 The next topic we take up is magnetism in nanomaterials and students who are very interested in learning more advanced topics in the area can consider the principles of nanomagnetism and this is a very compact book, but gives you lot of detail information about magnetism in nanomaterials. As before we will take up some basics of magnetism first followed by the changes which occur when we go to the nanoscale and we will discover some very interesting and what you might call properties which do not have an analog in the bulk scale like super paramagnetism, giant pharomagnetism etcetera. We will also see that the spin arrangement in certain magnetic nanostructures can be very very different from that of the bulk structures. If I use a phrase that magnetism is universal then this would not be an understatement. It has been found that coherent magnetic fields at the scale of galaxies and clusters of galaxies, the origin of this is not fully understood yet and there is still lot of work being done in this area, but even at the scale of galaxies and clusters of galaxies you find coherent magnetic fields. Earth itself is a magnet and earth's magnetic field has a it is not a very strong magnetic field has a strength of about 1 gauze and interestingly it reverses itself with an average period of about a million years. Of course, this is an average period and certain there could be smaller time period before reversal and certain time periods could be larger and as we know today perhaps the magnetic field as we talk now is itself undergoing a reversal. The strength of the earth's magnetic field is falling and this is perhaps signifies that the earth's magnetic field direction is actually reversing that means the north is going to south and south is going to north field. Magnetic nanoparticles are found in rocks and can actually be used to determine the earth's magnetic field. So, how do we know that the mean the earth's magnetic field has been reversing? These are because of magnetic nanoparticles which are found in some special rocks and from this we can actually find out the earth's magnetic field in strength and direction in the past. Such kind of magnetic nanoparticles have also been used in what we call plate tectonics where in they actually see how the plates move, how they rotate etcetera. Magnetotactic bacteria have again nanometer sized magnets which they use for alignment to the earth's magnetic field. In other words, magnetic nanoparticles are also found in the biological world and there are two examples we cite here. One is this magnetotactic bacteria, the other is what you might call the homing pigeon. Homing pigeons and many other birds also have clusters of nanoparticles and these are magnetic nanoparticles and the size of these nanoparticles is in the range of about 2 to 4 nanometer and typically it is found in the beak area and this is supposed to play a very important role in their homing ability or the ability to actually find direction and travel long distances without any other guidance. In other words, their guidance system is these magnetic nanoparticles and this magnetic nanoparticles in conjunction with the earth's magnetic field gives them the direction using which they can actually travel long distances and actually even use it for migration. So, we see that magnetism is practically universal, it is found in the cosmic scale, it is found at the earth's scale, it is also found in the biological world. Now, what is the origins of magnetism? So, anybody who has played around with magnets at a small age knows that when you have a current carrying conductor typically in the form of a solenoid then you actually have a magnetic field. So, there are these macroscopic origins of magnetism which comes from charge currents. The other magnetism which we will focus about in this course is the microscopic origin which is coming from the atomic scale. So, if you have a loop of area A carrying a current I, the intrinsic intensity of the magnetic field is given by the magnetic movement vector which typically given a symbol m or mu and directed from the north pole to the south pole such that the magnitude of m is given by I into A. Therefore, if you have a current carrying loop I of with an area A then the magnitude of the magnetic movement is given by I into A and has units of ampere meter square. This magnetic movement is a fundamental quantity in magnetism and is a measure of the strength of the magnet and its ability to produce and be affected by a magnetic field. So, it is this inherent strength of a magnet and if you have an external magnetic field then how is this magnetic movement respond, this magnetic material going to respond to the external magnetic field and how much magnetic field will this material produce is given by the strength m. We will also consider what you might call a volume version of this that is magnetic movement per unit volume in the next slide. We will also take up now certain other important quantities in magnetism apart from this magnetic movement vector m or mu which we already seen which is a measure of the strength of the magnet. The other important quantity is the magnetic field strength or otherwise known as the magnetizing force which is units of ampere per meter and it is a measure of the strength of an externally applied field. So, typically when I do a magnetization experiment I apply an external magnetic field and this field can be measured in units of H. The other important quantity I have mentioned typically is not the magnetic movement, but the magnetic movement per unit volume which is units of ampere per meter and this is given a symbol magnetization. So, magnetization is nothing but magnetic movement per unit volume and deliberately we are avoiding the use of the symbol mu though sometimes we will use the symbol mu for magnetize a magnetic movement. But as far as possible we will use try to use a small m to confuse it with an equivalent quantity which we will encounter shortly. M measures the materials response to an external magnetic field another magnetization is a measure of the response of a material to an external magnetic field, but we also have to note that m can exist even if H is removed. In other words magnetization can exist in permanent magnets even when there is no external field. So, this is well known to us that is why that is origin of permanent magnets and m is the magnetization induced by an external field H. Often we may use m or an equivalent quantity called the magnetic movement per unit mass which is given a symbol sigma and it has units of m per mass which is ampere meter square per kg. Apart from magnetization often an equivalent kind of a quantity is the magnetic induction or a magnetic flux density B which is defined as the magnetic flux per unit area and has units of Weber per meter square or volt second per meter square in more fundamental terms. And B is the magnetic flux density inside the material B is related to the previous two quantities we defined H and m by B is equal to mu 0 into H plus m, where mu 0 is the magnetic permeability of vacuum. Often a very fundamental question is asked which is the more fundamental quantity is it m the magnetization or B which is the magnetic flux density. Though in there are two schools of thought in one school of thought m is considered a more fundamental quantity and B is supposed to be a product of that, but in a certain other school of thought B is considered a fundamental quantity and all other quantities are derived from that. In this set of lectures we will assume most of the time will restrict our discussions to m that means we will assume that m is the more fundamental quantity and work around it. So, another important quantity which is known as the magnetic susceptibility chi and it is dimensionless kind of a number and it is the ratio of m to H. In other words if you have an external magnetic field H what is the magnetization it produces m. In other words it is a response of the material to an externally applied field its response in terms of the magnetization. Sometimes a subscript V is also added to the susceptibility because we are just to emphasize that the quantity we are talking about is what is known as the volume susceptibility. And the reason we need to do that is that sometimes X subscript m which is a measure which is also called the mass susceptibility and molar susceptibility are also used in literature. If I want to compare two or three different kind of magnetic phenomena and want to understand the response of a material susceptibility perhaps is a very good quantity and it helps us get a feel and physical picture of the type of magnetism in. We will see lot of details about this later, but typically if I have a paramagnetic material the magnitude and the sign of the susceptibility will tell me that it is a paramagnetic material. If I am talking about a phenomenon like diamagnetism again that sign and the magnitude will tell me of susceptibility will tell me that I am actually dealing with a phenomena like paramagnetism. And of course, technically susceptibility is a second order tensor and should be written with subscripts i comma j. And the equivalent quantity m by H for the case of B by H is called the permeability and it is again a dimensionless quantity. So, as I said in this lecture of lectures we will focus on m H and chi and we will play lesser importance to mu and B. Another important quantity we will deal with is the quantity known as magnetic anisotropy. Anisotropy implies that the property in question is not equivalent in all the directions. In for suppose I am talking about magnetization this implies that the magnetization may not be same in all directions and the magnetization may not be in the direction of the applied field. So, these are two important things that means that even though I am applying a field in a certain direction the magnetization may appear in a certain different direction which means there is an anisotropy. This also implies if you take a normal single crystal there are easy and hard direction for magnetization. And this of course, depends on the kind of magnetic crystal you are talking about for ion for instance the 0 0 1 direction is the easy direction of magnetization which changes to 1 1 0 and 1 1 1 for other materials like cobalt and nickel. Therefore, typically magnetic materials are anisotropic and the term used for this is called magneto crystalline anisotropy or sometime more casually as the crystalline anisotropy. There are other sources to magnetic anisotropy like the shape anisotropy the stress anisotropy etcetera. But without of course, objective if this quantity magnetic anisotropy is used we have to remember that this is arising because of what you might call the magneto crystalline anisotropy or the anisotropy coming from the crystal scale. So, let us summarize some of the important things. So, we have talked so far number 1 is that magnetism can come from two important sources from macroscopic currents or charge currents or from microscopic origins. The macroscopic currents is the more familiar one we are accustomed to whenever I have a conductor especially in the form of a loop. Then if I multiply the current into the area of the loop then I get a quantity called magnetization in other words this current carrying loop is a magnet having a north pole and a south pole. This magnetization which I have defined so far is a measure of the strength of the magnet. So, if I want to say how strong a magnet I will use this quantity called magnetization. The other important quantities in the field of magnetism are the magnetic field strength H, the magnetization M, the magnetic flux density B, the susceptibility chi which we said is a very important material parameter which we will discuss in detail later and the permeability mu and apart from this we also talk about the magnetic anisotropy. So, these are the important quantities in magnetism. Next we take up the atomic origins of magnetic movements and we will note that there are three atomic origins of magnetic movements one due to nucleus the other two due to the electrons. The one due to nucleus is because of what you might call the spin of the nucleus the one due to electrons can either be due to the spin of the electrons or can be because of the orbital motion of the electrons. Now, we have to be a little careful here because the spin is not like a rotating top this is the intrinsic spin of an electron and in some sense this is a classical way of looking at a quantum phenomena like spin or angular motion of an electron around the nucleus. Now, typically the nuclear spin and which is slow has a small contribution to the overall magnetic effect and this is typically ignored in most of our discussions in this set of lectures. But, we have to note that at very low temperature magnetization due to nuclear spin may become important and cannot be ignored. But, in these set of lectures we will essentially focus our attention on the magnetization appearing because of the spin of the electrons and that because the orbital motion of the electrons around the nucleus. Noting again that these are actually what you might call not the regular spin, but what you might call the quantum mechanical origin of a spin. Now, the magnetic movement due to a spin is equal to the magnetic movement due to orbital motion of course, here in the first Bohr orbit and is approximately expressed in terms of what is known as a Bohr magneton or mu subscript b. Of course, as I said the mu is also has a symbol m, mu b or m b is given by e h by 4 pi m and m is of course, the mass of the electron here and it has a value of about 9.27 into 10 power minus 24 ampere meter square. This quantity the Bohr magneton is actually fundamentally a very fundamental quantity. It is one of the fundamental constants of nature and this is equivalent to what you might call for charge in the case of an electron. So, the charge is one aspect of an electron. The other aspect of an electron is because of its spin or the angular momentum it has got and the associated magnetic movement with that spin has this value of the Bohr magneton and therefore, it is one of the fundamental constants of nature. Now, if you are looking at this equation it must become clear since the mass sits in the denominator and if I suppose I replace and I said that this what you might call this magnetization is basically because of an electron and instead of having the finding the magnetization for an electron. If I replace this mass with the mass of an for instance a nucleus then I would know since some nucleus is much, much more massive than a even for an hydrogen atom than an electron and since this quantity mass sits in the denominator the magnetic moment arising because of the nucleus is going to be a small quantity. And therefore, because of this reason that we can actually safely ignore the nuclear spin effect in most of the effects we will be talking about. Now, just to summarize this important slide that there are only three fundamental origins of magnetism and any kind of phenomena we will describe later has to come basically from these three origins of magnetic moments that means, the orbital motion of electron or the spin motion of electron or because of the spin of the nucleus. So, all of magnetism if you are going to be talking later like ferromagnetism, anti ferromagnetism etcetera, we will have to arise from these fundamental origins of magnetic moment. And additionally we have noted that there is a fundamental magnetic moment in nature which is called the Bohr magneton which is given by e h by 4 pi m subscript t which is the mass of the electron and it has a value of about 9.27 into 10 power minus 24 ampere meter square. Now, how do I start with this atomic magnetic moments and understand magnetism in bulk materials like a what you might call a polycrystalline specimen of iron. In fact, the material I could be describing could be more complicated material like what am I calling hybrid. So, I could actually be describing magnetism of hybrid and later on we will see that certain important phenomenon in the nano scale like giant magnetor resistance are phenomena which are found in hybrids. Therefore, I want to start with fundamental components of magnetism like the orbital motion and spin motion of electrons. Then I want to traverse a certain higher length scale and understand the magnetism of atoms. Then I want to perhaps understand the magnetism of molecules. Then I want to understand the magnetism of solids and finally, of course I want to understand the magnetism of hybrids. This is of course, a very arduous task. It is a very difficult task and perhaps there are many steps which will not be able to pause for us to take up in this course, but there are one important step we need to note that suppose I take this individual atomic magnetic moments and make an atom out of it. That means an atom now was many electrons. Therefore, there are going to be multiple electrons orbiting around the nucleus. There are going to be multiple spins involved. Nevertheless, all this finally is going to give rise to the magnetic moment of an atom and this magnetic moment could be 0 or it could have some positive value. More importantly, when I take these atoms and make a crystalline solid out of it, for now we restrict ourselves to crystalline solids. Then what happens to these magnetic moments? What happens is that this spin motion can couple with the lattice and this spin motion can couple with the orbital motion. In other words, the orbital magnetic moment may couple with the spin magnetic moment and finally, there could be a coupling between the orbital motion and the lattice. Because now I am taking these individual components of magnetic moments in the form of spins and orbital motion making an atom. Then I am putting together atoms in the form of a lattice and therefore, all these couplings are possible. It is to be noted most of the time this spin lattice coupling is weak. The spin orbital motion coupling is weak, but typically the orbital motion lattice coupling is strong. This is a very important point to note that implies most of the time the orbital, the contribution of the orbital motion to the magnetic moment of a crystalline solid is going to be small. Most of the time this coupling means that when I apply and as we said the magnetization is an ability to be affected by an external magnetic field and this will be only affected if the spin is able to respond to the external magnetic field. If this orbital motion lattice coupling is very strong, the response is going to not be that good. It is going to be sort of frozen and therefore, typically the orbital contribution to the magnetic moment of the solid is going to be small. Now, this strong coupling when we go down to nano particle sizes is going to get little weaker and we will notice because of this reason. Because now the lattice is very small it is a small crystal. We will actually see that when you reduce the dimension of a crystal the magnetic moment will actually increase. Even though the fundamental contributions to the magnetic moment are not changing this coupling is actually reducing and therefore, we will see that the magnetic moments of nano particles etcetera reduced dimension systems is going to be larger than their bulk counterparts purely coming from the fact that now these kind of couplings operate. Now, another question we can ask is that how can I understand magnetism and I am talking about magnetism means I am talking about various phenomena of magnetism like ferromagnetism, antiferromagnetism etcetera. Now, here we some what you might call it may take a broad overview or what you may call a simplified picture where we try to understand magnetism in three language of three terms. One is that there is actually direct coupling that means that the moments which are present in which are localized to an atom and the moments present in a neighboring atom talk directly to each other. And this kind of a coupling typically leads to what you might call a ferromagnetic effect or there could be a in other words an exchange coupling which leads to a parallel alignment of spins. In other words now have a net moment for an atom and now this is an atom and there is a net magnetic moment associated with this atom. And there is another atom in the lattice and there is a net magnetic moment of that and these two of course, in reality these wave functions of these two electrons actually overlap, but I am just drawing them separate. They may actually talk to each other directly and therefore, you may have an special alignment of these kind of magnetic moments. And this can be thought of as an origin of for instance ferromagnetism, but sometimes even in the same material like ion there are more than one way of visualizing the origin of this magnetism or the fact that it is a ferromagnet. And this is called the mediated interaction where in moments of magnetism arising from itinerant electrons in the bands of metals they actually mediate the interaction of magnetic moments between the frozen electrons. In other words there are localized electrons to this to this atom and there are localized electrons to this atom and those localized electrons and we will see a little more detail about these what kind of these electrons which are localized and they may have a net magnetic moment. And what might happen is that this mediation may actually be carried out by free electrons or what you might call the itinerant electrons. So itinerant electrons may actually mediate the coupling between what you might call the localized electrons which are actually responsible for what you might call the ferromagnetic coupling. There is another possibility which is known as super exchange and this is supposed to be responsible for anti ferromagnetism in materials like manganese oxide that local magnetic moments interacting with other local magnetic moments and this mediation is done by a non magnetic elements typically oxygen in these cases. This is called a super exchange phenomena and this is typically invoked when we want to understand anti ferromagnetism in oxides. Therefore, if I want to understand magnetism there are what you might call think in terms of what you may call a direct coupling. That means one magnetic moment localized on an atom is talking to another magnetic moment localized another atom. And when I am talking about localized magnetic moment I imply now it is a net resultant of all the magnetic moments coming some of them cancelling internally like this oppositely oriented spins etcetera leaving finally a few bore magnetons of magnetic moment per atom which is talking to a neighboring atom via direct coupling. Sometimes we may want to invoke what is called itinerant electrons in the bands which we know that in a metal we have these bands where there are free electrons and these itinerant electrons may actually be mediating what you may call the exchange between two atoms. And finally we have we invoke the concept of what is known as super exchange to understand what you may call phenomena like anti ferromagnetism in manganese oxide. So, we will have a little more to say about these the spin arrangement when we are talking about ferromagnetism or anti ferromagnetism in a later slide. So, to summarize this another important aspect of this slide that actually there is a very difficult journey involved in understanding magnetism going from the magnetism of the fundamental components to the magnetism of solids and hybrids. And here we restrict ourselves to crystalline solids. So, this is a difficult journey, but an important cog in this journey is the fact that there is a coupling between the spins the lattice and the orbital motion and out of the three the important coupling is a strong coupling between the orbital motion the lattice which is actually what you may call quenching the orbital magnetic movement which means that I am left with not the total potential magnetic movement for every atom in the crystalline solid, but the actual value is actually reduced with respect to the total magnetic movement. Two other important parameters are required when we want to understand magnetism in solids. One is the effect of external magnetic fields. In other words, if a material does not respond to external magnetic fields then I would not call it a magnetic material. And the second important effect is effect of temperature and a phenomenal like diamagnetism and polyparamagnetism are effects of external magnetic fields and do not arise independently from the atomic entities. So, far we have seen the three important atomic entities contribute to magnetism like important magnetism like ferromagnetism and anti ferromagnetism, but there are two of them which arise purely because of the effect of external magnetic field these are diamagnetism and polyparamagnetism. And we will see that we discussing more about this other forms of magnetism in this course rather than these two phenomena which are diamagnetism and polyparamagnetism. But nevertheless we have to note that external magnetic fields can itself be a source of magnetization in a material though often this is a very weak effect compared to some of the stronger effects like ferromagnetism. Effect of temperature we will see is very very important in other words if I heat a ferromagnet like iron above the curie temperature it will become a paramagnet. In other words and additionally we have to note that suppose I have a material which is in the like iron and or if you take a paramagnetic material like oxygen and I apply an external magnetic field the effect of an external magnetic field is to apply a torque on the magnetic movement which is now randomly oriented. This torque actually will need to what you might call precession of the magnetic field and not alignment of the magnetic field. And therefore, if I need alignment then this alignment itself is actually assisted by what you might call the thermal disordering effects of the spins. Therefore, effect of temperature and understanding magnetism is very very important. So, whenever I am talking about magnetism I need to know how a magnetic material is going to respond to an external magnetic field and how a temperature plays an important role towards understanding magnetism or the magnetic state of a material. Magnetism can if you want to understand the branches of magnetism there are three important branches. One we may call we just now pointed out diamagnetism which is a property of all matter. The second is what we have been talking so far that arising out of magnetic movements there is been an orbital motion of electrons. And this is what give rise to a familiar kinds of magnetism like ferromagnetism, ferrimagnetism, anti ferromagnetism and curie paramagnetism. But there is a third class which is arising purely because of the band structure in materials and these include the poly spin paramagnetism the band ferromagnetism and the band anti ferromagnetism. Most of these lectures we will actually be focusing on the central branch which is this branch. And we will not spend much time on the magnetism arising out of band structure of metals, but we will invoke the band structure of metals even to understand a simple phenomena like what you might call the non integral magnetic movements in a very simple material like ion. So, we have to invoke the concept of bands, but we can still understand magnetism in ferromagnetism in ion using some of the common what you might called localized movement concepts. While if the details actually involve bands we will briefly talk about diamagnetism though we will not take up the concept of diamagnetism in nano materials in any kind of a detail. So, in this set of lectures we will focus on the middle branch in which we will talk about those magnetic effects which can essentially be understood using what our familiar language of atomic magnetic movements. Though some of these effects are equally important and they also have to be kept in mind when you are talking about magnetism as a global phenomena. As I just pointed out diamagnetism is an property of all materials. So, it is ubiquitous in all materials and it is a response to an externally applied magnetic field and that means that there is no requirement on the type of atoms or the atoms to have any net magnetic moments. So, it is just a property of matter and therefore, they need not these atoms need not have net residual magnetic moments like in the requirement for the case of ferromagnetism. This is a weak negative magnetic effect the susceptibility is negative and of the order of 10 power minus 5. So, this is a negative susceptibility in other words the diamagnetic otherwise the reason the word dia is used because it opposes your externally applied field and usually this is masked by presence of other stronger effects like ferromagnetism. So, whenever we talk about a ferromagnetic material there is this underlying diamagnetic effect in this material also this weak susceptibility which is a negative susceptibility, but it is since its magnitude of the susceptibility is very small we usually ignore this and we only talk about the ferrom the effect which is coming from ferromagnetism. A simplified way of understanding diamagnetic effect which is something like a classical way of using is the lens law applied at the atomic scale. Where in lens law states that the change in magnetic field will induce a current in the loop of the electrical conductor which will tend to oppose the applied magnetic field. That means the material when you change the magnetic field will oppose by putting out a magnetic field in the opposite direction and this can be thought of as an electron velocity is a function of the energy of the electronic states the diamagnetic susceptibility essentially independent of the temperature. So, it is like schematically we shown that of course there are electrons moving in this direction with the velocity and there is an electron moving in this direction and these velocities are equal, but when I apply an external magnetic field there is an imbalance created and this imbalance will tend to cause. And this of course I am talking about a material which the spins are balancing out that means orbital and spin angular momentum or the spin magnetic contributions are cancelling out. Therefore, there is no residual magnetic moment that only then this effect is nicely observed, but when you apply a changing magnetic field you see that this m 1 becomes larger than m 2 and therefore, this material responds by putting out a magnetic field which is opposing your applying. Now, typically this kind of a effect is observed in closed shell electronic configurations and that means the spin and orbital moments cancel out each other. And there are nice examples of these for instance helium which are monatomic noble gases, polyatomic gases like hydrogen, then there are other materials like ionically bonded material like sodium chloride, magnesium oxide, covalently bonded materials like germanium and silicon are also diamagnetic and many organic compounds also are diamagnetic. But since we are not going to talk too much about this diamagnetic effect in nanomaterials, we will skip it for now except noting that this is an ubiquitous effect found in all kind of materials and the susceptibility in an image plot the soap we have already noted is called the susceptibility is negative and small. And later on we will be comparing it with paramagnetic susceptibility which has got a slightly larger value and positive. The next important magnetic phenomena is paramagnetism and this is important from two aspects because it arises when the atom or molecule has a net magnetic moment. Additionally paramagnetic effects can come from what is might call the band structure and we had noted that before that actually when you noted that you can either you can have paramagnetism coming which is called the curie paramagnetism or there is an additional form which is coming from the band structure which is called the poly spin paramagnetism. And this poly spin paramagnetism is also called the weak spin paramagnetism in the case of a paramagnet the net magnetic moments of an atom do not cancel out. And the material is then said to be paramagnetic and nice example of this is oxygen and which is a net magnetic moment of about 2.85 mu b where mu b we noted was the Bohr magneton which is a fundamental quantity which measures the magnetic moment or fundamental unit of magnetic moment. The point to be noted here is that even if there are many electrons in the atom which is the case of the oxygen atom most of the moments cancel out leaving out a few Bohr magnetons as the net magnetic moment. So, this is an important thing and that means that even though there are many electrons in any given atom only the net magnetic response of the material is only a few Bohr magnetons. In the absence of an external magnetic field these magnetic moments point in random direction and the magnetization of the specimen is 0. These materials are called paramagnetic because in normal circumstance suppose had a region where I enclose a gas like oxygen gas then I would not find any magnetization from this. Because even though the individual atoms are carrying magnetic moments what is happening is that they are all randomly pointing because of thermal motion of course motion of molecules in this case. And if you are talking about a solid material even then if it is a paramagnetic material this overall there is no there is a random orientation of the magnetic spins and that implies that there is no net magnetization for the material. Whenever we apply a field of course there is an aligning tendency for these magnetic moments and there are two things which we need to note is that there is of course the aligning phase of the magnetic field. But additionally the disordering tendency of temperature is constantly present this is very very important because later on we will be seeing that phenomena like super paramagnetism where the effect of temperature tends to win when we go down to small particle sizes. Even then that we are talking about in a material which is ferromagnetic in a paramagnetic material even though each atom has got some net magnetic moment there is no tendency for neighboring atoms to align parallel. So, there is that tendency is missing each atom is has a magnetic moment and when I apply a magnetic field only then there is a tendency for a parallel alignment. So, this is a paramagnetic material but later on when we talk about ferromagnetism there is a natural tendency for magnetic movements to actually align within the solid. So, always there is an external magnetic field which is trying to align the magnetic moments and there is this effect of temperature which is trying to disordering. But also we have seen that even in when we want to align the spins actually temperature plays an important role there because if you just apply a field on a magnetic movement then this will be only be precession and there will be no alignment. So, this alignment actually record little help from temperature. The combined effect of this magnetic field and the disordering effect of temperature only means that there is only partial alignment of magnetic movements. It means there is no complete alignment in a paramagnet a paramagnetic movements never completely get aligned at any reasonable positive Kelvin temperature. And the susceptibility of the paramagnetic materials usually small from oxygen at 20 degree Celsius has a susceptibility of 1.36 into 10 power minus 6 meter cube per kg. So, this so it seems that in paramagnetic material since there is no inherent tendency for magnetic adjacent magnetic movements coming from adjacent atoms to align the overall effect is that the susceptibility is very very small. And this susceptibility is a positive quantity because the alignment is along the magnetic field unlike the case of diamagnetism where the tendency is to oppose the externally applied magnetic field. Now, when you are talking about paramagnetism it is important to differentiate two kinds of paramagnets. Those which are always paramagnetic and wherein there we need not consider any other kind of details and those which are ferromagnetic, ferrimagnetic and anti ferromagnetic and which become paramagnetic on heating. Like you take an iron sample at room temperature it is ferromagnetic you heat it up then it becomes paramagnetic. This kind of a paramagnet is different from a paramagnet which is always paramagnetic and wherein there is no ferromagnetic ability on cooling down. Because when we later on we will when we in the next slide when we plot these quantities we will note that for a material which is just paramagnetic the theta will be having a 0 value which will define what is theta soon. And this obeys a law called the Curie law while a material which is ferromagnetic which became paramagnetic will obey at slightly different law which is called the Curie Weis law. So, I have to wonder if this material I am talking about which is paramagnetic is was always paramagnetic even on cooling or was it a ferromagnetic which or an ferrimagnet or anti ferromagnetic which a heated up to actually cause this paramagnetism. So, the effect of temperature which we told is an effect of actually causing disordering. And the when we plot the mass susceptibility for a material like so the effect of temperature on a paramagnet is actually different from the effect of a temperature on a diamagnetic material. Because the mass susceptibility for a diamagnetic material does not depend on temperature while the mass susceptibility of a paramagnet can be given by what is known as the Curie Weis law which is given as chi m the mass susceptibility is given by a constant c divided by t minus theta where t is the temperature and t theta also has units of temperature. C is also called the Curie constant and theta we shall see the value of theta will differentiate a material like we said which is always paramagnetic from a material which is ferromagnetic which even paramagnetic. And also can be used to differentiate a material which is anti ferromagnetic theta has units of temperature and is a measure of the interaction of the atomic magnetic moments. So, in a normal paramagnet there is no like oxygen there is no interaction of the atomic magnetic moments and this theta can also be thought of as an internal molecular or atomic field. So, this concept goes back to Weis. So, Weis had made two important contributions to magnetic the field of magnetism one of them was the introduction of this concept of molecular field and the other concept was the concept of domains. Because we ask a question that ion is actually magnetic all their atomic magnetic moments are parallel line, but you take any normal magnetic sample it does not attract any other magnetic sample. The reason is that it is divided into domains and therefore, the internally all the magnetic fields are cancelled and there is no net external field. So, this is one of the important concept proposed by Weis long time back and this concept is very similar to the concept of an exchange integral which is a quantum mechanical equivalent of this. So, this molecular field is an older concept, but the quantum mechanical equivalent is the exchange integral. This molecular field is actually a torque and this is not a field, but actually a force or a torque tending to align adjacent magnetic moments. And we said when there is a tendency for parallel magnetic moments to actually align or either parallely or anti-parallely then you get important magnetic effects like ferromagnetism or anti-ferromagnetism. Its value is very, very high. So, the order of 10 power 9 ampere per meter and this is much stronger than any continuous field produced in the lab. So, this molecular field though it is a force or a torque is much, much larger than anything which man has produced in a continuous field. There of course, very strong fields which can be produced in a very small time frame like once you can produce a mega gauss in a femtosecond, but if you want to have continuous fields which are much longer than the femtosecond time scale then it becomes difficult to actually maintain such kind of fields. Now, this equation chi m where t minus theta can be plotted for either we said a paramagnet where which is always a paramagnet in which case you observe a law called the Curie law and this is the variation of the mass susceptibility with temperature. In other words it is false asymptotically with temperature. On their hand there are other materials which have a positive value of theta and these are other materials where this concept of a molecular field has to be invoked and in such materials again the law is very similar to the Curie law and this is called the Curie Weis law which is what we wrote where there in this theta has a positive value and not a zero value. So, there are two possibilities here both the possibilities have a very asymptotic dependence on temperature of the mass susceptibility. In other words the susceptibility falls with temperature and let us note that if there is no interaction between the atomic magnetic moments then theta is zero and the Curie Weis law reduces to what is known as the Curie law and this is what you observe for a material like oxygen. The variation of susceptibility for a these kinds of behavior the two behaviors we have seen is shown in the curve below and negative. So, if the theta is positive what we are talking about is a ferromagnetic material which has become paramagnetic. In other words there is a internal molecular field which is tending to align these materials, but since you have heated it up the thermal disordering effects are one over and nevertheless the mass susceptibility of such a paramagnet follows similar to the Curie law, but is now shifted from the origin by a value which is theta which is now in the units of temperature. This theta becomes negative for anti ferromagnetic materials that means that you can use this value of theta itself as a discriminator if whether it is a just a paramagnetic material or it is an anti ferromagnetic material which became paramagnetic or a ferromagnetic material which became paramagnetic. Of course, whenever I am talking about a disordering temperature like Curie temperature this may not actually be sharp and we have to worry about other aspects like for instance there are local clusters which may for instance above the heat of ferromagnetic it is expected to become paramagnetic above the Curie temperature, but then there may still be local clusters which are still aligned and therefore the curve may not sharply fall down to 0 and we have to worry about other effects. Now, we come to the most important effect which is ferromagnetism and we know that of course, as we point out that you have a compass then this is aligned by a earth's magnetic field and this compass is nothing but a ferromagnet which is a permanent magnet. The three forms starting from the atomic magnetic moments which have a tendency to align giving rise to three forms of magnetism. One is ferromagnetism wherein all the magnetic moments have equal strength and they are aligned parallel. The circuit is the anti ferromagnetic wherein the magnetic moments have equal strength, but they are aligned in the opposite direction. The third possibility is ferrimagnetic which is similar to ferromagnetic wherein there is a net magnetic moment unlike the anti ferromagnetic case where everything cancels out, but this net magnetic moment is smaller because there is another anti parallel arrangement of another spin which is not of the same magnitude. Therefore, there is no net cancellation because the magnitudes of the two spins are different, but nevertheless you have a overall net magnetic moment which is similar to that of a ferromagnetic material. And it is important note that these effects like ferromagnetism, anti ferromagnetism etcetera involve no new types of magnetic moments. It is basically the atomic magnetic moments which we have been talking about before the spin and the angular motion of the electron. Two important ways of understanding ferromagnetism in metals is that one is assuming that the magnetic moments are localized to atoms and they talk to each other by a phenomenon known as exchange. The other is using what is the band structure of metals. So, we said that we will not be really going into deep in understanding the phenomenon of magnetism which is arising from purely from band structure of metals, but it is important note even to understand simple phenomena like magnetism in a night we may want to invoke the band structure of metals especially if you want to understand it quantitatively. Now it is clear that itinerant electrons or moving electrons or free electrons cannot give rise to magnetism. So, there has to be a local character to the magnetic moment which is which has to give rise to magnetism. The localized picture to atoms is actually conceptually easier and has been assumed in the molecular field theory of ways and in also in the Heisenberg's approach. It should be noted right at the outset that even in metals like iron most of the electrons behave as if they are localized and the number of itinerant electrons could be a small number. So, this is very important note though we are talking about bands and free electrons the character of the electrons may be that if you take a material like iron to give an example there are eight valence electrons which occupy the 3 D plus 4 S bands. And we had even schematically drawn these two bands like how this 3 D and 4 S bands look it said that the 4 S band is could be a kind of a and suppose I am now plotting my density of states rho and with energy. Then I noted the 4 S band is more like a broad band while the 3 D band is a more localized band and we said that these are what you might call. So, this is my 3 D band and this is my 4 S band. So, we noted that these are simplified pictures in themselves and we said that actually the 3 D band has a very kind of a density of states actually has a very more much more complicated variation. But the important point to note that these 3 D and 4 S bands filled parallel we had noted that even when we drew the picture of how atoms come together to form a band structure. Out of these 8 electrons only about 0.95 electrons in the 4 S band are truly free. So, this is very important note that though we are talking about bands and we are talking about itinerant electrons the truly free character of electrons is restricted about 0.95 out of these 8 electrons which are now residing in the 4 S and 3 D bands which fill parallel and most of the electrons which is about 7.05 are actually localized to the 3 D band because the 3 D band has a higher density of states as you can see that. And the narrow width of the 3 D band implies that an electron here has more a localized character. So, the width of the band itself can be used as a measure of the localized character of the electron. While the 4 S electron you can see that there are density of states is widely spread in energy. And this means that there are energy states available for electrons to actually be promoted they are easily accessible. But then the number of such electrons is small which are truly itinerant. In nickel if you see again the picture does not change much the 3 D plus 4 electrons is 10 in number out of which only about 0.6 is truly free and 9.4 of them which reside in the 3 D band are almost localized. So, these localized moments are what is contributing to what you may call the magnetic moments. But without understanding the band structure and how the bands will we will not be able to arrive at the exact magnetic moment. So, how do we understand the band theory to understand ferromagnetism. Therefore, we need to note that a correct theory of magnetism in metals has to involve bands. Because we are talking about outermost electrons which are which is what is contributing to this magnetic effect. And these electrons are typically not localized to atoms. However, as we have noted before most of the electrons especially in the 3 D metals which are elemental magnets. Because we know that 3 D like metals like cobalt, nickel etcetera are the elemental magnets. These 3 electrons are rather localized and free electrons do not contribute to the ferromagnetic behavior. So, in some sense these 3 D electrons in transition metals are neither fully localized or fully itinerary. So, because they are in a band now we have to talk about them being delocalized. But at the same time they are not that delocalized as you can talk about them as to be what you may call a totally free electron. The band theory is actually the theory which is able to explain the non integral values of magnetic moments per atom. Though often we have to go beyond even the standard band theory to understand how what is the correct value of these magnetic moments. Now, in iron the 3 D electrons are not fully localized and about 5 to 8 percent has some itinerary character as we have noted before. And these electrons in some theories actually are what mediate the exchange coupling between localized movements. So, we said that in a mental picture of magnetism either you can talk about magnetic moments talking to each other directly or being mediated by an by a hand what you might call a free electron slightly itinerant character of an electron. But in this second picture we can think of these some somewhat itinerant electrons or the itinerant character of the electron mediating the exchange. Now, if you look at the observed magnetic moment per atom in iron it is about 2.2 mu b and this can be only understood by a band structure and a band structure which depends on spin. So, if you look at the number of electrons occupying the up spin orbital and if you look at the number of electrons occupying the down spin orbital the total number is 7.05 and the difference between the 2 is 2.2 and putting them together we can see that actually you have an occupancy in the up spin is about 4.62 and the down spin is 4.2. So, in other words to explain it a little more clearly you have now orbitals which are now dependent on the spin character. In other words if I am talking about the forest orbitals here this is the energy versus number of states and you note that the forest orbital is a free electron orbital and the 3 d orbital is now split depending on the spin. So, this is the spin up orbital and this is the spin down orbital and this is the case of iron on the right hand side the same picture is drawn for nickel. You note that the 3 d and forest orbitals are filling parallely and knowing this number of electrons which are actually filling these 2 orbital the up spin and down spin orbitals I can actually go ahead and calculate the ions magnetic movement which turns out to be non integral because the complicated way in which the magnet orbitals are filled. So, we can use some kind of a simplified band theory to understand this what you might call the non integral value of magnetic movement though at this point of time we are not going to the lot of details of actually how this calculation is made what are the errors involved in this calculation. But, it is important known this fact that now we have a band structure which is dependent on spin that means the up spin orbital fills in a different way as compared to a down spin orbital and many of the important magnetic effects come because of this fact that the up spin orbital and the down spin orbital do not fill parallely and they are shifted with respect to each other in energy. So, the above discussions can be summarized as a few thumb rules for existence of ferromagnetism in metals. Number one the bands giving rise to magnetism must have vacant levels for instance the 3 d bands in iron cobalt and nickel for unpaired electrons to be promoted because you know that exchange coupling is actually promoting electrons to an higher level while they could actually have been in the lower level and pairing out with the pre existing electron protection coupling is actually promoting them to an higher level and unless there are free levels available above and there is going to be no magnetism arising from the band picture of magnetism. Close to the Fermi level the density of state should be high this ensures that when an electrons are promoted to the unfilled higher level the energy cost is small. And this is what you see the density of states in the d orbital is 3 the 3 d orbital is high as compared to the 4 s orbital. That means that the cost of promoting an electron from a level which is the lower level for instance in the 3 d band to the next level is actually small and you can see the density of states is actually high close to the Fermi level and this is the Fermi level. So, number 2 is that the density of states should be high so that the energy cost of the exchange in other words for the parallel alignment of the spin is small. The third thing if we are going to assume a picture of direct exchange then the inter atomic distance should be correct for exchange forces to be operated which is parallel alignment. And typically this is done using what is known as the Bethes later plot in which actually what they do is that they plot the ratio of the inter atomic separation with respect to the 3 d orbital distance to actually find that which kind of elements would actually give rise to ferromagnetism. But, in nevertheless the inter atomic distance is a important factor which gives rise to magnetism ferromagnetism. So, to understand ferromagnetism number 1 there should be vacant levels in the band which is giving rise to magnetic magnetism which is typically the 3 d band and ion cobalton nickel. Number 2 close to the Fermi level the density of states should be high so that the overall cost of this exchange coupling is not too much. Number 3 the inter atomic distance should be correct that means too much distance or too less distance does not give you ferromagnetism.