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Collective ion acceleration with electron rings

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Collective Ion Acceleration

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 84))

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Abbreviations

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

QS 2 :

Non-external axial focussing, QS 2=νZ 2−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

Z ~m :

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

É›:

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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  1. Uwe Schumacher
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© 1979 Springer-Verlag

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Schumacher, U. (1979). Collective ion acceleration with electron rings. In: Collective Ion Acceleration. Springer Tracts in Modern Physics, vol 84. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044497

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  • DOI: https://doi.org/10.1007/BFb0044497

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