Lecture 3 - Magnets and Transverse Dynamics I ACCELERATOR PHYSICS Lecture 3 Longitudinal Motion TT 2012 E. J. N. Wilson

Physics of the electron v. protons When we collide Protons we collide complex assemblies of three quarks Only two quarks interact Their available energy is on average only 10% of the total centre of mass energy We do not know which quarks they are Hence in some ways 100 GeV LEP is as good as 1000 GeV TEVATRON HENCE THE RETURN TO LEP AFTER SPS AND THEN LHC AFTER LEP

Newton & Einstein Almost all modern accelerators accelerate particles to speeds very close to that of light. In the classical Newton regime the velocity of the particle increases with the square root of the kinetic energy. As v approaches c it is as if the velocity of the particle "saturates" One can pour more and more energy into the particle, giving it a shorter De Broglie wavelength so that it probes deeper into the sub-atomic world Velocity increases very slowly and asymptotically to that of light

Center of mass v. Fixed target

Luminosity Imagine a blue particle colliding with a beam of cross section area - A Probability of collision is For N particles in both beams Suppose they meet f times per second at the revolution frequency Event rate Make big Make small LUMINOSITY

Phase stability PHS.AD5

Bucket and pendulum q q The “bucket” of synchrotron motion is just that of the rigid pendulum Linear motion at small amplitude Metastable fixed point at the top Continuous rotation outside

Analogy with gravity What keeps particles in the machine There is a solution to Hills Equation It is closed and symmetric It is closer to the axis at vertically Defocusing Quadrupoles Deflection is larger in F than D and cancels the force of gravity elsewhere. We could call the shape the “suspension” function. Fig. cas 1.4C

Dispersion Low momentum particle is bent more It should spiral inwards but: There is a displaced (inwards) closed orbit Closer to axis in the D’s Extra (outward) force balances extra bends D(s) is the “dispersion function” Fig. cas 1.7-7.1C

Dispersed beam cross sections These are real cross-section of beam The central and extreme momenta are shown There is of course a continuum between The vacuum chamber width must accommodate the full spread Half height and half width are:

Physics of the electron v. protons When we collide Protons we collide complex assemblies of three quarks Only two quarks interact Their available energy is on average only 10% of the total centre of mass energy We do not know which quarks they are Hence in some ways 100 GeV LEP is as good as 1000 GeV TEVATRON HENCE THE RETURN TO LEP AFTER SPS AND THEN LHC AFTER LEP

Transition - does an accelerated particle catch up - it has further to go Is a function of two, momentum dependent, terms b and R. and Using partial differentials to define a slip factor: This changes from negative to positive and is zero at ‘ transition’ when: GAMMA TRANSITION

Synchrotron motion Recall Elliptical trajectory for small amplitude Note that frequency is rate of change of phase From definition of the slip factor h Substitute and differentiate again But the extra acceleration is THUS

Synchrotron motion (continued) This is a biased rigid pendulum For small amplitudes Synchrotron frequency Synchrotron “tune”

A chain of buckets