 ...
 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. ). For the conversion factors to physical units, click on the Reduced Units... option under the Parameters menu item.
 ...see
 In order to see the highfrequency 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.
, the term ``quasiperiodic'' is somewhat of a misnomer, since significant nonlinear effects lead to frequencies of motion which are not clearly
discrete or welldefined.
 ...structure
 The geometry of the crystalline structure depends on the
Mathieu parameters, discussed in Sec. .
 ...equilibrium.
 Stable equilibria obtained from a timevarying 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 onedimensional 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. ).
 ...function
 The requirement that 212#212 is 213#213periodic 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#254periodic solutions for 255#255 is twofold degenerate for each q, since there exist both even (256#256 series) and odd (257#257 series) solutions.
 ...integer.
 These integralorder 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. .
 ...``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.
``PhaseSpace 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. )
and 389#389 (Eq. ) were deleted.
 ...parameters,
 Experimental exploration of Paultrapped particles was limited to stability region A as of 1989, although the higherorder 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 steadystate 447#447 is achieved when the cloud condenses into a crystalline phase, but
the perturbed ion cloud will not reestablish 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. : 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.
. 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 Paultrapped
crystal to the
LambeDicke regime, which requires that vibrations are small relative
to wavelengths emitted in conjunction with energylevel 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 timestandard.[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
micromotiondominated 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 twoion crystal is neither aligned
with the zaxis nor with the xy plane.[27]
 ...by:
 This
approximation is valid to within 504#504 for 505#505.[39]
 ...undamped
 Undamped crystals have highamplitude 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 zaxis.
 ...separation
 Click Parameters 548#548 TwoIon 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 ``builtin'' 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 15ion 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
alreadyapproximate series expressions
for the characteristic value curves used in constructing
this graph were truncated to attain ``wellbehaved'' stability plots.
 ...1989;
 See
Sec. for references to the articles.
 ...low
 Fourier amplitudes are 655#655 dB for the 10ion simulation vs. 656#656 dB for the 1ion simulation for the first dip, 5 dB vs. 657#657 dB
for the second dip.
 ...smooth
 Note the jagged ripples in the 10ion transform.
 ...regime.
 Typical temperatures for the Mathieu regime of
stability region A are 730#73020,000 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 spindependent 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.
 ...axiallyconfining
 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 ``treebased force calculation" can reduce
computational time to 771#771 on a sequential machine;
this method involves an adjustable error
term which controls the tradeoff between
speed and accuracy.[2,14,18] Note that the
impact of this treebased 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 LennardJones 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. is set to 800#800. For Penning trap simulations, 801#801.
 ...Angströms).
 Haile[19] is
an excellent source for more
information on LennardJones simulations.
 ...ions.
 When position or velocity
is initialized, higherorder (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 datataking 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, timeaveraged
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 LennardJones interactions is set to the radial separation corresponding to the minima of the potential well.
 ...interaction,
 The LennardJones 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
LennardJones 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