 So, the single particle motion in phase space the motion of a single particle is along curves of constant Hamiltonian. So, as we have seen the motion is along the curves of constant Hamiltonian and this curve is a ellipse, epsilon is a constant of motion and this is independent of s. Each particle in the absence of non-conservative forces has a constant invariant. So, remember all this is valid when we have conservative forces. So, only because we have derived this for conservative forces or linear forces. The shape and orientation of the ellipse is defined by the twist parameters alpha, beta and gamma. So, alpha, beta and gamma tell you about the shape and orientation of the ellipse and since these parameters are function of s, the twist parameters are functions of s, the form of the ellipse changes continuously. However, due to Liouville's theorem any particle starting on the ellipse will stay on it. So, if there is a particle on this ellipse it will remain on this ellipse only. Let us say we have another particle on another ellipse. So, it will remain on that only because you know that in phase space trajectories cannot cross. So, this will remain on this ellipse only. So, due to Liouville's theorem any particle starting on the ellipse will stay on the same ellipse. To describe the beam as a whole the beam envelope is defined. So, for so there are many particles each of them have an ellipse. If you want to define the beam as a whole you have to define a beam envelope. All particles on the beam emittance defining the ellipse follow trajectories defined by this expression where phi i is the arbitrarily phase constant for particle i. The beam envelope is determined by the beam emittance epsilon. So, beam emittance is the area of the outermost ellipse divided by phi. So, beam envelope or the beam size the maximum beam size this is determined by the beam emittance epsilon and the betatron function beta s. So, this under root epsilon into under root beta tells us about the beam envelope. The beam emittance is a constant of motion and it resembles the transverse temperature of the beam. The betatron function reflects the externally applied force from focusing magnets and depends upon the arrangement of the magnets. So, beta function or the betatron function beta it depends upon the value of k. So, it depends upon, so k depends upon the gradient or the field of the magnet. So, it is a function of the externally applied force from focusing magnets and it depends how you arrange the magnets whether it is a photo or a photo dough or some other arrangement. So, in this ellipse again this is the maximum beam size where this is the outermost ellipse and this is the maximum beam divergence again this is the outermost ellipse. So, all other particles lie on concentric ellipses inside this outermost ellipse. Now, now we had derived this expression from the Hilse expression from Hilse equation. So, we had we have beta double prime beta by 2 minus 1 by 4 beta prime square plus k beta square is equal to 1. Now let us put under root epsilon beta is equal to capital X where X is now the beam envelope or the beam size. So, we have seen that the beam envelope or the beam size is defined by both the constant of motion epsilon of the outermost ellipse and beta. So, here let us put under root epsilon beta is equal to capital X. So, we put this and if we substitute it in this expression and simplify this we get an expression in terms of capital X where capital X is the beam envelope or beam size. So, we have X double prime plus k X minus epsilon square X is equal to 0. Now, here you can see k is the force due to the quadrupole or whatever is the externally applied force and if you compare it with the equation of a single particle which is in small x. So, we see that these two expressions are similar except for an additional term in the envelope equation which is in terms of the emittance and it is defocusing ok. So, equation of motion for the beam size is similar to that of the single particle equation of motion except for the presence of additional emittance term which is defocusing. So, this is the envelope equation which describes the evolution of the beam envelope as a whole and this is the equation of the single particle under the influence of the externally applied forces due to magnets. Now, in real machines we barely have information about single particles what information we have is about the beam as a whole. So, we will have information whether what is the maximum beam size, what is the maximum divergence. So, you know that the maximum beam size is given by under root beta epsilon and you also know that the maximum. So, this is the maximum beam size and the maximum divergence is given as under root gamma epsilon. So, basically the maximum the information about the beam is contained in the twist parameter. So, if you know what is beta you know what is gamma you can also calculate alpha. So, you know how the and if you know alpha beta and gamma you know how the ellipse looks like the beam ellipse looks at at a particular location. So, let us say we have two locations 0.1 and 0.2 and this is how the beam envelope looks at beam ellipse looks at location 1 and this is how the beam ellipse looks at location 2. And in between we have some system we have some magnets we have some drift spaces. So, we have some we have a system we have some elements and the transfer total transfer matrix is given by m ok. Now, the general equation of an ellipse in xx prime trace space is given as gamma x square plus 2 alpha xx prime plus beta x prime square is equal to emittance where beta gamma minus alpha square is equal to 1 ok. So, at this location there will be some equation of the ellipse and at this location there will be some equation for the ellipse. Now, the equation of the ellipse can be expressed in matrix form as x transpose sigma inverse x is equal to epsilon where x here is the coordinate of the particle. So, x sorry this is x, x is written as xx prime. So, it is the position and the divergence of the particle x transpose can be written as xx prime it is a transpose of the matrix. Now, we define a sigma matrix which is only in terms of the twist parameters alpha beta gamma. So, we define a sigma matrix for the beam at any location as matrix beta minus alpha minus alpha gamma. So, sigma inverse inverse of this matrix can be written in this form ok. So, now let us write x transpose. So, we write x transpose here this is x transpose sigma inverse into x ok x transpose sigma inverse into x is equal to epsilon and we expand this we multiply this expression we get the equation of the ellipse. So, hence these two are identical this expression and this expression are identical. So, the equation of the ellipse can be expressed in the matrix form as this equation ok where x is the position coordinate sigma is the sigma matrix which is in terms of the alpha beta and gamma. So, it is giving you the entire information about the beam that means it is telling you about the ellipse of the beam at that location what is the shape and orientation of the ellipse. So, let us say at location 1 the beam is described by the sigma matrix sigma 1 and at location 2 it is described by the matrix sigma 2. So, this sigma matrix contains the entire information about the beam at these two locations it will tell you what is the maximum beam size at that location the maximum divergence of the beam at that location. So, beam at any location s in the lattice is determined by the twist parameters alpha beta gamma which is defined by the twist by the sigma matrix. So, this is the at location 1 this is the equation of the ellipse x 1 transpose sigma 1 inverse x 1 is equal to epsilon. Now, in between here let us introduce a unity matrix which is m transpose m transpose inverse and again at this location here let us introduce a unity matrix which is m inverse m ok where m is the transverse matrix that means x 2 is equal to m times x 1. So, m is the transfer matrix of this from region 1 to from point 1 to point 2. So, we have introduced two unity matrices here and now again rearranging. So, we have x 1 transpose m transpose we take this together and we take these three elements together. So, we have m transpose inverse sigma 1 inverse and m inverse and finally, we are left with m x 1 this is equal to epsilon. Now, x 1 transpose m transpose can be written as m x 1 transpose similarly this entire thing this entire matrix here can be written as m sigma 1 m transpose the whole inverse and this is m x 1 is equal to epsilon. Now, we know that x 2 is equal to m x 1. So, instead of m x 1 we write x 2 here. So, this is x 2 transpose let us call this thing as sigma 2 sigma 2. So, here this becomes sigma 2 inverse and this is again m x 1 is equal to x 2. So now, from this we get this expression which is simply the equation of the ellipse at this equation at this location at location 2. So, we started with the equation of the ellipse at location 1 and we have arrived at the equation of the ellipse at location 2 and comparing the two equations now we can see that sigma 2 which is the sigma matrix at location 2 is equal to m sigma 1 into m transpose. Now, m can have let us say m has elements m 1 1 m 1 2 m 2 1 m 2 2. So, this will depend upon the elements from 1 to 2 if it is just say just a quadrupole then it will be this will be the transfer matrix for the quadrupole. If it is a quadrupole followed by drift space another quadrupole then it will be the total transfer matrix for that element. So, we get this expression so now if you know the sigma matrix at location 1 and you know the transfer matrix for this system you can find out the transfer matrix at location 2. In other words if you know the beam size and divergence at location 1 and you know the transfer matrix of this element you can find out the beam size and divergence at location 2. So, now you need not calculate for the single particle for each of the single particles in the beam you can calculate for the beam as a whole in terms of the sigma matrices and you can find out what is the sigma matrix at the end of the elements. So, now this sigma 2 is in terms of alpha 2 beta 2 gamma 2 and sigma 1 is in terms of alpha 1 beta 1 gamma 1 and m is taken as n 1 1 n 1 2 n 2 1 and n 2 2. So, you can simply simplify this expression and you can write a convenient expression for alpha beta and gamma at location 2 in terms of alpha beta and gamma at location 1. So, here so, these are all m. So, this gives an expression which can be carried out by doing multiplication matrix multiplication above. So, transformation of a single particle and beam envelope. So, how can you transform a single particle if at this location at location 1 you know the coordinates of the single particle and you at location 2 you know the coordinates of the single particle and you know the transfer matrix the single particle can be transformed in this manner by the transfer matrix method. Now, if you have the information of the beam as a whole at the location at the first location and you have the information about the transfer matrix then you can find out the sigma matrix of the beam at location 2 by the sigma matrix method. So, where your x is the position coordinate and sigma gives you the information about the beam. So, in this way we can find out the transformation of the ellipse as a whole along various elements. For example, let us just see in a drift space we know that in a drift space there is no external force happening on the beam. So, the only the beam size will increase the beam divergence will not increase as we have seen in the case of the single particle transformation of the beam of the single particle transformation in phase space. So, here again we are talking of phase space or trace space. So, this is the ellipse at location at the initial location and as s is increasing as the beam is progressing in a drift space. So, since there is no external force the divergence remains the same. So, we can see here the divergence remains the same. However, the beam is expanding because the beam is like a ray of light and it tends to it tends to increase in size. So, we see that the beam size has increased here and here it has increased even more. So, we can see that the beam ellipse is transforming in the drift space. So, drift space is a field free region since no force is acting on the charge particle the beam divergence remains the same. For a divergent beam only the beam size increases as the beam travels through the drift space. So, this is analogous to the single particle which we have already seen in the last lecture. So, x increases while x prime remains constant. So, here again you can see here x increases x prime remains the same. Now, we can see the evolution of a beam through a photo channel. So, we have a focusing quadrupole followed by a de-focusing quadrupole again a focusing quadrupole de-focusing quadrupole in between there is the drift space. So, in the focusing quadrupole the beam gets focused. So, you will see that the so this is we have taken this at the middle of the focusing quadrupole the beam is getting focused here in the de-focusing quadrupole the beam gets de-focused. So, you can see the beam gets de-focused here. So, this is the beam envelope and this is given by under root beta epsilon. So, for a periodic lattice the ellipse repeats after one period. So, you can see here this is the beam ellipse at this location and how the ellipse is evolving as the beam moves in this focusing lattice. Now, at the end of one period we can see that the ellipse repeats itself. So, it is evolving along as s is changing, but at the end of one period the ellipse repeats itself. The location of the particle on the ellipse at the end of one period depends upon the phase advance per period. So, here this is the particle at let's say this is one particle lying on the outer most ellipse and at this location it is here as the ellipse evolves the particle moves along the ellipse. So, at the end of one period it will not come back to the same position. So, it depends upon the phase advance. So, now from at the end of one period it has moved from here to here this angle is known as the phase advance. So, the number of periods of the focusing lattice it takes for the particle to complete one full oscillation defines the phase advance. Now, in one period it has advanced the phase has advanced so much. Now, in n number of periods it will come back to the same location. So, that gives you the phase advance of the lattice. Now, a large beta represents a large beam size. So, the beam envelope is given by under root beta epsilon larger the value of beta larger is the beam size. Whenever beta reaches a maximum or a minimum alpha is equal to 0. So, you know that alpha is equal to minus beta prime by 2. So, wherever there is a maxima or minima of beta. So, there the alpha goes to 0 and the ellipse becomes an upright ellipse. So, there is a the beta is maxima here it is minima here. So, at this location alpha is 0. So, you can see an upright ellipse here you can see an upright ellipse here again here beta is maximum. So, alpha is 0 you can see an upright ellipse here. So, this is the in this picture you can see this is the beam envelope and the single particle trajectories are shown here. So, the single particle equation is the Hilse equation whereas, the envelope equation is being given by this expression. So, the envelope follows this expression here and the envelope has the periodicity of the lattice because beta has the periodicity of the lattice. However, the single particle does not have the periodicity of the lattice. So, for example, here this is the trajectory of one particle this is the trajectory of another particle. So, it depends the trajectory depends upon the initial position but all particles satisfy this equation this is in x and this is in y. So, you can see here if it is focusing in x here it is de-focusing in y this is for a photo channel. Now, we define phase advance as 360 degree divided by number of periods for one full oscillation of the single particle. So, in this case let us see the trajectory of the single particle now the trajectory starts for this particle starts from here and at this point it completes one full oscillation and this is in 6 periods of the lattice or the beam envelope. So, here one full oscillation of the single particle is in is in 6 photo periods. So, phase advance is defined as 360 degrees divided by 6 which is equal to 60 degrees. So, this is in x this is in y again if you see this here one full oscillation of the single particle is in 5 photo periods. So, 1 2 3 4 5. So, in 5 photo periods the particle completes one full oscillation. So, your phase advance is given by 360 degree by 5 which is equal to 72 degrees. So, the general knowledge calculate the general transfer matrix between two points in terms of the lattice functions alpha, beta and gamma at these points. So, we can write the this is the initial coordinates and this is the final coordinate. So, this is 0.0 this is 0.1. So, m s 1 s 0 is the transfer matrix from s 0 to s 1 and s this is a function of alpha, beta and gamma at s 0 and s 1. So, Hill's equation is this you can write the solution of the Hill's equation in the form of cos and sin where c 1 and c 2 are constants. Now, at point s is equal to 0 let the initial conditions be that let alpha be alpha 0 beta v beta 0 and phi is phi 0. So, you can calculate the values of the constants c 1 and c 2. So, the solution can be simply written as you put in the values of x 1 and the values of c 1 and c 2. So, the solution is simply this. Now, you can rearrange this and again you can take the derivative you can take the derivative and we see that these expressions are linear in x 0 and x 0. So, as the beam travels from s 0 to s 1 the above equations will give the value of x and x prime at s 1. So, you can write this from this expression you can write this in matrix form and you will get x at s 1 and x prime at s 1 is simply some matrix m x s 0 x prime s 0 and this matrix will come from these two expressions. So, the matrix is now given by this expression. Now, notice that this matrix depends only on the twist parameters at the two locations and the phase advance phi. So, this is the transfer matrix written here. This means that the transfer matrix between two points is purely determined by the lattice function at each point and the phase advance between the two points. So, if you have two locations and you want to find out the transfer matrix between two points you can find out it depends only on the twist parameters and the phase advance. So, this is known as the Courant-Sneider generalized matrix between two arbitrary points. Now, if you want to find out for one period now you know that the ellipse repeats after one period you have just seen that the ellipse repeats after one period. So, if you want to find out for one period now at the end of one period the twist parameters are identical. So, let us say alpha 1 is equal to alpha 0 let us call it alpha similarly beta 1 is equal to beta 0 is equal to beta and gamma 1 is equal to gamma 0 is gamma. So, substituting in the generalized matrix here. So, this matrix is simplified like this. So, the transfer matrix for one period is simply cos phi plus alpha sin phi beta sin phi minus gamma sin phi cos phi minus alpha sin phi. So, where again gamma is equal to 1 plus alpha square by beta and the phase advance for one period is given as phi 1 minus phi 0. So, this is the phase advance in one period. Now, let us consider a photo lattice again. So, the transfer matrix for one period of a photo channel can be written like this. So, let us consider half of a quadrupole followed by a drift space of length L then one full defocusing quadrupole followed by another drift space of length L and then half of another focusing quadrupole. So, we can write the transfer matrix this is the transfer and we are using here thin lens approximation. So, we can write the transfer matrix for half of a quadrupole focusing quadrupole transfer matrix for a drift space for a defocusing quadrupole again drift space and half of a focusing quadrupole. So, we multiply this and we get the total transfer matrix for this one period of the photo matrix. So, now the generalized matrix for one period is given by this we can compare the transfer matrix we can compare these two transfer matrices. So, this is from calculation and this we know is the generalized transfer matrix and from here we can find out the values of the beta function and the phase advance. So, comparing them we get cos phi is equal to 1 minus L square by 2 f square. So, you can find out the phase advance from here and the beta x and beta y can also be found out from here. So, beta x is the beta function in x and beta y is the beta function in y for this type of system. So, this is at the beginning and at the end of the this is the values of the beta function at the beginning and the end of the lattice which is the same because it is a periodic focusing here alpha is equal to 0. So, wherever we know that wherever beta is maximum alpha is 0 and so gamma will be equal to 1 by beta. So, this is the evolution of the beam along the photo channel. So, we see here that alpha is equal to 0 here and beta is maximum and at this location alpha is equal to 0 here beta is equal to minimum. So, this is the beam envelope. So, summarizing quadruples of reverse polarities are used in pairs for focusing the beam. We have seen different types of focusing lattice like 4doh, 4photodoh, solenoidal channel and others they are used for focusing the beam. For a periodic focusing system the general equation of you know the general equation in phase space is generally an ellipse. At any location s in trace space xx prime each particle will lie on an ellipse defined by the twist parameters alpha beta and gamma. The twist parameters are same for all the particles that is the ellipses for all the particles are concentric. The area of each ellipse depends on the value of the constant of motion which is its silent eye for that particle. The outermost ellipse defines the maximum beam size and the maximum divergence. And beam emittance which is a figure of merit of the beam that is defined as area of the outermost ellipse defined divided by pi. Now, single particle behavior can be studied either by solving the health equation. So, if you know the value of k you can write the equation of motion and solve it and find out the trajectory of the single particle. Alternatively, you can also do it by the transfer matrix method if the forces are linear. So, if you know the initial coordinates and you know the transfer matrix you can find out the final coordinates of the particle. The behavior of the beam envelope can be studied again by solving the envelope equation. This is the differential equation for the envelope and by using the sigma matrix method. So, if you know the beam ellipse at one location you can find out the beam ellipse at the other location if you know the sigma 1 and the transfer matrix at this location. So, in the next lecture we will see we will study about transfers dynamics with space charge. So, far we have considered only the forces due to external fields. Now, we will consider in the next lecture the forces due to the some forces in the beam itself the space charge forces. So, we will study the transfers beam dynamics with space charge.