 We started our discussion of the motion of a magnet in a magnetic field and we have seen earlier that if we keep a bar magnet in a magnetic field, it undergoes 2 N from motion. This is the magnetic moment and if put a magnetic field there then it undergoes 2 N from motion as we saw earlier. It is the bar magnet, the magnetic field and it undergoes oscillation. We call this motion oscillation. We also saw that if the magnetic moment arises from its angular momentum then it could relate the magnetic moment M minus angular momentum vector times the Bohr magneton in case of electron. Specifically we related this orbital angular momentum to the magnetic moment that should be kept in mind. Why does it undergo this 2 N from motion because it experience a torque. Torque is equal to vector product of the magnetic moment and the magnetic field. Now if a magnetic moment which arises from its angular momentum then what is it going to happen? That time it is not undergoing oscillation, it will undergo a special kind of motion what do you call a precession motion, precession. What is precession? All of you must have noticed that when you play at top and it is rotating at high speed you will notice that this axis of its rotation also keeps moving, so that type of motion is called precession. Let us try to understand little more detail what it is. Here this is a simple pendulum. If you leave it from here it undergoes this 2 N from motion. Now here at this point the gravitational force Mg pointing downwards. This is the distance r from this point where it is hung. So you see that this r times force produces a torque. So this torque is trying to rotate this body around it. The same time when it comes to the equilibrium position because of its inertia it continues to move in the other direction and it goes back here. So this also has an angular momentum which is causing it to move this way around this point here. So I can therefore see that a torque is applied here which is changing its angular momentum. What is the direction of this torque? That is here if you multiply vectorially r cross f this will be the direction that is it will point out of the screen. Now what is the direction of angular momentum that will be r cross with the linear momentum the V is in this direction so Mv will be the momentum pointing in this direction. So that also has the same direction L as the direction of the torque. So for a pendulum torque and the L angular momentum they point in the same direction. This is not same when a top is spinning around its axis. What is the difference? I understand that let us look at this picture. This spherical ball is sort of simplified version of a top and it is spinning around its axis given by this vector L which is also the direction of the angular momentum. So this is the direction of its rotation. Now the earth's gravitational field acting on it the force mg trying to bring it down. So the direction of the torque is essentially same as that which is the direction perpendicular to the plane. But the angular momentum the way it is shown here and the torque now are perpendicular to each other. So angular momentum is in this direction torque is in this direction. So that way because the torque is acting in the perpendicular direction it cannot change its magnitude all it can do is change its direction. So this red vector L now tries to move away in a perpendicular direction. So then it comes somewhere here but at every instant of time the torque acts always perpendicular to that. So it brings somewhere here then here then there then here. So effectively that causes the direction of this angular momentum vector follow this cone and that is the rational motion and that is the type of motion we see when the top is spinning so this axis starts moving. So exactly similar thing happens in case of magnetic moment kept in a magnetic field when the magnetic moment is arising from its angular momentum here. So I have said that the torque is given by this expression is perpendicular to both M and B. So M is in this direction B is in this direction. So the torque will be acting perpendicular to the M and B that is essentially here will be the direction which is vertically up. So M will therefore undergo exactly similar precessional motion the axis of M will follow this cone. So the magnetic moment which arises from an orbital motion or which is associated with an angular momentum will undergo a precessional motion in a magnetic field. This is the torque which is a vector product of M and B. Now can we calculate the frequency of this precession that is the motion that this M is undergoing what is the frequency of this motion and how is it related to the intensity of the magnetic field B. This is not very difficult to evaluate. We have seen earlier that this angular momentum small L vector is measured in units of plus constant by 2 pi and L is the dimensional quantity. The torque is nothing but the rate of change of angular momentum dL by dt is equal to M cross B. Then using this relation small L and this capital L I can write this but you have also seen in our derivation that magnetic moment and L are related in this way. So I can use this and put it here. That gives me the rate of change of angular momentum which happens to be now L cross B. This gives the precession of the angular momentum vector in the magnetic field the same way. Now L can be replaced by M in this fashion is in the same relationship and that gives me the dM by dt is equal to some constant here and you see that the minus sign can be removed by changing the order of multiplication. So if we define a constant beta E by h cross is gamma E then this relationship become dM by dt is equal to gamma E B cross M. So this is the equation of motion if you like to call it. This E stands for electron and we call this gyro magnetic ratio. Let us try to solve this equation of motion and then see if we can find out the frequency of the precession. We first define the direction of the magnetic field. So let us say the magnetic field is applied along this z direction. This is my B. So it will be B times the vector you call it k in a vector k and the magnetic moment is in this direction is precessing this fashion. So in that case if we expand this vector product I get equations of this kind. dMx by dt is gamma E Bmy dMy by dt is equal to minus gamma E Bmx dMz by dt is equal to 0. So here the last equation is very simple it means that Mz is constant. So Mz is constant that is this component this component is constant. So if this is the angle that this M makes with respect to z axis then this is the component which is m cos theta cos theta that this value remains constant. On the other hand the x and y component then this is the Mx this is My and you see as it moves the component changes its magnitude those are given by these two equations. Now these are very easy to solve and solution is this Mx is m sin theta times cosine of gamma E Bt My is minus m sin theta sin of gamma E Bt. So this is equation shows that angular frequency of precession is given by gamma E B. So this shows that frequency of precession is proportional to the magnetic field. So to show that I have made a small animation here B is the magnetic field and this is the magnetic field vector the green arrow. Now I will run it it is precessing magnetic field vector is precessing and we are increasing the magnetic field slowly the rate vector is increasing and if you notice it it is the frequency of precession is increasing gradually B is very high it is precessing very fast and now we are reducing the magnetic field precession is going down again. So this equation which relates the precessional frequency to the magnetic field this frequency of precession is called Larmor frequency. So whenever the magnetic moment has an angular momentum and already when it is kept in magnetic field it is always going to undergo a precessional motion of this kind. This was given by this person Larmor whose picture is shown here. If we go back to our derivation you will notice that we have defined gamma E to be this is the Bohr magneton by h cross. So exactly similar derivation can show that for nuclear magnetic moment relationship will be there is a gamma for nucleus so proton for example will be this will be nuclear magneton by h cross this is once again nuclear magneton and this is Bohr magneton a little more clarification necessary here. See we have arrived at this equation where the magnetic moment is related to the orbital motion if you go one step back we actually derived it by using a sort of circular loop of wire where some charges q was moving this is suddenly orbital motion and from that we arrived at this one. So in general the angular momentum can come from orbital motion spin motion or a combination of the two. So in general this relationship is modified to write G of electron Bohr magneton times let us say S for spin angular momentum this will be equal to similarly G of this is for orbital angular momentum G value for orbital motion G for spin motion and it so happened that this is actually equal to 1 and this is equal to 2.0023 something for free electron. But if it is combination of electron spin angular momentum and orbital angular momentum then this becomes some general G vector something like J let us say J is the total angular momentum this will corresponding G value. So we have seen that the magnetic moment arising from an angular momentum will undergo recessional motion. What happens now if we apply another magnetic field starts from here this is x y and z and the let us say the magnetic field pointing is this direction which is b let me call this now a very huge magnetic field let us call the Zeeman field and this magnetic moment is pointing its direction it is undergoing recessional of this kind. Suppose I bring a small magnetic field and try to see what happens to this recessional motion. If I apply that small field let us call it maybe I will call it b little small little field and apply along the z direction. So we will apply it around let us say K what will happen this huge magnetic field is already there and I am applying the small amount of field there. What will this magnetic moment see it will find that the field has changed just a little bit. So whatever the recessional frequency that it had given by the relationship that is let us call it that omega equal to gamma e b this small applied here we just get added to this one. So this will undergo just a little bit of change in frequency. So nothing exciting is going to happen therefore change omega a little bit. Now I apply let us say b little in the x direction here let us therefore I will call it I then what happens you might keep in mind that this is really very very small compared to this one this could be our let us say 1 tesla magnetic field 10,000 gauss or so and this could be hardly a few mille gauss or may be gauss little bit here though I have drawn it in the scale but this is tiny tiny compared to this one. So then what happens or what we expect it to happen this magnetic moment is really pressing at a very high frequency and this fellow is sitting here. So it will only see that presence of this only for a very very very short time because it is pressing very fast. So effectively therefore this is not going to disturb this at all no real effect because it is going on and on very very fast this little bit is therefore not going to do anything. Now let us change our tack little more suppose now this time instead of allowing it to stay static in the x direction I now allow it to move in this direction in the x y plane. So make it is little bit understandable let us clean up the figure and draw it better. So now my this small field b little now starts here and it also moves in this x y plane but a sudden angular frequency let us call it omega little not only one in the scenario this is going on at this way very fast rate and this fellow slowly rotating in this direction then what will happen again pretty much very little because as it is rotating so fast it finds once in a while this is near it at that time it is not going to perturb anything again it goes ahead and keeps pressing. So this is so when omega little is very very small compared to omega nothing nothing significant is going to happen. Now let us gradually increase the frequency of its rotation around the x y plane then imagine as it this frequency of the small field comes closer and closer to this precessional frequency here omega little is becoming comparable to omega here then what happens now this will find that even though it is very fast this fellow is also following it almost at the same rate. So this is sticking around like tailing it all the most of the time because they are not exactly same the effect may not be very seen very much but nevertheless this finds that another one is just following me almost all the time. So again this equation of torque will be applicable so it finds another field is little field is here so it will also try to precess around it so it will try to therefore bend from this this vector is going to come out of that. Now the extreme case when exactly equal to omega then wherever it is this is just near that exactly the same phase there when as it is rotating this is also rotating the same frequency therefore there will be always a constant phase relationship between the two that does not change. So it finds that in the vicinity there is a small but fixed magnetic field is present there so this will also therefore try to precess around this small magnetic field. Now what happens in the process therefore once it is try the precessors here this has to precess in this fashion so meson is complicated but not very difficult to visualize it is complicated motion consists of very very high frequency precession on this but a low frequency precession around this one but if that happens now this has to come out of this and then we go to the other direction now. So that is this vector now point in this direction now you see the magnetic moment of change direction of plus z axis to minus z from the low energy to high energy state it is absorbed energy and goes to higher energy state which is now if you relate it to the magnetic resonant transition it has changed its spin from plus half to minus half if it is for proton or for electron it is change from minus half to plus half the spin has flipped. So this is going to happen moment this condition is satisfied and it does not matter how small this magnetic field is compared to this one this could be very high but when these two have the exactly same frequency relationship then even a very very small magnetic field can completely change the orientation of the magnetic moment and it can cause it to change from one direction to other direction which is nothing but the change of spin flip or transition in magnetic resonance. In EPR we showed earlier that the minus half spin state is given by this plus half spin state is higher energy so here the low energy is here and high energy is here. So moment the small oscillating magnetic field is applied in the X Y plane this can cause transition from here to here it is the important requirement is this now therefore that this oscillating magnetic field has to be applied in a direction which is perpendicular to the extra magnetic field we saw that when it is parallel nothing really happens. Also the frequency has to be exactly equal to the frequency of its larmar frequency it is epsilon frequency so that is the resonance condition that need to be satisfied not just the energy gap here that is h nu this is the relationship we said earlier but we find a little more inside to what way the experiment has to be done then not only is the matching of this energy gap but the direction of application of the external magnetic field has to be perpendicular to the direction of the Ziemann magnetic field. If we keep increasing the frequency of this omega little more than this omega then what happens then again this magnetic moment vector will go out of sync with this omega little which is moving in the X Y plane. So if you keep on increasing this frequency of this omega little higher and higher this will go even further out of sync so this will rotate in the X Y plane at a very very rapid rate compared to this one so this will find therefore that most of the time there is nothing near it. So there will be again no precession of this around this omega little therefore there is going to be no effect again so you see that therefore that this condition that when these two frequency exactly same then even a small magnetic field can completely turn the magnetization from this direction to this direction and that is precisely the resonance condition. So with this we come to an end of this discussion.