Periodicity in three dimensions

All these surfaces have one important characteristic in common. The flat "points" are not located within any finite portion of the surface. Rather, the surfaces become asymptotically flat (e.g. the trumpet-shaped "ends" in the catenoid). As the number of flat points increases beyond four, the flat points are located at fixed identifiable sites and the surface closes up to b ome periodic in three dimensions. This distinction between one- or two-periodic and three-periodic minimal surfaces is a crucial one, since it implies that the average Gaussian curvature () of one-, and two-periodic minimal surfaces is usually zero, due to the overwhelming contribution from the... 

A crystal may be defined as a solid composed of atoms arranged in a pattern periodic in three dimensions. As such, crystals differ in a fundamental way from gases and liquids because the atomic arrangements in the latter do not possess the essential requirement of periodicity. Not all solids are crystalline, however some are amorphous, like glass, and do not have any regular interior arrangement of atoms. There is, in fact, no essential difference between an amorphous solid and a liquid, and the former is often referred to as an undercooled liquid. ... 

Note that the vectors a, b, c define, not only the unit cell, but also the whole point lattice through the translations provided by these vectors. In other words, the whole set of points in the lattice can be produced by repeated action of the vectors a, b, c on one lattice point located at the origin, or, stated alternatively, the vector coordinates of any point in the lattice are Pz, Qh, and Rc, where P, Q, and R are whole numbers. It follows that the arrangement of points in a point lattice is absolutely periodic in three dimensions, points being repeated at regular intervals along any line one chooses to drav/ through the lattice. 

It is now time to describe the structure of some actual crystals and to relate this structure to the point lattices, crystal systems, and symmetry elements discussed above. The cardinal principle of crystal structure is that the atoms of a crystal are set in space either on the points of a Bravais lattice or in some fixed relation to those points. It follows from this that the atoms of a crystal will be arranged periodically in three dimensions and that this arrangement of atoms will exhibit many of the properties of a Bravais lattice, in particular many of its symmetry elements. 

Surfaces are found to exhibit properties that are different from those of the bulk material. In the bulk, each atom is bonded to other atoms in all three dimensions. In fact, it is this infinite periodicity in three dimensions that gives rise to the power of condensed matter physics. At a surface, however, the three-dimensional periodicity is broken. This causes the surface atoms to respond to this change in their local environment by adjusting their geometric and electronic structures. The physics and chemistry of clean surfaces is discussed in section A 1.7.2. 

We have seen that the intensities of diffraction are proportional to the Fourier transform of the Patterson function, a self-convolution of the scattering matter and that, for a crystal, the Patterson function is periodic in three dimensions. Because the intensity is a positive, real number, the Patterson function is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

The supercell plane wave DFT approach is periodic in three dimensions, which has some disadvantages (i) thick vacuum layers are required so the slab does not interact with its images, (ii) for a tractably sized unit cell, only high adsorbate coverages are modelled readily and (iii) one is limited in accuracy by the form of the... 

Scattering from structures of any size is regular, i.e. takes place at well defined angles, only when the structures are periodic. The scattering is then usually called diffraction. The most important periodic structures suitable for WAXS investigation are crystals, which are periodic in three dimensions. 

The simplest approach is to introduce a gap in the 3D unit cell in a direction perpendicular to the surface. The simulation is still periodic in three dimensions, but the layers in the direction of the gap can be sufficiently well separated so that their interaction becomes negligible, as we shall see later, even for strongly ionic materials, a gap of around 10 A is often sufficient for this purpose. 

The viscous isotropic cubic phase, which is periodic in three dimensions, is produced with monogylceride-water systems at chain lengths above C14. This isotropic phase has a bicontinuous structure, consisting of a lamellar bilayer, which... 

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series ... 

A rigid material whose structure lacks crystalline periodicity that is, the pattern of its constituent atoms or molecules does not repeat periodically in three dimensions. See also metallic glass. 

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