 Hello we are going to continue our discussion on block equations. In our previous lecture we saw that the block equations in the presence of a magnetic field and an oscillating magnetic field which rotates in the xy plane the time dependence of the magnetization is given by these three differential equations. This is for the x component, this is for the y component and this is the z component. Let us briefly recapitulate how we arrived at these three differential equations. Here this is the Zeeman magnetic field pointing along z direction and B1 is the amplitude of the oscillating magnetic field which is rotating in the xy plane with an angular velocity omega here. So how we arrived at this one? We first have this time dependence of the magnetization in a magnetic field given by this equation. Now when only magnetic field that is present here is the Zeeman field then this give us this simple differential equation and then block introduces the relaxation of this three magnetization in this fashion that is Mx and My decrease with rank constant of T2 and Mz increases with T1 to bring the non-boltsman magnetization to the equilibrium magnetization here. Then the total magnetic field that the magnetization experiences is due to the external magnetic field B0 and the oscillating magnetic field B1 cos omega t in the x component B1 sin omega t is the y component this is total magnetic field then we get these three equations. Now here this x small x small y and small z these are the laboratory fixed x y z coordinate and in particular the B0 the Zeeman field is applied along the z direction. So to solve this one one can of course use the mathematical techniques of first order differential equation and solve them that is possible but somehow it is not very illuminating. So we are going to solve this differential equation in a different way using a special technique called rotating coordinate system this technique is very much used in magnetic resonance. Suppose I have got a coordinate system let us call it capital X, Y and Z and this is rotating in a certain direction may be let us I have say I have got this capital X and Y and Z a coordinate system which is rotating about an axis at a certain angular velocity omega. So in this rotating coordinate system how is the equation of motion of any vector going to change? Let us first learn that. Suppose I have got a vector F which is a function of time so this could be written as in terms of an usual coordinate system which is my let us say laboratory coordinate system which is x, y and z they are of course fixed with respect to space and this is the vector which has got three components in this coordinate system this is I in vector J and this is in vector K. So time dependence of this is given by the time derivative of each of this component here nothing new is happening here this is I, J, K and three little vectors fixed along this x, y and z direction so rate of change of this with respect to time is given by the corresponding rate of change of this three components of the vector along this three directions. But now we want to be more general and suppose now this coordinate system is rotating in a certain direction and angular velocity omega around this axis how is this time derivative going to look like with respect to these coordinates little x, y and z which are rotating now that is the exercise we are going to do now. So how does the motion of a vector appear with respect to a rotating coordinate system so it is a type of relative velocity or relative motion so let us have got the coordinate system again x, y and z and this is the vector pointing in certain direction so it is a question of relative motion if I sit on this coordinate system the vector itself will appear to rotate that is understandable that with respect to the vector this coordinate system is rotating means with respect to the coordinate system the vector is rotating so it is a matter of rotation of the vector how that appears with respect to a starting coordinate system so let us say that this is the direction of the angular velocity omega and from some origin this vector r is pointing. So as I said earlier the rotation of the coordinate system is as good as imagining that the vector r is rotating around the direction of this angular velocity omega. Suppose at the time t this was here and then it is rotating in this way so after some time this will appear here but this vector is different from this vector because the direction has changed may be it has not changed it is rotating in this way so the change here is I call it d r this is actually equal to r plus d r what would be the magnitude of this d r if this angle is theta then I draw a normal here this normal there so at time d t the change in the sort of amplitude of this is d r so what is d r by d t so from this angle this angularity is r cos theta and this angularity it is r sin theta so if the angular velocity is omega then at time d t this angle put it here this angle d alpha will be this is actually equal to d alpha this is the angular velocity at time d t this much angle is formed here so this will be given by this is the radius and this is the angle so this will be given by r sin theta d alpha so this is therefore is equal to d r by d t is r sin theta d alpha by d t which is equal to r sin theta omega that is straightforward so this rate of change of this is equal to r sin theta times the angular velocity so this gives the magnitude of this now we have to also worry about the direction of that this direction d r this increment vector is in the direction which is normal to r at every state of time and also normal to this direction of angular velocity omega if I write that properly using the vector algebra this will look like d r by d t is equal to this so this gives the correct magnitude as well as sorry r sin theta omega so this gives the correct magnitude as well as the sense of this direction now coming back to the our rotating coordinate system when this rotates I have another set of coordinate let us call it x y and z this they are rotating it is two certain axis angular velocity vector omega then this little each of them will have its own i j and k at this little vectors they keep on changing with time so I can therefore write using this equation d i by d t is this r vector is replaced by this in vector i similarly so once again that it should not get confused with respect to the fixed coordinate x y and z I have its own little i j and k which are fixed with respect to the laboratory fixed coordinate let us say but the other coordinate capital X and Y and Z they are rotating in certain direction with angular velocity omega then the unit vector associated with this rotating coordinate system is this i and j and k and they have got this set of time dependence so now coming back here now we can now make it more general by saying that this i j and k also depends on time in this fashion because for the rotation with respect to this certain direction with omega then I can write in more general way that f is i times f x j times f y k times f z but now this time limit that I got with respect to the fixed coordinate x y and z now if i and j k they are also changing because with rotation then I have to use another term here which is the rate of change of this with respect to time so this will be f of x d i by d t plus f of y d j by d t plus f of z d k by d t so this gives the complete time derivative where coordinate system is also rotating. Now I have already got this relationship this in terms of the angular velocity vector omega see if I write this here this will look like i t f s by d t plus here f x times this is omega plus i now we can collect all the terms that involve this omega this gives d f z by d t plus the omega term i f x j f y k f z see how all these things have come together and this is nothing but this vector itself so we can write this is equal to i d f x by d t j d f y by d t k d f z by d t plus omega cross f these three terms written as del f by del t this is omega cross f so this is the same as d f by d t so this is the equation we are trying to arrive at what is the significance of this this says that this part is the time derivative with respect to the rotating coordinates which are rotating around this axis with an angular velocity omega so where this i j k the unit vectors are rotating with respect to those rotating unit vectors this is the time derivative and this is with respect to the space fixed unit vector i j k so it also shows that if this term is 0 then with respect to the rotating coordinate system the three component of this vector f do not change so this is the equation we will find very useful so another way to look at it is that if I want to know the time derivative with respect to a rotating coordinate system that derivative plus this vector product together gives the time derivative with respect to the space fixed coordinate from here now we will try to see how the magnetization can be written in terms of rotating coordinate system and its time dependence can be derived in exactly say analogous fashion this is the differential equation creating the rate of change of magnetization in the presence of magnetic field here so you see from this I can straight away write if the that in a coordinate system which is rotating with an angular velocity omega this will look like this is the extra term that comes from here because the rotation coordinate system so this is the time independence of the magnetization in the rotating coordinate system and I get the extra term here and that is the way it is going to be so what does it mean this means minus omega cross m which is this case mean compare now this equation with this equation this is written in terms of the laboratory fixed coordinate system and this is in terms of a coordinate system which is rotating in an arbitrary direction with an angular velocity omega see they look essentially very very similar except that this effective B becomes different in the rotating coordinate system different from the magnetic field B that is present here so when the coordinate system rotates with an angular velocity omega the effective magnetic field becomes this so and then I can use the essentially the same type of time dependence as in the static coordinate system so with this now you see how easy to visualize the motion of this magnetization in a magnetic field so when the my B was B 0 k and I will now find out the time dependence of the magnetization if I choose a coordinate system which is rotating around the z axis with an angular velocity this then what happens that if I choose the frequency such a way that B effective becomes 0 then my Bm by dt becomes exactly 0 so that means that in the coordinate system which is rotating around the z direction with a frequency omega such that this is 0 or in other words my this implies that omega is actually equal to gamma B 0 then in that rotating coordinate system the magnetization does not change it appears static what is the consequence of that if the magnetization appears static in a coordinate system which is rotating at a frequency omega around the z axis then in a static coordinate system the magnetization rotates with the same frequency see how nicely come to the conclusion which we drew earlier is that magnetization precesses in the laboratory coordinate system with this frequency and when you look at the rotating coordinate system rotating with this frequency the magnetization appears static the picture is the same so this conclusion comes so easily by choosing a coordinate system which rotates in this particular fashion so this is the advantage of using rotating coordinate system we get better insight and expressions also look somewhat simpler so how will the block equation look like in the rotating coordinate system now what is the rotating coordinate system we should choose here here we chose the frequency which is the normal frequency but for this situation where this B1 is the micro magnetic field which is rotating in the xy plane in this fashion I choose a coordinate system which is rotating in the same angle of frequency as this one around the z direction that is it will be having an angle of frequency omega and direction is the z direction then what will happen to the B1 field in the laboratory coordinate system the magnetic field is rotating in this way but in the coordinate system now I have got this which is rotating in the same frequency as this one then this B1 field will appear static now so at time t equal to 0 if the B1 field is applied along this x direction and the coordinates will start rotating at time t equal to 0 from the x direction then in the rotating coordinate system the B1 will always be along the x direction there will not be any y component for that so we can therefore write it very easily from this knowledge that with when a coordinate system rotates I change the B by this and here in particular the B1 also has only one component it is the x and y component so here let us I modify right here so that the difference becomes easy to visualize this will become B0 minus omega y gamma E my this is the change of B0 value that is the way you come we got the answer earlier now this is the y component of the magnetization in the laboratory coordinate system but in the rotating coordinate system this is absent so this becomes 0 minus now I should call it now dm x by dt this is the capital dm x by dt please indicate that this is the rotating coordinate system similarly for y component d my by dt will be B0 will be replaced by this one yeah I should have changed this also this will be capital Y the rotating coordinate system this is the B1 magnetic field which is always present in the x direction this will be present here this is a mistake here t2 and finally this is capital Z this is again the y component of the rotating magnetic field which is not present in this coordinate system this is 0 this is t1 see how easily we can now transform the this laboratory coordinate system that magnetization evolves to a rotating coordinate system so I have got a slide here now capital X Y and Z are the rotating coordinate system and then now the total magnetic field appears only as the x component of the microwave and the Zeeman field is around the y I am sorry in the z direction and then this is the time dependence of the x y and z direction now we will try to get the steady state solution of the rock equation when the appear experiment is done all the time dependence have reached a steady state value and the spectrum is recorded as function of either frequency or magnetic field we get the steady state value of the magnetization that is directed by the spectrometer. We can solve for the steady state value of the magnetization from this three differential equation by setting this time derivative to be 0 dmx by dt is 0 dmy by dt equal to 0 dmz by dt equal to 0 then we do some algebra and the solution is given in this fashion this is called the steady state solution of block equation in the rotating coordinate system here we have made a small substitution gamma e b 0 is defined to be omega 0 and delta omega is omega 0 minus omega this omega is the angular frequency of the micro magnetic field and this is the frequency corresponding to the Zeeman field or Larmor frequency that is how it looks like. So now epr spectrometer can be set to detect the mx component of magnetization or my component of magnetization usually we look at the my component which is called the out of phase component because out of phase respect to the this rotating magnetic field which is present there which applied in the x direction so we are looking at the y1 which is 90 degree out of phase but one can detect the x component also which is the in phase component. Now how they will differ the appearance is given by this magnetization as a function of now the frequency of the micro magnetic field that is this omega so if I plot this that will give the epr line shape as I said that the usually the my component is detected so epr line shape will be given by the shape of this if you plot now my as a function of omega what you are plotting is this function in the that is the my in the rotating coordinate system this will look like this where this peak corresponds to omega 0 so this is called the absorption profile or absorption spectrum similarly if you plot the mx component this will look like this this is called the dispersion spectrum usually this is detected in the epr spectrometer my is proportional to if here if we neglect this term gamma square b1 square t1 t2 that is neglected that is very very small than 1 then it looks like this now this is exactly similar to the form y equal to 1 by 1 plus x square which is the Lorentzian so the epr absorption spectrum will therefore appear Lorentzian when this term is neglected here say when x is equal to 1 the value of y becomes half the same way here when omega 0 minus omega t2 is equal to 1 then the signal height becomes half of that signal height because half of that so here this will be half of that when this value is 1 by t2 similarly here also it will be 1 by t2 or in other words this totals this one if I call delta omega half corresponds to the half intensity of this one or full width at half maxima full width that half maxima corresponds to delta omega half this gives 2 by t2 so when the spectrum are recorded as function of frequency we can straight away get the spin-spin relaxation time from this relationship the second consequence of this block equation here see the intensity of this my is proportional to b1 which is the measure of the micro magnetic field when b1 increases the signal height will also increase is proportional to b1 for micro power is p then p is proportional to b1 square so when the epr signal is recorded at various setting of micro power the intensity will change as square root of power all these are true when we have neglected this factor this one but as you keep on increase the micro power this becomes more and more significant and it is possible that this may not be neglected then the line shape will start showing distortion so we call this signal when this because appreciable then this appears here as how much square p1 t2 so the signal will now try to become smaller and smaller this is becoming appreciable we have got b1 here but then as this because larger that this signal will start therefore initially keep going up and then because of this will start going down so we say that it is the system is undergoing partial saturation that is relaxation process is not able to maintain the population difference well so signal height goes down but this itself could be used as a tool to measure the spin lattice relaxation time how so let us call this factor because saturation factor so if I plot the signal height now as a function of various settings of b1 as I said initially will go up and then it will start coming down because of this one so by plotting the intensity as function of micro power I can get an idea of this value saturation factor so this is proportional to b1 square so from that experiment I get idea of this product now and then I can get t1 by knowing that from the unsaturated condition when the line shape is strictly Lorentzian power as small I can get t2 and partial saturation is achieved I can get t1 and t2 from this I can find out t1 also this technique of measuring the spin direction time is called continuous wave saturation technique now to measure the t1 and t2 from this line shape analysis one needs to keep in mind that the line shape should follow Lorentzian that means there should not be any unresolved hyper line then each hyper line must strictly correspond to one transition there is no residual line which are hidden here in that case one will get wrong value of t2 and if you get wrong value of t2 you will get of course wrong value of t1 because this is obtained from the product of t1 and t2 and also one has to have very accurate measurement of the micro magnetic field b1 that is not very easy to know unless one makes very careful estimate of b1 we can measure the micro power very accurately but how is that micro power forming a standing up inside the cavity and how the b1 field is experienced by the sample that needs very very careful measurement that is not a very easy task nevertheless this way of measuring t1 and t2 are possible if one takes care of these things. Now before concluding a little small note of importance that we do not do the experiment in terms of micro frequency we do the experiment at fixed micro frequency but vary the magnetic field. So we modify this block equation which are here in terms of fixed micro frequency the vary the magnetic field. So that is shown here the steady stress solution of block equation in the rotating coordinate system in terms of variable magnetic field here looks they are quite equivalent of course. So now to conclude that we have seen how the introduction of relaxation by block in the time dependent magnetization gives us to the line shape and it explains a host of things like saturation behavior how the t1 and t2 are intrinsically built into the line shape and one can learn from and one can measure those from the line shape analysis with this we come to end of this lecture.