 So in this lecture we will discuss about a new class of accelerators, basically the colliders. Colliders are the accelerator which actually collides the beam, means basically one collider consists two accelerators and in these accelerators the counter rotating beams are accelerated and then head on collision takes place and at these head on collision locations detectors are placed to record the events. So colliders may be linear accelerator based colliders or synchrotron based colliders. Presently all the colliders except one is based on the synchrotron. The very first collider is the linear accelerator based collider. Actually accelerators journey started with collision experiment. Recall the Rutherford experiment. In Rutherford experiment he bombarded alpha particle on the gold foil means some type of collision experiments was there and with this experiment he deduced the nuclear model of the atom and later he notices that if higher energetic particles would be available then nucleus also can be broken or nucleus can be split it. So by inspiring these kind of things Cockraft and Walton built the first accelerator that was the DC accelerator and using the beam of these DC accelerator they split it the lithium ion. So accelerator journey started with the collision experiments. However later on it was shown that in case of fixed target experiment means we have some accelerated beam of charged particle and we are hitting on the target rather than if we have two colliding beam we will get much much benefit out of the energies. So higher and higher energies and colliders building started. Why higher and higher energies are demanded? So there are two reasons about that first of that if we have higher energy means deep Broglie wavelength will be smaller deep Broglie wavelength is lambda is equal to h by p. So as p is higher means lambda is smaller. So we get a better resolution using high energy beam because each particle is associated with some deep Broglie wavelength and deep Broglie wavelength has been reduced due to high energy. So resolving power by analyzing those wave waves can be increased. The second is directly related to the e is equal to gamma mc square m is the rest mass of the particle. So if we have very high energetic particle it can be converted into a more massive particle mean more massive particle can be generated using very high energy beams. So now in this lecture we will see one of the important parameters which has to be optimized for making the collider and why really colliders are needed means how much benefit we are getting out of the fixed target experiments. To see why colliders are needed we recap some of the things of the relativity we want to know that these are the Lorentz transformation. This is unprimed frame and this is the primed frame and primed frame is moving with respect to there is a relative motion with the speed v in the x direction. So x coordinate will transform according to gamma x minus vt gamma is a relativistic gamma factor because there is no motion along the y and z axis so these axis will not change and time is also relative. In Newtonian mechanics time was absolute and this was the fundamental change by Einstein that time is also relative. So from one frame the other frame time changes by this formula there is gamma t minus vx by c square vx appears because motion relative motion is around the x axis. Now you can see that instead of v we can write down beta c so this transformation will be x prime is equal to gamma x minus beta ct. Similarly in the case of time here we can see that we can multiply this time with c. So this will be ct prime so here in RHS also we will multiply the speed of light c and here instead of a square this will be c because one multiplication of c is c and this will ct prime is equal to gamma ct again we will write v by c as a beta this v by c as a beta. So now you can see that transformation of x in this new notation is similar to the transformation of ct prime so ct prime can be used as the fourth coordinate and in this fashion we can make a four-dimensional space in which one dimension is by the ct prime a ct for ct prime so we can make three-dimensional instead of three-dimensional vector four dimensional in this space time. So now we take x0 as ct, x1 is equal to x, x2 is equal to y and x3 is equal to z so our equations can be written down as x0 prime is equal to gamma x0 minus beta x1 x1 this is the relation we obtained from here because x0 is ct so at the place of ct prime we have written x0 prime here it is ct so this will be x0 and x will be written as x1 similarly this equation will be written down as x1 prime is equal to gamma x minus beta x0 these two relations will be there for the y and z so in relativity we can form a four vector. Relativity can be understood without going into the four vectors however four vectors provide a powerful way of analyzing some problems and it also opens the way for understanding the gtr now we see the beauty of these four vectors if we multiply these four vectors together we get c square t square minus x square minus y square minus z square there is one rule that you have to put minus sign before the space like vectors space like components components and time like component will have positive sign means you can see that this has been initiated like this we can write down x m x t and m is the minkowski matrix by which we get minus sign so in this way inner product of the four vector is defined four vectors inner product definition is this c square t square minus x square minus y square minus z square any quantity which can be represented by four vectors in the relativity will have this formulation for their inner product now what what is the beauty of this inner product we transform this four vector inner product into the another frame so let us say that in primed frame it is defined like this c square t prime square minus x prime square minus y prime square minus z prime square now we transform it into another frame so at the place of 3 prime we will put ct minus beta x and at the x prime we will put gamma x minus beta ct y and z will be unchanged when we will open this c square t square beta square x square minus 2 beta cxt here we will get x square plus beta square c square t square minus 2 beta cxt you can see that this will be cancelled out by this and take the ct square common from this and this so you will get gamma square 1 minus beta square c square t square and similarly gamma square 1 minus beta square with x square because x square term is this and second x square term is this so we will get 1 minus beta square here also minus y square z square 1 minus beta square itself is 1 by gamma square so this will be cancelled out by this and this will be cancelled out by this so you will get c square t square minus x square minus y square minus z square means this inner product which is defined in this way is Lorentz invariant it remains constant under the Lorentz transformation and similarly for the energy or momentum we can define four vectors by taking the first component as e by c then space component exp by pz so there inner product will also be a constant under the Lorentz transformation or it doesn't vary under the Lorentz it is an invariant due to this invariancy we can put momentum and energy conservation law in a very compact form and it can be used when we study the collision in the colliders so this will be e square by c square p dot p the inner product of the four momentum e square by c square minus p square this p square is this space like component and this is the time like component so space like component again has minus sign by our definition and you can see that this e square minus c square minus p square is equal to m square c square because we have e square is equal to c square e square plus m square c four so this quantity becomes m square c square and m is the rest mass of the particle it is invariant so this quantity is invariant so this is a Lorentz invariant now in the beginning the studies were carried out using the bombardment of high energetic particles on the fixed target like the Rutherford Rutherford also bombarded particles on the fixed target that fixed target was the gold foil and later on high energetic particles ejected from the accelerator and then sent to the target and having the collision experiments however in 1956 the concept of colliding beam experiment emerged that we have to have instead of fixed target two colliding techniques means target is also beam so two beam which is having high energy will be good that's why collider is also known as this machine machines now what is the use of collider means instead of fixed target what benefit we get out of the quality suppose in the collision of two particles we have mass m1 and m2 and the center of mass energy can be written down as this is a mean this we have written down p square e square by c square minus p square this is the inner product of four momentum which we have calculated in the last slide for the two particles for the two particles total four momentum will be p1 plus p2 and we have to square out this this is the invariant means in the center of mass and left frame both will see this the same quantity so this will be one by c square e1 plus e2 square minus p1 plus p2 vectors and there is no in the colliding beams because these beams are moving in opposite direction so we have p1 momentum is equal to minus p2 of the moment means p1 is equal to minus p2 because both the beams are moving in the opposite direction and then they have will hide collisions so p1 p2 square will be this term will be zero because of the p1 is equal to minus p2 and we are having only this term in case if collision is taking place at same energy means both particles are having the same energy say e1 is equal to e2 is equal to e then we have this 2e so 2e square by c square is the invariant and it means energy is 2e 2e is the energy available in center of mass so if two particles having same energy e and are colliding in the collider collider then the center of mass energy will be 2e this center of mass energy is available for the new events so for new events means new creation of the particles for that total energy of 2e is available in this case now we compare this case with the fixed target experiment in fixed target experiment the second particle p2 which is in the target is fixed its momentum is zero so for square of the four momentum length of the four momentum will be e1 plus e2 square upon c square minus p1 square p2 is zero so this will be e1 by c square e1 square by c square this p1 square we have opened these brackets so the e2 square plus 2e1 e2 by c square now this is invariant for the first particle so and this is equal to m1 square c square we have seen this this is in irrespective of the lab or center of mass this is because the second particle is not moving so e2 will be only the rest energy so m2 square c square and the third at the place of e2 again we will put m2 c2 square and m2 c square c square c square will be cancelled out so this is the available energy in the center of mass frame in the case of fixed target experiment now again take the case of similar particles that is m1 is going to means rest mass are same suppose proton-proton collision or electron-electron collision so both the mass are same in that case this equation will be reduced to 1 by c square 2e0 square plus 2e0 now total energy will be square root of this which is available in the center of mass so total energy available is 2e0e0 has been taken out so e0 plus q so in the case of colliding beam the available energy was 2e while in the fixed target experiment energy available is this one this is much much lesser than the colliding beam energy we take an example for calculating this consider that as 7 TV proton it's another 7 TV proton 7 TV is the energy in the LHC large-eating collided at sun therefore available energy in the colliding beam facility will be 14 T 14000 g now consider that instead of 7 TV 7 TV we take a 14 TV proton and hits on the fixed target so available energy we can calculate using this formula 2e0 so 2 I have taken the 1g as the rest mass of the energy of the proton then again 1gb plus 1400 g it comes out only 167 g so instead of 14 000 gb we are getting only 167 gb in the fixed target experiment so compared to fixed target experiment colliding beam provides a very very very high energies means many fold massive particles can be generated in the colliders rather than in the fixed target experiment there is one comparison in this table suppose proton and proton hits with 7000 gb 7 TV which was the above example so in the center of mass 14000 gb is available and in the fixed target experiment it is only 114 gb here we have increased the energy to 14 TV of the single proton here it is calculated for the 7 TV suppose 100 gb electron impinges on the 100 gb electrons so in center of mass in the colliding beam facility 200 gb will be available while in the fixed target experiment only 0.32 gb will be available and if proton hits the electron one at the 30 gb and another at the 20 gb these are the numbers closely of the HERA collider in the Germany so center of mass energy will be available as 235 gb in the case of colliding beam and in the case of fixed target experiment this will be only 7.5 gb one should go through this reference it beautifully describes these things