 So, let us do a recap of the last lecture and this is the slide which kind of summarizes what we did in the last lecture. This is 2 dimensional NMR and this is based on the concept of segmentation of the time axis and the time axis is separated into 4 periods like this. The preparation period and then you have the evolution period T1 and the mixing period then you have the detection period T2. These are time variables which means you do lots of experiments, a series of experiments varying the values of T1 and the data is collected as a function of time during the detection period T2. Therefore, generate a 2 dimensional data matrix and of course in every case the preparation and the mixing they remain the same, they remain the same. So, you systematically increment the value of T1 and collect an FID. So, if you have to do signal averaging then you have to do for each value of the T1. So, you will have to start with a 0 value of T1, then you have 1 delta T1, then you have 2 delta T1, 3 delta T1 and so on and so forth. You collect a large number of FIDs, so you generate a 2 dimensional matrix of FIDs. So, there will be so many FIDs as many increments you use in this evolution period. And then we said that you have to do Fourier transformation, that you first do a Fourier transformation along the F2 dimension, then you do a Fourier transformation along the F1 dimension that means of the against the T1 period. The first this will be for the T2 time variable, other one will be for the T1 variable and this generates a 2 dimensional spectrum which is indicated like this. So, that is if you have a frequency here, a particular frequency which is indicated by this particular line here, so during the evolution period then during the mixing part of this magnetization of the spin is retained and part of the thing is transferred to another spin L, suppose I take with the k spin then I transfer part of it to the L spin and that appears as a cross peak here on this. And the whatever remains on the k itself which evolves during the period T2 appears as a peak here and this is the called the diagonal peak and this is the cross peak. Similarly, if I have a frequency omega L during the period T1 evolving and then during the mixing period for the same interaction there will be transferred to the k spin and therefore here you will have part of the magnetization on the k spin and part will be on the L spin. So they evolve with the respective frequencies, so the data you collect here will have 2 frequencies, so it generates after a 2 dimensional Fourier transformation a diagonal peak here and a cross peak here. So it produces a symmetrical spectrum like this. So now let us go into more details with regard to the mathematical operations which is important to understand the phenomena in greater detail because there is going to be more and more experiments coming in and this will depend upon what sort of preparations you do and what sort of mixings you do and depending upon you to generate the various kinds of data bodies and it is important to create a formalism or a general formal structure to analyze this kind of spectra. So let us first therefore look at 2 dimensional Fourier transformation, a 2 dimensional frequency spectrum which will represent by the frequencies f1 and f2 the 2 axes are represented as f1 and f2 will be generated from a 2 dimensional time domain data set which is represented as S, T1 and T2 by 2 dimensional Fourier transformation. So this is mathematically represented in this manner S, f1, f2 is equal to the Fourier transform of the 2 dimensional data body S, T1, T2 and now these 2 Fourier transformations are separately written here this one along the T1 axis, other one along the T2 axis. So this is the operators for the 2 and here you have the time domain data S, T1, T2. f1 and f2 represent Fourier transformation operators along the T1 and T2 dimensions respectively these have to be carried out independently. Let us look at that in somewhat more detail. So here you have the formula for this Fourier transformations S, f1, f2 is this is the same as before and this the 2 Fourier transformations are indicated here once more here. Now let us write this explicitly f1 is this integral minus infinity to infinity dT1 e to the minus i omega 1 T1 and f2 is this integral minus infinity to infinity dT2 e to the minus i omega 2 T2 these omega 1 and omega 2 are the Fourier transformation frequency variables along the T1 and the T2 dimensions and S, T1, T2 appears here of course at the end as the 2 dimensional data body. So these Fourier transformations integrals are calculated independently one after the other. Conversely, so if you want to get the time domain data here it is an inverse Fourier transform of the frequency domain spectrum S, f1, f2 this is the inverse Fourier transform f inverse and this again can be split into the 2 individual inverse Fourier transform f1 minus 1 f2 minus 1 of S, f1, f2. So put it in more explicit terms you have this is explicitly given as 1 by 4 pi square integral minus infinity to infinity df1 now this is the variable of the frequency spectrum and this is the Fourier transformation spectrum what in that when you do the forward Fourier transformation you have it as minus i omega 1. So here for the inverse Fourier transform you will have it as i omega 1 T1. So similarly here minus infinity to infinity df2 exponential i omega 2 T2 and here it is minus i omega 2 T2 on the time domain for forward Fourier transformation and here for the inverse transform you will have i omega 2 T2 and here you have the frequency domain spectrum S, f1, f2. Now generally the time domain function is a complex function. Let me try and explain this to you we have seen when we do data collection in the one dimensional experiment you have this frequency axis and if the magnetization is here and if this is processing along this during the detection period you have the detection here and here possible. So this component of the magnetization as it comes here it generates a cosine omega kT and this component produces the sine part and we write it as i sine omega kT for the k spin. If it is going with the frequency omega k then we will have the phi d which we are going to collect by collecting both these components will have these two terms cosine omega kT and plus i sine omega kT and to represent their orthogonal say this i is coming here. So therefore this is a complex signal. You do the same thing for T1 and T2 axis and therefore here in general this S, T1, T2 will be a complex function and we will write that as a real part and an imaginary part. So here you see this is the real part and this is the imaginary part. So in a two dimensional data matrix as well we will have a real part and also an imaginary part we will write in this two dimensional data body explicitly this is the real part SR T1, T2 plus ISI T1, T2 this is the imaginary part. So when you do a two dimensional Fourier transformation naturally you will also get a complex spectrum which will have a real part and also an imaginary part. So SF1, F2 will be written as SR F1, F2 plus ISI F1, F2 this is with regard to the spectrum although I use the same symbol here but notice this actually will be the discriminating factor. The variables here are F1, F2 variables here are T1, T2 therefore this is simply to indicate as the signal what you are going to measure in the two cases. Now this Fourier transformation in general can be written as a sum of two transformations this we have already seen this is FC minus IFS a general Fourier transformation is written as a sum of these two what are these? This is the cosine Fourier transform and this is the sine Fourier transform. So when we write the general variable as e to the minus i omega 1 T1 so this is you can write it as cosine omega 1 T1 minus ISI omega 1 T1 therefore this actually will have two components DT1 cosine omega 1 T1 and the second term will be DT1 sine omega 1 T1 and that will have an I factor. So therefore the first term will be the cosine will be the cosine transform and the second one will be the sine transform therefore we write here F the general Fourier transformation as FC minus IFS. Now we apply this formulation to both the domains. So this is for the first domain which is the T1 domain and this is for the T2 domain time axis so FC1 minus IFS1 and this is FC2 minus IFS2 and here we explicitly written this time domain signal as SR T1 T2 plus ISI T1 T2. So now what you do you do these operations explicitly independently all of them. So you will have this operating FC2 operating on this and on this likewise FS2 operating on this and on this similarly after that you get this FC1 operating on the result of those two and likewise FS1 operating on the results of those two. So you can combine these two together say FC1 FC2 SR T1 T2 and likewise you can say I multiply these and this I get plus I square that is minus FS1 FS2 SR T1 T2 and similarly you operate on the other side. So FC1 FC2 operating on ISI T1 T2 so and then you have FC1 minus IFS2. So individually you can calculate all of those. So the result of this what you get you get the real terms we do not have the I part and the imaginary terms as the frequency domain spectrum which has the I part. So which are the ones which gives the I part the real part FCC this, this and this. This use mere real part plus and multiplication of this, this and this so minus FSS SR T1 T2 this use the real part. Now FCS I T1 T2 that is so you have this one, this one and this one. So you get this minus I square right minus I square gives you plus 1. So therefore this, this and this also will be real. Therefore you have FCS SI T1 T2 which is operates on SI T1 T2 likewise you also have FSC that is this one, this one and this one. So because you must have one I from here and one I from here. So this one, this one and this one. So FSC SI T1 T2 all of these will be real and likewise if you see FCC operating on I. So this, this and this will give you FCC SI T1 T2 because this will have the I component here and therefore this is imaginary and similarly minus FSS that is this product and product with this. So this gives you plus I square then you have another I here and therefore you will get minus FSS SI T1 T2 and then you will have F minus FCS minus FCS that is this one and this one operating on this FCF, FC1 minus IFS2 operating on SR T1 T2 that gives you this term. So this is again imaginary because of this I and then you have minus FSC and SR T1 T2, SC that means it is this one, this one operating on SR T1 T2 this will have the minus I component therefore you get the minus sign. So FSC SR T1 T2. Now let us write these terms explicitly FCC SR T1 T2 is this integral minus infinity to infinity. So you have the cosine Fourier transform therefore this is DT1 cosine omega 1 T1 and this along the second dimension also it is a cosine Fourier transform therefore you have minus infinity to infinity DT2 cosine omega 2 T2 and this your time domain function SR T1 T2 and what about this FSS, FSS that is this one here so it is sine transform along both the dimensions therefore here you have DT1 sine omega 1 T1 and DT2 sine omega 2 T2 SR T1 T2. So this is also a real number. So likewise now if I to FCS so along the T1 dimension I have the cosine Fourier transform therefore this is DT1 cosine omega 1 T1 along the T2 dimension I have the sine Fourier transform therefore I have here DT2 sine omega 2 T2 SR T1 T2 and similarly this FSSC is along the T1 dimension I have sine transform and that is this one here DT1 sine omega 1 T1 along the T2 dimension I have the cosine transform therefore I have here DT2 cosine omega 2 T2 SR T1 T2. So I have written here all those explicitly for the real part of the frequency domain spectrum. Now you can do similar equations for the imaginary part of the Fourier transformations SI T1 T2 as well all of these you remember notice here are SR T1 T2 and so similarly you can write for the SI T1 T2 what terms that will come. Now the 5D is of course the transformations actually go from minus infinity to infinity but for time less than 0 there is no signal therefore this 5D will have zero signal therefore for T1 T2 less than 0 there is no signal transformations will have to be considered only for the range 0 less than T less than infinity. So that is so much the formalism for the Fourier transformation those are the definitions okay. Now let us look at at the end of this what sort of spectra we will get what sort of a peak shapes we will have what does the Fourier transformation yield. So you recall the Fourier transformations in the normal case 1 dimensional Fourier transformations we will have real and imaginary components and we will have different peak shapes. The peak shapes will be absorptive peak shapes and dispersive peak shapes so here also we can expect a similar thing. So what we will do is let us explicitly consider 2 particular transitions. Let us say we have an energy level diagram something like this various energy levels at various places and let us represent this energy levels with particular symbols let us call this energy level as T and this energy level as U and let us call this energy level as R and this energy level as S. There will be a transition from here to here and this will be represented by T U and there can be a transition from here to here this will be represented as R S transition. This is a T U transition and this is an R S transition. Now we assume that during the evolution period there is a particular transition there is a particular frequency T U and this T U we write it as omega T U and this transition we will therefore write it as omega R S. So let us go back and see what we are going to get considering a particular coordinate between levels R and S. Now R and S is taken in T 2 domain and T T U in the T 1 domain the time domain signal for this pair will be S R S T U T 1 T 2 and we have here well in fact omega T U is taken as in the T 1 dimension and omega R S is taken as T 2 dimension complex signal is written as e to the minus i omega T U T 1. Now this coherence this is the coherence right this is the coherence in the transverse plane this coherence decays and this decays with the transverse relaxation rates and these are the transverse relaxation rates. Lambda T U is a transverse relaxation rate for the transition T U and lambda R S is the transverse relaxation rate for the R S transition and so therefore this decay has to be included in the FID this is the free induction decay along the T 1 axis and this is the free induction decay along the T 2 axis. Now let us define this particular term Z R S T U is equal to S R S T U 0 0 this is the amplitude this is the amplitude for the after the Fourier transformation what we get for the frequency domain spectrum this is the amplitude and the frequency domain spectrum is now written as S R S T U omega 1 omega 2 and this is given by this expression and that actually comes from this particular integral as I can show you here this is 0 to infinity I have here e to the minus i omega T U T 1 into e to the minus lambda T U T 1 this is the decay part and then you have e to the minus i omega 1 T 1 dt 1. So pulling the terms so this will be equal to 0 to infinity here e to the minus i omega T U plus omega 1 T 1 into e to the minus lambda T U T 1 dt 1 so put this together once more so this is equal to 0 to infinity I have here e to the minus inside bracket I have i omega T U plus omega 1 plus lambda T U the whole thing is multiplied by T 1 and integral d T 1 so this is equal to I will write here this one is next step this is equal to e to the minus i omega T U plus omega 1 plus lambda T U T 1 divided by minus i omega T U plus omega 1 plus lambda T U and this whole thing is from 0 to infinity so if you want to expand this and this will be given as the first at the value of the infinity and then minus the value at 0. So this is explicitly writing it as e to the minus i omega T U plus omega 1 plus lambda T U T 1 divided by minus i omega T U plus omega 1 plus lambda T U at T 1 is equal to infinity minus the same expression e to the minus i omega T U plus omega 1 plus lambda T U divided by minus i omega T U plus omega 1 plus lambda T U and this is at T 1 is equal to 0 at T 1 is equal to infinity the term goes to 0 because e to the minus lambda T U T 1 is equal to 0 at T 1 is equal to 0 the numerator is equal to is equal to 1 therefore we get finally the integral is equal to 0 minus minus 1 upon i omega T U plus omega 1 plus lambda T U and therefore this is equal to 1 upon i omega T U plus omega 1 plus lambda T U. So this is the calculation of the integral and subsequently of course you can multiply this by lambda T U minus i omega T U plus omega 1 to the numerator as well as the denominator then you will get rid of the i part and then you will get expression in two different terms as indicated here. So we have here this is the first term this is the Fourier transformation with respect to the T 1 axis and this is the Fourier transformation with respect to the T 2 axis. Now here we have written here what is delta omega T U remember I wrote here explicitly omega 1 plus omega T U and delta omega RS is omega 2 plus omega RS delta omega RS is this omega 2 plus omega RS and this relaxation factor comes in here as well and this is the amplitude this is the amplitude of the Fourier transformation. So Z RS T U is now what you do you convert this into you multiply the denominator by and the numerators by this lambda T U minus i omega T U so therefore you get here omega delta omega T U square plus lambda T U square and you get two terms lambda T U divided by this minus i omega T U divided by this delta omega T U square plus lambda T U square. Similarly for the this term which is along the T 2 axis you get lambda RS divided by omega delta omega RS square plus lambda RS square minus i omega i delta omega RS divided by delta omega RS square plus lambda RS square. Now we recall from the discussions in the very first chapter that what are these line shapes. So here you are plotting as a function of the frequency if you plot this as a function of frequency what frequency omega T U these are the various frequencies which may be present in your spectrum and this omega 1 and omega 2 are the running variables of the Fourier transformation right. So for the various frequencies that are present so what do you get as a line shape in your spectrum. So if you plot this if you plot this as a function of frequency then you will see that this will actually generate a absorptive line shape and this is the same as what we have done earlier in the case of one dimensional Fourier transformation and this will generate a dispersive line shape because this is an i omega delta T U omega delta T U square plus lambda T U square and similarly this is a absorptive and dispersive components along the f 1 axis and now this is on the f 2 axis you have absorptive component and the dispersive component present. Now remember here we just put here as omega 1 omega 2 that is because we use the running variables omega 1 and omega 2 but in the frequency domain spectrum finally you may represent this as f 1 f 2 as well there is a running variable along the frequency axis. So now therefore now we have written use the symbols f 1 and f 2 here so I have here absorptive spectrum for the transition A T U for the coherence and a dispersive line shape for the along the f 1 axis for the same coherence T U and here I have ARS f 2 and minus ID RS f 2 this is the absorptive component and this is the dispersive component after the Fourier transformation. Now if I multiply this if you multiply this so what do I get A T U ARS and this will be real because there is no i component there and similarly this product these two terms product this will give me a real component again this is plus i square then therefore therefore it is minus DT U DRS and the cross terms this is minus IDT U and ARS this produces an imaginary term and this one against this will also produce an imaginary term therefore this satisfies what we said earlier that the frequency domain spectrum also has real part and an imaginary part. Now A T U ARS this one will now if you plot this it will have absorptive line shape along both dimensions and this will have a dispersive line shape along both dimensions now if you look at these two terms this will have mixed line shapes. The first term here DT U ARS this produces a dispersive line shape along f 1 and an absorptive line shape along f 2 and this term A T U DRS again produces a mixed phase and this is dispersive along f 2 and absorptive along f 1. So now if you look at the real overall real part this has both the absorptive and dispersive components so in principle if you collect the entire real part it will also have mixed line shapes so therefore now we have to choose what we want to have. So how do we choose it and what are the criteria how do we choose it okay now this is clear when you make a plot of this various line shapes so this is a line shape which is absorptive along both dimensions this is the first term that A T U ARS and here you have the dispersive line shape along the both the dimensions that is DT U and DRS that is this one here so if this both contribute to the real part of the spectrum if you collected both of these then of course you will have a mixture of the both the line shapes and so it will be mixed phase. Now here it is a more complicated situation that you have absorptive line shape along one axis and the dispersive line shape along the other axis. Now therefore now if I were to take a individually these line shapes and take their cross sections heights at various places your cross sections are plot the contours this peak will look like this and if we were to take the cross sections here at various levels we will have a peak shape which is like this it has the 0 at the center and it has lobes going out like this that is the characteristic of the dispersive line shapes right along both axis the dispersive line shape is of this type it has a 0 at the center and it has lobes on both the other sides and therefore this has a very broad signal and it is and the plus plus and minus indicate the positive and the negative signals and here it is a combination of the two and you have minus minus here and this is a very ugly line shape so typically we would like to have this so typically we would like to collect only the absorptive absorptive component of the line shape so that you have much better spectrum much better resolution in your two dimensional spectrum. So therefore we have to play around with the data acquisition and Fourier transformations so that in the end we generate a spectrum of this type which has absorptive line shapes along both the dimensions F1 and F2 okay so we stop here and quick recap that we have done today is the two dimensional Fourier transformations the theory of that one and we have seen how it generates various kinds of line shapes and how to optimize what we should do what we need is an absorptive line shape along both dimensions and we have to optimize our experiments so that we collect data in the appropriate manner and do a processing also in that manner so that we generate absorptive line shapes along both the frequency axis so we stop here and continue with the same in the future classes.