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
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.