ is the Boltzmann constant, and $T$ is the temperature.

The tritium flux $J$ diffusing through a material is expressed by Fick’s first law:

$$J = -D \nabla c$$

The diffusion coefficient $D$ has an Arrhenius-type temperature dependence:

$$D = D_0 \exp\left(-\frac{E_D}{k_B T}\right)$$

where $D_0$ is the frequency factor and $E_D$ is the activation energy for diffusion.

The permeability $\Phi$ is defined as the product of solubility and diffusion coefficient:

$$\Phi = D \cdot K_s = \Phi_0 \exp\left(-\frac{E_\Phi}{k_B T}\right)$$

where $E_\Phi = E_D + E_s$. Higher permeability means tritium passes through the material more easily.

Tungsten

Tungsten (W, atomic number 74) has been adopted as the divertor surface material for ITER. It has the highest melting point of all metals and is the optimal material for high heat load environments.

Physical Properties

Parameter	Value	Notes
Atomic number	74	High-Z metal
Atomic weight	183.84
Melting point	3695 K (3422 °C)	Highest among metals
Boiling point	5828 K (5555 °C)
Density	19.3 g/cm$^3$	Comparable to gold
Thermal conductivity	173 W/(m·K) @ 300 K	About 43% of copper
Specific heat capacity	134 J/(kg·K)
Coefficient of linear thermal expansion	4.5×10$^{-6}$ K$^{-1}$	Low
Young’s modulus	411 GPa	Very high
Poisson’s ratio	0.28
Electrical resistivity	5.3×10$^{-8}$ Ω·m

The thermal conductivity of tungsten depends on temperature and is expressed by the following approximation (300-3000 K):

$$\kappa(T) = 174.9 - 0.1067 T + 5.0 \times 10^{-5} T^2 - 7.8 \times 10^{-9} T^3 \quad [\text{W/(m·K)}]$$

It decreases to approximately 120 W/(m·K) at 1000 K and approximately 100 W/(m·K) at 2000 K.

Advantages

There are numerous reasons why tungsten is an excellent plasma-facing material.

First, its high melting point allows it to withstand extremely high heat loads. It can maintain structural integrity even when divertor surface temperatures reach 2000-2500 K. Sublimation is also negligible below 3000 K. The vapor pressure $p_v$ is expressed by the Clausius-Clapeyron equation:

$$\ln p_v = A - \frac{B}{T}$$

For tungsten, $A$ ≈ 29.5 and $B$ ≈ 92800 K, giving an extremely low vapor pressure of approximately 10$^{-5}$ Pa at 3000 K.

Second, its high atomic number results in a high sputtering threshold energy. The threshold for deuterium incidence is approximately 220-300 eV, and physical sputtering hardly occurs at normal divertor plasma temperatures (5-30 eV). This directly contributes to longer reactor wall lifetime.

Third, hydrogen isotope solubility is extremely low, resulting in low tritium accumulation. The solubility of hydrogen in tungsten is:

$$K_s = 1.3 \times 10^{-3} \exp\left(-\frac{0.90 \text{ eV}}{k_B T}\right) \quad [\text{H/(m}^3\text{·Pa}^{0.5}\text{)}]$$

At 500 K, this is approximately 10$^{-10}$ H/(m$^3$·Pa$^{0.5}$), about 10$^{-6}$ times that of stainless steel.

Recrystallization and Embrittlement

The main challenge with tungsten is recrystallization at high temperatures and the accompanying embrittlement.

Recrystallization is a phenomenon where dislocations accumulated through plastic deformation disappear and new crystal grains form. The recrystallization temperature $T_r$ is approximately 0.3-0.5 times the melting point $T_m$, and for tungsten, it begins at 1200-1500 K (900-1200 °C).

As recrystallization progresses, the following changes occur:

1. Grain coarsening: from initial few μm to several hundred μm
2. Decrease in hardness: loss of work hardening
3. Increase in ductile-brittle transition temperature (DBTT)

DBTT is the temperature at which a material transitions from ductile to brittle fracture. For as-worked tungsten, it is approximately 400 K, but after recrystallization, it can rise to 700-900 K. This means increased risk of brittle fracture during reactor shutdown (room temperature).

The degree of recrystallization progression is expressed by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation:

$$X(t) = 1 - \exp\left[-(k t)^n\right]$$

where $X$ is the recrystallized fraction, $k$ is a temperature-dependent rate constant, and $n$ is the Avrami exponent (typically 2-4). The rate constant $k$ has an Arrhenius-type temperature dependence:

$$k = k_0 \exp\left(-\frac{Q_r}{R T}\right)$$

where $Q_r$ is the activation energy for recrystallization (approximately 500 kJ/mol for tungsten) and $R$ is the gas constant.

To suppress recrystallization, oxide dispersion strengthened (ODS) tungsten is being developed. By dispersing nanoparticles of La$_2$O$_3$, Y$_2$O$_3$, TiC, etc., grain boundary migration is hindered, and the recrystallization temperature can be raised by 200-300 K.

Neutron Irradiation Embrittlement

Fusion reactors produce large amounts of 14.1 MeV fast neutrons, which cause irradiation damage to tungsten.

When neutrons collide with tungsten atoms, primary knock-on atoms (PKA) are generated. The PKA energy $E_{\text{PKA}}$ from collision dynamics is:

$$E_{\text{PKA}} = \frac{4 M_n M_W}{(M_n + M_W)^2} E_n \cos^2\theta$$

where $M_n$ and $M_W$ are the masses of neutron and tungsten respectively, $E_n$ is the neutron energy, and $\theta$ is the scattering angle. For 14.1 MeV neutrons, PKA energies up to approximately 0.3 MeV can be generated.

High-energy PKA sequentially knock out lattice atoms, forming cascade damage. The damage amount is expressed in displacement per atom (dpa), and for DEMO reactor divertors, 5-10 dpa per year is predicted.

The main effects of irradiation damage are:

1. Hardening: formation of obstacles by dislocation loops and voids
2. Embrittlement: increase in DBTT (100-300 K/dpa)
3. Decrease in thermal conductivity: increased phonon scattering due to lattice defects
4. Swelling: volume expansion due to He bubbles

Irradiation hardening is described by Orowan’s equation:

$$\Delta \sigma_y = \alpha M \mu b \sqrt{N d}$$

where $\alpha$ is the obstacle strength coefficient, $M$ is the Taylor factor, $\mu$ is the shear modulus, $b$ is the Burgers vector, $N$ is the defect density, and $d$ is the defect diameter.

Use in ITER

ITER’s divertor employs a full-tungsten design (Full-W Divertor). It consists of 54 divertor cassettes with a total weight of approximately 700 tons.

The divertor cross-section structure is as follows:

1. Tungsten monoblock: approximately 6 mm thick, 23-28 mm wide
2. Copper alloy (CuCrZr) heat sink: transfers heat to cooling water
3. Stainless steel structural material: mechanical support

The tungsten monoblocks are bonded to CuCrZr tubes via a copper interlayer rather than direct bonding. Gaps of approximately 0.5 mm are provided between monoblocks to relieve thermal stress.

The design heat load is 10 MW/m$^2$ in steady state, with instantaneous tolerance up to 20 MW/m$^2$. Transient heat loads from ELMs are planned to be controlled to 0.5 MJ/m$^2$ or less through Type-I ELM suppression techniques.

Beryllium

Beryllium (Be, atomic number 4) is a light metal adopted as the first wall surface material for ITER. Its low atomic number advantage and oxygen gettering effect contribute to maintaining plasma performance.

Physical Properties

Parameter	Value	Notes
Atomic number	4	Low-Z
Atomic weight	9.012
Melting point	1560 K (1287 °C)
Boiling point	2742 K (2469 °C)
Density	1.85 g/cm$^3$	Lightweight
Thermal conductivity	200 W/(m·K) @ 300 K	High
Specific heat capacity	1825 J/(kg·K)	Very high
Coefficient of linear thermal expansion	11.3×10$^{-6}$ K$^{-1}$
Young’s modulus	287 GPa
Poisson’s ratio	0.032	Very low

A notable property of beryllium is the combination of its low density and high specific heat. The volumetric heat capacity $\rho c_p$ is approximately 3.4 MJ/(m$^3$·K), higher than tungsten’s 2.6 MJ/(m$^3$·K), which moderates temperature rise during transient heat loads.

The thermal diffusivity $\alpha$ is:

$$\alpha = \frac{\kappa}{\rho c_p} = \frac{200}{1850 \times 1825} \approx 5.9 \times 10^{-5} \text{ m}^2/\text{s}$$

This is comparable to tungsten (approximately 6.9×10$^{-5}$ m$^2$/s).

Advantages

The greatest advantage of beryllium is its low atomic number. The radiation power loss $P_{\text{rad}}$ from impurities in plasma is dominated by bremsstrahlung for fully ionized ions:

$$P_{\text{rad}} = C_B n_e n_Z Z_{\text{eff}}^2 T_e^{1/2}$$

where $C_B$ ≈ 5.35×10$^{-37}$ W·m$^3$ and $Z_{\text{eff}}$ is the effective charge. Beryllium ($Z$ = 4) contributes orders of magnitude less to radiation loss at the same concentration compared to tungsten ($Z$ = 74).

However, at fusion core temperatures (10-20 keV), complete ionization does not occur, and line radiation must also be considered. The critical impurity concentration $c_{\text{crit}}$ (maximum concentration that can sustain burning) in core plasma is approximately 10% for beryllium and approximately 10$^{-5}$ (0.001%) for tungsten, a significant difference.

Beryllium has an oxygen gettering effect:

$$2\text{Be} + \text{O}_2 \rightarrow 2\text{BeO} \quad (\Delta G < 0)$$

It reacts with residual oxygen and water vapor in the vacuum vessel to form beryllium oxide (BeO), reducing oxygen impurities. This suppresses plasma radiation loss and improves burning efficiency.

Hydrogen isotope retention properties are also relatively favorable. The solubility of hydrogen in beryllium is:

$$K_s = 2.5 \times 10^{20} \exp\left(-\frac{0.21 \text{ eV}}{k_B T}\right) \quad [\text{H/m}^3\text{·Pa}^{0.5}]$$

Compared to carbon, the accumulation amount is smaller, making it appropriate for ITER’s first wall material.

Toxicity and Handling

Beryllium is toxic and requires strict safety management for handling.

Berylliosis is a chronic lung disease caused by inhaling beryllium dust. It is a delayed-type hypersensitivity reaction (Type IV hypersensitivity), and the risk of onset increases in genetically susceptible individuals (approximately 1-6% of the population).

Permissible exposure limits are defined by various national regulations:

• OSHA (USA): 2.0 μg/m$^3$ (8-hour TWA)
• EU: 0.2 μg/m$^3$ (workplace exposure limit)
• Japan Society for Occupational Health: 1 μg/m$^3$ (proposed value)

At ITER, processing and assembly of beryllium components are performed in dedicated facilities (Hot Cells), and workers wear protective clothing and respiratory protection equipment. When beryllium dust is generated inside the vacuum vessel, removal by HEPA filters and surface decontamination are required.

Steam Reaction

Beryllium reacts with steam at high temperatures to generate hydrogen:

$$\text{Be} + \text{H}_2\text{O} \rightarrow \text{BeO} + \text{H}_2 \quad (\Delta H = -286 \text{ kJ/mol})$$

This is an exothermic reaction, and during accidents (LOCA: loss of coolant accident), when beryllium walls are exposed to steam, pressure rise from hydrogen generation and temperature rise from heat release are concerns.

The reaction rate is strongly temperature-dependent and is expressed by the Arrhenius equation:

$$\dot{m} = A \exp\left(-\frac{E_a}{R T}\right) \cdot p_{\text{H}_2\text{O}}$$

where $\dot{m}$ is the reaction rate per unit area, $E_a$ ≈ 100 kJ/mol is the activation energy, and $p_{\text{H}_2\text{O}}$ is the steam partial pressure.

The reaction becomes significant above 600 °C, and ITER safety analyses evaluate hydrogen generation from this reaction. This is one of the reasons why ITER limits beryllium to the first wall (heat load 0.5-2 MW/m$^2$) and uses tungsten for the high heat load divertor region.

Use in ITER

ITER’s first wall consists of approximately 700 blanket modules (First Wall Panels). The surface area is approximately 700 m$^2$, and the total weight of beryllium is approximately 12 tons.

The first wall cross-section structure is:

1. Beryllium tiles: 8-10 mm thick
2. Copper alloy (CuCrZr) heat sink
3. Stainless steel (316LN) structural material

Beryllium tiles are bonded to copper alloy by hot isostatic pressing (HIP). The design heat load is 2 MW/m$^2$ steady state and 4.7 MW/m$^2$ instantaneous.

History and Challenges of Carbon-Based Materials

Carbon-based materials have a long history of use since the early days of fusion research. However, due to tritium accumulation issues, their use is now significantly restricted.

Graphite

Isotropic graphite was widely used in early tokamak devices.

Properties:

• Sublimation temperature: approximately 3900 K (no melting point)
• Thermal conductivity: 100-150 W/(m·K) (isotropic material)
• Density: 1.7-1.9 g/cm$^3$
• Atomic number: 6 (low-Z)

The greatest advantage of graphite is that it has no melting point. Even when surface temperature rises due to high heat load, it does not melt but gradually erodes through sublimation. This reduces the risk of material melt splashing during transient events such as disruptions.

The sublimation rate $\dot{m}_s$ is:

$$\dot{m}_s = \frac{p_v M}{\sqrt{2\pi M R T}} \quad [\text{kg/(m}^2\text{·s)}]$$

where $p_v$ is the vapor pressure, $M$ is the molecular weight, and $R$ is the gas constant. The vapor pressure of graphite is approximately 10 Pa at 3000 K, at which temperature significant sublimation begins.

Carbon Fiber Reinforced Composites (CFC)

CFC is a composite material with carbon fibers oriented in a matrix, enabling high thermal conductivity in one or multiple directions.

Properties:

• Thermal conductivity: 200-500 W/(m·K) (fiber direction)
• Density: 1.6-2.0 g/cm$^3$
• Tensile strength: 100-300 MPa

It was used in major tokamaks such as JT-60U, JET, and ASDEX Upgrade, demonstrating excellent performance in high heat load testing.

The CFC manufacturing process is complex:

1. Carbon fiber preform creation (2D or 3D weaving)
2. Resin or pitch impregnation
3. Carbonization (1000-1500 °C)
4. High-temperature heat treatment (2500-3000 °C) for graphitization
5. Densification cycle repetition

The final density and thermal conductivity depend on the number of densification cycles.

Chemical Sputtering

The greatest disadvantage of carbon materials is chemical sputtering. When hydrogen isotope ions are incident on the carbon surface, chemical reactions produce hydrocarbon molecules such as methane (CH$_4$) and ethylene (C$_2$H$_4$).

The reaction process is complex, but the main pathways are:

$$\text{C} + 4\text{H} \rightarrow \text{CH}_4$$ $$2\text{C} + 4\text{H} \rightarrow \text{C}_2\text{H}_4$$

The chemical sputtering yield $Y_{\text{chem}}$ has temperature dependence and is maximum at approximately 500-600 K (230-330 °C). This temperature dependence is explained by the competition between surface hydride formation and desorption.

$$Y_{\text{chem}}(T) = Y_{\text{max}} \cdot \frac{\exp(-E_1/k_B T)}{1 + \exp(-E_2/k_B T)}$$

where $E_1$ ≈ 0.2 eV is the activation energy for hydrogenation reaction and $E_2$ ≈ 0.7 eV is the activation energy for hydrocarbon desorption.

The chemical sputtering yield at peak temperature can exceed the physical sputtering yield by more than 10 times, significantly accelerating material erosion.

Co-deposition and Tritium Accumulation

Hydrocarbons released by chemical sputtering redeposit on cold surfaces (shadow regions) inside the vacuum vessel. This redeposited layer is called a co-deposited layer of carbon and hydrogen (including tritium) and contains high concentrations of tritium.

The hydrogen/carbon ratio (H/C) in co-deposited layers is approximately 0.2-0.8, and tritium accumulation increases with layer thickness. The deposition rate $\dot{d}$ is:

$$\dot{d} = \frac{\Gamma_{\text{dep}}}{\rho_{\text{layer}}} \quad [\text{m/s}]$$

where $\Gamma_{\text{dep}}$ is the carbon deposition flux and $\rho_{\text{layer}}$ is the deposited layer density. In JET D-T experiments, the tritium accumulation rate due to co-deposited layers was evaluated at approximately 0.5 g-T/shot.

If carbon materials were used in an ITER-scale reactor, the tritium accumulation limit (700 g) would be reached in several hundred shots, which was the main reason for abandoning CFC use in ITER.

Radiation Enhanced Sublimation

Under ion irradiation, sublimation is enhanced at lower temperatures than normal thermal sublimation. This is called radiation enhanced sublimation (RES).

RES becomes significant above approximately 1200 K and increases with ion flux $\Gamma$. The physical mechanism is that ion collisions cause carbon atoms near the surface to gain energy, making thermal desorption easier.

The additional erosion rate due to RES is:

$$Y_{\text{RES}} = Y_{\text{th}} \cdot \left(1 + f_{\text{RES}} \cdot \Gamma^n\right)$$

where $Y_{\text{th}}$ is the thermal sublimation rate, $f_{\text{RES}}$ is the RES coefficient, and $n$ ≈ 0.5 is the flux dependence exponent.

Current Status

Currently, carbon materials are mainly limited to use in experimental devices. In WEST (formerly Tore Supra), JET, and others, CFC was initially used, but conversion to tungsten is progressing.

JET implemented full tungsten-beryllium conversion in 2010-2011 as the “ITER-Like Wall” project. As a result, the tritium accumulation rate was reduced to approximately 1/20 of the CFC era.

Potential of Liquid Metals

The concept of using liquid metals as plasma-facing surfaces is being studied as an innovative option for future fusion reactors.

Advantages of Liquid Metals

Potential advantages of liquid metals compared to solid materials:

1. Self-healing: the surface regenerates as long as liquid is supplied even when eroded
2. Heat transport: convective heat transfer is possible through flow
3. Neutron irradiation: irradiation damage is not a problem (liquids have no crystal structure)
4. High heat load resistance: cooling effect through latent heat of vaporization

The evaporation flux $\Gamma_{\text{evap}}$ from a liquid surface is expressed by the Hertz-Knudsen equation:

$$\Gamma_{\text{evap}} = \frac{\alpha_e p_v}{\sqrt{2\pi m k_B T}}$$

where $\alpha_e$ is the evaporation coefficient (typically 0.1-1) and $m$ is the atomic mass. Evaporation suppresses surface temperature rise, providing a self-regulating function against heat loads.

Lithium

Lithium (Li, $Z$ = 3) is the lightest alkali metal with a low melting point of 454 K (181 °C), making it relatively easy to handle.

Physical properties:

Parameter	Value (liquid, 500 K)
Density	520 kg/m$^3$
Thermal conductivity	43 W/(m·K)
Viscosity	0.48 mPa·s
Surface tension	0.40 N/m
Vapor pressure	~10$^{-6}$ Pa

Lithium has the lowest atomic number among metals, minimizing radiation loss when mixed into plasma. It also has the property of absorbing hydrogen isotopes, and the wall pumping effect can reduce recycling at the plasma edge.

Experiments with lithium coating or liquid lithium have been conducted at NSTX (USA), LTX (USA), FTU (Italy), and improvements in plasma performance have been reported. Specifically:

• 50-100% improvement in energy confinement time
• Reduction in disruption frequency
• Reduction in impurity radiation

A challenge is lithium’s high reactivity. Reaction with moisture and oxygen in air:

$$2\text{Li} + \text{H}_2\text{O} \rightarrow 2\text{LiOH} + \text{H}_2$$ $$4\text{Li} + \text{O}_2 \rightarrow 2\text{Li}_2\text{O}$$

These reactions are exothermic and pose fire risks. Handling liquid lithium requires an inert atmosphere (argon, etc.).

Tin

Tin (Sn, $Z$ = 50) has a moderate atomic number and excellent chemical stability.

Physical properties:

Parameter	Value (liquid, 600 K)
Melting point	505 K (232 °C)
Density	6900 kg/m$^3$
Thermal conductivity	28 W/(m·K)
Viscosity	1.2 mPa·s
Vapor pressure	~10$^{-9}$ Pa

Tin’s vapor pressure is several orders of magnitude lower than lithium’s, offering the advantage of less evaporative contamination into plasma. On the other hand, due to its higher atomic number, radiation loss when contaminated is larger.

Experiments with liquid tin at Magnum-PSI (Netherlands) have confirmed resistance to high heat fluxes (10 MW/m$^2$ and above).

Capillary Force-Driven Systems

Systems utilizing capillary forces have been proposed to stably hold liquid metals on plasma-facing surfaces. Liquid metal is impregnated into porous metal (tungsten mesh, etc.) and supplied by capillary force.

Capillary pressure $\Delta p_c$ is expressed by the Young-Laplace equation:

$$\Delta p_c = \frac{2\gamma \cos\theta}{r}$$

where $\gamma$ is the surface tension, $\theta$ is the contact angle, and $r$ is the pore radius. For the lithium-tungsten combination, $\theta$ < 90° (good wettability), and capillary force supply can be expected.

Flow rate $Q$ from the Hagen-Poiseuille equation:

$$Q = \frac{\pi r^4}{8\eta L} \Delta p$$

where $\eta$ is the viscosity, $L$ is the flow path length, and $\Delta p$ is the driving pressure. If pores are too small, flow rate is insufficient; if too large, capillary pressure is insufficient, requiring optimal design.

Physics of Sputtering

Sputtering is the most fundamental process in plasma-wall interaction and is the main mechanism of material erosion.

Mechanism of Physical Sputtering

Physical sputtering is caused by momentum transfer from incident ions to target atoms.

When an incident ion (mass $M_1$, energy $E_0$) collides with a target atom (mass $M_2$), the energy $E_T$ acquired by the target atom from elastic collision dynamics is:

$$E_T = \frac{4 M_1 M_2}{(M_1 + M_2)^2} E_0 \cos^2\phi = \gamma E_0 \cos^2\phi$$

where $\phi$ is an angle dependent on the collision parameter and $\gamma$ is the energy transfer coefficient.

For a surface atom to be ejected, the acquired energy must exceed the surface binding energy $U_s$:

$$E_T > U_s$$

From this, the threshold energy becomes $E_{\text{th}} \approx U_s / \gamma$.

Sputtering parameters for representative materials:

Material	$U_s$ (eV)	$E_{\text{th}}$ for D (eV)
Be	3.4	9
C	7.4	12
Fe	4.3	26
Mo	6.8	60
W	8.7	220

Energy Distribution and Angular Distribution

The energy distribution $f(E)$ of sputtered atoms follows the Thompson distribution:

$$f(E) \propto \frac{E}{(E + U_s)^3}$$

This distribution peaks at energy $E = U_s / 2$, and the average energy is approximately $2U_s$. For tungsten, the average sputtered atom energy is approximately 17 eV.

The angular distribution approximately follows a cosine distribution ($\cos\theta$), but concentration toward the normal direction is observed for low-energy incidence.

Details of Chemical Sputtering

Chemical sputtering is a process in which volatile molecules are produced by chemical reactions between incident particles and surface atoms. It is most prominent in the carbon-hydrogen system and accelerates erosion of carbon materials.

Reaction process:

1. Incidence and implantation of hydrogen ions
2. Formation of carbon-hydrogen bonds (C-H, CH$_2$, CH$_3$, etc.)
3. Diffusion near the surface
4. Desorption of volatile hydrocarbons (CH$_4$, C$_2$H$_2$, etc.)

The temperature dependence of chemical sputtering yield is determined by competition between hydrogenation and desorption reactions.

Low temperature region ($T$ < 400 K): Hydrogenation proceeds but hydrocarbon desorption is rate-limiting Medium temperature region (400-700 K): Both reactions are active, maximum sputtering yield High temperature region ($T$ > 700 K): Sputtering yield decreases due to reduced surface hydrogen concentration

The Roth model describes the chemical sputtering yield as:

$$Y_{\text{chem}} = \frac{Y_{\text{max}} \exp\left(-\frac{E_1}{k_B T}\right)}{\left[1 + \frac{k_1}{k_2}\exp\left(\frac{E_2 - E_1}{k_B T}\right)\right]}$$

where $E_1$ ≈ 0.2 eV, $E_2$ ≈ 0.7 eV, and $Y_{\text{max}}$ ≈ 0.03.

Self-Sputtering

When sputtered atoms are ionized in the plasma and re-incident on the material surface, self-sputtering occurs. If the self-sputtering yield $Y_{\text{self}}$ exceeds 1, self-sustaining erosion occurs, leading to uncontrolled material loss.

For tungsten, the self-sputtering threshold is approximately 100-200 eV, and the energy at which $Y_{\text{self}}$ = 1 is approximately 400-500 eV. Under normal divertor plasma conditions, these values are not reached, but caution is needed during disruptions.

Neutron Irradiation Damage

D-T fusion reactions produce 14.1 MeV fast neutrons, which cause irradiation damage to plasma-facing materials.

Cascade Damage

When high-energy neutrons collide with lattice atoms, primary knock-on atoms (PKA) are generated. PKA sequentially knock out surrounding atoms, forming cascade damage.

The number of Frenkel pairs (vacancy-interstitial atom pairs) generated in a cascade $N_d$ is estimated by the modified Kinchin-Pease model:

$$N_d = \frac{0.8 T_{\text{dam}}}{2 E_d}$$

where $T_{\text{dam}}$ is the damage energy (PKA energy minus that consumed in electronic excitation) and $E_d$ is the displacement threshold energy (approximately 90 eV for tungsten).

Damage amount is expressed in displacement per atom (dpa):

$$\text{dpa} = \int \sigma_d(E) \phi(E) dE \cdot t$$

where $\sigma_d$ is the displacement cross-section, $\phi(E)$ is the neutron flux spectrum, and $t$ is the irradiation time.

ITER’s divertor is predicted to receive approximately 0.5 dpa per year, while DEMO is predicted at 5-10 dpa/year.

Irradiation Hardening and Embrittlement

Point defects (vacancies, interstitial atoms) formed by irradiation cluster through diffusion, forming dislocation loops, voids, and precipitates. These defects inhibit dislocation motion and harden the material.

The irradiation hardening amount $\Delta \sigma_y$ is described by the dispersed barrier model:

$$\Delta \sigma_y = \alpha M \mu b \sqrt{\sum_i N_i d_i}$$

where $i$ is the defect type, $N_i$ is the defect density, and $d_i$ is the defect diameter.

In tungsten, DBTT increases with irradiation hardening. The DBTT of unirradiated tungsten is approximately 400 K, but after 1-5 dpa irradiation, it has been reported to rise to 700-1000 K. This means increased risk of brittle fracture during reactor shutdown and startup.

Helium Generation and Bubble Formation

14.1 MeV neutrons undergo nuclear reactions with material atoms, generating helium and hydrogen.

Main reactions in tungsten:

$$^{184}\text{W} + n \rightarrow ^{181}\text{W} + n + 3n + \text{H}$$ $$^{186}\text{W} + n \rightarrow ^{183}\text{W} + \alpha$$

The helium generation rate is approximately 0.5-1 appm/dpa (appm = atomic parts per million) under DEMO conditions.

Helium has extremely low solubility in metals and is either trapped at interstitial positions or combines with vacancies to form He-vacancy clusters. At high-temperature irradiation, these grow into helium bubbles, and when accumulated at grain boundaries, they cause grain boundary embrittlement.

The internal pressure $p$ of helium bubbles from the equation of state of gases:

$$p = \frac{n k_B T}{\frac{4}{3}\pi r^3 - n b}$$

where $n$ is the number of helium atoms in the bubble, $r$ is the bubble radius, and $b$ is the excluded volume of helium.

The critical stress (stress at which bubbles cause grain boundary cracking) $\sigma_c$ is:

$$\sigma_c = \frac{2\gamma_{gb}}{r} - p$$

where $\gamma_{gb}$ is the grain boundary energy. As helium concentration increases, this critical stress decreases, increasing the risk of high-temperature helium embrittlement (HTHE).

Irradiation Swelling

The formation of voids and helium bubbles causes material volume increase. The volume change rate (swelling) $\Delta V / V$ is:

$$\frac{\Delta V}{V} = \frac{4}{3}\pi \sum_i N_i r_i^3$$

Swelling of 10-20%/year has been observed in austenitic steels, but for tungsten, it is generally predicted to be less than 1%. However, there are many unknowns about swelling behavior under high-temperature irradiation conditions (0.4-0.6 $T_m$), and research continues.

Tritium Retention and Permeation

The behavior of tritium in materials is important from both safety and fuel economy perspectives for fusion reactors.

Implantation and Diffusion

When tritium ions from plasma are incident on materials, they are first implanted near the surface (within the ion range). The ion range $R_p$ is calculated by TRIM/SRIM codes, and for 10-100 eV ions, it is on the order of several nm.

Implanted tritium moves into the material interior by interstitial diffusion. The diffusion equation:

$$\frac{\partial c}{\partial t} = D \nabla^2 c - \frac{c}{\tau_t} + S$$

where $c$ is the tritium concentration, $D$ is the diffusion coefficient, $\tau_t$ is the trap time, and $S$ is the implantation source.

In steady state, diffusion flux and implantation rate are in equilibrium:

$$J_{\text{diff}} = D \frac{dc}{dx} = \Gamma_{\text{impl}} (1 - R)$$

where $\Gamma_{\text{impl}}$ is the incident flux and $R$ is the reflection coefficient.

Trap Sites

Defects (vacancies, dislocations, grain boundaries, impurity atoms) exist in materials and capture hydrogen isotopes. The larger the binding energy $E_b$ of hydrogen at trap sites, the more difficult desorption becomes.

Representative trap energies (in tungsten):

Trap site	$E_b$ (eV)
Interstitial (dissolved state)	0
Single vacancy	1.0-1.4
Vacancy cluster	1.5-2.0
Dislocation	0.5-0.8
Grain boundary	0.5-1.0
He-vacancy complex	2.0-3.0

Trapping and detrapping are described by the kinetic equation:

$$\frac{dn_t}{dt} = \nu_c c (N_t - n_t) - \nu_0 n_t \exp\left(-\frac{E_b}{k_B T}\right)$$

where $n_t$ is the trap occupation concentration, $N_t$ is the trap density, and $\nu_c$, $\nu_0$ are the frequency factors for trapping and detrapping.

When vacancies are generated by irradiation, trap density increases and tritium accumulation increases. For ITER, it has been pointed out that accumulation may increase 10-100 times after 1 dpa irradiation.

Permeation

The tritium flux $J_p$ permeating through a material is:

$$J_p = \frac{\Phi \Delta p^{0.5}}{d}$$

where $\Phi$ is the permeability, $\Delta p$ is the pressure difference, and $d$ is the material thickness.

Permeability has an Arrhenius-type temperature dependence:

$$\Phi = \Phi_0 \exp\left(-\frac{E_\Phi}{k_B T}\right)$$

Permeability of representative materials (at 500 °C):

Material	$\Phi$ (mol/(m·s·Pa$^{0.5}$))
W	~10$^{-18}$
SS316	~10$^{-13}$
Cu	~10$^{-12}$
Be	~10$^{-15}$

Tungsten has extremely low permeability and also functions as a tritium barrier.

Permeation Barrier Coatings

To suppress tritium permeation to cooling systems, technology for applying barrier coatings to structural material surfaces has been developed.

Representative barrier materials:

• Al$_2$O$_3$ (alumina): permeation reduction factor (PRF) 100-1000
• Er$_2$O$_3$ (erbium oxide): PRF 100-1000
• TiN / TiC: PRF 10-100

The barrier effect is evaluated by:

$$\text{PRF} = \frac{\Phi_{\text{substrate}}}{\Phi_{\text{coated}}}$$

However, if coating film defects (cracks, pinholes) exist, the effect is significantly reduced. Bypass permeation flux through defects is:

$$J_{\text{bypass}} = \frac{\Phi_{\text{substrate}} \Delta p^{0.5}}{d} \cdot f_d$$

where $f_d$ is the defect area fraction. Even with $f_d$ = 10$^{-4}$, PRF is reduced to approximately 100.

Plasma-Wall Interaction

Interaction between plasma and wall materials not only affects material erosion but also significantly impacts plasma performance.

Recycling

A portion of fuel ions (D, T) incident on walls are re-released from walls. The ratio of re-release to incidence is called the recycling coefficient $R_{\text{rec}}$:

$$R_{\text{rec}} = \frac{\Gamma_{\text{out}}}{\Gamma_{\text{in}}}$$

At high recycling ($R_{\text{rec}}$ → 1), high-density plasma is maintained by fuel supply from walls, but edge plasma control becomes difficult. At low recycling ($R_{\text{rec}}$ < 1), plasma control becomes easier, but external fuel supply is required.

Low recycling through lithium coating has confirmed confinement improvement effects in experiments at NSTX and others.

Impurity Transport

Impurity atoms released by sputtering are ionized in the plasma and transported along magnetic field lines. The efficiency at which impurities generated in the divertor penetrate to the core region is:

$$\eta_{\text{imp}} = \frac{\Gamma_{\text{core}}}{\Gamma_{\text{source}}}$$

High-Z impurities such as tungsten are strongly bound to field lines due to high ionization degree, resulting in low penetration efficiency to the core. However, once they penetrate the core, even trace amounts affect plasma performance due to high radiative cooling rates.

The impurity screening coefficient $S$ is:

$$S = \frac{1}{\eta_{\text{imp}}} \propto \tau_{\parallel} Z^{0.5-1}$$

where $\tau_{\parallel}$ is the parallel transport time. The lower the temperature and higher the density in the divertor region, the greater the screening effect.

Plasma-Induced Erosion

Under plasma irradiation, special material erosion processes occur that do not happen in normal thermal equilibrium.

Arcing is a localized discharge phenomenon that occurs at surface oxide layers or impurity layers. Formation of arc traces (cathode spots) releases large amounts of material locally.

Melting and evaporation occur during high heat loads from ELMs or disruptions. When surface temperature exceeds the melting point, a melt layer forms, and droplets are scattered by $\mathbf{J} \times \mathbf{B}$ forces or evaporation recoil.

Brittle fracture occurs due to thermal shock below DBTT. When thermal stress from rapid temperature change exceeds material strength, cracks initiate and propagate.

Mixed Layer Formation

In environments where different materials (e.g., W and Be) coexist, mixed layers form on material surfaces.

When beryllium atoms deposit on tungsten surfaces, W-Be mixed layers form through interdiffusion. This mixed layer has properties different from pure tungsten:

• Lower melting point: W-Be system eutectic point is approximately 1500 K
• Changes in sputtering yield
• Changes in tritium retention properties

In ITER, beryllium from the first wall is predicted to be transported to the tungsten surface of the divertor, forming mixed layers. Based on experience from JET-ILW, this mixed layer thickness is estimated at several μm.

Research Trends in Future Materials

Advanced materials with properties exceeding current materials are being developed for fusion reactors beyond DEMO.

Tungsten Alloys

To improve the challenges of pure tungsten (recrystallization, embrittlement), various alloy systems are being studied.

W-Re alloys improve ductility and lower DBTT through rhenium addition. However, under neutron irradiation, Re is produced by transmutation, raising concerns about long-term property changes.

$$^{186}\text{W} + n \rightarrow ^{187}\text{W} \xrightarrow{\beta^-} ^{187}\text{Re}$$

W-Ta alloys and W-V alloys are also being studied, but many unknowns remain regarding behavior under irradiation.

Oxide Dispersion Strengthened (ODS) Tungsten

ODS tungsten with dispersed oxide nanoparticles of La$_2$O$_3$, Y$_2$O$_3$, etc. is effective for recrystallization suppression and high-temperature strength improvement.

The pinning effect by dispersed particles is described by the Zener equation:

$$D_{\text{lim}} = \frac{4 r}{3 f_v}$$

where $D_{\text{lim}}$ is the limiting grain size at which grain growth stops, $r$ is the dispersed particle diameter, and $f_v$ is the volume fraction. For dispersed particles with 50 nm diameter and 1% volume fraction, $D_{\text{lim}}$ ≈ 7 μm, suppressing coarsening of recrystallized grains.

Ceramic Materials

SiC/SiC composites are attracting attention as structural materials for future fusion reactors due to their low activation properties and excellent high-temperature strength.

SiC properties:

• Density: 3.2 g/cm$^3$ (lightweight)
• Melting point: approximately 2800 K (decomposition temperature)
• Thermal conductivity: 100-200 W/(m·K)
• Swelling under irradiation: low (less than 1%)

There are also concepts for using SiC as plasma-facing material, but there are concerns about chemical reactions with hydrogen isotopes (chemical sputtering).

High-Entropy Alloys

High-entropy alloys (HEA) containing multiple main elements at equal concentrations are expected to have properties exceeding conventional alloys.

For example, W-Ta-Cr-V system HEA has been reported to exhibit high high-temperature strength and radiation resistance. Lattice strain from mixing entropy $\Delta S_{\text{mix}}$ may suppress point defect diffusion and promote irradiation damage recovery.

$$\Delta S_{\text{mix}} = -R \sum_i x_i \ln x_i$$

where $x_i$ is the mole fraction of each element. For equal concentrations of 4 elements, $\Delta S_{\text{mix}}$ = 1.39$R$.

Materials Testing Facilities

Dedicated testing facilities are being constructed to evaluate material behavior under fusion reactor conditions.

IFMIF-DONES (Demo Oriented Neutron Source) is an accelerator-driven neutron source planning to conduct material testing at irradiation rates of 20-50 dpa/year.

ITER will have a materials test station (Toroidal Field Materials Test Station) installed inside the vacuum vessel to evaluate material behavior in actual plasma environments.

Data from these facilities will build the materials database essential for DEMO design.

Related Topics

• ITER Project - ITER plasma-facing components
• Tokamak Configuration - Explanation of divertor configuration
• Materials and Engineering: Overview - Overview of fusion reactor materials
• Structural Materials - Materials for blanket and vacuum vessel