 Now how collider actually looks like? So, this is one of the basic schematic of Intersecting the storage This was the first Hectron collider And it was operated in the Sun Actually, you can see that From the protons and proton Beam is sending here and protons and proton also beams sending here. So there are Two rings one in green color and one in a positive two are independent completely different accelerators And at these points these points these two accelerators Viking chamber crosses each other So at these locations Collision takes place and at these locations Detectors can be put to detect these collisions There may be another type of schematic Like we have in large electron collider at Sun here in the same vacuum pipe both the rings are both the beams are rotating and That's why It's a dipole looks like this These are the coils Around the dipole and they were two chambers One is for the first beam and another one is the for the second So both the beams are carried into the same channel and Then at certain locations collision are taking place Only energy is not sufficient For the collectors actually when we hit this fixed target in fixed target. There is a plenty of Molecules or atoms or particles are available because fixed target is high density material But beam is not so dense So events number of events means number of collision events reduces So there is another important factor, which is very useful that is related to the how much Events are taking place in per second and definitely for a particular event This rate how much event are taking place depends on the scattering cross section for that particular Supposedly we want to pinpoint some or pick some event Then what is the scattering cross section for that event? So that describes what is the event per second for that particular time? so There is proportionality constant L Which is known as luminosity in the case of colliders So in colliders Luminosity has to be increased if we have very high luminosity in this case So even with a smaller cross section of the events we get sufficient rate of events If we get sufficient rate of events that events can be detected Because Detectors cannot detect events if These are very very rare So far in the between the detection limit DR and DT by DR by DT has to be increased So this luminosity has to be increased Like in the case of synchrotron radiation sources We have seen that evidence has to be reduced in case of a spallation mutual source We have seen that beam power has to be increased and in the case of colliders Luminosity has to be increased So what is luminosity now? We see that Suppose there is a target right now. We are Taking fixed targets So there is a beam which is going on this target We take flux of this beam flux of this beam means number of particles per second As the file And the density of the target material is rho time And we are considering that This rho density is same throughout the target means target is homogenous And targets dimensions are much larger than the beam time Now the length travel by the beam in one second in the target take it is to be L This L is the length Taken by the beam in one second Now if cross section of the scattering is sigma Suppose we are interested in some Event and that has scattering cross section sigma Then total possible interaction in one second will be Rho target Means this suppose a single particle is passing through this target a single particle is passing through that target means it will interact with up to L length L Into sigma sigma is the scattering cross section into rho Sigma L sigma is the scattering cross section means it is area And L is the length and rho is the target materials density So total number of particles in the target In the path of this particle will be rho into L into sigma Now if we are taking Beams Cross section as a so if for one particle it is rho L sigma So for whole beam will be rho L sigma multiplied by the area of the beam So so many interactions are possible with sigma cross section Now for per unit time in per unit area we will have Phi rho target L sigma From this From this We will get this Now here you can see that This sigma Means dr by dt Is equal to L sigma So apart from this sigma this whole things define the Luminosity of the collider So luminosity of the collider A collision experiment i will not say collider because we have taken the fixed target So in this fixed target experiment What is the flux of the beam? What is the density of the target? And what is the length covered by beam in one second? That defines the total luminosity In case of target In case of colliding beam this target is also moving And rho is not so high So we have to take the density of both the beams And now if we will take density of both the beams means luminosity Depends on density distribution of the beam It is highly depending on that So now i will depend on the distribution function How the density is there So we can obtain from as of the previous line Total overlap integral of these two bunches What is the overlap integral? So overlap integral will be according to density What is the density of this first bunch? What is the density of this second bunch? And we can integrate on x, y, z and t because these are moving bunches So we have to integrate on time also And time has been changed as per in our accelerated ds0 So we have changed time from t to ds by beta So these integral will show how much humans can take place With particular cross-section theory This k here defines some kinematic coefficient because bunch are moving So when we solve this We will get rho 1x as x because density may Then there may be a distribution function of the density in this bunch There may be a distribution function in the perpendicular direction also So this is rho 1y And along the s also there may be a distribution function So that is rho 1s minus sv because it is moving along the s axis So in case of s we are getting s minus sv And similar distribution for the second bunch In case of second bunch there will be s plus sv Because it is moving in the opposite direction to the first bunch And again integrate So this will give you the total overlap of densities Distribution function And if we multiply by number of particles in bunches We get total events This is the distribution function Multiplied by number of the particles Then we get the total events Now if there are nb bunches This is for one bunch How many particles are there in the first bunch How many particles are there in the second bunch And many bunches may be there So this is the number of bunches And this is the revolution frequency Means in once evolution frequency How many bunches are there So this completely defines what is the luminosity of the collider So this number has to be increased for the collider Now you can see that This highly depends on the distribution function Density distribution function of the bunch So in most of the cases Analytical solution will not be available And numerical integration has to be given But for a simple case Gaussian distribution we can calculate it So take bunch as a Gaussian distribution It is a Gaussian distribution I have written here only for the case of s Similar distribution Gaussian distribution will be in x and y also Means in x also it is a Gaussian distributed In y also particles as Gaussian distribution And in s also particle I have Gaussian distribution So Gaussian distribution is defined as Exponential of minus s by 2 sigma square Sigma is the standard deviation of this distribution So when we will put at the place of rho 1x, rho 1y and rho 1s And here in this case we are considering Same beam, same bunch on the both side So second bunch will also have e raised to minus x square upon 2 sigma square That's why this beam has been omitted here Because both are having the same So for the first bunch we are having e raised to minus x square by sigma s 2 sigma square For second bunch also we are having e raised to minus x square by 2 sigma square So for it will become e raised to minus x square sigma s Similarly in the y case similarly in the s beam And this sigma s root 2 pi makes this denominator Now if we integrate e raised to minus 80 square From minus infinite to plus infinite The integration is root pi by k Using that relation we get finally the luminosity in this case Now you can see that in the case of Gaussian bunches Luminosity can be increased If number of particles per bunch can be increased If number of bunches can be increased And beam sizes has to be reduced Means very very small beam should be there When the collision is taken place And a very large populated bunch should be there It is from the common sense also If we have densely packed the bunch Number of collision events will be increased Means very large number of particles If it is packed into a very tiny bunch Then definitely if these bunches will be collide Collision events will be increased So this is the same thing we are saying mathematically But there are some problems Actually luminosity decreases in the practical cases Because we were considering as head on collision In accelerator Beam comes at certain angles First beam is coming through this direction And second beam is coming through this direction So this angle actually reduces the overlapping In the case of head on collision Overlap may be complete This will come here This will come here At certain even They may completely overlap But in the case of angled beam Complete overlap will not be there So overlap integral will be reduced Or we can say in other language That effective beam size has been reduced So this is the reduction in the beam size And this theta is there This is the theta This theta is here So by crossing angle Actually luminosity decreases So whatever we have calculated That is the maximum luminosity In actual collider This will be decreased Due to crossing There are more many more effects Which will decrease the Luminosity in particular collider The one is our glass effect What is our glass effect? Our glass effect is like this Suppose this is the S And suppose this is the collision point So I won't focus beam here Means this parameter beta will be minimum here At this location So suppose this is the quiz parameter beta So it goes like this It means sigma here at this location s is equal to s2 Or sigma here at this location s is equal to s1 Are different Means sigma or beam size Actually in collider is a function of s And we have not taken that We have taken the constant sigma throughout the s So in fact sigma changes with s And again this reduces the overlapping And if you want to make Beam size more focused Means if we reduce the beta more Then beta will go like this In the drift space On the both side So if we squeeze the beta Our glass effect increases Means sigma as a function of s Has a stronger dependence If we want to squeeze the beam more and more So there may be an offset also Offset means due to imperfections Actually no machine is perfect We have studied the perfect machines Means perfect dipole, perfect quadruple Beam is also perfect And these are nicely aligned along the design axis In real Actually There may be imperfections in the dipole magnets During manufacturing of fabrication In the quadruple also During placement There may be a misalignment also So these things deteriorates the parameters Which we have evaluated So in that case this luminosity will also be decreased So those all are effects Which decreases the luminosity And one has to control So that the luminosity should not decrease much Otherwise rate of events will be decreased So these are some of the list of colliders In the world Mostly there are circular colliders You can see This list is circular Only one collider SLC is linear accelerator based collider Now we have studied during the synchro-combed radiation chapter That when electron is subjected to some acceleration it regulates So pushing the electrons to very very high energy In circular accelerator is not possible Maximum energy achieved for electrons in circular accelerators 100 g That was in large electron positive collider That was in the same tunnel We are there Presently large electron collider is situated So beyond that As we increase the energies Emission of synchro-combed radiation becomes enormous And the requirement of RF follow becomes impractical So future collider in case of leptons Should be made linear However for heterons For heavy particles circular colliders can be Now in the case of heteron Heterons are composite particles Means proton itself has three quarks This is not a fundamental particle Proton is not the fundamental This is a composite particle And if heteron collision is taking place This becomes very complex Because we are colliding to composite particles So there may be a large number of events And to pinpoint some particular event Is just like to search the needle in the hole So that's why heteron colliders are known as discovery machines But leptons, electron positrons are fundamental particles These particles do not have any internal structure with them These are fundamental So far these are fundamental So their collision is clean collision So by tuning the energy of electron and positron The rate of event of particular collision can be increased So these are precision machines Laptons colliders are the precision machine Heteron colliders are the discovery machine However, leptons colliders cannot be built using the circular accelerator And if we go into built with linear accelerators The lens becomes nonx I will give an example There is a project plan For the international linear collider for electrons And positrons And in that collider The maximum energy which is nearly 1 Tb per beam And the length of that collider is going to be 14 kilometers So linear colliders are big So there is a way to get rid of this Instead of electron positron or proton colliders Make the neon colliders Neons are leptons These are fundamental These particles do not have internal structures So these are fundamental Neons are fundamental However, leons are 200 times heavier than leptons So it radiates less So we have leptons but less radiation So we can make circular accelerators Which can push the neons to very high energies So there is also a plan in the US to build a neon collider Tevatron now has been closed When LXC was started with its operation Hera is the only collider which collides electron and proton And in this course Now I will give you a very short summary That we have seen that different kinds of cyclic accelerators And in cyclic accelerators The basic condition is to maintain the synchronism And how we are achieving the synchronism On the basis of that different accelerators are categorized So cyclic now In this accelerator Revolution time is independent of the particles energy However, this can handle many heavy particles Because for the light particles gamma changes rapidly and synchronism picks up So for leptons we can accelerate in a tiny accelerator microchip Where the synchronism condition is achieved differently Then there are synchrotrons in which orbit is kept constant And the problem of large magnets of cyclotron can be solved Because on that constant orbit we have to put the magnets on it And when we analyze the Synchrotron or motion in the synchrotron we are in the Hill's equation And that Hill's equation can be solved in two different ways Both are leading to same solution One is the matrix method and other one is the parameterization of the optics In that we got three parameters alpha beta gamma Which is an industry's parameters and an important parameter And then we saw that how often momentum particles can be analyzed or their motion can be analyzed The solution of Hill's motion gives a vitatron oscillations And an often momentum particle with that vitatron oscillations there is a dispersion time In longitudinal plane RF cavity works as the focusing element That keeps the particles focused in the longitudinal direction also And therefore synchrotron oscillations are there in the longitudinal plane Then we saw three types of different accelerators Namely synchrotron radiation sources Proton synchrotron for espalation sources and a little bit of colliders In these three accelerators we picked one particular parameter In the lecture and we concentrated our lecture around that particular parameter For the synchrotron radiation sources we picked that parameter that what is the radiation damping and quantum citation and how we can define the beam emittance using that Why we picked that parameter because beam emittance is the most important parameter And whole design of synchrotron radiation sources Or efforts in design go far lowering down these beam emittance There are many more things in the synchrotron radiation sources that cannot be covered in this lecture And also in the case of proton machine We mostly concentrated towards how we can define the beam power And how it is different from the electron machine And we picked only one aspect again here that was the space charge A very simple picture was drawn by taking the uniform beam for the space charge calculation And we saw that beta electron tune has been changed in the presence of the space charge And this space charge can cause the beam growth and beam loss So this one important parameter was picked for this lecture There are many more parameters But anyhow in the limited time we cannot cover all these things And in the case of collider we again picked a single parameter that is luminosity And what was that luminosity we understood So in this case I complete this cyclic accelerator course