A three-dimensional particle in cell simulation code has been developed to study the photoelectron cloud instabilities in KEKB LER. In this report, the program is described in detail. In particular, typical simulation results are presented for the photoelectron motion in various kinds of magnetic fields. The simulation shows that a solenoid is very effective in confining the photoelectrons to the vicinity of the vacuum chamber wall and in creating a region free of photoelectrons at the vacuum pipe center. The more uniform the solenoid field is, the more effectively does it suppress the electron-cloud buildup. Multipacting can occur both in a drift region and in a dipole magnet, and the heat load deposited on the chamber wall due to the lost electrons is important in these two cases. Electron trapping by the beam field as well as by various magnetic fields is an important phenomenon, especially inside quadrupole and sextupole magnets. Our numerical results qualitatively agree with the experimental studies.

A photoelectron-trapping phenomenon has been found in the simulation of the photoelectron cloud. It is found that the photoelectrons can be trapped in the quadrupole and sextupole magnetic fields for very long time until it longitudinally drifts out of the magnets, even a long bunch train separation is not sufficient to clear up the photoelectrons. Therefore, such a kind of long time trapped photoelectron cloud can cause coupled bunch instability. The trapping phenomenon is strongly beam dependent, especially on the bunch length. There is no such kind of trapping if the positron beam does not disturb the photoelectrons during the whole process. There is also no trapping for positron bunch with bunch length longer than the period of the photoelectron's gyration motion at the mirror points. The trapping is a mirror field trap which is caused by beam disturbance. The trapping phenomenon and mechanism will be presented in detail.