[MUSIC] The atoms in a crystal arranged in a regular array of positions in space with a variety of possible crossline lattices. At zero temperature, the atomic coordinates are uniquely locked into spatial positions that minimizes their energy. Now, what happens at finite temperature? At finite temperature, the atoms fluctuate around the energy minimum positions leading to lattice vibrations that go in their thermodynamic behavior. Now, let's consider n atoms with positions r1, r2, etc, up to rn in a crossline lattice. Now, let's define the potential energy v of r that describes the energy for a given system configuration r. Now, energy minimum configuration satisfies the condition. The atomic positions r0 defines the regular crossline lattice. Now, let's expand the potential energy about the regular crossline lattice, and this leads to the expression. Now, the linear term in this expression is 0 by definition, and so the energy to lowest order is simply given by. Now note that in this equation, the matrix K can be diagonalized into normal modes given by the eigenvectors, but effective elastic constants given by the eigenvalues. Now, their overall 3n minus 6 degrees of freedom for sufficiently large n, we can simply say that there are 3n normal modes. Hence, the potential energy can be rewritten in the following way. Now in this equation, Siel is the magnitude of the lth normal mode. Now, the total energy of the system can now be rewritten in the following way. Now in this expression, pl and ml are the effective momentum and mass of the lth normal mode respectively. The total energy, E, is decomposed into normal modes with an individual mode energy in the form of a harmonic oscillator. These normal modes are called phonons. Now, what are these phonons? Well, phonons act as quasiparticles, that means that they're distinguishable and independent. What does independence mean? Well, independent simply means that they do not interact with each other, so this is a system of non-interacting phonons. The Hamiltonian, which is simply the sum of the kinetic and the potential energy of the lth phonon mode is simply given by. Now, remember the problem of the harmonic oscillator? The energy of a harmonic oscillator is simply given by the following expression. Remember in this equation, jl can take the values 0, 1, 2, 3, etc. And the phonon frequency, omega l, is giving us the square root of kl divided by ml. Now, the canonical partition function Q for this system is simply given by. Now here, each of the oscillator modes can be separated, and the overall partition function comes out as a product of the individual mode partition functions. Now from this we can write the hand hold free energy in the following way. Now the average energy is given from the following expression. Now, in order to evaluate the average energy, we need a model for what the frequencies omega l are. Now, we'll discuss two competing models to describe the phonon modes. The first model, called the Einstein Model, assumes that there is a single characteristic frequency of the crystal, defined as the Einstein Frequency omega E. The average energy for this model is simply given by. Now, for the Einstein model, the heat capacity is simply given by. Now, what happens to the heat capacity as T turns to infinity? The heat capacity converges to the limit 3NkB, which is what we would find from the Dulong–Petit law. Now, remember the equipartition theorem? The equipartition theorem states that each thermally active degree of freedom receives kBT amount of energy. Now, the Einstein model correctly reproduces this. Now, let's look at a second model, known as the Debye model. The Debye model treats the solid as some sort of an elastic material, sort of like a rubber. Now, the vibrational modes in an elastic solid correspond to sound waves. And the frequencies satisfy the relation omega equals ck, where c is the characteristic speed of sound in that solid. And k is a wave number given simply by m pi over l, where m can now take the values 1, 2, etc. Now, we can convert the sum over normal modes to a sum over k. Now in this equation, kc, is a cutoff wave mode that needs to be determined. This leads to the definition of a Debye frequency omega D, which is simply given as c times kc, the cutoff wave mode. Now, a complete conversion will include one longitudinal mode with the speed of sound, cl, and through transverse modes with speed of sound, ct. This gives the relation. Now, to find the Debye frequency omega D, we must enforce the constraint that these must add up to 3N. This solves for the unknown Debye frequency with the following relation. Now from this we can define Debye temperature, theta D, as the energy associated with the Debye frequency divided by the Bozeman constant. Now, this defines the temperature scale for vibrational fluctuations. Now, in the limit that T tends to infinity, the heat capacity from the Debye model also approaches the value that we would expect from the Dulong-Petit law. Now, we discussed two models, the Einstein model and the Debye model. Now, which one is the correct model? Well, in a certain sense, both of them are approximate models. But the one that best fits the experiment is the right model to choose. Now, both of these models produce the correct limit as T tends to infinity. Now, what happens as T tends to 0? Well, the heat capacity predicted by both the Einstein and the Debye models actually look very similar. It turns out that the low temperature heat capacity for non-conducting solids matches the Debye model a little bit better. The Debye model predicts that at the low temperature limit the heat capacity scales as N times T cubed. Now to summarize, we learned about phonons, which emerge as quasiparticles that represent the atomic vibrations. We learned two models to describe the frequency modes associated with phonons, the Einstein model and the Debye model. And we learned that both of them produce the right limit as T tends to infinity that is the Dulong-Petit law. And at low temperature, the Debye model predicts the experiments a little bit better for non-conducting solids.
