Coordinate System in Cheetah

In Cheetah, the coordinates of the particles are defined by a 7-dimensional vector

\[\vec{x} = (x, px, y, py, \tau, \delta, 1)\]

The first 6 values are the canonical variables in the phase space. The trailing \(1\) is augmented as in affine transformation, for convenient calculations of thin-lens kicks, misalignmenents, etc.

The phase space coordinates are defined in the curvelinear coordinate system with respect to a reference particle at a position \(s\) along the beamline:

  • \(x\) is the horizontal position in m

  • \(y\) is the vertical position in m

  • \(px\) is the normalized horizontal momentum, dimensionless

  • \(py\) is the normalized vertical momentum, dimensionless

  • \(\tau\) is approximately the longitudinal offset with respect to the reference particle in m

  • \(\delta\) is the energy offset over the reference momentum, dimensionless

The new variables are

\[ px = P_x / p_0 \]

\(P_x\) is the horizontal momentum and \(p_0\) is the reference momentum.

\[ \tau = c\Delta t = ct- \frac{s}{\beta_0} \]

\(s\) is the independent variable, denoting the position of the reference particle. \(t\) is the time when the particle arrives at position \(z\). In this notation, bunch head (particles arriving earlier than the reference particle) would have \(\tau<0\) as \(t<t_0\). The reference particle will have \(\tau=0\).

\[ \delta= \frac{E-E_0}{p_0 c} = \frac{E}{p_0 c} - \frac{1}{\beta_0} \]

Here, \(E\) is the energy of a particle. \(E_0\) is the energy of the reference particle, i.e. reference energy. \(p_0\) is the reference momentum.

Relation with coordinates in other simulation tools

OCELOT

https://github.com/ocelot-collab/ocelot

The coordinates are identical as the ones used in OCELOT.

Note that in OCELOT the energy has the unit of GeV, while in Cheetah the energy is in eV.

Bmad

https://www.classe.cornell.edu/bmad/

The transverse coordinates \((x, px, y, py)\) are identical to the Bmad coordinates.

The longitudinal coordinate in Bmad is defined as

\[ z_\text{(Bmad)} = -\beta c \Delta t \]
\[ \tau_\text{(Cheetah)} = -\frac{1}{\beta} z_\text{(Bmad)} \]

In Bmad, the sixth dimension (longitudinal momentum) is defined as the momentum offset over the reference momentum

\[ p_{z, \text{(Bmad)}} = \frac{p-p_0}{p_0}\]

Mad-X

https://madx.web.cern.ch/madx/

The longidutinal coordinate has an opposite sign

\[ \tau_\text{(Cheetah)} = - z_\text{(MAD)} \]

The longitudinal momentum is identical

\[ \delta_\text{(Cheetah)} = p_{t,\text{(MAD)}} \]

Conversion to trace space notation

In many literatures, the trace space, or slope notation is used.

\[ x' := \frac{dx}{ds} \]

In paraxial approximation, they are approximately the same \(px \sim x'\).

In general

\[ x' = \frac{p_x}{\sqrt{p^2 - p_x^2 - p_y^2}} (1+gx) \]

Note that \(p_s := \sqrt{p^2 - p_x^2 - p_y^2}\) is the longitudinal momentum, \(g = 1/\rho\) is the curvature of the trajectory.