 See, in our last lecture we have seen that William M. Macke Gotsmith made a very bold proposition that electron has an intrinsic angular momentum that is called electron spin angular momentum and that explained a host of experimental observation. So, here we say that now electron can have two types of angular momentum that is orbital angular momentum and spin angular momentum and according to Bohr's model orbital angular momentum takes only integral values L and spin angular momentum now according to William M. Macke Gotsmith takes only half. See component of this will be m of L minus L to plus L and component of this spin angular momentum will write S equal to minus half and plus half. So, angular momentum is a vector quantity it is certain direction and this way or that way if the spin component is minus half or plus half orbital angular momentum also is a vector quantity. So, this can take various components depending upon the value of this and correspondingly different directions. So, these two angular momentum which are vectors they can now be added to generate a net angular momentum and that can have various combination it can take integral values or half integral values depending upon this and let us say how this explains the various experimental observation. In last lecture I showed you the fine structure of spectrum of hydrogen atom and deuterium atom. Here 3S and 2P were connected by only one transition. Now this L equal to 0, 4, 3S and spin S as half together can give rise to total angular momentum of J equal to half. Similarly for 2P state L equal to 1 and S equal to half these two angular momentum combined together to give rise to value of J which is 3 by 2 and so these two now can have an energy level which is shown on the right hand side. So, earlier 3S and 2P are connected by one transition. Now because of this 2P can have two types of total angular momentum J equal to 3 by 2 and J equal to half now I have two transitions. So, these two transitions are actually what is seen in the spectrum given by deuterium and hydrogen. So, these are the fine structure and how easily this can be explained with introduction of this another angular momentum S which takes value of half the Ziemann effect. The splitting of spectral lines in the presence of magnetic field also can be now very easily explained by these two possible orientation of the magnetic moment in a magnetic field. The same is true in the case of the Stern-Gerlach experiment where the silver atom split into two and its origin is again the same that spin angular momentum takes two values plus half and minus half. It gives two component of magnetic moment and this causes the beam to split into two. So, you see how all these experimental results can be explained based on this existence of electron spin. So, electron has therefore an intrinsic magnetic moment. So, they behave like a tiny bar magnet. So, the bar magnet has a magnetic moment given by a mu sin mu and its direction goes from south to north and this is a vector quantity. Now, see the origin of magnetic moment and the angular momentum that we are intimately related. The relation is given by this mu Z is equal to G E beta E S Z where mu Z is the component of the magnetic moment vector in a given direction. G E is a proportionally constant called the G factor, beta E is Bohr magneton and S Z is the component of spin angular momentum. Magnetic field, magnetic field also is a vector quantity. It shows you the direction of the magnetic field lines that starts from north pole and goes to south pole. This is the way it is. So, magnetic field vector direction is given by this red arrow. So, I have a magnetic moment and a magnetic field. If I place them in a magnetic field then what happens? They will of course orient according to the allowed value of the angular momentum. If we take an ordinary magnet which is this north pole and south pole, if I place in a magnetic field it is this way north and south and all of you have experienced that when north facing north and south facing north this is not the energetically favorable system. What is favorable is that of this bigger magnet. So, this becomes the lower energy state. So, this is the state, this magnet is going to take naturally, but then if I have to turn it this way then I need more energy because these two will ripple. But classical magnet, if you are classical means the macroscopic magnet that normally you can see that can take any possible orientation. So, I can in fact have this also possible. So, energy of this will be somewhat intermediate between this and that. This is the least energy, the highest energy, this is the energy which is intermediate. The energy E is given as the scalar product of the vector B and the vector mu. So, this is of course dependent on the angle that this magnetic field vector makes with the magnetic moment of the magnet. So, classically any such orientation is possible, but for microscopic particle that has been now shown by Ulland-Mechan Gortzmann experiment, interpretation and splitting of silver particles that only two reindicances are possible here. So, because the angular momentum vector takes only two values, spin angular momentum specifically. So, even though all these are possible classically for this microscopic magnet, I only have these possibilities. Higher energy configuration or arrangement, other one is lower energy arrangement corresponds to m s equal to minus half or m s equal to plus half. So, this is now shown in this diagram here. So, for electron spin in a magnetic field has this type of two energy levels which are characterized by the m s equal to minus half or m s equal to plus half. This is the electron Ziemann splitting. Now, you see of course the splitting the energy difference between these two states with m s equal to plus half or m s equal to minus half is dependent on the strength of the magnetic. It is of course very obvious, if the strength of the field is small, the splitting will be small, if the strength is large, splitting will be large. The splitting changes linearly with the magnetic field. So, I said in the last lecture that NMR and EPR, electron parametric resonance and nuclear magnetic resonance are very similar in their characteristic. So, many nuclei have magnetic moment, in particular proton for example, which also has spin half system which nuclear spin is half. On the right hand side, the energy of a proton for example, when placed in magnetic field, the splitting is shown here or m i equal to plus half and m i equal to minus half. On the left hand side, the same splitting as shown earlier is also shown that the electron spin in a magnetic field splits into minus half and plus half, but the difference is in the strength of the interaction. Expressions are also very similar, for the electron spin it is mu z equal to minus g e beta i s z, for nucleus it is plus g n beta n i z. The difference is in the strength of their magnetic moments, that is of electron and the proton, that of a nucleus. So, we can write this is for electron magnetic moment, for nuclear magnetic moment. So, here the difference is actually in this and this. So, B is Bohr magneton and beta n, they are the units in terms of which the magnetic moment is measured. So, difference is in the magnitude. Where does the magnitude come into? How does it appear here? So, if I write here, this is actually e h by 4 pi mass of electron and this one similarly, will be equal to mass of proton. So, you all know that mass of proton is about 2000 times heavier than mass of electron and that appears in the denominator here. So, obviously the Bohr magneton is about 2000 times bigger than nuclear magneton. So, that is reflected here, typical electron magnetic moment will be 2000 times bigger than nuclear magnetic moment. So note this sign difference here. This says that the direction of the electron spin angle momentum vector is opposite to the direction of the magnetic moment of electron. Here they are the same direction. That is of course related to the charge of these two. Protons have positive charge, electrons have negative charge. So, we have this difference in the sign. Now, when you keep in the magnetic field, the interest in energy which is given here can be now written as in terms of the corresponding spin angular momentum. And the energy level is shown here. So, that they look very similar. The difference is only in terms of what spin state that low energy level correspond to. For e p r is the minus half state or n m r is the plus half state. Now, other difference is because of this difference of about 2000 in their values, the splitting of the energy level for the same magnetic field will be about 2000 times smaller than the splitting that the electron will see. So, similarity is here, but difference also here that principle is very similar, but magnitude of interaction is very very different. So, for typical magnetic field that is used spectrometer n m r usually appears in let us say megahertz region, but e p r in the similar magnetic field appear in 1000 megahertz region which is called gigahertz. So, we can write here megahertz is 6 hertz and 1000 megahertz is called this is typical and this is typical of. So, having obtained the splitting of energy levels now, I can look at their spectroscopy. How is it done? So, typical absorption spectroscopy involves shining light and seeing where exactly the light is absorbed. And this spectroscopy could be let us say electronics transition, household transition, rotatorial transition, almost all of this you must have come across. So, here this energy levels that is shown on the left hand side are actually property of the atoms and molecules, they are fixed there, all the energy levels are fixed. So, you cannot do very much about it, all you can do is the shine light of appropriate energy. So, the energy gap of the two levels match with the energy that goes inside. Say these are fixed energy levels and I shine light, the wavelength of the light matches with let us say this much then it can presumably absorb the light and I can get the absorption spectra. Now, if the energy of the light is matching with this one then it can also cause absorption and I can get another absorption here. So, these energy levels are fixed in a conventional spectroscopy, but here you have seen that energy level and their gap depends on the external magnetic field. So, they keep changing. So, they change the magnetic field, the energy levels change. So, I have to have a fixed energy gap and the shine light on that. How do I fix the energy gap? The delta E that is given here is now it will become G, beta and B, B is the external magnetic field. If B changes, the gap also changes. So, I can fix the energy gap by fixing the magnetic field and delta I could be equal to the external radiation and then which can be now this condition becomes the condition for matching the energy gap with the radiation energy. So, this fundamental relation we call the resonance condition that is satisfied for absorption to take place. So, I will write it once more here because this is going to be very fundamental to the spectroscopy. This is the incident radiation frequency and this is the magnetic field. You see they are proportional. So, this again can be contrasted the conventional spectroscopy where this energy gap is fixed and so we vary the frequency of the radiation to match the energy gap. Here now I have variable energy gap which is caused by the magnetic field B. So, I can to make this equation satisfied, I can vary this or vary this. This we will see later how actual experiment is done to look at the absorption spectrum. But what we expect about the spectrum, nature of the EPR spectrum when we do the experiment. So, this is the energy gap. So, if the delta E is fixed at a magnetic field B0, then I need to satisfy h nu to be equal to this. So, the spectrum as a function of magnetic field will have this sort of behavior that I suppose I keep the frequency constant h nu is constant. So, the energy gap is now varied by changing the magnetic field from lower to higher side. So, at this position then the energy gap delta is exactly equal to become h nu, then the absorption radiation takes place there. Nothing happens if the magnetic field is lower than B0 or higher than B0. Exactly at this condition this equation is satisfied. So, the spectrum should look like this one up down and then goes up. So, there is one line here. So, this is the electron parametric spectrum of an isolated electron kept in the magnetic field. In a sense it is very simple spectrum just only one line. But then the simplicity shows that if there is all there to it that if you have got one electron put a magnetic field gives one line, then this is not going to be very exciting because all the substance which gives EPR spectrum will give one line spectrum and what information can you get from there very little. So, it is not going to be very exciting or informative. Now, so let us see some real experiment spectrum and how they show their EPR spectrum. This is the radical which is called parabenzo semi quinone radical. I will write the structure and formula again here. If we start with this hydroquinone, this is hydroquinone and it just oxidize it in oxidize by air actually the oxygen of air. Very simple experiment dissolve it in alcohol and make it alkaline. Then in the alkaline medium this will convert to I should say this is alkaline medium. So, here this OH and OH in alkaline becomes O minus and it half of that gets oxidized. So, O dot. So, we call them semi quinone and this will be anion. So, we call it semi quinone and anion radical. The EPR spectrum is shown here and you see that gives 5 lines and lines have equal gap among them. But, intense is not same this follows certain pattern. We will see this thing pattern and try to figure out later why they are so, but it shows that the single unparallel letter which is present here. They are not giving single line EPR spectrum, but 5 lines are coming. So, something more is happening something more than what was sort of discussed and we expected to happen. Before we go to another example, here is a little digression. The EPR spectrum that we saw there is in the form of a derivative. In the previous experience of various types of spectroscopy that we have seen the absorption spectrum will always looks like this type of thing. So, intensity of light is observed in this fashion as a function of of course, it is a function of wavelength or frequency. Here we are plotting the spectrum as a function of magnetic field, but it is equivalently that we are doing the experiment that we are looking at the supposed to get absorption spectrum of this kind instead it shows derivative spectrum. So, why it is so is something again we will see later. So, we just right now take it from me that actually this absorption spectrum of this is of this kind that 5 line absorption line come and they are exactly equivalent to this one. So, what the spectrometer records is the derivative line, but internally the absorption spectrum is this. So, why the spectrometer gives the output in the form of a derivative is something we will take up shortly. Another example, the same semi known here if you start with you know instead of this if you start with this tertiary butyl group here then this becomes tertiary butyl group there. So, we call it this is the 2, 5 this is the 2 groups so we call it 2, 5 right tertiary butyl radical the appear spectrum of this is shown here and it gives only 3 lines. It is the same type of radical earlier give 5 lines now it is giving 3 lines. So, you see how now the appear spectrum is becoming interesting that it is something it is now giving in function about the structure of the radical. So, it is useful we take some more example here this is another radical the radical is called temporal as a short form, but the structure is shown here it is a prepared in group with 4 methyl groups attached to that and this N O where the O has one supply electron that unpaired electron give rise to the appear spectrum and the spectrum is shown here gives 3 lines of equal intensity now and the gap is same. Mind you that here the nitrogen nuclear spin is 1 nitrogen 14 nucleus we will see that that has something to do with these 3 lines which are appearing there. I said in the introduction that appear spectrum is very useful for characterizing transition metal complexes. So, here is an example of copper complex this is called copper diethyl diethiocarbomate and this is the chemical formula of this. So, this gives 4 and not quite 4 line but you see that one group here one here another there third one is there they have a funny looking structure this line is split here and that this line is out of split here but intensity seems to be quite different for different lines. So, again it shows that appear is able to give a lot more information than what we sort of anticipated the beginning. So, it is going to be useful technique to get the information of electronic structure. Now, this molecule or copper complex is made of naturally occurring copper and you know that naturally occurring copper has 2 isotopes nuclear spin 3 by 2 for both of them but mass number is 63 or 65. So, if one chemically purifies this isotope and have just pure 63 isotope then this spectrum looks like this. So, here see the simplicity now earlier we had funny looking spectrum of this structure there some structure there that has disappeared now instead of that we got 4 lines though 4 lines do not seem to have same intensities nevertheless there are 4 lines of this kind and intensity is strongest on the right hand side and because smaller and smaller but notice that width also becomes bigger and bigger as you go from right to left but here we have got enriched copper 63 nucleus the another complex of vanadyl this complex has 2 vanadyl nucleus and some complicated fashion. So, vanadyl nucleus spin happens to be 7 by 2 and here you see the whole lot of lines there 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 lines and intensity you can see quite unusual and it starts from the left hand side some sort of low intensity they become bigger and bigger intensity increases again goes down and down if you look at the width carefully the width becomes narrower and narrower in the middle and then again because broader gives the shape in this fashion. So, these lines that you see lots of lines these lines in a spectrum that you have shown these are called hyperfine lines 1 important characteristic of the line is that they depend on the nuclear spin state. So, that is a clue to something that origin of this thing that clue is that is copper 63 and 65 it is given as to quite different spectrum. So, this must have something to do with the interaction of the unpaired electron with the nucleus. So, this interaction is called the electron nuclear hyperfine interaction. So, at this point let us just summarize what we have seen now that EPR spectroscopy gives lots of lines in general and their characteristic of the nucleus that is present there are also characteristics of the type of radical which is given as to the EPR spectrum and from there therefore, we can learn a lot about the structure of the radical or structure of electronic structure of the transcendental complexes and so what not for most importantly now that it is the hyperfine line and the interaction of the electron with the nucleus that give us to the hyperfine line is the key to this summary of what we have learned today.