Lecture 23-N Spins in a Uniform Magnetic Field

Lecture 23-N Spins in a Uniform Magnetic Field

So, good morning students. Today we will talk
about Magnetic system of N Spins in a constant magnetic fields. I am going to take a constant and uniform
magnetic field. So, this is another system in the Gibbs canonical ensemble. The macrostate
in this system is given by a combination of three thermodynamic variables which are N,
the magnetic field B and temperature. So, the pressure in the previous ensemble
of the ideal gas has been replaced with a new variable which is magnetic field. Now
because this is a system of spins we are going to maintain the system of N spins in the constant
magnetic field. So, it is like if you consider pressure as a generalized force and you want
to keep pressure constant then you have to allow for exchange in volume because, PDV
is the work done on the system against an external pressure. So, this is the energy
scale ok. So, I will write this as the if you want to
write it the systematically, it would be better to sort of understand what we mean by a generalized
force and the corresponding conjugate extensive variable. So, in the previous canonical Gibbs
canonical ensemble where we maintained the system at constant pressure. So, the generalized
force that we subjected our system to was the pressure and we could do it by allowing
the system to exchange volume. So, the extensive variable that was allowed to fluctuate was
volume that is the conjugate variable, in the sense that PDV is an energy scale neither
is P force nor is delta V displacement, but the product of the generalized force into
generalized in into you know generalize displacement is always energy.
In this system I am going to maintain constant magnetic field and the extensive variable
conjugate to this magnetic field is the magnetization. So, the system will be having a fluctuating
magnetization. So, some of the fins can be instantly flip down or flip up in response
to a constant external magnetic field ok. So, as again B is magnetic field which is
not force, dimensionally it is not equal to force and neither is magnetization dimensional
equal into displacement. But the product of B into delta M has a dimensions of energy
like we have been saying some time now that generalized force are to be realized not as
force themselves they are thermodynamic variables that we are going to keep constant.
But conjugate to each of these generalized forces there exist an extensive variable such
that the product of force and the variable extensive variable is always energy. So, our
system is now a magnetic system which is under constant magnetic field. So, they are going
to be dealing with constant B and hence the system will be allowed to exchange allowed
to have fluctuating magnetization. So, let us think of draw small schematic that I have
a system that is in contact with reservoir as before and this reservoir has a two fold
task ok. So, the reservoir and this is my system the number of spins in my system is
constant that is easy because I am not allowing any particles to leave the system.
So, the box are has a volume has a has valves that does not allow exchange of particles
that is easy. Then I will be allowing the system to interact with the reservoir in the
sense that the system is allowed to exchange energy with the reservoir to keep the temperature
constant ok. So, this will keep the temperature constant. So, there is a give and take of
energy between the system and reservoir. In addition to this the system is allowed to
exchange or have its magnetization changed which will be a consequence of a constant
magnetic field exerted on the system ok. So, now if I write down the. So, this is a
system under Gibbs canonical ensemble which means the magnetization here that fluctuates
between the microstates can be represented as V mu and I am going to call it as a instantaneous
magnetization this instantaneous magnetization is nothing,
but that magnetization of that particular state that you are observing at any instant
and this is given as summation over all the spins that I have taken. So, i will run from
1 to N into the Bohr magneton or magnetic moment per unit spin into the excitation of
a particular spin. So, this is a spin half system, which means
the spins can be either along the magnetic field or anti parallel to magnetic field.
So, here for I have taken for very you know for the sake of convenience I have taken spin
half system, I can have my excitation to be only plus 1 or minus 1 means parallel to magnetic
field or anti parallel to magnetic field. Naturally in the presence of an external magnetic
field you would have more and more spins aligned in the direction of magnetic field because,
that is the that lower the energy of the total system.
So, alignment is preferred over non alignment and because of that I am going to write down
the energy scale of the problem or the total energy of the system as some internal Hamiltonian
and minus BM ok. So, this is the magnetization instantaneous magnetization ok. So, as you
can see I have to take the magnetic moment of the mu th microstate. So, I am going to
call as the energy total energy of the mu th microstate. So, that is my energy scale. Now our spins are non interacting to take
a very simple example which means there is no interaction between
spins. So, there is no internal energy in the absence of a magnetic field. So, that
basically makes my Hamiltonian in to be 0. So, there is no internal Hamiltonian
the corresponding equivalent would be in ideal gas you had taken the Hamiltonian as sum of
pi square over to M; now there is no such translational momentum here for the spins
they are sitting on you know their latest positions.
So, the only internal energy this spins can have is if they are talking to each other
the interaction energy and we are taking non interacting spins. So, there is no internal
Hamiltonian for the system the only energy the system will pick is by coupling to an
external magnetic field. So, the only energy scale is just minus BM and M is that magnetization
of the instantaneous microstate. So, then you can write down the probability distribution
function of the micro states as simply. Now microstate here is nothing, but collection
of all these excitations I am going to write it as collection of all these sigma is because
I have used the label sigma i’s ok. So, the particular values of sigma is that you
take constitutes a microstate ok. So, here i that we have taken is basically 1 of the
N particles in the system ok. So, this is the meaning of the excitation fine now this
can be written as you can write down this as e to the power
minus beta into the energy scale which is just minus B times M mu divided by the partition
function ok. So, this simply becomes e raise to beta BM mu upon the canonical partition
function fine. So, now we can compute what is the canonical partition function because
that is required to build the connection with thermodynamics. So, I can write down the canonical partition
function or the Gibbs canonical partition function as simply summation overall microstates
which is nothing, but the collection of all these sigma i’s e to the power beta B and
the magnetization is basically this expression ok. So, I have taken the magnetization as
microstate is nothing, but the symbol of a microstate is mu here.
So, we can just simply take this expression put it here. So, it will be nothing, but summation
over all the spins I am going to take mu naught outside and simply write down sigma j ok.
Now this sigma js are the values of sigma j that you have taken in the microstate fine.
So, I can write this as summation overall the micros summation overall the microstate
which is basically combinations of the sigma i’s e raise to mu 0 beta B into sigma 1
into mu 0 beta B sigma 2 all the way to e 0 beta B sigma N, because the exponent has
sum of terms if you spit up into products. Now you write down since there all independent
of each other always sums are independent of each other you can write this as nothing,
but a summation sigma 1 being plus 1 or minus 1 e to the power mu 0 beta B sigma 1 and then
you can write down summation sigma 2 as plus 1 minus 1 e raise to mu 0 beta B sigma 2 all
the way to the last particle. And if you see that these are all independent
of each other it is just amounts to write down just 1 of them and raising it to the
power M. So, the product of sums becomes sum of products
now you can right it as simply e to the power mu 0 beta B plus e to the power minus mu 0
beta B the whole to the power N. And this is nothing, but the expansion of twice for
cosine hyperbolic mu 0 beta B to the power N. So, I am going to write it as 2 to the
power N and cos hyperbolic to the power N fine.
So, this is the partition function for the NBT system that we have chasing and this opens
up the route to thermodynamics in the sense that I can now compute what is the average
magnetization of my system. Because, a system as a fluctuating magnetization the spins are
constantly in a between different microstates the spins are differently align. So, each
microstate has a different magnetic moment. So, you would be interested in the average
magnetization of the system because, magnetization of a micro microstate is a random variable
and when you are dealing with random variables you are interested its averages or moments. So, average magnetization
is then very easily computed. So, if you want to compute the average magnetization this
is nothing, but the average of the you know magnetic moment of each realization or each
microstate. So, you can compute this from the from statistical mechanics by simply summing
overall microstates M mu P mu. So, here you can write this mu as we already know that
this is nothing, but the collection of all the excitations.
So, if you specify the values of these excitations it constitutes a single microstate, if you
change this sigma is it becomes a new microstate. So, you sum over all such combinations and
sample this magnetic moment in this PDF and which is given as e to the power I have already
written the expression of the PDF is beta B M and this has to be divided by the partition
function. And well you know the trick here this is nothing,
but you have to pull out an M so, which means you have to take the derivative with respect
to beta B. So, this is d by dB of here exactly the partition function. So, this is my partition
function that I am going to write down. So, this is nothing, but 1 upon beta d by this
is not complete derivative. And you got a beta Z in the denominator ok. So, this is
nothing, but 1 upon beta d by d beta d by dB of ln Z ok. So, that is the value of the
average magnetization. So, I am going to call this as equation 2 and I am going to call
the partition function has equation 1. And so, I can just go ahead and take the logarithm
of this partition function compute M M. So, this is nothing, but I can write down as a
1 upon beta d by dB of ln Z would be lon Z would be ln 2 raise to N which is N lon 2
plus I have yeah N ln of cosine hyperbolic mu 0 beta B fine. So, you can easily see that
there is only the second term that is going to contribute. So, you can write down this as 1 upon beta
N comes out that makes the magnetization and extensive quantity. And the derivative of
logarithm of cos hyperbolic is nothing, but 1 upon cosine hyperbolic and the derivative
of cos hyperbolic is sin hyperbolic and I will have a mu 0 beta to complete the
derivative ok. Now you can write this as you can knock off a few terms
and write it as N times mu 0 into tangent hyperbolic of mu 0 beta B. So, this is my
magnetization for this N spins in the magnetic field B ok. So, now with the magnetization in the thermodynamic
limit we are in a position to compute the magnetic susceptibility
which is nothing, but derivative of magnetization with respect to field at 0 field. So, which
means that this is magnetic susceptibility at some temperature T, it tells you basically
how M can be expanded in powers of B for small value is a field. So, if M has a Taylor series
expansion around origin then this. Susceptibility is nothing, but the first sub leading term
in the in the in the Taylor expansion. So, here we can compute the chi of M at some
temperature t as simply the derivative of the magnetization that we have constructed.
So, you can complete the derivative is as a constant free factor tangent hyperbolic
becomes second hyperbolic square of mu 0 beta B and to complete the derivative I have a
mu 0 into beta outside ok. So, this is nothing, but mu 0 square N upon beta and the entire
derivative is taken at B equals to 0. So, this just becomes mu 0 N mu 0 square N upon
beta because second hyperbolic 0 is 1 is 1 upon cosine hyperbolic 1 0 which is already
1 this is basically the magnetic susceptibility. And there is a homework task that I would
give you in this lecture is to show how fluctuations of magnetization. Fluctuations of magnetization
which means fluctuation is represented as the second cumulant is related to susceptibility
magnetic susceptibility ok. So, that is your home work task you can do
this exercise and learn something about fluctuations in the magnetic system ok. So, you will be
you will be surprised to learn or you would be happy to see that these fluctuations again
have a very important meaning in canonical ensemble when quantities are not really constant.
So, let us meet in the next class and I will show you the result and then we will also
startup with the final ensemble of this course which is called as the grand canonical ensemble.
So, we will break here and we meet in the next class.