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 :
For the two-ion crystal on the z=0 plane (), substitution of leads to:
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 . 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:
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, 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.