Many of the units of physical quantities, such as frequency, length, voltage, resistance, and time are defined using physical laws or quantum effects which are considered to be universal and immutable. In contrast, the kilogram, the unit of mass, has continued to be based on a material artifact. The Avogadro constant is a fundamental physical constant that expresses the number of atoms or molecules contained in 1 mol of a substance. Therefore, an accurate determination of this constant would enable the kilogram to be defined based on an atomic mass. An interferometer equipped with a direct optical frequency tuning system has been developed for a determination of the Avogadro constant by the X-ray crystal density method. The volume of 1-kg silicon spheres is determined by this interferometer with a relative standard uncertainty of 3.0 × 10^-8, being the smallest uncertainty achieved for the silicon spheres to date. At present, an international project for the precise determination of the Avogadro constant is on-going. This interferometer is playing an important role in this project. An isotropically enriched silicon crystal is used in this project to further increase the accuracy of the Avogadro constant. This would realize the redefinition of the kilogram.