In a second investigation of crystal stability in region A as a function of the Mathieu parameters, q was raised linearly--- rather than instantaneously over periods of s. As shown in the results of Tab. , 2, 3, 4, 5, 8, 9 and 10-ion planar crystals were stable until the Mathieu instability was reached. In all of these planar-crystal cases, the radial geometry of the crystalline structure was preserved as the crystals exited the trap via axial instability (see file 4q455out.trp for illustration). Although the unbounded axial motion corresponds to an increase in kinetic energy, the radial component of kinetic energy did not increase. Had there occurred such radial heating, the radial crystalline structure would have melted. Hence, the axial and radial motions remain uncoupled, with Coulomb repulsion being balanced by the radial component of trap potential. On the axis, there is no proper phase transition or melting for unperturbed planar crystals as a function of the Mathieu parameter . Rather, adiabatically increasing at for a planar crystal can only lead to unbounded axial motion.
Nine and ten-ion non-planar crystals (files 9CrA.trp, 10CrA.trp) were such that, while they were nearly-planar, ions were slightly perturbed from the z=0 plane. Unlike fewer-ion planar crystals () and planar crystals of 9 and 10 ions, the radial order of non-planar 9- and 10- ion crystals was lost before or when the ions escaped via the Mathieu instability. While planar crystals in region A are stable through (MI=Mathieu instability), axial oscillations introduce radial perturbations which melt non-planar crystals. These radial perturbations are enabled by the coupling of axial and radial motions via Coulomb interaction. For a non-planar crystal, Coulomb interaction couples the axial and radial equations of motion such that a gain in axial kinetic energy yields a gain in radial kinetic energy. Thus, non-planar crystals ``melt'' when they are subjected to axial heating. This conjecture is supported by investigation of larger ordered structures (below).
Table: Behavior of planar crystals in response to linear increase in magnitude of applied oscillating voltage, . The Mathieu instability for occurs at and corresponds to unbounded axial oscillation.
When larger crystals (N=15, 64) were subjected to linear changes in , the crystals did not remain stable, but were thrust into the heating region well before the Mathieu instability. These larger crystals differ from the few-ion crystals of Tab. in that they were not planar crystals. They consist of 3 axial ``layers'' which are blurred but discernible. One layer is on the z=0 plane, one is below the z=0 plane, and one is above the z=0 plane (files 15CrA.trp, 64CrA.trp).
The radial projection of the 15-ion crystal appears as 5 central ions surrounded by 10 ions (file 15CrA.trp). Before the Mathieu parameter was increased, each ion had a definite (though subject to micromotion) radial and axial position relative to its neighbors. After was increased to , there was some migration among axial layers. However, no reproducible ``phase transition'' as a function of was found; axial ``melting'' occurred over a range of q-values for the configuration of 15CrA.trp. Furthermore, when the axially-melted 15-ion configuration was damped and allowed to re-crystallize, the new axial distribution was more symmetrical. Consequently, the of axial melting was increased. Thus, it appears that the axial ``melting'' of the 15-ion crystal is a function of the perturbations caused by inherent asymmetries of the multi-layer structure, rather than a function of the Mathieu parameter .
At the values for which axial melting was observed, the 15-ion configuration's radial positions remained stable and no heating was observed until was raised to higher values. This observation of stability in energy for a configuration perturbed from its crystalline state supports Blümel's assertion that a non-heating quasiperiodic phase surrounds the crystalline phase in region A.
Despite the existence of a quasiperiodic regime, multiple investigations demonstrated that the radial order ``melted'' in conjunction with a dramatic increase in kinetic energy well before the Mathieu instability was reached. The at which this phase transition occurred for the 15-ion configuration was not reproducible; it occurred within a range of approximately , apparently depending on the degree of asymmetry in the crystal and the rate of the change in Mathieu parameter. Upon melting, the ions continued to gain kinetic energy until they reached the Mathieu regime (see file 15melt.trp or Fig. to watch a melted crystal evolve through the heating phase to the Mathieu regime).
When a 64-ion crystal was subjected to a linear increase in q, the size and shape of the crystal were very much perturbed. The value of before
ramping was m, but decreased to m for and m for (file 64ramp.trp). Significant heating was
not observed until was raised to . The rates of heating observed
for the system reflected the two sides ( and ) of Blümel's heating curve. The heating rate increased, peaked, then decreased before kinetic energy stabilized in the Mathieu regime.