 Hello, I have said many times in this lectures that EPR spectrum appears in the form of a Lorentzian line shape. But I have never explained why that should be so. So today we are going to learn and see how the spin relaxation processes actually decide the line shape. Before we start, let us look at the concept of magnetization today little more carefully. Earlier I have used the term magnetization without defining it very rigorously. So for that let us consider not a single spin but a collection of spin something like avocado number of spins which are typically used in an experiment. So as this many spins are put in a magnetic field they will point along various directions given by their allowed angular momentum and allowed values of the magnetic moments. So we know that mu is a magnetic moment for an electron this is given as minus g e beta e s and so s can take various components in a magnetic field. So ms values are given as minus s to process changing in units of 1. So for a spin half system for example electron if I have got large amount of spins then ms takes l plus half or minus half and we know by now that in a magnetic field which is pointing along let us say this is the b magnetic field pointing along z direction. Many of these little spins will point in along the positive z direction and also many of them will point along the negative z direction this is x y and z. So this sum of all these will give rise to the net magnetic moment of all these particles. So magnetization is defined as total magnetic moment divided by the volume. Now total magnetic moment of course has to be obtained by adding all these individual magnetic moment of the individual particles. Now you see that if these two orientations for this one are equally populated then there will be as many number of spins which are going up and as many number of spin going down they will all equal. So net I remember to be 0 and net magnetization also be 0 and that is true for not just s equal to half if I have s of some other value then I can have this sort of energy level starting from ms equal to minus s to ms equal to plus s. So if all the levels are equally populated by the various spins then if I sum over all the magnetic moments here I will get the total magnetic moment average will be precisely 0. But in thermal equilibrium when these spins are distributed among various energy levels the distribution is governed by Boltzmann distribution. Now we all know that electrons are spin of particles so they are supposed to follow Fermi Dirac statistics they are called fermions that is absolutely correct. But the type of systems we have here that these particles which really do not interact very much they behave almost like independent particles. For such weakly interdict particles or non interdict particles the Boltzmann distribution works equally well. So we will use this distribution to find out how this various particles will be distributed among the various ms values. If the lower level will be more populated then the higher level this will be little bit less it will be still less this will be still less this is the least populated one. So now if I add all the magnetic moment the net result is not going to be 0. Therefore the system will have a net magnetization. So each of this ms values will have its own energy associated with that that is given by g e beta e if b0 is the magnetic field which is pointing along the z direction then g e beta e b0 sz is the energy corresponding to mz is the energy corresponding to the energy level given by the corresponding mz values. So this should be mz here and mz there. So we know that for Boltzmann distribution the number of particles is n1 by n2 all of this type of distribution where delta is the energy difference between the level which is n1 particle and this is n2 particle. So here the various energy levels are given by this set of expression and mz varies from one level to the other. So we can calculate the fraction of particle that these various levels will have given by let us call it spin population ms or I call it mz level is given by this is understandable that this is the sum of all the possibilities. So that is a normalization factor. So this ratio gives the fraction of population at a level which is characterized by the value of mz. So if we take some or all mz this is the total likelihood. Now if the number of particles in unit volume equal to n then I find out how many of this n are distributed in various energy levels here and each of them will have a magnetic moment given by expression of this kind. So I get the average value of the magnetization coming out for this n particles and since n is the number of particles in unit volume that average magnetization will essentially give rise to the magnetic moment, give rise to the magnetization. m is therefore given by this n times mz equal to minus s to plus s into the power this is v0 is missing here. So that is it. So if we can simplify this thing and get some sort of descent looking expression that will be the magnetization. How do I make it descent looking? Here the energy gap between this various states or of the order of this g beta v0 type of thing and for typical magnetic field of v0 let us say about 3000 gauss and g is typical to this factor that is this factor turns out to be less than 10 to the power minus 2. So of course and at temperature t is equal to some 300 Kelvin room temperature. So this is the typical condition we employ in the and according to the spectrum. So here this ratio is very small. So the exponential here can be expanded and we can keep only the first two term that is exponential x 1 plus x plus we keep only these two terms here. Then let us see how that expression simplifies. So then the magnetization m becomes and the denominator is mz equal to again minus s to plus s 1 minus g e beta i mz by kt. Now here when this summation is carried over this term here you see that this will take all the values from minus s to plus s in interval of 1. So this sum will be 0 in the same way when this is summation done over this term then again mz takes all the value from minus s to plus s this sum will be exactly 0. So this number 1 will be added 2s plus 1 times. So this is equal to square I forgot. This square comes from this reason that already there is one here and then other one comes from the expansion here. So you multiply these two so that this is equal to square of that one. This will be s plus 1. So this is n g beta i square b 0. I have forgotten kt here this one. Now here if now s is a whole number let us say 1, 2, 3 then mz will take value 0, 1 plus minus 1 plus minus 2 up to sum number plus minus s. These are the possible ms values. So when I take the square of that for both plus and minus the values are same. So they will appear in pairs and 0 of course does not contribute to this. So this is essentially this summation means that I take the sum of integers of this kind 1 square plus 2 square plus 3 square up to s square and this is known this summation is known. This is the summation of this percent this appears in pairs this summation will be 2 times. I do not need any more this one. So what I get from here these two cancels well and this gives 3. So this gives the simpler expression of magnetization is equal to this is s into s plus 1 by 3 kt. So this is the expression of magnetization. Here in this derivation I have taken the value of s to be a whole integer that is how we could use this expression here and then it simplifies to this. Now it is possible that s can be half integer also. So s equal to for example 3 by 2 then mz well I should call ms I suppose ms. This is the component of the magnetic moment vector here also. So ms will be minus 3 by 2 plus half plus 3 by 2 or it appears in pairs of this values plus 3 by 2 plus minus half. So here this expression cannot be used exactly as it is. So one has to change it to this half integral values and then do the summation. So I leave it to you as an exercise and see that you indeed can do that. Just cleverly manipulate this summation here and then everything will be very similar because this mz square appear there. So all the values will appear in pair and same. So you I leave it to exercise and see how you can do the summation here. Now it turns out that even if the s is half integer this is still valid the same expression is valid there. So this is true for all possible values of the spin angular momentum quantum number. So you see now that the moment the number of spins are kept in a magnetic field it develops a magnetization when kept in a magnetic field v0 there and of course now we know this because all the levels are not equally populated. So there is a term called susceptibility or magnetic susceptibility which is related to magnetization in this fashion called static susceptibility. When it kept a magnetic field v0 let us say I get magnetization with this kind then if you compare this with this static susceptibility is nothing but this n and this you should be able to recognize that this is nothing but the Koury law of the magnetic susceptibility and its temperature dependence. So if the external magnetic field is along the z direction then net this magnetization that we have calculated this will also therefore point along this z direction. So if I have another color this is the magnetization. So at equilibrium therefore this magnetization points along the direction of the magnetic field. Now if it so happens that something is done to the system and we disturb the orientation of the spin such a way that this magnetization does not point towards the direction and points some other direction. In other words what I am trying to say is that see at thermal equilibrium this magnetization which is a considered a vector quantity because it is a direction and also magnitude has only the z component it has no x or y component. So at equilibrium there is no x or y component but it is possible that we can disturb the spin system such a way that this magnetization can have x and y component also. This m now magnetization vector has this component. So once again at thermal equilibrium mz is this is equal to 0 is also equal to 0 and this takes the maximum value which I have derived here. Now I will call it as equilibrium magnetization I call it m equilibrium. How does one visualize this? Here again see these individual spins are pointing in all possible directions along this cone here. So if I take the their projection on the x y plane they will have all sorts of orientation here there will not be any preferred orientation of this. So naturally sum of this is going to be 0 for both x and y component. So any spin distribution which changes this magnetization from the equilibrium value to some other value can in general produce non-zero value for this and this and a value which is different from the equilibrium value. So here the spin relaxation process is going to restore this population to the thermal distribution here. It is going to restore the thermal distribution or the Boltzmann distribution that we saw earlier and that Boltzmann distribution gave rise to is equal to magnetization for z component no magnetization for this and this. So the magnetization which is if you write a this way a three component 0 0 m equilibrium and if I have got non-equilibrium magnetization let us say m x not equal to 0. So this relaxation processes which are there they will try to bring this to this. This is the job of the spin relaxation processes. Now here the difference between this change of magnetization from this change of magnetization here is quite significant. See the change of m z magnetization involves flipping of spin from one direction to other direction and that needs energy that causes needs some transition to take place. So either from here to there or there to here. So that energy has to be exchanged with the surrounding and the surrounding must give energy when spin flip takes place from this direction or energy must take over the energy when it goes in the other direction. So that involves exchange of energy that is for the change of m z component of the magnetization but for m x and m y is to bring back this sort of random distribution of the spins in the x y plane. All is necessary is that this various orientation re-energized among themselves. So that does not need any exchange of energy with the surrounding. So these two processes do that just rearrange all the spin orientation such a way the net m x and m y component 0. So we therefore characterize them by two different terminology and give different time constant for their processes. The time constant for these processes are given by certain time constant I will call them in a define them in a moment but importantly Felix Bloch he proposed that that this restitution of this magnetization from non equilibrium value to equilibrium value this process is a first order process which looks like this the m z y d t is equal to these are all first order chemical kinetics type of expression here that the time constant for this process is given by this t 2 for this and this and t 1 for this. So the reason for these two being different from this is I already mentioned that this involves energy exchange with the surrounding. So the processes or the mechanisms which makes this process to take place suppose quite different from the mechanisms which makes this process to occur naturally their time constant did not be same. We call this process transverse relaxation time and this is called longitudinal relaxation time also this is called spin-spin relaxation time and this is similarly called spin lattice relaxation time. The meaning is clear that energy exchange between the spin system and the surrounding is involved lattice is a general term to designate the surrounding or anything that is other than the spin system. We have seen earlier that the time evolution of a magnetic moment in a magnetic field follows this sort of relation where this m was the magnetic moment of a particle or some system. Now here we are using the same letter m to designate the magnetization because we are dealing with the collection of particles. Now since each of these little spins contributes to the total magnetization this exactly the similar relation holds good for the magnetization also where this is the magnetization. I expect that they should not have any confusion in going from here to there. We are using earlier this equation to describe the time evolution of a magnetic moment in a magnetic field. Now same exactly similar equation is used to describe the time evolution of a magnetization kept in a magnetic field. So, here look at this slide here is given here. So, when the magnetic field B is along the z direction then one can expand this term and we get three equations of this kind dmx by dt is minus gamma i B0 m y dm y by dt is plus gamma i beta B0 mx and dm z by dt is 0 that is mz component does not change. So, here of course it shows the simply the evolution of the magnetization in a magnetic field and we know that this is nothing but the precessional motion of the magnetization. Now to this we add this relaxation processes then the equation will look different of course here. All I have done is the add the this three terms here the corresponding three terms that this equation gives. So, that is it. So, here this shows the evolution of the magnetization vector mx, mi, mz component of them in this fashion relaxation terms are included here. But in the EPR experiment along with this B0 we also apply an oscillating magnetic field in the xy plane to cause the transition. So, this oscillating magnetic field rotates in the xy plane with a frequency omega. So, how do I show that this is my xy. So, this B1 is the oscillating magnetic field it starts at time t called 0 and on the x direction it starts rotating in the xy plane above the z direction at an angular frequency of omega. So, after time t the angle that will form here will be less than theta. So, theta will be equal to omega t. So, at this time x component of this is given by this and y component is given by this. So, here therefore, that B1 cos omega t i B1 sin omega t j and we thought of as the B1 vector. This is the one which is applied along the xy plane and it is moving around the z axis with an angular velocity omega. This is precisely the type of vector expression this will have. So, in this case magnetization sees two fields one is due to this other is due to this one which is appearing along the z direction. So, the total field B seen by the magnetization is given by therefore, i cos omega t this is x component B1 sin omega t this is y component k base 0 this is the z component. So, this is the total magnetic field that is experienced by the magnetization. So, again I can include that in the equation here and then find out the time dependence of this in the presence of the oscillatory magnetic field and that is done in this slide. So, the x, y, z are the laboratory coordinate and this total magnetic field in the laboratory coordination is given by this and that given when this expression is inserted here you get the time dependence of the Mx, My, Mz component of the magnetization. So, we will try to solve it using some special technique. At this stage let us summarize what we have done. We have taken a collection of spins and then when they reach thermal equilibrium the different energy levels have different number of spins. So, using that information we collected the static magnetization or equilibrium magnetization which appears here. Then we introduced this blocks idea of the first order chemical kinetics type of term which can restore the non-equilibrium magnetization to equilibrium magnetization and introduced these two different time constants for that. And finally we got this time dependence of the magnetization in the presence of all the magnetic fields that the magnetization sees. So, these equations are called the block equation for the time dependence of magnetization. These are very famous in magnetic resonance studies. With this we stop this lecture and we will continue our discussion in the next one.