2 Conventions
2.1 Physical Units
Throughout the computations, OPAL internally uses international units, as defined by SI, for all physical quantities. Some elements and field maps are defined in other units in the input file as specified in their corresponding manual entries.
| 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 | \(\beta\gamma\) 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) |
2.2 Symbols Used
The following symbols are used throughout the manual.
| Symbol | Definition |
|---|---|
| \(X\) | Ellipse axis along the \(x\) dimension, in m. \(X = R\) for circular beams. |
| \(Y\) | Ellipse axis along the \(y\) dimension, in m. \(Y = R\) for circular beams. |
| \(R\) | Beam radius for a circular beam, in m. |
| \(R^*\) | Effective beam radius for an elliptical beam: \(R^* = (X+Y)/2\), in m. |
| \(\sigma_x\) | RMS beam size in \(x\): \(\sigma_x = \langle x^2\rangle^{1/2}\), in m. |
| \(\sigma_y\) | RMS beam size in \(y\): \(\sigma_y = \langle y^2\rangle^{1/2}\), in m. |
| \(\sigma_i\) | RMS beam size in \(x\) (\(i=1\)) or \(y\) (\(i=2\)). |
| \(\sigma_L\) | RMS beam size in the Larmor frame for cylindrically symmetric beams and external fields. |
| \(\sigma_r\) | RMS beam size in \(r\) for a circular beam: \(\sigma_r = \langle r^2\rangle^{1/2} = R/\sqrt{2}\), in m. |
| \(\sigma^*\) | Average RMS size for an elliptical beam: \(\sigma^* = (\sigma_x+\sigma_y)/2\), in m. |
| \(\theta_r\) | Larmor angle, in rad. |
| \(\dot\theta_r\) | Time derivative of the Larmor angle: \(\dot\theta_r = -eB_z/(2m\gamma)\), in rad/s. |
| \(z_s\) | Longitudinal position of a particular beam slice, in m. |
| \(z_h, z_t\) | Position of the head and tail of a beam bunch, in m. |
| \(\zeta\) | Position label of a beam slice; for bunched beams \(\zeta = z_s-z_t\). |
| \(\xi\) | Position label of a slice image charge; for bunched beams \(\xi = z_h + z_t\). |
| \(K\) | Focusing function of cylindrically symmetric external fields: \(K = -\frac{\partial F_r}{\partial r}\), in N/m. |
| \(K_i\) | Focusing function in direction \(x_i\): \(K_i = -\frac{\partial F_{x_i}}{\partial x_i}\), in N/m. |
| \(I_0\) | Alfven current: \(I_0= e/(4\pi\epsilon_0mc^3)\), in A. |
| \(I\) | Beam current, in A. |
| \(I(\zeta)\) | Slice beam current, in A. |
| \(k_p\) | Beam perveance: \(k_p = I(\zeta)/(2I_0)\). |
| \(g(\zeta)\) | Form factor used in slice analysis of bunched beams. |
2.3 Elegant Multipole Conversion
OPAL-T uses the magnetic gradient in T/m, so the conversion is
\[ \frac{dB_y}{dx} = \frac{0.29979}{E[\mathrm{GeV}]} \, k_1. \]
A compact Python helper for conversion is:
def k1tog(k1, E=45):
"""convert K1 to gradient, E in MeV"""
g = 3.335e-3 * E * k1
return g