Welcome to lecture number 3. So, what are
we going to do is, we will start with where
we finished off in lecture number 2. So, let
us recall what we did in at the end of lecture
number 2 was we had said that if you know
that you have n electrons per unit volume
which is this n write here.
Is the n electron per unit volume, then as
you start filling, now you have a different
case state, we derived quantized case state
available and now we said that if you want
to take this n electrons and start putting
them from lowest energy up slowly, slowly
start putting them up. We will start looking
at this graphically also in little bit as
you start putting them up, then you will fill
up the last electron up to the energy level
E f, which is this energy.
This relationship between them this n and
this E f was derived has to be this pi there
as shown here. Similarly, we had said if you
want momentum, then if you want to do momentum
then this momentum was equal to h k f and
even this we can now in terms of this quantity
n in terms of n we can also determine momentum.
Once we determine momentum, then in terms
of this n, we can also determine what is the
fermi velocity? And as a momentum, we mean
momentum of the electrons which are at the
fermi energy.
That means, the highest highest energy which
electrons are having from lowest energy onwards
is up to that energy where electrons are filling
up. The velocity of those electrons then is
also given by this quantity which is where
we finished in the previous lecture. Now,
let us look at this little bit further that
we also said that, now that you know since
you know how to calculate this quantity n.
So, substitute n in to this equation of v
f and say v f we will find is about 10 to
power 8 centimeters per second.
That is the kind of velocity which you have
which means that you are in about one percent
of velocity of light. Remember in Drude’s
theory, this velocity turned out the velocity
of the electrons that where conducting turned
out to be 10 to the power 7 centimeters per
seconds, that is the difference we have. Gain
in free electron is what the free electronics
theory is saying so far. Now, we want to move
forward in this lecture three what we will
do is that we will look at four topics today
will look at the density of electron stage.
This is a side topic, this topics is going
to be useful when we start doing semiconductors,
then I am going to use the result from this.
Also after that we will start looking at electrical
conductivity which we derived in context of
Drude theory. Now, we will start looking at
the looking at that point of free electron
version, once we have done that I will show
you the difference between Drude theory and
free electron theory.
Finally, today I will show you that even free
electron theory has series limitation and
we have to see what next. So, that is what
this lecture number 3 would be today. So,
let us get going let us start with this density
of electrons state let us start with this
topic. So, what we derive was the less turn
around this equation for fermi energy we have
derive if we take this equation right here
if you take this equation and turn this around
then I would write this equation.
Then, I would write it as n as equal to I
will write n as equal to 1 by 3 pi square
two m e divided by h bar square power 3 by
2 E f power 3 by 2. So, what we mean by density
of states that was defined meaning of density
of states. This is the symbol I am going to
use. So, we will say that g of E d E is the
number of electron states in energy interval
of E and E plus d E. This is the number of
electrons that is number of electrons states
which are in energy level energy E and E plus
d E.
So, that is what the meaning of this density
of states says, now I like to pause for a
minute and clear up whole idea which may become
confusing later. So, let us clear it up right
now. First thing is, we could also define
something called as density of k states remember
what we have is the way you should think is
like this that there are allowed k energies,
the allowed k states, not k energy the allowed
k states.
So, if you have different k states which allowed
then you can ask the question what is the
density of these k states. So, therefore I
could have defined another quantity called
density of k states then since each of this
k states can take two electrons. So, if I
multiply by 2 these densities of k states,
then I would get what I what we call as density
of electron states. So, either you work with
density of electrons states or you can rather
density of k states as long as you remember
whether you multiply with this factor of 2
then you would be ok.
In this course what we will do is, we will
never deal with density of k states, we will
always deal with density of electron states
which means that there is a difference of
factor 2 only. That means, I have certain
days to k states, since each k state could
have taken 2 electrons it has been up, it
has been down electrons. Therefore, I will
always call that density multiplied by 2 as
the density of electron states; remember these
are density of electron state. It does not
mean that electron indeed is necessarily present
on those states is that the states are there.
Then that means, electron can occupy that
it is not necessary that electron does occupy
those states.
In fact, if you want to know whether electron
is there or not we will have to do something
else which I will do later, but notice I have
been silent so far about this aspect. I have
derived this relationship right there saying
that if I have n electrons up to what energy
they will fill. In that sense it is a hypothetical
question that if I have n electron then how
far they will fill that is the question. That
is the question I have answered I have not
said whether I have indeed put those electrons
in or not.
So, you can think it think of it like that
or you can think it think differently that
indeed all these states are indeed filled
with the electrons you remember that all this
calculation are revealed for 0 k. It is possible
we will see later that if we go for higher
temperature above 0 k, then some of these
electrons could go to even higher energies
leaving some of state vacant. In that case,
these formulas directly will not be valid.
So, that is why I am saying that you think
as if I have all this k states and therefore,
electron states multiplied by 2. If I had
n electrons, then how far they will fill is
the questions we have answered so far, and
indeed if they do occupy if that the probability
of occupying those states is 1; that means,
indeed all those states are occupied.
Then precisely, these would be the energy,
these would be the momentum, these would be
the velocity, fermi velocity etcetera of those
electrons. So, with this background, now is
very simple to derive what the, remember now
I can instead of writing this Fermi energy,
let me write another quantity called n prime
and I will write this same expression as like
this n prime. I am using only for explanation
in future; I will drop this and keep calling
back again and itself. So, there is no confusion
sorry let me remove this f here I just want
E, what I am trying to show is by exactly
same logic instead of saying earlier I said
I have n electrons, if I have n electrons
how far they fill and defined that is a fermi
energy.
But, you could think slightly differently
the way which will also think is that up to
energy E. I can fill total of n prime electrons
and if I go slightly d E energy higher then
I would have filled this n prime plus the
n prime electrons extra and we are seeking
what is that number of electrons. So, if you
can calculate that number of electron, then
you can immediately define our g E clearly,
then this quantity g E would be equal to simply
d n prime by d E why because d n prime would
be equal to g E d E.
This is the quantity we are trying to what
is that mean this is the number of electron
in energy, this is the number of electron
we read the definition, here is the number
of electron states in energy of this intervals.
So, I will explain n prime electrons are filling
up to energy E and if I go slightly higher
energy, the d E then the number increase the
electrons is d n prime. So, we are basically
trying to calculate d n prime by d for the
density of states. So, now once we have this
definition it is straight forward enough to
write this.
Density of states has been equal to just taking
the derivative of this quantity n d n d E
call it prime or do not drop this prime does
not matter really 1 over 2 pi square 2 m e.
At presume you can carry out this derivation
I have just written it out here and to save
time, 3 by 2 energy to power now half, which
is what density of state is. So, what density
of state goes as square root of energy? If
this is it this energy then g of E goes as
square root, it goes something like this.
Given this, now you stop at this topic, these
density electrons, density of electron states
just remember this expression I am going to
use as about 4, 5, 6 lecture down the line.
I will start using this when I start doing
semiconductors, but in context of free electron
theory, we derived this we given this definition
and from gives given this certain definition
which will use subsequently. So, now let us
move on to the next topic which is electrical
conductivity 
free electron version is now what you want
to do.
So, what do we have we know that h bar k is
momentum. So, if I divide this by mass of
electron I get velocity of any electron, I
get velocity of any electron in this in this
way now let look at this the way I am saying
is that if you have all this k states. If
you have all this k states that is a k x k
y k z and in this in there is a sphere up
to which electrons are all filling up. All
these electrons are filling up to these this
k f and this is what k f is up to here all
these electrons are filling up. Now, if we
notice that for every k in for fermi sphere,
there is a minus, there is a 
minus k vector also. What is that mean for
every velocity of electron, there is exactly
opposite velocity of electrons also.
So, net velocity net velocity is equal to
0, so that means, clearly if nothing then
this electron cannot cut material is this
as this no conductivity, we expected there
should be no net velocity if there was net
velocity then we would have correct. But,
clearly without application of electric field,
we can’t be having current. And therefore,
as we expected the net velocity must be 0,
but what happens if I apply high electric
field.
If we now apply electric field 
then what happens, we know that d P its rate
of change of momentum, now should be equal
to minus q times their electric field which
is a minus sign because a electron q is a
absolute charge 1, but positive number 1.6
into 10 to the power minus 19. So, erase that
number any way. So, this is the rate of change
of momentum which implies, I can write this
us h d k d t as equal to minus q times this
electric field e, what are that this mean.
So, in time t this fermi sphere starts moving,
this start moving how does it move? If I integrate
this expression right here, equation what
do I get, I get that k vector at any time
t then will be equal to k vector at time equal
to 0. That at time equal to 0 is when we apply
the electric field say t equal to 0 electric
field is applied in time, this k vector began
to move and this becomes equals to q minus
q E by h bar t.
This amount by which this k vector moves in
time t, now that in time as time increases
this k vector can keep increasing is like
shifting of origin of the fermi sphere. We
can think of this as follows, we can imagine
we have draw in a minute, but before that
you can think this is just a center the fermi
sphere changing from a different k position
moving from one k position other k position,
but the problem is now that t is changing,
this fermi’s starts keeps shifting.
So, this causes of problem same problem which
we had when we started that if I applied electric
field if there is nothing to dump the motion
then the current will continuously increases
for a given electric field which is contrive
to all observation. Same thing will happen
here if k continuously increases then therefore,
the same token this we will continuously increase.
And therefore, current will continuously increase
the current will also continuously increase.
So, we face with exactly same problem as we
started in the lecture number one who is where
we started Drude’s theory and we invoked
collusion which was really giving as stamping.
Hence, the velocity we found to be average
velocity was a fix velocity when we could
it collisions. Now, in this free electron
theory we will make more such statement as
to what the mechanism of mechanism of this
damping is, but we know that from experience
they must be some damping.
So, let us assume, let us invoke the same
time tau let say that by some mechanism there
is a relaxation time called tau the same relaxation
tau, we started using in Drude’s theory
may be because of pollution with ions. But,
same idea that in time tau that this k state
changes to that value to k state changes up
to this time tau, and then it is held at that
Fermi, then no more changes happened. You
think of tau of relaxation time, means if
you want to think same way as Drude, then
mean time between collisions. And therefore,
after tarp time tau on average is it a value
of k average value of k would have.
Then, in that case become as you write then
the same expression by saying k, then they
will become after applying electric field
k, then will become equal to k. Whatever it
was a 0 minus or plus, I should say minus
q times plus E, what is the expression was
minus q E by h bar minus q E by h bar down
here. So, if tau is the relaxation time, then
k average value k states where is a state
applying when apply electric field is then
k of tau, this quantity.
So, effectively therefore I can show in two
dimensions is easy to show you can imagine
this to be c. You can imagine this as a 3
dimensional figure, also what I will do is
let us say this is k x. Let us say this is
k y and let us say that I am going to draw
it at a circle or everything not as sphere
for a circle in two dimensions. You can keep
imagine, this is a 3 dimensional figure as
a sphere. So, what I do is I will draw a circle
here. Let us chose a different color, we will
chose a circle here like this. So, here is
a circle I have drawn here and this is at
t this is a t equal to 0 and what you have
done is what you have done is we applied a
electric field E.
Let us say, we applied a electric field in
x direction initially t equal to 0. We have
electric field equal to 0 and then at t equal
to 0, turn on electric field and what happens
then when we applied an electric field what
happens is a the picture will draw here right
here. This is k x, this is k y and let me
draw the same sphere first, the same sphere
which I had for I will draw, show it is a
dotted line. So, this is the sphere at t equal
to 0 which originally I draw in the previous
figure.
Now, I will show with another color pen, let
us say blue, let us say now I have a situation
where this is t is equal to tau and E field
which I have lied is let us say minus E magnitude
n x. I have direction we applied this field,
let us say if we applied this field then this
now see where the k will be every k point,
whatever the k point was a time equal to 0
in time tau. Its value would be that number
plus this number plus this number at I have
taken E to be minus field with every where
plus number here.
So, it is going to shift in the x direction.
So, let us draw that is speed also the circle
now in this case and you can imagine sphere,
then this after shifting will become something
like this right this whole thing as shifted
this center is shifted to here. Now, this
whole thing moves something like this now
something like this. So, this was at t equal
to 0 and this is at and this is at t equal
to tau. So, what you notice. So, what do you
see here. So, what has happened you my figures
are not nice you can imagine all of these
to be circles, but anyway. So, now what you
see now notice that there as in this particular
case.
For every vector k for every vector k I had
exactly equal minus k vector also and therefore,
what happened was for every vector k any vector
k I had a equal and opposite minus vector.
Therefore, net velocity was 0 where I when
if we go for 0 at that time the net velocity
will it trans for zero and therefore, no current.
But, now its notice in this blue circle which
is what it will be at t equal to tau when
I have applied electric field.
When I applied electric field now the net
velocity will not cancel, now notice again
a velocity, like this will have exactly component
like this. And all this velocity will cancel
out, but the electrons which have velocity
corresponding to k states which are near the
surface of this blue circle rose velocity.
Now, will not cancel out what is that mean
this very important fact this very important.
Therefore, since now the electrons the electron
velocity near fermi surface will not cancel
out meaning net velocity of electrons will
be due to those electrons, which are near
fermi surface. What is that mean? Let me repeat
what I said again I said that most of these
velocity like this k component will cancel
out this minus k component, but the k now
there will be those case, which will not cancel
out. And on those case will be the once which
lie on the surface which are not going to
cancel out any more.
So, what do we see that the electrons that
conduct implies the electrons that conduct
are once near the Fermi surface and have velocities.
Approximately as v f Fermi velocity which
we recall was 10 to power 8 centimeters per
seconds not 10 to power 7 centimeter per second
at root s we assume in Drude’s theory. Therefore,
you can see the first error which was in Drude’s
theory. Now, we can we have we have a better
estimates of what the velocity of electrons
that conduct should be now I will not derive
the expression, but only thing is that if
you go through a process of calculating this
average that velocity through this process
exactly, what the way it has been.
Describe if you carry out summation over l
if you average over all the velocity then
calculate what the net velocity is and you
also calculate, what will be the number of
electrons that will be what the number of
electrons, which will be is. Therefore, conducting
then you will find if you go through this
whole process, what I will do is write the
final expression that. In this case we find
that the conductivity will become equal to
1 by 3 q square the same, which is going to
has to appear and this becomes equal to then
tau I am going to chose the punting test g
itself.
This is the density of electrons near the
fermi energy for this expression for conductivity
becomes this which should make sense, it is
should have a dependence on the number of
electrons that are near the fermi energy.
It should depend on the velocity of the electrons
which is of the velocity of those electrons,
which conduct, which is the fermi velocity
this gives to say estimates of what the conductivity
of a material will be any way this was only
side topic I just since, we derive the same
expression for Drude’s theory. So, therefore,
I am done that for free electron theory, now
let us there is a time to take stop of what
is happened lets analyze what we have gain.
So, far and where the problem continues to
be. So, let us first start looking at what
have been gain Drude’s velocity verses while
talk about Drude’s per velocity Drude’s
theory verses free electron theory. Let us
do this first what is happened what are we
gain by doing this free electrons theory 
clearly find this case our estimates of things
will become better where ever velocity is
going involve where velocity is going involve.
Since, we know that the velocity is order
without between Drude’s theory and free
electron theory we found the disciplines of
order of magnitude in velocity.
Velocity being higher in case of free electron
theory, then we know that root theory wherever
velocity of electrons works will be involve
independently for that. It does not cancel
with something else then that case free electron
theory will make it better prediction. Then
Drude’s theory in order to really show you
advantage of brief free electron theory I
will take you through some other topic couple
of topics.
This means sound out of context in context
is electronic properties of material, what
I want to talk about this thermal conductivity
and thermal electric power we just show you
how free electron gives free electron theory
gives you better estimates then Drude’s
theory. So, let us let us look at two topics
in this case one is thermal conductivity 
in metals in particular I am going to talk
about this what is called as Wiedemenn-Franz
law. I will try to show you what have we gained,
now if we without any remember the conduction
thermal conduction in metals occurs by same
process by which electrical conduction occurs.
That means, the mechanism by which the heat
transport remember we said electrons are arrive
local thermal equilibrium.
So, now if electrons are moving now due to
electric field for example, if you wish that
gives you correct, but then this motion of
motion of electrons also carries heat. So,
the thermal conduction in metals also happen
through this m not same exact same motion
of electrons in that sense we are not talking
out of context some. When you talk of thermal
conductivity in metals we are at microscopic
level at atomic level, we are talking about
the same process which we talk for electrical
conductions.
So, if is you therefore, you are free to apply
the same theory to develop the idea of thermal
conduction also in which case by classic in
roots approach or classical approach 
without proof. I will write this thermal conductivity
or capital k to be equal to 1 by velocity
square tau times C v in quantity of Drude’s
theory. If we just derive then this quantity
is half derived as half b square tau times
specific heat C v this is the specific heats
of a material. This also can be written as
1 by 3 being the l being v times tau of what
being that l by 3 should here 1 by 3 v l c
v see the weight room number v times tau as
been equal to l therefore, we using here.
This quantity is derived like this second
thing also, I say there by classical approach
by same approach its specific heats specific
heat 
is derived as 3 by 2 n times Boltzmann constant
k b. There is a Boltzmann constant and estimate
of velocity is estimate of velocity is 
by half m m e v square is equal to 3 by 2
k b times t which you have seen before which
is same thing. This we have use before also
in when we are doing Drude’s theory I am
giving it two expressions right here.
This I am giving without proof that in root
context of root theory you can derive specific
heat of metals to be 3 by 2 n k time k Boltzmann
constant. You can derive thermal conductivity
by same transport mechanism to be equal to
1 by 3 v square tau c v or 1 by 3 v times
l time C v. For this you can derive by classical
theory Drude’s theory now there is a law
which is called Wiedemann-Franz law, which
says for metal k divided by sigma T will be
a constant and this is this law.
So, this is what this Wiededmann-Franz law
is, lets derive this expression got it now.
So, what is this quantity v equal to? So,
this we going to derive, so let us write down
this quantity k sigma by t will be equal to
the substitute k 1 by 3 v square tau c v divided
by conductivity. Remember what is conductivity,
conductivity we had derived as equal to n
q square tau by m e .That is what we derive,
this as therefore, you can write substitute
that in head. So, n q square tau by m e times
t. Therefore, then substitute here also for,
now we gone substitute, make two substitution
which is what we are going to substitute.
First we substitute for velocity this square
of velocity we should estimate from right
from here.
So, will say these square as being equal to
3 k b T by m e, that is what will substitute
in here and we will also substitute C v what
equal to of course, we have written there
C v is already there given there. So, let
us substitute this in here also, so 1 by 3,
we going to write and then for v square we
going to write 3 k b T by m e.
And then there is a tau here, then C v we
substitute in here as 3 by 2 n k b and this
whole thing divided by n q square tau m e
by T. Now, notice this tau and tau cancels
this t and this t cancels this m e and m e
cancels and this n and this n cancels. So,
what have we left with and lets allow the
3 to cancel this three. So, we have left with
essentially 3 by 2 k b by q whole square,
indeed is a constant Boltzmann constant divided
by this happens for charge electronic charge
1.6 into 10 power minus 19, indeed this is
constant which is what is observe.
In this Wiedemann-Franz law, which is x remittal
observation even in roots theory. This value
came out this constant which and this value
went plugged in. You can plugged in this value
what you get is the number which is like 1.11
into 10 to power minus 8 watts ohms or kelvin
square. Now, if you look at this number, this
number is about only half of what actually
is observed. So, it is pretty remarkable that
even Drude’s theory basically predicted
right result anybody thought. At this, Drude’s
theory is predicting this Wiedemann-Franz
law will because only half of what is experimentally
observed. So, the thing of it is this that
what happen was that electronic contribution.
The electronic contribution 
to specific heat 
is at least hundred times more than observed
and what is the electronic contribution remember
3 by 2 n k b. This is the classical theory
and this is the electronic contribution, the
specific heat and this number is at least
hundred times more than ever done any electrons.
Then what the electronic contribution to specific
heat is, so what happens, how it still work
does.
Now, let us go back if you go back notice
then also v square term in here there is the
v square term also in here and this v square
is what is helped this. And remember Drude’s
theory v also under estimates we under estimates
velocity by 10 times. We have v square and
over estimates of C v 100 times, therefore
2 error canceled out, and you got something
very close to what reality is in Drude’s
theory 10 times. Let me just repeat this part,
so this is in the Drude’s theory this is
in the Drude’s theory.
Now, let us see what happens in free electrons,
free electron theory in free electron theory,
the estimate of C v becomes equal to pi square
without proof. Again I just simply give you
by free electron theory what happens k b Boltzmann
constant, this fermi energy now 10 times k
b another post two in Drude’s theory. Where
it was C v equal to 3 by 2 n k b this factor
right here, then k b remember is same in both
of them classical theory, the free electron
theory. This factor if you look at even at
room temperature, this t at room temperature
if you look at this factor this pi square
by 2 k b t by E f, this factor at least hundred
times smaller than this 3 by 2.
Compare it 3 by 2 this factor is about hundred
times smaller which is why you are saying
that the electronic contribution to the specific
heat is at least 100 times more than observed.
This is what electronic contribution really
is which is being predicted well by this free
electron theory. Now, if you put this simultaneously
estimates this velocity if you estimates by
the velocity by 2 times E f by m e not where
is according to half m v square m e v square
mass of electron as equal to the fermi energy.
If we going that to fermi energy then you
estimate the velocity to be like this if you
estimate the velocity to be like this. Then
remember in classical theory we have estimate
this by as equal to 3 by 2 k b t and we have
under estimated this velocity 10 times. So,
now this velocity estimate is 10 for 8 centimeter
per second, as we have derived in free electron
theory earlier if you now plug in this number.
Then in this case, k by sigma t becomes equal
to pi square by 3 k b by q whole squared again
constant number even by free electron theory
we were getting k by sigma T with the Wiedemann-Franz
law. I have we say that this constant indeed,
we find derive by free electron theory also
that k divided by sigma T is constant. Now,
this substitute in the numbers this number
comes out as 2.44 into 10 to power minus 8
watts ohms per Kelvin square which is a even
better estimates of which is now correct estimates.
Of what we have what observed value, because
remember I said this was 1.11 in case of root
theory and it is more half of what it is actually
observe.
Now, this is beginning to predict even better
of course, it is a minor gain if you look
at another example which is thermo electric
power. Now, you will see what we are going
to do is, what we going to do is in case of
thermal conductivity thermal conductivity.
The two errors velocity and specific heat
canceled out each other, now we were going
to take a example where there is no velocity
term, but only C v term.
This means, now Drude’s theory will completely
go here where I will show to you the free
electron theory continues to predict. Now,
predict much better and that incase of thermo
electric power let us look at this, which
means if I take a hot bar, first on material,
then the number what will happen, if we keep
one end at T 1 temperature other one at T
2 temperature where T 1 is greater than T
2. Then what will happen, remember electrons
come to thermal equilibrium we had assume
that local thermal equilibrium. So, the once
which are near T 1 will have high energy higher
kinetic energy then the once which have other
end at T 2 end, which have lower kinetic energy.
So, there will be net flux of electrons going
from T 1 side to T 2 side, but that cannot
continue in differently because it can’t
be a current flow in this system. So, what
will happen as more and more electrons move
from T 1 sides to T 2 temperature sides. Then
a electric field will set up, this electric
field will oppose the motion of motion of
electrons from T 1 side to T 2, that mean
due to electric field you will have electron
going from T 2 to T 1. Due to thermal energy,
the net flow will be from T 1 to T 2 and in
equilibrium these two will balance out. So,
that there no current and this is well known
effect called See-back effect. Remember that
fewer the electric field that will develop
if you apply temperature gradient across the
bar.
Then, the field that will develop inside for
that the system comes in a steady state is
E equal to few times gradient in gradient
in temperature. That is the electric field
that is develops and this is called the thermo
electric power and this is well known See
back effect which you have seen in a various
situations thought in schools also. So, this
the thermo electric power, now if you go through
this again to have net 0 velocities, no current
flow, when there is temperature gradient electric
field also developing and both balancing.
So, there is no net velocity under those conditions
the expression which derives for q is the
q should be equal to minus C v divided by
3 n q that what this q should be thermo power.
Now, notice in thermo power we have only have
C v if we incorrectly estimated, we going
to get wrong thermo power and if you put it
right number then only we get thermo power
which is today, if you use classical estimates
of pacific heats. So, roots theory classical
estimates if you use estimate we get q which
is equal to minus 4.3 into 10 to power minus
4 volts per Kelvin, which is hundred times
more than observed value clearly.
Then, since estimates of free electron theory
of C v is hundred times less we will start
getting right numbers. So, that is really
the success free electron theory. This is
where the free electron theory has there is
an improvement from roots theory for classical
theory going to quantum mechanical free electron
theory, this is where we start seeing improvement.
Now, let us look at also failures of free
electronic and there are many, now notice
what about hall coefficient they call R h
was equal to 1 by n q. Now, notice that Drude’s
theory we found that does not work for metal
such as aluminum. Now, when you are done,
free electron theory what we have gained nothing,
we cannot say anything about hall coefficient
I mean no improvement, there is nothing is
only n appearing here that estimates have
not change. Therefore, we can’t and we found
aluminum the Halls sign of Hall coefficient
was positive the hall efficient was sorry
the hall coefficient, it appeared hall for
conducting.
So, therefore, in free electron theory also
we make no improvement in this regard same
following magneto resistance 
in this regard also. The field depended of
magneto resistance is again not predicted
give nothing done nothing. In this theory
which says that let me ask many questions
to you, now as we start discussing this there
too many thing which are which are problem.
How would you explain that carbon comes in
many form diamond, it comes in graphite. Why
is it carbon which is insulator why its graphite
which is conductor why that happen Wiedemann-Franz
law which is obeyed well at in room temperature.
But, we go to low temperature that even Wiedemann-Franz
law its cooperated well by this what is observed
this discrepancy between free electron theory
and free electron theory and what is berated
by what is observed. Similarly, this directed
constant of a material could be very, very
complex which is not by pirated by free electron
theory. Now, already talked you talked about,
I give you already hint that when you talk
about thermal when you talk about conductivity
in graphite or in diamond.
Now, both are carbon, this nothing we said
about differences between graphite and diamond
and therefore, the difference in conductivity
we just can’t bring it. Bring it in why
is a aluminum good conductor, but bismuth
and antimony which are also metal which are
not why boron is insulator. Why boron is insulator,
now these are the things which these are the
things, which are not coming, which are the
other observation. However, this free electron
theory we are not gaining anything out of
it why is it. So, you should already have
it to that our materials have special structure.
Then that materials are, they have a structure
also they underline atom that periodicity
in what is that mean, remember the true form
graphite is conductor diamond is not what
is the difference between them both are carbon-carbon
based materials. Now, all we have done is
in copper we have taken copper atom and consider
valence electron and went ahead and did both
this case.
But, you see there are underline latex, underline
periodicity, in these materials these periodicity
the electron wave which is coming. Now, you
notice this electron wave is traveling. Now,
there is interference of this electron wave
if the latex is perfect, then there could
be constructive interference, and this electron
wave could just simply go through, where as
if we break the periodicity of the latex somewhere.
Then you can have distractive interference
and therefore, you could have the electron
wave dying out and; that means, conductivity
being low. Now, all these there, therefore
in all the explain conductivity, we must take
into home what is the underline structure
of the material. So, what in the next lecture
I will do is I will cover two topic, one just
to build up the case, I will cover two topics
one is that will start with crystal structure
of material then after that I will take you
to something take you through, what is called
reciprocal lattice. Some of you would all
ready be familiar with it in context of X-ray
diffractions or electron diffraction.
So, you see will go through these two topic
first and then you will see the relevance
where we are heading what ultimately, I want
to do is now I am going to write down here
that what have we done. So, far we have said
that energy is equal to H Square by 2 m k
square, but ultimately I want to show you
is that there is something called E k diagram.
So, thought this E k diagram E k verses energy,
then clearly it look something like this 
it clear looks it look something like this,
this is this, this expression being ported
here.
What I am doing of course, k appears only
a distinct levels allowed values of k r. Only
these are the allowed values of k and corresponding
energy is ported here, but now these k also
totally pack that we draw then continues line
that it the line that the lack line I have
drawn is sufficient enough because this k
also. So, in this picture I show you by red
dot which a far apart, but in practice there,
so close there is no need to show those dot
they look like the black line which is continuous
line.
Now, what happens when what we have done so
far is that when we start filling of electron,
when we start filling of these electron we
start putting them two electrons in each of
these k states. In each of these k states
we start putting 2 electrons and when you
start putting them in we fill up to let us
say these energy and this energy we call use
E f. This energy we call E f that is the something
which we have done, I shown you the question
is what this quantity.
You also remember I have pointed out that
this is E k diagram, meaning this energy was
a momentum diagram k precociously represent
momentum as well. H bar multiplied by k represents
momentum. So, if I get a proper E k diagram
for each material, I should energy and momentum
there is 2 quantity of sufficient to do all
the dynamics we want to do whatever we want
to do.
With the dynamics of a electron you can do
if you have a E k diagram essentially for
a real material, I want to do this E k diagram.
In order to understand this E k diagram I
was understand k this k remembered in out
of plot like this, I have drawn at as a k.
But, the k really is a vector k is a big vector,
which is a in which direction should I draw
it how to draw with e k diagram this issue
we will start looking at almost 2, 3 lectures
on the line slowly will build the case in
order to use. So, we have to understand reciprocal
lattice because k belongs to reciprocal space
and in order to understand reciprocal lattice,
we must do real lattice, which we will start
next lecture.
Thank you.
