Statistical Mechanics Models for Materials

Statistical Mechanics of the Harmonic Oscillator. As has already been argued in this chapter, the harmonic oscillator often serves as the basis for the construction of various phenomena in materials. For example, it will serve as the basis for our analysis of vibrations in solids, and, in turn, of our analysis of the vibrational entropy which will be seen to dictate the onset of certain structural phase transformations in solids. We will also see that the harmonic oscillator provides the foundation for consideration of the jumps between adjacent sites that are the microscopic basis of the process of diffusion. 

In light of these claims, it is useful to commence our study of the thermodynamic properties of the harmonic oscillator from the statistical mechanical perspective. In particular, our task is to consider a single harmonic oscillator in presumed contact with a heat reservoir and to seek the various properties of this system, such as its mean energy and specific heat. As we found in the section on quantum mechanics, such an oscillator is characterized by a series of equally spaced energy levels of the form E = (n + )hu . From the point of view of the previous section on the formalism tied to the canonical distribution, we see that the consideration of this problem amounts to deriving the partition function. In this case it is given by 

The contribution of the zero point energy can be factored out of the sum leaving us with 

As a result, we are left with the insight that the free energy of such an oscillator is given by 

This equation will arise in chap. 6 in the context of structural transformations in solids when it comes time to evaluate the contribution made by vibrations to the 

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