Chains of crystalline-silicon nanospheres were formed by a self-organized process via an extension of the vapor-liquid-solid mechanism using gold as catalyst. The diameter of nanospheres and the spacing between nanospheres were controlled by changing the growth conditions: the diameter and the spacing were small when the gold deposit was thin and the growth temperature was low. The spacing between the crystalline-silicon nanospheres was kept nearly proportional to the diameter of the chain. In order to reveal the growth mechanism of the chain, we simulated the periodic instability in chain growth, taking account of the supersaturation of silicon in vapor phase and molten catalyst, the kinetic roughening transition at the liquid-solid interface, and the interface tensions. Simulated instability was consistent with the experimental observations.