Mechanics of the Harmonic Solid

The plan of the following sections is as follows. First, we undertake an analysis of the energetics of a solid when the atoms are presumed to make small excursions about their equilibrium positions. This is followed by a classical analysis of the normal modes of vibration that attend such small excursions. We will see in this case that an arbitrary state of motion may be specified and its subsequent evolution traced. The quantum theory of the same problem is then undertaken with the result that we can consider the problem of the specific heat of solids. This analysis will lead to an explicit calculation of thermodynamic quantities as weighted integrals over the frequencies of vibration. 

As noted in the introduction to the present chapter, the contemplation of thermal and elastic excitations necessitates that we go beyond the uninterrupted monotony of the perfect crystal. As a first concession in this direction, we now spell out the way in which the energy of a weakly excited solid can be thought of as resulting from excursions about the perfect crystal reference state. Once the total energy of such excursions is in hand, we will be in a position to write down equations of 

Our starting point for the analysis of the thermal and elastic properties of crystals is an approximation. We begin with the assertion that the motions of the i atom, will be captured kinematically through a displacement field Uj which is a measure of the deviations from the atomic positions of the perfect lattice. It is presumed that this displacement is small in comparison with the lattice parameter, uj C qq. Though within the context of the true total energy surface of our crystal (i.e. R2,. .., Rw)) this approximation is unnecessary, we will see 

As in chap. 4, we posit the existence of a total energy function of the form E toKRi R2, Rv) = where R denotes the entire set of ionic 

The component of displacement of the F ion is denoted Uia- The rationale for our strategy is cast in geometric terms in fig. 5.2 which shows fhe quadratic approximation to a fully nonlinear energy surface in the neighborhood of the minimum for an idealized two-dimensional configuration space. 

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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. 

What we have learned is that our change to normal coordinates yields a series of independent harmonic oscillators. From the statistical mechanical viewpoint, this signifies that the statistical mechanics of the collective vibrations of the harmonic solid can be evaluated on a mode by mode basis using nothing more than the simple ideas associated with the one-dimensional oscillator that were reviewed in chap. 3. 

Until now, our treatment has been built in exactly the same terms that might have been used in work on normal modes of vibration in the latter part of the nineteenth century. However, it is incumbent upon us to revisit these same ideas within the quantum mechanical setting. The starting point of our analysis is the observation embodied in eqn (5.19), namely, that our harmonic Hamiltonian admits of a decomposition into a series of independent one-dimensional harmonic oscillators. We may build upon this observation by treating each and every such oscillator on the basis of the quantum mechanics discussed in chap. 3. In light of this observation, for example, we may write the total energy of the harmonic solid as... 

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

An alternative approach involves integrating out the elastic degrees of freedom located above the top layer in the simulation.76 The elimination of the degrees of freedom can be done within the context of Kubo theory, or more precisely the Zwanzig formalism, which leads to effective (potentially time-dependent) interactions between the atoms in the top layer.77-80 These effective interactions include those mediated by the degrees of freedom that have been integrated out. For periodic solids, a description in reciprocal space decouples different wave vectors q at least as far as the static properties are concerned. This description in turn implies that the computational effort also remains in the order of L2 InL, provided that use is made of the fast Fourier transform for the transformation between real and reciprocal space. The description is exact for purely harmonic solids, so that one can mimic the static contact mechanics between a purely elastic lattice and a substrate with one single layer only.81... 

Figure 3.1 Energy levels and wave functions of harmonic oscillator. Heavy line bounding potential (3.2). Light solid lines quantum-mechanic probability density distributions for various quantum vibrational numbers see section 1.16.1). Dashed lines classical probability distribution maximum classical probability is observed in the zone of inversion of motion where velocity is zero. From McMillan (1985). Reprinted with permission of The Mineralogical Society of America.

The colinear collision problem of atom A colliding with a molecule BC was first attempted quantum mechanically by Zener [14,15] and then by Jackson and Mott [28] for the purpose of investigating thermal accommodation coefficients for atoms impinging on solid surfaces. An exponential repulsion was utilized, along with the harmonic-oscillator approximation. The distorted-wave (DW) method was employed to obtain a 1 — 0 transition probability of the form... 

The use of the analogy between (42) and Boltzmann statistical mechanics leads to a simple semi-quantitative description of radiationless processes in aromatic hydrocarbons, but a more accurate approach to the calculation of nonradiative decay rates has also been investigated in other contexts. For the case in which the vibrational modes are harmonic, but need not be parallel or have the same frequencies in the two electronic manifolds s and l, Eq. (40) is mathematically similar to expressions considered by Kubo and Toyazawa in discussions of optical line shapes in solids s°). In particular, they showed how the double summation in (40) can be expressed as a single definite (Fourier) integral of the form... 

This linear problem is thus exactly soluble. On the practical level, however, one cannot carry out the diagonalization (4.11) for macroscopic systems without additional considerations, for example, by invoking the lattice periodicity as shown below. The important physical message at this point is that atomic motions in solids can be described, in the harmonic approximation, as motion of independent harmonic oscillators. It is important to note that even though we used a classical mechanics language above, what was actually done is to replace the interatomic potential by its expansion to quadratic order. Therefore, an identical independent harmonic oscillator picture holds also in the quantum regime. 

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