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first-order approximation which describes the salient features of ion behavior in region A is no longer valid in this region since higher-order terms (corresponding to faster frequencies) are significant.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.