 Hello there, today we are going to discuss what are the various dynamical processes that go on when the resonance takes place. We will first discuss in general for any resonance absorption and then in particularly the differences that we need to keep in mind for EPR transition. For simplicity again let us consider only two level system, higher level and lower level, lower level and upper level let us say we call it. And let us see we have got a collection of particles which are distributed in these two level according to Boltzmann distribution, number of particle in upper level by number of particle in the lower level is given by where T is the temperature of system and this is delta E. Now if some radiation falls here which is matching with this energy gap and the selection also satisfied then what can happen? Of course it can absorb radiation and this particle which are lying here can go from here to there that is the absorption process is absorption let us call it absorption process. But there is some particles which are here they can also undergo transition from here to there because of this radiation why is that? Because the selection rule we have seen earlier let us say from upper to lower in the presence of some perturbation this gives the probability of transition from let us say lower to upper or upper to lower the form is the same. So probability of transition is actually the same thing whether it starts from here goes there or it starts from here or goes here that is the perturbation due to the radiation. So we call that induced emission or stimulated emission along with that the particles which are in the excited state can come to ground state by its own natural process let us call it some naturally process. What are the natural processes? In optical experiment for example EV visible experiment the natural processes are fluorescence emission phosphorescence emission those are radiative processes by means of which the particle in the excited state can come to ground state. Now these 3 things can happen in general in all experiment that involves transition in the presence of radiation. Now this fluorescence or phosphorescence these are spontaneous processes these processes can be actually they are nothing but the spontaneous process why we call spontaneous because this takes place even in the absence of any such external perturbation. In the experiment that involves magnetic resonance like NMR and EPR in principle this will of course take place this can in principle take place but there is a catch the catch is that the probability of this spontaneous process that drastically goes down when the wavelength of this experiment or the energy gap between the 2 becomes very small where does that come from? It comes from Einstein's we call A and B coefficient this gives the probability of spontaneous emission by the probability of induced emission is given by A by B which is this is the expression. So you see the spontaneous emission and induced emission these 2 ratio given by the frequency this frequency is related to the energy gap here and not only that this depends on the cube of the frequency. So as the energy gap becomes smaller and smaller this spontaneous emission probability becomes smaller and smaller very quickly. So whereas in optical experiment where fluorescence and phosphorescence becomes very common process for excited molecule to come to ground state when it comes to magnetic resonance experiments NMR or EPR this is so small that spontaneous emission probability virtually goes to 0. So no spontaneous emission in the microwave frequency. So in these 3 processes that we have here one is out so what we have now this is the lower level is the upper level so in the presence of radiation I have absorption process and this is the induced emission or stimulated emission if you like to call it stimulated emission and this is not present. What can happen now let us try to imagine the to start with before the radiation is applied the number of particles in the lower and upper level followed this Boltzmann distribution. So the number of particles here let us call it now N alpha and N sorry N lower and N upper. So if there was no radiation here present then N upper by N lower will be equal to delta E by kT and that is the Boltzmann distribution and let us put a superscript 0 here to indicate that that is what happens when nothing else is disturbing the system. In thermal equilibrium we have the Boltzmann distribution and total number of particles is equal to N0 alpha plus N0 upper and N0 lower. Now the rate of transition in the presence of radiation will depend on the probability times the number of particles which are present here for going up and similarly for this process the rate of stimulated emission will depend on the number of particles which are here and the probability of this. So I will have the if it is W downward and W upward these are the transition probability in the absence of this radiation that is important that this Boltzmann distribution maintain all the time. So if there are some transition which are taking place on its own which are not induced by this one and which is maintaining this distribution then what we must conclude from here is that rate of this downward transition that is the probability of going down times the number of particles in thermal equilibrium here which is N0 upper must be equal to rate of upward transition in the number of particles in the lower level. Once again these rates are the not the rate this is the probability for transition in the absence of any external perturbation then naturally if some transition taking place which is maintaining the thermal equilibrium and maintaining the population distribution according to the Boltzmann law. So in that case this must be satisfied the same as saying that this is satisfied automatically. Now if there is some disturbance to the population distribution then let us say that time Nu and Nl are the new number of particles here which are different from the thermal equilibrium values here. Then N alpha plus sorry Nl, Nr will also be equal to N the number of particles remain the same I may have come to a an assert saying that these are taking place here all the time even if there is no perturbation that may sound a little bit far-faced and ad hoc why do I need that. So to appreciate that let us once again go back to this situation where nothing else happens that is this has been removed and these two processes are taking place there. Then what can happen is that radiation will try to push this molecule from here to there and also push molecule from here to here. Now though in the beginning there are many particles slightly more number of particles here than there initially the rate of upward transition will be more than rate of downward transition. So if nothing else is happening then what happens very soon these two levels will get equally populated so that means we will not see any net absorption of radiation. So these two processes are taking place and then these two levels become equally populated so no net absorption of radiation so no absorption spectrum. So that is the problem that if there are no other processes involved here then very soon the moment the light is shown and resonance current is satisfied the absorption will disappear two levels will get equally populated that is the key problem that how can we see a net absorptive signal if these are the only two processes. So there must be some other process which maintains some sort of population difference here which can be sustained even in the presence of radiation and this population difference is the one which decides the amount of light that is absorbed and that gives the absorption spectrum. So you see that that is the requirement we have to fulfill so that we can explain that we yes indeed we see the absorption spectrum in a spectrometer because you see the absorption in the spectrometer we have to incorporate a process other than these two and that we call a same relaxation process. This relaxation process is not same as this spontaneous emission process of the kind that we have normally seen in optical experiment is something other than that. So this is the probability of downward transition and upward transition because of this natural relaxation processes and that is present all the time that maintains the thermal equilibrium so this is the relationship we get. Now if this population changes what way the system will try to develop a new steady state condition so that net absorption of radiation can be seen. So the absorption of radiation of course proportional to n alpha A minus n this difference this is the population difference see if the population difference is 0 then there is no absorption of radiation. So the thermal equilibrium these are the number of particles in upper and lower level so let us say n 0 in thermal equilibrium can be written as n 0 L minus n 0 upper level. So as much as here I have n 0 L plus n 0 is also equal to n the number of particles remain the same. Now when the transition takes place this population difference I can say n new population difference n alpha minus n L minus n U when one particle let us make transition from here to there then the population difference changes by 2 units because one comes from here to there the difference becomes 2. So the n by dt can be written as probability of this going downward transition times the number of particles in upper level lower level. So we are still considering the situation that how does this relaxation process maintain this population difference we have not yet applied the radiation yet. So only the difference is that if the population difference has become different from the equilibrium value and that is n L n U how this relaxation process try to bring back the population difference to the equilibrium value. So this is the probability for downward transition from here to here. So the number of particles which are here similarly the upper transition factor of 2 is because each transition changes the population difference by 2 unit using these equations one can write that this is by the thermal equilibrium condition from this relation this this this you can write this is in terms of the population difference n and the total number of particles if you substitute this here you get this value. Now this could be simplified as this is the total number of particles and this is the population difference at a given instant. Now if we define that average of the upward probability and downward probability is W then I get d n by dt is minus n all right. Now let us see here we use this condition which relates the upward rate in terms of n 0 L and n 0 U and downward rate in the process we can change this equation to look like this and from here we write d n by dt you expand this and this will give 2 W n. Now here now see that this is actually equal to this this is equal to this here this is the thermal equilibrium condition. So we might as well cancel that but instead let us do a little bit of mathematical trick. So you could as well write this equal to so this gives me W down that W up n 0 L minus n 0 U. So here using the same average value of the probability I can write this 2 W n 0 L minus n 0 U of course this term is there 2 W n so minus 2 W n. So this is equal to now this is nothing but the population difference in thermal equilibrium. So we call it n 0 so this 2 W n 0 minus n. So this is the expression for the rate of change of population difference in terms of the average transition probability due to the internal relaxation processes written in this fashion. So this shows that any deviation of the population difference from the equilibrium value can give rise to non-zero value of this one or in other words when this is same as equilibrium value this is 0 which is understandable when system has already reached thermal equilibrium there is no more change in the population difference. But anyhow let us see how it looks like if n is different from n 0 then this integration can be carried out very easily n 0 minus at time t equal to t is equal to given as n 0 minus n at time t equal to 0 into exponential 2 W t. So what is the significance of the equation at t equal to 0 this is the population difference now the deviation of the population difference from the thermal equilibrium value as t goes to very large value infinity then this goes to 0 so n 0 minus n goes to 0 or n tends to 0 n 0. So that is what the relaxation process is trying to do any non Boltzmann distribution that give rise to this n which is different from the equilibrium value of n 0 will be brought in to this value if we allow sufficient time to pass and these are brought about by whatever the internal relaxation processes which are working all the time. So now we can understand that why it is important that these are present there so that we can see the absorption spectrum as I said earlier if these were absent then these two processes will make these two levels equally populated and we cannot see any net absorption signal. So these are the most important requirement for observing the steady state spectrum which are we are seeing in the IPR experiment or in NMR experiment relaxation process are very important this shows that relaxation process also this rate of change of this restoration of the population deviation from the thermal equilibrium is a exponential process. So we can give a time constant to it so if we write it as I said 2 W is equal to 1 by t 1 then this could be written as n 0 minus n t exponential minus t over t 1. So this is the character is time constant which decides or which brings back the thermal equilibrium if the equilibrium is disturbed by whatever means. Now how do we visualize this relaxation processes we can think of now in particular in IPR experiment that we have got there is a collection of spins maybe I have got a number of spins which are put in a magnetic field and we have restored this equilibrium is satisfied the Boltzmann distribution in case of alpha by n beta this is the Boltzmann distribution. So all the spins are not pointing in the same direction some of them will be parallel to the magnetic field some will be in the opposite direction. So effectively therefore the total sample acquires certain magnetization magnetization I call M which is the average of all this if we take over all the particles there then this magnetization a certain value because this 2 levels are not equally populated. Now suppose I suddenly switch off the magnetic field then this magnetization has to disappear the all spins will get randomly oriented and that takes certain time. So that time is the time that we say here the system is trying to go from certain orientation to random distribution. So we can call this is the relaxation time for this magnetization to get destroyed and this will follow exponential time difference of this kind we can do the reverse experiment. So the spins are present I have got a number of spins are present and there is no magnetic field. So there is no particular direction of quantization so number of alpha spins are equal to number of beta spins there is no net magnetization. Now suddenly suppose I switch on the magnetic field then they are going to reach this distribution and magnetization is going to develop how much time it takes to bring up that magnetization that process also is going to happen in this fashion in exponential manner. The time taken or characteristic time for such process is called the relaxation time for this restoration of the magnetization. Now here you see that for this 2 level system whenever the transition has to take place from here to there it involves absorption of radiation but this relaxation process tries to maintain the equilibrium. So you always try to adjust its population such a way that thermal equilibrium is restored. So in particular if there are more particles here than here then when this has to reach thermal distribution some amount of energy has to be given out from this spin system to the surrounding. So this process therefore which restores the equilibrium distribution between the 2 level we call that spin lattice relaxation process. Now here lattice is a very generic term it implies that for reduced portion of particles there is an involved of energy exchange between spin system and the surrounding and the process which does that we call the spin lattice relaxation process and because it involves exchange of energy in the surrounding and the surrounding is given a very generic term as the lattice. This is called the lattice so not necessarily a crystal lattice in fact surrounding in not necessarily be the solvent it could be different degrees of freedom of the molecule itself. So this is given a characteristic time that T1 I have given here and you see that this W it is average of the probability of upward transition and the downward transition each of them involves exchange of energy average value of that is 1 called 2 times average value of this is 1 by T1. So they are more efficient the relaxation process shorter will be the T1. There is another kind of relaxation process involved in magnetic resonance to understand that again we go back to this idea of net magnetization in thermal equilibrium the spins point either along the direction of the magnetic field or the opposite direction to generate a net magnetization along the direction of the magnetic field. The direction of the magnetic field is z direction then at thermal equilibrium this magnetization as let us say m x m z this sort of these 3 components there but at thermal equilibrium there is no x component and no i component only z component is present there. Now suppose there is some disturbance to the spin system and net m x and m i component are created then what happens again there will relaxation process which will try to restore this m x not equal to 0 to this state. Similarly m i which is not equal to 0 will try to bring it to this state because these are again non Boltzmann distribution non thermal equilibrium to restore thermal equilibrium this m x m i component of the net magnetization must go to 0 that also involves not the energy exchange with the surrounding but the spin system must rearrange in the epsilon in such a way that all the m x and m i component goes to 0 that is also a relaxation process. We call that spin-spin relaxation and it is easily it is written by given a time constant written in terms of designated as t 2. Now finally we just examine what role it plays in deciding the signal of an absorption spectrum. Qualitatively what we are observing here are the result of three processes this is the alpha beta this is alpha and radiation comes in so this causes this absorption, stimulated emission and the relaxation which is trying to maintain the population difference. So this is the relaxation process so this is stimulated emission this is the absorption. So if now if this rate becomes very strong a very efficient compared to this one then these two population difference becomes smaller and smaller that is this process is not efficient enough to bring restore the thermal distribution. So the population difference will go down and effectively signal will also go down. So in magnetic resonance this efficiency of these two process can be decided by the this transition alpha perturbation beta this we have seen earlier lecture that this is nothing but B 1 field that is present there in the x y plane and square of that gives rise to the probability of this sort of transition there. So here this is related to the microwave magnetic field that is used to observe the EPR signal and so square of that B 1 square is proportional to power of the microwave frequency. So if the power becomes high this becomes more and more efficient and this may not be able to cope up with the stimulated absorption emission process. So signal will become smaller and smaller because these two level becomes more and more equally populated. So one has to be careful in deciding how much micro power to use to observe the EPR signal. For those systems where the relaxation is very efficient this will not be of much serious concern but for some systems when it is relaxation is not very efficient this could be a potential problem. So you see here therefore the relaxation process which are always present there plays a fundamentally important role in allowing us to see the EPR signal and without that we may not be able to see the spectrum at all if the relaxation is very inefficient with this we come to the end of this lecture.