Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 2

Single Particle Motion

Particle trajectories in electromagnetic fields

2.1 Lorentz Force Equation

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

$$\boxed{\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})}$$

Newton's second law gives the equation of motion:

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

Together with the position equation dr/dt = v, this forms a system of six coupled first-order ODEs for the three components of r and v.

2.1.1 Energy Conservation

The rate of change of kinetic energy is:

$$\frac{dE_{\text{kin}}}{dt} = \mathbf{v} \cdot \frac{d(m\mathbf{v})}{dt} = \mathbf{v} \cdot \mathbf{F} = q\mathbf{E} \cdot \mathbf{v}$$

The magnetic force contribution vanishes because:

$$\mathbf{v} \cdot (q\mathbf{v} \times \mathbf{B}) = 0$$

Fundamental Principle

Magnetic fields do no work — they only change the direction of motion, not the speed. Only electric fields can change particle energy.

This has profound consequences: magnetic confinement systems (tokamaks, mirrors) rely on B to confine particles without heating, while E-fields (RF waves, neutral beams) are needed for heating.

2.1.2 Relativistic Generalization

For relativistic particles (v ∼ c), we must use the relativistic equation of motion:

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

where γ = (1 − v²/c²)−1/2 is the Lorentz factor. The energy equation becomes:

$$\frac{d(\gamma m c^2)}{dt} = q \mathbf{E} \cdot \mathbf{v}$$

Relativistic effects become important when the kinetic energy approaches the rest mass energy:

$$E_{\text{kin}} = (\gamma - 1)mc^2 \gtrsim mc^2$$

For electrons: mec² = 511 keV. Relativistic effects are important in:

  • Runaway electron beams in tokamaks (E ≫ 10 MeV)
  • Laser-plasma accelerators (E ≫ GeV)
  • Astrophysical jets and pulsar magnetospheres
  • Inertial fusion targets (hot electron transport)

2.1.3 Hamiltonian Formulation

Using the electromagnetic potentials φ and A (where E = −∇φ − ∂A/∂t and B = ∇ × A), the Hamiltonian is:

$$H = \frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 + q\phi$$

where p = mv + qA is the canonical momentum(not the same as kinetic momentum mv). Hamilton's equations give:

$$\frac{d\mathbf{r}}{dt} = \frac{\partial H}{\partial \mathbf{p}}, \quad \frac{d\mathbf{p}}{dt} = -\frac{\partial H}{\partial \mathbf{r}}$$

This formulation is essential for identifying conserved quantities and for developing advanced techniques like action-angle variables and guiding center theory.

2.2 Motion in Uniform Electric Field

For a uniform electric field E = E0ẑ with no magnetic field:

$$m\frac{d\mathbf{v}}{dt} = q\mathbf{E}_0$$

The solution is simple uniform acceleration (like gravity in classical mechanics):

$$\mathbf{v}(t) = \mathbf{v}_0 + \frac{q\mathbf{E}_0}{m}t$$
$$\mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\frac{q\mathbf{E}_0}{m}t^2$$

The kinetic energy grows quadratically with time:

$$E_{\text{kin}}(t) = E_0 + q E_0 v_{0z} t + \frac{1}{2}\frac{(qE_0)^2}{m}t^2$$

2.2.1 Energy Gain and Acceleration Length

The energy gained over distance d is:

$$\Delta E = q E_0 d$$

This is the basis of particle accelerators. The characteristic acceleration length to reach energy E from rest is:

$$\ell_{\text{acc}} = \frac{E}{qE_0}$$

Example: Electron Acceleration

In a typical CRT or electron gun with E = 10 kV/m, an electron starting from rest reaches:

$$v = \sqrt{\frac{2qE_0 d}{m_e}} = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 10^4 \times 0.01}{9.1 \times 10^{-31}}} \approx 5.9 \times 10^6 \text{ m/s}$$

after traveling d = 1 cm, corresponding to Ekin ≈ 100 eV. This is ~1% of the speed of light, so nonrelativistic treatment is adequate.

2.2.2 Drude Model and Collisions

In a real plasma, collisions interrupt the acceleration. With collision frequency ν, theDrude model gives steady-state drift velocity:

$$m\frac{d\mathbf{v}}{dt} = q\mathbf{E} - m\nu \mathbf{v}$$

The steady-state (dv/dt = 0) drift is:

$$\mathbf{v}_d = \frac{q\mathbf{E}}{m\nu} = \mu \mathbf{E}$$

where μ = q/(mν) is the mobility. The current density is j = nqvd = σE, giving Ohm's law with conductivity:

$$\sigma = \frac{nq^2}{m\nu} = nq\mu$$

This simple picture breaks down in magnetized plasmas, where cross-field transport is much more complex.

2.3 Motion in Uniform Magnetic Field

For a uniform magnetic field B = B0ẑ with no electric field, the equation of motion is:

$$m\frac{d\mathbf{v}}{dt} = q\mathbf{v} \times \mathbf{B}_0$$

2.3.1 Parallel and Perpendicular Components

Decompose velocity into components parallel and perpendicular to B:

$$\mathbf{v} = v_\parallel \hat{\mathbf{b}} + \mathbf{v}_\perp, \quad \hat{\mathbf{b}} = \mathbf{B}/B$$

The parallel component equation is:

$$\frac{dv_\parallel}{dt} = \frac{1}{m}\hat{\mathbf{b}} \cdot (q\mathbf{v} \times \mathbf{B}) = 0$$

since (v × B) ⊥ B. Thus:

$$\boxed{v_\parallel = \text{const}}$$

The perpendicular equation, using B = B0ẑ and writingv⊥ = vxx̂ + vyŷ:

$$\frac{dv_x}{dt} = \frac{qB_0}{m}v_y = \omega_c v_y, \quad \frac{dv_y}{dt} = -\frac{qB_0}{m}v_x = -\omega_c v_x$$

2.3.2 Cyclotron Frequency

The cyclotron frequency (or gyrofrequency) is:

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

Note the sign convention: ωc > 0 regardless of charge sign. The direction of gyration (clockwise vs. counterclockwise when viewed along B) depends on the sign of q.

For electrons: ωce = eB/me ≈ 1.76 × 1011 B[T] rad/s
For ions: ωci = ZeB/mi = (me/mi)ωce ≪ ωce

2.3.3 Solution: Helical Trajectory

The perpendicular equations describe circular motion. Using complex notation v⊥ = vx + ivy:

$$\frac{dv_\perp}{dt} = -i\omega_c v_\perp \quad \Rightarrow \quad v_\perp(t) = v_\perp(0) e^{-i\omega_c t}$$

In real form:

$$v_x(t) = v_\perp \cos(\omega_c t + \phi_0), \quad v_y(t) = \mp v_\perp \sin(\omega_c t + \phi_0)$$

where the sign depends on q. Integrating to get position:

$$x(t) = x_c + r_L \sin(\omega_c t + \phi_0), \quad y(t) = y_c \pm r_L \cos(\omega_c t + \phi_0)$$
$$z(t) = z_0 + v_\parallel t$$

2.3.4 Larmor Radius

The radius of the circular motion (gyroradius or Larmor radius) is:

$$\boxed{r_L = \frac{v_\perp}{\omega_c} = \frac{m v_\perp}{|q|B}}$$

In terms of perpendicular energy E⊥ = ½mv⊥²:

$$r_L = \frac{1}{|q|B}\sqrt{2mE_\perp} = \frac{v_{th\perp}}{\omega_c}$$

where vth⊥ = √(2E⊥/m) is the perpendicular thermal speed. For T measured in eV:

$$r_{L,\text{electrons}} \approx 2.38 \times 10^{-6} \frac{\sqrt{T[\text{eV}]}}{B[\text{T}]} \quad \text{[m]}$$
$$r_{L,\text{protons}} \approx 1.02 \times 10^{-4} \frac{\sqrt{T[\text{eV}]}}{B[\text{T}]} \quad \text{[m]}$$

2.3.5 Magnetic Moment

The gyrating particle creates a current loop I = q/(2π/ωc) = qωc/(2π), which produces a magnetic dipole moment:

$$\mu = I \cdot \pi r_L^2 = \frac{q\omega_c}{2\pi} \pi r_L^2 = \frac{m v_\perp^2}{2B}$$

This can be written as:

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

The magnetic moment μ is an adiabatic invariant — it remains approximately constant when B varies slowly compared to the cyclotron period. This is one of the most important conservation laws in plasma physics (covered in detail in Chapters 5-6).

Typical Cyclotron Parameters

SystemB [T]fcefci (H+)rLe (10 eV)rLi (10 eV)
Earth's field5×10−51.4 kHz0.76 Hz3.4 km145 m
Tokamak5140 GHz76 MHz34 μm1.4 mm
Lab plasma0.12.8 GHz1.5 MHz1.7 mm72 mm
Magnetar1082.8×1021 Hz1.5×1015 Hz1.7 fm72 pm

2.3.6 Pitch Angle

The pitch angle α characterizes the helical trajectory:

$$\tan\alpha = \frac{v_\perp}{v_\parallel}, \quad \alpha \in [0, \pi/2]$$

Special cases:

  • α = 0°: Pure parallel motion along field line (no gyration)
  • α = 90°: Pure perpendicular motion (circular orbit, no parallel motion)
  • α = 45°: Equal parallel and perpendicular velocities

The pitch angle is intimately related to magnetic mirroring (Chapter 5).

2.4 E×B Drift

When both electric and magnetic fields are present with E ⊥ B, the particle exhibits a drift motion perpendicular to both fields.

2.4.1 Derivation: Gyro-Average Method

The equation of motion is:

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

Decompose velocity into rapid gyration plus slow drift: v = vgyro + vd. Averaging over a gyroperiod τc = 2π/ωc eliminates the oscillatory gyration:

$$\langle \mathbf{v}_{\text{gyro}} \rangle = 0$$

The time-averaged equation becomes:

$$0 = q(\mathbf{E} + \mathbf{v}_d \times \mathbf{B})$$

Solving for vd, cross both sides with B:

$$q\mathbf{E} \times \mathbf{B} = -q(\mathbf{v}_d \times \mathbf{B}) \times \mathbf{B}$$

Using the vector identity a × (b × c) = b(a·c) − c(a·b):

$$(\mathbf{v}_d \times \mathbf{B}) \times \mathbf{B} = \mathbf{B}(\mathbf{v}_d \cdot \mathbf{B}) - \mathbf{v}_d B^2$$

Since vd ⊥ B (the drift is perpendicular), the first term vanishes:

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

2.4.2 Physical Interpretation

Consider a particle gyrating in B = B0ẑ with E = E0x̂:

  • Upper half of orbit: Particle gains energy from E, increasing v⊥ → rL increases
  • Lower half of orbit: Particle loses energy to E, decreasing v⊥ → rL decreases

The asymmetric orbit (larger radius on one side) produces net drift in the E × B direction. Over one gyroperiod, energy gain and loss cancel, so no net energy change occurs.

Key Properties of E×B Drift

  • • Independent of q, m: All species drift together at vE = (E × B)/B²
  • • Perpendicular: vE ⊥ E and vE ⊥ B
  • • No energy gain: vE · E = 0 (drift perpendicular to force)
  • • Maintains quasi-neutrality: No charge separation current
  • • Can be large: For E = 1 kV/m, B = 1 T, vE = 1 km/s

2.4.3 Drift Velocity Magnitude

The drift velocity magnitude depends on the field ratio:

$$v_E = \frac{E}{B}$$

In SI units: vE [m/s] = E [V/m] / B [T]. The drift can exceed thermal speeds when:

$$\frac{E}{B} > v_{th} = \sqrt{\frac{2k_B T}{m}} \quad \Rightarrow \quad E > B \sqrt{\frac{2k_B T}{m}}$$

2.4.4 Plasma Rotation

In cylindrical geometry, a radial electric field Er with axial field Bz producespoloidal rotation:

$$\mathbf{v}_E = \frac{E_r}{B_z}\hat{\boldsymbol{\theta}}$$

This E×B rotation is crucial for:

  • Tokamak H-mode: Sheared Er creates velocity shear that suppresses turbulence
  • Magnetron devices: Crossed E and B fields control electron paths
  • Hall thrusters: E×B drift enables efficient ion acceleration
  • Penning traps: E×B rotation confines charged particles

2.4.5 Complete Trajectory

The full particle motion in crossed E and B fields is:

$$\mathbf{r}(t) = \mathbf{R}_{\text{gc}}(t) + \boldsymbol{\rho}(t)$$

where Rgc(t) is the guiding center position:

$$\mathbf{R}_{\text{gc}}(t) = \mathbf{R}_0 + \mathbf{v}_E t + v_\parallel t \hat{\mathbf{b}}$$

and ρ(t) is the gyration vector with |ρ| = rL, rotating at frequency ωc.

Example: Magnetron

A magnetron uses crossed E and B fields. Electrons drift in circles around the cathode at vE = E/B, interacting with RF cavities to generate microwaves. For E = 5 kV/cm = 5×105 V/m and B = 0.1 T:

$$v_E = \frac{5 \times 10^5}{0.1} = 5 \times 10^6 \text{ m/s} \approx 0.017c$$

This relativistic drift velocity enables efficient energy transfer to RF waves.

2.5 Guiding Center Approximation

When the Larmor radius is much smaller than the scale length of field variations (rL ≪ L), we can separate the rapid gyromotion from the slower drift of the guiding center.

2.5.1 Ordering Parameters

Define the small parameter:

$$\epsilon = \frac{r_L}{L} = \frac{m v_\perp}{|q|BL} \ll 1$$

This allows a systematic expansion in powers of ε. The particle position is decomposed as:

$$\mathbf{r}(t) = \mathbf{R}(t) + \boldsymbol{\rho}(t)$$

where:

  • R(t) is the guiding center position (varies slowly)
  • ρ(t) is the gyration vector with |ρ| = rL ∼ εL (varies rapidly at ωc)

2.5.2 Guiding Center Velocity

The guiding center moves according to:

$$\boxed{\frac{d\mathbf{R}}{dt} = v_\parallel \hat{\mathbf{b}} + \mathbf{v}_d}$$

where b̂ = B/B is the unit vector along the field, and vdis the sum of all drift velocities:

$$\mathbf{v}_d = \mathbf{v}_E + \mathbf{v}_{\nabla B} + \mathbf{v}_{\text{curv}} + \mathbf{v}_p + \cdots$$

Each drift is of order ε smaller than the thermal speed.

2.5.3 Parallel Dynamics

The parallel velocity evolves according to:

$$m\frac{dv_\parallel}{dt} = q E_\parallel - \mu \frac{\partial B}{\partial s}$$

where:

  • E∥ = E · b̂ is the electric field component along B
  • μ = mv⊥²/(2B) is the magnetic moment (adiabatic invariant)
  • s is arc length along the field line
  • ∂B/∂s is the gradient of B magnitude along the field

The second term is the magnetic mirror force, which converts perpendicular to parallel energy (and vice versa) as particles move along field lines with varying |B|.

2.5.4 Perpendicular Dynamics

The perpendicular energy E⊥ = ½mv⊥² evolves as:

$$\frac{dE_\perp}{dt} = \frac{d(\mu B)}{dt} = \mu \frac{dB}{dt} = \mu \left(\frac{\partial B}{\partial t} + \mathbf{v} \cdot \nabla B\right)$$

For static fields (∂B/∂t = 0), using v ≈ v∥b̂ + vd:

$$\frac{dE_\perp}{dt} \approx \mu v_\parallel \frac{\partial B}{\partial s}$$

Combined with the parallel equation, this conserves total energy:

$$\frac{d}{dt}\left(\frac{1}{2}m v_\parallel^2 + \mu B\right) = q E_\parallel v_\parallel$$

2.5.5 Validity Conditions

The guiding center approximation is valid when:

1. Small Larmor radius

$$r_L \ll L_\perp \quad \text{(spatial scale)}$$

Fields vary slowly over a gyroradius.

2. Slow time variation

$$\omega \ll \omega_c \quad \text{(temporal scale)}$$

Fields vary slowly compared to gyroperiod.

3. Small drift velocities

$$v_d \ll v_\perp \quad \text{(drift ordering)}$$

Drifts are perturbations to gyro-motion.

When these conditions break down (e.g., very strong E-fields, wave-particle resonances ω ≈ ωc), the full particle orbit must be computed numerically.

Power of Guiding Center Theory

The guiding center approximation dramatically simplifies plasma analysis by:

  • • Eliminating fast cyclotron oscillations (reduces stiffness of ODEs)
  • • Revealing conservation laws (μ, canonical momentum)
  • • Enabling analytical understanding of confinement
  • • Providing closure for kinetic equations

Most modern plasma simulations use guiding center equations rather than full particle orbits.

2.6 General Force Drifts

Any force F perpendicular to B causes the guiding center to drift. This provides a unified framework for understanding particle motion in complex field geometries.

2.6.1 General Drift Formula

For an arbitrary force F with component perpendicular to B, the drift velocity is:

$$\boxed{\mathbf{v}_F = \frac{1}{q}\frac{\mathbf{F} \times \mathbf{B}}{B^2}}$$

Derivation: In the guiding center frame moving with velocity vd, average the equation of motion over a gyroperiod:

$$\langle \mathbf{F} + q\mathbf{v} \times \mathbf{B} \rangle = 0$$

The gyration velocity averages to zero, leaving only the drift:

$$\mathbf{F} + q\mathbf{v}_d \times \mathbf{B} = 0$$

Cross with B and use (a×B)×B = −B²a when a⊥B:

$$\mathbf{F} \times \mathbf{B} = -q B^2 \mathbf{v}_d$$

2.6.2 Gravitational Drift

For gravity F = mg, the drift is:

$$\mathbf{v}_g = \frac{m}{q}\frac{\mathbf{g} \times \mathbf{B}}{B^2}$$

This drift is charge-dependent: electrons and ions drift in opposite directions, producing a current density:

$$\mathbf{j}_g = n(q_i \mathbf{v}_{g,i} + q_e \mathbf{v}_{g,e}) = \rho_m \frac{\mathbf{g} \times \mathbf{B}}{B^2}$$

where ρm = nmi + nme ≈ nmi is the mass density. This gravitational drift current can drive instabilities in stratified plasmas.

Example: Flute Instability

In linear pinches and θ-pinches, gravitational drift drives the flute (interchange) instability. The current jg generates perturbation fields that amplify, causing vertical plasma displacement. This severely limits confinement in simple mirror configurations.

2.6.3 Polarization Drift

When the electric field varies in time, the E×B drift velocity changes, requiring acceleration. Theinertial force F = −m dvE/dt produces the polarization drift:

$$\mathbf{v}_p = \frac{1}{q}\frac{(-m d\mathbf{v}_E/dt) \times \mathbf{B}}{B^2} = -\frac{m}{qB^2}\frac{d\mathbf{v}_E}{dt} \times \mathbf{B}$$

For time-varying perpendicular E:

$$\mathbf{v}_p = \frac{m}{qB^2}\frac{\mathbf{B} \times \frac{d\mathbf{E}}{dt}}{B} = \frac{1}{\omega_c B}\frac{d\mathbf{E}_\perp}{dt}$$

The polarization drift is mass-dependent and creates a polarization current:

$$\mathbf{j}_p = n(q_i \mathbf{v}_{p,i} + q_e \mathbf{v}_{p,e}) \approx \frac{\rho_m}{B^2}\frac{d\mathbf{E}_\perp}{dt}$$

This current is important in RF heating and wave propagation in magnetized plasmas.

2.6.4 Centrifugal Drift in Curved Fields

Particles moving along curved field lines experience centrifugal force:

$$\mathbf{F}_{\text{cf}} = \frac{m v_\parallel^2}{R_c}\hat{\mathbf{R}}_c$$

where Rc is the radius of curvature. This produces the curvature drift:

$$\mathbf{v}_{\text{curv}} = \frac{m v_\parallel^2}{q B^2}\frac{\mathbf{R}_c \times \mathbf{B}}{R_c^2}$$

Using b̂ · ∇b̂ = Rc/Rc²:

$$\mathbf{v}_{\text{curv}} = \frac{m v_\parallel^2}{qB^2}(\mathbf{b} \times \nabla) \times \mathbf{b}$$

In toroidal devices like tokamaks, curvature drift is a major source of radial transport.

2.6.5 Gradient-B Drift (Preview)

In inhomogeneous magnetic fields, the magnetic moment μ experiences a force F = −∇(μB), producing:

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

This drift is perpendicular to both B and ∇B, and is covered in detail in Chapter 5.

2.6.6 Summary of Drifts

Drift TypeFormulaq-dependenceCurrent?
E×B(E×B)/B²IndependentNo
Gravitational(m/q)(g×B)/B²q−1Yes
Gradient-B(mv⊥²/2q)(B×∇B)/B³q−1Yes
Curvature(mv∥²/q)(Rc×B)/(Rc²B²)q−1Yes
Polarization(m/qB²)(dE/dt×B)q−1Yes

Key Distinction

E×B drift: All species drift together (q-independent) → no current, maintains quasi-neutrality.
Force drifts: Depend on q, m → electrons and ions drift differently → generate currents → drive instabilities and transport.

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