Map Methods for Accelerators Alex Dragt University of Maryland Abstract An overview is given of accelerator design with special emphasis on the Large Hadron Collider, the world's largest accelerator scheduled to become operational in 2007, and on the proposed International Linear Collider. An overview is also given of how map, Lie algebraic, and symplectic integration methods are used in accelerator design. € Introduction Particle accelerators have broad applications including manufacturing, biology, medicine, chemistry, material science, nuclear physics, and high-energy physics. They range in size from small (almost table-top) cyclotrons to the Large Hadron Collider (LHC) under construction at CERN and the proposed International Linear Collider (ILC). Worldwide, accelerators number in the several hundreds. Synchrotron light sources and free-electron lasers are particularly common. Accelerators use magnetic fields to guide and confine a charged-particle beam, and the electric fields in radio frequency (RF) cavities to accelerate the beam. Magnets include dipoles, quadrupoles, solenoids, sextupoles, octupoles, etc. They can be normal or superconducting (1.9 degrees Kelvin). RF cavities can also be normal or superconducting. The various beam-line elements (magets, intervening drift spaces, and RF cavities) are arranged sequentially to form what is called a lattice. Accelerators come in two basic geometries: linear (single-pass) machines called Linacs, and circular (multi-pass) machines. To achieve very high energies for electrons or positrons, only linear machines can be used. High-energy electrons/positrons in a circular machine would radiate energy, due to synchrotron radiation, faster than it could be re-supplied by RF cavities. (Circular motion implies continuous inward acceleration, and accelerated particles radiate.) For heavier particles (muons, protons, ions) synchrotron radiation energy loss is much less, and the use of circular machines is advantageous. In a circular machine particles can be made to pass through the same RF cavity multiple times thereby gaining energy on each pass. • The action of each beam-line element can be described by a (symplectic) map M. Charged particle motion is Hamiltonian, and Hamiltonian flows generate symplectic maps. Elements that Produce Linear Maps • DIPOLE MAGNET • Provides bending • QUADRUPOLE MAGNET • Provides focus and defocus Elements that Produce Nonlinear Maps • SEXTUPOLE MAGNET • Provides quadratic nonlinearity/correction • OCTUPOLE MAGNET • Provides cubic nonlinearity/correction