Divertor

The divertor is a critical component in fusion reactors that handles heat and particles from the plasma while controlling impurities. By engineering the magnetic field configuration, plasma particles are directed to specific locations where exhaust and heat management can be efficiently performed. The divertor represents one of the most challenging technical issues in achieving practical fusion reactors, involving a complex interplay of materials engineering, plasma physics, and thermal-hydraulic engineering.

Basic Concepts of the Divertor

Historical Development

The divertor concept was first proposed by Lyman Spitzer in the 1950s. Early tokamaks used limiters—protruding wall surfaces that defined the plasma boundary—but the problem of impurity contamination led to a transition toward magnetic divertors.

Research on divertor configurations began in earnest with large devices such as JT-60 (Japan) and JET (Europe) in the 1970s, and since the 1990s, compatibility with H-mode operation has become a major research theme. Today, virtually all tokamak devices employ divertor configurations.

Fundamental Roles of the Divertor

The divertor has three primary functions.

Impurity Control

When plasma particles and neutrals collide with plasma-facing walls, sputtering ejects wall material particles. If these impurities enter the plasma, low-Z (light element) impurities dilute the fuel, while high-Z (heavy element) impurities increase radiation losses and drain energy from the plasma.

The radiative power loss due to impurities is expressed using the radiation power coefficient $L_Z(T_e)$:

$$P_{\text{rad}} = n_e n_Z L_Z(T_e)$$

where $n_e$ is the electron density, $n_Z$ is the impurity density, and $T_e$ is the electron temperature. The radiation power coefficient depends strongly on atomic number $Z$, and radiation intensity in high-temperature plasmas ($T_e \sim 1$ keV) increases dramatically with increasing $Z$. For example, the radiation power coefficient of tungsten ($Z = 74$) is approximately 100 times that of carbon ($Z = 6$) at the same electron temperature.

The sputtering yield $Y$ depends on ion energy $E_i$ and incidence angle $\theta$, approximated by the empirical formula:

$$Y(E_i, \theta) = Y_0(E_i) \cdot \cos^{-f}(\theta) \cdot \exp\left[f(1 - \cos^{-1}\theta) \cos \theta_{\text{opt}}\right]$$

where $Y_0$ is the sputtering yield at normal incidence, $f$ is a material-dependent parameter, and $\theta_{\text{opt}}$ is the angle at which maximum sputtering occurs. The sputtering threshold energy for tungsten is approximately 200 eV for deuterium ions, and reducing the plasma temperature near the divertor plate below this value significantly suppresses effective sputtering.

Since the divertor is located away from the main plasma, the amount of impurities generated at the divertor that enter the main plasma can be suppressed. The impurity screening efficiency $\eta_s$ quantifies this effect:

$$\eta_s = 1 - \frac{\Gamma_{\text{imp,core}}}{\Gamma_{\text{imp,div}}}$$

where $\Gamma_{\text{imp,core}}$ is the impurity flux to the core plasma and $\Gamma_{\text{imp,div}}$ is the impurity flux generated at the divertor. Well-designed divertor configurations achieve $\eta_s > 0.99$.

Helium Ash Exhaust

Alpha particles ($^4\text{He}$) produced by DT reactions heat the plasma and then slow down and accumulate within it. This helium ash dilutes the fuel and must be efficiently exhausted.

For fusion power $P_f$ (MW), the helium particle production rate $\dot{N}_{\text{He}}$ (particles/s) is:

$$\dot{N}_{\text{He}} = \frac{P_f}{E_\alpha} = \frac{P_f}{3.52 \times 10^6 \times 1.60 \times 10^{-19}} \approx 1.78 \times 10^{18} P_f$$

where $E_\alpha = 3.52$ MeV is the alpha particle energy. To sustain fusion power, an exhaust rate equivalent to this production rate is required.

The relationship between helium concentration and fuel dilution is expressed as follows. When helium concentration is $f_{\text{He}} = n_{\text{He}}/n_e$, the effective fuel density is:

$$n_{\text{DT,eff}} = n_e (1 - 2f_{\text{He}})$$

Since fusion power is proportional to $n_{\text{DT}}^2$, a helium concentration of 10% results in approximately 36% power reduction. Steady-state fusion reactors target $f_{\text{He}} < 5\%$.

The helium residence time $\tau_{\text{He}}^*$ is a parameter representing helium exhaust efficiency:

$$\tau_{\text{He}}^* = \frac{N_{\text{He}}}{\dot{N}_{\text{He}}}$$

This is often discussed as the ratio to the energy confinement time $\tau_E$: $\rho_{\text{He}} = \tau_{\text{He}}^*/\tau_E$, where $\rho_{\text{He}} < 5$ is considered a condition for self-heated plasma.

Exhaust pumps installed in the divertor region remove these helium particles from the reactor. From the pumping speed $S$ (m$^3$/s) and neutral particle pressure $p$ (Pa), the exhaust particle flux is:

$$\Gamma_{\text{pump}} = \frac{S \cdot p}{k_B T_n}$$

where $T_n$ is the neutral particle temperature. ITER is designed with a pumping speed of approximately 100 m$^3$/s at the main divertor.

Heat Handling

Heat transported from the main plasma through the scrape-off layer is received by the divertor plates and transferred to the cooling system. In fusion reactors, this heat load is extremely high, making it the greatest challenge in divertor design.

The heat flowing into the divertor is the heating input $P_{\text{heat}}$ minus radiation losses:

$$P_{\text{div}} = P_{\text{heat}} - P_{\text{rad,core}} - P_{\text{rad,SOL}}$$

In fusion reactors, $P_{\text{heat}} \sim 100$ MW or more, and this heat must be handled over an area of only a few m$^2$.

Magnetic Configuration and Divertor Geometry

Scrape-Off Layer and Separatrix

Tokamak plasmas are divided into closed and open magnetic surfaces. The outermost closed magnetic surface is called the separatrix, and the plasma outside it is called the scrape-off layer (SOL).

The separatrix is defined as a contour of the poloidal flux function $\psi$. The tokamak magnetic field is:

$$\mathbf{B} = B_\phi \hat{\phi} + B_p \hat{\theta} = \frac{1}{R}\nabla \psi \times \hat{\phi} + B_\phi \hat{\phi}$$

On the separatrix, $\psi = \psi_{\text{sep}}$ is constant, and this value is controlled by divertor coil currents.

Plasma in the scrape-off layer is transported along field lines and eventually reaches the divertor plates. Plasma particles are neutralized at the divertor plates and released into the divertor chamber as neutral particles. These released neutrals are re-ionized and return toward the divertor plates in a recycling process, reaching a pressure equilibrium.

X-Point and Poloidal Divertor

The point where the separatrix is formed is called the X-point (null point). At this point, the poloidal magnetic field becomes zero:

$$B_p = |\nabla \psi|/R = 0$$

and field lines cross in an X-shaped pattern. The poloidal magnetic field near the X-point is:

$$B_p \approx B_{px} \cdot \left(\frac{r}{a}\right)$$

where $B_{px}$ is a parameter related to the magnetic field gradient at the X-point.

The position of the X-point is controlled by the arrangement of poloidal field coils to form the divertor configuration. The X-point position $(R_X, Z_X)$ is determined by the ratio of divertor coil current $I_{\text{div}}$ to plasma current $I_p$.

Single Null Configuration (SND)

The single null configuration is the most basic divertor configuration with only one X-point. Usually, the X-point is placed at the bottom (lower single null, LSN), making it easier to support the divertor structures with gravity.

The poloidal flux distribution in the single null configuration is expressed as a superposition of magnetic fields from the plasma current and divertor coil currents. The poloidal flux from the plasma current is:

$$\psi_p(R, Z) = \frac{\mu_0 I_p}{2\pi} \sum_{n=0}^{\infty} a_n P_n(\cos\alpha)$$

where $P_n$ are Legendre polynomials and $\alpha$ is the angle from the magnetic axis.

Characteristics of the single null configuration:

• Simple structure, easy to manufacture and maintain
• Good access to divertor plates
• Asymmetric heat load distribution between inner and outer divertors
• Heat load on the outer divertor is approximately twice that on the inner

The heat load asymmetry arises from field line geometry. On the outer (low-field) side, field lines spread more widely, so the heat flux reaching the same area is reduced, but considering toroidal symmetry, the total heat load is distributed more toward the outer side.

Double Null Configuration (DND)

The double null configuration is a symmetric configuration with X-points at both top and bottom.

$$\psi(R, Z) = \psi(R, -Z)$$

Due to this symmetry, both X-points have the same flux value.

Advantages of the double null configuration:

• Heat load is distributed among four divertor legs (upper and lower)
• Heat load on each divertor is approximately half that of SND
• H-mode confinement performance may be better in some cases
• Good symmetry of bootstrap current distribution

Challenges of the double null configuration:

• Difficult to control the positions of two X-points
• Heat load balance between upper and lower divertors is unstable
• Remote maintenance is complex
• Difficult access to the upper divertor

The magnetic balance $\sigma_{\text{DND}}$ is defined as:

$$\sigma_{\text{DND}} = \frac{\psi_{\text{X,up}} - \psi_{\text{X,low}}}{\delta\psi}$$

where $\delta\psi$ is the flux difference between the separatrix and magnetic axis. Maintaining $|\sigma_{\text{DND}}| < 0.01$ is necessary for stable double null operation.

ITER employs a single null divertor from the perspective of reactor structure and remote maintenance. However, double null configurations are being considered for future power plants to distribute heat loads.

Scrape-Off Layer Physics

Basic SOL Parameters

The scrape-off layer (SOL) is the region outside the separatrix where plasma exists. The behavior of SOL plasma is a key factor determining divertor performance.

The SOL width is characterized by the heat flux decay length $\lambda_q$. This represents the decay of heat flux with distance $r$ from the separatrix at the midplane (at the height of the magnetic axis):

$$q_\parallel(r) = q_{\parallel,0} \exp\left(-\frac{r}{\lambda_q}\right)$$

The Eich scaling is widely used as an empirical scaling:

$$\lambda_q = 1.35 \times B_p^{-1.05} [\text{mm}]$$

where $B_p$ is the poloidal magnetic field at the midplane (T). For ITER, $\lambda_q \approx 1$ mm is predicted, meaning large power is concentrated in a very narrow heat flux channel.

Parallel and Perpendicular Transport

Transport in SOL plasma differs greatly between the parallel (along field lines) and perpendicular directions.

Parallel heat conduction is described by Spitzer electron thermal conduction:

$$q_\parallel^e = -\kappa_\parallel^e \nabla_\parallel T_e = -\kappa_0 T_e^{5/2} \nabla_\parallel T_e$$

where $\kappa_0 \approx 2000$ W/(m$\cdot$eV$^{7/2}$). In high-temperature plasma, heat conduction is very efficient, and temperature gradients along field lines are small.

When parallel heat conduction becomes too high, flux limiting occurs:

$$q_\parallel^{\text{FL}} = \alpha_e n_e c_s T_e$$

where $c_s = \sqrt{T_e/m_i}$ is the sound speed and $\alpha_e \sim 0.1$–$0.3$ is the heat flux limiting coefficient.

Perpendicular transport is dominated by turbulence and described by anomalous diffusion coefficient $D_\perp$:

$$\Gamma_\perp = -D_\perp \nabla_\perp n + n V_{\perp}$$

Typical values are $D_\perp \sim 0.1$–$1$ m$^2$/s, more than two orders of magnitude larger than classical values.

Two-Point Model

The two-point model is a simplified model for analyzing SOL temperature and density distributions. It relates plasma parameters at two points: the midplane (upstream, u) and the divertor plate (target, t).

From pressure balance:

$$n_u T_u = 2 n_t T_t$$

where the factor of 2 accounts for sheath effects at the divertor plate.

From energy balance, integrating along field lines:

$$q_\parallel L_\parallel = \int_0^{L_\parallel} \kappa_0 T^{5/2} \frac{dT}{ds} ds = \frac{2\kappa_0}{7}(T_u^{7/2} - T_t^{7/2})$$

where $L_\parallel$ is the connection length (field line length from midplane to divertor), typically tens of meters.

For $T_t \ll T_u$ (conduction-limited regime):

$$T_u = \left(\frac{7 q_\parallel L_\parallel}{2\kappa_0}\right)^{2/7}$$

Combined with particle balance, the relationship between divertor temperature and midplane density is obtained:

$$T_t \propto \frac{q_\parallel^2}{n_u^2}$$

This relationship shows that divertor temperature can be lowered by increasing density, forming the basis for detachment operation.

Sheath Boundary Conditions

When plasma contacts the divertor plate, electrons being faster than ions cause the divertor plate to charge negatively, forming a potential drop (sheath).

The sheath potential drop is:

$$\phi_s = \frac{T_e}{2e}\ln\left(\frac{m_i}{2\pi m_e}\right) \approx 3 T_e$$

Ion flow velocity at the sheath boundary must be at least the sound speed (Bohm criterion):

$$v_i \geq c_s = \sqrt{\frac{T_e + \gamma T_i}{m_i}}$$

where $\gamma$ is the ion adiabatic index.

Heat flux to the divertor plate uses the sheath transmission coefficient $\gamma_{\text{sh}}$:

$$q_t = \gamma_{\text{sh}} n_t T_t c_{s,t}$$

where $\gamma_{\text{sh}} \approx 7$–$8$ includes contributions from electrons and ions.

Recycling

Ions reaching the divertor plate are neutralized and released as neutral particles. The process where these neutrals are re-ionized in the plasma and return toward the divertor is called recycling.

The recycling coefficient $R$ is defined as the ratio of returning particle flux to incident particle flux:

$$R = \frac{\Gamma_{\text{return}}}{\Gamma_{\text{incident}}}$$

At steady state, $R \rightarrow 1$, but it varies in time due to wall absorption and desorption.

In high recycling regions, neutral particle density $n_0$ increases, and the ionization particle source increases:

$$S_{\text{ion}} = n_e n_0 \langle\sigma v\rangle_{\text{ion}}$$

Simultaneously, charge exchange reactions transfer ion energy to neutrals, contributing to plasma cooling:

$$Q_{\text{cx}} = n_i n_0 \langle\sigma v\rangle_{\text{cx}} \cdot E_i$$

Heat Load Concentration and Countermeasures

Quantitative Evaluation of Heat Loads

Heat loads on divertor plates differ significantly between steady-state and transient conditions.

Steady-state heat flux $q_{\perp}$ can be calculated from the power entering the SOL $P_{\text{SOL}}$:

$$q_{\perp} = \frac{P_{\text{SOL}}}{A_{\text{wet}}} = \frac{P_{\text{SOL}}}{2\pi R_t \cdot \lambda_{q,t} \cdot f_x}$$

where $R_t$ is the major radius at the divertor plate position, $\lambda_{q,t}$ is the heat flux width at the divertor, and $f_x$ is the flux expansion factor.

Flux expansion is given by the ratio of poloidal magnetic field at the midplane to that at the divertor plate:

$$f_x = \frac{B_{p,\text{mid}}}{B_{p,t}} \cdot \frac{R_t}{R_{\text{mid}}}$$

Near the X-point, $B_p \rightarrow 0$, so theoretically $f_x \rightarrow \infty$, but in practice, 10–30 is the limit.

Heat load calculation for ITER steady-state operation:

• $P_{\text{SOL}} \approx 100$ MW (heating input minus radiation losses)
• $R_t \approx 5.5$ m
• $\lambda_{q,t} \approx 5$ mm (1 mm at midplane spreads to approximately 5 times at divertor)
• $f_x \approx 20$

This gives:

$$q_{\perp} \approx \frac{100 \times 10^6}{2\pi \times 5.5 \times 0.005 \times 20} \approx 29 \text{ MW/m}^2$$

This value greatly exceeds material limits (approximately 10 MW/m$^2$), making heat load reduction through detachment essential.

ELM Heat Loads

In H-mode plasmas, Edge Localized Modes (ELMs) occur, periodically releasing energy accumulated in the pedestal region.

The energy loss of Type-I ELMs is:

$$\Delta W_{\text{ELM}} \approx 0.05–0.1 \times W_{\text{ped}}$$

where $W_{\text{ped}}$ is the stored energy in the pedestal.

The energy influx to the divertor during an ELM is:

$$\epsilon_{\text{ELM}} = \frac{\Delta W_{\text{ELM}}}{A_{\text{wet}}}$$

For ITER, $\epsilon_{\text{ELM}} < 0.5$ MJ/m$^2$ is the allowable limit. Exceeding this causes tungsten melting and evaporation.

Lower ELM repetition frequency $f_{\text{ELM}}$ means larger energy per ELM, making ELM control important:

$$\Delta W_{\text{ELM}} \cdot f_{\text{ELM}} \approx P_{\text{ELM}}$$

Disruptions

Disruptions are phenomena where plasma confinement is suddenly lost, and stored energy flows into the divertor within a few milliseconds.

Energy density during thermal quench:

$$\epsilon_{\text{TQ}} = \frac{W_{\text{th}}}{\pi R^2} \cdot f_{\text{asym}}$$

where $W_{\text{th}}$ is the thermal energy and $f_{\text{asym}}$ is the concentration factor due to toroidal asymmetry.

Thermal quench in ITER:

• $W_{\text{th}} \approx 350$ MJ
• Arrival time $\tau_{\text{TQ}} \approx 1$–$3$ ms
• Asymmetry factor $f_{\text{asym}} \approx 2$

This results in instantaneous heat flux reaching several GW/m$^2$, and tungsten surface melting and evaporation are unavoidable. Disruption mitigation systems (such as shattered pellet injection) are essential.

Detachment Operation

Detachment is an operating mode that significantly reduces heat and particle loads on divertor plates. When divertor plasma temperature is sufficiently lowered, the plasma becomes “detached” from the divertor plates.

In normal attached conditions, high-temperature plasma directly contacts the divertor plates, imposing large heat and particle loads. In the detached state, divertor plasma temperature drops to a few eV or less, and energy is dispersed through radiation and recycling.

The physics of detachment is based on the following processes:

1. Formation of ionization front: High-density operation causes the ionization region to move away from the divertor plate
2. Volume recombination: Three-body recombination occurs in low-temperature, high-density plasma
3. Momentum loss: Momentum is lost through charge exchange with neutrals

The onset condition for detachment is when divertor temperature falls below approximately 5 eV. At this point, volume recombination rate begins to exceed ionization rate:

$$n_e n_i \langle\sigma v\rangle_{\text{rec}} > n_e n_0 \langle\sigma v\rangle_{\text{ion}}$$

The degree of detachment $D_{\text{det}}$ is defined as:

$$D_{\text{det}} = \frac{\Gamma_{t,\text{attached}}}{\Gamma_{t,\text{actual}}} = \frac{n_u^2 / T_t}{\Gamma_t / c_s}$$

$D_{\text{det}} > 3$ indicates partial detachment, and $D_{\text{det}} > 10$ indicates full detachment.

Radiative Cooling

Radiative cooling is a technique that intentionally increases radiation losses from the divertor plasma to reduce heat loads on divertor plates.

This is achieved through impurity gas injection. Commonly used impurities:

• Nitrogen (N): High radiation efficiency with small impact on core plasma
• Argon (Ar): Higher radiation efficiency, but requires control due to high Z
• Neon (Ne): Intermediate characteristics
• Krypton (Kr): Very high radiation efficiency, effective in small amounts

Radiated power is proportional to impurity concentration $c_Z = n_Z/n_e$:

$$P_{\text{rad}} = c_Z n_e^2 L_Z(T_e) V$$

With radiative cooling, the energy balance in the divertor region becomes:

$$P_{\text{div}} = P_{\text{cond}} + P_{\text{conv}} = P_{\text{rad,div}} + P_{\text{target}}$$

By increasing radiation efficiency $\eta_{\text{rad}} = P_{\text{rad,div}}/P_{\text{div}}$, $P_{\text{target}}$ can be reduced. ITER targets $\eta_{\text{rad}} > 0.7$.

Controlling the position of the radiation region is an important challenge. If the radiation region penetrates beyond the X-point into the main plasma, confinement performance degrades. The radiation front position $z_{\text{rad}}$ is adjusted by controlling impurity injection rate and density:

$$z_{\text{rad}} = z_X - \Delta z(c_Z, n_e, P_{\text{SOL}})$$

Challenges of Detachment Operation

Detachment operation has the following challenges:

• Radiation region control: If the radiation region penetrates beyond the X-point into the main plasma, confinement performance degrades
• Compatibility with density limit: High-density operation approaches the Greenwald density limit, increasing disruption risk
• Stability: Control techniques are needed to maintain detachment for long periods
• Inner-outer asymmetry: Different detachment behavior at inner and outer divertors

The Greenwald density limit is:

$$\bar{n}_e < n_G = \frac{I_p}{\pi a^2} \times 10^{20} \text{ m}^{-3}$$

where $I_p$ is in MA and $a$ is in m.

ITER plans partial detachment operation to keep steady-state heat loads below 10 MW/m$^2$. This requires approximately 95% radiation efficiency, and feasibility demonstrations are ongoing.

Particle and Impurity Exhaust

Exhaust System Configuration

The divertor exhaust system is responsible for neutral particle removal and maintaining plasma purity.

Main components:

• Cryopumps: Capture gas by condensation on liquid helium-cooled panels
• Turbomolecular pumps: Mechanically exhaust gas
• Exhaust ducts: Connect divertor chamber to exhaust pumps
• Neutral beam shields: Protect pumps from plasma

Exhaust conductance $C$ (m$^3$/s) is determined by geometry:

$$C = A \sqrt{\frac{k_B T}{2\pi m}} \cdot K$$

where $A$ is the opening area and $K$ is the Clausing factor ($K < 1$ for long ducts).

Helium Exhaust Requirements

Efficient helium ash exhaust is essential for steady-state fusion reactors.

Required effective pumping speed for helium $S_{\text{He}}^{\text{eff}}$:

$$S_{\text{He}}^{\text{eff}} = \frac{\dot{N}_{\text{He}}}{n_{\text{He,div}}} \cdot k_B T_{\text{div}}$$

Pumping speed needed to maintain helium partial pressure $p_{\text{He}}$ in the divertor chamber:

$$S_{\text{pump}} = \frac{\dot{N}_{\text{He}} k_B T_{\text{pump}}}{p_{\text{He}}}$$

ITER targets $\dot{N}_{\text{He}} \approx 10^{21}$ s$^{-1}$ and $p_{\text{He}} \approx 1$ Pa, requiring $S_{\text{pump}} \approx 100$ m$^3$/s.

Fuel Cycle and Processing

Gas exhausted from the divertor is sent to the tritium processing system.

Fuel cycle flow:

1. Exhaust from divertor (DT + He + impurities)
2. Hydrogen isotope separation
3. Tritium purification
4. Fuel pellet production
5. Re-injection into plasma

The relationship between helium concentration in exhaust gas $c_{\text{He}}^{\text{ex}}$ and concentration in plasma $c_{\text{He}}^{\text{plasma}}$:

$$\eta_{\text{enrich}} = \frac{c_{\text{He}}^{\text{ex}}}{c_{\text{He}}^{\text{plasma}}}$$

The larger this enrichment factor $\eta_{\text{enrich}}$, the more efficient helium exhaust becomes. Typical values are $\eta_{\text{enrich}} \approx 0.2$–$0.5$.

Tungsten Divertor Design

Properties of Tungsten

Tungsten (W) is the most promising armor material for divertors. Key properties are shown below:

Property	Value	Unit
Atomic number $Z$	74	-
Atomic weight	183.84	g/mol
Melting point $T_m$	3422	$^\circ$C
Boiling point	5930	$^\circ$C
Density	19.25	g/cm$^3$
Thermal conductivity (20$^\circ$C)	173	W/(m$\cdot$K)
Thermal conductivity (1000$^\circ$C)	108	W/(m$\cdot$K)
Thermal expansion coefficient (20$^\circ$C)	4.5	$\times 10^{-6}$ K$^{-1}$
Young’s modulus (20$^\circ$C)	411	GPa
Young’s modulus (1000$^\circ$C)	356	GPa
Ductile-to-brittle transition temperature (DBTT)	200–400	$^\circ$C
Recrystallization temperature	1200–1350	$^\circ$C

Sputtering threshold energy $E_{\text{th}}$ for tungsten:

• Hydrogen isotopes (D, T): approximately 200 eV
• Helium: approximately 100 eV

Sputtering yield versus incident ion energy $E$:

$$Y(E) = Q \cdot s_n(E) \left[1 - \left(\frac{E_{\text{th}}}{E}\right)^{2/3}\right] \left[1 - \frac{E_{\text{th}}}{E}\right]^2$$

where $s_n(E)$ is the nuclear stopping power and $Q$ is a material-dependent parameter.

Monoblock Structure

The monoblock type has a cooling tube passing through the center of a tungsten block. It is the primary structure adopted for the ITER divertor.

Monoblock design parameters (ITER):

• Tungsten block dimensions: 28 mm $\times$ 28 mm $\times$ 12 mm
• Cooling tube inner diameter: 12 mm
• Cooling tube outer diameter: 15 mm
• Cooling tube material: CuCrZr alloy
• Cu interlayer thickness: 1 mm

The temperature distribution within the monoblock is obtained as a solution to the steady-state heat conduction equation:

$$\nabla \cdot (k \nabla T) = 0$$

Surface temperature $T_s$ at surface heat flux $q_0$:

$$T_s = T_{\text{coolant}} + \frac{q_0}{h} + \frac{q_0 t_{\text{Cu}}}{k_{\text{Cu}}} + \frac{q_0}{k_W}\left(r_o \ln\frac{r_o}{r_i} + t_W\right)$$

where $h$ is the heat transfer coefficient at the cooling tube inner surface, $t_{\text{Cu}}$ and $t_W$ are the copper layer and tungsten thicknesses, and $r_i$ and $r_o$ are the inner and outer diameters of the cooling tube.

Advantages of the monoblock type:

• Heat flux to the cooling tube is uniformized, preventing local overheating
• Even if the armor-cooling tube joint separates, the armor material is unlikely to fall into the reactor
• High resistance to thermal stress
• Less stress concentration due to axisymmetric thermal expansion

Thermal Stress Analysis

Temperature gradients from heat loads generate thermal stresses. Thermoelastic stress is:

$$\sigma_{\text{th}} = \frac{E \alpha \Delta T}{1 - \nu}$$

where $E$ is Young’s modulus, $\alpha$ is the thermal expansion coefficient, $\nu$ is Poisson’s ratio, and $\Delta T$ is the temperature difference.

For tungsten, with $E = 400$ GPa, $\alpha = 5 \times 10^{-6}$ K$^{-1}$, $\nu = 0.28$, at $\Delta T = 500$ K:

$$\sigma_{\text{th}} \approx \frac{400 \times 10^9 \times 5 \times 10^{-6} \times 500}{1 - 0.28} \approx 1.4 \text{ GPa}$$

This exceeds the yield stress of tungsten (approximately 750 MPa), causing plastic deformation and cracking.

Thermal cycle fatigue life $N_f$ is estimated by the Coffin-Manson law:

$$\Delta \epsilon_p N_f^b = C$$

where $\Delta \epsilon_p$ is the plastic strain amplitude, $b \approx 0.5$, and $C$ is a material constant.

Tungsten-Copper Joining

The thermal expansion mismatch between tungsten (CTE: $4.5 \times 10^{-6}$ K$^{-1}$) and copper (CTE: $17 \times 10^{-6}$ K$^{-1}$) creates large stresses at the joint.

Thermal expansion mismatch stress:

$$\sigma_{\text{mismatch}} = \frac{E_{\text{eff}}(\alpha_{\text{Cu}} - \alpha_W) \Delta T}{1 - \nu}$$

At $\Delta T = 500$ K, stress of approximately 2 GPa is generated, causing interface delamination.

The following joining technologies have been developed as countermeasures:

• Functionally graded materials (FGM): Interlayers with continuously varying composition
• Brazing: Active brazing materials such as CuAgTi between copper and tungsten
• HIP (Hot Isostatic Pressing): Diffusion bonding at high temperature and pressure
• Cast bonding: Direct casting of molten copper onto tungsten

Cooling Technology

Principles of High Heat Flux Removal

Removal of heat flux exceeding 10 MW/m$^2$ requires highly efficient cooling technology.

Heat transfer at the cooling tube inner surface is characterized by heat transfer coefficient $h$:

• Forced convection: $h \sim 10^4$ W/(m$^2\cdot$K)
• Subcooled boiling: $h \sim 10^5$ W/(m$^2\cdot$K)
• Critical heat flux (CHF) exceeded: $h$ drops sharply (transition to film boiling)

Critical heat flux defines the limit of heat removal capability, and exceeding it causes cooling tube burnout.

Heat Transfer Enhancement with Swirl Tape

Inserting swirl tape (twisted tape) inside cooling tubes generates swirl flow and improves heat transfer.

Swirl tape characteristics are expressed by twist ratio $y$ (number of tube diameters required for 180-degree twist):

$$y = \frac{\text{pitch}}{2D}$$

$y = 2$–$4$ is typical.

Heat transfer enhancement effect of swirl tape insertion:

$$\frac{h_{\text{swirl}}}{h_{\text{smooth}}} = 1 + \frac{A}{y^B} Re^C$$

where $Re$ is the Reynolds number and $A, B, C$ are experimental parameters. Typically, 2–3 times heat transfer improvement is obtained.

Critical heat flux is also improved:

$$\frac{CHF_{\text{swirl}}}{CHF_{\text{smooth}}} \approx 1.5–2.5$$

ITER Divertor Cooling System

ITER divertor cooling system specifications:

Item	Value
Coolant	Pressurized water
Inlet temperature	100$^\circ$C
Outlet temperature	150$^\circ$C
Pressure	4.2 MPa
Flow velocity	10 m/s
Total flow rate	Approximately 1000 kg/s
Heat removal capacity	Approximately 200 MW

Reynolds number in cooling tubes:

$$Re = \frac{\rho v D}{\mu} \approx \frac{850 \times 10 \times 0.012}{2 \times 10^{-4}} \approx 5 \times 10^5$$

This is sufficiently turbulent, providing high heat transfer.

Advanced Cooling Concepts

The following advanced concepts are being researched for future high heat flux removal:

Helium gas cooling:

• High-temperature operation possible ($>$600$^\circ$C)
• Improved thermal efficiency
• No chemical reaction risk with water
• Low heat transfer coefficient ($h \sim 10^3$ W/(m$^2\cdot$K)) requires extended heat transfer surfaces

Porous media insertion:

• Insert metal porous bodies (foam) in cooling channels
• Significant increase in heat transfer area
• High pressure loss

Microchannels:

• Increased surface area through fine channels
• High heat transfer coefficient
• Complex manufacturing

ITER Divertor Detailed Specifications

Divertor Cassettes

The ITER divertor consists of 54 cassettes. Each cassette comprises the following components:

• Inner Vertical Target (IVT): Inner divertor plate
• Outer Vertical Target (OVT): Outer divertor plate
• Dome: Neutral particle shield below the X-point
• Cassette body: Structural support and cooling water piping

Main cassette dimensions:

• Toroidal length: approximately 660 mm
• Poloidal height: approximately 3000 mm
• Weight: approximately 8.8 tonnes/cassette

Plasma Facing Units

Plasma Facing Unit (PFU) configuration:

Item	IVT	OVT
PFUs per cassette	16	22
Monoblocks per PFU	Approximately 60	Approximately 60
Total monoblocks per cassette	Approximately 960	Approximately 1320
Total monoblocks (entire ITER)	Approximately 52,000	Approximately 71,000

Monoblock manufacturing precision requirements:

• Flatness: 0.1 mm or less
• Bonding defect area: Single defect 5 mm$^2$ or less, cumulative 10% or less
• Cooling tube wall thickness variation: $\pm$0.2 mm

Heat Load Conditions

ITER divertor design heat load conditions:

Condition	Heat flux	Allowable cycles
Steady-state operation	10 MW/m$^2$	3000
Low-frequency transients	20 MW/m$^2$	300
Type-I ELM (after mitigation)	0.5 MJ/m$^2$	$10^6$
Disruption (after mitigation)	8 MJ/m$^2$	100

At steady-state heat flux of 10 MW/m$^2$, surface temperature is calculated to be approximately 1200$^\circ$C and cooling tube temperature approximately 300$^\circ$C.

Remote Maintenance

Divertor cassettes can be exchanged by remote operation.

Maintenance procedure:

1. Evacuate tokamak vacuum vessel
2. Insert cassette handler from lower port
3. Cut cooling piping (orbital weld joints)
4. Release cassette mounting bolts
5. Transfer cassette to hot cell
6. Install new or refurbished cassette
7. Re-weld piping and perform leak test

One cassette exchange is estimated to take approximately 1 month, and full cassette replacement approximately 1 year.

Advanced Divertor Concepts

Conventional single null and double null configurations are approaching heat load tolerance limits, so advanced divertor concepts that achieve more effective heat load distribution are being researched.

Snowflake Configuration

The snowflake configuration forms a second-order null point near the X-point. At a normal X-point:

$$B_p = 0, \quad \nabla B_p \neq 0$$

In the snowflake configuration, the additional condition:

$$B_p = 0, \quad \nabla B_p = 0, \quad \nabla^2 B_p \neq 0$$

expands the region where the poloidal field is zero, increasing flux expansion.

In an ideal snowflake configuration, the poloidal field near the X-point varies as:

$$B_p \propto r^2$$

(second order, compared to $B_p \propto r$ normally), roughly doubling the heat flux width.

Types of snowflake configurations:

• SF+ (Snowflake plus): Two X-points separate toward the plasma interior
• SF- (Snowflake minus): Two X-points separate toward the plasma exterior

Experimentally demonstrated at TCV (Switzerland), NSTX (USA), and shown to be effective for detachment stabilization.

Super-X Configuration

The Super-X configuration extends the outer divertor leg significantly outward.

Design principles:

• Increase target major radius $R_t$ of outer divertor
• Lengthen connection length $L_\parallel$
• Reduce total magnetic field strength, increasing flux expansion

Heat flux reduction effect:

$$q_t \propto \frac{1}{R_t \cdot f_x \cdot \lambda_{q,t}}$$

Doubling the target major radius halves the heat flux. Additionally, increased connection length enhances radiative cooling effects.

Super-X configuration experiments are ongoing at MAST-U (UK), and significant reduction in detachment threshold compared to conventional configurations has been reported.

X-Divertor

The X-divertor is a configuration with the X-point lowered close to the divertor plate.

Characteristics:

• Divertor chamber volume is reduced
• Improved neutral particle confinement
• Radiation region naturally moves away from divertor plate

Because the X-point position is low, there is greater freedom in poloidal coil placement.

Long-Leg Divertor

The long-leg divertor is a configuration with increased distance from X-point to divertor plate.

Advantages:

• Increased connection length $L_\parallel$ increases radiation time
• Expanded neutral particle interaction region
• Detachment stabilization

Relationship between connection length and radiation loss:

$$P_{\text{rad}} \propto L_\parallel n_e^2 L_Z$$

Doubling the connection length doubles radiation losses at the same impurity concentration.

Closed Divertor

The closed divertor physically isolates the divertor chamber from the main plasma region.

Design elements:

• Divertor chamber enclosure by baffle plates
• Confine neutral particle recycling within divertor chamber
• Maintain high neutral particle pressure

Relationship between neutral particle pressure $p_0$ and recycling efficiency:

$$p_0 = \frac{\Gamma_{\text{ion}} k_B T_n}{S_{\text{pump}} + S_{\text{leak}}}$$

where $S_{\text{leak}}$ is the leak from divertor chamber to main plasma. In closed divertors, minimizing $S_{\text{leak}}$ achieves high exhaust efficiency.

Liquid Metal Divertor

Liquid metal divertor concepts are being researched to avoid solid material erosion and lifetime issues.

Liquid Lithium

Liquid lithium (Li) is the most researched liquid metal divertor material.

Properties:

Property	Value	Unit
Melting point	180.5	$^\circ$C
Boiling point	1342	$^\circ$C
Density (200$^\circ$C)	515	kg/m$^3$
Thermal conductivity (200$^\circ$C)	46	W/(m$\cdot$K)
Viscosity (200$^\circ$C)	0.56	mPa$\cdot$s
Vapor pressure (500$^\circ$C)	0.001	Pa

Advantages:

• Self-healing: Evaporated material recondenses and replenishes
• Low Z ($Z = 3$): Small plasma radiation loss
• Hydrogen retention: Favorable for tritium recovery
• Gettering effect: Absorbs impurities

Challenges:

• Plasma contamination from evaporation (vapor pressure increases at high temperature)
• Safety (reactivity with water and air)
• MHD pressure loss (conducting fluid in magnetic field)
• Tritium retention

Evaporation rate $\Gamma_{\text{evap}}$ is given by the Hertz-Knudsen equation:

$$\Gamma_{\text{evap}} = \frac{p_{\text{vap}}(T)}{\sqrt{2\pi m k_B T}}$$

Maintaining surface temperature below 400$^\circ$C keeps evaporation rate at practical levels ($< 10^{19}$ m$^{-2}$s$^{-1}$).

Liquid Tin

Liquid tin (Sn) has relatively low vapor pressure and is attracting attention as an alternative to liquid lithium.

Properties:

Property	Value	Unit
Melting point	232	$^\circ$C
Boiling point	2602	$^\circ$C
Density (300$^\circ$C)	6890	kg/m$^3$
Vapor pressure (1000$^\circ$C)	0.1	Pa

Advantages:

• Low vapor pressure: High-temperature operation possible
• Chemical stability: Mild reaction with water and air
• Availability and low cost

Challenges:

• High Z ($Z = 50$): Plasma contamination requires attention
• Alloying with tungsten
• Low surface tension

Capillary Porous Structure (CPS)

Capillary Porous Structure (CPS) is used to retain liquid metal on a solid substrate.

Design principles:

• Mesh or sintered tungsten or molybdenum
• Pore size: 10–100 μm
• Capillary forces retain liquid metal

Balance of capillary force and gravity:

$$\Delta p_{\text{cap}} = \frac{2\gamma \cos\theta}{r} > \rho g h$$

where $\gamma$ is surface tension, $\theta$ is contact angle, $r$ is pore size, and $h$ is liquid film height.

With CPS lithium retention at 50 μm pore size, liquid films of several cm can be held against gravity.

Liquid Metal Divertor Experiments

Liquid metal divertor experiments are progressing at the following facilities:

• LTX-β (USA): Liquid lithium limiter experiments
• EAST (China): Liquid lithium divertor experiments
• FTU (Italy): Liquid tin limiter experiments
• Magnum-PSI (Netherlands): High heat flux plasma irradiation tests

Full-scale liquid lithium divertor experiments are planned at NSTX-U (USA).

Divertor Challenges for Future Reactors

DEMO Divertor Requirements

DEMO (demonstration reactor) has stricter divertor requirements than ITER.

Parameter	ITER	DEMO
Fusion power	500 MW	2000–3000 MW
Thermal power	600 MW	2500–3500 MW
Divertor heat load	10 MW/m$^2$	$>$10 MW/m$^2$
Operation time	400 s	Continuous
Neutron fluence	0.3 MWa/m$^2$	50–100 MWa/m$^2$
Availability	25%	$>$50%

Main technical challenges:

1. Increased heat load: Accommodate increased power density
2. Continuous operation: Thermal cycle fatigue, material degradation
3. Neutron irradiation damage: Tungsten embrittlement, thermal conductivity reduction
4. Tritium breeding: Compatibility with blanket
5. Remote maintenance: Exchange in high-activation environment

Tungsten Degradation from Neutron Irradiation

Fast neutron irradiation degrades the mechanical and thermal properties of tungsten.

Main irradiation effects:

• Irradiation hardening: Increased hardness due to lattice defects
• Irradiation embrittlement: DBTT increase (200$^\circ$C → 800$^\circ$C or higher)
• Thermal conductivity reduction: 30–50% reduction
• Swelling: Volume expansion (several %)
• Rhenium production: Transmutation via (n, 2n) reactions

Irradiation damage is expressed in dpa (displacements per atom). In DEMO, approximately 20 dpa/year irradiation is expected, reaching approximately 40 dpa at a design life of 2 years.

Thermal conductivity degradation:

$$\lambda_{\text{irr}} = \frac{\lambda_0}{1 + \alpha \cdot \text{dpa}}$$

With $\alpha \sim 0.02$–$0.05$ dpa$^{-1}$, thermal conductivity may decrease by more than 50% at 40 dpa.

Directions in Material Development

Material development for future reactor divertors:

Tungsten alloys:

• W-Re alloys: Improved ductility
• W-TiC alloys: Improved high-temperature strength
• W fiber-reinforced composites: Improved crack resistance

Self-healing materials:

• Self-recovery of irradiation defects
• Nanostructure control

Advanced manufacturing technologies:

• Additive manufacturing (3D printing)
• Functionally graded materials (FGM)
• Nanoparticle dispersion

Divertor Design Optimization

Directions for future reactor divertor design optimization:

1. Radiative divertor: Advanced detachment operation
2. Increased area: Reduction of heat flux density
3. Advanced configurations: Super-X, snowflake, etc.
4. Liquid metal: Overcoming solid material limits
5. Helium cooling: High-temperature operation and high-efficiency power generation

Divertor design trade-offs:

$$\text{Performance} = f(\text{heat load}, \text{material life}, \text{maintainability}, \text{cost})$$

No solution satisfies all requirements, and system optimization is necessary.

Summary

The divertor is one of the most critical components for achieving fusion reactors. It must maintain long-term operation under severe heat and particle load environments while simultaneously achieving impurity control, helium ash exhaust, and heat handling.

Current divertor technology is being demonstrated at ITER, but the following challenges must be solved for future power plant realization:

• Improved heat load removal capability in steady-state operation
• Maintaining material performance under neutron irradiation environment
• Stabilization and control of detachment operation
• Design optimization considering remote maintenance
• Practical application of innovative concepts such as liquid metals

Intensive research and development is underway at research institutions worldwide to address these challenges.

Related Topics

• Tokamak Configuration - Tokamak magnetic configuration and divertor
• Plasma-Facing Materials - Material properties of tungsten and others
• ITER Project - International Thermonuclear Experimental Reactor
• Superconducting Coils - Poloidal field coils
• MHD - Plasma magnetohydrodynamics
• Plasma Instabilities - ELMs and confinement degradation