 Welcome to the fourth lecture on this course on NMR spectroscopy. In the last class we looked at the solutions of the block equations which represent the motion of the spins in the presence of the RF field and also the main magnetic field. We also looked at the modifications of the block equations which had to be done or which block himself did including relaxation effects. We saw there are two kinds of relaxation times are involved. One is the so called spin lattice relaxation which is also called longitudinal relaxation which brings the populations back to equilibrium. In other words, it brings a z component of the magnetization back to equilibrium and this is represented by the symbol capital T1. The other relaxation time is called the transverse relaxation time and that is represented by T2 is also sometimes called spin relaxation time. The solutions of the block equations were obtained under steady state conditions going into the rotating frame of the RF. Why do we go into the rotating frame? Because we actually observe the nuclear spin systems motions in the presence of the RF and we sit to the RF and looked at it. The steady state solutions can be obtained in the rotating frame and we indicated that as U and V to represent the magnetization along the axis of the RF, U and V represents the orthogonal component to it. So we will continue from there now. The experimental setup can be designed to observe either the U component or the V component. Under conditions of low RF power such that gamma square h1 square T1 T2 is far far less than 1. This condition is called as no saturation condition. Under these conditions, this term can be neglected in the expressions for U and V and the expressions will simplify V is equal to minus M0 gamma h1 T2 divided by 1 plus T2 square omega i minus omega naught square and omega i is the frequency of the nuclear spin procession and omega naught is the RF frequency. U is given by M0 gamma h1 T2 square into omega i minus omega naught divided by 1 plus T2 square omega i minus omega naught whole square. These actually represent the line shapes that we observe. If you plot this component here as a function of the frequency omega naught because this is what we vary. The precessional frequency is fixed depending upon the nuclear spin and the magnetic field. The RF frequency is what we can vary. So as we vary the RF frequency omega naught, the V function looks like this. So this is an absorption line and we generally observe this component. The U component goes in this manner, it is so called dispersive line shape which has a positive component and a negative component and a long tail. Only one uses this for drawing the NMR spectra but sometimes this is also used in cases where you want to monitor the field drifts and things like that. Especially for locking purposes which we talked about earlier. For those purposes, these dispersive line shapes are used because it is easy to see the deviation from the resonance condition. You see here under the resonance condition this signal has the maximum and this signal has the minimum. This is 0 here. Any deviation in the magnetic field causes it shift along this line and therefore it produces a certain signal for the spectrometer to correct so that it brings a field back to the value it should be. But for the spectra, this is the most convenient one and this has a non-zero integral. If you take the integral of this line shape, it is non-zero whereas the integral of this line shape is 0. So having said so about the line shapes and how the NMR signals are observed, we now return to the question of relaxation to look at it in somewhat more detail. We go into the mechanistic aspects. What brings about the relaxation? We said there are two kinds of relaxation, spin lattice relaxation or longitude in a relaxation T1. What it has to do with the population changes? We said it has to do with the magnetization along the Z axis. Obviously it has to do with the population changes between the levels, the transitions between levels. If there are transitions, the population will change and therefore the spin system will relax. The second one is the transverse relaxation which is the spin spin relaxation. This has to do with the transverse components of the magnetization. Remember in the block equations, we had this T2 appearing in the equations for MX and MY and this was appearing for the equation for MZ. The transverse relaxation has to do with the loss of phase coherence between the spins in the transverse plane. We said if there is no phase coherence in the transverse plane, there is a distribution of spins all along in the transverse plane. Therefore the total cancellation of all the components, therefore at equilibrium there is no transverse magnetization. But if somehow you are able to bring in a certain coherence between the spins which are moving in the transverse plane, it results in a certain net MX or the MY component. That means there is a phase coherence between the spins. The appearance of the transverse magnetization implies phase coherence between the spins. Therefore anything that causes loss of this phase coherence contributes to the transverse relaxation. Now how does this happen? How do the transitions occur and how does the phase coherence is lost? How is the phase coherence lost? Let us look at the ensemble of the spins. We have lots of spins, we have a particular spin and there are many other spins which are in the lattice and all of them are processing around the magnetic field. If this is the direction of the magnetic field, all of them are processing in the magnetic field in the form of a cone like this here indicated. They can be present anywhere on the surface of the cone as indicated by the hypothesis of random phases which we described earlier. And since each of them is a magnetic moment, it is a magnetic dipole, it produces its own field and this is represented by the flux lines given here. The flux lines go in this manner and all of these will have such kinds of flux lines. Now if the spin precesses around this cone, this flux lines also will move. So they will also move, this is the cone here moving, the flux lines are there around it. And with that all the flux lines also will move, which means and the flux lines meaning what this is the field. So the field has therefore a z component and a transverse component and as the flux lines move, the transverse components also keep moving, they keep oscillating. So therefore the motion of the spins in the solution causes fluctuations in the flux lines of the individual magnetic dipoles. Now when there are many magnetic dipoles like this, there will be an interaction between them, each one of them interacts with the other. Suppose this nuclear spin comes close here to this one, then of course this nuclear spin sees the flux lines of this and it will experience a different magnetic field. Depending upon whether it is here or here or anywhere here, it sees a different magnetic field. So the z component of the magnetic field keeps fluctuating depending upon whether this spin is here or here or somewhere and likewise the transverse component of the magnetic field of this will also influence the transverse component of the spin. So here there is a fluctuating magnetic field in the transverse component. So the interaction between these two spins is responsible for what we will call as is the process of the relaxation. We will see how that happens. Just for sake of magnitudes, if we consider a distance of r here, notice this one is actually mu, this is not m here. So this is a magnetic moment. The field due to magnetic moment at a distance r typically is in certain range plus minus 2 mu by r to the power q. Mu is the magnetic moment and if the two are unit dipole, if we take as a unit magnetic moment here, the field created by this magnetic moment mu at this particular site is 2 mu dot r divided by r cube. So when there is another magnetic moment here, this will be interaction between the two to mu 1 dot mu 2, that will be the interaction. So for a proton, suppose this is a proton, the field created by it at a distance of one angstrom is approximately 57 gauss. This is not a small number, this is a huge number. So if this spin keeps moving or this one keeps moving, the field created at the site of this will keep on varying in all the three dimensions, all the x axis, y axis and the z axis. Imagine an ensemble which is all fluctuating here in the solution. So there is a continuous fluctuation in the magnetic fields at the site of this particular nucleus and this magnetic field has components along the z axis and the x axis and the y axis. We call this field as a H local field. The H local field varies along the z axis, the x axis and the y axis. So if I went to plot the transverse component of this H local field, the x and the y components, you can take any one of those, does not matter and you plot it as a function of time and what this time represents? This represents the motions of the spins in the solution. Notice that if the tumbling spin, if the molecule has a vector like this, if it orients like this, the magnetic moment still remains like this only, magnetic moment does not become like that, it still remains like that only. So therefore, the fluctuating magnetic field, these cones are always in the same direction for every particular spin, alpha state or like this, beta state or like this. Now so if I went to plot the local field, the transverse magnetization created at a particular site by the fluctuating from positions of the other spins, then the field seen here fluctuates like this in a random manner. So what is the implication of this? Does it have any frequency components? In fact, it has the frequency components. Any fluctuating thing can be described by certain frequencies. If there is a time dependent fluctuation, it will have a frequency component. It may have one frequency component, multiple frequency components. So typically one can write this as fluctuation as a summation of multiple frequencies, sum of frequencies omega i with respect to amplitudes a i. Let me just demonstrate this to you here. This is one particular frequency here. Now to this I add another frequency and this is the sum of the two frequencies. I add a third frequency, this is the sum of the three frequencies. I add a fourth frequency, this is the sum of four frequencies, fifth frequency, sum of five, sixth one, sum of six and so on. See what is the consequence, what we have seen here? As we are adding regular frequencies with different amplitudes a i's, I am generating a time dependent function which looks quite random. So the noise which I showed earlier as the h of t is the summation of all these cosine frequencies here, they all look like this. Now recall back the RF field which we had applied is actually a function like this. We wrote it as 2h1 cosine omega naught t. So that means here there are many RF frequencies present in this random fluctuations of the field components. The h local fluctuation creates a multiple RF fields here. I want to write it in a more formal way in a more general manner. To analyze that we call it as the summation of the cosine term which we said is basically what we call as the Fourier transform. The Fourier transform explicitly is written in this manner. This is the time dependent fluctuation in the x-y plane and then you write this Fourier transform equation here and this becomes the frequency dependent part. So this is how you analyze what all frequencies are present in h of t. So there will be multiple frequencies here. So and these are also be very randomly fluctuating quite frequent, positive and negative frequencies of can be present and then if I define a term which is called j omega which is an ensemble average of all these frequency components here we take complex conjugate here. Why do I take this? Because this itself the average will be 0. Because the average of h of t itself can be 0 and this will be 0. But the square of this will not be 0. So why do we need to take the square? Because when you look at the RF the 2h1 cosine omega t when you look at the RF the h1 was an amplitude of the particular frequency and the square of that represents the RF power. So likewise here in the summation what we had ai cosine omega it in the previous equation ai represents the amplitude and here the ai represents the amplitude the square of this represents the power. So what is the implication of that? The square of the amplitude of the frequencies present here it represents the power distribution in the fluctuating magnetic fields. So this is therefore called as spectral density function. Spectral density means it is a distribution of power at different frequencies in the fluctuating ensemble. Without going into the mathematical details of how this actual expression can be derived we will take the result for this course. That will be quite an involved calculation we will take that result here and here is an ensemble average of the fluctuating fields in the xy plane and we have this function called 2 tau c divided by 1 plus omega square tau c square. Tau c is a time constant called as correlation time. This actually indicates the correlation between the spins when they are moving in the ensemble. Is there any phase relationship between the spins when they are moving? Is there any sort of a relationship or a memory of the phase of the spins when they are moving in the solution in the ensemble? So this correlation time actually represents the kind of a correlation that may be present between two spins in the ensemble. In other words this correlation time characterizes the fluctuations. How much is the phase coherence between the spins? How much is the time between the two molecules hitting each other colliding with each other? Every time there is a collision there can be change in the phase memory of the spin system and therefore there can be all these kinds of contributions to the fluctuations in the magnetic fields. And that is characterized by a particular time constant called as tau c and this obviously has a units of time and then if you want to plot this function as a function of frequency it will represent a distribution like this. Notice and this is my RF frequency here and it can keep changing. You can have various kinds of powers depending upon what sort of a frequency you want to consider. If you have to consider frequency here it has this much power, if you have a frequency here it has this much power, frequency here so and so forth. So now I consider three particular situations. When my spectrometer frequency or the RF frequency is represented by omega naught, omega naught tau c much much larger than 1 that means this tells you also how much is the correlation time. There can be a situation when omega naught tau c is much much less than 1 or omega naught tau c is approximately equal to 1. In this plot only one half of this curve is plotted compared to the previous figure. Here you had this half and this half on both sides of omega naught 0 frequency or whatever and only this half is plotted. Now before we actually go into this have a look at this equation. What is this tells? We have to take an integral of j omega or d omega it turns out that this is a constant. In other words the total area under the curve of the spectral density function is constant because the integral represents the area under the curve. The total area under the curve is a constant which means the total power available in the ensemble from the magnetic dipoles fluctuating is constant. Now if you look at this curve omega naught tau c is far greater than 1 which means compared to the spectrometer frequency tau c is such that this product is much larger than 1 then the function behaves like this. That means at certain power level this almost approach is 0. All the power is concentrated in this much frequency distribution only in this much area of frequencies. On the other hand if omega naught tau c is far far less than 1 that means this tau c is much much less very rapid motions very fast motions then this condition will be satisfied. Now you see the power is distributed over a whole range of frequencies all the way from and then eventually much higher frequency that goes down to 0. For the intermediate case the power lies in between these two. What is the consequence of this? So if I have to consider a transition in my spin system suppose the transition has a particular frequency here then of course this much power is available for the transition to occur for the spin system. If this condition is satisfied if this condition is satisfied then this much power is available. If this condition is satisfied then at the same frequency this much power is available. So therefore depending upon the amount of power you apply of course you will have a different signal intensity. On the other hand if you have a situation something like this here if you are at this particular frequency when you are looking for omega i's omega i frequency right when you are looking at omega i which is at somewhere here how much power is available at this frequency that is if you have a condition of this type satisfied then it is this much power available if this condition is satisfied this much power is available but if this condition is available 0 power is available. So therefore depending upon how much is the correlation time in your sample there can be variations in the efficacy of transition at your respective frequencies. So therefore let us consider a switch so if I have more power then of course I have a transition occurring then the relaxation will happen faster and you have relaxation time changing. The same thing is now indicated by this particular curve. If I were to plot the T1 T1 as a relaxation time as I said which causes transitions at the given frequencies now if I have omega naught tau c much much less than 1 then as a function of tau c if I plot the T1 continuously decreases here as the tau c increases and then it reaches a value minimum value and then it starts increasing from here onwards for omega naught tau c greater than 1 it starts increasing all this has to do with the power that is available at a particular frequency where you are looking at. This can be understood from the previous curve here. Suppose I have a frequency omega i which is here where I am looking for transitions. If this condition is satisfied I have practically no power here so therefore it will not cause a transition at all if it does not cause a transition then the T1 relaxation time will be very very high because it takes a long time for the system to come back to equilibrium. If I have this condition satisfied in my spin system then this much power is available at that particular frequency so accordingly the relaxation time will be determined depending upon how much power is available it will determine how much time will be required for the system to come back to equilibrium. If we look at this situation suppose this is the situation when we have omega naught tau c is equal to 1 then you have the maximum power available at that particular frequency that means it will cause the fastest relaxation of the spin system that means the T1 time will be the smallest. In other words this indicates how the T1 time goes through a minimum and this will be the minimum time and this will be the maximum time and this will be somewhere in between. So that is the indicator here if you see here you have a minimum time here and then it comes down to a minimum and then it starts increasing again. You go on either side of tau c of this point either side you go the T1 increases T1 therefore goes through a minimum here whether it is omega naught tau c much less than 1 much greater than 1 omega naught tau c much smaller than 1 it keeps on increasing. Now omega naught also can change if I change the omega naught what will be the implication of that the omega naught tau c is equal to 1 will be reached at a different value of tau c there if I increase my omega naught the spectrometer frequency I make it from 100 megahertz to 200 megahertz then the tau c will be accordingly reduced so that this condition will be satisfied at a lower correlation time likewise if I take it 300 megahertz it will come down even further it will be satisfied here. So therefore what is the slow motion at one frequency can become fast motion at another frequency and vice versa. For example if I am considering a situation here then this will be in the fast motion limit for the black curve but in the slow motion limit in the blue curve which is a 300 megahertz. Therefore this explains why the T1 relaxation time depends on the spectrometer frequency it varies with the spectrometer frequency and one has to measure relaxation times at different frequencies to understand about the system better. Now we talked about the relaxation time behaviors how much time it will take and what the how does it depend upon the spectrometer frequencies. We can go into little bit more details with regard to the mechanisms now we said it is the dipole-dipole interaction which causes the relaxation. So let us continue and look at how there is the transitions are brought about you remember the expression for the transition probability when we consider the RF induced transitions. We said here the RF is applied along the x-axis the interaction between the RF and the magnetic dipole causes this transition probability and we had this term here mu dot h1 or it is simply ix1 was there this thing had come here the perturbation that is causing the transition is the dipole-dipole interaction between two magnetic dipoles. So therefore here the perturbation will be represented by this dot product i1 dot i2 where i1 and i2 represent the spins spin operators for the two spins that are interacting. If you want to expand this in more explicit form this will have the ix1 ix2 iy1 iy2 iz1 iz2 and these states are the two states in the same manner as we considered earlier we are considering transitions between m prime and m with some operator algebra of angular momentum theory we will realize that this particular term is the one which is responsible for the transitions this causes the transition between the states m prime and m this will not contribute to the transition between the states m and m but it is these two terms which contribute to the transition between the two states therefore if we ignore this we will have only this part and now we do little bit more algebra with the angular momentum operators this will be non-zero for delta m is equal to 0, 1 and 2 this is in contrast to what we said for RF induced transition where we had only one RF and the other one was a static field here both are RF kind of things and therefore here we have transitions possible even when the two states have delta m is equal to 0 or 1 or 2 earlier in the case of RF induced transition we had the so called selection rule delta m is equal to plus minus 1. So here also there are selection rules but different from what is RF induced transition here you can have 0 plus minus 1 and plus minus 2 we will face this situation when we consider multiple spins or spins with higher i values and so on so forth. So this implies that 0 and double quantum transitions will be caused by lattice fluctuations. Now going into the block equations a little bit more detail once more now with respect to the H local field H local is the one which is created by the fluctuating magnetic moments at magnetic dipoles. We write that explicitly for the spin systems here consider this magnetic moment here and it is rotating on the z axis this we will all write it in the z axis presses around the z axis the H x, H y and H z are the components of the H local. Notice therefore there is a precession here and any fluctuation here will cause a precessional frequency change. Any fluctuation will here cause a change in the frequency again and that leads to the loss of phase coherence of the components of this mu in the transverse plane. Therefore if you look at here the Mx and the M y it can be mu x this is the summation of all the mu's. Now you see here you have the H y component appearing here, H z component appearing here for this also H z is appearing here, H x is appearing here for M z you have only H x and H y. Among all these three H z is the one which is a slowly varying component because there is no frequency here it changes randomly as the system as the spin systems move closer or further and things like that but the change in the frequency shows up in the H x and the H y components. Therefore H x and H y are rapidly varying functions where a H z is the slowly varying function. So therefore the rapidly varying components H x and H y contribute to both T 1 and T 2 this represents the T 2 relaxation this represents the T 1 relaxation. Mx, M y appear in both all the three components therefore they contribute to both the T 1 and T 2 whereas the T 1 relaxation is contributed by the H x and the H y components only. Now without going into the much greater details of this calculations I will give you here the final equations with regard to the relaxation rates. Now the relaxation rates are represented by R 1 and R 2 R 1 is the inverse of T 1 and R 2 is the inverse of T 2 this is proportional for the spin systems I value the nuclear spin this is the inter proton distance or inter nuclear distance R and this represents the spectral density distribution here. Notice here this has two terms tau c upon 1 plus omega 0 square tau c square and this term has 4 tau c divided by 1 plus 4 omega 0 square tau c square. Notice here this one is 2 omega 0 tau c whole square. Now what is 2 omega 0? 2 omega 0 actually represents the double quantum transition. Omega 0 represents a single quantum transition 2 times omega 0 represents the double quantum transition therefore for the R 1 the double quantum transition probability also appear we are not going to the details of this theory but we will just take it for granted this will be discussed later on when we talk about advanced topics later and here it is a single quantum transition and for the T 2 we have here a frequency dependent term omega 0 tau c square and a frequency independent term in other words this is like a 0 frequency. So a slow motions like 0 frequency if you treat they contribute to the T 2 rapid motions contribute to the T 1 slow motions contribute to the T 2 slow motions of course will also contribute to the T 2 I mean the rapid motions also will contribute to the T 2 in addition the slow motions and the rapid motions contribute to the T 2 or the R 2 value. So if you were to plot this R 1 and R 2 R 1 as a function of tau c we can just simply show it in this manner this is tau c if we plot here R 1 the R 1 goes like this and then it decreases and the R 2 goes like this and it keeps going this is R 2 and this is R 1 why does this R 1 go down further that follows from the previous equation. If you look at this equation here with omega 0 tau c square is far far less than 1 here if you look at this one here is far far less than 1 and again ignore this condition and then it will be proportional to the tau c. So as tau c increases it will go on R 1 goes on increasing but if this condition is such that omega 0 tau c is far far larger than 1 then it almost approaches 0. So this is consistent with our previous equation it goes through a minimum T 1 goes through a minimum and then goes up which means R 1 goes through a maximum and then goes down whereas here this implies that R 2 continues to increase regardless of the omega 0 tau c condition because this is a term which is independent of omega 0 tau c and that is indicated here with increasing tau c R 1 increases initially and then approaches 0 then of course it is in between there is a maximum there is a maximum which will come as indicated you in the curve and then approaches 0 for omega 0 tau c far far less greater than 1 and while R 2 increases monotonically this typically happens when the motions are extremely slow when the motions are very slow there will be 3 contributions to it one is the fluctuating the magnetic power that is available will be different and if the extremely slow then it is different frequencies superimposed different lines superimposed resulting in the line broadening and that is called inhomogeneous line broadening that contributes to the large line widths and this typically is observed in case of solid state NMR and also in systems of very large molecular weights okay. So the final topic though when the experiment is optimized with no saturation the maximum obtainable signal intensity is proportional to N into I plus 1 mu omega 0 square notice it is not just proportional to the omega 0 the field it is proportional to the square of the magnetic field or the square of the frequency under the conditions of T 1 and T 2 being nearly the same this is seen from the curve which I indicated there for low tau c's for a low tau c's T 1 and T 2 curves are actually overlapping and in terms of the field it is given by this essentially translation from here your mu cube dot H naught square mu cube into H naught square for the I values are indicated here for the difference. So it is proportional to the cube of the magnetic moment of a nuclear spin and it is proportional to the square of the magnetic field applied. However this does not include the noise which is contributing typically one does not see the square dependence once is a 3 by 2 dependence because the noise goes as a square root and therefore you typically see a 3 by 2 dependence on the magnetic field. And here is some calculations to show what are the relative sensitivities of the different nuclei. So the proton has the maximum sensitivity it is 100% the relative resonance frequency if it is 100 then the relative sensitivity of this one is taken as 1 with the respect to this we put down here the relative sensitivities of different nuclei. Euterium is just 0096, carbon 13 is 0159 and this is N15 is 0.001, F19 is 0.326, P31 is 0.0663. So after proton the one which is most sensitive is fluorine 19 and after that you have the other nuclei typically we come across protons, the carbons and the nitrogen 15s. Notice here the number of spins is treated to be is considered to be the same. Natural abundance is not included in this this is the relative sensitivity simply based on the mu, mu to the power cube you remember and this is one the garamagnetic ratio of proton and carbon 13 is 1 by 4. So the 4 to the power cube is 64 therefore this is nearly 1 by 64 of this. Now if you want to include the natural abundance of this then you got what is called absolute sensitivity, absolute sensitivity is proportional to the product of natural abundance and relative sensitivity. Therefore this goes down quite substantially and that is why often one says the NMR technique while it is so elegant it is also a very insensitive technique and you require large amounts of samples for recording quality NMR spectra. So with that we come to a close and this chapter we close here this concludes the first chapter of NMR spectroscopy which has to do with the basic concepts and then we go along analyzing the details of the NMR spectra in the other other lectures. Thank you.