Before investigating crystal stability with and without viscous damping,
it is essential to recall Blümel's
two previously stated assertions for crystals in region A: (1)
micromotion is decoupled from random thermal motion; (2) there exists a
non-heating quasiperiodic phase for a range of 
immediately above
the largest 
of the crystalline phase. The first statement implies
that there is no inherent heating mechanism
for a crystal in stability
region A. Without a heating mechanism for drawing
energy from the oscillating field, the crystal cannot absorb the energy required to place its 
in
the domain of chaotic
heating. The second statement is significant because if there exists such
a quasiperiodic, non-heating regime, then deformations of a crystal
(due to micromotion) with large 
(i.e., near the upper-bound 
for
the crystalline phase) will thrust the crystal into a non-heating
zone rather than into the heating chaotic region. Preservation of the crystal
requires that it remain in non-heating regions of phase space, since addition
of energy via excursion into the chaotic, heating regime would eventually
melt the crystal.
Thus, even in the absence of damping, crystals in region
A should be stable.
In order to test the stability of undamped crystals in region A, crystals
consisting of 2, 4, 5, 8, 15, 32, 64, 65, and 128 Mg
ions were formed with
the aid of viscous damping, where the drag coefficient was 
. Initial conditions for all configurations consisted of velocities distributed according to Maxwell-Boltzmann statistics at 15 K and a face-centered cubic
lattice with a lattice distance of 
m. When
the crystalline phase was fully
established, damping was turned off (Tab. 
).
In the 2, 4, and 5 ion simulations, the crystals were stable--- the relative positions of ions and the crystal geometries were preserved.
In the eight-ion case, the geometry of the crystal changed when the damping was turned off (Fig. ). Before
the damping was turning off, the crystal lay on the z=0 plane with no ion occupying the center of the trap; after the damping was turned off, the crystal remained on the z=0 plane, but there was one ion in the center with the remaining 7 ions surrounding it. When damping was resumed, the configuration with one central ion surrounded by 7 ions remained stable. The existence of
two stable damped configurations suggests that there exists a local minima of
potential for the damped crystal corresponding to the first case (no crystal occupying the center); when the damping is turned off, the micromotion of the
ions introduces large enough perturbations from the local minima for them
to ``escape''
the local minima and re-equilibrate in a lower potential minima corresponding
to the body-centered configuration. Indeed, simulation with TrapApp shows that the first configuration ( 8CrA.trp) has more potential energy associated with Coulomb interactions (
meV) than does the second configuration ( 8CrA2.trp, 
meV). However, the Coulomb coupling
constants of both configurations are the same: 
. This equality
of coupling is due to a decrease in kinetic energy (proportional to the decrease in interaction potential) for the body-centered configuration of 8CrA2.trp.
The ordered structures formed with 15, 32, 62, 65, and 128 were not ``perfect crystals'' in that they were not absolutely symmetric. Nevertheless, there was clearly an ordered pattern in these many-ion simulations. In the radial plane, these ordered structures appeared as concentric rings; a projection in the xz or yz plane reveals a ``stack'' of layers of crystals (as opposed to the planar (z=0) structure of the previously investigated few-ion crystals). Despite
the asymmetries and large ion numbers of these crystals, the radial geometry of undamped many-ion crystals simulated in stability region A remained qualitatively stable with and without damping: radial order was preserved
with no heating.
Table: Test of stability of undamped crystals in stability region A. source files: 2CrA.trp, 4CrA.trp, 5CrA.trp, 8CrA.trp, 15CrA.trp, 32CrA.trp, 64CrA.trp, 65CrA.trp, 128CrA.trp.
Figure: Geometry of an 8-Mg
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.