Motion of Charged Particles

A plasma is a collection of countless charged particles. The first step toward understanding its behavior is to know how a single particle moves in electric and magnetic fields. This page explains, in order, the helical motion that wraps around a magnetic field, the drifts by which a particle slowly slides sideways, confinement by a magnetic mirror, the banana orbits characteristic of a tokamak, and finally gyrokinetics, which underpins modern numerical computation.

Start with Intuition (High School)

A charged particle moving through a magnetic field feels a force that is perpendicular to both its direction of motion and the direction of the field. This is called the Lorentz force. It is the very force whose direction you determine with the palm-based “Fleming’s left-hand rule.”

The key point is that this force always points directly sideways relative to the direction of motion. Being pushed sideways continuously, the particle cannot travel in a straight line and instead traces out circles. It is just like a hammer-throw athlete pulling on the wire so that the hammer swings around in a circle: the magnetic field plays the role of an invisible wire that keeps the particle turning. This circular motion is called cyclotron motion, and the radius of the circle is called the Larmor radius.

The stronger the magnetic field, the harder the wire is pulled, so the circle becomes smaller. The faster the particle, the larger it must circle to keep from losing to the centrifugal effect, so the circle becomes larger. Inside a fusion device the magnetic field is very strong, so this circle is only a few millimeters for ions and about one hundredth of that for electrons. Since the device itself is several meters across, the picture is that a particle stays almost glued to where it is, moving while wrapped around a thin thread called a field line. This is the starting point of what it means to “confine with a magnetic field.”

However, a particle can move freely in the direction along the magnetic field (along the thread). So overall it traces a spring-like helix, wrapping around a field line while advancing along it. Furthermore, when another force acts or the field has regions of varying strength, the center of this helix itself slowly slides sideways. This sideways slip is a drift, and it is the single most important phenomenon in designing a fusion device.

Understand the Physics (Undergraduate)

The equation of motion for a charged particle can be written with the Lorentz force.

$$m\frac{d\boldsymbol{v}}{dt} = q(\boldsymbol{E} + \boldsymbol{v}\times\boldsymbol{B})$$

Here $m$ is the mass, $q$ the charge, $\boldsymbol{v}$ the velocity, $\boldsymbol{E}$ the electric field, and $\boldsymbol{B}$ the magnetic field. The second term on the right, $q\boldsymbol{v}\times\boldsymbol{B}$, is the force from the magnetic field, and it is perpendicular to both $\boldsymbol{v}$ and $\boldsymbol{B}$.

With no electric field and only a uniform magnetic field, the particle undergoes uniform circular motion in the plane perpendicular to the field. Its angular frequency is the cyclotron frequency.

$$\omega_c = \frac{|q|B}{m}$$

Because it is inversely proportional to the mass $m$, in the same magnetic field a light electron circles far faster than a heavy ion. The mass ratio of a deuterium ion to an electron is about 3670, so electrons rotate roughly 3700 times faster than ions. The radius of rotation is the Larmor radius (also called the gyroradius).

$$r_L = \frac{mv_\perp}{|q|B}$$

Here $v_\perp$ is the velocity component perpendicular to the magnetic field. As a rough guide, in a plasma at a temperature of 10 keV and a magnetic field of 5 T, the Larmor radius is about 0.05 mm for electrons, about 4 mm for deuterium ions, and about 5 cm for the 3.5 MeV alpha particles born in fusion. Being 2 to 4 orders of magnitude smaller than the device size is the reason magnetic confinement works.

Instead of following the particle’s detailed helical motion, the approximation of following only the motion of the center of the helix (the guiding center) is called the guiding-center approximation. It is valid when the Larmor radius is much smaller than the scale over which the field varies, and it averages away the fast cyclotron rotation, extracting only the slow drifts.

The most fundamental drift is the $\boldsymbol{E}\times\boldsymbol{B}$ drift (E cross B drift). When there is an electric field $\boldsymbol{E}$ perpendicular to the magnetic field, the guiding center slides sideways at the following velocity.

$$\boldsymbol{v}_{E} = \frac{\boldsymbol{E}\times\boldsymbol{B}}{B^2}$$

Neither $q$ nor $m$ appears in this expression. That is, both electrons and ions move in the same direction at the same speed, regardless of the sign of the charge. Because all particles move together, no current is produced. Physically, it arises because the Larmor radius changes between the half orbit accelerated by the electric field and the half orbit decelerated by it, so the circle shifts.

When the field has regions of varying strength, the grad-B drift arises.

$$\boldsymbol{v}_{\nabla B} = \frac{m v_\perp^2}{2qB^3}\,\boldsymbol{B}\times\nabla B$$

Because the Larmor radius is small on the strong-field side and large on the weak-field side, the circle is distorted and the guiding center slides sideways. Since $q$ appears in the expression, electrons and ions drift in opposite directions, and a current is generated.

When field lines are curved, a particle running along them feels a centrifugal effect, and the curvature drift arises. Writing the radius of curvature as $R_c$,

$$\boldsymbol{v}_{R} = \frac{m v_\parallel^2}{qB^2}\,\frac{\boldsymbol{R}_c\times\boldsymbol{B}}{R_c^2}$$

which is caused by the centrifugal effect of the parallel velocity $v_\parallel$. This too contains $q$, so it causes charge separation. These two drifts are the fundamental reason confinement cannot be achieved with a purely toroidal magnetic field alone.

Deepen the Theory (Graduate)

Guiding-center motion has a hierarchy of adiabatic invariants, quantities that are approximately conserved. The most basic one is the magnetic moment.

$$\mu = \frac{m v_\perp^2}{2B}$$

As long as the magnetic field changes slowly compared with the Larmor period, $\mu$ is conserved as the first adiabatic invariant. This corresponds to the action variable $\oint p\,dq$ being invariant for the cyclotron motion.

The magnetic mirror can be derived from the conservation of $\mu$. As a particle advances into a strong-field region, $v_\perp$ increases by the amount that $\mu B$ increases, and by conservation of total energy $v_\parallel$ decreases. If $v_\parallel$ reaches zero at some point, the particle is reflected. Whether it is trapped is determined by a pitch-angle condition set by the ratio of the weak-field side $B_{\min}$ to the strong-field side $B_{\max}$; the larger the mirror ratio $R_m = B_{\max}/B_{\min}$, the wider the range of angles over which particles can be trapped. Conversely, particles with a large parallel velocity and a small pitch angle are not reflected and escape, which is the loss cone. The device built on this principle is the mirror device.

In a tokamak, the magnetic field is strong on the inner side of the device (the side closer to the center of the torus) and weak on the outer side. Because of this nonuniformity, particles with a large pitch angle are reflected in the weak-field region on the outer side before completing one lap of the torus, and they shuttle back and forth between there and the strong-field region on the inner side. The fraction of trapped particles is set by the aspect ratio, reaching around 70 percent in a standard tokamak.

Viewed on a surface along the field lines (the poloidal cross section), the outbound and return orbits of a trapped particle are shifted by the grad-B drift and the curvature drift, forming a crescent shape. This is the banana orbit. The width of a banana orbit reaches several times the Larmor radius, and this finite orbit width strongly affects transport. The back-and-forth (bounce) motion of trapped particles also has a corresponding second adiabatic invariant (the longitudinal invariant).

The collisionality, the ratio of the collision frequency to the bounce frequency, determines the transport regime. The core of a fusion plasma is in the weakly collisional banana regime, where a trapped particle can shuttle back and forth many times while completing its banana orbit. The diffusion that arises here is neoclassical transport, which is tens of times larger than simple classical diffusion because of the banana-width effect. See transport for details. Neoclassical theory also predicts effects such as the bootstrap current, in which a pressure gradient spontaneously drives a current.

These drift-kinetics lead to gyrokinetics, a more complete kinetic description. In gyrokinetics, the fast cyclotron motion is analytically averaged to reduce the number of degrees of freedom by one, and the evolution of the distribution function is followed in a 5-dimensional phase space (the 3-dimensional position of the guiding center, the parallel velocity, and the magnetic moment). This makes it possible to treat slow phenomena such as turbulence numerically without being disrupted by the short timescale of the cyclotron motion. The single-particle drift picture serves as the physical foundation that supports these large-scale simulations. It stands in a complementary relationship with MHD, the macroscopic fluid approximation.

Research Frontier (PhD)

The single-particle, guiding-center picture is classical, but modern problems built on it are being actively researched.

The confinement of the fast alpha particles born in a fusion reactor is an important topic. Alpha particles have large Larmor orbits and banana orbits, and moreover they resonate with waves in the plasma to excite an instability called the Alfven eigenmode. This can carry fast ions outward anomalously quickly, and because it bears on heating efficiency and the load on the reactor wall, it is studied as a nonlinear particle-wave interaction.

In a stellarator, the magnetic field is inherently 3-dimensional, and left alone the particle orbits fail to close, inviting large neoclassical transport. So optimization for quasi-symmetry and omnigenity, which design the field configuration so that guiding-center orbits are well confined, is being pursued, and configurations obtained by numerical optimization are being verified on real machines.

Turbulent transport simulations based on gyrokinetics are a field where code-to-code comparison and validation against experiment continue. Large-scale computations that simultaneously handle kinetic electrons, electromagnetic effects, multiple ion species, and realistic geometry are demanded, and work is advancing on reducing computational cost and on developing reduced models that use machine learning to rapidly surrogate transport.

On the theoretical side, the breaking of adiabatic invariants is also a subject of study. In regions where the field varies rapidly in time, or in resonant regions where orbits bifurcate, the conservation of $\mu$ breaks down, and particles can jump onto unexpected orbits. Such nonadiabatic processes and chaotic orbits are considered important for understanding fast-ion losses and heat influx to the divertor. In the literature, terms such as guiding-center, gyrokinetics, neoclassical, banana orbit, and Alfven eigenmode appear frequently.

Check Your Understanding

Q1. When the magnetic field becomes stronger, what happens to the circle (the Larmor radius) traced by a charged particle?

A It becomes smaller, because the Larmor radius is inversely proportional to the field strength B It becomes larger, because the Larmor radius is proportional to the field strength C It does not change, because the Larmor radius does not depend on the field strength D It stretches along the field and becomes an ellipse

Q2. Why does the E×B drift produce no current?

A Because electrons and ions move in opposite directions and their currents cancel B Because the drift velocity depends on neither charge nor mass, and electrons and ions move in the same direction at the same speed C Because only the electrons drift while the ions stay at rest D Because the drift velocity is parallel to the magnetic field and there is no sideways flow

Q3. When the magnetic moment μ is conserved and a particle advances into a strong-field region, what happens to the parallel velocity?

A It increases, because the stronger the field the more the particle is accelerated B It does not change, because the parallel velocity is constant regardless of the field C It decreases, because as the perpendicular velocity increases, conservation of total energy reduces the parallel velocity D It is not converted into perpendicular velocity and passes straight through the field lines

Q4. Which is the correct cause of the banana orbit in a tokamak?

A Because the field is strong on the inner side and weak on the outer side, particles with a large pitch angle are trapped, and the grad-B drift and the curvature drift shift the outbound and return orbits B Because the E×B drift alone leaves the orbit unclosed and makes a crescent shape C Because the particle is completely glued to the field line and does not move D Because the field is perfectly uniform, making the orbit straight

Q5. Why can gyrokinetics reduce the dimension of phase space?

A Because it ignores the 3 spatial dimensions and treats only velocity space B Because it approximates the particles as a fluid and does not follow individual orbits C Because it analytically averages the fast cyclotron motion to remove one degree of freedom, reducing 6 dimensions to the guiding center's 5 dimensions D Because it assumes the field does not change in time and removes the time axis

Related Topics

• Plasma Physics Overview - Introduction to plasma parameters
• Debye Shielding - Collective shielding effect
• MHD - Fluid description of a plasma
• Transport - Neoclassical and turbulent transport
• Tokamak - The leading method of toroidal magnetic confinement
• Mirror Device - Confinement by a magnetic mirror