a hydrated hydrogen ion in aqueous solution, H₃O⁺. In solution the free hydrogen ion (that is, the nucleus of the hydrogen atom—the proton) is bound to the water molecules in the solution, forming mainly the hydronium ion: H⁺ + H₂O = H₃O⁺. Because of the small size of the proton (10⁻¹³ cm; other atoms have radii on the order of 10⁻⁸ cm) a strong electrical field is created, and there is covalent bonding between the proton and the unshared pair of electrons of the oxygen atom in water. The formation of the hydronium ion is analogous to the formation of the ammonium ion, NH₄⁺; it has been established that the crystal hydrate of perchloric acid, HClO₄H₂O, has an ionic crystal lattice that is isomorphic with ammonium perchlorate, NH₄⁺C10₄⁻. In crystals the H₃O⁺ ion is called oxonium (as distinct from the hydronium ion, H₃O⁺, which is in solution). Because of the association of water molecules, the hydronium ion binds a larger amount of water. The resultant hydrated hydronium ions have the formulas H₅O₂⁺, H₇O₃⁺, and H₉O₄⁺.