 Right, so today we will look at the fluid magnet analogy in a little more detail and I will introduce the so-called mean field critical exponents. So just to recall to you quickly what we had said, we said if you have a collection of spins each of which can point in one of two directions up or down in the simplest case then we had a simple formula for the magnetization in an applied magnetic field H and the magnetization per particle I denoted it by M and this quantity was equal to tan hyperbolic Mu H over K Boltzmann T where Mu is the magnetic moment, magnitude of the magnetic moment of each particle, pardon me, I defined it as M over Mu was equal to this and now of course we can plot this M versus H diagram. So here is H, here is M and it runs from between minus 1 and plus 1 in this fashion and the isothermal susceptibility which is the derivative at 0 field out here, this quantity chi T was equal to Delta M over Delta H at constant T at H equal to 0, this quantity was equal to Mu over K Boltzmann T which is proportional to 1 over T which is the curie law for paramagnets. Now the question is how are you going to include interaction between these various magnetic moments leading to the phenomenon of ferromagnetism itself but while we do that and I am going to do that in a theory called the mean field theory, let me qualitatively draw graphs and show you what the actual results are so that you can see immediately where we are headed for okay and the graphs are as follows again the so-called fluid magnet analogy which said that for a fluid you had P versus V and these isotherms were things like this below the critical point and then you had a critical isotherm of this kind and above the critical point you had things like that so this point this inflection point is V C P C and of course for T C so this is T equal to T C isotherm corresponding to that we have the M versus H diagram here in this fashion so this region above T C corresponds to these isotherms here then let me just say what the result is this critical isotherm T equal to T C with an inflection point corresponds to what is called a critical isotherm here like this and this is for T equal to T C this is for T greater than T C just as this is for T greater than T C and this is for T less than T C of course you immediately see a difference which is that while the pressure here at the coexistence region changes with the temperature that does not happen here they are all at the same point here and now you can ask what happens if you go below T C for this system here this model will not show anything because it is equivalent to the ideal classical ideal gas in other words there is no phase transition P V equal to RT there is no condensation because there is no interaction between the molecules assumed the moment you put an interaction between molecules then condensation can happen if there is an attractive interaction and there is we know in real life an attractive interaction at long range and that is what is responsible for this condensation below T C here the same thing happens here and we will see this explicitly what happens actually is that you come across come till here and then you discontinuously jump out here so this is an isotherm this thing here for T less than T C and all the isotherms for T less than T C are of this kind they have a discontinuity on the H equal to 0 line and we jump out so in a sense all these points these flat portions are all compressed onto one line okay in this case so this is the connection between those 2 fellows now you can ask what is the susceptibility well I defined it delta M over delta H at H equal to 0 you could ask is there a susceptibility in the ferromagnetic phase now why do I call this the ferromagnetic phase because even when I switch off the field there is a remnant magnetization that is the characteristic of a ferromagnetic phase so if I switch off the field from the positive direction at that temperature the system will show this magnetization but if I switch it off from the negative direction from opposite sense of the field it would show this value just as out here if I decrease the pressure to this point you would be here if I decrease it on the gas phase you would be here and exactly the same way so it looks like the analog of the liquid phase and the gas phase here it does not it has more symmetry it is a magnet with the magnetization positive versus a magnet with magnetization negative here the analog of the fluid homogenous phase out here is the paramagnetic phase when there is no magnetization at all now this becomes yeah yeah so we will have to see what this TC is and what the difficulty is we are going to see this in some detail okay what happens is in a nutshell his question is does thermal noise start doing something at TC or below TC and the answer is yes it is the other way about the interaction tends to align things beyond a certain temperature the disorder effect wins and the ordering effect does not that is essentially in a very crude way what happens there is always thermal noise present thermal fluctuations there is typical strength of a unit of thermal fluctuation is in energy units is KT now if the characteristic energy associated with ordering is much less than KT then you are in the so-called high temperature regime which is disordered and if it is small compared if it is large compared to KT then of course you are in the low temperature regime okay so ultimately as we will see today the strength of the interaction decides where this critical point where the critical temperature is this is always thermal noise of course many other crazy things happen at the critical point and that is why it is so hard to understand in elementary terms one of the things that happens is and this is the modern way of understanding the correlation length of the system diverges at the critical point and we will in fact in fact establish that we will see how it diverges today okay so this picture has to be kept in mind the PV and the m versus h then the next one was in the fluid case we had the fluid I am doing here so we had the T here and we had P and remember that this was liquid this was gas and this is the critical point so this point is TC this is the critical pressure PC the question is what is the analog of that here I have already said that the two phases liquid and gas correspond to magnet up and magnet down in this case and the fluid phase above the critical point the critical region corresponds to a single phase there the one phase region corresponds to the paramagnetic region here so what is the analog of that graph look like well it is like this I will draw it here of course there P and T are non-negative so only the first quadrant is involved but here you certainly have P will run all the way from 0 upwards but M of course can be positive or negative H can be positive or negative is a function of this field and the question is what is the analog of the PT curve it is got to be the H versus so what do you think is the analog well let us remember what happened here we know that the slope slope of coexistence line DP over DT equal to Delta S over Delta V by the Maxwell relation and then I argued that generally Delta V between vapor and liquid is very very large so I was large denominator and the slope is very small in magnitude it is also always positive because if that stands for S entropy of gas minus entropy of liquid divided by volume of gas minus volume of liquid specific volumes then both numerator and denominator have necessarily to be positive and the slope is small here the same thing happens but now you have the symmetry between up and down whatever you got here with a certain switching off the field here exactly the same thing with a minus sign appears here this is reflected here in this is completely symmetric you can see in this graph that if you change the sign of H you change the sign of M so it is completely symmetric in this situation therefore this graph has no choice but to be a straight line here it cannot be in any particular direction because if it were up it would violate this up down symmetry so it is got to be like that and then at T critical and that is the coexistence line between a phase which is not liquid but ferro up let me just denote it by this and this is ferro down and on this side just as you have a homogeneous fluid phase out here on that side you have the paramagnetic phase so you already see by symmetry that the slope here dh over dt of the coexistence curve must necessarily be 0 right I just argued this by words saying that look whatever happens up here when we reverse the sign of the field the magnetization changes and it is got to be so but we can now ask no no no let us just blindly apply the Clausius-Clapeyron equation let us just blindly apply the analog here then why should it be 0 dh over dt that graph must have a slope which is delta s divided by delta what what is the analog of the volume the magnetization and if this is 0 what does it mean well delta M is finite because this is M for 0 field and this is M from the negative side so delta M the denominator is finite it is this difference but the numerator is 0 because how after all do you get this graph how do you get a point here you have a specimen like this in which everything can point only up or down then if this is remember this is an average magnetization over the whole sample so it is clear that if you were at absolute 0 of temperature you applied a field infinitesimal field in the up direction everything would align in the up you would be here similarly if you applied a negative field you would be here the reason you have this gap here that is not gone all the way up there is because of thermal noise thermal fluctuations so what you have is large islands of up up up and then there are smaller regions of down down etc. As you bring the field down towards 0 this on the average there will be more pointing up than down and you have a net up magnetization as you lower the temperature further it takes less and less field in order to align so you are going this intercept is going to move up ditto for the other side move down towards minus 1 now it is clear by the symmetry that the entropy of the system which is a measure of the disorder in the system is equal on both sides because whatever domains you had here for up field and down domains in the up field exactly the opposite happens in the case of a negative field so these would become down and these would become up here so it is not surprising that the entropy is 0 change in entropy is 0 the moment that happens the slope here has to be 0 because the denominator does not vanish the numerator vanishes by this symmetry so that is a rigorous argument that this curve coexistence curve between up and down has to be flat here it is a good thing it is flat because you see otherwise look at what will happen out here if I start with a liquid at atmospheric pressure say and I heat it I increase the temperature it will boil once you cross 100 Celsius so it will go across so it will certainly boil because you can go across this line out here you could also boil get it into a gas form by dropping the pressure down this way or anywhere across which cuts this line here on the other hand if you start here with fellow up and you just increase the temperature it is going to go up like this it is not going to transform into a magnet down without changing the field you have to change the field to do that on the other hand you can boil water you can boil water it becomes gas without changing the pressure but you cannot make an up ferro magnet down ferro magnet without changing the field just by increasing the temperature so that itself tells you that this graph has to be flat by symmetry we are now going to talk about para to ferro that is a different matter out here what is going to happen here for that I need the third of these graphs and the third graph was of course in this case in this case it was the density T versus row graph and it was roughly like this this was the coexistence region this here was the gas phase this is the liquid phase this is the critical point and this region was a coexistence region in the van der Waals isotherms if I quench something suddenly to a point here it would phase separate into a part liquid and part gas saturated vapor above it the exact ratio of how much happens can be found by so-called lever rule which I am not getting into here but the fact is the two coexist here on the other hand there is no coexistence region here at all because let us look at what the analog of that graph is so on this axis I draw temperature I draw magnetization here and since M can be plus or minus I draw this then here is TC first let us look at what happens if I have 0 field then depending on whether I switch it off from above or from below above TC I have 0 at TC you just about taking off from 0 and then these intercepts go towards the saturation value so this means they are going to go like this and this branch corresponds to the other part of it so let me call this M not to show that it is the remnant magnetization in the absence of a field so M not is either this or this at different temperatures as I go towards absolute 0 this tends to 1 in the units we have chosen and this tends to minus 1 out here the reason we call it a continuous phase transition as opposed to a discontinuous phase transition is that when you when you boil liquid here the density changes discontinuously there is latent heat given to the system and there is a discontinuity across the order parameter but that this is that discontinuity disappears as you get to the critical point in exactly the same way this intercept for M not starts off smoothly from the origin and goes up but it is 0 outside of the ferromagnetic phase in the in the paramagnetic phase but in the ferro phase it starts from this point does not jump certainly and then it goes up and that is the other branch going down here now you could ask suppose I looked at these intercepts for small positive values of the field I do not switch off the field I keep it very small and positive then out here of course you would have for large thing you would have for this field you would have this much magnetization as you lower the temperature you are going to get more and more of it and you are not crossing any discontinuity or anything like that so this would in fact start like this and go up like that it would cannot be more than the saturation value therefore it saturates on the other hand in a perfectly symmetrical way on the negative side it would do this as you come down in the field to 0 and then go towards negative fields the magnetization jumps discontinuously to this point and goes here so you have a jump from here to here and then it goes for the down okay. So there is no coexistence region here this region does not because what happens essentially is that the free energy to plot the free energy will have an unstable minimum unstable extreme at that point okay since we are going to talk about Landau theory let me say that right away if you plot the free energy the corresponding free energy f as a function of m magnetization then in the so-called paramagnetic region this is in the absence of an external field so f a function of m h t and now I set h equal to 0 since at h equal to 0 the only stable solution is 0 magnetization 40 greater than t c and that is a stable thermal equilibrium state this free energy must look like this 40 greater than t c on the other hand when you come to this point here the critical region this free energy you still have 0 as a solution so it gets extremely flat in this fashion in fact the leading term starts with the fourth power rather than the second power so it is not a parabola but it is got higher derivatives as 0 as well and when you come down below t c it does this that is the way this parabola with the single maximum single minimum it breaks up finally it degenerates into a curve which is not a parabola anymore near the origin but a fourth power it starts and then for t below t c even below t c this higher order minimum splits into 2 minima and a maximum in between this is the m equal to 0 solution that is unstable so you see this becomes unstable this solution but you have 2 stable minima as solutions they correspond respectively to this point and this point since this happens smoothly and continuously as you cross t c it is called a continuous phase transition there is no discrete jump finite jump in the value of the order parameter pardon me yeah this is at h equal to 0 so it must be plotted it that pardon me well okay I am being a little loose here in principle what happens in landau theory is that you write it as a function of the thermodynamic variable phenomenologically and then you say the thermal thermodynamic equilibrium state corresponds to an extreme of this f so I have written a functional relay as a functional of this m and said that the minimum is at the average value okay yeah so that is why I put an m there right okay I mean yeah I know I I should put expectation values or something we will do that when we need it we are going to be careful without notation okay all right so those 3 figures and these 3 figures are sort of 1 to 1 correspondence today what we will do quickly now is I want to show you that this is a square root type of singularity that is this curve here in this region the m is like the square root of t minus t c exactly the same thing in this case parabolic then I want to show that this graph here corresponds to the reduced pressure being a cubic function of the reduced volume so remember I already defined p minus p c over p c equal to p and v minus v c over v equal to v in a little abuse of notation here and I said in this region focusing on this region and expanding it it looks like this v p let us just write modulus cubic curve I want to show exactly the same way that at this critical isotherm I would like to show that h is proportional is it goes like m cube on this critical isotherm the curve that goes like this 3 is a critical exponent and I want to show that in both cases especially in this all right now the point about critical phenomena is that these exponents this 3 this half which we still got to establish we will also establish that this is not the correct formula that you will get the susceptibility which goes not like 1 over t but 1 over t minus t c so this is going to be changed to something which goes like 1 over t c minus t this quantity as you come down from the paramagnetic region there is an exponent 1 in exactly the same way the analog there is the compressibility will also diverge on this graph like 1 over t minus t c in the critical region now you could ask one question which is what happens to the compressibility as I go from below t c below the critical temperature well nothing much happens there it still diverges at that point the question is can I define susceptibility in this case at all can I define because the function is discontinuous remember it is defined in this fashion so just to get it clear I just draw 3 graphs now in the para phase you had no problem in the critical isotherm you had this and I said that it diverges it blows up it should because this is not linear it is a cubic curve so slope in becomes infinite at that point but below the critical point my isotherms look like this the function becomes discontinuous can I still define a slope and a susceptibility for it the function is not continuous can I define a susceptibility why not why not I can take a right derivative here and I can take a left derivative here why not they are not sure if they will or not but if there is symmetry in the problem they do in this problem there is m to m h to minus h there is so it is perfectly alright you can therefore define a susceptibility so here is an example for function as a finite discontinuity it may still have derivative at that point they have a left derivative and a right derivative and they may even be equal so it is perfectly alright alright so these are things which we have to deal with now so what we will do is while we have a model here we need to improve this model and that is what I am going to do now so we are going to take this model here and write it in a more sensible fashion and see whether we get the results we want or not here goes in all of condensed matter physics magnetism has always served as a kind of paradigm especially spin models served as a kind of basic fundamental paradigm for various modeling all sorts of phenomena and among them the simplest model is a so-called Ising model which has served as a model for not just magnetism but binary alloys plus a whole lot of today a huge number of physical phenomena can be modeled by the Ising model but I will try to motivate it in the simplest manner possible what you start by saying is that if you apply an external magnetic field on a system of magnetic moments these moments would try to align along the field lowest energy configuration is along the field but in addition to that the moments may be interacting with each other okay what is the simplest interaction between two magnetic moments so if I have a few moment mu 1 and a moment mu 2 in this fashion you have to ask what is the potential energy of interaction between them now this is not a trivial matter because this is going to depend not only on the distance between them these are point dipoles so you must imagine their physical extent is actually atomic in size so it is very small but it not only depends on this thing here but it also depends on the relative orientation of mu 1 and mu 2 with respect to the vector joining them and you have to construct a scalar now one term in that scalar if you remember from electrodynamics would be of the form mu 1 dot mu 2 over R cubed and then there is another term which goes like mu 1 dot R mu 2 dot R divided by R to the power 5 with a different coefficient so in general it is a fairly messy but we are now concerned about situation where you can only point up or down nothing more than that we also want to ask what leads to magnetism what leads to ferromagnetism on the face of it it looks like this kind of interaction cannot lead to ferromagnetism because when you have two dipoles this and this the north pole north pole they would tend to align like this because that is the lower energy configuration so on the whole when you average you are going to get 0 because plus and minus would just cancel out right so dipolar ordering generally tends to be towards a state of zero magnetization anti ferromagnetism at best certainly not ferromagnetism the origin of ferromagnetism is much more subtle and this is what Heisenberg discovered one of his biggest discoveries he discovered that purely due to the fact that electrons are fermions and obey the Pauli exclusion principle when you bring two atoms together their electronic clouds would tend to repel each other there is a statistical repulsion and you can now ask what happens to the energy levels of this combined system then you can write down after a lot of rigmarole and effective Hamiltonian which tends to favor under suitable conditions a parallel alignment which is a purely quantum mechanical effect it is called the exchange interaction and this exchange interaction depends on the overlap of wave functions of electrons belonging to neighboring atoms if you like and depends on other details of the system there are situations there are some substances for which the ordering tends to be anti the ordering is anti parallel for lower energy but in most of the ferromagnetic substances it is always plus up here so essentially an s1 dot s2 kind of interaction with the minus coefficient which tells you that the parallel ordering is favored leads to ferromagnetism it is not a direct dipole-dipole interaction because that is classical that is long range it dies down in this fashion etc this exchange interaction comes about because if you have one atom here with an electron which has got a wave function like that and another one here with an electron like this this overlap integral leads to that effective so called exchange constant since these are bound electrons the wave functions die down exponentially with distance so this interaction which leads to ferromagnetism is a very short range interaction essentially acts only on nearest neighbors in a lattice for instance it is exponentially short in range but that suffices to produce long range order in a big crowd if each person acts suddenly one person starts looking in one direction and he or she can only affect his neighbor and he starts looking this way too then this short range interaction can spread over the entire crowd and soon everyone gets aligned so short range interactions if sufficiently strong can lead to long range order okay that is an important lesson and that is what happens in ferromagnetism so to cut a long story short in our model in the Ising model the Hamiltonian looks like this now we start writing a Hamiltonian then we write a free energy and so on and so forth so we assume that the system is on some kind of lattice for example so lattice with sites labelled by i j etc these are site labels and we are only interested in the magnetic properties so I write down only the magnetic degrees of freedom I write the Hamiltonian only as far as the magnetic moments are concerned that is all that we are concerned with so the Hamiltonian H is equal to minus I put an external field H and then a summation over i let us say s i is of the spins or moments from now on I am going to get rid of this mu little mu I am going to subsume it in whatever parameter you so this is going to be pure number which I am going to take to be plus or minus 1 the mu has gone into this out here the temperature is going to come out externally separately so all that has been done and I have chosen proper units etc it just makes the notation much simpler so I have this minus this is under a uniform magnetic field H across the entire lattice minus a term which describes the interaction which is nothing to do with the external field at all and we are interested in ferromagnetic ordering so that is minus 2 summation j which depends on 2 sides at a time the spins of 2 sides at a time so i and j j i j the strength of the interaction or overlap integral depends on this i and j out here and then an s i s j since I have assumed only one direction of magnetization one axis everything is a scalar here otherwise I would have to put vectors this would be operators then you have a quantum mechanical three dimensional problem for the spins which would be horrendous we are now doing just what is needed the essential part of the physics as far as magnetism is concerned for our purposes at the moment I will assume that each s i equal to plus or minus one and I am going to take traces now if I have n of these spins then since each s i is plus or minus one I have 2 to the n possible microstates whenever I say trace I mean trace over all the possibilities okay and of course we will put in the Boltzmann factor very shortly this thing is called an exchange integral and I already said that since this is a quantum mechanical effect and the interaction range is very short i and j have to be nearest neighbors otherwise you do not have an interaction at all that is generally denoted by putting a bracket like this this means i how many nearest neighbors do you have on a line on a linear lattice to what about a square lattice well a square lattice will have so if this side has a nearest neighbor 1 2 3 and 4 a d dimensional lattice hyper cubic lattice 2d in 3 dimensions you have one on top one below one behind one front etc you have 6 so you have 2d nearest whatever it is it is a finite number that is very crucial it is a finite number otherwise you are finished you will see as in a second you will see that is not true you have finished why do I say that you have finished if it is not true because you take the expectation value of this guy you have to average over all the sites over all possibilities this fellow will tell you for a given value of this j i j if both are plus it gives you a number 1 if both are minus it gives you 1 1 is plus the others minus it gives you minus 1 etc and you have to add over all these things but the point is if this is summed over all i and j 1 to n capital N and this is the interaction some fixed number let us suppose that it is the same number no matter what the range what the difference between i and j is then you see it is clear that this term is of order capital N squared because I pull out the j outside it is a finite number I have to sum over i and j and it is going to be of order capital N squared that violates the extensivity this has got to be linear in the system size okay. So an infinite range interaction with the same interaction strength is going to finish you off it has to drop down so that is not even a thermodynamic system once I put nearest neighbors there is no problem it is a finite number I go one step further and say this guy is a constant across the nearest neighbors in other words this interaction is the same as this is the same as this is the same as this and what is the job the task is to say oh this whole thing is put in a heat bar and I am going to find the partition function is trace this trace stands for saying that the set si is equal to plus or minus 1 every si takes values plus 1 or minus 1 and I have to find this whole thing now why is this such an impossible task by the way you can do this in one dimension on a linear chain with some suitable boundary conditions but you cannot do this in higher dimensions so easily even if you stop this field you still cannot do it in higher than two dimensions okay in 1944 once I have solved this problem of finding this exactly for this problem without this on the square lattice that is a two order force it is a very very complicated proof but now it is a standard proof it is a long it is a long derivation but it can be done explicitly and then the idea is to find out what happens when the number becomes infinite n becomes infinite as long as n is not infinite you cannot get any singularity because remember oh by the way I should also say the free energy is minus k Boltzmann t log z and then once you have the free energy all thermodynamic quantities can be computed their variances everything can be done in equilibrium statistical mechanics what is the complication the complication is this because you have that interaction you have to enumerate all possibilities the possibilities of this will affect what happens here but the possibilities of this will affect what happens here and therefore here and so on so it is an extremely complicated problem and there are very very few exactly solved statistical mechanical problems this is called the two dimensional easing model no one has succeeded in doing it with the field present and no one has succeeded in doing anything beyond two dimensions and three no one knows how to do this is probably not doable in this form anyway okay. Now our task is going to be much simpler we are simply going to argue that each magnetic moment in the ferromagnetic phase will see an effective magnetic magnetic field due to the fact that all its neighbors tend to be aligned in the direction of the field so this term is going to be replaced by an effective term we want to eventually find the expectation value of this to find the internal energy because that is the expectation value of the Hamiltonian with respect to this weight function the expectation value of this is essentially the magnetization what I called M so let me write that down definition in the units I have chosen I put mu I have subsumed it inside this constant inside the field etc so it is the magnetization per particle which is this quantity the internal energy will be the expectation value of this guy which is expectation here and then expectation here so the problem will arise because we do not know how to compute this expectation the M comes out here and by the way this is independent of I because in an infinite lattice every magnetic moment is the same as every other moment on the average but you have this quantity here that is bad news because it is correlated so what this assumption of mean field theory does is the same as the van der Waals approximation in the gas case is to say that this thing here if you look at a particular site I look at this particular site I then J runs over all its nearest neighbors and as far as this is concerned I can replace this SJ here by the expectation value of this SJ so if I make that approximation then and then I take expectation values but of course there is nothing to take expectation value here because already an average it factors out in this fashion what has been the effect of that the effect of that is to write this as equal to minus summation I sum H I S I an effective field at every point the minus sign where H I equal to external field plus J times summation J equal to nearest neighbor of I so essentially it is as if I put this spin here in a field which is not only the external field H but also the internal field due to the fact that these guys have got a magnetization on the average so it is been put equal to SJ here so it adds to the external field some effective field this is called the molecular field approximation this is the old famous vice molecular field approximation this changed this field to this but now we can go further we have already said J must be a nearest neighbor of I so let us put in let us say we are working on a D dimensional hyper cubic lattice because one of the things that we learn in critical phenomena is that things are very sensitive dimension dependent how many spatial dimensions are you in the effect of thermal fluctuations depends very sensitively on what dimensionality you are in we will see that in one dimension no ordering is possible because thermal fluctuations destroy this order completely in two dimensions it is possible in these class of models as you go in higher dimensions it gets more and more possible so the power of thermal fluctuations or quantum fluctuations for that matter to restore symmetry and stop ordering gets lower and lower as you increase the dimensionality because the number of neighbors increases and tends to order the system so all of phase transitions is a competition between the disordering tendency of entropy and the ordering tendency of the internal energy okay so you have F this is just a heuristic argument you have U minus TS this will tend to order that will tend to disorder and the competition between the two will decide the minimum of F that you are at low temperatures sufficiently low temperatures this is low and you win and the system has a minimum controlled by the internal energy on the other hand at sufficiently high temperatures that wins and you have a disordered state okay so that is really all that is happening but that is saying it in very very bare bones the rest of it is matter of detail to work things out but it is worth remembering this so what does this become there are 2D nearest neighbors and each of them has the same exchange constant D J and this is M itself so this is equal to H plus 2 D J M H yeah but it was for each side so H effective now let us go back and put this in our model by the way I wrote our model down by averaging the value of Mu up and down using the Boltzmann factor we could have written a partition function for it each of them has partition function e to the beta Mu H plus e to the minus beta Mu H the partition function for the full system is that to the power n that quantity is twice cos beta Mu H to the power n and then I can take the free energy minus N KT log Z and we could have done that too for instance you have said the magnetization M equal to it is after all a Legendre transform remember that you have the free energy which has a minus S D T minus a V D P or P D V depending on which free energy you are talking about since we are using H as the control parameter we are using P it is the analog of the Gibbs free energy so there is going to be a minus M D H precisely so I want you to take this guy and show that this is turns out to be unhepabolic exactly I just argued saying that if it is up the magnetic moment is Mu which down its minus Mu we took the average with the Boltzmann factor and did it directly but even go through the rigmarole by the way this is helpful because we can write a formula for the susceptibility which we are going to use so chi T equal to Delta M over Delta H at constant T and 0 field and 0 field so we can write it from this you write one more thing this is equal to minus 1 over N Delta M over Delta H so the susceptibility cannot be an extensive variable it is like a bulk modulus or something cannot depend on the system size this fellow is an extensive variable that is why I got an N here this is an intensive like I promised the second derivative of a free energy would always be some response function this case susceptibility we are going to use that in a minute so we have this and now let us put it back in our equation of motion so it says perpetually M equal to tan hyperbolic Mu over K T H plus 2 DJ that is our magnetic equation of state this is our Van der Waals equation if you like in this mean field here first order of business let us find out what is TC in this let us find out what is the spontaneous magnetization so spontaneous magnetization what are those intercepts M not is given by tan hyperbolic to be a I got rid of this H Mu I subsumed it in this in my notation so let me write it as 1 over so tan hyperbolic 2 DJ M over K Boltzmann M not I put H equal to 0 by the way this is an equation which is a transcendental equation you got a power FM here algebraic function you got a transcendental function so you cannot solve it in explicit closed form only numerically you can solve this but we can solve what have for what happens when H is 0 that those are those intercepts in the M versus H graph well it is equal to this and the question is does this thing have any solutions or not let us plot both sides so I plot M not the left hand side is M not itself the 45 degree line and this one is a tan hyperbolic guy now the tan hyperbolic has a slope tan hyperbolic X goes like X near the origin so it is linear near the origin saturates to plus minus 1 and 2 possibilities arise one of them is that the graph looks like this then the only solution is out here but if the slope at the origin exceeds 1 then the graph would look like this and then you have one solution 2 solutions and 3 solutions depending on whether the slope is greater than or less than 1 so you have M not equal to 0 is the only solution as long as the slope at the origin which is 2 dj over k Boltzmann t is less than 1 or t greater than tc which is equal to 2 dj over k Boltzmann so we found a critical temperature in this problem as you expect it has got to be proportional to the exchange constant J the greater this is the higher the temperature at which disorder will set in and it depends on the dimensionality through the number of nearest neighbors if you are more of them in a higher dimensional lattice even at a lower temperature you a higher temperature you would still have order now it is easy to show that these 2 solutions correspond to precisely those minima I showed in the free energy curve and this corresponds to an unstable group so we have our first formula which says tc is this guy here now we put that back here so I can write this magnetic equation of state as equal to tan hyperbolic h over k Boltzmann t plus now 2 dj over k Boltzmann is tc so M tc and now you see what happens as you lower the temperature further and further this as you lower the temperature the slope guy goes up more and more so these points are moving further and further away and that is exactly what happens when you plot this M0 versus they start at this point but they move further and further out towards plus 1 and minus 1 so this is our equation of state this is like volume this is like this is temperature and this is like pressure but the equation of states says p equal to something or the other so we would like to solve for this and see what is this thing look like I want h as a function of M and t that is not hard to do so let us write that as just one more line and then we will resume from here so this is equal to tan hyperbolic h over a beta let us just call it well okay h over k Boltzmann t plus tan hyperbolic M tc over t divided by 1 plus just use the formula for the tan hyperbolic of a plus b and then let us so this is equal to M so let us solve for this it says tan hyperbolic h over k Boltzmann t times 1 minus M tan hyperbolic M tc over t this guy moved this there is equal to M minus tan hyperbolic that is the solution so you have an so h is tan hyperbolic inverse of this whole mess okay let me see if that is correct minus M tan hyperbolic that is okay and then that is equal to M minus tan hyperbolic that is okay so now we are interested in the critical region we are interested in exactly what happens at the critical point now I am going to stop here but I am going to tomorrow assume that you have already done this algebra so show that as you approach tc from below show that it goes like a square root then on the critical isotherm that you can do very easily by putting tc equal to t then all this goes away and you have to show that h is proportional to t cube M cubed second with may be a different coefficient so all you have to do is to solve this self consistently in each case and what this means is that the susceptibility will look like this chi t versus t and this is tc it diverges at tc so we will start from this point.