# 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.