September 2025 - Coulomb Crystals

Coulomb crystals are a class of spatial ordered structures of matter which ignores entirely the quantum nature of single particles. They consist of same sign charged ions, which then are only subject to classical electrodynamic interactions between confined charged particles. The change in electronic structure and properties of matters at electron-densities low enough that the wavefunctions of their DeBroglie wavelengths don't overlap. The electron can then be thought of taking up a particular position in space, rather than the quantum-mechanics interpretation which features overlapping probability densities. Coulomb crystallization of 2D electron-gases were demonstrated within GaAs/GaAlAs quantum wells. 3D Coulomb crystals of electrons have not been produced yet. They are called Wigner-crystals specifically. Coulomb crystals are composed of positively charged particles.

The condition for Coulomb crystallization can be formulated in terms of the plasma coupling parameter Γ = Q²/(4πϵ₀ak_BT) ~ E_Coul/E_kin. For infinitely large 3D identical charged particle systems (one-component plasmas, OCPs), the thermodynamical properties scale with Γ. At Γ ≥ 175, or using the local single ion oscillation frequency ω_plas at Δr_exc/a < 0.1

with Δ_exc is the root-mean-square single particle excursion around its equilibrium position due to its kinetic energy. For valence electrons in usual solids, at absolute 0-temperature, the quantum mechanical excursion of the individual electrons makes it impossible to fulfill Δr_exc/a < 0.1 and delocalizing electrons leading to specific energy band structures emerge as a consequence. More An approximation for quantum melting sits at

where M is the mass of charged particles. In contrast to short-range interacting particles leading to closed packed crystal lattices, the ground state of a infinite Coulomb crystal has bcc packing. Hcp and fcc structures correspond only to slight excitation. Effective harmonic confinement potentials and finite size one-component systems have constant particle density at low temperatures.

In Penning traps or Paul traps, the highest practical reachable particle density of single-charged ions is at around 10¹⁵m^-3, which would have a ~ 10 μm, which, if satisfying the inequality and definition of Γ requires that T ≤ T_CC ~10 mK. Standard laser Doppler cooling can satisfy this. The production of ion Coulomb crystals thus becomes possible.

The lowest energy state of infinite constant density is bcc, while that of finite size are more dependent on the particle number (and the trapping potential). For the isotropic confinement case with confining force in terms of a force constant κ, F_trap(r) = -κr the over-all structure is spherical due to the symmetry, and up to a few thousand particles, the total potential energy is minimized by configurations consisting of concentric shells with a specific magic number of ions as temperatures falls to zero. Each of the shells consists of closed 2D structures which is like a hexagonal lattice. The minimum energy state for an infinite 2D planar crystal. With adequate numbers of ions, the ground state configurations are essentially impossible to predict. Metastable configurations increases roughly exponentially with the number of particles forming the cluster. The MD simulations with projection images of crystals in lab environments are a reliable method to obtain insight to the particle numbers and densities. The general case of harmonic confinement with an isotropic potential in 2D with different force constant in the third one, the equipotential surfaces are described by concentric spheroids. When the ion number is large enough that the extension in all 3D is much larger than the spacing of the individual ions, it can be expected that the shape of the crystal should be very similar to the case of a confined cold charged fluid with a constant charge density. There is an analytic expression for the outer shell. Due to Coulomb repulsion, the shapes deviate from the equipotential surfaces of the trapping potential, though the outer contour adheres to it. Negative Poisson coefficients can be created through trapping conditions.

If two ion species with the same charge-to-mass ratio is crystallized. Both ion species are subject to the same effective potential, and could be expected to mix. This holds true for simulations. For small finite-systems, common shell-structures are formed again, though with common shells consisting of slightly displaced sub-shells of the species.

Coulomb Crystals have vibrational (phonon) modes, with 3N normal modes, corresponding distinct eigenmode frequencies. These are determined primarily through simulations in smaller crystals. For the larger spheroidal crystals information on mode structures and frequencies can be gained by approximating the real granular Coulomb crystal by a cold charged liquids with the same geometry and finding its eigenfrequencies and eigenmodes. As long as the characteristic length scale for the (l, m)-modes is much larger than the ion-ion spacing within the Coulomb crystals, these modes are also analogous. Said modes can be excited in Coulomb crystals experimentally. Subset modes can be found more easily.