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The equations of motion for space-charge dominated beams in synchrotron are derived. We find that the space-charge force generates an identical defocusing function to the betatron motion and dispersion function. The self-consistent envelope equation obeys the Kapchinskij--Vladimirskij-type equation similar to that of the linear transport system. We employ these results to analyze the stability of the crystalline beams, and discuss the implication on the high intensity proton driver for the neutron spallation sources.
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Lee et al. (1998) studied this question.
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