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.

Thu Dec 3 17:27:26 JST 1998