Direct Observations of Structural Phase Transitions in Planar Crystallized Ion Plasmas

Abstract

Laser-cooled 9 Be ϩ ions confined in two-dimensionally extended lattice planes were directly observed, and the images were used to characterize the structural phases of the ions. Five different stable crystalline phases were observed, and the energetically favored structure could be sensitively tuned by changing the areal density of the confined ions. The experimental results are in good agreement with theoretical predictions for the planar (infinite in two dimensions) one-component plasma. Qualitatively similar structural phase transitions occur, or are predicted to occur, in other experimentally realizable planar systems.The one-component plasma (OCP) has been a model of condensed matter in statistical physics for over 30 years, and it is used to describe such diverse systems as dense astrophysical matter (1) and electrons on the surface of liquid helium (2). Laser-cooled trapped ions (3) are an excellent experimental realization of the OCP. The phase structures of spatially homogeneous (infinite) (4) and cylindrical (infinite in one dimension only) (5) OCPs have been explored previously. Here, images of individual ions that were confined in twodimensionally extended lattice planes are presented and used to characterize the structural phases; the observed structures agree well with the predictions of an analytic theory for the planar OCP.The OCP model consists of a single charged species embedded in a uniform, neutralizing background charge. In Paul (6) or Penning (6, 7) traps, which are used to confine charged particles, a (fictitious) neutralizing background is provided by the confining potentials. The thermodynamic properties of the infinite classical OCP are determined by its Coulomb coupling parameter, ⌫ ϵ, which is the ratio of the Coulomb potential energy of neighboring ions to the kinetic energy per ion; ⑀ 0 is the permittivity of the vacuum, e is the charge of an ion, k B is Boltzmann's constant, T is the temperature, and a WS is the Wigner-Seitz radius [defined by 4(a WS ) 3 /3 ϭ 1/n 0 , where n 0 is the ion density]. The onset of shortrange order for the infinite OCP is predicted (8) at ⌫ Ϸ 2, and a phase transition to a body-centered cubic (bcc) lattice is predicted (8, 9) at ⌫ Ϸ 170. With an OCP in a planar geometry (infinite in only two dimensions), boundary effects are predicted to cause the formation of a variety of additional structural phases, such as the hexagonal close-packed (hcp) and face-centered cubic (fcc) phases (10, 11). Qualitatively similar structural phase transitions occur, or are predicted to occur, in other planar systems with varied interparticle interactions, such as plasma dust crystals (12), colloidal suspensions (13), semiconductor electron bilayer systems (14), and hard spheres (15).The crystallization of small numbers (total number N Ͻ 50) of laser-cooled ions into Coulomb clusters (16) was first observed in Paul traps (17). With larger numbers of trapped ions, concentric shells (18) were directly observed in Penning (19) and Paul traps (5,20). Recently, Bragg diffractio...

Rapid crystallization of externally produced ions in a Penning trap

“…When the laser cooling reduces the kinetic energy of the confined ions sufficiently, they "freeze" in the effective potential given by the trap and their mutual Coulomb interactions, as has been demonstrated in numerous experiments; see, for example, [7,8,[20][21][22][23][24].…”

“…The geometric properties of ion plasmas and the formation of ion Coulomb crystals have been described in detail, for example, in [7,8,[20][21][22][23][24]46,[49][50][51][52]. For the present ion numbers of the order of 1 × 10 3 to 1 × 10 5 ("mesoscopic"), and aspect ratios (axial extension to radial extension) of typically α 1, the so-called planar-shell model is a good approximation to describe the geometry of the confined plasmas.…”

“…While in a real plasma the number of shells depends on the radial position in the crystal and decreases towards the edges, this model describes the plasma throughout as a series of S parallel planes at axial positions z i with area ion number density σ i . For the sake of simplicity, it does not explicitly account for correlations between shells, in which case one would also expect lateral offsets in the ion positions between neighboring shells [23]. Figure 5 depicts the geometry and the involved quantities.…”

Ion Traps as a testbed of classical and quantum statistical mechanics

“…An important step in these studies was the observation of structural phase transitions in trapped ion Coulomb crystals [17]. This work has triggered a steady flow of theoretical, numerical and experimental effort to characterize and to measure the various crystal phases and the nature of the phase transitions between them [18].…”

Ultrasensitive detection of force and displacement using trapped ions

“…Scattered fluorescence is detected using an imaging system connected to a CCD or photomultiplier tube. For these experiments we focus on ion crystals [24][25][26] with n≈100 ions in a two-dimensional planar array (perpendicular to the direction of the detection laser), having a diameter ∼300 µm. (Fig.…”