In this video we'll talk
about the electron gas. Electrons or I should say metals are excellent conductors
of heat and electricity. It was experimentally observed
quite a long time ago that the electric current
in a metal involves the flow of electrons
and not ions. There was a simple model proposed by Drude in
1900 that assumes that when the atoms
of a metal are brought together to form a metal, the valence electrons
become detached and are free to move within
the metal lattice. If we assume that the electrons do not interact with one another, then collectively they must
act like a monatomic gas. So Drude assumed that, if the electrons act
like an ideal gas, then each one would contribute three-halves kT to
the total energy, so that the energy would be equal to N_0 which is the number of density of free electrons or number of free
electrons I guess. Yeah a new unit, volume times three-halves kT and
that would make the specific heat three-halves k. But that doesn't work very well and a full quantum
analysis is really required. Electrons satisfy the
exclusion principle, and they'll fill a lattice
and increasing energy order. This chart shows the
potential energy for a metal and it's low within the metal and
then high at the edges. So it's like a particle
in a box situation. So but because electrons have to satisfy the Pauli
Exclusion Principle, if the temperature
was absolute zero, if each of the electrons had
zero translational energy, then they would stack up in that box one electron at a time, up to a certain level, and that level is called
the Fermi energy. So this plays a role in how the electrons
are going to behave. You'll recall from our
introduction to quantum mechanics, that one of the experiments
was the photoelectron effect. That was where if light is
directed onto a metal surface, then only photons that
have enough energy to overcome the so-called
work function can remove electrons
from that surface. Well, U-naught in this image
is the depth of the well. In other words have
an electron had enough energy to
get over U-naught, then it could escape the metal. In practice, if this well is
filled up with electrons, that means that the
most energetic electron has energy equivalent to Mu_0. So the difference
between Mu_0 and U0 is the energy required
to remove the electron. Now for a Fermion the electron, the quantum state
distribution is equal to the first equation there N_k, the number of electrons and
quantum state k is equal to 1 over Epsilon k
minus Mu over kT plus 1. So in the continuum limit, the single electron
energy distribution is given by this
simple function of 1 over e to the Epsilon
minus Mu over kT plus 1 and a plot
of this is shown. As you increase the energy, well as you increase the
temperature at zero, the electrons have to be in the ground in the
lowest allowable state. As you increase the temperature, there's a probability
that they will have that the number of particles above that Fermi level is positive. Okay. So we can get the
Fermi energy if we integrate the energy distribution
at zero degree K. It has to be equal to
the number of electrons. So if you just solve
this equation, you can get Mu_0 the Fermi
energy as a function of N over V. So N over V is the free
electron number density to the two-thirds power. If you run the numbers, values are typically 1-5 ev, to run the numbers you got to know N over V or
the number density, the free electron number density. So that turns out to depend on the specific metallic atom, and there's a table in
the next slide that shows some experimentally
determined values. Bottom line is, mostly
they're about one. So here is this table and it lists and you can see
most of them are ones, some are twos, some
are three and four. But thereof order one. So if you convert
Fermi energy into temperature recalling
that one ev is 11,600 degrees Kelvin in this odd set of energy units
that we've talked about, then the Fermi temperatures are typically very much larger than temperatures that you're
going to be deal with metallic compounds
because most metals melt at 2,000 Kelvin. If the Fermi temperature are 1-5 ev that means they're going to be 11,000
and higher degrees Kelvin. So the electron gases behave as though they are at
very low temperatures. You can get the
fundamental relation for the electron gas at zero
degrees Kelvin it's easy. It's the ideal gas results. This is the Helmholtz
function version of the fundamental relationship. Higher temperatures it's a
little more complicated for the simple reason
that the electron gas does not satisfy the
Maxwell-Boltzmann limit. For copper for example, the limit is 1.42 times 10 minus 3 which
is much less than 1. Therefore, the full
Fermi-Dirac treatment has to be followed. That is usually done by expansion techniques, series
expansion techniques. This shows the
Helmholtz function and the internal energy function. Most importantly, this
is the specific heat. So in the case where the Fermi temperatures is much larger than the temperature, then the specific heat is
approximately equal to Pi squared over 2 times k times T divided by the Fermi temperature. So this is going to be
roughly the ratio of the Fermi temperature or the ratio of the temperature
to the Fermi temperature, times the ideal gas value
approximately speaking. So it's a fairly small number. In the exercises
there are problems to calculate what this is and you do that you'll get the idea that
it is fairly small. So that's it for this video. Thanks for listening.
Have a great day.