which has been modeled as a quantum harmonic. The heat capacity of the entire solid is then given by , where the total number of degree of freedom of the solid is three (for the three directional degree of freedom) times , the number of atoms in the solid. One thus obtains which is algebraically identical to the formula derived in the previous section. The quantity has the dimensions of temperature and is a characteristic property of a crystal. It is known as the Einstein temperature. Hence, the Einstein crystal model predicts that the energy and heat capacities of a crystal are universal functions of the dimensionless ratio . Similarly, the Debye model predicts a universal function of the ratio , where is the Debye temperature. Limitations and succeeding model In Einstein's model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a power law, which matches experiment. This modification is called the Debye model, which appeared in 1912. See also Kinetic theory of solids References External links . Condensed matter physics Albert Einstein