1. The Physics of Accelerators C.R. Prior Rutherford Appleton Laboratory and Trinity College, Oxford CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory

2. Contents Basic concepts in the study of Particle Accelerators ( including relativistic effects ) Methods of Acceleration –linacs and rings Controlling the beam –confinement, acceleration, focusing –animations –synchrotron radiation CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory

3. Introduction Basic knowledge for the study of particle beams: –applications of relativistic particle dynamics –classical theory of electromagnetism (Maxwell’s equations) More advanced studies require –Hamiltonian mechanics –optical concepts –quantum scattering theory, radiation by charged particles CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory

4. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Applications of Accelerators Based on –possibility of directing beams to hit specific targets –production of thin beams of synchrotron light Bombardment of targets used to obtain new materials with different chemical, physical and mechanical properties Synchrotron radiation covers spectroscopy, X-ray diffraction, x- ray microscopy, crystallography of proteins. Techniques used to manufacture products for aeronautics, medicine, pharmacology, steel production, chemical, car, oil and space industries. In medicine, beams are used for Positron Emission Tomography (PET), therapy of tumours, and for surgery.

5. Basic Concepts  Energy of a relativistic particle where c=speed of light = 3x10 8 m/s is the “relativistic  -factor” For v<<c, Rest energyClassical kinetic energy CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Relativistic kinetic energy

6.  Momentum  Connection between E and p: so for ultra-relativistic particles  Units: 1eV = 1.6021 x 10 -19 joule. E[eV] = m 0  c 2 /e where e=electron charge  Rest energies:Electron E 0 = 511 keV Proton E 0 = 938 MeV CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Basic Concepts

7. Velocity v. Kinetic Energy Low energies: E  E 0 +½m 0 v 2 High energies: CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Basic Concepts At low energies, Newtonian mechanics may be used; relativistic formulae necessary at high energies

8.  Equation of motion and Lorentz force  Acceleration in direction of constant E-field  Constant energy and spiralling about constant B-field. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Basic Concepts Radius Frequency

9. Methods of Acceleration  Linacs Vacuum chamber with one or more DC accelerating structures with E-field aligned in direction of motion. –Avoids expensive magnets –no loss of energy from synchrotron radiation –but many structures, limited energy gain/metre –large energy increase requires long accelerator CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory

10. Fermilab linac

11.  Cyclotron Use magnetic fields to force particles to pass through accelerating cells periodically R.F. electric field Constant B, constant accelerating frequency f. Spiral trajectories For synchronism => approximately constant energy, so only possible at low velocities  ~1. Use for heavy particles (protons, deuterons,  -particles) CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration ~ B

12.  Isochronous Cyclotron Higher energies => relativistic effects =>  no longer constant. Particles get out of phase with accelerating fields; eventually no overall acceleration. Solution: vary B to compensate and keep f constant. Thomas (1938): need both radial (because  varies) and azimuthal B-field variation for stable orbits. Construction difficulties. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration

13.  Synchro-cyclotron Modulate frequency f of accelerating structure instead. McMillan & Veksler (1945): oscillations are stable.  Betatron (Kerst 1941) Particles accelerated by rotational electric field generated by time varying B: Theory of “betatron” oscillations. Overtaken by development of synchrotron. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration

14.  Synchrotron Principle of frequency modulation but in addition variation in time of B-field to match increase in energy and keep revolution radius constant. Magnetic field produced by several dipoles, increases linearly with momentum. At high energies: B  = p/q  E/qc E[GeV]=0.29979 B[T]  [m] per unit charge. Limitations of magnetic fields => high energies only at large radius e.g. LHC E = 8 TeV, B = 10 T,  = 2.7 km CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration

15. (i) storage rings: accumulate particles and keep circulating for long periods; used for high intensity beams to inject into more powerful machines or synchrotron radiation factories. (ii) colliders: two beams circulating in opposite directions, made to intersect; maximises energy in centre of mass frame. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration Example of variation of parameters with time in a synchrotron: Important types of Synchrotron: B  E

16. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Confinement, Acceleration and Focusing of Particles  Confinement By increasing E (hence p) and B together, possible to maintain cyclotron at constant radius and accelerate a beam of particles. In a synchrotron, confining magnetic field comes from a system of several magnetic dipoles forming a closed arc. Dipoles are mounted apart, separated by straight sections/vacuum chambers including equipment for focusing, acceleration, injection, extraction, experimental areas, vacuum pumps.

17. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Confinement ISIS dipole (RAL)

18. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Confinement Hence mean radius of ring R>  e.g. CERN SPS R=1100m,  =225m Can also have large machines with a large number of dipoles each of small bending angle. e.g. CERN SPS 744 magnets, 6.26m long, angle  =0.48 o Injection Extraction Collimation R.F. Dipoles Focusing elements Dipoles

19. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Acceleration  Acceleration A positive charge crossing uniform E-fields is accelerated according to Simple model of an RF cavity: uniform field between parallel plates of a condenser containing small holes to allow for the passage of the beam

20. Revolution period when v  c. Revolution frequency Hz. If several bunches in machine, introduce RF cavities in straight sections with oscillating fields h is the harmonic number. Energy increase  E when particles pass RF cavities  can increase energy only so far as can increase B-field in dipoles to keep constant . Magnetic rigidity CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Acceleration

21. Bunch passing cavity: centre of bunch called the synchronous particle. Particles see voltage For synchronous particle  s = 0 (no acceleration) Particles arriving early see  < 0 Particles arriving late see  > 0  energy of those in advance is decreased and vice versa. To accelerate, make 0 <  s <  so that synchronous particle gains energy CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Acceleration Bunching effect

22. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Acceleration Not all particles are stable. There is a limit to the stable region (the separatrix or “bucket”) and, at high intensity, it is important to design the machine so that all particles are confined within this region and are “trapped”. 

23.  Weak Focusing Particles injected horizontally into a uniform magnetic field follow a circular orbit. Misalignment errors, difficulty in perfect injection cause particles to drift vertically and radially and to hit walls.  severe limitations to a machine. Require some kind of stability mechanism: Vertical stability requires negative field gradient. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory i.e. horizontal restoring force is towards the design orbit. Weak focusing Overall stability: Focusing

24.  Strong Focusing Alternating gradient (AG) principle (1950’s)  A sequence of focusing-defocusing fields provides a stronger net focusing force.  Quadrupoles focus horizontally, defocus vertically or vice versa. Forces are proportional to displacement from axis. Focusing CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory  A succession of opposed elements enable particles to follow stable trajectories, making small oscillations about the design orbit.  Technological limits on magnets are high.

25. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Focusing Fermilab quadrupole Sextupoles are used to correct longitudinal momentum errors.

26. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory  Thin lens analogy of AG focusing: x f F-drift-D system:  Thin lens of focal length f 2 /l, focusing overall, if l  f. Same for D-drift-F (f  -f ), so system of AG lenses can focus in both planes simultaneously. Drift: Focusing

27. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Focusing Matched beam oscillations in simple FODO cell Matched beam oscillations in a proton driver for a neutrino factory

28. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Focusing Typical example of ring design: basic lattice beam envelopes phase advances phase space non- linearities Poincaré maps

29. Electrons and Synchrotron Radiation Particles radiate when they are accelerated, so charged particles crossing the magnetic dipoles of a lattice in a ring (centrifugal acceleration) emit radiation in a direction tangential to their trajectory. After one turn of a circular accelerator, total energy loss by synchrotron radiation is Proton mass : electron mass = 1836. For the same energy and radius, CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Synchrotron Radiation

30. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory In electron machines, strong dependence of radiated energy on E. Losses must be compensated by cavities Technological limit on maximum energy a cavity can deliver  upper band for electron energy in an accelerator: Better to have larger accelerator for same power from RF cavities at high energies. To reach twice LEP energy with same cavities would require a machine 16 times as large. Synchrotron Radiation e.g. LEP 50 GeV electrons,  = 3.1 km, circumference = 27 km. Energy loss per turn = 0.18 GeV per particle

31. Radiation within a light cone of angle For electrons in the range 90 MeV to 1 GeV,  is in the range 10 -4 - 10 -5 degs. Such collimated beams can be directed with high precision to a target - many applications. CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Synchrotron Radiation

32. Luminosity CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory Area, A Measures interaction rate per unit cross section - an important concept for colliders. Simple model: Two cylindrical bunches of area A. Any particle in one bunch sees a fraction N  /A of the other bunch. (  =interaction cross section). Number of interactions between the two bunches is N 2  /A. Interaction rate is R=f N 2  /A, and Luminosity CERN and Fermilab colliders have L ~ 10 30 cm -2 s -1. SSC was aiming for L ~ 10 33 cm -2 s -1

33. Reading E.J.N. Wilson: Introduction to Accelerators S.Y. Lee: Accelerator Physics M. Reiser: Theory and Design of Charged Particle Beams D. Edwards & M. Syphers: An Introduction to the Physics of High Energy Accelerators M. Conte & W. MacKay: An Introduction to the Physics of Particle Accelerators R. Dilao & R. Alves-Pires: Nonlinear Dynamics in Particle Accelerators M. Livingston & J. Blewett: Particle Accelerators CERN Accelerator SchoolLoutraki, Greece, Oct 2000 C.R. Prior, Rutherford Appleton Laboratory