 Welcome to the second lecture on NMR spectroscopy. In the first lecture we covered the basic concepts such as the nuclear spin, nuclear magnetic moment and how they interact with the magnetic field in a brief manner. We also talked about the angular momentum, spin angular momentum of the nuclei which is an intrinsic property of the nuclei and the magnetic resonance phenomenon appears as a consequence of the presence of both angular momentum and the magnetic moment which are related in a linear fashion. We may be repeating a little bit of what we said last time for sake of continuity. We said the nuclear angular momentum is quantized which means it has specific values which are determined by the nuclear spin quantum number i which is also called just the nuclear spin and the z component of the angular momentum is also quantized. This is by choice. It could have been x or the y but one has chosen to take it as z magnetization which is quantized. If you have a nuclear spin i then the z component of the magnetization which is represented by another quantum number called m that takes a certain number of values specifically 2i plus 1 values and it starts from minus i to plus i with integral increments. I can have a value 0 half integrals or integrals this depends on the nuclear spin. For most of chemistry and biology applications we will be dealing with spins which have i is equal to half. These are like proton, carbon 13, nitrogen 15, phosphorus 31 and so on. When there is no magnetic field applied all the orientations of the nuclear spins have the same energy. Their energy levels are degenerate so they can be oriented in any direction in space because of the superposition principle of the wave functions in quantum mechanics. However, when a magnetic field is applied the degeneracy of these different orientations gets lifted because of the interaction of the nuclei with the magnetic field. This is explicitly given by this equation here. The interaction energy is given by minus mu dot H naught where mu is the magnetic moment and H naught is the field applied. For a spin half there are two orientations as I indicated to you. These states are represented as alpha and beta. Alpha is the state which is whose z component is parallel to the magnetic field and the beta is the state where the spins are oriented anti-parallel to the magnetic field. This is the beta state. The alpha state has a lower energy for positive gamma and in most cases we are dealing with positive gamma except nitrogen 15 which has a negative gamma value. And the beta state has a higher energy here. The spins will get distributed between these two energy levels and this is determined by the Boltzmann statistics. Now what is Boltzmann statistics? Typically if you consider a state i if there are many states, if you consider a state i represented by this index i the probability that the nuclear spin is in that state is given by this expression exponential minus E i by k t where k is the Boltzmann constant and t is the temperature and z is a function called as the partition function. z is defined in this manner which is the summation of this sort of terms E j by k t where j is an index which runs over all the states all the 2 i plus 1 states. Now the magnetic moments are generally is too so small for nuclei that these energies are extremely small with interaction energy is quite small compared to the value of k t. So therefore in this expansion which goes up to infinity these higher order terms do not contribute much at all. So typically one can neglect this higher order terms and that is called as the high temperature approximation. E j by k t is far, far less than 1 and therefore you can neglect the higher order terms in this expansion for z. So what happens then? So the p i the probability that the spin is in the i state which has energy a i is given by 1 minus E i by k t divided by the summation 1 minus E j by k t. Now in the denominator this E j by k t is still very small compared to 1. Therefore I can further neglect this term in the denominator and then I will get p i is equal to 1 minus E i by k t divided by summation j 1. So what does that mean? This is simply equal to the number of states in your system. For a spin half there are only 2 states as we mentioned which we represent as alpha and beta. So if I put this energy terms explicitly the probability for the alpha state is given by 1 plus mu H naught by k t divided by 2 and p beta is 1 minus mu H naught by k t divided by 2. This is the interaction energy for the alpha state and this is for the beta state. This is typically true for protons, carbon thirteens and nitrogen 15 it will be rather around because the gamma is negative for nitrogen 15. Now so in the ensemble the spins will be distributed between the 2 states alpha and beta. Now look at this we said that the magnetic moments are oriented with respect to the magnetic field its magnitude is not equal to the z component of the magnetic of the magnetic moment. Therefore you also said there is an angle between the magnetic moment orientation and the z field and therefore they all the ensemble spins are distributed in a cone here. All the alpha spins are oriented in the surface of the cone here and likewise the beta spins are oriented at different points in the cone here. There is no specific preference for any specific position on this surface of the cone for any spin. Any orientation or any position on the surface of the cone is equally allowed and they have the same energy. This is called the hypothesis of random phases because if I take a want to take a projection of this orientation on the on the x y plane the component will have an angle here and that is a phase of the magnetic moment the magnetization. Therefore if any phase is allowed then we say it is hypothesis of random phases. Now look at this for all the spins which are present in the surface of the cone here we take projections on the x y plane the z component will be the same for all of them the x y component will be distributed in the plane x y plane like this all over in this plane and they will be randomly oriented which means any orientation in the x y plane has an opposite orientation in the x y plane. Therefore those magnetization components will cancel out this happens for both to the alpha state as well as for the beta state whereas the z component will all co-add they will add here and they will also add here. But since the population in the beta state is lower than the population in the alpha state there will be no complete cancellation of these two opposite orientations along the z axis. Therefore there will be a net magnetic moment which is given by m naught and that stays along the positive z axis which is the direction of the magnetic field. So I represent this as m naught is equal to p alpha minus p beta times mu which is the magnetic moment and is also you have to multiply by f number n which is the total number of spins in the ensemble. So this represents the total magnetization if we do not have this then we have what is called as the mean magnetic moment of the nucleus spin system. Now how does the resonance absorption of energy happen? Let us look at it in the classical way when we apply a radio frequency now we apply the radio frequency which is given by 2 H1 cosine omega t. H1 is the amplitude of the radio frequency and omega naught is a frequency of the radio frequency and this is represented by 2 H1 cosine omega t. Therefore this is an oscillating wave as indicated by this and this is applied in the transverse plane. And when there is an interaction between the RF and the magnetic moment there will be absorption of energy. When the absorption of energy happens what should happen? There should be redistribution of the spins between the alpha and the beta states which means there will be some transitions which will happen from the alpha state to the beta state. It will flipping of the magnetic moment from the alpha state to the beta state. It can also happen from the beta state to the alpha state but the number of spins that will flip from the alpha state to the beta state will be more when there is absorption of energy. That interaction means that because it is flipping from alpha state to the beta state it can also happen in the reverse direction but the total in the net there will be more spins which will be flipping from the alpha state to the beta state and that amounts to absorption of energy. Let us understand it in a classical sense in a slightly different manner. We said we applied an RF which is represented by the term 2 H1 cosine omega t in the XY plane. Assume that it is along the X axis we put it along the X axis here. Now, we also know that the cosine term can be expanded in this manner. 2 H1 cosine omega not t is given by H1 exponential e to the i omega not t plus H1 e to the minus i omega not t. What do these represent? These two represent rotating fields in the XY direction in the XY plane. If this represents a rotation in this direction this represents a rotation in the opposite direction. Now, the spins themselves are rotating like this in both the states in both the alpha and the beta states the spins are rotating like this these are going like this and these are going like this. Let us look at that here. This is the magnetic moment which is spinning in this direction and these are the components of the two RF. The one RF which is going in this direction the second RF component which is going in this direction these are the two rotating fields. Now, suppose I were to sit on this RF component and look at the spin what does it see? What do I see? I will see that the spins are processing with the frequency omega i minus omega not. So, they are going in the same direction this direction and this direction are the same. Therefore, the net frequency of the precession will be the difference between these two frequencies. Whereas, if I am sitting on this one the net frequency will be the addition because the two are going in the opposite direction. Notice these are in mega hertz this is also in mega hertz. Therefore, this frequency addition will also be in mega hertz which means the spins here will not be able to see this field properly at all. This will be in kilo hertz as you come closer and closer it will become smaller and smaller and when you reach a condition these two frequencies are the same then it is called as resonance under those condition the RF will be stationary as seen by the nucleus or if I see sitting on this RF I will see the nucleus is stationary then if it is stationary which means there is no field along the z axis the only field that is present will be the field along the x axis and that is in the transverse plane. So, what will happen now? The nuclei will try interact with this RF field because that is a magnetic field which is there which is a small magnetic field the interaction leads to changes in the energy of the system. What is the meaning of changing in the energy of the system? It means the redistribution of the populations once more there will be some transitions occurring from the alpha state to the beta state beta state to the alpha state. But if the energy of the system has to change there has to be a net absorption of energy which means there will be more spins going from the alpha state to the beta state as indicated in the previous slide. So, this is a classical description to show how the nuclear spins interact with the RF field. Therefore summarizing this the principle of NMR can be shown in this particular slide this may be a repetition from the previous lecture but nevertheless it actually consolidates what we were trying to say. If you have in the absence of the magnetic field the nuclear spins are in a particular state here they have the same energy the two orientations of the spins which has i is equal to half are the same and when you apply the magnetic field the two energy levels become non-degenerate and the separation between the energy levels will depend upon the magnetic field because the interaction depends on the magnetic field and they will be distributed between the two states and now the energy absorbed will depend upon how much is the excess population here compared to this and this distribution is determined by Boltzmann statistics as I already mentioned. If I supply energy to this system now we look at it in the quantum mechanical sense earlier we looked at it from the point of view of the classical description now this is the quantum description of the energy level diagram. So, if I supply energy which corresponds to this energy difference there will be absorption of energy at this particular frequency and this is called as the signal this is the signal in NMR the intensity of this signal will be proportional to the energy absorbed and therefore it will be proportional to the population difference between these two states. Now this is a schematic of the NMR spectrometer modern NMR spectrometers. So, we have a magnet here these days superconducting magnet in the previous case what I showed here in the schematic this was an electromagnet or a permanent magnet this was the case in the early days when the superconducting magnets were not available all the NMR experiments were done with these electromagnets and today the magnets are superconducting magnets this is the revolution in the technology you can go to very high fields and the homogeneities of the magnetic fields are very good therefore you get good signal with good intensity and very sharp lines and then you have the sample is sitting inside this magnet here at the center and this is called as the and there is a sample probe and you have a transmitter controller which supplies the radio frequency energy and then the signal that comes out from there is detected by this detector this is RF transmitter and you have a shim system here the shim system means you have to adjust the homogeneity of the magnetic field you have to have homogeneous magnetic field the H O naught field which we said has to be the same over the entire sample volume. So, this has to be adjusted and therefore there are various kinds of coils here which produce currents producing magnetic fields to correct the variations in the magnetic field distributions in the sample volume and this is the detector of that is called a field frequency lock this detects if there is any instability on the field because of various kinds of disturbances outside the magnet if there is a small disturbance because of some movement of some object then the field will get disturbed and then of course there will be countercurrents applied in this shim system and which will bring the field back and that is called as the field frequency lock. So, these are the basic components of the NMR modern NMR spectrometers of course there are many many more complexities here that we do not need to discuss here. So, now we return to this question what causes redistribution of the populations when field is applied we said when a RF is applied the RF interacts with the spins magnetic moment and causes transitions we are now when we applied a main magnetic field RF is not applied how do the transitions occur how do the populations redistribute themselves between the two fields alpha and the beta obviously there must be transitions happening which causes the redistribution of the populations. Let us represent this in the following manner let us call this state as alpha state and this is the beta state the number of spins here is N alpha the number of spins here is N beta and there is a transition occurring here with from alpha state to the beta state and w alpha beta is a transition probability for whether transition from alpha to the beta and likewise there can be transitions which are coming from the beta state to the alpha state. Now you notice when we go from the equilibrium to the situation when there is redistribution of the populations here in the absence of the field the two energy levels were degenerate and that was here okay and this was equal and when the field is applied the beta states went up here and the alpha states went down here if there is more population here there are more spins which have gone down than the number of spins which have gone up therefore the system is actually lost energy right because there are more spins which have lost energy a few fewer spins have gained energy which means this is in the net there is a loss of energy where does this energy go who takes this energy unless there is somebody to take this energy the process will not happen and that brings us into the concept of the lattice there is a lattice in your sample everything other than your particular spin system is called the lattice and lattice has all kinds of energy levels in it so every time there is a transition absorption there is a release of energy from here the lattice takes up that energy and causes the transition upward transition here if there is a downward transition here of the spin system there will be upward transition here of the spin system if there is an upward transition of the spin system here there is a downward transition of the spin system therefore the spins and the lattice are coupled there is energy exchange between the spin and the lattice and which allows the redistribution of the populations in the spin system okay now how does this happen let us look at this in a little bit more quantitative manner dn and this written if there is a perturbation of the spin system which the spin the number of spins in the two states are not at equilibrium and they would like to tend to equilibrium through these transitions so therefore one can write here the rate equations for these transitions if I write n alpha minus n beta is the difference in the population between the two states then dn by dt if I call this dn alpha minus dn divided by dt is equal to this equation here two times w beta alpha n beta minus w alpha beta n alpha this represents the number of transitions coming down there represents the number of transitions going up and therefore this variation goes by this rate equation now we define here some terms n is the total number of spins which must be equal to n alpha plus n beta and these are the populations of the alpha and the beta states any point in time therefore this is obviously time dependent that is why you have this time dependent equation here and these are the equilibrium populations of the two states n alpha naught and n beta naught represent the equilibrium populations of the alpha and the beta states we write n is equal to n alpha minus n beta then this equation can be recast in this manner dn by dt is equal to w beta alpha plus w alpha beta times n naught minus n this is simple algebra problem here if you recast this putting in these equations you will get this equation the solution of this equation can be obtained readily and that gives you n minus n naught is equal to n minus n naught at time t is equal to 0 multiplied by exponential minus t by t 1 what is t 1 we will soon see now if at time t is equal to 0 the system is un-magnetized then n at time t is equal to 0 is 0 and therefore this equation reduces to n of t is equal to n naught into 1 minus exponential minus t by t 1 so t 1 is given by 1 by 2 times t w 1 by 2 times w and w is the average transition probability that is w beta alpha plus w alpha beta divided by 2 so now we look at the kinetics of resonance so we looked at how the system attains equilibrium because of transitions which are supported by coupling to the lattice now let us look at the transition induced by the RF now the RF induces transitions from both alpha to the beta and beta to the alpha and these two rates will be the same this comes from the principles of quantum mechanics when you calculate the transition probability then that transition probability is represented by p then dn by dt is given by p times n beta minus n alpha where RF and p is the RF induced transition probability one can also calculate what this p is p is obviously proportional to the amplitude of of the RF field so it is proportional to the power therefore it is given by p is equal to 1 by 4 gamma square h 1 square this actually follows from the rule of the transition probability formula of the transition probability RF induced transition probability now we have we already know n alpha is equal to n plus n by 2 n beta is equal to n minus n by 2 so you recast this you get dn alpha by dt is equal to half dn by dt and that is equal to minus pn because this is minus pn and dn by dt is equal to dn alpha by dt is half dn by dt is equal to minus pn which means dn by dt is equal to minus 2 p 2 pn and this gives you the solution n of t is equal to n at 0 e to the minus 2 pt so this shows you how the population distribution changes with the RF that you apply at long time if you continuously apply the RF this obviously goes down to 0 and therefore your population difference also goes down to 0 in which case you will not have any energy absorbed. Now we can write a similar equation for the energy absorbed energy if e is the energy of the system then the d e by dt represents the change in the energy of the system that will amount to energy absorbed and that is given by the change in difference in the energy multiplied by the population difference multiplied by the transition probability. So d e by dt is equal to n p times delta e and if you obtain the solution of this then you get the energy of the system is given by delta e into n naught 1 minus e to the minus 2 pt delta e is a fixed number because that is the energy difference between the two states n naught is a fixed number because this is the time t is equal to 0 whatever was whatever was the population difference between the two states and this is the turn which is dependent on time. So this describes how the energy will be absorbed how the energy of the system will change as the RF is applied. So if you plot this equation then you will find that d e by dt goes in this manner exponential decay and the e in the system it goes in this manner and reaches a saturation at some stage. After this there is no change in the energy of the system which means there will be no energy absorbed. Now if at this point the system no energy is absorbed there will be no signal but in an MN experiment we always find that there is a signal is present when you are apply the RF energy is continuously absorbed so something else also must be happening. So far we included the RF induced transition in these equations. So there is something which is compensating which brings back the population difference and allows the system to continuously absorb the energy and that is the relaxation. We had ignored the relaxation till now we had simply had d n by dt is equal to minus 2 pn. Now we have to add a term n minus n naught divided by t1 this defines the transitions occurring because of relaxation the system tries to recover from the perturbation tries to approach the equilibrium population distribution and this happens with the rate constant 1 by t1 as we described earlier. Now at equilibrium or steady state d n by dt will be 0 which means the population difference remains the same and there is continuous energy absorbed. And if I call that steady state population as n dash from the previous equation I will get n dash is equal to n naught divided by 1 plus 2 p t1 p is the power and t1 is the relaxation time which is called as the spin lattice relaxation time. Now here I want to say if 2 p t1 is approximately equal to 0 which means apply a very low power p is the power as I mentioned it is related to the power then n dash is equal to n naught. In other words if the power is very low I reached the equilibrium population very soon so it will always remain as though it is at equilibrium therefore there will be continuous absorption of energy. If 2 p t1 is approximately infinity which means I apply very high power which means very high power then n dash becomes equal to 0. So, in that case this will be called saturation the signal will get saturated after that we will not be able to observe the signal. Therefore what one needs to do we have to apply very low power to be able to observe the signal continuously over a long period of time. In fact I must tell you a story here that Gorter who actually did this experiment earlier he actually win lost the Nobel Prize because he missed the signal because he used the system which had very long t1 the t1 is very long which means even at a small power p 2 p t1 is very large. So, therefore immediately he was reaching a condition of saturation therefore there was no signal coming this is unfortunate he had chosen a sample therefore it turns out that the sample choice is also very important. So, you have to be lucky to be able to observe signal and you have to choose the right kind of samples. So, we will stop here and continue with further topics in the next class. Thank you.