## Task A: Computation of Charged Particle Beam Transport

A new method, employing Lie algebraic tools, has been developed for the computation of charged particle beam transport and accelerator design. It represents the action of each separate element of a beamline or accelerator, including nonlinear effects, by a certain operator. These operators can then be combined, following well-defined rules, to obtain a resultant transfer map that characterizes the entire system.

Lie algebraic methods may be used for particle tracking around or through a lattice and for analysis of linear and nonlinear lattice properties. When used for tracking, they are both exactly symplectic and extremely fast. Tracking can be performed element by element, lump by lump, or any mixture of the two. (A lump is a collection of elements combined together and treated using a single net transfer map.) The speed for element by element tracking is comparable to that of other tracking methods. When collections of elements can be lumped together to form single transfer maps, tracking speeds can be orders of magnitude faster.

In addition to single particle tracking, Lie algebraic methods may also be used to determine how particle phase-space distribution functions evolve under transport through both linear and nonlinear elements. These methods are useful for the self-consistent treatment of space-charge effects and for the study of how moments and emittances evolve. They also provide a means for characterizing beams in terms of invariant eigen-emittances.

Lie algebraic methods also provide powerful means for carrying out analytic computations. Such computations include the calculation of first, second, and higher order chromaticities; first, second, and higher order dispersion; the dependence of tune on betatron amplitude; nonlinear lattice functions; nonlinear phase-space distortion; transfer map normal forms; nonlinear structure resonances; nonlinear beam-beam effects; and nonlinear invariants.

Finally, Lie algebraic methods can be used to give an explicit representation for the linear and nonlinear properties of the total transfer map of a system. This information can be used to evaluate or improve the optical quality of a single-pass system such as a beam transport line or linear collider. For example, it is possible to design high-order achromats, and aberration-corrected telescopes and final focus systems. For a circulating system such as a storage ring, explicit knowledge in this case of the one-turn transfer map can be used to predict the linear aperture of the system without the need for long-term tracking. It is hoped that eventually knowledge of this map can also be used to make predictions about the dynamic aperature of the system.

All these map methods, originally pioneered and developed by our Maryland group, are now beginning to find wide use in Accelerator Physics. Indeed, one of the three working groups of LHC95 (an International Workshop on Single-Particle Effects in Large Hadron Colliders held in 1995 at Montreux, Switzerland) was devoted to the subject of maps and was chaired by Alex Dragt.

Currently the program MARYLIE 3.0 implements these methods through third order. More advanced and higher order versions of MARYLIE are under development.

Lie algebraic methods are also applicable to many other areas including the design of high resolution electron microscopes and microprobes; the design of low energy wide acceptance spectrometers; light optics; and many other problems in the field of dynamical systems. Work is also being carried out in some of these areas.

## Task B: Computation of Electromagnetic Fields and Beam-Cavity Interactions

Recently interest in high current linacs for uses such as the production of tritium, the transmutation of waste, fission drivers etc., make it essential that we understand the formation of beam halos first seen at LANL over 20 years ago. We are studying various resonant mechanisms involving mismatch of the beam to the channel. In addition we are exploring the stability of a mismatched beam to perturbations in charge distribution.

The next generation of linear colliders involves higher current beams of low emittance and short bunch length. The image currents produced by such beams generate wakefields which are capable of inducing instabilities in the beam. Accurate methods are needed to calculate these wakefields and the corresponding impedances for iris loaded cavities.

The next generation of high energy proton rings (e.g. LHC) enters a parameter region where synchrotron radiation becomes increasingly important. Insertions into the beam pipe are needed to remove the heat generated and to shield the beam pipe, usually operated at liquid helium temperatures, from the fields due to the circulating beam. The present plan is to insert a liner with holes into the beam pipe. Methods are needed to calculate the wakefields induced by such a liner, as well as the penetration of the fields due to the circulating beam into the region between the liner and the beam pipe. Of particular interest is the effect of small holes at finite wavelength.

The penetration of electromagnetic fields through small holes in a thick wall enters into a variety of problems of current interest. These include the coupling of wave guides to cavities, the construction of an accurate equivalent circuit for an iris loaded cavity, the design of beam position monitors, and the generation of wakefields by holes in a liner within a beam pipe. Methods are needed to extend the calculation, first done by Bethe in 1944 for circular and elliptical holes in a zero thickness wall for long wavelengths, to the case of thick walls and shorter wavelengths.