 Let us begin the last academic session of the day, but before I begin this topic on superconductivity, I want to make a mention a point. I am going to be talking about what are known as conventional superconductors. There are two reasons why I would not touch what are known as high temperature superconductors. The first and the foremost is that I am not knowledgeable about high temperature superconductors. The second and which is probably a more appropriate reason is that theoretically high temperature superconductors are not yet properly understood. There are no standard theories. There have been certain theory, but no standard theories as yet. And being a theoretical physicist I found understanding high temperature superconductors is not my cup of tea as yet. So what we will do however is in the December session we will get some people working on high temperature superconductors to give you a talk on there. So my talk will be entirely restricted to the what are known as the conventional superconductors which in principle actually were discovered back in 1911. So therefore what I am talking about is not really modern physics. In 1911 the Dutch physicist Camerling Owens he was essentially looking at low temperature behavior of mercury and he wanted to solidify mercury and as he went doing this you know I mean he was looking at various physical properties that mercury had as the temperature was decreased. Now what he found is that when the temperature dropped below 4.15 Kelvin in fact incidentally he was the person who was working on reduction of temperature also. So as the temperature went to 4.15 degrees solid mercury suddenly lost any vestige of resistivity that it had. And from them started the whole subject of superconductivity as we understand it. It was not until something like 40 to 50 years later that people tried to even have a theory of the conventional superconductors the Bardeen Cooper and Schiffer provided the necessary theory which is still the standard theory of conventional superconductors for which Bardeen Cooper and Schiffer were given a Nobel Prize and in fact Bardeen is the only person who have received two Nobel Prizes in physics. There have been other people who have received two Nobel Prizes but never in physics. Usually they have been in different subjects but of course there is one person who has received two Nobel Prizes in chemistry but we are not discussing chemistry here okay. So yes. No, no, no, no, no. No the only person to have received two Nobel Prizes in physics is Bardeen okay. One for discovery of superconductor or theory of superconductivity, other one he shared a Nobel Prize with Bratton and Shockley for discovery of transistor okay. There is another gentleman whose name is Sanger who received two Nobel Prizes in chemistry okay. There have been other people like for example Madame Curie got two Nobel Prizes but one in physics one in chemistry not in the same subject. History there are only two people who have received two Nobel Prizes in the same discipline. Linus Pauling got it in chemistry and peace but let us not discuss history of Nobel Prizes okay. So let us look at the phenomenology of superconductivity. What are the basic things that we understand about superconductivity? Now one is vanishing of DC resistance of a sample. As the temperature goes down to a temperature which we called today as the critical temperature, the resistance of the sample become 0. Now I want to make this point very clear because this seems to be very well misunderstood. It is not that resistance has become vanishingly small but is still there okay. The resistance has actually become 0. It is conducting electricity without any resistance okay. And the thing is of course it has to be verified experimentally. Today the verification is to one part in 10 to the power 15 that the resistance is actually equal to 0. So DC resistance of the sample goes and most of these okay are behave like an ordinary metal above DC. And for example now this way you are talking about when there is no external magnetic field applied. And well AC resistance also remains 0 up to a critical frequency. A much better test of whether something is a superconductor or not is not what is its resistance because of the reason that I will explain to you just now. A much better test of a superconducting phenomena is an effect known as Meissner effect. The Meissner effect essentially says that a superconductor does not allow flux to enter the specimen. Now actually this problem is lot more serious than that. Is superconductor a perfect conductor which obeys electrodynamics laws of laws? The answer is no. And I will just explain why there is a difference between superconductor and what you can call as a perfect conductor. The perfect conductor is something which where my conductivity is infinite or the resistivity is 0. So let us look at what is this difference. So suppose I am looking at a perfect conductor. Now one of the things which we have done that we know that we have a Faraday's law which says del cross of E is equal to dB by dt with a minus sign. Now if I have a perfect conductor then I would have sigma going to infinity which means since the current must remain finite the electric field is rigorously equal to 0 for a perfect conductor. Now if the electric field is 0 the curl of that is also 0. Now if the curl of that is 0 by Maxwell's equation that curl of E equal to minus dB by dt it tells me dB by dt is 0. If dB by dt is 0 then B is constant. This is clear. So a consequence of perfect conductivity, perfect conductivity is that B must remain constant. Now let us look at that problem in the following way. Suppose I take a metal at an ordinary temperature it is not a superconductor ordinary metal and I do the following thing. I apply a magnetic field I subject this metal to a magnetic field. Now magnetic lines of forces will obviously penetrate the sample. So the picture that I would have will be something like this. Now what you do is this that suppose you now reduce the temperature. Suppose you now reduce the temperature at a particular temperature T below Tc the substance will become a superconductor. Clear? But I have just now said B field must remain the same because dB by dt equal to 0 means B equal to constant. Now in this case you notice there was B inside the specimen and as you reduce the temperature the flux was expelled out of the specimen. So what I am trying to tell you is that supposing you take a superconductor and subject it to a magnetic field the magnetic field will not penetrate that sample understandable. But having a metal at a normal temperature subject it to a magnetic field so that the magnetic lines of forces penetrate the sample and now you reduce the temperature I expect according to the Maxwell's equation the magnetic field B to be constant. But the magnetic field B is not constant but is exactly equal to 0 again. So therefore a superconductor not only does not allow any flux to get inside it if it is in a superconducting state. If any magnetic flux was already within it on becoming a superconductor those fluxes will be expelled out and it is the second thing which is against the dictates of the Maxwell's equation because Maxwell's equation says B must remain constant. So therefore perfect conductivity is not the test of superconductivity. Meissner effect which says that superconductors are perfect diamagnets they do not allow you know what is a diamagnet you remember the cause of diamagnetism is basically Lange's law which says if you have a changing magnetic flux you there will be an equivalent induced electric field and this electric field will oppose the changing in the flux that you are trying to make. Perfect diamagnet means that the current is strong enough that it does not allow any flux change to happen inside. So Meissner effect is a statement that superconductor is a perfect diamagnet. Now another interesting consequence comes there of supposing I have taken a metal which has a hole in it now the metal has a hole in it I subjected to a magnetic field. Now when I subjected to a magnetic field the magnetic lines of forces they not only go penetrate the sample but a part of it is also go through the hole in the sun. Suppose now I reduce the temperature. Now when the this ring becomes a superconductor of course this now what you do is it becomes a superconductor the flux will be expelled out. Now these fluxes which will be expelled out part of it will remain because nothing can stay within the specimen. So the magnetic field lines which were closer to the outer edge of the sample they will go outside they will be expelled outside those which are closer to the inner edge of the sample they will simply go to the hole. They will simply go to the hole and now I do something else I withdraw the magnetic field. Now if I withdraw the magnetic field these flux lines have to go to infinity right the magnetic field vanishes. But in order that the flux lines can go to infinity there is no other way for them other than to traverse through the material of the sample and the ring being superconductor is not going to allow. So in other words these flux which were put into the hole when the material was superconducting but when the material was there was a magnetic field on withdrawal of the magnetic field those flux lines get permanently trapped ok. You do not have a magnetic field but the flux flux lines are there obviously they will close on themselves. Now they cannot go to infinity they are closing on themselves and the quantum of flux is known to be in the units of H C over 2 E. It is this 2 E instead of E which had originally you know been an indicator of that it is always the electronic charge comes in pairs there and we will see why as you go about ok. Now the point is this that suppose I have a superconductor at a temperature T less than T C. Now of course, I can increase the temperature and destroy the superconductor but I can do something else. I keep the temperature the same and I apply a magnetic field increase in temperature destroys superconductivity but so does application of a magnetic field. As you apply a magnetic field there would be a particular value of the magnetic field strength H C at which the superconductivity will go away. Now if the value of the magnetic field that you have to apply to remove the superconductivity at a temperature T equal to 0 is H C 0 then the field that you have to apply at different temperatures has a dependence on temperature as 1 minus T square by T C square. Obviously if T is equal to T C you do not need any magnetic field to destroy superconductor. The other thing is so most of the elemental superconductors typical examples being mercury lead tin, niobium they are they follow this principle that is you apply a magnetic field at a given temperature when they are superconductor and as you go on increase the strength of the magnetic field you will find a limiting value or a particular critical value of the field critical field at which the superconductivity will help. Now such material they are known as type 1 superconductor type 1 superconductors are those where the magnetic field destroys superconductivity in this fashion. There are many alloys there are many alloys which have a slightly different behavior and that behavior is given by this expression they are called type 2 superconductors. Now in type 2 superconductors remember I maintain that Meissner effect is the test of superconductor and what is Meissner effect? No magnetic flux should be permitted to enter a specimen. Now what we do is this that these materials you keep it at a particular temperature and you increase the magnetic field. Now you will find that as you increase the magnetic field when so below a particular magnetic field has a particular value the material remains a superconductor, but when it exceeds a value value which I call as H c 1 what do you find is there is some expulsion of flux, but it is incomplete expulsion of flux. Remember that we said Meissner effect says that on becoming a superconductor or become superconductor does not allow anything to get in. So what we are talking about is this that I am coming from that side that if this strength is large enough which we will call as H c 2 the material will become a normal metal. But if it is not large enough, but greater than some value H c 1, but less than that value H c 2 then what one finds is there is expulsion of flux, but there is a partial expulsion of flux. This can be called as a mixed state. The reason I cannot go into because what happens there is certain tubes of fluxes they actually can enter the specimen and these have been also known as vortex test state. What happened to the current between H c 1 and H c 2? That is a mixed state. So therefore, the it remains partly superconducting. See it is basically a situation in which I have a superconductor. I told you that the current is not a test that I am looking at. That it is partly a superconductor because its resistance is fairly low, but that is totally immaterial to me. I am looking at does it expel magnetic flux or not. The next property which is of importance is I have only talked about properties which are important in our discussion is if you look at the specific heat of a superconductor. You remember specific heat of a metal. Specific heat of a metal has two components. The extreme low temperature it is linear in temperature T. But there is a you know if it is not that low then there is also a lattice contribution which goes as T q. You must have done that several times for your student that linear in T plus T q due to phonons. So, I am not talking about phonons at all. I am talking about only electronic contribution which takes place at extremely low temperature. So, I expect for a normal metal for a normal metal the behavior of the specific heat the electronic part of the specific heat to be a straight line. Higher temperature side is this side. Now what happens is this is the normal specific heat. Now as you reduce the temperature the substance becomes a superconductor. What is found is that this specific heat suddenly jumps to a higher value almost factor of 3. But after that as the temperature decreases further it does not follow a straight line but follow an exponentially decaying behavior. And the specific heat behavior is goes like this that is e to the power minus delta by 2 k. Now this reminded people of the fact that the specific heat for semi metals if you like also had this type of a behavior. And exponential behavior of the specific heat the delta there is what is normally called as the activation energy in case of semi metals. That is the minimum amount of energy that the electrons has to overcome in order to go to the conduction line. And so therefore this behavior that it is exponentially decaying was already known in a different context. And it was known that the material where the specific heat decays exponentially that happens because in the energy spectrum in their electronic energy spectrum there is a gap. There is a gap between the field states and the nearest unoccupied state to which the electrons must be promoted in order that they can carry entropy and things like that. So this was our indication that the superconducting spectrum had a gap and typical energy gap is not very large. Typical energy gap is a few milli electron volts and that is the reason why superconductivity gets easily destroyed at very low temperature because as you know the room temperature is just about 25 milli electron volts. And we are talking about energy gaps which are much less typically you know few single digit values. So this is what we talked about that the if a superconducting specimen is subjected to an electromagnetic field these of photons or whatever electrons or whatever is there you need a minimum energy of 2 delta and I will come to why you need 2 delta in order to overcome the energy gap. Now these 2 delta become very important. See if there is a energy gap of delta then you expect that is the amount of energy I will require in order to overcome the barrier. But you see what is happening there is this that if a photon is absorbed and overcomes an energy gap delta in this picture that we have the I have to break it up into 2 electrons and not 1 electron because our state is very funny here there is a pairing of 2 electrons and which we will come to little later. These were known as the Cooper pairs. So only when H nu exceeds 2 times the energy gap the breaking up of the Cooper pair will be possible because the Cooper pairs absorb that amount of energy and then only they can be separated ok. Let us go through some elementary theories now. First is there are 2 or 3 theories couple of them fairly elementary and after that of course BCS theory which is the most important theory. First is superconducting transition is obviously reversible. I can decrease the temperature it becomes a superconductor increase the temperature it becomes a metal again ok and I am talking only about type 1 superconductor. Type 2 is lot more difficult to understand. So in type 1 superconductor so let me write down what is the Gibbs free energy. You all know the expression for the Gibbs free energy. So Gibbs free energy is your u minus T s there is a PV term but of course since our PV terms remains the same on superconducting and normal side I have not written it unnecessarily and suppose I have applied a magnetic field. If I apply a magnetic field in addition to the u minus T s term the Gibbs free energy will pick up a term like H times m. That is the energy in the magnetic field. So my change in the energy u d u is given by T d s plus h d m. So if you substitute there you find that d g is s d t minus mu times m d h ok. Now if I have a perfect diamagnet then B is equal to 0 therefore m is equal to minus h. Remember that for a perfect diamagnet the susceptibility is minus 1. Now in that case I can get an expression by integrating this of the Gibbs free energy as a function of the applied magnetic field. So Gibbs free energy as a function of the magnetic field is some g s 0 plus mu h square by 2. This is the same as B square by 2 mu if you are using B ok. So that is the energy of the system in the when it is in superconducting state. Now supposing I want to find out the when will the two phases the normal phase and the superconductive phase will be in equilibrium. I know at temperature T c the two phases are in equilibrium. So when my temperature or alternatively at any given temperature if I apply a magnetic field of strength h c the superconductivity will be destroyed it will go to a normal state. So therefore normal state is in equilibrium with a superconducting state at h equal to h c. So therefore my normal state Gibbs free energy is given by the same expression with g s 0 which is this one plus mu h c square by 2. So you notice this what we are trying to say is the normal state Gibbs free energy minus the superconducting state Gibbs free energy is equal to the energy extra energy that you need to supply in order to destroy superconductivity by a magnetic field you know which is obviously makes a that if I apply a magnetic field strength is h c the extra energy that I am providing it is mu times h c square by 2 or B square by 2 mu whatever you want to talk about. And so therefore my difference in the Gibbs free energy is given by that expression. I can calculate the entropy from by differentiating the Gibbs free energy these are basic thermodynamics and what we find is there is a difference between the entropy of the normal state and the superconducting state and that quantity S n minus S s happens to be this minus it is just a differentiation of this quantity. So therefore since it is found that the critical field varies with temperature in such a way that d h c by d t is always negative what we make a statement is this that superconducting state has a smaller entropy than a normal state as you know entropy is a measure of the order. So therefore superconducting state is more ordered than a normal state. Now the other thing to notice is this that at t equal to t c at t equal to t c my h c is equal to 0. So my S n is equal to S that is superconducting entropy is the same as the normal entropy. So as a result there is no latent heat at t equal to t c. Since there is no latent heat the order of transition is not first order transition. It turns out that the discontinuity is the specific heat shows that order of transition is a second order transition. Incidentally most of all the phenomena that we know the only example of a second order phase transition which is familiar to us is that of superconducting. The water ice is the first order water ice or boiling of water is the first order phase transition because there is a latent heat involved. This is a process where there is no latent heat involved, but the gradient of that it is discontinuous and as a result since the specific heat is discontinuous I have that this is a second order phase transition ok. So let us come to some other theory. First is we will take very elementary theories which was one of the first to come in. This was known as London equations. London has nothing to do with the name of the city that but their name of two brothers who had same surname obviously and some fifths London and some other London their equation. Now at that time people did not understand much about why such a funny phenomena is occurring. So what London or actually I should say London's they thought was that there are two types of electrons. Now mind you this is a purely phenomenological exercise it has no microscopic basis. So they said that look I am thinking though we postulate there are two different types of electrons in a material. One is known as superconducting electrons ok. There are two types of fluids ok they actually turns out the theory of super fluidity and superconductive we have a lot of similarity. So there is this type of fluid which is called as superconducting electrons and there is this normal electrons. The difference between them is that the normal electrons when they travel through a material they are subject to the usual resistance and things like that. But the superconducting electrons they do not satisfy Ohm's law. So see so I assume that normal electrons are subjected to Ohm's law. Superconducting electrons they move but obviously they have to satisfy a force equation. So for them the ordinary Newton's law is valid. The if there is an electric field the simply mass times dv by dt is the charge times the electric field. So what is what is what they do is what is shown here that this here I plot electron density with temperature T. Now the electron density normally does not vary very much with temperature T there is a minor variation because when you increase the temperature some extra conducting electrons may be available but forget about that but in this model look at this dotted figure. So I start with at T equal to 0 certain number of superconducting electrons and as that temperature reduces the fraction of the superconducting electrons go on decreasing until at T equal to Tc the superconducting electron density becomes 0. And the only electrons that I have now are the normal electrons and they are subject to of course resistance. This model has a lot of you know I mean incorrect hypothesis about it because if you believe this it would mean that as in the superconducting state as long as temperature T is less than Tc its resistivity would depend upon how far it is from Tc and not what we find here that it is rigorously equal to 0 ok. So let us look do a very simple arithmetic. So arithmetic is this I have my number density which is simply n n n means normal S is superconduct this is actually very trivial arithmetic. And by multiplying with the charge times the velocity I get an expression which says the current density J consists of two parts there is a normal current density and there is a superconducting current density I have not put in vectors always I have been careful. So normal fluid is dissipated which is it satisfies Ohm's law Jn Jn is sigma E Js does not satisfy that instead what Js does dJs by dt is d by dt of minus n times Vs there is a electronic charge missing there which I think I have brought back here ok. So it is minus any d by dt of V ok and so this is dV by dt mass times dV by dt would be the force on it. So this is what I have written down that so this is equal to ns minus e and another minus e makes it e square ok and I need a divided by m because it is dV by dt not dP by dt so therefore divided by m times e. So this gives me that electric field is given by d by dt of not just Js but some lambda times Js which is obtained by bringing that m and ns e square to this side. So this is known as the first London's equation ok fortunately there are 2 brothers so they decided to have 2 equations there ok. So let us go to second of the London's equation. Now in order to do that I go back to it is very fortunate that I have just finished electromagnetism there because we keep on coming back to electromagnetism. So I again wrote down del cross e equal to minus dV by dt but since I am dealing with H vector non magnetic so I write it as minus mu 0 dH by dt ok. I just now said e is d by dt of lambda Js so this is what I have written down there and right hand side has remained the same. Recall that Js is also del cross H right this is one of the Maxwell's equation del cross B equal to mu 0 J so del cross H is equal to Js. So therefore if I substitute for Js del cross H this equation becomes this capital lambda that I have written d by dt of del cross del cross of H equal to minus mu 0 dH by dt both the sides there is a d by dt there. So therefore the equation that I get is del cross del cross H is minus mu 0 by this capital lambda times H. In some sense we are at London equation but if you now choose that look my del dot of H equal to 0 then I get into that this gives me an equation which is del square H equal to a constant which I have defined as 1 over this 1 over lambda square H which this lambda with a subscript L is known as the London penetration depth. So this is the second London's equation ok this is what I did I had del cross del cross H equal to minus 1 over lambda square H and so therefore this is what happens. Now if you wanted to convert this into an equation in J then reproduce for del cross H your J and this becomes that your J and the vector potential are linear with each other. This is this is called London gauge the divergence of A and the normal component of A becoming equal to 0. You need normal component of A equal to 0 because you do not want any current to flow through across the superconducting boundary to normal state alright. So I have got an equation. Now let me give you an example of how this equation helps us in understanding Mycena effect. So notice this that the equation that I gave you was this del square H equal to 1 over lambda square H. The geometry that I have given here is the following that consider a material superconducting material with which occupies the entire semi-infinite region x greater than 0. In other words x equal to 0 is its boundary surface and it extends all the way to infinity and then I have applied a magnetic field in the z direction. There is a geometry the cross section of this slab is perpendicular to the plane of the blackboard x direction perpendicular to x direction. The magnetic field is in the direction outward direction. Now in that case because the material is semi-infinite the only variation that can happen has to happen in the x direction because y direction and z direction there is no variation possible. So therefore, my del square H can be essentially written as a d square H by dx square because that is the only direction in which H can vary. So therefore, the this equation becomes d square H by dx square equal to 1 over lambda square H. Solution of this equation is very simple H of x equal to H 0 e to the power minus x by lambda. So, in other words if the field at x equal to 0 is H 0 then the magnetic field strength as you get through a distance of lambda which has been calculated earlier becomes 1 over e eth of x value. And I have told you several times this 1 over e keeps on coming because that always is a measure of how far a field penetrates. This happens whether you are calculating the skin depth in a metal or whatever. So therefore, according to London there is a quantity there is a distance lambda after which the magnetic field or magnetic flux is effectively screened. Magnetic flux is effectively screened. So, this depth to which the magnetic field penetrates inside the sample is what is known as the London penetration depth. And mind you that I can actually calculate this because everything there is known to me ok and compare it against these. The typical penetration depth as one sees in the samples as is given here that is about for aluminum it is about 500 angstrom and it is a very small depth to which the field effectively penetrates. I would be just discussing BCS theory in a very qualitative way because firstly BCS theory is not something which you can teach to your students at that level. Secondly BCS theory requires understanding of many-body theory in order to work it out. But so therefore, I will just give you a few pointers. The BCS theory assumes very interesting there is an attractive interaction between the electrons. Now obviously, you say that is nonsense electrons are similar charges and they must be subject to Coulomb's repulsion, but there is a problem that this is not the direct Coulomb repulsion. So, the way it happens is that when an electron enters let us say metal. Now this electron also sees ions in it. Now the electrons are typically travelling with velocities of the order of Fermi velocities. Now when the electron gets in the ion polarizes. Now ion will polarize, but by the time ion has actually woken up to the fact that there is an electron nearby by that time because of the high velocity of the electron electron has run away. But the typical response time for the ion to realize that the electron has gone away is somewhat larger. We are still talking about small time scales, but basically it is something like this, bad example, but somebody comes hits you and runs away by the time you have realized you have been hit you are still having a little bit of a pain. So, that pain is still continuing on the ions, at that stage if another electron comes it finds an ion already polarized. See the previously the job of the electron that came in was to polarize the ion which it did, but it did not experience any effect of it it ran away because it is moving away very fast. But another electron which is now coming it finds an the ion to be already polarized and because it has a positive charge it can reduce its energy using the positive charge cloud of the ion. So, effective result of this thing would be that because of the presence of the ions my two electrons which we are talking about now have an effective attractive interaction you see it something like this. The attraction is not because the electron is attracted to the ion which it is, but electron 1 is attracted to the ion the same ion attracts the electron 2. So, this results this if you work it out the results in an effective interaction which shows that the two electrons have an attractive and if this trend overcomes the direct Coulomb repulsion then we will have a net attraction between the electrons which are well net attraction between the electrons. Now, what is the thing why did we bring up the question of indirect interaction. Indirect interaction because it is electron-electron interaction which is normally repulsive, but it is an electron-electron interaction mediated by phonons the ionic motion. So, electron interacts with ion-ion interacts with another electron as a result the two electrons have a resultant this interaction. The role of the ion was actually realized long back. Sir, if the second electron comes before the ion you know fills the effect of the first one so it gets attracted why it stops at 2 I mean if the third. No, no, no we are just at this moment saying that suppose our world had two electrons only BCS will bring in more electrons ok, but it is a pairing mechanism which the BCS takes. They say the point is that if you bring in more and more electrons it is not very clear that the Coulomb repulsion the number which is going on increasing can be overcome effectively, but the ion is still the same. So, the and in any case three body interactions are always known to be either non-existent or they are extremely weak. So, we are talking about electron-electron interaction mediated by a phonon and that is effectively why I am doing that I am coming to in a second ok. So, what is it that I want now and why did I bring in that that is the next question. One of the properties that was known about superconductors from the beginning was that if you took isotope of the same material different isotopes of the same material then you found out what is the variation of the critical temperature with isotopic mass. Normally you do not expect the critical temperature to depend upon isotopic mass because isotopic mass means more neutrons. So, neutrons being charged neutral objects you do not expect them to have any role in the whole business. But however because of the fact that Tc this is known as isotope effect because of the fact that Tc is known to go as 1 over square root of m. I believe with square root of m I have probably written it wrongly right. Anybody remembers it? 1 over square root of m ok yeah this sort of looking at it I realize I must have made a mistake yeah, but that is not important what power appears what is important is in the process the mass of an ion came into the picture. So, in other words ion is playing a role in the superconducting process. Now, this incidentally was the PhD topic of a young man known as Cooper. Cooper did the following problem. In fact, I would urge all of you to read Cooper's paper and I will tell you why it is such a simple paper ok that all of you will be able to understand it without any help from anybody and the paper goes like this. What he did is this he said suppose I have a field for machine right this is something which you know that I have a conductor with a background of field for machine and suppose I put in two electrons in the background of two for machine. Now, where does the for machine come into the picture? You see field for machine means the states there are not available to another electron to be occupied it is a field machine. So, he says that suppose I put two electrons there normally what would happen? The electrons should occupy an energy higher than the top of the pharmacy right I mean or they will go to the conduction band and things like that. But he said that suppose I gave it an attractive interaction and his paper makes it very clear I do not care how weak that attractive interaction is as long as the interaction is attractive. What he found is that if the attraction is there between the two electrons put in the background of field for machine the for machine becomes unstable with respect to the two electrons. In other words the two electrons actually have a bound state which has an energy lower than the Fermi energy. Remember the bound states are not scattering states. So, they are in different things and he worked it out. He worked it out you know with a very trivial ordinary Schrodinger equation not a very difficult one two electrons with an attractive interaction. What he did is this I am not going to work it out but I will show you what happened. So, he said that supposing R1 and R2 are the coordinates of these two he expanded the wave function in terms of the momentum of both. And then he went over to a center of mass frame so that you had total momentum and the relative momentum. Now since he is trying to minimize energy he says I will choose only those solutions for which the total momentum is 0. Now if you do that then the wave function that he wrote down can be written like this sum over k all possible values uk e to the power ik dot R1 minus R2 this is purely written in the relative coordinates and relative momentum. Now let us examine that wave function. Now suppose I split this wave function into two one for which uk is even in k suppose u minus k equal to uk and another part for which it is odd namely u minus k equal to minus uk. Then you realize if uk is even then this exponential will give me cosine function uk and u minus k they are the same. So, I will have the two terms and I will get cosine k dot R1 minus R2. On the other hand if uk is odd then of course I have e to the power ik dot R minus R1 minus e to the power minus ik so I get sine function. Now if two electrons are to be in the same point in space then their wave function cannot be the sine function because sine k dot R1 minus R2 when R1 is equal to R2 is 0. So, therefore, this space part of the wave function has to be an even function even function meaning thereby symmetric function it has to be k dot R1 the cosine k dot R1 minus R2. So, if you interchange R1 and R2 the function does not change as you know that the total wave function is a product of the space part of the wave function and the spin part of the wave function. If the space part of the wave function is known to be even symmetric then the spin part of the wave function must be antisymmetric because the total product wave function must be antisymmetric. So, what he did is to say he has a pair of electrons whose wave function this space part is symmetric and they form a singlet. Singlet is s equal to 0 which is known to be an antisymmetric function. So, this is what this is what is known as Cooper pairs. So, Cooper pairs are basically pairs of electrons which always are together they have opposite spin polarity and what the BCS theory does is basically to look at it as a collection of Cooper pairs he because he has many more. So, therefore, he would need a many body theory for that. So, this is the basic idea I thought the Cooper pair can be talked about BCS theory becomes a little more difficult to work and it certainly lot more time taking is BCS theory itself you normally takes one or two lectures so, I could not do that ok. The last thing that I will do any question till now yeah. As far as ions stand basically phonon this is phonon mediated interaction ok. So, phonon is the entire lattice quantized energy. Yes. So, you are talking about the atom or the ion which is playing the important role. That is the same thing actually. So, the thing is this that the lattice vibrations which are basically ionic vibrations ok when they are quantized they are called phonons. So, I brought in the presence of an ion all right, but what makes them couple whether it is vibrating or not is a different matter ok. What I am saying is how does phonon enter into the picture. So, in order to do that I said that how does ion enter into the picture to start with is the vibration of the ions which will give rise to normal modes the classically there will be normal modes of vibration when you quantize them there your phonon. So, there is no difference between what you are saying and I am saying yeah. I am simply saying that you need the presence of the ions right am I making sense. Ok. So, is it like that the upper there formation is like this but it is the fear of electrons which are there. The electrons which are there whose interaction is mediated by the presence of phonons ok. Basically the way it happened to ok bad example, but let me tell you this that supposing we are playing a ball passing game. Now, I send a ball to you you send a ball to him right. Now, he does not ever send it he talks to you you talk to me, but in the process the communication we are bound because I cannot get away now ok. So, this is the effect that because you have in our in fact the picture is essentially that what you find is that between an electron and a phonon there is an interaction which will be shown ok. You must have seen those diagrams and then between the phonon and another electron there will be another spring type of things shown there, but the effect is to bind these two people together. Sir, generally those are very good conductor at room temperature is not superconductor because Cooper pair formation is not possible and those having a not good conductor the room temperature might be a superconductor. So, what is the fundamental difference why Cooper pair formation is not. Did you come a little late to this session? I had made one announcement right in the beginning. I said high temperature superconductors I do not understand ok. In fact, very few people understand it. The theories are not yet standard and the theory that is applicable to the current cuprite superconductors are not BCS type. It is not a Cooper pair type. People have tried to do it, but they have not succeeded. There are other types of theories that are coming out now ok. So, therefore, that is one question which it is not that I cannot give you a sort of a some sort of a guess, but it would certainly not be correct for the simple reason theoretically the high temperature superconductor though there are hell of a lot of experimental evidences the theory wise they are not yet as good. That is why it is good to say what I am discussing today is connected with conventional superconductors which typically are superconducting and that too I have mostly talked about type 1. In type 1 temperatures I think the maximum is about 9 degrees Kelvin or so. Type 2 conventional superconductors is about 23 degrees of that order, but what you are talking about are much higher temperature superconductors. So, I I do not know the answer to that question. Where do you find is the I do not have a liquid where did you find negative ions in a solid? Yes, the electron is coming and it strikes on an ion and that ion may be positive or negative or both. Where will you find a negative ion inside a solid? You know ions are positive and negative both you know sorry sorry sorry. I use the word ions to indicate the structure of positive ions that are there in a solid. Ions that you are talking about are different you are thinking in terms of something like a sodium chloride chloride salt or things like that. No, we are talking see there are electrons what is the structure of a solid? I have positive charges my ions are positively charged. Accordingly you second you looked on changes its charge from charge cloud you know you told that. I said that it polarizes it. Yeah, polarized that one. It polarizes that and we have seen the polarization affects the electric field. Ion is polarized that. Ion is polarized. Ion is polarized. Ion is polarized. Now once the ion is polarized it would have normally affected that electron. Yeah. But that electron because of it high speed has already gone away. That becomes phonon perhaps you looked on after collision. Phonon is the vibration of the ion. After interaction the electron that we get or the particle that we get that is also called as phonon you know. Phonons are lattice vibrations only. It has no no connection with electrons okay. Phonons are just quantized name of lattice vibration and when you say lattice vibration you mean the vibration of the ions the massive ones. And because they are massive ones they are lot more sluggish okay. Okay thank you. The last thing that I want to do talk about is a very interesting property. It is called Josephson effect. And I am talking about it because it has become very important technical device in this case. So the Josephson effect takes place in the following way that imagine a thin layer of insulator sandwiched between two superconductors on either side okay. Now if the thickness of the insulator is small because of the spread the fact that the wave function of the superconductor spreads a little bit what you can find is there can be a tunneling of the type that you have already done from one side to another. But this is very interesting. The interesting thing is that if you have such a material system sandwich connected with the wires on the outside don't apply any voltage. Don't apply any voltage. You find that in the presence of a 0 voltage there is still a current. Now that can happen provided this system is acting like a battery that is electrons from one side is penetrating through the insulating section and going to the other side and completing a circuit. Now that is called DC Josephson effect. Now more interesting thing happens when you apply a DC voltage. If you apply a DC voltage what you find is that the current starts oscillating. You don't expect that right. You apply a DC voltage you don't expect the current to oscillate. That is called AC Josephson effect. So let us look at the theory I am giving you is not a rigorous quantum mechanical derivation. But this was worked out by Feynman and you can still find it if you people have the Feynman's. I believe there is a second volume or third volume has a chapter which says a seminar on superconductivity. Anybody remember having anybody has seen it? You have seen where is it is it in the electrodynamic section on the quantum mechanics book? Volume 3 okay so it is in the quantum mechanics part. Title of that chapter is a seminar on superconductivity. Please read it because only a teacher like that could have given you this type of a derivation which is very interesting. Yes all right let us look at that problem. Now look at it. Supposing I denote the wave function on one side with a psi 1 and the other side with psi 2. Now I want to write down what is the time dependent Schrodinger equation. So it is ih cross d psi by dt is equal to the Hamiltonian. But where is my Hamiltonian? Okay my Hamiltonian is just because of the fact that there is a coupling term which couples me to the other side. I told you it is not rigorous but it is very interesting. So what he does is to say that there is a coupling strength which he writes as h cross t t is coupling and he says this coupling strength is related to the wave function on the second side. So ih cross d psi 1 by dt is t times psi 2 and likewise ih cross d psi 1 2 by dt is the coupling strength times psi 1 the coupling has to be symmetric. Now look at these two equations and now remember that these wave functions are very gross wave function in the sense the so what is a wave function? I know that psi star psi is basically a probability density. So I write down psi as square root of n 1 times a phase factor right wave function has a amplitude part and a phase part then the amplitude has to be square root of the density. So this is what we did psi 1 is square root of n 1 e to the power i theta 1 psi 2 is square root of n 2 e to the power i theta substitute this there substitute it there and write down this remember that both n and theta can vary with time. So when you put it there d by dt of this there is a d by dt of square root of n which gives you half 1 over square root of n and then there is a term d theta by dt. So if you trust my algebra then this is what I get I get d psi 1 by dt equal to half square root of n 1 e to the power i theta 1. So this is what I get these two equations. After that just do some arithmetic with this multiply the first one by this second one by this you find equations of this type 1 by 2 d n 1 by dt plus i n 1. So this is the equation that I get and you will find that there will be a phase difference theta 2 minus theta 1 will come in. Now what we do is this we say write down this equation separate their real and imaginary part. The arithmetic or the mathematics that fine one is given is to be seen to be believed that you can get as complicated things as Josephson junction by almost doing very trivial arithmetic. So these equations on separating real and imaginary part become like this. Now look at this the only way this has come in now is d n 1 by dt is proportional to root n 1 n 2 sin delta and d n 2 by dt is minus that has to happen because if there is an increase in the density on one side that has to lead to decrease in the density on the other side and it is n which is connected with the current. So therefore, I get an expression of this time supposing my n 1 and n 2 are approximately the same I find that my current which is related to n 1 minus n 2 goes as j equal to j 0 sin delta. Now remember delta is a constant so there is no time variation there. So I have a DC current due to the fact that I have simply connected by where there is no source of voltage. I can repeat this calculation by putting in a voltage term and that is again very easy to do because what he does there is to say all right go back to these equations again and now in addition to this term put a voltage term V psi appropriately dimension. Now if you do that then what you find is the following you find that the current which was earlier j 0 sin delta that gets this expression. But now the time enters the phase factor so as a result the current oscillates that is called AC just okay any question any doubts yeah. Capital T is the coupling constant between the two sides see you have you are writing down what is ih cross d psi 1 by dt now that has to be the that is equal to the Hamiltonian of that. So therefore, what we are saying is the h psi equal to e psi so I am saying that that is equal to t times psi 2. It is an adjuvant. Because at 0 degree Kelvin phonons are absent so superconductivity will be absent in the material at 0 degree Kelvin. Remember I sort of told you that the one doesn't talk about what happens at 0 degree Kelvin because of the third law okay entropy of everything whether it is normal state or the superconducting state everything will go to 0 okay. So 0 degree Kelvin is not something that we talk about because even the normal state is still ordered that is the only state I have okay thank you very much.