**Figure:** Radial and axial positions of 32 Paul-trapped particles--- (* left*): Radial projection of positively charged aluminum particles experimentally observed by Wuerker, et al. (* center, right*): Computer simulation of 32 Mg ions, showing (* center*) instantaneous radial (XY) and (*
right*) axial (XZ) positions. * source file*: ` paul32cr.trp`.

In 1959, R.F. Wuerker, H. Shelton, and R.V. Langmuir used the theory of ``alternating gradient focusing'' and the ``strong focusing principle'' to confine electrically charged particles in a quadrupole trap to which oscillating and static electric potentials were applied. Charged aluminum particles with dimensions of about m were observed to form a crystalline array or a radially-concentric crystalline structure which melted and reformed as a function of the applied potentials and vacuum pressure of the electromagnetic trap.[39]

**Table:** Experimental parameters for the 1959 work of Weurker, Shelton, and Langmuir.

Since the pioneering experiments of Wuerker, et al., electromagnetic traps have been used to confine thousands of ions as well as single atoms and even single elementary particles.

- Motivation for Ion-Trapping
- The Quadrupole Potential of Ion Traps
- The Oscillating Paul Trap Potential
- Motion in the Paul Trap: the Mathieu Equation
- Classes of Solutions to Mathieu's Equation
- Stability Regions for the 1-Dimensional Mathieu Equation
- Stability Regions for a Single Ion in a Paul Trap
- Approximate Solution of Mathieu Equations in Region A
- Higher-Order Stability Regions

Fri May 12 10:36:01 EDT 1995