Accelerator Physics - Beam Dynamics

Beam Dynamics

See also: Particle beam, Strong focusing, Beam emittance, and Radiation damping

Due to the high velocity of the particles, and the resulting Lorentz force for magnetic fields, adjustments to the beam direction are mainly controlled by magnetostatic fields that deflect particles. In most accelerator concepts (excluding compact structures like the cyclotron or betatron), these are applied by dedicated electromagnets with different properties and functions. An important step in the development of these types of accelerators was the understanding of strong focusing. Dipole magnets are used to guide the beam through the structure, while quadrupole magnets are used for beam focusing, and sextupole magnets are used for correction of dispersion effects.

A particle on the exact design trajectory (or design orbit) of the accelerator only experiences dipole field components, while particles with transverse position deviation are re-focused to the design orbit. For preliminary calculations, neglecting all fields components higher than quadrupolar, an inhomogenic Hill differential equation

can be used as an approximation, with

a non-constant focusing force, including strong focusing and weak focusing effects

the relative deviation from the design beam impulse

the trajectory curvature radius, and

the design path length ,

thus identifying the system as a parametric oscillator. Beam parameters for the accelerator can then be calculated using Ray transfer matrix analysis; e.g., a quadrupolar field is analogous to a lens in geometrical optics, having similar properties regarding beam focusing (but obeying Earnshaw's theorem).

The general equations of motion originate from relativistic Hamiltonian mechanics, in almost all cases using the Paraxial approximation. Even in the cases of strongly nonlinear magnetic fields, and without the paraxial approximation, a Lie transform may be used to construct an integrator with a high degree of accuracy.