Early History of Lattice Dynamics

The first example of lattice dynamics was the theory of the linear chain consisting of equidistant mass points with spring forces acting between them. 

In 1819, DULONG and PETIT [1.1] published the results of their specific heat measurements of thirteen solid elements at room temperature. From these measurements, they deduced that the product of the specific heat and the atomic weight was approximately a constant C = 3R = 5.96 cal mole K . BOLTZMANN in 1871 demonstrated that the law of Dulong and Petit follows from his equi-partition principle. However, in 1875, WEBER [1.2] found that the atomic specific heat of silicon, boron and carbon are considerably lower than the Dulong-Petit value. For example, the atomic specific heat of silicon, boron and diamond were found to be 4.8, 2.7, and 1.8 cal mole K , respectively, at room temperature. Subsequent specific heat measurements at T 300 K revealed that the specific heat of solids decreased rapidly with decreasing temperature. Classical theory does not explain this behaviour. 

In 1907, EINSTEIN [1.3] proposed a simple model to account for the decrease of the specific heat at low temperatures. He took the atoms of a cys-tal to be independent oscillators, each having the same frequency and able to vibrate isotropically. He then quantized the energy of the oscillators in accordance with the results obtained by PLANCK for radiation oscillators. 

The mean energy of a crystal of Nj atoms at temperature T is then given by 

In Fig.1.1, is shown as a function of T/e where the Einstein temperature 0g is an abbreviation for hw /kg is zero at T = 0 and rises asymptotically to the Dulong-Petit value 3R when T 0. For high temperatures, therefore, quantization is unimportant and the specific heat has the same value as if each degree of freedom of the system had energy kgT/2 at lower temperatures, however, there is a pronounced deviation from the law of equipar-tition. According to PAIS [1.4], Fig.1.1 is the first graph dealing with the quantum theory of solids. 

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