 Today, we are going to learn how EPR spectroscopy can enable us to study anisotropic interactions. That is this today's topic is anisotropic interactions and EPR. What is anisotropic interactions? Anisotropy is something which means that some property depends on the direction and there are many properties that depends on the direction. One very familiar example is the way some substance behaves to a force that you apply from outside. Here is a simple example. We have this ruler, I will apply some force on this and try to bend it. Now if I apply force this way, see how easy it is to bend it but if I had to apply force in this fashion then it is very very difficult. In other words, this is an object, the way it responds to my force depends on in which direction I am applying the force. So this response of this object is anisotropic. In a similar manner the beams on the ceiling are given this sort of shape that is this side is wider and this side is thinner and the load is carried along this direction. So the example that I gave on the ruler is essentially applicable here also. The way this beam responds to the stress is not same in all directions. In many degree of resonance many phenomenon are anisotropic. We have not considered them so far. All we have assumed is that electron Zeeman interaction for example is isotropic and the way we wrote is that Hamiltonian was written as G beta i this fashion. So this was considered a scalar quantity. We also considered the isotropic hyperfine spreading constant which is written as a times i dot s. This is also a pure number but we did mention that electron nuclear dipole interaction is a directional dependent interaction which we have neglected so far. We will see those things here and then how far of this thing can be applicable or needs to be modified. At the outside we should keep in mind that a paramagnetic species which is undergoing sort of rapid tumbling in a solution for example then it experiences all possible orientations very quickly. So what property will measure for this sort of system is an average property. So all the directional dependences will not be seen in the experiment except to a certain extent the average value of this directional dependent properties will be seen there. So to get the true directional dependent properties we must stop this motion as far as possible. So one way to do is to study the solid state system or in powder form where the tumbling motions are almost gone or have a solution but freeze it to a very low temperature so that it is a frozen solution sample and again the tumbling motions are substantially stopped. We have seen many times earlier that angular momentum of a electron can come from either spin motion or orbital motion and both orbital motion and spin motion can give rise to the respective magnetic moment. The magnetic moment for these two motions are written in this fashion. This is the magnetic moment due to the spin motion e beta e s is the spin angular momentum where g is really 2.0023 and something like this and similar to orbital motion this could be exactly similar fashion we write g of l is 1. So here both the equation says that this vector and this vector are parallel to each other similarly this and this are also parallel to each other. Now if parametric species has both this angular momentum then they can combine together according to the quantum magnetic rules of additional angular momentum let us say this j is the total angular momentum made up of this l plus s then this will also produce certain net magnetic moment and I can write by adding this to something like me of j could be written as now here by analogy of these two equations I am tempted to write that something like let us say this angular momentum and some sort of g factor let us call it g of j and this is written strictly on the basis of the similarity that s gives mu s they are parallel to each other l gives mu l again parallel to each other. So this equation is fine it is addition of these two but can I write an expression of this kind that mu j that is magnetic moment arising out of the total angular momentum is something some constant here this gives the property constant between these and this is possible if this is true of course then these and these are going to be parallel and we are going to see in a moment that this net magnetic moment does not become parallel to the net angular momentum let us see how that is possible so using this one let us say I have got this is okay let us take this one first that is l vector and this gives the angular momentum of this magnitude and corresponding magnetic moment which points in opposite direction because of the negative sign g l is equal to 1. So let us say this is my mu of d2 l then I have the spin angular momentum let us say this is the magnitude of that and the direction and this angle is size that these two can be added according to the laws of addition of angular momentum given by quantum mechanics. Now the corresponding magnetic moment due to s will be twice the size of this one because this is value of this is 2 so this is mu of s the net of this and this is given by this addition rule of vectors. So this gives the vector j similarly the net of these two angular momentum is given by this addition this is the mu of j now it is obvious from here that because this is twice this one this vector cannot be parallel to this one therefore I cannot write equation of this kind that is I cannot have a sum g factor which gives a magnetic moment associated with the net angular momentum this cannot be a pure number that is not possible. So what is the remedy how do I proceed that this is not possible we can do it two ways we can use this total angular momentum expression the way we have written here and the corresponding magnetic moment. So g l is one so let us write one now we do not have to write that g of s and with this I can write the interaction of this magnetic one moment in a magnetic field in this fashion now this is very nearly equal to 2.0 or 2.3 let us write 2 which is approximately quite acceptable this is the way one can write the Hamiltonian which is fine. Another way of doing it is to say that because this cannot be parallel to each other I cannot use a pure number. So how about using a 3 by 3 matrix here mu of j is something I write in this fashion where this is a 3 by 3 matrix let us call it g x x then this and this need not point to the same direction. So both are acceptable only the approach is different here the approach is that we just treat this interaction by assuming that the interaction could be written as using this type of g matrix that this could be replaced at this one. So here we treat this s to be some sort of effective spin angle momentum and the interaction can be written in the same fashion that is we do not explicitly think of the contribution of orbital angle momentum the effect of this is to make this to non-linear so that is taken care of by changing g from a pure scalar number to a 3 by 3 matrix and this therefore is considered to be an effective spin angle momentum and that gives the same interaction which is given by this one. We will take this approach because it is sort of easy to visualize what is happening but nevertheless both are equivalent approach there. So here when I define this of course we need to define a coordinate system so this could be some coordinate system to start with where the parametric species is kept here this is the x y and z. Now in this product we have to keep in mind how these three items are multiplied this is shown here. So as I said earlier that mu is written as the product of this 3 by 3 matrix with this effective spin angle momentum given as to this sort of multiplication. So here B is the magnetic field written as a row vector because that is why the multiplication has to be carried out for compatibility of the rules of compatibility with the rules of matrix multiplication. Now with this one s is taken to be the effective spin so in a magnetic field this will give rise to two energy level e1 and e2 so this could be written as energy of this will be g of effective and this will be similarly plus g of effective total energy delta e is g of effective beta e b. So what is the g effective now so once again if you look at it that these two terms together give some sort of effective magnetic moment which is interacting with this magnetic field but we can look at it differently also that is we look at it as this is an effective magnetic field which is interacting with the spin angle momentum and that is the way the energy levels can be thought of that. So this gives effective g value for a given value of the magnetic field b here and the energy level of speed both are equivalent actually how we look at it. So this gives the effective spin that interacting with the magnetic field here or this gives the effective magnetic field which is interacting with the spin. Now with that idea now if this is the way it is should be able to get some expression of this in terms of these things total energy square this is a scalar quantity can be written as a scalar product of this with itself so that is done here this delta e square is product of this with this of course here to again the make sense of the multiplication of vectors and this matrices this one is to take the appropriate transpose of the matrix here that is done here. So then by simplifying the g effective square with these two constants gives comes out to be this here the b x b y b z are the three components of the magnetic field. So if b is let us say i b x is the three components and the magnitude of this is given by this then one can write this three components in terms of the direction cosines of this b with respect to the external coordinate system that we had earlier. So l x l y l z are the direction cosines of b in the similarly here. So then b square comes out then this quantity g f effective square is actually is this which is written here g effective square is the direction cosines of the magnetic field and square of this g matrix. So the square matrix defined by taking the product of these now if you see these matrix and is the transpose of this matrix. So no matter whether g is symmetric or not these will always be a symmetric matrix that is very important. So then because g square is a symmetric matrix now we can choose suitable coordinate system. So initially I had some coordinate system whatever it is in the laboratory and experiment is done and this is the paramanic center sitting here and we just took this x y and z the g square matrix here was defined. Now because it is symmetric matrix I can have a special coordinate by suitable rotating this one so that the g square matrix become a diagonal matrix. So these special coordinate let us capital X capital Y and capital Z are the special coordinate with respect to the molecule so that this becomes diagonal g square matrix is diagonal. So if it is square root of this 3 then I get g x x g y y g z z these are called the principal components of the g matrix. So here if the magnetic field is now pointing towards either this special coordinate or the principal axis capital X capital Z then you see the equations becomes very simple now that the magnetic moment along that particular direction x becomes beta i g x x s x which is what you know always get that this type of thing. But then this the important thing is that here the magnetic field is pointing to one particular direction of the molecule that is the principal axis similarly for s y and s z and if the corresponding energy of transition will be if the magnetic field is along the x direction in h nu is beta g x x v x similarly for the others. See in general the magnetic moment can be written in terms of this principal component g x x g y y g z in this fashion where capital X capital Y and capital Z are the unit vectors which are pointing towards this principal axis this is shown here. So these are the principal axis let us say b is pointing along this then effective g square which is from this expression that diagonal element appears to be this. So, these are the three direction cosines of b with respect to this principal coordinate. So this shows that the interaction depends on directions and depending upon the value of this g x x g y z z one can get complicated appear spectrum as the angle changes the effective g value changes therefore the spectrum will also change. Now how different this three values are that will tell us about the symmetry. So we have got three principal component if all three are same then we call this an isotropic system g is isotropic this is what we have been actually using so far in our previous examples g does not depend on the direction. Other possibility let us say two of them are equal third one is not then we will call this system to have an axial symmetry and if all three are different then we can say this is a rhombic symmetry in a sense no symmetry. So, far as the g values are concerned to get a interactions of this kind the molecule has to be kept in a present solution or in a crystal. So, right now let us consider the situation in crystals. So, crystals have certain symmetries see the parametric center is sitting in this crystal they will experience the local symmetry. Now local symmetry could be cubic symmetry here or it could be a uniaxial where there is a axis one axis of symmetry. So, that is if you rotate in this fashion it will be at least three fold axis of symmetry should be there or rhombic symmetry no symmetry axis higher than C 2 here. So, when I said a rhombic symmetry means no symmetry that is not quite correct this is the crystal can have C 2 symmetry, but there is no symmetry axis higher than C 2. If there is higher than C 2 then C 3 or C 4 then it will be either here or here. For example, here this one is C 4 axis of symmetry this is C 4 no doubt, but it will also diagonal if you take it will have C 3 axis of symmetry. Now for cubic symmetry a cubic symmetry is essentially will give rise to this isotropic g value these are various subclasses of cubic symmetry. It is a cube well in a indeed a cube all this lengths are same the way the atoms are placed or it could be octahedral or could be tetrahedral these are all subclasses of cubic symmetry and all of them have an isotropic g value. Now we take some examples of if you are in crystals in a single crystal all radicals are oriented in the same way. So, by changing the orientation and the resonance position of the magnetic field will change for all the radicals in the same way as you rotate the crystal all the radicals which are inside will experience the same magnetic field. Now here is an example of a parametric center trapped inside the sodium chloride crystal lattice. Here this block or the square is the place where a chloride atoms should have been there, but that has disappeared to make a vacancy and instead a single electron is residing here. So, these are some defect centers or color centers type of names given to this. So, this is the center of parametricism. So, here this belongs to this octahedral symmetry. You can see the sodium atom and one at the top of the bottom. So, here this is purely octahedral. So, that belongs to this three values to be same and so g value will be isotropic. What is the experimental observation that if I take this system let us say crystal and do the experiment as a function of magnetic field. So, this will give a spectrum somewhere let us say here and you keep rotating the crystal around certain axis and this will not move. It will be precisely at the same place that is the same thing what I am been trying to say that g value is isotropic and it does not depend on the orientation of the crystal with respect to the magnetic field. So, this remains constant. So, this is the place which gives me the value of v 0. Then the g value will be correspond to this one and this is not going to change no matter which direction the magnetic field is applied. In contrast there is another such vacant center here magnesium oxide crystal. Again there is a defect is created where magnesium atom has disappeared. It was done by irradiating this crystal with x ray. Now, so that is magnesium ion has disappeared and the same time this O 2 minus oxides were there from one of those this one electron was knocked out. So, this O 2 minus because O minus. Now, in the process the position of this thing has got slightly moved from the normal usual position here. Usual position is very similar to this octahedral symmetry, but because this has gone and so therefore, the little repulsion between this and this. So, that repulsion pushed this away from that. So, this therefore is no more cubic symmetry. So, it has a symmetry axis along this direction. Now, this the g value is dependent on the direction. That is if you do the same experiment here and rotate the crystal the position is going to change in certain way and wherever it changes I get a effective g at that particular orientation at that magnetic field. So, once again this is the symmetry axis C 4 axis is here and let us say B magnetic field is applied along this direction which makes an angle theta and this is the perpendicular direction. So, here if the crystal is rotated around this C 4 axis then because the symmetry axis the interaction is not going to change. So, this will be staying right here wherever it is. On the other hand here the interaction depends on the angle. So, this angle is varied in the y z plane then the effective g is going to change that will be seen as the change of the line position. This is the result here. So, you see that angle is varied from 0 to 100 degree. So, this is the position of the EPR signal in the magnetic field unit. So, it starts with 3233.1 gauss and as the angle is changed this becomes smaller and somewhere at 90 it reaches minimum again it goes back here. So, for it starts from here goes down down down again goes back. So, for each orientation one can therefore, determine the g value and this is shown here as the angle is varied g changes from 2.0033 to 2.0386. Now, this is 0 and 1 degree is nothing but the magnetic field is pointing along the direction of the symmetry axis. So, we call the g value to be g parallel value and 90 degree is this direction. So, that is the g perpendicular value. So, this expression can be fitted to an expression of this kind g parallel and g perpendicular expression and the angle is theta. So, that is for axial symmetry. Now, for ombic symmetry where all 3 are different the expression for effective g is given here depends on all the 3 principal component of g and also the angle that the magnetic field makes with respect to the principal axis. Now, to determine this if your spectra recorded by rotating the crystal successively in x y y z and x z planes then for this measurement each of them will give a curve of this kind then the corresponding elements of this g square matrix I have got this elements could be determined from this measurement and then one can now diagonalize it to get the principal component of the g matrix and then also the angle of the principal component of the g matrix that gives rise to the anisotropic interaction. Now, that is the way one can study the single crystal and find out the corresponding anisotropic interactions, but it is not always easy to form single crystal. So, the experiments are difficult in that sense. So, what one does is that one can freeze a liquid solution if parametric sample and then try to see if one can get similar results from studying the appear spectrum of a frozen solution or powder for example, if the crystal is not formed one can do appear on powder samples, but then main problem is that in powder and frozen solution the magnetic moment of parametric particles will point to all possible directions with respect to the external magnetic field. So, we have a randomly oriented spin system. So, here we need to find out that how the random orientations are reflected in the spectrum of the appear signal when recorded in case of powder for crystal for example, as the orientation changes this magnetic field position change for powder or frozen solution similar there will be magnetic field direction this is the signal. So, as the different orientations are available in case of powder. So, this will also certain range of allowed magnetic field as we have to find out whether all such magnetic field positions are equally likely or there are some selectivity. It is very obvious that all orientations are equally likely. So, we are going to see that that is even though all orientations are equally likely this all magnetic field positions are not equally likely. So, first let us find out therefore, that how many magnetic moment will point to certain direction here for a let us say the case of actually symmetric g. So, these are we have to find out how many molecules will point to a this direction given by r theta and phi and r remains constant theta changes from theta to d theta here and since the actually symmetric the g does not depend on the phi. So, for all values of phi that happens to be the this particular band here. It is easy to see that area of this band is 2 pi r square signed with the d theta. But total surface area is 4 pi r square. Therefore, the probability of finding a vector pointing towards the shaded region is given by the this divided by this which is signed theta d theta by 2. Now, we so when. So, these vectors which are pointing in this direction will give the same in the neighborhood of same magnetic field in the appear spectrum. So, where are they going to appear maybe they will appear somewhere here let us say. Now, if now this if theta is changed now the resonance position is going to change somewhere here. So, when we cover all the possible orientation how this distributions are going to be appearing at different magnetic field region. So, that is p theta d theta which is already found out to be this proportional to some distribution of intensity as a function of magnetic field b to d b. So, p b is just d is divided by this one. So, we have to find out this relationship b is equal to h nu by g beta e. Now, you have seen that for axially symmetric g this is a relationship. So, just do the algebra that is necessary to simplify this. So, b is given by this relation and that, but g itself is given by this. So, if we do the differentiation the p b happens to be in this fashion. So, this let us take a closer look at that. That means all values of magnetic field orientation are not allowed here it can range from let us say g perpendicular value to g parallel value. This is the range of allowed b and as the cos theta changes from 0 to 90 degree. And for each angle the b is given by this relationship. So, further angle theta I find b that b is a put here and here to find out the value of the probability. Here see the moment cos theta becomes 0 this shoots up. So, this goes to very very large value otherwise this smoothly comes to maximum b parallel. So, of course, this here this going to infinity is not very serious issue because this transitions width are supposed to be 0. But once we are give finite width then they all sort of broadens up and the sharp edges are gone. So, this is given here the blue has a very small width of transient one gauss. Then as width becomes bigger and bigger see how they become less rugged. Now here the g perpendicular value is here correspond to this magnetic field and g parallel will correspond to this magnetic field. So, we can get the corresponding values from this powder spectrum as well. But in practice we do not get the absorption spectrum we get the derivative spectrum. So, we have to take derivative of this one and that is the way this looks like. So, here so if you take derivative of this this will actually go up and come down and here again that is going to be. So, this point is corresponding to here and this point corresponds to this one. So, g perpendicular will be corresponding to this magnetic field position and g parallel corresponds to this magnetic field position. So, that is the way it is going to. So, these are the example for the magnetic field oxide vacancy and g perpendicular is here g parallel here. Now for rhombic symmetry we have to do the same calculation, but little bit more complicated. Instead of the shaded area now here you have to see the probability the vector points to a certain only small region here that r square shantitha d theta d phi somewhere here. Then as theta and phi varied how this similar distribution is going to change with the condition that g square is given by this relationship. So, here the absorption profile looks like this the given by the dotted line with a where the transition is no width, but with the finite width this colored line is the appearance of the spectrum. Now again if you do that derivative then the signal will look like something like this. This is the g x x g y y g z n g z z is the second to be the one which is farthest from the other two. So, here the g z z and g y is the intermitted values and here is an example that this c o 2 dot this is one radical which is trapped in the magnetic oxide powder and this is the g x x somewhere here g y y and g z z. So, with this now we take the next item which is electron nuclear dipolar interaction. We have seen this is the interaction which is highly directional dependent on the isotropic part is given by this sort of expression dipole double interaction is this. So, both of them have this type of s dot i type of interaction here this also is equal to s dot i. So, we can as well add this isotropic interaction dipole double interaction to get a single expression which looks like this. I is the nuclear spin A is now a matrix 3 by 3 matrix and S is the spin angular momentum and A here now you automatically a symmetric because of the way this D is defined. But here if you take the average of this diagonal element this turns out to be 0 that is average dipole interaction is 0. This again in tumbling motion is fast in solution one does not see this, but it is not true in frozen solution or in crystals. So, matrix A is symmetric hence it again similar to this we should be able to diagonalize the matrix by choosing suitable axis these are the called the principle axis of the hyperfine matrix. Now, average of this 3 principle component will be nothing, but A 0 obviously, because that is the way these are added that we add this constant isotropic value. So, that is the average value is the sum of all this and anisotropic values are now given by these 2 parameters B 0 gives how far the z is different from the other 2 and C is how this other 2 x and y are different from each other. So, B 0 and C 0 all called the anisotropic parameters and in particular B 0 is called uniaxiality parameter and C 0 is called the rhombusity parameter. Now, you can single crystal in principle one can find out the orientation dependence of the u-spectrum and axially symmetric is see this equation looks very similar to g anisotropic and in frozen solution again the same type of relationship can be derived as we did for g anisotropic. Now, here are some type of patterns we can get the axially symmetric I is half if C 0 that is axially symmetric now B 0 defines how far it is deviating here uniaxiality parameter here. So, you see B is 0 means it is really anisotropic. So, you get 2 doublet line now as B 0 changes which changes the xi various types of patterns. So, these 2 patterns particularly you can recognize that they are very similar to axially symmetric g type of thing. So, each of the hyperfine transition I get similar pattern as the xi becomes smaller they come closer and closer and closer and they give all sorts of pattern of this kind. It is also possible that g matrix and a matrix are both present the isotropic g and isotropic a. So, they can have this type of appearance then here isotropic g that is all 3 transitions here the pair of transition corresponding to ax a y a z as a center value of all 3 of them are same. So, g is isotropic on the other hand g is axially symmetric then I can get this 2 center of this and this is corresponding to a perpendicular center of this and this is the distance between this and this is a parallel and center of this is g parallel. So, this and this are different these are different therefore, but values are also different, but here another place isotropic a where a 0 is same for all of them there is a gap between the corresponding lines are same, but the g is not isotropic. So, these are 3 different values of g. So, that way one can get various types of combination and identify the isotropic contribution and the parameters of g and a from this powder pattern as well. So, we have seen how the e p s studies on single crystals powder or in frozen solution one can determine all the isotropic interactions arising from g and a. This slide shows the 2 textbooks from which we have taken much of our material in this lecture.