conventions.asciidoc

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Table of Contents

• 1. Conventions
  • 1.1. Physical Units
  • 1.2. Symbols used
  • 1.3. Elegant Multipole Conversion

1. Conventions

1.1. Physical Units

Throughout the computations, OPAL internally uses international units, as defined by SI (Système International), for all physical quantities (see Table 1). However, some elements and field maps are defined in other units in the input file, as is specified in their corresponding description in the Manual.

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} (see Units)
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)

1.2. 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.

1.3. Elegant Multipole Conversion

OPAL-t uses gradient in T/m so the conversion is dB_y/dx=0.29979/(E[GeV])*dBy/dx[T/m])

A Python code for conversion:

 def k1tog(k1, E = 45):
    """convert K1 to gradient, E in MeV"""
    g = 3.335E-3 * E * k1
    return g
// EOF