Shallow Lattice Research Articles

Adiabaticity and Localization in One-Dimensional Incommensurate Lattices

We experimentally investigate the role of localization on the adiabaticity of loading a Bose-Einstein condensate into a one-dimensional optical potential comprised of a shallow primary lattice plus one or two perturbing lattice(s) of incommensurate period. We find that even a very weak perturbation causes dramatic changes in the momentum distribution and makes adiabatic loading of the combined lattice much more difficult than for a single period lattice. We interpret our results using a band-structure model and the one-dimensional Gross-Pitaevskii equation.

Bragg resonances and Zener tunneling in quasiperiodic two-dimensional optical lattices and photonic crystals

Nonresonant Zener tunneling in decagonal quasiperiodic structures in two spatial dimensions is defined by its relation to Bragg resonance and is studied by direct numerical simulations and an analytical approach. It is shown that, in the shallow lattice limit, the tunneling dynamics about the Bragg resonances is described by the multilevel Landau-Zener-Majorana models, which capture the essential peaks of the complicated Fourier spectrum. The results have applications to dynamics of cold atoms and Bose-Einstein condensates in quasiperiodic optical lattices, light propagation in quasiperiodic photonic crystals, and ultrasonic experiments with quasiperiodic structures.

Magnetism and Hund's rule in an optical lattice with cold fermions

We demonstrate that a two-dimensional (2D) optical lattice loaded with repulsive, contact-interacting fermions shows a rich and systematic magnetic phase diagram. Trapping a few (N ⩽ 12) fermions in each of the single-site minima of the optical lattice, we find that the shell structure in these quantum wells determines the magnetism. In a shallow lattice, the tunnelling between the single wells is strong, and the lattice is non-magnetic (NM). For deeper lattices, however, the shell filling of the single wells with fermionic atoms determines the magnetism. As a consequence of Hund's first rule, the interaction energy is lowered by maximizing the number of atoms of the same species. This leads to a systematic sequence of NM, ferromagnetic (F) and antiferromagnetic (AF) phases.

Microwave photoresistance of a high-mobility two-dimensional electron gas in a triangular antidot lattice

The microwave (MW) photoresistance has been measured on a high-mobility two-dimensional electron gas patterned with a shallow triangular antidot lattice, where both the MW-induced resistance oscillations (MIRO) and magnetoplasmon (MP) resonance are observed superposing on sharp commensurate geometrical resonance (GR). Analysis shows that the MIRO, MP, and GR are decoupled from each other in these experiments.

Nonlinear dynamical bifurcation and chaotic motion of shallow conical lattice shell

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.

Bistable Solitons in Single- and Multichannel Waveguides with the Cubic-Quintic Nonlinearity

We consider spatial solitons in a channel waveguide or in a periodic array of rectangular potential wells (the Kronig-Penney (KP) model) in the presence of the uniform cubic-quintic (CQ) nonlinearity. Using the variational approximation and numerical methods, we find two branches of fundamental (single-humped) soliton solutions. The soliton characteristics, in the form of the integral power Q (or width w) vs. the propagation constant k, reveal a strong bistability with two different solutions found for a given k. Violating the known Vakhitov-Kolokolov criterion, the solution branches with dQ/dk > 0 and dQ/dk < 0 are simultaneously stable. Various families of higher-order solitons are also found in the KP version of the model: symmetric and antisymmetric double-humped solitons, three-peak solitons with and without the phase shift π between the peaks, etc. In a relatively shallow KP lattice, all the solitons belong to the semi-infinite gap beneath the linear band structure of the KP potential, while finite gaps between the bands remain empty (solitons in the finite gaps can be found if the lattice is much deeper). But in contrast to the situation known for the model combining a periodic potential and the self-focusing Kerr nonlinearity, the fundamental solitons fill only a finite zone near the top of the semi-infinite gap, which is a manifestation of the saturable character of the CQ nonlinearity.

Bistable Solitons in Single- and Multichannel Waveguides with the Cubic-Quintic Nonlinearity

We consider spatial solitons in a channel waveguide or in a periodic array of rectangular potential wells (the Kronig-Penney (KP) model) in the presence of the uniform cubic-quintic (CQ) nonlinearity. Using the variational approximation and numerical methods, we find two branches of fundamental (single-humped) soliton solutions. The soliton characteristics, in the form of the integral power Q (or width w) vs. the propagation constant k, reveal a strong bistability with two different solutions found for a given k. Violating the known Vakhitov-Kolokolov criterion, the solution branches with dQ/dk > 0 and dQ/dk < 0 are simultaneously stable. Various families of higher-order solitons are also found in the KP version of the model: symmetric and antisymmetric double-humped solitons, three-peak solitons with and without the phase shift π between the peaks, etc. In a relatively shallow KP lattice, all the solitons belong to the semi-infinite gap beneath the linear band structure of the KP potential, while finite gaps between the bands remain empty (solitons in the finite gaps can be found if the lattice is much deeper). But in contrast to the situation known for the model combining a periodic potential and the self-focusing Kerr nonlinearity, the fundamental solitons fill only a finite zone near the top of the semi-infinite gap, which is a manifestation of the saturable character of the CQ nonlinearity.

Design and static behaviour of a shallow lattice wooden dome

The design and an experimental study of a shallow lattice wooden dome constructed as the roof of a pavilion for the World Flower and Greenery Expo'90 held at Osaka, Japan are described. The dome had a number of special features. Among these were its unusual shape, special connectors and a very small rise-to-span ratio. The determination of the design loads and the evaluation of the buckling capacity of the structure are briefly discussed. Experiments on the actual structure were performed by applying a point load at the centre node. It was observed that the dome exhibited a localized deformation pattern and the load-displacement relationship was characterized by torsional buckling of the centre node, with increasing load-bearing capacity at the postbuckling stage. Numerical results are also described. It was found that torsional buckling of a joint may occur when the joint has a rigid connection and the inplane bending rigidities of the connecting members are small as compared with the corresponding out-of-plane bending rigidities.

Computation Models and Numerical Solution Procedures for Nonlinear Analysis of Single Layer Lattice Shells

Single-layer space frames consisting of reticulated surfaces with assumed pin or rigid jointed members are widely used in practice. This is accounted for by the fact that they meet the need to cover large spans without the use of intermediate supports, while at the same time, yielding a very small structural weight per unit of area covered. Because of the in-plane membrane action of shallow single-layer lattice shells associated with their initial curvature, such structures are able to carry considerable out-of-plane loads relative to the weight of the structure itself. However, in the case of shallow single-layer lattice shells with small rise-to-span ratios, the response is highly nonlinear even when the stresses in the members are small enough to ensure that the material properties fall within the elastic range. These structures become unstable at some critical value of the load and any design based on an elastic linearized analysis is potentially unsafe. Shallow single-layer lattice shells are characterized by a strong nonlinear behaviour with considerable softening, so they are extremely sensitive to out-of-plane instability and are affected by general and local instability. As far as the pre-buckling range is concerned, an extensive literature dealing with effective solver techniques exists for the numerical solutions of structural problems, but only a few algorithms have been put forth with a view to tracing the nonlinear response from the pre-limit range into the post-limit one. The main purpose of this paper is to review these methods and to discuss their performance. The author's ASEF-N computer program and selected numerical examples are presented.

Single Layer Shallow Lattice Domes: Analysis, General Behaviour and Collapse

This paper discusses some aspects of the pre- and post-collapse analysis and behaviour of shallow lattice (reticulated) domes. Use is made of a nonlinear computer program to study the behaviour associated with changes in both structural geometry and material properties, as well as that associated with bifurcation and limit point buckling. This program has been described in detail elsewhere, and apart from a discussion of the use of static condensation in the nonlinear range, only a brief summary of the theory used in the program is given here. The program is used to predict both the pre- and post-buckling response of a 42-member shallow braced model dome under two types of loading and the numerical results are compared with those of an experimental study. The influence of initial imperfections on the general behaviour is discussed briefly, and some guidelines are set out to aid the numerical determination of the true load-deflection response of shallow lattice domes.

Collapse of Shallow Lattice Domes

A numerical and experimental study of the collapse and postcollapse behavior of shallow lattice domes is presented. The numerical analysis is based on an updated Lagrangian, materially and geometrically nonlinear, displacement‐based finite‐element program. The formulation incorporated in the program can detect plasticity, instability, and finite deflection effects in the dome structure. Warping effects are ignored. The solution of the nonlinear stiffness equations is designed so that critical points on the equilibrium path are identified in the results. A method is used to obtain the lowest bifurcation path of a perfect lattice dome, once it is established that there is a bifurcation point on its primary equilibrium path. Limit point behavior is detected automatically. A shallow model dome was tested experimentally in a purpose‐built test rig. The experimental results presented are compared with the corresponding numerical predictions and good agreement between them is demonstrated.

Role of shallow impurities and lattice defects in nucleation of electron-hole droplets in Si

Contribution of lattice defects and shallow impurities to electron-hole droplet (EHD) formation has been confirmed by the study on photoluminescence due to EHD's in various kinds of silicon crystals. Sample dependence of the EHD luminescence peak energy shift in the variation of the excitation intensity is explained in terms of the surface energy contribution to EHD binding energy and concentrations of EHD's and impurities.

Self-Diffusion along Dislocations in Single-Crystal Au Films

Self-diffusion measurements over the temperature range 247-352\ifmmode^\circ\else\textdegree\fi{}C have been made on single-crystal Au films of $2\ensuremath{-}\ensuremath{\mu}$ thickness grown epitaxially onto (001) MgO substrates, using ${\mathrm{Au}}^{195}$ radioactive tracer and rf sputter-etching techniques for serial sectioning. The penetration profiles revealed shallow lattice diffusion and a much deeper tracer penetration down dislocations; the former is significantly enhanced by diffusion in dislocations. The activation energy ${Q}_{d}$ and the combined preexponential factor ${A}_{d}{D}_{d}^{0}$ for self-diffusion along dislocations are found to be 1.16 \ifmmode\pm\else\textpm\fi{} 0.02 eV and 5 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}16}$ ${\mathrm{cm}}^{4}$/sec, respectively. From the enhanced lattice diffusion observed in the first stage of the penetration profiles, a dislocation density of \ensuremath{\sim}${10}^{11}$ lines/${\mathrm{cm}}^{2}$ could also be evaluated for the single-crystal films. The data compare well with other fcc metals and appear to favor a vacancy diffusion mechanism.