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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 - PowerPoint PPT Presentation
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Theory on Electron Cooling
He Zhang
CASA Journal Club Talk, 12/03/2012
He Zhang ---2---
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
• YA. S. Derbenev and A. N. Skrinsky The Effect of an Accompanying Magnetic Field on Electron Cooling
• YA. S. Derbenev and A. N. Skrinsky The Physics of Electron Cooling• A. H. Sorensen and E. Bonderup Electron Cooling• H. Poth Electron Cooling: Theory, Experiment, Application• V. V. Parkhomchuk and A. N. Skrinsky Electron 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. Dendy Plasma Dynamics
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Theory on Electron Cooling
Basic Idea
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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
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Binary Collision Model
Coulomb scattering formula
Momentum lost
Mean energy lost through electron gas
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Binary Collision Model
Friction force
If electrons are moving with
If electrons have a velocity distribution
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Binary Collision Model
Diffusion coefficients
If electrons have a velocity distribution
Relation between Friction and Diffusion coefficients
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Binary Collision Model
An example: a spherical Maxwellian electron velocity distribution
Rewrite the friction force formula
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Binary Collision Model
Now plug in the electron velocity distribution:
with
Using the error function and integrate by parts
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Binary Collision Model
The friction force:
Similarly one can calculate the diffusion coefficients:
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Binary Collision Model
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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
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Binary Collision Model
Under a longitudinal magnetic field
• Larmor resonance
• Two classes of the collisions• Fast collision
• Adiabatic collision
•
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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
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Binary Collision Model
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Binary Collision Model
• When and
• When and
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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
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Dielectric Plasma Model
From Poisson Equation
We get
For point charge
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Dielectric Plasma Model
Electron plasma at rest with no magnetic field
Because of the symmetry, is directed along
Using
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Dielectric Plasma Model
with
Comparing with the bi-collision formula
Agree!
is determined by the minimum impact parameter
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Dielectric Plasma Model
Electron gas at rest with finite magnetic field
Dielectric function
Friction force
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Dielectric Plasma Model
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Dielectric Plasma Model
Thermal electron gas with finite magnetic field
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Cooling of Positive and Negative Ions by Magnetized electrons
• Whenelectrons will be push back bynegative ions
• Extra • Extra friction force
•
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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.
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Summary
• Friction and diffusion
• Binary collision model
• Dielectric plasma model
• Solve Fokker-Plank equation to get
