 So, we have been discussing 2D spectra, two dimensional Fourier transformation. Let us do a quick recap of what we did in the last class. So, we said if your time domain data if you write it as t1 t2, this is of the form is proportional to e to the minus i omega t u plus lambda t u t1. This comes from the evolution during the t1 period and that during the detection we have also this e to the minus omega rs plus lambda rs t2. If this is the kind of a FID what is it we are going to get? The first term comes as a result of evolution during the t1 period and we see that this is the complex data which is collected e to the minus i omega t u t1 is a complex data and e to the minus lambda t u t1 represents the decay of the signal due to relaxation. Similarly, e to the minus i omega rs t2 is the complex signal collected during the t2 period and e to the minus lambda rs t2 is the decay of the signal due to the relaxation in the t2 period. Now, if we had to do a 2 dimensional Fourier transformation of this then what we get? We have 0 to infinity this integral here. So, e to the minus i omega t u plus omega 1, omega 1 is the Fourier transformation variable plus lambda t u t1. So, this is the omega 1 is the thing which is coming from the Fourier transformation variable and for the second Fourier transformation similarly, we will have e to the minus i omega rs plus omega 2 plus lambda rs t2. This i multiplies omega rs plus omega 2 and then you have d t2. So, let us consider the just the transformation of this one at the integral of this is given by minus 1 by i omega t u plus omega 1 plus lambda t u and the exponential that term remains as it is i omega t u plus omega 1 plus lambda t u t1 and this is taken the limit from 0 to infinity and you can write a similar expression for this as well. Now, what happens here if you see this one e to the minus lambda times t1 at the limit is infinity this actually decays to 0 and we will only have this one is oscillating function. So, therefore, we will only have contribution coming from when t1 is equal to 0. So, therefore, so this the first term will be proportional to you have 0 that first integral will be 0 plus 1 by i omega t u plus omega 1 plus lambda t u and this 0 is because that when the t1 is equal to infinity this goes to 0 and you will have similarly from the t2 transformation you will have 1 upon i omega rs plus omega 2 plus lambda rs. So, this is the product of these two will be the result of the two dimensional Fourier transformation. Now, so therefore, you can write this as the proportional to lambda t u will multiply bottom and top by the minus of this components here lambda t u minus i omega t u plus t1 you multiply by that you get here t u square plus lambda t u square minus i omega t u divided by delta omega t u square plus lambda t u square. So, this is the first term multiply this by a similar term coming from this. Now, this is now a and what is delta omega t u delta omega t u is this this is delta omega t u where omega 1 is the running variable. So, if you were to plot this as a function of frequency you will see that this is an absorptive signal lambda a t u and this is the dispersive signal d t u and similarly you will have an absorptive signal and a dispersive signal this is i and a dispersive signal d rs. So, now let us say you when you take the product. So, you will get a t u a rs minus d t u d rs minus i d t u a rs plus a t u d rs. So, you have absorptive line shape absorptive line shape and this is the whole thing is the real part and this whole thing is the imaginary part. So, we are going to collect the real data if you collect the real data therefore, you will see it is a mixture of a t u a rs minus d t u d rs and this results in a mixed line shape. So, when you have a complex signal as e to the minus i omega 1 t 1 omega t u t 1 and e to the minus i omega t u t 2 this is what we are going to get we get mixed line shapes and that is what we demonstrated by peak shapes and this is I am going to repeat that to you. Now so, here you have the line shapes indicated here this is the first one is the a t u a rs that is you have absorptive or absorptive line shape along both the frequency axis and this is pure dispersive line shapes along both the frequency axis because if you take a cross section here see it is a dispersive line shape whichever section you go with this is omega 1 and omega 2 directions are indicated here and if you take this one this is a mixture of this and this. So, that is a rs and a rs omega 1 and a rs omega 2 minus d rs omega 1 d rs omega 2 that is the real part the total real part is represented by this and this is a mixed phase or is also called as the phase twisted line shape and this sort of line shapes are not or desirable and we would like to have line shapes of this type and this is what we would like to achieve. So, now see how one can achieve. So, typically in order to do this we must analyze there were time dependent data in a little bit more formal way the evolution we have seen here in the earlier case we considered the evolution e to the minus i omega 1 t 1 omega t u t 1. So, such a kind of a thing this is actually a phase factor the phase of the signal is getting modulated by evolution during the t 1 e to the i omega t u t 1 and the detected signal is modulated by this term which is the evolution in the t 1 period and we call this as the phase modulation of the detected signal. So, and the evolution you can have another type of modulation where the evolution in t 1 modulates the amplitude of the detected signal. For example, if your detected signal is of this type cosine omega t u t 1 instead of this then this actually contributes to the amplitude of the detected signal f t 2 and this is called as amplitude modulation. So, we will see that if you have an amplitude modulated signal you will get a pure phase pure absorptive spectra. Now, several methods have been designed to obtain pure absorptive spectra and the most common is to perform real Fourier transformation with respect to t 1 and then it will also depend upon what sort of a data you will have. So, therefore what we will do we will consider the following here we consider an amplitude modulated signal. So, this is my amplitude modulation, this is the amplitude modulation cosine omega t u t 1 e to the minus lambda t u t 1 is the signal from the t 1 domain and the signal detected is of course, complex signal e to the minus omega r s t 2 e to the minus lambda r s t 2. Therefore, the detected signal its amplitude is modulated by evolution during the t 1 it is not the phase remember with respect to the previous case where it was e to the minus i omega t u t 1 which was a phase modulation and that resulted in mixed line shapes. So, now we do we take this sort of NFID and do a real cosine Fourier transformation of this data and that is given by this. So, you have double integral here s r s t u t 1 t 2 cosine omega 1 t 1 that is e to the minus i omega 2 t 2 d t 1 d t 2. So, this is the real Fourier transformation that is the cosine Fourier transformation and s r s t u t 1 t 2 is given by this. When you do this I have already shown you how to do the integration and things like that we will not do that again here. So, I will just write here the finally what we will get as result after the two-dimensional Fourier transformation. So, you will get here absorptive signal a t u omega 1 plus a t u minus omega 1 and this side here because it is a minus e to the i omega 2 t 2 we get here as before a r s omega 2 minus i d r s omega 2 this is a absorptive signal and this is a dispersive signal. So, on this side we have the purely absorptive signal and here I have a mixture of absorptive and dispersive components and they are in the real and the imaginary parts of the spectrum. So, therefore I now if you are detecting only the real part of the spectrum you are recording only the real part of the spectrum then of course what I will have here that is because these things can be stored separately on your computer the real and the imaginary components can be stored separately in the computer and you can pick up only the real part. So, if we pick the real part what I have here is a t u omega 1 plus a t u minus omega 1 and here a r s omega 2. So, therefore I will have a pure absorptive, but now you see I will have two frequencies one will be at omega t u and the other will be at minus omega t u and that is because of the minus sign here and this however is artificial we have actually only one frequency and that is omega t u, but I get two frequencies omega t u plus and minus omega t u. So, this we do not want to do this and though we will have to find methods to get rid of this. So, this can be avoided by doing quadrature detection along the t 1 axis. So, we have seen this in normal one dimensional NMR when you take a normal cosine FID and a Fourier transform it you get two signals at plus minus omega and in the same manner here also it is same thing is happening. So, therefore by doing quadrature detection now along the t 1 axis we can get rid of this and select only one of the frequencies that means we can discriminate between the positive and negative frequencies and cancel out one of those and keep only the one which is required for you. Now, but since you are not actually collecting the data in the t 1 axis how do we achieve this? We are actually collecting data only during the t 2 period how do we achieve this quadrature detection along this thing. Now, here there are many ways one can do it and that is along the F1 dimension there is a difficulty because data is not collected actually during the t 1 period. For long the F2 axis there is no problem we collect the data as before in the normal one dimensional NMR and we can do Fourier transformation, collect the two things two components separately and add and subtract as we saw in the case of normal Fourier transform NMR and we get the pure absorption signal and the discrimination of the positive and the negative frequencies. So, here we will have to use different strategies for achieving this and this is achieved by manipulating the way the increments in t 1 are given. So, they are adjusted along with the receiver phases in your experiment. There are three different methods to do that we are not going to go into the theory of this but we will only see the way these things are done. So, here the first method the quadrature detection method the first method here is called time proportional phase incrementation. When you are collecting 2D data we are doing number of experiments by increment the value of the t 1. So, the t 1 starts with a particular value at time t is equal to 0 we call that the first experiment and we will have that is called as the t 1 0 point and then after that we keep incrementing it systematically and we have suppose we are doing N experiments total. Total number of experiments if it is N and we divide this into groups of 4 experiments. So, in how we manipulate the increments in these individual 4 experiments. So, if I totally collecting N experiments then I divide this into N by 4 groups. So, therefore this is indicated by the index K. So, the index K here goes from 0, 1, 2, 3 up to N by 4 minus 1 where N is the total number of experiments. So, when K is equal to 0 I will have the first 4 experiments 1, 2, 3, 4 when I will have the K is equal to 1 then I will have the next 4 experiments 5, 6, 7, 8 and K is equal to 3 I will have the next 4 experiments 9, 10, 11, 12 and so on. So, we collect the data in that manner. So, we divide this into groups of 4 experiments all this total increments which you are doing in the T1 dimension they are divided into groups of 4. That is why it is always said this is 4 K plus 1. So, what when it is K is equal to 0 I will have this 1, 2, 3, 4 when K is equal to 1 what do I get I get 5, 6, 7, 8 when K is equal to 3 I will have here K is equal to 2 I will have here 9, 10, 11, 12 K is equal to 3 I will have 13, 14, 15, 16 and so on so forth. That is the way the total number of experiments are done. Now, so when we are doing this how the increments are given the increment is given in a way it is proportional to the experiment number itself. So, for the first experiment if your T1 value is a particular value starting value if it is for the time T is equal to 0 K is equal to 0 we will have a particular T1 value. So, this is ideally this is 0, this T1 0 is typically 0 however sometimes for practical considerations a small delay has to be given because of the hardware considerations and things like that practically a few microseconds may be given there. So, and then when you have that when K is equal to 0 of course these things are all you have 4 experiments here you have first experiment there is no increment then the second experiment there is an increment of delta there is a third experiment there is an increment of 2 delta for the fourth experiment there is an increment of 3 delta. So, you have as you are the 4 experiments the increments are going on in one as delta 2 delta and 3 delta. Now, along with that we also change the phase when the increments are being given we change the phase how are these increments this is that is why it is called as time proportional phase incrementation. The phase of incrementation of the phase of one we change the phase of the excitation pulse this is the pulse phase. The pulse phase the one which creates a transverse magnetization that is the first pulse during the preparation period you also create transverse magnetization and the pulse which creates a transverse magnetization we are talking about the phase of that pulse and that pulse the first experiment if it is x for the second experiment when there is increment is delta the phase is also increment by 90 degrees you get y and for the third experiment the phase goes to minus x and fourth experiment it goes to minus y and the receiver phase is always kept constant here x, x, x, x. So this increment this incrementation of the phase is represented by this terminology here time proportional phase incrementation as a time is getting incremented you also increment the phase of the pulse this is the excitation pulse. Now what is the value of delta? The value of delta here is 1 by 2 times SW1 SW1 is your total spectral width in the T1 dimension. So you have 2 times this is a normal mixed criterion for representing the frequencies that are present in your spectrum the last frequency that are present in your spectrum. So you always take this as mixed criterion delta is equal to 1 by 2 times SW1. So let me repeat here so for the purpose because otherwise this is a very important concept. So you do if you are doing n experiments here you group them into groups of 4 and in each group the increment is systematically done as 0, delta, 2 delta, 3 delta and the pulse phase is incremented along with that as x, y minus x minus y and the receiver phase is kept constant here it is not changed x, x, x, x and this results in discriminate it will allow you in fact what it will do is it will have only one kind of a sign in your frequency spectrum and that is the way it achieves the problem of positive negative frequencies in the spectrum. So your offset your carrier is still placed in the center of the spectrum you notice that one is to put a carrier at a particular portion in the spectrum the same thing is valid in the T1 and the T2 periods. So if you are doing quadrature detection in the T2 axis the same offset is available in the T1 axis as well. So if you put the carrier in the middle of the spectrum you have this strategy to shift the artificially shift the carrier to one end of the spectrum so that becomes a single channel detection in the T1 dimension and therefore the problem of positive negative frequency does not arise there and that is how you circumvent this problem of positive negative frequencies in the time proportional phase incrementation method. So we will not go into those theoretical details here we will simply take it that with this sort of a strategy we will get over the problem of this positive negative frequencies in the spectrum finally. This is the first method then you have the second method called as states method once again here also you group this number of experiments into groups of 4 so if you are doing n experiments you divide them by 4 you have that many experiments n by 4 minus 1 k is the index which runs through these groups. So what is the way this is done here now once again if k is equal to 0 this is the first group I will have here experiments 1 2 3 4 and what is the increment the T1 0 remains the same as before but now what is the increment here the increment here is 2 delta the this definition of delta being the same but the increment that is given here is 2 delta and for the second experiment the increment remains the same but the pulse phase is changed to y here if the pulse phase was x this is y and the receiver phase is not changed it remains x and x now for the third experiment okay k is equal to 0 what you do now is you have 1 2 the third experiment here for k is equal to 0 of course there is no this thing here the delta is not there so you have T1 0 x and y for the third experiment k is equal to 0 then I will have the increment 2 delta okay so let me correct it say clearly once more when k is equal to 0 there is in the increment is not due to delta is not there we only have the T1 0 we have T1 0 and this is 0 for the third experiment there is this first increment that increment is now 2 delta and the phase of the pulse is x and here the once again we do 2 experiments for the for the fourth experiment the increment is the same 2 delta but the phase of the pulse is shifted to y and receiver phase is the same so let me repeat in these 4 experiments you have the first 2 experiments for k is equal to 0 you have the T1 0 is the increment and which is usually 0 but it can be small as I mentioned before for the second for the third and the fourth experiments your increment is 2 delta T1 0 plus 2 delta and the phase of the pulse is changed from x to y and that is here as well okay so this is the difference between those from from TPPI notice here that in TPPI we had the increment as 1 by 2 times sw1 and here the increment is 2 delta that is 1 by sw1 so therefore the increment here is higher okay this phase remains the same okay now let us look at one more method which is called as states TPPI this is the combination of states and TPPI methods and it goes in the following manner so once again groups of 4 so we start with let us say k is equal to 0 I will have the experiments 1 2 3 4 and k is equal to 1 I will have the experiments 5 6 7 8 and so on so forth as as before so for the here for the first 2 experiments what we do this goes in the same manner as at the states that we have this the T1 0 and no increment here for the third and the fourth experiment I will have the increment 2 delta 2 delta here okay because k is equal to 0 I will have here 2 delta so 2 experiments with the same increment but now the phase is 1 first and the second experiments it is x y now for the third and the fourth experiments it is minus x minus y so therefore you see you have combined the phase incrementation here with as in the TPPI method okay so you have the incrementation here is 2 delta 2 delta first 2 experiment the same increment the third and the fourth experiments also the same increment but the phase is changed compared to what it was in the states method so if I go from k is equal to 1 then I will have here experiments number 5 6 7 8 okay so therefore accordingly you will have different experiment numbers okay so these other things remain the same okay so therefore the states TPPI and the states method are most commonly used and they eventually they achieve the same result of discrimination of the positive negative signals right so now having said this now we have to discuss the various kinds of 2D experiments we have generally covered the principles of obtaining 2D spectra how to record 2D spectra how to obtain pure phase experiments and this will allow us to discuss the different experiments and there are various kinds of 2 dimensional NMR experiments and we will discuss these things in the coming classes but briefly I would like to indicate here of course this is a topic which will go on for quite a long time and the known 2D NMR experiments can be broadly classified into three categories these are called resolution and separation experiments correlation experiments multiple quantum experiments and so on so forth okay so therefore this we will stop here and we are going to go into the individual experiments of these ones in greater detail one by one in the coming classes