 So, what we have done so far in magnetic materials is that how an atom gets magnetic moment, what are the reasons for that? How to calculate atomic magnetic moment? So, that defines paramagnetism of an atom, when it is extended to a solid it gives rise to paramagnetic susceptibility. And we also saw what happens when it is a molecule, the game is slightly different as we have seen the contradiction between hydrogen atom and hydrogen molecule versus oxygen atom and oxygen molecule. This we have seen this the effect of or response of the magnetic field that we have been talking about is somewhat different when it comes to a metal that is what somebody was asking in the morning which I answered that is giving rise to different paramagnetism called the Pauli paramagnetism which I am not able to talk today. Let me see whether I can include in the last lecture. So, now we are in a position to see what happens to the magnetism of a solid, the real bulk solid that we know. So, basically we are looking at the solid state magnetism. Our journey started with electronic part, went and we built up an atom consisting of electrons, the contribution giving rise to atomic magnetic moment, the Hund's rules giving you the atomic magnetism atomic magnetic moment. When we went to molecules where we saw that molecular orbitals, bonding versus anti-bonding, filling, degeneracy and Pauli's exclusion principle all these things together determine what kind of magnetism whether it is going to be paramagnetic, the molecule is going to be paramagnetic or diamagnetic this we have seen. Now, the next obvious thing is to see what happens in the case of solid state bulk magnetism. Here we as I think mentioned earlier we have to distinguish between metals and insulators. In this lecture I will only talk about insulators. In the case of metals the problem is as we have seen in the morning you have this bands, you have this free electron kind of picture coming there and the magnetism the response of the magnetic field is somewhat different. So, let us proceed with what happens in the case of magnetism or the magnetic response of an insulating material like ferrite for example. So, now we are trying to look at you know what happens in the case of paramagnetism in the case of an atom. Now the same thing can be extended to a solid insulating now onwards solid means insulating solid most of the oxides are actually insulating solids. So, in the case of Hund's rule we found that it is a ground state that we obtained from the rule of J equal to L minus S or L plus S depending on the situation less than half will more than half will case, but that was a ground state. Now in reality we will be interested in finding out at finite temperatures what is the magnetic response for that. So, that is what. So, this is the ground state that we want obtains when you actually have the paramagnetic kind of a situation. When you apply a magnetic field as we have seen this ground state J split this is a degeneracy of 2 J plus 1 that is 2 J plus 1 different M J values all of them have the same energy in the absence of a magnetic field. When you apply a magnetic field the degeneracy is removed you have M J levels 2 J plus 1 such values will be there and this is what is known as this is known as a ground state to multiply because these all levels are actually degenerate in the absence of a magnetic field and hence this is called the ground state multiplied. So, the Hund's rule actually gives you the quantum number corresponding to the ground state multiplied. This multiplied splits under the influence of an applied magnetic field like the one which is shown here. So, at finite temperatures what is going to happen is these levels get occupied and the magnetization or the susceptibility will be determined by what kind of population what kind of distribution that we talked in the case of solid state also what kind of a distribution is going to show give you the physical property here which is in this case susceptibility magnetic susceptibility. To do that I have just given all the steps you can go through it some of the important things I will talk about it. So, the general principle is like this you start with what is known as a partition function you have studied the partition function partition function is defined like this e to the power of the energy of a particular level divided by k B T and this is summed over all the possible levels in this case the levels are designated in terms of quantum number m j the values of which are minus j to plus j. So, this summation is nothing but a partition function and you can make a substitution like this for our mathematical convenience and once I know the partition function actually I can calculate many things. So, in this case the most important thing is the average value of m j because you have different populations and different m j levels. So, what is the at a given temperature equilibrium situation as we mentioned in the earlier lectures what is the average value of m j because average value of m j will give you the average value of the magnetization and then you can proceed with the calculation of magnetic susceptibility. So, m j the average value can be calculated like this using this partition function related formula this actually can be shown that it is 1 by z times dz by dx where z is a partition function. So, I can substitute all these finer details of mathematical steps I am not showing. So, if there are n at ions per unit volume this now is a solid not just one atom you take a unit volume you have n atoms. So, this has to be actually multiplied this is small n. So, this actually gives you the magnetic moment because the m j as you have seen in the case of atom m j value will determine what is the magnetic moment only other thing is that the g factor and the mu b factor in this case n will come g will come mu b will come as expected instead of a single m j value because of the thermal averaging you have to worry about an expectation value or the average value of m j that is what is shown here. So, when I substitute I get this expression which is not very difficult all the steps are given here. So, little bit of mathematical manipulation some of these expansions are needed I will not worry about all these things we can work out the details ultimately this gives you this magnetization is actually given in terms of a function this b j y this is called a Brillouin function this Brillouin function has a particular form as shown here this is a hyperbolic function that is shown here and the magnetization the so called the saturation magnetization m is given by n times g times j times mu b j is the maximum value it can take along a particular direction because when you tell magnetization it means that you are talking about a particular direction usually the direction of the quantization direction or the magnetic field direction so that is what is this one. So, this is the magnetization the system is going to have show when you are applying a magnetic field to a magnetic solid of this kind. So, Brillouin function has certain properties that is what is shown here once I know the magnetization I can find out the susceptibility is m by h or I have been using h and v quite interchangeably here there is no problem of confusion with the Hamiltonian. So, I will better I will use h in place of b h stands for the applied magnetic field. So, this gives me this expression this you can see that j into j plus 1 is coming which of course, it came as an eigenvalue for this j square operator. So, this is given as n this is a number of ions per unit volume number of ions because it is a solid now mu effective square this is called a mu effective square it is a kind of maybe it is a magnetic moment which actually is defined here this is given by g times square root of j into j plus 1 mu b. We have been talking about the z component of the magnetic moment as g times j times mu b, but this is the full moment once I show earlier I showed you this is a length of the full arrow corresponding to the total angular momentum and now the corresponding magnetic moment this one. So, this is a full magnetic moment which will never get aligned along the applied field direction because of this quantization rule and this is called effective magnetic moment. What you actually measure as the component that aligns with respect to the magnetic field that is the z component of this one which of course, will be g times the component of this will be j and this mu b. So, the whenever you tell that you are measuring the saturation magnetization what you are measuring is g times j times mu b and of course, in this case n times that is a number of ions. So, this is one has to keep in mind this effective magnetic moment is actually defined by this. Now, using this so, we can see this particular form this is n mu effective square divided by 3 k b t k b is a Boltzmann constant. So, here we can see that this has a form chi equal to c by t where if I define c as this one so, n effective square divided by 3 k b t I think it should be small l n or capital n consistently there is a mistake I will correct it. If you take a one mole of the substance naturally the number of ions that will be present will be Avogadro number. So, n becomes n a the Avogadro number mu effective square divided by 3 k b t. So, this is a chi n and once you can actually plot and this is something which happens when you assume that the paramagnetic solid insulating solid all these magnetic moments of these different ions they are not interacting they are purely isolated they do not talk to each other they are completely independent there is no coupling between them that is why you are simply summing over without really worrying about any kind of interaction among them. So, that is what is given by this one, but in reality what happens is there will always be an interaction that is what happens when you take into account that interaction this the so called curie law which actually is given by this one modifies to what is known as a curie voice law where it becomes c by t minus theta p theta p essentially is having dimensions of temperature and it kind of gives you a measure of the interaction among the different atomic trionic magnetic moments. So, this is a case when there is no interaction when here we actually make it more realistic having some because you have dipole sitting within a volume. So, definitely there will be at least a dipolar interaction magnetic dipolar interaction that is what is represented by this theta p having dimensions of temperature. Now, if you see if you compare the effective magnetic moments so what we can do is I think I will just go one step. So, what happens if I do a temperature variation of susceptibility measurement one can see that it will have the 1 by t form that is what is chi versus t it will be in this form. If I take 1 over chi versus t it will be just proportional and in the ideal curie situation when there is no interaction you will see that this will exactly go through the origin, but if it is not if there is an interaction it will actually have a next intercept. So, it will go something like this with a positive intercept here that is actually showing how much interaction is there among the atomic or ionic magnetic moments and of course, if you plot chi t versus t it will be a constant. Now, what one can do is that if I plot for example, this way from the slope of this actually I can find out what is a curie constant c from the curie constant actually I can find out what is my mu effective because curie constant essentially is determined by the other than the constant it is determined by the mu effective. So, such a calculation one can actually measure experimentally measure the this plot which I showed you and from that I can experimentally calculate and tell what is a effective magnetic moment. I can also do one thing I can take the electronic configuration and I can find out what is my l a s j assuming that the Hund's rule is correct here and try to find out what is a j value then I find out substitute a value and see that it is n times g times square root of j into j plus 1 mu b if I do I can do a calculation. So, that is I can do a Hund's rule calculation and compare it with an experimentally observed mu effective then if I do this for one set of materials that is the rare earth materials rare earth based materials which I talked about that is the one row which is below the periodic table the so called four of elements if I do what is your what you are getting is something like this. You can see that the Hund's rule calculation that is what is the square symbols are showing and the experimental ones experimental means I measure chi the susceptibility and from that I find out a slope from that I get curie constant from the curie constant I find out what is the effective magnetic moment if I do that is the circle. You can see that very good match in most of the cases except for one or two cases very good excellent matches there between the two what does it mean this tells you something it means that at least in this series of elements it does not really matter whether you are in an atomic situation or in a solid case especially in an insulating case why is that this is very important this is because as I mentioned sometime I think the earlier lecture yesterday or before yesterday what is happening is these rare earths the magnetic electrons which are the four of electrons the unpaired electrons are all in the 4 f shell which are very much inside they are not in a talking term with the other ionic environments. So, what happens is they are kind of independent they are able to retain the atomic character in the way it happens in an isolated atom. So, if you take different atomic ionic moments even though they are part of a solid yes if it is in the rare earth insulating materials it is more or less identical to what happens in the case of a single isolated rare earth atom ion. So, the atomic character is more or less retained even when it has formed a solid in the case of rare earth compounds that is what is shown by this excellent agreement between the Hund's rule calculation and the experimental determination. Remember I also always told that Hund's rule should be used only in the case of atomic magnetism when it comes solid you should not use, but here you see that that statement has to be modified at least in this particular series of elements. The reason why I still insist on telling that Hund's rule must be used only in the case of atoms you will see very soon. So, in general better not use the Hund's rule as the tool to calculate the magnetic effective magnetic moment of a solid it will give you problem in general in this case being a very very special case you do not see that. So, some of the values are shown here the molar susceptibility values are shown here the diamagnets and paramagnets as I showed you the values are very small. Now, so this is what is shown in the earlier plot like the match between the theory and experiment being very good the same thing is shown here. So, what is shown here is l value as value j value the Hund's rule calculation and the experiment and theory you can see that very good match between the two. So, this column and this column the agreement is excellent in the case of the so called rare earth elements rare earth compounds solid compounds insulating rare earth compounds. The interesting point is the other series which is magnetically important from the periodic table is the transition metal series where iron cobalt nickel those things are there I can have compounds insulating compounds solids come using these elements like for example ferrite Fe2O3 is an example of that MNO is an example of that it is an insulating compound it is made up of or the magnetic element in that case is transition metal non-dryer earth. What is happening is I have written all the details here, but I will talk about this. So, same thing if I do you can see that the experimental moments are shown here the Hund's rule calculated moments are shown here you can actually see that so this is the Hund's rule calculated value this is experimental value you can see that there is a big difference in most of the cases there is a big difference for example you see here it is 0.77 the experimentally it is 3.88. So, this is a Hund's rule j value using the Hund's rule j value I am comparing with the experimental value there is a big problem. So, it is unlike earlier case where agreement was excellent when I calculated the j and found the comparison between experiment and theory here it is exactly different. On the other hand interestingly if I do if I do not worry about orbital I if I take that L to be 0 everywhere if the orbital angular momentum or the orbital magnetic moment contribution is completely killed if I assume that way if I instead of j that means if instead of j if I simply take as s in which case my g factor must be 2 because pure spin case g factor is 2 as we have seen if I do that the calculation will give me this thing so instead of j times j into j plus 1 I will use 2 times square root of s into s plus 1 j becomes s. So, if I do that I will get the last column and you see the comparison between the last column and the last but one column 1.70 1.73 3.85 3.87 you see the comparison 0.77 was far away from 3.85 whereas when I do not take the orbital into account it is actually 3.87 very very close to 3.85 which is experimentally observed. So, you can see in all these elements what is happening is the system is behaving as if it is not able to contribute to the magnetism from the orbital part. That means orbital angular momentum is somehow disappearing that because of that orbital magnetic moment is also not able to contribute to the total magnetic moment of the solid. It is a big big difference compared to atoms big big difference compared to rare earth compounds insulating compounds. So, in that is why I was making this statement earlier the Hund's rule using this Hund's rule to calculate or talk about or compare the susceptibility is curie constant and effective magnetic moments can in general go wrong and that is what you are seeing in this example. In the case of rare earth somehow it is working the somehow I will tell you that in the case of transition metal it is completely collapsing. Now, before we try to see what is the difference between the two why this happens with the disagreement is there in this case and the disagreement is not there in the rare earth case you can see what happens what is the electronic structure what happens to the magnet the so called the magnetic electrons in these two series of compounds. As I mentioned just now in the case of rare earth materials the magnetic electrons are the four of electrons four f electrons which are very much inside the shell. On the other hand in the case of transition metal the magnetic electrons are the 3D electrons in the series they are much more exposed than the four f electrons and this is the only thing that one can think at the moment. So, the difference between the four of series and the 3D series which of course is very important that as we are seeing in the disagreement versus agreement one should immediately think about the difference in the electronic part electronic configuration wise the difference is this one one is it is and remember as it is more and more close inside core that means the atomic character is going to be predominant once you are moving away from the nucleus as you are going out and out that means 3D is an outer case which means that atomic character is not expected to be correct that is what is seen here. So, when the atomic character is not there you cannot use J as your quantum number it is actually contributed only by S and not by L further. So, this what does it mean? It means that when you have transition metals whether you have a situation the number wise whether you are total L is becoming 0 or not if it is a transition metal irrespective of the number of 3D electrons you are going to have L equal to 0 somehow L is made to be 0. So, that it is a pure spin magnetism that is coming into picture that is why when you take J it does not happen it does not agree, but if you take purely S alone you are able to get a good agreement between the theory and experiment. This is called quenching of orbital angular momentum or the quenching of orbital magnetic moment. So, it is a pure spin case that you are seeing in the case of transition metal compounds especially in insulators. This does not happen in the case of rare earth compounds as we have seen J was working all right. So, this is a very important difference compared to the transition metal between the transition metals and the rare earth compounds both insulating as I mentioned. So, having said this now we have some idea about two different classes of paramagnetic solids all are insulators one is rare earth waste one is transition metal based and there is a difference between the two in one case there is a problem of quenching of orbital angular momentum in one case it is not the rare earth case there is no such quenching. Now, let us go and look go to the next thing which is more applied more important more useful materials the ferromagnetism. How do you see that? As I mentioned earlier a ferromagnetic material can be kind of thought out both in the form of paramagnets being building blocks using those paramagnetic entities you can create ferromagnetism and the characterizing features of ferromagnetism are these they are all very important they are somewhat connected also first of all they they have what is known as a spontaneous magnetization spontaneous magnetization means in the absence of a magnetic field also unlike a paramagnet paramagnet does not show any magnetism unless you apply a magnetic field. But here ideally you are supposed to get a spontaneous magnetization that means magnetization in the absence of a magnet applied magnetic field. They have what is known as a curie temperature below which the material is ferromagnetic and above which it is paramagnetic it is supposed to have. So, somebody must be there internally so that the spontaneous magnetization can be explained that somebody is nothing but what you call initially people started calling as an internal field later people started calling it as a molecular field finally now we know that it is nothing but and what is known as an exchange field I talk about it later. So, that is something which is characterizing a ferromagnet then you have to see what is an exchange interaction ferromagnets are known to have magnetic anisotropy anisotropy means it is a direction dependent properties they have domains which actually tell you that even though you talk about spontaneous magnetization as a first point in reality you do not see the spontaneous magnetization because they have domains which actually does not they do not allow the spontaneous magnetization to be seen in a bulk scale and the most important property of ferromagnets namely the hysteresis is the last point all these properties are some or other connected and they all characterize a ferromagnetic material this is a usual plot of M versus T and this is a TC where the magnetization essentially changes abruptly and here it is ferromagnetic here it is paramagnetic. So, what is shown here is magnetization and remember usually a ferromagnetic state is characterized in terms of magnetization not a susceptibility. Susceptibility is something which is used essentially to characterize paramagnets as long as the material is in the ordered region the best thing to talk about best parameter to talk about is the magnetization which is a magnetic moment per unit volume as we defined earlier that is what is shown here. Now, let us quickly go and see what happens why is this particular set of materials are so called ferromagnets have the so called internal field or the molecular field. What people thought in the beginning is that you have this the magnetic dipoles as in the case of paramagnets only thing is that they are close by maybe now they have that magnetic dipole dipole interaction which actually aligns them that is a usual thing that you would have studied in electromagnetism also when you put two dipoles there is an interaction the minimization of energy will actually align them to be in the same direction. So, people thought that must be the dipole-dipole interaction that must be responsible for getting this kind of an order that is a so called ferromagnetic order which is essentially represented by the value or the strength of this curie temperature Tc the higher the curie temperature the ferromagnetic strength is larger. So, people thought try to explain the Tc values in terms of this interaction that is so called the magnetic dipole-dipole interaction and it was found that you cannot explain Tc's of the order of 1000 Kelvin for Fe and it has a Tc of something like 1050 Kelvin which is very large about the room temperature that is why it is room temperature ferromagnetic. So, people realized that this kind of an interaction is completely not one cannot use this interaction to account for the Tc of these ferromagnets. So, this is completely ruled out this is a very very weak interaction which at the most can give one Kelvin Tc nothing more than that and we are talking about 1000 Kelvin. Remember this is a magnetic interaction what really is responsible for this large Tc values or the strong ferromagnetic interactions in solids is what is known as exchange interaction is very important. I will not really go into a solid to explain this this is again quantum mechanics which you have been talking about in various forms. Let us take the simple example of hydrogen molecule which can be extended to solids hydrogen molecule is this we have a hydrogen atom here another hydrogen atom this is one nucleus this is the corresponding electron similarly for the next hydrogen atom you have this one. You can take this whole system and write down forgetting about the nuclear part I can write down the two electron system considers as a two electron system I can write down that total wave function which actually consists of the spatial coordinates that is R1, R2 and the spin coordinates as I told you yesterday it is S1 and S2. You can write this in the form of purely orbital part that is here and the purely spin part this is a spin wave function this is a orbital wave function. And remember they are electrons are fermions when you exchange two fermions the property that is very very important from quantum statistics is that the wave function should change the sign. So, when you change R1 to R2 to R2 to R1 the wave total wave functions sign should change this psi remember this is capital psi this is small psi the total psi should change its sign when I flip between the two. And remember these two electrons are indistinguishable because it is a quantum system. So, this is something which is very important. So, the total wave function must be anti symmetric that is what is meant by the sign change anti symmetric this has to be guaranteed. So, this can happen in two ways because you have a product you can it can happen in such a manner that suppose this is not changing the sign when you change R1 to R2 that is fine. But then this has to change sign you have to have a plus minus combination or a minus plus combination. So, that the net result is minus. So, what happens one can actually write down I can write down the spatial part of the wave function using the single electron wave functions that is psi A1 that is first electron associated with the first nucleus second one associated with the second one. And I can make a anti symmetric combination that means I am putting a minus sign here and I take this R1 and R2 R flat. So, what does it mean physically? It means that your R1 and R2 these two electrons since they are quantum particles you cannot tell that this is R1 and this is R2 they are not distinguishable they are identical as we saw yesterday. So, if you are telling that at some instant R1 that is first electron is with nucleus A you cannot rule out that the first electron is sometime with nucleus B. So, that is what is represented by psi A1 psi B2 R2 and the reverse one here the Ulta one here. And when you are putting a minus sign here this makes the total wave function anti-symmetry because if you change you are the changing the sign of the wave function. When this is anti symmetric the corresponding spin part must be symmetric a symmetric spin wave function corresponds to what is known as a triplet. A triplet actually corresponds to a magnetic case in this case it is just two electrons once you bring in magnetism that corresponds to a magnetic state. On the other hand I can actually do the same thing with a plus sign which means that these when you flip between R1 and R2 between A and B the sign of the wave function is not changing the total wave function is not the sign of the orbital part is not changing. In that case definitely the spin part has to change in sign in such a case it is called a singlet the singlet means it is non-magnetic. So, you can see in this case is triplet in this case is a singlet. Now what happens is you have two electrons you have some separation between the two in the normal case when you have two electrons classically speaking if you want to find out what is the interaction between the two the electrostatic interaction between the two because I am not interested in the magnetic interaction now why I am interested in the electrostatic interaction between the two the electrostatic repulsion between the two the classical Coulomb interaction. If I do I have to find out what is the E square divided by R 12. R 12 is a separation between the two electrons. So, I have to calculate E square by R 12 that is a Coulomb repulsion. In quantum mechanics if I want to do that I am in the quantum mechanical regime I have to calculate the expectation value for that term which is E square by R 12. So, I have to find out E square by R 12 for this state and for this state. When I calculate that is a very important point when I calculate the expectation value of E square by R 12 which actually happens to be as we did earlier it happens to be the energy shift due to the perturbation calculation. As I told you the first order correction is simply the expectation value of that extra term. The extra term right now is E square by R 12 term. So, when I calculate for these two terms I see that it is different and this difference gets reflected between the two here. So, if I straight away if I go and all these steps are written here if I have S 1 and S 2 are half each you can have two situations. When they are symmetrically combining that gives rise to your S equal to 1 that is a triplet state because I can have values of M s 1 0 minus 1 that is my it is triplet 3 and if it is 0 it gives you only the singlet. So, the electrostatic energy for this state and electrostatic energy for this state is different. This is not what is expected in classical electromagnetism. So, when I can actually calculate and show what happens, but better to show this picture. So, when there is no electrostatic interaction between the two electrons you have some energy the separate energy added together hydrogen atom 1 hydrogen atom 2 take the energies add them together that is what is represented by this horizontal line. Now, if you bring in the usual classical electrostatic interaction without really worrying about wave function and things like that I have this charge I have this charge E and E square by R 1 2 R 1 2 is a separation between the two I get a normal Coulomb term E square by R 1 2 term which is positive is a repulsive so energy increases. So, that is why this is at a higher level. Now, if I do the same thing quantum mechanically which is a requirement at this point of time because of the difference between the two states what we found is when I take the triplet state the energy is going to be slightly smaller compared to this one. So, that means in this case when my J is called the exchange term this J as long as J is positive what you are seeing is you are having a preference of triplet state over the singlet state and remember singlet is non-magnetic triplet is magnetic. So, what you are seeing is an interaction which you have completely forgotten earlier a quantum mechanical analog of electrostatic interaction is responsible to stabilize a triplet state over a singlet state. A triplet state in general is nothing but a magnetic state. So, in this case you are seeing an example where your magnetic state is stabilized compared to a non-magnetic state because of an electrostatic interaction. It is non-magnetic interactions very very important point. Remember I started with a magnetic dipole-dipole interaction which a purely magnetic interaction it has no role in determining what kind of a magnetic order at a high temperature situation is going to happen. On the other hand the real issue the real reason why you are getting a magnetic state stabilized over a non-magnetic state that is a triplet state is stabilized with respect to a singlet state in the case of a simple example like hydrogen molecule is the electrostatic interaction quantum mechanically. This quantum mechanical electrostatic interaction or it is a spin dependent electrostatic interaction or a spin dependent coulomb interaction. This is a spin dependent coulomb interaction that is what is stabilizing magnetism here over the non-magnetic state. This kind of a thing can be expected for a solid also only thing is that you have to expand little more not just two electrons two atoms you have to expand it in a much much larger number of neighbors and so on. So, this idea is very important. So, the first thing is electrostatic interaction spin dependent electrostatic interaction is responsible for the ferromagnetism. On the other hand in a general sense if my j is negative one can show that the anti ferromagnetism will be lower energy state compared to a ferromagnetic state. So, the ground state will be determined by the sign of your exchange term j actually the exchange integral I will not go into the details it is essentially the exchange term j. So, this is a usual coulomb term which is therefore, both the states equally that is why it is equally displacing the energy to the positive side but then there is a positive shift or a negative shift depending on the spin state the total spin not a spin state in the sense that the total spin triplet has lower energy singlet has higher energy. So, this is a preference of magnetic state over the non magnetic state this is nothing but ferromagnetism if you expand for a solid. So, that is what is shown here. So, when you have s equal to 1 you have three possibilities they are all symmetric possibilities and when s is equal to 0 you have only m s equal to 0 that is a anti symmetric possibility triplet versus singlet. See the hydrogen molecule which we talked about earlier it is a diamagnetic material hydrogen molecule unlike hydrogen atom. The reason is for a strong bonding hydrogen atom is a very strongly bonded molecule when you want a strong bonding you need to have the wave functions to be in such a manner that they are bonding you needed a bonded orbital when it is bonded it is a symmetrically orbital it must be symmetric which means that your spin wise it must be anti symmetric anti symmetric means it is a singlet state that means it is something like this a bond well bonded situation demands a singlet state that is why this is non magnetic hydrogen molecule is non magnetic. So, this idea can be extended for solids without much of a difficulty. So, very important point is the first thing first characterizing property of a ferromagnet namely spontaneous magnetization which can be explained by the molecular field which can be called the exchange in due to the exchange interaction is mainly it is really because of the the electrostatic contribution spin dependent electrostatic contribution is responsible for this thing it is somewhat unexpected because one would have expected a magnetic contribution to give rise to this one that is not the case. So, electrostatic interactions play a big role in determining ferromagnetism it is a very very kind of surprising information another surprising information comes. Other very important property as I mentioned is magnetic anisotropy what is magnetic anisotropy direction dependence as I mentioned if I take a crystal magnet I apply the magnetic field along different crystallographic directions as you would have seen in the crystallographic topics for example, in the case of Fe it is much easier to magnetize along 1 0 0 direction as you can see the saturation is very fast whereas, 1 1 1 it is difficult why is it happening. So, the same material the crystal is behaving differently for different directions you need very high magnetic field for certain directions or magnetism wise certain access are acting as if they are very soft and certain other access are hard or this in this case 1 0 0 direction is called an easy axis of magnetization compared to 1 1 1 which is actually a hard axis of magnetization. So, this is easy axis this is hard axis there is a difference between the two. One can actually write down the energy the so, there is a built-in anisotropy energy the energy required to magnetize along different directions this can be written in terms of this constants as I have written here for a cubic crystal you can see that one can write in this form where alpha 1 alpha 2 alpha 3 are nothing but the direction cosines of this magnetization vector with respect to the access ABC in the cubic case ABC are all the same, but they are mutually perpendicular. So, this direction cosines of course, the square of the direction cosines sum is 0 1. So, it becomes this first term is this one second term is given by this third term is given by this. So, this tells you if you depending on the direction of the magnetization you can have different energies that is what is represented by this. So, basically tells you that different directions if you try to magnetize you need to do different amounts of work or the magnetic field needed to saturate will be different depending on the direction as you have seen in the figure. For a hexagonal system the same thing can be talked about here again we write the expression for the so called anisotropy energy from different directions. The direction in this case is easier because here you have a unique axis what is a unique axis in these systems the so called uniaxial crystal systems like hexagonal tetragonal systems you have a C axis which is different from the AB plane. So, the magnetization is always written with respect to that C axis the angle is taken as theta. So, you write the energy in terms of this theta either in the I mean in the form of sin square theta and sin to the power of 4 theta. One can find out which is easy axis and which is hard axis by the minimization of the energy in this case that is what is written here. What are the again very important what are the reasons why they for this direction dependence what is the problem? So, this again gives you some very important physics of these materials there are two reasons for this remember we are talking about a solid we are talking about a ferromagnetic insulating solid like the ferrite material that is a very common example that they are always people can give. So, what is happening? The two reasons are where I start to give the answer and then explain one is what is known as a crystalline electric field second is a spin orbit coupling. So, the dependence of magnetic anisotropy is very strong on the hexagonal and tetragonal systems compared to the cubic systems I will come to that. So, will first let us worry about the two main reasons for that one is a crystalline electric field second is spin orbit coupling what is the meaning of this remember I am talking about ferrite let us for the sometime let us take a ferrite as an example Fe 2 O 3 I am taking the Fe 2 O 3 example till I finish this section it is an insulator we can think like a ferromagnet does not matter whether it is exactly ferromagnet or not but the idea is good to work what is going to happen is you have a d electron there all the d electrons are there that is why you have a paramagnetism it is interacting exchanges there it is becoming ordered what is going to happen is you have these d electrons d electron we have the wave functions corresponding to d electrons there is an orbital degeneracy of 5 there is a spin degeneracy of 2 total degeneracy of 10 that is why you tell that it can take a maximum 10 electrons d electrons d shells can take 10 electrons maximum this is an isolated case isolated atom case now such an atom is actually surrounded by oxygen neighbors in this case oxygen is an neighbors oxygen is positive or positively charged because it is an ion it is a solid ionic solid in some sense you so you have positive charges surrounding this atom which is magnetic so you can expect an electrostatic interaction or electrostatic field produced by this negative ions of oxygen oxygen is negative negative ions of oxygen which are the neighbors will produce an electrostatic field what is going to happen because of this electrostatic field what is shown here first let us see what happens here in the absence of these neighbors this negative oxygen ions what is going to happen is that you have a 5-fold orbital degeneracy that is what is shown here all of them have degeneracy of 5 now if you are actually having the electrostatic interaction you will see that because of this electrostatic interaction one can actually apply a perturbation here I will not go to the details this degeneracy is partially removed and you will see that you have a 3-fold degeneracy which is pushed down and a 2-fold degeneracy which is up if you have a slightly different symmetry you can get exactly Ulta you have a double doublet here and a triplet here and you see this is atomic situation where you are talking about this one or this one is an atomic situation where your Hund's rule is fine but now your energy scheme for the d electrons is not what was as in the case of an atom it is quite different now because of the presence of these neighbors which are negative ions oxygen negative ions so the energy level scheme to deal with now is not this but this and our Hund's rule definition Hund's rule description everything was with respect to this so naturally it is going to fail that is why you have seen one of the reasons why there was a discrepancy between the calculation of effective magnetic moment for the 3D case as the table suggested this is not a case in the case of rare earth compounds as we have seen so the 3D case this interaction is going to be strong why because your 3D shell is more exposed compared to the 4F shell and this electrostatic charges which are there on this oxygen ions this is going to create this electrostatic field and this electrostatic field is acting as a perturbation and it is going to change the energy level scheme from this situation to this situation in the case of one symmetry in a different symmetry is this one in either case this is not good this is not correct this is a very very important difference compared to an isolated transition metal atom and a transition metal compound insulating so what what is the real situation so what really happens is I will first show you the familiar picture that you all seen you would have seen similar pictures like this so this is called a d x y orbital d y z orbital and things like that these orbitals are the one which actually are used in molecules or in solids these molecules these charge clouds have certain direction dependence these direction dependence and then these direction dependence as are there and you have an electrostatic interaction coming from the negative ions of oxygen certain directions get preferred over the others because of this particular nature of charge clouds because these are all charges negative charges so if there is a corresponding electrostatic interaction naturally certain directions will be preferred over the other that is what is actually shown quantum mechanically here I will not go into all the details of this one so basically your wave functions when you are in a solid which is having an interaction of this kind the wave functions of this d electrons must be modified and they must be not in the usual form of psi n l m like 3 2 2 it should be the linear combinations of this kind which give rise to the probability densities or the charge densities as shown this one again you have to get 5 5 orbital states are there so you should get 5 after the linear combinations also but this linear combination business becomes an important thing because of this presence of this perturbation which is an electrostatic perturbation because of the oxygen level again you see the role of electrostatic interaction is becoming very very important in determining the magnetism like the exchange interaction exchange interaction is purely an electrostatic interaction we do not expect it but again here you see another electrostatic interaction which is responsible for creating the stabilization of certain energy directions over others so what happens now so because of this direction dependence it is not isotropic it is direction dependent and hence depending on the situation of the electrostatic sign one can show that certain things are preferred over the other the energies are changing certain things are stabilized over the others that is why something is pushed down something is pushed upside and not only that the degeneracy 5-fold degeneracy has been removed and as I showed you 3 versus 2 or 2 versus 3 not only that now if you see this linear combination combined wave functions shown here in the form of dxy or dyz or dxx for that matter they are all real functions you see there is an i here so because of this they are all real functions unlike our psi nlm functions of hydrogen atom they are all complex there is an e to the power of iml5 functions that will make it complex now that thing is not there they are all real function that is what is plotted here all these things are real functions there is nothing complex here there is nothing imaginary here what is a immediate effect of this the immediate effect of this is what is showed here I want to find out in magnetism context I am always interested in finding out what is the expectation value of angular momentum because I know that that is what is related to the magnetic moment corresponding orbital magnetic moment so I want to find out what is the expectation value before I calculate like one of the tutorial problems which I did before I calculate this I can find out what is going to be the value what is the expectation value of lz I have to find out I can actually I have take I can take the form of these wave functions it is typically of the form xfr yfr and zfr these are the wave function form the p p orbitals but what is going to happen they are all real and what is my operator my operator has an i operator is imaginary and remember any result of a quantum mechanical measurement must be real expectation values and eigen values must be real I am calculating the expectation value of lz lz is the z component of orbital angular momentum since my orbital part the wave function part is completely real and I am having an imaginary operator here I cannot get anything other than 0 so my orbital angular momentum is going to be 0 in this case and hence the orbital angular momentum is said to be quenched and hence the magnetic corresponding magnetic moment also is quenched that is why I am forced to put j equal to s when I deal with transition metal compounds as we have seen in that table. So, what really happens here what is unexpected that is what is happening here if you are doing the same thing for an atom which of course I did there your wave functions are not real and hence there is no question of this becoming 0 unless it is an l equal to 0 state that is a very special case but in general it does not become 0 but now if you see this is actually becoming 0 it has to be 0 provided the orbitals are real and that is what is actually demanded by the presence of this perturbation namely the electrostatic interaction. So, this is a very very important aspect of this one this explains the two things this effect of this crystal field the so called crystal field the electrostatic field produced by the oxygen negative ions in this example this is called crystalline electric field crystalline means because they are all arranged in particular positions in the solid it is an electrostatic field hence it is called the crystalline electric field this has two things one is it gives rise to the direction dependence of the magnetic moment second is at least first order it actually shows you that it cannot have any orbital contribution to the magnetic moment or orbit angular momentum. So, what does it tell so this direction dependence comes into picture and now because of this spin orbit coupling that is a second term whatever orbital contribution it has got the spin orbit coupling the spin also will be forced to follow the orbital and gives rise to the certain specific preferences of directions over the others which will tell you that in the as in the case of other FE example 100 is preferred over 111. So, the direction dependence is purely due to electrostatic interaction produced by the ions which are surrounding this particular magnetic ion the effect of this one can see very interestingly here. So, you can see this is a 5 fold degeneracy because of the crystal field the splitting has happened with 3 versus 2 what you can see here is that you can have various situations if I try to put it here if I have 3 electrons 5 electrons for example, if I take FE 3 plus I have to accommodate 5 electrons in the 3D shell there are possibilities one possibility is that I put they are they are anyway degenerate I can there are the orbital degeneracy of 3 I can put 3 up spins no problem no violation of Pauli's principle I put 3 here remaining 2 I have 2 possibilities I can put either here so if I am ready to expense some energy. So, I can go here higher level and put it here or if I want to put it in the lower level only then I cannot make them up spins they have to make them down spins only. So, this if I it has to be here these next 2 the last 2 will be down spins which means that this will kill the 2 up spins essentially the magnetism will be reduced. So, that is why it is called a low spin configuration whereas, here if you see these 3 are up these 3 are up that is giving rise to a high spin case if it is an atom it is a pure atom these things are not really true I just have a 5 fold degeneracy and as we have seen the 5 boxes I can put it is a half filled situation I can put all the 5 up. So, again something like this but energy is going to be different but you can see this is a high spin case this is a low spin case this is purely because of the electrostatic interaction namely the crystal field that we talked about. Now, what is the next thing is domains many of you know about it domains are needed to explain the fact that you have no magnetization in the bulk scale even for a ferromagnet the reason I will straight away go to the reason since we are running out of time what happens is if I take a ferromagnet and if I assume that everything is one domain the magnetization has to be spontaneously there and the magnetization will be something like this this is not a stable system energy wise this is not a stable system it will prefer to be something like this just to give an example the reason for this is this has got what is known as a very large magnetostatic energy the system does not want to have large magnetostatic energy system tries to reduce the magnetostatic energy that is done by splitting into multiple domains that is what is shown here but in that process what you have to get these lines are nothing but the domain volts these domain volts will give you a slight increase in energy that is why if you okay I am telling that this is not favorable this is favorable then the immediate question is can you have infinite number of domain volts I cannot do that because I want to reduce the magnetostatic energy for that I am constructing these volts and separating them out but remember like what I am making a cabin here if I want to get two cabins I have to make a wall construction of a wall costs some energy so if I even though I am getting gaining something if I am increasing energy in that process it will not survive after some time similarly here if you are trying to make more and more domains to reduce the magnetostatic energy beyond a point the domain volts that you are creating or force to create will have some energy associated with it it will try to increase energy and finally you will see that there is no energy gain and this process will stop there so that you always end up with a finite number of domains so this is the calculation of the so called magnetostatic energy or it is also called the demagnetizing energy the demagnetizing energy I have a very simple example here to show so what happens is that when you take a magnet you magnetize it it has got a south pole and the north pole the lines of force as shown here something like this if you see here so this is a magnet material you have taken you magnetized with the help of a magnetic field and the field has been removed assume that you got north pole here south pole here so the lines of force are coming like this now inside the material the lines of force are something like this now you remember your original magnetic field has been applied like this so that you got this now because the poles are developed like this there is an internal field that is happening the so called a demagnetizing field which is in this direction HD which is opposite to the applied field which actually cause this magnetization so when you magnetize any object any ferromagnetic when you make it what is going to happen is the process itself the very process itself actually creates a field which actually is going to be opposite to the original field this is true only within the material so within the material it is subjected to a reverse field so the system is always like that when you are trying to magnetize it you are always leaving the system with a demagnetizing field as long as you have poles and as long as you have poles only you will tell that it is magnetized so the system will always have a demagnetizing field the moment it is born the enemy is always born along with its birth the moment it is born the enemy is also born you cannot separate them out the demagnetizing field is an enemy which will be born along with the magnet itself here the magnet mean anything that is magnetized so whenever you are telling that I am trying to magnetize something that means you are actually trying to overcome the demagnetizing field so the demagnetizing field is essentially proportionate to the magnetization within a minus sign so this proportionality constant is called the demagnetizing factor so that is minus n d times m at any instant if the magnetization is m prime let me call it as the demagnetizing field at that extent of time will be minus m d times m prime now if you want to increase the magnetization little more by dm I have to do a work that work is given by this against this one so the minus becomes plus so when I have I am telling that I have a material which is magnetized to a value m which means that I have done a work which is actually the integral of this which is nothing but half n dm square this is the magneto static energy which actually can be also called as the demagnetizing energy which is a thing which actually is an unstable thing only way this can be reduced to zero or minimize is to reduce the magnetization so that is why any magnet when it is made to have some magnetization m the tendency is always to reduce the magnetization that is why people tell whenever you are using bar magnets handle it very carefully assume that you are not doing any temperature increasing temperature nothing happens but still there is an unstable situation there the system has an intrinsic tendency to reduce this m that is to make the energy less which energy less the magneto static energy less so this is a very important aspect of this any magnet that you have this I already explained so creation of domain walls will create the increase energy so beyond a point this will not be entertained because of this demagnetizing part when the poles are there when you have free poles this is always creating these demagnetizing fields so when you have a thin film you have a thin film generally what happens is we are trying to magnetize it it is much easier to magnetize like this because if you are in this direction the poles will be coming here these poles are far separated and the effect of demagnetizing field is very small compared to the situation when you are magnetizing like this when the poles are separated by a very small distance typically if you angstrom or something like a few microns so when these poles are so close by the effect of demagnetizing field within the material day and the magnetizing field is only within the material is very strong that means it is very difficult to magnetize like this you need extremely much larger fields to have the magnetization like this compared to this one so any thin films usually magnetization like this is easier compared to this one unless there are some other effects when for example you have a magneto crystalline anisotropy which is very strong then this is okay but in usual cases this is something which is usually preferred this is a direct immediate effect of the demagnetizing fields I am coming to the end let me so when you apply a magnetic field what is really happening is you are having a multi domain structure giving no magnetization in the beginning 0 as you increase the magnetic field the domains actually the domain which has got the magnet intrinsic magnetization in the same direction as applied field that grows in size with the help of these domain walls movement and ultimately you see that the material becomes single domain that is what is corresponding to the saturation the magnetization of a ferromagnet always saturates at some high field that is what you see here this again is very important as I mentioned this is a hysteresis when I actually make a complete field cycle it starts with a very magnetization increases it saturates then when you reduce this one as I was mentioning just now the domain wall motion which is responsible for the change in the magnetization as a function of the magnetic field increase or decrease or reversal this is going to be an irreversible process because of various factors that are present in the material I will not be able to go to the details it can be intrinsic defects artificially created defects many other situations this will make the magnetization reversal or the domain wall motion kind of irreversible giving rise to this hysteresis loss which is a loss which actually appears as heat in the material so if you cycle it after sometime you see that the magnet gets heated it is not a good thing for many applications but for certain applications like permanent magnets you need to store energy it has to give the field you need to get large remnants that means the magnetization at the end of the zero that is after reducing the magnetization the first magnetization cycle you want the material to have the maximum magnetization of course it is unstable that is true but you need to have that or similarly you need to have a very large quiescivity that is the field required to kill the magnetization completely so these two things should be as large as possible in the case of permanent magnets the bar magnets that we use for many of our experiments but applications where you are actually you force to cycle the thing like for example a transformer where the field is cycling every I mean 50 times every second the 50 hertz frequency of the line we have there this should not be the case it should be very thin so that the energy losses are minimal this is called a hysteresis loss which has to be minimized for any application where cycling of this magnetic field is involved transformer is a case with only the frequencies only 50 hertz nothing ferrites are used for microwaves and other things where the frequencies are much larger gigahertz and other things where you can assume what kind of an energy loss will be there if the hysteresis losses are very much so this hysteresis area must be very small in such cases so one can define soft and hard magnetic materials depending on the values of this one when the hysteresis losses are I mean the loop is very big then you call magnetically hard materials otherwise soft materials I think all these things are self explanatory this is the last part so what happens is I will only qualitatively mention so what happens is you started with a material which actually is a ferromagnetic particle as I told you it always prefers to be in multi domain system than in the single domain case but when you start reducing the size when the size is reduced what is going to happen is at certain time one can show that it the size has come to a value where it is not in a position to support two domains two domain that means a domain wall because the domain wall may be taking more energy so that it is not energetically an advisable situation for having two domains with the help of a domain wall so this once this state is arrived the system prefers to be in a single domain status than having a multi domain situation the calculations are shown here please work it out if you have any doubt you can actually ask me because we are running out of time so when the size of the particle is generally less the tendency for this multi domain formation is reducing and the system has a tendency to be a single domain that is what is shown in this calculation so this shows you that one actually gets so when the particle size is reduced in that manner first it to get into what is known as a single domain material system where the whole material the whole material particle is just one domain now if you reduce further this is quite unstable against thermal energy and anything any magnetic moment which is unstable against thermal energy is a paramagnet that is a characteristic of a paramagnet but remember now this is not a one single ionic magnetic moment or atomic magnetic moment this is a particles like a nanoparticle and this is forced to behave like a paramagnet but originally it belongs to a ferromagnetic family only because the size has been reduced it has become like this so this is called a super paramagnet a super paramagnet is slightly below the size of a single domain particle and the interesting point is this is very important so what is expected is the coercivity which is a very important parameter of hysteresis that I mentioned as you start with a ferromagnetic system where you have this multi domains as you reduce a size what you see is that coercivity increases and when it reaches a so called single domain situation you have only one domain there are no the domain walls domain walls are regions where you have all kinds of magnetization directions which are not good as far as this anisotropy is concerned the coercivity is concerned so the coercivity increases reaches a maximum at this region if you reduce the size further the system enters the so called super paramagnetic region anything that is paramagnetic generally no hysteresis is there and slowly the hysteresis related properties will come down and that is what you are showing here so hysteresis the coercivity increases reaches a maximum at the single domain situation and then decreases and become zero in the completely super paramagnetic regime this is the end of this talk I think we will take a couple of questions so my question is how the internal field can be calculated in paramagnetic material I mentioned in one of the slides yeah yeah in fact I will tell how it can be measured in both the cases the internal field I will not call as internal field in the case of a paramagnet I will use our internal field for a ferromagnet only for a paramagnet I will tell the interactions among the magnetic moments that strength can be determined by the theta value chi equal to c by t minus theta what is known as the paramagnetic curie temperature that is called the theta so once I measure theta theta is a measure of the interaction strength of these paramagnets I will reserve the word internal field for ferromagnets where the interactions are much stronger it is not the dipole dipole interaction or anything it is a exchange interaction the spin dependent coulomb interaction that I talked about much stronger the measure of that internal field in the case of a ferromagnet is nothing but the curie temperature the curie temperature is more means the internal field is stronger if it is low means that the internal field is weak so the curie temperature is the best thing to measure the internal field strength of a ferromagnet for a paramagnet I do not want to call it as a internal field but the interaction strength I will call that is represented by your theta thank you sir I have one more question please how the domain area domain wall area you get increased when the applied field is increased yeah so it is not a domain relation yeah it is not the domain wall area actually it is a volume what happens is when you apply the magnetic field definitely there are many domains which have different directions randomly oriented there will be at least one domain whose intrinsic magnetization direction will be coinciding with the applied field so because when you have minus mu dot h term minimized that particular the domain the size will increase so that the energy minimization happens like the Zeeman term so to reduce the Zeeman energy minus mu dot h kind of term the domain which has got exactly same direction of magnetization as the applied field that will increase the size of when when that increases it has to be at the expense of some other domain so the unfavorable domains directions will decrease how does it happen remember the domain and domain wall everything is made up of ionic ions and their movements so basically the domain walls kind of move in such a manner so that this favorable domain size increases at the expense of others this remember this is not a smooth movement this is a jerky movement this is an irreversible movement that is why the jerkyness actually you can see in the form of if you magnify it any magnetization plot there is a very small fine jump that is called the Barkhausen jumps and the irreversible nature is reflected in the hysteresis loop so the way it actually goes is the domain wall movement is quite restricted by various contributions defects impurities externally made things internal intrinsic things everything kind of impede the domain wall motion in certain applications where you need to have large hysteresis areas like permanent magnets one artificially creates the so called pinning centers you create artificially domain wall pinning centers so that you get a broader hysteresis loop in other extreme when you are looking for an extremely soft magnetic material you will try to get a clean system absolutely no defects no pinning center it is ideal case but one can only minimize it sir a material can be easily magnetized through easy direction yeah 100 plane right in the case of it is difficult to be magnetized along hard direction yeah so on which basis we select this direction is direction changes with respect to material or not it is it is we we can we can choose it is the nature it is determined by the material it changes from material to material that is why in the example that I showed you is exactly for Fe if I take nickel exactly Ulta it is 100 is hard and 111 is easy what is determining this is to some extent it is determined by the crystal structure it is determined by the ionic presence what is the symmetry of the ions the crystal field that is acting on it this will determine what kind of a direction is favorable what is not favorable it is quite dependent on the crystal structure for that matter since this question is very important let me add to that if you are taking cubic crystals Fe and nickel are cubic systems the difference between 100 or 111 even though I showed an exaggerated figure the difference actually is small compared to a case when you have a hexagonal system hexagonal system trying to magnetize in the AB plane versus the C plane the difference is huge why because this is a uniaxial system the crystal structure itself is anisotropy cubic case is a symmetric case more or less symmetric right cubic is more or less symmetric so you cannot expect too much of an anisotropy in a cubic system which is naturally a symmetric case whereas when you have an anisotropic crystal structure the magnetic anisotropy is also going to be very large because of this reason any magnet that you are seeing any bar magnet that you are seeing is always made of a material which is having a uniaxial crystal structure it is never made of a cubic system it is not possible because it will not be stable because your hysteresis area will be smaller even though I told the exaggerated figure but the difference will be small in the case of a cubic system so the difference between easy and hard axis is dependent very much on the crystal structure sir I have a doubt that you talked about this electron electron interactions sir yeah whether this electron electron interaction plays a vital role in visceral coupling electron electron interaction plays a vital role in visceral coupling or a visceral coupling has this electron electron interaction no no I am looking at only the electrostatic interaction between the electron and electron I took the example of hydrogen molecule I was trying to calculate the coulomb interaction between these two electronic charges quantum mechanically if I do classically it is a simply e square by r12 term I am doing it using the wave function I am trying to calculate the expectation value of that one because in quantum mechanics I have to calculate the expectation value why because that is nothing but a perturbation change that expectation value of this term will give me the corresponding energy shift so I was simply calculating the expectation value in quantum mechanics for this particular term the coulomb term it is a very simple coulomb term sir in NMR studies yeah we are mainly considering the spin-spin coupling sir yeah NMR is yeah that is why I just talked about magnetic resonance in the earlier lecture yeah tell me sir but in this spin-spin coupling we talk about geminal and both the visceral also let me tell you whether this electron electron interaction changes the value chemical shift value of course it will change it will change that is I mean different topic altogether but in the case of NMR shift right it is definitely going to change depending on the yeah so because what happens is there you are talking about a hyperfine coupling you are going to talk about a nuclear versus electron coupling the hyperfine coupling definitely it is going to change yes sir that is what there is this electron electron plays a role in both geminal and visceral yes yes that is my question yes but in my case I am only interested in the the usual electrostatic term okay sir thank you so but it is an indirectly it is related to the spin-spin interaction yeah of course sir thank you yeah is there any relation between cell energy and magnetic static energy when orbital radii will increase but the difference between magnetostatic energy and relation between cell energy and magnetostatic energy when radii will be increased of atom what is shell energy sub-cells are cell and sub-cell energy okay shell okay no no no no shell is yeah shell is purely an atomic property no no so definitely no we are actually not talking about that so we are talking about the magnetostatic energy is basically a solid property magnetic static energies you are talking about when it is magnetized all the atoms we do not really take the atomic picture into account there you are having a net magnetization the bulk magnetization what is the energy associated with that just because it is magnetized in a particular direction and is there any unstable part associated with that is there any energy that has increased because of the process of magnetization which actually is true here that is what we are calculating there is no atomic uh interpretation for that no sir one more question yeah how to find the electron density when sub-cell sub-cell size will increase suppose spdfc sub-cell is given yeah so how to find the electron density or probability of an electron density you have to use some kind of I mean x-ray kind of spectroscopy you have to use not an image imaging you can do but it is I think is very difficult indirectly you can use some kind of a spectroscopic technique so that you can actually probe the electron density these days of course lot of imaging techniques are also available so one has to do that but then the problem is it cannot be a very depth information that you can get it will be more of a I mean you can do even x-ray to some extent can give you indirectly this information of electron densities associated with even bulk systems one can do that so basically what you have to do is you have to take x-ray diffraction pattern and you have to do a proper refinement once you can do a refinement the standard refinement is what is known as a reitwilde refinement if you are an expert in that one you can get the information of actual electron density map so you can get that but you cannot tell spd and or using that one for that what you have to do is you have to do a spectroscopy in the sense that you are you can use x-ray spectroscopy not the diffraction technique you have to use an x-ray spectroscopy or things like that so you can actually find out the binding energies and get some information about that otherwise telling that this is s electron density this is t electron density I doubt very much sir I just want to make the analysis between grain and a domain sir yeah okay in grains there are defects like dislocations point effects like that yeah is there anything in domains also dislocation defects like that no no no no grain is a crystallographic defect right you talked about long range periodicity in the morning so that defects associated with that gives rise to the polycrystalline I mean the long range it is not possible but in the local I mean dimensions it is possible that gives rise to the polycrystalline sample so polycrystalline has many crystallites that is called a grain and they are separated by grain boundaries the domain is purely a magnetic concept where is to my knowledge there is nothing like a defect in that sense the domain wall is the only thing which actually can be treated as a defect in the sense that within the domain the magnetization is completely spontaneously magnetized I mean in the same direction the domain any domain is spontaneously magnetized but the domain wall regions are bad as far as this is concerned because they I mean again remember domain walls as a finite width they have an associated energy and so on but that is the only thing that actually can create some problem for example as I told you when you reduce a particle size as you are seeing the cohesivity is increasing why the role of this domain walls is decreasing that is the row the domain wall region is a one where you have all kinds of directions and so on so which is actually a reverse thing a magnetization not in a particular direction that is why it is going against the anisotropy idea so it is becoming kind of isotropic so that is why once once you are getting rid of this domain walls slowly you are improving the cohesivity that is why it increases it reaches a maximum at the single domain condition there you have only one domain that is no domain wall you have reached the maximum cohesivity that is possible then of course it comes down because it is entering into a paramagnetic kind of a region so as far as this question is concerned only problem is the domain walls nothing else but domain wall motion when it happens they get impeded they get pinned by the crystallographic defects like the dislocations and other thing that you are telling okay sir thank you sir you explained well about that super paramagnetism what are the experiments as a record to conduct super paramagnetism to prove super paramagnetism yes sir what are the experiments are required to confirm that super paramagnetism yeah it is actually little difficult to confirm to confirm that is not very easy one thing is of course you have to use various techniques to do that one is you can actually take an mh plot the hysteresis plot you should get you should not get any hysteresis that is number one that does not again prove that it is one what one can do is that you can actually find out the effective magnetic moment effective magnetic moment typically will be very large because this is a particle not an atom it is it is expected to it can be something like 100 more magnet on 200 more magnet on this is one possibility you can actually do certain ac techniques for example an ac magnetic susceptibility can be done ac magnetic susceptibility dynamic susceptibility not the dc there again it is not a confirmatory test you can get some information what you can do is that you know something about the system that you are working then you have to combine all these results together you will be able to make a meaningful I mean guss that is all I will tell so otherwise tell that is exactly super paramagnetic it is not a very simple thing because there are other interactions which will have more or less same kind of features to rule them out it is not very easy the domains region size are in the microscopy scale yeah let us 10 power minus 4 to 10 power minus 6 how did we confirm sir that domains the size of the domains yes yeah size of the domains can be measured I mean then various techniques are available there is a Kerr microscopy is available which of course is not very simple it is an optical technique because the polarization gets shifted by the magnetic field direction so that that is there higher solution TEM is also able to show this one there is a I think there is an attachment needed the simplest thing that people talk about is what is known as a bitter pattern where you do is what you take is if you take a clean ferromagnetic surface you pour a ferrofluid on that the force that this acting on this ferro this ferrofluid particles is different in the domain size region and in the domain wall region so this actually traces out the domains and you from there you can get a guss of these domain size sir in the case of the super paramagnetism you explained well what about the surface effect sir you explained well about the crystal field effect and everything you said what about the surface effect surface effect is there in the case of nano particle that is an issue because what happens is there is a abrupt discontinuity there that is why if you remember I have shown you a parallel direction of a plane and a perpendicular direction of plane if the for example there is a sudden jump from material medium to the air so there can be problems there there can be I mean roughness and all other contributions at the surface which are actually can kind of log these magnetic moments so that is why in certain cases in spite of the fact that the demagnetization contribution demands that the plane of easy magnetization should be I mean the easy magnetization direction should be the plane but in some cases due to the other contribution the surface contribution that you are talking about it is possible to get the perpendicular thing but the problem is it is not much of a physics that is known on that however the surface actually contributes is not very clear the surface effects have become important only now when the nano thing came till then nobody really worried too much about it so if you ask me I cannot give an explanation on the basis of crystal field theory on this one because it is not known very clear surface effects are known to show some effects but what is a physical basis for that is not very clear at the moment thank you sir thank you for your wonderful answers sir thank you sir my question is we start from the ferromagnetic materials yeah and go to paramagnetism you explained that there are two ways one by increasing the temperature and second by decreasing the dimension of the domains yeah in one case we reach at paramagnetism in second case we reach at superparamagnetism yeah now how the property will changes is this paramagnetism and superparamagnetism are different if they are different so what are the physical property that changes okay in this transition second question is is there is interaction between spin and electrostatic field no 0 that is 0 that is straight away 0 the spin is influenced only by magnetic field that is let me answer the that question first so spin is influenced only by the magnetic field electrostatic field has no role in that one coming to the other question there is a small difference I talked about the ferromagnetism to paramagnetic transition that is a magnetic transition is temperature induced okay so there the temperature changes there is a thermodynamically it changes from one phase to the other ferromagnetic phase to the paramagnetic phase that is right so that is because the order is reducing it actually goes from the high I mean ordered phase to a low ordered phase the super paramagnetism is different you are it is not a phase transition in that sense it is you are starting with a material you are playing with the material you are reducing the size that is your job not the nature's job it is you are reducing the size of the particle and because of the domain structure and the domain wall structure below a point it is not able to contain one even one domain wall which means that it cannot support domains in that region it has to take only single domain and further if you reduce the energy the magnetic energy becomes comparable because the size of the particle has become so small that magnetic energy becomes comparable to thermal energy whenever you have magnetic energy becoming comparable to thermal energy that is precisely paramagnetism the random alignment of magnetic moments is paramagnetism so we tell even though it is birth was in a ferromagnetic family you have reduced the size that is our job and you have reduced it you have killed some interactions you have made it smaller and smaller now you have reduced the magnetic part to such a low value so that the thermal energy is able to compete with it and hence you are telling that you have to treat it definitely this is not in a ferromagnetic particle now you have to call it as a paramagnetic particle but the problem is it is not a usual paramagnetism that we know of this one important difference is the magnetic moment is much larger as I mentioned to the earlier question it can be 200 300 more magnet ones whereas atomic moments are maybe for maximum 5 6 more magnetons so this is what we are making artificially of course when it happens the system has to energetically be stable and the nature is doing that job but this is not a phase transition like the ferromagnetic paramagnetic phase transition that we talked about so they are they cannot compare okay thank you sir why cooper pass are called bosons sir can you explain me yeah what is the spin up in electron it is half what is the spin of a cooper pair it is 1 1 or I mean it I mean when you add half and half it can be 1 or 0 a 1 or 0 right so 1 or 0 you say integer spin integer spin must be a boson integer spin must be a boson as per the spin statistics theorem sir I have one more question what is a permanent magnet have an advantage over an electromagnet a very relevant question the advantages are like this electromagnet is very heavy compared to a permanent magnet of the same field you need to have a cooling system you need to have a current power depends on the power if the power is not there rural places you cannot have an electromagnet in India today whereas a permanent magnet as a name suggests it is permanent but to get the psi I mean you need a small small pieces you have to have a particular arrangement you have to get a meaningful field and other things you need to work out the design in such a manner so that the region where you require the field you get the required field it is not very easy but these days the problem is you can get the magnets but the magnet cost is also going up the main difference of course I will tell is the size because electromagnet is huge I mean it is not something which you cannot which you cannot transport I mean going from one place to other it is not easy I will I will not really compare that way I will tell it depends on the application for example to conduct an experiment in a lab I will definitely prefer an electromagnet if the power situation is all right okay sir thank you sir