 So, we were discussing about the nuclear spins in a magnetic field and let me just briefly recap that what we saw nuclear spins in a magnetic field. Consider a spin with I is equal to half, we said it has two possible orientations in space and if I want to represent them like this, so one possible orientation like this other one is in like this and this is the orientation with we call it as alpha state and this is the beta state and here m is equal to half and here m is equal to minus half. The magnetic field is applied along the z axis and therefore the alpha state has a lower energy and the beta state has a higher energy, so this the different orientations have different energy values and that energy is given by E is equal to minus mu dot H naught where mu is the magnetic moment H naught is applied magnetic field in terms of, so this can be called as mu z H naught minus mu z H naught or this is minus gamma m H cross H naught, so therefore obviously when alpha is equal to plus half, m is equal to half then the alpha state has a lower energy compared to the beta state and there is a population difference between these two and that is population difference the alpha and the beta states have different populations which is dictated by Boltzmann statistics, so I want to represent it like this, so this is alpha state there are more spins here and there is this is the beta state there are less spins here, so this population difference is responsible for the NMR signal intensity. So this dictates the NMR signal intensity, how it will, we will see that later. So now let us look at this in a different way and this is what actually Felix, this was the approach which Purcell had taken, this is a quantum mechanical description of the interaction of the spins with the magnetic field, Felix Bloch had looked at it in a different way and he said okay, he took analogy with the magnetic moments in general when you apply a magnetic field to a magnetic moment what happens? Suppose for example I have a compass needle, a compass needle is a tiny magnet, so suppose I have a compass needle like this which has orientation if you put it in a magnetic field what happens to the compass needle, so everybody knows about this, so it will reorient itself move it like that and orient itself with respect to the magnetic field and this is used to determine the directions you know as north-south directions because it orients itself with respect to the earth's magnetic field. So the similar analogy is with this analogy we can describe about the nuclear magnetic moments as well, however only to a certain extent and why is that so? Because this is the bar magnet, this is the large magnet, it can orient itself, this is the classical description, orient itself aligned itself parallel to the anti-parallel to the earth's magnetic field. In the case of the nuclear spins however that kind of a motion is not possible, there is a definite orientation of spins as I mentioned to you they are either here or here, we cannot change this angle, this angle if I call it theta, we cannot change this angle. So okay, now what will happen then, why does the compass needle move when you apply magnetic field? Because it experiences what is called as a torque, the north pole and the south pole of the tiny magnet will experience forces like this and since there is a distance between the to north and the south pole it has a certain length, this results in the motion of the compass needle in a particular direction. Now here also of course there will be a torque, torque will be present here but it cannot orient itself with respect to the z axis like this, it cannot become exactly parallel or anti-parallel to the field. What does it do? In such a case the classical description is this spin will start exhibiting a kind of a motion that it will keep going round like this and similarly here this also will keep going like that and of course the origin of this lies in quantum mechanics, we will not go into that detail but the classical description is that this kind of a motion is called as precessional motion and it goes by the name Larmor precession, Larmor is the name of the scientist who actually described this quite early and therefore this motion is called as Larmor motion and this goes in a particular direction, it goes in this clockwise direction and the frequency of this motion if I want to write it as omega naught, this is related to the magnetic field by this kind of equation H naught is the strength of the magnetic field and gamma is your magneto-gueric ratio. So now is there a parallel between this and the energy level diagram what we had? Energy level diagram also showed, let me draw that here energy level diagram, we had this energy here E is equal to minus gamma m H cross H naught. So for this one this is minus m is equal to half minus H cross H naught by 2 and here it is gamma H cross H naught by 2 because m is equal to plus half here and minus half here and what is the energy difference between these two? So this is minus energy difference delta E is equal to minus gamma H cross H naught you see. So this is very similar to what we have here, omega naught is the frequency and therefore this is equal to minus omega naught H cross. So therefore the classical description of Larmor precession and the quantum mechanical description of energy level diagrams are identical. The energy separation between the two levels is in terms of the frequency it corresponds to the Larmor precessional frequency. So this is a very important concept with regard to the precessional motion and we are going to use this quite significantly extensively in the subsequent description of resonance absorption of energy and so on. When these ones are executing the motion all the spins go in the same direction whether it is alpha state or the beta state they both go in the clockwise direction and the minus sign indicates that the minus sign means that there is a clockwise description conventionally the anticlockwise motion is considered positive and the clockwise description is considered as negative. So now the Boltzmann statistics I already described now there is a population distribution as I said. Now if I were to sit on the spin if I were to sit on the spin and look at the magnetic field so what happens the what happens the the nuclei will not see the field at all and that is what is called as going into the rotating frame of reference. So if I go into the rotating frame of reference means you sit on the RF as you sit on the spin and look at the magnetic field okay if you sit on that and then look at the magnetic field it looks as if you are not rotating right so it is for the observer the emotional frequency is with respect to the observer. So if the observer is sitting on that thing and look at this then of course it is not moving right so that is called as the rotating frame of reference. A classical analogy for this is that suppose there are two persons sitting on two different trains there is one train going here train one train two both are going in the same direction and if they are going with the same speed if they are going with the same speed then if person who is sitting on this looks at the person who is sitting on this they feel as if they are not moving okay so that is called as the frame of reference changing the frame of reference. So if they are going with the same speed and the same direction there is as good a stationary if it is stationary which means that okay they are not moving in the same manner when you go into the rotating frame of reference there is no H0 field because it is not moving in the rotating frame of reference we sit on the R sit on the spin and look at the magnetic field if there is any magnetic field perturbation then you do not see the field because it is stationary if the analogy is here so if you are one train the other train if it is both two persons are sitting on the two different trains and if the both the trains are going in the same direction so this is train one one and this is train two if they are going in the same direction with the same speed then the two persons will feel that they are not moving with respect to each other if the of course they have different frequencies different speeds then they will see the difference and this becomes an important concept when you are discussing about the absorption of energy and so on let us talk about now an important concept here at this point in time and that is called as the spin lattice relaxation what is it all about okay we said that for the two two orientations we have the one state here alpha other state is the beta there are certain populations here and this state the population difference will depend upon Boltzmann statistics Boltzmann statistics means that okay the probability of the spin being in the alpha state is equal to p is p alpha e to the minus delta e divided by kt and similarly p alpha by p beta this is the ratio of the populations is given by e to the minus delta e by kt delta e is the energy separation between them and that of course we will depend upon the magnetic field okay and this is equal to e to the minus gamma h cross h naught by kt okay and that was the energy difference what we saw and kt t is the temperature so if I change the field if h0 is changed if h0 is changed then the populations will have to readjust so that means that there have to be transitions from one state to the other state forward transition and backward transitions is both way it has to happen what brings about these transitions and if a particular spin has to lose energy where does the energy go okay so this means lose energy and where does it go if it has to gain energy if a spin has to gain energy where does it come from okay so this is the question what is the source therefore here there is a concept which is called as the lattice so this brings us to a concept called as lattice which is everything else which other than your actual spin system so it is a large number of magnetic dipoles which are present in your system that is this may be solvent molecules or it may be other things so they will all have magnetic dipoles and this constitutes what is called as the sink okay so I have here the spin system here and here it is the lattice lattice is all kinds of energy levels because it has various kinds of magnetic moments and various kinds of interactions possible therefore the lattice is there which is a parallel with respect to the spin system so this is your spin system and this is the lattice so if an absorption has to have to happen transition like this and it has to have a transition like this here so energy required will be gained by this and if a loss has to happen if it has to lose energy where does the energy go it has to go here the lattice will take this energy so therefore it is a kind of a sink or the resource so therefore lattice and spin are constantly interaction interacting therefore this is spin lattice coupling is responsible for for population adjustments and we call this as the relaxation process and this is there so is characterized by a time constant called spin lattice relaxation time and it is represented as T1 the same thing is responsible for the adjustments of the populations starting from 0 so in the absence of the magnetic field all your spins were here in one particular state right so here H0 is 0 and when H0 is non-zero is not equal to 0 then we generated two states and the readjustment of the populations took place okay so therefore how do I characterize this without going into the theoretical details of this let me just give you the how does the population difference change suppose N alpha is the population of the alpha state alpha state and N beta is the population of the beta state and if I define N is equal to N alpha minus N beta if N naught is equal to N alpha naught minus N beta naught this is the equilibrium population difference then we can write an equation for how the system will approach the equilibrium as in this manner N is equal to N naught 1 minus exponential minus T by T1 okay so therefore you see T1 is a time which it determines how fast the system will reach so N naught is the equilibrium population difference so the rate at which it will approach the equilibrium population difference is given by this equation now obviously if T is equal to infinity N is equal to N naught so depending upon the value of capital T1 it will approach the equilibrium value slowly or rapidly and this becomes an important factor okay as we will see later on okay so now we will see how we do an NMR experiment so this is so far as the spin systems are concerned what happens to the spin system when the presence of the magnetic field how there is we considered with spin is i is equal to half but the same things will apply for other spin systems as well i is equal to 1 or whatever spin systems we consider the same principles will be valid so now we consider the NMR experiment that is the resonance absorption of energy what is resonance absorption of energy we said we have the population difference resonance absorption of energy we can look at it in two different ways easier one is to look at it in the quantum mechanical picture again we will consider i is equal to half first we have the two states and this energy difference is equal to delta E and this we say delta E is in the in the radio frequency regime that means in the megahertz so it may be 100 megahertz 200 megahertz and once you go into the gigahertz it will go into the microwave regime so therefore this energy difference is in the radio frequency regime why is it so because it depends upon the gamma the gamma and the H naught gamma H naught is the omega naught that is the frequency of precision and that will turn out to be for the range of values of gamma what we have it will be in the radio frequency region and this is the highest for proton for proton because it has the highest value of gamma highest value of gamma so typically when we always say we have 100 megahertz spectrometer 200 megahertz spectrometer we always refer to the proton absorption frequency okay so what does that mean now if apply a RF here radio frequency which corresponds to this energy difference whose frequency corresponds to this energy difference then absorption of energy will take place by the spins to go from the alpha state to the beta state and the RF is what RF is is it is given by 2 H 1 cosine omega t and this is the amplitude amplitude of RF radio frequency and this is the frequency so if this energy RF energy matches this energy separation delta E then there will be maximum absorption if omega is equal to omega naught is equal to omega omega naught was what that was the energy frequency corresponding to the energy difference between the two states and omega is the RF frequency if omega naught is equal to omega this is called as the resonance condition so this is nuclear magnetic resonance so you have the RF which has a frequency which of course you can change this frequency this is in your electronics control once you have a magnetic field H0 magnetic field the processional frequency is determined by the magnetic ratio and the H naught that we cannot change and the obviously you do not know where this one is so therefore you have the RF which you can keep changing RF frequency can be changed can be changed to achieve the resonance condition okay when that happens this leads to absorption of energy absorption of energy in the quantum mechanical picture because it will cause transitions let us also look at this in a classical picture the classical picture again we go back to the singles spin half system i is equal to half we said in the classical picture these ones are processing like this these are also processing like this and this frequency of precision was given by omega is equal to minus gamma H naught and this is H naught now this is the transverse plane i have here the transverse plane this is the z axis here remember this is the z axis and here i have the x and the y components x and the y axis here the RF has to be applied in the transverse plane so i have applied here at RF is the radio frequency RF is the radio frequency this is applied is the electromagnetic spectrum right so you have applied this so i have here 2 H1 cosine omega t okay so i can break this up into two components H1 e to the i omega t plus e to the minus i omega t divided by 2 this is the expansion of the cosine function so what does this e to the i omega t meaning e to the i omega t means a rotation along the this axis in this this direction this is e to the i omega t and the other one is the rotation in this axis e to the minus i omega t this is the rotation in this direction now out of this which one do we consider we consider the e to the minus i omega t because this goes in the same sense as your nuclear spins are processing okay so this is the one which we consider because if you consider the e to the i omega t it goes in the opposite direction therefore there is no time for the spins to interact with the RF so this is remember this H1 amplitude is the magnetic field so this is the magnetic field the magnetic component of the RF which we are looking at the electromagnetic spectrum has both electrical a component and the magnetic component we are talking about the magnetic component because we are talking about nuclear magnetic resonance magnetic interactions therefore we are here have the magnetic component of it and interaction will happen with the magnetic component which is the H1 and therefore if the spins have to interact with H1 we have to consider the frame in which we are sitting and then how we look at the RF okay so all right so okay so let me let me stop here and we will continue for the next class from here