Hello. In this video, we're going to start talking about solids. On this slide, I show pictures of the three great names in solid analysis. Einstein, of course who you're all familiar with, Debye, and Flory. We'll not talk about Flory's work. He is famous. The first theories of amorphous solids. We'll talk about crystalline solids and we'll talk about the Einstein model and the Debye model. So solids are materials in which the constituents atoms and molecules are so closely packed together, and so tightly bonded that although under a shear load, they might deform, but they won't deform continuously like a fluid. They will deform a little bit, but not continuously. We all know examples of solid materials; metals, minerals, ceramics, polymers, semiconductors composites, various biomaterials, wood, tissue, and so on. We classify solids according to whether they're crystalline or amorphous, and I haven't included here biological but biological materials obviously are tremendously important. So the study of solids, at least non-biological solids, falls under the purview of solid state physics, sometimes called material science. This is generally beyond the scope of what we're going to talk about. But simple crystalline structures can be analyzed using some of the tools that we've developed, and so it's worth talking about them, and two examples are the Einstein monatomic solid and the Debye monatomic solid. We'll start with the Einstein monatomic solid or crystal. Suppose we construct a crystal out of atoms so that every atom sees exactly the same environment. It's surrounded by other atoms in a regular rectangular array. The number of atoms is fixed, so we would probably use the canonical ensemble. If we also assume that each atom only moves a little bit and the motion is harmonic, then what we have is a system of distinguishable independent harmonic oscillators. For harmonic oscillator, the potential energy can be given as shown here. U is equal to a U Naught plus one-half, some constant a times x squared plus y squared plus z squared, where x, y, and z are the x, y, and z coordinates of the particle or the atom as it oscillates. U of zero is the equilibrium zero-point potential energy and a is the force constant in this potential energy model. They'll be a function of the specific volume and that will give the volume dependency in any analysis. The energy of the system can therefore be written as u equals NU0, U of zero over two plus the sum of the energy of all of the oscillators. The one-half under U of zero is to avoid counting each interparticle interaction twice. So e_nj is the vibrational energy of the nth oscillator when the system is in the j'th quantum state. The sum is over all oscillators of which there 3n not particles because there are three coordinate directions for each of the n particles. So the canonical partition function is equal to the sum over e to the minus Beta U_j. If we sub in U_j, then we get the first expression, but we have a sum of e to a sum and that can be rearranged into a sum of a product as shown. The sum of the products may be factored into a product of sums such that the partition function is equal to e to the minus n Beta U over two U of zero over two times a product over all the oscillators of a quantity lowercase q_n. Q_n is a partition function for the vibrational modes and it's equal to the sum over e to the minus Beta Epsilon sub mj where Epsilon sub nj is the energy of the oscillator n in it's j'th quantum state. So qn is just the vibrational partition function that we already developed for ideal gases. So it can be written as shown. It's one over 2 sin Theta e over 2t where Theta e is a characteristic vibrational temperature, the Einstein characteristic vibrational temperature. Typically, Theta e is around 300 Kelvin in the characteristic vibrational. Frequency is about six times 10 to the 12 per second. So then the partition function becomes equal to the expression on the top, and we can obtain the fundamental relationship here, the Helmholtz version of it. Then from that, we can get the properties. We can get the internal energy. If we take the derivative with respect to t, we can get the specific key. We'd get the entropy and so on. If we plot, well, and we can plot all three of these. We can plot the specific key, the energy and the entropy and they're shown on this graph. We can get the limiting values for the specific key as t goes to zero and t goes to infinity. These are measures of how good a theory does for specific key and we'll see that at least when the temperature goes to zero, one obtains a different result from the Einstein analysis than from the Debye analysis. So that's it for this video. In the next video, we'll go through the Debye analysis. Thanks for listening and have a great day.
