Centered lattices

Figure 3.13. Top Model of an ideal (100) surface of a face-centered crystal (fee) lattice. Center and bottom Model of a vicinal surface of an fee cut at 12° to the (100) plane a) with straight monatomic steps and (Z ) monatomic steps with kinks along the steps. (From Ref. 11, with permission from Pergamon Press.)...

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

Deduction of lattice centering and translational symmetry elements from systemic absences... 

Systematic absences (or extinctions) in the X-ray diffraction pattern of a single crystal are caused by the presence of lattice centering and translational symmetry elements, namely screw axes and glide planes. Such extinctions are extremely useful in deducing the space group of an unknown crystal. 

Consider now a 2 -axis parallel to the b axis. As shown in Fig. 9.4.4, the coordinates x and z are irrelevant for the (OkO) planes, and the systematic absences (OkO) absent with k odd implies the presence of a 2i-axis parallel to b. Note that this is a very weak condition as compared to those for lattice centering and glide planes, and is already covered by them. Since the (OkO) reflections are few in number, some may be too weak to be observable, and hence the determination of a screw axis from systematic absences is not always reliable. 

The systematic absences due to the various types of lattice centering, screw axes, and glide planes are given in Table 9.4.2, which is used in the deduction of space groups. 

Figure 1. 205T1-NMR spectrum (solid line) at 5 K. The intensity is plotted in a linear scale. The thin solid line depict the histogram at particular local fields of the Readfield pattern. The dotted line represents the simulation spectrum convoluted with Lorentzian broadening function. The filled circles show the frequency dependence of 205 f,1 1 at the T1 site. The inset shows the image of the field distribution in the vortex square lattice center of vortex core (A), saddle point (B) and center of vortex lattice (C).

Table 2 Systematic absences related to lattice centering and translational elements of symmetry...

Bravais in 1849 showed that there are only 14 ways that identical points can be arranged in space subject to the condition that each point has the same number of neighbors at the same distances and in the same directions.Moritz Ludwig Frankenheim, in an extension of this study, showed that this number, 14, could also be used to describe the total number of distinct three-dimensional crystal lattices.These are referred to as the 14 Bravais lattices (Figure 4.9), and they represent combinations of the seven crystal systems and the four lattice centering types (P, C, F, I). Rhombohedral and hexagonal lattices are primitive, but the letter R is used for the former. 

Since every unit cell in the crystal lattice is identical to all others, it is said that the lattice can be primitive or centered. We already mentioned (Eq. 1.1) that a crystallographic lattice is based on three non-coplanar translations (vectors), thus the presence of lattice centering introduces additional translations that are different from the three basis translations. Properties of various lattices are summarized in Table 1.13 along with the international symbols adopted to differentiate between different lattice types. In a base-centered lattice, there are three different possibilities to select a pair of opposite faces, which is also reflected in Table 1.13. 

The introduction of lattice centering makes the treatment of crystallographic symmetry much more elegant when compared to that where only primitive lattices are allowed. Considering six crystal families Table 1.12) and five types of lattices Table 1.13), where three base-centered lattices, which are different only by the orientation of the centered faces with respect to a fixed set of basis vectors are taken as one, it is possible to show that only 14 different types of unit cells are required to describe all lattices using conventional crystallographic symmetry. These are listed in Table 1.14, and they are known as Bravais lattices. ... 

Primitive rhombohedral lattices, i.e. when a = b = c and a = p = y 90° are nearly always treated in the hexagonal basis with rhombohedral (R) lattice centering. In a primitive... 

All reflections should be first checked for general systematic absences (conditions) caused by lattice centering, which are shown in the first... 

A increasingly popular method for closing the moment-transport equations is to assume a discrete form for the phase-space variables. Taking the velocity NDF as an example, the velocity phase space can be discretized on a uniform, symmetric lattice centered at Vp = 0. For illustration purposes, let us assume that A = 16 lattice points are used and denote the corresponding velocities as Ua. The formal definition of the discrete NDF is... 

Figure 1 Illustration of the discretization of space by a cubic lattice centered on grid point i. The potential < > is located on the solid circles and the dielectric e is placed on the open circles. The difference equation resulting from this is...

The symmetry of the diffraction pattern is known as the Laue class of which there are 11 types (i.e. excluding the nature of the lattice centering). 

Host lattice Center Pump wavelength (ftm) Tuning range (Mm) Maximum power (watts) Operational lifetime... 

In the course of the phase transition the space group A2/m changes into its subgroup P2i/ loosing certain symmetry elements (lattice centering, mirror plane, twofold axis). Below the pha transition the FA-rings as well as the counterions gain a limited freedom for rotations. 

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