Long-range Translational Order Research Articles

Path-integral Monte Carlo study of the hard-disk solid.

We present a study of the properties of the quantum two-dimensional hard-disk solid at finite temperatures using path-integral Monte Carlo simulations. We performed extensive simulations on systems with 64, 144, 256, 576, and 1024 disks. We used the image approximation for the high-temperature density matrix and ignored exchange effects. Despite the discontinuous nature of the hard-disk interaction, a long-wavelength harmonic theory provided a good model for the solid. This theory was used as a guide to study the long-range translational and orientational order of the solid at low but finite temperatures where the zero-point motion and finite-temperature excitations are both important. The properties we have computed include the particle displacements, structure factor, angular order, and density of topological defects. We compare these properties with those of the same system in the classical regime and at absolute zero, thus gaining some understanding for the crossover from classical to quantum behavior.

Orientational order in lipid monolayers: A one-dimensional model

Lipid monolayers at low temperatures and large spreading pressure exhibit long-range translational order and the hydrophobic alkane chains of the lipid molecules are then stretched out in an all-trans configuration. The orientational order of the remaining rotational degree of freedom of these alkane chains is studied by means of Monte Carlo and molecular dynamics simulations as well as analytical calculations, but considering a one-dimensional rather than a two-dimensional model. Depending on the interactions among the chains, the ground state of the model is either a ‘‘uniformly tilted’’ state (all chains are tilted by an average angle +θ̄ or −θ̄ relative to the direction perpendicular to the lattice direction) or a ‘‘no-tilt’’ state (θ̄=0). At nonzero temperature, the uniform tilt state disorders by soliton-like kink–antikink excitations, in addition to phonon-like excitations, which exist also in the no-tilt state. In the continuum limit the treatment can be related to the Krumhansl–Schrieffer theory of the one-dimensional φ4 model. However, the quantitative predictions of the latter are in pronounced disagreement with the numerical results, in the temperature regime investigated. By our Monte Carlo techniques, we are able to study the orientational correlation function over the full temperature range, except extremely low temperatures. It is shown that for this model and temperature range the molecular dynamics technique requires much more computer time in order to obtain similarly reliable results than our discretized Monte Carlo approach.

Close packings of uniform disks with perfect orientational order.

Computer-generated close packings of uniform disks around rhomboidal seeds are studied. We explain why these packings have perfect long-range orientational order without long-range translational order. A variety of structures are formed that are either deterministic or nondeterministic. Simple geometrical considerations enable us to predict the nature of these structures from the characteristics of the seeds. Certain deterministic structures grow by propagating a branching network of defect arms that is fractal. The fractal dimension D is a universal function of the acute angle of the rhomboidal seed. All of our structures are free of topological defects; i.e., they can be continuously deformed into perfect triangular lattices without long-range reconstruction. We discuss the implications of our work for the general theory of packings, for fractal, branching structures, and for close-packed models of glasses.

Quasicrystals. I. Definition and structure.

In a recent paper, we introduced the concept of quasicrystals [Phys. Rev. Lett. 53, 2477 (1984)], a new class of ordered atomic structures. Quasicrystals have long-range quasiperiodic translational order and long-range orientational order. In the present paper and the following one, we discuss the details of our analysis of the mathematical and structural properties of quasicrystals. We begin with a general overview of our analysis. We then discuss our computation of the diffraction pattern of a quasilattice, using as an example the case of icosahedral orientational symmetry. We demonstrate that two quasilattices with the same orientational symmetry and quasiperiodicity which are not locally isomorphic will have diffraction patterns with different peak intensities. Finally, we describe some examples of computer modeling of atomic quasicrystals.

Solid versus fluid, and the interplay between fluctuations, correlations, and exchange in the fractional quantized Hall effect.

In this paper, we examine a series of solid wave functions for electrons moving in two dimensions under a magnetic field, with all of them staying in the lowest Landau orbital. We discover that whereas the direct energy of the system is almost optimized by the correlated motion of electrons as in the magnetophonons, the effect of exchange is to soften the transverse branch, allowing larger fluctuations of the electrons from their lattice sites. These fluctuations are large enough to destroy the true translational long-range order of the solid and change it into mere algebraic long-range order. Thus our phonons possess a gap without violating Goldstone's theorem. This gap is of the same order of magnitude as that exhibited by Laughlin's fluid wave function, but the ground-state energy for the algebraic solid turns out to be slightly lower. Finally, we suggest a plausible connection between the odd-denominator rule and fractional statistics, which does not require long-range positional order and is applicable both to the solid and the fluid.

Density of states for a two-dimensional Penrose lattice: Evidence of a strong Van-Hove singularity.

The density of states for a tight-binding Hamiltonian on a two-dimensional Penrose lattice is computed numerically by the continued fraction recursion method. The result shows evidence of a strong Van Hove--type singularity which is remarkable for a system possessing no long-range periodic translational order. By a method of finite-size scaling extrapolation the exponent \ensuremath{\alpha} and amplitude C, where \ensuremath{\rho}(E)\ensuremath{\sim}${\mathrm{CE}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, are estimated to be \ensuremath{\alpha}=0.09\ifmmode\pm\else\textpm\fi{}0.05 and C=exp(3.8\ifmmode\pm\else\textpm\fi{}0.2).

Aggregates of amphiphilic molecules in lyotropic liquid crystals

Aamphiphilic molecules are made of two parts with different chemical affinities. They build interfaces which limit the contact between these two parts. We shall focus our attention on the case of soap molecules in the presence of water, in which the interfaces separate the paraffinic chains from the water and delimit aggregates of various shapes. Classically the aggregates can be infinite, with flat or cylindrical interfacial curvatures, and packed with long-range translational order in the well-known lamellar, cubic and hexagonal phases; or they can be finite, quasi-spherical and packed without any long-range order in the micellar phase. However, recent investigations show that a long-range ordering of finite aggregates is possible. There is one example of translational order, in which the micelles build a cubic structure, and much more examples now of orientational order. In the latter case the micelles, oblate or prolate spheroids, are dispersed in the solution with their axes nearly parallel to each other, a nematic phase is formed. We shall discuss the possible factors controlling the shapes of the aggregates and their interactions.

Liquid crystals of disc-like molecules

Liquid crystals of disc-like molecules fall into two distinct structural types, the columnar and the nematic. The columnar phase in its simplest form consists of discs stacked aperiodically in columns, the different columns constituting a two-dimensional array, whereas the nematic phase is an orientationally ordered arrangement of discs without any long-range translational order. After a brief review of the structures of the various mesophases, two topics are considered in somewhat greater detail. (i) An extension of McMillan’s mean field model of smectic A to liquid crystals of disc-like molecules. The translational order is now assumed to be periodic in two dimensions. Calculations show that when the lattice is hexagonal, or departs from it only slightly, the transition from the columnar to the isotropic phase may take place either directly or via a nematic phase, depending on the model potential parameters Interpreting a to be a measure of the chain lengths as in McMillan's model, the theoretical phase diagram is shown to be in broad agreement with the experimental trends. As the asymmetry of the lattice is increased the theory predicts the occurrence of a smectic A phase as well. The new smectic A phase is biaxial. (ii) A theoretical study of fluctuations in the columnar liquid crystal. It is shown that the Frank elasticity of the liquid-like columns stabilizes the two-dimensional order, a result that was in fact envisaged by Peierls and Landau in the 1930s. The meansquare fluctuation of the lattice as well as the Debye—Waller factor show a certain dependence on the linear dimensions of the sample.

Low-temperature laser Raman spectroscopy of synthetic carbonated apatites and dental enamel

A range of synthetic carbonated apatites and human dental enamel have been studied at low temperatures (50 K) by laser Raman spectroscopy. The widths of the Raman spectral bands of hydroxyapatite were significantly reduced at low temperatures with the appearance of several peaks unobserved at room temperatures. The number of observed phosphate and hydroxyl ion Raman peaks was consistent with the space group P63. Spectra of carbonated apatites were not further resolved at low temperatures and Raman peak broadening due to loss of long-range translational order in the apatite structure was linearly correlated with the carbonate content of the sample. The same Raman peak broadening occurred in the spectrum of dental enamel. The number of carbonate bands in the Raman spectrum of carbonated apatites indicated that carbonate exists in two crystallographically distinct sites of different symmetry within the apatite structure.

Structure function of smectic liquid crystals

The behavior of the structure function is investigated for single-domain smectic A and C liquid crystals in the vicinity of the reciprocal lattice vectors in connection with the absence of translational long-range order. The influence of an external orienting magnetic field, the dimensions of the system, and the temperature is discussed.

Single colloidal crystals

Colloidal suspensions of highly charged, monodisperse polymer spheres exhibit long-range (crystalline) translational ordering in appropriate conditions of charge, number density, counterion concentration and temperature1–5. These ‘colloidal crystals’, which can conveniently be made to have lattice parameters comparable to or greater than optical wavelengths, offer unique opportunities for the study of the collective static and dynamic behaviour of strongly interacting spherical particles. For example, such systems may readily be probed by relatively simple but powerful light scattering spectroscopic techniques6,7. Furthermore, the ordering itself offers intrinsic advantages, both experimental (for example, fluctuations normally observable only about the k-space origin appear about Bragg spots, with reduced stray light and multiple scattering effects), and theoretical (the simplicity of calculating on the basis of a known rather than only statistically defined structure is well appreciated from experience with atomic solids and liquids). To exploit these ordered structures fully, reliable methods of producing orientated single crystals suitable for light scattering and other optical studies must be developed. For light scattering from bulk samples the primary requirement is that the ratio of interparticle spacing to diameter, a/d, be sufficiently large. This reduces interference by multiple scattering and renders the particle scattering factor nearly constant through those scattering angles [|k|<(2→5)2π/a] where the most useful information concerning crystal structure and dynamics appears. We now describe a technique whereby single body-centred cubic (b.c.c.) colloidal crystals, well suited for light scattering and having particular orientations, may be produced.