 Quantity that is used to characterize a distribution and compare a beam with another, this is known as the RMS emittance. So, we have studied so far the total emittance which is the area of the outermost ellipse divided by pi, then 50%, 90% or 99% emittance. Now we will define an RMS emittance, so this is a quantity that can be used to compare one beam with another beam. So what do you mean by one beam? So one beam could be uniform distribution and another beam could be Gaussian distribution. Now Luegel's theorem is satisfied when there are no dissipative forces, no particles are lost or created and no small impact parameter binary coulomb collisions between particles are present. So when these conditions hold, the volume of the 6-dimensional phase space defined by any fixed density contour of the beam is invariant. So according to Luegel's theorem, the volume of the hyper ellipsoid in 6-dimensional phase space is conserved. Now RMS emittance is defined as function of the second moments of the distribution. The horizontal and vertical RMS emittance can be statistically defined from these relations. So the RMS emittance in the x direction can be defined in terms of the second moments. So we have x square average multiplied by x prime square average minus xx prime average square and then we take the under root. Similarly, the y RMS emittance can also be defined. The ratio between the full emittance, the full emittance is the area of the ellipse containing all the particles. The full emittance and the RMS emittance, it depends upon what is the distribution. So let us see the importance of RMS emittance here. Now let us consider a particle distribution in phase space representing a beam that lies on some line that passes through the origin. So let us assume for any x, the divergence, the divergence x prime of the particle is given as x prime is equal to some constant C x to the power of n, where n is a positive number. So C is a constant. The second moments of this distribution can easily be calculated as we have seen here because the distribution is given and the squared RMS emittance is given by, so you can calculate the squared RMS emittance using this formula so we get this expression. Now when n is equal to 1, let us put n is equal to 1 here. Now when n is equal to 1, the line is straight and so when n is equal to 1 here, we see that it is a straight line and the RMS emittance here, the RMS emittance value these two terms are equal and this becomes equal to 0. Now when n is not equal to 1, the relationship is non-linear. So you see that this relationship is non-linear. The line in phase space is curved and the RMS emittance in general is not 0. So we see that this is for n not equal to 1, you could have a distribution or a relationship between x prime and x which is like this. Now if you see the RMS emittance, you put n is not equal to 1 here, the RMS emittance is in general not equal to 0. But if you see the area of both the distributions for n is equal to 1 and n is not equal to 1, the area inside this is 0. So the total emittance is 0 whereas the RMS emittance is 0 here, the RMS emittance is not 0 in the second case. So thus even when this phase space area is 0, if the distribution lies on the curved line, its RMS emittance is not 0. So RMS emittance depends only on the true area occupied by the beam in phase space but so it depends not only on the area occupied by the beam, it also depends upon the distortions produced by non-linear forces. So it is a very useful quantity for describing the beam or it is a good figure of merit for describing the beam. Now let us discuss some beam distributions. The simplest distribution is the KV distribution. The KV distribution is defined, it is defined for a DC beam in 4-dimension phase space. So whenever we talk of DC beams, we talk of only the 4-dimensional phase space which is x, x prime, y, y prime. So all the 2-dimensional projections of the KV distribution are uniform. So you take any 2-dimensional projections that means you take the x, x prime, y, y prime, x, y, x prime, y prime, x, y prime or x prime, y all the 2-dimensional projections are uniform. So therefore this distribution has a property that the charge density across the beam is constant. So since the 2-dimension projections are constant, so charge density is constant in any 2-dimensional phase space and transverse phase charge associated with the cell fields are linear. So if you have already calculated, if you have a uniform charge distribution in 2-dimensional phase then the force due to that is linear. So the forces associated with the cell fields are linear functions of the particle's position in the beam that is they vary linearly with the radius. Now where the axis of the hyper ellipsoid are parallel to the coordinate axis, the KV distribution has a simple form of delta function of the transverse emittences epsilon x and epsilon y. So it has a form of the delta function. So you can write the KV distribution like this, it is a delta function of x square by Ax square where Ax is the beam size in x, y square by Ay square where Ay is the beam size and y, Ax square x prime square by epsilon x square plus Ay square by prime square by epsilon y square minus 1. So the particles fill uniformly only the surface of the hyper ellipsoid in 4-dimensional phase space. So therefore the 4-dimensional volume is 0. So it is like you can imagine it is like a balloon or a ball with only the particles lying only on the surface of the ball or balloon. So if you take any 2-dimensional projection, so it is like a 4D ellipsoid which is like a shell. So the particles, the charged particles, the particles of the beam they lie only on the surface of the ellipsoid. So the 4D ellipsoid shell projects into the xy plane as a 2D uniform ellipse. Such a distribution is actually physically unrealistic. However, it is useful because the force due to this distribution is linear. So it helps to calculate the space charge and handle it in a simplified manner. You can analytically calculate the space charge forces and try to understand the effect of the space charge forces. Other types of distribution are the water bag distribution. You could have a 2D water bag, a 4D water bag or a 6D water bag. A 2D water bag distribution is uniform in two dimensions. So you take any two-dimension xx prime, yy prime, xy. So two-dimension projections are uniform. The 4D water bag, again here, the 4-dimensional hyper ellipsoid volume, xx prime, yy prime is uniform like an elastic bag filled with water. So you take a ball or an elastic bag filled with water. So the entire volume, charge is distributed uniformly in the entire volume. And if you take any 2D projections, they are parabolic. So we will derive this in a moment. You can also have a 60 water bag distribution. Here the 6-dimensional hyper ellipsoid volume is uniform. So the charge is distributed uniformly in the 6-dimensional hyper ellipsoid volume. So in this case, the 2D projections are cubic. We can also have a Gaussian distribution. The 4-dimensional hyper ellipsoid volume is populated with a Gaussian density and therefore particle density is also Gaussian in the two-dimensional projections. So this is true for two-dimensional longitudinal distribution also. So you could have a distribution in which the 4-dimensional hyper ellipsoid volume is populated with Gaussian density and the 2D longitudinal phase space is again distributed, particles are distributed in a Gaussian fashion. Now in both the above non-KV distributions, so whether it is a water bag or it is the Gaussian, the transverse space charge forces associated with the cell fields are non-linear functions of the particle's position. So only when you have a uniform distribution, the space charge forces are linear otherwise the cell fields are non-linear. So this shows the projection of various distributions in the XY space. So this is a KV distribution where the charge density is uniformly distributed. This is a 4D water bag distribution where the two-dimensional projections are parabolic. So here the beam density is parabolic and this is a Gaussian distribution where the charge particles are distributed in a Gaussian manner. So let us consider the 4D water bag distribution. So let us consider the beam to be specified by a four-dimensional hyper ellipsoid. So x square by a square plus x prime square by b square plus y square by c square and y prime square by d square is equal to 1. Now in a four-dimensional water bag the particles are uniform in the 4D space xx prime yy prime. So we can write the charge density in xx prime yy prime is equal to some constant k. Okay so let us normalize this. So we take the integral over dx dx prime dy dy prime and equate it equal to 1. So the charge density is equal to k and from here we can calculate the value of k. This comes out to be 2 upon pi square by abcd. So the charge density is equal to 2 upon pi square abcd. So we see that in the four-dimensional volume it is a constant. Now we want to find out its projection in two-dimensional space okay so rho xx prime. So rho xx prime we can find this out by integrating it with respect to dy and dy prime and applying the appropriate limits. So when we do this we get so we can put in the value of the distribution here and if you calculate this, this will come out to be rho xx prime is equal to 2 by pi ab1 minus x square by a square minus x prime square by b square. So thus we see that the two-dimensional projection is parabolic. So the particles occupy they are uniformly distributed in the 4D volume but in the two-dimensional the distribution is parabolic for the case of 4D water bag. So similarly we can find out the projection of all the distributions in two-dimensional phase phase. So just summarizing here the now ratio of the full emittance and the rms emittance. This is equal to 4 for a four-dimensional kv distribution while it is equal to m plus 2 for a m-dimensional water bag distribution. So just summarizing here we have a kv distribution this is defined in four-dimensions. The space charge here is linear and the ratio of the full emittance that is the area occupied by all the beam all the particles of the beam to the rms emittance is 4 and the in real space the distribution is a constant. Similarly a 2D water bag it is defined in two-dimensions. So any two-dimensional projections are uniform the space charge again is linear here and the ratio of the full emittance to the rms emittance is 4 here. Again the charge density is constant or uniform in real space. The 4D water bag distribution is defined for four-dimension space the space charge is non-linear the distribution in two-dimensional phase space is parabolic and the ratio of the full emittance to the rms emittance is 6. The six-dimension water bag in which the charge density is uniformly distributed in a six-dimension hyper ellipsoid. So here the space charge is again non-linear the ratio of the full emittance to the rms emittance is 8 and the two-dimensional projections are cubic which can be calculated or derived in a similar manner as we derived for the 4D water bag. For the Gaussian distribution whether it is a two-dimension or four-dimension the space charge is non-linear the emittance the ratio of the full emittance to the rms emittance can be calculated it is greater than n square and the projection in two-dimension phase space is Gaussian. Now let us just again come back to the envelope equation. So we had derived the envelope equation for from the single particle equation. So this is a single particle equation without space charge. So here ks is the external force acting on the beam due to the quadrupole. So this is a single particle equation and this is this is a Hilse equation this periodic with so l is the length of the period and we saw that the solution of this equation can be written in this manner and from and if we substitute this solution in the Hilse equation we can derive the envelope equation. So this equation can be simplified and we put under root epsilon beta is equal to x where x denotes the beam envelope or beam size. So we get the envelope equation this is without space charge and this is the single particle equation without space charge. Now equation of motion for beam size is similar to the single particle equation of motion except for the presence of an additional emittance term which is defocusing. So this is the envelope equation and this is a single particle equation they are similar except for an additional term in the envelope equation which includes the emittance and this is a defocusing term. Now let us try to calculate the envelope equation and the single particle equation with space charge. So let us consider a beam moving in the S direction where the individual particle satisfy the equation of transverse motion. So this is the equation of transverse motion now we have an additional force here. So this is the force due to the quadrupole now this is the external force due to the this is the force due to the space charge of the beam itself. The linear external force is given by minus KSX and FS is the space charge force term which in general is non-linear and it includes both the self-electric and self magnetic forces which we have seen. The quantity FS is related to the space charge electric field ES by this expression. Now we are interested in finding the equation of motion of the RMS beam size. So basically the envelope equation in presence of space charge. So we write the equations of motion for the second movements of the distribution. So thus we take x square average and we find out the d by ds of that so that is equal to 2xx prime average and d by ds of xx prime average is given by this expression. So here we can substitute the value of x double prime from this equation. So we get and again we can find out dx prime square average by ds so it is this expression. So again we can put in the value of x double prime from the equation of motion and we get this. So where the averages are taken over the particle distribution. Now the first two equations lead to the equation of motion for the RMS beam size. So let us call A as the RMS beam size which is square root of x square average. So now using the x square average by ds is equal to 2xx prime average we have a a prime is equal to xx prime average. So differentiating this equation and using equation 2 so equation 2 using this equation we get an expression in terms of now the RMS beam size A. So A second derivative of A minus this term and minus this term is equal to 0. Now this term is the RMS emittance. So numerator of the second term is the square of the RMS emittance. So substituting equation 1 we get this. So now we have an expression in A which is the RMS beam size. So this is now the new envelope equation. So this is the RMS envelope equation and it expresses the equation of motion of the RMS beam size in the presence of space charge. Now here the second term is a de-focusing it is a focusing term and the third term is the emittance term. So this is similar to the equation without space charge. The emittance term is negative and is analogous to a repulsive force acting on the RMS beam size as before. The last term is now due to the repulsive space charge term. So because now the beam has space charge. So we have an additional term here in the envelope equation which is similar to this term. Now let us talk of continuous elliptical beam. So this is derived for the earlier equation is derived for a round beam. Now we have generally the beams in phase space are elliptical. So envelope equations for continuous beams with arbitrary density profiles that have elliptical symmetry in XY space. So electric field components for the uniform density distributions are. So we can derive as before the electric field components for uniform density distributions now with elliptical boundaries. So we get in the electric field in the x direction as this and in the y direction as this. So notice that they are linear in x and y. So rx and ry are the semi-axis of the ellipse related to the RMS beam size y, rx is equal to 2ax and ry is equal to 2ay. Now substituting this equation in equation 6 which is the envelope equation. So substituting it here we get the RMS envelope equation for uniform density beam which are given by this. So now we have 2 equations in x and y because the beam is now elliptical. So in this the quantity k is called the generalized perviance of the beam. So k is the generalized perviance and it is given by this expression. It depends upon the beam current and it depends upon the beam velocity. So higher the beam current, higher the perviance, higher the beam velocity, smaller the perviance. So it is a measure of the space charge of the beam. So higher the current, higher is the space charge and higher the velocity, lower is the space charge. So we know that at higher velocities the magnetic field component cancels out some part of the electric field component of the space charge. So here i is equal to q into n into v. This is the current expressed in terms of number of particles per unit length. So n is number of particles per unit length. These equations were first derived by Kapschensky and Ladermisky for a stationary uniform beam in a quadrupole focusing channel and these are known as KV envelope equations. So it was later shown by Lappostel and Sackerel that these equations are valid not only for uniform density beams but for all density distributions with elliptical symmetry. So even though they have been derived for uniform beams, they are valid for all density distributions. So thus the form of envelope equation is independent of the density profile of the beam. To calculate the RMS beam trajectories, even in the presence of space charge forces, we can replace the actual beam distribution which may not be known in advance with an equivalent uniform beam. So whether it is the Gaussian distribution or a 4D water bag or a KV distribution, you can replace this by an equivalent uniform beam. In the same current and the same second moments as the real beam. So an equivalent uniform beam is so it could have a different distribution but the same current and same second moments as that of the real beam. And then this equation is still valid. So it is convenient to work with an equivalent uniform beam because as we have seen the space charge field for a uniform beam with elliptical or ellipsoidal symmetry is easily calculated and it is linear. So this is where the advantage is that the space charge force is linear. And if you replace the beam with an equivalent beam, whatever the distribution the equations are still valid. So this is a very useful result. The ratio of the space charge to the emittance term. So this is the space charge term and this is the emittance term. In this equation this can be used to determine when space charge is important compared with the emittance in determining the RMS beam size. So both these forces are defocusing but so we can try to find out which is more dominating whether it is the space charge term or the emittance term. Now for a round beam you can take A is equal to AX is equal to AY. So the beam is emittance dominated. So this term dominates when KA square by 4 epsilon square is much much less than 1 and the beam is space charge dominated. So this term dominates when KA square by 4 epsilon square is much much greater than 1. Now a space charge dominated beam can be compared with coal plasma where collective effects are more dominant. So the effect of the distribution as a whole are more dominant. Whereas an emittance dominated beam is dominated by random or thermal effects. Accelerators that are characterized as high current machines can be designed to avoid the space charge dominated regime by increasing the focusing force to reduce the RMS beam size. So you can increase the focusing force here to reduce the RMS beam size. So if this is very high in order to compensate for this the focusing forces can be increased so that your RMS beam size comes down. So let us summarize what we have learned today. The force due to beam self-charge this is known as space charge force. The coulomb effects in Lenard they are usually most important in the non-relativistic beam at low velocities because for relativistic beams the self-magnetic forces increase and produce a partial cancellation of the electric coulomb forces. So we see that we have seen that the force due to the electric field of the beam is repulsive whereas the force due to the magnetic field of the beam is attractive and force due to the magnetic field increases as velocity increases. So this cancels out some part of the repulsion due to the electric field and so at higher velocities the space charge effects are not so important they are more important at lower velocities. The most mostly used quantity to characterize a distribution and to compare a beam with another one is known as the RMS emittance. In the presence of space charge the envelope equations are modified to include a space charge de-focusing term. These equations were first derived by Kapcinski and Ladermisky and for a stationary beam in a quadrupole focusing channel and these are known as the KV envelope equations. So these are these KV equations are very useful because they are valid not only for the uniform density distribution for all but for all density distributions with elliptical symmetry. Only thing is you need to now replace the replace it with an equivalent beam. To calculate the RMS beam trajectories even in the presence of space charge forces we can replace the actual beam distribution which may not be known in advance with an equivalent uniform beam having the same current and same second moments as the real beam. So with this we complete the transverse dynamics of particles in the Linnak and in the next lecture we will study about the longitudinal dynamics of the beams in the Linnak.