Lecture 3 - Magnets and Transverse Dynamics I

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

2. 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

3. 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

4. Center of mass v. Fixed target

5. 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

6. Phase stability PHS.AD5

7. Bucket and pendulum 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 q

8. 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

9. 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

10. 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:

11. 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

12. 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

13. 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

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

15. A chain of buckets