Periodicity two-dimensional

Surface states can be divided into those that are intrinsic to a well ordered crystal surface with two-dimensional periodicity, and those that are extrinsic [25]. Intrinsic states include those that are associated with relaxation and reconstruction. Note, however, that even in a bulk-tenuinated surface, the outemiost atoms are in a different electronic enviromuent than the substrate atoms, which can also lead to intrinsic surface states. Extrinsic surface states are associated with imperfections in the perfect order of the surface region. Extrinsic states can also be fomied by an adsorbate, as discussed below. [Pg.293]

Figure B3.2.12. Schematic illustration of geometries used in the simulation of the chemisorption of a diatomic molecule on a surface (the third dimension is suppressed). The molecule is shown on a surface simulated by (A) a semi-infinite crystal, (B) a slab and an embedding region, (C) a slab with two-dimensional periodicity, (D) a slab in a siipercell geometry and (E) a cluster.

Fig. 31. Two-dimensional periodic orbits for vibration coupled antisymmetrically to the reaction coordinate. Caustics (C) and take-off points (T) are indicated.

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... [Pg.42]

The last twenty years have seen a rapid development of surface physics. In particular, the properties of clean perfect surfaces (with two-dimensional periodicity) are henceforth well known and understood. In recent years, the focus has been put onto surfaces with defects (adatoms, steps, vacancies, impurities...) which can now be investigated experimentally due either to the progress of old techniques (field ion microscopy or He diffraction, for instance) or to the rapid development of new methods (STM, AFM, SEXAFS...). [Pg.371]

We start with some elementary information about anisotropic intermolec-ular interactions in liquid crystals and molecular factors that influence the smectic behaviour. The various types of molecular models and commonly accepted concepts reproducing the smectic behaviour are evaluated. Then we discuss in more detail the breaking of head-to-tail inversion symmetry in smectic layers formed by polar and (or) sterically asymmetric molecules and formation of particular phases with one and two dimensional periodicity. We then proceed with the description of the structure and phase behaviour of terminally fluorinated and polyphilic mesogens and specific polar properties of the achiral chevron structures. Finally, different possibilities for bridging the gap between smectic and columnar phases are considered. [Pg.200]

Single slab. A number of recent calculations of surface electronic structures have shown that the essential electronic and structural features of the bulk material are recovered only a few atomic layers beneath a metal surface. Thus, it is possible to model a surface by a single slab consisting of 5-15 atomic layers with two-dimensional translational symmetry parallel to the surface and vacuum above and below the slab. Using the two-dimensional periodicity of the slab (or thin film), a band-structure approach with two-dimensional periodic boundary conditions can be applied to the surface electronic structure. [Pg.52]

Since the interaction energy /s with the finite substrate is not easy to handle due to the loss of two dimensional periodicity assumed in the derivation of Eq. 5, we simplify the substrate interaction. We considered that the substrate was a sandwich of the amorphous and the crystalline layers, and the attractive potential Uo(z) works at any point (x,y), while the translational barrier Ui(z) cos(2nx/k) only works on the crystalline substrate. The implicit assumption is that the atomic densities of the crystal and the amorphous are not so different. [Pg.59]

Fig. 27. Nonfiltered AFM image of a lignoceric acid monolayer on a scan area of 9x9 nm2. Note a two-dimensional periodic structure with locally disordered molecular arrangements which is marked by a circle.

The passive film is composed of metal oxides which can be semiconductors or insulators. Then, the electron levels in the passive film are characterized by the conduction and valence bands. Here, we need to examine whether the band model can apply to a thin passive oxide film whose thickness is in the range of nanometers. The passive film has a two-dimensional periodic lattice structure on... [Pg.383]

Two-dimensional periodic organization of eukaryotic and prokaryotic ribosomes occurs under special conditions in vivo (Byers, 1967 Kress et al., 1971 Taddei, 1972 O Brien et al, 1980) or in vitro (Barbieri, 1979 Clark et al, 1982). The two-dimensional sheets have been analyzed by image-reconstruction techniques (Kiihlbrandt and Unwin, 1982 Clark et al, 1982). [Pg.32]

Figure 1. A two-dimensional periodic pattern eomposed of black squares...

For diffusion in the open two-dimensional periodic Lorentz gas with parallel absorbing walls separated by the distance L, Eq. (95) shows that the diffusion coefficient is given by [38]... [Pg.113]

Figure 8.9 Schematic illustrations of spin states in a two dimensional periodic material. Circles indicate individual atoms and the dotted lines show a single supercell. In (a), all electrons are paired on each atom. In the remaining examples, a single unpaired electron exists on each atom. Examples of a ferromagnetic state, an antiferromagnetic state and a more complex mag netic state are shown in (b), (c), and (d), respectively.

For a solid surface with two-dimensional periodicity, such as a defect-free crystalline surface, all the measurable quantities have the same two-dimensional periodicity, for example, the surface charge distribution, the force between a crystalline surface and an inert-gas atom (Steele, 1974 Goodman and Wachman, 1976 Sakai, Cardino, and Hamann, 1986), tunneling current distribution, and STM topographic images (Chen, 1991). These quantities can be expanded into two-dimensional Fourier series. Usually, only the few lowest Fourier components are enough for describing the physical phenomenon, which requires a set of Fourier coefficients. If the surface exhibits an additional symmetry, then the number of independent Fourier coefficients can be further reduced. [Pg.353]

The two-dimensional periodicity can always be described by two primitive vectors ai and a2, as shown in Fig. D.l. A periodic function F has the property... [Pg.353]

Fig. D.l. A surface with two-dimensional periodicity, (a) in real space, the function has two-dimensional periodicity, which is indicated by two primitive vectors, ai and 02. (b) In reciprocal space, two primitive vectors bi and ba are introduced.

As we have discussed previously, any function with two-dimensional periodicity can be expanded into two-dimensional Fourier series. If a function has additional symmetry other than translational, then some of the terms in the Fourier expansion vanish, and some nonvanishing Fourier coefficients equal each other. The number of independent parameters is then reduced. In general, the form of a quantity periodic in x and y would be... [Pg.361]

With these strategies, various large two-dimensional periodic aggregates have been obtained. They can be classified in two main groups planar structures built... [Pg.245]

The first theoretical analysis of this problem was proposed by Landsberg et al. [44-46] following a concept developed in electrical engineering and formely used by Smythe [47]. The calculation was carried out for an interface composed of purely insulating and purely conducting areas, the latter being small disks of diameter d arranged in a two-dimensional periodic array. [Pg.227]

Students will show that their periodic table includes all the information presented in a two-dimensional periodic table. [Pg.179]

The importance of the two dimensional periodicity on the transmission properties is demonstrated in Figure 5, which presents the transmission probability of electrons as a function of the photoelectron energy for layers of Cdar (dashed), Cdbr (dotted) and of mixed layers (solid) for three (Fig. 5A) and nine (Fig. 5B) layers. As is clearly evident, the electron transmission through the mixed layers is significantly less efficient than that through the Cdar or Cdbr layers themselves. Moreover, the spectmm for the mixed layers is much closer to the relaxed type (Fig. 2). [Pg.77]

The surface atoms of a solid have a definite ability of forming bonds with molecules of the gas phase, which increases the concentration of these molecules near the surface [32-34]. A homogeneous surface implies a two-dimensional periodic lattice with equivalent sites. Each site of the... [Pg.354]

Figure 26 A schematic picture of the Mn2P22 layer band structure as derived by first turning on local Mn-P interactions and then the two-dimensional periodicity and Mn-Mn interactions. The unit cell contains two Mn and two P atoms, so in reality each of the levels in the first two columns should be doubled.

Plane waves are often used when two conditions are satisfied 1) many (but not necessarily all) atomic layers of the surface have a two-dimensional periodicity, and 2) either a plane-wave incident electron beam is present or angle-resolved electron detection is applied. Computation based on the plane-wave expansion are often much more efficient than those based on the spherical-wave expansion. This explains their frequent use even in problems that do not involve strict two-dimensional periodicity, as with disordered overlayers on an otherwise periodic substrate. [Pg.57]

Simplifications can be brought about whenever the surface structure has symmetries. Point-group symmetries help moderately to reduce the matrix dimensions. On the other hand, two-dimensional periodicity can help drastically by reducing the number N to the number of atoms within a single two-dimensional unit cell with a depth perpendicular to the surface of a few times the electron mean free path. For surface crystallography this is, however, not yet sufficient, because surface structural determination requires repeating such calculations for hundreds of different geometrical models of the surface structure. [Pg.64]

Bita I, Yang JKW et al (2008) Graphoepitaxy of self-assembled block copolymers on two-dimensional periodic patterned templates. Science 321 939-943... [Pg.192]

The precise features of real catalysts at a microscopic scale are rather unknown but in all cases the main interactions occur through a surface. Two different theoretical models are often used to describe the electronic and other microscopic features of a surface. On the one hand, there is the solid state physics approach in which a surface is considered as a slab of a given thickness, finite in the direction perpendicular to the surface and infinite in the two other dimensions with perfect two-dimensional periodical symmetry. On the other hand, one has the cluster model approach which represents the surface with a finite number of atoms and the surface-adsorbate interaction as a supermolecule this is essentially a quantum chemical approach. It is important to realize that both approaches are crude representations of physical reality because real surfaces are far from being perfect, usually... [Pg.150]

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