Brillouin zone

About: Brillouin zone is a research topic. Over the lifetime, 13849 publications have been published within this topic receiving 383077 citations.

Light Scattering from Binary Solutions

Abstract: The spectrum of the light scattered by a binary solution is calculated from thermodynamic fluctuation theory and the linearized hydrodynamic equations appropriate to a two‐component fluid The spectrum consists of three peaks Expressions are obtained for the positions and widths of the two‐side, Brillouin peaks In general the central, unshifted Rayleigh peak is found to consist of a superposition of two Lorentzians that involve the combined dynamical effects of heat conduction and diffusion The condition is stated under which it is possible to separate the central peak simply into two contributions, one arising from diffusion and one from thermal conduction For many binary systems this separation is justified In these cases measurement of the spectrum of the scattered light should prove to be an attractive alternative means of measuring the diffusion coefficient of binary solutions

Applications of the Bloch Theory to the Study of Alloys and of the Properties of Bismuth

Abstract: In a recent years many binary alloys have been examined by means of X-rays and their structures have been determined at various compositions. It has been shown in this way that a pure phase consists of a single structure, whilst in the regions of mixed phases the alloy forms a simple mixture of two crystal types. Sometimes the range of composition of a pure phase is extremely narrow, as, for example, in the e-phase of the Cu-Sn alloy. The composition within this phase corresponds almost exactly with the formula Cu3Sn. Examples of this kind have naturally led to the conception of intermolecular compounds, and, since the formation of a compound in chemistry is described in terms of valency bonds, it is natural that in the existing literature attempts to discuss the reason for the formation of various phases, and the properties of the alloys in these phases, have been based mainly upon the conception of the homopolar bond. It does not seem , however, that this conception is very successful in the interpretation of the formation and properties of metallic alloys. It is advantageous therefore at attempt an examination of alloys from the standpoint of he Bloch model. In Bloch’s theory a stationary state of an electron in a crystal is specified by the three components of a vector k which may be regarded as the average momentum associated with the state; k is not, however, the average velocity times the mass of the electron, but is defined in such a way that the rate of change of k is proportional to the external force acting on the electron. Since the difference in k between two successive stationary states is in general exceedingly small, the vector k may be regarded as varying continuously from state to state. It is now a wellknown result of the theory that the energy of a state is not a continuous function of position in k space, but is discontinuous across certain planes. The positions of these planes are determined by the crystal symmetry of the metal, and they divide k space into the various “Brillouin zones.”

Polarization Stop Bands in Chiral Polymeric Three‐Dimensional Photonic Crystals

Abstract: Chiral 3D photonic crystals are an interesting subclass of 3D photonic crystals. For example, large complete 3D photonic bandgaps have been predicted for high-index-contrast silicon square-spiral structures; corresponding experiments using glancing-incidence deposition, interference lithography, or direct laser writing have been published. In addition to complete gaps or stop bands, theory also predicts polarization stop bands, i.e., stop bands for just one of the two circular polarizations. Such polarization stop bands can give rise to strong circular dichroism, which can potentially be used for constructing compact “thin-film” optical diodes. In this report, we fabricate high-quality polymeric 3D spiral photonic crystals via direct laser writing. The measured transmittance spectra of these low-index-contrast structures reveal spectral regions where the transmittance is below 5 % for one circular incident polarization and above 95 % for the other—for just eight lattice constants along the propagation direction. The experimental data are compared with scattering-matrix calculations for the actual finite structures, leading to good agreement. For what conditions do we expect strong circular dichroism? For circular polarization of light, the tip of the electricfield vector simply follows a spiral. The pitch of this spiral is just the material wavelength k. Thus, intuitively, we expect a chiral resonance from spiral photonic crystals if the pitch of circularly polarized light matches the pitch of the dielectric spirals, i.e., the lattice constant az. This condition, k/az = 1, corresponds to the edge of the second Brillouin zone, i.e., to a wave number kz = 2p/k= 2p/az. Recall that the edge of the first Brillouin zone is at kz = p/az. Thus, one does not anticipate a strong chiral response around and below the fundamental stop band (or bandgap), but rather at higher frequencies. Theory for high-index silicon-based structures confirms this intuitive reasoning. We have repeated similar calculations for low-index-contrast polymeric structures, revealing essentially the same trends. The parameters of the 3D spiral photonic crystals to be discussed below are the result of an optimization with respect to circular dichroism. The samples in our experiments are made by direct laser writing, which essentially allows for the fabrication of almost arbitrarily shaped 3D photoresist structures. Details of our process based on the commercial thick-film resist SU-8 can be found in the Experimental section and in earlier work. Our structures are mechanically supported by a 2D network of bars at, or close to, the top of the 3D crystal. As the spirals are not at all mechanically connected to their neighbors, very unstable low-quality structures would result without this grid. Furthermore, all the structures for optical experiments are surrounded by a thick massive wall (see Fig. 1a), which aims at reducing the effects of strain on the 2D grid caused by photoresist shrinkage during development. Here, we use a round (rather than a rectangular) wall in order to evenly distribute strain inside the wall. Through numerical calculations (see below), we have confirmed that the distortion of the optical properties by the 2D network is only marginal. Most importantly, the network does not introduce any chirality. A small gallery of selected electron microscopy images is shown in Figure 1, which gives first evidence that the sample quality is very good. Figure 1a gives an overview of the sample to be optically characterized below. The sample parameters are: in-plane lattice constant axy = 1.3 lm, pitch az = 1.3 lm, spiral diameter d = 0.78 lm, volume filling fraction 34.7 %, lateral diameter of the spiral arms darm = 380 nm, ratio between the axial and the lateral diameter 2.7, and N = 8 lattice constants along the z-direction. These parameters were extracted from the close-up cross-sectional view in Figure 1a. To demonstrate the versatility of our approach, Figure 1b exhibits a cut of a structure with axy = 1.5 lm, az = 1.5 lm, and N = 4. Because the focused-ion-beam cut was stopped in between two rows of spirals, the stabilizing network mentioned C O M M U N IC A IO N

Phonon spectrum in a plasma crystal.

Abstract: The Fourier spectra of longitudinal and transverse waves corresponding to random particle motion were measured in a two-dimensional plasma crystal. The crystal was composed of negatively charged microspheres immersed in a plasma at a low gas pressure. The phonons were found to obey a dispersion relation that assumes a Yukawa interparticle potential. The crystal was in a nonthermal equilibrium, nevertheless phonon energies were almost equally distributed with respect to wave number over the entire first Brillouin zone.

Electronic structure and hybridization effects in Hume-Rothery alloys containing transition elements

Abstract: We present a systematic study of the electronic structure of Al-based Hume-Rothery alloys containing transition elements performed with the use of the linear muffin-tin orbital in atomic-sphere approximation method. Our analysis focuses on the formation of the pseudogap at the Fermi level leading to the stability of materials containing transition-metal elements in small concentration. From the self-consistent calculated density of states, we observe a strong deviation from the two classical limits: (a) the Friedel-Anderson virtual bond state's model and (b) the nearly-free-electron diffraction by some Bragg planes in the usual Hume-Rothery picture of simple metal alloys. Transition-metal atoms have a crucial role on electronic structure via the combined effect of the sp-d hybridization and of a strong interaction between the Fermi surface and a predominant Brillouin zone.