Space charge effect on the parametric resonance of betatron oscillations

The effect of space charge forces on the parametric resonance of betatron oscillations is discussed. Used as initial equations are those derived by Vladimirsky and Kapchinsky for beam envelopes. Conditions have been found, when beam sizes vary insignificantly. In selecting parameters of collective accelerator adgezators, the effective external field fall off index which takes into account corrections brought in by fields reflected by a chamber but not the coulomb correction, should be taken less than 0.25. Small deviations of a beam from the matched one result in a high amplitude of envelope oscillations. Limiting an upper limit on the fall off index in collective accelerators would limit the electron ring current. Therefore, provision should be made for fast passage of the parametric resonance