conventions · Wiki · snuverink_j / OPALManualWiki

Table of Contents

1. Conventions
- 1.1. Physical Units
- 1.2. Symbols used

1. Conventions

1.1. Physical Units

Throughout the computations OPAL uses international units, as defined by SI (Système International).

Table 1. Physical Units
Quantity	Dimension
Length	`\mathrm{m}` (meters)
Angle	`\mathrm{rad}` (radians)
Quadrupole coefficient	`\mathrm{Tm^{-1}}`
Multipole coefficient, 2n poles	`\mathrm{Tm^{-n + 1}}`
Electric voltage	`\mathrm{MV}` (Megavolts)
Electric field strength	`\mathrm{MV m^{-1}}`
Frequency	`\mathrm{MHz}` (Megahertz)
Particle energy	`\mathrm{MeV}` or `\mathrm{eV}`
Particle mass	`\mathrm{MeV c^{-2}}`
Particle momentum	`\mathrm{\beta\gamma}` or `\mathrm{eV}`
Beam current	`\mathrm{A}` (Amperes)
Particle charge	`\mathrm{e}` (elementary charges)
Impedances	`\mathrm{M \Omega}` (Megaohms)
Emittances (normalized and geometric	`\mathrm{mrad}`
RF power	`\mathrm{MW}` (Megawatts)

1.1.1. OPAL-cycl

The OPAL-cycl flavor see Section [cyclotron] is using the so called Cyclotron units. Lengths are define in (mm) frequencies in (MHz) momenta in (\beta\gamma) and angles in (deg), except RFPHI which is in (rad).

1.2. Symbols used

Table 2. List of Symbols used and their definition.
Symbol	Definition
`X`	Ellipse axis along the `x` dimension [m]. `X=R` for circular beams.
`Y`	Ellipse axis along the `y` dimension [m]. `Y=R` for circular beams.
`R`	Beam radius for circular beam [m].
`R^*`	Effective beam radius for elliptical beam: `R^* = (X+Y)/2` [m].
`\sigma_x`	Rms beam size in `x`: `\sigma_x = \langle x^2\rangle^{1/2}` [m]. `\sigma_x = X/2` for elliptical or circular beams (X=Y=R).
`\sigma_y`	Rms beam size in `y`: `\sigma_y = \langle y^2\rangle^{1/2}` [m]. `\sigma_y = Y/2` for elliptical or circular beams (X=Y=R).
`\sigma_i`	Rms beam size in `x` (i=1) or `y` (i=2): `\sigma=\langle x^2\rangle^{1/2}` or `\langle y^2\rangle^{1/2}` [m].
`\sigma_L`	Rms beam size in the Larmor frame for cylindrical symmetric beam and external fields [m]: `\sigma_L = \sigma_x = \sigma_y`.
`\sigma_r`	Rms beam size in `r` for a circular beam: `\sigma_r =\langle r^2\rangle^{1/2} = R/\sqrt{2}` [m].
`\sigma^*`	Average rms size for elliptical beam: `\sigma^* = (\sigma_x+\sigma_y)/2` [m].
`\theta_r`	Larmor angle [rad]
`\dot\theta_r`	Time derivative of Larmor angle: `\dot\theta_r = -eB_z/2m\gamma` [rad/sec].
`z_s`	Longitudinal position of a particular beam slice [m].
`z_h,z_t`	Position of the head & tail of a beam bunch [m].
`\zeta`	Used to label the position of a beam slice in the beam [m]. For bunched beams: `\zeta = z_s-z_t`.
`\xi`	Used to label the position of a slice image charge [m]. For bunched beams: `\xi = z_h + z_t`.
`K`	Focusing function of cylindrical symmetric external fields: `K = -\frac{\partial F_r}{\partial r}` [N/m].
`K_i`	Focusing function in `x_i` direction: `K_i = -\frac{\partial F_{x_i}}{\partial x_i}` [N/m].
`I_0`	Alfven current: `I_0= e/4\pi\epsilon_0mc^3` [A].
`I`	Beam current [A].
`I(\zeta)`	Slice beam current [A].
`k_p`	Beam perveance: `k_p = I(\zeta)/2I_0`
`g(\zeta)`	Form factor used in slice analysis of bunched beams.