 Hello and welcome to lecture 10 of module 2 of this course on Accelerative Physics. Today, we will learn about transverse dynamics of beams with space charge. But before that, let us just quickly revise what we learnt in the previous lecture. So we studied that the quadruples of reverse polarities are used in pairs for focusing the beam because we have seen that quadruples focus in one direction and defocus in the other direction. They are used in pairs for focusing the beam. Different types of focusing lattices like 4-do, 4-4-do-do, solenoids etc. these are used for focusing the beam in linax. So yesterday we saw that inside the drift tube linax there are drift tubes and in these drift tubes quadruple magnets are put for focusing the beam. They could be arranged in any configuration like 4-do, 4-4-do-do or even a 4-4-4-do-do configuration. For a periodic focusing system, the general equation of any particle is an ellipse. So we derived this in the previous lecture. At a given location S in trace space XX prime, each particle will lie on the ellipse defined by the twist parameters alpha, beta and gamma. The twist parameters are the same for all parameters that is ellipses for all the particles are concentric. So alpha, beta, gamma tell you about the shape and orientation of the ellipse. They tell you about the beam properties, the beam size, the beam divergence, whether the beam is diverging or converging and so on. The area of each ellipse depends upon the value of the constant of motion which is epsilon i for that particle. The outermost ellipse defines the maximum beam size and the maximum beam divergence. The beam emittance is defined as the area of the outermost ellipse divided by pi. We also saw that the single particle behavior can be studied by solving the Hill's equation which is a single particle equation. So it includes the force due to the quadruple magnet or whatever the case may be, whatever element is there. We can also study the single particle behavior by the transfer matrix method. So if we know the transfer matrix of the element and we know the initial coordinates of the particle before the elements and we can find out the final coordinates of the particle at the end of that element. The behavior of the beam envelope can be studied by solving the envelope equation which we derived in the previous lecture or by sigma matrix method. So sigma matrices are basically in terms of the two parameters alpha, beta and gamma. So they will give you the entire information about the beam at a particular location. So if you know the sigma matrix at the initial location and you know the transfer matrix for the element in between, you can find out the final sigma matrix and hence the beam size, beam divergence at the end of that element. Now in all this analysis, we have studied the effect of only external elements like quadruples etc. on the beam. We have not studied the effect of the force due to the beam itself. So if the beam is a high current beam, so there will be many particles of the same charge in the beam and these particles being of the same charge will tend to repel each other because of which there will be a repulsive force acting on the particles due to the beam itself. So this is known as the space charge force and today we will study about the transfer beam dynamics including the space charge forces. So this is particularly important for high intensity beams and high intensity beams are required for several applications. One is the accelerator driven system. So I discussed about this in the first lecture. So we need a high current beam here. Another application is spallation neutron sources here also, a high intensity beam hits a target and spallation neutrons are produced which are used for doing experiments. So in order to have a high yield of neutron we should have a high current proton beam. So both of these applications require very high intensity beams. So high intensity beams are high current beams and since the beam current is high and particles in the beam are of the same charge they will experience coulombic repulsion. This results in degradation of the beam quality so the beam emittance will increase. Increase in beam size resulting in beam loss. So since these forces are repulsive the beam size tends to increase and if the beam hits the aperture beam will be lost. Beam loss at high energies can result in activation of the accelerating structures preventing hands on maintenance of the accelerator. Now if beam is lost at higher energies this can activate the structure so that during operation nobody is allowed to enter the beam the accelerator tunnel. But after the accelerator is shut down people enter inside to for maintaining the structure. So if there is beam loss at higher energies this can activate the accelerating structure so that it can become radioactive and even after the beam is shut down the accelerator is shut down people will not be able to enter inside for hands on maintenance of the structure. So beam loss at higher energies is undesirable. So the force due to the beam self charge is known as space charge force. Now in order to understand the space charge force in high intensity beams let us consider a very simple model. Let us consider a cylindrical uniform beam so let us say we have a cylindrical uniform beam with circular cross section. The beam current here is I the beam radius is A and the beam is moving with the velocity V. So due to symmetry the electric field here will be in the radial direction so we will have an electric field which is in the radial direction and a magnetic field which is in the azimuthal direction. So there will be only two components of electric field and magnetic field. Here rho is the charge density and J is the current density. So the electric field the radial component of electric field can be calculated so we know that from Maxwell's equation divergence of E is equal to rho by epsilon 0. Applying Gauss's law we can write the volume integral of divergence of E is equal to the surface integral of E dot ds. So applying this over a cylinder of radius R and length L we get this expression and by simplifying this further we get the radial component of electric field as this expression. So we see that it depends upon the beam current higher the beam current higher is the radial component of electric field it depends upon R the distance from the center and also on the beam size. Similarly we can calculate the magnetic field due to this cylindrical uniform beam with circular cross section. So there will be only theta component of magnetic field so again using Maxwell's equation curl of B is equal to mu 0 J and applying the Stokes theorem we can write B R d theta is equal to integral of this is equal to curl of B dot dA. So applying this over a cross section of radius R we get B theta 2 pi R is equal to mu 0 R square pi beta C rho again simplifying this we get B theta is equal to I by 2 pi epsilon 0 C square R by A square. Now force acting on a charge particle due to this uniform cylinder so we can calculate using Lorentz force so the force in the radial direction will be Q E R minus V Z into B theta. So we have already calculated the E R and B theta values if we substitute in the Lorentz force we get this expression so and if we simplify this we get this expression for the force acting on any charge particle due to this uniform cylindrical charge of cross section. So here we see that the there are two components to this force one is due to the electric field the other is due to the magnetic field. The component of force due to the electric field is repulsive and due to the magnetic field is attractive you can see here because there is a minus sign here so it is attractive whereas the component due to the electric field is repulsive. So we all and we can also see that the force is linear in R. So the force is linear in R and the coulomb effects in Linux are most important in non-relativistic beam at low velocities. So as the beam becomes relativistic and beta approaches 1 so the magnetic field the force due to the magnetic field will cancel the coulombic repulsion due to the electric field. So the coulomb effects or the space charge effects are more important at lower energies for high current beams. For relativistic beam the self magnetic forces increase and produce partial cancellation of the electric coulomb forces. So at higher energies or let us say for electrons because we know for electrons the beam becomes relativistic at lower values of kinetic energy these space charge forces are not so important they are more important for the high current beams, ion beams at lower energies. So moving charges here they produce mutually repulsive electric field and attractive magnetic field. Now we have calculated for a uniform charge distribution. Now if we calculate for a non-uniform charge distribution let us consider a Gaussian beam distribution for a circular cross section. So the charge density is given by this expression. So again using Maxwell's equation so divergence of E is equal to rho by epsilon 0. Now rho we substitute the distribution we Gaussian distribution here and we can calculate the force in the radial direction using qer by gamma square as we have calculated previously. So we get this expression. Now here notice that for a non-uniform Gaussian beam the space charge radial force is no longer radial in R. So this is a non-linear force now. So the force depends upon the distribution the type of distribution for a uniform charge distribution it is linear for a non-uniform distribution like a Gaussian distribution it is non-linear. So this is a uniformly distributed beam in real space in x and y. So this is x and this is y and here we see that the force is linear in R. This is a Gaussian distributed beam so that the charge density is high at the centre and falling as the radius increases. So for a Gaussian distributed beam the force is non-linear. So we can summarize the result here. The coulomb effects in Linux are usually most important in non-relativistic beams at low velocities because for relativistic beams the self magnetic forces increase and produce partial cancellation of the coulomb forces. So they are important at lower energies only. The net effect of the coulomb interaction in a multi particle system can be separated into two contributions. So one is the space charge field the result of combining the fields from all the particles to produce a smoothed field distribution which varies appreciably over distances that are large compared with the average separation of the particle. So this is what we have just calculated the effect of the entire charge distribution on a single particle. Then the second is the contribution arising from the particulate nature of the beam which include short range fields describing binary small impact coulomb collisions. So between adjacent particles there are coulombic collisions so these are short range forces. So typically the particles in the Lenard bunch exceed 10 to the power of 8 particles there are a large number of particles in a bunch and the effects of the collisions are very small as compared to the effects of average space charge field. So we usually consider the force on the charge particle due to the entire beam distribution. The force due to the coulombic repulsion of adjacent particles this is short range and usually very small and can be ignored. Now let us see how the beam can be described. Every particle in the beam can be described by three position and three momentum coordinates. So we usually talk of the beam in phase space so we so each particle can be described by three position and three momentum coordinates. Then each particle is represented by a single point in the six dimensional phase space of coordinates and momentum. In practice it is convenient to work with two dimensional phase space projections. So even though we are describing the beam in six dimensional phase space so we take projections in two dimensional phase spaces and we work with them. So the normalized phase space projections are x and px by mc, y and py by mc, z and pz by mc. So x, y, z are the coordinates and px, py and pz are the momentum components. Instead of a transverse momentum it is convenient to measure the divergence angle. So we have already seen in the previous lectures that it is convenient to work in trace space x and x prime rather than x and px. So we take the divergence angle dx, y, ds which is same thing as x prime, dy by ds which is same thing as y prime. So plots of x prime and xx prime and yy prime are known as trace space or unnormalized phase space projection. So you can see here this is the projection in xx prime, this is the projection of the beam in yy prime, this is in real space x and y and this is in the z direction in terms of phase and energy. So in longitudinal phase space position and momentum relative to the synchronous particle can be used. But more often these are replaced by phase and energy phi and w. So you can see here this in z direction instead of z, z prime we have used phase and energy and this is with respect to the synchronous particle. The beam phase space contours in a linac have the approximate shape of an ellipse. So we have derived this in the previous lecture we saw that whenever there are linear forces the beam or any charged particle in phase space travels in a elliptical path. So this is due to the predominance of linear focusing forces quadruples, the focusing force due to a quadrupole is linear in most accelerators. With linear focusing the trajectory of each particle in phase space lies on an ellipse which is called the trajectory ellipse. So these are the trajectory ellipse for various particles. So each particle has the same twist parameters alpha, beta and gamma. What differs is the value of the constant of motion which is epsilon i. Due to the tendency of linac beams to exhibit approximately elliptical phase space distributions it is conventional to define for each two dimensional projection a quantity called emittance which is proportional to the area of the chosen beam ellipse. So we define an emittance. So we saw in the previous lecture that the outermost area of the outermost ellipse divided by pi is known as the emittance. So just quickly revising what we have already done the beam phase space contours in a linac have the approximate shape of an ellipse the general equation of the ellipse is given by gamma x square plus 2 alpha xx prime plus beta x prime square is equal to x island. So here alpha, beta, gamma are the twist parameters and they are related as gamma is equal to 1 plus alpha square by beta. So these twist parameters tell you about the shape and orientation of the ellipse. They also tell you about the beam properties. So the maximum beam size is given by under root beta epsilon. This is the maximum beam size. This is the beam half width. Similarly the maximum beam divergence xm prime is equal to under root gamma epsilon. So this is this value it is the beam half divergence. So the motion of the particles is a long constant Hamiltonian and then emittance is defined as the area of the outermost ellipse divided by pi. So here emittance corresponds to the constant of motion of the outermost particle. So quality of the beam is quantitatively described by its emittance. So emittance is a figure of merit of the beam which is closely related to the area of the two dimensional projections of the hyper ellipsoidal volume occupied by the particles in six dimensional phase space on x, p, x, y, p, y and z, p, z plane. So the beam occupies a hyper ellipsoid in the six dimensional phase space x, p, x, y, p, y, z, p, z. So we take projection and emittance is defined for the projection in two dimensional phase spaces. The transverse emittance is normally expressed in millimetre millirad since x prime and y prime are preferred to px and py. So we normally use instead of the phase space the trace space so the emittance is taken in the trace space x, x prime. So the units are millimetre millirad. The longitudinal emittance defined only for bunched beams. So if it is a DC beam, if the beam is not bunched then the beam is continuous in the z direction. So the emittance is not defined in the z direction. The longitudinal emittance is defined only for bunched beams. This is normally expressed in nanosecond kV or degree kV. In many cases an ellipse can roughly approximate the emittance boundary. Sometimes beams do not have proper elliptical boundaries. So the area of such an ellipse represents the full emittance of the beam in that plane and its size is determined by the outermost particle of the distribution. So whenever there are linear forces the trajectories of the particles in phase space they lie on an ellipse but if there are non-linear forces the particles can lie outside the ellipse. So you can still use an ellipse to roughly approximate the emittance boundary and then in this case area of such an ellipse represents the full emittance of the beam in that plane and its size is determined by the outermost particle of the distribution. But sometimes it can happen that let us say some very few particles can go far away from the main beam. So here as you can see in this picture, so this is taken from the code called trace fin. So you can see here most of the beam is in the central region but a few particles have now are far away from the central core of the beam. So if non-linear forces are present then projections of the phase space of a real beam may have complex shape and poorly defined boundary. A few particles may go very distant from the beam core and form halo. So this is what is known as the beam halo. So if you calculate the emittance, the containing all the particles this value will come out to be very large. So in such cases the emittance can be defined as 1 by pi times the area delimited by an isodensity contour containing some large fraction of the particle. So you can define a 90% emittance or a 99% emittance, so you can define an ellipse which contains let us say 90% of the particles or 99% of the particles. So an ellipse which contains 90% of the particles, 99% of the particles and that is called the 90% emittance or 99% emittance. So here you can see this for this beam this is the 50% ellipse. So the area of this ellipse will be the 50% emittance. This is the 90% ellipse. So containing 90% of the particles in the beam so this will be the 90% emittance and this is the ellipse containing 99% of the particles and this will correspond to the 99% emittance. So generally beams do not have well defined boundaries. So in the presence of non-linear forces the beam need not be an elliptical beam. So one method for assigning an emittance is to choose a specific density contour in phase space such as at 50%, 90%, 99% of the maximum density. It can be shown that under certain conditions such emittances are conserved. For example when Lewis theorem is satisfied in the 6 dimensional phase space or when forces in the 3 orthogonal directions that is x, y and z they are uncoupled. Now let us talk of beam distributions. We have already seen a uniform distribution and a Gaussian distribution. So beam of charge particles can be conveniently represented by means of a distribution function f of the charge in 4 dimensional phase space or 6 dimensional phase space. So 4 dimensional phase space x, p, x, y, p, y is generally used when you have a DC beam. So you need not define the beam in the z direction. So in that case a 4 dimensional phase space is sufficient to define the beam. When you have a bunched beam in that case 6 dimensional phase space is used x, p, x, y, p, y and z, p, z. So motion of the particles can be described as that of a set of points in 6 dimensional phase space. So we can have a distribution function in 6 dimensional phase space which can be defined like this or in 4 dimensional phase space as like this f is equal to f which is a function of x, p, x, y, p, y and in this case 6 dimensional phase space x, p, x, y, p, y, z, p, z. So motion of the center of mass of the beam is described by first moments. So first moments are you can take average over x, average over x prime, y, y prime of the particle distribution f, x, x prime, y, y prime which is then defined statistically as so average x averages taking the integral of x multiplied by the distribution function f and taking the integral over dx, dx prime, dy, dy prime here f is normalized to 1 and similarly for the other x, x, x prime average y, y and y prime. The other important information is contained in the second moment. So second moment is defined as average over x square, average over x, x prime, xy, xy prime and so on. So of the particle distributions which is defined statistically in a similar way. So it is defined again just like we have defined the first moment. So average of x square is the taking the integral of x minus average x square multiplied by the distribution function f and integrating over dx, dx prime, dy, dy prime. Similarly the second moment of x, x prime it can be defined in this manner and so on for the other 8. The root mean square values of such quantities are defined as x rms is taken as so you simply take the square root of the second moment and you can define the root mean square or the rms of such quantities.