 So, we continue from the description of the application of the RF and we said there are two components rotating magnetic fields one going in the direction which is the same as the precessional motion and the other one is going in the direction which is opposed to the precessional motion and we consider that particular component which is in the same direction. Now what we do, so to understand this interaction we go into the rotating frame. Which rotating frame we go? Rotating frame of RF. In other words, so I had this RF which was going in this direction we consider this one, this is H1 e to the minus i omega t. The spins were like this, they were going like this and this is also going in the same direction. So, therefore now if suppose I sit on the RF, rotating frame of RF means I sit on the RF and look at the spins. So, when omega is different from omega naught, this frequency is omega naught, when omega is not the same as omega naught, what will happen? What do I see if I sit on omega and look at omega naught? What will be the precessional frequency? So, omega naught precessional frequency if I call it as omega r in the rotating frame is equal to omega naught minus omega. So, it is like going in the two different trains, one of them is going with the one frequency other one is going with another speed. So, it is the difference between the two speeds is what we see that they are going with this particular speed. So, similarly if I sit on the RF and look at the spins it will appear to me that the spins are moving with the frequency which is omega naught minus omega. And if I want to consider this as a magnetic field just as we said omega naught is equal to minus gamma H naught. So, omega r will also be equal to minus gamma another field which I call it as HR and this will be the field in the rotating frame. Now, as I said we can keep on changing this omega, omega is in our control. So, we can keep changing omega slowly change omega and see you match the resonance condition. At the resonance condition what happens? At resonance omega naught is equal to omega. So, what will happen? The spin is stationary, the spin is stationary. If the spin is stationary omega naught is equal to omega there is no processional motion and therefore the field is 0 HR is 0. So, this will imply HR is equal to 0. So, then what is the field that it sees? The spin sees 0 field along the z axis and they also see H 1 in the transverse plane. So, if I draw it here so I have the spin here and I see an H 1 field here. So, this spin will interact with this H 1. So, it has to orient itself with respect to the H 1. So, spins we will try to orient with respect to H 1. What does that mean? What does that mean? So, they will have to change the redirection I will change the colour here. They will have to move towards the H 1 because H 1 is in the transverse plane. It moves in the transverse plane. So, therefore the magnetization moves in the transverse plane. The net magnetization away from z axis. This is the consequence of the interaction between the RF and the fields. So, this implies we have created XY magnetization which was 0. Which was 0 in the beginning at equilibrium. Once we apply the RF field we have created XY magnetization. So, let me draw that here. So, initial magnetization was here and when I apply the RF I created some magnetization in the transverse plane. If I continuously apply this RF entire thing will have to orient itself with respect to the RF. But this will take a long time and that will take a long time but we do not apply it for that long. So, what happens is therefore the magnetization will start processing in the transverse plane. So, this transverse magnetization it is not an equilibrium situation. Transverse magnetization is not an equilibrium situation. Because equilibrium situation was magnetization was only along the z axis. So, this is how the absorption of energy happens and we create transverse magnetization. And if we look at this transverse magnetization, now transverse magnetization will start processing in the transverse plane. This will start rotating in the transverse plane. Transverse meaning XY plane and therefore and if you have a detector in this transverse plane this rotating magnetization will induce a signal. The rotating magnetization induce a signal. Induce a signal meaning what? It is an electrical signal. It is a voltage. So, this is the voltage in the detector kept in the transverse plane. This is my NMR signal. So, this is the NMR signal. So, you see we saw how an NMR signal is observed. Quantum mechanical picture told you that if you supply energy it will lead to absorption of energy. What does absorption of energy mean? Absorption of energy means the magnetization which was along the z axis in the absence of the RF will tilt away from the z axis and creates some transverse magnetization. This transverse magnetization will have to recover back to equilibrium and while doing so it is processing along the transverse in the transverse plane. It rotates in the transverse plane and this induces a voltage into the detector. Because always the rotating magnetization induces an electrical voltage. This is a common law of physics here. So, therefore we get the NMR signal. So, that is how we get signal. But now this signal will also decay. The signal will also decay. Why? Why? Because the magnetization has to recover to the z axis and therefore there is a decay here. This decay is characterized by a time constant known as transverse relaxation time and we often represent this as T2. So, this is also called as spin-spin relaxation time. So, therefore we have introduced these two concepts here that one is the spin lattice relaxation time and the other one is the spin-spin relaxation time. Spin lattice relaxation time is also called as the longitudinal relaxation time. So, the T1 is also called as longitudinal relaxation time. Why? Because it has to do with the populations. The populations dictate the z magnetization because this is related to z magnetization and therefore it is called as the longitudinal relaxation time and the transverse relaxation time has to do with the spins in the transverse plane. Now, there is one more concept which we need to understand here that is transverse magnetization represents what? It represents a kind of a phase coherence between the spins. Transverse magnetization which may represent as Mx or My represents phase coherence among the spins. You remember we talked about the hypothesis of random phases. The phase has to do with the angle a spin magnetization component makes along with the x axis or the y axis. So, that was a phase. So, in the equilibrium situation this because of the randomness of the orientations of the spins there was no correlation between the phases of the individual spins and therefore there was no transverse magnetization. The phases were random and therefore there was no transverse magnetization. In the case of absorption of energy it creates a transverse magnetization it creates a phase coherence among the spins. Therefore T2 also represents loss of phase coherence. So, these are important concepts which will be very necessary as we go along in the later stages. Look at one more concept that is which we are suppose we are not at resonance but off resonance situation and that means that omega naught is not equal to omega. So, therefore omega r is not equal to 0. So, how do we represent this? So, we have let us write here this is the hr corresponding to the omega r and this is my z axis and let us say this is my x axis and I have here the h1 which is going with the frequency omega. Of course, it is not stationary now it is going with the frequency omega. So, this is processing here and hr is the field along the z axis in the rotating frame because we are sitting on the RF and looking at it. And now what is the field effective field? The effective field is a vector addition suppose I have this vector here like this hr is represented like this then I will have here an addition vector addition which is this and this is called as h effective. The effective magnetic field the h effective is the effective magnetic field in the rotating frame. So, I can represent it in this manner. So, I have here h effective and this is my this is hr this is h1. Of course, this effective magnetic field also keeps rotating because the h1 is rotating therefore effective magnetic field also will keeps rotating along the in the in the transverse plane. It keeps changing depending upon the magnitude of h1 it will have a different orientation. So, long as the omega is rotating frame frequency is non-zero it will keep rotating but its orientation can change depending upon the magnitude of h1 and hr. So, this is the off resonance condition and depending upon that we have different creations of the transverse plane the magnetization in the transverse plane. So, now we have this brings us to basics of the NMR spectrometer. So, NMR spectrometer consists of a magnetic field. So, we have the magnetic field. So, let us say so we have this north source magnetic field and we have the sample here and then we have a coil around which is the RF use the different color. So, I have a coil here which is the RF RF coil which gives out the radio frequency energy the magnetic field is there so then I will have a connected with that is a detector which detects the magnetization. So, this is the basic component of the NMR spectrometer there are so many other things which are which are important here of course the technicalities we will not go into that. So, now let us look at kinetics of resonance absorption kinetics of resonance absorption. So, what does that mean absorption of energy we have put in the RF right RF is causing absorption of energy. So, we consider the two states here the RF induces transition from here to here but notice it can also induce a transition from here to here as well the radio frequency can cause upward downward transitions with equal probability. So, we represent that probability as P this is the RF induced transition probability. So, now we can write rate equations for the transfer of energy in both ways when it absorbs energy let us say I have this population here N alpha and I have the population here N beta. So, if I write here dN alpha by dt what is it dependent on it is dependent on the transition probability and the populations. So, this is the first order kinetics then the chemical kinetics you already have studied this. So, minus P N alpha because it causes depletion of the populations plus P N beta when it causes the transition from beta state to alpha state it increases the population there. Similarly, I can write for N beta by dt this is P N alpha minus P N beta. So, therefore, this can be combined into one equation. So, we will not derive these all of these things there but I will just give you the basics there and we will go into the solution. So, if I write d by dt N alpha minus N beta. So, what will I get d alpha by dt this is I call it as dN by dt this is the population difference and it turns out that this will be equal to minus 2 P N simple algebra here. So, we will generate this sort of an equation dN by dt is equal to minus 2 P times N and what is N? N is equal to N alpha minus N beta. Now, if we solve this differential equation we will I am not going to solve it here but I will give you the result what it will come out to be. So, the result is the solution of this is N is equal to N 0 at time t is equal to 0 e to the minus 2 P t 1 e to the minus 2 P t N 0 is the population difference at time t is equal to 0 and at time t this will be e to the minus 2 P t. So, what happens at t is equal to infinity? So, N will become equal to 0 at t is equal to infinity but of course, you do not continuously pump in energy for so long t is equal to infinity meaning you are putting the rf continuously for so long you do not put in energy for so long you put it on for a certain time and it is dictated by the transition probability P at this one. So, therefore, depending upon the value of P how much is the population difference. So, therefore, you will see that if this is very large if this time constant is very large then you will have this energy population difference going to 0. If N is equal to 0 there will be no absorption of energy therefore, the signal has to vanish. So, if I want to plot this here N versus time it will be like this. So, this is the initial population difference at time t is equal to 0 then eventually it will go down to 0 and this depends upon the value of P relative values of P and t at what time you are looking at. However, we always see some signal in NMR experiment we always see some signal. So, what does it mean? So, how do we reconcile this we always in NMR signal notice here where we ignored one thing we ignored the fact that there is what is called a spin lattice relaxation. Spin lattice relaxation is trying to bring this population difference back when you induce transitions by applying the rf it causes the transfer of population from the alpha state to the beta state and when the two populations become equal then alpha N alpha minus N beta is equal to 0 when there will be no absorption of energy. So, if I want to plot this same thing in the form of energy the form of energy I say I will say here energy here rate of absorption of energy dE by dt this will also follow the same pattern here there will be no absorption of energy as the population start getting equal. So, eventually it will go to 0 then there will be no signal. However, in an NMR experiment it does not happen that how long you keep the sample in your magnet you keep on getting the signal and this happens because there is what is called as the spin lattice relaxation. Since we do observe signal continuously there must be some mechanism for re-establishing the population difference and this is the spin lattice relaxation which we had ignored. Therefore, we will have to modify your equation. So, to include this we modify the equation as dN by dt is equal to minus 2 PN this is the original what we had then we have this other term which is N minus N naught divided by T1. So, if you want to continuously observe the signal without N becoming 0 then what we should say we look at the steady state where N is not 0. So, at steady state N is equal to N dash and dN by dt is equal to 0. So, when we do this we get we after solving that equation when we set this we get N dash is equal to N naught divided by 1 plus 2 P T1. You see now if depending upon what is the value of T1 versus the value of P. So, you get a steady state population difference N dash and so long as N dash is non-zero if N dash is not 0 we have signal. Therefore, 2 times P times T1 becomes an important factor in determining what is the NMR signal. In fact, there is a story here Gorter who actually discovered this phenomenon in the very beginning. However, he had he did the experiment in the beginning but he could not detect the signal for the simple reason that he chose a sample which had a very high T1 value. When the T1 value is very large what happens if 2 P T1 is far, far, far larger than 1 then N dash goes to 0 even with very low RF power. N dash goes to 0 for very low RF power and why is RF power because RF power dictates the value of P. Depending upon how much value of RF power we have. So, let me write here that there is also an equation for P. P is equal to 1 by 4 gamma H1 square gamma square H1 square and H1 is my amplitude. So, therefore, if H1 is very, very even when it is very small when, when it is small if 2 P T1 is very, very large then N dash goes to 0 and unfortunately Gorter chose a sample which had a very large T1 very long T1. So, therefore, with very low power he was actually saturating the signal N dash was going to 0 therefore, he could not see the signal at all. So, unfortunately he missed the Nobel Prize. So, I think we will stop here.