## Research

Particle accelerators are the largest and among the most expensive scientific instruments in existence. Their theory, design, and understanding requires a host of tools and methods ranging from applied physics to pure mathematics. Charged-particle beams are not isolated systems; thus, they exhibit fascinating nonlinear dynamics and chaotic behavior. They represent probably the most complex low-dimensional Hamiltonian system of practical interest. There is also a wide range of practical applications of accelerators and beams.

### Related Links

### Context of Our Research

The context of our research spans both electron beams and hadron beams. Concerning electron beams, our focus is principally on sources, i.e., electron injectors. Of special interest are (1) high-brightness sources that will feed 4th-generation light sources (**4GLS**) and/or a linear electron-positron collider **(****ILC**), and (2) high-average-current sources that will feed accelerator drivers for high-average-power free-electron lasers (link). Concerning hadron beams, our focus is on both high-average-current accelerators such as is required for copious production of spallation neutrons (**SNS**), and on the Rare Isotope Accelerator (**RIA**) envisioned for nuclear-physics experiments of astrophysical interest, wherein beam strippers (thin foils) can greatly perturb the beam dynamics.

### Main Research Activities

Our main research activities center on the following general topics:

- the production of high-brightness electron and ion beams,
- high-resolution electron-beam diagnostics, and
- theory and simulation of nonequilibrium beams.

Our program has been primarily theoretical and has entailed the development of cross-disciplinary techniques of nonlinear dynamics and their application to charged-particle beams. These techniques are related (but not limited) to the validity of the continuum limit in N-body simulations of beams and galaxies, the existence of chaotic orbits in both time-independent and time-dependent N-body systems, chaotic mixing in these systems, the validity of the continuum limit (Vlasov-Poisson formalism), and noise-enhanced halos.

### Laboratory Experiments

Our work includes laboratory experiments involving novel beam diagnostics that have been done at the Fermilab/NICADD Photoinjector Laboratory (**FNPL**). We are also collaborating with the University of Maryland in planning experiments on the fundamental dynamics of space charge in beams that will be done at the University of Maryland Electron Ring (**UMER**). As we explain briefly below, and more thoroughly in a recent paper, these experiments relate to violent relaxation during the evolution of galaxies and thereby comprise a program of laboratory galactic dynamics. We (especially Prof. Philippe Piot) are also building an in-house Beam Diagnostic Laboratory, one that will include an electron gun for testing and commissioning new instrumentation. We are grateful for help from John Lewellen at Argonne and Kevin Jordan at Jefferson Lab for their assistance in this endeavor.

### Rare Isotope Accelerator

A major component of our research program involves the design of the proposed Rare Isotope Accelerator (**RIA**). RIA has the highest priority for a new construction for the Nuclear Physics community. We are involved in the modeling and design of several components of the accelerator complex: from finding novel tuning methods of approximately 400 independently phased superconducting resonators to allow acceleration of every element from protons to uranium with optimal parameters, design of high resolution isotope separators with large acceptance, high power beam dumps, to ion optics for gas catchers. The large phase spaces require precision modeling of dynamics, including higher order aberrations, and the use of sophisticated codes like COSY Infinity, an arbitrary order charged particle optics code based on differential algebra, and a general purpose nonlinear dynamics code.

### Space-Charge Algorithm

Our** ****research** has revealed that the hierarchies of temporal and spatial scales are critically important drivers of the evolution of beams with space charge; we have found that details do matter. Consequently, we have embarked on an intensive effort to develop a new space-charge algorithm that faithfully preserves these hierarchies while still enabling efficient computations. The underlying methodology is multiresolution analysis, e.g., application of wavelets.

### Symplectic Dynamics

The study of Hamiltonian systems in general led to the development of seemingly two different branches of mathematics: the theory of dynamical systems and symplectic geometry. Both fields have undergone dramatic recent development and it is becoming clear that there is a common core which could lead to a new field called "symplectic dynamics". We (especially Prof. Bela Erdelyi) are investigating this connection. One of the best test beds of this new field is the accelerator (or particle beams in general).

### Summary

In summary, we are developing new theories and methods to describe, simulate, and gain further insight into the behavior of particle beams. The methods range from abstract mathematics from symplectic geometry and topology, through advanced numerical methods such as differential algebra and symplectic integration, to experiments. We hope you find the description of our research and the results of our publications to be of interest.