 We have been discussing electron spin polarization in photochemical systems we have learnt how the single phase hyperfine independent polarization appears and the spectra appear in this fashion either is totally down or totally up nevertheless both of them are very much away from the thermal distribution. And now we are going to discuss and learn the mixed phase hyperfine dependent polarization the example of that is shown here in the slide this is the familiar acetone isopropanol experiment in the presence of evilite it produces identical pair of radicals and if the experiment is done in the direct detection mode the time observed appear spectrum after 0.5 microsecond shows this pattern the low field lines are emissive high field lines are absorptive and there is no relation whatsoever with respect to the thermal appear signal where the middle line is expected to be most intense and other lines will be smaller and smaller intensity. Here this sort of signal comes when these conditions are satisfied one is that reaction can take place either singlet or triplet state this is a triplet mechanism that we discussed earlier the triplet mechanism necessarily requires a triplet state triplet state. So if the reaction takes place in the triplet state or singlet state this polarization mechanism that we are going to discuss can work and the reaction need not be very fast compared to the spin load reduction of the triplet state but here that condition was necessary that reaction has to be very fast and be comparable to the triplet spin load reduction state. In Tm the polarization was created when the radical was created at the same time when we get this sort of signal these conditions are not satisfied. So examine if the electron spin polarization can be generated during the evolution of the radical pair that is that after radical say A and B reacts something like this happens. So during the generation process I get triplet mechanism. So if these conditions for Tm is not satisfied then can the polarization develop as the radicals are evolving may be that is the way the other mechanism works. So we will have to understand how the radicals evolve after they are produced. So this mechanism is called radical pair mechanism and we will see that the formation of the radical pair is the key to this mechanism that radicals are created in pair and they are evolving in each other's presence. So when they are created they evolve in each other's presence they ultimately becoming free what do I mean by that. The moment they are created let us say in a solution these radicals are surrounded by the solver molecules and they are influencing each other when they are very close to each other but after some time once they go away from each other and each of them get solvated I call them the set of free radicals free from each other's presence and EPR spectrometer actually sees these radicals. So going from here to there what sort of interactions can produce spin polarization. So we now make some basic statements which are so obvious that we will accept them to be true. For example radicals are created in pairs and in the vicinity of each other surrounded by the solvent molecules they must separate before they are detected in the EPR spectrometer and number 3 is very important they have an overall spin state of 0 or 1 that is they could be either singular radical pair or triplet radical pair. What do you mean by that is A and B when they together surrounded by solvent it has got a spin of let us say some unpaired electron this some unpaired electron. So overall spin state can be constituted which will be some combination of this and this. So alpha and beta so alpha and beta can be either here alpha or beta this can be here alpha or beta. So overall spin state will depend on the relative orientation of this spin over this spin. So this could be a singular radical pair or triplet radical pair depending upon the relative orientation of these two spins. So that is number 3 and number 4 is that if they are in the vicinity of each other this radical can as well react this radical of this sort of spin that they come together and then if they are opposite then they can form bond that is the basic requirement for forming a covalent bond that two spins must be opposite. So here the same simple principle is applied here that if A and B have opposite spin that is if they are singular radical pair then they can react and disappear as a diamagnetic product. On the other hand if they have parallel spin like this then even they come closer they may not react and go away from each other and go back to the bulk of the solution. So that is very important physical expectation that we have about the evolution of the system that radical pair will undergo this evolution such that the triplet radical pair does not react what a singular radical pair does. See each of these conditions are very very simple and understandable but nevertheless they will have profound effect on the ultimate outcome of the spin distribution in the radicals. And last point is that there are some interactions that can change a singular radical pair to a triplet radical pair and vice versa. This is not very obvious but we will therefore investigate this in detail little later. The point I am trying to make here is that this radical pair no matter what the spin state is the number 5 says that a triplet radical pair which starts with this one they can change over to this to become a singular radical pair or a singular radical radical pair comes in this fashion. Some interaction can keep the spin to produce triplet radical pair why it is happening or why it can happen is something we will see very soon. How do you visualize the singular radical pair and the triplet radical pair? This is the spins multiplicity of the overall pair of radicals. So I must visualize them as a combination of two spins of the two radicals. So here a magnetic field of the spectrometer this spins are processing in this fashion radical 1 radical 2 they are color coded. So when I say the radical pair has a singular spin state then they have this sort of a function alpha 1 beta 2 beta 1 alpha 2 net m s 0 and net s square also 0 if you see here the way these are exactly pointing in the opposite direction. So there is no net z component here in the direction of magnetic field this is therefore singular radical pair. In contrast to that let us say look at this one here here around R 2 both point in the same direction along the magnetic field. So here if you see the total component of the electron spin along this direction I get 1 unit of angular momentum. So this is a triplet radical pair with m z equal to plus 1 and m square will be 2. So wave function for this is alpha 1 alpha 1 similarly the triplet radical pair with m z minus 1 will have this sort of spin but the triplet state with a 0 component of magnetization along this direction has this set of spin orientation because net is given by this green arrow this is a m square which 2 the green is present here green is there here there is no green because m square is 0. So the way these are oriented around R 2 net component along this direction is 0 but it has still a net component in this direction. So this is the triplet radical pair in fact this is the main difference between singlet radical pair and triplet radical pair both of them have got m z equal to 0 but this has no m at all no magnetic movement here there is one there is one this this unit of this magnetic moment is present here. So how can the inter combination of the singlet r p and the triplet r p take place. So the way this relative orientation are it is not easy to visualize that this can flip over to this or this on the other end these two reasonably similar what do you mean reasonably similar is of course that one spin is in this fashion other is this way. So this is singlet r p and second one is just this and this triplet r p only difference is the relative phase of these two. So presumably inter combination between these two and these two does not look very very unlikely and in fact we will see that is what is happening here on the other end between singlet to these triplets are very unlikely because spin has to really flip from this direction to that direction that is not so likely compared to that here the phase has to change from this to this. The difference in the s and t 0 states is in the relative phases of the spin vector. So we see if such a inter conversion between s and t 0 is possible or not what is necessary for that conversion to take place all that is necessary is that this phase should change to this now may not equally in this direction and they are precessing in this way. So how can the relative phase change from this to this very simple if they precess at a just little bit different frequency. So if the precessional frequency is different for this to radical then this relative phase can change to this or this phase can change to this that is what is claimed here now. If the precessional frequencies of the two spin vectors of the radicals are slightly different just slightly different their relative phases will change time then a singlet can change to a triplet and vice versa. So now more pictorially one can understand of this s t 0 mixing process this is a singlet and triplet 0 state mixing process. Once more when the relative orientation of this two radicals exactly opposite I get singlet radical pair with a singlet wave function alpha 1 beta 2 minus beta 1 alpha 2 and when these two are in the same direction I get triplet radical pair with the wave function of the triplet state is alpha 1 beta 2 plus beta 1 alpha 2. So here you see the difference in sign is nothing but depicting the difference in their relative phases of this. When the phases are somewhat in between then one can call it a mixed state. So this is a wave function with a mixture of singlet radical pair state and triplet radical pair state and this is a coefficient and since this is not a static situation because they keep on this is in this fashion and the frequency is a different they can keep on changing their relative phases. So intermediate phase will be a mixture of singlet and triplet radical pair and the coefficients are going to be dependent on time that is very important. Now comes the wave the radicals are generated right at the time let us say pulse of laser comes and the radicals are created that at that instant the spin state of this will be decided by what the precursor states were there. See if the precursor state or singlet state reacting then the nascent radical also have all the electrons paired in a sense that this is alpha there will be beta this is alpha this is beta that is overall spin state of the radical pair will be also singlet state. On the other hand if the reaction was in the triplet state suppose A was a triplet state then overall spin state of this radical pair will also be triplet. So mathematically that could be set as the initial condition for reactions from a singlet precursor C s at time t equal to 0 is 1 that is at the triplet component here is 0. Similarly the reaction from a triplet precursor is C s component is 0 and C t component is 1. But these coefficients are the initial condition and in course of time of course these values are going to change. So all that is necessary is the difference in the precessional frequency. So which one decides the precessional frequency difference? Individual precessional frequency given by the normal frequency type of thing this is the resonance condition for EPR transition or in terms of precessional frequency it is gamma v and these and these are related. So if the g factors are different for the two radicals then the precessional frequency will also be different. So all that is required here is that the two radicals that I have here has slightly different in the g values. So in other words the spectrum may look like appearing at two different places in the magnetic field. This is b this is radical number 1 and radical number 2 because this is decided by the g value. So this should not appear exactly same place just a little bit away from each other. So that is pictorially shown here. So radical 1 appears here let us say is value g value is g 1 radical 2 has g value of g 2 they are appearing at different place. So delta g is not equal to 0. So this is enough for this two radicals to have different precessional frequency. But in practice most of the organic radicals which you have seen that g values are very nearly equal to free electron g value 2.00 something. So this effect is not very large effect is very small. It is not a very significant factor to produce the type of signal that we see here. So what other way the precessional frequency can be different? We bring in now the role of hyperfine interaction. Let us say radical 1 has no hyperfine line. So it appears at g equal to this g 1. Radical 2 also appears at the same place that is delta g is 0 for both of them. But it has a one spin half nuclear. So they give doublet line one line here other line there. So this axis is the magnetical axis but one can think of this as also as a frequency axis. So if this line is appearing in this place then between this and this is the difference in their precessional frequency. Similar between this and this is the difference in their precessional frequency. Therefore the difference of the precessional frequency of radical 1 and 2 will depend on what the nuclear spin state of this radical is. If it is plus half it will have one type of difference. If it is minus half it will have other type of difference. It is so happened that the magnitude is same but nevertheless the sign is different. So different combination of nuclear spin states give different precessional frequency difference. We can extend the argument further and say let us say R1 and R2 both have hyperfine line. This is also doublet and R1 also is doublet of the different value. So this is smaller coupling constant is bigger coupling constant. Now there are 4 possible arrangement for the radical when they encounter each other. So radical 1 can have alpha or plus half nuclear spin radical 2 can have plus half spin state that means this transition is here for R2 this here. So difference in precessional frequency is this much between this and that. Similarly here both could be minus then the difference in precessional frequency is between this and this. Let me do it here. This is the same delta G is equal to 0. So one possibility between this and that. This is one difference in precessional frequency when this has got alpha spin for the nucleus. This much is the difference in precessional frequency. But then the radicals can have all possible nuclear spin state plus half and minus half both of them. So other combination could be that between this and this. But this radical comes with alpha spin and this radical comes with beta spin and this much is the difference in precessional frequency. Same is true for the other one from between this and that. So you see that it depends on what sort of nuclear spin states are involved in this radical pair here or there. The precessional frequency is going to depend on that. So higher the difference in precessional frequency more effective will be this sort of singular to triplet radical pair mixing. More quickly it will go from this to this. So more efficient mixing will take place for those pair where the difference in precessional frequency larger. So outside line, pair of line will experience more such mixing, more efficient mixing of this kind. So further there more is the difference and more effective will be the singular to triplet interconversion. One of the condition was that when the radical pair has singlet nature there is finite probability that they can react and disappear as a diamagnetic product. But when they have got parallel spin then they will survive. Again here the difference in their survival. Now if the radical pair started with a singlet radical pair that is that reaction took place a singular state initially they were in the opposite phase. So those radicals which get converted to the triplet nature they will survive and those which are not they will die away. So survival probability for a singlet radical pair depends on how quickly they evolved to a triplet radical pair. They will be seen as the free radical free from each other and detected in the spectrometer. On the other hand if the reaction took place in the triplet state then at the time of the creation both the radicals have got parallel spin. So radical pair is a triplet nature then this sort of interconversion will converts them to a singular state and they will disappear. So whatever surviving probability that these two radical have they will be seen in the IPA spectrometer and I can get a spectrum. So the difference is again the way the precursor are the precursor decides how the final fate is going to be. In one case the one which is evolving to the singular state will die if the reaction started from the triplet state. In other case those who are starting from the singular radical pair whatever is evolving in the triplet state will survive. But in the case of other way round is that whatever is not evolving they will survive. Again summarize the dynamics of this one. So the precursor decides the evolution in this fashion that for singlet precursor at the time of creation these two radicals have opposite spin. We call singlet RP and for triplet precursor the initial the spins of the two radicals are parallel we call them triplet radical pair and then this ST0 mixing takes place by this sort of difference in the precision frequency. So the wave function here has this mixture of singlet radical pair and triplet radical pair and these coefficients depend on time. So survival probability depends on to the extent that the triplets have been produced that is square of this one. And initial conditions is that CS0 is one CT0 is 0 for this singlet precursor and for triplet precursor exactly the opposite CT0 is 1 CS0 is 0. So here radicals that evolve to triplet give appear signal. Here radicals that do not evolve to singlet give appear signal and you see that exactly opposite effect is everywhere and that is what we see in the experiment also that in the singlet state react we get one type of this sort of behavior and the triplet reacts we get other type of behavior. And from this radical pair when they separate away from each other these radicals are with a stirred signs for the polarized radical those are seen in the experiment. Now more mathematically this qualitatively we understand why it is undergoing this singlet triplet interconversion. Now we can quantitatively calculate that by knowing their precision frequency. So Hamiltonian of the radical pair will be Hamiltonian of the radical individually plus this exchange interaction when they are nearby. Where is that when they are formed in the then vicinity of each other they can influence each other and form a triplet pair or singlet pair depending upon the exchange interaction between them. So that is given here. Here you see the other magnetic interaction here is that this is the Zevan interaction of the first radical and its own hyperfine interaction electron nuclear hyperfine interaction this is for the second radical its Zevan interaction and electron nuclear hyperfine interaction this is the exchange interaction. So total wave function is a combination of the singlet state radical pair and its own nuclear spin function which could be dependent on the nuclear spin state. Similarly for triplet radical pair function and the nuclear coordinates of the corresponding radical pair. Remember they define the polarization to be n beta minus n alpha by total number of spin. So we can you know this can be easily calculated from this sort of average value of the Sz angular momentum that gives alpha spin and minus Sz gives the beta spin. So that is p will be equal to 2 times the average Sz. Now the way now this equations are written that polarization of radical 1 at time t is 2 times Sz that average value of the z component of the spin angular momentum of radical 1 and this is given by this one wave function and this is the Sz operator. Similarly for the radical 2 again this is the similar expression. Now the way these are you see they are exactly opposite p 1 t is equal to minus p 2 t and no net electron spin polarization takes place there. It only shows that the spins simply get distributed such a way that one becomes this way other becomes that way. There is no net polarization. Now we can use the time dependent throwing a equation to see how the wave function is going to evolve and how the coefficients are going to evolve that is the way the equation is. So here the term which are looking very unfamiliar is actually this one but if you carefully this is quite meaningful because this is the precessional frequency difference there is a omega of first radical omega second radical this one difference in precessional frequency. So that is decided by the delta G value we have already encountered that delta G decides the precessional frequency difference and also it is decided by the hyperfine coupling constant of various nuclei and the corresponding mi value here these are the different mi values which pair is present in a particular radical pair. So that is for first radical and second radical so this is perfectly understandable now. Now mixing is possible only when this singlet triplet energies are almost degenerate that is J is very nearly going to 0 otherwise this mixing will not take place because singlet energy level is here triplet is there unless they come closer they will not be able to mix. So the way the radicals are evolving that aim is that when they are in the vicinity of each other they have to go away from each other being a radical solvents separated from the other one radical B again separated from each other this they have to physically go away from each other. Now for that to happen lot of solvents must collide with each other this random Brownian motion will try to push them away from each other so there will be lot of interactions that diffusive motion that A and B will let me go away and they will again come back this sort of multiple re-unquanted will take place and every time they are nearby too close then the splitting increases they do not mix when they become comparable they again mix so this mixing takes place continuously for quite some time before they go away from each other. So during that process then whatever survival probability that is there for the triplet radical pair they will be seen to produce the EPR signal and you see that interaction is very much dependent on predominantly by the nuclear hyperfine coupling constant and the which pair of lines are involved there so outer lines will have maximum difference in potential frequency they will show maximum effect in the polarization. So these are further mathematics which I have seen only written the final result random diffusive motion and multiple re-encounter is involved there before the radical separate A from each other the polarization of radical one for this A and B A B is that what sort of pair you have in mind say this is nuclear spin A and nuclear spin B whatever that is this type of pair A and B that depends on the magnitude of this and also sign of this one plus it depends on the initial condition sign is understandable because one pair if it is looking at the polarization of this one for this to this is positive for other one here for this to this is negative so polarization is going to be opposite also depends on the magnitude of that in a square root of fashion and the initial condition the singular state is the reacting state or the triplet state is the initially reacting state that we have already understood earlier that how it is going to be different for the initial state here initial state decides the overall outcome S G n means the sign of this one whether this is positive we have one type of polarization is negative is other type of polarization and this is already true that polarization of one radical is exactly opposite of the other one so this is the final expression of the polarization of radical one for a given pair of spin state sign of this and the J value as a sign of this and this together J s decides the triplet is higher or the singlet is higher for the radical pair and initial condition and square root of this magnitude of that one so summary is that polarization depends on the square root of the precessional frequency difference is all depends on the sign of the product of this and this this depends on the precursor the singular state or triplet state usually triplet state is higher than the singular state because this is the usual bond formation the two radical a and b is are coming together then when they have got this singlet nature that is opposite spin they form a bond of this kind here another if they are triplet that is spin parallel a and b then they will not form a bond because in energetically not possible and in all this ultimately the net result is that you see the how the precursor state decides the sign of this exactly opposite fashion that if the triplet is reacting the spectrum will look like e a type that is this type of thing low field line are emissive and high field lines are absorptive this is for triplet precursor similarly if it is a e which will look like this type of and presumably this low field lines are absorptive and high field lines are emissive this is for singlet precursor that is the summary of this radical pair mechanism. Now a little more quantitative discussion that as this polarization depends on the square root of this difference in precessional frequency the outer lines will be more polarized than the inner lines but the degeneracy such that the outer lines have a lower intensity than the inner lines so they go in the opposite fashion. So, net result will be the observed intensity will be a product of the degeneracy factor and the square root of the precessional frequency difference and the sign. So, this complex way the ultimately the spectrum will look like what we see here the outer lines are more polarized than the inner lines but the inner lines have more contribution from degeneracy. So, together this will be the overall appearance of that but then you have considered two mechanisms the radical pair mechanism is we have just discussed and earlier we have discussed triplet mechanism. So, in reality it is not exclusively this or that it is a combination of both the mechanisms that usually will always operate but then dm may or may not take place reason being that the reaction has to complete with the spin lattice reaction of the triplet and which is very fast but rpm will always take place as the individual radicals have to diffuse away from their partners that is this is the sort of gemini radical pairs are created and they have to diffuse away from each other that will always with the case whether the triplet reacts or the signal state react. So, this rpm will always take place tm may or may not take place only thing is that for tm to take place triplet state must react and also the rate of reaction has to be very fast and be comparable to the spin lattice relaxation rate of the triplet state then only tm will dominate. So, here is the example where the tm dominates now but then rpm is also there if you see very carefully that this height of this line is more than the height here this is a corresponding line but for a true tm this will be same as this the way the absorptive spectra is this line should have same intensity as this one this intensity should be same intensity of this one but this is not true. So, there is a little bit of net absorptive signal is present here it is dominantly tm and little rpm that is why this line is somewhat smaller and so this is also consistent with this prediction that for ea it will be having this set of polarization this line shall be emissive this line will be absorptive. So, that little absorptive component is causing this to become smaller even the quinone radical here these lines have slightly bigger intensity than these lines here I mean this is also slightly bigger than that one. So, there is a residual amount of net absorptive signal is present here. So, it is also so dominantly tm and little rpm other than here if it is pure rpm then the two radicals are identical. So, delta g is 0 precisely for both of them. So, intensity for this if we calculate now. So, radical 1 if I calculate in this fashion something like this radical 2 is exactly same same. So, for the this middle high power line precision frequency is precisely 0. So, here it is the if the values are delta omega this have to be minus delta omega, but here it is actually delta omega is equal to 0. So, intensity should be precisely 0 here, but that is not true again for rpm this will be exactly anti symmetric in this type of thing that whatever intensity I see here I should get exactly opposite value in other side. So, signal will be exactly anti symmetric because the values are in this fashion, but that is not happening here. So, these lines are somewhat more absorptive than these lines that means it also has a residual absorptive signal present here which might be due to a little bit of triplet present here though there are controversies whether this indeed triplet or not, but we will not get into that debate right now at least it shows that it is not a pure rpm which is operating here. So, now we have really come to an end of this discussion on electron spin polarization in photochemical system and we have seen that many interesting things can happen and just a pure observation of such polarized DPR spectrum gives us a wealth of information about the detail reaction dynamics of the system not just the chemical reaction. Chemical reaction often very simple one does not need this sort of technique to establish that, but the power of this is that it gives insight into the detail evolution of the spin system and the chemical dynamics if you want to call it that is starting from here how it goes to the singular state or a triplet state and then whether the inter system crossing is fast enough or not reaction is taking place at the comparable rate or not. I may not be able to measure most of these properties, but I can make a firm conclusion by looking at the spectrum which looks like this or which looks like this. So, all these are the power of this experimental observation and the mechanism which most definitely say something about the photo physical processes and photochemical processes and the relative rates of evolution that are all built into this appearance of the spectrum and that is not a very trivial conclusion. Most of the time inter system crossing it is a so fast that it is very difficult to measure them and EPR spectroscopy here is very slow in that sense it looks at the signal in the time scale of hundreds of nanoseconds by the everything is over it just so happened the relaxation time of the radicals which are less of few microseconds maintains the history of the evolution of the spin system. So, all these detail interactions which eventually produce the radicals which are free from each other they have been sort of maintained as a memory and the EPR spectrum reveal this history of the evolution of the photo physical and photochemical nature of this particular chemical reaction that is where the power and beauty of this technique is that is all.