Computers in Physics

Computers in Physics Molecular Dynamics Inside Ion Traps Susan L. Fischer, Wolfgang Christian, and Denis Donnelly Citation: Computers in Physics 10, 123 (1996); doi: 10.1063/1.4822369 View online: http://dx.doi.org/10.1063/1.4822369 View Table of Contents: http://scitation.aip.org/content/aip/journal/cip/10/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rotational dynamics of a diatomic molecular ion in a Paul trap J. Chem. Phys. 143, 204308 (2015); 10.1063/1.4936425 Molecular dynamics simulation of inertial trapping-induced atomic scale mass transport inside single walled carbon nanotubes Appl. Phys. Lett. 102, 083108 (2013); 10.1063/1.4793533 Adsorption of water molecules inside a Au nanotube: A molecular dynamics study J. Chem. Phys. 128, 174705 (2008); 10.1063/1.2907844 Molecular dynamics simulations of collisional cooling and ordering of multiply charged ions in a Penning trap AIP Conf. Proc. 576, 118 (2001); 10.1063/1.1395264 Molecular beam studies of trapping dynamics J. Vac. Sci. Technol. A 8, 2699 (1990); 10.1116/1.576653

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Susan L. Fischer andL.

MOLECULAR DYNAMICS INSIDE ION TRAPS

z

Susan L. Fischer and Wolfgang Christian '-----x,y

Department Editor: Denis Donnelly donnelly@siena. edu

W

e have developed an interactive molecular-dynamics simulation capable of modeling electromagnetically trapped ions on a high-end personal computer running Microsoft Windows. The executable program Trap, 39 sample configuration files, and a manual are available on our World Wide Web server at the address http:// www.phy.davidson.edu.This program Figure 1. Schematic ofelectrode structure ofquadrupole ion trap. Typical dimensions are ro =2500 has pedagogic value for introducing stuJ.iI11, Zo = 1770 J.iI11. dents to ion-trap dynamics. In addition, Trap demonstrates phenomena that are relatively unexplored or subject to dispute among different and oscillating electric potentials, and Penning traps, which research groups, including the stability and behavior of manyemploy a static electric potential and a static magnetic field. ion crystals, dynamics in higher-order stability region B, and Both these traps have been used successfully for confining particles to small (on the order of 1 um) regions of space. the nature of heating in a Paul trap, The stability regions are described below, To achieve ion confinement in a Paul trap, an oscillating electric potential difference is applied to an electrode strucA complete discussion of all the features of Trap is not possible in this article; we have chosen instead to present a ture whose axially symmetric geometry is described by two detailed analysis ofthe stability of many-ion Wigner crystals hyperbolas (see Fig. I). The resultant potential distribution, in a Paul trap in Mathieu stability region A, as well as given by illustrations of the four dynamical regimes of ion configuraUo + Vo cos (D I) 2 2 2 tions, Interested readers then can download the complete V(z, r) = 2 2 (2 z + (r 0 - r )) , (1) work over the Internet.l-' r o+2z 0

Paul traps The Trap program models Paul traps, which utilize static

Susan L. Fischer, a 1995 graduate ofDavidson College, isagraduate student in mechanical and aerospace engineering in the Center for Energy and Environmental Studies, Ptinceton University, Princeton, NJ 08544. For her work on this ion-trapping simulation, she was named one of the student winners in the sixth annual CIP educational software contest. E-mail: [email protected] Wolfgang Christian isaprofessor ofphysics and the director ofIhe &ientific Computation Center atDavidson College. Davidson, NC 28036. He isone of CIP's department editorsfor Books. E-mail: [email protected]

alternates between a focusing potential well in the axial direction with a defocusing inverted radial potential and a focusing potential well in the radial direction with a defocusing axial potential. By focusing particles in one direction and then in the other, the dynamic Paul-trap potential creates a stable equilibrium, whereas the static quadrupole potential is capable of only unstable equilibrium.'

Single-ion behavior Consider the simplest case of a single ion. The equations of motion for a single ion of mass m and charge ne in a Paul trap are

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-

—x

2ne (Uo + Vo cos(.(21))

d2 m dt2

= 0 , (2)

r 2 + 2z2

2z

beo,

where U0 is the applied static potentia1,4 Vo is the amplitude of the applied oscillating potential, Q = 2rc.f is the frequency of the oscillating potential, and r0, zo are the electrode dimensions as depicted in Fig. 1. Each dimension (x, y, z) of the second-order differential equation for a single ion's motion in the Paul trap can be expressed as a one-dimensional Mathieu equation,

_

dz.2 (*+

..... ..:,

0

cos(2z))

—a z

1.5

2

2

2

m (r 0 -I- 2z 0 )

- q z -

2

gr

axial stability -

bei,

411;111

bOi ,.

bo2,

/

be 0 ,.

Figure 2. Approximate zones of the two known Paul-trapping stability regions. Note that the exact boundary depicted for region B is inaccurate to the extent that the series approximations for the characteristic value curves (especially beo, and bek) depart from their theoretical behavior in the proximity of region B: beor should converge to boir and bei, to bo22.

8neUo

2

qr -

.1

-0.5

(3)

through the transformations

radial stability

S

(0, 0.454)

-2

ar—

S

.

-1.5

d2

iz

0.5

4neVo 2 (r o + 2z 0)

2 2

(4) where ar, qr describe the dynamics in the radial (x—y) plane and az, qz describe axial motion. Stability regions for the one-dimensional Mathieu equation (3) correspond to regions of a, q parameter space for which there exist bounded solutions for Paul traps can achieve particle confinement when the Mathieu parameters a and q fall in the intersection of the radial and axial stability regions. Figure 2 was generated using series approximations of the curves bounding radial and axial stability.5,6 These intersections are known in Paul-trapping literature as stability regions A and B. Each dimension of the motion of a single Paul-trapped particle is described by a series of harmonies of the driving frequency Dperturbed by a "secular" frequency ,tir2/2, whose magnitude is a function of the Mathieu parameters:

Figure 3. Axial position and x-position as a function of time for a single particle trapped in stability region A with a = 0 and q, = 0.1790. Download configuration file paul1A.trp.

cases, the damping that arises from laser cooling or collisions with a cold background gas:7

+

C2,, cos ((2 n + 1.Q) "r+ q) •

(5)

n= —

In stability region A, Trap demonstrates that a single ion' s motion is approximated by a fast, small-amplitude micromotion at frequency Q superposed on a slower, 4 large-amplitude secular motion (see Fig. 3). In stability region B, the single particle's motion is more complicated (see Fig. 4), but is still described by a discrete series of harmonics in accordance with Eq. (5).

Multiple-ion dynamies In the context of a many-ion configuration, an ion's equation of motion must take into account the forces exerted by the field of the trap, the Coulomb interactions, and, in many

d 2,

AL,

in -- kx 1 y + z K) — r trap — FCoulomb— F damp = 0

dt

F coulomb —

(n e)2

N

ireo j= 1,j i

(6)

A

1Pi./

2 '

(7)

The Coulomb interaction introduces a nonlinear term that couples the motions of different particles as well as one particle's x, y, and z motions. Unlike the single-ion scenario of the previous section, the many-ion configuration cannot be solved analytically. Trap can, however, explore many-particle simulations through a highly visual, interactive numerical simulation. The numerical methods employed are described in "Numerical Methods and Program Structure," p. 127.

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4,501e051t1ohytino10

MReg5A.trp demonstrates that a collection of repulsive ions can be confined in a Paul trap without damping. Neither the kinetic energy of the ions nor their root-mean-square distance from the trap center increases. Either effect would eventually "boost" ions out of the potential welf.

Crystal formation In stark contrast to the Mathieu re-2.5gime, in which the ion trajectories are virtually uncorrelated, an ordered phase 05 15 It2 Tignticroltcondsr is possible in which ions occupy definite positions relative to each other. Figure 4. Motion of a single Paul-trapped particle in stability region B. Left: Axial position versus This phenomenon of Wigner crystaltime. Right: x-position versus axial position. Download configuration file paull B.trp. lization of particles ionized with the It is convenient to classify many-ion dynamics into four same polarity occurs when the potential energy due to repuldynamical regimes, discussed extensively by B1mel.8,9 The sive Coulomb interactions is large relative to kinetic energy. different characteristics of these regimes arise from differApplying viscous damping (representative of laser coolences in the amount of interaction between particles and the ing or contact with a cold thermal reservoir) to a Trap simudegree of damping in the system. Trap demonstrates configulation of Paul-trapped ions results in Wigner crystals. The rations in all four of these regimes in stability region A, as qualitative structure of these crystals and, in the two-ion case, discussed in the following sections. the orientations and equilibrium spacings of the crystallized ions demonstrate excellent agreement with experimental reMathieu regime sults and theoretical prediction (see Fig. 5). Beyond confirmIn the high-energy Mathieu regime, the interparticle ing experiment and theory, Trap allows the researcher to spacing is large, and the Coulomb interactions and damping transcend limitations imposed by the experimental apparatus are negligible. Thus, the equations of motion for N ions in the and inspect parameters that might be difficult or impossible Mathieu regime are well-approximated by N uncoupled, unto explore in the lab. For example, the x y projection of 32 damped equations of the form of Eq. (2), which, as discussed crystallized Mg + ions shown in Fig. 5 is similar to Wuerker's earlier, can be understood in terms of the Mathieu equation. resultl° and shows a definite ring structure, but an x z projecThe hallmarks of the Mathieu regime are demonstrated tion shows unexpected layering. On the other hand, crystals by a configuration that depicts five undamped Mg+ ions containing fewer ions (up to 8 or 10 in this case) assume planar Paul-trapped in stability region A, where each particle' s moconfigurations. tion is similar to that of Fig. 3. To demonstrate that Coulomb Trap also demonstrates that the distinction between plainteractions are essentially zero, the user may download file nar and nonplanar crystals is critical with regard to crystal MReg5A.trp, turn off particle interaction, and note that the stability as the Mathieu parameter q increases at constant a = system' s behavior does not visibly change. The simulation of 0. Simulated undamped planar crystals composed of 2 to 10 —

—

e

Figure 5. Radial and axial positions of 32 Paul-trapped particles. Left: Radial projection of positively charged aluminum particles experimentally observed by Wuerker et al. (Ref. 3). Center and right: Computer simulation of 32 ions, showing (center) instantaneous radial (x—y) and (right) axial (x—z) positions. Download configuration file pau132cr.trp. Each particle of Wuerker et al. 's picture appears as a line, rat her than a dot, because it is subject to micromotion, also demonstrated by running Trap.

me

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ions are stable until the Mathieu instability at q = 0.454, a = 0 is exceeded, at which point the magnitude of the crystals' axial oscillations increases exponentially. Even when the axial motion is unbounded, the radial geometry of the planar crystals is preserved. Although the unbounded axial motion corresponds to an increase in kinetic energy, the radial component of kinetic energy does not increase. The resultant crystal stability is due to the fact that the radial and axial motions are uncoupled for planar crystals, since the Coulomb forces have no axial components. Simulated nonplanar crystals consisting of 9, 10, 15, and 64 ions are not stable through the Mathieu instability but consistently melt before or at the Mathieu instability. The q-values at which melting occurs do not appear to be reproducible; rather, they depend upon the degree of axial asymmetry and the rate of change of q. Simulations on Trap support the conjecture that in region A, melting of crystals in conjunction with increasing q is a function of coupling of the axial and radial motion via the Coulomb force, the degree of axial asymmetry, and the rate at which q is increased. Trap does not provide evidence for the existence of a well-defined phase transition as a function of q. Such a transition was postulated but subsequently refuted in the 1980s.8> 11

and the "quasiperiodic" phase. The chaotic heating regime applies to configurations with interparticle spacing between that of the Mathieu regime and that of the crystalline phase (download files 15melt.trp and cloud9A.trp). In the heating regime, nonlinear Coulomb interactions in the presence of an oscillating potential permit absorption of kinetic energy from the oscillating electric

Heating phase While the uncorrelated behavior of the high-energy Mathieu regime is easily explained in terms of negligible Coulomb interaction and Wigner crystallization is intuitively understood as a consequence of strong damping, which forces the crystal into a "lowest-energy" configuration, there exist two dynamical regimes whose behavior is not readily anticipated. Trap provides a means of visualizing and investigating these two dynamical regimes—the chaotic heating regime

-Time (rnicrosetonds)

x10 2

Figure 6. Running averages of energy and T rms for a melted 15-ion crystal that evolves through the heating regime to the Mathieu regime. Note the large gain in kinetic energy and the increase in rmis for the heating phase. When the Mat hieu regime is reached, rr„,, and KE oscillate about stable average values. Download configuration file 15melt.trp.

Figure 7. Top: Axial (z) motion of an ion in the "quasiperiodic" phase Bottom: Fourier transform of axial position of an ion in the "quasiperiodic" phase. Download configuration file cloud5A.trp.

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potential.v 12 As depicted in Fig. 6, the increase in kinetic energy is accompanied by an increase in interparticle spacing (rrms)' Hence, an undamped heating configuration evolves to the nonheating, virtually uncorrelated Mathieu regime, in which interparticle spacing and kinetic energy oscillate about stable averages.

Quasiperiodic phase The fourth dynamical regime, known in the literature as the "quasiperiodic" regime, holds important implications with regard to stability ofcrystal structures. In stability region A, this nonheating phase is characteristic of configurations with interparticle spacings slightly larger than those that characterize crystalline structures. Trap simulates several nonheating configurations of crystals that have been perturbed into the quasiperiodic phase (download file c1oud5A.trp). As shown in Fig. 7, there is no heating mechanism in this dynamical regime, and the Fourier spectrum is dominated by frequencies corresponding to secular and micromotion oscillations. The existence of this nonheating phase for configurations corresponding to perturbed crystals in region A suggests stability of undamped crystals in region A, since the crystals remain in a nonheating regime of phase space despite slight perturbations induced by the micromotion. Indeed, Trap simulates crystals of 2 to 128 ions whose qualitative structures are preserved in the absence of damping.

Interactive study of traps In conclusion, the Trap program provides an interactive visualization of the conditions under which particle confinement can be achieved in an ion trap. Simulations agree with theory and experiment on the orientation and spacing of simple crystals. In agreement with Bliimel, but in contradiction to Hoffnagle et al., small planar crystals are observed to be stable as the Mathieu instability is approached. When large crystals are simulated in region A, nonplanar geometries couple the axial and radial equations of motion via the Coulomb interaction. These crystals melt well before the Mathieu instability is reached. Stability region B was not dis-

Numerical Methods and Program Structure A molec ular-dynamics simula tion invo lves four main components: • initialization-define relevant parameters, assign initial positions and velocities; • equi libration-al1owthe system to evolve to the point that it "forgets" its initial values, which may have been unrealistic (for example, random assignment of initial positions might put two ions unreaso nably close together); • evolut ion- use a finite-difference algorithm to model the time evolu tion ofthe syste m; monitor values of interest (ene rgy, ion's position , and so forth); • analysis- represent data graph ical1y; perform further numerical ana lysis ofthe results. While this basic structure is common to any molecular-dyna mics simulation, severa l factors render ion-trappi ng simulations different from-and, in gene ral, more computational1y intensive than- those discussed in Haile's and Allen and Tildesley's books. 1.2 First of all, the Coulomb interaction falls into the category oflong-range interactions; fonnal1y, this means that the force term is proportional to 1111, where i < d, and d is the dimen sion of the space occupie d by the syste m. Loosely speaking, long-range forces arc those for which the interaction of every particle with every other particle (as opposed to only interactions with nearest neighbors or with particles within a critical radius, rc ) must be taken into account. Notor ious for its computational intens ity, the long-range interaction problem is general1y O(N2) on a sequential machine. Furthermore, because Paul and Penning traps involve an external potential that is a function of space, "tricks" suggested by Al1en and Tildesley for dealing with the long-range potential by using periodic boundary conditions are not valid. Another difference between this simulation and general molecular-dynamics (M D) simulations arises from time scale. A small time sca le is characteristic of MD simulations, which typical1y model phenomena that fluctuate on the order of 100-1 000 ps. However, ion trapping involves terms that fluctuate on a range of significantly different time sca les. For example, the posit ion of a sing le Paultrapped ion in stability region A consists of a fast micromotion superposed on a slower secular motion. In order to see what is happening on the larger time scales without losing accuracy with respect to the short-time-sca le behavior, it is necessary to use a large number of time steps with small dt, where dt is the increment between steps. Two finite-differe nce algorithms can be selected in the Trap program : the velocity-Verlet method and the fifth-order Gear predictor-corrector algorith m. Using this fifth-order Gear predictor-corrector algorithm requires eight arrays of dimension 3 x numlons, whereas the velocity-Verlet algorithm requ ires only three such arrays. The increased memo ry requirement and complexity of the Gear algorithm might, at first glance, render it less attractive than the velocity- Verlet algorithm. However, the Gear algorithm has an enormous advantage over the velocity-Verlet algorithm: it requires only one calculation of the interactio n force per tirncstep, whereas the veloc ity-Verlet algorithm makes two calls to that function at each update. Since the calculation of interact ion force is O(N1 ) , the speed of the velocity-Verlet algorithm suffers for large N. Furthermore, the Gear algorithm can achieve a higher degree of energy conservation than the Verlet algorithm with a longer timcstep, According to Allen, "based on energy conservation, Gear 's algorithm is about one order higher in accuracy than Verlet' s.'? Penning-trap simulations on TrapApp illustrate the superior stab ility of the Gear algorithm: one can readily construct a given configuration and timeste p such that the velocity-Verlet algorithm will gain energy while the Gear predictor-corrector algorithm remains stable.

References I. H. M . Haile, Mo lec ular Dyn amics Simu lation (Wiley , New York, 1992). 2. M. P. Allen and D. J. T ildesley, Computer Simulation ofLiquids (Clarendon Press, Oxford, England, 1987).

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cussed in this paper, although the relevant configuration files can be downloaded from our Internet server. In region B we observe dynamics corresponding to the Mathieu regime, the chaotic heating regime, and the ordered phase. However, simulated crystals in this region are not stable in the absence of damping, and a nonheating quasiperiodic phase is not observed. Rather, undamped crystals heat into the Mathieu regime.

Acknowledgement The authors thank Dr. Laurence S. Cain and Dr. Robert Cline for many helpful suggestions and for testing the computer simulation. One of us (WC) would like to thank Dr. Herbert Walther and the DAAD for generous support ofstudy at the Max-Planck-Institut fur Quantenoptik.

E.

References

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[ohn Howard, former Chief Scientist, Air Force Geophysics Laboratory

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for physicists' desks....lt represen ts a lot of excellen t wo rk." - Cimth ia

Carter, Department of Ellergy (PrL'Vio/ls/y tilled: The Physicist's COlI/pallio1l )

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1. Our documentation was written in LaTeX, and we offer it on our Web site in a number of different formats including raw TeX, dvi, and PostScript. For those wishing to browse the documentation before downloading, we provide an online hypertext-markup-language (HTML) version using the latex2html Perl script written by Nikos Drakos (see Ref. 2). 2. Nikos Drakos's latex2html perl script is available at the URL http://cbl.leeds.ac.uk/nikos/tex2html/doc/latex2html/latex2html.html. 3. R. F. Wuerker, H. Shelton, and R.V. Langmuir, 1. Appl. Phys. 30, 342-349 (1989). 4. Positive values of Udindicate the end caps are at a higher potential than the ring electrode. 5. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1947). 6. National Bureau of Standards, Tables Relating to Mathieu Functions, prepared by the Computation Laboratory ofthe National Applied Mathematics Laboratories (Columbia University Press, New York, 1951). 7. Trap provides two damping algorithms: a simple viscous drag force that is proportional to velocity and a Monte Carlo collision that employs user-specified mean free path and temperature to simulate collisions with a background gas by periodically assigning randomly directed, energetically appropriate velocities to atoms. 8. R. Bliimel, C. Kappler, W. Quint, and H. Walther, Phys. Rev. A 40, 808-823 (1989). Bliimel demonstrates the origin ofion heating by numerical calculation ofthe work done by an oscillating field acting on Coulomb point charges. 9. R. Bliimel, E. Peik, W. Quint, and H. Walther, in Quantum Optics V, edited by J. D. Harvey and D. F. Walls (Springer-Verlag, Berlin and Heidelberg, 1989), pp. 152-175. 10. R. F. Wuerker, H. Shelton, and R. V. Langmuir, J. Appl. Phys. 30, 342 (1989). 11. J. Hoffnagle, R. G. DeVoe, L. Reyna, and R. G. Brewer, Phys. Rev. Lett. 61,255-258 (1988). 12. G. Werth, Contemporary Phys. 26, 241-256 (1985). 13. R. Blumel, Phys. Rev. A 51, 620-624 (1995). 14. G. Werth, Progress in Atomic Spectroscopy, Vol. C, Chapter 5, pp. 151-175.

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