 So, now we have come to the very important part of magnetic interaction known as exchange interaction. Basically, exchange energy in a magnetic lattice, usually write it like this H exchange is nothing but j ij which is an exchange integral si dot sj, but si and sj are the local spins or magnetic moments in a system. So that means we have this lattice I showed you in a figure earlier also and this can be between nearest neighbors or it can be between next nearest neighbors and that exchange energy. Please note the term exchange energy is given by this is the exchange integral it is known I will show why it is exchange integral si dot sj in a magnetic lattice this is responsible for the alignment in the order state. So you can see this ij as I showed you just now that can be nearest neighbors often that works next nearest neighbor it can be directional also. So I will discuss later at least briefly take you through the various magnetic interactions that are possible in a solid. I must say this nature of interactions are so varied and so interesting that magnetic neutron diffraction is possibly the most popular tool to understand magnetic interactions and the structures and right from the beginning of neutron scattering till date it has been used extensively to understand the interaction. But the exchange energy there are atomic spins so the question comes is it a dipole-dipole interaction because I can understand that in a lattice at a side there is one magnetic dipole and then another side maybe nearest neighbor maybe nearest neighbor there is another dipole and is it a dipole-dipole interaction and the answer is no magnetic exchange energy is not a dipole-dipole interaction because the interaction between two magnetic dipoles is very weak goes as 1 by r to the power 3 and also it comes to if it comes to energy it will be few kts whereas the room temperature is equivalent to 30 millilectron volt and that can be few MEVs. So this is so much smaller than room temperature that if it is only dipole-dipole we will not have any magnetic alignment at room temperature we cannot have whereas iron gates aligned very early iron curie temperature the alignment from paramagnetic to ferromagnetic phase is very high iron, cobalt, tickle they are all room temperature that is because the root of exchange interaction is not in dipole-dipole interaction but it is electronic so I will come to that now so it is electronic and it is much stronger than magnetic dipole-dipole interaction. So I have written down the I have just taken the simplest example hydrogen like molecule so you have got an atom A and an electron 1 and atom B and electron 2 so the interaction energy between these two as I have written down it is the interaction between electron electron nucleus nucleus nucleus electron and nucleus nucleus so I will write it down as I shown here so VAB between two sides so I have got atom A and atom B this is A and B and this is one electron so hydrogen like but this clarifies the concept so it is equal to E square then it is 1 R AB plus 1 by 1 by 1 by 1 by 1 by 1 by 1 by 1 by 1 by 1 by R 1 2 minus 1 by R A 1 minus 1 by R B 2 can you go back R AB R 1 2 B 1 A 2 sorry I made a mistake here this is the distance between the electrons and A that is already the A 2 and B 1 so electron nucleus nucleus nucleus electron electron nucleus electron nucleus but this is purely coulombic in nature our J exchange it is due to electron interaction but not purely coulombic it comes from exchange interaction it comes from the the exchange between electrons between the two I can say it is due to Pauli exclusion principle that exchange of electrons between one and two gives me the exchange energy so the coulombic term is there and the exchange term is there and I have written down this exchange term so please note that electron one is there then I have V AB then we have exchange electron one goes to two two goes to one and this is integrated over interspace this gives me the exchange energy interestingly the origin of this lies in Pauli exclusion principle you can see here all I have done is actually I have just put electron A at one and electron A at two and I have taken two states where the electrons have been exchanged this is possible for identical particles and in this case following fermi direct statistics a similar thing we have also you might have seen it when you write down the stator determinant for a n electron system so J I J is calculated it is a calculation of this integral where we have exchange we have psi A star one psi B star two which is exactly what you see here then the interaction energy and between these two states where the electrons are exchanged and this interestingly there is no typical classical force term that I can bring in to explain the exchange energy it is simply because of the Pauli exclusion principle of two identical electrons at two atomic sites I can exchange the two and create a new state and this J is basically the evaluation of the between the two states of the exchange interaction V AB if you remember even in neutral diffraction I did that earlier when I took a plane wave state another plane wave state and between the two I put the interaction potential it is exactly same only here the state of the system is psi A to psi B one which is after exchange and psi A one psi B two before exchange complex conjugate and it is I can say in the typical formula quantum mechanical formula and this term is appearing due to pure exchange interaction coming due to Pauli exclusion principle this J I J calculation is involved what I showed you earlier the actual physical extension of the electronic wave function from here you can see that this J I J can be calculated for any system depending what are the electrons that are responsible for the exchange and then that has to be much higher than what is can get from a magnetic dipole-dipole interaction so this is the origin and magnetic interaction is much much less than KT but J is can be as comparable or higher compared to KT and that dictates the alignment so this is what I showed you and interestingly because this depends on the overlap integral for the electronic wave functions so this J it can change sign depending on whether we can take the atoms closer or further so the if you consider the exchange integral J E for low values it is negative and then it is positive now if J E is sorry is less than zero it is less than zero because at the way I have written you can see if it is less than zero then SI dot SJ should be negative to lower the energy and antiferromagnetic alignment is preferred whereas above a certain distance because the electronic overlap integral changes sign it becomes positive and when J is greater than zero we get ferromagnetic order so I just mentioned here simple calculations can show that if RAB in case of iron it is 3.26 cobalt 3.64 nickel 3.94 so these are all ferromagnetic because we have gone above zero part of the exchange integral whereas chromium where it is 2.6 it is not a ferromagnetic material because the J is negative in its case now let me quickly introduce you to the bulk properties of the magnetic material that is important to know because the bulk measurements will tell you whether it is a ferromagnetic or it is a ferrimagnetic or it is an antiferromagnetic in a very what should I say in a very macroscopic manner so let me just quickly write down the field that we calculate the field inside a magnetic material is given by the applied magnetic field plus into the magnetism induced in the system if it is M then we also know that susceptibility of a material is nothing but M by H that is the magnetic moment induced per applied field so that when I write that then B is becoming equal to H plus 4 pi chi H equal to 1 plus 4 pi chi H equal to 1 plus 4 pi chi H equal to mu H there where mu is the permittivity of the medium so these are very good relationships you are well aware of but I just write it for completeness sake so that when later we discuss there should be any confusion so now coming back to ferromagnets and you saw the chi the chi for a ferromagnet follows something called Curie Weiss law Curie Weiss law says the chi with temperature goes as C upon T minus T C and this T C is known as Curie temperature so you can see at T equal to the susceptibility diverges to infinity and for antiferromagnetic order chi goes as C upon T plus T N where T N is known as Neal temperature so neutron diffraction is done below Neal temperature or Curie temperature for the order state and we also do neutron diffraction above the ordering temperature and let me just tell you that in case of ferromagnets because it is chi is equal to C upon T minus T C if I plot C or 1 by chi 1 by chi is equal to T minus T C by C so you can see that it is a straight line 1 by chi plotted at this temperature cuts the axis somewhere you can see it is T by T C T C by C it is a constant which dictates the Curie temperature not exactly equal to Curie temperature Curie temperature it cuts and at Curie temperature it comes to 0 and similarly for a antiferromagnet because chi is equal to C upon T plus T N so the 1 by chi plot hits it the negative side of the axis and this is a T N minus something T so if I write though I am not allowed to write temperature is negative we need absolute scale but it cuts at minus T N and we get the relationship T minus T N T plus T N so this is 1 by chi plot for a ferromagnet and antiferromagnet like chromium MNO and ferromagnet ferrimagnet we will have other ordered states we will have this kind of behavior so we can measuring the susceptibility we can at least understand whether it is a ordered state is a ferromagnet or an antiferromagnet but on the other hand if it is a ferromagnet state the ordered moment as we raise temperature you can see the magnetic moment it undergoes a second order phase transition here M is the order parameter for a magnetic phase transition from ferro to para so this is the para magnet is a ferromagnet in para magnet it is 0 for the sample whereas in the magnetic state you can see the magnetic moment rises to a high value the second order phase transition the sources are mentioned here so now we do experiments here as well as here we shall get to when I discuss the neutron diffraction and before that let me give you the idea about the hysteresis loop because even before we do any kind of diffraction studies when you make a sample the first thing we can do is to measure the hysteresis loop so the hysteresis loop though we have I am sure you have done it let me just bring it to you so here it is B versus H so first if I take a piece of iron let us say it is non-magnetic because its demands are all misoriented when you heat it when you sorry not heat it I am sorry when you raise the temperature the with the applied magnetic field the induced magnetic field also increases it reaches a saturation saturation saturation but when I reduce the field the magnetization decrease decreases but when I reach 0 0 magnetic field it does not go back to 0 from where it started because I have already aligned the domains and they remain aligned now if I start applying a magnetic field in the negative direction again I can force it to go to 0 at some negative field H and then the same thing repeats in the other half and this is how the hysteresis loop looks like and we know that the area of the hysteresis loop B or M equals to m dot th if you say m is equal to chi H chi H so chi H dh and the area under the curve is half chi H square this is the in one cycle this is the energy spent in the system and this is energy actually if you take a ferromagnetic material through cyclic magnetic fields you will find this in terms of heating of the sample so this is the hysteresis loop of a ferromagnetic so if you increase from 0 magnetization goes to saturation magnetization goes back to a value which is known as remnant this is remnant field where you just bring the magnetic field to 0 applied magnetic field but still some value of m remains that is why remnant after that when I go to negative it goes to 0 magnetization which is known as a cohesive field that means to bring the to bring the magnetization to 0 the cohesive field then when you go increase in the negative side just this part is followed again it goes to negative side saturation and then this cycle repeats so this is the whole cycle known as magnetic hysteresis loop hysteresis because you lose power every time you go through one cycle of this hysteresis loop and it is evident when you do this that is how the transformer if you have some magnetic material inside a transformer it gets heated up but now this hysteresis loop also has several properties I have drawn one for a ferromagnetic which is a hard ferromagnetic if I consider which is a soft magnetic material then this loop will be much smaller or smaller so that means the difference between a hard ferromagnetic and a soft ferromagnetic is that for a hard ferromagnetic it is larger for a soft ferromagnetic possibly it will be much smaller reason being the area is smaller for this soft ferromagnetic and so the heat dissipated is smaller than the hard ferromagnetic if it is a diamagnetic material then the induced magnetization small but will be in the negative direction so in case of a diamagnetic material h versus m or b what a plotting it should show a linear inverse direction that means for h positive it is negative and for h negative it will be positive so it will oppose the applied magnetic field and for antiferromagnets where the average s is 0 for an antiferromagnet for an antiferromagnet again it should show a very narrow loop very narrow loop settle on the 0 so there are various cases where even this loop can shift left and right and you can say there is something called interface exchange interaction so those will come to later but the fact is that even from the hysteresis loop of a material you can at least the order state you can make some idea whether it is ferromagnetic or antiferromagnetic or a soft magnet or a hard magnet from this so this is the integrated diagram of various kinds of materials for a paramagnetic material it is almost similar like diamagnetic but the fact is that paramagnetic materials do not oppose the field and as you apply field the magnetic energy increases and somewhat tries to align the magnetic moments in the direction of the applied field so you get a loop which looks like the green loop so I have shown you the first experiment that one will be doing when one makes a sample that is measurement of the magnetic hysteresis loop and you can see that there is a saturation magnetization sometimes the magnet is difficult to align and in that case your hysteresis loop even if you apply a high field will not reach saturation and before that you have might be doing coming back in negative in the field direction because this is this is a hard magnet and even large magnetic fields are required to align them compared to what you have may be in your experimental facility so next lecture I will briefly mention the various magnetic exchange interactions which are important for understanding the alignment in our system so there are direct exchange which I discussed just now but there is something called super exchange there are indirect and double exchange interactions importantly there is something called RKKY or Ruderman Kittel-Kassoua interaction which is through conduction electrons and there is an interesting kind of interaction what known as DM or Gialoshinsky-Moria interaction and dipolar exchange interaction I will briefly take you to some of these interactions which are important in case of neutron diffraction and the final structure so with this I end this module