🔆Plasma Physics Unit 3 – Single Particle Motion

Single particle motion in plasma physics examines how individual charged particles behave in electromagnetic fields. This study is crucial for understanding plasma dynamics, as it forms the foundation for collective behavior in ionized gases. Key concepts include gyroradius, gyrofrequency, and pitch angle. These describe particle trajectories in magnetic fields. Drift mechanisms and adiabatic invariants are also essential, explaining particle motion in varying field conditions and their applications in plasma devices.

Study Guides for Unit 3

3.1

Charged particle motion in electric and magnetic fields

3 min read

3.2

Drifts and adiabatic invariants

3 min read

3.3

Magnetic mirrors and particle trapping

3 min read

Key Concepts and Definitions

Plasma consists of ionized gas containing charged particles (electrons and ions) that exhibit collective behavior
Single particle motion focuses on the trajectory and dynamics of individual charged particles in electromagnetic fields
Gyroradius (Larmor radius) represents the radius of the circular motion of a charged particle in a uniform magnetic field
Gyrofrequency (cyclotron frequency) describes the angular frequency of a charged particle's circular motion in a magnetic field
Pitch angle defines the angle between a particle's velocity vector and the local magnetic field direction
- Determines the ratio of perpendicular to parallel velocity components
Magnetic moment quantifies the magnetic dipole strength of a charged particle orbiting in a magnetic field
- Relates to the particle's perpendicular kinetic energy and the magnetic field strength
Drift velocity represents the average velocity of a charged particle perpendicular to the magnetic field due to various forces or field gradients

Fundamental Equations of Motion

Lorentz force equation: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ $F = q (E + v \times B)$ describes the force experienced by a charged particle in electric and magnetic fields
- $q$ is the particle charge, $\vec{E}$ is the electric field, $\vec{v}$ is the particle velocity, and $\vec{B}$ is the magnetic field
Newton's second law: $\vec{F} = m\vec{a}$ relates the net force acting on a particle to its mass and acceleration
Equation of motion for a charged particle: $m\frac{d\vec{v}}{dt} = q(\vec{E} + \vec{v} \times \vec{B})$ combines the Lorentz force and Newton's second law
Gyroradius equation: $r_L = \frac{mv_{\perp}}{|q|B}$ $r_{L} = \frac{m v _{⊥}}{∣ q ∣ B}$ calculates the radius of a particle's circular motion in a uniform magnetic field
- $m$ is the particle mass, $v_{\perp}$ is the perpendicular velocity component, and $B$ is the magnetic field strength
Gyrofrequency equation: $\omega_c = \frac{|q|B}{m}$ determines the angular frequency of a particle's gyration in a magnetic field

Types of Particle Trajectories

Gyromotion (cyclotron motion) describes the circular motion of a charged particle perpendicular to a uniform magnetic field
Helical motion combines gyromotion with a constant velocity parallel to the magnetic field, resulting in a helical trajectory
Trapped particle motion occurs when a particle is confined within a magnetic mirror or a potential well
- Particles bounce back and forth between regions of high magnetic field strength or potential barriers
Passing particle motion refers to particles with sufficient parallel velocity to overcome magnetic mirrors or potential barriers
Chaotic motion arises in complex magnetic field geometries or in the presence of strong electric fields or collisions
- Particle trajectories become irregular and unpredictable
Runaway motion happens when an electric field accelerates particles to high energies, overcoming collisional drag forces
Pinch effect describes the self-confinement of charged particles due to their own magnetic fields generated by the current they carry

Guiding Center Motion

Guiding center approximation separates the fast gyromotion from the slower drift motion of a charged particle
Guiding center represents the average position of a particle over one gyration period
- Follows the magnetic field lines while experiencing drift motions
Parallel motion along the magnetic field lines is determined by the parallel component of the Lorentz force
Perpendicular drifts arise from various forces or field gradients acting on the guiding center
- Examples include $E \times B$ drift, gradient drift, and curvature drift
Conservation of magnetic moment $\mu = \frac{mv_{\perp}^2}{2B}$ holds for slowly varying fields, leading to adiabatic invariance
Mirror force $F_{\parallel} = -\mu \nabla_{\parallel} B$ $F_{∥} = - μ \nabla_{∥} B$ acts on the guiding center due to the gradient of the magnetic field strength along the field lines
- Causes particles to bounce back and forth between regions of high magnetic field strength

Drift Mechanisms

$E \times B$ $E \times B$ drift: $\vec{v}_E = \frac{\vec{E} \times \vec{B}}{B^2}$ $v_{E} = \frac{E \times B}{B ^{2}}$ arises from the perpendicular electric field and causes particles to drift perpendicular to both $\vec{E}$ $E$ and $\vec{B}$ $B$
- Independent of charge, mass, and energy of the particles
Gradient drift: $\vec{v}_{\nabla B} = \frac{mv_{\perp}^2}{2qB^3} \vec{B} \times \nabla B$ $v_{\nabla B} = \frac{m v _{⊥}^{2}}{2 q B ^{3}} B \times \nabla B$ results from the gradient of the magnetic field strength
- Depends on the charge and energy of the particles
Curvature drift: $\vec{v}_R = \frac{mv_{\parallel}^2}{qB^2} \frac{\vec{R}_c \times \vec{B}}{R_c}$ $v_{R} = \frac{m v _{∥}^{2}}{q B ^{2}} \frac{R _{c} \times B}{R _{c}}$ occurs when particles follow curved magnetic field lines
- $\vec{R}_c$ is the radius of curvature vector, and $v_{\parallel}$ is the parallel velocity component
Polarization drift: $\vec{v}_p = \frac{m}{qB^2} \frac{d\vec{E}_{\perp}}{dt}$ arises from time-varying electric fields perpendicular to the magnetic field
Diamagnetic drift: $\vec{v}_D = -\frac{\nabla p \times \vec{B}}{qnB^2}$ $v_{D} = - \frac{\nabla p \times B}{q n B ^{2}}$ is a fluid drift caused by the pressure gradient in a plasma
- $p$ is the plasma pressure, and $n$ is the particle density
Gravity-induced drift: $\vec{v}_g = \frac{m\vec{g} \times \vec{B}}{qB^2}$ results from the gravitational force acting on charged particles

Adiabatic Invariants

Adiabatic invariants are quantities that remain approximately constant under slow changes in the system parameters
First adiabatic invariant (magnetic moment): $\mu = \frac{mv_{\perp}^2}{2B}$ $μ = \frac{m v _{⊥}^{2}}{2 B}$ is conserved for slowly varying magnetic fields
- Relates the perpendicular kinetic energy to the magnetic field strength
Second adiabatic invariant (longitudinal invariant): $J = \oint mv_{\parallel} d\ell$ $J = \oint m v_{∥} d ℓ$ is conserved for particles bouncing between mirror points
- Integral is taken along the guiding center path between mirror points
Third adiabatic invariant (flux invariant): $\Phi = \oint \vec{A} \cdot d\vec{\ell}$ $Φ = \oint A \cdot d ℓ$ is conserved for particles drifting in a closed loop
- $\vec{A}$ is the magnetic vector potential, and the integral is taken along the drift path
Adiabatic invariance breaks down when the system parameters change rapidly compared to the characteristic timescales of particle motion
- Leads to non-adiabatic behavior and violation of the invariants

Applications in Plasma Devices

Magnetic confinement fusion devices (tokamaks, stellarators) rely on single particle motion concepts to confine and heat plasma
- Particles follow helical trajectories along magnetic field lines and experience various drifts
Particle accelerators use electric and magnetic fields to accelerate and guide charged particles to high energies
- Understanding single particle dynamics is crucial for beam focusing, steering, and stability
Magnetron devices employ crossed electric and magnetic fields to generate microwave radiation
- Electron motion in the device is governed by $E \times B$ drift and cyclotron resonance
Ion thrusters and Hall thrusters utilize single particle motion principles to generate thrust for spacecraft propulsion
- Ions are accelerated by electric fields and experience $E \times B$ drift in the presence of magnetic fields
Plasma processing reactors (etching, deposition) control the motion of charged particles to achieve desired surface modifications
- Particle trajectories are influenced by electric and magnetic fields, as well as collisions with neutral gas molecules

Problem-Solving Techniques

Identify the relevant forces acting on the charged particle (electric field, magnetic field, gravity, collisions)
Write down the equation of motion for the particle, considering the appropriate force terms
Determine the initial conditions (position, velocity) and boundary conditions (field geometries, device dimensions)
Simplify the problem by making appropriate assumptions or approximations (e.g., neglecting collisions, assuming uniform fields)
Solve the equation of motion analytically or numerically, depending on the complexity of the problem
- Analytical solutions may involve integrating the equations of motion or using conservation laws
- Numerical solutions require discretizing the equations and using computational methods (e.g., Runge-Kutta, Boris algorithm)
Analyze the particle trajectories and calculate relevant quantities (gyroradius, drift velocities, energy, pitch angle)
Consider the effects of adiabatic invariance and identify any non-adiabatic behavior or breaking of invariants
Interpret the results in the context of the specific plasma device or application, and draw conclusions based on the analysis