Figure: Electrode structure of quadrupole ion trap. Typical dimensions are m, m. source: Winter, H. et al., Am. J. Phys., Vol. 59, No. 9, 1991.
Paul traps, which utilize static and oscillating electric potentials, and Penning traps, which employ a static electric potential and a static magnetic field, have both been used successfully for confining particles. The electrode surfaces of an ideal Paul or Penning trap are hyperboloids of revolution about the z-axis (Fig. ). The ring electrode is described by the equation:
and the end caps are defined by:
Applying an electric potential difference A between the ring electrode and the end caps yields a potential distribution of (taking the ring electrode as ground):
and an electric field of:
Inspection of the potential and electric field expressions reveals that: (a) the and components of the electric field are independent of each other and are linear functions of r and z, respectively; (b) the potential is that of a harmonic oscillator in one direction, and an inverted parabolic well--- resulting in unbounded exponential displacement--- in the other direction. Consider the electric field acting on a particle of mass m and charge ne:
harmonic oscillation for neA>0
exponential displacement for neA>0
Thus, the static quadrupole potential alone, like any other static electric potential, is clearly incapable of confining particles in three dimensions since its saddle potential shape can produce, at best, only unstable equilibrium. However, modifying the apparatus by applying an oscillating electric field (as in the case of the Paul trap) or an axially-aligned magnetic field (as in the case of the Penning trap) produces a situation in which particles can be confined.