...
Advisor: Dr. Wolfgang Christian, Professor of Physics, Davidson College

...directory.
Install.EXE automatically places BWCC.DLL in your WINDOWS directory.

...atoms
The atoms interact as point charges via the Coulomb potential. For different potentials, select Parameters150#150 Interaction Type.

...display
With the exception of time (displayed in 155#155s) all quantities are displayed in reduced units (see Sec. gif). For the conversion factors to physical units, click on the Reduced Units... option under the Parameters menu item.

...see
In order to see the high-frequency peaks, the transform must be scaled to decibels (this is the default option).

...s.
In order to preserve the phase of the oscillating potential, the simulation starts at the same time that the configuration was saved, rather than at t=0.

...regime;
As remarked upon in Sec. gif, the term ``quasiperiodic'' is somewhat of a misnomer, since significant nonlinear effects lead to frequencies of motion which are not clearly discrete or well-defined.

...structure
The geometry of the crystalline structure depends on the Mathieu parameters, discussed in Sec. gif.

...equilibrium.
Stable equilibria obtained from a time-varying potential are also demonstrated by a vertically oscillating inverted pendulum and by a ball on a rotating saddle.[39,34]

...potential
Positive values of 189#189 indicate that the end caps are at a higher potential than the ring electrode.

....
Because only the amplitude of the oscillating potential is of interest with regard to the one-dimensional Mathieu equation--- the phase, and thus the sign, is arbitrary--- some sources[39] write 203#203. However, the convention 204#204 will be used in this text for consistency with regard to defining stability regions as a function of the Mathieu parameters (Sec. gif).

...function
The requirement that 212#212 is 213#213-periodic with respect to 214#214 allows for harmonics with frequency 215#215, 216#216. Recall that 217#217

...set
The set of characteristic values of a yielding 254#254-periodic solutions for 255#255 is two-fold degenerate for each q, since there exist both even (256#256 series) and odd (257#257 series) solutions.

...integer.
These integral-order Mathieu functions correspond to the previously mentioned case of 261#261.

...literature.
For example, some sources[39] cite the second term in the series for characteristic value curve 262#262 as 263#263, whereas this term more commonly appears[28,29,25] as 264#264. Also, some sources, including the National Bureau of Standards, express the Mathieu equation in a different form (with different parameters) than the canonical form of Eq. gif.

...``micromotion.''
Blümel points out that the term ``micromotion'' is sometimes inappropriate since when parameters are near the transition to Mathieu instability, the amplitude of the ``micromotion'' is approximately 322#322 of the amplitude of the secular oscillations.[4]

...small
Recall that this evalulation holds only when q is small. Inspection of stability region A illustrates that ``small q'' implies ``small a''.

...frequency
This approximation fails by 341#341 for 342#342 [39]

...micromotion,
Recall that 356#356.

....
Many chaotic systems exhibit resonance at subharmonics and superharmonics of some driving frequency as they approach chaotic behavior as a function of some control parameter.

...trap.
The curious reader is referred to: Jiebeng, W. and Z. Xiwen. ``Phase-Space Analysis of the Ion Cloud in the Second Stability Region of the Paul Trap.'' International Journal of Mass Spectroscopy, Mar. 11, 1993, Vol. 124, No. 2, p. 89; and Zhu, X. and D. Qi. ``Characteristics of Trapped Ions in the Second Stability Region of a Paul Trap.'' Journal of Modern Optics, Feb. 2, 1992, Vol. 39, No. 2, p. 291.

...approximations
The highest order terms in the published series approximations for the 388#388 (Eq. gif) and 389#389 (Eq. gif) were deleted.

...parameters,
Experimental exploration of Paul-trapped particles was limited to stability region A as of 1989, although the higher-order stability region B was postulated.[4]

...negligible.
To demonstrate that Coulomb interactions are essentially zero, turn off particle interaction and note that the system's behavior does not visibly change. To disable particle interaction, select Parameters from the menu bar, then choose Interaction 413#413 no interaction.

...chaos
The aperiodicity of deterministic chaos arises from nonlinearities which can be expressed in the equations of the system, rather than from random perturbations or ``hidden'' variables.

...possible
Experimentally, imperfections in the electrode structure and the resultant departures from the ideal quadrupole potential introduce nonlinearities in the electric field which also contribute to heating of the ions.[24]

...,
If the rate of energy removal is greater than the rate of energy gain at 443#443, the cloud condenses to the crystalline phase rather than reestablishing equilibrium in the chaotic regime.

....
A steady-state 447#447 is achieved when the cloud condenses into a crystalline phase, but the perturbed ion cloud will not re-establish equilibrium as a chaotic cloud in C+.

...instability.
As pointed out by Blümel, the source of the discrepancy might lie in the different methods by which the researchers introduced ``noise'' to the system. Hoffnagle's group modeled ``noise'' as spontaneous changes in position, while Blümel's group used the more physically plausible model of spontaneous changes in momentum.[4]

...phase,
The strong degree of coupling in the quasiperiodic regime is reflected by the Coulomb coupling parameter, 477#477, introduced in Sec. gif: 478#478 for cloud5A.trp.

...Wigner
E.P. Wigner. Trans. Faraday Society, 34:678(1938).

...energy:
In TrapApp, the ``running'' average coupling parameter refers to the quantity designated by Eq. gif. The ``instantaneous'' coupling parameter is 482#482.

...particles.
As noted by D. Segal and R. Thompson, vibrations due to micromotion preclude the possibility of cooling a Paul-trapped crystal to the Lambe-Dicke regime, which requires that vibrations are small relative to wavelengths emitted in conjunction with energy-level transitions. Only a single ion ``confined to the very centre of the quadrupole trap can be cooled sufficiently'' to serve as the basis for a time-standard.[35]

...ions
Viscous damping (which is employed for crystal formation) in the absence of random thermal motions arising from ``collisions'' with a background gas results in micromotion-dominated kinetic energy since transient oscillations are damped out in such a simulation.

...regime
Recall that in the Mathieu regime, the dynamics of N particles are essentially described by N uncoupled equations of motion. Hence, interaction is negligible in the Mathieu regime.

...wells.
A recent article by Moore and Blümel discusses a more accurate model which includes terms of up to second order in micromotion amplitude, rather than neglecting the micromotion altogether and considering only the secular motion. This article predicts the existence of a ``transition region'' in which the two-ion crystal is neither aligned with the z-axis nor with the x-y plane.[27]

...by:
This approximation is valid to within 504#504 for 505#505.[39]

...undamped
Undamped crystals have high-amplitude micromotion relative to their damped counterparts.

...frequency
For a crystal on the z=0 plane, the radial secular frequency is the relevant frequency, while the axial frequency is relevant for a crystal on the z-axis.

...separation
Click Parameters 548#548 Two-Ion Equilibrium Sep. to display theoretical equilibrium spacing, 549#549.

...separation.
The mean separation distance data option is under the State Data menu.

...mechanism
While there is no ``built-in'' heating mechanism for crystals in stability region A, collisions with a background gas can introduce deformations which perturb 566#566 into the chaotic heating region.

...symmetric.
If the damping phase of the simulation had run longer (and possibly at lower viscosity to avoid getting ``stuck'' in a metastable state), it is possible that ``perfect'' crystals would have formed. However, the running time of the simulation is 576#576, where N is the number of ions, so simulations are exceedingly slow for very large N.

...instantaneously
Strictly speaking, the discretization of time in quanta of the time step dt requires that 618#618 is actually raised by a series of small instantaneous changes. However, dt is sufficiently small that we can consider changes in q as a continuous linear function, rather than as a series of step functions.

....
As with the 15-ion case, this value was sensitive to the asymmetries of the system and should not be interpreted as a reproducible transition point.

...approximately
Recall that the already-approximate series expressions for the characteristic value curves used in constructing this graph were truncated to attain ``well-behaved'' stability plots.

...1989;
See Sec. gif for references to the articles.

...low
Fourier amplitudes are 655#655 dB for the 10-ion simulation vs. 656#656 dB for the 1-ion simulation for the first dip, 5 dB vs. 657#657 dB for the second dip.

...smooth
Note the jagged ripples in the 10-ion transform.

...regime.
Typical temperatures for the Mathieu regime of stability region A are 730#730-20,000 K.[36]

...#quadPotSec#1542>.
A magnetic bottle, created, for example, by a nickel ring centered about r=0 in the z=0 plane, is sometimes added to the apparatus for investigation of spin-dependent quantities whose observation requires an inhomogeneous magnetic field.

...oscillation.
Since the static field must confine the ions' axial positions, the Penning trap--- unlike the Paul trap--- cannot simultaneously confine both negatively charged particles and positively charged particles.

...axially-confining
For axial confinement, 733#733 is chosen such that 734#734

...limit.
A very large magnetic field justifies the approximation of ``averaging out'' the cyclotron motion, since the cyclotron motion is decoupled from the plasma interactions when 758#758, with 759#759 being the plasma frequency.[12]

...machine.
Implementation of a ``tree-based force calculation" can reduce computational time to 771#771 on a sequential machine; this method involves an adjustable error term which controls the trade-off between speed and accuracy.[2,14,18] Note that the impact of this tree-based algorithm on a 772#772molecular dynamics simulations is exactly analogous to the revolutionary impact of the Fast Fourier Transform (FFT) on Fourier analysis: a 773#773 algorithm can reduce to 30 s of CPU time what would require 2 weeks of CPU time via a 774#774 algorithm![33]

...magnitudes.
Allen and Tildesley[1] and Haile[19] provide excellent explanations of reduced units.

...units:
For Lennard-Jones simulations, the length unit, 779#779, is defined as the equilibrium spacing between atoms and the energy unit equals the depth of the potential well.

...step
For Paul trap simulations, c of Eq. gif is set to 800#800. For Penning trap simulations, 801#801.

...Angströms).
Haile[19] is an excellent source for more information on Lennard-Jones simulations.

...ions.
When position or velocity is initialized, higher-order (third, fourth, and fifth) derivatives associated with the Gear algorithm are set to zero so that ``phantom forces'' from the previous simulation do not perturb the new configuration.

...gas.
Because damping is performed one time for each data-taking cycle, changing the number of finite difference steps per data cycle may significantly alter the effects of damping.

...ions
Although temperature is generally interpreted as a macroscopic, time-averaged parameter, we can derive the instantaneous ``temperature equivalent'' of our microscopic collection of ions by expressing temperature of the ions as 826#826 where KE denotes kinetic energy, N is the number of ions, and k is Boltzmann's constant.

...scale,
This microscopic approach differs markedly from the macroscopic approach of shamelessly rescaling ion velocities for the desired temperature.

...interaction.
The reduced length unit for Lennard-Jones interactions is set to the radial separation corresponding to the minima of the potential well.

...interaction,
The Lennard-Jones interaction is defined by 857#857, where 858#858 is the depth of the potential well and 859#859 is the separation distance for which interaction potential is minimized. For more information about the Lennard-Jones interaction, the reader is referred to Haile.[19]

...``continuously''
One data point is taken per steps per data time steps, where the number of time steps between data points is set in the Parameters 860#860 Numerical.. dialog box.

...command.
In order to choose which data to collect, use the mouse sequence State/Particle Data 869#869 Select Data to Collect.

Wolfgang Christian
Fri May 12 10:36:01 EDT 1995