Download Solids: The free electron gas:

Solids:
In a solid a few loosely bound valence electrons (outer most and not in
completely filled shells) become detached from the atoms and move in the
material subjected to the entire crystal lattice rather than each respective
atomic core.
We will follow the book and study two models.
In the first one we consider electrons as free in a three dimensional box
with infinite walls
In the second we will assume they fill the periodic potential of the
crystalline lattice and use the so called Bloch theorem to understand
the energy levels.
The free electron gas:
Consider a solid as a rectangular box with dimensions Lx, Ly and LZ, and
that the electrons experience the potential associated with impenetrable
walls:
The Schrodinger equation
Which separates in cartesian coordinates. We can then
Assume and solve for each of the coordinates.
This yields
To satisfy X(0)=0 and to satisfy X(Lx)=0
and the total energy
To visualize what happens when we have many particles, it is useful to
imagine de 3D-k space with a grid where each block is generated with line,
drawn at k(x,y,z)= /L(x,y,z),/L(x,y,z),/L(x,y,z)…….
In each Intersection point on each block we can put a distinct particle state
Therefore each state occupies a volume in K space:
Suppose our solid contains N atoms with each atom contributing q
electrons and our solid Is in its collective ground state. (No
thermal excitations).
If the electrons were bosonic particles all will occupy the 111 state. But
electrons are fermions, obey the Pauli exclusion principle and only two can
occupy the same quantum state (two since they have S=1/2.
We can say that electrons will fill up one octant of a sphere in k-space (see
picture). The radius kF of the sphere is determined by the number of
electrons
We have N atoms with each atom contributing with q electrons. Each pair need
a volume 3/V
: Is the free election density.
kF: the radius of the sphere in u space is called the Fermi wave vector and
ћkF the Fermi momentum.
The boundary that separates the occupied and unoccupied
states is called Fermi Surface.
The energy of the highest occupied state is called the Fermi energy
To calculate the total energy of the system we need to calculate the
density of states
The density of slates times DE, (E) dE, is the number of electrons
with energy between E and E+dE
The total energy is then
This energy can be thought as the internal energy
of a gad, and we can associate a pressure to it known as
the Fermi pressure
Let's do an example.
If N identical spin 1/2 particles are subjected to
a one dimensional simple harmonic oscillator potential, what is the Fermi
energy, What is the ground state energy.
EF is the energy of the last occupied state.
We consider the two possible cases