 So, we are going to start a slightly different topic now. So, this is somewhat connected with the topic that you have been exposed to in the last 4 or 5 days, something extra also will be there. So, the topic is the title of these theories of talks, it is given as a basic introduction to magnetism in materials. I understand that a large number of you actually belong to engineering colleges. So, you must be teaching various branches of engineering. So, that way this course I have made in such a manner that this is kind of appealing to different sections. Magnetism as you know it is very applied strictly speaking, but it is also very rich as far as the fundamentals are concerned. So, I have made a mix of these two. So, that means I will cover considerable amount of fundamental physics of magnetism in various materials and I will also give you a flavor, at least a flavor of some of the applications. In fact, magnetism has a large number of applications. So, I will touch upon some of the applications which will be again appealing to at least some of you. This is the way the lectures are planned. I also understand that you had a course on probably just finished on electromagnetism. There is some difference between the contents of electromagnetism and the magnetism that I am going to talk about. What you have seen in electromagnetism is basically the magnetic effects of current. You have a steady current in a wire, you know that it is producing a magnetic field by a tenservert law, you have studied Ampere's law, you have already seen you have I mean of course, you knew earlier also. I am not going to talk about those things. In fact, I understand that some of the questions regarding domains and so on, they were asked to the other instructor. These are the things which I am going to cover in the next three or four lectures on magnetism. So, I will borrow certain results of classical electromagnetism, but this will be purely on the magnetic effects produced in a magnetic material. So, this is basically the magnetism in materials. That is the first difference that you should note compared to the classical electromagnetism which you have just seen in the last four or five days. So, let me just tell you how we are going to do this course starting today and it will be there for the next week. So, I will be giving four lectures on magnetism. As I mentioned, I will heavily depend on classical electromagnetism and quantum mechanics which you have studied again almost it has come to the end right now. So, both these things will be used very much to understand the basic magnetism in materials. As all of you know and this is something which is not new as far as magnetism in materials is concerned broadly speaking we have classification of this kind you have diamagnets, you have the paramagnets, then you have set of materials which are ferromagnets, antiferromagnets and ferrimagnets. These last three varieties they are actually called magnetically ordered materials ferromagnets antiferromagnets and ferrimagnets they are actually they belong to the class of magnetically ordered materials which have large number of applications. Most of the applications that we are going to talk about or which you know they are all based on these magnetically ordered materials. But before we really reach here you have to have some idea about diamagnetism or diamagnetic materials and paramagnetic materials. This is needed not only in the context of magnetic materials we are going to have some discussion on superconductor so there also these informations will be quite useful. This is a periodic table where all the elements are shown as I have written strictly speaking if you take the definition or the classification of magnetic materials which I showed just now all elements in the periodic table are actually magnetic in some sense. If you include a diamagnetism or anti-paramagnetism also every element in the periodic table is magnetic. However in real situations when we talk about magnetic materials we generally worry about only ordered magnetic materials which means that ferro, ferri and antiferromagnetic materials to some extent we will talk about the paramagnets also. So diamagnetic materials are generally not treated in that sense we tell that only very few elements in that periodic table are actually magnetic. But if you include the diamagnetism also into the definition of magnetic materials one of the classes of magnetic materials then you have all the elements in the periodic table being categorized as magnetic. So just to give you a basic idea of how the magnetic moments are there or the magnetic materials are can be further looked upon. The first thing is diamagnets they have no permanent dipole moment we talk about every time we talk about magnetic moment what we mean is magnetic dipole moment. So diamagnets they do not have permanent magnetic dipole moments all other categories starting with paramagnets they have intrinsic permanent magnetic moments that is what is shown here these circles what you are seeing they are actually the atoms and these arrows represent the magnetic moment. We will see what are the origin of these magnetic moments and so on later but this atom and the corresponding magnetic moment they are shown here. So here I can show because they have permanent moment whereas in the first case in the diamagnets there is no permanent magnetic moment and hence I will not be able to show this kind of arrow. And as you know the case of paramagnets what you are seeing is these moments are randomly oriented we can see there is no particular order whereas in the magnetically ordered materials namely the furrow, antifurrow and furry magnets as you can see there is an order in the first case the furrow magnets you can see where they are all in the same direction the magnetic moments are in the same direction you call them furrow magnets. If they are antiparallel the adjacent neighbors are antiparallely oriented as far as some magnetic moments of these atoms are concerned it is a very ideal one dimensional situation that is shown here for our simple understanding or simple description one can have three dimensional and two dimensional magnetism we will not worry about right now just to give you a schematic one dimensional picture is enough. So this is the antifurrow magnetic arrangement ideal antifurrow magnet you can think that at a very very low temperature this is something which is possible. Then you have furry magnets furry magnet is something like a antifurrow magnet but the difference is you have the magnetic moments of there are two species of atoms or ions there and their magnetic moments and magnitudes are not the same and they are antiparallely aligned that means this is up this is down but they are not exactly like the antifurrow magnets because they are equal here it is not and if you take on the bulk scale if you see you can see that because of this non cancellation of these magnetic moments up and down they actually behave in some sense they actually behave like their furrow magnets. So the famous ferrites which we will be talking about later actually belong to this category furry magnets. So these are the three magnetically ordered materials and yes you can see to construct any of the three ordered materials the building block that you need is a paramagnet that is what is shown here. So using paramagnets giving different kinds of ordering you can get furrow magnets antifurrow magnets and furry magnets. So furry magnets is not made up of one kind of atom one kind of ion it is consisting of two that is why the magnetic moments the magnitude of magnetic moments are different. There is another way of classifying just since it is an introduction lecture right now I will talk about general features of magnetic material then we will go to the details. Another way of classifying which actually is done is something like this this again will help you to understand the physics of magnetism little better if you try to see these classifications also. Broadly again you can talk about magnetic materials are something which is metallic metals and their alloys they form a huge chunk of applied magnetic materials which are very very important. Then you have insulating magnetic materials like the magnetic oxides which again are very important in some sense for example ferrites belong to this category I generally call it as magnetic ceramics. So they are insulating they are not metallic this forms another very important category. Another way of looking at it is you have magnetic thin films which again is very very important for many of the applications these days not only talk about thin film magnetic thin films you also talk about magnetic multi layers where you have not just one layer you have many layers so that is what is the multi layers. Then as all of you know all of you are all of us are actually attracted by the developments in the nano science and nano technology magnetism also is contribution is there. So you talk about nano structured magnetic materials which are very very important today I will be talking about it in the remaining lectures later on. Then again a very exciting theme that is emerging or it has emerged very recently magnetism of molecules you talk about very complex molecules there some of them actually contain magnetic elements or magnetic ions. So that actually gives rise to many interesting magnetic and related properties they are actually called molecular magnets so that is another category which is very important I will talk about very basic thing about it later. Then you also have ferrofluids basically you have a magnetic fluid so you can apply the magnetic fluid you can change the fluid properties like viscosity will not be worrying too much about it in this series of lectures but this is something which is again as many of you are engineers. So this is something which is very important mechanical engineering and other things where ferrofluids are actually offering very important contributions. So this is also basically a ferromagnetic system that is in the fluid form yet another classification that is based on the structure is this one you have single crystals I am sure you all of you have heard about single crystals anyway they will be having a course on solid state physics crystallography very soon. So single crystals are the one where you have long range periodicity of atoms throughout the crystal so that is called a single crystal sometimes this may not be completely long range you have regions where this periodicity of the atoms is there but then there is that is broken for some region then again you have single crystal like regions. So you have such polycrystalline samples which actually consisting of a large number of single crystalline regions separated by not very crystalline environment then you have the other extreme of single crystals which is amorphous there is absolutely very low ordering crystalline ordering in the bulk scale that is called amorphous materials then as I mentioned you have the nano structured materials where the crystallinity is only for a very very short distance of something like nanometers. So this is another way of classifying any material system definitely for magnetic materials also. So this also will help us in understanding some of the special magnetic properties or related properties in these things as a function of the crystallinity. So single crystals have certain advantages polycrystalline magnetic materials have certain other advantages nano structured materials have certain other advantages as we are going to see. Coming to the applications this we will do towards the end of the four lecture series only but just to give you an idea again as the first lecture we should talk about get a broad picture of the applications also because very soon we are getting into the finer aspects of the physics of magnetism but before that let us see what are the main applications of magnetic materials that everybody talks about. First and foremost these are so called soft magnetic materials I will talk about I will define what is soft what is hard later. Obviously when you have a soft magnetic material you have a hard magnetic material also you have a very big class of high frequency materials. You have magnet optic materials actually the name we did that title is actually very self-explanatory you can understand what is actually going to happen magnetism and optics. So materials where there is a connection between the two you have magnet or resistive materials that means electrical resistivity is actually dependent on the magnetic state of the material that is giving rise to magnet or resistive phenomena and those materials are actually called magnet or resistive materials. You have magnet or stricative materials which is very important for many of the applications including the many of the engineering applications mechanical engineering for example very very important. Magnet of stricative materials I will be talking about it later where the dimension of a material changes as a function of the applied magnetic field very important application there. So that is magnetostriction. Magnetic refrigerant material which again is more important from the point of view of cryogenics and heat pumps and so on where you actually have a system where by changing the magnetic field you are actually changing the or playing with the entropy of the system which actually is basically the principle of refrigeration which of course is used in other forms also. Then you have something which is very recent everybody knows about the electronics all of us are using electronics left and right for various applications that we are using every day. But now we have one step further that is what is known as spin tronics which actually is a spin based electronics. Magnetic materials certain type of magnetic materials are very very important or the they are the active components of these so called spin tronic materials I will call them as half metallic ferromagnetic materials. So they are also very important as far as applications are concerned we have to worry about their properties and so on. Again materials for bio and medical applications which have come to the limelight very recently lot of work is being done today. Materials magnetic materials for bio applications and medical applications I may be talking about very little in the later part of the remaining lectures. So these are some of the very very important applications where magnetic materials play a very very important role. So this very clearly tells you that magnetism is really an applied field. Now what we are going to do is we will talk about as I mentioned some basic physics of magnetism because if you really want to appreciate or try to understand evaluate the importance or the role of a particular magnetic material for an application. You should know what is required in a particular application you should also know what is expected out of a material for that you should know the magnetism basically the physics of magnetism of those materials need to be understood. So I will give you since I do not know whether you have really gone through a course some of you have probably gone through a course on magnetism but most of you would not have gone through a course in magnetism. So I thought I will give you a very very elementary idea of basic physics of magnetism. Here I will use the ideas of quantum mechanics and electromagnetism that you have studied but of course I will have to use a little more quantum mechanics which of course is unavoidable I am sure you will be able to appreciate it I will make it very simple and now in this part you will see that some amount of derivations are there. So all these things are given in the slides some of the things I will explain in very great detail but if you after going through if you still feel that there is a problem please write to me I will try to explain it in the remaining lectures. So you have to really work out these derivations because if you want to really appreciate you have to really work out each and every step that actually is shown here in these derivations because otherwise you will not be getting a clear picture of this you may be getting a qualitative picture what I am looking for is a little more of a quantitative picture of magnetism in these systems. So let us try to see forget about applications for a some time we will purely look at the physics of magnetism in these materials. Before I do that first let us define certain very important characterizing parameters which are required to talk about any magnetic material without this you will not be able to proceed further. So the first thing that we talk about as I mentioned as I shown you in the picture the first thing is the atomic magnetic moment which is represented as mu usually these symbols as standard symbols are not changed. Then this is the atomic property as I have written atomic magnetic moment the circle I have shown you with an arrow that is representing the mu. We will talk about we will see what is mu made up of and other things very soon. As far as solids are concerned because most of the applications as you know are with respect to magnetic solids the most important parameters are one is the magnetization magnetization is M which actually is defined as the magnetic moment magnetic dipole moment mu summed over a volume and divide by the volume. So basically magnetic moment per unit volume is defined as magnetization. Then you have a magnetizing field which actually I call it as H the field that is applied with the help of a magnet and so on. And the response of the material is actually represented in terms of the magnetization quantity M. So the response can be found out in terms of this magnetic susceptibility. In fact is what I mean is magnetic susceptibility even though I did not write here because you have you would have heard about the electric susceptibility also in the context of dielectric for example in the other course. So here I mean the magnetic susceptibility chi is defined as M by H. M is a response of the material to an applied magnetic field H. Then you also have another field which is very often used very important is called magnetic induction which is again a field as I mentioned this is B and the relation between B and the applied field H is something like this I am showing you two different units because in magnetism practically speaking most of the time we will be using CGS units not the way we generally use always SI units here most of the time for convenience we use CGS units. So I am showing both these units that at least many of the definitions I will show relations I will show for both the units. So in the SI units the relation between B and H is something like this mu naught is a real permeability of free space mu naught times H plus M and in the case of CGS units is nothing but H plus 4 pi M. You can see in both these relations M comes into picture M means the magnetization of the medium. So B actually is referred to the field that is seen within the magnetic material the material medium. So that is what is shown by B and the permeability which is again a very important parameter which we will be using when we talk about some of the applications is the ratio between B and H. So B by H is the permeability permeability is usually given as mu but since I have already used mu for the atomic magnetic moment I did not write it here. So mu I am reserving for the atomic magnetic moment. So these are the most important parameters which are needed to describe magnetic properties of solids especially. Of course as I mentioned without the atomic magnetic moments which are the building blocks there is no way you can proceed with and see. So for example your M is made up of your mu. So these are some of the relations later on most of the time I will stick to CGS units but I just wanted to show you how the difference comes. So here I have shown you both SI units definition of magnetic induction B as well as the CGS units. Now the real question comes. The question is where is this magnetism coming from? I talked about dipole moment atom and so on. So if you want to understand how the magnetism comes when I am telling that this is a magnetic material what does it mean? To understand that the best thing to do is to classify or to go in a sequential manner. That sequential matter is what is shown here. You first talk about the electronic magnetism because when you tell something is magnetism I am going to the details very soon I actually take into account the magnetic moment associated with the atom electrons. So the first and foremost thing to worry about is the magnetism or the magnetic moment coming from the electron. So when the electrons of a material are giving rise to magnetic moment that makes the system magnetic in a very simple way. So one has to start with the understanding of what is meant by electronic magnetism. Once you get an idea about electronic magnetism we can go to the next building block that is the atom which actually is made up of electrons. Z number of electrons where Z is atomic number gives you an atom. So once I know some rules I can actually find out what will be the contribution to the magnetism of an atom given the fact that I know the contribution that is coming from a single electron. Then from the atom I can actually go to the molecules as I mentioned molecules are very interesting as far as magnetism is concerned especially these days. So we will have a very simple understanding of what is happening what is different when you talk about magnetism in a molecule very simple examples. Then we will go to the actual applied materials which are actually solid materials bulk the so called bulk materials I call it as bulk bulk versus thin film thin film or nano they are having lower dimensionality bulk means really a three dimensional material. So the most of the materials we will be talking about will be bulk that is what is meant is a solid state magnetism. So atomic molecular then we go to the bulk magnetism. Then as I mentioned these lot of interest is there for nano magnetism and ferrofluids. So one has to go in this order if you want to really appreciate and understand how a given material gets magnetism. So this is a plan that we have to execute. So let me tell you what is going to be done by me. So I am planning something like four lectures on magnetic materials as I mentioned two of the lectures that is today's lecture and one more lecture I will worry about magnetism in atoms molecules and insulating magnetic solids the magnetic ceramics kind of materials there is a reason for that you will see later why this classification I have to make it different. So the two lectures I will worry about the mainly I will focus on the atomic magnetism, molecular magnetism and magnetism of oxides let me put it in very straight manner. Reminding two lectures I will do after I do two lectures on solid state physics because of certain conveniences. So after finishing my two lectures of magnetism I will go to the solid state part where I will give you a very brief introduction about the solids especially metals what is the difference and so on very very different picture that you see there. Those pictures are actually needed to explain the magnetism of metals and their alloys. So that will be reserved after we finish the solid state part. Then towards end the last lecture I will also worry about little bit about the applications of materials. So that will complete our discussion about magnetic materials. So first two lectures will be mainly worrying about magnetism of atoms molecules and magnetic ceramics because the mechanism is quite different between the insulating materials and metals or the alloys. So you will see that very soon. So this classification is very essential and very important otherwise it will be complete mess up. I do not want to do that. So let us classify like this for understanding the basic magnetism in these different classes of materials. Again as I mentioned this lecture was scheduled for this time because I know that quantum mechanics has been done to a certain level because a meaningful discussion of magnetism is not possible without quantum mechanics. In fact, we do not have time to really go into the details but there is a very important statement that a purely classical treatment cannot explain any kind of magnetism in a solid in a material. So quantum mechanics is essential. Without quantum mechanics there is no way you can explain magnetism. But as I mentioned earlier certain relations, certain understanding, certain level of understanding can be done even using some classical picture and there are certain approximations and other things I am going to show you these things later. But I will also show the classical ideas wherever it is possible and how these classical ideas or classically use the derivations or the relations, how they are actually resembling the correct quantum mechanically derived relations. That is a very interesting point as generally happens in many situations. This is something happening here also. So classical physics also gives you some ideas but we should always keep in mind that quantum mechanics is essential for a clear real understanding of magnetism in materials. The magnetism that you have studied in the form of biot and servo and other amperes and so on that is a purely a classical picture. But what you need here when you talk about magnetism in a material what we need is purely quantum mechanics. Still as I mentioned sometimes the classical picture also works fortunately and all those things I will show you because since you have been already exposed to classical electromagnetism and quantum mechanics it is of interest to see how these things actually go together at least in some cases. So that is a very interesting some of the examples will be interesting for you. So I already talked about the magnetism being thing coming from electrons. So the question is what is the idea about magnetic moment of fundamental particle? Actually speaking electrons, protons and neutrons I have considered these to be the fundamental particles all of them have magnetic moments. Magnetic moment that means a nucleus also has a magnetic moment because protons and neutrons have magnetic moment. But when we tell something is magnetic this element iron is magnetic when we tell we do not really look at the magnetism of the nucleus what we look at is the magnetism of electrons only there is a reason for that I will come to that. So when somebody discusses about magnetism in materials what we are looking is only at the magnetism are associated with the electrons. No discussion regarding the magnetism of the nucleus which actually comes because the fact that magnetic moment is associated with the protons and neutrons that is completely ignored for reasons I will explain. However since this is a general course on magnetism I should also mention you that in certain situations you will only worry about the magnetic moment of nucleus. The best example is what is known as nuclear magnetic resonance which actually used in MRI. MRI is something which everybody knows today it is a very important medical tool today. So MRI works on the principle of nuclear magnetic resonance as the name suggests it is associated with the nuclear magnetic moment where what you are actually looking at is the nucleus the magnetic moment of the proton and neutron there are certain combinations which will give you the magnetic moment for the entire nucleus otherwise they will cancel. But in certain cases depending on the relative number of neutrons and protons in the nucleus you can have non-zero nuclear magnetic moments they become the so called NMR active nuclei and they are essentially very important for fundamental studies and applications like MRI. Proton our water nucleus is a very hydrogen nucleus is a very important example for MRI because what you are using is a proton magnetic moment that is what is the NMR active nucleus in MRI. So we will not worry about in our course we will not worry about that aspect. So our discussion will be purely with respect to the electrons only. Before we go certain extra quantum mechanical information is needed as I mentioned and as I mentioned when we talk about magnetism of atoms that is the electrons and then the atoms that is the first thing that we have to do the lightest element that we have in our periodic table is hydrogen which all of you have studied. So let us very quickly go through some of the very important in quantum mechanical information regarding the hydrogen atom. I will talk about little bit about the wave function and we will talk about energy. This is nothing new you have already studied so I have to just recollect these informations for you. So we have the wave function you have been talking about recently also in this course also we have been talking about wave function and so on probably hydrogen atom was not done but you have studied hydrogen atom wave function actually is represented in terms of three quantum numbers which are the principal quantum number N, the orbital quantum number L and the magnetic orbital quantum number AnL. So the wave function is written in terms of psi NL ML these quantum numbers are needed for us and very important operator for us we have talked about operators in the quantum mechanics course you must be knowing it. The operator associated with the orbital angular momentum orbital angular momentum is what is represented here as L the corresponding operator is written as L the operator symbol you must be knowing now. So this is a very important relationship this is an actually the Eigen value equation which you have been hearing. So the L square operator operating on this wave function gives you an Eigen value of L into L plus 1 h bar square times the wave function itself. Similarly, if I take the z component of the angular momentum operator that becomes Lz operator Lz operator operating on this one the hydrogen wave function isolated hydrogen atom wave function gives me the magnetic orbital quantum number EmL that is the Eigen value and it is multiplying the total wave function. So this is a spatial part of the wave function that you have been seeing in the case of particle in a box and so on we will be doing some problems in the next class anyway. So this is for the hydrogen atom which is slightly different from the other examples that you have been talking about like the particle in a box and things like that but we do not have time to go into the details I assume that you have a fair amount of idea regarding the hydrogen atom this is actually discussed in physics and chemistry courses. So this is something which is very important the connection how or what is the physical importance of this quantum number that is very important here that is what is shown here I am talking only with respect to the orbital angular momentum and the z component of that that is what is it coming here one is in the form of L into L plus 1 h bar square sorry there is an h bar here EmL h bar times the wave function that gives you the z component part when you talk about hydrogen I mean atom or electron in a hydrogen atom you have to also worry about that there is a spin part associated with that because electron has a spin also. So the spin of course I am not going to define what is spin the usual classical picture of that something spinning and so on please get rid of those concepts spin is just an additional quantum number correspondingly there is an additional operator it is you treat it as an additional angular momentum. So I have to consider another wave function which actually will bother about only the spin part this is purely the spatial part the orbital part that you have been seeing I have a similar part to represent the spin degrees of freedom and spin part I call it as a spin wave function and this is a function of two quantum numbers s and ms for single electron s is the corresponding quantum number s is half and this ms has values of plus or minus half and the relations are something similar to what is given here. So a square operator operating on this wave functions spin wave function gives you s into s plus 1 h bar square times the wave function this is again eigenvalue equation the z component of that the angular momentum spin angular momentum z component acting on this spin wave function gives you ms h bar this ms is the magnetic spin quantum number again multiplying the spin wave function. So this is talking about the orbital part and telling that this is the eigenfunction corresponding to the orbital angular momentum and z component of the orbital angular momentum for an isolated hydrogen atom isolated is very important this the same thing as far as a spin angular momentum is concerned. So please take spin in this course please take spin only as an angular momentum as something to be added to the orbital part I will not be able to explain more because of the lack of time so please take spin as additional angular momentum for the electron. Now other thing which of course all of you know is regarding the hydrogen atom energy. Energy the Bohr picture itself Bohr model gives you this energy expression which is minus 13.6 minus is very important it is a bound system minus 13.6 by n square electron volt is the energy associated with the hydrogen different energy levels the n is a principal quantum number which has already appeared in the wave function. And the orbital degeneracy you have studied when you have a given value of n the orbital quantum number l cannot take any value it has to take values from 0 to n minus 1. And once I fix my l value the ml value that it can take is actually from minus l to plus l with a gap of 1 that gives you 2 l plus 1 because there is a 0 also there is there are 2 l plus 1 combinations for of ml for a given value of l. And l can take values from 0 to n minus 1 so if I add all those things I see that this works out to be n square which means that this all these n square states have the same energy even though the wave function will be different because you have different l values different ml values but the energy is the same and when the energy is the same and the wave functions are different you have a very important term or the terminology called the degeneracy. So, you have an orbital degeneracy of n square associated for the hydrogen atom similarly you have a spin degeneracy which is something similar to this which is 2 s plus 1 and as I mentioned s is a quantum number half that gives you 2 into half plus 1 gives you 2. So, if I take these two things together the total degeneracy associated with an orbital having a principal quantum number of n in the case of hydrogen atom is actually twice n square 2 n square if I tell you 2 n square you know that 2 n square is nothing but the number of electrons that can be accommodated in a given shell or orbital having principal quantum number n. So, the number of electrons that can be accommodated in a given orbital or a shell is actually nothing but is the total degeneracy associated with that particular quantum number n this is something which we need as we go along this remember this is purely for an isolated hydrogen kind of an atom let us take hydrogen itself and that is what is given by 2 n square please see the difference you talk about orbital degeneracy you talk about spin degeneracy total degeneracy is made up of both these contributions and the product gives you the total degeneracy of 2 n square. Now, let us come to our having this in our mind let us look at the most basic issue how one can actually attribute magnetic moment to the electron how does an electron get magnetic moment magnetic moment of an electron in a very simple language arises due to the two angular moment I talked about just now one is because of the orbital part of the angular momentum the other thing is a spin angular momentum as I mentioned in general the electron has both I showed you the corresponding wave functions and the other relationships. So, these two contributions give rise to the electronic magnetic moment. So, in general the electronic magnetic moment has two contributions one is orbital magnetic moment other thing is a spin magnetic moment there is a relationship there is a constant of proportionality between the angular momentum and the corresponding magnetic moment and this proportionality constant is called a gyromagnetic ratio which many of you have heard about it. So, the connection between the magnetic moment and angular momentum. So, angular momentum is very important that is why I talked about just now because any discussion of magnetic moment of an electron you need to bring in angular momentum both orbital and spin and hence this talk about magnetic moment and angular momentum various properties of angular momentum very very important. And the connection the ratio between the two ratio between the magnetic moment and the corresponding angular momentum is nothing but a gyromagnetic ratio. I will come to the details very soon before we do that how do we find out the net contribution of orbital and spin magnetic moment because I said mentioned you have both the contributions in general for that a very simple quantum mechanics I will very simplified give it to you is you need to know a very important thing about addition of angular momenta. In quantum mechanics when you have two angular momenta namely J1 and J2. So, these are the two angular momentum quantum numbers and correspondingly I have two operators J1 operator and J2 operator where J1 corresponds to the first angular momentum quantum number J1. That means when I do J1 operation this J1 will be the remember this is an operator this is a quantum number. So, this J1 is a quantum number which appears as I mentioned earlier. So, if you want to add these two the quantum mechanical rules are very clear you will get the many resulting states when you add these two angular momenta J1 and J2. The resulting states have angular momenta given by not just one value you have many states possible those states are given in terms of the new angular momenta starting with J1 plus J2 that is sum of the two sum minus 1 sum minus 2 and so on you go down reducing every time by 1 ultimately you will reach J1 minus J2 take the positive value the modulus. So, you basically you have various states with various angular momenta possible when you add two angular momenta J1 and J2. So, these are various states possible when you add J1 and J2 if I substitute the actual values numerical values of J1 and J2 which I will do later on you can see what are the resulting states possible. So, usually you get a large number of states definitely 2 3 4 states possible when you add J1 and J2 unless you are one of them is 0 all other cases you will get many terms of this kind which are actually the resulting states coming because of the addition of two angular momenta. If you have 3 of course then you have to do it again this is something which is very important information which we will need when you try to construct the magnetic moment of various electrons and so on or for a given electron itself how the orbital and spin add together and give you the total magnetic moment. Same rules are applicable for spin angular momentum also as I mentioned as far as this course is concerned most of the rules of orbital angular momentum and spin angular momentum will be identical and hence S1 and S2 if I take two spin operators of two electrons and electrons S1 and S2 will be half using the same argument of S1 plus S2 to S1 minus S2 I can get two states only one corresponds to half plus half is 1 the other thing is half minus half is 0 nothing else is possible. So, I have two states S equal to 1 and S equal to 0 S equal to 1 state is actually called a triplet state because corresponding to S S is an angular momentum as I mentioned. So, it has a corresponding magnetic component associated with that which is MS. MS as I mentioned the values are from minus S to plus S like minus L to plus L with a gap of 1. So, it is 1 to minus 1 with a gap of 1 is 1 0 minus 1. So, you have three states arising out of S equal to 1 and hence this is called a triplet. When you have another thing that is S equal to 0 there is nothing else possible there. So, MS only value that is possible here is MS equal to 0 and hence this is called a singlet. As far as magnetism is concerned this distinction is very very important little early to tell, but I am telling you that as far as magnetism is concerned a singlet state is not good singlet represents non-magnetism we are not interested in that, but we should know why the difference comes as we go along we will see this difference. So, the spin multiplicity is an important issue here it is a singlet here it is a triplet this difference actually gets reflected in the magnetic character as we go along. So, spin is a very important property as far as magnetic materials are concerned this is something which is very different from the classical electromagnetism you have talked about. There you would not talk about spin at all this was purely an electric current generating magnetic field, but here it is the magnetic material you have a spin contribution which is very very important in determining what kind of a magnetism at kind of a magnetic moment your electron has or your atom has or your solid has. So, this is something which is the singlet versus triplet please keep in mind this is a very important step as far as our future things are concerned. Related to that another very important because when we talk about magnetism there are various interactions one has to worry about we cannot neglect. In that context one of the very important aspects is what is known as a spin orbit coupling you remember I talked about spin angular momentum orbital angular momentum and associated with angular momentum you have a magnetic moment. So, you have correspondingly spin magnetic moment orbital magnetic moment. So, when I talk about this that means basically I have one magnetic moment another magnetic moment both associated with the same electron. Is there a coupling between the two that is what is called a spin orbit coupling. So, spin orbit coupling is essentially a magnetic interaction this is a very very important interaction. So, how do you see that is a very deeper meaning associated with this. So, as I mentioned electron is there take a very classical picture more kind of a picture electron is there electronics moving in a circular orbit for example let us assume that way it has got the orbital angular momentum and spin angular momentum and hence it has got orbital magnetic moment and spin magnetic moment. Let us assume that we are in the frame of reference of this moving electrons we are going to see relativity very soon. So, assume that I am moving with the electron that means in that frame of reference electron is not moving what will be moving will be the proton the nucleus hydrogen is the in the case of hydrogen nucleus is nothing but a proton. So, what is going to happen if I travel sitting on the electron I will see that the electron is stationary I am stationary and the proton is actually circulating. So, it gives rise to a proton current. So, this proton current as you have seen in the electron magnetism will give rise to a magnetic field this is a relativistic effect I will not go into the details. It will produce a magnetic field but remember when I tell that I am sitting in on the electron and moving only the orbital angular momentum is not there because it is not moving but there is other contribution namely the spin angular momentum which is independent of these spatial coordinates or the spatial motions. So, it has got a spin angular momentum which is independent and the spin angular momentum gives rise to the spin magnetic moment. So, what is going to happen if I look at the picture there is a proton current as seen by the electron frame of reference and at the site of the electron there is a magnetic field produced by the proton current and my spin magnetic moment which is independent of the spatial motion as I mentioned is actually influenced by the magnetic field produced by the proton current. So, you have a dipole which you have seen in the other code you have a dipole mu which in this case is purely the spin dipole moment interacting with the magnetic field produced by the proton current. So, the interaction energy as you have seen is your minus mu dot B term only difference is your mu is not a total magnetic moment but only the magnetic moment associated with a spin path and hence I write it as minus mu s dot B. So, this is the interaction which is called a spin orbit interaction spin orbit coupling which is very very important in discussing many other things. So, this is something which you cannot avoid in many situations this coupling will be very strong and in many magnetic materials this is an important contribution. So, please understand the issue associated with this there is some relativity here some electromagnetism here. So, the fact here is that electron has two kinds of magnetic moments because of the two kinds of angular momenta when you are travelling with the electron you are killing only the orbital path the spin path is there the spin path interacts with the protonic regenerated magnetic field giving rise to this one. This has got consequences which you will see later. Now, having said that let us try to find out what is the total magnetic moment of an electron we are coming to the basic issue which I started off how does the electron get a magnetic moment. As I mentioned the magnetism comes because of the orbital angular momentum and the spin angular momentum and the proportionality constant is a gyromagnetic ratio which is written here I have given the value straight away it is E divided by 2 m c c is a speed of light in vacuum this c will not be there if I am using SI units but since I already told you that I am mostly sticking to CGS units my c will be here. So, this is the gyromagnetic ratio and you can see a minus sign here this is to tell that your electron is negatively charged. So, this minus will not be there if I talk about the proton this is representing the relation between electronic orbital angular momentum and a corresponding electronic magnetic moment. This is the same thing for the spin part spin angular momentum relating to the spin magnetic moment only difference they you have a gyromagnetic ratio here except the fact that there is an extra 2 here which of course I will not go into the details there is a 2 here. So, the total magnetic moment comparing the 2 which again need some discussion. So, I will give you the straight away the result the total magnetic moment can be written in terms of this gyromagnetic ratio E by 2 m c and a new factor comes into picture which is a G factor G which of course some of you have heard about it and a new quantum number in place of your L operator and S operator I have a new operator J coming here J is nothing but your L plus S operator L is orbital angular momentum S is a spin angular momentum. And this is addition of 2 angular moment I mentioned and I told you that do not distinguish between the orbital and spin. So, the usual rule I mentioned earlier is 2 here that means the resulting states are given by quantum numbers the total angular momentum quantum numbers J given by L plus S, L plus S minus 1 so on it goes on down to L minus S. So, that is what you see here exactly the same addition of angular momentum I talked about sometime back. So, only factor which is coming here is G this called a G factor 1 can show with a little bit more of quantum mechanics 1 can show the expression for G is given in terms of these quantum numbers L, S and J, L of course you know what is L, you know what is S the quantum number and J. Please do not confuse between the operators and quantum number this when I tell addition of 2 angular momenta J vector operator equal to L vector operator plus S vector operator. What I am giving you here this is just 1 it is always plus I am adding 2 angular momentum the resulting states are represented in terms of the corresponding quantum numbers which are just the numbers that is just J no operator no vector nothing it is just the number they are the one which actually have multiple values. And these quantum numbers are appearing here in the determination of G as given by this expression 1 can derive it, but not in this course with 4 lectures. So, I can write this magnetic moment as in this form. So, G times the gamma the gyromagnetic ratio times this is the magnitude of this vector J. So, this you take this whole total angular momentum and magnetic moment if I write the magnitude alone. So, the magnitude of this can be written in this form that is magnitude of mu this G and this will be same and you have the magnitude of J vector. This is given by as I showed you J square gives you J into J plus 1 h bar square. So, you get the J magnitude will be square root of J into J plus 1 h bar and this I can write. So, this h bar is you can see here this h bar is a constant as you know this h bar will be joined here. So, it becomes e h bar divided by 2 m c m is a mass of the electron please remember m is a mass of the electron. So, what I do is I define another constant e h bar divided by 2 m c in CGS units if it is Si units it will be e h bar divided by 2 m c will not be there. This I am going to call it as a new constant called a Bohr magneton that is called mu B and in terms of this I can write the magnitude of the total magnetic moment of an electron under these conditions as G times the G factor times the Bohr magneton times square root of the quantum number J into J plus 1 that is what you are seeing here. So, mu B is the Bohr magneton if you remember h bar is the unit of angular momentum. Similarly, mu B is a unit of magnetic moment. So, magnetic moment one way to define the magnetic moment like length is in sub centimeter or time is in units of second. The magnetic moment the most convenient unit for magnetic moment especially the atomic magnetic moment of course, for the solid salts also later on is the Bohr magneton which is mu B. So, this is the total magnetic moment an electron can have when it has got orbital contribution and spin contribution. When I am telling that if it has got both the contributions as you know for example, in a shell given by L equal to 0 the so called S state S electron. S electron you know that L equal to 0 in such a case there is no orbital contribution. The contribution to the angular momentum will be purely from the spin part and hence the magnetic moment will be purely a spin magnetic moment I will come to that. So, this in general when you have both the contributions I use a quantum number J and the total magnetic moment is given by this. Now, if I am looking at the Z component of the magnetic moment that means, I take the projection of this total magnetic moment along Z axis. This what I have to do is I have to take the dot product of this this thing about the Z axis that is what is the angular momentum J dotted with Z cap. Z cap is a unit vector along the Z direction. So, this gives me G times this of course, is there J dot Z is nothing but the Z component of my total angular momentum which actually as I have shown you the Eigen value equations which is nothing but M J the projection the quantum number the magnetic quantum number corresponding to the total angular momentum which is M J H bar. So, that is what is M J H bar. So, on substitution I see that this is minus G times if I do not worry about the magnitude it is G times G bar magneton times M J. Now, you see the why it is called magnetic quantum number because it is going to determine if your M J is 0 this is going to be 0 the magnetic moment along Z axis is going to be 0 because G is a constant in a given case mu B is a constant of course and your M J is going to determine what kind of a magnetic moment along the Z axis is going to happen for this particular electron. So, this gives you the Z component of the total magnetic moment of a single electron in a situation like that of a hydrogen atom. Now, this picture is very important what is shown in the earlier slide regarding the total magnetic moment where the magnitude is given by square root of J into J plus 1. Remember J represents the total angular momentum quantum number and the Z component is given by M J which I already showed you in the Eigen value relations these two things are shown in this picture. This is the so called Z axis which in atomic physics is called the axis of quantization. This is called the axis of quantization which is the axis along which we generally apply a magnetic field here we are not done so far. There is no magnetic field applied so far in this case, but this is the axis of quantization which is Z axis. This is the magnetic moment I talked about which acts the value of G times mu B times J into J plus 1. Do not worry whether it is capital J or small j generally when you are talking about single electron we use small j otherwise it is multi electron we talk about capital J do not worry about it. So, the condition is that the quantum mechanical conditions are very important here this vector this is a vector the total magnetic moment has to be aligned in such a manner with respect to the axis of quantization that this should always give projections along the this one where your the projections are nothing but your M J values. M J values cannot be anything M J values are determined by your J values for example, if my J is 2 my M J values are 2 1 0 minus 1 minus 2. So, which means this theta that is shown here cannot be anything only certain values of theta that is theta is the angle between the so called total magnetic moment direction and the axis of quantization. You cannot have any kind of an angle any arbitrary angle will not work because that will violate my original definition of what is my M J with respect to my J that is gone. So, my theta is quite restricted this is a very important thing in quantum mechanics or quantum theory. In fact, it does not need quantum mechanics even in quantum theory days which is little earlier than quantum mechanics this was found out. So, what you see is this theta is not a classically continuously varying theta this theta is very very discrete discrete in the sense that this discreteness in theta should give you discreteness in your M J value consistent with your J values. Once your J is fixed your M J is fixed in some sense this is something very important number 1. Number 2 you see that your theta cannot be 0 if your theta is 0 because this is not a good number whereas your M Js are all good numbers either integers or half integers and 0 it cannot have any funny number whereas this is a square root here. So, if this is completely along this direction that means your theta equal to 0 that means your total magnetic moment of a single electron is completely aligned along the axis of quantization or if I assume that I applied a magnetic field here that means it is completely aligned along the applied magnetic field. Then it is violating the condition that my M Js are all integers or half integers. So, when I tell that the moment is completely aligned along certain direction of the magnetic field it does not mean that my theta is 0 theta cannot be 0 this is a very important thing. Maximum component that you can get along the axis of quantization all along the applied field is a maximum value of my M J and the maximum value of M J that I can get for a given value of J is J itself. As I gave you an example if I take J equal to 2 I can get the values of M J as 2 1 0 minus 1 minus 2. So, the maximum M J I can get is 2 whereas if I take this one is square root of 2 into 3 square root of 6 which is not an integer which is not a half integer. So, this complete alignment making this theta equal to 0 is not possible here. So, this angle theta will always be non-zero. However, if you are looking at a classical picture very classical world if you see what is happening is your classical ideas can be generated then you go to very large quanta numbers. If your quanta numbers are very large let us say my quanta numbers of this J here are very large. Then if I have very large quanta numbers the square root of large quantity times that quantity plus 1 the J into J plus 1 for example J is 100 100 into 100 times 100 plus 1 square root is not quite different from my 100 here which is a maximum value. So, classically speaking you are justified in taking a continuous variation of theta that means your magnetic moment can align in any with a with respect to the magnetic field in any angle that is ok. But in quantum mechanics when you are looking at very small quanta numbers this is a big difference and you have to make the condition that theta is discrete number 1 and theta can never be 0 number 2. This is a very very important picture as far as the connection between angular momentum and magnetic moment of a single electron is concerned. So, this is some picture some summary of what is happening as far as the magnetic moment of a single electron is concerned. Now, just as I mentioned suppose I have a orbital there is there is no spin as I mentioned an s electron 1 s electron of hydrogen atom s orbital has l equal to 0 p orbital has l equal to 1 d orbital has l equal to 2 the orbital angular momentum part. So, if I take s orbital electron my orbital angular momentum is 0 I have only pure spin part for which my g factor is actually 2 if I substitute the value you can see that it is 2 you can actually do a very simple exercise. When I substitute and find out what is a total magnetic moment it will be purely spin that is what is written as s here mu is mu s I substitute and I can show I already have shown you this you can see I will get one more magneton. So, a pure spin magnetic moment that is if I take the hydrogen atom ground state electron 1 s electron the magnetic moment associated with it is just one more magneton as I mentioned more magneton representing the unit of magnetic moment here. So, this is a very important numerical result now comes a related issue. Now, if I bring in a magnetic field as I mentioned there is an interaction between the magnetic moment and the magnetic field the usual interaction is called a Zeeman interaction the interaction of any magnetic moment with an external magnetic field is called a Zeeman effect there is a Zeeman term the interaction energy is given by u is minus mu dot b which already we have used this when I substitute the minus is here already there was some minus so this becomes plus g times mu b times j this represents the magnetic moment mu dotted with b the magnetic field you applied this gives me j dot b remember b is generally taken along the z axis. So, it gives me g times mu b times m j m j representing the magnetic quantum number associated with my j as I mentioned. So, this what if my m j is 0 that means this magnetic moment is not there for example, if j equal to 2 there will be 5 levels possible that is 2 j plus 1 plus 1 is corresponding to 0 otherwise you have plus m j minus 1 and minus this thing that is 2 1 and minus 1 minus 2 and 0 come that is why you have 5 states. So, there are all 5 different states and now in presence of a magnetic field you can see the energy the interaction energy u is determined by your m j different m j s will give you different energy values here. So, that means you see that you will see different 5 different energy levels separated. Whereas, if this magnetic field was not there if this c-man field was not there what would have happened I would have got these states 2 1 0 minus 1 minus 2 that is still meaningful, but all those 5 states would have got the same energy or I would have told you that it has got a degeneracy of 5 or a 5-fold degeneracy was present when the magnetic field the z-man field was absent. So, when I apply a magnetic field to a situation like hydrogen what I am doing is actually removing the degeneracy or lifting this degeneracy associated with the z-man interaction which is given by minus mu dot b is a very simple z-man effect is a very elegant demonstration of this magnetism of this electrons and of the atom in this case hydrogen atom electron means the atom itself because there is nothing else that we can account for. So, the degeneracy of 5-fold degeneracy associated with j equal to 2 m different example is lifted by the magnetic field. So, magnetic field when you apply to the atom later we will see in the case of solid also generally the degeneracies are removed different energy states will come in the picture this mj is going to determine what kind of an energy we are going to have and mj changes from 2 to 1 to 0 the energy is also going to change which is not the case when there is no magnetic field. So, z-man effect is historically very very important in the discussion of any atomic magnetism. Now, we are ready to just see what happens to the I will just finish with this atomic magnetism you know what is a contribution from electron we have seen just now. An atom is made up of such electrons hydrogen is special because I have only one such electron. But if I talk about an atom in general I have many electrons which are contributing to this there will be a summation of all these contributions together will give you the atomic magnetism. So, this has to be treated in a very general sense because I want to give you a quantum mechanical flavor of this in the light of the quantum mechanics that you have studied plus the electron magnetism that you have studied so that you have a little formal introduction to magnetism of an atom. How does an atom in general when it is subjected to a magnetic field gives rise to various responses not just one response in general an atom kept in a magnetic field is subjected to various responses one response you call it as diamagnetism another response you call it as paramagnetism and so on. There are many effects I will worry about diamagnetic response of the atom not from the solid will come through that debate. And the paramagnetic response these two things if you want to appreciate in a very very basic level one has to look at let it more carefully a little more quantum mechanically that is what is to be done. I think I will give some time for getting some questions based on whatever I have told you till now. 1016. Gages away I go ahead. So, my question is for the nuclear magnetic moment is negligible as compared to electron magnetic. Yeah. Are there any situations where the nucleus spin due to different atoms they add up and become significant more than the electronic. Yes, yes very very important question. See if you see the relation the constant of proportionality between angular momentum and magnetic moment the so called gyromagnetic ratio has a form I have already given you it is E by 2 m the m is the mass of that particle. So, when I consider electron I will put the mass of the electron. As you know the mass of the proton or mass of the neutron it is typically 2000 times more than the mass of the electron. So, your numerator is increasing by a factor of something like 2000. So, the quantum number is more or less same order right. So, what happens is the corresponding magnetic moment will be much much smaller. So, when you compare electron versus proton or proton versus neutron we do not have to worry about the neutron contribution and the proton contribution. But your question is very important because of this thing neutron has a magnetic moment I forgot to mention since you are raising I will talk about it. Neutron has a magnetic moment is very very important because this allows you to find out the magnetic properties using a different technique. What is that technique? That technique especially when sitting in bomb bay view how to talk about it that technique is called a neutron diffraction. Neutron has a magnetic moment and this magnetic moment can interact with the magnetic moment of atoms of your solid or your nanomaterial or your thin film magnetic thin film. This interaction can be studied by sending like the way you send x-rays. You can actually send neutron beam and study like you have x-ray diffraction. It x-ray diffraction has certain problems certain shortcomings which you cannot help it. Neutron diffraction technique overcome some of these things and neutron diffraction is a very very important tool today to understand magnetism specifically. So there we use this neutron magnetic moment but then I generally talk about it I will not worry about the magnetism of the nucleus the neutron or proton. The main difference is because of the large mass difference between electron and proton. 1157 go ahead. Why the earth is acted as a natural magnet? Basically it has got all these magnetic elements that we talk about all is in the earth only right. Earth contains all these things whatever magnetic elements which I show the thermo magnetic elements in the periodic table everything is essentially there in the earth under the earth. So that must be one of the reasons people tell that there is a circulating current I am not very sure about it but my feeling is that these things which are there in the core of the earth must be producing this magnetic moment. The about the rotation of this one I do not know frankly I do not know. So earth is a very giant magnet. Thank you sir. 1210 go ahead. So what is the difference between the magnetic and non magnetic materials? Okay that is what I told you in the beginning there is nothing like if you really bring in my classification that is dia, para, ferro and other things there is nothing like non magnetic. I say that is why I have written in one of the captions I have written that in some sense everything is magnetic. Why this confusion what is coming in from your question is coming is because of this. Usually we look at from the point of view of applications. For applications you look for materials where at least you have a permanent moment that means you have to rule out diamagnetism. You have to start with paramagnetism and the ordered materials of the three varieties. So if you do not include a diamagnetism and some people do not include a paramagnetism also they only include ferro, ferris, antiferro kind of materials then they will tell okay these are magnetic other things are non magnetic. So that is why to avoid any confusion my classification has 5 and if I take all 5 into account I cannot tell anything non magnetic everything that is why I showed you a particularly the periodic table everything is magnetic and that is why I have put a caption also everything is magnetic nothing like a non magnetic in this one. So usually what people are telling non magnetic is mostly the diamagnetic. There is no difference between the two I mean the way you look at it. Okay sir thank you another question sir. So how that velocity of light is related with the magnetic field? It is a very important thing in fact that is what is creating this new unit difference you see is coming extra in all these things remember c square is 1 over square root of mu naught epsilon naught. So the material properties are related to the speed of light. So this is a historically a very important physically and historically there is a very important connection between the two. So basically it is a relativistic effect that is coming into picture why the c is coming there. Please write to me so that I don't forget it we will answer that through our email yeah. Okay sir definitely but that c is not related with the electrons sir the electron mass that is m and c totally they are different. Yeah that is because see if I use SI units the c will not be there even in this case it is just if the gyromagnetic ratio will be simply e by 2 m. Okay sir thank you. One three one three go ahead. There is a charge particles such as an electron experience a force from its own magnetic field. In this case for example I talked about how I actually showed you the spin orbit coupling. Okay in some sense what you are asking is this electron has essentially two components to the magnetic moment right one is a spin part and is orbital part. When I talk about spin orbit coupling I brought this same issue. The orbital part is responsible for the creating the magnetic field but what we have done is to bring in the relativistic effect here we have actually taken the electron to be the frame of reference in which the rest frame. So we have seen that essentially electron orbital angular momentum since we are treating it as a rest frame the proton is the one which is giving rise to the current which actually gives rise to the magnetic field which actually is felt by the spin part of the electron. So in that sense the same electron another contribution to the magnetism which is coming from the spin part is influenced by this magnetic field usual Zeeman kind of a mu dot d effect that is what gives rise to the spin orbit coupling. So in that sense your question is right. So that interaction happens specifically in this case. Okay sir thank you. 1, 2, 5, 5 go ahead. Sir you classify the magnetic diapara and then you group them and then orderly you go for that side this one sir that is diapara okay and then ferromagnetic if it is a magnetic moment in the single direction and the anti ferromagnetic in the opposite direction. But in case of ferrimagnetic it is in opposite direction but uneven magnetic moment. Then how it is under orderly materials. Yeah. So what happens is I think I have another picture where this is schematic I showed you the one dimension thing. So what you have is there is an order it is not that this moment is somewhere here it is actually aligned exactly anti parallel to the first one this moment is in this direction plus z this is minus z this is plus z there is an order here not like this one where there is absolutely no order this is a paramagnet there is no order this is randomly oriented. If I tell that there is an order here I should accept that there is an equal amount of order here. So for example I as far as I worry about entropy which actually determines order or disorder I have to treat this exactly in the same way as the ferromagnet or the anti ferromagnetic. There is no difference as far as order is concerned among the three the nature of the order is different but the degree of order is the same. Can you ask again if you have some more clarifications needed. Sir one more question sir. Yeah. Suppose when we are calculated the magnetic moment values it is does it match with the theoretically observed magnetic moment values actually what are the factors affecting these things sir. So in the case of an atom there is no problem it matches very well in the case of solid you will see that it will not match generally it will not match in the next lecture I will show you where it matches where it does not match and the reasons. There I will make the difference between that is why classified into two things one is metallic systems and in the insulating systems the contributions to this discrepancy between the experiment and theory these differences are different in the two classes of materials that is why I made them separate. So I will show you the actual comparison and explain the reasons the reasons are different in different systems I will talk about all of them your statement is right in general in the case of a solid it will not match they will not match but I am in the atomic case there is no problem I think the time is running out as I told you in the beginning in my lectures there are certain derivations which are mostly written by me but please try to work it out if there is any problem with those derivations also please write I will try to clarify please do this much.