Hello, today we're going to talk about
the Debye Crystal as a follow-up to our discussion in the last
video of the Einstein model. The Einstein Crystal or
solid model works pretty well, except that interparticle
interactions are ignored. It treats every oscillator as the same. In fact, at significant temperatures, a solid acts more like a multibody
system with 3N degrees of freedom. In such a multibody problem, all the vibrations can be expressed in
terms of so-called normal coordinates. Which correspond to the normal
modes of vibration, each of which oscillates
like a harmonic oscillator. In consequence, there's a wide distribution of
natural frequencies in a real solid. Although the total number of modes that
can exist at any one time has to be equal to 3N, where N is the number of atoms. So that the normalization
criteria must be as given. In other words, the integral over g(nu)
d nu has to equal 3N, and that means there's an essential upper limit to
the integration which we call nu sub m. We'll discuss the meaning
of this in a later slide. So we can write the partition function
as shown in the bottom equation. The question is,
of course, what to use for the distribution function of frequencies? Einstein used a delta function. Debye, however, assumed that he could
determine the distribution using elastic wave analysis and
ignoring quantum effects. The wave analysis results in a wave
velocity distribution that is essentially the same as the translational energy
distribution from the particle in a box problem. The bottom line is that we
can write the frequency distribution function as
shown in the first equation, where v sub t and
v sub l are the transverse and longitudinal wave speeds in the solid. If you've ever done
an experiment involving ultrasound propagation in solids,
you'll recall that most ultrasound devices will measure both the
transverse and longitudinal wave speed. Defining an effective wave speed
as 3 over a v naught cubed, equal to 2 over v transverse cubed
+ 1 over the longitudinal cubed. Then you can write the frequency
distribution function is 12 by v, the volume v, over the effective wave
speed cubed times the frequency squared. So if we use this normalization criteria,
then we can actually, now that we have an expression for g(nu), we can we can calculate what nu sub m is,
and it's equal to this expression. And nu sub m is called the cutoff
frequency because it's a maximum frequency that you can have given
the normalization criteria. And so
the distribution overall frequency range is given by the expression
on the previous slide. 9N over the cutoff frequency
cubed times nu squared, as long as the frequency is
less than the cutoff frequency. And then of course it
would be 0 after that. So this just shows a plot
of both the Debye and Einstein frequency distribution functions,
Einstein being a delta function and Debye this increasing function,
until it reaches the cutoff point. And we'll see in the next slide that we can define a Debye temperature
in terms of the cutoff frequency. So plugging all this into
the fundamental relationship, we get the Helmholtz function. The Debye frequency is
defined as h nu sub m over k. And we can go on to calculate,
for example, the internal energy, the entropy,
the specific heat, and the specific heat is written
the specific heat over k. So the specific heat per
molecule essentially is written as 3N times a function D,
a function of T over theta D. And that function is called the Debye
function, and it's something you can easily calculate it in MATLAB or
Mathematica, something like that. It's an easily calculated function. You'll often find it
tabulated in textbooks. We mentioned in the Einstein Crystal
discussion that we can determine the limits at 0 temperature as
the temperature goes to 0 temperature. And as the temperature goes to
infinity for the specific heat. We can do the same for
the Debye Crystal, and you notice that the two results
are somewhat different. And they are actually
quite a bit different as you really approach absolute zero. But the high temperature
limits are the same. So here for, I think this is for silver, is a plot comparing data with
the Einstein and Debye theories. And you see that the Debye theory
does work quite a bit better than the Einstein theory. You can infer the Debye temperature
from specific heat data, obviously, or from the elastic constants. The Debye temperature tends to
decrease with increasing atomic mass. And that's because the frequency of
a harmonic oscillator is proportional to a over m to the one-half. Typically values of theta D,
as we showed in a couple of slides ago, are fairly low, usually of the order
of room temperature or less. And that's due to the relatively
low force constants in large masses in monatomic crystals. So in general, Debye theory fits fairly
well for monatomic crystal solids. It does not work for
non-crystal solids, and they are really beyond the scope
of this set of courses. Thanks very much, and have a great day.