 Good afternoon. So we will do some solid state physics part now which actually is a continuation of what you have been seeing by the lectures of professor Shiprasad regarding the crystal structure, various types of crystal structures and also the structural determination using mostly XRD, the X-ray diffraction technique. So with this basic ideas we are in a position to go and see something on solid state physics. Not only that even for the topic that we are doing in magnetism as I told you during those lectures, most of the time the materials which we want for applications are mostly solids and we need to understand the physics behind that. But if you remember when I did magnetism I talked about, initially I talked about atoms. Now the question is what is the difference between the physics of atoms and physics of solids? As we have seen in the case of atoms we have this discrete energy levels. We have seen these various energy levels, discrete energy levels we will call it 1s, 2s, 2p and so on. They are all discrete energy levels of course they have some kind of a fine structure but they can still be treated as very discrete energy levels. However, when you go to a solid, when you go from an atom to a solid that means you have put a large number of atoms, atoms of the order of Avogadro number if we are putting that becomes a solid. In that case what is going to happen to these energy levels? These energy levels will not be as discrete as you are seeing now in the case of atoms. On the other hand they will actually if I assume that this is the distance of separation between the atoms. As this distance decreases that is what is going to happen when you are making a solid out of atoms, these energy levels slowly they broaden and I am just showing one of them. So they broaden and as the distance of separation decreases that means they become closer and closer, these energy levels which were there earlier for the atom they actually become more and more wide and this I can call it as the band width. So this is the band structure idea that you know in the case of solids which is not there in the case of atoms. In the case of atoms you have fine structure split energy levels but still they are discrete. However, when you make a solid out of atoms you have these kind of bands and the band idea is coming only in the case of solids. Now one more point is there in this case again something related to what I was answering to a question yesterday. As far as these core electrons are concerned even when it actually goes from an atomic state to a solid state these kind of broadening will be very very small effectively will be very very small. If you go to higher and higher energy levels of the atom they are the ones which are actually going to get broadened as in this case. So the so called core energy levels for example oneness is a core level that is essentially the atomic character will be retained even when it actually is in the solid. So the broadening happens only for higher energy levels this is what is actually going to determine magnetism in some sense which we have been talking about yesterday we will come back to that in the other lectures. So that way this is important for the magnetism discussion also. So the core electrons I mean core levels are more or less unaffected whereas higher energy levels actually get broadened because of the more and more interactions that these atoms are subjected to when you are having a solid constructed out of an atom. With this basic idea so as I mentioned the when we have solids broadly we can classify them to be of two kinds for this discussion. One is the solids which are actually non-metallic and other set is metallics. So we have metallic solids and non-metallic solids. In the non-metallic category I will include insulators, semiconductors all these things. In the case of metals we have good conductors where we have large number of free electrons and so on. In this discussion we have only very limited time. So I will only talk about the physics of metals or only one part of solids we will be able to discuss in this lecture series. So I will straight away start and show you some of the very important properties of metallic solids not just solids in general but metallic solids. So that is why I have given the title as free electron theory and the band theory. We will start with the free electron theory and then we go to the band theory which I was just telling how the energy bands are happening when you have the solid of various atoms. Before we do that so basically you are going to deal with a large number of electrons. So the question that is coming to our mind immediately is how you have various energy levels and you have large number of electrons. How these electrons are arranged or distributed in the allowed energy levels? This is a very important point because this is going to determine what kind of a physical property this metallic solid is going to show. So the electron distribution among the various energy levels is a very very important thing to do before we really take up this matter. The other issue one has to keep in mind which I have already done is as I told you when you are constructing a solid out of atoms you see that the complexity is much more because you are having single atom in atomic physics. In the case of solid state you have a very large number of atoms to deal with. So the problem should have been extremely extremely difficult. It is almost impossible to solve but for a particular issue that was what I was actually talk to you in the last two lectures of solid state physics namely the crystal structure. What is the advantage you have in the case of solids over the atoms which actually helps you or helps in minimizing the complexity that is expected when you have solids compared to atoms. The extra advantage that you have in the case of solids is nothing but the symmetry. So you have seen various crystal structures, various symmetries associated with that this was done earlier. So why the crystal structure and the symmetries are very important this is what is going to be seen as you go along in this lecture series. So that is why first we do the crystal symmetry, this crystal structure then we will go to the actual theory behind the electronic properties and so on of in fact for other properties as well of the solids. So the symmetry of solids gives you a very important advantage as well as the theoretical understanding of various physical properties of solids are concerned. So two things one is we should have an idea about the symmetry of the solid various crystal structures their symmetry is associated with various structures that is number one that is done. And the other thing is what I was talking about just now the way in which different electrons are accommodated among the various energy levels consistent with the given solid this is to be understood before we really try to attack the problem of energy bands or things of that kind. So first let us see but let us ask this question. So as I mentioned you need to know what kind of a distribution that you have for these electrons. Of course we will not worry about the other part the electrons only we will worry about I will tell you why we are neglecting the other contribution slowly those things will come in picture I will come to those details as we go along. So the question is how do we fill particles in general before we really discuss what happens in the case of electrons. Let us have a very preliminary idea very qualitative at least a qualitative idea regarding the distribution of particles in general among the allowed energy levels in a particular case. So as I have written this kind of this distribution the nature of distribution that is possible is going to be determined by the nature of the particles as I am going to show you. And in this case you have two varieties one is a so called a classical particles with which we are all very familiar and the quantum particles. So the distribution of classical particles and quantum particles these distributions are quite different as you go from the classical nature to the quantum mechanical nature. In fact in the quantum mechanical nature there are two at I will show you. So the branch of physics which actually deals with these things or gives an answer to these kind of distributions in addition to many other things is the statistical mechanics. So we need to get a very elementary idea of statistical mechanics if you want to understand the physics aspect of metals or solids in general. So the crystallography is one idea that is very important the symmetry related information is very important and then the ideas of statistical mechanics also important in understanding these properties. So since you have already seen the crystal structure related issues I will only talk about some ideas about the statistical mechanics part. To give you an idea about this first let us take a very simple picture of what is meant by so to see the difference between the particles which are distinguishable and particles which are indistinguishable. Indistinguishable particles are also called identical particles. So electrons for example they are identical particles you cannot name electrons ABC. So they are identical particles you can have indistinguishable I mean distinguishable particles like what I have shown here in this figure I am taking two red balls and one blue ball or stands for red B stands for blue. So the question is and they are given three boxes you have three balls two of them are red these two red balls are exactly identical you cannot separate them out and you have another blue ball. You have to arrange them in three fashions three different distinguishable arrangements you have to make using these three boxes. As it is obvious you have only three possibilities as shown here you can have an R R B combination R B R combination B R R combination. Suppose these R and R these two red identical balls were having some difference you would have got many more such distributions or many more such possibilities. If all the three balls were of different color then you have a large number of combinations possible large many kinds of possibilities where X would have been possible. On the other extreme if all the three balls were let us say red in color then you would have got only one arrangement you cannot tell this is different from this is different from this this is not possible. So you have only one kind of an arrangement if you have only all the colors to be the same. So that means the it is a very simple picture but it gives you a very important information what is it information depending on the nature of the particle depending on whether the particles assume that these balls are particles if these particles are identical and when they are not identical the number of distributions that is possible for a given set of energy levels here the boxes are essentially the energy levels this number is going to be quite different. So this is what you are seeing from this picture. So the number of possible ways in which you can arrange particles among various energy states is critically determined by the nature of these particles. First of all whether they are distinguishable or not if they are distinguishable you have many more possibilities if they are identical you have very very small number of possibilities or distributions. This is a very simple idea but it gives you the very important difference between indistinguishability versus distinguishability and in quantum mechanics quantum particles are identical they are indistinguishable and as we are going to see we are going to deal with electrons you are actually discussing the identical particles in a very big way so that is why this becomes this difference becomes very important for us. So particles in general particles available in nature in general can be classified as identical particles and distinguishable particle strictly speaking you have only this kind particles which are identical under certain conditions the so-called classical conditions they behave as if they are indistinguishable and when they are distinguishable the usual classical particles when you have a dilute gas kinetic theory of gases what we do we use the distribution which all of you know is nothing but Boltzmann distribution. So the distribution that one talks about when you are talking about atoms or particles which are distinguishable the so-called classical particles when you tell it is distinguishable when you are telling it is distinguishable you mean that you are in the classical regime then you use the Boltzmann distribution which you have used in various forms. In fact Maxwellian distribution of velocities finally gave you the equipartition theorem which you used in other course. Coming to the quantum particles the so-called identical particles there are two categories one is the Bose particles or bosons other thing is fermions or the Fermi particles. So electrons as I am going to show you belong to this category of fermions and you have bosons simplest example of course is photon which actually is a boson. So quantum particles you have two categories one is bosons one is fermions. So let us see further what is the difference. So in that sense we can have three kinds of particles actually only two but if you bring in the classical also into picture under certain conditions these quantum particles behave as if they are classical particles. If you bring them that means you are treating them as distinguishable particles. The main thing to worry about in this case is that in the case of classical particles they do not obey the Pauli's exclusion principle. Pauli's exclusion principle need not be worried about in the case of distinguishable particles classical particles. What are the example any gas that can be approximated to ideal gas conditions where you assume that you can track the various particle atoms. So you call this atom A this atom B and so on you can track the atom every time especially when it is very dilute and the temperature is very high. This approximation is valid you will really get into an ideal gas situation there you are justified in calling it as a classical gas and this Pauli's principle will not be coming into picture you can use the Boltzmann's statistic without much of a problem. What happens when you have other particles? So quantum particles type 1 as I mentioned they are called bosons they are indistinguishable identical they do not obey Pauli's principle examples are helium 2 4 this is an atom and as I mentioned the photon. So the characterizing thing here is that if you have the spin of the particle to be an integer including 0 then the particles are bosons there is a very important theorem which is connecting the two the spin and the nature of this particle I will not go into those details but whenever the spin of a particular particle is an integer then it has to be a boson. On the other category is that you have quantum system particles the particles which are called fermions they are again indistinguishable they are identical but they obey Pauli's principle Pauli's exclusion principle has to be followed here these particles are characterized by spin which are actually half integers for example half starting with that is the smallest number possible smallest half integer example electron proton neutron we talked about it earlier also all of them have spin half helium 2 3 this is an isotope of helium which actually has s equal to half because this 3 is coming here so then its spin is half here so this is also a fermion. So you have quantum particles you have bosons and fermions bosons have integer spins whereas fermions have half integer spins. So this is something which is very different between the two and when the temperatures are very high energies are very high both these particles both bosons and fermions behave as if they are kind of classical behave as if they are distinguishable then you can approximate them to be following the classical distribution function which is actually the Boltzmann distribution which all of you have done in the kinetic theory of gases but the kinetic theory of kind of a treatment as you have seen in the modern physics lectures will fail when you come down to low temperatures and things like that because the quantum features wherever the quantum features set in the Maxwell Boltzmann kind of a description will fail. Now our thing is we are more worried about fermions because we are trying to see what happens to the electrons in the case of solid or in the case of metals and electrons being fermions as mentioned here we have to we will only worry about of course this is also an interesting topic bosons we will not go into those details we will right now worry only about the fermions the physics of fermions the distribution of fermions for example as I mentioned earlier if you have set of energy levels you are given some fermions like for example electrons how do they arrange themselves among the various energy levels this is an important issue for us to understand the properties of solids. So we will only worry about fermions from now onwards fermions also we will mainly worry about electrons just to give you an idea about this what I have shown here is you are taken you are given two fermions let us take two take two electrons and you are given three energy levels and you are also given another constraint that the total energy of this arrangement whatever you are going to choose must be equal to 2 epsilon. So you are given three energy levels the value of the first energy level is 0 the ground state the second is epsilon but remember this epsilon is given as a doubly degenerate level as we talked about yesterday. So that is why the two lines are there the double degeneracy is here and then you have another level which is non degenerate that means the degeneracy is one which has got an energy of 2 epsilon. Now you are given two electrons and you have to make sure that the total energy is exactly 2 epsilon. So how do you do that one possibility is this one you put one electron here or one fermion here let us worry about electrons here this is one here. So this is zero energy level so this will not contribute to anything to the energy and the other one can be put in this one here so that total energy is 2 epsilon because this is 0 and this is 2 epsilon times 1 is 2 epsilon. The other possibility is you can put you can ignore the ground state you can ignore this state also you can take two of them and all the both of them and put here and here there is no problem the Pauli's principle is not violated even though they are supposed to follow the Pauli's exploration principle because this is a fermion but it is not violated because there is a degeneracy of 2 here. So as long as a degeneracy is there you do not have to worry about it it is taking here one is here one is here here again you can see that one electron in energy epsilon the other electron in energy epsilon the total energy is 2 epsilon so our original constraint is satisfied you do not have any other choice in this case if you do not change your total energy so once the total energy is fixed total number is fixed and the total energy levels are fixed the degeneracies are fixed then there is no other possibility in this case. So like suppose I have if you see both these things are written as x yx because they are identical I am calling as fermions means they are identical they must be there should not be any symbol that distinguishes them and hence if I tell one as x the other thing must be x if I call the first one as a there is nothing like first and second you have to call both of them a both of them x both of them y on the other hand if this same thing was done in a classical manner it was treated these things were treated as a classical particle I should have been calling one as a other thing as b then as in the case of the ball example you would have seen you would have got many more more possibilities. Similarly if I have if I did not have the Pauli's exclusion principle condition when it was a boson then again the number will be different I am not going to those pictures but one can actually construct something like this I will put some of these things in the lecture slides when I upload it. So the point is depending on the degeneracy depending on the energy levels depending on the total energy constraints so total number of particles is a constraint here total energy is a constraint here degeneracies are fixed in the problem energy levels are fixed in the problem then so one can actually find out how various electrons can be distributed among various energy levels the total number has to tally. So this becomes an important issue as far as filling electrons in the areas energy levels of a solid is concerned also. So this idea how the electrons get distributed among various energy levels is actually determined or given by the ideas of statistical physics. So what is a point this is what I have been mentioning you are given a fixed amount of energy that is a constraint number of particles is fixed degeneracies are fixed energy levels are fixed. Then what is a distribution corresponding to this very important the equilibrium situation that means the situation where the system is in equilibrium we can have various possibilities but we are looking at the equilibrium situation distribution function corresponding to the equilibrium scenario. So when the system is in equilibrium what kind of distribution how many electrons will be in the first level how many will be in the second level how many will be in the nth level this information is very crucial especially when it is regarding the equilibrium situation because we all everything we are going to discuss for equilibrium situations. So equilibrium distribution function is very important. So what we are trying to look at is you want to find out what kind of a distribution function I am going to get for fermions in this case electrons at a given temperature when all the other things are fixed like for example your energy total energy energy level degeneracy all these things are fixed what kind of a distribution function one can get this is going to be very crucial in determining the physical properties. This is true for any property including the magnetic property that we have been talking about. So this kind of a picture ultimately gives you this idea the what is this idea this is given by this expression I have cut short some of the things to save our time. So n i that is a number of electrons that I am taking only electrons number of electrons that is accommodated in an energy level i i th energy level is related to g i that is g i is the degeneracy of this particular level e to the power of alpha alpha is a constant I will come to the very soon e to the power e i divided by k t plus 1. So t is a temperature k is a Boltzmann constant e i is or epsilon i is a energy level of that particular i th level using this what is more often described is this n i the ratio of n i divided by g i that means the number of particles in a particular energy level divided by that particular degeneracy of that level. So that is called the distribution function I have written as f f d f d stands for Fermi Dirac distribution function which actually holds good for fermions like electrons as we are going to see we are to worry about only electrons electrons being fermions the distribution function which is relevant for these fermions is nothing but a Fermi Dirac distribution function that is why this f d stands for Fermi Dirac distribution function which actually tells you at a given temperature the equilibrium configuration corresponds to this particular function called a Fermi distribution function. It has got a particular nature 1 by e to the power of alpha e to the power of epsilon i divided by k B t k or k B both represent the Boltzmann constant plus 1. Just to give you an idea about this do not worry about this alpha alpha factor is essentially to take care that the total number is concerned if you give me 1000 electrons I should finally add up together and give you 1000. So that is essentially played by this alpha quantity that we will see later you can see a plus sign here this is for the fermions just to show you one important difference this is basically like a probability I can think this is a probability distribution. Suppose I have I do the same thing for bosons the main difference that is going to happen when I do the whole theory I would have got a similar expression the main difference would have been that I should have been getting a minus sign here. So in the case of Fermi Dirac distribution function it is a plus 1 here in the Bose-Einstein distribution which will describe the Bose particles or the bosons it will be minus 1 here and why is it plus 1 here very simple argument is this one you see what is the main difference between the two kinds of particles here the Pauli's principle must be valid. So you can see because of this plus 1 here this is always a positive quantity this can never exceed 1 as I told you this is essentially you can treat it as a probability this probability if it goes to more than 1 that means you have two for example if you are getting this to be 2 that essentially means that you have two particles surely there in a particular energy level that means you are actually violating the Pauli's principle. So that is avoided violating the Pauli's principle is why completely avoided by this plus sign this plus sign will always make sure that your denominator is more than 1 which means that this fraction will always be less than 1 at the most it will be 1 which actually tells you that the particle is actually there. So this role of this plus sign here this plus 1 is to basically make sure that the Pauli's principle is satisfied it is not violated in the case of bosons as I have written down earlier this condition is not there the Pauli's principle condition is not there and this minus sign is alright. In fact you have many more particles many particles coming and occupying a particular energy level very interesting physics there is possible in the case of bosons but definitely not for electrons and fermions. So this plus sign the physical significance of plus sign in the context of Pauli's principle must be appreciated. So this is the very important Fermi distribution function for determining how various electrons will get distributed among the electronic energy levels of a solid at a given temperature corresponding to the thermal equilibrium situation because the properties that we are going to calculate will be all for equilibrium properties. So this is the other requirement to study solids. So first thing was the symmetry second is this distribution function of solids I mean electrons in the case of solids. So this gives you a very important clue to go ahead and see to do the problem of how electrons behave in the case of a solid especially a metallic solid. Before we do that still we have to do something more. So as I mentioned we will only worry about properties of metals that means good conductors. So as I mentioned mostly we are dealing only with metals. Metals are very well known for various reasons many of the applications are dependent on metals including magnetism in many many applications as I have shown you in the other lectures and whatever is remaining also many of these metallic materials are actually magnetic and they are very useful. Many of the permanent magnets for example which you are using they are all metallic systems not may not be a single metal but a metallic alloy where the theory can be essentially extended. So metals are very important in that way how do you characterize a metal how do you tell that this is a metal and this is not an insulator. A metal is characterized by its positive temperature coefficient of resistance what does it mean? That means if you increase the temperature metals will definitely show an increase in the electrical resistance this is a very very important property of metals. So the positive temperature coefficient of electrical resistance is a very very important characteristic of a metal they are also known to have very high electrical and thermal conductivities because in general they have large number of free electrons. The free electrons are very large simple example is sodium for example this as I have been talking earlier sodium is a metal because it has got very large number of free electron I mean free electron per atom. So this is a huge thing when you have a solid this contributes very much in the case of electrical conduction and thermal conduction. So metals are actually both good thermal conductors and electrical conductors. So if you look at historically how the things were developed the first thing that was done was what is known as a free electron theory. Free electron theory in 1897 was the discovery of electron so things started from there onwards. The first theory which was called a free electron theory came in 1900 and the theory was purely a classical theory was given by Drude and this is called the Drude model of metals which actually is purely classical no quantum mechanics of course quantum mechanics was not there at that time. What was done there was at that time thermodynamics classical statistical mechanics these things were known and the kinetic theory of gases as I mentioned just now was known the theory of kinetic theory of gases was well known. So basically Drude was trying to use the ideas of kinetic theory of gases to the solid what is the connection the connection is this in the case of kinetic theory of gases you have the dilute gas we have atoms there moving around freely. In the case of metals take a monovalent metal like sodium every atom contributes one electron monovalent means one electron is given. So basically what Drude assumed was these electrons which are free one electron from sodium they essentially behave as if they are completely free they will not be worried about the positive ions which are there when you take one electron out of sodium you are left with a sodium positive ion you do not worry about it. So you only worry about these if you are 100 atoms you are having you have 100 electrons to be worried about forget about all other things your container you have to worry about container is nothing but a crystal in this case the crystal sodium crystal you have to worry about how these 100 electrons will behave how they will distribute and things like that. So this is the analogy between the kinetic theory of gases classical idea and the Drude's classical idea of a metallic solid. Obviously it will not be correct as you can see today but you can see certain things were still explainable at that time even with this one. So what are the essential assumptions of Drude model as I mentioned again it is a classical model it is a free electron model there are two things one is classical classical means there is no quantum ideas taken into account free electron means it will not worry about the positive ions which are left behind the sodium ions are not at all treated in the problem they are completely forgotten. So it is purely the free electrons which are taken into account it is a free electron model classical and free electron model that is what is the Drude model is. What are the assumptions that I already mentioned but still it is listed here positive ions are assumed to be immobile they are not contributing. The valence electrons the last electron the 11th electron gets completely detached and it wanders freely in the crystal we do not so the electrons are there there is there can be an interaction between the electron and electron one electron and another electron this is also ignored. So you do not worry about electron electron interaction you do not worry about the electron core interaction this is again an electrostatic interaction is possible there because you have a positive ion core core and you have an electron that is also ignored. Only way the electron this electrons when they move only way they see the core the job of the core is basically to scatter these electrons. So it was assumed that its only role of this positive ions is to scatter these electrons as they move otherwise they are completely free. So these are the assumptions of these were not the correct things they were the assumptions of under the Drude model and when he worked out these assumptions were used to derive something that is what I am coming to. Then typically if I try to work it out typically the free electron density number of electron per unit volume in a metal if I take it is something like 10 to the power of 22 cubic centimeter it is a very large number compared to atomic physics the number is very large. So you are basically dealing with a large number of electrons of the order of Avogadro number as I mentioned electrons put in a very small centimeter size crystals metals. Then you have two I mean concepts which are very important again these concepts were there again in the case of kinetic theory of gases. One is when these things move around they get scattered by the positive ions they one can define what is known as a relaxation time that means the time over which they are completely free from the scattering when they move around it is scattered by one positive ion. Then before it actually encounters another collision another scattering there is a time lifetime over which it is free that is called a relaxation time tau this actually this is tau. And if you can multiply this relaxation time with the meaningful velocity or the speed you can actually define what is known as a mean free path which again is defined in the case of kinetic theory of gases mean free path is the distance over which an atom can move before it encounters another collision something similar here this is the distance over which this electron can move around so that it does not suffer a collision during that distance. So then what we are going to see is how you can actually talk about electrical conductivity which is a very very important very basic property in this case of metals. So what you have is you have an electron which is actually moving then you do not have an electric field you see that very at time t equal to 0 there is no electric field. So what happens is these velocities with which they are moving they will all be random and this does not give rise to any current as you all know unless you apply an electric field by with the help of a source of emf you do not establish a current even in the best conductor possible. So this is because there is a thermal velocity which actually is averaging to 0 which does not give rise to any particular directional flow of flow of these charges and hence you do not get a current. If you want to get current what you have to do is you have to apply an electric field when you apply an electric field like this shown here simple mechanical kind of things here. So when you apply an electric field because of the negative charge the electron moves in the opposite direction. So assume that this electric field has been applied just after the collision this electron has suffered. So assume that at that time its velocity was v0. Now this actually is acted upon by a force e time c the acceleration is divided by m and over the next time period t this is actually having a drift velocity this is the drift velocity that it is going to happen just because this applied electric field is directional and this is giving some particular direction for the charge flow to happen. Whereas this one will essentially average to 0 if you take a over a finite time interval this will not contribute because this is a thermal average velocity which will actually become 0. Whereas the contribution that is coming from the applied electric field will survive as we are going to see in the next one. So one can actually find out the average of velocity is the average of the first term and the average of the second term. So the average of the first term as I mentioned will go to 0 because of the random nature whereas a second one can be written like this these are all constants only thing that is coming is essentially the average of the time. What is the average time here average time over which this is able to survive the collisions that is nothing but by our definition that is nothing but the relaxation time. So the average of time is nothing but the tau here so this becomes minus ee tau divided by m this is called the drift this has dimensions of speed as you know this is called the drift velocity. So this drift velocity which actually is non-zero unlike the v0 gives rise to the current and that is why when you apply an electric field connector source of emf to a piece of wire metallic wire you establish a current in that wire. And going further if I have n number of electrons per unit volume I can find out a charge density the current density j actually is given by this expression some problem has happened I will try to correct it when I do that. So this gives rise to n e square by m tau this actually what is not appearing here is this tau. So this is n e square by m tau times e this is the current density which actually is obtained by taking the number of charge carriers times a charge that is e multiplied by the speed will give you the current density because charge when it is flowing is rise to the current. So when I substitute this for vd here I get this expression this is nothing but my ohm's law j equal to sigma e this sigma is missing here sigma is j equal to sigma e is nothing but your ohm's law which actually can be shown that is exact like v equal to i times r. So j can be converted as i divided by a a is the area of projection and sigma is nothing but 1 over rho rho is electrical resistivity and as you know rho is related to the resistance r is given by rho times l divided by a. So one can rearrange this and you can show that this j equal to sigma e where j is a current density sigma is electrical conductivity e is the applied electric field can be written in terms of i and the applied voltage v then you will get v equal to i times r where r is electrical resistance not a resistivity that is nothing but the ohm's law. So j equal to sigma e is exactly same statement of ohm's law the way you write v equal to i r. So this is the ohm's law which has found to be alright in many metals when it was initially tried for experimental conformation. So j equal to this I will not worry about it I already mentioned here. So j equal to sigma e is a thing which is working and to some extent the this idea was possible to be explained with the help of this druid picture. If you look at this the if you take the actual value the experimental value of the resistivity of copper this is something like 0.3 micro ohm centimeter this is the unit of resistivity at 77 Kelvin. As I told you earlier it will be more at the higher temperature 270 that is the room temperature it is around 1.56. So this is the experimentally observed electrical resistivity. As I mentioned I can define the mean free path as I mentioned earlier I know the relaxation time. So if I am able to multiply this with a meaningful velocity that is what I was using at that. A meaningful velocity I can actually get some characteristic length over which the scattering does not happen. What is that characteristic length that is nothing but the mean free path. So what is the velocity that I can use? The main thing I can use is the RMS velocity. RMS velocity again we know from the equiparty from the kinetic theory of gases we know what is RMS velocity. RMS velocity is obtained by equating the kinetic energy to the equipartition value this is as shown here. This gives us very famous the famous relation V RMS is given by square root of 3 kT by m where k is a Boltzmann constant m is a mass of the atom there electron here. So what Drude did was he used exactly same idea here and try to see whether I can get V RMS using this at a given temperature and then try to see what happens. Typical value of this mean free path relaxation time was 10 to the power of minus 14 second that means when I apply multiply with the RMS speed we got RMS speed is something like this one. So if I multiply the 2 I get typically the mean free path to be 10 angstrom. 10 angstrom you know how much is the distance in the case of a solid typically the inter atomic distance is 1 angstrom to 2 angstrom. So it is able to have something like 10 angstrom it is able to go without having any scattering right. So 10 angstrom separation it is able to go without really and on the average it is able to survive the any collision. So this is something which is so if you have more and more mean free path value that means it is a good better and better conductor. So this is something which is obtained purely classically purely classically purely using the classical kinetic theory of gases simply borrowing those ideas to the electron system and in fact some people you are calling it as a classical free electron gas F E G free electron gas. So because in analogy with the kinetic theory of gases it was called free electron gas till it is called free electron gas in some sense. So this free electron classical free electron gas was the idea used by Drude and he got something like 10 angstrom as the upper limit of mean free path. How does the experimental kind of value show? These are the problems to be worried about. So Drude has got something definitely something is there. The problems are like this. The conductivity is found to depend on temperature whereas if you see it is not really explained using this Drude picture. Much larger mean free path more than 10 angstrom were seen in many experiments. If you prepare the sample very carefully carefully prepared samples the mean free path actual experimentally observed mean free paths were much larger than this 10 angstrom which was predicted by the Drude's model. Other issue was associated with the issue of Drude picture was this one very important again. It was observed that the contribution to the electronic specific I mean the specific heat of metals it was much smaller than what was assumed for a free electron classical free electron gas. I am going to the details of this. So as far as specific heat of a solid is concerned there was a problem. Mean free path was concerned there was a problem when we simply apply the classical Drude picture. Obviously that is expected because we know today we know that we can appreciate these differences definitely must have existed. But now we are trying to see how these things were found out how these discrepancies were solved one by one so that today we have the correct theory which actually developed over a century. So what is the problem with the heat capacity or the specific heat I use both in the same meaning. As was done earlier you know equipartition theorem when you have a solid you have when you are giving heat the heat can be absorbed by the positive ions. So here you cannot ignore the positive ions like the way Drude did that is not possible. So these things can absorb energy and you have the free electrons which can absorb the energy. So the heat capacity when you talk about at a given temperature and if you are using the classical kinetic theory kind of a model you have to use the picture of these positive ions which is a so called a lattice which of course you know by this time. The lattice which will take some energy so that the heat capacity contribution from the lattice is taken into account then there is a free electron contribution. There is a difference in the calculation between the two. So if you these are all three dimensional systems as you know when you take a molar substance one mole of a substance the total energy at a temperature T is given by you have seen this. This is 3 times Na that is the total number of degrees of freedom and these atoms these ions the lattice actually vibrates a vibrator as equipartition theorem gives you a half k B T for kinetic energy and half k B T for potential energy. So the total for each degree of freedom is k B T or k T that is why this is multiplied with k T. So this gives you 3 RT Na times k B is your universal gas constant as you know. So this gives you the ionic part of the lattice part I will like to call it as a lattice part for a solid lattice makes more sense than the ionic term. So the lattice contribution is given by 3 RT. Now if the material the solid was purely an insulator this would have been the only term but in our case it is a metal it is a conductor. So you have the free electrons also coming into picture they also are in a position to take the energy but they are free they are not like a vibrator the lattice part. So they are free electrons there is no anything that is free there is no potential there is no binding. So you have only kinetic energy contribution kinetic energy contribution from the equipartition theorem is only half k B T. So here again it is 3 Na the number of degrees of freedom because I am taking monovalent metal one mole if I take one mole of sodium for example I have 3 times the regatta number that is the total number of degrees of freedom where it can move the velocities are in three dimensions I mean these three dimensions and then only kinetic energy contribution will be there and hence I cannot multiply with k B T but I have to multiply with the half k B T. So I will get a 4 I mean this is 3 and this is 3 by 2 so this is 3 and this is 3 by 2 so it becomes 4 by 4.5. So and this Na times k B is r so 4.5 is the heat capacity which is a derivative with respect to temperature. So heat capacity is independent of temperature as this problem you would have seen earlier also this is a problem of just when you have lattice part alone this is a usual Debye picture. So where you have the this result actually is the do long and pettits behavior where it is constant at any temperature. Now what is extra is you are adding the electronic part also but the idea is not changing. So you have a constant heat capacity independent of temperature which actually is not really true if you do an experiment. So this again is completely wrong when you compare the experiment and the Drude assumption. So Drude fails in various things the heat capacity description fails completely large mean free path it fails completely. So definitely one needs to one needs to go and see what is a solution. This was done later immediately after Drude's model came somebody else came into picture and tried to do the problem that is the somerfield model very well known model as written here the difference you can immediately make it out. What is that? Earlier it was classical free electron model Drude model the somerfield model tells you that it is quantum mechanical free electron model that means the assumption regarding the free electron model that is correct that is same no change what is modified is classical has become quantum mechanical that is what is the somerfield model is. So all the assumptions of the Drude model were retained except that the treatment has become quantum mechanical. So if you look at the problem from a quantum mechanical point of view like the way you have seen the quantum mechanical discussions here the crystal the metallic crystal which where you have one one mole of this crystal you can consider that it is a box and you have this electrons put there you have got a number of electrons to be accommodated in that one in that box a 3D box and the quantum mechanical 3D box how it behaves it has got energy levels it has got degeneracies discrete all these things we know. So and we also know by this time we also know the Fermi Dirac statistics as it started today. So you know the quantum mechanics of a three dimensional box which is actually going to represent your crystal and you know the distribution function that will be followed by the electrons the fermions the Fermi Dirac statistics we know. So we have to combine these two so that we can find out how quantum mechanical things are different not the way the kinetic theory can tell you kinetic theory gives you a continuous distribution of speeds and things like that which actually is failing that is why Drude model is failing at least partially. So to correct that we are using the quantum mechanical picture in the lines of the discussion that you had last week and adding to that we have the Fermi Dirac statistics combining these two we are going to see what happens to the electron distribution what is the difference that is going to happen from the classical kinetic theory regime to this quantum mechanical regime that is why you are exposed to this quantum mechanical one-dimensional box two-dimensional box degeneracies and things like that. So what is the problem before you proceed there is a problem so pi square is not coming here so we know this energy levels I am not going to spend any time so these are three-dimensional box energy levels are given by this. So these energy levels are to be worried about so we have the corresponding degeneracies as I mentioned so if I take an electron and you have box length if you take the box length to be something like 0.1 meter you see that this is something like 3.76 into 10 to the power of minus 17 electrode volt this is a very very small value. So if I put this in a proton in the same box you see that it is something like minus 20. So what you are seeing is that energy separation that you have between energy levels when you are using a box of macroscopic dimensions L equal to 0.1 meter is a macroscopic dimension not the angstrom dimension that you have been using for the quantum mechanics problems earlier. So what you are actually seeing is when you so the formula same formula if you substitute macroscopic quantities values for this the length of the box what you are seeing is the energy levels become very close by the separation that you saw the discreteness that you are able to appreciate slowly disappears because these separations are actually coming down because of the fact that you are actually substituting a macroscopic value for L and a crystal size a actual crystal is macroscopic in dimensions it is millimeter centimeter meter it is not in units of angstrom. So naturally this separation is coming to be very very small when the separation becomes very very small it actually becomes like a continuum in some sense like a continuum the separations are very close by as you know in many cases as quantum number changes in the case of hydrogen atom if you remember as the quantum number increases the separations become very very close by energy levels becomes very very close by something like that is what is happening here because the energy levels all of them are very close by. So this discreteness that we know in the case of an atomic situation that you have been dealing with the quantum mechanics also here is going to change. So when your crystal is a realistic macroscopic crystal so or you are putting this avocado number of electrons in a macroscopic macroscopic crystal dimensions the energy levels are not the way we have been seeing in the case of a textbook problems that we have been doing in quantum mechanics they are much closer. So when it is much closer this idea of degeneracy is going to be difficult because when they are separate I can tell this as I showed you how the filling happens this is a degeneracy of 2 this is a degeneracy of 1 and things like that. But now when they are very close by this idea of degeneracy is not really a good out. So whenever something is more or less continuous what we have is the density you have mass density you have seen in the other course charge density the probability density and things like that wherever there is a continuous variation you talk about the density. So here what you have is these states energy states are very continuous to describe them now we need what is known as a density of states. So we will not worry about the degeneracies degeneracy is what we will be completely removing and we will purely bring in the density of states. So when you are trying to tally the total number which has to be conserved I will not use the concept of degeneracy I will simply bang on the density of states. So density of states must be chosen such that it has to take care of the whole number conservation. So this is something which is very important which is very different compared to the usual quantum mechanics 1d or 3d box problem. So this is a realistic issue. So this is what we have energy levels are given by this pi square h bar square n square divided by 2 ml square where n square stands for nx ny and nz squares and what are these nx ny nz they are nothing but integers they must be integers they are all n pi by l condition that was followed so that you have a complete wave inside the box which is a rigid box potential outside is infinity in z is 0. So nx remember nx ny nz are integers the very important part of our discussion now which I am going to represent as n square n square stands for nx square plus ny square plus nz square. Now I want to find out how many states are there in a small remember I am trying to talk about the density density of states that means I want to find out how many states are possible in a given range of energy. Let us say between e and e plus d e the usual calculus that we do between e and e plus d e how many states are possible that is going to give me an information about the density of states if I divide that number by the volume I will get that actually the density of states. So to find out the density of states I need this information what I am critically looking at is the number of states between energy e and e plus d e but before I do that if I look at this expression I can see that e is essentially determined by by n n squares because for a given problem my box is fixed and electron mass is fixed l is fixed all other things are constants. So what I have to do is I have to look at a picture this actually a three-dimensional picture but I can show only three I mean in two-dimensional way it is very important please listen very carefully. What I am trying to find out is I want to find out the number of states in the energy range between e and e plus d e for that before I do that I have done what I have done is I have taken an integer space what is integer space I as I told you I am constructing a space with n x n y n z as the axis and I know that the allowed values for all the three are only integers that means these things are all discrete something like a lattice is something like a lattice very perfectly placed at different locations. So you can see if it is actually three dimensions like a cube that is shown here and what is the separation I have shown you one cube here what is the volume of this cube you remember n x n y n z the separations are all one n x can take value positive values only it can take if you remember it can take value 1 2 3 4 similarly for n y n z which means that distance between two adjacent points along any three axis will be one unity that means the volume of this cube is one unity this I am calling a cell and I you have to treat this problem as a three-dimensional picture what I am looking at is the number of states between E and E plus D the same thing I can tell the number of states is nothing but number of points each point in this picture talks about or represents a particular energy state each point in this picture represents an energy state I am trying to count them but where I am doing the counting I have to do the counting between a certain range of energies between E and D plus D which is equivalent to telling that I have to count the number of points between n and n plus dn as shown here you see here I have taken a spherical surface spherical shell of radius r and radial thickness dn so I am I have to actually count the number that is coming within this this is what is to be found out if I want to tell finally how many states are there between E and D plus D I am repeating this important information what I am looking for is the number of states between E and D plus D for that it is enough if I calculate the number of points that lie in the range between n and n plus dn that is what is shown by this green circular part that I can find out because once I know the radius and I know the radial thickness I can find out the volume of this red I mean green portion that is nothing but p 4 pi n square dn as all of you know so the volume of this part is this green part is 4 pi n square dn but how do I know how many points are there in that one because I am looking at that number how many points are there that points will tell me how many states are there but how do I know that for that you look at this thing each cube has a volume of one unity and how many points these points how many such points actually belong to this one cube it is just one because if I put it I mean at the end of course does not really show but if this cube was anywhere inside each of these corners will be actually shared by 8 such cubes and there are 8 such corners for any given cube which means that the actual contribution of points to a given cube this is something similar to the simple cubic structure which you saw today morning or yesterday so actual lattice point if I use borrow those terminologies it is actually one so this volume of unity corresponds to one state one point one state so if I know the total volume of this which actually is 4 pi n square dn and if I divide by the volume corresponding to one point I can get a number of points so I have to divide this 4 pi n square dn volume divide by the volume corresponding to one point but volume corresponding to one point is the volume of the cube which is actually one unity so what I have to do is I have to take this volume and divide it by one one the denominator one is the volume corresponding to one point which in this case is the volume of this cube which actually is one so that is what is shown here so this is and as I mentioned earlier even though the whole sphere is shown but remember there is a restriction in this picture your nx, ny, nz must be all positive you cannot have negative values that is assumption because you have to have a standing wave pattern you have to have physically allowed numbers so what does it mean I am not justified in taking the whole volume but I have to take only one eighth of the this path where all the three quantum numbers nx, ny, nz are positive like you have a two dimensional case your first quadrant is positive in every sense similarly here only one eighth will contribute to all positive scenario so to find out the actual number I should take one eighth of this thing and divide it by the volume of one cell which is actually one so you do not see it here so one eighth of this is pi n square dn divided by 2 so in terms of this new variable n the number the density of states which I am going to call it as gn gn dn is nothing but pi n square dn by 2 this is the density of states that I am going to calculate for electrons put in like this I can convert the variable from n to e which is not a very difficult thing at all so that I can get an expression for g e d e which tells you that the number of states per unit volume correspond and the states coming in the range between e and e plus d here it is n and n plus dn that will get translated to e and e plus b before we do that we should have to worry about one more thing so far in the quantum mechanics discussion you had in the last week you would not worry about spin I only use spin when we talked about magnetism so spin is not taken into account in any of this discussion that we had in the earlier picture which is purely the particle in a box extension so spin is not taken into account so every orbit so that means these are all orbital states that we have every orbital states I can put two spins one spin up and one spin down which means that to calculate the actual number of electrons that can be put in I have to take this density of states and it has to be multiplied by two or make it double so that every state like this can take two spins up spin and a down spin so the number of electrons that is possible is obtained the number density of electrons will be actually multiplied this by 2 so this will be pi n this two will go and it becomes pi n square dn so g n dn is pi n square dn if I change the variable from n to e I can get an expression again I will correct it so these are all constants you can see this I will go to the next one finally it comes rise coming to the something like this the density of state g e d e is actually proportional to e to the power of half so that is what is shown here in the next one oh this this is simplification I will correct it you can see if I plot the density of states that is g e as a function of energy for a three dimensional system a 3d solid you can see that it has got this parabolic e to the power of half behavior as you can see so this is very important for a solid which is three dimensional so this density of states we need to calculate the distribution as we are going to see the next slide so I think because of this problem I will leave the remaining time for questions Dronacharya college how to find the Fermi level when material behaves like a dielectric okay oh so that I think I will not be able to cover in this course please write to us I will give the reply so what you are asked basically ask me is an insulating case so something like a semiconductor so I thought of doing some semiconductor but I do not think we have time so what I started is only with metals even that I have not reached here we have reached there so that is a different treatment I will not be able to answer I mean in a couple of minutes like this please put the question in the email what are super electrons okay so the way I understand your question what you are asking is what is the connection between free electrons and super electrons so super electrons are essentially I mean talked in the context of superconductivity I have not talked about here in this thing here I was talking about in fact I did not reach there also trying to see what is the role of Fermi surface what is the meaning of Fermi level what is the importance of electrons very close to the Fermi level how those electrons actually take part in various properties like electrical conduction hall effect and magnetic properties and other things so I have not come to that level so I have just reached there now I have actually show you I have shown you the density of states now I actually I have to fill the states with the density of state that I have derived so I have to reach and I have to make sure that the total number of electrons are accommodated so that will give me an upper limit that is called the Fermi level so and I am going to show you the Fermi energy that is the highest occupied energy is typically 4, 5, 6 electron volts which actually is much larger than the other energy slugs like the thermal energy so very small fraction of electrons which are close to the Fermi level they are able to participate in any of these properties like electrical conduction and things like that so what you are asking is regarding superconductivity the electrons which are actually getting affected so that they become a cooper pair there again these electrons which are very close to the Fermi level within a small thickness I will come to the details when I explain that only those electrons will be able to form part of this cooper pair so they are called super electrons in that context but that is not the language that I will be using when I discuss a normal metal so the electrons which are very close to the Fermi energy they are the ones which actually will come into picture in your super electron definition. My question is related to yesterday's lecture as it is said that superconductors are quickly diamagnetic and superconductors need critical temperature does it mean that diamagnetic are dependent on temperature? Diamagnetic susceptibility actually for a normal magnetic material it is not it is completely temperature independent as I showed you the derivation yesterday but superconductivity is a special case super I mean see the superconducting state once it is in the superconducting state of course it is diamagnetic with a susceptibility of one which is essentially temperature independent but you have to start the material in the superconducting state and then define this diamagnetic susceptibility there so you cannot tell that just because at some temperature it is becoming normal and then it is not a perfect diamagnet there is a temperature dependence that is not a fair comparison so when I when we tell diamagnetic susceptibility is independent of temperature we do not really mean superconductor superconductor is a very very special case of a diamagnetic case so there of course it has to be superconductor otherwise a whole discussion does not be valid it will not be there so not in that context you are you cannot take a temperature range where it is superconducting and normal and then tell that diamagnetic susceptibility is temperature dependent that is not a fair statement I think I saw your question in the email same thing was asked I think in the email so there is no contradiction in that sense definitely not yes university BDD how the energy level distribution can be explained for the materials which exhibits polymorphism okay so different crystal structures right right so materials with the different crystal structures same material in different crystal structures how the energy levels will be different that is the question okay so this is yeah so this cannot be answered in a general sense depends on see definitely as we are going to see in these lectures probably by tomorrow second lecture I will reach there the crystal structure plays a very important role in determining what kind of a band structure this particular solid will have so naturally when the crystal structure changes let us say from cubic to something else definitely this is going to get reflected in the band electronic band structure and that is why the physical properties are different for different crystal systems even when the material is the same but unless you really specifically take a particular system and study one cannot give a general statement okay this will be like this and when the crystal structure is the other thing it will be changing to this that is not possible but in general there is definitely a very very critical dependence of crystal structure on the electronic properties which will actually determine most of the physical properties I can give magnetism example if you take a cubic thing and the same thing if you make in some other crystal structure definitely there is a physical property difference that is because the magnetic property difference that is because of the electronic structure has completely I mean quite a bit changed from the first crystal structure to the second one so there is definitely a dependence how the dependence will come it depends on the particular material system that you are studying that is essentially the connection between the structure electronic structure and the crystal structure that is what I was talking in the beginning that is why you first talked about crystal structure importance of the symmetry because symmetry will determine energy levels degeneracies and all other things naturally there will be a dependence that depends in dependence is crucial in determining all the properties of that material okay sir but if you take a particular temperature as in a transition temperature which will change its structure whether that particular temperature itself the energy level distribution will change of course it will change that is why it is undergoing a phase transition yes yes yes in fact the crystal structure sometimes changes sometimes doesn't change but in both the cases there can be considerable physical property change because of the electronic structure changes yes yes can we achieve Bose Einstein condensation for free electron gas you tell me the answer my slide had a I mean one of the slides like it was shown like this so can you tell whether it is possible or not I am asking the question back to you yeah I am asking what is your thinking what is your thinking what do you feel why do you ask this question because we treat this free electron model as a nearly free free electron gas model yeah so you are you are saying under statistical mechanics Fermi Dirac statistics is applied here yeah but the distribution is suppose for can I interrupt you what is the statistics that you use for Fermi Dirac particles particle identical and indistinguishable and what is the maximum value for the Fermi Dirac function I showed you that plus one importance of this plus one in the denominator so what does it give you so it gives you the probability of maximum is one it will violate the Pauli's principle otherwise right and what is Bose Einstein condensation you have many particles it is only valid for Bose on more than one particle in a given state is needed for condensation for BC we can bring the bring all the Bose on I mean Bose on to lower quantum state no but we are talking about can we bring all the electrons such that no that is what I am telling because when you are telling that it is a fermion like electron you have to satisfy the Pauli's principle at any cost you cannot do anything with that one energy state one particle nothing else so there is no question of condensation there there is no question of condensation forget about condensation you cannot put two you can put only one that is the issue that is a fundamental issue between the two the difference between the two statistics two nature of I mean two different types of particles fermions versus bosons one more question yeah out of that you are talking about free electrons suppose if you are taking any atom both the number of electrons and protons are same suppose or if you say the whole and electron combination suppose is there any free electron means it creates hole in the at the same case and what about the whole conduction there okay so remember one thing I was giving this example of sodium always I talked about sodium which is a metal and when I tell free electron I consider only one electron per sodium atom so the 10 electrons of sodium atom will become part of the lattice the positive ion core I will take only one electron of sodium per atom and this I am trying to fill it up and see where it is going that gives me the fermi level finally we will reach there tomorrow I mean that discussion and I did not come across I mean we did not reach the point where we have this concept of holes before we really get in the concept of holes we need to do many things in fact in this picture that we are talking about now the so called free electron picture you will not even encounter energy bands so it needs some time for us to reach there then we can discuss what is meant by hole first of all we have to define what is hole for that we need to really get into the band properties we are not reached there we are in the free electron picture where you do not have any band concept you will have your e versus k diagram going all the way in the quadratic manner there is no no concept of energy bands coming there so only after you get into the concept of two different bands this excitations for holes and other things will come I will try to show you something towards the end of this lecture tomorrow but right now it is too early so in the free electron picture this is very important two things one is we are completely ignoring the core electrons we are treating them as part of the core the lattice the other thing is we we will not encounter any kind of that is one problem of the free electron picture finally even quantum mechanical treatment the somerfield model also fails as I am going to show it fails one point at is here where you do not get the energy bands you cannot talk about insulators semiconductors metals that you can do if you use purely the free electron model why it happens what is ignored in free electron model I talked about it even when we do quantum mechanical treatment those assumptions were retained from the earlier classical model so that is a problem so this has to be corrected then we can see what happens when you get energy bands and the excitations and so on then we can define what is a hole I will do that and then we can talk about this from yesterday's lecture yeah I asked this question yeah please why do the magnetic reversals happen what would happen if they did not occur further see I mean I did not talk about it in yesterday's lecture if it is a paramagnetic material if I apply a magnetic field and if I reverse the magnetic field nothing is going to happen if it is a ferromagnet it is actually going to have some effect because if I reverse the magnetic field it is going to have some properties retained that I am going to discuss in my third or fourth lecture anyway so that is what is very making the ferromagnets important in various applications for a paramagnet it is completely reversible it will simply come back it does not matter there is no hysteresis what is actually happening when you talk about this reversibility is the hysteresis in the case of paramagnet there is no hysteresis whereas in any ordered system you will have hysteresis actually is a loss in some sense in many senses it is actually bad in some cases it is needed so only in the context of ferromagnets you talk about this hysteresis or the problem of reversibility or irreversibility in a paramagnet no but I yesterday talked about only paramagnets