Collective ion acceleration with electron rings

  • Uwe Schumacher
Chapter
First Online:
Part of the Springer Tracts in Modern Physics book series (STMP, volume 84)

Keywords

Electron Ring Coupling Impedance Azimuthal Magnetic Field Betatron Oscillation Betatron Tune 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Important Symbols

A, A

Vector potential

Aθ

Azimuthal vector potential component

Ãr, ÃZ

Radial and axial amplitude

a

Small radial ring half axis

aβ

Radial betatron amplitude

B, B

Magnetic field strength

BC

Magnetic field strength at compressed ring

Br, Bθ, BZ

Magnetic field strength components in cylindrical coordinates

b

Small axial ring half axis

bo, b1

Magnetic field ratio abbreviations

bβ

Axial betatron amplitude

c

Speed of light, c=(2.99792458±1.2·10−8)·1010 cm sec−1

E

Electric field strength

Ef

Final energy

EH

Holding power

Ei

Ion energy

Ein

Initial energy

Em, Emax

Maximum electric field strength

Er, Eθ, EZ

Electric field strength components in cylindrical coordinates

Et

Total ion energy, Et=Ei+Mic2

ΔE/E

Relative electron energy spread

e

Elementary charge, e=(1.602189±5·10−6)·10−19 A sec

Fr

Radial force component

f

Ion loading fraction, f=ZNi/Ne

fo

Electron revolution frequency

G

Growth

Gi, g

Abbreviations

h

Distance of the beam from a wall

I

Current

Ie

Electron current

k

Boltzmann constant, k=(1.38066±5·10−5)·10−3 JK−1

k, l, m

Integers

M

Mode number

Mi

Ion mass

m

Electron mass, m=γme

m

Mode number

me

Electron rest mass, me=(9.10995±5·10−5)·10−28 g

Ne

Total electron number

Ñe

Electron line density

Ni

Total ion number

Ñi

Ion line density

n

Magnetic field index

nb

Beam density

ne

Electron density

P

Radiation power

p

Momentum

p⊥

Transverse electron momentum

P⊥

Canonical angular momentum

Bθ

Initial value of the canonical angular momentum

Q1, Qi

Coupling coefficients between electrons and ions

QS2

Non-external axial focussing, QS2=νZ2−n

R

Major ring radius

RS

Surface resistance

r

Radius

ro

Classical electron radius, ro=μoe2/4πme=(2.817938±7·10−6)·10−13cm

SE, SM

Radius of electric or magnetic images divided by R, respectively

Sk

Coherent oscillation frequency, Sk=(1−νr)ωce

Te

Electron temperature

Ti

Ion temperature

t

Time

Ũ

Electron beam potential with respect to the cathode

U, V

Parameters describing the electromagnetic fields related to the collective instabilities

V

Potential

v

Particle velocity

Vr, Vω, vZ

Velocity components in cylindrical coordinates

x

Radial deviation from major ring radius, x=r−R

Z

Ion charge

Zo

Impedance of free space, Zo=(μo/ɛo)1/2=(ɛoc)−1=μoc=376.732 Ω

Zm~

Coupling impedance (m-th mode)

z

Axial position

α

Ratio of azimuthal to axial magnetic field component

αc

Radiation loss parameter

α1, α2

Parameters describing the holding power function

β, β, βe

Electron speed relative to the speed of light c

βt, β⊥

Electron speed transverse to B divided by c

β∥

Electron speed parallel to B divided by c

γ

Relativistic factor

γc

Relativistic factor of the electrons in the compressed state

γ∥, γ⊥

Relativistic factor with respect to the magnetic field direction

ɛ

External electric field strength

ɛo

Dielectric constant (permittivity), ɛo=(μoc2)−1=(8.85418782±7·10−10)·10−8 Asec·(Vcm)−1

ɛ1,E; ɛ1,M

Image field coefficients, electrostatic and magnetostatic, respectively

η

Ratio of the holding power to the maximum electric field strength in the electron ring

η

Chromaticity

Θ

Azimuth

Θ

Angular velocity

λ

Wave length

λc

Critical wave length

λ1

Abbreviation

μ

Abbreviation, μ=νBidker/γ

μo

Permeability, μo=4π · 10−9 H/cm=1.25663706144·10−8 Vsec · (Acm)−1

ν

Betatron tune

νBudker

Budker parameter

νr, νZ

Radial and axial betatron tune

ξ

Deviation from equilibrium axial position

σr, σZ

Radial and axial small ring dimensions, standard deviations

Φ

Magnetic flux

ω

Angular frequency

ωce

Electron gyrofrequency

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© Springer-Verlag 1979

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  • Uwe Schumacher

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