 In previous lectures we have seen that what are the requirements of running a cyclic accelerator and the major requirement was to keep this synchronization. Now in this lecture we will take a detailed look on the cyclotron. Consider that there is magnetic field B here and magnetic field direction is out of the screen perpendicularly. Now consider a charge particle which is passing through this magnetic field. So, this is the trajectory of the particle and at this point it enters into the magnetic field. Now as it enters into the magnetic field magnetic field imparts a force given by q v cross. So, q v cross B means a force in the perpendicular direction to v as well as B. So, this force on a positively charged particle will be in this direction and because of the perpendicular force this magnetic field will advance the trajectory of the particle. Now we are considering that in this region where the magnetic field is non-zero it has constant magnitude means the value of magnetic field does not change over the space and direction is also constant means everywhere the direction is out of the screen in the perpendicular direction. So, in this magnetic field when particle passes through this magnetic field it feels a perpendicular force means a centripetal force and its trajectory becomes a circular trajectory and because we are considering constant magnetic field means this circular trajectory has that radius r constant everywhere means radius does not change as particle proceeds. So, it is a perfect circle made by the trajectory. When we consider that this magnetic field is constant over the space it means we say this is a dipolar field. So, dipolar field means it is constant over the space if we change the coordinates x, y, z in the magnetic field the magnetic field does not change. This type of magnetic field can be generated by taking two poles very long surfaces. Suppose this is a north pole and this is a south pole and this is a very long broad surfaces. So, in between there will be a uniform magnetic field that is why this is known as dipolar field because this can be generated using two poles. Now, here because magnetic field imparts the centripetal force. So, centripetal force is equal to the magnetic force. Here magnetic force should be q v cross b however we are having that v perpendicular to p. So, v cross b will be just v b because sin 90 is 1. So, this is written as q b b and we know that centripetal force is m v square by r. However, because we are considering energies which may be higher than the rest mass energy or may be comparable to the rest mass energy. So, this equation or formulation of the centripetal force must be corrected relatively and that is why this gamma has been introduced here. Now, remember here just multiplying mass with gamma does not produce correct result of this. This is a special case when force is in perpendicular direction to the velocity means if acceleration is in perpendicular direction to the velocity then only this can be done. If velocity and accelerations are not perpendicular then we have to obtain explicitly the expression of the force. You will see this kind of things when we will study the syncopon radiation sources in detail. Now, this v here is cancelled by this v. So, we have gamma m v by r is equal to q. Gamma m v is just the momentum p. So, this is gamma m v upon q v r will be and gamma m v is p. So, radius of this circle, this radius of this circle which is made by the trajectory of the charge particle is given by p by q. Means r is directly proportional to p momentum of the particle. A higher momentum particle will have larger radius of curvature. Means it will make a larger circular particle because it will bend less by the magnetic field because it is a higher energy. So, radius of curvature will be larger. And r is again inversely proportional to means if we increase the beam the radius of curvature will decrease. So, a higher magnetic field will make a shorter circle for the given charge particle and because r is directly proportional to p means magnetic field separate the trajectories according to the momentum of the particle. That is why this kind of dipolar magnetic field is also known as momentum analyzer. Now, we the very first thing for successful acceleration is to maintain the synchronization. So, we calculate what is the revolution time of this particle when it passes through the magnetic field. So, if there is a magnetic field and particle trajectory makes a complete circular path or complete circle and let the radius of this circle be r. So, length will be dipolar. Now, we have seen that in last slide that r can be calculated if we know the magnetic field and value of the charge particles charge and v. So, this value of r can be kept here from this value r is gamma mv by qv this gamma mv by qv. So, we just it will be 2 pi mv upon qv. So, this will be the length of this orbit. Now, we want to calculate the revolution time. So, this length will be divided by the v. So, this 2 pi gamma mv qv divided by this v. So, this will give you the revolution time. Now, this v cancels out. So, 2 pi gamma m by qv. So, revolution time you can see here depends on the rest mass of the particle which is constant quantity q which is again a constant quantity b magnetic field which we are applying because we are applying a constant magnetic field. So, we can see that b is also constant for a given configuration and 2 pi is also constant. Only t revolution changes with gamma. So, at lower energy regime where gamma remains constant how gamma changes with beta we can plot here suppose on the axis we are plotting beta means how the speed is changing of the particle and here on the y axis we are plotting the gamma. So, gamma minimum value of gamma is 1. So, it goes like this. So, in the initial part gamma almost remains constant. So, when gamma remains constant t revolution does not change with energy. It is a very beautiful result that as energy increases the length of the orbit increases and because the energy increases speed also increases. So, increasing speed just compensates the increase in orbit length and that is why t remains constant. And this is true only in the low energy regime where gamma remains constant. When gamma changes this t revolution will increase with energy. So, for running a cyclotron if we want to maintain the synchronism with a constant frequency RF field this is valid only in the low energy regime. So, how the cyclotron looks like there are two big poles north and south this makes these poles make dipolar magnetic field perpendicular to this orange structure. This orange structure is known as D inside this D there is vector and charge particle freely can move freely inside this D and when there is a gap between these two D there is the applied electric field. So, whenever particle crosses this field it gets energy from this electric field and due to this magnetic field which is constant over the D the particle makes a circular path. So, we can see here that particle starts its journey from this point. So, it gets some energy from the field and then because of the magnetic field it makes a circular path inside the D and again it crosses this electric field then again it gets some energy from this electric field and it makes a larger circular path because of the increased energy magnetic field is constant. So, when energy increases the radius of curvature of the path will also increase. So, a larger circular path will be there in the next fall after here again particle crosses this gap between the D and gets the energy from the electric field and it makes even larger circular path. In this fashion energy increases and the radius also increases and on the desired level of energy we can extract the particle from the cyclotron and this is how B looks like from the side this is the side view. So, North Pole and South Pole and we have a constant magnetic field inside this area. Now, here we have calculated that T revolution is basically 2 pi gamma m upon Q B and omega revolution will be just 2 pi upon T revolution. So, omega revolution will be Q B by gamma m. So, this revolution frequency is known as cyclotron frequency. If gamma is not changing much we can say that omega revolution is almost constant because we have applied constant magnetic field Q is constant and for even charge particle. Now, how cyclotron works we will see it in this plot you can consider that this is a D means whenever particle will cross this line its energy will increase. In this plot we will plot the radius of curvature for each orbit and in this plot we will plot the energy. So, whenever the particle will cross this line its energy will be stacked and in this plot we will show the revolution frequency of that particle. So, now we will see this in movie. So, these are the orbit on each crossing radius has been increased and energy is also stepping up and here radius of curvature is also becoming larger and larger. While here you can see that revolution frequency is almost constant and because of this constant revolution frequency synchronism can be maintained by a constant frequency RF light. However, when gamma will change much here you can see that gamma changes a very little bit. If gamma changes with a large value this synchronism will be improved and acceleration will take place. So, what is the highest energy possible in a cyclotron? If we make the cyclotron magnet with a radius r means up to the r, r is the largest radius the particle can go up to that orbit which is having the radius of radius r. So, t is equal to half MV square we are using non-relativistic formulation here because we are considering that we are working in that region where the gamma remains constant. So, naturally we are working in the non-relativistic region that is why kinetic energy can be written down as half MV square because v is equal to omega r omega is the revolution frequency. So, v can be written down as omega r so it will be half m omega square r square. Now, at the place of omega we can put qv by m keeping the gamma as one because we are considering that gamma is constant. So, this half mv square means omega is equal to qv upon gamma m and we are considering that gamma is approximately 1 no change in gamma then omega will be approximately equal to qv upon m and this value has been written down. So, kinetic energy will be half m q square v square r square by m square this m will be cancelled out by m. So, we have half q square v square r square by 2. Now, at the place of q we can write down q as z means if we are accelerating some ion so charge on the ion will be z into electronic charge and mass of the ion will be a into mass. So, a is the mass number and z shows the charge state of that. So, at the place of q we keep z square e square and place of m we keep a m. So, here e is the charge on the electron and here m is the mass of the probe. So, these quantities are constant and for a given configuration of cyclotron b is also constant and r we have taken the largest possible radius in that cyclotron. So, r is also constant for a given cyclotron. So, these quantities are constant and z and a depends what ion we have selected and what is the charge state of that ion. So, kinetic energy can be written down as k into z square y. So, this k is known as k value of the cyclotron. If we are accelerating the proton then z becomes 1 and a also becomes 1 in that case t is equal to k means k shows you what is the maximum possible value of a proton in a given cyclotron if gamma remains unjust. So, this is the maximum possible value. So, we say that k 130 cyclotron in the variable energy cyclotron center at Kolkata means proton can be accelerated theoretically maximum up to 130. However, we have seen that gamma changes a little bit. So, if we consider that little bit change. So, phase of RF where the particle arrives after one turn will change a slight. So, when this change accumulates over the turn button and when it change in phase becomes sufficient particle cannot accelerate beyond that limit. So, in practically the value of proton or energy which we can achieve using a cyclotron for the proton will also be lesser than this k. k is also known as bending power of the cyclotron. Now, we have considered that particle is entering into the magnetic field where the V is in perpendicular direction to the magnetic field B means particle velocity vector is making 90 degree angle with respect to magnetic field B means it is nicely following a trajectory which we wish. This kind of particle is known as synchronous particle because this is an ideal particle which shows what should be the ideal trajectory or descent trajectory in the accelerator. However, in B there is a large number of particles and all these particles may not follow exactly the descent trajectory. There may be deviation in the velocity vector, there may be deviation in the from the descent trajectory itself. So, consider that a deviated particle in velocity means a particle is not having only perpendicular component of the velocity also it has some parallel component of the velocity than the field. This is the component V perpendicular and this component decides what should be the radius of curvature of the orbit which it will follow under the given magnetic field B. While this V parallel there will be no force because V and V for this component is in same direction. So, in this direction particle will advances with this V component velocity. So, particle will make a helical path means particle will go in vertical direction continuously while making circular path. So, this circular path is decided by this V perpendicular and advances in the vertical direction is decided by this V parallel. Now, if it will be the case after few turns particle will hit the surface of the T's because there are D inside which particle always circular trajectory. So, after certain time it will hit either the above surface or the below surface depending with what is the direction of this V perpendicular, a V parallel. So, in that case we have to have some mechanism to keep these particles confined in the vertical plane also. How we can do that? We have considered that we have a nice dipole magnet means pole faces are parallel to each other. This is the pole face of the north pole and this is the pole face of the south pole both are exactly parallel to each other and we have nicely constant magnetic field in this space. Now, suppose a particle is deviated in vertical direction from the central axis this dotted line shows the central axis. So, a particle which is deviated from the above or below from the central axis is still the force is exactly parallel to x axis or parallel to this central axis. Suppose particle is going inside the screen in the perpendicular direction V is in this direction. So, force will be in this direction for a positively charged particle. Now, we want to bring some vertical force which will keep the particle confined in case if particle has certain velocity component in the perpendicular direction. So, instead of making pole face perfect parallel make some curve in the pole faces. So, the gap between the pole increases as we go away from the center. So, here the gap is high here gap is low because here gap is low means magnetic field is stronger here and magnetic field is weaker here. So, as we go outside from the center magnetic field becomes weaker and weaker means we are introducing a gradient in the magnetic field. What will happen now for a particle which is deviated in the vertical direction from the central axis in this configuration of the magnetic field we see it. Now, suppose a particle is again deviated here now here because the magnetic field name as curvature. So, magnetic field direction will be decided by the tangent at this point. So, at this point magnetic field is in this direction. So, this is marked here this is the magnetic field direction is still we are considering that particle is going in the screen in the perpendicular direction. So, force will also be perpendicular to V as well as perpendicular to this is a magnetic field. So, force now has vertical component also because force will be in this direction. So, this will be the horizontal component of the force this will be the vertical component of the force. So, now we have vertical component of the force. Now, here you can see that when the particle is below the axis the magnetic field line will look like this and force will be in this direction. So, horizontal direction of the force is still in the same direction while the direction of the vertical component of force has been reversed. So, if particle is going above or below it feels a force towards the central axis means a confinement is possible for the particles in case of vertical displacement or vertical velocity if we introduce some gradient in the dipolar magnetic field. So, if we introduce gradient in the dipolar magnetic field of a cyclotron then vertical focusing is also possible. This kind of focusing which is obtained using the gradient in the dipole magnet is known as weak focusing. In later chapters we will see what is the strong focusing when we will consider the quadruple magnet.