Principles of Plasma Heating

To sustain fusion reactions, plasma must be heated to temperatures exceeding 100 million degrees (approximately 10 keV). This chapter provides a detailed explanation of the physical principles behind major plasma heating methods, their technical implementations, and the characteristics and limitations of each approach.

Necessity of Heating and Basic Concepts

Fusion Reactions and Temperature

For fusion reactions to occur, positively charged nuclei must overcome the Coulomb repulsion and approach each other to within the range of the nuclear force (approximately 1 femtometer). The energy required to overcome this Coulomb barrier reaches several hundred keV even for deuterium-tritium (D-T) reactions.

However, due to quantum mechanical tunneling, reactions can occur at lower energies. The reaction cross-section for D-T reactions peaks around 100 keV, but since particles in a plasma follow a Maxwell distribution, high-energy particles with 10-20 times the average energy contribute to the reactions. As a result, a plasma temperature of 10-20 keV (approximately 100-200 million degrees) becomes the target operating temperature for fusion reactors.

Energy Balance and Heating Power

In a steady-state plasma, heating power balances energy loss power:

$$P_{\text{heat}} = P_{\text{loss}} = \frac{W}{\tau_E}$$

where $W$ is the stored energy in the plasma and $\tau_E$ is the energy confinement time. The stored energy in the plasma is:

$$W = \frac{3}{2}(n_e k_B T_e + n_i k_B T_i)V = 3nk_B T V$$

Assuming $T_e = T_i = T$ and $n_e = n_i = n$, the required heating power is:

$$P_{\text{heat}} = \frac{3nk_B T V}{\tau_E}$$

For example, substituting ITER’s main plasma parameters ($n = 10^{20}$ m$^{-3}$, $T = 10$ keV, $V = 840$ m$^3$, $\tau_E = 3$ s), the required heating power is on the order of 100 MW.

Classification of Heating Methods

Plasma heating methods are broadly classified into the following categories:

1. Ohmic heating (Joule heating): Resistive heating by plasma current
2. Neutral Beam Injection (NBI): Injection of high-energy neutral particles
3. Radio-frequency heating: Resonant interaction between electromagnetic waves and plasma particles
   • Ion Cyclotron Resonance Heating (ICRH)
   • Electron Cyclotron Resonance Heating (ECRH)
   • Lower Hybrid Heating (LHH)
4. Alpha particle self-heating: Heating by fusion reaction products

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Ohmic Heating (Joule Heating)

Basic Principle

Ohmic heating is a method that utilizes Joule heat generated when current induced in the plasma flows against electrical resistance. In tokamaks, plasma current is induced by magnetic flux changes in the central solenoid coil, and this current heats the plasma.

The Joule heating power density per unit volume is:

$$p_J = \eta_\parallel J_\parallel^2 + \eta_\perp J_\perp^2$$

where $\eta_\parallel$ and $\eta_\perp$ are the electrical resistivities parallel and perpendicular to the magnetic field, and $J_\parallel$ and $J_\perp$ are the corresponding current density components. In tokamaks, the plasma current flows mainly in the toroidal direction (roughly parallel to the magnetic field), so $\eta_\parallel$ is dominant.

Spitzer Resistivity

The electrical resistivity of high-temperature plasma originates from Coulomb collisions between electrons and ions. The classical resistivity of a fully ionized plasma was derived by Lyman Spitzer in the 1950s. The Spitzer resistivity is:

$$\eta_\parallel = \frac{\pi Z_{\text{eff}} e^2 m_e^{1/2} \ln\Lambda}{(4\pi\varepsilon_0)^2 (k_B T_e)^{3/2}} \cdot \frac{1}{3\sqrt{2\pi}}$$

Numerically:

$$\eta_\parallel \approx 5.2 \times 10^{-5} \frac{Z_{\text{eff}} \ln\Lambda}{T_e^{3/2}} \quad [\Omega \cdot \text{m}]$$

where $T_e$ is the electron temperature (in eV), $Z_{\text{eff}}$ is the effective charge number, and $\ln\Lambda$ is the Coulomb logarithm (typically 15-20).

As evident from this equation, plasma resistivity is inversely proportional to the 3/2 power of temperature. This determines the fundamental limitation of ohmic heating.

Neoclassical Resistivity

In actual tokamak plasmas, particles trace closed orbits called “banana orbits” due to the inhomogeneity of the toroidal magnetic field. These trapped particles do not contribute to current transport, effectively increasing the resistivity.

The neoclassical correction to resistivity is:

$$\eta_{\text{neo}} = \eta_\parallel \cdot C(\nu_*^e)$$

The correction factor $C$ is expressed as a function of the collisionality parameter $\nu_*^e$:

$$\nu_*^e = \frac{\nu_{ei}}{\varepsilon^{3/2} \omega_{te}}$$

where $\nu_{ei}$ is the electron-ion collision frequency, $\varepsilon = r/R$ is the inverse aspect ratio, and $\omega_{te} = v_{te}/qR$ is the transit frequency.

In the banana regime ($\nu_*^e \ll 1$):

$$C \approx \frac{1}{1 - 1.95\sqrt{\varepsilon} + 0.95\varepsilon}$$

For a typical tokamak with $\varepsilon \approx 0.3$, $C \approx 2$. This means neoclassical effects approximately double the resistivity.

Temperature Limit and Heating Efficiency

Ohmic heating power is determined by the product of resistivity and current squared:

$$P_\Omega = \int \eta J^2 dV = \eta_\parallel I_p^2 / A_{\text{eff}}$$

where $I_p$ is the plasma current and $A_{\text{eff}}$ is the effective cross-sectional area.

As the plasma heats up, resistivity decreases, so heating power at constant current diminishes. The maximum temperature achievable by ohmic heating alone is determined by the balance between heating power and radiation/transport losses. Typically:

$$T_e^{\text{max}} \approx 2\text{-}3 \text{ keV} \quad (\text{approximately 20-30 million degrees})$$

This falls far short of the 10 keV or more required for fusion, demonstrating the necessity of auxiliary heating methods.

Ohmic Confinement Scaling

For plasmas with only ohmic heating, an empirical scaling law for energy confinement time (Neo-Alcator scaling) is known:

$$\tau_E^{\text{OH}} \propto \bar{n}_e a^2 R q$$

where $\bar{n}_e$ is the line-averaged electron density, $a$ is the minor radius, $R$ is the major radius, and $q$ is the safety factor.

This scaling law shows that confinement improves with increasing density in ohmic heating plasmas. However, the Greenwald density limit prevents unlimited density increases.

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Neutral Beam Injection (NBI)

Basic Principle

Neutral Beam Injection (NBI) is one of the most widely used plasma heating methods. High-energy neutral atom beams are injected into the plasma, ionized within the plasma, and then transfer energy to plasma particles through Coulomb collisions.

Since charged particles are deflected by magnetic fields, they cannot directly reach the plasma interior. Therefore, beams are generated and injected through the following process:

1. Ion source: Generate hydrogen or its isotope (D, T) ions
2. Accelerator: Electrostatically accelerate ions to the desired energy
3. Neutralizer: Neutralize ions in a gas cell or plasma neutralizer
4. Deflection magnet: Separate and remove residual ions
5. Beam dump: Absorb non-neutralized ions
6. Beamline: Transport neutral beam to the plasma

Types of Ion Sources

Positive Ion Sources

Conventional NBI systems have used positive ion sources (H$^+$, D$^+$, T$^+$). Positive ions can be easily generated by plasma discharge or filament discharge, achieving high current density beams.

Characteristics of positive ion sources:

• Ion generation efficiency: Over 90%
• Extraction current density: 100-500 mA/cm$^2$
• Neutralization efficiency: Achieved through charge exchange in gas cell

The neutralization efficiency $\eta_0$ strongly depends on beam energy, and from the cross-section relationship:

$$\eta_0^+ = \frac{\sigma_{10}}{\sigma_{10} + \sigma_{01}}$$

where $\sigma_{10}$ is the neutralization cross-section and $\sigma_{01}$ is the re-ionization cross-section.

For positive ions, neutralization efficiency drops rapidly when beam energy exceeds about 50 keV/amu (100 keV for deuterium):

Beam Energy (keV/amu)	Neutralization Efficiency
20	80%
40	60%
60	40%
80	20%
100	5%

Therefore, positive ion source NBI is suitable for small and medium-sized devices but insufficient for large devices like ITER.

Negative Ion Sources

To efficiently neutralize high-energy beams, negative ions (H$^-$, D$^-$) must be used. For negative ions, neutralization by photo-detachment is possible, maintaining high neutralization efficiency even at high energies:

$$\eta_0^- = 1 - \exp(-n_g \sigma_{\text{det}} L)$$

where $n_g$ is the gas density, $\sigma_{\text{det}}$ is the detachment cross-section, and $L$ is the neutralizer length.

Challenges and solutions for negative ion source NBI:

1. Low negative ion production efficiency

   • Surface production method: Secondary electron capture from low work function surfaces with cesium (Cs) addition
   • Volume production method: Dissociative attachment from vibrationally excited molecules

2. Simultaneous electron extraction

   • Electron removal by magnetic filter
   • Typical $e^-/H^-$ ratio: $<1$

3. Beam optics complexity

   • Space charge effect compensation
   • Multi-stage acceleration systems

ITER’s NBI system is designed to use 1 MeV deuterium negative ion beams, supplying a total of 33 MW injection power from two units each delivering 16.5 MW.

Beam Energy and Penetration Depth

The attenuation of neutral beams in plasma is determined by ionization reactions. The attenuation of beam intensity $I(x)$ is:

$$\frac{dI}{dx} = -\frac{I}{\lambda}$$

where $\lambda$ is the ionization mean free path. There are three main ionization channels:

1. Charge exchange (CX): $\text{H}^0 + \text{H}^+ \rightarrow \text{H}^+ + \text{H}^0$
2. Ion impact ionization: $\text{H}^0 + \text{H}^+ \rightarrow \text{H}^+ + \text{H}^+ + e^-$
3. Electron impact ionization: $\text{H}^0 + e^- \rightarrow \text{H}^+ + 2e^-$

The total ionization rate is:

$$\langle\sigma v\rangle_{\text{ion}} = \langle\sigma v\rangle_{\text{CX}} + \langle\sigma v\rangle_{\text{ii}} + \langle\sigma v\rangle_{\text{ei}}$$

The ionization mean free path is:

$$\lambda = \frac{v_b}{\langle\sigma v\rangle_{\text{ion}} n_e}$$

In high-temperature, low-density plasmas, charge exchange dominates, and the relationship between beam energy $E_b$ and penetration depth is approximately:

$$\lambda \propto E_b^{1.5}$$

To reach the plasma center, $\lambda > a$ (minor radius) is required. For ITER ($n_e \sim 10^{20}$ m$^{-3}$, $a \sim 2$ m), this requires beam energies of about 1 MeV, which is why negative ion source NBI is adopted.

Critical Energy and Energy Partition

Fast ions ionized in the plasma lose energy through Coulomb collisions. The energy transfer rates to electrons and ions differ, and the beam energy at which both become equal is called the critical energy $E_c$:

$$E_c = 14.8 A_b T_e \left(\sum_j \frac{n_j Z_j^2}{n_e A_j}\right)^{2/3} \quad [\text{keV}]$$

where $A_b$ is the mass number of the beam particle, $T_e$ is the electron temperature (keV), and $n_j$, $Z_j$, $A_j$ are the density, charge number, and mass number of each ion species.

For a pure deuterium plasma:

$$E_c \approx 18.6 T_e \quad [\text{keV}]$$

The physics of energy partition is described by the slowing-down equation:

$$\frac{dE}{dt} = -\left(\frac{E}{E_c}\right)^{3/2} \frac{1}{\tau_s^e} E - \frac{1}{\tau_s^i} E$$

where $\tau_s^e$ and $\tau_s^i$ are the slowing-down times for electrons and ions. Upon integration, the energy partition ratio between electrons and ions is:

$$\frac{W_e}{W_i} = \frac{1}{3}\ln\left[1 + \left(\frac{E_b}{E_c}\right)^{3/2}\right] / \left[1 - \frac{1}{3}\ln\left(1 + \left(\frac{E_c}{E_b}\right)^{3/2}\right)\right]$$

• $E_b \ll E_c$: Predominantly ion heating ($W_i/W_{\text{total}} > 0.8$)
• $E_b \gg E_c$: Predominantly electron heating ($W_e/W_{\text{total}} > 0.8$)
• $E_b \approx E_c$: Energy distributed equally between electrons and ions

Injection Geometry and Current Drive

The injection direction of NBI significantly affects heating characteristics and current drive efficiency:

Perpendicular Injection

Injection perpendicular to the plasma current. Provides uniform heating throughout the plasma but does not contribute to current drive.

Tangential Injection

Injection along the toroidal direction of the plasma. Depending on the injection direction:

• Co-injection: Same direction as plasma current. Contributes to current drive.
• Counter-injection: Opposite direction to plasma current. Reduces current.

Neutral Beam Current Drive (NBCD) efficiency is:

$$\eta_{\text{CD}} = \frac{n_e R I_{\text{CD}}}{P_{\text{NBI}}} \quad [\text{A/W/m}^2]$$

Typical NBCD efficiency is about 0.02-0.05 A/W/m$^2$, playing an important role as a non-inductive current drive source for steady-state operation.

Plasma Rotation and Momentum Injection

An important secondary effect of NBI is momentum injection into the plasma. The injected fast ions transfer momentum to plasma particles through collisions, driving plasma rotation.

Typical toroidal rotation velocities are:

$$v_\phi \sim 100\text{-}300 \text{ km/s}$$

This plasma rotation plays an important role in suppressing MHD instabilities (especially resistive wall modes) and suppressing turbulent transport.

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Ion Cyclotron Resonance Heating (ICRH)

Resonance Condition and Basic Principle

Ion Cyclotron Resonance Heating (ICRH) is a method that heats ions using electromagnetic waves in the ion cyclotron frequency range.

Charged particles in a magnetic field undergo cyclotron motion in the plane perpendicular to the magnetic field lines. The ion cyclotron frequency is:

$$\omega_{ci} = \frac{Z_i e B}{m_i}$$

Expressed in frequency:

$$f_{ci} = \frac{\omega_{ci}}{2\pi} = \frac{Z_i e B}{2\pi m_i} = 15.2 \frac{Z_i}{A_i} B \quad [\text{MHz}]$$

where $B$ is the magnetic field strength in Tesla and $A_i$ is the mass number.

For representative values, in a magnetic field of $B = 5$ T:

Ion Species	Cyclotron Frequency
Hydrogen (H)	76 MHz
Deuterium (D)	38 MHz
Tritium (T)	25 MHz
Helium-3 ($^3$He)	51 MHz
Helium-4 ($^4$He)	19 MHz

The resonance condition for electromagnetic waves to efficiently transfer energy to ions is:

$$\omega = n\omega_{ci} + k_\parallel v_\parallel$$

where $n$ is the harmonic order ($n = 1, 2, 3, ...$), $k_\parallel$ is the parallel component of the wave number, and $v_\parallel$ is the parallel velocity of the ion. The second term on the right represents the Doppler shift.

Harmonic Resonance

In addition to the fundamental resonance at $n = 1$, harmonic resonances at $n = 2, 3$ can also be used. The absorption strength of harmonic resonance is proportional to:

$$\left(\frac{k_\perp v_\perp}{\omega_{ci}}\right)^{2(n-1)}$$

where $v_\perp$ is the perpendicular velocity of the ion.

Characteristics of harmonic resonance:

• Fundamental ($n = 1$): Absorption even for low-energy ions
• Second harmonic ($n = 2$): Requires finite Larmor radius effects, efficient for high-energy ions
• Third harmonic and above: Weak absorption but usable for special applications

Minority Ion Heating

In a pure single-ion-species plasma, absorption at the fundamental cyclotron resonance is weak, making efficient heating difficult. This is because the left-hand circularly polarized component (co-rotating with ions) is cut off.

Minority ion heating scheme solves this problem. A small amount (a few percent) of a different ion species is added to the main plasma, and electromagnetic waves are injected at the resonance frequency of the minority ion species.

Typical examples:

• Hydrogen heating in deuterium plasma: D(H) scheme
  • Injection at $\omega = \omega_{cH}$
  • Hydrogen has twice the cyclotron frequency of deuterium
• Helium-3 heating in deuterium plasma: D($^3$He) scheme
  • $^3$He has $\omega_{c\text{He3}} = (2/3)\omega_{cH}$

The absorbed power for minority ion heating is:

$$P_{\text{abs}} \propto n_{\text{min}} \left(\frac{Z_{\text{min}}}{A_{\text{min}}}\right)^2 \left|E_+\right|^2$$

where $n_{\text{min}}$ is the minority ion density and $E_+$ is the left-hand polarized electric field component.

The optimal minority ion concentration is typically 2-10%. If the concentration is too low, absorption is weak; if too high, it approaches a single-ion species and absorption efficiency decreases.

Mode Conversion Heating

When the minority ion concentration increases, wave propagation characteristics change, and “mode conversion” from fast waves to Bernstein waves occurs. The Bernstein waves generated by mode conversion efficiently transfer energy to electrons.

The condition for mode conversion to occur is:

$$\left(\frac{\omega_{pi,\text{min}}}{\omega}\right)^2 > \frac{\omega_{ci,\text{maj}} - \omega}{\omega_{ci,\text{min}} - \omega}$$

In this regime, electron heating using electromagnetic waves in the ion cyclotron frequency range becomes possible.

Antennas and Couplers

ICRH electromagnetic waves are injected from loop antennas installed at the plasma edge. Important antenna design parameters:

1. Toroidal spectrum control: Control of $k_\parallel$ spectrum by phased arrays

   • Symmetric spectrum ($[0,\pi,0,\pi]$): Pure heating
   • Asymmetric spectrum ($[0,\pi/2,\pi,3\pi/2]$): Current drive

2. Coupling efficiency: Optimization of antenna-plasma distance and density gradient

3. Thermal and voltage resistance design: Technology required for high-power (MW-class) operation

The ITER ICRH antenna is designed to operate in the 9-55 MHz frequency range and inject up to 20 MW of power.

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Electron Cyclotron Resonance Heating (ECRH)

Resonance Condition

Electron Cyclotron Resonance Heating (ECRH) is a method that directly heats electrons using millimeter waves in the electron cyclotron frequency range.

The electron cyclotron frequency is:

$$\omega_{ce} = \frac{eB}{m_e}$$ $$f_{ce} = \frac{\omega_{ce}}{2\pi} = \frac{eB}{2\pi m_e} = 28.0 \times B \quad [\text{GHz}]$$

At $B = 5$ T, $f_{ce} = 140$ GHz, corresponding to millimeter waves with a wavelength of about 2 mm.

Due to relativistic effects, the cyclotron frequency of high-energy electrons decreases:

$$\omega_{ce,\text{rel}} = \frac{\omega_{ce}}{\gamma} = \frac{\omega_{ce}}{\sqrt{1 + p^2/(m_e c)^2}}$$

where $\gamma$ is the Lorentz factor and $p$ is the electron momentum.

The resonance condition including harmonics is:

$$\omega = n\omega_{ce,\text{rel}} + k_\parallel v_\parallel \quad (n = 1, 2, ...)$$

O-mode and X-mode

The propagation characteristics of electromagnetic waves in plasma depend on polarization and propagation direction:

O-mode (Ordinary mode)

Linear polarization with the electric field vector parallel to the magnetic field. Propagation condition:

$$\omega > \omega_{pe}$$

The cutoff is determined only by the electron plasma frequency $\omega_{pe}$.

X-mode (Extraordinary mode)

Polarization with an electric field component perpendicular to the magnetic field. Propagation conditions are more complex:

• Low-frequency cutoff (L-cutoff): $\omega = \omega_L$
• High-frequency cutoff (R-cutoff): $\omega = \omega_R$
• Upper hybrid resonance (UHR): $\omega = \omega_{UH}$

where:

$$\omega_{L,R} = \frac{1}{2}\left[\mp\omega_{ce} + \sqrt{\omega_{ce}^2 + 4\omega_{pe}^2}\right]$$ $$\omega_{UH} = \sqrt{\omega_{ce}^2 + \omega_{pe}^2}$$

Practical Injection Mode Selection

Mode	Injection Direction	Frequency	Characteristics
O1	Perpendicular to field	$f_{ce}$	First harmonic, high density capability
X2	Perpendicular to field	$2f_{ce}$	Second harmonic, localized heating
X3	Perpendicular to field	$3f_{ce}$	Third harmonic, ultra-high density capability

O-mode has a high density limit ($n_e < n_c = \omega^2 \varepsilon_0 m_e / e^2$) and is advantageous for high-density plasma applications. X-mode has higher absorption efficiency but has density limitations due to cutoffs.

Heating Position Control

The greatest feature of ECRH is the ability to precisely control the heating position. Since the resonance condition $\omega = n\omega_{ce}$ is determined by magnetic field strength, any position within the plasma can be heated by adjusting the magnetic field distribution and beam injection angle.

In tokamaks, the toroidal magnetic field varies as $B \propto 1/R$ in the major radius direction, so:

$$R_{\text{res}} = \frac{n e B_0 R_0}{2\pi m_e f}$$

With fixed frequency $f$, the resonance position is uniquely determined in the major radius direction.

Applications utilizing this characteristic:

1. Localized current drive (ECCD): Current profile control at specific positions
2. MHD instability control: Stabilization of neoclassical tearing modes (NTM)
3. Internal transport barrier formation: Local thermal transport control

Gyrotron Technology

As the high-frequency source for ECRH, vacuum electron tubes called gyrotrons are used. Gyrotrons generate high-power millimeter waves by utilizing the interaction between electron cyclotron motion and electromagnetic waves.

Gyrotron operating principle:

1. Inject electron beam from electron gun into axisymmetric magnetic field
2. Electrons undergo cyclotron motion while spiraling along magnetic field lines
3. Interact with electromagnetic waves in the cavity resonator and release energy
4. Extract millimeter waves through waveguides

Performance of modern gyrotrons:

Parameter	Typical Value
Output power	1-2 MW
Frequency	77-170 GHz
Efficiency	40-50%
Pulse length	CW (continuous operation) capable

ITER plans to use 24 gyrotrons at 170 GHz with 1 MW output, supplying a total of 20 MW ECRH power.

Electron Cyclotron Current Drive (ECCD)

When the ECRH beam is injected into the plasma from a non-perpendicular direction, an asymmetric velocity distribution is generated in the absorbed electrons, driving current. This is called Electron Cyclotron Current Drive (ECCD).

ECCD efficiency is:

$$\eta_{\text{ECCD}} = \frac{n_e R I_{\text{CD}}}{P_{\text{EC}}} \approx 0.02\text{-}0.05 \quad [\text{A/W/m}^2]$$

Characteristics of ECCD:

• Localized current profile control is possible
• Particularly effective for NTM control
• Transport improvement through magnetic shear control

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Lower Hybrid Heating and Current Drive

Lower Hybrid Frequency

Lower Hybrid Waves (LHW) are electromagnetic waves in the frequency range between the ion cyclotron frequency and the electron cyclotron frequency.

The lower hybrid frequency is:

$$\omega_{LH} = \omega_{pi}\sqrt{\frac{1}{1 + \omega_{pe}^2/\omega_{ce}^2}}$$

In the high-density limit ($\omega_{pe} \gg \omega_{ce}$):

$$\omega_{LH} \approx \sqrt{\omega_{ce}\omega_{ci}} = \sqrt{\frac{m_e}{m_i}}\omega_{ce}$$

For typical tokamak parameters ($B = 5$ T, deuterium plasma):

$$f_{LH} \approx 1\text{-}5 \text{ GHz}$$

Wave Propagation and Absorption

Lower hybrid waves are absorbed through interaction with parallel electron motion. From the dispersion relation, the parallel component of the phase velocity is:

$$v_\phi = \frac{\omega}{k_\parallel}$$

The condition for efficient electron Landau damping is:

$$v_\phi \approx 3 v_{te}$$

where $v_{te} = \sqrt{2k_B T_e/m_e}$ is the electron thermal velocity.

Converting this condition to wave number gives the accessibility condition:

$$n_\parallel = \frac{ck_\parallel}{\omega} > n_{\parallel,\text{acc}}$$

where the accessibility limit $n_{\parallel,\text{acc}}$ is:

$$n_{\parallel,\text{acc}} = 1 + \frac{\omega_{pe}}{\omega_{ce}}$$

Since $\omega_{pe}$ is large in the plasma core, $n_{\parallel,\text{acc}}$ also becomes large, making it difficult for lower hybrid waves to penetrate to the center. This is the fundamental limitation of LHW.

Waveguide Grill Antenna

Lower hybrid waves are injected into the plasma from waveguide arrays (grills). Grill design parameters:

1. Number of waveguides: Control of $n_\parallel$ spectrum
2. Phase control: Adjusting $n_\parallel$ by phase difference between adjacent waveguides
3. Coupling efficiency: Matching with plasma edge density

Typical grill specifications:

• Frequency: 3.7-5 GHz
• Number of waveguides: 8-32
• $n_\parallel$ range: 1.5-2.5
• Maximum power: Several MW

Lower Hybrid Current Drive (LHCD)

The most important application of lower hybrid waves is current drive. When waves are absorbed by electrons, momentum is also transferred, driving plasma current.

LHCD efficiency:

$$\eta_{\text{LHCD}} = \frac{n_e R I_{\text{CD}}}{P_{\text{LH}}} \approx 0.1\text{-}0.3 \quad [\text{A/W/m}^2]$$

This is high efficiency compared to other current drive methods, making it an important element for steady-state tokamak operation.

Characteristics of LHCD:

• High current drive efficiency
• Effective for current profile control in the plasma periphery
• Difficult to penetrate to the hot plasma core
• Used for current profile tailoring

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Alpha Particle Self-Heating and Ignition Conditions

Principle of Alpha Particle Heating

In the D-T fusion reaction:

$$\text{D} + \text{T} \rightarrow \alpha (3.5 \text{ MeV}) + n (14.1 \text{ MeV})$$

the 3.5 MeV alpha particles produced are confined by the magnetic field and transfer energy to plasma particles through Coulomb collisions. This process is called “alpha particle self-heating.”

The alpha particle heating power is:

$$P_\alpha = \frac{1}{4} n_D n_T \langle\sigma v\rangle_{DT} E_\alpha \cdot V$$

Assuming $n_D = n_T = n/2$:

$$P_\alpha = \frac{1}{16} n^2 \langle\sigma v\rangle_{DT} E_\alpha \cdot V$$

where $E_\alpha = 3.5$ MeV and $\langle\sigma v\rangle_{DT}$ is the fusion reaction rate coefficient.

Alpha Particle Slowing Down

The energy of 3.5 MeV alpha particles is much higher than the plasma temperature (10-20 keV), so they behave as “fast particles.” The energy relaxation process of alpha particles is:

$$\frac{dE_\alpha}{dt} = -\frac{E_\alpha}{\tau_s}$$

The slowing-down time $\tau_s$ is:

$$\tau_s = \frac{3\sqrt{\pi}}{4} \frac{m_\alpha m_e}{Z_\alpha^2 e^4 n_e \ln\Lambda} \left(\frac{k_B T_e}{m_e}\right)^{3/2}$$

Numerically:

$$\tau_s \approx 0.1 \frac{T_e^{3/2}}{n_e} \quad [\text{s}]$$

($T_e$ in keV, $n_e$ in $10^{20}$ m$^{-3}$ units)

For alpha particles to efficiently heat the plasma, $\tau_s$ must be sufficiently shorter than the energy confinement time $\tau_E$. This is satisfied under normal tokamak conditions.

Energy Partition

The energy partition from alpha particles to electrons and ions is determined by the critical energy:

$$E_{c,\alpha} = 14.8 \times 4 \times T_e \left(\frac{n_e Z^2}{n_e \times 2}\right)^{2/3} \approx 33 T_e \quad [\text{keV}]$$

For $T_e = 10$ keV, $E_{c,\alpha} \approx 330$ keV, and since $E_\alpha = 3500$ keV $\gg E_{c,\alpha}$, alpha particle energy is primarily transferred to electrons:

$$\frac{W_e}{W_{\text{total}}} \approx 0.7\text{-}0.8$$

However, energy is also eventually distributed to ions through electron-ion energy exchange.

Ignition Condition

Plasma “ignition” refers to a state where alpha particle self-heating alone can compensate for energy losses, maintaining fusion burn without external heating.

Ignition condition:

$$P_\alpha \geq P_{\text{loss}}$$ $$\frac{1}{16} n^2 \langle\sigma v\rangle_{DT} E_\alpha \geq \frac{3nk_B T}{\tau_E}$$

Rearranging:

$$n\tau_E \geq \frac{48 k_B T}{\langle\sigma v\rangle_{DT} E_\alpha}$$

Ignition condition at $T = 10$ keV (near maximum reaction rate):

$$n\tau_E \geq 1.5 \times 10^{21} \text{ m}^{-3}\text{s}$$

This is more stringent than the normal Lawson criterion, and is expressed as the “triple product” condition including plasma temperature:

$$n\tau_E T \geq 3 \times 10^{21} \text{ m}^{-3}\text{s keV}$$

Fusion Gain Q

The fusion performance of plasma is evaluated by the fusion gain $Q$:

$$Q = \frac{P_{\text{fusion}}}{P_{\text{ext}}}$$

where $P_{\text{fusion}} = 5 P_\alpha$ (including neutron energy) and $P_{\text{ext}}$ is the external heating power.

• $Q = 1$: Breakeven
• $Q = 5$: Alpha heating comparable to external heating
• $Q = 10$: ITER target
• $Q = \infty$: Ignition

In burning plasmas ($Q > 5$), alpha particle behavior significantly affects overall plasma stability and performance.

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Comparison of Heating Methods

Heating Efficiency

Energy transfer efficiency to plasma for each heating method:

Method	Input-to-Plasma Efficiency	Power Supply-to-Input Efficiency	Overall Efficiency
Ohmic	100%	-	-
NBI	70-90%	30-50%	25-40%
ICRH	80-95%	50-70%	45-65%
ECRH	90-99%	35-50%	35-50%
LHW	60-80%	40-50%	25-40%

“Input-to-Plasma Efficiency” indicates the effective power absorption rate into the plasma, and “Power Supply-to-Input Efficiency” indicates the conversion efficiency from the power supply to each heating device.

Current Drive Efficiency

Comparison of non-inductive current drive efficiency:

Method	Efficiency $\eta_{\text{CD}}$ [A/W/m$^2$]	Drive Location
NBCD	0.02-0.05	Global
ECCD	0.02-0.05	Local
LHCD	0.1-0.3	Edge
FWCD	0.01-0.03	Core

LHCD has the highest efficiency but is limited by difficulty in penetrating to the plasma core. For steady-state operation, multiple methods are used in combination.

Cost and Complexity

Rough estimates of capital investment and operating costs for each heating system:

Method	Capital Cost	Operating Cost	Technical Complexity
NBI	High	Medium	High (beam source, neutralizer)
ICRH	Medium	Low	Medium (vacuum tube technology)
ECRH	High	Medium	High (gyrotron)
LHW	Medium	Low	Medium (waveguide technology)

Controllability and Flexibility

Operating control characteristics of each method:

Method	Response Speed	Position Control	Modulation	Feedback
NBI	Seconds	Difficult	Possible	Limited
ICRH	Milliseconds	Medium	Possible	Possible
ECRH	Microseconds	Excellent	Possible	Excellent
LHW	Milliseconds	Limited	Possible	Possible

ECRH offers fast response and precise position control, making it optimal for real-time control of MHD instabilities.

Guidelines for Method Selection

Each heating method has unique characteristics, and appropriate selection based on objectives is important:

When ion heating is required

• NBI (under $E_b < E_c$ conditions)
• ICRH

When electron heating is required

• ECRH
• LHW
• NBI (under $E_b > E_c$ conditions)

When current drive is required

• Core: ECCD, FWCD
• Edge: LHCD
• Global: NBCD

When plasma control is required

• MHD control: ECRH/ECCD
• Rotation drive: NBI
• Current profile control: LHCD, ECCD

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Heating Systems in Experimental Devices

JT-60SA

Heating system of Japan’s superconducting tokamak JT-60SA:

Method	Power	Notes
NBI (positive ion)	24 MW	85 keV
NBI (negative ion)	10 MW	500 keV
ECRH	7 MW	110/138 GHz
Total	41 MW

ITER

Heating system of the international fusion experimental reactor ITER:

Method	Power	Frequency/Energy
NBI (negative ion)	33 MW	1 MeV D$^0$
ICRH	20 MW	40-55 MHz
ECRH	20 MW	170 GHz
Total	73 MW

With these heating systems, ITER targets $Q = 10$ (500 MW fusion output / 50 MW heating input).

Outlook for DEMO Reactors

For commercial fusion power demonstration reactors (DEMO), more efficient and reliable heating systems are required:

1. High-efficiency power supplies: To reduce recirculating power
2. Long-life components: To improve annual availability
3. Remote maintenance capability: Maintenance in activated environments

In particular, higher efficiency gyrotrons (target 60% or more) and long-duration operation (1000 hours or more) are technical challenges.

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Frontiers of Heating Physics

Fast Particle Physics

Fast ions generated by NBI and ICRH, as well as alpha particles, can excite various plasma instabilities:

• Alfven Eigenmodes (AE)
• Fishbone instability
• Toroidal Alfven Eigenmodes (TAE)

These instabilities can scatter fast particles, reducing heating efficiency and causing localized heat loads on the first wall.

Fast particle confinement and stability is an important topic in burning plasma research.

Advances in Harmonic ECRH

In addition to conventional fundamental and second harmonic, research on third harmonic ECRH is advancing. Third harmonic has weak absorption but has potential for application to high-density plasmas, which may become important for future high-density operation scenarios.

Integrated Heating Scenarios

Development of “integrated heating scenarios” that optimally combine multiple heating methods is progressing. For example:

• Current ramp-up: ECRH + LHW
• Main heating phase: NBI + ICRH
• Steady-state maintenance: ECCD + LHCD + NBCD
• Shutdown: Controlled power reduction

Research on automatic adjustment of optimal heating according to plasma state in conjunction with real-time control systems is also ongoing.

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Related Topics

• Charged Particle Motion - Fundamentals of cyclotron motion
• MHD Instabilities - Instabilities related to heating
• Plasma Transport - Physics of energy confinement
• Tokamak - Implementation examples of heating methods
• ITER - Example of large-scale heating systems