 So, we will have the second lecture on solid state physics now, afternoon the magnetism lecture also will be there that is the last lecture today. Before I start, there were couple of questions asked yesterday through email. One of them is very important, I thought I will just answer that one and then I will proceed with the lecture. So, the question is like this. So, I talked about three different statistics, one is classical statistics that is Maxwell Boltzmann's statistics, then for the quantum particles we talked about the Fermi Dirac statistics and the Bose-Einstein statistics. So, the question that was asked is even though we are talking about quantum statistics, the Boltzmann constant is appearing that is K v is appearing. So, how is this possible? That was a question because Boltzmann is known for the classical part and how come the constant in his name is appearing in the quantum statistics both in the case of Fermi Dirac and the Bose-Einstein statistics. I actually wanted to discuss it yesterday, but I did not get time. So, I kind of avoided, but since this question has come it is a very important question. So, I thought I will answer it. You remember when I talked about distribution functions, I very specifically mentioned that the distribution function we are calculating for the equilibrium situation. So, we are only interested in the equilibrium case, equilibrium state and how do you know that something is equilibrium? I can have different energy levels you give me and a certain number of particles you are giving, quantum particles let us take and we are we can put in various possibilities are there. So, different distributions are possible, but we wanted to find out only the distribution corresponding to the equilibrium situation. So, how do you know that equilibrium is there? There the equilibrium actually is given by the state where the entropy is maximum. The entropy is maximum gives you the state which is actually in equilibrium and what is the connection to that? The entropy is actually given by there is a very important relation like this which is connecting the entropy as and I am calling this as the possible configuration which I talked about yesterday like different balls and different energy levels. So, different values of W are possible in general and the entropy is related to this W through this manner and essentially S is proportional to the natural logarithm of omega or W and the proportionality constant is nothing but the Boltzmann constant. This is very important one of the most important contributions of Boltzmann. This is true whether you are talking about classical physics or distribution or you are talking about quantum distribution. So, using this only we are actually making sure that the system is actually in equilibrium. So, yesterday we only talked about W, we did not really talk about the case when it actually relates to the entropy. We only made the statement that we are always interested in the equilibrium, but the equilibrium is guaranteed by maximizing this S which actually is related to the W through this relationship KB. So, because of this, this is used in the case of Fermi Dirac, this is used in the case of Bose Einstein as all of us know it is used in the case of classical distribution also. So, because of this the KB will keep coming in the quantum distributions also. So, this is a very very important point which actually I did not specifically mention yesterday. Another very important aspect of this equation is if you see this W that we talked about different possibilities, different configurations and so on actually the number of microstates since I could not get time. So, I did not really use that terminology. This is purely a statistical physics quantity, this W whereas entropy as you know is a thermodynamic quantity. So, this relation actually is connecting a statistical physics quantity with the thermodynamics quantity. That way it is very fundamental and this is the reason why you are getting KB in all the distribution functions irrespective of whether you are actually having a quantum system or a classical system. So, this was a very important question asked by somebody. Now, we will proceed with our discussion on the density of states. So, we were talking trying to derive the density of states for a 3 dimensional solid and so what we did was I will quickly go through this. What we did was we actually went into a new space that is a n space so called integer space. We found out this volume and try to find out how many points each point actually represents a state, how many points are accommodated within this green path and from here we change the variable from n to e. So, that we can actually find the number of states lying in the range between e and e plus d e. This happens to be proportional to e to the power of half as shown here. So, that is what is plotted in this graph here. You can see the density of states is proportional to e to the power of half. The parabolic density of states is shown here. One very important point in this case is that we have derived for 3 dimension one can actually do for 1 dimension as well as 2 dimensions. If you do the same thing for a 1 dimensional solid what you find is that it is e to the power of minus half dependence and for a 2 dimensional case which is somewhat physical in many situations one can actually approximate the system to be 2 dimensional. There what is interesting is that the density of states actually is independent of this energy. So, it is a constant value whereas, in 3 dimensions it is parabolic. So, 1 dimension it is e to the power of minus half, 2 dimension it is independent of energy and in 3 dimensions it is parabolic that is e to the power of half. So, if you remember we are trying to see how certain number of energy electrons typically Avogadro number of electrons if you are worried about one mole of a monovalent metal then we are looking at something like Avogadro number of electrons how they can be accommodated in very closely spaced energy levels of this box. So, that is because of the closely spaced nature only we are having this density of states. The other issue that has to be worried about is the distribution function as we talked earlier. So, the distribution function comes into picture. So, that also has to be looked at it. So, one part is the density of states the other thing is what is a probability that given a state what is the probability that there is an occupancy of that or not. So, that is actually given by the density of the Fermi Dirac distribution because it is electrons it is a Fermi Dirac distribution function that as we have mentioned yesterday it is here as was just now mentioned K b always appears here. So, here so, this can be modified in this form by a substitution that alpha is written as minus E f divided by K b t where E f is called a Fermi energy do not confuse with Fermi level for semiconductor. So, I will come to that little later this is called a Fermi energy. So, in this substitution this becomes something like this and I told you yesterday that the role of alpha is basically to make sure that the total number is conserved. If you have given 100 electrons finally, after all this distribution when I add up together I should get 100. So, that is actually done by this alpha quantity or in this case it is done by this E f Fermi energy. Looking at this expression one can easily see a t equal to 0 a t equal to 0 if you see that epsilon is our energy is less than E f E or epsilon does not matter when energy is less than E f what we can see is that this value of this function actually becomes 1 can easily substitute and see. On the other hand if the energy is more than E f this is actually 0 what does it mean that means if the energy of the system energy of the electron is less than the Fermi energy the orbital is definitely occupied that is what is reflected by the fact that the probability is 1 the Fermi direct distribution is essentially probability that is 1 means it is definitely occupied and for any energy above this Fermi energy the this the probability is actually 0. So, it is not occupied that is what is seen from this one. And if I take at a non-zero temperature that E is not equal to 0 one can see this value of this distribution function is exactly half when I have my E equal to E f one can have various possibilities but if I take E equal to E f the physical significance of E f is when E equal to E f at any finite temperature the distribution function actually takes the value of half that is what is seen here in the example substitution you can get these three values. So, let us look at the case only at E equal to 0 first we will talk about T equal to 0. So, what does it mean number of electrons in the range between E and E plus D E we know that is nothing but the density of states and this has to be multiplied with the probability of occupancy of each and every level that is possible in this range. So, that is why it has to be product of electrons that is in the range between E and E plus D E is nothing but the product of the density of states and the distribution function. So, this G E multiplied with the Fermi Dirac distribution gives you the number of electrons in the range between E and E plus D E. But we know that at E equal to 0 just now we have seen that the Fermi Dirac distribution is actually 1 and hence this f is actually 1. So, it becomes any D E is nothing but G E D E itself. So, at E equal to 0 again the total number. So, I know how the distribution is happening this is the number in the range between E and E plus D E and I know that the energies are starting from 0 to the maximum value of Fermi energy E F. So, I should satisfy this condition that the total number that is given to me n is the total number of electrons in the case of a volume V or if I am taking the molar quantity this n is nothing but the vagadro number. So, this must be E equal to the integration of this G E D between the limit 0 that is a lower limit of energy and the upper limit of energy has to be E F. So, when this integration is done I should get the number that has given that you gave me in the beginning that is n. This is a constraint that has to be taken into account this constraint actually is taken by the value of E F. So, this using this actually I can get G E D we know all this expression earlier we have seen by substituting this and integrating I can find the value of n the quantity n in terms of this E F when I do the integration for this particular function not very difficult because e to the power of half is not difficult to integrate I get this function. So, this is this relation is actually connecting the Fermi energy which is a upper limit of energy at t equal to 0 and the number that is accommodated the total number. So, this is something like this what is written here or if you want to write in terms of E F I want to write E F in terms of the total number that you have given me this is nothing but this. So, n by V is nothing but the number density. So, this is 3 pi square n small n which is called the number density that electron density that is given by this. So, what happens I have shown you two pictures here one is a picture which actually is not really true, but I just wanted to give you the information. These energy levels which are actually seen in the case of a particle in a box. So, they are separate they are discrete there is a separation between them. But as I told you yesterday when the size of the box is of macroscopic dimensions these level differences are very very small and essentially it will be like this. So, just to appreciate I first show you this and then I will go to this one. So, you have these are all orbital states and as we saw every state can take care of two spins one up spin and down spin that is m s is plus half m s is minus half for everything that is why we multiplied the when we derived the density of states we multiplied with two finally that is what is shown here every orbital state can take two electrons one up spin electron and one down spin electron. When I tell up spin down spin do not think that s is different s is half I am taking up spin represents plus half down spin represents minus m s is minus half m s plus half and m s minus half s is always half for an electron. So, this is how it looks like, but since these separations are so close by this is something like this. So, actually the filling is something like this you start remember you are only talking about the valence electrons of sodium kind of situations only the valence electrons. So, you start from 0 value energy is 0 and it goes on up to a maximum value E f or E f is reached when all the n electrons are completely filled and this any state that you can think of in this region in this rated region they are completely filled you cannot put one more electron there because they are all filled like this actually. So, if you want to put one more any place if you want to put one more electron it has to be either an up spin electron or a down spin electron. In either case when you put it what you see is that there is a conflict the Pauli's principle will be violated because if you bring in up spin it will create it will clash with this one if you bring in down spin it will clash with this one. So, that way there is no vacancy absolutely there is no vacancy up to this energy level E f. So, that is the sanctity of E f because all the states especially t equal to 0 I am talking about t equal to 0 only absolute 0 there is absolutely no vacancy below E f and it is absolutely empty above this region. So, that is why it is written here it is empty here completely and here it is absolutely there is no space available here which means it is again important if I want to add one more electron into the system I want to add one more electron into the system I can only do it here. That means that energy the energy of injection if I want to call that must be at least equal to E f so that it can come here otherwise no there is no way I can push an electron into this one it is energetically not possible for bringing another adding another electron into this region and only way is to have energy at least equal to E f. This picture is extremely extremely important and this is the main difference between the classical way of looking at the whole picture and this tells you what is the problem when you do it the system this free electron gas as a classical gas. The kinetic theory which is used for a kind a usual classical gas fails miserably because of this problem when you have fermions electrons occupying various energy levels of a crystal this part is absolutely different compared to the classical treatment which we know there is no such thing there is nothing like a minimum of energy that is required for putting another electron and things like that this kind of a complete filling here and a completely vacant states above this idea does not come at all in the case of a classical picture and this makes the difference between the classical description the drood description and the summer field description. I am not telling that summer field description is completely alright but still definitely there is a big difference between the drood picture which is a classical picture and the summer field picture which is represented by it. So, just to give you some idea about the numbers take sodium as I was talking about the actual density is this we know the atomic weight. So, n by v if I calculate we can see that this is coming to be something like this. If I substitute this in the expression if one can get that E f is something like 3.16 electron volt as I mentioned yesterday typically for metals it is in the about 5 electron volt. Very important thing to remember at this stage is the thermal energy K B T at room temperature which is something like 300 or 273 Kelvin the thermal energy is something like 27 milli electron volt. You see the difference you have Fermi energies shown in this earlier picture this is something like 3 or 4 or 5 electron volt and your thermal energy is very very small orders of magnitude smaller. So, this is a problem in the classical picture when we talked about the way it can actually take the energy from the applied heat like for example the heat capacity that we talked about there was a problem in as far as a heat capacity discussion was concerned. There is a big difference between the experiment and theory what is the reason? The reason was that we thought it should be a constant throughout independent of temperature, but what we see what actually experiment tells you is that it is not it starts with the 0 value and initially it increases linearly and reaches a high value. The assumption in the classical picture was that every electron that is there in the system will be able to absorb the energy that you are giving, but that is completely wrong in the light of this picture that we have now and especially with these numbers in front of us. So, you can see this is a Fermi energy that is typically 3 electron volt for sodium and what you are giving in the form of heat the energy that you are giving is something like a few milli electron volt 30 milli electron volt. So, what does it mean and I told you that this these are all completely filled and these are all completely empty these states are completely empty. So, if you are thinking that in electron like here or electron sitting here electron sitting here in this picture they are all absorbing the energy that you are giving that is actually not true at all because if they are absorbing energy they have to go to higher state take an electron here if it has to absorb this energy of 30 milli electron volt it has to go slightly above, but slightly above there is no vacancy available and there is no question of absorbing the energy when it is not in a position to get excited. Only certain number of electrons have the advantage of taking the energy and go to higher energy states that is the these are the electrons which are actually on the surface of this this line the top the energy the electrons which have actually very close to the Fermi energy electrons the correct statement is electrons whose energies are close to the Fermi level Fermi energy only those electrons will be in a position to absorb the thermal energy that you are giving so that they can get excited. So, that is why there is no wonder that your classical picture is overestimating the heat capacity because classically you thought everything like a kinetic theory of gases everything is able to absorb energy and it can actually take a large number of a large amount of heat. So, the heat capacity is large which is not true because here only a very very small fraction of electrons are in a position to take the energy that you are giving so that they get excited which means the classically you are doing a complete overestimation of the energy that can be absorbed by the system is quite different quantum mechanically and classically. So, this tells you why the heat capacity variation actually starts from 0 at when your thermal energy that you are giving is 0 practically nothing is able to absorb no electron is able to absorb the heat and hence it is starting with 0 and slowly it increases and reaches a reasonably high value as the temperature increases. So, this is a very important thing to be noticed from this picture that actually we got after lot of things worried about. Another picture before we proceed further is that so if you see this is completely free electron that we are talking about which means that the energy is purely kinetic and kinetic energy as we have seen it is purely proportional to k square because momentum is given by h bar k. So, the kinetic energy is h bar square k square by 2 m m is a mass of the electron. So, that means energy is proportional to k square I can show that as a parabola like this. So, this E is energy and k is a wave vector. So, in terms of in this picture correspondingly I can tell there is an upper limit of k that is called a Fermi wave vector that is kf. So, up to this it is completely filled all the k values up to this is they are completely filled and above that there is no k that is occupied by an electron. Another way of showing this is if I take a k space I will there was a question about reciprocal space I will come to that this is some actually connected with that what I am showing here you see we are not describing the electron in the real space in any of these discussions. We are always talking about the k values of electron either energy or the k values of the electron most of the time we are talking about the k values of the electron I am that is what I am doing right now. I am telling that this is a electron with the k value 0 this is the electron with slightly more k value and this goes on and in the t equal to 0 case the maximum k that an electron can take is kf. So, if I use a k space this k space is something like our integer space what we do used yesterday where nx n by nx where z where the axis correspondingly you remember k and n are related k is nx kx is nx pi by L where L is our dimension along the x axis and so on. So, when I tell nx n by nz space that is nothing but kx ky kz space that is actually the reciprocal lattice reciprocal space that you have seen in the other lecture. So, many times it is convenient to work with a space which is not the real space near lattice real lattice but the lattice where the separations are reciprocal of the distance it is if it is d it is 1 over d. So, why it is happening I will come to that little later but remember this picture or this picture what you are seeing is s or the nx n by nz the picture that I showed you to derive the density of state they are all basically they are all the reciprocal lattices or the reciprocal spaces which has got a very strong connection with the real lattice that we all know. So, this is another way of showing that all states are starting from 0 the states are occupied and when you reach kf it is completely filled. So, that means this sphere nothing a real sphere this is in the k space. So, it is completely occupied that means any k that you can take within this sphere because the case any point you consider that will have a corresponding k value that is completely occupied you can definitely see an electron corresponding to that or having that particular k value and no electron will be available for k values corresponding to these places that is just outside this. So, this is called a Fermi surface and this one and this is called a Fermi sphere this is absolutely true only in the case of t equal to 0 because t equal to 0 things are very clean. So, t equal to 0 you can definitely make the statement that absolutely all the states within this or below this are completely filled not even one we can see is there and any state outside this or any state outside here they are completely empty electrons can come if they are you are injecting electrons externally because the system has already taken all the electrons into account they have been completely filled whatever as they are there we have completely filled it. So, there is no question of an extra electron for the sphere you inject something externally there is a. So, the question is what is EF is it does it have a temperature dependence strictly speaking there is a small temperature dependence is here but usually we do not worry about it. So, the expression is given as you can see most of the time this can be ignored which means that EF at t equal to 0 is essentially treated as EF at t equal I mean finite temperature also by the way this also has another name which is known as a chemical potential. So, actually is also called a chemical potential some of you may be using this terminology. So, EF at t equal to so chemical potential at t equal to 0 is actually called a Fermi energy. So, in general EF as a function of t is called chemical potential it actually has a small temperature dependence when the value you talk about at t equal to 0 it is called a Fermi energy. So, that is what is written here to a good approximation we can still use this EF 0 at t equal to 0 proceed to describe the total number even a temperature not equal to 0. We starting going from t equal to 0 to t not equal to 0 the total number is still the same this is completely different in the case of a semiconductor when the number of charging carriers will be a function of temperature but as far as a metal is concerned that is what we are discussing the total number of free electrons is going to be same irrespective unless the temperature is not very large the number is going to be the same I mean I am not talking about the thermionic emission and things like that. So, when the temperatures are not very high the total number of electrons must be the same that means I should have this condition satisfied only differences I have to take the product of density of states and the distribution function now the distribution function is not 1 I am going to show and the limits it will not be e f but it has to be from 0 to infinity because of this what is the problem? The problem is if I take t not equal to k 0 that is not at absolute 0 the distribution function takes this particular shape as you can see at t equal to 0 there is a discontinuous jump below t f it is 1 above t f it is 0 at t equal to 0 but when the temperature is not 0 that is what is written t greater than 0 the distribution function takes values initially very low temperatures very low energies of course it is 1 and then slowly it decreases and it takes to higher values that means these are all empty regions and these are all filled regions. So, the sanctity of e f in that sense has gone so now it cannot tell that below this is completely filled and above this is completely empty that you cannot tell but another sanctity you can bring in that is what we defined earlier at any temperature other than absolute 0 the Fermi energy e f can be defined as that energy where this value of this distribution function is 1 so you cannot draw anyway it has to be corresponding to half so any temperature this curve has to cut the value of half that is what is shown here so because of this ideally this tail will go on and actually the when you do the integration you have to actually do the integration starting from 0 to infinity so that is why in this expression you are multiplying this from 0 to infinity so you are taking all the things into account and wherever this even if really does not go to the value of infinity but correspondingly the distribution function will be very very small if it becomes 0 this integral will not be contributed by that region so there is no harm in putting this the limits to be from 0 to infinity and find out this product wherever this term actually goes to 0 the whole thing because it is a product the whole thing goes to 0 so as long as the temperature is not 0 the total number is given by 0 to infinity the product of density of states and the distribution function corresponding to the t is not equal to 0 but here we do not really worry it take the distribution as far as the distribution function is concerned we simply take the same thing that we take for the E f equal to 0 so this difference one should note what happens when t equal to 0 and t is not equal to 0 Fermi energy in different quantities in various units we talk about Fermi velocity for example you take E f and equate it to half m v m v half m v square so that corresponding v whatever we get is called a v f Fermi velocity Fermi wave vector I talked about it typically 10 to the power of 10 meter inverse you remember this is wave vector so it is 2 pi by lambda so lambda is in the denominator why it is actually called reciprocal space the k space is called reciprocal space lambda is in the denominator so lambda actually gives you the dimensions of length and it becomes meter inverse Fermi temperature Fermi temperature is what you do is you take E f equate it to k B T f and so E f divided by the Boltzmann constant will give you a temperature that is typically 60,000 Kelvin what is the importance of this number T f when your T f is this kind of thing what is the meaning of T f if the temperature that you are giving to the system is of this order then what it means is that the system is actually behaving like the case of a kinetic theory of gases all the electrons are completely free and they are not really bound in some sense the way the box will tell you let us go back to this box this is very important so what is happening is if you are telling that the temperature that you applied is of the order of Fermi temperature that means there is nothing like a bonding here otherwise what happens is remember we are talking about only valence electron the sodium electrons the 11th electron but because of this particular feeling following the Pauli's example which is a very very important consequence or ingredient of Fermi Dirac statistics you if you see this one as I mentioned earlier we are trying to give energy to the system most of the electrons are not able to take the energy and get excited this is a quantum mechanical restriction the Pauli restriction therefore even though we started with the completely free electrons the so called valence electrons because of this particular kind of feeling because of this particular statistics certain a large number of electrons are essentially bound electrons they are behaving as if they are bound electrons bound means they are not in a position to absorb energy and go to higher energy states only a very very small fraction is possible to take the energy and go to higher energy levels so you are making that initially valence free electrons you have you are taking only a very small of them to be unbound or free and remaining the predominantly large fraction is actually kind of bound electrons this picture is absolutely away from the classical treatment classical treatment kinetic theory of gases it treats all of them to be completely free everything will absorb energy if you are giving that is why it always overestimates the heat capacity whereas in reality in the case of a quantum mechanical free electron gas something like what we are seeing now the number of electrons which are in a position to absorb energy whether the energy is thermal electric or magnetic whatever be the form of the external agency that is giving the energy the number of electrons which are in a position to absorb this energy is very very small therefore you will naturally overestimate a classical result and you will get the correct results using this in that context what is the physical significance of this Fermi temperature if the temperatures that we are giving as I mentioned is generally very small if the temperatures are very small what is going to happen is the fraction that can actually get excited is very very small that is typically given by K B T divided by E F that is the ratio and you know what is the value of K B T it is something like milli electron volts and E F is electron volts so the ratio you can find out this is the fraction that it is the fraction of electrons which actually get receive the energy and get excited but when your temperature is the real term I mean the applied that temperature to which the solid is subjected is of the order of Fermi temperature this ratio will be very large almost all the electrons will be in a position to receive energy and get excited that is a classical region that is a classical situation so Fermi temperature actually gives you the temperature where you can actually use a classical treatment you do not have to really use a quantum mechanical treatment and since the temperatures that we generally employ are much much lower than the Fermi temperature you do not apply something like 10 to the power of 6 Kelvin by the time the solid is gone you have to use quantum mechanics you have to use quantum statistics you have to use the properties purely quantum mechanical so this is a very very important thing as far as why quantum mechanics is important what is the difference that quantum mechanics makes to the solid it is a very very important point so the point is even though you are thinking that all the electrons are free because I started with all the valence electrons of sodium it is not true because of the Pauli's principle because of the particular statistics the number of electrons which are in a position to act as if they are free that number is extremely small slightly it increases in fact increases more or less linearly with increase in temperature but it starts with the really zero value that is why the heat capacity electronic heat capacity start from zero value and linearly increases and it takes a very high temperature to get into the classical kind of value of 0.5 R this is a very very important aspect of this discussion so I am giving some other fermi energies of different metals typically as I showed you in the range of 5 electron volts I already mentioned this free electron gas which is represented as I mean abbreviated as F E G free electron gas quantum mechanical free electron gas because it is free is reflected by this fact there is no other energy potential energy is not this p square by 2 m p is h bar k that is what is here that is why you have the parabolic dependence between E and K what are the observations observations are something like this fermi energies already mentioned at least two orders of magnitude larger compared to the thermal energy at room temperature already mentioned I think I will not repeat it this is what I was telling so you heat capacity actually almost actually starts from zero in reality it will be there will be small value but it goes like this and takes very high temperature to get into this classical regime electrical conduction if you see using the same picture this is the fermi sphere I talked about it so in the absence of an electric field as we discussed yesterday these velocities of various electrons they are all randomly there so the net result will be zero because all of them they are completely going in the random direction so there is no net directional flow of charge there is no current whereas when you apply a magnet electric field you can see that there is a displacement of this whole fermi sphere because electrons are negative so it will go against the direction of the applied electric field and you can see that the number it will not exactly cancel each other which gives rise to a net flow of charge or there is a net current in the direction which is opposite to I mean the direction of current will be in the same direction as the fermi sphere so electrical conduction as I mentioned all these responses electrical conduction is a response heat capacity or receiving the thermal energy is a response something like a thermal response apply magnetic field that is a magnetic response in all these cases as I mentioned earlier the external agencies are ready to give the energy but the system the electron system only very large very small number of electrons are in a position to take that energy and get excited to the higher energy states because of the fact that the states below EF are completely filled especially at E equal to 0 so what are the issues fermi velocity actually is one can find out but what is found even with this kind of a picture what is found is that the mean free path one can find out using this picture is found to be more than what one can get in the Drude model gives you a very small mean free path but and experimental findings as I showed you yesterday the mean free paths are much larger definitely the this picture the quantum mechanical picture gives you a better or a larger mean free path value compared to the Drude picture in real case it is not really so you are to still do something so that you can actually find out much larger mean free path which are actually encountered in experiments I just wanted to give you another picture here so this is the picture I give you but I am adding another symbol here that is why this is the picture that I go earlier and I was only talking about EF earlier so this EF is the highest energy electron so electron here will have the maximum energy in this particular case so if you give that much energy does it mean that the system or give little more energy will the electron come out no if you want this one electron to be out of this metal as you know what is nothing but thermionic emission and things like that there are many ways of getting an electron out of the metal what you have to do is you have to not only give this energy corresponding to EF you have to also give this even these energies which are the highest energy possible for this case they have to kind of cross this barrier of something like phi which is called a work function they have to you have to supply which is corresponding to the work function so that this barrier is crossed and it actually comes out of the crystal so this is what actually is given in the form of heat in the case of thermionic emission you can use an electric field which actually becomes a field emission which actually is used in instruments like a CM and TEM so where one has to if you use electric field then becomes field emission this is thermionic emission if I use heat of the thermally phi essentially represents the work function of the metal which actually is used in the case of photoelectric effect discussion so this phi that is this one so all these cases the assumption is that the most energetic electron that is electrons with the value energies of EF they are the ones which are subjected if you are thinking of exciting these electrons which are here this your phi will not be enough it needs much larger energies to get these electrons out so when you are telling that I am getting electrons out the electrons which are actually coming out are the ones which are here even for them you need to give an energy of the order of phi another very important aspect of metals and of course for semiconductors is Hall effect Hall effect again is a very long story it actually was discovered in 1879 by H Hall later on many versions of Hall effect came into picture interesting things happen I will not be able to go into those details today here in this lecture so I am sure all of you have seen a what is known as a Hall probe Hall probe is something which actually is routinely used as a small strip which actually is used to measure magnetic fields if you have an electromagnetic you want to know what is a magnetic field actual magnetic fields not with the calculation of current and things like that you simply insert in the magnetic field region and you will be able to measure the field so what is the principle of Hall effect we are not really looking at from the applied point of view but we want to see how Hall effect actually contributed to the development of this understanding the quantum mechanical understanding of physics of metals that is what we are interested so the Hall effect has a very famous geometry you have a slab which is in our case we are taking a metal in generally semiconductor why I will come to that later of this kind you pass a current along x axis a DC current a magnetic field is applied along z axis what you see is that there is an voltage developed across the third axis three mutually perpendicular axis along the third axis you see that there is a voltage that is generated and what is interesting why it is interesting especially in the case of semiconductors as you must be knowing is that depending on the sign of the charge carriers whether it is electrons or holes we will talk about it later whether what is a hole and things like that depending on whether the charge carriers the charge in nature of the charge the direction of this electric field or the polarity of the voltage developed along the y direction is different and that is how one can actually distinguish between n type semiconductor and the p type semiconductor for example we are not going to those details right now so how do what does it what happens really this is a picture so when the magnetic so current was there initially established and when you apply a magnetic field you have the Lorentz force coming into picture q v cross b q is the electronic charge so first let us assume that we are taking the positive charge carriers not electrons let us just call the positive charge carriers are there let us not worry about what is positive charge carriers so that means v axis along your current is along x axis means v axis along x axis because the carriers are positively charged so the Lorentz force is given by this q v cross b so v is along x axis so it is q vx v is along vz so this becomes i cross j i cross k cap is nothing but minus j cap that is what is written here so what happens to this one because of this negative sign because of this magnetic field as you know magnetic field always reflects the charge particles in a direction which actually is perpendicular to the magnetic field the plane containing the magnetic field and the velocity direction so what happens is in this case v is along x axis and v is along k z axis so the deflection is going to be along the j axis so the positive charges will come to because of the negative sign the positive charges will come to the negative side negative y side so this is y positive y so it goes to the negative y so the positive charges actually are deflected and they come like this so electrons we are actually moving in this direction they get deflected and they get kind of deposited on the lower phase the bottom surface of this and since the system is electrically neutral the corresponding negative charges must get appeared here which means that this positive charge collection and this negative charge collection has to give you it has to produce an electric field as shown here so you have an electric field which has come in the picture along the third mutually perpendicular direction because of the effect of the applied magnetic field there is no electric field otherwise of course electric field was there to establish a current along x axis we are not really bothered about it how long this can go on but before that we can see what happens if it was negatively charged cart this one exactly same thing only thing is that the charge is negative same idea is there so what happens is the charges negative charges I will not go to the explain because it is all the steps are here the charges will get deposited here positive charges have to come here so that the electric field is now pulled up so it is in opposite direction the electric field has come the polarity has changed as I mentioned between n type and p type that is what is happening in this case so basically there is an accumulation of charges in the y direction and how long this will go on so what is going to happen is you had a steady current in the beginning you apply the magnetic field that has produced this accumulation of charges in the third direction now the electric field also has come into picture because this accumulation will not be without the producing an electric field so this electric field will actually try to compensate with the magnetic force so the condition is that the electric field that is produced along y direction is E y this will compete with this at equilibrium so they must be equal at equilibrium so that means this condition is satisfied so initially when the magnetic field is applied because of the deflection of the charge carriers the free flow of charges along the x direction so that there is a steady current along the x axis will not happen or in other words there is an increase in the resistance electrical resistance along the x axis but this cannot continue further after some time once this equilibrium is reached satisfying this condition what is going to happen is the original resistance will be covered and the increase in resistance will go away so the steady current will further happen so it will be the initial current will be restored after this condition is satisfied so this gives us in the steady state condition you have an expression connecting the induced electric field which is called a haul electric field anything that happens along the y direction is haul that is haul electric field this is the input vx the current was given the velocity that is the input the vz that you have given the magnetic field is an input so this is the condition so from this I can rearrange so this is an actual experimental setup one uses in many of the labs teaching labs you also may be having this in your labs so what you actually measure is not electric field but potential difference between the these two sides as I showed you positive charges here negative charges here and so on so what you do is we actually define what is not as a haul coefficient which actually is the ratio of this response e y is the response jx the current density along x axis is an input bz as I mentioned is an input you are trying to take a ratio of the response the hauled response divided by the ratio of the hauled response and the input parameters namely jx jx is essentially ix right ix by the corresponding area of cross section will be jx times the magnetic field applied so this is bz I can further simplify this so I know this expression I have this expression connecting j and e so I when I substitute I put these things I see that the haul coefficient this is already we have seen earlier this expression for jx is nq vx and so on so I can get this expression for haul coefficient as 1 by nq remember q is a charge carrier in general so haul coefficient depends on the number of charge carriers actually inversely proportional this actually tells you why when you do a haul effect measurement or when you take a haul probe the way I mentioned to measure the magnetic field you do not really take a metallic piece you always take a semiconductor the material medium that is used in the haul probe is always a semiconductor the reason is there is because of this inverse relationship with n metal the n is very large the electron charge density is very large n is nothing but a charge density is very very large so if you use a metal this is going to be very very small and the response will be very feeble on the other hand if I use a semiconductor semiconductor has typically smaller values of n compared to metals that means this quantity will be larger so one can use a semiconductor so that measurable effects can be seen and hence use a semiconductor to use in a haul probe in a device but in theory as far as physics is concerned metals also give a haul effect and the haul effect observed in metals gave lot of interesting ideas as we are going to see so the sign of this also will give you what kind of a charge carrier you have as I mentioned earlier so what is expected so just let us go back so if I do this one if I take this R H also here if I take 1 by n q and R H that is what is here 1 by e has been taken as q here because in a real metal q is nothing but electronic charge when I do this and if I do compare the observed values of this quantity for various metals what is found is so ideally I expect when I do this it must be 1 when I take this R H also here with a proper sign it must be 1 see what happens in the case of sodium what is expected is because it is a mono valence 1 electron per atom so what I expect is 1 I am getting 1.2 when I take sodium potassium it is again close to the expected value what I expect is the valency thing but it is more or less agreeing with the valency that is what we generally expect the problem comes for copper copper is 1.5 it is not very close to 1 1.1 is ok but 1.5 is not ok gold again it is 1.5 what is expected is 1 let us go further problems are much more serious if you take beryllium what is expected is 2 what you are getting is minus 2 minus 0.2 problem with the magnitude as well as the sign what we expected is a positive sign because it is negative charges we know that the charges responsible for any of these things are negative with the negative because electrons and the minus sign is already taken so I should get positive values as we have seen earlier but here beryllium is negative and not only that the value is much smaller magnitude is much smaller magnesium same problem indium same problem aluminium words so I mean and of course indium also instead of 3 you are getting 0.3 it is 1 to 10 with a wrong sign kind of wrong sign as per this is all taken from this solid states very famous solid state physics book some of you would have used it also so which means there is some problem the picture that we have used to derive the quantities of hall coefficient or the hall they explain hall effect it is not working alright ok for monovalent metals it is fine it is ok but when you go to little more complex atoms things are not good you can see there is a sign problem there is a magnitude problem that these things have to be addressed so let us see what are the problems so quantum mechanical free electron model that is a somerfield model it is able to explain the heat capacity issue because why it is very low in the beginning increasing linearly that is why the problem is some divalent and trivalent metals what we expect is that sodium has one valence electron so I expect certain conductivity if I take a divalent metal I expect two electrons to be completely free from the atom so I expect a larger conductivity but that is not happening it actually is less than monovalent in some cases so this cannot be explained by the somerfield model the model quantum mechanical free electron gas model is not able to explain that there are issues associated with impurities how the electrical resistivity changes with the various crystal purities that is a temperature variation these things are not clearly explained because why this model Hall effect is much more miserable there are issues with sign there are issues with magnitude at least in certain elements completely out of the picture as far as somerfield model is concerned so which means somerfield model even though it is a proper quantum mechanical model is again having problems the problems may be much better than the case of Drude model which is a classical model but somerfield model is again not the final point somerfield model also needs some serious corrections some serious thought has to be put in about what really is a problem with this model what we have done is we have taken the free electron model and instead of classical treatment we have given a quantum mechanical treatment is it solving all the problems answer is no it is not solving that is very clearly seen in this slide Hall effect I will take as a very important point which actually tells you is that something is seriously wrong even now with the somerfield model so quantum mechanical treatment alone like this is not going to solve the problem completely before I take up this issue before I take up this issue I wanted to just mention one more point which actually is very relevant if you remember somebody asked this question also earlier if you remember the discussions we had regarding the integer space and other things originally the starting point was the particle in a box particle in a box what we do is we have a stationary wave so that the nodes are there at the walls so that you condition that the length is equal to n lambda by 2 or is equivalently telling that l this is k equal to n pi by l the condition which we use basically they are not travelling waves they are standing wave kind of patterns which are used to get into these quantum mechanical boundary conditions the wave function requirements what happens if I want to use a travelling wave kind of a situation if I want to I want to treat the electrons as a travelling wave instead of a trapped stationary wave I want to use a travelling wave solution will some of the things change just because that treatment is done this is very important and that is what is written here what happens if I assume a travelling wave solution instead of a stationary standing wave solution the way we have done I want to change from the stationary state situation to a travelling wave situation for the same problem so let us see what happens this actually gives rise to a very important concept in solid state physics namely the periodic boundary condition what is periodic boundary condition is this I will again start with a simple one dimensional solid having length l there are atoms arranged so I will not worry about it assume that the length is I mean as in a real crystal the length is finite l what I am going to assume because I want to have a travelling wave not a standing wave for that first thing that I have to do is that this travelling wave must travel throughout entire length but if the length of this thing is finite then obviously there will be this ends will be there the length is finite means you will always encounter the ends as you are seeing there is an end here end point here there is an end point here but I do not then I cannot really talk about it is not meaningful to talk about a travelling wave in such a case what I will do is I am going to assume that I make this as a loop which I can do or physically what I am actually thinking is the end effects are to be ignored the ends are not really significant then I can think this to be equal to this where there are no ends it is a loop if I think this kind of a thing and the total length is l the same thing is retained here one condition I must always satisfy what is that condition this condition condition 1 it is for one dimension only you take any point and call it as x the wave function at that point that is psi x must be when I travel a distance of l I should reach the same point so when I do that that means the wave function must not change so that means I have to satisfy this condition psi x equal to psi x plus l if I do this thing I just do one round or in fact two rounds three rounds things should not change this condition has to be imposed definitely if this is imposed how do we impose this condition if I take this kind of a picture that is psi x is proportional to e to the power of i k x this condition can be satisfied because e to the power of i k l will give you one that means this can be satisfied in that case if my k prime is k plus 2 n pi by l where n is now positive and negative plus or minus 1 plus or minus 2 plus or minus 3 like that for any of these values for n and if I satisfy this condition then I can always satisfy this periodic boundary condition this equation 1 is actually representing what is known as a periodic boundary condition so boundary once you come back you will see that there is no change in the wave function of e to the power so this is a very important condition so you are seeing that your k values allowed k values are 0 when n is 0 plus or minus 1 plus or minus 2 this is something which is different from the earlier picture where you had only positive values in the standing wave pattern your n values are all positive that is why yesterday we had to take only 1 8 of the sphere so how does it give you the further the allowed k values are always separated by this so the values are 0 plus or minus 2 pi by L plus or minus 4 pi by L and so on so if you take adjacent values allowed values of k which are consistent with the equation 1 the separations are always 2 pi by L 2 pi by L next is 4 pi by L or other side it is 0 and next is minus 2 pi by L like that so interval is 2 pi by L 2 pi by L corresponds to a reciprocal space because n is in the denominator again any k will be a reciprocal space as you are seeing here so you are having now k values are representing states so these states are separated by k values of 2 pi by L where L is the length of the crystal so if I want to find out the density of states initially let us take a one dimensional situation so if I have the length of this k space corresponding to the original real space having a length L why if I want to find out how many such states are possible I have to find out how many such k points are possible this can be found out by dividing the total length of the k space corresponding to the one dimensional lattice divide this by the range separation between adjacent k values that is nothing but my delta k so if I take the length of the k space and divide by the adjacent separation that is 2 pi by L in one dimension I will give get the density of states if I do as I mentioned finally I should get an e to the power of minus half dependent minus half dependent so using this picture I can actually do what happens in the case of three dimensions so instead of the lengths I will be worried about this sphere that we talked about yesterday same sphere instead of n integer I am taking k but as I told you k and n are the same except for some constants so what you have is as I mentioned yesterday 2 is degeneracy, spin degeneracy factor 4 pi square dk now I do not have to take this 1 eighth business why because in this case my n is allowing plus and minus because travelling wave there is no such condition that it has to be positive only so there is no question of multiplying this with 1 8 that is not n but remember your separation in that picture the stationary wave picture was pi by L n pi by L here it is 2 n pi by L so there is a 2 which has already come into picture so if I substitute this here so delta k will be so our range in one dimension is 2 pi by L in three dimension it will be the cube of this so dividing by this will be equivalent to multiplying with L cube by 8 pi cube when I do this I get exactly same result as you got yesterday you can see even though we are not dividing by 8 we are not taking the one octant because we now we have to allow both positive and negative k values n values it is not going to change the density of state expression because the range of delta k encountered now is twice as in the case of standing wave solution there it was n pi by L now it is 2 n pi by L and hence you exactly get the same result so it does not matter for you whether you use a travelling wave solution like this one or the standing wave which we generally do in the case of a particle in a box remember yesterday all the discussion was purely using the particle in a box extension three dimensional particle in a box extension where you started the degeneracy and other things but today if this picture is quite different I am not assuming a box I am assuming that I need a continuous flow, I need a continuous path so I make a ring so basically what is meant by ring, ring means I can ignore the boundaries boundaries do not contradict much so that gives us this picture which exactly gives you the same expression for the density of states as we encountered in the case of a standing wave treatment so this is a very very important point again as we refer density of states you should know that there are these are the two ways in which you can actually find out the density of states both are having their own specific points which are important in their respect so this finishes one part of our story which of course I wanted to do it yesterday not finish it so what we have seen so far in the procedure and till now is that a quantum mechanical treatment is definitely needed to get a meaningful discussion, meaningful understanding of even simple solids like sodium kind of situations we need to understand further already there are some signatures which actually tell that this treatment is again not complete we need to go further and see what is happening so as I mentioned we have discussed this part the quantum mechanical free electron theory was discussed FET certain aspects we have seen they are okay they are reasonably explained certain things are not okay as you have talked about whole effect for example if you ask what are the main problems of this free electron theory including the quantum mechanical I am only when I am not talking about Drude model Drude model is over the somer field model which is a quantum mechanical free electron model I am only worried about that now whether it is all right completely or not that is the question so the main issue is that still it cannot explain things like the resistivity variation heat capacity in some cases and so on all as we have seen just now sign problem and magnitude problem there are much bigger much larger issues it is not very clear even at this stage how you are able to get much longer much larger mean free parts the conductivities are telling us experimentally observed conductivities are telling that electrons are actually not encountering the kind of scattering that we expect even from the quantum mechanical picture it is it is encountering much less resistance much less scattering that again is not coming out of this picture the other issue is again a very broader issue is this picture the way things stand I do not think or we can think that there is no way we will be able to encounter or we will be able to classify metals versus semiconductor versus insulators that is a very important thing I do not we do not think that this is going to be happening using this treatment so something is seriously missing in this somer field picture so to understand that let us see what are the things which we have ignored that is a way to proceed further right you should know what is ignored whether that is a problem or something else is a problem what is ignored most importantly is this as I mentioned in the beginning yesterday a solid is more complicated than an atom but it has also got certain advantages what is the advantages an ideal solid ideal crystal has what is known as discreteness in position or the long range periodicity or long range order atoms cannot sit anywhere they have to sit at designated places that is why you have a lattice concept there or it can you can tell that it has got a translational symmetry if I go from one place by certain multiple of this the so called lattice displacement I should reach the same environment so the translational symmetry that is a correct word to use is the most important aspect of a solid which actually has not been taken into account at all we completely ignored that then there is other issue of as I mentioned yesterday there is an electron positive ion core interaction an attractive potential is there which we did not take into account another thing was again ignored was an electron-electron repulsive interaction because we ignored the second contribution and third contribution that is why we called it as a free electron picture let us not worry about the third contribution let us see whether we are able to manage we are able to incorporate the discreteness path and the electron positive ion core interaction the so called attractive interaction if you are able to make definitely they are going in the right step then and we are making it more realistic this is an issue which these are the two issues which you have to be taken into account they are somewhat connected but as you are going to see there is some difference which we will see so we have to worry about the first two neglected contributions one by one so that we can see whether some of the problems that we are not able to answer at this point of time can be answered soon the question is are the electrons really free so the if you see the classical to quantum mechanical conversion did not yield much of a result it definitely gave you something but not complete the other issue is what is common between the classical treatment and the quantum mechanical treatment is treating the electron as free so that has to be doubted very much now are the electrons really free obviously the answer is no because we have we know that this positive ion cores are there and they will not be silent so we expect that electrons are not really free but at the same time we know that the mean free paths are much larger much longer relaxation times are there so definitely something is missing the question now is we know as we have seen the electrons which are free are represented by the plane wave solution e to the power of a times e to the power of i k x solutions where a is the normalization kind of a constant there so this wave function should be affecting the periodicity that is something which is missing completely periodicity aspect has not come into picture at all then you are calling it as a solid which is not really justified the actual main signature of solid has not come into picture that is why you are always calling is free electron gas free electron gas where is a solid coming into picture solid has not really come into picture so the periodicity is the first thing that has to be brought in if you want to really tell that ok I am working on solids so how do you do that so now what we have to do is instead of treating the electron to be completely free electrons have to be treated in what is known as a periodic potential you see this positive ion cores are sitting everywhere and electron is coming so these are all attractive centers lattice points they are all attractive means wells you have many wells arranged in a very very systematic manner you are having pits very regularly dug up so and electron is coming naturally what we think is that there are many of these pits these electrons should get trapped there and the conductivity must be very very small but that is not true in reality that we will see that is the whole question of talking about the dynamics of electron in the periodic potential the title of this file so we have to see what is the effect of this electron this positive attractive potential wells in the electron motion so how this is going to modify the motion of the electron for that we have to know what is the wave function of the electron so what is meant by periodicity what is meant by translational invariance is this as something similar to the periodic boundary condition we have seen remember periodic boundary condition the periodic is simply the full length of the crystal here what it means is that the positive ions cores which were so far they are all here positive and yesterday we only worried about this negative charge electrons here but these electrons are moving like this that you have this positive ion core of course we have taken the scattering into account but much more than that is needed they are all arranged like this this separation A is called lattice constant which actually can be determined from your XRD technique which was discussed yesterday so in the in presence of this kind of translational invariance one cannot actually use the e to the power of i k X plane wave solution this has to be modified this modification was done by a great physicist block this gives a blocks theorem which actually tells that in place of a completely plane wave solution for the so called free electron the wave function must be of course you can take the plane wave part but do not multiply with a constant you have to multiply with another function which is called UX function here but this UX has to satisfy certain condition the condition is that the original lattice has a periodicity represented by this the periodicity of the potential here the lattice potential is here it has got a periodicity of A this periodicity must reflected in the wave function also this periodicity must get reflected in the wave function also means this job must be done by this new function UX UX has to represent this reflect this thing so how do we do that so your wave function has changed from the plane part so your original plane wave something like this this is the free electron plane wave which we have been talking about now it gets modulated the multiplication actually some modulation modulated by a factor which is a function of X not a usual 1 over square root of L kind of a function which we are having earlier the length of the crystal was L in the earlier case it to the function which the constant which was multiplying the plane wave part was 1 by square root of L so that it is normalized in place of that now you have a UX function so UX function is such that this also must represent the same periodicity of the lattice or same periodicity of the lattice potential so exactly the way VX plus N A must be equal to VX UX plus N A must be equal to UX so if you are travel the distance of N times A you should get the same thing for the function the way you started off so that is what is represented by this so this is a very important point as far as the modification is concerned so you cannot anyway some very simple mathematical steps are here so one can actually see that one can prove that this actually gives you the same picture for the physically what is more important is a probability density as you know when you change from N X to X plus N A you see that the probability density does not change provided you satisfy this condition so I am stopping for some questions so what we are doing is from the free electron picture we are moving away not completely yet we are actually trying to bring in the concepts or we are we want the way function to get reflected in the fact that there is a solid problem the problem is that there is a discreteness it is not a gas problem any longer now it has to be reflecting the periodicity of the lattice then only you are justified in telling that I am working for the solid this aspect is brought in with the help of Bloch's theorem modifies the plane wave solution the wave function becomes little more complicated this has got lot of implications as we are going to see in the next lecture I am adding one more lecture to this one even though we were given two lectures because of this thing we are adding one more lecture to that where we see that this modification given by Bloch's theorem is going to give you lot of answers which are not able to be answered with the help of pure free electron summer field lecture so we will proceed with this tomorrow again today afternoon we will talk about usual magnetism lecture we will continue right now whatever time we have I think 10 minutes 10 15 minutes we can ask the questions whatever you have there are many question people are asking this question how a neutron can have a magnetic moment because it is electrically neutral the problem is not very simple answer is not very simple but the understanding is or the explanation is something like this assuming that the neutron is something like a made up of a proton anton and electron that is a usual simple picture that always we give just because the charges are actually cancelling each other that is why it is electrically neutral there is no reason to think that the angular moment are actually cancelled each other so somehow the charges are cancelled but the angular moment are not cancelled the spin angular moment are not cancelled against each other which gives rise to even the electrical neutrality but a finite angular momentum for the neutron particle so charge cancellation does not guarantee that angular momentum cancellation has happened this is the simplest explanation that we can give and again related to this many people have been asking this question what is the use of neutron diffraction I think I reply to some of them very important thing is I think I mentioned neutron has a magnetic moment just now I mentioned that it has got an angular momentum because of that it has got a magnetic moment this magnetic moment can interact so if you send a neutron beam of course the neutron beam must be having wavelength comparable to the separation of the solid that is the lattice constant which we just now discussed typically a few angstrom one angstrom two angstrom that means the neutron must be thermalized the neutron which are coming out of the nuclear reactors cannot be as such used because these are much larger wavelengths will not be fitting into the diffraction limit so what you do is you thermalize it that means you reduce energy so that the wavelength increases to these levels they can be used exactly x-rays or electrons you can use them for diffraction so this magnetic moment parts will interact with the magnetic moment if the material is magnetic so you will get the magnetic structure this is additional information but if the material is not magnetic and if any the material is magnetic also what happens is x-rays are going when the x-rays actually scatter or when you get x-ray diffraction pattern what really happens is that x-rays are electromagnetic radiation they are actually scattered by the electron cloud so if your material has small z number and large z number atoms for example iron and hydrogen hydrogen as you know has one electron Fe has a larger number of electrons so naturally the contribution to the scattering from Fe will be much much larger you will not essentially see hydrogen at all so this is a problem of x-ray diffraction whereas so the reason is the scattering power of x-rays by the atom actually is dependent on z light elements will not be scattering powerfully whereas this problem is not there in the case of neutrons neutrons are essentially scattered kind of independent of the z value so whether it is light element the scattering is not proportional to z value so even hydrogen and Fe is there usually you can actually see where hydrogen positions are there where Fe positions are so it is very good tool from the structural point also so in general for a magnetic solid it can give you the magnetic structure and the usual crystal structure the crystal structure part comes because of what is known as a nuclear scattering magnetic structure comes because of the magnetic scattering so there is a magnetic scattering there is a nuclear scattering nuclear scattering will always happen whether your material is magnetic or not this is something like the x-ray scattering except that your scattering power will not have the z problem magnetic scattering will happen just because your material is magnetic and the magnetic moment of the neutron is there non-zero so this is what is giving rise to the neutron diffraction being a very important tool of thermal neutron which is not possible anywhere in India as I mentioned only in Bombay it is possible if you go to BARC you can do this experiment diffraction as you know is an elastic process you can also have an inelastic process with x-rays generally we do not talk about it but neutron you also talk about an inelastic process what happens is the energy kinetic energy conservation will not be there you can actually find out what happens to the excitations if you have excitations in the material you send a beam and you see how it is absorbed or released this will give you information regarding the excitation any excitation of the material can be studied with inelastic processes using an inelastic x-ray scattering will be very difficult there are problems whereas neutron is a very good thing to do the inelastic part so inelastic neutron scattering as well as elastic neutron scattering which is nothing but neutron diffraction they are very important tools in material science today Patil hello good morning sir sir my question is regarding the in reference of semiconductor what is the physical significance of forbidden region what is the physical significance of forbidden region with respect to superconductivity or the superconductor semiconductor forbidden region sir for a p-n junction you are talking about p-n junction yes semiconductor no no you are talking about just one semiconductor or you are talking about p-n junction p-n junction see we are as I mentioned we are reaching there slowly let us wait for one more lecture because we will reach there because you have to have the energy bands you have to talk about the holes you have to talk about energy gap then only you will reach there so I have to cut short I mean a lot of discussion to reach there just wait for one more lecture slowly the block picture will slowly give you some idea about how the bands can actually come and the band gap can come slowly we are coming but not exactly but slowly we will see what is the problem so give me one lecture so that we will be able to answer it little more fruitful thank you very much sir IPS Academy Indore good morning I have one simple question what would be the effect of magnetic field on the Fermi level of the metals okay very important question I should cover in magnetism also but I will answer now as I mentioned the Fermi level the Fermi energy for example is very large 5 electron volts 6 electron volts these energies are very very large compared to the thermal energy or the magnetic energy or the electrical energy that you generally can give in a lab so the energy that you can give with the help of magnetic field is very very small so but any magnetic field when you apply you get the usual whatever I have been talking about you can actually get a diamagnetic response and a paramagnetic response if you remember I talked about curie paramagnetism and the usual diamagnetism in the context of our magnetism discussion but there I was very careful I talked about an isolated atom and then we are slowly going to do it in the afternoon for the insulating materials I will not cover the metals part at that time but in this lecture I was only covering metals so what is going to happen when I combine the two if I apply your question is very important if I apply a magnetic field to a metal the free electrons that I am talking about there is no core electron in that sense everything is free I mean the I am really worried about free electrons because free electrons also are behaving most of the free electrons are behaving as if they are core as I mentioned so there is a very small number of free electrons which will be able to receive the magnetic energy these electrons are at the top of the layer of this Fermi level Fermi energy not the Fermi level let us call Fermi energy they will absorb they have they will also undergo both these responses the usual diamagnetic response and the paramagnetic response the paramagnetic response gives rise to what is known as a Pauli susceptibility Pauli susceptibility is paramagnetic susceptibility which is positive this susceptibility this Pauli susceptibility which is paramagnetic is unlike the Curie susceptibility Curie susceptibility has a very strong temperature dependence is chi equal to C by T whereas this conduction electron contributed paramagnetism namely the Pauli paramagnetism is independent of temperature because your band structure as I mentioned is not going to I mean not the band structure the filling is not going to change very much with the temperature so the temperature has very little role here so the temperature dependence will be very very small for this paramagnetism so do not think that all paramagnetism are temperature dependent no Pauli paramagnetism which is a paramagnetism shown by conduction electrons in metals it is completely temperature independent that is the paramagnetic part there is also a much weaker diamagnetic contribution that is actually called a Lando diamagnetism which is actually one third of paramagnetism as I told you paramagnetism itself is very weak then this diamagnetism is one third of that so usually people do not worry about the diamagnetism of metals but people generally take care of the Pauli paramagnetism if it is a metal definitely the question is very very important that is because these free electrons have a magnetic moment of one magnet as I showed you it is a purely spin there is see when somebody ask this question why free electrons do not have orbital part there is no orbital orbital means there is a potential there is a kind of a fixed path free means it is completely free only angular momentum it has got is intrinsic angular momentum so it is free so it has but still because of the intrinsic thing it has got a spin angular momentum that is what is giving rise to the the Pauli paramagnetism one more question what is the difference between quantum mechanical Hall effect and Hall effect that is what I was telling it is little difficult to explain in one or two minutes but I will since I mentioned I will talk about it the normal Hall effect is what I have been talking about in this lecture what I showed you is normal Hall effect there are two more things both of them actually got Nobel Prize I think this Hall also got Nobel Prize so three Nobel Prizes for Hall effect totally so that shows the importance there is one quantum mechanical I mean the so called quantum mechanical Hall effect is what is known as the integer Hall effect integer Hall effect and then later came what is known as a fractional quantum Hall effect fractional quantum Hall effect is much later integer quantum Hall effect and this one so basically what happens is you see that the resistance variation or the voltage variation in the setup takes care in the form of jumps very simple argument where this explanation so what happens is they are very sharp this has some connection with superconductivity also I will not be able to explain everything here so this resistance for example the Hall resistance what we talk about the Hall resistance takes jumps they take jumps see what you do is you actually vary the magnetic field there the magnetic field is not constant here our magnetic field is fixed along that axis in the integer quantum Hall effect and the fractional quantum Hall effect we vary the magnetic field as we vary the magnetic field the Hall resistance actually jumps very very discreetly very sharp in fact it is so sharp that they are actually used as I mean resistance standards or voltage standards so sharp so this happens this explanation comes basically what is done is we take a two-dimensional picture so the picture is you take a two-dimensional electron gas for which the density of states is independent of energy as I mentioned three dimension it is parabolic two dimension it is independent of energy so you have constant density of states but when you apply a magnetic field interestingly what happens is this picture changes and how I mean you have a Fermi sphere you are actually getting into what is known as Landau levels that I cannot explain in short time so the way the Landau so when you increase the magnetic field the Landau levels keep increasing and the Fermi sphere which I showed you in this case will be fixed so as the these Landau levels slowly go out they develop and so on then what happens whenever these Landau levels cross this surface you see that there is a change this change happens not only in this case all the other physical properties also it happens this gives rise to integer quantum Hall effect then bringing in some other interactions like for example the spin orbit interaction you can see that it actually happens in the fractional also so that becomes fractional quantum Hall effect strictly speaking even the Hall effect that I discussed we need a proper explanation needs quantum mechanics but somehow this is actually treated something like a classical thing and the other two are treated much more quantum mechanical the real thing because reason is because you need a quantum mechanical two dimensional free electron gas explanation to see what happens when the magnetic field is applied so basically the sharp changes in the Hall resistance that is what is giving rise to the quantum effects there thank you PA power engineering college what is the use of entropy used in thermal systems sir okay I will talk about it in my magnetism application of magnetism lecture entropy is a very important thing for example if you talk about heat pumps you talk about refrigeration all these cases you are actually exploiting the entropy entropy is very important because what many of these things what you are doing is you are actually changing your playing with the entropy so you can generate heat or you can absorb heat so that is a very simple way of looking at it from the physics point of view entropy is important because as I showed you today it is actually determining what is equilibrium situation equilibrium distribution is given by the case when the entropy is maximum that is from a fundamental point of view from the application point of view whenever you talk about heat exchanges like what you see in case of a heat pump or a refrigerator I will talk about refrigerator specifically in the context of magnetic materials that is more of physics also you what you are actually playing is playing with the magnetic entropy entropy has various contributions one form is the magnetic contribution so you can play with the magnetic contribution by changing the magnetic field so entropy is basically is telling you what kind of heat exchanges you can have with the system so let me finish the magnetic some lecture on that you can ask further questions on that ok