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By taking into consideration both the effects of space‐charge and large‐amplitude scalloping, an analytic solution for periodic‐field beam focusing is obtained through use of successive approximation and Fourier series expansion. The stability limit is then determined by studying a perturbation of the obtained solution. The perturbation results in a Hill's differential equation for the incremental beam radius. This equation yields a stability criterion for periodic‐field beam focusing. It is found that instability starts at some degree of beam scalloping and ends at a higher one, and thus forms an unstable band.