A systematic investigation of the vibration of a lattice containing isotopic impurities of a single kind is made in terms of the Green's function, and a general expression for the squared-frequency spectrum is obtained from a solution of difference equations, which can be applied to any number of dimensions. For a fixed spacial configuration of impurities, the results yield the spectrum of the main continuum and of the localized modes. When the number of isotopes is few, the frequencies of the localized modes, the criterion for their appearance, and the distribution of in-band frequencies are studied in detail, and the results are applied to a simple cubic lattice. The localized modes due to a substitutional impurity atom are also studied. In the case of a random distribution of isotopes, the spectrum is obtained in a closed form with use of a graphical method for the perturbation calculations. It is shown that (1) the perturbation due to lighter isotopes causes an impurity band or a tail of the spectrum into the forbidden region, and (2) when heavier isotopes are present, the spectrum in the low frequency part is increased, while in the high frequency region it adds a tail. The criteria for the appearance of the impurity band and for its merging into the main continuum are obtained. For a simple cubic lattice, the construction of the spectrum is performed.