**Figure:** Failure of series approximations of characteristic values to converge. According to Mathieu theory, .
However, the published series approximations lack the precision
needed to force convergence of the even and
odd characteristic value curves.

**Figure:** Approximate zones of the two known Paul-trapping stability regions. Note that the exact boundary depicted for region B is inaccurate
to the extent that
the series approximations for the characteristic value curves (especially and ) depart from
their theoretical behavior in the proximity of region B. should converge to and to .

**Figure:** **x**-position vs. axial position for a Paul-trapped single ion in stability region B. * source file*: paul1B.trp.

Although stability region A was, as of 1989, the only parameter region in which Paul traps had been operated,[4] theoretically there exist other parameter regions in which confinement could be achieved. A literature search on a database of articles published between 1989 and April 1995 revealed a wealth of research associated with Paul traps, but only two articles concerning higher-order stability regions in the Paul trap.

In order to locate the **a**, **q** parameter regions of higher-order stability, one must graph the characteristic curves for larger values of **q**. However, as shown in Fig. and Fig.
, the series approximations even for the lowest-order characteristic values as a function of **q** fail for **q** as low as about two (be) or three (be). This inadequacy of published approximations for characteristic values
for **q** parameters
attainable in a Paul trap prevents one from accurately graphing (via the published series approximations) the boundaries of higher-order regions in which ion confinement is achievable. However, by truncating the series approximations for characteristic values and , an approximate graph of stability region B was generated (Fig. ).

Stability region B corresponds to larger values of **a** and **q** and faster (relative to the frequency of the applied potential) secular oscillations. As illustrated in Figs. , , , the
first-order approximation which describes the salient features of ion behavior in region A is no longer valid in this region since higher-order terms (corresponding to faster frequencies) are significant. Fourier transforms of axial and radial motions of a single ion in region B (Figs. , ) show that, although the motion is more complicated in that it is composed of a greater number of non-negligible frequency components, the constituent frequencies * are* discrete. Thus, axial and radial motions are periodic--- rather than chaotic--- for single ions in stability region B, in accordance with the fact that the governing equations are linear differential equations.[23]

**Figure:** Fourier transform of **x**-position of a Paul-trapped single ion
in region B. Peak frequencies (MHz): . * source file*: paul1B.trp.

**Figure:** Fourier transform of axial position of a single Paul-trapped ion in region B. Peak frequencies (MHz): . * source file*: paul1B.trp.

Fri May 12 10:36:01 EDT 1995