 Good morning, as you know the topic was given to me is some basic ideas of magnetism in materials. Materials mostly as you are going to see will be on solids and I will mention as we go along how the thing is done. In some sense as you can see from this very standard thing the periodic table it contains a large number of elements of course, the number keeps going I think even today. If you actually can see this strictly speaking every every element that you can see in this periodic table is magnetic if you bring in diamagnetism also. So, any material that you consider definitely is magnetic if you bring in diamagnetism also. But in certain materials would be special materials what happens is certain other magnetism will be more prominent in that case the diamagnetism is not seen. So, that is why if you see here there are only three elements which are actually iron cobalt and nickel which are actually having a curie temperature above room temperature or they are ferromagnetic at room temperature all other things you can see many of them are diamagnetic and many of them have curie temperatures much lower than room temperature. Still the surprising thing is that even with a small number of I mean essentially three elements as you can see most of the applications that we are using every day in this room outside in our daily life left and right whatever we are using many things actually contain magnetic elements or magnetism is there. So, magnetism has become very indispensable as I am going to show you in various technologies, various applications and it is going and growing every day. So, that way that is a very interesting thing only with these three elements and most of the applications are at room temperature, but still we are able to manage with these three elements. Fact is not really true because even though there are large number of so called non-magnetic elements which are actually the diamagnetic kind of elements you can get lot of choices if you can combine these three elements with other things. So, you can generate a large number of compounds materials by combining them and that is what is actually working in many of these devices. Another thing that you can see in fact I have another slide which I again tells you that what are the magnetic states of various elements you can see very few elements are the so called paramagnetic elements as you can see this is in the atomic state what is shown here and many of them are diamagnetic and very very few actually three room a ferromagnetic at room temperature. Still we are able to get a large the large number of magnetic compounds magnetic materials and they have all kinds of interesting applications and these applications are actually growing every day as I am going to show you ok. So, just let us start with a very very basic classification the classification is something like this this is you will see in any standard textbook you will be seeing this broadly they can be classified as diamagnets as I mentioned which actually is not really of interest as far as applications are concerned, but still that has to be considered. Then you have paramagnets then you have ferromagnets, antiferromagnets and ferrimagnets these three categories are called ordered magnetic materials or they are magnetically ordered whereas diamagnets and paramagnets are not ordered and for most of the applications you need ordered magnetic materials which means that you will be mostly using ferromagnets and very small extent antiferromagnets and a very large extent ferrimagnets. So, the ferrites for example, they belong to ferrimagnetic materials which as you know have lot of applications. So, the difference is they are not ordered these three are ordered of course, this this list can be made longer, but that will be a very special thing I will not go into those details I will make it very broad classification like this. Just to give you the idea how these things are different you start with diamagnets diamagnets are essentially cases where you do not have any permanent magnetic moment what is the meaning of permanent it will be seen very soon. So, there is no magnetic moment as it is unless you apply a magnetic field you apply a magnetic field you start seeing a moment in other cases what you see is that you apply a magnetic field you start seeing the ordering, but in this case diamagnets there is no moment in the first place you create the moment and as long as the magnetic field is there that will be there once the magnetic field is removed the moment is also not there. So, that is why I have written clearly that there is no permanent magnetic moment and permanent magnet this as we are going to see it will happen in all electronic orbits wherever there is an electron and there is an orbit this diamagnetic contribution will be there whether it is going to dominate or not is a different issue, but the diamagnetism will always be present if there is an electron and this is not a permanent situation the moment is a function of the magnetic field. Now, coming to the paramagnets you can see this what is shown here where the circle is an atom. So, these are different atoms and the arrows show the magnetic moment the total magnetic moment and you can see that there is no ordering and they are randomly oriented like this they are actually paramagnets, but the difference from the first case is that these are all permanent moments irrespective of the magnetic field that is present they are all having a magnetic moment. In the first place they have a magnetic moment only the magnetic field does is to align them depending on the magnetic field strength and the temperature otherwise they have permanent magnetic moments. If these so this becomes a building block these paramagnets actually becomes a building block for getting the ordered materials namely ferromagnets, antiferromagnets and ferrimagnets. If the order is something like this that means everything is in the same direction then it becomes a ferromagnet if it is up down up down where these arrows actually represent the total magnetic moment I will tell you what is total magnetic moment right now let us call it as total magnetic moment. So, if the ordering is like this this is also an order only thing is that in the bulk scale if you see there you can actually see a magnetic moment whereas in the second one there is zero magnetic moment totally and the ferrimagnets you what you are seeing is essentially it is an antiferromagnetic kind of an arrangement but what happens is you have two different kinds of atoms or different kinds of ions their magnetic moment there the magnitude is not the same but they are oppositely aligned. So, even though the alignment is antiferromagnetic you will have a net magnetic moment at the end. So, that is the difference between ferromagnets, antiferromagnets and ferrimagnets. So, these three come under the category of ordered magnetic materials. Before we really try to see each of these categories we will also talk about some of the other ways of classifying them this is another classification this is also important to understand magnetism in some sense that is why I am giving you this classification also basically I can classify them as metallic like for example iron, cobalt or their alloys all of them come under the category of metallic magnetic materials many of the permanent magnets that you see are made of such alloys. So, they come under the category of metallic magnetic materials then you have magnetic ceramics that means they are electrically insulating but they are magnetic best example is ferrite ferrites are all magnetic ceramics they are their electrical resistivity is very very high whereas in the first case it is very low. Then other this difference is very important because in the first case metallic materials the electrical resistivity resistivity is very low and as you must be knowing if you are actually subjecting to a cycle of magnetic field or if you are alternating the magnetic field you will always end up with eddy currents as you have studied right. So, this resistivity being very small where eddy current losses are going to be very large whereas in the case of ceramic materials like ferrites even though the frequency is very high like microwaves and so on there is no such loss as far as eddy current is concerned. So, that is why any of the high frequency applications what you see is the magnetic ceramic materials ferrites are very important because they are they do not have this problem of eddy current losses and they are always preferred for high frequency applications. So, this electrical resistivity difference is important. Then comes from the point of view of application as well as from the point of view of fundamental physics this classification of thin films and multi layers multi layers is somewhat a recent phenomenon but thin films of course for a many many decades it is known because of the magnetic recording and so on magnetic thin films and multi layers are very important. These days of course we cannot skip this thing that is nanostructured magnetic materials as in the other fields. Magnetism also shows a lot of change when the material is reduced in dimension from the bulk the so called bulk dimension to a nanoscale dimension you can actually see wonderful properties sometimes good sometimes bad both are studied and this is a very very important topic of research today the magnetism in nano materials because of various applications this is very interesting today. Another very new somewhat new topic is molecular magnets magnetism in molecules again very very interesting lot of chemistry people are actually doing because of the interest that they have. So, molecular magnets are very very important today. Then comes all these things are essentially in the solid form but now some of you must be knowing magnetism actually is extending to the other phase also that is liquids where you have ferromagnetic suspension of ferromagnetic particles in the liquid medium by which you can actually by applying a magnetic field you can change the liquid properties like viscosity and so on there are many many properties I am just giving a flavor of that. So, ferro fluids again is a very very important topic of course India has contributed a lot to this field and it is going on today. So, this is another way of classifying various magnetic materials mainly look with respect to their application in some sense. Another still another way of classifying this is more with respect to the crystalline nature of that first is the single crystals as you must have studied you must be knowing single crystals means you have a completely a long range order throughout the material which of course is not possible and you will always have some kind of a defect thermodynamically it is the only situation that is possible but still you can have a very good long range order throughout the material. So, that becomes single crystals but again as you know that size of those crystals that you can grow in the completely single crystalline form is very limited very very small. So, practically for any application this is almost ruled out because you cannot get sizeable crystals and hence one cannot really work with that. So, most of the time you work with polycrystalline samples most of the applications that we use magnetic materials all such materials most of the materials are actually polycrystalline in nature which means that you have grains and grain boundaries grains means you have some kind of a order crystalline order I am not talking about a magnetic order here the crystalline order over some region and then you have some problem and then you have other grains. So, the grains and separating them the grain boundaries again very important thing in determining the magnetic properties. So, that is other class which actually is easiest to think to prepare because single crystals are extremely difficult to prepare polycrystalline is alright. Then comes amorphous amorphous again these days applications are coming of course it was known earlier also but this again is increasing today amorphous materials is extreme of the other extreme end of single crystalline there is absolutely no long range order throughout long range crystalline order and this gives rise to amorphous materials. But as single crystals amorphous materials also are very very difficult to prepare. In single crystal preparation as you must be knowing you need an extremely slow cooling so that you give enough time for the atoms to go into regular sites and settle down. So, you need an extremely slow cooling rate on the other hand for amorphous materials you want them to arrest in totally random sites for that you have to have an extremely fast cooling. So, this is an extremely slow cooling this is an extremely fast cooling that is why something in between that is a polycrystalline sample this is the easiest thing to prepare. Then of course various methods of nanostructured materials are possible one thing is from the one of the ways is to have an amorphous thing and then give a annealing you heat it locally the crystals will grow and usually if you can control the size over with the crystallinity grows ease of the order of nanometers and then you have these nanocrystals embedded in an otherwise amorphous matrix. So, that way amorphous and nanostructured are in some sense connected but of course as you know there are other methods of preparing nanomaterials. In fact, most of the time use other materials but this is also one technique by which you can prepare nanostructured materials. Coming to another way of classifying them this is purely based on applications which of course all of us are using and this is something which is kind of everyday happening thing. So, the first is the soft magnetic materials Soft magnetic materials means this is not soft with respect to the mechanical properties it is with respect to the magnetic properties I am going to show you later. This is basic as you must be knowing things where you need large magnetic flux like in the case of a transformer all such applications what you are using is soft materials soft magnetic materials. In fact, the core that is used in transformer is called soft iron soft iron because it is magnetically soft. So, that is why because it can actually link the flux very well. So, the permeability is very high such materials are called soft magnetic materials. Then comes the other end that is a hard magnetic materials where you do not have this condition you have exactly the opposite condition where once it is magnetized it has to work indefinitely that is what is nothing but your magnet permanent magnets the bar magnets other things that we use every day in our labs and so on they are basically made up of hard magnetic materials exactly opposite of what is there in the soft case. Then comes high frequency materials obviously Farais will come as the first material here as I mentioned they are preferred because high frequencies the eddy current is a problem and hence you should take materials where the electrical resistivity is very very high materials of this insulating category ceramic category are preferred and they are called high frequency materials. Then you have magneto optic materials for various magnetic reading systems magneto resistive materials where the magnetic property actually the is coupled to the electrical property and hence by applying or changing the magnetic state what you can do is that you can change electrical resistivity and such properties and such materials are called magneto resistive materials very important for applications because of magnetic recording is done with the help of this magnetic reading is done with the help of this. So, this is very important in terms of magnetic recording industry magneto resistive materials GMR heads and other things you must have heard about it that is coming from this property or this particular application is being exploited there. Then comes magneto strictive materials where the dimensions of the material change as a function of magnetic field you take a ferromagnetic rod you apply a magnetic field the length increases that is important for various sensors and so on such this phenomenon is called magnetostriction it can be a length it can be volume various possibilities are there such things are actually very important for applications and those materials are actually called the magneto strictive materials these magnetostrictive materials were known in 80s that is a very very hot topic and the strain the change that happens is typically very very small of the order of some 10 to the power of minus 6 of the original thing that is very small but still it was used in many applications today you what you see is there a lot of developments in this one the conventional materials are not used you have lot of new materials coming one very important discovery in this aspect today is what is known as ferromagnetic shape memory materials I do not have time to go into those details but I will mention here because such materials due to a different property what you get the volume change or the length change that you get when you apply the magnetic field is enormous compared to this one it is enormous and such materials are actually called ferromagnetic shape memory materials so they are kind of in some sense they are able to substitute for these materials in a big way very similar to piezo materials yes absolutely absolutely so in fact originally the magnetostrictive materials were thought of as substitutes for piezoelectric materials they were actually used in transducers basically magnetostrictive transducers because what happens is as I mentioned when you apply magnetic field the length changes so if you if you apply an alternating magnetic field in the rod for example vibrates because it length increases and decreases it vibrates so what are you doing you are actually applying electrical energy that you are converting to magnetic energy this magnetic energy is producing this vibration that means you are converting again to mechanical energy so there is there are two steps of energy conversion one from electrical to magnetic and then magnetic to mechanical so reverse also is possible in submarines and so on so called sonar applications where people use these transducers to find out the presence of other things because this waves the way it comes they give the correct signal by undergoing this magnetostriction the in the reverse sense so they were exactly able to replace the piezoelectric transducers so magnetic transducers became kind of substitutes for the piezoelectric thank you yeah you use the magnetic and piezoelectric material for the production of patra sound is there also yeah so what ranges of frequencies are to be produced I don't remember the exact values right now I have to check right typically this I think the frequencies are generally very high is it very efficient to use the magnetic materials than the piezoelectric okay so that you can always argue I mean people working in piezoelectric always will tell that that is better magnetostrictive people will tell that is better there are certain pros and cons in both so that way it's where you can't tell that okay this is the best or the other thing is that you cannot tell there are certain at where for like one big problem since you are asking I'll tell you is that these materials they're highly corrosive many of them are highly corrosive compared to piezoelectric ceramics so what happens is with the time there is a problem so this is a problem with most of these these many most of the good magnetostrictive materials are actually metallic systems so naturally they will have this corrosion as a big issue I am not telling that that is not correct I mean I'm not refuting that what I'm telling is I'm just giving you the possibilities of these materials where the possibilities have been already worked out I am as I mentioned just now I am not telling that piezoelectric is bad I can't make that statement but there are certain plus points and certain minus points with both of them choice depends on the kind of application the kind of surrounding that you have the kind of application that you are looking for I am only telling you what are the possibilities of these materials and these these are not just taken to just show you but they are actually used these applications are being used magnetostrictive transducers are there in fact BRDO had a big funding for such materials and I understand that such materials are being used such transducers are being used in submarines so yeah basically they are put in the submarines so the DRDO's program was to used in submarines okay then comes magnetic refrigerant materials which again is a very new thing this is something which is thought of as a substitute for the conventional cooling mechanisms I will not go to those details because all these topics by itself will take at least a few lectures to finish then comes as again as you know you have this spin tronics which is coming a big way replacement for the conventional electronics lot of things are going on very exciting things are going on where you are actually going to use in the conventional electronics you actually exploit only the charge of the electron but in the case of spin tronics you are actually looking at the spin also the spin also is taken into consideration and this plays a lot of role and gives you lot of new things and that is becoming a very important applied as well as fundamental research area that is the so-called spin tronic materials which actually easily giving rise to what is known as half metallic systems identification of such half metallic systems is very important today then again as some of you must be knowing lot of magnetic materials are being considered I can only tell considered I do not know whether it is really there in real sense many materials are being considered as bio materials and for medical applications some of these nano materials as you must be knowing for various treatments of cancer and tumor and other things people are thinking I will only tell that they are being thought of in this one because the trials are going on I do not know what is the latest today but generally there is a thought in that direction as well. Now I will come to some of the basic physics aspects of this if you want to get some idea one has to understand this one otherwise it becomes a very qualitative discussion I do not want that. So basically the characterizing the most important characterizing parameters are shown here the atomic magnetic moment as I showed you with the arrows in the earlier picture they are generally designated as mu the magnetic moment this is an atomic property each atom has a magnetic moment if the certain conditions are satisfied then for solids the parameters of interest are one is the magnetization which actually is a vector quantity the magnetization is essentially defined as magnetic moment per unit volume you have many atoms you take unit volume and find out what is a magnetic moment. So that is why it is called magnetic moment volume it is magnetization the susceptibility various kinds of susceptibility one can define but I give a general definition susceptibility is how much magnetization you can produce by applying a magnetic field. So basically it is a ratio of the response to the input so that is what it is written as M by H. H is a magnetizing field the field you apply magnetization that is produced because of the magnetic field is M. So M by H is a susceptibility the magnetic induction is B which is written as V naught times H plus M this is SI units fortunately or unfortunately in magnetism we usually stick to CGS units because of its convenient mathematical convenience. So in the CGS thing I have written specifically it is H plus 4 pi M then permeability is again important quantity which is defined as B by H for example a soft magnetic material I mentioned earlier it is supposed to have a very high permeability because your B by H is should be large in such cases and this permeability is a very important thing as far as a good soft furrow magnet is concerned. I should also mention here that the susceptibility strictly speaking is a property which is usually used to characterize a paramagnet whereas permeability is something for a furrow magnet. So susceptibility is for paramagnets essentially and permeability for furrow magnets I will come to the details anyway. Now one if you want to understand what is the magnetism due to where is the magnetism coming from then it is a very very important thing and the best way as far as I am concerned is to look at like this because if you take an atom so what is contributing to the magnetic moment you take an atom that is the simplest thing that you can do there you have electrons protons neutrons what are the constituents which are actually contributing to the magnetism why not protons and neutrons that is not really true proton has a charge electrons protons and neutrons contribute to the magnetic moment all the three particles irrespective of whether they have charge or not all of them have magnetic moment neutrons also have magnetic moment that is why neutron diffraction is a very important technique of course you need neutrons it is possible in India only in BARC but that is purely because neutrons have a magnetic moment so neutrons have a magnetic moment protons also have a magnetic moment that means nucleus also has a magnetic moment under certain conditions and certain conditions they will cancel each other but otherwise in principle nucleus also has a magnetic moment electrons also have a magnetic moment but when we tell that something is magnetic we usually mean the magnetism that is coming from electrons nuclear magnetism is also important in some sense because nuclear magnetic resonance which is basis of MRI magnetic resonance imaging where you are not looking at the magnetism produced by the electrons it is by the nucleus so essentially the proton a hydrogen so when you are discussing magnetic materials we are not looking at the nuclear part but that does not mean that the nuclear magnetism is not there nuclear magnetism is there it is a very important valid issue but in our case it is not going to be significant as I am going to show you the reason so the best way to understand is to look at how an electron gets a magnetic moment then using that try to find out an atom which consists of a large number of electrons generally how different electrons are contributing to the so-called atomic magnetism using atoms first thing that I can do is to create a molecule like I have hydrogen I can my hydrogen molecule that is so I should see how magnetism in a molecule can be explained then the extreme case is I put Avogadro number of atoms of that kind and it becomes a solid and under that condition what will be the magnetism in that so this is the sequence with which one should understand how the magnetism actually can be explained and of course this is as I mentioned this is also there but this is I am not going to talk about today but one has to keep this in mind because the mechanism is going to be different compared to other cases okay let us take these things one by one electronic magnetism the question is how does an electron get a magnetic moment very basic question the probably the most basic question that one can think of when we are talking about magnetic materials as you must be knowing electron magnetic moment essentially comes due to two reasons only two reasons one is a so called orbital angular momentum the other thing is has been angular momentum I do not like this concept of telling that there is a charge which is moving around and it is like a circular loop of current that is giving a magnetic moment that idea is not really true in a quantum mechanical sense that is not true so you just attribute to the angular momentum this electron has an orbital angular momentum it has a spin angular momentum and both contribute to the electronic magnetic moment how is it coming so it comes like this orbital magnetic moment I write it as I mentioned mu is a magnetic moment so this is mu L that is due to the orbital part there is a minus sign here because electron E is taken as positive the electronic charge is negative in CGS units this is related to the orbital angular momentum L similarly I can write this S that is a spin part so there is orbital part and this is a spin part with some quantum mechanics which I do not have time to explain the total magnetic moment can be related by connecting the two with the help of this g factor which of course you must be aware of this say land a g factor which is defined like this so I can write the total magnetic moment mu as minus g times this is called a gyromagnetic ratio e by 2 m c and this j becomes a total angular momentum so you can see why we are ignoring the nuclear part from these equations you see the relation connecting the angular momentum and magnetic moment is the that the relation is critically dependent on this gyromagnetic ratio where the mass of the particle is in the denominator neutrons and protons the mass is something like 2000 times greater than that of the electron and hence this gyromagnetic ratio will be very small does not mean that it is 0 but it is compared to electron this ratio is going to be small and hence we ignore the nuclear contribution so you have only these contributions and this gives you your j and you must have gone through in quantum mechanics course your j is nothing but j vector is l vector plus s vector this is a place where you have people generally get confused this is a quantum mechanical addition of two vectors l and s this has multiple values not a single value so your j corresponding states that are resulting from this addition of l and s it is multi valued this is j the resulting states are designated in terms of j the j values actually go from l plus s to l minus s either there are different values depending on the values of l and s so you can see there are many values possible and so this one has to keep in mind this here it is a vector vector is l plus s always but the values of this addition the quantum numbers j quantum numbers are from l plus s to l minus s of course this is a magnitude so i get the total magnetic moment that is coming from the these two contributions orbital and spin so i can write the total magnetic moment in this form and with the help of a new constant and remember your j square l square or s square when they operate on these eigen functions the corresponding eigen values are for j square it is j into j plus 1 h bar square as you have studied in quantum mechanics i do not have to spend time so that is why the j you will get square root of j into j plus 1 h bar so this with the help of this substitution which is e h bar by 2 m c which is called the Bohr magneton which is the unit of magnetic moment i can get an expression for the magnetic moment the total magnetic moment as g times mu b times j into j plus 1 square root that means if i know the j value if i know the l value and s value and if i know what value i am looking at i can find out the magnetic moment provided i know g because i can only find out g in terms of magnetic Bohr magneton this is the magnetic moment of a single electron just one electron and this is the total magnetic moment but if you want to find out the magnetic moment along a particular direction usually the axis of quantization is taken as z axis you will be finding out mu z that will be you have to take this projection of this along the axis of quantization that is why i have written as j dot z z cap z cap is a unit vector along z axis so that will give you the corresponding quantum number j I mean operator the corresponding quantum number is j z that has a quantum number associated with this m j so that is why the along a particular direction namely the z axis it will be g times mu b times m j this is the moment that is that you can see that you can measure by applying a magnetic field along a particular direction so this is a total magnetic moment this is a component of that magnetic field coming from one electron this is what is the picture is this is a total magnetic moment of an electron so I can write this one only thing is that is capital J does not matter so this is the moment that is coming along a particular direction the so called axis of quantization and this I will come to that later do not worry about this I am just showing the picture only because in quantum mechanics as you know these vectors cannot have any kind of orientation with respect to z axis your theta cannot be continuous theta has to be discreet and your relationship between the two is always discreet and that is why this picture is shown so theta cannot be anything here the spin magnetic moment of electron this is the expression so it is applicable to proton also because all these things are applicable to protons except that your m will be the mass of the proton what about neutron then neutron also this is true I mean what you are telling is that there is no charge here see this expression is not coming from because of the charge this actually is a fundamental constant this one I mean the derivation of this is not very easy to do right now so this expression the gamma gyromagnetic ratio expression holds good so the magnetic moment if you calculate whether the proton or neutron if your angular momentum is the same more or less same magnetic moment you have so that problem this the gamma factor is independent of whether it has got a charge or not whether it is electrically neutral or not that is not an issue yeah so okay that is the only difference yeah neutron usually the magnetic moment mainly comes from the spin part but still you will use this kind of an expression only so this e doesn't mean that I can use this expression only when the case where there is an electrically not neutral charge that is not an issue this gyromagnetic ratio expression comes from a more fundamental derivation so that way same things can be used as he said mostly that in the case of neutrons the contribution is from the spin yeah your c indicates the velocity of yes yes in the cgs unit you have this thing because your mu naught is not coming and c is coming always in the cgs unit magnetism because of your relation between mu naught epsilon naught is 1 over c square that is okay so if I use the same argument for a free electron when it is a free mean there is no orbital moment there is no orbital angular momentum so that is 0 so the g factor for a pure spin case is 2 and by substituting it that is what is written here I get the magnetic moment as this much and the component of magnetic moment as one more magneton a conduction electron in a metal for example which is completely free if that is completely free the magnetic moment associated with that is one more magneton which is the magnetic moment unit more magneton now so we have some idea about the magnetism that is produced by one electron now the question is using this how to find out what will be the contribution of the magnetism from an atom which consists of a large number of electrons that is why the question is here like how do different electrons contribute to the net magnetic moment of an atom so how do we do that this of course the little quantum mechanics is involved I will just put it I will upload it but I mean if you are not comfortable I will not go into the details I will only tell the basic pics here so basically what is to be done you want to find out how the magnetism arises in an atom so the best ways to find out is you take the atom and try to see what happens when you apply a magnetic field when you apply a magnetic field what you as you have studied in quantum mechanics course you have a Hamiltonian initially now you are applying a magnetic field so that means Hamiltonian is going to get extra terms what is done is that is here I will not go into all the details because that will take lot of time so you have an extra term which is coming due to the applied magnetic field but as you must be knowing when you write down the Hamiltonian you do not use the magnetic field or electric field use the corresponding potentials because this is essentially the potentials and the kind energies and the kinetic energy that is coming into picture so the magnetic field actually is represented by the corresponding potential what is that potential you must have studied in electromagnetism the electrostatic field is represented by the electrostatic scalar potential phi and magnetic field is represented by its magnetic vector potential a where the relation between b and a is b is del cross a you must be remembering so that is what is coming this a is the vector potential associated with the magnetic field b that is applied so the whatever magnetic field I apply that is represented in the Hamiltonian by this vector the vector potential a and then what one sees is that the momentum again I will not be able to go into the details the momentum gets modified usually we will be writing the kinetic energy as p square by 2m plus the usual potential energy for example hydrogen atom p square by 2m plus the v0 will give you the full Hamiltonian now if it is subjected to a magnetic field the momentum changes to p plus a by c in cgs units square divided by 2m and potential energy remains the same so what happens if I expand this is purely quantum mechanics it needs lot of time I will just give the idea so you expand this what you are looking at is because of this what is the extra term that is coming due to the supplied field so what you find is you expand this you do not consider the initial terms the p square by 2m plus v0 term you do not worry about it that is anyway there what is extra and try to find out what is that extra term that is what is written here in the last step here if you see this is one term here the other term here very important thing to note here is that the signs of these two terms h0 is to be ignored because that is anyway there in the original Hamiltonian the magnetic field effect is represented in terms which actually contain a that means this term and this term in this term the sign is positive everything is square so it is always positive quantity this is something which is negative which again consists of a that means the magnetism is affected by these two terms the second term and the third term now what are these two terms representing that is the reason I am talking about it so what one can do is I can take a magnetic field to be completely along z axis then using the definition that v is del cross a I can write the vector potential in this form one can do that and then little bit of arrangement not very difficult actually I will give you the slides one identifies that you get a term lz that is orbital angular momentum along the z axis lz component is coming here and arranging this I get that the extra term is 1 s is e square me z square now times x square plus y square and minus mu b lz so two terms are coming and these two terms have opposite signs what does it indicate the sign which is positive here actually is indicating the diamagnetism as I mentioned which is always present and the second term is what is representing the paramagnetism that you see so when you apply a magnetic field to an atom you see two main contributions coming out of this perturbing the Hamiltonian you are actually changing the Hamiltonian because of the applied field the applied field is represented in our equations with the help of magnetic vector potential and using this quantum mechanics here one sees that there are two contributions coming these are the two contributions if I only worry about these two magnetic field effects these are the two contributions which are important now so this is positive quantity that you see that it is one term and there is another term that is here now okay this little bit of things you can ignore so this is the same term so what I do is this is a general expression suppose I take a electron spin also into account I should write mu b l times b I should write plus 2s because the earlier expression does not consist of a spin so that is also taken into account here and assuming that there are many electrons because I am talking about an atom so this is for various electrons of the atom the summation i stands for various electrons and I can expand I get this kind of an expression what is done here is this these two extra terms of the Hamiltonian will give rise to an energy shift using the perturbation approach I must have studied so by applying these two terms or these two terms are being treated as a perturbing terms one can find out what is the energy change that energy change is what is written in the first order delta e1 that gives you this term and this term as you can see the energy change in this case is positive in this case is negative so now this is what is written this term actually is I am calling it as a diamagnetic contribution and this is called the paramagnetic contribution given by the curie paramagnetism I will come to the details now what happens is if I take a solid for example in an atom you do not talk about susceptibility suppose I have atom of this kind giving rise to a solid then I can actually define a susceptibility the definition of susceptibility is you take the second derivative of this quantity of course with a minus sign this extra energy that is nothing but a susceptibility so this being positive the negative second derivative will be negative that is why the diamagnetic susceptibility is negative whereas here it is already negative it will give you a positive susceptibility so positive and negative susceptibility of paramagnetism and diamagnetism essentially comes because of the general treatment that you are doing for an atom subjected to a magnetic field so that is what is shown here one should you can work it out once I upload it you can just work it out so some interesting situations are suppose all the orbits are completely filled then what happens is you are L and S will completely be 0 the total completely filled shells there is no orbital contribution there is no spin contribution so that means you are one term is not going to contribute this term will be completely gone this is 0 and then the expectation value will be 0 you do not have to worry about it then you will see that you will have only diamagnetism that is what is given by this so all the completely filled shells magnetism is purely purely from diamagnetism because there is no contribution to paramagnetism that is possible that is coming from this result that does not mean that if it is not completely filled the first term is 0 so any shell if you have at least one electron definitely there is a diamagnetism contribution as given by the first term first term is never going to be 0 so diamagnetism is always there but if the number of electrons in the I mean the second term is not really 0 that will dominate the first one and you will call it as the paramagnet not the diamagnet so even paramagnets will have some contribution coming from the diamagnetism of all the shells that is something which is a very important point to note okay so now one can actually find out the diamagnetic susceptibility that is what is written here as I mentioned first the derivative with respect to field V stands for this is actually for the solid because by using various atoms I can actually make a solid take this differential of this extra energy coming due to these terms and differentiate with respect to the field so that and this is per volume that is why it is magnetization and once again derivative will actually should be B that gives you the susceptibility which is going to be negative as you can see here because everything else is positive so this gives rise to the negative susceptibility of diamagnets as all of you must be knowing in the same way I can treat the so the some of the features of diamagnetism are written here due to essentially due to the closed shells because other things you do not see if you want to see you have to look at the closed shells susceptibility is negative magnetic moments are induced by the magnetic field otherwise there is no permanent moment the atoms will not have any unpaired electrons generally the susceptibility the diamagnet susceptibility is very very small compared to one examples are hydrogen molecule is a standard example the susceptibility is very large since you are going to have a talk in the afternoon on superconductors superconductors have some connection here between magnetism and super conductivity lot of connection ideally the superconductors are the one where the susceptibility is exactly minus one so that is why they are called the perfect diamagnets nothing like superconductor as far as a perfect diamagnet is concerned because compared to one very very small values here it is actually one so I will come to that anyway it will be mentioned today in the afternoon talk I will not be able to go into the details so under certain conditions you can actually get minus one under the superconducting trend in the superconducting state what is the meaning of this this is nothing but a masoner effect what is represented by this statement is a masoner effect because whatever you apply it will completely expel it that happens only below the certain temperature after that superconductivity is gone so this this statement is in a purely superconducting state when the masoner effect is possible so this is I am not telling that this is always minus one I am not making that statement these two are independent statements in the case of superconductors you have the possibility of getting minus one I am not talking that is minus one is throughout I am not making that statement again what I want to tell you is that the masoner effect is actually a demonstration of this statement you whatever you apply exactly it will be killed if you apply five units it will produce minus five so the ratio is minus one that is what I was telling this is why the field is completely expel by the superconductor okay this actually tells you how this equation actually is valid in many situations this one it actually is proportional z and this charge and also the mean square radius if you have larger and larger diameter radii orbits it is going to be more that is what is shown by this effective z time square the susceptibility is going to increase the diamagnetic susceptibility now comes to the other term we will take the second term suppose you look at the second term unpaired electrons so definitely if the electrons are all paired up then there is no second term but suppose there are unpaired electrons then you can get that the second term also contributing to the energy change that is what happens in many situations it happens like for example sodium it happens you have one electron unpaired in the last drill the valent drill but the problem is in the atomic sense it is fine sodium is fine sodium should be paramagnetic going by this definition but sodium atom is never treated as an individual thing because you talk about sodium no the question that you started arguing about this existence of diamagnetic term and paramagnetic right using only first order calculation so there will be hard orders yes I had one more lecture I would have talked about it there is the second order correction can be done second order correction gives you another susceptibility which is known as a van black susceptibility for paramagnets so it is a very good question the second order correction is there it is important in some sense but here we are not taking because of time okay so if you want to look have nonzero second term definitely you should have this one but situations like sodium when you have an outermost electron as the unpaired electron it is not going to survive when it is actually forming a bond to make it as a solid so atomic case sodium is fine but sodium solid if you take this one electron is not unpaired it will be bonded with other electrons sodium and then you will not see in that case again you will see essentially all filled orbits that is why sodium metal as it is will not be a good paramagnet but fortunately in two sets of elements in the periodic table namely the transition metal series where 3d, 4d and 5d and similarly in the case of rare earths and actinides the 4f and 5f series where you have unpaired electrons the possibility of getting unpaired electrons in one of the inner shells infused in the inner shell even after the bond is formed even if after the solid is formed even after the molecule has formed there is no problem the unpaired electrons will remain as unpaired electrons in the inner shells so that is what is happening that is what is making these two series very important as far as magnetism is concerned but this comes if you see this comes from the second term that we talked about in the general formalism so the examples are so this paramagnetism that is why because of this question only I made it as curie paramagnetism because if you talk about paramagnetism there are at least three varieties of paramagnetism one is what he is asking if you do a second order correction there are other there is at least another very important contribution I will not talk about those things that is why here what I am talking about is purely curie so atoms must have at least one unpaired electron I will come to that what is a Hund's rule and the moments are permanent susceptibility is positive as you have seen but is small examples are transition metal ions lanthanide ions as I mentioned hydrogen atom sodium atom oxygen atom oxygen molecule also I am not writing hydrogen molecule here you see the problem I wrote hydrogen molecule as an example of diamagnetism but in the same line I am writing oxygen molecule as an example of paramagnetism see the problem here so this is a usual mistake that people make people think that if I have H becomes H2 okay it becomes completely filled and I lose paramagnetism oxygen also should happen it is not the reason is I will show you very soon so these are the things some of the examples so what happens is I talked about Hund's rule let us take an atom where there are two electrons when you have two electrons in the diesel so I will take D electrons because that is the most important series if I have what is the ground state you have to find out because you have two electrons means each of the electron will have L value and S value that means I have to find out the resultant L and resultant S that is what is found out with the help of set of rules called the Hund's rules what is especially you are looking at the ground state if you want to find out the ground state what do you do you take the shell I will go to the example that is better I have two electrons to fill that is a 3D2 case two electrons in the 3D shell there is an order in which these electrons must occupy these ML values L equal to 2 for a diesel and the values are 2 1 0 minus 1 minus 2 that you must be knowing so if I want there is a way in which this has to be filled the first electron will get filled here it will be an up spin and the next one will come and its color is not so it is coming here as an up spin the point is if the first is up the second also will be up so that the maximum value of MS is obtained MS is the corresponding quantum number for S and they should not go to the lower ML values as far as possible only rule they have to follow is a Pauli's principle that means two up spins should not go to the same box a two down spins also should not go to the same box but as far as possible they should be up and that means they have to take only one per this one and goes on so if I have five all five up will be there up to this if I have sixth one it will be one it will be down but that has to come to the two place so this is what is shown as Hund's set of rules which actually tells you what is a ground state value for the total j because the total j is what is going to determine the total magnetic moment the total magnetic moment as I told you is determined by your j the total angular momentum so j has to be found out but remember very very important thing that many people make this mistake Hund's rule when you apply it only talks about the ground state ground state as an individual atom is what is given by Hund's rule in all other cases if you use you have to be careful you can make lot of mistakes so this gives you the idea of finding out I have written the values so the way if you fill I can find out my j and for less than half fill because half fill case is 5 here it is only 2 it is less than half fill so the j is given by l minus s one of the values allowed when you add an l and s so it is l minus s the values are given here I can find out the magnetic moment in this case to be 4 by 3 more magnetons if I have 3D2 configuration atom isolated no other interaction I talked about oxygen and hydrogen comparison the mistake is when you are trying to find out magnetism of molecules you cannot use a normal molecule the atomic orbitals what one has to do is one has to use a so called molecular orbitals that is what is shown here in the case of hydrogen the molecular orbitals this is the bonding orbital this is the anti-bonding orbital molecular orbital here it can take 2 electrons one up one down that means it is diamagnetic so hydrogen molecule is diamagnetic if it is an atom only one will be there it is paramagnetic whereas if I go to oxygen things are little more complicated because of 8 electrons the molecular orbitals are something like this I will not go to the details what you see is certain degeneracies are there that will allow two spins to be in the same thing and you can see there is no complete canceration and hence oxygen molecule is paramagnetic and not diamagnetic now comes the most important thing if you have any doubts you can ask me okay I will go to hydrogen hydrogen see you have one electron so when you are making hydrogen molecule you have to worry about you are actually creating molecular orbitals using atomic orbitals so the molecular orbitals of hydrogen are like this this is the bonding orbital of sigma that is sigma 1 s this is the sigma star one so you have to accommodate two electrons those two electrons will be accommodated easily here that is a ground state because one is up another thing is down that is a minimum thing so this becomes diamagnetic do the same thing for oxygen oxygen the construction of molecular orbitals is not very straightforward because you have more because you have 1 s 2 s 2 p I am not showing 1 s 2 s and 2 p these are the two bonding orbitals here these are they once here now if you try to how many electrons you have you one as you forget there are six plus six there are twelve electrons to be accommodated in various things two here two here like that if you see and remember these two are pi you can see here pi 2 p y and pi 2 p z they are degenerate they have the same energy so there I can to put two up spins no problem because they are the quantum numbers are different Pauli's principle will come in the picture only if all four quantum numbers become identical right so I can put two up spins and two down spins actually sticky no problem so I have here and then remaining are two electrons are remaining these two electrons both of them can be up and it can be here so that means I have a all these things will cancel out but there is something which is not getting cancelled so it becomes paramagnetic now yeah yes yes very interesting and in fact you can actually extend to nitrogen oxygen NO is nitric oxide right so one can actually compare exactly same thing use the same kind of scheme you can find out nitrogen actually will be diamagnetic nitri NO NO is in between it will be less paramagnetic than oxygen same scheme can be used same concepts can be used for exactly same concept can be used any diatomic homo nuclear molecule one can use the same concept because why I thought I mentioned is molecular or magnetism is a very important thing today so you should I mean this is just a basic for that since we are everything we are explaining based on only hydrogen and oxygen yeah yeah that's what I want of course nitrogen then I just wanted to compare usual this concept of telling that okay hydrogen is hydrogen atom is paramagnetic when it becomes molecule it becomes diamagnetic that argument is not true always oxygen is a simple example where it is not true oxygen atom is paramagnetic oxygen molecule also is equally paramagnetic so nitrogen again varies changes NO for example is paramagnetic because of this you will have unpaired electron one of them okay now let us come to more realistic thing that is a solid state magnetism magnetism in solids here again that's very big challenge because as I mentioned you have two essentially two kinds of materials one is metal other thing is insulators do again due to time constraint I will not talk about anything on the metals I will only talk about magnetism in insulators like ferrites that I talked about so magnetism in insulators like ferrites it's slightly easier to talk also because magnetism in metals is more complicated and it needs a lot of time to have some meaningful discussion so I thought I will not be able to do this time so what is to be worried about as I mentioned when you have paramagnet so we are trying to look at paramagnetic solid if you want to find out what I am looking at is essentially not just at ground state I want to find out what happens as a function of temperature because we are going to slowly to see how the applications are coming and other things so what happens is if you are trying to find out what happens to paramagnetism of a solid and insulating solid at non-zero temperatures what happens is you have a ground state whatever we defined earlier when there is no magnetic field it is has a degenerate they are all in terms of the j value total j angular momentum so this has a degeneracy of 2j plus 1 as you must be knowing any value a j value have a has a degeneracy of 2j plus 1 when you apply a magnetic field all these 2j plus 1 fold degeneracy will be removed and you will end up with 2j plus 1 levels for example if I have 2 degeneracies 2 I will end up with 5 levels that is what is shown here so depending on the value of j you will get many more lines many more energy levels when you the magnetic field is applied and when you are looking at 0 Kelvin of course the ground state only will be populated and you can ignore these things suppose you are applying a magnetic field and the temperature is not 0 what happens is there is a finite probability of occupancy of these levels also so one has to really apply a statistical mechanics and find out what will be the contribution from all these things if everything was here no problem I know the mj value of this one I can find out what is a magnetic momentum but now at a finite temperature these things are all populated and I mean of course with a decreasing number one has to do a statistical mechanical treatment and I will not go to the details one can do that not difficult thing you have studied anyway one can do that and that gives me the magnetic moment or the magnetization as a function of temperature or strictly speaking the susceptibility as a function of temperature that is what is the Curie susceptibility or the famous Curie law which actually gives you a constant divided by T and that constant is called the C the Curie constant C by T so what is contained in C C is contained in this one C is given by n the number of molecules or the number of atoms in the volume that you are taking mu effective is the j into j plus 1 term divided by 3 kb so this is C by T this is what is called as a Curie law or the Curie susceptibility that generally we use in any paramagnetic situation so Curie constant if you know because once you do the susceptibility versus temperature I will get a Curie constant the Curie constant gives you this factor the so called defective magnetic moment that is what I showed in the picture the full length of this arrow represented by g times square root of j into j plus 1 what is the role of polarizability with this case what is polarizability polarizability which polarizability linear non-linear no no okay okay so what you are asking is the like in the case of electric field when you apply electric field you can actually get non-linear terms also into account like for example in the dielectric case but what happens is even in the case of dielectrics usual dielectrics they are all linear in certain cases of course you have non-linear terms magnetism also you can get but remember magnetism is a much weaker effect than the electrostatic electric field effect so magnetism more or less the linear terms will work I am not telling that other terms are not there your question is important but usually in basic ideas we do not really worry about the higher order terms only very very special cases if you are looking at very very carefully you will take the non-linear terms into account then you have a non-linear susceptibility but usually no usually they are much weaker but much weaker than the electric field electric field itself you generally do not take most of the dielectrics are linear dielectrics okay okay but the question is very important okay so that gives me the query constant and that from the query constant I can get some idea about this one so once you know the query constant you can find out various things this is a scheme actually which tells you how because four of electrons I mean four of atoms are all paramagnetic so you can see how the experiment and theory are matching you find out the query constant and from that you can find out the effective moment you can actually calculate the effective moment using the value of j and substituting it you will see that there is a good match between the two as far as rarer circuits they are all rarer the four of elements are concerned they are all fine they are able to manage this very well and the experiment and theory are I mean both of them are matching very well so this is a query law C by T as I mentioned where C is given by this if I take Avogadro number instead of this small n if I take Avogadro number then it becomes the molar susceptibility otherwise it becomes a volume susceptibility you can get mass susceptibility you can get molar susceptibility if you take n the Avogadro number it becomes naturally the molar susceptibility with which many of us work okay this is the thing chi versus T is 1 by T dependence if I take 1 by chi versus T it is linear ideal paramagnets this should go through origin as shown here but if there is some interaction given by the query voice interaction there is small interaction between the individual atomic magnetic moments then you will see that this is slightly deviated not going through the origin this of course is the chi T if I take it will be something like this so this is query versus query a query wise law query as I mentioned query and query wise there is a separation if I plot inverse one will go through the origin one will not go through the origin it will have a positive intercept along the temperature axis that is because you are taking into account a small interaction between the two okay some of the values are given here just to give you an idea again the values are very small compared to one yeah this as long as I mean I will tell the other way as long as it is valid I will call them as a query paramagnet if it is not again coming back to your earlier question if I have a van blak susceptibility this will not be followed I will get a temperature independent susceptibility that is why I did not talk about the second order correction okay so another very important point is this one I will straight go here so what is shown here again what is shown in the picture only the plot if you take these are all the rare earth ions 4F ions so 4F as I told you there it is well inside so the outside effects are not very important they will retain essentially like an atom and if you see what is shown here is the calculated effective magnetic moment this is experimentally observed magnetic moment experimentally observed means from the query constant I find out that is the last thing and the next one is the one which is calculated by taking J earlier find out G value and so on substitute and get this one if you see in this case there is a good match between the two which means that the Hund's rule which actually led to the final query expression all these things are all right that is why I get a good match between the two because it started with Hund's rules then we extended to query wise form and I got the values I found out the query constant and all these things so if these two are matching that means my application of Hund's rule is correct my application of query law is correct that is what is shown here so this is the very good agreement between the two as far as rare earth ions are concerned when they are in a paramagnetic solid the problem comes now a similar situation should be expected in the case of transition metal ions like 3D ions but on the other hand what you see here is this one same thing these are all the ions transition metal ions 3D and 4D you can see the match is not very well between the two the match is not at all good which means that in this case the application of Hund's rule that leads to the query law is not really true in reality that is what is seen by the disagreement between the two so what could have gone wrong that answer is written here very very important what happens and not only that if you try to see if I take original Hund's rule and if I make my L equal to 0 and proceed as if it has got only a spin part then my calculation and my experiment will exactly match in the case of rare earths I have to use my J I have to take that means I have to take L and S it will match with the experiment but when it comes to transition metal ions paramagnetic ions if I take L to be 0 everything is fine if I do not take that there is a disagreement between theory and experiment which tells something what does it tell that is what is written I am sure you have heard about this name this is what is known as orbital angular momentum is quenched what does it mean when in this atom the transition metal atoms especially when they are in a part as a part of a solid like for example ferrite if you take Fe 2 plus or Fe 3 plus when they are surrounded by oxygen ions positive ions they produce an electric field electrostatic field is produced and this electrostatic field I will not go to the details will somehow kill the orbital angular momentum essentially they are arrested they are not able to have an orbital angular momentum which means that they are left with only a spin angular momentum that is why when you put L equal to 0 it works if you do not put L equal to 0 it does not work so what is happening here is in the case of transition metal ions when they are part of a solid especially in the solid like ferrites the orbital angular momentum is not surviving then the question is why not this happened in the case of rarer ions the reason is in the case of rarer the so called magnetic electrons unpaired electrons are in the 4 f shell which is well inside so they are not affected by the presence of oxygen or any other fellow outside whereas in the case of transition metal the 3d is much more outside so this 3d is more susceptible to changes that is happening in the surrounding and hence only in the case of 3d this problem comes whereas 4f you do not see that in the case of 4f because of this reason in the case of 4f magnetism j is called as a good quantum number total angular momentum is the good quantum number whereas in the case of transition metal the spin is a good quantum number and j is not a good quantum number j is a bad quantum number so that is what you see the comparison here yeah this is what is obtained from the curie constant c you measure what you measure is c yes by finding out the susceptibility versus temperature you measure c you calculate c from c i will find out what is the effective magnetic moment so this highlighted part is this one is the expected from one is this is experimental and this is the expected using your j into j plus one so in this table if i look into this fe three plus almost in the middle yes you are saying that you have l equal to zero yeah and you are getting five point eight two and five point nine two you are you are putting me to spend more time oh no no no i am sorry i just noticed it no no no very very good question very very good question in fact that if you see i have skipped a few things in between let me go back because of your this one so what happens is see this question this point is very very important let me come to this what happens is if you see fe three plus you see that the agreement is very good five point eight two five point nine two unlike other cases what is the problem the problem is fe three plus is a special case fe three plus even if i take the orbital to be i mean uh into picture it is exactly half filled case when it is exactly half filled your l is zero anyway so it is when you put a hoods rule you take l i am using the normal thing but my this is exactly zero so l is zero it will behave as if it has quarter only yes and whether you use it as l or j doesn't matter because your l is anyway zero so l equal to zero is a special case so you should not look at the the the three plus fe three plus case to conclude anything fe three plus is a special case in other cases only the problem is more important you have to look at the case where l is non zero but yeah it acts as if l is zero okay but in that respect if you take a look into the uh the upper element mn three plus yeah in that case the results is almost like i mean i can say that they are very good same situation mn three but l is not zero there no it's two that's what i'm telling you the differences are changing uh yes it's zero eight the point is the point is if i put exactly l equal to zero and j equal to s i get a good match between the theory and experiment in some cases see what when you are looking at you have to look at the whole series you can't just look at one so when you especially if you are looking in a case where it is close to zero you don't really know what is really happening to find out what is really happening you should take a case which is far away from l equal to zero so that is why i will more importantly i will worry about earlier ones where you see that the j equal to s only gives you the match between experiment and theory that is the issue okay now comes uh the probably the last part so you now you have paramagnets we know how the paramagnets are formed how they have the susceptibility how they behave when they are part of a when it is a rarer when it's a transition method so now using these atoms paramagnetic atoms as i mentioned they can be building blocks and these building blocks can be used to make magnetically ordered materials three types but i will take only the ferromagnets so that is what is here when you tell something is ferromagnet these are the things which you should never forget they are the most important characterizing features so that is what is written all of them are actually connected also one is they are supposed to have compared to paramagnets these people have what is known as spontaneous magnetization spontaneous magnetization means even in the absence of a magnetic field they must be able to show magnetism that means it is not completely disordered random as in the case of paramagnets it's something else because that's the only way this is different compared to paramagnets otherwise why should you call it as a with a new name so first thing is they have spontaneous magnetization then as you know they have a curie temperature below which they are ferromagnetic above which they are paramagnetic then comes they have so the reason for this is attributed to originally as an internal field which is playing the role of an external field then it was more clear later it was called a molecular field precisely they actually that is called an exchange field which is responsible for curie temperature and spontaneous magnetization then comes they are known to have what is known as exchange interaction very very important purely a quantum mechanical property which I will not be able to go into the details then you have anisotropy I will talk about it you have domains as you must be knowing and you have hysteresis which again you know something about so these are the properties one cannot forget when you talk about ferromagnets this of course is the well known picture that showing the temperature dependence of the magnetization where the temperature a particular temperature at which it changes becomes almost zero that temperature is called the curie temperature but remember I told you that they are characterized by so called spontaneous magnetization spontaneous magnetization means in the absence of a magnetic field they should show magnetism is it happening does it happen it take piece of iron does not attract anything does not have any magnetism but we call if he has a ferromagnet right if he is a ferromagnet but it does not attract anything by itself unless you first bring it to a magnet then of course it starts attracting otherwise no so what is the reason yes there only the thing is you have to bring in the concept of domains that I am going to show the other important thing is magnetic anisotropy magnetic anisotropy what does it mean again it is not very clear what happens is if you have a single crystal of a ferromagnet if you try to apply the magnetic field along different directions you see that in certain along certain directions the magnetization is very fastly picking up in certain directions it is very slow for example here if it is applied along one of course it is a single crystal if it is applied along one zero zero direction the magnetic field I see that magnetization easily comes and has a particular path and after a point it saturates it is completely constant here whereas if it is applied along one one one crystallographic direction you see that it is very difficult and it actually comes something like this it takes lot of time a lot of magnetic field as if there is a hindrance by itself within itself which is not allowing the external magnetic field to do the job so that is why it takes more field to get into the saturation mode whereas in the first case it is which means there is some property which is internally there which actually is going to be dependent on the direction there is a directional influence of an internal property that is coming into picture that is what is preventing this otherwise everything should have equally followed the saturation path that is not happening yeah you know in optics what happens you have refractive index ordinary refractive index and extraordinary what happens there is also a crystal so what happens is along different directions the properties are different why I am sticking to a single crystal because if it is a polycrystal I cannot have a unique direction of one zero zero or one one one so only in the case of a single crystal I make this statement so direction dependence can be talked about only in the case of a single crystal so in fact almost all the properties will show some difference with respect to the direction that is what you are showing you are seeing here so because of that reason in this case I will call one zero zero as an easy axis of magnetization whereas one one one is called a hard axis of magnetization what is the reason for this very quickly I will give you the answer because I do not have time so thing is electron a magnetic electron is part of a solid like for example a 3d electron part of a solid what you see is the electronic cloud is something which like this this you have seen so you can see there is a directional dependence here and assume that you have positive ions like oxygen ions sitting there it is a solid ferrite for example what happens is certain directions especially single crystal certain directions you can see that these positive ions and this these are all negative electron clouds they will be closer by that means there is a strong interaction attraction there if you apply a magnetic field there the magnetic field will not be very powerful to change it to align in a different direction that is what I was telling that there is an internal mechanism which is preventing the magnetic field from aligning the system so that is what is happening here if it was a sphere if the charge density was a sphere this problem would not have been there precisely this happens if I be going back to your Fe3 plus case if you take an Fe3 plus compound you see that there is no difference between 1 0 0 and 1 1 1 the reason is the magnetic electron the charge cloud is spherical there when it is spherical this kind of a difference I cannot tell everything will be same so very very important thing if you are looking for properties like this where you want deliberately you want the difference between 1 0 0 and 1 1 1 never choose a material where the number of electrons is exactly half filled half filled always will give you a spherical charge distribution L equal to 0 L equal to 0 as you must be knowing SPDF configuration is S S is spherical so it does not give you any anisotropy this is what is anisotropy so any anisotropic behavior if you want you want for example as I mentioned a good permanent magnet a good hard magnet must have large difference between these two that means you do not look for such L equal to 0 material for such material but they will be very good for soft magnets so that is the contrast okay so I told you that this oxygen in this example gives rise to this extra interaction which is purely inside and the charge cloud electron charge cloud interaction with the positive ions of oxygen that is an electrostatic interaction and that is very strong compared to any magnetic interaction that you can give from outside so this interaction is what is giving rise to just to give you an idea that there is a splitting here you can see so if your magnetic field is showing some effect it has to really see whether this can be overcome or not if this this do not worry about the simple and all this separation between this e g and t 2 g basically they are two different sets of orbitals this if you have a very strong electrostatic interaction this difference will be large so this is what actually is produced by so called crystal field the oxygen interacting with this electron cloud giving rise to an electrostatic interaction that is called an electrostatic field produced by the crystal that is why it is called a crystal field so you see the thing here just wanted to show you what is the effect of this one suppose you have this kind of a thing and if you go from one end to the other you see for example if it is Fe3 plus which has a d5 so you have this case it is like this if you have d6 I have to put one more I am putting one more electron there are two possibilities either it can go here or it can be here suppose this difference is very large the sixth electron will try to be here but it has to be a down spin electron which means that the total number will come down because it will cancel with one suppose on the other hand if I have this separation smaller compared to the magnetic energy or the change in energy when you make one spin up as spin down you see that this will not go here and everything will be here and you get absolutely no spin no contribution that means this is giving rise to a high spin the other thing is giving rise to a low spin this is what actually in many chemistry people call it as a high spin complex low spin complex in many of their molecules so that is essentially coming even though it is a molecule you have a crystal field kind of an interaction like the one which actually splits these two things and the competition between magnetism and electrostatic interaction that is what is going to determine whether it is going to be a low spin or a high spin case this is what I asked you earlier a ferromagnet as it is will be split into various domains like this within the domain it is spontaneously magnetized that is why we write as a first statement the ferromagnets must have spontaneous magnetization but they are split into various domains and different domains are randomly oriented as far as the magnetization is concerned something like the one shown here so bulk scale you do not see any magnetization so a iron piece does not attract anything because it actually has to reduce the energy what is known as the magnetostatic energy that is what is shown here if it is a single domain like this it is having a large magnetostatic energy system does not prefer to have a large energy it will always try to reduce energy as usual and it gets into a from a single domain system it actually becomes a multi-domain status and hence you do not see any spontaneous magnetization in the bulk scale this of course it is dependent on various I mean things so the I think I will not worry about it I do not have time oh just to give you one idea because of this reason usually if you take a thin film the easy magnetization direction lies in the plane it is easy to magnetize within the plane rather than making like this if in a multi layer like the one shown here you can magnetize like that easily but in a thin film usually the magnetization will be this one magnetizing in the other direction is difficult because that is going to be a hard axis of magnetization whereas within the plane it is going to be an easy axis of magnetization somehow yeah so this is what I was talking earlier when you apply a magnetic field what happens is these domains will change their orientation and that is what actually give rise to the initial increase and finally when all the domain aligned in the direction of the applied field you see that essentially it has a single domain that means the magnetization will no longer increase even though it is to the saturation this is again a very important property of ferromagnets this of course is a hysteresis which all of us know various things this is called the remnants that is after the field is reduced also you see that the magnetization will not come to zero unless you apply a reverse field what is known as a coercive field it does not really come to this one this is the case of a bulk what happens is that as coming to your question when the size is reduced you take a ferromagnet tries to I mean try to reduce a size when the size is reduced to a very very small dimension what is going to happen is the size has become so small it is not able to contain two domain walls I mean or one domain wall or a two domains so the size has become so small it is not energetically favorable to support two domains in which case the system will try to be in the single domain against our original assumption that a ferromagnet always has domains so the size matters and the size becomes smaller it becomes a single domain particle favorably energy wise so when you have this one because of that a very immediate consequence of this is the variation of cohesivity what happens is when you have multiple domains the cohesivity is generally very low as you can see but as you reduce the particle size you see that it is increasing and it reaches a maximum at a particular minimum value and if you reduce further if you reduce a size further what happens is it is actually almost losing its ferromagnetism and it enters to what is known as a superparamagnetic state I do not have time to go into so this becomes a superparamagnetic region so that is why it is called unstable it is this is a superparamagnetic region this is the single domain maximum cohesivity region then this is the usual bulk region any questions yeah not easy not easy because you should know what is a crystal field you should know the symmetry you should know the site symmetry then you have to find out the crystal field you have to find out the Hamiltonian in with respect to that you have to diagonalize it you have to find out the eigenvalues you have to see which is a ground state which is a excited state what is the energy difference as I showed you right so that only can tell you how strong the anisotropy will be it is not very easy thing okay calculation wise it is not very easy I mean okay so just by knowing the configuration of it is not possible to identify you can only tell at least to first order we can tell l equal to zero if it is half filled it is it's not okay but which will be hard axis or which will be easy axis it's not I mean you have to do the full calculation yes you have to do a full calculation or do an experiment okay so in experiment can you just tell what is the indicator experiment what you have to do is that you have to take a single crystal and you have to apply the magnetic field along different directions as I plotted and see where it is easily saturating that becomes easy access okay yes otherwise one can also do a very complicated thing what is known as a torque magnetometry so we can actually find out that will give you the anisotropy constant from again you can actually do some calculation and find out what is the easy axis and hard axis not very easy okay so identify the easy axis is not