Let us resume our discussion on phase transitions
and I quickly recap what we did last time?
We took a simple single component 
system and for this system we do the P V diagram,
the P T diagram, the T V diagram and our focus
was to recall in the P T plane. Our focus
was the critical point and this was the liquid
phase and this is the gas phase. The focus
was on this phase transition here from liquid
to gas and corresponding to this. We found
that in the V versus T diagram roughly speaking
we had some kind of curve like this, which
did not display too much symmetry, schematically
something like this as T c and this as V c
just as this is critical point T c and this
is the critical pressure P c.
And finally in the P T plane, the V T plane
and then the P V plane. We found that there
were high temperature isotherms then the critical
isotherm, which went off like this with an
inflection point and then isotherms of this
kind which corresponded on this side. In the
gas phase and this side to the liquid phase
and this was T greater than T c this is the
critical isotherm T equal to T c and T less
than T c. Now, what I would like to focus
on is the fact that near this critical point.
You have some kind of universal behavior in
a very deep sense many systems behave independent
of the actual interactions or details. They
behave identically near the critical point
and this was one of the fundamental observations,
the theory of phase transitions and critical
phenomenon. We will try to see why that is
so and we will try to see what is universal
about that point? So, T c this was P c.
The first point to observe is that in a phase
transition of this kind. When you have a critical
point here and this curve of course, discontinuous
transitions ends abruptly at this point. It
is possible to start with some thermodynamic
equilibrium state here and take it by a process
which goes through process at equilibrium
all the time. So, you could take a path of
this kind and come to that point there. So,
you converted a gas into a liquid without
encountering the sharp transition at any point,
this is possible because this curve is ended
on the other hand. If you look at the liquid
solid curve keeps going for ever in principle
there is no way of doing this and the reason
this curve does not end in a critical point.
Because the solid has crystalline symmetry
it is symmetric under the group of transformations
corresponding to the phase group of the crystal
a set of rotations and translations and so
on.
On the other hand, the liquid is homogeneous
and isotropic it has the same properties in
all directions and therefore this has a much
greater degree of symmetry than the solid.
The solid has order crystalline order long
range translational and rotational order.
On the other hand the liquid does not have
this order at all. It has at best some transient
short range order so; order and symmetry are
kind of opposites. The more ordered the phase
the less symmetric. It is in the sense that
the set of transformations under which it
remain unchanged is smaller and smaller. Normally,
intuitively we would associate order with
symmetry it is just the other way above the
most disordered state.
I think that is completely random it is in
fact the most symmetric, because it looks
exactly the same in every direction. There
is no change at all. So, statistically both
the liquid and gas phases are isotropic phases.
They are homogeneous, if you have an infinite
expansive liquid or solid or gas it looks
exactly the same, but a crystal is not like
that at all. You have a discrete symmetry
so, what happens is that the Euclidian symmetry
of a liquid. The factor is exactly the same
on the average under all rotations, under
all translations is broken down into a discrete
symmetry called the space group of a crystal.
We come back and talk about the broken symmetry
little later in the course, but that is the
reason why? Either you have symmetry or you
do not have symmetry, discrete symmetry. You
cannot do this in a continuous way that is
the reason why this curve does not end in
a critical point. On the other hand between
liquid and gas there is no symmetry breaking,
it is exactly the same kind of isotropic behavior
on both sides. Therefore since there is no
symmetry breaking involved in this phase transition.
It is perfectly acceptable to prefer it, but
of course the other thing that happens is
that the line of discontinuous transitions
meets another line and that is quite a common
occurrence.
So, many possibilities occur the most significant
ones being that a line like this. If for example,
for water it goes in other direction could
actually end on the physical boundary at T
equal to 0 or at T equal to zero, P equal
to 0 either of them equal to infinity whatever.
So, either of phase transition line ends on
a physical boundary in this plane or it hits
another phase transition, first order transition
line or it ends in a critical point. There
are other exotic possibilities can occur,
but for the simple system we are looking at
these are the only ones and of course there
is also a triple point here. So, three phases
could co-exist. Now what we would like to
do is to show that this system behaves more
or less like a magnet does and a fluid and
magnet have close analogies.
So, I would like to build up this fluid magnet
analogy to show you that near the critical
point, which is called the Curie point in
the case of Ferro magnet, Para magnet transition.
You have identical behavior with the fluid
and this was the start of the modern critical
phenomenon. When this was recognized in a
systematic way for that we need a model of
magnetism. This model here is some kind of
Vanderwaal’s what we used here and then
we fixed this Vanderwaal’s and corrected
it by adding the tie line and so on and so
forth. But all sudden done a liquid is a very
complicated system and then it depends on
the interactions during a detailed theory
of liquids is quite hard.
We are not interested in that, we are interested
in finding out what a critical behavior is?
So, let us do it by looking at the fluid magnet
analogy, but before I build the analogy I
have to tell you what a magnet is? And refresh
your memory about what a Para magnet is? So,
let us look at very simple model Para magnetism
to start with then we go on to Ferro magnetism.
The naive ways of looking at things we are
going to assume that the system, we are looking
at consist of elementary magnetic dipole moments.
We know that many atoms have permanent magnetic
dipole moments or whatever it is the reasons
are quantum mechanical. We will come back
to that in the next course at the moment,
we are only concerned with the fact that you
may have a system, which has a lot of magnetic
dipole moments oriented at random in the absence
of external magnetic field. The moment you
put a external field on these dipole moments
tend to orient themselves along the direction
of magnetic field, because that is what minimizes
the potential energy and then you have some
kind of possible order. So, what is our model?
Our model consists of N dipole moments atomic
dipole moments.
So, you have one of them like that one of
them pointing like this and so on. They are
all independent of each other and in the simplest
approximation; we assume that these fellows
are all sitting in a system in a thermal bar
at some temperature T. For example, it could
be in a solid and then the solid is maintained
at a particular temperature and these dipole
moments are oriented at random completely.
That is the initial state and it is supposed
to be in thermal equilibrium. The energy of
a dipole moment once I apply an external field
H this is what I call magnetic field, it is
actually the auxiliary field.
I should talk in terms of the magnetic induction
vector P, but you know that in linear materials
these are related. So, let me call it just
H modulo of some constants.
Then the potential energy of a dipole mu in
a field H is equal to minus mu dot H. So,
it is clear that if it is along the field,
you have the least amount of energy minus
mu H and if it is opposite erected in this
fashion energy is maximum plus. Now, the system
has put in thermal equilibrium with a heat
path and you ask, what is the actual magnetization
along the direction of the field, the average
value of the magnetization? We can do this
using the statistical mechanics we do it in
number of ways but let’s look at the simplest
case possible.
We make the assumption that this dipole moment
can either be along the field or perpendicular
or anti parallel to it just two possibilities.
Simplest assumption in other words mu dot
H is mu H cos theta and I assume that cos
theta is a plus one or minus one. The reason
I do this is because later we will see that
some of these in many cases is magnetic dipole
moments would arise from for instance. The
magnetic dipole moment of an electron, which
is related to the spin of the electron? Then,
if you measure the spin of the electron along
any given direction, it has only two possible
Eigen values plus or minus half H cross and
the reason for this is quantum mechanical.
So, I am going to anticipate and let us look
at a very simple model such that mu dot H
is either plus mu H or minus mu H in that
case so, two energy levels. So, if the field
is like this and the dipole moment is like
this corresponds to energy minus mu H. On
the other hand if it is in this form this
epsilon plus, we will relapse this assumption
and assume all possible angles and so on later
on. But to start with just to get our bearing
state this would be a very simple model, what
happens next? This immediately implies that
the probability well we can compute the magnetization
directly.
So, let us just do that the magnetization
M this is the average value of the magnetization
thermodynamic quantity M. Since, I am only
interested in the magnetization along the
direction of the field always then we call
it a scalar number. This is equal to since
these are all independent of each other the
net magnetization just adds up there is no
interaction between these dipole moments.
I have not switched on any dipole interaction.
This is equal to n times the average magnetization
of each of these dipole moments well, if the
dipole is in this fashion up there. When it
contributes an amount mu, but the probability
with which it does show is proportional to
e to the power minus beta times. The corresponding
energy that was our rule in canonical, the
energy was minus mu H.
So, this becomes e to the beta mu H 
divided by a normalization, which we are going
to do a partition function. We will put that
in plus, if the magnetization is in the opposite
direction it contributes an amount minus mu
and the probability of it is e to the minus
beta mu H divided by the total probability.
There are only two possibilities this and
that therefore this is just the sum of these
two terms e to the power bet m u H.
So, coming back here we will get back to these
curves little later, it says that the average
magnetization 
of the sample of N elementary dipole moments
M equal to N mu tan hyperbolic beta mu H,
because it is this minus that over this plus
that and that is equal to sine hyperbolic
over cos hyperbolic and that is the tan hyperbolic.
This gives us a magnetic equation of state,
because as you can see this is equal to N
mu tan hyperbolic over k Boltzmann. I call
the magnetic equation of state, because the
fluid case. You have a pressure a volume and
a temperature and in the magnet case you have
the temperature of course instead of the pressure.
You have the magnetic field H and the response
of the system as you change the magnetic field
is a magnetization. So, this is the analogy
this is the way you go from one to the other
and it is also clear. Just as P dV is the
amount of work done is the contribution to
d U distance that appears in the first law
of thermodynamics.
Similarly, you have a contribution here instead
of the form H T M. If you do a Lagrange transform
it becomes M D H, but it is like V D P and
P D T. So, this is the analogy we are going
to use the pressure is replaced by the magnetic
field. The volume by the magnetization and
the temperature remains as it is the relation
between P V and P is in equation of state.
Here is a relation between M H and T and you
need to know how much of the material is there
and that is done by the number N of magnitude.
This is our simple equation of state magnetic
equation of state for a Para magnet substance
whatever consequences well you can plot one
against another. So, let us do that of course
we have to be careful to realize that unlike
P and V, which have to be non negative M and
H could be negative or positive, that is along
the given direction will be opposite to that
plus or minus. So, if I plot let us look at
the M versus H diagram so, I plot H here and
M here and what this curve looks like well.
I can normalize it by saying that let me look
at the magnetization per atom or per dipole
moment and the unit is mu. So, I write this
for convenience as M divided by N mu and then
it is just the tan hyperbolic function as
a function of H. Now we know that the tan
hyperbolic function cannot exceed one in magnitude.
So, this is minus one that is plus one will
not exceed that and for small values of x
tan hyperbolic x is proportional to x. So,
this thing here is proportional to mu H over
k T therefore, there is small linear region.
But then it goes and saturates that side and
then on this side and that saturates to minus
one and this region is in linear region M
increases proportionally only in that region.
So, that it becomes non linear and it saturates
what is the definition of susceptibility.
How do you define the magnetic susceptibility
call it high. So, what is the definition of
the magnetic susceptibility so, we used this
definition right M is equal to M divided by
H M divided by H M equal to the H proportional
etcetera.
This is the definition of susceptibility that
you learn in high school, but that is not
quite right, because it is clear here. That
this is a non linear graph so, dividing M
by H does not make much sense as H increases
it simply goes to zero. Because M remains
fine and h becomes infinity. So, that is not
the right definition of the susceptibility
this is a linear relation it assumes that
M is proportional to h, but that is only true
near the origin. So, the correct way of saying
it is it is equal to M divided by H in the
linear region in this region, but then where
does this region stop. How do I decide where
it stops pardon me to what accuracy do I want
it?
I want it to arbitrary accuracy I want it
till arbitrary accuracy. So, what should I
do? What take the limit what is the meaning
of taking the limit? What is the operation
which takes the limit takes a little M and
you take a little H and then you take the
limit. You differentiate it to find the slope,
where slope at the origin.
So, the right definition of the susceptibility
this equal to delta M over delta H partial
whatever, we are supposed to keep constant
the temperature constant at a given temperature.
So, we keep the temperature constant and of
course we keep the number of particles constant
also just like that. As we defined the compressibility
as delta V over delta P so on. Keeping temperature
constant, but we call that the isothermal
compressibility we have kept the temperature
constant.
So, what should I call this the isothermal
susceptibility 
is delta M over delta H evaluated at H equal
to zero. That is important only then does
it make sense. It is the slope of this graph
that is all it is, but what is that equal
to present equation of state? So, what is
that equal to I will plug it in so, what is
that equal to what is tan hyperbolic x? Look
like as x goes to 0 goes to x. So, what is
tan hyperbolic ax look like, where a is constant
so, what is the isothermal susceptibility?
It is equal to N mu square over k Boltzmann.
If I define it per unit, per atom or per magnetic
moment dipole then I divide by M whatever
it is. That is not the relevant part the relevant
issue is that it is inversely proportional
to the temperature.
What do you call that law the fact that the
Para magnetic susceptibility is inversely
proportional to the temperature? It is called
Curie’s equation. This is Curie’s so;
you have correctly got Curie’s equation,
which was this Curie spear, who for the first
time recognized first person to recognize
the very important thing, which became very
important later on. The role of symmetry in
deciding the properties versus materials role
of symmetry in deciding, how condense matter
behaves? Now, that of course later on became
a very big subject and today, of course it
is combined with many other things. We talked
about broken symmetry and the role of symmetry
a little later.
But, this was the person who recognized Curies
law he met a tragic end early in his life.
Stepped out in cold Paris in afternoon and
his scarf got caught under the wheels of the
horse carriage, which was passing by at this
time later on. He never recovered from this
for long time, so many other things you did,
but the fact is that responsible for understanding
the role of symmetry first.
So, we have Curie’s law, which goes by one
over T. Now, what is that imply as you lower
the temperature? What would happen? What would
happen to this curve? It would still saturate,
but the slope increases. Therefore, it would
start doing this and theoretically, what would
happen, when you go to absolute zero? At absolute
zero, this curve would start doing this it
would simply become an extremely sharp curve
like this.
Does that happen with real materials most
often know, it does not do that, because many
other things start happening and what is the
one thing that you think on physical terms
is going to happen that is what you neglected
something there? We neglected the interaction
between the dipoles completely. Now the reason
we have got away with it is, because at sufficiently
high temperatures imagine this guy is inside
a lattice and it is being kept at temperature
T. So, it is being kept around due to the
vibration of the lattice that is the heat
path. In this case then of course if that
kicking k t is much larger than the typical
interaction energy it is ok, because these
fellows do not see each other’s interaction
very much, but as you lower the temperature
and k t becomes very small. Then the interaction
between these dipoles becomes more and more
favorable and all phase.
Transitions of this kind are dependent on
the fact that some kind of free energy is
minimized and if you look at the free energy
f which is U minus T S U. As we have seen
in statistical mechanics is just the expectation
value the average energy is E bar or the expectation
value of the Hamiltonian. I will use the script
H for the Hamiltonian, because I have used
H for the magnetic field. So, there is some
interaction between the two constituents of
the system. That is sitting in H the script
H the Hamiltonian and this here tells you
what is going on in real life? It so, happens
that at sufficiently low temperature the ordering
tendency of the internal energy tends to minimize
potential energy overcomes the effect.
So, F is dominated by this quantity as you
go to lower and lower temperature the T becomes
smaller and smaller and S also becomes smaller,
because S is k log omega and omega is number
of accessible microstates and that is decreasing
as the temperature decreases. So, in the low
temperature phase you would expect the internal
energy to dominate and decide the minimum
of three energies. In general that could be
an ordered phase at high temperatures. On
the other hand the effect of this is overcome
by the effect of the very large accessible
microstates and a increasing temperature.
So, that the entropy term governs the free
energy behavior of the free energy and that
is generally a disordered state.
This is the reason why underline reason why
in general you find that the high temperature
state is a disordered state. The low temperature
state is ordered state it is just the competition
between U and T S the Hamiltonian does not
change the interaction between the independent
constituents does not change. The interaction
between two water molecules remains exactly
the same. Whether it is a steam phase or a
water phase or it is a solid ice phase does
not change. What does change? However is the
number of accessible microscopes these changes
and the average energy change internal and
external?
So, this is important for the member that
is not the interaction is changing the fundamental
interaction does not change at all. It is
simply the effect of statistics as the temperature
goes and of course many complex possibilities
can occur. We will see some of them, but this
is something we have to keep at the back of
our minds. So, it is clear that in this case
that the interaction is switched on. Then
this cannot be the picture right down the
absolute zero. There would come a state, when
the effect of these interactions between these
moments will start dominating and then what
could you expect?
If it is dipole what would you expect? I have
a dipole moment here. I have a dipole moment
here and their interaction must be taken into
account. What is the minimum energy configuration
for this type both are like this? It is clear
that parallel, anti parallel is lower potential
energy is like this. This is certainly true
for a dipole interaction; it is more complicated
than that, because this interaction depends
on the directions of the two dipoles at a
given distance. This has one energy, that
has another energy, and this has yet another
energy and even if these two are parallel,
this energy is different from this energy.
So, it is clear? That this interaction is
not isotropic, it is going to depend on the
interaction of the directions of both these
guys and so on. But in a naive way, if you
say this and this in two magnets bar magnets.
It is clear? That the dipole interaction goes
one over r cube kind of things mu one dot
mu two over r cube to start with that would
favor this another one. After that would go
up, and then down, and then you end up with
a configuration, which you will have an up
and down and an up and down on an average
the magnetization will be 0. So, if you relied
on the dipole interaction inside the bar magnet,
you will not have a bar magnet at all. These
atomic dipole moments were anti parallel would
order such that it is parallel and is finished.
So, what do you think is happening? But we
do see permanent magnets, what do you think
is responsible is a clear indication that
the phenomenon of magnetism is more subtle
than one imagines permanent magnetization.
It is not connected to the dipole interaction.
It is connected to a quantum mechanical effect
called the exchange interaction, which actually
favors this over that and in fact overcomes
this interaction the dipole classical dipole.
It is a very strong dipole interaction so,
nearest neighbors would tend under favorable
conditions to align parallel to each other
and this interaction switches off exponentially
fast.
So, magnetic dipole moment here another one
here, the interaction would die down exponentially
with a distance whereas, the classical dipole
interaction would favor anti parallel alignment.
And would be a long range interaction, because
it goes like one over r cube distance, but
it is a very week interaction. There are dipole
or magnets, there are magnets, where this
becomes an important affect and so on. But
in general the naive picture this would not
be to Ferro magnetism to anti Ferro magnetism.
So, therefore we will keep that also in the
back of our minds that underlying some kind
of interaction. We must include which favors
parallel alignment and that happens we will
see how that happens?
It turns out that this picture is not quite
right or rather you do not have to wait to
absolute zero. For this to happen at a finite
temperature called the Curie temperature,
it happens and below that the picture would
in fact be this jump this continuously and
you have a phase transition. But for the moment
I want to call attention to the fact that
we have not switched on interactions. And
therefore you are still in the Para magnetic
phase, which means that if you switch off
the field the magnetization goes to zero.
Now let us relax this assumption that we have
made that the magnetic moment can be either
just be parallel to the field or anti parallel,
assume that it can be in any direction.
Then what does this magnetization look like?
to compute. What does M look like in this
case. This is equal to… Now remember that
if I have a field in this direction and I
have a magnetic moment dipole moment. In this
direction, the energy epsilon e to the minus
cos theta minus mu dot H; and now, if I choose
the direction of the magnetic field to be
along the polar axis, then spherical polar
coordinates the field can the magnetic moment
can be in any direction. What so, ever and
where ever it is the angle between the two
is theta, that is the polar angle the interaction
the energy is minus mu H cos theta. Now let
us compute the magnetization what is this
equal to and I am interested in the magnetization
in the direction of the field.
So, let me just put a scalar here take it
to the z, what does this become? It is equal
to again there is N, because there is no interaction
mu times. This is the contribution that this
guy would give along the direction of the
field. The contribution will be mu cos theta
times the probability that the angle is there,
that is equal to e to the power beta mu H
cos theta. Now, you have to integrate over
all theta, you have to integrate over all
directions and normalize. So, this is equal
to an integral over all solid angles all directions
divided by an integral e to the power beta
mu H cos theta. That is the average magnetization.
I have not put these angular brackets, because
I have already used a capital letter M, it
is a thermodynamic variable so; there is N
factor here. This is the contribution from
a dipole pointing like this to the magnetization.
In this direction that is mu cos theta multiplied
by the Boltzmann factor e to the minus beta
times the energy, which is equal to minus
mu H cos theta integrated over all possibilities
and normalized.
You recognize this quantity is in fact, if
I raise it to the power N, it is the partition.
Now what is d omega, I have included the magnetic
interaction, I am not looking at the kinetic
energy or translation energy of these dipole
moments. Only the magnetic portion that is
all I am interested in. Now what is this equal
to? What is this kind? What can we do this
with you? We should do this integral what
is d omega? This is equal to…
So, integral d omega really stands for integral
d cos theta minus 1 to 1 0 to 2 pi. It is
sine theta d theta from zero to pi, but I
like to write it as d of cos theta from minus
one to one, that is what it is makes it easier?
So, let us do that.
This gives a 2 pi factor from the free integration
integral minus 1 to 1 d of cos theta. There
is N mu and then there is a cos theta divided
by the 2 pi, the 2 pi cancels out and then
you have minus 1 d cos theta e to the beta
mu H cos theta. This is the integral and what
does that work out? What does this guy work
out?
So, it is M over N mu equal to this thing
minus 1 to 1 d cos theta, by the way can I
equate this to zero on the top? Because it
is minus 1 to 1 d cos theta and there is a
cos theta sitting here. Yes. It is an odd
function. Can I equate it to 0? No, because
this guy is sitting here. That is a mix function,
I cannot do that right. So, I have an integral
let me call that let U equal to cos theta
so minus one to one d u e to the beta mu H
mu divided by integral minus one to one d
u e to the beta mu H mu, because once you
do this integral and then say this signal
is the derivative of this denominator with
respect to beta mu H.
Let us just do the integral so this is equal
to integrate by parts. Therefore, you have
u e to the beta mu H mu over beta mu H and
slow at this minus 1 to 1 minus integral minus
1 to 1. By I can do this integral directly
e to the beta mu H minus divided by beta mu
H, which cancels. This is equal to plus minus,
if I integrate this. I get exactly the same
null set, but this factor and that is equal
to cot hyperbolic. That is this is cos hyperbolic
over sine hyperbolic so, it is cot. So, this
is the function we get its different from
the function we got earlier, which was a tan
hyperbolic whatever it is.
So, let us compare these functions earlier,
we got tan hyperbolic some x where x equal
to mu H over k Boltzmann T. That has simply
been replaced by cot hyperbolic x minus one
over x, where x is the same value mu H over
k Boltzmann T. Now the time hyperbolic function
we plotted we found out from minus one to
one and pass through the origin linear, but
this is a different story. We have to plot
this function as a function of x.
So, what does this look like? Here is x and
here is cot hyperbolic x minus one over x.
What does it look like? What does it look
like Does it diverge is it finite at the origin.
What does it look like? If tan hyperbolic
x goes like x near the origin cot hyperbolic
x must go like one over x and that neatly
cancels the singular part cancels and you
are left. Now, with rest of it and that is
a regular function and what would it do well
tan hyperbolic x saturates to plus minus one
at infinity? We call the way this function
looked as x saturated term like this and therefore
we got x versus cot hyperbolic x.
This stayed out all the time it went like
this so, now both are odd functions and now
you have to subtract the one over x part.
And when you do that you discover that once
again you have a function, which behaves pretty
much like this. And this function has a name
it is called the Langevin function occurs
again and again Para magnetism. It is called
the Langevin function. Langevin was the first
person to give the theory of Para magnetism
he is a friend of great physicist, friend
of Madam Curie’s, and one of the founders
of the subject non-equilibrium statistical
mechanics. So, this is what the function looks
like? Pretty much same as before except we
got to decide, what the slope of this function
at the origin is?
Now, remember that cot hyperbolic x blows
up at x equal to 0 and it is an odd function
in singularities of one over x. So, it must
go like one over x plus, what is the first
correction going to look like. Near x equal
to 0 and then you have to tell me what is
going to be the next term near the origin.
This is certainly one over x it will cancel
the minus 1 over x and what is the next term?
What will it be proportional to what power
of x it will be proportional? It is not more
singular so, it cannot be one over x square
one over x cube or anything like that x. It
cannot be proportional to x to the 0 why odd
function.
It has to be proportional to x could be proportional
to x cube, but that will be an accident. If
it started so, the next term is in fact proportional
to x. Some a x plus may be b x cube plus dot
dot dot A and B and so on are real constants
could be positive negative. We do not know.
Would you expect that to be positive or negative?
A positive what is a can we find out and of
course do this painfully?
So, let us write cos hyperbolic h equal to
cos hyperbolic x over sine hyperbolic x. Now
what is cos hyperbolic x power series in x
1 plus what is the definition of cos? Cos
is one minus x square by two factorial plus
x over 4. So, what is cot x one plus everything
is plus one plus x square by two and so on.
So, this is one plus x square by 2 plus order
x 4 and what does sine hyperbolic x do x plus
x cube over three factorial etcetera. So,
there is x, which I take out then this becomes
one plus x square by six plus higher terms.
So, this whole thing goes like 1 over x times
this is 1 plus x square by 2. This is 1 plus
x square minus 1 so that is one minus x square
over six to start with therefore, you get
one plus half x square minus one sixth x square
and what is half minus one-sixth one-third?
So, this goes like 1 plus x square over three
plus higher order terms and therefore this
is 1 over x plus x over three plus higher
order terms.
Therefore, this quantity here, now that we
found magnetization lets write it down M over
N mu equal to cot hyperbolic mu H over k Boltzmann
T minus k Boltzmann T over mu H. This quantity
is function goes as H goes to 0 to what term?
What is the leading function leading quantity?
It goes like mu H over three k Boltzmann three,
which came from half minus one sixth that
came one over two factorial minus one over
three factorial. It came from the exponential
function and turned out to be this. So, this
would imply that the isothermal susceptibility,
in this case is equal to one third of the
earlier N mu square over three k Boltzmann
T.
How come we got a one k Boltzmann T initially
and now we get a three what is going on? It
is a different model. We assume first it goes
up or down and now we are saying that it can
be in all directions and so on. So, what has
changed the number of dimensions has changed?
We have said that this magnetic moment lives
in one line go either up or down. Now, we
are saying it can live in three dimensions
so, really this one, this three has to do
with the fact. That you are in three dimensional
spaces, although it came from someone over
2 factorial minus 1 over three factorial really
is the reflection of the fact that you are
in three dimensions.
You trace it back carefully you discover that
this guy here, the Curie law comes from the
number of dimensions in which you have allowed
magnetic moment to move a test would be to
ask. Now, we did it for one, we did it for
three what should we do it for 3? Do it for
two. We will say what if the magnetic moment
moves only in the vertical plane or something
like that a plane containing the magnetic
field. Let us do that just to see there is
a reason for doing all this, because phase
transitions turn out to depend these exponents
on the properties. Critical properties turn
out to depend on two universal two basic objects
one is the number of dimensionality of the
space.
So, if your system is one dimensional two-dimensional
three dimensional things change drastically.
The other has to do with whether the quantity
you are looking at is a vector or scalar or
tenser. What is the number of components it
has these seem to be the universal quantities
upon, which critical depend independent of
interactions. We will see why that is so?
So, let us see if our guess is right. Let
us assume heck of it see whether we can do
this for a planar magnet.
So, what would you now say here is H, and
here is the magnetization mu and I am going
to assume that it moves in this plane and
nothing more right. And I would like to find
the magnetic susceptibility. I would like
to find the magnetization, let us see if we
can do this would you expect you can do this?
Yes. We expect we should be able to do this.
We did it one; we did in three so, why not
in two. So, what should I write M divided
by the N mu the average is equal to and now
I have to integrate over all possibilities
and I integrate. I have to integrate over
this angle theta, I continue to call it theta,
but what is the range of theta 0 to 2 pi or
minus pi to plus pi does not matter, because
you are in this plane it is like a plane polar
angle term, it is not a polar angle term in
three dimensional. So, it is 0 to 2 pi 0 to
2 pi and what is the measure? It is just theta.
There is no sine theta d theta, it is just
d theta and then cos theta e to the beta mu
H cos theta divided by integral 0 to 2 pi
d theta e to the beta mu H cos theta. This
is the integral we have to do.
That we think we can do this well the denominator
looks easier. Can we do it? How to do this
integral? We can always do it numerically,
if you give me values of beta mu H, I put
that in and I can do it, but now I am trying
to find out what is the functional dependence.
Unfortunately this is not a easy integral
to do and the reason is that there is no d
cos theta. If there had been a sine theta
d theta I would have done it in short, but
you cannot do this otherwise. So, what is
this integral equal to? What is this? It is
a Bessel function this guy here is related
to j 0. So, you see already strange things
can happen to change the dimensionality. The
fact that this measure has gone the sine theta
has gone in given dimensional space it made
it much worse.
So, it is not a trivial thing one can evaluate
it in terms of Bessel functions and so on,
but our purpose is not that. I want to find
out what the magnetic susceptibility is? And
I want to find out if you get N mu square
over two k T. That is my target, I would like
to find out, but what is the susceptibility?
It is the slope at the origin, which means
I need to find this quantity to what order
N the magnetic field linear. I need the linear
behavior. So, let me expand it and do everything
to first order. Let us expand so, we put everything
and try to expand everything to first order
in H. I am only interested in that.
So, this is approximately equal to 0 to 2
pi d theta cos theta and the first term is
one plus beta mu H cos theta plus dot dot
dot divided by integral 0 to 2 pi d theta
and one plus beta mu H cos theta plus etcetera.
What is d theta cos theta 0 to 2 pi that 0?
You have to get rid of that immediately and
I get a beta mu H 
and then a cos square theta plus higher orders
divided by what is in the denominator? This
term goes away d theta the first order term
in H goes away and you just get a 2 pi in
the denominator, what is this integral equal
to? This is equal to beta mu H, if I write
this as cos square theta as one plus cos 2
theta divided by 2 and the cos 2 theta parts
will just vanish. You just get a two in the
denominator, but you also get a 2 pi.
So, indeed in this planar model indeed susceptibility
this T is equal to N mu square over 2 k Boltzmann
T planar. So, sometimes you have to notice
these facts this two this three looks very
innocuous and so on. It comes from various
places here, it came from cos square theta;
fact that it is one plus cos 2 theta over
2 and so on, but underlying this whole business
is the fact. That it is really telling you
this guy is really telling you how many dimensions
the magnetic moment is free to move?
So, that is the reason I did the one dimensional
cases the easy model, but it has implications
a numerical purely numerical, but this is
worth bearing in mind. In all cases it is
proportional to one over T. That is the Curie
law and that is the law for Para magnetism,
but we know it cannot be valid till T goes
to 0. So, now we got to put in the fix, we
got to put in the fact that you have Ferro
magnetism, you have an interaction. How should
we do this? You have to write the model for
Ferro magnetism that is much harder to do
than this elementary models. Incidentally
you can ask the following question.
If I really took into account magnetic interactions
that quantum mechanical interaction between
dipoles, then what does it look like? What
would be the interaction? How do I find the
Eigen values energy values and so on? You
get essentially for what is called spin half;
you get the maxima function for higher spins.
You get what is called the function, which
would have similar behavior very similar.
Eventually you would end up with a Curie law
some constant multiplying that is not so,
important. What is crucial is to ask? How
do you change this to Ferro magnetism? And
just as in the ideal gas we go from there
to a real gas. We have to put in interactions
in a realistic way, but we did this by a short
cut of saying that this is the phenomenology
called the Vanderwaal’s equation.
Similarly, here there is a phenomenon called
the Weiss molecular field. It is the analog
of the Vanderwaal’s equation and the fix
is very simple. You could take any one of
these models; let us take the simple one dimensional
model.
So, we have M equal to N mu tan hyperbolic
mu H over k Boltzmann T and the fix consists
in the following this is called the Weiss.
Remember, all this was done in pre history
before we understood quantum mechanics properly.
Before Iceberg came out with the idea, that
this is a exchange interaction so on. The
idea was that in a real medium the Ferro magnetic
medium. Each dipole moment sees not only the
external field, but also the internal magnetic
field. Due to the other magnetic dipole moments
and this would act like an effective field.
It gets added to this stage so, it says replace
H by H effective, which is equal to H plus.
The part which says if this is the external
field H and you have a dipole moment here
and it is surrounded by other fellows of this
kind. These are the fellows produce on your
given dipole moment this guy and effective
field, which tends to align. In this direction
it adds to the external and what is it going
to be proportional to it will be proportional
to the magnetization itself?
The idea is proportional to how much of it
is already magnetized? So, there is a portion
which goes like M multiply by some constant
lambda, but you have to be little careful
in doing. This I would like this lambda to
be some constant, which is for a unit can
be evaluated, but this H is an intensive quantity
supplied from outside. The M is an extensive
quantity, if I increase the size of the system
M increases. So, I must normalize this M to
the magnetization per unit volume or better
still per unit magnetic dipole moment. So,
let me call it by N mere make it a little
intensive quantity like H itself, you should
add an intensive and extensive quantity.
If you do then the constants multiplying,
it will become of order N. So, I put this
in and ask what happens? Now then my magnetic
equation of state is M equal to N mu tan hyperbolic
plus lambda M by N times mu over k Boltzmann
T. That is 
much harder equation, because it is not an
explicit equation for M as a function of H
and T, because M appears on this side also.
And you cannot solve this equation in closed
form for M, because it is a transcendental
equation an algebraic function M on the left.
There is an exponential function and hyperbolic
on the right s. So, no closed form solutions
can be given, but you can certainly do this
numerically or graphically. And we in fact
we are interested in asking what happens to
this equation for small values of H near the
origin. We would like to see what is the magnetization
at all? What is near the origin? What does
it look like?
If you do that you immediately realize when
H is equal to zero switch it off. Then in
Para magnet the magnetization goes to 0, but
in a Ferro magnet with this equation of state
we have M naught equal to N mu. I will denote
the magnetization by M 0, when the magnetization
in the absence of the field. I will call it
M naught this is equal to tan hyperbolic mu
lambda M naught over N, that is it. This is
again a transcendental equation and you have
to ask is there a solution at all. M naught
equal to 0 is always a solution, because the
right hand side vanishes the left hand side
vanishes and that is the Para magnetic solution.
Because it says in the absence of an external
field then magnetization of the sample is
also zero. This is how a Para magnet behaves,
but a Ferro magnet does not do that. A Ferro
magnetic curve has a hysteresis curve in general.
So, what it does is to behave in this fashion
to plot H versus M from this side instead
of the Para magnetic behavior like this. What
are Ferro magnetic nulls after you stabilize
it is behavior is that you start with a sample
and you switch on the field. It will saturate,
but then when you switch the field back to
do this go there and then it would do this.
So, that the hysteresis loops therefore, when
you switch off the field from positive side,
there is a remnant magnetization. That is
what I called M naught and if you do it from
the negative side there is a remnant magnetization
from the opposite direction.
So, you have to allow for the possibility
that there is this solution rather than that
solution and that should come out of this
equation. Now how do you solve this equation?
You would solve it graphically; we will do
this next time. What would happen is that?
If I plot it M naught the left hand side is
a 90 degree, 45 degree line M naught equal
to N naught. The right hand side is N mu lambda
M naught over N k Boltzmann T, that quantity
for M naught approximately 0 is N mu square
lambda M naught over N k B T.
So, it goes like here M naught is equal to
0 the N cancels out and we have mu square
lambda M naught by k B T. Now, that is a straight
line of slope mu square lambda M naught over
k B T. So, if T is very high this slope would
look like this and the curve would then saturate,
but if T is sufficiently low the curve would
look like this. And then saturate and then
you have a root here and a root here other
than the root at the origin. That is how you
get permanent magnetization and when would
this happening? It would start happening as
soon as the slope exceeds 1.
So, you have a non trivial non-zero solution,
when the slope mu square lambda over k B T
is greater than unity, which implies that
T is less than mu square lambda over T. That
is your critical point below that you have
these new solutions. Then of course you have
this new solution, but now we have to look
at the free energy and show that solution
is unstable and this is how Ferro magnetism
appears in a simplest phenomenon. So, the
whole idea is below a certain physical below
a certain physical temperature. The cooperative
tendencies of all these magnetic dipole moments
to line up will dominate over the disrupting
tendency of entropy or fluctuations and you
get order. But above the critical point the
disorder wins and the order is the energy
interaction is the strong enough to maintain.
All this is how the phase condition occurs
between the Para magnet and the Ferro magnet?
Now, we got to take all this and put it back
and see where the connection is with. We will
do this in the next class.
