Lawson Criterion

The Lawson criterion defines the plasma parameter conditions necessary for a fusion reactor to produce energy in a self-sustaining manner. It was proposed by British physicist J.D. Lawson in 1957. This criterion is one of the most important metrics in fusion research and has been used as a yardstick for measuring fusion development progress for over half a century.

Historical Background of Lawson

John David Lawson conducted research around 1955 at AERE Harwell (Atomic Energy Research Establishment) in the UK to evaluate the feasibility of nuclear fusion. Although fusion research was being conducted secretly at the time, Lawson derived the minimum conditions for a fusion reactor to be viable from a purely thermodynamic perspective.

Lawson’s originality lay in developing a general argument that did not depend on any specific confinement scheme. He analyzed the energy balance of the plasma in detail and expressed the conditions for fusion output to exceed losses as the product of density and confinement time, $n\tau$.

His paper “Some Criteria for a Power Producing Thermonuclear Reactor,” declassified and published in 1957, became a foundational document that set the direction for fusion research.

Fundamental Principles of Energy Balance

The most important aspect in fusion reactor design is understanding the energy balance. By analyzing the energy flowing into and out of the plasma in detail, the conditions for reactor viability can be derived.

Plasma Energy Density

Consider a thermal equilibrium plasma at temperature $T$. When ions and electrons exist with densities $n_i$ and $n_e$ respectively, the thermal energy per unit volume is:

$$w = \frac{3}{2}n_i k_B T_i + \frac{3}{2}n_e k_B T_e$$

The electrical neutrality condition $n_e = Z n_i$ (where $Z$ is the ion charge number) holds. For DT plasma, $Z = 1$, so $n_e = n_i = n$. Also, in thermal equilibrium, we can assume $T_i = T_e = T$. Then:

$$w = \frac{3}{2}n k_B T + \frac{3}{2}n k_B T = 3n k_B T$$

In plasma physics, temperature is often expressed in energy units (keV), simplifying $k_B T \to T$:

$$w = 3nT$$

Here $T$ is in keV and $n$ is in m$^{-3}$.

Input Power to the Plasma

The main mechanisms for supplying energy to the plasma are as follows.

External heating power $P_\text{heat}$ is the power artificially injected into the plasma:

• Neutral Beam Injection (NBI): Injecting high-speed neutral atom beams into the plasma
• Ion Cyclotron Resonance Heating (ICRH): Irradiating with radio frequency at the ion cyclotron frequency
• Electron Cyclotron Resonance Heating (ECRH): Irradiating with microwaves at the electron cyclotron frequency
• Lower Hybrid Current Drive (LHCD): Heating and current drive using lower hybrid waves

Ohmic heating $P_\Omega$ is the Joule heat generated when plasma current $I_p$ flows in a tokamak:

$$P_\Omega = \eta j^2 V = R_p I_p^2$$

Here $\eta$ is the electrical resistivity, $j$ is the current density, and $R_p$ is the plasma resistance. According to Spitzer resistivity:

$$\eta \propto T^{-3/2}$$

In high-temperature plasmas, the resistivity becomes extremely small, so ohmic heating becomes ineffective at temperatures above a few keV.

Alpha particle heating $P_\alpha$ is the power that heats the plasma from alpha particles (helium-4 nuclei) produced in DT fusion reactions. This will be discussed later.

Power Loss from the Plasma

Let’s examine in detail the main mechanisms by which energy is lost from the plasma.

Thermal conduction loss $P_\text{loss}$ is the loss due to plasma particles and energy flowing out from the plasma region. In magnetic confinement schemes, transport across magnetic field lines (neoclassical transport, turbulent transport) is the main cause.

The energy confinement time $\tau_E$ is defined to characterize this loss:

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

Here $W = wV = 3nTV$ is the total stored energy of the plasma. Per unit volume:

$$p_\text{loss} = \frac{3nT}{\tau_E}$$

Bremsstrahlung loss $P_\text{br}$ is the phenomenon where electrons emit electromagnetic radiation when accelerated (decelerated) in the electric field of ions. The bremsstrahlung power density derived from classical electromagnetism is:

$$p_\text{br} = C_\text{br} n_e n_i Z^2 \sqrt{T_e}$$

For DT plasma ($Z = 1$, $n_e = n_i = n$):

$$p_\text{br} = C_\text{br} n^2 \sqrt{T}$$

The bremsstrahlung coefficient $C_\text{br}$, including quantum mechanical corrections, is:

$$C_\text{br} \approx 5.35 \times 10^{-37} \text{ W} \cdot \text{m}^3 \cdot \text{keV}^{-1/2}$$

Bremsstrahlung increases rapidly in proportion to $Z^2$ when impurities (high-Z elements) are mixed into the plasma. This is one reason why impurity control is important in fusion reactors.

Synchrotron radiation loss $P_\text{syn}$ is the power radiated when electrons rotate around magnetic field lines (cyclotron motion):

$$p_\text{syn} = C_\text{syn} n T^2 B^2$$

This becomes a non-negligible loss in high-temperature, high-magnetic-field plasmas, but most of the radiation is reabsorbed by the plasma itself, so the net loss also depends on the wall reflectivity.

Line radiation loss $P_\text{line}$ is the line spectrum radiation accompanying electron excitation and de-excitation of impurity ions. It strongly depends on impurity concentration and species, and becomes particularly important in the plasma periphery.

Steady-State Energy Balance Equation

In steady state, the input power to the plasma and the loss power balance:

$$P_\text{heat} + P_\alpha + P_\Omega = P_\text{loss} + P_\text{br} + P_\text{syn} + P_\text{line}$$

In high-temperature plasmas, ohmic heating can be neglected, and assuming a pure DT plasma, synchrotron radiation and line radiation are not the main terms, so simplifying:

$$P_\text{heat} + P_\alpha = P_\text{loss} + P_\text{br}$$

Written per unit volume:

$$p_\text{heat} + p_\alpha = p_\text{loss} + p_\text{br}$$

Fusion Reactions and Output

Fusion Reaction Rate

For the DT reaction, the number of reactions per unit volume per unit time (reaction rate density) is:

$$R_{DT} = n_D n_T \langle\sigma v\rangle_{DT}$$

Here $\langle\sigma v\rangle_{DT}$ is the reactivity (the product of reaction cross-section $\sigma$ and relative velocity $v$ averaged over a Maxwellian distribution).

For a 50:50 mixed DT plasma ($n_D = n_T = n/2$):

$$R_{DT} = \frac{n^2}{4} \langle\sigma v\rangle_{DT}$$

Temperature Dependence of Reactivity

$\langle\sigma v\rangle$ is a strong function of temperature. For the DT reaction, an approximate formula is:

$$\langle\sigma v\rangle_{DT} \approx 1.1 \times 10^{-24} T^2 \text{ m}^3/\text{s} \quad (10 \lesssim T \lesssim 20 \text{ keV})$$

For a wider temperature range, the Bosch-Hale approximation is commonly used:

$$\langle\sigma v\rangle = C_1 \theta \sqrt{\frac{\xi}{m_r c^2 T^3}} \exp(-3\xi)$$

Where:

• $\theta = T / (1 - \frac{T(C_2 + T(C_4 + TC_6))}{1 + T(C_3 + T(C_5 + TC_7))})$
• $\xi = (B_G^2 / 4\theta)^{1/3}$
• $B_G$ is the Gamow factor
• $C_1, C_2, ...$ are coefficients determined for each reaction
• $m_r$ is the reduced mass

For the DT reaction, $\langle\sigma v\rangle$ reaches its maximum value of approximately $8.5 \times 10^{-22}$ m$^3$/s at about $T = 64$ keV, but in practice, operation is often at around 10-20 keV. This is due to the following reasons:

1. Bremsstrahlung loss increases at higher temperatures
2. Due to plasma pressure constraints ($\propto nT$), density must be reduced at higher temperatures
3. Confinement performance depends on temperature

Fusion Power Density

The energy released per DT reaction is $E_f = 17.6$ MeV. Of this:

• Neutron: $E_n = 14.1$ MeV (4/5 by mass ratio)
• Alpha particle: $E_\alpha = 3.5$ MeV (1/5 by mass ratio)

The fusion power density per unit volume is:

$$p_f = R_{DT} \cdot E_f = \frac{n^2}{4} \langle\sigma v\rangle E_f$$

Substituting numerical values:

$$p_f = \frac{n^2}{4} \langle\sigma v\rangle \times 17.6 \text{ MeV} = \frac{n^2}{4} \langle\sigma v\rangle \times 2.82 \times 10^{-12} \text{ J}$$

Alpha Particle Heating

Neutrons are electrically neutral and escape directly from the plasma, depositing their energy in the blanket. On the other hand, alpha particles have a charge of $+2e$ and are confined by the magnetic field, decelerating in the plasma and transferring their energy to it.

The alpha particle heating power density is:

$$p_\alpha = R_{DT} \cdot E_\alpha = \frac{n^2}{4} \langle\sigma v\rangle E_\alpha = \frac{E_\alpha}{E_f} p_f = \frac{1}{5} p_f$$

The slowing-down process of alpha particles proceeds as follows:

1. Alpha particle energy immediately after production: 3.5 MeV
2. Slowing down due to collisions with electrons (mainly in the high-energy region)
3. Slowing down due to collisions with ions (in the low-energy region)
4. Thermalization to become helium ash

The critical energy $E_c$ is the energy at which electron slowing-down and ion slowing-down are equal:

$$E_c \approx 14.8 \, T_e \left(\frac{A_\alpha}{A_i}\right)^{2/3} \sum_j \frac{n_j Z_j^2}{n_e A_j}$$

For DT plasma, $E_c \approx 33 T_e$ keV. When $T_e = 10$ keV, $E_c \approx 330$ keV, and the 3.5 MeV alpha particle initially heats mainly electrons, then ion heating becomes dominant below $E_c$.

Fusion Gain Q

The most important indicator of fusion reactor performance is the fusion gain $Q$:

$$Q \equiv \frac{P_f}{P_\text{heat}}$$

This is the ratio of total fusion power $P_f$ to externally injected heating power $P_\text{heat}$.

Relationship Between Q Value and Various Conditions

The state of a fusion reactor can be classified by the value of $Q$.

$Q = 0$: No fusion reaction is occurring (pure heating experiment).

$0 < Q < 1$: Fusion reactions are occurring, but output is smaller than input power. Many experimental devices are in this range.

$Q = 1$ (critical plasma condition, scientific breakeven): The state where fusion output equals the injected heating power. However, this is a comparison with the net power injected into the plasma, not the “energy balance” as seen from the wall outlet.

$Q = 5$: This value corresponds to the condition where alpha particle heating power $P_\alpha = P_f/5$ equals the external heating power $P_\text{heat}$. This is the region where self-heating begins to dominate.

$Q = 10$ (ITER’s target): The condition to achieve 500 MW fusion output with 50 MW external heating. Alpha particle heating (100 MW) is twice the external heating, enabling the study of burning plasma physics.

$Q \to \infty$ (ignition condition): The state where fusion reactions are sustained without external heating. Since $P_\text{heat} = 0$, mathematically $Q = \infty$.

Engineering Q Value

When considering actual fusion power generation, the engineering Q value $Q_E$ is also important:

$$Q_E = \frac{P_\text{electric,out}}{P_\text{electric,in}}$$

Here $P_\text{electric,out}$ is the electrical output and $P_\text{electric,in}$ is the total power consumption including auxiliary equipment.

If the conversion efficiency from fusion output to electricity is $\eta_\text{th}$ (thermal efficiency, about 30-40%) and the heating system efficiency is $\eta_\text{heat}$ (about 30-70%, depending on the method), then:

$$Q_E \approx \eta_\text{th} \cdot Q / (1/\eta_\text{heat} + \text{other losses})$$

$Q_E > 1$ is required for economical power generation, which is estimated to require approximately $Q \gtrsim 20-30$.

Detailed Derivation of the Lawson Criterion

Here we rigorously derive the Lawson criterion.

Starting Point: Energy Balance

We start from the steady-state energy balance equation:

$$p_\text{heat} + p_\alpha = p_\text{loss} + p_\text{br}$$

Expanding each term:

$$p_\text{heat} + \frac{n^2}{4} \langle\sigma v\rangle E_\alpha = \frac{3nT}{\tau_E} + C_\text{br} n^2 \sqrt{T}$$

Expression Using Q

From the fusion power density $p_f = \frac{n^2}{4} \langle\sigma v\rangle E_f$ and $Q = P_f/P_\text{heat}$:

$$p_\text{heat} = \frac{p_f}{Q} = \frac{n^2 \langle\sigma v\rangle E_f}{4Q}$$

Substituting this:

$$\frac{n^2 \langle\sigma v\rangle E_f}{4Q} + \frac{n^2 \langle\sigma v\rangle E_\alpha}{4} = \frac{3nT}{\tau_E} + C_\text{br} n^2 \sqrt{T}$$

Combining the left side:

$$\frac{n^2 \langle\sigma v\rangle}{4} \left( \frac{E_f}{Q} + E_\alpha \right) = \frac{3nT}{\tau_E} + C_\text{br} n^2 \sqrt{T}$$

Using $E_\alpha = E_f/5$:

$$\frac{n^2 \langle\sigma v\rangle E_f}{4} \left( \frac{1}{Q} + \frac{1}{5} \right) = \frac{3nT}{\tau_E} + C_\text{br} n^2 \sqrt{T}$$

Solving for $n\tau_E$

We solve the above equation for $n\tau_E$. First, dividing by $n$:

$$\frac{n \langle\sigma v\rangle E_f}{4} \left( \frac{1}{Q} + \frac{1}{5} \right) = \frac{3T}{\tau_E} + C_\text{br} n \sqrt{T}$$

Moving the term containing $\tau_E$:

$$\frac{3T}{\tau_E} = \frac{n \langle\sigma v\rangle E_f}{4} \left( \frac{1}{Q} + \frac{1}{5} \right) - C_\text{br} n \sqrt{T}$$

Multiplying both sides by $\tau_E/3T$:

$$1 = \frac{n \tau_E \langle\sigma v\rangle E_f}{12T} \left( \frac{1}{Q} + \frac{1}{5} \right) - \frac{C_\text{br} n \tau_E \sqrt{T}}{3T}$$

Solving for $n\tau_E$:

$$n\tau_E = \frac{12T}{\langle\sigma v\rangle E_f \left( \frac{1}{Q} + \frac{1}{5} \right) - \frac{4 C_\text{br} \sqrt{T}}{1}}$$

The second term in the denominator represents the effect of bremsstrahlung, and more precisely:

$$n\tau_E = \frac{12T}{\langle\sigma v\rangle E_f \left( \frac{1}{Q} + \frac{1}{5} \right) - 4 C_\text{br} \sqrt{T} / n}$$

In the approximation neglecting bremsstrahlung:

$$n\tau_E = \frac{12T}{\langle\sigma v\rangle E_f \left( \frac{Q+5}{5Q} \right)} = \frac{60 Q T}{(Q+5) \langle\sigma v\rangle E_f}$$

Special Cases

Breakeven condition ($Q = 1$):

$$(n\tau_E)_{Q=1} = \frac{60 T}{6 \langle\sigma v\rangle E_f} = \frac{10 T}{\langle\sigma v\rangle E_f}$$

Ignition condition ($Q \to \infty$):

$$(n\tau_E)_\text{ign} = \frac{60 T}{5 \langle\sigma v\rangle E_f} = \frac{12 T}{\langle\sigma v\rangle E_f}$$

We can see that the ignition condition is about 1.2 times more stringent than the $Q = 1$ condition.

Rigorous Derivation Including Bremsstrahlung

We perform a complete derivation including bremsstrahlung loss. Energy balance:

$$p_\alpha - p_\text{loss} - p_\text{br} = -p_\text{heat}$$

For the ignition condition ($p_\text{heat} = 0$):

$$p_\alpha = p_\text{loss} + p_\text{br}$$ $$\frac{n^2}{4} \langle\sigma v\rangle E_\alpha = \frac{3nT}{\tau_E} + C_\text{br} n^2 \sqrt{T}$$

Dividing by $n$ and collecting terms for $n\tau_E$:

$$n\tau_E = \frac{3T}{\frac{\langle\sigma v\rangle E_\alpha}{4} - C_\text{br} \sqrt{T}}$$

The denominator must be positive, which means:

$$\frac{\langle\sigma v\rangle E_\alpha}{4} > C_\text{br} \sqrt{T}$$ $$\langle\sigma v\rangle > \frac{4 C_\text{br} \sqrt{T}}{E_\alpha}$$

At low temperatures the reaction rate is too small, and at high temperatures bremsstrahlung increases, so there exists a temperature range where ignition is possible. The lower temperature limit satisfying this condition is about 4 keV.

Minimum Value of $n\tau_E$

For the ignition condition, viewing $(n\tau_E)_\text{ign}$ as a function of temperature $T$, it has a minimum value. This is found by solving:

$$\frac{d}{dT}(n\tau_E)_\text{ign} = 0$$

For the DT reaction, the minimum is at $T \approx 25-30$ keV:

$$(n\tau_E)_\text{ign,min} \approx 1.5 \times 10^{20} \text{ m}^{-3} \cdot \text{s}$$

Fusion Triple Product

Definition and Significance of the Triple Product

In actual fusion devices, density $n$, temperature $T$, and confinement time $\tau_E$ cannot be controlled independently. They are interrelated, and increasing one may decrease another. Therefore, the triple product $nT\tau_E$ (also called the fusion product) is used as a comprehensive indicator of plasma performance.

In the Lawson criterion $n\tau_E = f(T)$, there is a temperature range (10-20 keV) where $f(T)/T$ is approximately constant regardless of $T$. In this case:

$$n\tau_E T \approx \text{const.}$$

For the ignition condition:

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

Or in SI units:

$$n\tau_E T \gtrsim 5 \times 10^{21} \text{ m}^{-3} \cdot \text{s} \cdot \text{K}$$

Physical Meaning of the Triple Product

Each factor in the triple product has the following physical meaning.

Density $n$ [m$^{-3}$] determines the frequency of reactions. Since the fusion reaction rate is proportional to $n^2$, higher density leads to more reactions. In magnetic confinement schemes, $n \sim 10^{20}$ m$^{-3}$, while in inertial confinement, compression can reach $n \sim 10^{31}$ m$^{-3}$.

Temperature $T$ [keV] determines the “quality” of reactions. To cause fusion reactions, kinetic energy sufficient to overcome the Coulomb repulsion is required. Higher temperature means larger reaction cross-section (reactivity $\langle\sigma v\rangle$), but losses also increase, so an optimal temperature exists. For DT reactions, 10-20 keV (100-200 million degrees) is the optimal range.

Confinement time $\tau_E$ [s] represents the energy retention capability. It indicates how long the energy stored in the plasma is retained; the longer it is, the more efficiently fusion reactions can be sustained. Magnetic confinement aims for $\tau_E \sim 1-10$ s, while inertial confinement has extremely short $\tau_E \sim 10^{-11}$ s, but compensates with density.

Strategies for Achieving the Triple Product

To increase the triple product, each parameter must be improved, but they are not independent.

When increasing density:

• Reaction rate increases as $n^2$ (favorable)
• Bremsstrahlung loss also increases as $n^2$ (unfavorable)
• Confinement time has density dependence (varies by device and mode)

When increasing temperature:

• Reactivity increases (favorable, but saturates above optimal temperature)
• Bremsstrahlung loss increases as $\sqrt{T}$ (unfavorable)
• Thermal energy increases (relaxes $n\tau_E$ requirement)

To extend confinement time:

• Increase device size (approximately $\tau_E \propto a^2$ correlation)
• Increase magnetic field (use of superconducting magnets)
• Improve plasma operating mode (H-mode, etc.)

Difference Between Ignition Condition and Q=1 Condition

The ignition condition and $Q = 1$ (scientific breakeven) are often confused, but they are physically different concepts.

Scientific Breakeven ($Q = 1$)

$Q = 1$ means:

$$P_f = P_\text{heat}$$

This is the state where fusion output equals the heating power injected into the plasma. However:

• External heating is still required
• Alpha particle heating power $P_\alpha = P_f/5 = P_\text{heat}/5$ is only 1/5 of external heating
• The plasma is not self-sustaining

The energy balance is:

$$P_\text{heat} + P_\alpha = P_\text{loss} + P_\text{br}$$ $$P_\text{heat} + \frac{P_\text{heat}}{5} = P_\text{loss} + P_\text{br}$$ $$\frac{6}{5} P_\text{heat} = P_\text{loss} + P_\text{br}$$

Ignition Condition ($Q = \infty$)

The ignition condition is:

$$P_\alpha \geq P_\text{loss} + P_\text{br}$$ $$P_\text{heat} = 0$$

This is the state where the plasma can be maintained by alpha particle heating alone. In this case:

• No external heating power is required (for maintaining steady state)
• All heating is performed by alpha particles
• The plasma is self-sustaining

Calculating the ratio between both conditions, neglecting bremsstrahlung:

$$\frac{(n\tau_E)_\text{ign}}{(n\tau_E)_{Q=1}} = \frac{12T/(\langle\sigma v\rangle E_f)}{10T/(\langle\sigma v\rangle E_f)} = \frac{12}{10} = 1.2$$

Thus, the ignition condition is only about 20% more stringent than the $Q = 1$ condition. However, in practice, as ignition is approached, temperature rise due to self-heating occurs, creating control difficulties.

Controlled Burn

In practical fusion reactors, it is expected that controlled burn will be performed with some external heating maintained, rather than operating just at the ignition condition. Operating at $Q = 20-50$ enables:

• Easy control of burning power
• Prevention of thermal runaway
• Optimization of plasma density and temperature profiles

Role of Self-Heating (Alpha Particle Heating)

Alpha Particle Confinement and Transport

For alpha particle heating to work effectively, the generated alpha particles must be confined in the plasma until they slow down and thermalize.

The Larmor radius (cyclotron radius) of a 3.5 MeV alpha particle is:

$$\rho_\alpha = \frac{m_\alpha v_\alpha}{eB} \approx \frac{\sqrt{2 m_\alpha E_\alpha}}{2eB}$$

In a tokamak with $B = 5$ T, $\rho_\alpha \approx 5$ cm. If this value is sufficiently smaller than the plasma radius $a$, alpha particles are confined while rotating around magnetic field lines.

However, alpha particle losses occur through the following mechanisms:

• Ripple loss: Deviation from orbit due to toroidal field ripple (undulation)
• Instability-driven loss: Collective instabilities caused by alpha particles (fishbone instability, toroidal Alfven eigenmodes, etc.)
• Collision with first wall: When the bottom of the banana orbit reaches the first wall

In large devices like ITER, an alpha particle confinement rate of over 90% is expected.

Alpha Particle Heating Efficiency

If the energy transfer efficiency from alpha particles to the background plasma is $f_\alpha$, the effective alpha particle heating power is:

$$P_\alpha^\text{eff} = f_\alpha P_\alpha$$

Factors causing $f_\alpha < 1$:

• Loss before thermalization
• Energy transport due to instabilities
• Energy outflow to the wall

The ignition condition is modified to:

$$f_\alpha P_\alpha \geq P_\text{loss} + P_\text{br}$$

If $f_\alpha = 0.9$, the required $n\tau_E$ for ignition increases by about 10%.

Burning Plasma Physics

In the region where $Q > 5$, alpha particle heating exceeds external heating, and the plasma exhibits characteristic properties as a “burning plasma.”

The distribution function of alpha particles is not isotropic but has a non-thermal distribution according to the slowing-down process:

$$f_\alpha(E) \propto \frac{1}{E^{3/2} + E_c^{3/2}}$$

This high-energy alpha particle population can:

1. Drive instabilities through resonance with Alfven waves
2. Contribute to internal transport barrier formation in negative shear regions
3. Affect plasma rotation and momentum transport

ITER aims for $Q = 10$ and will be the first device to experimentally study this burning plasma physics.

Helium Ash Accumulation

When alpha particles thermalize, they become “helium ash” (thermalized helium) having lost their 3.5 MeV kinetic energy. This accumulates in the plasma, causing the following problems:

• Fuel dilution: Under the constraint $n_D + n_T + n_{He} + n_e =$ constant (pressure constraint), an increase in helium means a decrease in DT fuel density
• Increased bremsstrahlung: Helium ($Z = 2$) has larger bremsstrahlung than hydrogen isotopes ($Z = 1$)

Helium ash exhaust is one of the important functions of the divertor system. For steady-state operation, the same amount of helium as produced by fusion must be exhausted.

Approaches to Achieving the Lawson Criterion in Various Confinement Schemes

Tokamak

The tokamak is currently the most advanced magnetic confinement scheme and is the device closest to achieving the Lawson criterion.

The confinement time scaling in tokamaks is empirically derived from databases of many devices. The representative IPB98(y,2) scaling is:

$$\tau_E = 0.0562 \, I_p^{0.93} B^{0.15} P^{-0.69} n^{0.41} M^{0.19} R^{1.97} \kappa^{0.78} \epsilon^{0.58}$$

Here $I_p$ is the plasma current [MA], $B$ is the toroidal magnetic field [T], $P$ is the heating power [MW], $n$ is the line-averaged density [$10^{19}$ m$^{-3}$], $M$ is the average ion mass number, $R$ is the major radius [m], $\kappa$ is the elongation, and $\epsilon = a/R$ is the inverse aspect ratio.

Directions for confinement improvement that can be read from this scaling:

• Increase device size (approximately $\tau_E \propto R^2$)
• Increase plasma current
• Reduce heating power (degradation scaling)

Also, by transitioning to H-mode (high confinement mode), the confinement time improves to about twice that of L-mode.

ITER is designed with $R = 6.2$ m, $B = 5.3$ T, $I_p = 15$ MA, and aims to achieve $\tau_E \approx 3.7$ s, $n \approx 10^{20}$ m$^{-3}$, $T \approx 8.8$ keV, targeting $Q = 10$.

Stellarator

Unlike tokamaks, stellarators form the confining magnetic field with external coils alone, so no plasma current is required and inherently steady-state operation is possible.

The stellarator confinement time scaling (ISS04) is:

$$\tau_E = 0.134 \, a^{2.28} R^{0.64} P^{-0.61} n^{0.54} B^{0.84} \iota_{2/3}^{0.41}$$

Here $\iota_{2/3}$ is the rotational transform at $r/a = 2/3$.

Main characteristics:

• Strong dependence on magnetic field strength ($B^{0.84}$)
• Large dependence on size ($a^{2.28}$)
• No disruptions (advantageous for steady-state operation)

Wendelstein 7-X in Germany, as an optimized stellarator, aims to achieve $\tau_E > 1$ s. However, stellarators are difficult to manufacture due to their complex coil geometry and are behind tokamaks in development.

Inertial Confinement Fusion (ICF)

In inertial confinement schemes, the Lawson criterion is expressed in terms of areal density $\rho R$, where $\rho$ is the mass density and $R$ is the fuel pellet radius.

The inertial confinement time is given by the hydrodynamic disassembly time:

$$\tau \sim \frac{R}{c_s}$$

Here $c_s$ is the sound speed. Substituting density $n = \rho/m_i$:

$$n\tau \sim \frac{\rho R}{m_i c_s}$$

The Lawson criterion for inertial confinement is:

$$\rho R \gtrsim 3 \text{ g/cm}^2$$

To achieve this condition, the fuel must be compressed by several thousand times ($\rho/\rho_0 \sim 10^3$).

In December 2022, NIF (National Ignition Facility) achieved 3.15 MJ of fusion output for 2.05 MJ of laser energy, demonstrating scientific breakeven (in the laser-to-target sense). However, there remain significant challenges when including laser generation efficiency in the overall efficiency.

Other Schemes

Various alternative confinement schemes are being studied, including FRC (Field-Reversed Configuration), spheromak, and Z-pinch. These generally have the potential for compact, low-cost reactors, but demonstration of confinement performance has not progressed as far as tokamaks.

Recently, small high-field tokamaks using High-Temperature Superconducting (HTS) magnets (such as SPARC by Commonwealth Fusion Systems) have attracted attention. By increasing the magnetic field, under the constraint:

$$\beta \equiv \frac{p}{B^2/2\mu_0} = \text{const.}$$

higher pressure ($p = nT$) can be achieved, enabling device compactification.

Historical Progression of Achieved Values

Dawn Era (1950s-1970s)

In the early days of fusion research, plasma confinement itself was a major challenge.

1958: Fusion research was made public at the Geneva Atoms for Peace Conference 1960s: Approximately $n\tau_E \sim 10^{16}$ m$^{-3}$・s 1968: Soviet T-3 tokamak achieved $T \sim 1$ keV, $n\tau_E \sim 10^{18}$ m$^{-3}$・s

The success of T-3 established the tokamak as the mainstream approach to magnetic confinement.

Large Tokamak Era (1980s-1990s)

From the 1980s, the three large tokamaks TFTR (USA), JET (Europe), and JT-60 (Japan) were constructed, and fusion research made great progress.

TFTR (Tokamak Fusion Test Reactor):

• 1994: Achieved 10.7 MW fusion output in DT experiments
• $n\tau_E T \sim 6 \times 10^{20}$ m$^{-3}$・s・keV

JET (Joint European Torus):

• 1997: Achieved 16.1 MW fusion output, $Q = 0.67$ in DT experiments
• 2021-2022: 59 MJ fusion energy at $Q \approx 0.33$ (5-second average of 11.8 MW)
• $n\tau_E T \sim 1.2 \times 10^{21}$ m$^{-3}$・s・keV

JT-60 / JT-60U (Japan):

• 1996: Achieved ion temperature of 520 million degrees (45 keV) (world record)
• Although using DD experiments, demonstrated performance equivalent to $Q_\text{eq} \sim 1.25$ in DT equivalent
• $n\tau_E T \sim 1.5 \times 10^{21}$ m$^{-3}$・s・keV (world record)

These devices demonstrated performance approaching the breakeven condition ($Q = 1$).

Modern Era (2000s onwards)

From the 2000s, emphasis has been placed on ITER construction preparations and more detailed physics understanding.

KSTAR (Korea, 2008-):

• Superconducting tokamak
• Demonstrated long-duration maintenance of high-performance plasma

EAST (China, 2006-):

• Fully superconducting tokamak
• 2021: Maintained 120-million-degree plasma for 101 seconds

JT-60SA (Japan, 2023-):

• Successor to JT-60U, superconducting
• Development of advanced operation scenarios complementing ITER

Overview of Triple Product Progression Graph

The achieved value of the fusion triple product has improved by about 10 orders of magnitude over the past 50 years:

Era	Representative Value [m$^{-3}$・s・keV]	Main Device
1965	$\sim 10^{16}$	Early tokamaks
1975	$\sim 10^{18}$	PLT
1985	$\sim 10^{19}$	TFTR, JET (early)
1995	$\sim 10^{21}$	TFTR, JET, JT-60U
2025	$\sim 10^{21}$	JET, JT-60SA
2035 (projected)	$\sim 3 \times 10^{21}$	ITER ($Q = 10$)

This rate of improvement is comparable to “Moore’s Law” in semiconductors.

Target Values for ITER and Future Reactors

ITER Design Targets

ITER is an international project aimed at demonstrating the scientific and engineering feasibility of fusion energy.

Main parameters:

• Major radius $R = 6.2$ m
• Minor radius $a = 2.0$ m
• Toroidal magnetic field $B = 5.3$ T
• Plasma current $I_p = 15$ MA (inductive operation), 9 MA (steady-state operation)
• Heating power $P_\text{heat} = 50$ MW
• Fusion output $P_f = 500$ MW (target)

Operational targets:

Operation Mode	Q Value	Burn Time	$n\tau_E T$
Inductive	10	400 s	$\sim 3 \times 10^{21}$
Steady-state	5	>1000 s	$\sim 2 \times 10^{21}$

ITER’s achievement of $Q = 10$ corresponds to a burning plasma state where alpha particle heating power (100 MW) is twice the external heating (50 MW).

Prospects for Achieving Conditions at ITER

Conditions required for ITER to achieve $Q = 10$:

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

Design values:

• $n \approx 10^{20}$ m$^{-3}$ (about 85% of Greenwald density limit)
• $\tau_E \approx 3.7$ s (H = 1 for IPB98(y,2) scaling)
• $T \approx 8.8$ keV (optimization of $\langle\sigma v\rangle$)

This gives an expected $n\tau_E T \approx 3.2 \times 10^{21}$ m$^{-3}$・s・keV.

Requirements for the Demonstration Reactor (DEMO)

The demonstration reactor (DEMO) planned as the next stage after ITER will be the first fusion reactor to actually generate electricity.

DEMO requirements:

Parameter	ITER	DEMO
Fusion output	500 MW	2000-3000 MW
Q value	10	25-50
Burn time	400 s	Steady-state (continuous)
Net electricity	None	300-500 MW

For DEMO, $Q \gtrsim 25$ is estimated to be necessary, which corresponds to:

$$n\tau_E T \gtrsim 5 \times 10^{21} \text{ m}^{-3} \cdot \text{s} \cdot \text{keV}$$

An improvement of about 1.5-2 times the confinement performance demonstrated at ITER is required.

Future Commercial Reactors

From an economic perspective, commercial fusion reactors will require:

• $Q_E > 1$ (net electricity generation)
• High availability (>80%)
• Low cost (<$100/kW construction cost)
• High reliability

Physically, operation at $Q \sim 30-50$ is anticipated, which is less stringent than the ignition condition, but technology to maintain stable burning for long periods is required.

High-Field Approach

Recently, advances in High-Temperature Superconducting (HTS) magnets have opened the possibility of small high-performance reactors using stronger magnetic fields ($B \sim 12-20$ T) than before.

The fusion power density is:

$$p_f \propto n^2 \langle\sigma v\rangle \propto \beta^2 B^4$$

Since it is proportional to the fourth power of the magnetic field, doubling the field gives 16 times the power density at the same $\beta$. This has the potential to significantly reduce device size.

SPARC (Commonwealth Fusion Systems) is a high-field tokamak with $B \sim 12$ T, aiming to achieve $Q > 2$ around 2025.

Lawson Diagram

Structure of the Diagram

The Lawson diagram plots temperature $T$ (keV) on the horizontal axis and $n\tau_E$ (m$^{-3}$・s) on the vertical axis.

On this diagram:

• Constant Q curves ($Q = 1$, $Q = 10$, $Q = \infty$, etc.)
• Achievement points of various devices

are plotted, allowing visual understanding of the progress of fusion research.

Shape of Constant Q Curves

The constant $Q$ curves are derived from the complete Lawson criterion including bremsstrahlung:

$$n\tau_E = \frac{12T}{\langle\sigma v\rangle E_f (1/Q + 1/5) - 4C_\text{br}\sqrt{T}/n}$$

Characteristics:

• Low temperature side ($T < 4$ keV): Bremsstrahlung loss dominates and the curve rises steeply
• Intermediate temperature range (10-30 keV): The curve has a minimum and is nearly flat
• High temperature side ($T > 50$ keV): Due to saturation of reaction rate, the curve rises again

The ignition condition ($Q = \infty$) curve is at the top, and curves move downward as $Q$ decreases.

Triple Product Plot

As an alternative representation, a triple product plot with $T$ on the horizontal axis and $n\tau_E T$ on the vertical axis is also commonly used. In this case, the constant $Q$ curves become nearly horizontal in the 10-20 keV range, making the target values to be achieved intuitively understandable.

Ignition condition: $n\tau_E T \gtrsim 3 \times 10^{21}$ m$^{-3}$・s・keV Breakeven ($Q = 1$): $n\tau_E T \gtrsim 2 \times 10^{21}$ m$^{-3}$・s・keV

Future Outlook and Challenges

Physical Challenges

Physical challenges toward achieving the Lawson criterion include the following.

Confinement improvement: Understanding and control of turbulent transport, formation and maintenance of internal transport barriers, peripheral plasma control (ELM mitigation, etc.).

Burning plasma physics: Confinement and thermalization of alpha particles, control of alpha particle-driven instabilities, operation scenarios under self-heating.

Steady-state operation: Efficiency improvement of non-inductive current drive, helium ash exhaust, steady-state handling of heat and particle fluxes.

Engineering Challenges

Even if a plasma satisfying the Lawson criterion is achieved, many engineering challenges remain for viability as a power plant.

Materials: Resistance to 14 MeV neutrons, resistance to high heat loads, tritium breeding and recovery.

Systems: Large-scale and high-field superconducting magnets, remote maintenance technology, tritium safety systems.

Economics: Reduction of construction costs, improvement of availability, optimization of power generation efficiency.

Beyond the Lawson Criterion

The Lawson criterion indicates the viability conditions for a fusion reactor, but meeting it alone is not sufficient for a practical reactor.

• Sustaining stable burning (disruption avoidance)
• Controlled power adjustment
• Establishment of fuel cycle
• Economic viability

A design that comprehensively satisfies these requirements is necessary for the realization of future fusion power generation.

Summary

The Lawson criterion is a fundamental indicator showing the minimum physical conditions necessary for fusion reactor viability.

1. Derived by J.D. Lawson in 1957, a condition on $n\tau_E$
2. Today, the triple product $n\tau_E T$ is more commonly used
3. Ignition condition is $n\tau_E T \gtrsim 3 \times 10^{21}$ m$^{-3}$・s・keV
4. About 10 orders of magnitude improvement achieved over the past 50 years
5. ITER aims to achieve $Q = 10$ and demonstrate burning plasma physics
6. Future reactors require $Q \gtrsim 30$ from an economic perspective

Fusion research has come a long way and is now on the verge of achieving the Lawson criterion. Following burning plasma experiments at ITER, fusion power generation is expected to become a reality in the latter half of the 21st century.

Related Topics

• What is Fusion - Fundamental principles of fusion reactions
• Fusion Reactions - Details of DT reactions, DD reactions, etc.
• Tokamak - Representative magnetic confinement scheme
• Stellarator - Steady-state confinement with external coils
• ITER - International Thermonuclear Experimental Reactor project
• JT-60SA - Japan’s large tokamak device
• SPARC - High-field compact tokamak
• Plasma Physics Overview - Basic properties of plasma
• Plasma Heating - Details of external heating methods