The equilibrium spacing, 
, of a two-ion crystal is obtained by equating
the repulsive Coulomb force with the restoring force of the appropriate
secular oscillator (depending
on whether the crystal lies on the
z-axis or on the z=0 plane)
and keeping in mind that, at
equilibrium, each particle's
displacement from the potential minima is 
:[4]
For the two-ion crystal on the z=0 plane (
), substitution
of 
leads to:[4]
For the crystal on the z-axis (
), substitution of
leads to:
Note that these values for 
are approximations based on the model of Sec. 
for behavior in low q areas of stability region A.
Thus, since the estimations used for 
, 
are inaccurate by 
for 
, the estimated values of 
lose accuracy when q exceeds 
. Moreover, Blümel suggests that the micromotion--- which was ignored in the derivation of 
--- leads to slightly larger experimental equilibrium spacings. Simulations with TrapApp support Blümel's
conjecture that micromotion produces a very slight increase in
equilibrium spacing. In the
undamped
simulations of Tab. the equilibrium spacings
of axially-aligned and radially-aligned (on the z=0 plane)
two-ion crystals exceed theoretical predictions by
and 
, respectively.
The separation distance of the two-ion crystal will oscillate about its mean
equilibrium value of 
.[4] The frequency of these oscillations can be deduced from a Taylor series expansion of the sum of the force of the secular oscillator and that of the Coulomb repulsion:
where D represents the distance between ions. The force of
the secular oscillator on an ion at
a distance 
from the potential minima is:
Hence, 
. Ignoring higher-order terms and making use of the identity that, by definition of 
as equilibrium spacing, 
leads to:
Plugging in the approximation for 
(Eq. or , depending on the relative strength of secular potentials) and noting that the ``real-time'' secular frequency is given by 
yields:[4]
Thus, in two-ion configurations for which these approximations are valid, the frequency of oscillations of an ion about its equilibrium position, 
, is related to the secular frequency by a factor of 
. When damping was turned off on the axially-aligned two-ion crystal of axialCr.trap, the x-positions of the ions oscillated at 
MHz, a factor of about 
times the axial secular frequency of 
MHz. When damping was turned off on the two-ion crystal of intrCrys.trp, the mean separation of the ions oscillated at 
MHz, approximately a factor of
times the radial secular frequency of 
MHz.
The theoretical predictions for orientation and
equilibrium spacing of the
two-ion crystal can be easily tested in TrapApp through comparison of the theoretical separation to the mean value of
separation. Table shows the equilibrium spacings of two-ion crystals on the z=0 plane and z-axis, respectively, for
various values of the viscous damping coefficient. Note that for small values of damping, mean separation decreases with increased damping, but for larger
damping coefficients, mean separation increases as damping increases. This increase of mean separation with higher damping coefficients suggests that, as predicted in a recent article by Blümel,[6] exceeding a certain
critical damping parameter may induce melting of crystals. Although this
phenomenon was not thoroughly investigated in this thesis work,
TrapApp does demonstrate that very large damping parameters cause melting of crystals.
Table: Simulated two-ion equilibrium spacings for a crystal on the z=0 plane ( left) and a crystal on the z-axis ( right). Mathieu
parameters--- left: 
, 
, right: 
, 
. Theoretical equilibrium spacings--- left: 
m, right:
m. source files: intrCrys.trp, axialCr.trp.
