Hello, this is our fifth class in this course
on physics of materials and in today’s class,
we will begin our first attempt to create
a model, for the electronic properties of
material
specifically, we will focus on conductivity.
As we discussed in the last class, there is
a
good reason, why we are spending, why we will
spend so much time on the electronic
properties, especially conductivity simply
because this lot of technological interest
in this
property, and there is also a lot of scientific
interest because of the range of values that
conductivity has.
As I mentioned last class between the best
conductor, and the worst conductor of
electricity that you can find, there is a
range of values of about twenty seven orders
of
magnitude, which is more than any other property,
more than the range of values, you
will find for almost any other property, that
you will you are likely to measure for a
material. Therefore, there is a lot of scientific
interest on the property that is
conductivity. So, for this reason we will
spend some time, trying to build a model for
conductivity and build several models actually,
for conductivity.
And try to see what is the insight that, we
can gain into how various constituents of
the
materials, interact with each other and help
as get a model and therefore a prediction,
on
what the conductivity of a material should
be. I must also that in our discussion so
far,
we have specifically not included the special
class of materials, which are referred to
as
superconductors that is something we will
look at towards the end of this course.
So, when I talk of twenty seven orders of
magnitude, it does not include superconductors
it is what we would call so called normal
conductors. So, standard conductors, that
we
are more familiar with a typical metal at
room temperature and so on. The concept of
superconductivity is very different, there
are special aspects or special features that
are
prevalent, when we say that a material is
demonstrating superconductivity. It is very
different from the normal mode of the standard
mode of conduction that we see in the
typical materials at room temperature. So,
therefore, we would like to treat that as
a
separate topic.
The kind of theory that explains a difference,
from what we would use for conductivity
at room temperature for most metallic systems.
So, I am excluding that, I just wish to
alert you to that we have excluded it for
now, towards the end of the course, we will
have
a class more than one class perhaps, on just
the aspects of superconductivity. So, for
now
we will focus on conductivity in the sense,
that we are more commonly aware of and
specifically, in the case of metal that there
is a positive coefficient of thermal coefficient
of resistivity.
So, that is the conductivity that we are interested
in when, we raise the temperature of the
metal the resistance of metal or the manner
in which obstructs, the flow of electrons
actually becomes more prevalent. So, this
is what we would like to explain. So, we will
do this, we will begin our first attempt to
come up with a model by looking at something,
that we have possibly you have been familiar
with, from your high school days. Which is
that we describe metals and as being a situation
or state of a materials, where you have a
free electron gas.
So, we hear this description so, we say that
this what we are
often told that metals behave as though, there
is a free electron gas inside the system and
so, our attempt to describe metallic systems,
our first attempt to describe metallic
systems will take advantage of this description
or will at least work around this
description, this is the general framework
within which we will operate.
So, what do we mean by a free electron gas.
So, the way it is describe to us and which
is
the general format in which, we will continue
to use this description is, that in a metal
we
have all the atoms that going to towards making
a metal, those atoms actually have some
valence state and to that extend they release
an electron, which stays within that material,
but is no longer belonging to a specific atom.
So, if there are thousand atoms present there
and each of them has a valence 1 plus 1 or
valence state of plus 1, then each of them
releases one electron. So, the thousand atoms
now together release one thousand electrons,
these one thousand electrons are now free
to run across, the entire extent of that.
So, this general state we then referred to
as a free
electron gas state, where all those electrons,
those thousand electrons which are now
freely running across extend of that solid
are together, referred to as a free electron
gas.
So, for example we are just saying that, we
have an ionic core here. So, we just have
a
schematic of two dimensional ionic core and
let say all of these have released electrons.
So, these electrons are actually running across
this entire area, this is what the ionic core
remains wherever they are and the electrons
are running across the extent of the solid.
So, this picture taken together is what we
are referring to as the free electron gas
picture.
So, in this system originally when we think
of a metallic system, we have those atoms
sitting at regular lattice positions as we
call them. So, these ions continued to sit
those
regular lattice positions, but they have released
one electron, which stays within this
solid.
So, to use the term the kind of terminologies
use for it, the electrons are still localized
within the extent of the solid. In other words
they cannot ran away from that solid only,
if only we extract the electrons somehow and
remove it away from the solid and take it
to infinity, where it no longer can interact
with this, only then it is escaped. So, those
electrons have not escaped, they are sort
of stuck within the boundary of the solid.
So,
some boundary is there. So, the electrons
now stay within this boundary. So, for our
purposes they stay within this boundary, but
at the same time, they are not attached to
any particular ion, any particular ionic core.
So, an electron that might have been released
from this ionic core no longer actually,
specifically belongs to this ionic core, it
runs across the entire solid. So, what happens
now you have positively charged ionic cores,
which in principle should repel each other
and actually try to get away from each other.
So, that is the general tendency because
their like charge they are push each other
away, the electron cloud as we describe it
of
this free so, called free electrons they provide
the negatively charged atmosphere, within
which these positively charged ions can now,
sit stable at some equilibrium distance with
respect to each other.
So, this is how the picture of the solid is,
you can I mean loosely we can think of it
you
know some kind of a box within which we have
contained a gas, something like that you
think of. So, that description is not a very
far from what we are attempting to use here.
So, these electrons are free to run. So, this
is the general idea that we use to wish to
use.
So, what we like to do is to see if given
that, this is the picture or to use the term,
this is
the model that we would like to that, we believe
that the solid actually has, we would like
to put some equations, numbers, relationships,
to this model. Based on our best
understanding of how these particles interact
with each other and see if from all those
equations, relations and numbers that we throw
in to the system. We are able to get some
prediction for how the conductivity of the
solid given these, these general picture of
the
solid how will the conductivity be. So, what
is the kind of conductivity that, we get from
it.
So, what we are actually doing is, we are
actually focusing on this word gas. So, free
electron gas. So, gas is the word that we
wish to focus on with respect to this particular
description. Why I wish to highlight that,
is that we are actually quite familiar with
again
from our high school days or even from our
early college days. We are very familiar with
this concept of an ideal gas.
So, we have this concept of an ideal gas,
this we are very aware of we have a good sense
of what it represents. We have a feel, we
are very familiar with what equations go along
with an ideal gas and so on. So, but we are
at least quite conscious of what it is and
how
we can, what properties it may display, what
can we extract from it, what predictions we
can make about it? So, what we are going to
do is actually take the rules or the behavior
of an ideal gas, the general behavior of an
ideal gas, all the rules that we associate
with
an ideal gas, all the behavioral trends that
we associate with an ideal gas and impose
that
on this free electron gas.
So, we are going to take the ideas rules and
the concepts that, we associate with an ideal
gas impose it on this picture of a solid,
metallic solid which contains a free electron
gas
using this idea, where this is independently
developed and by imposing it on this system
here we make a predictions. So, from based
on this we are able to make some predictions
so, predictions. So, we are that is our final
goal that is what we intend to do, we would
like to make predictions.
So, based on this combination, we would like
to make predictions. So, of course, the
minute we so, to step back and see what we
are doing is, we are taking rules associated
with a gas and enforcing it on a solid. So,
we need to be very clear, on what we are
attempting to do. We are taking rules associated
with a gas and in fact not just any gas,
rules associate with associated with the so
called ideal gas and imposing it on something,
that we have never imagine does a gas when
a solid is given to you. When a solid piece
of metal is given to you never, you never
intuitively think of it as a gas.
So, in fact in our mind it is clearly etch
that a solid is not a gas, that is the very
definition
of a solid as a state of matter, but we are
going take the rules of an ideal gas, impose
it on
a free electron gas on a solid and then make
some predictions. Where we are actually not
stretching ourselves too far is because we
are not actually imposing it on all of that
solid.
We are actually imposing it on just the free
electron gas, the free electrons the so called
free electrons, which are now free to run
across the entire extend of the solid, it
is that set
of electrons which are now freely running
across the solid and therefore, having some
characteristics similar to that of the of
freely running molecules of a gas, which have
some similarity. So, based we are taking that
similarity taking advantage of that
similarity to make this extra operation.
So, now given this we will add little more
we will probe this idea, that we can put this
ideal gas behavior on a free electron gas,
a little more to see. How justified we in
doing
this, I have just tried to indicate that there
is some similarity between this ideal gas
and
so, called free electron gas. So, we will
try and put explore that idea little bit more
to see,
how justified we are in extending this idea
of a ideal gas, to a free electron gas and
we
will also see, if there is any reason why
we need to be cautious about this extra operation.
So, what is the aspect of extra operation
that we should be little bit concern about.
So,
that at least upfront we realize that we can
anticipate some limitations, when we try and
do this kind of an extra operation alright.
So, we will first look at the good news in
terms
of these two of this extra operation.
What is right about this extra operation.
So, to speak we will now see in terms of a
solid,
what is said that we are trying to, how we
can justify this alright? Now, let take a
solid
which has a simple cubic crystal structure.
So, we will just draw that here right. Now,
at
the corners of the simple cube, there are
atoms. So, the general picture that we have
is
that, we can put atoms which are roughly of
this dimension, we just make those that
atoms touch.
So, that is what I am trying to do here, for
clarity I will just stick to the atoms with
the
front face of this cube, there are atoms on
all of those corners also the other corners,
which are there I have only put four here,
there are four more behind. So, just for clarity
we will just stick to these four atoms for
now, and not worry about the other four atoms.
So, the atoms have radius r and this crystal
structure has a latish parameter a. Now, we
are familiar with this idea of a packing fraction.
So, this is something that we are familiar
with so, we will just very briefly run through
the calculation.
So, the volume 
of cube is simply a cube, when you have a
simple cubic structure, you
have eight atoms associated, I mean which
are placed at the edges of this cube, but
given
that each of them is now shared by all of
the neighbors. If you actually run through
how
many neighbors each of them is shared with,
you will end up finding that on average per
cube there is one atom, that is what you will
end up finding on average, you will have
one atom per cube because you have 1, 2, 3,
4 neighbors here and 4 neighbors on top.
So, 1 by 8th of this atom belongs to this
cube similarly, 8 atoms are there. So, on
average
one atom so, the volume of an atom associated
with this cube is simply. So, volume of
atom associated with the cube. So, 4 by 3
pi r cube is the volume single sphere will
assume spherical atoms are all spherical and
then you will get 4 by 3 pi r cube. Given,
that the dimensions of this cube, relate to
the dimensions of this atom such that, the
atom
just about touch each other when they make
this cube, we see that r is simply a by 2
therefore, r equals a by 2.
x
So, if we look at packing fraction, what we
have here is packing fraction is simply the
percentage or the fraction of the volume of
this cube, which is occupied by this atom.
So,
that is the packing fraction of this system.
So, if you look at we have packing fraction
so,
I put it down here. So, a by 2 I have substituted
for r. So, you have a cube by 8 and a
cube is the volume of the, of the cube. So,
if we simplify we actually have so, this is
2 so,
this is actually pi by 3 into 2, pi is 3.14.
So, we just want idea of the value we are
not
really interested with the exact precise value,
but so approximately this is 1 by 2 because
this is also about 3, this is also 3.
So, approximately 1 by 2 so this is approximately
50 percent, actual value comes to
about 52 percent or something 51, 52 this
is just order of magnitude, we got 50 percent
fine. So, if you take a simple cubic structure
and we run through this calculation, we get
about 50 percent fine, this is one of the
less structures. So, to speak so we will actually
look at a more packed structure for a moment
here. So, if you look at a so called close
packed structure, which is like the face centered
cubic structure F C C.
So, in face centered cubic structure, if you
just look at the face of the cube,
for based on symmetric consideration it is
enough, if we look at just one face of the
cube
for our purposes. So, face centered cubic
structure looks something like this, if you
look
through the calculations of how many if you
again see that, you know every corner atom
is actually shared by 8 other cubes. So, 1
by 8th of every corner atom and there are
8
corner atoms, there are 4 here there are 4
in the other corners. So, into 8 so one atom.
And then the face centered atoms each of them
is shared by the adjacent neighbor. So,
half of this face centered atom belongs to
this cube. So, there are 1, 2, 3, 4, 5 and
6 face
centered atoms. So, 6 into half equals or
half into 6 to speak to this convention, half
into
6 equals 3 atoms. So, per cube there are 4
atoms now present alright and if you look
at
the relationship, this is root 2 times a this
is so, this diagonal here is root 2 times
a. So,
root 2 times a because it is simply square
root of a square plus a square. So, 2 a square
root so, that is root 2 a that is equal to
r plus 2 r plus r so equals 4 r.
So, again if you go through this packing fraction
calculation, what we have is packing
fraction here your denominator is a cube,
still the volume of this cube that is a cube,
there are 4 atoms times 4 by 3 pi r cube and
r is related like this it is a by. So, this
4 will
come down here so, 2 root 2 so, put it down
here this is root 2 a by 4 whole cube. If
you
simplify this again we will just see here,
I will just write it here. So, that we can
cancel
out, what we need to cancel out.
So, this is 2 root 2 a cube by 4 into 4 into
4 by a cube. So, we can just cancel these
things
out by a cube goes here this 4, this 4, this
4 and this 4 will go. So, we will have 2 here
and once again we will just approximate and
cancel out 3 and pi this is just an
approximation just give us give ourselves,
the order of magnitude so, we actually have
this is approximately 1 by root 2.
So, this is what we will end up getting 1
by 1.4 1 4 is what you will end up getting
as the
packing fraction. So, this is 4 atoms their
volume 4 by 3 pi r cube or you can even say
you know root 2 by 2. So, 1.41 4 by 2 so this
is roughly about 75 percent. So, this root
2
by 2, if you just look at it as root 2 by
2 root 2 is 1.414. So, you divide by 2, 70
to 75
percent, this is just an order of magnitude.
So, we get of simple cubic you get about 50
percent packing fraction which means, about
50 percent of space is occupied by atoms
so, to speak in much more packed system, you
have 75 percent of the atoms occupying
the space.
Now, if you look at it if you look at all
the metallic systems, that you if you look
at the
other one the B C C, which we will not go
in to the calculations, it comes out to about
62
percent, no actually 68percent, 68 percent
is the packing fraction that you have for
B C
C, these are all approximate numbers. So,
all I wish to point out is that you have numbers
of this order 50 percent, 68 percent, 75 percent
approximate numbers of the packing
fraction. If you actually down through the
correct calculations, you may get more precise
numbers, but this is what you are looking
at.
So, what you need to understand is that you
know these are for example, solid objects
that you find, solid objects like this. So,
this is a metal metallic solid object that
we have
so, this is just a rod of a metal and this
is a part of a gear, part of a gear it is
a failed part.
So, this is being actually some analysis is
being going down on this part. So, this is
a part
of a gear and this is a spring, very strong
powerful spring so, strong that is very difficult
even press with your hand. So, this is the
spring.
So, these are all solid metallic systems metallic
objects that you can locate and you can
find easily find. So, when you lift them when
you hold them in your hand when you look
at them, it is difficult to imagine that you
know anything like 50 percent of it, 25 to
50
percent of it is actually open space it is
vacant. So, when you say a packing fraction
is 50
percent, when you say a packing fraction is
50 percent. It means 50 percent of the
volume of that system is occupied by atoms,
the other 50 percent is empty. So, this is
not
something that intuitively occurs to as, when
you look at a solid object.
When you look at this solid object, if in
case this had been simple cubic based structure,
there are very few metallic systems that fall
in the category, I believe polonium is one
of
them, but if this where such a structure 50
percent of this, half of this would be vacant
empty space, that is not something that we
intuitively imagine when you see it, even
if
you say quarter of it is empty again, it is
not something that is intuitively known. So,
but
that is the fact when you actually do run
through the calculations, you find there is
that
much of vacant empty space present within
this solid object.
So, when you go back to our picture, that
we say that you know there are ionic cores
and
then, the rest of the volume of that solid
is actually being freely occupied by all those
free electron gas. The electrons that have
been released by those ionic cores and actually
once the atom actually releases the electron,
it is radius will actually slightly decrease.
So, we will actually have to use the ionic
radius here. So, in fact in principle you
can
think of it as even more space has been available
for the electrons.
So, even more than 50 percent may be available
for those electrons to run across the
entire volume of that solid. So, therefore,
from the perspective of a packing fraction,
when you recognized how much of that solid
is actually vacant, and from that
perspective it is not a very unreasonable
thing to treat, the free electron gas as behaving
in the manner that the ideal gas. So, where
we actually look at the fact that an ideal
gas,
the atoms run across the volume of the container
that and they have a lot of free volumes
to run across. When you make that comparison
to that solid even though in principle, we
know upfront that solid is a different state
of matter.
Even though we know that, when you realize
that 50 percent of the solid is vacant or
could be vacant depending on that crystal
structure, 50 percent of it could be vacant
and
those electrons which are very, very tiny.
Please remember an electron is extremely tiny
relative to the molecule of a gas, when you
take that in to account and it you assume
that
it is freely running across, that half the
volume of that solid from that perspective
alone,
this link between an ideal gas behaviors does
not seen very far. So, therefore, at least
to
start off with it is not an unreasonable thing
to extend, the ideal gas ideas to a free
electron gas, which happens to exist within
a metallic system.
So, that is the first part of what we would
to highlight at the same time, I think it
is
necessary to recognize the reasons why such
an extra operation, should be taken with
degree of quotient? Now, look at the reason,
why we need to take this from an ideal gas
behavior to a free electron gas behavior inside
a solid with some degree of quotient. So,
to do that what we need to consider is that,
we get need to get a feel for is the number
density of particles so, we will just see
what we mean by that.
So, let us take for example, an ideal gas
what we are told of an ideal gas or what we
are
aware of an ideal gas is that at S T P. So,
0 degree c and one atmosphere pressure
standard temperature and pressure 0 degree
c and one atmosphere pressure. An ideal gas
will occupy, one mole of an ideal gas occupies
22.4 liters. So, that is the volume
occupied by one mole of an ideal gas. Now,
we will run through some numbers one mole
implies 6.02 3 Avogadro number into 10 power
23 atoms or gas molecules so whatever.
So, one mole is 6.02 3 into 10 power 23 atoms,
in this case let us assume an atomic gas.
So, we will have this some ideal gas we have.
So, many atoms are present in one liter so,
in 1 I am sorry in 22.4 liters at S T P. So,
let us see the number density that we are
looking at. So, we have 6.023 into 10 power
23 atoms in one liter and one liter is, 1000
liters is 1 meter cube. So, we will run these
numbers on a perimeter cube basis. So, 1 into
10 power I am sorry 22.4 into 10 power minus
3 meter cube.
So, 22.4 liters 1 mole is contained in 22.4
liters 1 mole has so many atoms. So, so many
atoms are contained in 22.4 into 10 power
minus 3 meter cube fine. So, this is
approximately, if you look at it we are looking
at approximately 3 into 10 power 25
atoms per meter cube, 3 into 10 power. So,
this is 2.24 into 10 power minus 2. So, that
would make it 10 power 25 and 2 and 3, 2 and
6 would give a 3 there. So, this is
approximate just because we are only interested
in the order of magnitude really.
So, 10 power 25 atoms per meter cube is what
we are looking at 3 times 10 power 25
atoms per meter cube so, this is for an ideal
gas. Now, we will look at the other
possibility, let us take a solid, in fact
a metal for which we are to impose this kind
of
trying, where we this ideal gas behavior to
the free electrons present within that system.
So, an example that we will take of a solid
system would be silver, we will just take
silver it is density is approximately 10.5
grams per centimeter cube, normal it is atomic
mass is approximately 107 atomic mass unit’s.
So, 107 AMU and normally valence,
commonly demonstrated by it is plus 1. So,
valence state is plus 1.
In other words, if you have a block of silver,
it is reasonable to treat it as though, every
atom in that silver in that block has released
one electron for this so, called free electron
gas. Now, given this information we will not
worry, we will just use this density and
atomic mass to give us an idea of the volume
we are dealing with. And we will take use
of the fact that again, one mole of this substance
would have released per atom released
one electron to this free electron gas. So,
we will have an idea of the number of
electrons, we will have an idea of the volume
their contained it and therefore, we will
get
a number density.
So, since this is a solid all we are saying
is that, we will just go this is some many
grams
per centimeter cube. So, we will convert this
to grams per meter cube because we would
like it in atoms per meter or particles per
meter cube. In this case we will get it as
electrons per meter cube. So, to do that we
have 10.5 into so, this is centimeter cube
10
power minus 2 each centimeter is 10 power
minus 2 meters. So, 10 power minus 6 meter
cube. So, this is it will become 10 power
6 grams per meter cube that would be in the
denominator so, it will go to the numerator
so, many grams per meter cube we will have
of silver.
And if you divide this by its atomic mass
107, that will give us the number of moles
of
silver per meter cube number of moles, of
silver per meter cube and if you multiply
this
by Avogadro number this now, gives us 10 power
23, this now gives us the number of
atoms of silver per meter. So, this is the
density of silver, this is the atomic mass
therefore, number of moles per meter cube
and ten times the Avogadro number, gives
you the number of atoms of silver per meter
cube and assuming that every atom has
released one electron effectively, the same
number will be there, same number of
electrons per meter cube will be available.
If we just run through this calculation, what
we see here is again this is similar kind
of
number so, this is we can treat this as 10
power minus 1. So, this is 10 power 5. So,
this
is approximately 6 into 10 power 5 plus into
10 power 23. So, 10 power 28 electrons per
meter cube. So, if you just run through these
calculations we get 6., 6 something into 10
power 28 electrons per meter cube. So, if
you do not worry about 6 and 3 essentially,
we
see that in an ideal gas, we have about 10
power 25, in this case atom per meter cube
and
in a solid, we have 6 power 10 power 28 electrons
per meter cube. In other words that is
the 3 orders of magnitude increase, in the
number of particles per meter cube fine.
So, in a solid the electrons are one thousand
times more densely packed, than the atoms
in a gas. So, that is a difference of 3 orders
of magnitude thousand times more densely
packed electrons relative to the atoms in
a gas. So, this is the reason why we need
to be
cautious, when we extend an ideal gas behavior
to these so, called free electrons, which
are present the free electron gas, that we
believe exists inside a solid.
So, there is a different in thousand, 3 orders
of magnitude or thousand times more
densely packed electrons are present within
this solid, relative to what is present here.
The reason why this is something that we need
to be about is because in an ideal gas, we
make statements and which we will do even
now, that the particles do not interact with
each other. So, they are isolated enough from
each other, even though I mean they do
interact with each other, but they do not
each other. There is a we basically say that
they
colloid with each other, but after that collisions
there is no further interaction between
the particles.
So, they do not influence each other between
those collisions. Now, the more densely
you packed those particles, the less reasonable
that statement becomes. So, the more
closely the particles are packed, there is
a greater chance that they will actually interact
with each other, even between the collisions
therefore, the kinds of things that we say
for
an ideal gas. Now, slowly becomes less and
less reasonable to say about system where
you have electrons, which are much more closely
packed thousand times more closely
packed.
So, we have actually two pieces of information,
one is that most of the there is a lot of
vacant space present within the solid. And
therefore, on that basis it is reasonable
and
interesting to extend ideal gas behavior to
study electrons in a solid, at the same time
even though there is that much space available.
The electrons present within the solid are
about thousand times more densely packed than,
the atoms present in an ideal gas
therefore, that provides us with a reason
to be about such an extra operation.
So, both these things are there, we need to
keep both of them in mind. So, it is not
unreasonable, but at the same time we need
to be cautious about this whole process. So,
with these ideas in mind, we will just put
down now the kinds of rules, that we will
expect our free electron gas to obey and based
on those rules, we will then develop our
theory for a free electron gas and therefore,
predict the conductivity of the material.
Especially, a metal based on our of ideal
gas behavior to a free electron gas. So, we
will
just put down the rules today, we will then
have to take some tangential deviation to
develop few of the ideal gas concepts, that
are of interest to us and then impose them
on
the free electron gas.
So, we will go into our ideal gas behavior
a little bit into in the next class, and then
we
will come back again in the class after that
and impose that on the rules, that we are
now
going to put down and see how the equation
step for metals, is based on some
assumptions. So, the first assumption that
we put down is that undergo, electrons
undergo collisions with each other. So, electrons
undergo collisions with each other,
which are instantaneous and these lead to
scattering. So, the first thing that we say
about
electrons, as they behave as a part of free
electron gas is that, they undergo collisions
with each other and as a result of these collisions
the electrons scatter.
So, these scatter they go all around the place
before they bounce of other electron. So,
this is one thing, these rules that we are
going to put down, you will see that many
of
these rules are very similar to what is being
stated, when you develop the ideal gas
behavior. So, to the equation for the ideal
gas behavior and that is the reason, why I
highlighted that upfront between scattering,
between collisions, Interactions with other
electrons 
and ionic cores 
is neglected.
So, between collisions interactions with other
electrons and interactions with ionic cores
is neglected, in it is details, in the details.
And I think this needs to be explained a little
bit, what we are saying is between once an
electron is collided with another electron
and
bounces off, till it makes the next collision
with some other electrons, which is randomly
moving through the solid, in between there
is a certain amount of time that it is travelling
between two collisions in that time, we are
not specifically writing any detailed
equations on how it is being influenced by
other electrons around it, being influenced
by
the ions around it.
So, however we do put in an averaged term.
So, we do put in some kind of
an average resistive term to account for this
general interaction, but we are not specifying
anything in any grade detail. So, therefore,
some aspect of this ideal gas kind of
behavior, we are still maintaining. So, the
detail of the interaction is neglected, but
some
averaged interaction is being accepted or
accounted for. So, that is what is important
there is something that, we call the mean
free time between collisions, the mean free
time
between collisions is represented by tau.
So, this is the average time between
collisions. So, an electron as I mentioned
bounces of one electron of another electron,
then travels for some time, then hits the
third electron.
So, this is going on for all the electrons
within the system at any given time, the duration
between two collisions will vary from for
all the electrons. So, we may travel a longer
distance before they hit the next electron,
some may immediately hit the next electron,
and some may immediately hit the next electron
and so on. So, this happens so, we
cannot actually since, there are so many electrons
within a solid, we cannot actually
individually note down the numbers for every
single electron or attempt to put numbers
down for this.
So, what we do is there is something called
the mean free time on average, this is the
time that an electron travels between two
collisions, it will have a collision on average,
it
travel this time before it has the next collision
that average time is tau and that is
independent of the position and velocity of
the electron. So, in other words and this
is
reasonable, because by definition this is
mean free time this is just a mean average.
So,
clearly velocity is higher the chances that
it another electron may be higher, but that
is
that is all accounted for the fact, based
on the fact that we using the mean free time,
this
is the third assumption electrons attain equilibrium
with their surroundings,
using these collisions with other electrons.
So, therefore, the collisions are important,
you cannot just completely neglect the
collision. So, the collision is so, that is
the reason why we are putting these rules
down.
We wish to highlight that they are the electrons
are free to run across the extent of that
solid there is a lot of vacant space within,
the solid and the electrons are free to run
through the solid and as they do so, they
colloid with other electrons.
So, those collisions are instantaneous in
other words there is no time that we are
associated we are not saying that, you know
when the electrons hit each other for one,
second their next to each other, we just say
instantly they hit bounce off. So, that is
instantaneous and this causes scattering.
So, the electrons run all around the place,
because of those collisions, between the collisions
the interaction between those
electrons is being neglected the interaction.
So, there will be some repulsive interaction
between the electrons, that is being neglected
there will be some attractive force between
the ions present in the solid and the electrons
that is also being neglected and that is not
unreasonable because as we mentioned. You
know even between the ions there is a force
of repulsion, but the electron cloud
effectively which has the same amount of charge
as those ionic cores.
The total charge of all those ionic cores
is the same as the total charge of that electronic
cloud. So, these sort of cancel each other
which is true even for the electrons. So,
even
for a single electron that is running around,
there is a large ionic core that is present
within the system, large set of ionic cores
and there is also a large electronic cloud.
The
total negative charge and the total positive
charge more or less, cancel each other out
expect for this one electron charge. So, largely
it is running in a more or less neutral
system surrounding.
So, therefore, the interaction is being neglected
in these details, but an average resistive
term is being included, which we will see
when we actually put it is value down, and
there is a certain mean free time between
collisions, which is independent of position
and
velocity of the electrons because it is the
mean free time, as we describe and finally
the
reason we need to focus on the collision because
that is the manner, when the two
electrons colloids they exchange energy. So,
there is an exchange of energy. So, one is
gaining energy losing energy, overall energy
is concerned, but there is an exchange of
energy and that is how the electrons attain
equilibrium with their surroundings.
So, when you take a block of metal and place
it in a room, that has a certain temperature,
let say this block of metal was cold, you
take it out and keep it in room, which has
a
higher temperature, it is only because of
these collisions with various electrons and
so
on, that slowly the heat begins to permit
into the system. The electron which very close
to the surface of the metal, sense the temperature
first they gain energy. We just
deflected by their velocities, how they move
and so on and they go on colloid with other
electrons, the ionic cores also participate
in the process.
So, which we are not addressing at the movement,
but for our purposes we will focus on
the electrons at the movement. Later we will
see what the ions do, but we will focus right
now at the electrons because the properties
we are going to focus on depend on those
electrons, the specific property that looking.
So, they attain equilibrium with their
surroundings, because of the collisions and
therefore, it is important to keep track of
the
collision, we cannot completely neglect those
collisions.
So, that is the basic idea and so, these are
the broad rules, that we will take they have
lot
of similarity with the rules that ideal gas,
ideal gases have and that we impose on an
ideal
gas or we expect an ideal gas to demonstrate
and the very same rules. Now, we will now
impose on the solid and on the electrons in
the solid and see what kind of behavior, those
electrons demonstrate and based on this, based
on this model we will now be able to
predict specifically, the electronic conductivity.
We will also predict the thermal
conductivity, these are two major properties
that we predict and we will see some
interesting relationships between them.
So, it is with is background that we will
halt here. In our next class since, we are
going to
use some ideal gas behavior and specifically,
we will use specific quantities or
relationship from an ideal gas behavior, which
will be relevant here, we will develop
those specific quantities briefly, then we
will come back to these assumptions impose
those quantities on these assumptions, I am
on that basis we will develop the electronic
conductivity, that will be two classes from
now and 3 classes from now, we will look at
thermal conductivity. So, that is how it looks
going forward from here. Thank you. .
.
