s of simulation.
ions, showing ( center) instantaneous radial (XY) and (
right) axial (XZ) positions. source file: paul32cr.trp.
m, 
m. source: Winter, H. et al., Am. J. Phys., Vol. 59, No. 9, 1991.
, 
and 
, 
to converge. The published series approximations are inaccurate for ``large'' values of q.
and 
.
This region is defined by the intersection of the
lowest-order stability regions of the Mathieu equations
for axial and radial motions.
. Note that: ( a) both motions consist of a small-amplitude, high-frequency ``micromotion'' superposed on a larger-amplitude, slower secular motion; ( b) the frequency of axial secular oscillations is twice that of secular oscillations on the x-axis; ( c) the frequencies of axial and x-position micromotions are roughly equal.
source file: paul1A.trp.
= 0, 
, 
MHz. Fourier amplitudes are expressed in decibels. The theoretical frequencies were calculated from the application of Eq. 2.12 to the approximations (developed in Sec. 2.8) of motion in Region A. source file: paul1A.trp. Note: since the resolution of the FFT depends on the time step, dt, you may need to change the time step of paul1A.trp to generate a transform of this resolution.
.
However, the published series approximations lack the precision
needed to force convergence of the 
even and 
odd characteristic value curves.
and 
) depart from
their theoretical behavior in the proximity of region B. 
should converge to 
and 
to 
.
. source file: paul1B.trp.
. source file: paul1B.trp.
of the chaotic regime, non-negligible Coulomb interactions give rise to heating, and higher-density configurations (smaller 
) correspond to higher heating rates; ( 2b): in region C
of the chaotic regime, higher-density configurations correspond to slower heating rates; ( 3): the quasiperiodic regime is characterized by quasiperiodic trajectories and the absence of heating; ( 4): the crystalline phase.
for a melted 15-ion
crystal which evolves through the heating regime to the Mathieu regime.
Note the gain in kinetic energy and increase in 
for the
heating phase. When the Mathieu regime is reached, 
and <KE>
oscillate about stable average values. source file: 15melt.trp.
m. source file:
cloud5A.trp.
ion crystal in stability region A with
( left) and without ( right) damping. The left-hand configuration
represents a metastable state with slightly higher energy than the
configuration on the right. source file: 8CrA.trp.
s) of total, potential, and kinetic energies are displayed. The total energy of about 
eV is partitioned
equally among potential and kinetic components, indicating a temperature of approximately 1660 K. source file: paul10B.trp.
, 
, 
, 
, 
, 
, 
. Peak axial frequencies (MHz): 
, 
, 
, 
, 
, 
, 
, 
, 
, 
. source file: paul10B.trp.
ions evolving from a crystalline configuration to
the Mathieu regime upon removal of viscous damping.
source file: 2CrB.trp.
ion crystal in stability region B. left: XZ projection shows that the crystal is not planar--- it
measures about 
m in the radial direction, 
m
is the axial direction; center: XY projection shows radial order; right: x-position vs. axial position, showing periodic variation in axial placement. source file: 8CrB.trp.
m peak-to-peak) for a 32-ion ordered structure in stability region B. These figures have the same scale and represent the same ordered configuration at different phases of
the micromotion. source file: 32CrB.trp.
ion. Note that the cyclotron motion (represented by fluctuations in the radial position) has the largest frequency and smallest amplitude, while the magnetron motion (represented by the x-oscillations) has the smallest frequency and largest amplitude. The axial oscillations have a frequency between those of the magnetron and cyclotron motions. source file: introPen.trp.
ion. source file: introPen.trp.