Charged Particle Motion in Electric and Magnetic Fields

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Charged Particle Motion in Electric and Magnetic
Fields

• Introduction to PlasmaPrinciples and
  ApplicationsProf. Reuven Boxman
• Electrical Discharge and Plasma Lab
• Tel Aviv University

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Why Study Particle Motion?

• Presence of free mobile charged particles
  differentiates plasma from usual gas
• Motion of charged particles gives plasma its
  special characteristics
• Conductivity
• Interaction with E-M waves
• Usefulness in Applications
• Light sources
• Plasma processing
• Etc.

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Hierarchy in Studying Charged Particle Motion

• Single particle motion in known, static fields
• This Lecture!
• Continuum Approach
• Fluid Model (collision dominated)
• Collective Motion
• Next Lecture!
• Motion in A.C. fields
• Interaction with E-M wave
• Last lecture today
• Influence of particle motion (i.e. current) and
  density (i..e. charge) on E and H
• Self-consistent solutions
• MHD, (EHD)
• Tomorrow

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Charged Particle Motion - Text

• J. Reece Roth
• Industrial Plasma Engineering
• Volume I Principles
• Chapter 3 (pp. 54-116)
• Institute of Physics Publishing
• Bristol and Philadelphia, 1995.

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Single Charged Particle Motion in Static Electric
and Magnetic Fields

• In General

y
E
Consider Static E only
x
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Motion in Static Electric Field

• Consider particle at rest, at origin, at t0

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Motion in Static Magnetic Field

• No Motion ? No Force
• Motion parallel to B
• (i.e. in z direction)
• No magnetic force no influence

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Motion in Static Magnetic Field contd

• Motion ? to B
• i.e. in x-y plane

• No Motion ? No Motion
• Solve bottom eq. for vx,
• sub into top eq
• Vice versa

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Motion in Static Magnetic Field contd

• Solution of last equation in form

n.b. speed vo constant
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Motion in Static Magnetic Field contd
n.b. rotation direction depends on sign of q
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MOTION IN COMBINED ELECTROSTATIC AND
MAGNETOSTATIC FIELDS

• E and B in same direction (e.g. y)
• If no initial motion in x or z directions
• B has no effect
• Acceleration in y direction per previous
  electrostatic solution

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Motion in Combined Electrostatic and
Magnetostatic Fields-contd

• E and B in same direction (e.g. y) but with
  initial motion in x or z direction
• Independent solutions for y-direction and for x-z
  plane
• Acceleration by E in y-directions, identical to
  previous solution
• Circular motion in x-z plane, as in previous
  magnetostatic case

• Superimpose spiral motion in y direction, with
  increasing pitch

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Motion in Crossed Electrostatic and Magnetostatic
Fields
Solve 1st eq for vy, sub into 2nd
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
Initial Condition particle at rest at xy0
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
Example H ion in crossed field
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
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Motion in Crossed Electrostatic and Magnetostatic
Fields, contd
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General Motion in Combined, Uniform, Static E, B
Fields

• Resolve E into two components, E?and E?? (with
  respect to B direction).
• In direction, no influence of B usual
  electrostatic acceleration
• In ? direction, crossed-field motion
• Drift velocity, independent of charge,
• Superimpose ? and components

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Crossed-Field Magnetic Insulation

• Consider vacuum diode
• Electron emitted from cathode
• If Em/qB2ltgap, no electron reaches anode
• Magnetically insulated

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Effect of Collisions on Crossed-Field Motion

• Charged particle collides at random with other
  particles in volume
• e.g. electrons with background gas
• Simple model for collision charged particle
  looses all of its energy, starts from 0

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Collisions in Crossed-Field, contd

• Guiding Center Motion
• vxE/B
• Amplitude of y oscillations 2Em/qB22E/?cB
• Collisions occur randomly - Every collision,
  advance in y direction half of amplitude on
  average
• vy?cE/(?cB)
• Collisions allow magnetically confined charged
  particles to escape
• Transverse B increases effective path length for
  electrons from cathode to anode, can increase
  ionization probability
• Used in magnetron sputtering to lower pressure

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Summary

• Charged particles accelerated by electric field
• Circular motion in plane normal to magnetic field
• Crossed field complex motion, overall direction
  in EXB direction
• Collisions disrupt ordered motion, allow drift
  across magnetic field lines