Charged Particle Motion

Understanding how individual charged particles move in electromagnetic fields is fundamental to plasma physics and fusion research. This single-particle picture provides insights into plasma confinement, heating, and transport.

Equation of Motion

The motion of a charged particle in electromagnetic fields is governed by the Lorentz force:

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

where $m$ is the particle mass, $q$ is the charge, $\mathbf{v}$ is the velocity, $\mathbf{E}$ is the electric field, and $\mathbf{B}$ is the magnetic field.

Cyclotron Motion

Basic Cyclotron Motion

In a uniform magnetic field with no electric field, charged particles execute circular motion perpendicular to the field. This is called cyclotron motion or gyromotion.

The cyclotron frequency (gyrofrequency) is:

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

For electrons and ions in a 5 T magnetic field (typical for tokamaks):

• Electron cyclotron frequency: $\omega_{ce} \approx 8.8 \times 10^{11}$ rad/s
• Deuterium ion cyclotron frequency: $\omega_{ci} \approx 2.4 \times 10^{8}$ rad/s

The cyclotron radius (Larmor radius or gyroradius) is:

$$\rho_L = \frac{v_\perp}{\omega_c} = \frac{mv_\perp}{|q|B}$$

where $v_\perp$ is the velocity component perpendicular to $\mathbf{B}$.

Thermal Larmor Radius

For thermal particles with perpendicular velocity $v_\perp \sim v_{th} = \sqrt{2k_BT/m}$:

$$\rho_L = \frac{\sqrt{2mk_BT}}{|q|B}$$

In a 10 keV, 5 T tokamak plasma:

• Electron Larmor radius: $\rho_{Le} \approx 0.1$ mm
• Deuterium Larmor radius: $\rho_{Li} \approx 4$ mm

The ion Larmor radius being larger than the electron Larmor radius has important consequences for plasma confinement and transport.

Guiding Center Motion

The guiding center is the center of the circular gyration. On timescales longer than the cyclotron period, particle motion can be described as the movement of this guiding center. This adiabatic approximation greatly simplifies the analysis of particle dynamics.

Particle Drifts

When additional forces or field gradients are present, particles drift perpendicular to both the magnetic field and the force. The general drift velocity for a force $\mathbf{F}$ is:

$$\mathbf{v}_d = \frac{\mathbf{F} \times \mathbf{B}}{qB^2}$$

E x B Drift

In crossed electric and magnetic fields, all particles (regardless of charge or mass) drift with:

$$\mathbf{v}_E = \frac{\mathbf{E} \times \mathbf{B}}{B^2}$$

This drift is independent of particle charge and mass, so both electrons and ions drift together, not causing any current.

Gradient-B Drift

In a non-uniform magnetic field, the drift due to the gradient in $|\mathbf{B}|$ is:

$$\mathbf{v}_{\nabla B} = \frac{mv_\perp^2}{2qB^3}(\mathbf{B} \times \nabla B)$$

This drift depends on the sign of the charge, so electrons and ions drift in opposite directions.

Curvature Drift

When magnetic field lines are curved, particles experience a centrifugal force that causes:

$$\mathbf{v}_c = \frac{mv_\parallel^2}{qB^2}\frac{\mathbf{R}_c \times \mathbf{B}}{R_c^2}$$

where $\mathbf{R}_c$ is the radius of curvature vector and $v_\parallel$ is the velocity parallel to $\mathbf{B}$.

Combined Gradient-B and Curvature Drift

In vacuum magnetic fields, these drifts often occur together. The combined drift can be written as:

$$\mathbf{v}_{gc} = \frac{m}{qB^3}\left(\frac{v_\perp^2}{2} + v_\parallel^2\right)\mathbf{B} \times \nabla B$$

Magnetic Mirror Effect

Principle

When a charged particle moves along a magnetic field that increases in strength, it experiences a force opposing its motion:

$$F_\parallel = -\mu \nabla_\parallel B$$

where $\mu = \frac{mv_\perp^2}{2B}$ is the magnetic moment, an adiabatic invariant.

Mirror Ratio

A particle is reflected (mirrored) if it enters a region where:

$$B > B_0 \frac{v^2}{v_\perp^2}$$

The mirror ratio $R = B_{max}/B_{min}$ determines the trapping condition. Particles with pitch angle $\alpha$ (angle between velocity and magnetic field) satisfying:

$$\sin^2 \alpha > \frac{1}{R}$$

will be trapped between mirrors.

Loss Cone

Particles that are not trapped escape through the “loss cone” in velocity space. In tokamaks, this effect creates trapped and passing particle populations that behave differently.

Importance for Fusion

Magnetic Confinement

The small Larmor radius compared to device size (by factor of $10^3-10^4$) is what makes magnetic confinement possible. Particles are “tied” to magnetic field lines.

Particle Drifts and Confinement

The gradient-B and curvature drifts in a purely toroidal field would cause vertical separation of electrons and ions, leading to charge accumulation and loss. This is why the poloidal magnetic field component is essential in tokamaks.

Banana Orbits

In tokamaks, trapped particles execute “banana” shaped orbits due to the combination of parallel motion, mirror reflection, and drifts. The banana width:

$$\Delta_b \approx \frac{q \rho_L}{\sqrt{\varepsilon}}$$

(where $q$ is the safety factor and $\varepsilon$ is the inverse aspect ratio) is larger than the Larmor radius and affects transport.

Heating Methods

Understanding cyclotron motion enables resonant heating methods:

• Ion cyclotron resonance heating (ICRH) at $\omega_{ci}$
• Electron cyclotron resonance heating (ECRH) at $\omega_{ce}$

Related Topics

• Plasma Physics Overview - Introduction to plasma parameters
• Debye Shielding - Collective screening effects
• MHD - Fluid description of plasmas
• Tokamak - Application of confinement principles