Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 5

Magnetized Plasmas

Cyclotron motion, magnetic moment, and drift velocities

5.1 Magnetic Moment and Adiabatic Invariance

A charged particle gyrating in a magnetic field acts as a magnetic dipole with an associatedmagnetic moment. This fundamental quantity is crucial for understanding particle motion in inhomogeneous magnetic fields and is the basis for magnetic confinement.

5.1.1 Definition and Physical Picture

Consider a particle with charge q, mass m, gyrating with perpendicular velocity v⊥in a magnetic field B. The gyration constitutes a current loop:

$$I = \frac{q}{\tau_c} = \frac{q}{2\pi/\omega_c} = \frac{q\omega_c}{2\pi} = \frac{|q|^2 B}{2\pi m}$$

where ωc = |q|B/m is the cyclotron frequency and τc is the gyration period. The area enclosed by the orbit is:

$$A = \pi r_L^2 = \pi \left(\frac{mv_\perp}{|q|B}\right)^2 = \frac{\pi m^2 v_\perp^2}{q^2 B^2}$$

The magnetic moment of a current loop is μ = IA:

$$\mu = IA = \frac{|q|^2 B}{2\pi m} \cdot \frac{\pi m^2 v_\perp^2}{q^2 B^2} = \frac{m v_\perp^2}{2B}$$

Noting that the perpendicular kinetic energy is E⊥ = ½mv⊥²:

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

5.1.2 Magnetic Moment as Action-Angle Variable

In Hamiltonian mechanics, the magnetic moment is an adiabatic invariant - a quantity that is conserved when system parameters change slowly compared to the characteristic timescale of the motion. For gyration:

$$J = \oint \mathbf{p} \cdot d\mathbf{l} = 2\pi \mu$$

where J is the action integral. When B varies on timescales τ ≫ τc = 2π/ωcand length scales L ≫ rL, the action J (and hence μ) is conserved to very high accuracy.

Mathematically, the conservation can be expressed as:

$$\frac{d\mu}{dt} = \frac{d}{dt}\left(\frac{mv_\perp^2}{2B}\right) \approx 0 \quad \text{when } \frac{1}{\mu}\frac{d\mu}{dt} \ll \omega_c$$

5.1.3 Force from Magnetic Moment

A magnetic dipole experiences a force in an inhomogeneous field:

$$\mathbf{F}_\mu = \nabla(\boldsymbol{\mu} \cdot \mathbf{B}) = -\mu \nabla B$$

where we've used μ antiparallel to B for positive charges (right-hand rule for current loop). This force is directed away from regions of strong field - the basis for magnetic confinement.

The parallel component of this force affects v∥:

$$F_\parallel = -\mu \frac{\partial B}{\partial s} = -\frac{m v_\perp^2}{2B}\frac{\partial B}{\partial s}$$

where s is the coordinate along the field line. This is the mirror force that reflects particles in converging magnetic fields.

5.1.4 Accuracy of μ Conservation

The magnetic moment is conserved to order ε where:

$$\varepsilon = \max\left(\frac{r_L}{L_B}, \frac{\omega_c}{\omega_{var}}\right)$$

where LB = B/|∇B| is the magnetic field scale length and ωvar is the timescale of field variation. After N gyrations, the fractional change in μ is:

$$\frac{\Delta\mu}{\mu} \sim N\varepsilon^2$$

First Adiabatic Invariant

The magnetic moment μ is the first adiabatic invariant. There are actually three adiabatic invariants corresponding to three characteristic motions:

  • • First: μ (gyration, τc ∼ ωc−1)
  • • Second: J = ∮ mv∥ ds (bounce motion between mirrors)
  • • Third: Φ = ∮ A∥ ds (toroidal drift in closed systems)

Each is conserved when perturbations are slow compared to the associated timescale.

Numerical Example

ITER-like parameters: Electron with E⊥ = 10 keV, B = 5 T

$$v_\perp = \sqrt{\frac{2E_\perp}{m_e}} = \sqrt{\frac{2 \times 1.602 \times 10^{-15}}{9.109 \times 10^{-31}}} \approx 5.93 \times 10^7 \text{ m/s}$$
$$\mu = \frac{m_e v_\perp^2}{2B} = \frac{9.109 \times 10^{-31} \times (5.93 \times 10^7)^2}{2 \times 5} \approx 3.2 \times 10^{-17} \text{ J/T}$$

Even in strongly varying fields (LB ∼ 1 m, rL ∼ 1 mm), ε ∼ 10−3, so μ conserved to ~10−6 per bounce!

5.2 Grad-B Drift

In a spatially non-uniform magnetic field, the gyrating particle experiences different field strengths at different points along its orbit. This inhomogeneity produces a systematic drift perpendicular to both B and ∇B, known as the grad-B drift or∇B drift.

5.2.1 Physical Mechanism

Consider a particle gyrating in a field with ∇B ⊥ B. As the particle orbits:

  • • On the side toward stronger B: radius decreases (rL ∝ B−1)
  • • On the side toward weaker B: radius increases
  • • Net result: orbit center drifts perpendicular to ∇B

The Lorentz force magnitude varies around the orbit:

$$F_\perp(\theta) = qv_\perp B(\theta) \approx qv_\perp\left[B_0 + (\nabla B \cdot \mathbf{r}_L)\right]$$

where θ is the gyrophase angle. Averaging over a gyroperiod gives the drift.

5.2.2 Derivation of Drift Velocity

Using the general drift formula for a force F:

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

The force from the magnetic moment gradient is Fμ = −μ∇B. Taking the perpendicular component:

$$\mathbf{F}_{\perp} = -\mu \nabla_\perp B$$

The grad-B drift is then:

$$\mathbf{v}_{\nabla B} = \frac{-\mu \nabla_\perp B \times \mathbf{B}}{qB^2} = \frac{\mu}{q}\frac{\mathbf{B} \times \nabla B}{B^2}$$

Substituting μ = mv⊥²/2B:

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

5.2.3 Properties and Characteristics

Charge Dependence

Unlike E×B drift, ∇B drift is charge-dependent: v∇B ∝ 1/q. Opposite charges drift in opposite directions, creating a current:

$$\mathbf{j}_{\nabla B} = \sum_s q_s n_s \mathbf{v}_{\nabla B,s} = \frac{n(m_i T_e + m_e T_i)}{B^3}\mathbf{B} \times \nabla B$$

Energy Dependence

More energetic particles drift faster: v∇B ∝ E⊥. This causes energy-dependent transport and can lead to particle loss from high-energy tails.

Direction

Drift is perpendicular to both B and ∇B. In cylindrical geometry with radial ∇B, the drift is azimuthal (toroidal in tokamaks). In tokamaks, grad-B combined with curvature causes vertical drift.

5.2.4 Grad-B Drift in Common Geometries

Cylindrical Z-Pinch (B = Bθ(r))

Field decreases with radius: ∂B/∂r < 0

$$\mathbf{v}_{\nabla B} = \frac{mv_\perp^2}{2qB^2}\frac{1}{r}\frac{\partial(rB)}{\partial r}\hat{z}$$

Particles drift vertically, causing end losses.

Tokamak Poloidal Field

Toroidal field Bφ ∝ 1/R creates radial gradient

$$\nabla B = -\frac{B_\phi}{R}\hat{R}, \quad \mathbf{v}_{\nabla B} = \frac{mv_\perp^2}{2qBR}(\hat{R} \times \hat{\phi}) = \frac{mv_\perp^2}{2qBR}\hat{Z}$$

Vertical drift causes charge separation unless compensated by plasma shaping.

Numerical Example: ITER

Deuterium ion: E⊥ = 10 keV, B = 5 T, R = 6 m, ∇B ~ B/R

$$v_{\nabla B} \approx \frac{E_\perp}{qBR} = \frac{1.602 \times 10^{-15}}{1.602 \times 10^{-19} \times 5 \times 6} \approx 3.3 \times 10^3 \text{ m/s}$$

This vertical drift of ~3 km/s must be compensated by plasma elongation and shaping.

5.3 Curvature Drift

When magnetic field lines are curved, particles moving parallel to the field experience acentrifugal force in the non-inertial guiding center frame. This produces the curvature drift, which is closely related to grad-B drift and often occurs simultaneously in realistic geometries.

5.3.1 Physical Origin

Consider a particle moving with velocity v∥ along a curved field line with radius of curvature Rc. In the guiding center frame (moving with the parallel motion), there is a centrifugal acceleration:

$$\mathbf{a}_c = \frac{v_\parallel^2}{R_c}\hat{R}_c$$

where R̂c points outward from the center of curvature. The associated force is:

$$\mathbf{F}_c = m\frac{v_\parallel^2}{R_c}\hat{R}_c = mv_\parallel^2 \boldsymbol{\kappa}$$

where the curvature vector is defined as:

$$\boldsymbol{\kappa} = (\mathbf{\hat{b}} \cdot \nabla)\mathbf{\hat{b}} = \frac{\hat{R}_c}{R_c}$$

where b̂ = B/B is the unit vector along the field.

5.3.2 Curvature Drift Velocity

Using the general drift formula for an arbitrary force F:

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

Applying this to the centrifugal force:

$$\boxed{\mathbf{v}_c = \frac{mv_\parallel^2}{qB^2}(\mathbf{B} \times \boldsymbol{\kappa}) = \frac{mv_\parallel^2}{qB^2 R_c}(\mathbf{B} \times \hat{R}_c)}$$

Like grad-B drift, curvature drift is charge-dependent (∝ 1/q) and energy-dependent (∝ v∥²).

5.3.3 Relation Between Curvature and Grad-B

In many geometries, field line curvature and field gradients are intimately connected. From Maxwell's equation ∇ × B = 0 (in current-free regions):

$$\nabla \times \mathbf{B} = \nabla B \times \mathbf{\hat{b}} + B\nabla \times \mathbf{\hat{b}} = 0$$

This gives the relation:

$$\nabla \times \mathbf{\hat{b}} = -\frac{\nabla B \times \mathbf{\hat{b}}}{B}$$

For a curved field in vacuum, ∇B is related to the curvature. This means grad-B and curvature drifts often have the same form and add constructively.

5.3.4 Combined Drift

The total drift from both effects is:

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

This can be written in terms of total kinetic energy W = ½m(v⊥² + v∥²):

$$\mathbf{v}_d = \frac{1}{qB^2}\left(W + \frac{W_\perp}{2}\right)(\mathbf{B} \times \boldsymbol{\kappa}) = \frac{W + \mu B}{qB^2}(\mathbf{B} \times \boldsymbol{\kappa})$$

Tokamak Application: Vertical Drift

In a tokamak, the toroidal field Bφ ∝ 1/R creates both radial gradient and toroidal curvature:

$$\boldsymbol{\kappa} = -\frac{\hat{R}}{R}, \quad \nabla B \approx -\frac{B}{R}\hat{R}$$

The combined drift points vertically (∝ R̂ × φ̂ = Ẑ):

$$v_{\text{vert}} \approx \frac{m(v_\perp^2/2 + v_\parallel^2)}{qBR}$$

This charge-dependent vertical drift causes charge separation, generating radial electric fields and driving plasma rotation. It must be balanced by plasma shaping (elongation, triangularity) to prevent vertical instabilities and disruptions.

Numerical Example

10 keV deuteron in ITER: B = 5 T, R = 6 m, vth ≈ 106 m/s

$$v_d \approx \frac{m_i v_{th}^2}{|e|BR} = \frac{3.34 \times 10^{-27} \times 10^{12}}{1.6 \times 10^{-19} \times 5 \times 6} \approx 7 \times 10^3 \text{ m/s}$$

Vertical drift ~7 km/s. Without compensation, this would cause charge separation on millisecond timescales!

5.4 Polarization Drift and Time-Varying Fields

When electric fields vary in time, particle inertia prevents instantaneous response to changes in the E×B drift. This lag produces the polarization drift, which is crucial for wave propagation and plasma dynamics on timescales comparable to the cyclotron period.

5.4.1 Derivation

The E×B drift velocity is:

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

For time-varying E, particles must accelerate to follow this drift. The equation of motion perpendicular to B is:

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

Taking the time derivative of the E×B drift and cross with B:

$$\frac{d\mathbf{v}_E}{dt} = \frac{1}{B^2}\frac{\partial \mathbf{E}}{\partial t} \times \mathbf{B}$$

The inertial correction gives the polarization drift:

$$\boxed{\mathbf{v}_p = \frac{m}{qB^2}\frac{d\mathbf{E}}{dt} = \frac{1}{\omega_c^2}\frac{d\mathbf{E}}{dt}}$$

where we used ωc = |q|B/m. Note that unlike E×B drift, polarization drift is:

  • • Mass-dependent: vp ∝ m, so ions drift ~1800× faster than electrons
  • • Frequency-dependent: vp ∝ ω, important for high-frequency waves
  • • Charge-independent in direction: Same direction for both species

5.4.2 Polarization Current

The mass-dependent drift creates a polarization current:

$$\mathbf{j}_p = \sum_s q_s n_s \mathbf{v}_{p,s} = \frac{\sum_s n_s m_s}{B^2}\frac{\partial \mathbf{E}}{\partial t} \approx \frac{\rho_m}{B^2}\frac{\partial \mathbf{E}}{\partial t}$$

where ρm ≈ nimi is the mass density (ion contribution dominates). This can also be written as:

$$\mathbf{j}_p = \epsilon_0 \omega_{pi}^2 \frac{\partial \mathbf{E}}{\partial t}$$

This looks like a displacement current and modifies wave dispersion relations.

Physical Significance

  • • Wave propagation: Polarization drift allows electric field oscillations perpendicular to B, enabling electromagnetic waves
  • • Low-frequency limit: For ω ≪ ωci, polarization drift is small; E×B dominates
  • • High-frequency limit: For ω ≳ ωci, polarization becomes important; modifies ion dynamics
  • • Plasma oscillations: Restoring force for plasma oscillations at ωpi

5.5 Magnetic Mirrors and Particle Trapping

When a charged particle moves along a magnetic field that increases in strength (converging field lines), the conservation of the magnetic moment μ leads to particle reflection - the magnetic mirror effect. This is a fundamental mechanism for particle confinement in both laboratory and astrophysical plasmas.

5.5.1 Mirror Force and Reflection

For a particle with conserved magnetic moment μ moving through a region of increasing B:

$$\mu = \frac{m v_\perp^2}{2B} = \text{const} \quad \Rightarrow \quad v_\perp^2 \propto B$$

Since total kinetic energy W = ½m(v∥² + v⊥²) is conserved (no work done by Lorentz force):

$$\frac{1}{2}mv_\parallel^2 + \frac{1}{2}mv_\perp^2 = \frac{1}{2}mv_\parallel^2 + \mu B = \text{const}$$

As B increases, v⊥ increases and v∥ decreases. At the mirror pointwhere B = Bm, the parallel velocity vanishes (v∥ = 0) and the particle reflects:

$$\frac{1}{2}mv_0^2 = \mu B_m \quad \Rightarrow \quad B_m = \frac{mv_0^2}{2\mu} = \frac{v_0^2}{v_{\perp,0}^2}B_0$$

where subscript 0 denotes initial values. The parallel force responsible for slowing is:

$$F_\parallel = -\mu \nabla_\parallel B = -\frac{mv_\perp^2}{2B}\frac{\partial B}{\partial s}$$

5.5.2 Mirror Ratio and Loss Cone

For a magnetic mirror with minimum field Bmin and maximum field Bmax, define themirror ratio:

$$R_m = \frac{B_{max}}{B_{min}}$$

A particle starting at Bmin will be reflected if it reaches Bmax before v∥ → 0. Using μ conservation:

$$\frac{v_{\perp,max}^2}{v_{max}^2} = \frac{v_{\perp,min}^2}{v_{min}^2} \cdot \frac{B_{max}}{B_{min}} = \frac{v_{\perp,0}^2}{v_0^2} R_m$$

Reflection occurs when v⊥,max² = vmax² (i.e., v∥ = 0), which requires:

$$\boxed{\frac{v_{\perp,0}^2}{v_0^2} = \sin^2\theta_0 > \frac{1}{R_m}}$$

where θ0 is the pitch angle (angle between v and B). Particles with small pitch angles θ0 < θc are not confined and escape. The loss cone angle is:

$$\boxed{\sin\theta_c = \frac{1}{\sqrt{R_m}} \quad \text{or} \quad \theta_c = \arcsin\left(\frac{1}{\sqrt{R_m}}\right)}$$

The confined fraction of an isotropic distribution is:

$$f_{conf} = \frac{\int_{\theta_c}^{\pi-\theta_c} \sin\theta \, d\theta}{\int_0^\pi \sin\theta \, d\theta} = 1 - \frac{1}{\sqrt{R_m}}$$

5.5.3 Applications

Earth's Radiation Belts (Van Allen Belts)

Charged particles (mostly protons and electrons) are trapped in Earth's dipole field, bouncing between mirror points near the magnetic poles.

$$R_m \approx 20-100, \quad \theta_c \approx 5°-15°$$

Bounce period: ~0.1-1 s. Particles can be trapped for years.

Simple Mirror Machine

Early fusion concept: axisymmetric magnetic field with Bmax at ends. Abandoned due to excessive end losses through loss cone.

$$R_m \sim 2-10, \quad f_{loss} = 1/\sqrt{R_m} \sim 30\%-70\%$$

Minimum-B and tandem mirror concepts improved confinement.

Solar Coronal Loops

Plasma confined in magnetic flux tubes connecting photosphere to chromosphere. Mirror effect confines hot coronal plasma.

$$R_m \sim 2-5, \quad \text{Confinement time: hours to days}$$

Tokamak Trapped Particles

In toroidal geometry, |B| varies with R: B ∝ 1/R. Particles with appropriate pitch angles are "trapped" on banana orbits.

$$R_m = \frac{B_{max}}{B_{min}} \approx \frac{R_0 + a}{R_0 - a}, \quad f_{trap} \approx \sqrt{\varepsilon} \sim 30\%$$

where ε = a/R0 is inverse aspect ratio.

Numerical Example: Earth's Magnetosphere

Electron at L = 4 (4 Earth radii): Beq = 0.4 G, Bpole ≈ 8 G

$$R_m = \frac{B_{pole}}{B_{eq}} = \frac{8}{0.4} = 20$$
$$\theta_c = \arcsin(1/\sqrt{20}) \approx 13°$$
$$f_{conf} = 1 - 1/\sqrt{20} \approx 78\%$$

78% of particles with isotropic pitch angle distribution are trapped and bounce between hemispheres.

5.6 Summary of Drifts

Complete perpendicular velocity in magnetized plasma:

$$\vec{v}_\perp = \vec{v}_E + \vec{v}_{\nabla B} + \vec{v}_c + \vec{v}_p + \vec{v}_{gyr}$$

Charge-Independent Drifts

  • • E×B drift: vE = (E × B)/B²

Charge-Dependent Drifts (can drive currents)

  • • Grad-B drift: v∇B = (μ/q)(B × ∇B)/B²
  • • Curvature drift: vc = (mv∥²/q)(B × κ)/B²
  • • Polarization drift: vp = (m/qB²)dE/dt
  • • Gravitational drift: vg = (m/q)(g × B)/B²

These drifts are fundamental to understanding plasma confinement, transport, and stability in magnetic fusion devices.

← Plasma ParametersNext: Adiabatic Invariants →