Coulomb Crystals Research Articles

Haptic rendering through a dynamic atomic unit approach: toward nonlinear, viscoelastic, anisotropic behavior

A cold plasma produced by laser excitation of trapped atoms at very low temperature creates an unusual ionic configuration of Coulomb crystal type before it starts to expand under pressure. A simple model is constructed to describe such a state at the time of plasma formation. The potential of the ions and a threshold shift in the electron energy are estimated. The shift is found to be ΔP≃CP/a, where a is the Wigner–Seitz radius and CP is a universal constant CP≃11±5, all in atomic units. The experimentally observed anomalies in the ion velocity and its dependence on the plasma density at very low temperature have been treated in terms of CP, as vI≃CP/Ma. The effective electron temperature can also be defined as Te≃ΔP/2.

Theory and simulation of strong correlations in quantum Coulomb systems

Strong correlations in quantum Coulomb systems (QCS) are attracting increasing interest in many fields ranging from dense plasmas and semiconductors to metal clusters and ultracold trapped ions. Examples are bound states in dense plasmas (atoms, molecules, clusters) and semiconductors (excitons, trions, biexcitons) or Coulomb crystals. We present first-principle simulation results of these systems including path integral Monte Carlo simulations of the equilibrium behaviour of dense hydrogen and electron– hole plasmas and molecular dynamics and quantum kinetic theory simulations of the nonequilibrium properties of QCS. Finally, we critically assess potential and limitations of the various methods in their application to Coulomb systems.

Pair distribution of ions in Coulomb lattice

The pair distribution function g(r) ≡ g(x, y, z) and the radial pair distribution function g(r) of ions in body-centred-cubic and face-centred-cubic Coulomb crystals are calculated within the harmonic-lattice (HL) approximation in a wide temperature range, from the high-temperature classical limit (T wp, wp being the ion plasma frequency) to the low-temperature quantum limit (T wp). In the classical limit, g(r) is also calculated by the Monte Carlo (MC) method. MC and HL results are demonstrated to be in good agreement. With decreasing T, the correlation peaks of g(r) and g(r) become narrower. At T wp they become temperature independent (determined by zero-point ion vibrations).

A semiempirical mass formula for spherical Coulomb crystals

In analogy to the Bethe–Weizsäcker formula for nuclei we establish a semiempiricalmass formula for the energies of Coulomb crystals including Madelung, surface,compression and curvature energy terms. The surface tension and theincompressibility are extracted as well. The coefficients are fitted at data obtainedfrom molecular dynamics calculations for spherically confined systemsof charged particles with particle numbers up to about 100 000 undertheir mutual Coulomb repulsion and under constant radial focusing. Thesystems are cooled down until the excess energy stays constant up to10−8.For particle numbers below about 10 000, they settle in icosahedral structureswhose faces are rounded off. At about 10 000 particles there is a transition to bodycentred cubic (bcc) crystals where particles are arranged in parallel layers. Theicosahedra have lower surface energy but slightly higher Madelung energy of−0.8949,in contrast to the bcc crystals which have the lower Madelung energy of−0.8959but essentially larger surface energy. The icosahedra are more compressible thanthe bcc crystals.

Rotation of coulomb crystals in a magnetized inductively coupled complex plasma

Under suitable conditions, micron-sized dust particles introduced into inductively coupled argon plasma form a stable microscopic crystal lattice, known as a Coulomb (or plasma) crystal. In the experiment described, an external axial magnetic field was applied to various configurations of Coulomb crystal, including small crystal lattices consisting of one to several particles, and large crystal lattices with many hundreds of particles. The crystals were observed to rotate collectively under the influence of the magnetic field. This paper describes the experimental procedures and the preliminary results of this investigation.

Non-stationary Coulomb crystals in linear Paul traps

Ion Coulomb crystals form when trapped ions are cooled below a certain criticaltemperature. In radiofrequency (RF) traps, such crystals are usually stationary(i.e. non-rotating) with shapes similar to those of cold, homogeneously chargedliquids when the number of ions is sufficient for three-dimensional structures toappear. ‘Dynamically stable’ structures can, however, also be created either bydirectly applying a torque to an initially stationary crystal or by choosing trapparameters just within the stability region where stationary crystals will melt dueto parametric resonances between the RF trap field and ion plasma modes.

Pair distribution of ions in Coulomb crystals

The pair distribution function g( r) of ions in body-centered-cubic and face-centered-cubic Coulomb crystals is calculated using the harmonic-lattice (HL) approximation in a wide temperature range, from the high-temperature classical regime ( T⪢ℏω p, ω p is the ion plasma frequency) to the low-temperature quantum regime ( T⪡ℏ ω p ). The radial pair distribution function g( r) is calculated by averaging g( r) over orientations of r . In the classical limit, g( r) is also obtained from extensive Monte Carlo (MC) simulations. MC and HL results are shown to be in good agreement. With decreasing temperature T, the correlation peaks of g( r) and g( r) become narrower and finally freeze at T⪡ℏ ω p being solely determined by zero-point ion vibrations.

Nano-beam production by RFQ with ion-trap injector

Nano-beam production by RFQ with ion-trap injector

Effect of the electron gas polarizability on the specific heat of phonons in Coulomb crystals.

The effect of the background polarizability on the thermodynamic properties of a Coulomb crystal of ions is studied. The response of electrons is treated using the Thomas-Fermi (TF) and random phase approximations (RPA). For the case of ions fixed at their lattice sites, the energy of bcc and fcc crystals is calculated to first order in the screening parameter (kappa(TF)a)(2) (kappa(TF) is the TF wave number and a is the ion sphere radius). It is shown that in the RPA there exist domains of parameters (mass density rho and charge number Z) where energy of fcc crystal is lower than that of bcc. The effect of ion vibrations is studied using harmonic lattice approximation. It is shown that phonon modes are nearly identical in the RPA and in the TF approximation. The latter allows one to apply the Ewald technique to the construction of the dynamical matrix, which speeds up all calculations considerably. The main thermodynamic quantities of phonons are calculated as functions of the quantum parameter T(p)/T (where T(p) is the ion plasma temperature) and the screening parameter. The electron polarizability leads to a moderate increase of the phonon thermodynamic quantities as compared to the case of one-component plasma with rigid background (by approximately 30% at kappa(TF)a=0.8). Zero-point motion of ions modifies the aforementioned domains where fcc has lower energy than bcc for static ions. The effect is profound at small Z but leaves the domains unaltered at larger Z. The thermal vibrations of ions at T greater, similar T(p) eliminate completely the domains where fcc is thermodynamically preferable at T=0. The related model of Yukawa-Wigner solid is briefly studied. It is shown that neither bcc nor fcc crystal structures are stable in this model.

Self-consistent three-dimensional model of dust particle transport and formation of Coulomb crystals in plasma processing reactors

Dust particle transport in low-temperature plasmas has received considerable attention due to the desire to minimize contamination of wafers during plasma processing of microelectronic devices and for their use to study nonideal plasmas. Dust particles in radio frequency discharges form Coulomb crystals and display collective behavior under select conditions. In this article, we discuss results from a self-consistent three-dimensional model for dust particle transport in plasma processing reactors. The consequences of varying the bias voltage of the capacitively coupled discharge, plasma density, particle diameter, and the number of particles on the propensity for Coulomb crystal formation in argon plasmas will be discussed. We found that a single one-layer lattice spontaneously breaks up into separate lattices as the substrate bias is increased due to a redistribution of plasma forces. At high substrate biases, a void occurs in the plasma crystal which tends to close upon addition of electronegative gases such as O2 and Cl2 to argon. The interparticle spacing in the lattice depends on the number of particles in the lattice due to gravitational compressive forces; and on the plasma density due to the change in shielding length.

Formation process of large ion Coulomb crystals in linear Paul traps

We report on the results on various aspects of the formation of ion Coulomb crystals in linear Paul traps, for plasmas containing from a few hundred to several thousand laser-cooled ${}^{24}{\mathrm{Mg}}^{+}$ ions. The presented studies include observations of changes in emitted fluorescence, changes in plasma aspect ratio, and the formation of shell structure, during the transition from the cloudlike to the crystalline state. The competition between rf heating and laser cooling is observed to result in a very dramatic transition from the cloudlike disordered state to a liquidlike state. After the liquid state is reached ordering into shell structure takes place as a continuous process at even higher plasma coupling parameters. This is in contrast to earlier observations of few-ion plasmas in hyperbolic rf traps.

Stability of Coulomb crystals in a linear Paul trap with storage-ring-like confinement.

We report experiments on the stability of ion Coulomb crystals in a linear Paul trap with storage-ring-like confinement. The transverse dynamics of charged particles in a trap of this type is analogous to that of a fast beam traveling through a channel with periodic, magnetic alternating gradient confinement. The experimentally observed stability conditions for stationary crystals comply remarkably well with current theory of crystalline plasmas and beams.

Dispersion relations of longitudinal and transverse waves in two-dimensional screened Coulomb crystals.

Dispersion relations of longitudinal and transverse waves in two-dimensional (2D) screened-Coulomb crystals were investigated. The waves were excited in 2D crystals made from complex plasmas, i.e., dusty plasmas, by applying radiation pressure of laser light. The dependencies of the dispersion relation on the shielding parameter, the damping rate, and the wave propagation direction were experimentally measured. The measured dispersion relations agree reasonably with a recently developed theory, and the comparison yields the shielding parameter and the charge on particles.

Particle interaction measurements in a Coulomb crystal using caged-particle motion.

A technique for characterizing the particle interaction potential of a Coulomb crystal is developed. The mean-square displacement (MSD) is measured, showing both caged- and superdiffusive-particle motions. By subtracting the center of mass of neighboring particles in computing MSD, only short-wavelength particle motions are retained. This yields the lattice Einstein frequency, which contains information about the interparticle forces and potentials. Video measurements of particle motions in a complex (dusty) plasma are used to demonstrate the technique.

Effect of a Longitudinal Radio-Frequency Field on Crystalline Beams

Effect of a Longitudinal Radio-Frequency Field on Crystalline Beams

Laser-generated waves and wakes in rotating ion crystals.

Locally excited plasma waves are generated in a Coulomb crystal by "pushing" with radiation pressure on a rotating cloud of laser-cooled 9Be+ ions. The waves form a stationary wake that is directly imaged through the dependence of the ion fluorescence on Doppler shifts, and theoretical calculations in a slab geometry are shown to accurately reproduce these images. The technique demonstrates a new method of exciting and studying waves in cold ion clouds.

Formation of Coulomb crystals in a capacitively coupled plasma

Dust particle transport in low-temperature plasmas has received considerable attention due to the desire to minimize contamination of wafers during plasma processing of microelectronic devices and as their use to study nonideal plasmas. Observations of dust particles in radio frequency discharges have shown that particles form Coulomb crystals and display collective behavior. Images are presented here of Coulomb crystals simulated in a capacitively coupled discharge. An abrupt splitting of a single hexagonal lattice into two lattices occurs with increasing substrate bias.

Threshold lowering effects on an expanding cold plasma

A cold plasma produced by laser excitation of trapped atoms at very low temperature creates an unusual ionic configuration of Coulomb crystal type before it starts to expand under pressure. A simple model is constructed to describe such a state at the time of plasma formation. The potential of the ions and a threshold shift in the electron energy are estimated. The shift is found to be Δ P ≃ C P / a, where a is the Wigner–Seitz radius and C P is a universal constant C P ≃11±5, all in atomic units. The experimentally observed anomalies in the ion velocity and its dependence on the plasma density at very low temperature have been treated in terms of C P , as v I≃ C P/Ma . The effective electron temperature can also be defined as T e ≃ Δ P /2.

Potential of a dust grain in a nitrogen plasma with a condensed disperse phase at room and cryogenic temperatures

The first numerical study is presented of the self-consistent potential of a dust grain in a nitrogen plasma with a condensed disperse phase at room and cryogenic temperatures and at high gas pressures for which the electron and ion transport in the plasma can be described in the hydrodynamic approximation. It is shown that the potential of the dust grain is described with good accuracy by the Debye potential, in which case, however, the screening radius turns out to be larger than the electron Debye radius. The difference between the radii is especially large in a plasma with high ionization rates (about 1016–1018 cm−3 s−1) at room temperature. It is found that, in a certain range of the parameters of a nitrogen dusty plasma, the parameter describing the interaction between the grains exceeds the critical value above which one would expect the formation of plasma-dust structures such as Coulomb crystals. For a plasma at cryogenic temperature (T=77 K), this range is significantly wider.

Thermodynamic functions of harmonic Coulomb crystals.

Phonon frequency moments and thermodynamic functions (electrostatic and vibrational parts of the free energy, internal energy, and heat capacity) are calculated for bcc and fcc Coulomb crystals in the harmonic approximation with a fractional accuracy < or equivalent to 10(-5). Temperature dependence of thermodynamic functions is fitted by analytical formulas with an accuracy of a few parts in 10(5). The static-lattice (Madelung) part of the free energy is calculated with an accuracy of approximately 10(-12). The Madelung constant and frequency moments of hcp crystals are also computed.