 Hello and welcome to lecture 8 of module 2 of this course on Accelerator Physics. So today we will learn about the transverse dynamics of beams in Linux. So far we have learned about the acceleration of charge particles, how we can accelerate them using time varying fields. So if you want to accelerate to very high energies you can use time varying electric fields, unlike DC accelerators where the final energy is limited by how high a voltage you can achieve here, how high you can go in energy depends upon how long you can make the Linux or how many gaps you can use. So there is no limitation on the maximum value of the energy using time varying accelerating fields. So we can use time varying electric fields associated with the electromagnetic waves in a high Q cavity for acceleration, we have seen these this in the previous lecture how we can do that. So TE, MNP and TM, MNP modes are excited in the cavity, these indices M, N and P in the TM and TE, MNP they describe the field pattern in the cavity. So we have seen the field pattern in detail particularly in a pillbox cavity. We have also seen how the TM010 mode in the pillbox cavity we have EZ field that is constant along the length of the cavity and V theta component, the fields are varying in time. So this EZ field in the pillbox cavity can be used for acceleration. If you put drift tubes inside this cavity, the structure the resulting structure is called the drift tube linear, this can be used for acceleration of charged particles. We learnt about the figure of merits of the cavity, the quality factor, shunt impedance and so on and the aim of cavity design is to maximize the quality factor and the shunt impedance. We also saw that hollow empty wave guide cannot be used for acceleration because here even though the TM wave is possible, the phase velocity is greater than the velocity of light and so synchronism of the particle with the wave is not possible. However, if you use a periodically loaded wave guide that can be used to accelerate charged particles because here you are able to slow down the phase velocity of the wave. For a normal conductor accelerator the largest power loss is between the RF cavity and the beam. We have seen typical efficiencies of normal conducting accelerators are in the range of 20 to 30 percent. So whatever power is fed into the cavity out of that only 20 to 30 percent goes to the beam, the remaining is dissipated in the cavity walls. The RF surface resistance of superconducting cavities is less than the RF surface resistance of the normal conducting cavities by a factor of 10 to the power of 6. So the RF power dissipation in superconducting cavities is very low. So superconducting cavities offer a huge advantage for accelerating charged particles because now the entire RF power can go for acceleration of the beam. And also we can use large accelerating gradients in a superconducting cavities. So now having learned about the various methods of acceleration in a linear accelerator today let us learn about the transverse dynamics of particles. So transverse dynamics refers to the dynamics of the beam in the direction transverse to the direction of motion. So we have the particle moving in the z direction. So there will be the particles have some thermal velocity in the x and y direction. Now we have already seen that to qualify as a beam vx and vy should be much much smaller as compared to vz. So they are very small as compared to vz but there is a finite quantity. So because of this the beam tends to diverge. So there is a motion, there is a velocity in this direction, there is a velocity component in this direction, the beam tends to move like this. So the beam tends to diverge and over long distances this beam size, the distance from the axis could become very large and could lead to beam loss. So the beams tend to diverge in the transverse direction due to inherent divergence of the beam. So the beams have some inherent thermal energies because of which they will diverge. They also diverge due to effects of RF field. So these are for example these are two drift tubes. So you will have the electric field in this region here like this and the field is not linear with z. There will be always some fringing fields as you can see here. So there is a radial component here as well as the longitudinal component here. So because of this the beam will get some kick in the radial direction. Now from here it may look like that in the two halves of the gap the radial components cancel out but actually if you see the field is varying with time in the gap. When the particle enters the gap and sees this field which is bringing the particle towards the axis, the field is smaller and when it sees this field where the radial field is taking it away from the axis the field is larger. So the net effect could be that the beam is defocused. So because of this also the RF field also the beam gets defocused in the transverse direction. Then there are space charge effects. Now if the beam current is very high that means the beam has large number of charged particles and these are charged particles of the same charge. So let us say we have a proton beam. So if the beam current is very high then there are lots of proton ions that are confined in this area. So because they are of the same charge they will tend to repel each other and because of that the particles will exert force on the other particles in the outward direction. So this is known as space charge effects. So it is generated by the coulomb repulsion between the particles in the beam. So let us say you have a long uniform infinite cylinder of radius r. So this let us say this is the whole beam and the force acting on any charge inside this beam is given by this expression and we can see that it is radially positive. So the charge particle will be pushed out of the beam. So in other words the beam size will increase. Now if beam size, increase in beam size if not control can lead to beam loss. So we need to focus the charged particles. So these charged particles can be considered just like a ray of light. So like here it is just like a ray of light. A ray of light is focused using lenses. Here we will use since these are charged particles and they respond to electric fields and magnetic fields we will use magnetic fields and electric fields for focusing. So focusing the beam means bringing the charge particles in the beam close to the axis. So if the charge particle is going like this by focusing the beam we have to bring it back to the axis just like in the case of a ray of light being focused by a converging lens. So we use magnetic fields. So we use magnetic quadrupole and solenoids for focusing. We also use electric fields and electric quadrupole or Einzel lenses for focusing. We have also seen in the first lecture where for a beam with high velocity magnetic field is more efficient for focusing whereas for low velocity beams both electric and magnetic fields can be used. So we had seen that the force acting on a charge particle is given by Qe plus V cross V this is the Lorentz force. Here the force due to the magnetic field is Q V cross B and the force due to the electric field is F is equal to Qe. So let us consider a high energy beam so V is close to the velocity of light. So the force due to the magnetic field is given by Q into C into B. Now let us take B as 1 Tesla which is a reasonable field we get FM is equal to Qc. Now suppose I want to produce the same amount of force using an electric field on this charge particle. So Fe is equal to Qe and this is equal to Qc. So if you calculate the value of E using this you get E is equal to 300 million volts per meter. So this is an extremely high electric field and it is not possible to generate such high electric field. So for high energies it is better to use magnetic fields for focusing. If however the energy is not so high so it is just about 1 percent the velocity of light then the electric field required comes out to be 3 million volts per meter which is like a reasonable number. So magnetic field is more efficient for focusing at higher velocities. At lower velocities you can use both magnetic fields and electric fields. So let us see a magnetic quadrupole first. So this picture shows a magnetic quadrupole. So the quadrupole is used to focus the beam in the transverse direction. So this red thing that you can see is the beam. So the beam passes in this region and here it experiences the force due to the magnetic field of this quadrupole. The magnetic quadrupole has 4 magnetic poles 90 degrees apart. So you can see 4 poles here and they are 90 degrees apart. The adjacent poles are of opposite polarity. So if this is a north pole, this is a south pole, this is again a north pole and this is a south pole. So these poles could either be electromagnets. So in this case they are electromagnets so you see that there are coils wound around here and depending on the direction of the flow of current in these coils this will be a north pole or a south pole. You could also use permanent magnets. So these poles could also be permanent magnets. So let us see how the magnetic quadrupole focuses. So this shows the layout of a magnetic quadrupole. So you have 4 poles, a north pole, south pole, north pole, south pole. Now the magnetic field lines they originate from the north pole and terminate on the south pole. So everywhere I have drawn these lines. Now in, so this is the transverse plane and this is the beam, this in the center is the beam and the direction of the beam is vz out of your screens. So now let us see the fields. We can resolve the magnetic fields in each of these 4 quadrants. So this is first quadrant, this is second quadrant, this is third quadrant and this is fourth quadrant. In each of the 4 quadrants we can resolve this magnetic field in the x and y direction for simplicity. So if we see in the first quadrant, so this magnetic field in the x direction is in the positive x direction and in the positive y direction. Here it is in the negative, in the second quadrant it is in the negative x direction and negative y direction and so on. Also if you see the magnetic field, if you see the variation of magnetic field by with x it is linear. So you can see from here this is x and let us see in the first quadrant for example at x is equal to 0 at this point. There is no component of, so magnetic field is only in the x direction, by is equal to 0. Now as you are moving or as x is increasing. So here the y component keeps on increasing. So if you see the variation, the variation of by with x is linear. Similarly if you see the variation of bx with y, so let us see here. So this is y is equal to 0, y is equal to 0 is here and if you see at y is equal to 0. Here the magnetic field is only in the y direction, the x component is 0. As you go, increase y, so the by component is, as you increase y, the bx component is now increasing. So here it is 0 and slowly it is increasing. So if you see the variation of bx with y, it is again linear. So we can write by is proportional to x and by is equal to gx and similarly bx is proportional to y and bx is equal to gy, where g is called the magnetic field gradient. Okay now let us see how the charge particle gets focused using this type of magnetic field produced by the quadrupole. So this is the beam at the center and these are the directions of the magnetic field in x and y in each of the four quadrants. So there are some particles here in the first quadrant, some in the second quadrant, some in the third quadrant and some in the fourth quadrant. Now the force acting on the charge particle is given by q v cross v, v is in the z direction that means coming out of the screen. So the charge particles are moving in the z direction which is out of your screen. So using the right hand rule, we can find out the direction of the force acting on the charge particles due to these magnetic fields. So according to the right hand rule, the thumb shows, if you take your right hand, the thumb shows the direction of velocity, the fingers point in the direction of your magnetic field. Then the direction perpendicular to your palm will give you the force on the charge particle. So using this, if we calculate the force on the charge particles in each of the four quadrants, so here we see that in the first quadrant, the magnetic field by will produce a force fx which is in the negative x direction. Similarly the magnetic field component bx will produce a force fy which is the positive y direction and similarly in the other three quadrants. So this is the direction of force acting on the charge particle in each of the four quadrants. We can see from here that this beam due to this configuration of quadrupole or this configuration of fields is pushed towards the axis in the x direction whereas in the y direction it is pushed away from the axis. So such a quadrupole focuses in the x direction and de-focus is in the y direction. So thus we can see that a magnetic quadrupole focuses in the x direction and de-focus the beam in the y direction. Now let us reverse the polarity of the quadrupole. So now let us make this as north pole, this is south, this is north and this is south. So again now the magnetic field lines will start from the north pole and terminate on the south pole and we can calculate or we can just draw here the directions of the magnetic field in each of the four quadrants due to this field configuration. And again using the right hand rule we can find out the force in the acting on the charge particle in each of the four quadrants. Now here we notice that when we have reversed the polarities of the quadrupole, this is now focusing in the y direction and de-focusing in the x direction, ok. So this quadrupole will focus the beam in the y direction and de-focus the beam in the x direction. So from here we have seen that any quadrupole will focus the beam in one transverse direction but in the other transverse direction the beam will get de-focused. Now here we have seen that Bx is proportional to y and By is proportional to x. So we can calculate the force due to this magnetic field. Now Fx is equal to minus q Vz By, ok and we know that By is equal to Gx and similarly Fy is equal to q Vz Bx and we know that Bx is equal to Gy. So substituting here we get Fx is equal to minus q Vz Gx and Fy is equal to q Vz Gy, ok. Now for a given beam going through a given quadrupole, this q Vz into G is a constant. Q is the charge, Vz is the velocity so the charge particle is moving with a fixed velocity and G is the gradient which is a property of a given quadrupole. So this is a constant. So we see that the force is linear in x and the force is linear in y. So in a quadrupole the force acting on the charge particle in the x direction is proportional to x and the force acting on the charge particle in the y direction is proportional to y. So now let's see if we want to focus the beam in both the directions in x as well as y directions, ok. So we have, so in order to focus the beam in both the directions we use two quadruples, quadrupole 1 and quadrupole 2. So they are of opposite polarity. So for quadrupole 1 if this is the north pole, for quadrupole 2 this will be the south pole. So there are two quadruples. The first quadrupole is a focusing quadrupole in x direction and defocusing in y direction. The second quadrupole since it is of the opposite polarity is a defocusing quadrupole in the x direction and focusing quadrupole in the y direction, ok. And remember that fx is proportional to x and fy is proportional to y. Now let's see the motion of the particle in x and y separately through the system of two quadruples. In x direction the charge particle now goes through the focusing quadrupole first. So here its displacement from the axis in the x direction is x1 at the quadrupole. So the force acting on the charge particle here is fx1 which is proportional to x1, ok. So this being a focusing quadrupole will focus the beam and then it passes through the defocusing quadrupole which will defocus the beam. So this defocusing in this defocusing quadrupole the distance from the axis is x2. So the force acting on the charge particle is fx2 which is proportional to x2, ok. Now notice that x1 is greater than x2, ok. So fx1 is greater than fx2. So in the x direction the force due to the focusing quadrupole is more than the force due to the defocusing quadrupole. Hence the net result will be that the beam will be focused in the x direction. Now let's see the behavior of the beam in the y direction. In the y direction we have the defocusing quadrupole first. So the beam, so the charge particle passes through the defocusing quadrupole and gets defocused. At the defocusing quadrupole the coordinate of in y or the displacement of the charge particle in the y direction is y1. So the force acting on the charge particle is fy1 which is proportional to y1. So this particle gets defocused and then it enters the second quadrupole and the second quadrupole is focusing in y. So here the displacement in the y direction is y2. So this focusing quadrupole now focuses the beam. So here the force acting on the charge particle is fy2 which is proportional to y2. Now here again y2 is greater than y1, ok. You can see from here that y2 is greater than y1. So fy2 will be greater than fy1 or in other words the force due to the focusing quadrupole will be more than the force due to the defocusing quadrupole. So the net result is that the beam will get focused in the y direction. So thus we see that the beam is now focused in both x and y direction. So if a charge particle or a beam passes through two quadruples of opposite polarities. So a single quadrupole will focus in one direction, defocus in the other direction but a combination of two quadruples of opposite polarities will focus the beam in both the directions. So there is net focusing force in both the directions x and y. Next is solenoid focusing. You can also use a magnetic solenoid for focusing. So this is the picture of a solenoid. So solenoid you have coils here, you can see these are coils. So these are coils inside and this is the path for the beam to go through. So this is a solenoid that is attached in a beam line, this is the path for the beam to go through. There are coils like this, ok. So these are the coils, this is showing you the z direction and this is the radial direction. So these are the coils and due to these coils there is a magnetic field which is reduced like this. So at the center of the solenoid you will have a BZ field, ok. However at the ends of the solenoid because the magnetic field lines are closed there will also be some radial component of field, BR will also be there because magnetic field lines are closed. So at both the ends there will be some radial component of magnetic field. So at the entrance of the solenoid we have a radial component of the magnetic field. Now this is the beam which is coming in the z direction. So the beam interacts with the radial component of the magnetic field. So the force is now VZ into BR. So this will be in the perpendicular direction, so that is in the theta direction. So what happens when this VZ interacts with BR the beam gets a force in the theta direction. So it starts spiraling like this. So it gets a theta component of velocity. So the beam enters the solenoid with a theta component of velocity. Now inside the solenoid we have a BZ component of magnetic field. So this V theta again interacts with the Z component of the magnetic field and we get a radial component of force. So this radial component of force focuses the beam and brings it close to the axis. So that is solenoid focusing is a consequence of interaction between the azimuthal velocity component. So this azimuthal velocity component induced in the entrance fringe field region and the longitudinal magnetic field component in the central region. At the ends of the solenoid the field flares out and there is an interaction between the radial field component and the axial velocity component. So there is an interaction between the radial field component and the axial velocity component producing an azimuthal force. So in this way the particle acquires an azimuthal velocity component in the entrance fringe field region. So it gets a V theta component. In the central region particles travelling parallel to the field are unaffected but those with an azimuthal velocity component they will experience a force causing them to describe an orbit that is helical in space and circular when viewed from the end of the solenoid. So in this way a solenoid focuses the beam and the solenoid will focus the beam in both the transverse directions because it is azimuthal symmetric. Now let us talk of phase space and trace space. Now generally we are interested in the behaviour of the beam in phase space. So phase space is what position and momentum coordinates. So in X, P, X or Y, P, Y. So what is PX? PX is M into VX. This you can write it as mass can be written as gamma times the rest mass and velocity can be written as dx by dt. Now this dx by dt can be written as dx by dz into dz by dt. What is dx by dz? It is X prime. So it is also called the divergence of the particle or the divergence of the beam. So you can write this as X prime and dz by dt is VZ. So VZ is beta into C, okay, VZ is beta into C. So this can be written as beta gamma MOC into X prime. Now if the velocity of the charge particle is constant, if it is moving with a charge with a constant velocity that means there is no acceleration. In that case beta gamma is constant, beta gamma is not changing. The rest mass is the rest mass of the charge particle and C is the velocity. Again these are constant for a given charge particle. So in the absence of any acceleration, beta gamma MOC is constant for a given charge particle. So here now PX is proportional to X prime. So we can work in trace space. Trace space is X X prime space of the phase space. So we need not go to phase space, phase space and trace space will be identical, okay. So now the coordinates of any particle in trace space are given in the matrix form as X X prime. So we will be working in trace space over the next few slides. Okay, let's see now the behavior of a beam in a field-free region. So this is known as a drift space. So field-free region means there is no field, no electric field, no magnetic field. So let's say the charge particle is here at point A. So we are analyzing its motion in the x direction and this is the z direction, okay. So let's say at in the beginning the charge particle is at location X1, X1 prime, okay. So you can write it as X1 is equal to X1, X1 prime. And then after traversing a distance L, it goes to new coordinates X2 is equal to X2, X2 prime, okay. Now we want to find out the values of X2 and X2 prime. Now since no force is acting on the charge particle, it will move with a constant divergence. So what is a divergence? So here the position coordinate is X1 and divergence. That means the angle it makes with respect to the z-axis is X1 prime. Since no force is acting on it, this angle here will remain the same. So this will be again X1 prime, okay. So now we can write this as X2. So this is the coordinate X2 is equal to X1. So this is X1 plus L times X1 prime. So you can write this as X1 prime. So this is theta. So this is tan theta is equal to X2 minus X1 upon L, okay. And since theta is small, so theta this can be written as equal to X1 prime. So from here you can write X2 is equal to X1 plus LX1 prime. So this expression can be written here like this. And also as I said X2 prime, since there is no change in the divergence, X2 prime is equal to X1 prime, okay. So now this equations can be written in matrix form as X2 X2 prime is equal to 1 L01 X1 X1 prime. So solve this matrix you will get these two equations, okay. Now this matrix if you see it depends only upon the length of the drift space. So this is known as the transfer matrix of the drift space. So if you know the initial coordinates at the beginning of the drift space and you know the length of the drift space, you can find out the final coordinates at the end of the drift space. So X2 is equal to MD times X1. So this is known as the transfer matrix of the drift space.