Principles of Plasma Heating

What physics can we use to raise the core of a fusion reactor to the staggering temperature of roughly 100 million degrees? This page walks through, in order from everyday analogies to the research frontier, why such extreme heat is needed and what physical principles underlie the main methods of heating a plasma. The engineering details of real heating devices are left to Heating Systems; here we focus on the physics of why heating works at all.

Start with Intuition (High School)

Fusion is a reaction in which light atomic nuclei collide, merge, and release a large amount of energy. But every atomic nucleus carries a positive electric charge, so when they try to approach each other, the electric force pushes them apart strongly. Picture the resistance you feel when you try to press the like poles of two magnets together. To make nuclei fuse, they must collide with enough momentum to overcome this repulsion.

Colliding with force means making the particles fly around at high speed. The temperature of a gas is a measure of how vigorously the particles inside it are moving. Raise the temperature and the particles move faster, increasing both the chance of a head-on collision and the chance of breaking through the repulsion to fuse. Fusion needs about 100 million degrees precisely to overcome the electric repulsion between nuclei with the vigor of particle motion.

So how do we heat something that hot? The methods fall broadly into three ideas. The first is to pass an electric current and heat by resistance. Just as the nichrome wire of an electric heater glows red, we run a current through the plasma itself and heat it with Joule heating. The second is to shoot in high-speed particles from outside. We inject a bullet stream of already extremely fast particles, and like billiard balls they hand off their momentum to the surrounding particles. The third is to shine electromagnetic waves on the plasma, like a microwave oven. If we send in a wave tuned exactly to the natural rotational rhythm the particles in the plasma have, the particles accelerate more and more, just as pushing a swing in time makes it go higher.

And once fusion truly gets going, the reaction’s own heat begins to warm the reactor by itself. If this cycle takes hold, the plasma keeps burning even when external heating is switched off. Think of lighting a campfire with a match at first, but once the wood catches, the fire sustains itself. In fusion, this self-sustaining state is called ignition.

Understand the Physics (Undergraduate)

Let us first pin down “100 million degrees is needed” a bit more quantitatively. Temperature $T$ is a measure of the average kinetic energy of the particles, and in plasma physics it is conventional to express temperature in electron volts (eV), a unit of energy. The conversion is $1\ \mathrm{keV} \approx 1.16 \times 10^7\ \mathrm{K}$, so about 100 million degrees corresponds to roughly 10 keV. The reaction cross section, which represents the probability of a fusion reaction occurring, rises steeply with temperature, and for the D-T reaction (the reaction of deuterium and tritium) it reaches a practical reaction rate in the region from 10 keV to several tens of keV. How high a temperature, held for how long, at what density is required to obtain net output is set by the Lawson criterion.

Let us look at the first heating method, ohmic heating. In a tokamak, the magnetic flux of the central solenoid is varied in time, and like the secondary winding of a transformer it induces a current $I_p$ in the plasma. This current generates Joule heat through the plasma’s electrical resistance. The heating power is expressed as $P_\Omega = \eta j^2$, where $j$ is the current density and $\eta$ is the plasma’s electrical resistivity. Here is the crucial point: the plasma’s resistivity (the Spitzer resistivity) falls with temperature, with the relation $\eta \propto T^{-3/2}$. The hotter a plasma gets, the more easily it conducts electricity.

Because of this temperature dependence, ohmic heating becomes less and less effective the more you heat. As the resistance falls, so does the Joule heat. As a result, ohmic heating alone can reach only about 1 keV to 3 keV, far short of the 10 keV that fusion requires. Filling the gap beyond this is the job of additional heating supplied from outside.

The second form of additional heating is neutral beam injection (NBI). The principle is as follows. First, ions such as deuterium are accelerated to high energy by an electric field, then passed through a neutralizer cell where electrons are attached, turning them into a beam of electrically neutral, high-speed atoms. Why neutralize them? Because as charged particles they would be bent by the strong magnetic field around the reactor and never reach the plasma center. Neutral atoms fly straight in without being bent by the magnetic field, are stripped of their electrons within the plasma to become ions again, and from there share their energy with the surrounding particles through repeated Coulomb collisions.

The third is wave heating, which uses the phenomenon of resonance. In a magnetic field $B$, a charged particle feels the Lorentz force and moves in a circle, winding around the field line (for details, see Motion of Charged Particles). The angular frequency of this rotation is called the cyclotron frequency and is set by $\omega_c = qB/m$, where $q$ is the charge and $m$ is the mass. If we send in an electromagnetic wave whose frequency matches this rotational rhythm, the wave’s electric field keeps pushing the particle in the same direction, so energy is transferred efficiently. It is the same resonance principle as a swing. Ions, which have large mass, have a low frequency (in the tens of MHz band, ion cyclotron resonance heating, ICRF), while electrons, with small mass, have a high frequency (millimeter waves above 100 GHz, electron cyclotron resonance heating, ECRH).

Deepen the Theory (Graduate)

The resonance condition for wave heating cannot be described by a simple match of the cyclotron frequency alone. A wave propagating through the plasma appears, as seen by a moving particle, to have its frequency shifted by the particle’s own motion. The resonance condition that incorporates this takes a form including the Doppler shift,

$$\omega - k_\parallel v_\parallel = n \omega_c$$

Here $\omega$ is the wave frequency, $k_\parallel$ is the wavenumber along the field line, $v_\parallel$ is the particle velocity in that direction, and $n$ is an integer harmonic number. The left side means the wave frequency as seen in the coordinate frame moving with the particle, and it can be read as: resonance occurs when this equals an integer multiple of the cyclotron frequency. $n=1$ is the fundamental resonance, and $n=2$ and above are harmonic resonances.

In ion cyclotron resonance heating (ICRF), there is the complication that a plasma with only a single ion species absorbs the wave weakly. So a small amount of a different ion species (such as hydrogen or helium-3) is mixed in, and minority heating, which uses the cyclotron resonance of that minority species, is widely employed. The minority species selectively receives energy from the wave and becomes fast ions, which in turn heat the main ions and electrons through Coulomb collisions, a two-stage mechanism.

A major advantage of electron cyclotron resonance heating (ECRH) is the locality of the heating. The position where resonance occurs is limited to where $\omega = \omega_c(r) = qB(r)/m$ is satisfied, and because the tokamak’s magnetic field varies greatly in the radial direction, the resonance surface is fixed in a spatially narrow region. Using this property, heating power can be injected pinpoint at a targeted radial position, which can be applied to the local control of MHD instabilities discussed below.

These heating methods simultaneously serve the role of current drive. To run a tokamak in steady state, the plasma current must be maintained without relying on transformer action. If the distribution function of the charged particles is distorted asymmetrically along the field line by waves or particle beams, a net current is produced. Lower hybrid current drive (LHCD) uses the lower hybrid wave, which lies between the ion and electron cyclotron frequencies, to selectively accelerate the tail of the electron velocity distribution and drive current. It has high current drive efficiency per unit power, but the wave penetrates poorly to the plasma center, making it suited to controlling the current profile in the outer region. Electron cyclotron current drive (ECCD) is inferior in drive efficiency, but because of the locality of the resonance it can suppress magnetic islands such as those of the neoclassical tearing mode (NTM) by targeting them locally, making it an important stabilization tool.

Last is alpha particle self-heating. The D-T reaction, $\mathrm{D} + \mathrm{T} \rightarrow \alpha + n$, releases 17.6 MeV, of which 3.5 MeV is carried by the alpha particle (the nucleus of helium-4) and 14.1 MeV by the neutron. The neutron, being electrically neutral, is not confined by the magnetic field and escapes into the blanket, but the alpha particle, being charged, is confined by the magnetic field and deposits its energy into the plasma through Coulomb collisions, heating it. Using the fusion gain $Q$ (the fusion output divided by the external heating input), a measure of heating efficiency, the alpha heating power is one fifth of the total fusion output, so the ratio of alpha heating power to external heating power $P_{\mathrm{ext}}$ is $P_\alpha / P_{\mathrm{ext}} = Q/5$. $Q=5$ corresponds to alpha heating equal to external heating, and $Q \to \infty$ corresponds to ignition, the self-sustaining state with zero external heating. The $Q \geq 10$ that ITER aims for means obtaining 500 MW of fusion output for a 50 MW input, the regime of a burning plasma where alpha heating carries twice the external heating.

Research Frontier (PhD)

Today, the frontier of heating physics is converging on “the physics of fast ions and burning plasmas.” The fast ions generated by NBI and ICRF, and the alpha particles born in fusion, have energies far higher than the thermal background plasma. How these fast ions (energetic particles) interact with the background plasma and how they are confined has become a central question in designing burning plasmas.

Especially actively studied are the Alfvén eigenmodes driven by fast ions. When the velocity of the fast ions resonates with the velocity of Alfvén waves in the plasma, waves including the toroidal Alfvén eigenmode (TAE) are destabilized. These modes can eject fast ions out of the region where they should be confined, potentially causing anomalous transport of alpha particles. If the alpha particles escape to the wall before contributing to heating, the self-heating is undermined, and the local heat load may also damage the wall, so this is studied as a problem directly tied to the viability of a burning plasma.

On the theoretical and numerical side, because the fast ion distribution deviates greatly from thermal equilibrium, a kinetic description that explicitly treats velocity space is indispensable. Gyrokinetics simulations that follow the time evolution of the distribution function, and hybrid models that couple MHD with kinetic effects, are advancing predictions of the nonlinear saturation of Alfvén eigenmodes and of fast ion transport. At issue is how accurately the confinement of alpha particles in ITER and future demonstration reactors can be extrapolated.

The integrated optimization of heating and current drive is also an important theme. To realize a steady-state tokamak, external current drive must be combined with the bootstrap current that arises spontaneously from the pressure gradient, so that a desirable current profile and pressure profile are established at the same time. NTM suppression by ECCD, real-time feedback control of the heating position, and the construction of high-performance operating scenarios that combine multiple heating methods to form transport barriers are being pursued from both integrated simulation and experiment. In the literature, keywords such as energetic particle physics, alpha particle confinement, Alfvén eigenmode, burning plasma, and integrated modeling appear frequently.

Check Your Understanding

Q1. Why does fusion require temperatures as high as about 100 million degrees?

A Because unless it is hot, neutrons inside the nucleus cannot turn into protons B Because atomic nuclei are positively charged and repel each other strongly, so violent collisions are needed to overcome that repulsion C Because high temperature increases the mass of the nuclei and makes them easier to fuse D Because at low temperature the magnetic field cannot confine the plasma

Q2. Why can ohmic heating alone not reach the temperature required for fusion?

A Because the plasma resistivity falls with temperature, so the hotter it gets the less Joule heat is produced B Because the plasma current naturally decreases with temperature C Because at high temperature the magnetic flux of the central solenoid can no longer be varied D Because the resistivity rises with temperature and melts the device

Q3. In neutral beam injection, why are the accelerated ions deliberately neutralized before injection?

A Because neutral atoms can be accelerated to higher energy than ions B Because as charged particles they would be bent by the strong magnetic field and never reach the plasma center C Because neutralizing raises the particle speed and lets them penetrate deeper D Because staying as ions would charge the device wall and cause a discharge

Q4. Why can electron cyclotron resonance heating (ECRH) be used for the local control of MHD instabilities?

A Because the electron cyclotron frequency is lower than the ion one, allowing the whole device to be heated uniformly B Because changing the frequency of the electromagnetic wave heats the entire plasma simultaneously C Because the position satisfying the resonance condition is fixed in a narrow region by the spatial variation of the magnetic field, so heating can be injected pinpoint at a specific radius D Because electrons are not confined by the magnetic field and can freely move the heating position

Q5. In a burning plasma with fusion gain Q = 10, how many times the external heating power does alpha particle self-heating carry?

A 0.5 times B 1 time C 2 times D 10 times

Related Topics

• Motion of Charged Particles - The basics of cyclotron motion and resonant heating
• Lawson Criterion - How far you must heat to obtain net output
• Heating Systems - The engineering implementation of heating systems