This article presents a relationship between the size of an ion crystal in a Paul trap and the radio frequency potential applied on the central ring electrode. Using a simple and elegant derivation it has been shown that the distance of an ion in the crystal from the trap center is proportional to power of the applied radio frequency voltage. The validity of this power law has been demonstrated on ion crystals having up to 13 ions. A spring-mass model has been presented to predict structure of ion crystals in Paul traps operating in the Dehmelt regime. Structures are obtained by minimizing the total potential energy stored in an ion ensemble. The total potential energy is taken to be the sum of the electrostatic potential due to ion–ion interaction and the potential energy stored in the springs. Structures of crystals having up to 13 ions predicted by our model have been verified by comparing them with results obtained by direct numerical simulations.
BlockMDrakoudisALeuthnerH, et al. Crystalline ion structures in a Paul trap. J Phys B: At Mol Opt Phys2000; 33: L375.
7.
RaizenMGGilliganJMBergquistJC, et al. Ionic crystals in a linear Paul trap. Phys Rev A1992; 45: 6493.
8.
SzymanskiBDubessyRDubostB, et al. Large two dimensional Coulomb crystals in a radio frequency surface ion trap. Appl Phys Lett2012; 100: 171110.
9.
NaegerlHCBechterWEschnerJ, et al. Ion strings for quantum gates. Appl Phys B: Lasers Opt1998; 5: 603–608.
10.
BirklGKassnerSWaltherH. Multiple-shell structures of laser-cooled 24 ions in a quadrupole storage ring. Nature1992; 357: 310.
11.
DrewsenMBrodersenCHornekærL, et al. Large ion crystals in a linear Paul trap. Phys Rev Lett1998; 81: 2878–2881.
12.
AltWBlockMSeibertP, et al. Spatial separation of atomic states in a laser-cooled ion crystal. Phys Rev A1998; 58: R23–R25.
13.
RahmanA. Structure of a one-component plasma in an external field: a molecular-dynamics study of particle arrangement in a heavy-ion storage ring. Phys Rev Lett1986; 57: 1133.
14.
Lozovik YuEMandelshtamVA. Coulomb clusters in a trap. Phys Lett A1990; 5: 269.
15.
HasseRWSchifferJP. The structure of the cylindrically confined Coulomb lattice. Ann Phys1990; 2: 419.
16.
DanielHE Dubin. Minimum energy state of the one-dimensional Coulomb chain. Phys Rev E1997; 55: 4017.
17.
BulutaIMKitaokaMGeorgescuS, et al. Investigation of planar Coulomb crystals for quantum simulation and computation. Phys Rev A2008; 77: 062320.
18.
ItanoWMBergquistJCWinelandDJ. Coulomb clusters of ions in a Paul trap. GSI Report1989; 89-10: 241.
19.
CasdorffRBlattR. Ordered structures and statistical properties of ion clouds stored in a Paul trap. Appl Phys B: Lasers Opt1988; 3: 175.
20.
BriannaRH. Cold ion chemistry within Coulomb crystals. Mol Phys2019; 14: 117.
21.
HeazlewoodBRLewandowskiHJ. Chemistry using Coulomb crystals. In: Emerging trends in chemical applications of lasers. Chapter 17. 2021, pp.389–410.
22.
KrohnOALewandowskiHJ. Cold ion–molecule reactions in the extreme environment of a Coulomb crystal. J Phys Chem A2024; 128: 10.
23.
DiproseJARichardsonVReganP, et al. Spatial and temporal detection of ions ejected from Coulomb crystals. J Phys Chem A2024; 128: 19.
24.
ThompsonRC. Ion Coulomb crystals. Contemp Phys2015; 56.
25.
GhoshPK. Ion traps. Oxford: Oxford Science Publications, 1995.
26.
McLachlanNW. Theory and applications of Mathieu functions. 1964.
27.
MarchRE. An introduction to quadrupole ion trap mass spectrometry. J Mass Spectrom1997; 32: 351.
28.
PaulW. Electromagnetic traps for charged and neutral particles. Nobel Lect1989; 8.
29.
WangYFranzenJ. The non-linear resonance QUISTOR part 1. Potential distribution in hyperboloidal QUISTORs. Int J Mass Spectrom Ion Process1992; 112: 167–178.
30.
WangYFranzenJWanczekKP. The non-linear resonance ion trap. Part 2. A general theoretical analysis. Int J Mass Spectrom Ion Process1993; 124: 125–144.
