 Good morning, in the previous classes especially those by Professor Ashutosh Kumar when he discussed about analysis of NMR spectra you have learnt a lot about spin-spin coupling constants. How do they affect the appearance of the NMR spectra? How is it useful to identify groups in your molecules and how it is useful to determine the structures of molecules? Often it also becomes necessary to record spectra without spin-spin coupling, how to do that? So we need to do something special to remove the spin coupling and this is called as a spin decoupling. So we are going to look at that aspect now, how we can decouple the spins and get simplified spectra. Essentially those spectra will have only chemical shifts and you can count the number of peaks to identify the number of spins and so on. This is particularly useful in the case of carbon NMR where you can simply count the number of carbons in your molecule if you have recorded a spin decoupled NMR spectrum. In organic chemistry this is an extremely useful technique. You will record spin coupled spectra for identification of the functional groups, what kind of carbons they are and you record proton decoupled carbon spectra to identify the number of carbons in your molecule. It is also useful in homonuclear NMR to identify pairs of protons which are coupled to each other. If there is one proton which is distinct and it is coupled to some other one, if you remove the coupling then there will be changes in the spectra and you can identify the pairs of protons which are coupled to each other. This is again a very useful thing to find out the structures of molecules. How to do spin decoupling, this is an important topic, it is lot of discussion can be made on this one. But we will look at the basic concepts, how is routinely used, how it can be applied to different kinds of situations. Spin decoupling, you may recall the coupling between two nuclei, let us say A and M, two spins A and M is written as JIA.IM, this is the interaction between the two spins A and M, this is the coupling interaction and J is the coupling constant. These two may be separated by one bond, two bond, three bond and so on and so forth which has been discussed by Ashutosh Kumar in his lectures and J is the coupling constant between the two spins A and M. Notice this is the dot product here. If we can make this dot product go to 0, then essentially this interaction will not appear in your spectrum. You cannot change J, when you say spin decoupling you are not changing J, you are only seeing to it that this does not appear in your spectrum. So to do that we can manipulate this part of the interaction, this is IA dot IM. So how should we do it? Let us look at the situation at equilibrium. At equilibrium we know that the magnetization is along the Z axis that is the direction of the magnetic field. So your magnetic field is here and your protons or the spins are oriented in the direction of the magnetic field at equilibrium. Both the spins A and M are oriented with respect to the magnetic field along the positive Z axis. We say that the two spins are quantized along the Z axis along the direction of the field. These two spins have their independent chemical shift and the coupling constants. So typically such a kind of a thing will give you two lines for the A and two lines for the M and this will be your A chemical shift and this will be the M chemical shift. This is A and this is M, this is A. So if one can somehow orient this M spins along a different direction, for example if we do a kind of a spin lock on the M spin so that this magnetization M magnetization comes along the Y axis and the A magnetization continues to remain on the Z axis. In other words the A and the M spins are now quantized along orthogonal axis. How does one do this? This is done by what is known as the spin lock. You apply an RF on the M spin. So here you have the spectrum recorded after applying a normal 90 degree pulse and you apply the F, collect the FID, Fourier transformation will give you two lines for the A spins and two lines for the M spin. Now you apply an RF here, very selective RF which means it is a low power RF and it saturates the M spins completely. It not only just saturated actually is little higher power than what is required for saturation and that is indicated by this RF here. This is applied at the middle of the M doublet. So at the chemical shift of M you apply this low power RF so it perturbs only the M spin and does not disturb the A spins at all. So therefore what happens to the magnetization which was along the Z axis. The M magnetization if you apply the RF along the Y axis here M spin along the Y axis this magnetization will rotate like this. It will rotate like this and eventually come down and orient itself along the Y axis as indicated here. So therefore that is the way you try to achieve the orthogonal orientations of the two spins and this is what we said. We want to quantize the two spins A and M spins along orthogonal axis. So now you see as a result of this if you record the spectrum here you apply the M spin log decoupling. This has to be applied for a long time. So long as your FID is there you have to be applied all along. When you are collecting the data this should be on. So after the 90 degree pulse therefore the coupling is not seen in this FID therefore the A doublet will collapse into a singlet and which will have twice the intensity because both these intensities are now merged into this. On the other hand the M spin will not have any intensity because that is saturated and it will be 0. So this is how you achieve decoupling. You apply an RF while on the M spin while you are acquiring the data. We can also achieve decoupling by using what is called as the spin echo which we have discussed earlier. How does it allow us to get spin decoupling? How do we achieve spin decoupling by the spin echo? You recall your vector picture of the spin echo how the spin echo works. I remember in the spin echo we had a 90 degree pulse here and a 180 degree pulse on both the spins A and X. We consider A and X spins and the 180 degree pulse was applied here on both the A and the X spins. In this particular case we are talking about A and M spins. In the normal spin echo you apply the 180 degree pulse on both A and M. However if you want to do decoupling what you do is you apply this 180 degree pulse only on the M spin. You do not apply to the A spin at all. So what is the consequence of that? Let us look at these stages here. This stage of the magnetization here is 1, 2, 3, 4 and 5 and these vector diagrams tell you what is the orientation of the magnetization at these individual time points. At time point 1 both the spins are oriented along the Z axis as was indicated before. Now when you apply the 90 degree X pulse both the spins will get rotated on to the negative Y axis. They are both here. Now let us look at the A transitions. As we did in the case of the spin echo we looked at the A transitions A1 and A2. There were 2 transitions A1 and A2. You are sitting in the rotating frame of the A spin which means you are sitting at the chemical shift of the A spin and look at the 2 transitions how they are moving. So we said earlier also the A2 which is the higher frequency goes in this anticlockwise direction. So this goes in this rotates in this direction and A1 which is the lower frequency compared to the chemical shift. So that is the slower one that goes in this direction. After time tau they have separated by this much amount. The 2 transitions have defaced by this much amount. Now when you apply the 180 degree pulse on the M spin as here indicated here what happens? Once again you recall our discussion on the spin echo. When you apply to one of these spins M spin the 2 transitions change their labels A1 becomes A2 and A2 becomes A1. So therefore this becomes A2 this becomes A1 and the A2 continues to go in this manner. A1 continues to go in this manner their original sense because these are slow and fast moving transitions. So after the next time tau that is at this point these ones would again move by the same amount this and this will come here, this will come here and therefore they come back here along the minus y axis as we started off here. Which means the coupling has had no effect on the movement of the 2 transitions. Therefore at this point it is as though there has been no coupling there is only simply chemical shift chemical shift of the AS spin which has been rotated by 90 degrees from the z axis to the minus y axis. In other words during this entire period the coupling has been removed this is this spin has been decoupled. The same thing happens if you applied a 180 degree pulse on the AS spin only and not on the M spin. If you did that then what happens is these 2 transitions will move here this A1 will move here and A2 will move here and they continue to go in this direction and here and eventually they will orient themselves along the positive y axis. So in either case they are refocused the 2 transitions are refocused and you will not have any effect of the coupling on the observed magnetization at the time of the echo. The spin J coupling modulation is removed in other words if you apply a 180 pulse on any one of the spins and not on both the spins the coupling effect is removed and you achieve what is known as the spin decoupling. This is required in many multiples experiments notice here we are not actually collecting the data. In the previous case we are collecting the data and here the RF has applied while you are collecting the data here RF has applied on the M spin and you can collect the data there. So that the A transitions are seen as decoupled from the M transition but that would mean that you should be able to apply the RF power selectively on the M transitions. This becomes a particularly useful thing when you are doing heteronuclear experiments because if this is the proton channel and this is the carbon channel then there is no difficulty in applying a 180 pulse on the carbon channel here or if you are doing the carbon channel here in the proton channel here there is no difficulty in applying a 180 pulse on one of the channels so that they from here to here there is no coupling evolution at all. You may collect the data here with coupling or without coupling it does not matter. So in many of these multidimensional experiments and other pulse sequences which we will see such kind of combinations of evolutions with and without spin coupling will be coming out. And that is an important part of multiples design. So so much for the spin decoupling and you will see many examples of this in various multiples experiments which we will be discussing at a later time. Now we switch to a different topic now which is measurement of relaxation times. We have discussed this relaxation times earlier in the first few lectures and we showed the importance of T1 and T2 relaxations in determining the evolution of the magnetization recovery of the magnetization along the z axis and decay of the magnetization in the transverse plane in the previous lectures. And these are important parameters of individual spin systems and we must know how to measure the T1 and the T2 of the given space systems. First common technique for measurement of T1 is the so called inversion recovery experiment. We had already looked at some application of this sort of a strategy in water suppression okay but we will look at that same thing again and how one can use it to measure the T1 relaxation times. The pulse sequence is pretty simple so we have here a 180 degree pulse in the beginning which is applied to all the spins. These are hard pulses. This is a hard pulse which is applied to all the spins. Then you wait for a time tau and then at the end of it you apply a 90 degree pulse and after that you collect the data as in the FID. So these are the time points 1, 2, 3, 4 and the data collected is actually 5. This tau is a period which is adjustable and depending upon how you adjust it you can have different kinds of signals in your NMR spectrum. Let us look at that in little bit more detail. So here is the initial situation. Again consider two transitions here. We are not considering coupling at this point but it does not matter. So we are considering two transitions which are belong to two different spins. And when you apply a 180 degree pulse to this transition what happens? The magnetization is rotated from the Z axis to the minus Z axis. So both of them have come here. Now these ones will start recovering and as they start recovering notice these are along the Z axis now. So there is no precession. Since it is along the Z axis the transitions will start going up along the negative Z axis and eventually they will go to the positive Z axis but they are not in the transverse plane therefore there is no precession. Now depending upon the time tau they would have recovered to different extents along the Z axis. For small tau for example then both these are still on the negative Z axis not quite as intensity as this to begin with small tau they are slightly different and if you apply a 90 degree pulse and measure this then what we will get? We will get the two signals but with a negative intensity because these are on the negative Z axis here. When you are applying the 90 degree pulse they are along the negative Z axis therefore they will have a different sign in your spectrum. Positive Z axis magnetization leads to positive signals. Negative Z axis magnetization leads to negative signals that is the initial point. Similarly if you have a very large tau this is the situation. If all of them have come back to the Z axis they are recovered completely along the Z axis they both will be here as they started off here they will be here. And if you apply 90 degree expulse to this or the Y pulse or whatever then magnetization gets rotated along to the minus Y axis and this is what we observe in your spectrum both positive signals you are going to observe this is in contrast to this. So this is complete recovery and this is the initial very small value for the intermediate values there will be intermediate situations. For example for this the two spins here have different relaxation times the cyan one has already gone partly along the positive Z axis the orange one is still lying along the negative Z axis. Therefore when you apply the 90 X pulse the cyan one moves to the negative Y axis here whereas the orange one here moves to the positive Y axis here. Therefore when you Fourier transform this you will get this as positive and this one as negative. So this will vary from different spin to spin depending upon their relaxation times. What is recovering along the Z axis is dictated by the spin lattice relaxation time T1. Therefore if you can monitor this as a function of tau look at the intensities of each one of these as a function of tau it naturally allows you to estimate the relaxation times of the individual spins. So mathematically of course we have seen this when you solve the block equations then you have or even when you are considering the recovery of the populations during the initial phases and this is the same equation which is put here. So the rate of change of the Z magnetization is proportional to the deviation from equilibrium and it is dictated by the time constant T1. Now in the inversion recovery experiment at time T is equal to 0 MZ is equal to minus M0 and MZ at time is equal to infinity is equal to M0. Now you solve this equation you get the solution for MZT as MZT is equal to M0 into 1 minus 2 e to the power minus T by T1. So at time T is equal to infinity you can see here that this term actually goes to 0 e to the minus infinity and therefore it is 0 and then MZ will be equal to M0 and that is what is here. Now what is infinity? What is time T is equal to infinity? This depends upon how much you want the magnetization to recover or come closer to the M0 value. Typically if you see that if this is factor of 5 if 5 times T1 for example then it will happen that almost 99.9% of the magnetization has recovered and it will be almost this will be equal to 99.9% of M0. So therefore you see after that it does not make any difference. So it reaches the plateau here at this point already most of the magnetization has recovered after that if you give more time it does not matter at all because it is going to reach the maximum value which has already reached. Now you get a curve if you plot this MZ as a function of T that is tau here is the same as a T here then you get the recovery going like this at T is equal to 0 what it is? This one is equal to 1 and therefore this is minus 2, 1 minus 2 is minus 1 therefore you have it equal to minus 1 here MZ is equal to minus M0 and then it will slowly recover to the equilibrium value at large values of tau. So this is an experimental spectrum to demonstrate this. So you have some particular molecule and you can you have done the experiments at various values of tau. So here it is 0.005 seconds this is the couple and this is 0.005 this thing is extra 0 here. So there and then it is 0.1 second, 0.3 second, 0.4 second I think I should remove this extra 0 this 0 is extra 0.005 then you have 0.05, 0.1, 0.3, 0.4, 0.75, 2 and 5 seconds. You can see by 5 seconds almost entire magnetization has recovered but not completely though. So it is still long way to go but nevertheless you can see the differences in the recovery rates of the individual spins. The one which is in the magenta color so this one here this one here is recovering faster than this one. This is a slow recovery this is a faster recovery but even though this has not reached the complete recovery still it has to still go longer for it to complete for the complete recovery. And if you plot this as a function of tau as we discussed earlier you get a curve which is like this. Now you fit this to that equation which we show earlier you get the value of t1. Now at particular value of t that is tau not recall a tau not mz is equal to 0 got it passes through the minimum when it is recovering from the negative z axis to the positive z axis it goes through the 0 at some point in time. So at that time point mz is equal to 0 and if you put that mz is equal to 0 then you get this equation here e to the minus tau not by t1 is equal to half and this tau not is called as the null that means at that point the magnetization is 0 and from here also we can get t1 is equal to tau not divided by ln2. So this is the quick way of finding out what is the approximate value of t1 for a given proton or for any given spin. So for accurate measurement of course you have to do it for various values of tau although a priori you do not know what value of tau you should use to get a tau not but quickly one does the variations and see where the rough tau null will be and you can estimate the value of t1 and accordingly then you choose the various tau values. You must choose the tau value so that you have the right point of points for you to fit your data to the equation given. Now so this equation allows a quick estimation of the T1 relaxation times. Alternatively you can also put the above equation in a different form m not minus mz is equal to 2 m not e to the minus t by t1. Now you take the logarithms on both sides so you take the logarithm the ln m not minus mz is equal to ln 2 m not minus t by t1. If you plot ln m not minus mz versus t you will get a linear equation linear curve that is a straight line. So the slope of that straight line yields you the relaxation time t1 and the intercept will give you ln 2 m not. So you will have to collect this data for various tau values and you can plot this ln m not minus mz against the tau values. Now measurement of T2 relaxation time, T2 relaxation time is now the transverse relaxation time. This is the FID, the FID is going like this. So this is the decaying and this decay is due to two factors as we discussed earlier. One is the Spinsken relaxation time T2 and the other one is a contribution from the field inhomogeneities. Field inhomogeneities in your sample cause different lines to process at slightly different frequencies and that inhomogeneity appears as the line width factor or it can treat it as a separate contribution to the T2. Therefore the measured relaxation rate here or the time constant what we get here is generally represented as T2 star, 1 by T2 star is equal to 1 by T2 which is a true relaxation time plus 1 by T2 which is a contribution from the field inhomogeneities. The field inhomogeneity is caused defacing and that leads to the defacing means it leads to the cancellation of the magnetizations and therefore it appears as intensity has gone down. So therefore this has to be removed if you want to measure the T2 precisely. So how do we do this? So this has to this is the observed decay what we will get from here. Here we make use of the spin echo. You remember the spin echo refocused with the field inhomogeneities. So this is the pulse sequence for the spin echo once again if we have seen that the echo amplitude is collected as is unaffected by the field inhomogeneities and therefore whatever decay what we will measure here in the echo amplitude as a function of this tau will be truly due to the spins in relaxation time T2. So what you do is your echo amplitude is collected as a function of tau that is the peak intensity in the NMR spectrum represents the echo amplitude for the individual spins. When you take this FID and Fourier terms from it you will get different lines right and the different lines the intensity of each line represents the echo amplitude for that particular frequency. Therefore we measure the peak intensity for all the spins and plot it as a function of tau then you will fit it to an equation of this type M echo is equal to M0 into the e to the minus T by T2. Notice the field inhomogeneity effects are gone and therefore here you have the exclusively the T2 coming up as a result of the spin echo amplitude changes as a function of tau this allows precise measurement of T2. However, there is one difficulty and that is during this period tau suppose the molecules shift from one place to another place and if there is a field inhomogeneity there. So it is you have it has to completely refocus whatever it has decayed here has to completely refocus here means the precessional frequency of the particular spin has to remain the same in this tau as well as in this tau. But if during this tau the sample is in the first tau if the molecule is here and in the second tau if it is here and in the fields are different in the two places then the refocusing will not happen refocusing from this precision of this during this tau will not be completely refocused here therefore the field inhomogeneity effects will not be removed so some of that contribution will be present therefore this diffusion has to be avoided. So therefore how do you achieve this and this is dependent obviously on the magnitude of of your tau value how much time you give the car Purcell made a modification to the pulse sequence they said okay let us not do this spin echo in a simple way like that as indicated but let us do it in the following way you give a 90 degree pulse tau 180 tau echo tau 180 tau echo and you go on doing this and at the after a certain number of such echoes you do the data collection. And now from here to here it is a echo right but what you do it is you keep this tau period extremely small very very small something like about 1 millisecond or sometimes even less than a millisecond. So if you keep that small then there is not enough time for this molecule to diffuse from one portion of the sample to another portion of the sample therefore complete refocusing will happen here and after the echo it will again decay the tau is kept the same 180 pulse tau and again an echo. So once again there is no time for the molecule to diffuse therefore the echo amplitude at each point is not modulated by the field inhomogeneity at all okay. So therefore then you say how do I get different time points if I want to measure the as a function of tau as in the previous case what we have to do is so we have to repeat this keeping small value of tau as required here repeat the number of echoes. Suppose for 10 milliseconds I have to do 10 echoes and after that 10 echoes you collect the data one data point for 20 milliseconds I have to do 20 echoes. So after 20 millisecond whatever echo comes I collect that data point. So like that 30 milliseconds 40 milliseconds 50 milliseconds if you have to do is simply increase the number of echoes here keeping this tau the same as a result the diffusion portion will not alter your data but you still get different time points for the measurement of your T2's. So the tau value is kept at the minimum to avoid diffusion effects and the number of echoes is varied so as to get different time points for the exponential fitting procedures for T2 estimation this yields a more reliable value of T2. This is the experimental demonstration of a particular sample. So you have here the normal 1D spectrum this is done with the 0.2 seconds 0.4 seconds 0.6 seconds 0.8 seconds and 1.0 seconds notice here these are all the total tau values it does not tell you how many echoes have been collected here that is not relevant for you we keep the tau value so small so that the diffusion effects are removed and the total number of echoes is adjusted so that I get a time point which is equal to 0.2 seconds here it is 0.4, 0.6, 0.8 and so on and so forth. And now you are able to fit this in intensities of all of these transitions for the individual ones you can fit this and you get a relaxation fitting like this and you get a T2 value. Obviously for the different transitions here you get different values of the T2's as you can see they are all decaying at different rates. So this is because of the different T2 values. So I think we will stop here so we have completed here the discussion on relaxation times and the measurement of the relaxation time how one can achieve good quality spectra and how one can remove the coupling and that was the main theme of the today's talk we will stop here.