 Welcome to the third class on NMR spectroscopy. In the last class we looked at resonance absorption of energy by a spin system. Absorption of energy depends upon the RF, the power that we apply and also the spin lattice relaxation time T1. There is a factor which we derived called saturation factor and that is equal to 1 plus 2p T1. This is the saturation factor. We showed that in the steady state the population difference between the two levels is equal to N0 divided by 1 plus 2p T1 where N0 is the equilibrium population difference between the two states. So if 1 plus 2p T1 is extremely large then N dash will tend to 0 and then it will lead to saturation and there will be no signal observed. And in fact I said that was the reason why Götter-Miskanabel price he chose a sample which has such a large T1 value that even at very low powers the signal was getting saturated and he was not able to observe that. So therefore choice of the sample becomes important and here p is the transition probability which in turn is proportional to the power that we apply in the RF. We now go forward and ask are there any further restrictions in the absorption of energy. Some principles will have to be understood. In other words are there any selection rules for absorption of energy. Indeed there are some selection rules and this comes as a result of quantum mechanical treatment of the interaction between the spin system and the applied RF. This is called first order perturbation theory and the p what we used earlier can actually be calculated using this first order perturbation theory. We will not go into the details of this calculations. We will simply take this formula says p is equal to gamma square h1 square times a modulus square of a matrix element between two states m dash and m between which we are considering the transition of the spin system. Here ix is an angular momentum operator x component of the angular momentum. It comes here as ix because we have assumed that the RF is applied along the x axis. If it were applied along the y axis then Iy operator will come but that does not matter so far as the conclusions with regard to the p are concerned. Now to calculate this course one has to go into the theory of angular momentum operators. We will not go into the details of that one. We will simply take the result which says that this element which is called the matrix element here of the operator ix between the states m dash and m this vanishes unless the difference between m and m dash is equal to 1. What are m and m dash? m are azimuthal quantum numbers of the spin i. We said m takes values from minus i to plus i 2 i plus n values if the i for a given i. So if i is equal to half then we have m is equal to plus half and minus half these two states possible. So what is the implication of this? If I say m minus m dash modulus has to be 1. This implies only single quantum delta m is equal to plus minus n transitions are permitted. Let us look at this in little bit more illustrative manner. Let me write here i is equal to half. I have two states alpha and beta these for m is equal to half and this is m is equal to minus half. Suppose i is equal to 1 then I have three states here m is equal to minus 1, m is equal to 0 and m is equal to plus 1. If i is equal to let us say 3 by 2 how many states are there here? 2i plus 1 is 4 therefore there will be 4 states notice all of them are equally spaced. The energy levels are equally spaced and here m is equal to 3 by 2, m is equal to half, m is equal to minus half and m is equal to minus 3 by 2. What is the implication of the selection rule what we showed just now? That the transition between this is always allowed because here the delta m is equal to plus minus 1. This transition is allowed, this transition is allowed delta m is equal to plus minus 1 for this but this one is not allowed. So delta m is equal to plus minus 2 not allowed. Let us look here. So here there are 4 energy levels we can draw many transition possibilities here. This is m delta m is equal to 1 allowed, this is also allowed, this is also allowed all of these correspond to m is equal to delta m is equal to plus minus 1. But if you took here this is not allowed likewise this one is not allowed and similarly this one is also not allowed. So delta m is equal to plus minus 2, delta m is equal to plus minus 3 not allowed. These are the selection rules, these are the selection rules for RF induced transitions. So we say when delta m is equal to plus minus 1 these transitions are called as single quantum transitions, delta m is equal to plus minus 2 are called as double quantum transitions, delta m is equal to plus minus 3 are called triple quantum transitions. There is also of course delta m is equal to 0 which do not come across here in this kind of spin system they will come later. Those are called as 0 quantum transitions those also will not be allowed by RF. See the same thing is stated here transitions between m is equal to m 1 to minus 1 and vice versa is not allowed which is what I explained just now. Now if an absorption of energy happens the energy should correspond to the actual value of the energy difference. If this energy difference is delta E and if I call this is equal to H times nu where nu is the frequency of absorption then your nu has to be equal to delta E by H which is a single frequency. However it does not happen this way it does not happen always that absorption of energy occurs exactly one single frequency. It happens over a range of frequencies and these contribute to what is called as the line width. So you see here the absorption of energy spans a certain range of frequencies. Signal will have a shape like this and see this is the central frequency at which energy should have been absorbed. But there is of course absorption of energy if the frequency is slightly different is here or here or here and but of course the amount of energy absorbed will be different. So therefore it generates a line which has a width and if I take at the half height of this line then this is typically called as the line width. What is the reason for this line width? This again comes from quantum mechanics. There is what is called as uncertainty principle in quantum mechanics which says that delta E times delta T which refers to the uncertainties in the energy and the time and that is approximately equal to H cross. This is the intrinsic principle which quantum mechanics defines. So we will have to follow this. Delta E is the uncertainty in energy value of a state and delta T can be taken to be the lifetime of the spin in the state. A spin when it undergoes a transition from one state to another state obviously its lifetime in the particular state is changing. It has a well defined lifetime and therefore it undergoes a transition. So depending upon what is the lifetime of the state then your energy value is not precisely defined. If this is extremely high then this will be very small then the energy value will be very precisely defined. But if this is small then this energy value is not very precisely defined therefore there is a certain uncertainty in the energy which means absorption of energy can take place at multiple frequency values. So therefore in an ensemble of spins some will absorb energy at one particular frequency, some will absorb at a slightly different frequency and so on. And because of that you get a line width. What are the factors which influence these line widths? I indicated here for the two spins for the two level system the alpha and the beta states have a certain width in the energy. Have a certain width here for each of these states and because of this there will be certain width in the frequencies absorbed or or emitted. So therefore this leads to what this we called as the line width. What factors determine the line width? We will list here some of the important factors. The first thing is spontaneous emission. This is a very common mechanism for line widths in spectroscopy. This arises because of interaction of electromagnetic radiation RF is electromagnetic radiation with matter. This is a very general quantum mechanical phenomenon. It depends upon how much is the strength of your RF, how many photons are there in a particular volume of the sample, how does it interact with the matter. And depending upon that you have spontaneous emission happening and that means it limits the line way, limits the lifetime of the state and therefore the uncertainty in the energy and you get a line width contribution. However this is extremely weak in NMR. One could calculate this for the given kind of energies we use and the given kind of powers we use. This will roughly of the order of 10 to the power minus 20 minus 23 something like that and therefore we can simply ignore it. This is not a major contributor to the line width. Width due to spin lattice relaxation. We said earlier that there is a spin lattice relaxation which allows the spins to come back to equilibrium whenever there is a perturbation. So these transitions are always happening. These are lattice induced. There are interactions between the lattice spins and the spin system of your interest. Because of these the energies are fluctuating and that limits the lifetime of the state and this contributes to the width of the line. Similarly the spin-spin interactions within the given spin system there will be spin-spin interactions which also cause fluctuations in the energy values. It cause transitions between the states and this also results in the line width. Magnetic field in homogeneity effects. Different portions of the sample experience different fields hence absorb energy at different frequencies. We noticed in our spectrometer we have the sample in a tube which is put in the center of the magnet. We assume that all the spins in your sample see the same magnetic field. Now this depends upon how homogeneous is your field. If your magnetic field is not very homogeneous over the sample volume different spins in the sample experience different kinds of magnetic fields and therefore their processional frequencies are going to be different. This results in width of the line. This is magnetic field homogeneity effects. There are other interactions which get averaged out in solution state spectra and we do not wish to consider that here. Overall considering the line width the expression for P is modified in this manner. P is equal to 1 by 4 gamma square h1 square g nu. This is the factor which we introduced now to take care of the line widths. So that which means the transition probability is slightly modified depending upon what the frequency is and that shows up as the line width the line shape function. This is called g nu is called as the line shape function. Now let us return to Larmor Precision. This we have already seen Larmor Precision happens in the presence of the main magnetic field the nuclear spins process around the magnetic field direction h0 with the frequency which is dependent on the magnetic moment and the field. Now what happens when the RF is applied? How does it influence the Larmor Precision? You notice here if we are on resonance consider that we are on resonance or we are slightly away from the resonance what because of the line width also line width you already considered we have this h0 field applied along the z axis and we applied the RF applied along the x axis. You could have applied along the y axis also does not matter but we consider here the RF applied along the x axis and it has a magnetic field which is h1 remember we called our RF as 2 h1 cosine omega t and the rotating component is h1 e to the minus i omega t. So when we are considering this along the h1 along the x axis h1 field we have a h0 field here there will be an effective field which is a vector addition of this and this and the effective field will be here. Now the spin system sees this effective field spin system does not see only the h0 field but sees the h1 field as well therefore there is an effective field which is in this direction therefore the spins will have to possess around this effective field this is the cone the cone which was here earlier now gets tilted to go like this all the spins are now on the surface of the cone described by this circle. So what is the implication of this? The magnetization now gets tilted because effective field is tilted remember the magnetization is along the magnetic field axis over a long period of time. Now this will be tilted along this axis which means in this double rotary frame there will be certain component of magnetization on the x or the y axis. Of course there will be z axis component but there will be x and the y components we call this x and the y components as transverse magnetization. And what is the implication of transverse magnetization? We said earlier when the system is at equilibrium there is no transverse magnetization because all the transverse components cancel out because of the hypothesis of random phases. Now by implication if there is a nonzero transverse magnetization it will imply that some of the spins here have acquired a sort of a phase coherence. The ones which were all going randomly like this now they have come together in a particular fashion they come closer they come move together this we call it as a phase coherence and the spins move together then it would mean there is a phase coherence and to that extent the cancellation will not happen there will be a transverse component of the magnetization. So therefore generation of the transverse magnetization implies generation of a phase coherence among the spins in the ensemble. These are the important implications as we will see later. Therefore now we have x and the y component of the magnetization in the presence of the RF we also have z component of the magnetization in presence of the RF. So to describe this motion Bloch wrote a set of equations. He wrote a set of equations in a phenomenological manner for the precision of the spins considering the effective magnetic field that is given by this we have the H 0 which is the field along the z axis then you have the RF which is H 1 e to the minus i omega t putting it together we had the H effective then he wrote this equation dm by dt is equal to gamma times m cross H effective. This basically represents a force which is experienced by the magnetization in the presence of the field this is the torque basically it represents a torque when the magnetization interacts with the field. So he wrote this equation in a phenomenological manner it tells you that chain of change of magnetization is proportional to the cross product of magnetization and the H effective both are vectors here. Now m is a vector representing the total magnetization which has components mx, my, mz along the x, y, z directions and H effective consists of a static field H 0 which is along the z axis and the rotating field H 1 e to the minus i omega t and this is it is rotating it continuously generates x and y components oscillating x and y components. Put this in the more formal way putting into the various components here the same equation is recast in this manner d by dt is imx plus jmy plus kmz ijk are the unit vectors along the x, y and z axis respectively. So typically you write a vector in this manner taking into account the various components. Similarly you write for the effective field you write this iHx plus jhy plus khz and ijk again are the unit vectors along the x, y, z axis respectively. Now we take cross product of this without going into the explicit mathematics details there we will just write the end result. What is Hx? Hx is H1 cosine omega t omega naught t and Hy is minus H1 sin omega naught t because this is the processing field H1 e to the minus omega t means it is a field which is going around in the x, y plane. So it generates x and the y components which are oscillating in time. So therefore the x component varies as cosine omega naught t and the y component varies as sin omega naught t and they will taking into the consideration the sense of the rotation we have put here the minus sign. So Hx is H1 cosine omega naught t and Hy is minus H1 sin omega naught t and Hz is simply H naught. So expanding that previous equation you get explicit expression for the Mx, My and the Fz components of the magnetization. Notice here you have My H naught and Mz H1 sin omega naught t for the Mx component and for the My component you have Mz H1 cosine omega naught t minus Mx H naught and for the Z component you have minus Mx H1 sin omega naught t minus My H1 cosine omega naught t. It is interesting to see that where the x and the y components have the dependence of the H naught here because it leads to the processional. These are processional processional related and then the Mz component appears here. Whereas the Mz here depends on the x and the y components only on this side and the H1 sin omega naught t and H1 cosine omega naught t. These are the two components of the RF field along the x and the y axis respectively. Notice here the relaxation effects are not included. This is simply the RF is applied and the H0 is the field and there is no relaxation effect because the relaxation is happening. The relaxation happens to change the Z component of the magnetization and relaxation must also happen for the x and the y components because if the spins are having a certain phase coherence at a particular point in time this phase coherence will not last forever. It will slowly change as the system evolves and therefore there will be decay of this phase coherence therefore there has to be some kind of a time constant to characterize this. So therefore once again block modified this equations to include the relaxation effects. He wrote for dMx by dt introduced this term Mx by t2 likewise for the y component he introduced the term My by t2 and for the z component introduced the term Mz minus M0 divided by t1. This represents the deviation from the equilibrium value which is M0. This represents the transverse components of the magnetization created by the application of the RF and notice here we have a different relaxation time here. This is called t2. t2 is called as the transverse relaxation time and t1 is called as the longitude in the relaxation time because this has to do with the z component of the magnetization. These have to do with the x y components of the magnetization and therefore these are called as transverse relaxation time. Sometimes they are also called as thin relaxation time here t2 and this we will discuss later. Now to solve these equations it is a long effort. However we make some simplifications in this expressions by going into what is called as the rotating frame because we observe sitting on the RF. We make a measurement sitting on the RF with respect to the RF. Therefore if we go into the rotating frame and look at the magnetization components how they behave and that will give us the greater insight into the behavior of the spin system as time passes. If we make this transformation here we define a new axis x prime, y prime, z axis remains the same. We make a transformation here Mx is called as u cosine omega naught t minus v sine omega naught t. In My is minus u sine omega naught t minus v cosine omega naught t. This is the basically coordinate transformation. u and v are components of the magnetization in the rotating frame parallel and perpendicular to the RF direction. If the RF is applied along the x axis we are sitting on the RF and what is the component of the magnetization on the direction of the RF and that we call it as u and the one which is orthogonal to that we call it as v. So y prime component is v the x prime component is u dash and there is angle between them is omega naught t because it is with this frequency the RF is moving. The consequence of this is the expressions become very simple you get three expressions for u, v and z. This is some algebra of course we have skipped this algebra here those one can work it out. So at steady state most important thing for us to look at is the steady state solutions of this. At a steady state all the derivatives will have to vanish u by dt is equal to 0 dv by dt is equal to 0 and dmz by dt is also equal to 0. Once you put in that you get expressions for u, v and mz. These are all proportional to the frequencies here omega i is the frequency of precision and omega naught is the frequency of your RF. The u is no proportional to this expression as indicated proportional to the power amplitude of the RF and proportional to the t2 value and there is a factor here gamma square h1 square t1 t2 we will see what it means and this factor is common in all of these and mz represents the z magnetization how does it recover to equilibrium as a function of time. And in this case of course we are talking about the magnetization in the rotating frame. So therefore we have now got the solutions of the equations of block in the rotating frame these result in the description of the resonance phenomenon. We will go into the details of the line shapes how they allow us to measure the line shape in the in the next class and we take forward from there with regard to the relaxation phenomena. So we will stop here. Thank you very much. If you have any questions keep it and we will try and answer those.