 So, welcome to today's lecture and we will continue with polarization transfer and we will again go little more detail in nuclear overhouser effect. So, if you recollect in the last class we looked at what is actually nuclear overhouser effect and how experiments are done and what we can get from nuclear overhouser effect related spectroscopy. So, to summarize we looked at the population difference of two states for a particular nuclei give rise to an intensity and if we compare NMR with other electronic spectroscopy the difference between the NMR the difference between two states is very small therefore it is one of the least sensitive technique. But among all the nuclei that are available for NMR proton is the highest sensitive because its magnetic moment is higher and the population difference between two states it quite a large. We also looked at to enhance this polarization that we had for enhancing the polarization between these two states say alpha states and beta states. If there is another spin which is coupled to this we perturb this by a radio frequency pulse and if these two spins say I spin or a spin if they are two coupled if we perturb I spin the effect of that will be seen on a spin. Now this can have a homonuclear effect or heteronuclear effect. Heteronuclear effect is generally used for measuring the distance between two coupled spin and this coupled spin what I mean by coupled through a space they are closer in a space close proximity in a space. In heteronuclear case this is used for transferring polarization from the higher sensitive nuclei to the lower sensitive nuclei like from proton, proton to carbon 13 and these kind of concept is very much used in heteronuclear experiments that we will be seeing in the next classes. Now then we moved ahead and defined two class of NOE one was called continuous wave like irradiation NOE which is called a steady state. So we had seen that we irradiated with a continuous wave pulse and then we excited all the spins by 90 degree and we detected it. So we need to do two experiments one experiment where we irradiate and the next experiment where we do not irradiate and then we take the difference between these two states and then if we take the difference the signal intensity will increase or decrease and this is called NOE effect. So we looked at the NOE will be given by I minus I0 divided by I. So this is of the saturated spectrum, this is non-saturated spectrum and then this you with the non-saturated spectrum gives the enhancement factor which is the NOE. So we looked at the another way to do it is by irradiating with a soft pulse and then here is time for mixing the spins then we apply a 90 degree X pulse. This was 180 degree X pulse and then measuring the FID. So here if we vary the mixing time we can actually quantify the distance between two spins in this transient NOE because this gives you the estimate how much your perturbation effect is moving. So we looked at these all qualitatively and then we went ahead and try to understand what is the origin of this NOE effect and I left you with a question that by doing those treatment of the population difference where the NOE effect is coming from. So we will continue from that states. So we looked at the population between these states P1, P2, P3 and P4 has that term is called D and then we measure that in X there is an enhancement of 2D. So if you remember the previous lecture here the enhancement was 2D between these two states. So we will continue from there and we try to look at the effect that is coming because of irradiation of spin A. So actually we should remember that there is another phenomena which is all the time active and this is called relaxation process. So in any of these phenomena where we excite the nuclei they always want to come back to the equilibrium state and that phenomena is called relaxation and this relaxation actually drives the system towards equilibrium state. Now so in this case as we have discussed earlier there are three kind of transition probability. Here two for A spins so this is called omega 1 single quantum transition again omega 1 single quantum transition for 2A spin and 2 for X spin single quantum transition and single quantum transition. However, there was another two which is this is called double quantum transition. So here both spin flips like alpha-alpha state this goes to beta-beta states and here alpha-beta states goes to beta-alpha states so this is zero quantum transition. Now as we are irradiating the spin A so single quantum transition which is because of omega 1 for A spin will be gone because we are irradiating it. Now what is happening to the X states? So single quantum transition for X spin cannot cause any population transfer why? Because we have irradiated a single quantum transition if you look at for X spin like here d by 2 and d by 2 that is the difference is in d here and here. So from each of them is d and that is why total is 2d. So that is not causing much population difference between states 1 and 3 and 2 and 4. So are actually already at equilibrium difference. Now so this is also not causing any enhancement but let us look at something else and let us look at this double quantum transition which is omega 2. So omega 2 effect that is coming because of double quantum transition actually causes something and that is here. This is basically at this they introduce as a parameter called T. Now that is the steady state population will happen because this double quantum relaxation is active so it will change the population of state here P1 and P3. Now so omega 2 changes the population and because this relaxation is faster so this happens faster than the saturating field and because of that there is a net gain of the state 1 and 4. So if you look at here previously this is minus plus T and this is minus T so that is the net gain we are talking because of fast relaxing double quantum transition that is the changes the population. So now this that is what the transition changes. So X will have P1 minus P3 plus T and X2 will have P1 minus P3 plus T and that is what actually the enhancement that is what the net increase is happening because of the transition of double quantum. Now similarly there is another phenomena so this is because of double quantum transition and relaxation there is another phenomena that is called zero quantum transition. So zero quantum as we know that it is a spin flip flop alpha beta going to beta alpha. So because of this zero quantum another state will happen and that happens between state 2 and state 3 so here should be minus. So now the population transfer occurs between state 2 and 3, state 3 has actually lower population than the equilibrium value and state 2 will be having higher population thus there is a intensity of X transition is proportional to the population difference as we know that what will be the distance so X1 will be P1 minus P3 minus T1 and X2 will be P1 minus P3 minus T. So if we looked at here what is happening we are gaining some intensity by T because of this and in this case we are losing some intensity because of the zero quantum transition. So there is a net gain in intensity and there is a net loss in intensity so NOA is causing to effect either intensity can increase or intensity can decrease and why this is happening not because of single quantum transition but there is a another phenomena which is active at that time because of relaxation property double quantum and zero quantum. So this double quantum seems to be enhancing the intensity of these states and zero quantum is decreasing the intensity of the state. Now X transition have lost some intensity compared to these unperturbed states in the previous case so thus X transition either gain intensity or loses intensity the gain in intensity is called positive NOA and loss in intensity is called negative NOA. Now the magnitude of the NOA as we know the proportional to the difference in the two relaxation rates. So which is actually referred to cross relaxation rate. So what are those two relaxation rate? This omega 2 which is between state 1 and 4 and then here omega 0 which is between state 3 and 2. Now so therefore this cross relaxation rate depends upon difference between omega 2 and omega 0 and that actually gives rise to different kind of NOA positive or negative. So where is positive NOA and where is negative NOA? So positive NOA are generally occurs in small molecules small organic molecules why negative NOA is a common feature of macromolecule. Now this is very interesting and I will leave you with this thought that why small molecule shows positive NOA and micromolecule shows negative NOA. So let us on the basis of this let us try to think little bit what is happening in macromolecule and what is happening in a small molecule. So macromolecule generally what we mean is like a protein. So this is suppose a protein. Now a protein has thousands of proton atoms sorry this is active nuclei there are thousands, ten thousands of proton atoms whereas let us take a small molecule something like this where we have different protons here. Now if you look at macromolecules what is happening because of this abundant active nuclei the redistribution of magnetization happens very fast or very efficiently by this zero quantum phenomena and actually that happens within the like distribution of the magnetization. So because of this zero quantum phenomena distribution of magnetization leads to negative NOA. However for small molecule redistribution of population happens but here it is mostly dominated by double quantum transition. So if zero quantum is there there is a negative NOA if double quantum transition is dominating factor of redistribution of magnetization then it is positive NOA. So for macromolecules like we have in proteins we have negative NOA for the small molecules like a small organic molecule we have positive NOA. So let us move ahead that was a qualitative description of NOA how NOA can arise and how they are coming what kind of NOA we have and we looked at the two kinds of NOA steady state experiment or transient NOA experiments and the outcome in one case was positive and negative. So let us little bit rigorously let us try to understand what is the origin of this NOA. So as we understood that NOA arises due to interplay between relaxation and population distribution upon perturbation. Perturbation we are perturbing one spin like that we had seen earlier. So if we have two coupled spin say I spin and S spin we are perturbing one spin either by continuous wave radio irradiation or by pulse radio irradiation and then there is a redistribution of population happens and there are already existing relaxation mechanism that actually causes the NOE that is what we have looked for in the last class. Now we also looked that time evolution of this population is a crucial process and that actually further leads to the distribution and readjustment of the population. So let us look at generalized phenomena where we have to consider a system which is actually n level system. So say here is the ground state and then there are n level system of these different levels and then transitions are happening here. So what will be the population and let us look at what happens upon perturbation so that we can understand what is the origin of NOE phenomena. So if we start with this and we start perturbing the system because for any perturbation is very important the recovery of population of say any state I you will be governed by the following equation I will take time little bit to explain what actually this equation means. So here rate of change of population of ith spin with time. So this is governed by few of the parameter like here this is the transition probability. So transition probability between i and j. So what I mean again I will draw this little diagram. So this is transition probability between say ith state and jth state and what is the population of pj at any time and what was the equilibrium population minus the population of i state at any time and that will be subtracted with p i0 that is the equilibrium population. So this two together is subtracted from here and then you actually so here you sum over transition probability of all jth state and here sum over all here again jth state. So the population difference between jth state and ith state you sum over that multiply with the transition probability that gives you rate of change of population with time. So as we discussed p0 are the equilibrium population and pj is the population at any time t. So if you calculate this equation we can find it out rate of change of population with time and this is called master equation. The basis of actually on the basis of this we can find it out what is the decay or what is the change in the population of any state that we showed in generalized equation between any two states like this state and this state or this state and this state we only need to know what is the population at the any time t, what is the equilibrium population of these two states and what is the transition probability and then one need to sum over all the jth and if we do that then we can calculate that dpi with time. So let us take a simple example like 2 weakly coupled spin that is AX spin. So if we see the energy level that we had discussed earlier. So this is alpha alpha state with the population p1 then we have alpha beta state population p3 and here beta alpha state with population p2 and beta beta state with population p4. These are different transition probability so here is a single quantum so this from here to here it is a single quantum transition from alpha alpha state to beta alpha state again single quantum transition. So we have a here 2 for A spin and 2 for X spin single quantum transition then we have a double quantum transition like alpha alpha state going to beta beta state or p1 going to p4 that is a double quantum transition and here it is a zero quantum transition alpha beta state going to beta alpha state. So these are all transition probability and the population of various state. Now suppose we have a MA magnetization associated with A spin and MX is the Z magnetization associated with the X spin. So the related population of p1 and p4 p1 and p4 can be given by something like this. So MA that is that is Z magnetization of A state will be half of p1 this plus this minus this and this. So that will be for A spin similarly for because why A spin here if you look at this transition and here we are changing from alpha to beta and here it is going again alpha to beta. So similarly for X spin one can write half of the population of p1 plus p3 minus p2 minus p4. So we have the magnetization coming from A and X for two spin system. Now so population change then we can put that in master equation that we have described here and as we discussed we need only three things. First the transition probability, second population at any time t and the equilibrium population. So if we put that in the master equation for these two spin system one can write it dp1 that is change of population of state 1 will be given by the transition probability of A spin the single quantum transition plus single quantum transition of the X spin plus double quantum transition and this is the population difference at any time t with p10 that is equilibrium population. So we have to sum this and actually for omega 2 that is p4 and the equilibrium population of p4 then again single quantum transition for X for p2 states and single quantum transition for A of p3 states. So if we sum all those we can get what is the change of the population with time for like p10. Similarly one can find it out by putting all those and sum over the all states for p2. So again here it will be single quantum transition for this zero quantum transition and that will be multiplied with the population of p2. I just go and explain you this is the p2. So here zero quantum transition means transition happening from alpha beta state to beta alpha state. So if you take all this and again here single quantum transition here single quantum transition for X here single quantum transition for A and zero quantum transition for this spin third. So if you take this we can get the rate of change of population of state 2 p2. Similarly for p3 and one can get for p4. So now we got the population change with time for all these states. So now we can define the equilibrium magnetization that is actually mA0 and mX0. So this is equilibrium magnetization for A spin and this is for X spin. So as we said the magnetization depends upon what is the population in different state in the equilibrium state. So that will give us the equilibrium magnetization of A spin. So here if you look at what actually it is in a simple term it is a population of p1 and p2 minus p3 and p4. So 2 of these states minus 2 of this state will give you equilibrium magnetization for A spin. Similarly one can write for X spin. So A transition are this and this. X transition is for this and this. So we just need to take the difference appropriately so that we can get so for p1 this is the case. So here and here that is the transition happening and if we go back one can see p10 plus p20 minus p30 and p40 for A and similarly for X spin we can get it. So then rate of change of magnetization one can write it with time. So for A spin mA and for X spin mX one can find it out by putting all these equilibrium magnetization. So here 2 into omega 1 of A that is transition probability for A spin plus 0 quantum transition probability plus double quantum transition probability and that is difference of magnetization for A spin at any time t minus equilibrium population and that is added with the double quantum transition minus single quantum transition and the difference in the magnetization of X spin. So similarly one can write it for X spin one can find it out this will be the like with transition probability this is the magnetization. So now here one can find the rate of change of magnetization for both spin A spin and X spin. So then let us simplify this complex equation and for that let us define that here sigma A one can write it this term whole this term one can write it sigma A and sigma X we can write it whole this term. So we can simplify and then define this sigma sigma is omega 2 minus omega 0 that is double quantum transition minus 0 quantum transition. So if we define this then the previous equation that we had simply doing little bit of algebraic representation we can simplify this equation. So rate of change of magnetization of A spin can be written like rho A. So if you look at the previously here we have defined this as a rho A minus this this is sigma A minus this. So that is what we have written here rho A minus sigma rho A minus magnetization of A and equilibrium magnetization and sigma A X into this. Similarly for rate of change of magnetization of X spin one can write like this. So here we have two term rho A and sigma A X here rho X and sigma A X. So now if you look at here what is happening this is autocorrelation function because they are correlating with itself. So A is correlating with A. So one can define in T T this rho A and rho X are auto relaxation rate for spin A and spin X and this is called cross relaxation rate. Self correlation means like it is correlating with itself what was the magnetization or what was the population at time t with time t plus delta t but cross relaxation correlating with other spins. So that is what here is sigma X. So sigma X is called cross relaxation rate between two spin A spin and X spin. Then one can write actually a generalized equation for rate of change of magnetization like this we can write dm with respect to the change of magnetization m with time one can write for r and m. So r one can define r and m are basically here m is a column vector. So here like we can write here column vector and then vector representing deviation from the magnetization from the equilibrium value. So like previously here this is the ma and one can write the r is a matrix of relaxation rate. So we called it either auto relaxation or cross relaxation rate. So we can write the magnetization product with the relaxation rate like for two spin one can write this m will be column vector matrix for ma minus m0a mx minus m0x and one can write relaxation matrix that is here the auto correlation rate of A minus cross correlation rate of A X minus cross here cross correlation rate of A X divided by cross correlation rate of A X minus auto correlation rate of X. So that means the change of magnetization of any spin now you can define in two terms one is relaxation matrix term and then actually magnetization deviation in the magnetization from its equilibrium state. So if we can define then we can generalize the equation what happens upon perturbation how the population gets redistributed or in another term how magnetization value changes. So if we can see that then one can find it out what will be the case for a steady state NOE. So here in steady state NOE what we were doing we had a two spin one spin was A another spin was X one spin was saturated and we are looking at the effect of that saturation on the other spin. So now if you perturb one spin in the steady state NOE just to remind you we are doing experiments like this we are saturating this we are applying a pulse and looking at the response of that perturbation. So this is saturation pulse. So if we do that what is going to happen of magnetization of the other spin if you irradiate so for a steady state NOE if you spin X what is happening because of irradiation of spin A that we are going to look at in detail using this generalized magnetization form that we had in few lectures. So and how this gives rise to enhancement in the magnetization or reduction in the magnetization that actually results in the value of the positive NOE or negative NOE. So that is going to be topic for our next class. So I hope that you understood this and if you have a question please come back this is little bit mathematical so we will go slowly and we try to develop the concept of origin of NOE in a more regress manner more mathematical manner and we will continue with steady state NOE and transient NOE how the enhancement comes because of the perturbation of a copper NOE Ki with X. Thank you very much.