 going to start a new chapter, Fourier transform NMR spectroscopy. This has to do with how we actually record the NMR spectrum. You have already seen in the previous lectures some details about the characteristics of the NMR spectra. There are parameters called as chemical shift coupling constants. For example, if you have a molecule such as this which is ethanol one of the simplest molecules on which actually the chemical shift was discovered by Dr. Dharmati when he was in Felix Bloch's lab. If you look at this molecule this has 3 protons here, 2 protons here and 1 proton here. All these 3 protons have different absorption frequencies. The CH3 group absorbs signal at this chemical shift. The CH2 group this absorbs frequency at energy at this chemical shift, this frequency and the OH group appears somewhere here. Notice there is also fine structure in each of these lines. This happens because of the spin-spin couplings between the protons of the different types. This has been discussed previously. Now the question is how do we measure this? How do we know? What are the frequencies present in your NMR spectrum? In the early days the method was called slow passage experiment. That means you have a particular magnetic field, you sweep the frequency, reach the resonance condition, here you get the signal here, reach the resonance condition, here you get the signal here. Then you reach the resonance condition, here you keep the signal here. So this is keeping the field constant, you sweep the frequency. Alternatively what you can also do is keep the frequency constant and sweep the field. So one by one you will reach the resonance condition for the individual lines. It was more convenient to sweep the field rather than sweeping the frequency in those days because the magnets are electromagnets and you can simply change the current through the electromagnet and that will change the field. Now how fast you can sweep the field? You recall your discussion with regard to the relaxation time. Whenever we change the field, there will be changes in the energy levels, there will be changes in the populations. Therefore the population changes have to follow the field changes in order that you represent your equilibrium situation appropriately. The system has to be in equilibrium. You notice here that the population changes follow this kind of an equation where the population difference here, n between two levels for a two level system is n naught into 1 minus 2 exponential e to minus t by t1, where t1 is called the spin lattice relaxation time. Therefore if the t1's are of the order of seconds then you cannot be sweeping this field too fast otherwise you will not be reaching an equilibrium situation at every point you try to measure the resonance. So typically if I have to sweep the field at the rate of 1 hertz per second then the time per sweep of a spectrum of with 1000 hertz, what is the meaning of 1000 hertz here? If I am looking at the proton spectrum, typically the proton spectrum has a range of 10 ppm. So the 10 ppm on 100 megahertz spectrometer corresponds to 1000 hertz. If it were a 500 megahertz it will be 5000 hertz. I am considering just now the 100 megahertz which is 10 1000 hertz. If we do it at this rate this will take me 16 minutes approximately for one sweep. Now when you actually look at the spectrum, the signal to noise ratio, what is signal to noise ratio? That is if we were to take the peak height here, take the signal and the measure the peak height and then you take somewhere here where there is no signal but there is only noise. When you have the noise then you have a peak to peak separation between the noise. So you have noise like this. So you take the peak to peak separation from here to here that is the separation for the noise. So if you have such a kind of a thing then you measure the peak height above a mean noise level and measure the peak to peak separation in the noise and take the ratio of that if you multiply by a factor of 2.5 this happens because of the stochastic reasons and that is called as your signal to noise. And typically this is very small because the as I mentioned in the previous classes the NMR spectrum spectral sensitivity is actually quite low. Now if I want to increase the sensitivity signal to noise ratio so that I can actually measure it with confidence what should I do? This is the technique called as signal averaging. What you do is you collect sweep the frequency several times sweep the spectrum several times and add them together and this is an additive process therefore once you add them the signal adds as the number of scans whereas the noise adds by square root of the number of scans. Therefore the signal to noise ratio goes as the square root of the number of sweeps co-edit. So for example if I want to enhance my signal to noise ratio by a factor of 10 then I have to sweep 100 times to get this enhancement by a factor of 10 which means if it were 16 minutes per scan then I will need 1600 minutes and this is a huge amount of time. It puts a lot of pressure on your spectrometer conditions this spectrometer has to be stable for that long period of time the current should not vary the field should not vary and then you have the stability of the samples. The samples have to be stable for that long period. If your signal to noise is very poor alternatively one would then require is use very high concentrations of samples but high concentration of sample may require that the sample is properly soluble in yours in your solution. If the solubility is very low then high concentrations cannot be achieved then it is such a situation it is difficult to observe low abundance nuclear such as carbon 13 and nitrogen 15. Carbon 13 you remember is 1.1 percent natural abundance N15 is 0.37 percent natural abundance such nuclear it becomes very difficult to observe. So what do we do? So there is a new strategy here instead of sweeping the field as we did earlier can we do the following keep the field constant apply all the frequencies in one go generate so many frequencies is in one go so that one or the other of the frequency matches the resonance condition and you will have excitation of all the spins in one go. This is termed as pulse excitation how is it achieved? You have the RF frequency going let us say this is the time here this is the time axis and the RF is a sine wave the cosine wave whatever you want to call it it is going like this continuously but what we do now is you apply this RF only for a short period of time you just start here and cut it off here. So therefore you apply the RF only for a short period and we call this as the pulse you applied for a period tau and what is the consequence of this? The consequence of this now your time axis is time profile of your RF looks like this you have no RF here then suddenly you applied an RF for certain time and then you put it off again then you again 0 here. So your time profile is like this as indicated under the red what does it imply? How do I get this sort of a profile now? This amounts to applying a large number of frequencies in one go and what is the distribution of these frequencies and what are the amplitudes of these frequencies which you apply when you generate in this manner and that is indicated by this figure it shows that this is the main frequency omega naught and in addition to this it generates a larger number of other frequencies with different amplitudes some are higher than omega naught some are smaller than omega naught and at this time this at this frequency it is excitation is 0 this is 0 then it becomes negative here and so on. So you generate a larger number of frequencies the consequence of such a kind of a treatment and all of them have different amplitudes. In other words if I were to take individually all of these frequencies with the respective amplitudes all of these and superimpose all of them co-add all of them then you will generate a time domain profile which is like this this is called as the Fourier transform of the pulse. So there is a relationship between the this time domain and this frequency domain and that relationship is the Fourier transform. So by doing this applying for a short period of time you generated a larger number of other frequencies. Now how long this should be if you look here the tau was the period for which this pulse was applied the radio frequency was applied and this comes to 0 the excitation comes to 0 at 1 by tau. So suppose tau is 1 microsecond then this will be like 10 to the power 6 1 by tau is 10 to the power 6. So compared to this we have 10 to the power 6 hertz that means 1 megahertz range is covered here 1 megahertz range is covered here 1 megahertz range is covered here that is the kind of an excitation you have although they have different frequencies different amplitudes at different frequency values. Now what do we do with those many frequencies we do not need so many frequencies we have another condition that is that if I want to have a proper excitation of my spin system I must have the same power for all of these. Why do I say this let us look at recall the discussion from the first chapter you remember the transition probability induced by the RF was written as P is equal to 1 by 4 gamma square h1 square h1 is the amplitude of the RF that is applied and gamma is the gyromagnetic ratio and the transition probability P is proportional to the square of the square of the amplitude of the RF. Now if I want to excite all the spins similarly then I must have the same power applied to all of them. So how do I achieve that? So what do I do I select only a small portion here this small portion which has a nearly similar amplitudes and therefore their transition probability induced by at these frequencies will be the same then I can compare the intensities of my signals in the NMR spectrum. So this is a requirement therefore I have to do what is called as filtering I filter the frequency response to keep only these many frequencies and throw the rest. Now you notice this is quite sufficient because if this is megahertz this will be of the of kilohertz roughly of few kilohertz. Now few kilohertz is few thousand hertz and remember in 100 megahertz the proton spectrum was 1000 hertz. So even if you go to 5 kilohertz 5000 hertz I will be able to excite this with the reasonable uniformity if I filter out this kind of things. So depending upon what value of tau I choose I will have a distribution of powers and accordingly I can choose how much should be the spectral width. Now what is the effect of this kind of an RF on the spin system? Let us go back and look at the nuclear precision in the RF rotating frame. You remember the nuclei are processing around the H0 field in this manner with the frequency omega i and omega naught is the RF frequency that we have applied. Now if I were to sit on this omega naught RF frequency and look at the spins then the spins will be processing with the frequency omega i r which is equal to omega i minus omega naught. Now we consider the H1 field as well. If I am sitting on this omega naught then the H1 field is stationary in that one. Now I have two frequencies one is the omega i minus omega naught which is the rotation frequency omega naught is the precessional frequency, omega i is the precessional frequency of the spins and omega naught is the RF frequency. Now if I want to convert this frequency into magnetic field then what I do? I will divide this frequency by gamma omega i minus omega naught by gamma then this will be the effective field along the Z axis. So in the rotating frame I have the effective field which is H i r in the absence of the H1 just in the rotating frame. If I have the H1 as well now we have a new effective field which will be the vector addition of H i r and H1 and that will be this H effective. The magnetization of precess kills away from this Z axis and it will have to orient itself with respect to the H effective and it will start processing around this H effective in this manner. So what is the consequence of this? We calculate here what is an effective field in a more quantitative terms here. The H i r in the rotating frame effective field is given by omega i minus omega naught whole square plus gamma H1 square to the power half divided by gamma. This is basically a vector addition of the H i r and the H1 field vector addition of these two. Now this will be oriented with respect to the Z axis and that is given by a certain angle tan theta which is H i r divided by H1 and now the precessional frequency here what will be the precessional frequency? The precessional frequency will be depend on this H effective, gamma times that H effective. So that is given by this here the precessional frequency in the rotating frame now will be minus gamma H i r effective. Now suppose I choose gamma H1 is far far larger than omega i minus omega naught how do I do this? Because H1 is in my control H1 is the amplitude of the RF which I am applying therefore I apply with a very high power if I choose a very high amplitude here. Suppose this is much larger than this omega i minus omega naught that means this term is much larger than this term for the whole range of frequencies that are present in your spectrum then I can ignore this compared to this therefore then H i r effective simply becomes equal to H1. Then rotating frame frequency will be minus gamma H1 the precessional frequency will be minus gamma H1 for every spin because this has been this is much smaller compared to this and we have one frequency of precision and that is given by minus gamma H1 because H1 is not the effective field. What is the consequence of this? Remember my equilibrium magnetization was M naught in the absence of any other perturbation for a two spin system or whatever we have the M naught which is along the z axis. When I apply the RF suppose I apply that along the y axis then I said the H1 is here the magnetization will have to orient itself with respect to this axis because now this is the field this is the effective field. So, the magnetization will eventually have to rotate here how does it do it? It goes out like this moves like this and eventually after a long time it will come back and orient itself with respect to the H1 field parallel to the H1 field. This will take a long time because this depends upon the relaxation phenomena how fast the system can move and this will eventually come down to reach this gamma axis. This is the effect of the RF notice here that the RF has produced transverse magnetization. Earlier the magnetization was along the z axis the magnetization will get reoriented itself with respect to the RF field. Now suppose I do not give you enough time for the system to move from here to here. I stop somewhere in between I stop before it while it is moving I suddenly stop the RF because I know I am applying a pulse I suddenly stop the RF then it would stop there the magnetization whatever it has it has moved it will stop there and that is indicated by this how much angle it has covered when applied for a short time tau. If the precision frequency is omega r omega r times tau will be the angle covered by the magnetization so we call that as a rotation angle. This rotation angle is omega r tau which is equal to minus gamma H1 tau. If this is arranged in such a way that theta is equal to 90 degrees then the magnetization would simply move from here to here. So this is the effect of the RF pulse. So let us continue looking at the effect of the RF on the magnetization. We said that the RF rotates the magnetization away from the z axis and the rotation is given by this equation here theta is equal to omega r tau where omega r is the precessional frequency in the rotating frame and tau is the pulse width that is for which the time for which we apply the RF and this is given by minus gamma H1 tau and if this is 90 degrees theta is equal to 90 degrees then I get a 90 degree pulse. See this is a 90 degree pulse we can look at it in the particular sense of rotation we maintain the same sort of rotation while describing the various pulses. Now if theta is equal to 180 degrees I produce put the magnetization along the minus z axis this is also called as inversion. If theta is equal to 270 then the magnetization goes here this is along the minus x axis. Remember the RF was applied along the y axis 90 degree pulse rotated the magnetization here 180 degree rotated here and the 270 degrees takes all the way like this and brings it down to this axis. So in principle you can apply a pulse of any angle you do not need to be only 90, 180 or 270 you can also apply a 45 degree pulse or a 10 degree pulse or a 20 degree pulse or whatever and you simply have to adjust the time tau for this and that is the one parameter which typically one has to adjust when you record your NMR spectrum. Now what happens after the pulse after the pulse there is no perturbation the magnet assume that you have put a magnetization along the x axis here you applied a 90 degree pulse and the magnetization has come along the x axis. Now it will have to recover back to equilibrium it has to go back to the z axis because now there is no RF therefore the only field which is the present is the H0 field which is along the z axis therefore the magnetization has to go back here this will now this is now in the transverse plane so it has to recover as it recovers it starts precessing here and precesses and starts recovering along the z axis. Both T2 relaxation and the T1 relaxation will be operative here so the magnetization takes a spiral pathway here it goes like this this this and then eventually it will come back along the z axis so this is indicated here and eventually it will align itself along the z axis. As they are rotating magnetizations here there are components along the x and the y axis these are fluctuating magnetization components therefore if we are to look at these components since they are fluctuating they actually induce a certain kind of a voltage if you put a receiver here it will in a coil it will induce a voltage likewise here so therefore rotating magnetization components will induce a signal in your detectors if you keep them along the x and the y axis and that constitutes your signal. The precessing magnetization components induce a signal in the detectors and if there are many many frequency components which are present each one of them will introduce a signal in your detectors and this will be the y component detector and this will be the x component detector you will have therefore the signal total signal which is induced in your detectors will if I want to represent as G of t this is a n cosine omega nt plus summation b n sin omega nt and this is essentially a Fourier series the put it in the kind of a continuous form this will be G of t is equal to 1 by 2 pi integral f omega e to the i omega t d omega therefore this is the Fourier transform relationship between this time domain function and the frequency domain function. However there is one thing we have not included here that is the relaxation the frequency components are precessing in the x y plane at the same time as I said they are relaxing therefore the transverse components which are in the along the x or the y axis they decay with the relaxation time t 2 therefore you will have to multiply this G of t by e to the minus t by t 2 this is f of t f of t is equal to G of t e to the minus t by t 2 and f of t is called as the free induction decay why it is called free induction decay because is result of free precession there is no perturbation it is induction because rotating magnetizations induce voltage in your detectors and it is decay because the signal decays because of the transverse relaxation. Now therefore the f of t and the f of omega are the Fourier phase if I do a Fourier transformation along this I got call it as f t gives me a frequency domain spectrum I can also do a inverse Fourier term which is called as I of t often one does this as well from the frequency domain spectrum you go to the time domain signal and that is called as inverse f t. So this is indicated in a more explicit manner here you have the FID which is a superposition of various frequencies in this case considering only one and you have the exponential decay factor this is e to the minus t by t 2 we remember we multiplied by this factor and that is to take care of the relaxation. So when I multiply by this factor then I generate an FID which is like this otherwise it would have been simply going like this when I multiply by the e to the minus t by t 2 because of the decay it goes like this now if I do a Fourier transform of this time domain function I get this signal in frequency domain spectrum this is a time axis in seconds now this is the frequency axis if I represent this as Hertz. So now if I have earlier I showed you one frequency decaying now I have another frequency which is a black curve I have two frequencies here I have a red frequency and a black frequency both are present at the same time and therefore of course what we will observe will be the superposition of this and if you do a Fourier transformation of this I get two frequencies here this is how you unravel the frequencies that are present in your FID. I explicitly show you here this is how the FID will look like if you have two lines separated by a certain number if I have four lines I will have the FID which is looking like this looking like this of course you cannot figure out what the frequencies that are present here what if you have 100 lines or 1000 lines then you have several lines which are present then your FID may look like this now if you Fourier transform this one then you will get your frequency domain spectrum notice what we have got here how long is this this will be obviously determined by your t 2 time exponential minus t by t 2 and t 2 is of the order of what that is of the order of seconds it will never go beyond a few seconds in most cases it will be few hundreds of milliseconds and notice here this in this particular case a spectrum of some large molecule this goes up to about by about 400 milliseconds it has already come down to 0 nearly close to 0. So, depending upon the t 2 the length of the FID is determined this was the major breakthrough why is it a breakthrough now you collected the entire spectrum in few hundreds of milliseconds how much time did it take for you to excite it you applied the pulse for about a few microseconds maybe 1 microsecond and immediately after that you collect the signal for about a few hundred milliseconds and you do Fourier transformation which is done off the line you can do it on your computer and then you generate a frequency domain spectrum which is complete representation of your NMR sample and that is the sensitivity enhancement you achieve here is a comparison of the Fourier transform spectrum with the CW spectrum here is a CW spectrum that is the slow passage spectrum recorded in one scan in 500 seconds you scan through the whole spectral region and it took 500 seconds now this is an experiment which is signal averaged 500 times because you took only one second to record one FID now you add 500 FIDs that is equivalent to doing 500 scans you add 500 FIDs and then you take a Fourier transform then you get a spectrum which is looking like this look at the signal to noise enhancement that is a major breakthrough in fact initially many people did not believe this was achieved by Richard Ernst and W Anderson and people initially did not believe this and therefore obviously did not go to high high high profile journal so to say I mean I remember Richard Ernst telling me that it was rejected three times first of the so-called high profile journals never mind but eventually it was published in review of scientific instruments and you can see the signal to noise enhancement per unit time and therefore what it entails it entails that you can use low concentration of the samples low concentration of the samples can be used the concentrations are often limited by solubility availability viscosity changes at high concentration etc and nuclei with low natural abundance such as carbon 13 nitrogen 15 can be studied signal averaging which is a must in these cases can be easily performed you can add as many signals as you want before you Fourier transform collect the FIDs as many FIDs as you want and then you do Fourier transformation in the end once we will see it in more detail as you look at the theorems of Fourier transforms now due to enhanced speed of data acquisition short lived species which are half lives of seconds only can be readily studied in the earlier case you could not have to study this kind of species if you do it would take 16 20 minutes for you to record one spectrum then short lived species short lived species would have decayed or disintegrated by then and you would never get a proper spectrum of such kind of molecules the dynamic processes can be investigated and data can be collected as a function of time dynamic processes meaning there are various kinds of exchange phenomena that happen in molecules or there are rotational processes that happen in your molecules and all of these can be studied because you can actually record data as a function of time these have opened up enormous applications of NMR in various areas of chemistry and biology these of course will be discussed at a later stage. Now here is an example this is the carbon 13 NMR spectrum at natural abundance recorded by the Fourier transform NMR here is the spectrum which is of this molecule and this has a coupling constants indicated here there are carbons there are 3 carbons 3 types of carbons and these are split because of the couplings between the various carbons protons carbon proton coupling and this is the triplet pattern because of the carbon couplings a carbon proton couplings there is a process called as a spin decoupling which you can do if you we will discuss that at a later stages and if you do that all the couplings will be removed and then we will get only the chemical shift sides of the 3 carbons of the carbon that are present here all these carbons appear at one place and these 2 carbons appear at one place this one carbon appears at one place how do we figure this out you figure this out by looking at the intensities of these signals now these intensities are proportional to the number of nuclei presented that particular chemical shift these 4 carbons are equivalent and therefore it has an important information about the chemical structure of the molecule these 2 carbons are equivalent and whereas this carbon is a single carbon and therefore this intensity is twice the intensity this intensity is 4 times this intensity therefore this is enormous information about the structure of your molecule which you are studying likewise if you took this molecule you have the CH3 group here and the carbonyl here and you have the CH2 here and the CH3 here so there are different kinds of carbons now all of these carbons are now separated they appear at different chemical shifts notice in the carbon frequency it goes all the way from 0 to 200 ppm the carbonyl appears at 180 ppm close to that area between 160 to 180 this chemical shift is very characteristic of carbonyl carbons and these are aliphatic carbons there are the CH3, CH2 and the CH3 all these 3 carbons are separated here and each one of them is one this is appearing with a shorter intensity because of some relaxation attenuations which happen when you are not optimizing for the intensity measurements in your carbon spectrum how much time you give between 2 scans will determine how much the intensities can be calibrated so nonetheless this gives you the number of carbons present in your molecule and obviously they these ones that can be considered to have nearly similar intensities and these one one one each and that of course is indicating to you on the basis of the chemical shift which carbon is what so thus even at natural abundance you are able to acquire data which is of great value for structure characterization this is how you entered the realm of chemistry in a big way you could record carbon 13 spectra which will tell you how many carbons are present in your molecule then look at the coupling constants then you can say what kind of a pattern it is so for example this one will tell you that this is the proton this is a CH2 carbon because it has a coupling to 2 protons and therefore it appears as a triplet this of course you would see in the we have seen in your analysis of spectra if you have a CH2 group it produces a triplet in 1 is to 2 is to 1 ratio that is the great application of Fourier transform NMR there is one other important consequence of Fourier transform NMR that is often not talked about but this is the most important in some sense recall again what is Fourier transform NMR how you perform the experiment you apply an RF pulse here RF pulse which is an excitation pulse which is applied for a short period of time tau which may be of the order of 1 microsecond 2 microseconds or 5 microseconds or whatever and then you are collecting the FID here when there is no RF when there is no perturbation if this is representing excitation this is representing detection along the time axis you have separated the excitation and the detection this is called a segmentation of time axis excitation and detection are separated in time and this has important implications for variety of further developments in multidimensional NMR spectroscopy or various kinds of pulse techniques which have been developed subsequently these ones we will describe in greater detail with that I think we will stop here thank you