To trap a single ion in a Paul trap, both the axial (z) and radial (x, y)
motions must be bounded. Thus, the stability regions for the Paul trap are defined by the intersection of a, q, parameters for which both the radial
and axial motions are bounded. The obvious requirement that a and q fall in the stability regions of radial and axial motions is a severe constraint which profoundly reduces the size of the parameter space in which Paul traps can achieve particle confinement. To get a feel for the degree of limitation imposed by stability in both the radial and axial dimensions, refer to Fig. 
. Note that for 
and 
, where 
and 
are the integers associated with the even and odd characteristic values, respectively, the characteristic value curves do not cross the a=0 axis. Since 
, the regions in which radial and axial stability regions both associated with higher-order characteristic values overlap will be zero. Consequently, the only possibilities for stability in both the axial and radial directions occur when one or both of the stability regions is of low (first or second) order.
Keeping in mind from Eq. that 
and 
, one can use the series approximations of the characteristic values to graph the lowest-order region for which radial and axial motions are bounded (Fig. ). This parameter space is known in Paul-trapping literature as stability region A. The boundaries of this figure represent 
and 
where 
and 
are the 
even characteristic values for r and z, respectively, and 
and 
are the 
odd characteristic values for r and z. The series approximations for the characteristic values used in graphing Fig. are:[28,39]
Figure: Stability region A as a function of 
and 
.
This region is defined by the intersection of the
lowest-order stability regions of the Mathieu equations
for axial and radial motions.
