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Theory on Electron Cooling. He Zhang CASA Journal Club Talk, 12/03/2012. Theory on Electron Cooling. This talk is based on the following references:. YA. S. Derbenev and A. N. Skrinsky The Kinetics of Electron Cooling of Beams in Heavy Particle Storage Rings

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theory on electron cooling

Theory on Electron Cooling

He Zhang

CASA Journal Club Talk, 12/03/2012

slide2

Theory on Electron Cooling

This talk is based on the following references:

  • YA. S. Derbenev and A. N. SkrinskyThe Kinetics of Electron Cooling of Beams in Heavy Particle Storage Rings
  • YA. S. Derbenev and A. N. Skrinsky The Effect of an Accompanying Magnetic Field on Electron Cooling
  • YA. S. Derbenev and A. N. SkrinskyThe Physics of Electron Cooling
  • A. H. Sorensen and E. BonderupElectron Cooling
  • H. Poth Electron Cooling: Theory, Experiment, Application
  • V. V. Parkhomchuk and A. N. SkrinskyElectron Cooling: Physics and Prospective Applications
  • V. V. Parkhomchuk and A. N. Skrinsky Electron Cooling: 35 years of development
  • J. D. Jackson Classical Electrodynamics
  • F. Yang Atomic Physics (in Chinese)
  • R. O. DendyPlasma Dynamics

He Zhang

slide3

Theory on Electron Cooling

Basic Idea

He Zhang

slide4

Theory on Electron Cooling

Two models:

  • Binary collision model:
    • Collisions between ions and electrons
    • Statistical effect
  • Dielectric plasma model:
    • Electromagnetic wave travelling through the plasma
    • Response of the plasma

He Zhang

slide5

Binary Collision Model

Coulomb scattering formula

Momentum lost

Mean energy lost through electron gas

He Zhang

slide6

Binary Collision Model

Friction force

If electrons are moving with

If electrons have a velocity distribution

He Zhang

slide7

Binary Collision Model

Diffusion coefficients

If electrons have a velocity distribution

Relation between Friction and Diffusion coefficients

He Zhang

slide8

Binary Collision Model

An example: a spherical Maxwellian electron velocity distribution

Rewrite the friction force formula

He Zhang

slide9

Binary Collision Model

Now plug in the electron velocity distribution:

with

Using the error function and integrate by parts

He Zhang

slide10

Binary Collision Model

The friction force:

Similarly one can calculate the diffusion coefficients:

He Zhang

slide12

Binary Collision Model

Another important case: disk-like velocity distribution

  • Deeper potential and larger friction force in longitudinal direction
  • Force can be calculated using the following approximation

He Zhang

slide13

Binary Collision Model

Under a longitudinal magnetic field

  • Larmor resonance
  • Two classes of the collisions
    • Fast collision
    • Adiabatic collision

He Zhang

slide14

Binary Collision Model

No-magnetic component (same as before)

Magnetic or adiabatic component

  • Cannot use the same formula due to the loss of transverse freedoms for the electrons
  • Diffusion coefficients can be calculated

He Zhang

slide16

Binary Collision Model

  • When and
  • When and

He Zhang

slide17

Dielectric Plasma Model

  • Electron beam is treated as a continuous fluid (plasma)
  • A moving ion inside the electron plasma will induce a field
  • Define the dielectric function as

He Zhang

slide18

Dielectric Plasma Model

From Poisson Equation

We get

For point charge

He Zhang

slide19

Dielectric Plasma Model

Electron plasma at rest with no magnetic field

Because of the symmetry, is directed along

Using

He Zhang

slide20

Dielectric Plasma Model

with

is determined by the minimum impact parameter

Comparing with the bi-collision formula

Agree!

He Zhang

slide21

Dielectric Plasma Model

Electron gas at rest with finite magnetic field

Dielectric function

Friction force

He Zhang

slide23

Dielectric Plasma Model

Thermal electron gas with finite magnetic field

He Zhang

slide24

Cooling of Positive and Negative Ions by Magnetized electrons

  • When

electrons will be push back by

negative ions

  • Extra
  • Extra friction force

He Zhang

slide25

Fokker-Planck equation

  • probability for a particle at velocity to have a change of velocity during time .
  • distribution function at velocity space

using the Taylor expansion of ,

and , we get

  • Knowing and , can be solved.

He Zhang

slide26

Summary

  • Friction and diffusion
  • Binary collision model
  • Dielectric plasma model
  • Solve Fokker-Plank equation to get

He Zhang