We saw in the last session how the physical
properties of solids depend rather strongly
on this symmetric properties; however there
is 1 class of solids in which the details
in the lattice structure does not play a very
big role on the physical properties, these
are metals such as gold silver copper and
so on. This metals their behavior can be understood
on the assumption that each atom donates in
the case of mono valid metal one-electron
to the metal, and this electron is usually
the outermost 1 in the metal atom.
For example, in the case of sodium which has
an atomic number 
atomic number is the number of electrons of
leaven z is the usual symbol. There are leaven
electrons in the sodium atom of which 10 of
them go into this so-called 1 s 2 electrons
go into 1 s shell and then 2 electrons go
into the 2 s shell and then six electrons
go into the 2 p sell. So, this is the so-called
close shell structure out of these leaven.
Electrons ten of them go into the close shells
leaving only one-electron in the outermost
three s shell. So, this the electron which
is the outermost in the metal atom this is
just an example similarly 1 can look at any
metal atom and then analysis the outermost
electron is available for conduction, because
it is rather weekly bound to the parent atom
and therefore, can be ionized rather readily.
And this becomes this outermost electron is
known as the conduction electron because this
gets ionized and this electron becomes available
and is free to wander around inside the metal.
So, it is the very much like the atoms and
molecules in an ideal guess.
So, one speaks of an electron gas in this
case. So, this means that even though the
metal is the conduction this system a condensed
matter a solid, but the electrons inside are
behaving very much like the atoms. And molecules
in an ideal gas be one made ask what happens
to the coulomb repulsion between the 2 electrons
pars of electronics this intel electron repulsion
is rather week in comparison to the attraction
between the electrons and the positive ions
which are left behind after the ionization.
So, 1 has 1 neglects the first approximation
inter electron are coulomb repulsion. If overlooked
ignore to start with. So, that one can think
of free electrons which are not strongly interacting
free electron for free in the sense that the
free-for conduction to carry electricity insight,
that is why metals are such good conductors
of electricity only thing is these electrons
the ideal gas molecules are atoms are classical
particles.
Whereas electrons are quantum particles which
obey electrons are known as fermions fermions
means they obey so-called for me the dirac
statistics rather than not Maxwell Boltzmann
statistics. So, there collective behavior
is not describe by classical Maxwell Boltzmann
the statistics which are which is obey it
by ideal gas molecules and atoms, but in then
we discuss the properties of this electron
gas we have to take count of the fact that
they obey fermi dirac statistics. So, we have
to use fermi dirac statistics in order to
describe.
They are collective behavior this is are other
important factor up the fermi direct distribution
function I will write it as a f t for short
the fermi direct distribution function 
as the following firm f of e that is the distribution
function which describes how the electrons
are distributed into the various energy states.
So, this system of electrons has different
energy levels the electron energies are different.
And therefore, the electrons occupy these
energies and the way we are distributed energy
is given by this function 1 by a exponential
e by the e f by kb t plus 1, where kb is the
Boltzmann constant and the e f is known as
the fermi energy k b is the universal constant
as you know. And the fermi energy is a characteristic
of the metal now everything will depend the
collective behavior will depend on the statistical
distribution how the electrons are distributed
an energy.
Therefore let us look at how this function
looks, this function is plotted in this figure.
Let us look at the figure on the left side,
which gives the value this function plotted
at absolute 0 lets discussed the behavior
of the metal at absolute 0 and then we can
go to finite temperatures this simpler. So,
you can see that this function looks like
this. So, if I as the fermi energy here than
this is 0 this is one. So, that is the behavior
of this function at absolute 0. So, this is
at 0 k what is this there the physical meaning
of this picture is that if you look at all
state's all energies which are less than the
fermi energy. If you look to the left that
this ef in the graph on the energy access
all this states are occupy with a probability
of unity. In the sense that this mean that
they are fully occupied the states are all
completely occupied by the electrons none
of the state's is empty all the states below
e f are all occupy by electrons this. Whereas
all the state above ef the fermi energy are
completely empty they have 0 probability the
f of e is 0.
So, the probability of occupation of the state
about the fermi energy at absolute 0 is 0
that mean that they are completely an occupy.
This is because the electron obey what is
known as Pauli is crucial principle that is
way they are fermions this means that it if
1 state are 1 energy level is occupied by
an electron then another electron cannot be
found in the same state it is excluded from
occupying the same state. So, each state is
occupied by an electron, and you have all
the states below e f occupy well this picture
is slightly modified at finite temperature,
but we will come to that a little latter the
it is this is enough for as…
Now, So, if this is. So, at 0 kelvin the fermi
level e f is the highest occupied state now
of course, then I say that each state is occupied
by one-electron according to the Pauli principle
what I mean is that we do not consider this
spin of electron if you consider the spin
of the electron. Then we know that electron
can have has a spin of half and. So, can occupied
2 states with parlor or anti parallel spin
beer given direction in space therefore, these
2 electrons with opposite spins both have
the same energy. And so each of the states
can be occupied by through electrons with
opposite spins without violating only principle.
You will remember this, but this spatial energy
of these electron is simply given by the kinetic
energy, which is h cross square k square by
2 m there h cross is h by 2 pi h is the planck’s
k is the wave number which is equal to 2 pi
by lambda we already talked about lambda broglie
wavelength. And m is the mass the electron.
So, the energies of the electron in the state
is given by the wave vector k, and this is
just the kinetic energy h cross square k square
by 2 m.
Now, if we why do we need this we need this
information united calculatefor example, the
number of electrons conduction electrons inside
a metal the metal is now somewhat like a box
with in which this electron gases free to
wonder around, but the electron is not a allow
to escape out of the better. So, that is a
the only constraint on the electrons. So,
within the metal they are free to wander around
very much like the atoms and molecules of
a gas.
So, next we should to calculate the concentration
of electrons which is the number of electrons
for unit volume can I say electron I only
mean the conduction electrons in order to
find this t we have to also consider in addition
to the distribution function. We have to consider
what is known as the density of states 
this is because the distribution function
tells us how the electrons are distributed
in the energy, but as you can see from this
equalization connecting the energy and the
wave vector are the wave number it is the
wave number which decides that the electrons
state.
So, we want to also know how the hold me a
conduction electrons are distributed indifferent
states corresponding to a given energy. So,
this is given by what is known as the density
of states. So, it tells us how many states
are available in a differentia energy interval
agents to a given energy. So, if I look at
a particular energy. So, I would loot look
at an infinite this month d e around a given
energy. So, if I take this infinity dismal
energy interval we d f e is the density of
a state function which described the d f e
d e gives the the number of steps in this
energy interval.
So, we wish to find out this density of states
together the distribution function ff e and
the density of the state function d f e d
e together will determine the average this
statistical properties are this electron gas
how do you find this density of states function
we will just discuss this next.
So, we have we know that the energy is just
p-square by 2 m because the electron have
only kinetic energy. So, d e is p d p by m
now if I look at the momentum space and if
I regard the fermi energy as corresponding
to a value which corresponds to an isotropic
fermi surface which means it is fermi surface.
So, it is the spear with radios e f in energy
space. So, that is the fermi spear 
and all the inside state inside this or occupy
by electron gas. So, we know that radios the
volume of this will differential volume is
four pi p square d p.
This is the differential volume in momentum
space position and momentum together define
a state in statically physics. So, this is
the differential volume in momentum space
if we take the number. Now, we count the number
or state in this interval by writing d f e
d e e equals v the actual volume in real physical
space the position space multiplied by the
differential volume in momentum space times
2 the factor 2. In order to take it account
this spin and divide this by the volume of
f s cell in phase space sell means state each
cell each of the cell correspond to 1 state
of the electron as we discussed already according
to the pauli principle.
Now, this volume of a cell in phase space
is given by the so-called uncertainty principle
in quantum mechanics as I already told you
the electrons are quantum particles. So, we
have discussed as statistical behavior according
to the quantum statistics and fermi dirac
statistics is a quantum statistics. Now the
main feature of quantum behavior comes from
the so-called uncertainty principle due to
Haisonbag this principle states that the product
in uncertainties of position. And momentum
is of the order of the planks constant and
therefore, this is in 1 by mention therefore,
we can write the volume of a cell in phase
space as the product of the uncertainty you
cannot look it a particle beyond this accuracy
in quantum mechanics. So, this is the minimum
uncertainties.
So, this is the spatial and momentum, extent
the extension in space and a real space and
momentum is given is order of the planks constant.
So, if we use the same argument for all the
three dimensions and in real space and all
the three components of linear momentum then
they get this volume is h cube this is h for
each dimensions. So, there are three dimensional.
So, each time it is multiplied. So, get h
cube and that is the uncertainty to which
you can locate a given state in phase space
in quantum mechanics therefore, that corresponds.
So, this space the phase space is quantum
mechanics is course grain and this is the
volume occupied by a a state in phase space.
So, you derive the total volume available
physical volume v times the differential volume
in momentum space times the factor 2 due to
spins state and divide the whole thing by
h cube.
That will give the number are state which
is intern to the given by d f e d e. So, this
gives as the way to calculate density of state
function therefore, we just substitute p square
is just 2 m e.
So, 4 pi. So, p is 2 m e to the power half.
So, here I have p square d p which I can write
as p times p d p p d p already I have as m
d e. So, 2 m e to the power half times m p
e therefore, substituting here d f e d e 2
v into four pi into p square d p which is
here to m e to the power half into m d e pi
h cube. So, this some simplification gives
you the density of space function as four
pi v.
So, you can see that this is the density of
state function which is plotted these together.
So, you can see the density of state goes
as e power half.
Now, we have a all the we have developed all
the things that we need 2 evaluate the electron
concentration at 0-kelvin let us do this now.
So, the number of electrons in unit volume
is the total number divided by volume and
that will be the total is what is given by
this. So, I have 1 by integral of e f of e
d e comes 0 to e f this is because of ff e
gives you how the electrons are distributed
in energy. And d f e gives you use you how
the energies are distributed at in states
and therefore, the product of these 2 integrated
over an energy interval from 0 to fermi energy
up to the all states are completely occupied
rest of them are completely empty. So, it
is enough if I integrated over all the energies
from 0 to e f let me write e f at 0 in order
to remind ourselves of the fact, that we have
calculating it at 0 kelvin. So, if I substituted
the this 1 by b 0 to e f, then I have this
four pi v into 2 m to the power three by 2
by h cube e power half d e and I have removed
f of e because this is going to be 1 for all
the states. So, it is going to have a value
1 at absolute 0.
So, we can write all these constants can be
back out v cancels for 4 pi 2 m to the power
3 by 2 by h cube e f 0 to the power 3 by 2
into 2 by 3 as a result of integration.
So, that gives me final result as 
using this and using ef as we already saw
e f is nothing, but the kinetic energy up
to the wave vector at fermi energy h cross
square k f square by 2 m therefore, we get
n s. So, that gives you rather compact relationship
between the fermi wave vector at absolute
0 and the electron concentration n now me
assume that because for example, in a metal
like sodium each atom donates 1 electron to
the conduction band. So, if we have a moral
a gram atom of this solid then this will contain
as is well known and number of atoms. So,
that correspond to the number of electrons
donate at a conduction electron and do not
donated.
So, if you take number and divide by the atomic
wait.
So, that gives me in 1 gram atom are kilogram
atom correspond to n a number of atoms and
I have each atom gives you 1 electron in a
monatomic solid. So, this is the number of
electrons. So, this correspond to your weight
of a where this is atomic weight therefore,
the number of electron is just in 1 mole is
n a by a times.
They have to we are forgetting the fact that
we have also row because it this gives you
the number half electrons per unit mass, and
then we have to multiply this by the density.
So, we can calculate n and therefore, calculate
the fermi wave vector and inurn the fermi
energy. So, we have way of calculating the
fermi energy from this formula.
So, the fermi energy values calculated in
this way are also shown in the table. So,
the fermi energy at 0 kelvin the values are
given in an electron volt which is a convenient
unit in a the case of atomic physics. This
gives you the energy of an electron when is
accelerated through a potential of 1 volte,
now the various metals aluminum copper gold
potassium silver sodium.
You can see the fermi energy the various from
something like eleven points six electron
volts to the three-point to electron volts
in any case all of the order of electron volts.
So, this is a very important idea because
if you converted by using Boltzmann constant
to equal intemperate this will be a the order
of ten to the power of four kelvin. So, a
very high temperature. So, the energies the
fermi energy corresponds to be a very high
temperature the corresponding temperature
in temperature units. Now we will use these
concepts to calculate an important thermal
property namely be electronic specific heat
this means that this electron gas, if you
inject some heat and energy into it they electron
absorbs this energy. And that temperature
goes up though these this specific heat is
defined as the rate of change of the mean
internally energy of the conduction electron
at 0 k with respect to temperature.
So, the electronic specificate can be calculated
that by calculating the average energy of
this electron gas which is simply done by
again at absolute 0. This is done by integrating
from 0 to e f 0 of e d f e f f e d e by 0
to e f 0 of e f e f of e d. So, this gives
going by the same procedure they arrive at
the result this is three fifth of e f 0.
The average energy is three fifth of fermi
energy at absolute 0 now the electronics specific
heat having got the average energy, we can
simply differentiate this expression fermi
energy with respect to temperature that gives
you the electronic specific heat. Now we will
first qualitatively see what kind results
that we are going to get or this calculation
the electronic specific heat, since we said
that the e f 0 is the order of ten to the
power four kelvin in temperature units.
So, if you thermal exited usually thermal
excitation is at best of the order of hundred
kelvin. So, it is a very small if you go back
to the energy distribution this is e f 0 if
you go back to this this is of the order of
this in temperature e units is that the the
order of ten thousand kelvin, but our energy
excitations thermal energy excitation is only
at the order of a hundred kelvin. So, it is
small temperature window. So, the exhibition
is going to shift states from this to this,
but the state's are all remember that they
are all completely occupied and are subject
to the electron for subject pauli is crucial
that mean if state is completely occupying
already you can put another electron.
So, this exhibition from this one occupy state
to another occupy state in this whole range
is not going to be possible even though you
give with excitation, the electrons cannot
be exited fermi occupied state here into another
occupied state which is not empty. So, this
is only pass only at this edge this is not
possible and the thermal exhibition is one
only possible here here. So, this is of the
order of ten to the power four Kelvin, and
this is order of ten square. So, this is only
possible across this fermi energy if it bring
it here.
So, it can excite across in to 1 of the empty
state here. So, only the a fraction of the
electron which occupy a stated within this
ten hundred kelvin in the neighborhood of
the fermi energy only they will be able to
get excited. So, what is this fraction they
are k b t by k b t r k b t f corresponds to
all the states the electrons not the energy
scale and response b t corresponds to the
energy of the thermal excitation.
So, this is the fraction which is t by t f
that is the fraction of electrons 
excited exited thermally and each of them
has an exhibition of order of k b t therefore,
the excitation energy is as the order of k
b t square by t f, so d by d t I of this use
this specific. So, these are the order of
k b t. So, this tells me that the electronic
specific is the order of k b t it is proportional
to the temperature t. So, the electronic heat
capacity r c e electronic specific heat is
plus equal to the constant times the temperature.
So, that is the basic results that we get
for the electronic specific heat of the conduction
electronic gas, so you get this is the value
at absolute 0.
So, this is shown graphically. So, in figure
of course, you will never be able to measure
the electronic heat capacitance alone it will
be also the specific with includes the contribution
from the electron. And also from the lattice
of ions are the atoms in the solid and that
as we will see later is given by the may be
theory specificate and that temperature depend
is a t q dependence. So, the overall behavior
is given off by a relation of this time c
total and therefore, if you brought c by t
verses p-square that would be a straight line
that is what is shown in figure. So, from
the intercept of this we can get the heat
capacity question gamma, we have discussed
everything at absolute 0, but the question
arises w what happens? Then you have an electron
at a finite temperature the this the bit more
difficult to calculate, we will not go through
the details at this calculation here, but
I will just roughly.
It what happens by looking at the distribution
function. So, at a finite temperature the
fermi dirac distribution function this at
a finite temperature which is not 0. So, the
distribution function, now deviate from the
behavior at absolute 0, this is at 0 kelvin
as we have already seen, and at finite temperature,
this reduces to something like this. That
is the behavior here at the not equal to 0
at any finite temperature it the f f e decreases
from the value 1 to something like of fermi
energy. And then it goes on beyond it is non-zero
even beyond the ef this means that some of
this state or empty even before even below
the fermi energy, and some on the state or
above the fermi energy are occupy and these
the number occupied states goes on increasing.
So, this is the behavior and this will modify
the fermi energy the fermi energy will be
a function of temperature. Now and the heat
capacity equation which is still found to
be a linear function of temperature, but but
the constant the linear heat capacity question
is slightly different. We will not calculate
this this is the overall behavior now electron
gas also processes many other interesting
properties the electron have a magnetic moment
because that that is been and this contributes
to magnet. So, this is known as Pauli paramagnets
or this is a spin susceptibility this is the
magnetic susceptibility which arises from
the fact that the electron Aspin’s, and
therefore a magnetic moment. So, this is another
important characteristic of the conduction
electron gas in metals in addition the metals
most important characteristic is that metal
is a very good contact to have electricity.
Now we would like to have an expression far
the electrical conductivity of a metal.
And how it depends on for example, temperature
and not only the electrical conductivity the
a metal is also a good conductor of heat.
So, we would like to know how the thermal
conductivity is determined by the behavior
electron gas, we also know that there is such
a thing called thermo electric power the phenomenon
the thermo electricity in which metal junction
used in order to produce the thermo electricity
and e m f. So, we would also like to know
how the thermoelectric power of a good conductor
is determined by the behavior the electron
gas these are things we will discuss in next
lecture.
