.
In the last lecture we saw that there was
this kinetic theory of gases and we saw that
the specific heat of these gases by taking
the assumptions of the kinetic theory of gases.
We could get a value of 3 by 2 R and the question
there we posed is that now from gaseous phase
if you bring the particles close together,
you will get a solid and if you measure the
specific heat of solids which was done by
Dulong and Petit in 1819; then he obtained
a specific heat not 3 by 2 r, but it was a
value of 3 R it was constant .
And it was 3 R not 3 by 2 R ; so, this was
a first sign that something else is happening
in the solid which is unlike that of a gas.
And what is one of the most obvious things
that you can think of? In kinetic theory,
the particles are non interacting they are
hard spheres, but when you bring the particles
close together; they start interacting.
So, you have to bring in the role of interaction
that was true and as we will see that that
is important . But apart from that there were
other things and other mysteries which came
up. Because at that point from 1819 onwards,
people started finding ways to lower the temperature
of the solid and you could go down by 1911;
from 1819 to 1911 from room temperature of
300 k you could go down to 4 k; this was the
year in which Kamerlingh Onnes liquefied helium
.
He obtained liquid helium which was at 4.2
Kelvin . So, you could go from room temperature
down to 4 k a wide temperature range and people
were interested what happens to the properties
of the solid. For example, this thermal property
which is the specific heat; if we measure
it as a function of temperature does it remain
constant at 3 R or does it vary? And was found
was that over a certain window of temperature;
it is constant at 3 R , but then it starts
varying below a certain temperature, it has
a temperature dependence .
And this temperature dependence turns out
to be cubic ; the specific heat has a cubic
temperature dependence. If you go to very
low temperatures below 4 Kelvin and as you
go lower and lower; which people were able
to do then instead of a cubic temperature
dependence, you start getting a linear temperature
dependence the specific heat becomes linear.
So, from a linear specific heat if you are
starting from low temperature the specific
heat is initially linear .
Then the specific heat develops a curve which
is has a cubic temperature dependence and
then as it goes to high temperature; it becomes
constant . So, the thermal property of the
solid turns out to be very complex and we
will look at all these different regimes in
the course . But this is just to motivate
you that various effects and phenomenas are
coming into the picture . And just to tell
you that this temperature dependence of the
specific heat that you see; you cannot explain
it using classical theories. You cannot use,
explain it using something like kinetic theory
of gases where you consider classical particles
.
You have to invoke quantum mechanics to understand
these temperature dependences and that will
be what will be covered in this course also
. So, let us go forward from here; I have
talked to you a little bit about and given
you a motivation about what happens to thermal
properties, when you go to a solid and you
see unusual things happen; what about other
properties?
And here we come to the 
and we will spend some time and effort into
trying and understand being the property of
this solid; the electrical conductivity of
a solid .
Now, a solid has atoms and we will essentially
study a metal . So, what is the metal? A metal
has atoms 
which are such that the outermost electron
of the atom is; the outermost electron in
that atom is loosely bound to that atom and
it gives up that electron; the electron is
free to wander around inside the solid . So,
what you have are positive ion cores and there
are these electrons which are wandering across
the lattice. And this was sort of known and
it was known that inside the metal ; the charge
conductors are charge carrying particles are
electrons .
The metals have weakly bound electrons; the
metal ions or the metal atoms have weakly
bound electrons; these electrons are delocalized
they are sort of not bound to the atom as
such, but they spread out across the solid
and they wander around the solid. So, this
is the model of a metal you have positive
ion cores and you have electrons which are
delocalized across the metal ok .
And people wanted to understand the behavior
of conductivity of the metal ; namely what
they wanted to know was how does the electrons
move through the solid when an electric field
is applied to this solid or a metal .
So, the question was that you have electrons
inside the metal which are the charge carriers;
how do these electrons actually move through
the metal? Can we understand that? And this
was one of the very important topics in the
early 1800s and which continues; I mean part
of the theories which were developed here
approximately they are still valid and people
still use it to a good amount .
And this was an important part of solid state
physics to understand the behavior of electrons
moving through the metal . How does it conduct
electricity?
So, the earliest experiments which wanted
to study this was your very well known Ohm's
law experiment . What was the Ohms law experiment
the Ohms law experiment was quite reasonably
simple. Say you take a piece of metal; it
could be a piece of copper or whatever silver,
aluminum whatever; .
You connect to leads to the ends of this piece
of metal; you connect it to a battery 
and when you connect it to a battery, you
can change the voltage which is given by the
battery and you can measure how much is the
voltage drop across the metal and how much
is the current I which is flowing through
the metal .
As you change the voltage; as you apply the
voltage to the metal, how does the current
through the metal change? And what did Ohms
law find? Ohms law found, Ohms found that
if you measure the voltage as a function of
currents ; then the behavior is linear. The
voltage and a current are directly proportional
to each other or the voltage is written as
R into I, where R is the resistance of the
metal; the slope of this curve is equal to
R .
Now, you can quickly sort of convert this
Ohms law expression V is equal to R into I
and you can write it in terms of electric
fields and current densities. If E is the
electric field 
and J is the current density; then you can
recast this expression in terms of electric
field and current density; where electric
field is volts divided by the length of the
metal over which the voltage is being applied.
So, suppose you have a wire of length L if
this is a wire of length L and V is the voltage
applied across this wire then E is the electric
field which is volts per unit length . And
if the cross sectional area of this wire is
A and if current I is flowing across this
wire .
Then the current density J is defined as current
per unit area of cross section; where A is
the area of cross section of this wire. And
now using the expressions for V and I in terms
of E and J; you can rewrite the Ohms law as
E into L is equal to R into A into J or E
is equal to R A over L; where A is the area
of cross section of the wire L is the length
and this gives us a quantity which is called
the resistivity of the metal or of a material
where rho is equal to resistance over L.
This just comes from the earlier expression
and we can define also the conductivity sigma
which is 1 upon rho . So, you can rewrite
J as sigma times E or E is rho of J .
And these are important parameters which measure
it does not these quantities you have already
removed the effect of the length and the area
of cross section . So, you can compare materials;
you can compare different materials although
they can have different lengths and different
areas of cross section. If you measure their
conductivity or their resistivity then which
is equal to R A divided by L . And this sigma
which is 1 upon rho is the conductivity of
the material of the metal . And the important
thing about sigma and rho is that; they do
not depend upon the geometry of the sample
what is the area, what is the length you have
already taken into account that effect.
So, you can start comparing different materials
and looking at the behavior of this quantity
. And it turns out that this resistivity can
vary significantly; there is a huge variation
in the resistivity . So, just let me give
you an example; we can take a look at first
the materials like copper which has a resistivity
; copper has a resistivity of 1.7 into 10
to the power of minus 6 ohm centimeter. Gold
has a resistivity of slightly higher; in fact,
this is pure copper it has a very high; its
resistivity is very low.
Gold has slightly higher resistivity 
and if you take let us say bismuth ; then
the resistivity can jump up typical values
are of this order. That if you look at the
resistivity of these materials these different
materials then there is a variation in the
resistivity; these are as far as metals are
concerned.
But if you go to insulators; then the resistivity
could go orders of magnitude high . You can
go from values as large as 10 raised to 6
and the metals will be at 10 raised to minus
6 . So, this pan actually orders of magnitude
if you go to an insulator; its resistivity
could be sitting as 10 raised to 6 ohm centimeter
. Whereas, if you take a metal the metal could
be sitting at 10 raised to minus 6 ohm centimeter
.
And on this axis I have just drawn the temperature
and on the temperature scale; the resistivity
of an insulator and a resistivity of a metal
can behave very differently. If you lower
the temperature the resistivity of an insulator
can actually show an increase; whereas, for
a metal the resistivity can actually decrease
. So, you have things like an insulator 
and you have things like a metal and there
is a huge difference between the resistivity
why is it so different?
What is it that makes the solids have such
huge variations in the properties of the resistivity
or even if you think if you want to think
in terms of conductivity; they will be correspondingly
the huge difference in conductivity. Why does
this difference in conductivity come about?
It is still an electron which is flowing through
the material, but what makes it so different?
Furthermore, towards the end of this course
you will again take a look at the behavior
of resistivity as a function of temperature.
And there are certain materials whose resistivity
could be constant or varying with temperature,
but below a certain temperature the resistance
drops to almost 0; below measurable limits.
The resistivity actually falls below 10 raised
to minus 25 ohm centimeter and these are called
as superconductors.
They are not metals they are not insulators,
but they are fantastic conductors with very
very low resistivity and the resistivity of
these materials can be very low . So, you
can have insulators, you can have metals,
you can have superconductors, you can have
semiconductors all of them have electrons
as conductors, but yet the resistivity and
the temperature dependence is very significantly
.
You will study some of these as we go along,
but let us come back to metals and try and
understand the behavior of metals. And let
us try and understand the conductivity of
metals; the electrical conductivity of metals.
