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.

Fri May 12 10:36:01 EDT 1995