Abstract: Suppose only the focussing and the Coulomb force act on an ion, where the focussing force is linear and the Coulomb force is nonlinear. When the two forces are balanced, the critical spacing between ions can be obtained. In this paper the radiation energy loss and the cooling of the onedimensional crystallization beam are discussed. The coherent transverse equation of particle motion has been reduced to Duffing equation with the hardspring properties in the frame of Newton mechanics. This nonlinear equation has been solved exactly by a Jacobian elliptical function, and the motion period of a particle has been expressed exactly by the firstkind complete elliptical integral. The energy loss and the beam cooling by the electromagnetic radiation of a charged particle in a storage r