Beam Parameters

  The BEAM command makes it possible to define a beam in terms of the conventional Twiss parameters. A beam is defined by many parameters described in Sec.3.3. Here, we will give formulas how to generate a beam using these parameters. An important point is that the beam is defined on a plane s=constant rather than t=constant. Thus, the longitudinal structure of the beam appears as the time structure. Note that t is larger at the bunch tail.

The parameters are

Reference point of the Twiss parameters. (m)

Reference energy. (eV)

Twiss parameters.

Geometric emittance. (radm)  

R.m.s. bunch length. (m)

R.m.s. relative energy spread.

Gaussian tail cut off.

Orbit slope. (rad)

Crab angle. (rad) 

Beam roll in the x-y plane. (rad)

Coherent energy slope (1/m).

First, generate particle variables in usual accelerator coordinate:







where () is a Gaussian random number of zero mean and unit standard deviation cut at (), (j=1,2) is a random number of exponential distribution () cut at and a uniform random number in .

These variables are then transformed to







Finally, the energy-momentum is given by







where is + for right-going beam and - for left-going (note that right/left appears only here) and m is the relevant particle mass in units of eV/c.