 So, we talked last time about the absorption of energy by the spin system when we apply an RF. So, considering the relaxation also we said okay there is a steady state population difference between two energy levels, two energy levels we are considering for a spin half system alpha and beta and this n dash is the steady state population difference between the two and that is given by n naught divided by 1 plus 2 P T1 and where P is the RF induced transition probability and T1 is the spin lattice relaxation time is because of this spin lattice relaxation while the RF tends to reduce the population difference the T1 tends to restore this population difference therefore as a result of the balance between these two we have a continuously nonzero population difference and therefore we get the NMR signal okay we also said P is equal to 1 by 4 gamma square H1 square where H1 is the amplitude of the RF and gamma is the gyromagnetic ratio. So, in H1 square this defines the power while H1 is the amplitude H1 square defines the power of the RF how much power you apply though the more the power higher is the transition probability you tend to reach the saturation okay. So, therefore this 1 plus 2 P T1 is this term is called as the saturation factor because if this is too high if the 1 plus 2 P T1 is very very large then of course n dash may go down to 0 in which case of course you will lose the signal that is why this is called as the saturation factor 1 plus 2 P T1 if which is very very large then you have the signal disappearing and therefore this is called as the saturation factor okay. So, now from here we go forward and to see now next we ask what kind of transitions the RF will induce so what transitions the RF will induce. In other words suppose we have a spin system which has many energy levels so far we are considering only 2 energy levels so therefore there is no issue but suppose I have a system which are many levels like this and this will depend upon the value of I because we said if there are 2 I plus 1 states for a particular I that is considering if there is a one spin with one type of spin which has the I value that 2 I plus 1 states if there were multiple spins then of course you will have a net I value and which will of course be very different and you will have many different orientations possible many different states possible for the combination of this for the various states what sort of transitions are possible here. So, these are various values of m here I call it as m what is the magnetic quantum number so let us say this has a 3 by 2 half minus half 1 by 2 or 5 by 2 and so on so forth we have various kinds of spin states which are possible what sort of transitions are allowed so in other words what are the allowed transitions the same thing is to say what are the selection rules does the RF induced transition between every 2 levels so it turns out that this comes out from quantum mechanics by the perturbation theory in quantum mechanics quantum mechanics perturbation theory we will not go into that detail perturbation theory it turns out that RF can induced only those transitions for which delta m is equal to plus minus 1 so it cannot reduce transition between m is equal to if I have a let us say for i is equal to 1 i is equal to 1 what are the states possible I will have i is equal to 1 m is equal to 1 this is m is equal to 1 and then you have the 0 here and you also have m is equal to minus 1 so delta m is equal to plus minus 1 meaning what what states are possible we can have a transition between these 2 levels we can have a transition between these 2 levels so these are both delta m is equal to plus minus 1 so but this transition this transition is not allowed this will not be allowed because that black one has delta m is equal to 2 and that is not allowed and that is typically known as the double quantum transition wherever there is delta m is equal to plus minus 1 this is called as single quantum transition so this is so far as the energy absorption is concerned now it leads us to the next question will the energy absorption happen precisely only at that frequency or there is a certain variation allowed so E is the NMR spectrum if absorption of energy happens exactly at one frequency if this is omega does it happen exactly at one frequency or some other frequencies which are nearby will also cause a transition will also cause absorption of energy what does that mean if I have a certain kind of for possibility of having a transition for these energy levels also then they may have different intensities alright then I will get a profile of line like this so the frequency here if I apply an RF frequency here it will have an amplitude absorption of energy will be this much if this exactly a resonance condition it will have a highest amplitude if it is frequency is here then I will get this much of amplitude there and therefore it will generate absorption of energy different RF excitation efficiency will be different this width here at half height this width here at half height this is called as the line width so the absorption of energy does not occur only at one frequency but there is a certain tolerance so it there is a certain tolerance on what should be there for the frequency for absorption to take place this is called as the line width now what does this line width depend upon so this is something which one has to see we can look at that so the line width what are the factors width influence the line widths ok now consider let us say a two level system so we have the exact frequency one value but there are certain other values we said it will have absorption of energy possible and we are going to observe a signal which is looking like this and this is the central value omega naught and I will have this width which is here what does it depend upon basically this come from a principle known as Heisenberg uncertainty principle what it says is in quantum mechanics in certain in particles of this type there is a definite uncertainty in the energy values and there is a certain uncertainty in the time if there is a certain uncertainty this product is approximately equal to h cross uncertainty in the energy there is an uncertainty in the time now what is the meaning of uncertainty in time uncertainty in time means this is the lifetime of the state so we can take this uncertainty in time to imply lifetime of the states of the spins in the states so what does that mean if a particular spin is in the particular state let us say this is alpha and this is beta will it remain there forever no it does not remain there forever there is always a certain continuous transitions going on from up and down alpha to beta beta to alpha these transitions are continuously going on these are induced by the lattice are induced by the RF or whatever so there is a kind of a lifetime for the spin in a particular state and that is called as the lifetime and therefore if that is delta T accordingly if this condition has to be satisfied there is a certain uncertainty in the delta E also therefore that uncertainty reflects in this width when there is an uncertainty in the energy level so it is either here or here or here or here some uncertainty in the energy value then it will cause absorption of energy at slightly different frequencies and that is what is called as the line width now therefore whatever affects the lifetime of the individual states that will cause a line width so therefore alteration in the line width so one reason is alteration in the in the lifetimes we will not go into the details as to what all these things can happen of course this is reflected in what we already seen in the spin lattice relaxation we have seen this in the spin lattice relaxation okay this will alter the line width and the phase coherence changes that is that we will see afterwards and then there is a second reason what can cause the line width this is the field inhomogeneity field inhomogeneity what is the meaning of that suppose I have a sample like this I have the solution here the various spins are presented various places here now apply I am applying a field which is H naught suppose this field keeps fluctuating the value of a particular spin of a value of field at this point is slightly different from the value of the magnetic field at this point then this fellow will absorb energy at one frequency this fellow will absorb at another frequency okay therefore the result will be so this may give a freak an absorption at this position and this may give me an absorption at this position these are slightly changed slightly different this plus this will give me a free a light broader line okay so therefore this contributes to the line width because both the things are present the same spin absorbs energy at two different frequencies because of the field inhomogeneity okay so this is a different mechanism earlier one was the lifetime of the individual states has changed and this is an indifferent mechanism this can be controlled by adjusting the homogeneity of the sample very well homogeneity of a magnet very well this can be controlled so this is an important contributor to the line widths there are various other reasons why there can be changes in the lifetimes and these are all based on the quantum mechanical principles it is sufficient to know whatever alters the line lifetimes of the individual states that will cost contribute to the line widths okay so now let us look at the description of the spin system from a different viewpoint and this is how Felix Bloch described this magnetic resonance phenomenon in a classical way in a phenomenological way and he set up a set of equations called as Bloch equations these are known as Bloch equations okay so he described the movement of the magnetization in the presence of the RF how it will be in a very generalized manner. Let us say we have the axis like this here x, y, z and we have a generalized representation of the magnetization which is let us say oriented somewhere like this this is my magnetization medium now this is a vector the magnetization is a vector right so the magnetization will as vector means I have components Mx, My and Mz these are the components along the 3 axis there is also this effective field and that is like this so I have this H0 field along this axis this is the main field then I have applied RF which is the H1 along this axis let us say then therefore the effective field is the vector addition of these 2 fields I will get a field H effective so if the H effective is oriented in this manner of course since RF is a rotating field this H effective also keeps rotating H effective also keeps rotating so therefore this has a time dependent component and there is a time independent component. Now H0 is constant is constant and the RF is H1 e to the minus i omega t this is the time dependent component here and this is the rotating field therefore the H effective also keeps rotating so therefore it has these components let us represent this H effective with 3 components as okay now this is effective here H effective here this has the components Hx, Hy and Hz of these Hz is constant Hz is equal to H0 this is constant and Hx we write it as H1 cosine omega t and Hy we write it as minus H1 sin omega t because it is going in the minus e to the minus i omega t e to the minus i omega t is what cosine omega t minus i sin omega t therefore so this Hx remember this this is goes to the this one here Hy is down there so therefore we have 3 components and this 3 components are we can write explicitly like this so I have the m vector m vector has the components Mx, My, Mz and the H effective is H0 then H1 cosine omega t and minus H1 sin omega t now notice here this H0 is along the Z axis here Mx, My, Mz they are in a different order this is along the Z axis this is along the Z this is along the X this is along the Y Hx, Hy okay I wrote here Hz, Hx and Hy okay so let me write again down here Hz, Hx, Hy so Block wrote a set of equations to describe the moment of the magnetization vector in the presence of such a kind of a H effective so I will write those equations here directly and of course you also consider the relaxation phenomena so dMx by dt is equal to gamma My H0 plus Mz H1 sin omega t minus Mx by t2 now t2 is the relaxation time which I had introduced to you earlier this is the spin-spin relaxation time or it is a transverse relaxation time d My by dt this is the rate of change of the Y component of the magnetization because remember these are time dependent phenomena because of the RF which is time dependent and the relaxation is also it changes the magnetization a time dependent manner therefore this these all of these equations will have time dependence so this is gamma Mz H1 cosine omega t minus Mx H0 minus My by t2 and this is obviously these Mx and the My components are the transverse components therefore they depend upon the transverse relaxation time t2 now what about the Z component dMz by dt this is given by gamma to minus Mx H1 sin omega t minus My H1 cosine omega t minus Mz minus M0 divided by t1 so the first is the transverse components which are present here and the last relaxation part which is there this is dependent on the t1 relaxation time Mz is the magnetization at any point in time M M0 is the equilibrium magnetization so depending upon the deviation of Mz from M0 it will have a relaxation going on there is a magnetization Mz is going to change. So these are the block equations in the so called laboratory frame but the NMR experiment is done actually in the rotating frame NMR experiment is done in the rotating frame is in the rotating frame that is as if you are sitting on the RF and looking at the magnetization how they it is spinning so therefore the main RF frequency will get subtracted out the time dependence will go away in the rotating frame meaning you sit on the RF and look at the magnetization in the rotating frame therefore you can transform the equations into the rotating frame I am not going to give the details of that one but I will only give the results transform the equations to the rotating frame rotating frame is you have this the RF is applied the RF is moving here this is the H1 and the component of H1 which is let us say you are sitting on it and looking at this is the H1 here and what is on that axis if the H1 is rotating in this field I will have that particular axis I am sitting over here and there is also a the orthogonal component which is going like this it will have an orthogonal component this is the normal x y z components then I will have sit on the RF and look at it in the same direction suppose it is moving like this so I sit on it I have this axis here and then there is the other axis which is perpendicular to it so that perpendicular axis I am writing like this okay so I call this axis as U and this axis as V okay so I measure the magnetization in this two in this magnetization components along the U and the V axis okay so therefore U is the magnetization component in the direction where the RF is applied and U is the orthogonal component so U is along the direction of the RF and V is the orthogonal component so this is the mathematical operation how to transform it so you have the magnetization M which you can write in the laboratory frame as Mx, My, Mz or you can write in the rotating frame as U component V component and the z component which is Mz only z component does not change and that is the U component so now I make this transformation after doing the transformation I will write the block equations and this will be du by dt plus U by T2 plus omega naught minus omega into V is equal to 0 then I will write similar one for dV by dt plus V by T2 minus omega naught minus omega into U is equal to minus gamma H1 Mz and for the Mz this does not change because this is along the z axis it is a constant term this dip does not depend upon the rotating frame dMz by dt plus Mz minus M naught divided by T1 now it is dependent on T1 notice here minus gamma H1 V is equal to 0 so these are the three equations in the block equations in the rotating frame. So now at steady state what happens at steady state as I said we are going to measure the signal continuously so therefore at steady state what will be the situation at steady state all the time derivatives will be 0 at steady state du by dt is equal to 0 and dV by dt is equal to 0 if you substitute this into those block equations then you will get U is equal to M naught into gamma H1 T2 square omega naught minus omega divided by 1 plus T2 square omega naught minus omega square plus gamma square H1 square T1 T2 and V will be minus M naught gamma H1 T2 divided by 1 plus T2 square omega naught minus omega square plus gamma square H1 T1 T2 and Mz I will write the Mz here Mz is equal to M naught 1 plus T2 square omega naught minus omega square divided by the same term 1 plus T2 square omega naught minus omega square plus gamma square the denominator is the same in all the cases T1 T2. Under the conditions gamma square H1 square T1 T2 is far far less than 1 that is this represents no saturation remember saturation is dependent on the power H1 so H1 pair H1 square is very small compared to the T1 and the T2 when you take the product is very small then you will get the simplified equations for U and V then you get V is equal to T2 divided by 1 plus T2 square into omega naught minus omega square and U is equal to omega naught minus omega divided by 1 plus T2 square into omega naught minus omega square. These are the components which we are actually going to measure these are the transverse components we are going to measure this. Now if I want to draw this curve I want to plot V versus omega see if I plot V versus omega this is omega here this is V here how does it look this will look like this and this is omega naught center is omega naught if I plot U versus omega how does this look this will look like this down like this so this is called as the absorptive signal and this is the dispersive signal. So these are the two basic line shapes in your NMR spectrum. So but most of the people use this absorptive line shape because it is a very it does not have the positive negative components and it is sharp it does not have the tails as this U component has. So this is the measurement so therefore NMR signal NMR spectra you will see this term this is what we will observe in your NMR spectrum. So I think we can stop here for this class and we can go over to the other things in the next class.