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.