Provided are a sodium ion secondary battery and a lithium ion secondary battery capable of undergoing a reversible large-capacity charge/discharge reaction. The sodium and lithium ion secondary batteries each have a positive electrode, a negative electrode, and an electrolyte. The active substance of the positive or negative electrode of these secondary batteries is a single-phase polycrystal represented by the following chemical formula: NaxTi4O9 (2≦x≦3), preferably Na2Ti4O9, having a one-dimensional tunnel type structure, and belonging to a monoclinic crystal system. This polycrystal is obtained by filling a container made of molybdenum or the like with a raw material containing a sodium compound and at least one of a titanium compound and metal titanium, and firing at 800° C. or more but 1600° C. or less.