 Here we are making the same beta function as a function of beta form phase. If we are changing the beta form phase, how the beta function maximum value of beta function changes in the photo cell and this is minimum value of the beta function which occurs at the symmetry point, how it changes with the variation in beta form phase means again if I plot the photo cell here figure, so red figure shows the beta value here and blue plot shows the beta values of this point. So, how this is the beta max here and here is the beta max, now you can see here in the near 76 degree, this is nearly 76 degree, you have minima in this curve means minimum value of beta function occurs if you have 76 degree phase advance in the photo cell, so preferable phase advance for keeping the vacuum chamber aperture minimum is 76 degree in the case of the photo cell. Now, this is the example of periodic solution of a photo cell, this is the schematic shown here, continuous same cells are repeated and this is the plot of beta functions, so if we choose the matched solution means a periodic values of beta and alpha, then this kind of nicely variation in the beta functions shown, now you can see here this beta value of this at this location of the beta function is same as the value of this location and it is again the same here means values of beta functions are repeated. Instead of this, if you choose some initial beta and alpha other than this value, then periodic solution will not be there and some other solution will be there and this will look like this, now you can see that maximum value of beta is also varying here, so beam sizes will not be smooth in this case and they will vary, how these two solutions can be used practically we will see. Suppose at a location we are plotting the ellipse in the phase plane, at certain location say s is equal to s1, optical parameters are gamma 1, alpha 1 and beta 1 and the invariant of motion say it is emittance, so this gives you an ellipse here like this and you can see this particular point is the injection point of the accelerator, injection point of the accelerator means a beam of charged particle will be injected in this accelerator at this location. Now, suppose a previous accelerator from which we are getting the beam to inject in that sense the beam suppose this is the accelerator orbit here in which we want to inject the beam and there is a transport line by which beam is coming inside this accelerator. So, this transport line will also have its certain alpha beta depending on the magnetic arrangement and this optics circular optics of the synchrotron will also have alpha beta here this is the s is equal to s1 which we have chosen and here we are seeing that our risk parameters are hiding the value alpha 1 beta 1 gamma 1. These risk parameters are the periodic solution for the synchrotron optics. Now, suppose the transport line also have the solution in which at the end point of transport line alpha beta gamma has the same value as of this line I have all these values alpha beta gamma then we have both the accelerators or beam or accelerator has the matching. In this case suppose this is the beam which is coming this is the spread of the beam in the phase space if same alpha beta gamma here is at the end of the transport line we get such kind of ellipse of the beam in this sensor. So, accelerator optics produces this ellipse and beam is coming in this ellipse via transport line. So, both are matched. So, this shows that this beam will nicely follow the periodic solution of beta function which we obtained which we have shown the upper photograph of beta functions. Now, suppose this alpha beta gamma at the end of transport line do not match with the values of these alpha beta gamma of the synchrotron. Then transport line will send the beam with different orientation this is again I am plotting here in another this is the ellipse drawn by alpha 1 beta 1 gamma 1 parameters at the injection point of the synchrotron. And now suppose transport line sends the beam with different alpha beta gamma means ellipse of that beam will be misoriented with respect to this ellipse means beam will come like this. Now, after some point this particle will follow the contour of these optics now because this is in the synchrotron and these optics will be in these optics this will follow this ellipse this particle will follow this ellipse. So, after some time you can see that ellipse may be like this and ellipse may be like this. So, at a certain location also means in this case on each turn you will see beam has the same ellipse because we have a periodic solution. And in this case after each turn the ellipse of the beam has been rotated means on each turn when passing the beam through s is equal to s 1 at that location we are not getting the fixed beam size. Instead we are getting the beam size variation. So, Mishma's solution gives you the beam size variation in each turn while the periodic solution gives you a same fixed beam size at a particular location. Suppose we are looking now at this location beam size may have different values but that will remain fixed in each turn in the case of matched solution. And in case of the mismatched solution even at a certain location beam size will not remain fixed on each turn. So, this is the problem with mismatched solution. So, always we have to find out a periodic solution and we try to match the beam ellipse with the optic ellipse. So, we have a matched beam inside the accelerator. So, this kind of example was called mismatched solution and this is the matched solution. So, far we talked about only the particles which are having the same momentum as of the design moment. But in practice we have a bunch of the particles means many particles are there and each particle may have slightly deviated momentum than the design moment. How the motion of these particles which have deviated momentum can be analyzed is now. Now, consider three part this is a dipole magnet. This is a dipole magnet and this is the design trajectory which is plotted here and design momentum is p0. Means this design trajectory is designed considering the momentum as p0. Now, this is the design path through this bending magnet. Suppose on this path initially two more particles are coming. One particle is having momentum higher than p0. So, definitely this dipole magnet will bend less or you can say radius of curvature will be larger for this higher momentum particle. So, it will be deviated from the design path and it will go like this. If a particle has momentum less than p0 then it will bend more or radius of curvature will be smaller and this will go like this. So, initially all the particles were going together but as it passes through the dipole magnet the dipole magnet disperses the particles according to their movement. It is just like the prism does for the white light. In white light we have different energy for different waves or colors and each color is separated by the prism. Similarly, each momentum is separated by the dipole magnet. How this we can relate we have formulation r0 is equal to p0 by QB. Radius of curvature of a particle having momentum p0 charge Q and is passing through the magnetic field B. So, this is the radius of curvature r0. Now, if instead of p0 we have p0 plus delta. So, new radius of curvature will be r and you can take p0 out. So, p0 to B plus 1 plus delta P by P0 and this gives you r is equal to 1 plus r0 1 plus delta. This delta is equal to delta P by P means fractional deviation in the moment. This is denoted by delta. So, for a particle which has fractional momentum deviation of delta the radius of curvature will be r0 1 plus delta. So, for a positive delta P positive delta P means momentum higher than the design one r will be increased and for a negative delta P means a particle which is having momentum lesser than the design momentum r will be increased. How far the trajectory of a momentum particle is away from the design path? This is quantified by a new parameter which is known as dispersion. Now, you can see that at this location the central path is the design path. So, at this location this is the deviation in the path. At this location this may be the deviation for this. At this location this is the deviation. At this location this is the deviation means deviation is varying with s means this deviation is a function of s. See delta x as a function of s and it is proportional to delta. We have seen that r is equal to r0 1 plus delta and this r is producing this deviation. So, this deviation will be linear in delta and this quantity which is relating this deviation with this delta is known as dispersion. Similarly, if we take the derivative delta x prime means what is the angle here with respect to design trajectory or angle here with respect to design trajectory that is delta x prime that is derivative of the D multiplied by momentum offset. This delta P by P fractional changing momentum is also known as momentum offset and the particle which is on the correct momentum or the design momentum is known as on momentum particle and the particle which is away from the design momentum is known as off momentum particle. Now, suppose these particles travels downstream to this optics. So, these trajectories will look like this they will go in this direction this will go in this direction. These are dispersing away from the design orbit. So, we want to focus it. So, we have a focal quadrupole which can focus it. So, these particles will come. Now, again you can see this deviation has been reduced downstream to this quadrupole means quadrupole can be used to change or modify the value of dispersion. So, whenever a dipole will be there dispersion will be generated and using these quadrupole magnets the values of dispersion can be modified. Consider initially dispersion is 0 in the optics. 0 dispersion means all the particles if we collect different particles according to their momentum means we can select the particles having the momentum p0. We can select the particles having the momentum p1. We can select the particles having the momentum p2 and all these particles centroid are on the design trajectory. Then we see that there is no dispersion means all the particles are having same centroid and they are exhibiting betaton oscillations around the same orbit. Then we say dispersion is 0 means orbit is not differentiated according to the moment. So, if now we can learn something about that suppose a particle is having initially x is equal to 0 x prime is equal to 0 and delta is equal to 0 means this particle due to x is equal to 0 and x prime is 0 means it is on the design path. Design path itself shows the origin means it defines the origin of our coordinate system. So, design path always has 0 0 coordinates and there is no momentum offset means this particle is having exactly the design momentum then it will move definitely on the design path nicely. It will not exhibit any kind of betaton oscillation it will follow nicely the design path actually the path of this particle will define the coordinate system. Now any one of these either x or x prime becomes non-zero or both both can be not 0. Still we have no momentum offset means correct momentum but non vanishing initial condition in x or x prime. This particle will exhibit betaton oscillation which we have seen and this around the design orbit around the design orbit and the displacement at any location of this particle from the design orbit will be given by this formulation. This is invariant of the motion this is beta at that location where we want to find out the displacement and this is the phase advance of betaton oscillations. So, this will give you the displacement of this particle with respect to design orbit. Now suppose again a particle is going on the design orbit particle is going on the design orbit means x is equal to 0 x prime 0 but this time its momentum is wrong means it is not having the momentum exactly as the design one rather than it has certain tilt of p until the dispersion occurs it will nicely follow the design orbit. However as the dipole magnet comes in the path of this particle it will deviate from the design orbit and now the position of this particle with respect to design trajectory will be defined by the dispersion at that location dispersion is also a function of s. So, now displacement of this particle with respect to design orbit will be dispersion multiplied by delta p by p. Now if I have considered a particle which is having all non vanishing condition means it is neither initially on the design orbit and also had wrong movement. So, now it will exhibit betaton oscillation because of this non vanishing nature condition and because of this it will follow a dispersive orbit means this kind of particle will exhibit betaton oscillations around the dispersive orbit not around the design orbit means center of betaton oscillation for this particle will not be on design orbit rather than its center of oscillation will be on that orbit which is decided by this dispersion into delta p by p. So, displacement of this particle total displacement of this particle from the design orbit will be betaton oscillation part plus d delta. At a certain location say s is equal to s 1 this d is fixed and on each turn the location or displacement of this particle will change according to betaton oscillation that is why its center has become d delta for the oscillation. Now we can see in phase plane initially x is equal to 0 x prime is equal to 0 and delta is equal to 0 this defines the origin of this our coordinate system this is the design orbit initially non vanishing x and x prime however delta is still 0 this kind of particle makes an ellipse after turn betaton and this ellipse is d 2 betaton oscillations. Now we consider the third case where initial conditions were vanishing x is equal to 0 x prime 0 and delta was not 0. So, it will pass through some deviation from the design orbit this is the deviation this deviation is d delta prime and this deviation is d delta means in x the deviation is due to dispersion multiplied by delta p by p and in x prime deviation is due to d prime sorry this will be d prime delta is fixed because we are studying only the magnetic optics and magnet does not change the energy of the particles so delta will remain fixed. When we will study about the rf cavity particles motion through the rf cavity then delta will change and now consider the last case where x is also not 0 x prime is also not 0 and delta is also not 0. Then the ellipse of this particle due to betaton oscillation will be around this central point this is the central point and around which betaton oscillation is taking place. This increase in beam size occurs only when dispersion is non-zero at deprecation if dispersion is 0 all the particles are exhibiting betaton oscillation around the design orbit and beam size does not depend on the momentum set. Now as for individual particle we have x is equal to x beta and beta from part is this and this is due to dispersion so for beam we have to calculate the rms values of it this is for a single particle for collection of the particles we have to obtain the rms values of this so this will give you the beam size so beam size will be this is the rms value of the first quantity that is the betaton part that is admittance rms admittance multiplied by beta at that location and this point is due to dispersion and momentum offset. Now suppose initially there is no dispersion and its derivative is also zero so all the particles irrespective of their momentum are exhibiting betaton oscillations around the design so this is the case and these particle passes through some optics and exits from that optics and during exiting of from that optics again dispersion and its derivative but inside that optics there were dipole magnets so in that case dispersion was also not zero and dispersion prime also worked means at the entry of this optics dispersion is zero and dispersion prime is also zero and this optics has been arranged in a fashion so that on exit d is also zero and d prime also zero means there is a special arrangement of the magnets to achieve this condition if d is zero at this point and d prime is also zero means derivative is also zero then there will be no dispersion downstream to this because we are considering only linear dynamics so first order derivative is zero it is sufficient to tell us in nonlinear dynamics we have to make sure that second derivative is also zero but that is beyond the scope of this course so in this course we are restricting ourselves in the linear transverse beam dynamics so if d is zero here and d prime is also zero here then downstream to this point there will be no dispersion so such a special arrangement of this optics or arrangement of the magnets by which this optics has been made which offers again dispersion zero and its derivative also zero is known as acrobat means initially dispersion is zero and its derivative is zero and on exit the arrangement of magnet gives a dispersion and its derivatives so this kind of optics is known as an acrobat and this is a very useful arrangement of magnets in making the synchrotron optics and various kind of transverse planes now you can see that once dispersion has been generated using the dipole magnet water pole magnet can also modify the dispersion but there is a problem it cannot make dispersion zero and its derivative zero at both simultaneously either you can make dispersion zero or its derivatives not simultaneously by the particle so for making the acrobat we need another dipole also so only quadrupole cannot make the acrobat why i am saying this statement we can see now suppose this is a dipole magnet i am plotting this axis is s and this is dispersion i am plotting dispersion with this direction vertical direction so initially there is no dispersion so this zero then dipole occurs so dispersion will increase and again here is the drift space so dispersion will go like this now suppose we put a quadrupole at this location so it will it can focus the beam so beam focusing means all the particles which have deviated momentum will also focus so dispersion will be reduced but after some point when dispersion becomes zero again dispersion will grow because derivative has not been vanished now if we put another quadrupole here and try to vanish the dispersion derivative it is not possible why not possible because dispersion is passing through the center of the particle at the center of the quadrupole will be zero so this cannot modify the dispersion or quadrupole cannot modify or change the value of dispersion if dispersion is zero inside the quadrupole so you cannot make dispersion and its derivative simultaneously zero only using quadrupole magnets so for making an acrobat you need more than one dipole magnet we see it suppose this is the dipole and this is the dispersion generated by the dipole so at exit d is equal to d1 and d prime is d1 prime and this is the second dipole magnet after which we want again dispersion zero and its derivative zero means we want this if this condition is met then this optics will be an acrobatom now after this point how the dispersion will go like this it will go like this so when it will enter inside this dipole magnet there is no chance that I know exit d is equal to zero and d prime is zero because it will increase further by this dipole magnet so if this is zero here what should be the initial condition here for d and d prime so this dipole makes a chromatic condition at the exit so if this is the condition means these values of dispersion and d1 prime are there then a chromate can be built so for bringing this value here we can use the quadrupole so now quadrupole magnet changes the direction of d and it brings a particular value of d and d prime at the entry of the dipole magnet and then dipole magnet makes d and d prime both so after exiting this optics there is no dispersion and it's there so this is known as a chromate because this a chromate has been built using two dipole magnets it is known as double band acrobatom we can use three dipole magnets to make the acrobatom then this is known as triple band acrobatom and more than three dipole magnets if we are making the acrobatom then it is known as multi band chromate so we stop here this lecture and again the references are same in the next lecture we will see chromaticity in the synchro form and we will introduce one new magnet which is known as sextopole magnet