Epilogue

Webster's dictionary defines an epilogue as ``a short poem or speech to the audience at the end of a play,'' but this addendum is merely intended to fulfill the roles of conclusion, speculation on where one might (armed with TrapApp) go from here, and acknowledgement.

To bring this thesis to a close (and firmly imprint its salient points in your memory), let us briefly survey what we have seen:

1. Beginning with a mathematical description of the Paul trap's electric field, we have illustrated, in terms of stability regions of the Mathieu equation, conditions under which particle confinement can be achieved in a Paul trap.
2. An approximation for particle motion in stability region A has been described mathematically and demonstrated via simulation.
3. Higher-order stability region B was graphed, and particle confinement was achieved for simulations in this parameter region.
4. The four dynamical regimes (enumerated by Blümel[4]) relevant to Paul-trapping were explored in region A.
   • TrapApp showed stability of undamped crystals in region A.
   • Simulations demonstrated agreement with theory on the orientation and spacing of two-ion crystals.
   • In agreement with Blümel[4,5], TrapApp simulated stability of few-ion crystals as the Mathieu instability of was approached on the axis (zero DC voltage).
   • When many-ion crystals were simulated in region A, non-planar geometries coupled the axial and radial equations of motion via Coulomb interaction. These non-planar crystals melted--- generally well before the Mathieu instability--- as the Mathieu instability was approached at .
   • TrapApp modeled non-heating melted crystal configurations in region A, providing computational evidence of Blümel's assertion that a quasiperiodic regime exists between the crystalline phase and chaotic heating regime.
5. In stability region B, we observed dynamics corresponding to the Mathieu regime, the chaotic heating regime, and the ordered phase. However, crystals in region B were not stable in the absence of damping, and no non-heating, quasiperiodic regime was observed. Rather, undamped crystals heated into the Mathieu regime, in accordance with Blümel's speculation that crystals in region B require extensive cooling.
6. TrapApp demonstrated the validity of Penning trap stability conditions and illustrated that trajectories described by Penning-trapped particles consist of axial harmonic oscillation and, in the radial dimension, small-amplitude cyclotron motion superposed on a slow magnetron motion.

This thesis has not exhausted the resources of TrapApp. In light of this untapped potential, I would like to suggest a few issues one might wish to pursue:

• search for a two-ion alignment transition region, recently predicted by Moore and Blümel,[27] in which the two-ion crystal is neither aligned with the z-axis nor with the z=0 plane.
• exploration of dynamical regimes observed for a collection of Paul-trapped ions interacting via the screened Yukawa potential. Blümel[4] postulates that dynamics observed in such a system would be qualitatively the same as for a system interacting as Coulomb point charges.
• stability of crystal structures as the Mathieu instability is approached by ramping at constant .
• search for higher-order stability regions (beyond A and B).
• an investigation of cooling-induced melting in the Paul trap; this phenomenon was recently investigated by Blümel through analytical computation and numerical simulation.[6]

Finally, I'd like to thank everyone who has made this work possible and profitable. Many thanks are due to my advisor, Dr. Wolfgang Christian, for valuable insights, for good and numerous sources, for help with everything from pointers to mean free paths, and for composing the graph objects which enabled me to finally (after a summer of blind work on Fortran 90) see my ions. The Davidson College Physics Faculty provided valuable feedback on the program and drafts of the thesis; in particular, I am grateful to Dr. Laurence S. Cain and Dr. Robert Cline for thouroughly proofreading (and correcting) the thesis. I'd also like to thank Dr. Ken Hawick, a researcher at the Northeast Parallel Architectures Center, who advised my summer (1994) work on TrapApp's ancestor and introduced me to the wonders of Numerical Recipes. Finally, I acknowledge my debt to the many scientists whose works on ion trapping and molecular dynamics I have relied upon.