The Coulomb coupling parameter, 
(also known as the plasma parameter[39] or the correlation parameter[12]), of a collection of charged particles
is defined as the ratio of potential (due to Coulomb
interaction) energy to kinetic energy:
For 
greater than unity, the strong correlation of the system
raises the possibility of Wigner crystallization. The precise
values of the Coulomb coupling parameter at which phase transitions occur for
a one-component plasma of infinite extent has been the subject of extensive
quantitative study.[12,32] For 
, a transition
from the gaseous phase to liquid phase is predicted, with solidification into
a body-centered cubic lattice occurring at 
or
.[12,17]
Several factors complicate attempts to predict phase transitions in Paul
traps as a function of the Coulomb coupling parameter. To begin with, the
conditions under which phase transitions occur in a finite system are heavily
dependent on the boundary conditions of the system.[17] Thus, for a
small system with realistic boundary conditions, the precise value of
at phase transitions may be of little interest, since it is
system-dependent rather than ``universal.''
Further difficulty in applying 
to Paul-trapped ions arises
from the micromotion induced by electric-field oscillations. Since the
amplitude of the micromotion is proportional to the displacement from the
center of the trap (Sec. 
), any ion crystal will---
since no
more than one ion will occupy the exact center of the trap, where micromotion
vanishes--- be subject to micromotion. Because the micromotion contributes
significantly to a crystal's total kinetic energy, the value of the
traditionally defined coupling parameter (Eq. ) will be low
when applied to a collection of Paul-trapped particles.
Indeed, Blümel asserts[5] that the upper limit for the traditionally-defined Coulomb coupling parameter is 
for a
collection of Paul-trapped particles (see Tab. for simulated coupling parameters in stability region A).
The energy of micromotion vibrations is not random thermal energy;
the micromotion is periodic and predictable. Additionally, the
observation via computer simulation (Sec. )
that small crystals in region A are stable even in the absence of damping (which is required for their formation)
demonstrates that the crystalline phase of region A is a non-heating region
of phase space for small crystals. As put forth by Blümel, `` a Coulomb crystal does not
necessarily convert deformation energy into interval heating''.[5] This decoupling of kinetic energy due to micromotion from
kinetic energy of random thermal motion suggests that a more appropriate
form of the Coulomb coupling parameter would be:
where 
excludes the regular periodic micromotion and
includes only random thermal motions. Blümel suggests adjusting the
traditionally defined coupling parameter, 
, by generating a Fourier transform of kinetic energy, calculating
the fraction of kinetic energy due to micromotion, and adjusting 
accordingly. However, the micromotion so severely dominates the transform of
kinetic energy that this approach is not feasible for a simulation of collisionless crystallized ions (Fig. ).
Figure: Fourier transform (2048 data points) of kinetic energy of an undamped two-ion crystal on the z-axis in stability region A: the dominating contribution to kinetic energy is the micromotion. (The transform is not scaled to decibels.) source file: 2CrA.trp.
Another possibility is to use the kinetic energy of the center of mass, since the symmetric crystal deformation induced by micromotion does not affect the position of the center of mass. This method also failed when applied to an undamped crystal in stability region A: the kinetic energy of the center of mass was infinitesimally small and the program crashed due to numeric overflow.
In light of these results, it is clear that interpretation of the Coulomb coupling parameter, when applied to Paul-trapped ions, must be fundamentally
different from interpretation of 
as applied to the unbounded one-component plasma. The coupling parameter associated with Paul-trapping experiments is sensitive to the initial conditions and to configuration specifications and is relatively low even for strongly correlated systems, since the micromotion places an upper limit on 
.
Nevertheless, the coupling parameter is a useful measure of the degree of interaction in a system. For example, the default, undamped configuration of TrapApp, with consists of four Mg
ions in the Mathieu regime corresponds to 
. This value of 
is characteristic of configurations in the Mathieu regime. Damped and undamped crystals in region A, on the other
hand, correspond approximately to 
, in agreement with Levi[24] and Blümel[5] (Tab. ).
Table: Coulomb coupling parameters for damped and undamped crystal
simulations in stability region A.
