Introduction to the Physical System

Confinement of particles to small regions of space for long periods of time makes possible many high-accuracy experiments whose results are on the cutting edge of research in physics. The pertinent fields of research include molecular spectroscopy, quantum electrodynamics, plasma behavior, the size of elementary particles, and heavy ion storage [13][14][4][12]. Ions, or electrically charged particles, can be confined for these types of experiments by electric and/or magnetic fields without the difficulties introduced by collisions with physical walls.

Both the Paul trap and the Penning trap have been used successfully as ion traps. The geometry of the traps is shown in Fig. 1: an electric potential difference is applied across electrodes whose surfaces are defined by hyperboloids of revolution about the -axis, typically with [13]. The Paul trap has an oscillating electric potential of:

where , , and are spatial coordinates; represents time; is in the radio frequency range; is a constant (DC) electric potential; and is the amplitude of an oscillating (AC) electric potential.

The electric potential of the Penning trap is constant with respect to time and depends only on position:

In addition to an electric field, the Penning trap contains a constant magnetic field directed along the -axis.

The force acting on a single ion in a Paul trap is given by the scalar product of the ion's charge and the electric field; the electric field equals the negative gradient of the potential:

where î, j, and k are the unit vectors along the -, -, and -axes, respectively, and is the charge of the ion. The oscillating polarity of the Paul trap's electric potential results in a force which alternately focuses the radial and axial (-axis) displacement of the ion.

A single ion in a Penning trap is influenced both by its constant electric field, which restricts the ion's axial position, and by a magnetic force whose magnitude and direction are given by:

where v is the vector representing the ion's velocity and B is the magnetic field, which is a vector directed parallel to the -axis.

When a collection of ions is trapped, one must also consider the forces of interaction between ions. The Coulomb force describes the interactive force on a charge in a collection of point charges:

where is a constant, and are the position vectors of particles and , and are the particles' charges, and is the unit direction vector from particle to .

Trapped ions display a variety of interesting behaviors, depending on parameters including: the charge of the ions, the mass of the ions, the temperature at which the experiment is conducted, the size of the trap, the amplitude of electric and/or magnetic fields, and the frequency of field oscillation (in the case of the Paul trap). For example, an ion may oscillate periodically or chaotically with respect to the -axis. Similarly, the radial position of an ion may vary periodically or chaotically. When atoms are closely coupled - that is, when the energy associated with ion interactions is large compared to thermal energy - crystals may form. In other cases, the ions may escape the trap.