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A self-consistent model for trapped-ion-cloud formation around a dust particle immersed in low-temperature low-density plasma is presented. The model is based on the solution of an integral balance equation for trapped-ion density, the free ion and electron densities, and the Poisson equation for electric potential distribution. It is shown that, in a weakly collisional regime in plasma, trapped ions form a spread “shell” with the maximum located at some distance from the negatively charged core (dust particle) proportional to an ion Debye length. The estimated polarizability of such dusty “quasi atoms” in an external electric field is very high, and their induced dipole moments can be responsible for the alignment of dust particles, the formation of chain and multilayer structures, and the coagulation of dust grains.