 In last two lectures, we have seen how to solve the Hill's equation and one of the methods was using the matrices and another method was such a way that we have obtained three optical parameters namely alpha, beta and comma. Now we will extend our studies on the Hill's equation and these parameters. So let us start. Now in a synchrotron basically first of all we define a basic cell. Basic cell means a basic optics which will be repeated many times to make a closer path. There may be a variety of basic cells or optics by which the basic cells has been made. So we take one example. One example like this there is a drifting space then there may be a focusing lens. Focusing lens means horizontally focusing lens then there may be a defocusing lens because we have to focus the beam in vertical plane also then we may have some dipole magnet then another focusing and defocusing magnet. In this fashion we can define our basic cell. There may be a variety of basic cells. So this may be one of the type of basic cell and then these basic cells are repeated many times to make a close path. So this is a basic cell and then by repeating we make the complete drink. Means in a synchrotron the optics is periodic means this period of the optics has been repeated many times. So from this starting point and again here will be the starting point of the next cell means these point are actually having the same environment. Similarly this point and corresponding to the point in the next cell will have the same environment means we have optical periodicity in the synchrotron. Even if we make synchrotron using only a single basic cell then after one turn particle traverses through the same environment of the optics means after one turn it will traverse through the same optics again and again. So even though even there is basic cell equivalent to the length of the circumference of the synchrotron in that case also we have periodicity in the optics in a synchrotron. That periodicity means that that period will have the length equivalent to the circumference of the synchrotron. So in synchrotron particle motion takes place in a periodic environment of k means k become the periodic. So we can write down k s is equal to k l plus s. Here l is the length of the period length of the in in accelerator diagram this period is known as super period length of the super period. This length may be a length of the super period or maybe the circumference of the synchrotron. Even in linear accelerators or transport line some basic cells can be repeated many times to make the complete linear accelerator or transport line in that case this k s will again be periodic. So now we will solve this equation under this condition that when k is periodic. If differential equation has a coefficient which is periodic in independent parameter then solution can be written down using the flow theorem. What flow theorem says it says that x s which is a solution can be written down as w s e raised to iota 5 and this w s will have the same periodicity as of the k s means we will have w s also as w s and s plus l. The same theorem you might have studied in solidistic physics course namely block theorem in which you solved Schrodinger equation in the Kröding-Pen model under the periodic potential. Now in case of our notation we have written down x s is equal to the root beta s cos 5 plus 5. So in this case beta s which is at the place of w s here will be periodic. So this equation solution can be written down using the periodic condition and then beta s can be written down as beta s plus l alpha s is equal to alpha s plus l and when these two parameters will be periodic definitely the third parameter gamma will also be periodic. Remember here that this is a particular solution which we have obtained. Non-periodic solution can also exist means if I take some initial arbitrary value of beta and alpha is still beta and alpha can be obtained after some length s s1 or s2. However our unique values of beta alpha which will give you the periodic solution. There may be infinitely many solutions in which we have selected one solution which has periodicity in alpha and beta. This solution in accelerator jargon is known as matched solution and in accelerators if initial beam is defined using the values of these beta alpha gamma or matched parameters then it will nicely behave in the periodic lattice. What does that mean? We will see it. Now in accelerators suppose we have chosen one point beta one alpha one at location s is equal to s one and another point say this is s is equal to s two here the values are beta two alpha two. So if we have chosen our solution periodic then this beta alpha one and beta two alpha two are unique in this case. Means we can write now beta one s is equal to beta one s plus l. Here beta two is equal to beta two s plus l. It may be a case here also that we choose some two arbitrary points s is equal to s one and s is equal to s two in some non-periodic lattices. Non-periodic lattices may be of the transport line where same cell is not repeated to make the complete optics. In that case we cannot have periodic solution of the beta and alpha. In that case of non-periodic lattice beta two and alpha two will depend on the initial values of beta one alpha one and as we change the initial values of beta one alpha one we will get another values of beta two alpha two. So in this case beta two alpha two will not be periodic. Now how beta function looks like in a periodic lattice? Suppose we have a single cell like this. This is the basic cell which will be repeated four times because this is making 90 degree angle from final trajectory to initial trajectory. Means if it will be repeated four times it will make a complete closed loop of 360 degree. So this is a basic cell. In this basic cell these green elements are the quadrupole magnets and this red element is the dipole magnet. So here schematically it is shown in this picture. Here you can see that focusing quadrupole is above the line and defocusing quadrupole is shown below the line. Dipole magnet is showing by a rectangle covering above and below both area around this line. So this is the basic symbols which one uses to show the accelerator optics or charge particle optics. Now you can see here it is a matched solution means beta and alpha at this point must be equal to beta and alpha at this point. Means here you can say this is the beta s. Here you can say this is the beta s plus l. l is the length of this optics or period then these two beta values must be equal. Here you can see that in black color this is the beta x it is the value at the starting and it is the value at the end. So these are the same. Similarly this is the beta y its value in the start of the optics is this and here it is also this. So this is the matched solution for this optics. Now you can see that as beam size increases this beta is increasing means beam size is increasing or particles amplitude of beta translation is increasing means at this location the maximum amplitude of a particle is lower than at this location. Now as the quadrupole appears it focuses the beam. So again downstream to this quadrupole the amplitude of beta translation has been reduced. Here it is a defocusing quadrupole. So downstream to this quadrupole amplitude of beta translation has been increased and again there is a focusing quadrupole this is a focusing quadrupole. So again downstream to this quadrupole amplitude of the beta translation has been increased. You can see under this particular case of matched solution at the focusing quadrupole beta has the maximum value. Again at the focusing quadrupole beta has largest value local maximized area. Again here you can see at the location of quadrupole local maximized area. Similar is the case for the vertical beta function also. Here you can see vertical beta function will have local maxima at the location of defocusing quadrupole. Here again you can see defocusing quadrupole is there. So local maxima of the vertical beta translation function is there. Now suppose we selected a location s is equal to s0. This is the location and particle is revolving in this synchrotron. This is the orbit of the synchrotron and particle is revolving around this. If particle is having non-vinitial initial condition means x or x prime is non-zero initially then it will exhibit the beta-tron oscillations around this orbit. This will be like this beta-tron oscillations of a particle. Now on passing through this detector on each turn if we record x and x prime of particle how the motion will look like. The amplitude of beta-tron oscillation at this location will depend on the value of beta-1 here and it will make a complete ellipse after many turns and that ellipse or geometry of that ellipse depends on the beta-1 alpha-1 and what are the initial condition we have chosen x and x prime which defines the invariant of the motion a. So after turn by turn the recorded position and angle you will see in this model. Now you can see that after each turn in x and x prime space there are some coordinates on which particle is coming and here we are plotting the angle after each turn. So you can see angle is oscillating like a sinusoidal oscillation and here x is also showing some oscillatory nature which is like the simple harmonic oscillator means here we are drawing the position of the particle here we are drawing the angle of the trajectory of the particle with respect to the sine trajectory. In this plot the a is a square has been plotted so it has a constant value over turn by turn. So turn by turn behavior of a particle at a certain location we are looking this behavior at this location at this detector we are recording x and x prime. So we are getting the ellipse defined by alpha beta gamma of that point and if we will see only x this will behave like this sinusoidal oscillation simple harmonic oscillator like this and x prime also will be like the simple harmonic oscillation at a certain location. Keep in mind that we are talking about this kind of oscillatory behavior if we are recording position at a particular location s is equal to s1. If we record say these parameters at another location say s is equal to s1 then again the value of beta alpha at this location will decide the ellipse. So ellipse may have different orientation and different elongation only this a will be constant. So area of that ellipse will be same as the area of this ellipse and at that location also if we will see the projected angle of position of from this ellipse they will execute the simple harmonic oscillations. There is a different way for looking this invariant of motion also. Instead of single particle executing motion and we are recording turn by turn its position and we can have different initial conditions in the beginning. Means we may have many particles having different sets of x and x prime but these sets are chosen in a way that this leads to same a square. Means we are choosing the values of x and x prime in such a way that they are producing same a square at a given location say s is equal to s0. Then again we will see that these initial conditions will make an ellipse. So how this ellipse will be generated can be seen in two ways. Both are equivalent. One is taking many initial particles many particles with different initial conditions with the same a square or having a single particle and we are recording its position and angle on each turn. So in both cases we will have ellipse defined by alpha beta and a square. A square is constant and alpha beta is position dependent. So say s is equal to s1. Again we are reiterating here that at different locations say s is equal to s2 the parameters of that location alpha and beta will be different and those different parameters will be different ellipse. It may be in this orientation also. It may be like this perfectly aligned with the axis or this orientation also. Only the condition is that the area of all these ellipses will be the same. Now suppose if we change only alpha we keep the beta constant. Beta constant means maximum amplitude of the particle at that location is a constant only we change the alpha and how we change the alpha we switch the sign of alpha. Here alpha is minus 1 here alpha is plus 1. So you can see that the tilt of this ellipse has been reversed here x x prime and this is the major axis and this is the minor axis. So here tilt is here and here you can say tilt is from here. So tilt is reversed. Why this is the case because alpha is minus half d beta by ds. So as you reverse the sign the d beta by ds has reversed itself. Now if d beta by ds is positive d beta by ds positive means beta is increasing with length. Beta is increasing with length means maximum amplitude of beta term oscillation is increasing as we are going downstream. So as we are going downstream and amplitude of beta term oscillation is increasing means we are talking about a diverging. So if d beta by ds is positive so alpha will be negative because alpha is defined as minus of d beta by ds. So minus alpha this minus alpha defined a diverging. Similarly alpha with positive value shows a converging means a focused spin. In this case d beta by ds will be negative means amplitude of beta term oscillation will decrease as we will go downstream the optics. So sign of alpha whether it is positive or negative shows you whether the beam is diverging or converging. Now if we choose the initial condition differently that the a square of these initial conditions are different. Means we have set two sets say x1 and x1 prime. This is the first set of the initial condition and say this is corresponding to the value of a1 square and which is choose another set of initial conditions say x2, x2 prime. And in this case you can say this leads to another invariant of motion say a2 square. Then how the motion will look like? So if we record now position and angle of these two particles on each term at a certain location as then the above particle or above initial condition will make its own ellipse with certain area defined by a1 square and another ellipse will be defined by this a2. However because we are recording position at a certain location say s is equal to s1 and the optical parameters at that location is same for both the particles say beta alpha. So tilt or orientation of the ellipse will be same for both the particles. Now we will see how the motion is evolved on each term in this mode. So in left side you can see that the value of lower a, suppose this is the value of a1 invariant of motion and this is value of a2. So particle with a smaller value of invariant of motion will always remain inside the larger value of invariant motion. Means two trajectories having different a1 and a2 in the phase space will not cross each other. While in the right side you have seen that these two vectors which shows instantly instantaneous position and angle of the particle at the location chosen locations say s is equal to s1. Here the displacement of these two particles can have common value at certain instants during the betaton oscillations. So in real space you can have common x for both the particles at particular time means these trajectories in real space can cross each other if you plot it. However in phase space these two particle trajectories will never cross each other. Suppose in phase space if it cuts suppose this is the first trajectory we are having here x and here x prime this is the phase space this is the trajectory say particle one and this is the trajectory of the particle two if it cuts here see and we choose initial condition of these particles at this coordinate say this coordinate is x1 x1 prime. So if we choose the initial conditions as x1 and x1 prime we cannot define whether this is the particle one or particle two means there is no uniqueness in the solution. So this is not possible with any differential equation. So these trajectories in phase space cannot cross each other.