 Suppose there are in a beam many particles and each particle has its coordinates xx prime and these coordinates will lead to different values of invariant of motion a1, a2, a3, etc. So we choose the particle which has largest invariant of motion and this particle bounded all the times all other particles inside this means never this particle will come outer of this ellipse. So this largest area which is defined by the outermost particle at certain location is known as beam emittance. So we have different many initial conditions and we choose which initial condition in a beam leads to largest invariant of motion and that invariant of motion largest value of invariant of motion is known as beam emittance. So this beam emittance is very important parameter in an accelerator physics. Suppose there are many particles inside this ellipse these all particles will remain inside this ellipse. So and now the maximum amplitude of this outermost particle outermost means a particle which is on the outermost ellipse. So we can define the maximum amplitude of this outermost particle and this is the maximum amplitude in angle for this outermost particle. So this shows the maximum angle possible by this beam and this is the maximum displacement possible by the particles in this beam. So this shows you the beam size and this shows you the beam divergence. So by knowing the outermost ellipse you can define the beam size and beam divergence and such quantity which is defined as beam emittance now has information for both the parameters of beam that is the beam size and divergence. And now you know that the area of this ellipse will remain constant throughout this reconference of the accelerator means at any location we will measure the area it will seem as of the other locations. So if beam size changes divergence changes accordingly so that area remains constant. In some distribution of particles beam has certain distributions of the particles and the number of particles may be very large. So in certain distribution of the particles in a beam there may be a very faint boundaries for diluted boundaries you cannot define a clear cut boundaries in this case like here we are having a clear cut boundary. Now in this case of distribution we don't have such boundary. So how the beam emittance can be defined in such distribution. So in any distribution suppose this is the distribution of the particles in the displacement means at the center here particle has highest density then density lowers down and lowers down. We can take one sigma, two sigma, three sigma kind of displacement in the beam this sigma shows the standard deviation of this distribution means at one sigma what is the distance and what is the corresponding ellipse in this distribution. So we can say one sigma ellipse is this if we make two sigma ellipse this may be like this. So in this case emittance is defined by showing that how much particle we are containing the beam ellipse. Now beam as I have said that beam emittance defines the beam size as well as divergence. The maximum amplitude of outermost particle gives you the beam size. So if we take beam emittance at the place of invariant of motion of single particle and beta function of that certain location say s is equal to s1. So this gives you the beam size. In case of single particle this gives you the maximum displacement x max is equal to a under root beta. Now at the place of a we have beam emittance so this gives you the beam size complete. Similarly in the angles we have x prime max is equal to a under root gamma beam divergence will have emittance multiplied by gamma and take the root. So this is the case for single particle and this is the case for complete beam. So in complete beam you have beam size defined by emittance beta root and divergence emittance gamma. You means it may either be horizontal x plane or vertical y plane. If you have clear cut boundary of the beam distribution then this sigma u shows you clear cut boundary beam sizes and beam divergence otherwise if clear cut boundaries are not available in the beam distribution then you have to define that up to what extent we are showing this emittance and according to that up to that point this is the beam size and up to that point this is the beam divergence. Now during the study we have seen that beta was defined by this differential equation. Now instead of beta we have seen that if beta is combined with beam emittance it gives you the beam size. So why not we change this equation in the equation of beam sizes that will be much more transparent for understanding the beam behavior. So let us do that. So beta u is defined like sigma square by u because we have seen that sigma u is under root emittance u beta u. So this gives you the beta u is equal to sigma square u upon emittance u. u means either x or y, formulation is same for both the planes and beta prime u will be 2 sigma u prime emittance u, emittance cannot be differentiated because it is a constant. Here if you can see that when we differentiate sigma u we get sigma prime u is equal to d sigma by d. It is not equal to the sigma u prime which shows the beam divergence. The beam divergence is defined by sigma u prime means what is the spread in the angles and it shows you the what is the derivative of beam size with respect to s. These two quantities are different keep it in mind. So we again differentiate beta because we want beta double prime also in this equation. So we know now beta, beta prime and beta prime will look like this. This is just the differentiation of this quantity and here again we are omitting the subscript u keep it in mind that this formulation is valid for horizontal as well as vertical. Only the thing is that when you are calculating for the horizontal plane put the values correspondingly means beta and emittance for the horizontal plane and when you are calculating for the vertical plane put the values of beta and emittance for the vertical plane. So now we put this beta double prime, this beta prime and this beta in this equation. What we get? 2 sigma square by epsilon on emittance and then beta double prime this is this sigma sigma double prime plus sigma prime square of the emittance minus 4 sigma sigma prime. So we have now from this equation to this equation. You can see that 4 will be cancelled out and this after rearranging you will get sigma double prime plus k sigma is equal to emittance square upon sigma 3. So now you can see in the equation of beam sizes it is again you can see that this is like the simple harmonic oscillation sigma double prime plus k sigma but this is an additional term and this additional terms comes because of this emittance. So this emittance shows you an extra force term. This force term is known as emittance force and if you will put k is equal to 0 means drift space even though sigma double prime will not be 0 and sigma double prime will then be defined by this emittance force. Now sometimes if you know the beam distribution we have seen that there may be a difficulty in obtaining geometrically the largest ellipse means defining the emittance using the geometry may be difficult. However beam emittance can be defined statistically in that case very easy. If you know the beam distribution we can obtain what is the rms values of the positions, what is the rms values of the angles and what is the rms value taken together x and x prime means a correlation. So this is the second movement for the two dimensional distribution. So basically emittance is a measure of second movement of any beam distribution. So if we calculate the second movement of any distribution you will get the emittance. This is the statistical definition of the emittance and generally known as rms emittance. So emittance can be defined geometrically by taking the area of the largest ellipse or you can define the emittance using the statistical formulation if you know x and x prime of each particle. So by taking out this xx prime square is common you will have here in the second term xx prime square upon under root xx prime square and this is the r. So this r is defined by the alpha if alpha is 0 then this r will be 0 and at that location your emittance will be just like sigma and sigma prime. So now this beam size is defined by emittance beta. So beta has the physical meaning defining the beam sizes, gamma has the physical meaning defining the beam divergence and alpha has the physical meaning that it correlates between the beam size and divergence at that location. So tilt of the ellipse. If alpha is 0 means ellipse is not tilted, ellipse is not tilted means the major axis and minor axis of the ellipse coincide with the coordinate axis. This is the case when alpha is 0. So alpha is 0 means d beta by ds is 0 and d beta by ds is equal to 0 means beta has either minima or maxima at this location means either beta is like this so here is the minimum so you will get d beta by ds is 0 here or it may have a maxima like this so here again you will have d beta by ds is equal to 0. So let the location of maxima and minima alpha becomes 0 and ellipse axis means ellipse major axis and minor axis coincide with that of the axis of coordinate axis. Now we see some interesting examples and calculate what is the emittance in that case. Suppose there is a beam which behaves nicely, nicely means all the trajectories of particles are coming parallel to the optic axis or design path and it is a focusing lens so it passes and nicely focused at a point here. So we see what is the phase space distribution of these particles. So if you draw it on x and explain phase space all the particles are having 0 angle with respect to design trajectory so it has a horizontal line here this is the elements and area of this line is 0 so emittance is 0 in this case. After focusing suppose we plot phase space here at this location we see that when x is increasing the angle is also increasing but with minus sign so x here it is increasing and angle is also increasing with minus sign. Similar is the case when we go in the negative x if value of negative x is increasing here the angle is also increasing but in the positive side. So we get again a tilted line here so physically you can say alpha is non-zero here again you can see that there is a simple straight line showing this beam here so emittance is 0, emittance is 0 here, emittance is 0 here it should be because emittance remains constant. This kind of beam is known as laminar beam in which particles are coming parallel to the optic axis or making angle linearly with the displacement. In real practice there is no laminar beam this laminar beam exists only in text books. In real practice you always have non-liminar beam in this case suppose if we plot somewhere here we will have certain x and corresponding x prime and such kind of distribution will be there and after focusing lens the distribution will be changed or you can say the orientation is like this but still if you will find the area or emittance statistical emittance in this case this will remain constant. Now we see that we have defined emittance geometrical area of ellipse and during the solution of equation of motion we have seen that this area remains constant and then we defined emittance statistical which corresponds to second moment of the particle distribution in the position and angles second moment whether this quantity remains constant or not we have not seen that. So taking a simple example we check whether this quantity remains constant or not. So let us suppose that our initial point is here and then beam progress in this way in this direction beam propagates in this direction and reaches here and this length is L take it as a simple drift and emittance here is emittance 1 and emittance here is emittance 2. We have to see whether emittance 2 is equal to emittance 1 or not if it is there we say that in a statistical definition also emittance remains constant. So we check it. So suppose here you can say x 2 is equal to at this point will be x 1 plus L x 1 prime and x 2 prime is equal to x 2 x 1 what we have did we have applied the transfer matrix of the drift so we can write down x 2 and x 2 prime in terms of x 1 and x 1 prime. So now at the location of s is equal to s 2 we calculate the rms emittance. So what will be the rms emittance it will be emittance 2 is equal to we can put it square here this will be x 2 square x 2 prime square minus x 2 x 2 prime square this is the emittance at the location 2. Now put all these variables in terms of the initial condition so at the place of x 2 we will write down x 1 plus L x 1 prime square and x 2 prime is equal to x 1 prime so x 1 prime square minus x 1 prime and x 1 plus L x 1 prime and square if we calculate this they should come out with a equivalent to the emittance at location 1. So let us do that this will be x 1 square plus twice L x 1 x 1 prime plus L square x 1 prime square and x 1 prime square just I have opened the bracket here again we will do the same kind of thing so you will get x 1 x 1 prime plus L x 1 prime square and it is square. So again we will see that this will be multiplied by this so we will get x 1 square x 1 prime square now this multiplied by this will give you plus twice L x 1 x 1 prime here x 1 prime square now multiply this with this you will get L square x 1 prime square x 1 prime square similarly we square this term so you will get minus x 1 x 1 prime square plus L square x 1 prime square x 1 prime square plus twice L x 1 x 1 prime x 1 prime square now you can see that this is cancelled by this and this is cancelled by this and we have x 1 square x 1 prime square minus this term this is x 1 x 1 prime square and this is emittance 1 at the location so emittance at the location 2 is equal to emittance at the location 1 so statistical emittance also remains constant now if particle gains some energy we were talking about the cases when we are having the constant energy we did not mention the energy in our equation of motion means all the particles are having with the same energy now if it passes through the cavity the energy changes whether the emittance remains constant in that case or not we will see this now suppose consider a particle which is having momentum vector like this in this direction so this shows you the transverse momentum px and this is the longitudinal momentum px and angle of trajectory of this particle is defined by px by px now this particle gets some acceleration due to RF cavity when it traverses through the RF cavity the electric field of that cavity increases the energy now electric field in the RF cavity always in the direction of orbit or path means it increases the momentum in the direction of propagating means px so px component has been increased while px remains same and again you can see the vector p as larger magnitude because energy has been of gained through the RF cavity but px remains same so now angle of trajectory of this particle with respect to design trajectory will be defined like this px by ps plus qv by c so denominator has been increased numerator is same means value of x prime has been reduced when value of x prime has been reduced means emittance has been reduced so during acceleration the emittance we defined does not remain constant it remains constant only if energy is constant this can also be understood in this way x prime is equal to px by ps so ps is approximately equal to p because we are working in the paraxial approximation so whole momentum is equal to the longitudinal momentum so at the longitudinal momentum we can write down gamma beta cm during acceleration gamma beta increases so again when gamma beta increases x prime reduces so again we can see in this fashion that when gamma beta increases x prime decreases so if we multiply emittance with gamma beta then it can remain constant so as gamma beta increases that increases the numerator denominator both because we have multiplied emittance with the gamma beta then emittance remains constant so if we multiply emittance with gamma beta then this new term is known as normalized emittance so normalized emittance remains constant even though we are accelerating the particles now we see the finally how the ellipse we have seen that at different location ellipse will have different orientation different elongation we take one such example when particles are passing through the drift space that is the simple example so suppose at location first location the ellipse is like this right the whole all the particles on the periphery of this ellipse has been propagated up to this point son how the ellipse will look like because this is a drift space so maximum angle will not change because particles trajectory is angle will remain constant because there is no force angle can be changed only if we apply the force so particles go in the street in the direction already they have so x prime will remain constant so x prime will be like this but as particles are going like this you can see that after certain distance the maximum amplitude may increase the maximum amplitude may increase means here the maximum amplitude force this here it has been increased to this way so ellipse will be bounded between these points now draw an ellipse between these boundaries keeping the area same as of this ellipse because this is now elongated so width of this ellipse would be reduced so this is the ellipse at this location there is this ellipse or these particles go further downstream to this point and reaches here so what will be the ellipse when it reaches here again it will be bounded vertically in the same boundaries angle angles of these particles do not change and the maximum amplitude is further increased so maximum amplitude of width at translation displacement is further larger than this value so ellipse will be more tilted like this more tilt and more elongated but less width to have the constant in this fashion we can draw ellipse so I left one exercise for all of you that you draw the ellipse is just before a focusing lens and just after the focusing lens so just before take the ellipse in this orientation because it is the diverging beam and after quadrupole or focusing lens suppose beam has now convergent behavior so how the ellipse will be there just after the quadrupole it is left as an exercise for so references are seen and in next lecture you will see more on these