Abstract: The mode density of the lattice vibration (phonon density of states)，i.e.，the mode number in the unit frequency interval，is a physical quantity reflecting the density of the phonon in the space of the wave vector. In order to calculate accurately the lattice heat capacity varying with temperature，one must calculate the density of lattice vibration by using more accurate method，and then grasp the thermodynamic properties of materials. Some general textbooks on the part of the explanation are obscure and difficult. This paper starts from the physical meaning of the lattice vibration，analysis and derivation of 1D，2D and 3D different systems of lattice vibration mode density formula，then the lattice heat capacity formula of Debye model for different systems is presented.

Key words: lattice vibration, phonon density, Debye model, lattice heat capacity