Crystals lattice translations

Laue realized in 1912 that the path length differences PD, PD2, PD3 for waves diffracted by atoms separated by one crystal lattice translation must be an integral number of wavelengths for diffraction (i.e., reinforcement) to occur further, he showed that this condition must be true... 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

The space group of a crystal structure can be considered as the set of all the symmetry operations which leave the structure invariant. All the elements (symmetry operations) of this set satisfy the characteristics of a group and their number (order) is infinite. Of course, this definition is only valid for an ideai structure extending to infinity. For practical purpose, however, it can be applied to the finite size of real crystals. Lattice translations, proper or improper rotations with or without screw or gliding components are all examples of symmetry operations. 

Figure 16-19. A Tt-stacked arrangement of Ooci-OPV5-CN" molecules in the crystal lattice (view at a slight angle with respect to the long molecular axis). B lop-view (lop) and side-view (bottom) of two nearest-neighbor Oocl-OPV5-CN" molecules in the crystal. related by a translation along the ci-axis. The distance of 3.5 A corresponds to the shortest distance between two atoms on different molecules (sec Fig. 16-18 for details).

Translational symmetry is the most important symmetry property of a crystal. In the Hermann-Mauguin symbols the three-dimensional translational symmetry is expressed by a capital letter which also allows the distinction of primitive and centered crystal lattices (cf. Fig. 2.6, p. 8) ... 

Do not confuse crystal structure and crystal lattice. The crystal structure designates a regular array of atoms, the crystal lattice corresponds to an infinity of translation vectors (Section 2.2). The terms should not be mixed up either. There exists no lattice structure and no diamond lattice , but a diamond structure. 

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

For the application to a crystal lattice Eq. (51) can be generalized to include the possibility of translation of a point in space by writing... 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

In a crystal lattice where each atom contributes one atomic orbital, and where these orbitals are related to each other by the translations characteristic of the lattice, the molecular orbitals must belong to irreducible representations of the group of these translations and hence form so-called Bloch orbitals. 46)... 

Several types of symmetry operations can be distinguished in a crystalline substance. Purely translational operations, such as the translations defining the crystal lattice, are represented by I 1, n3, with nu n2, n3 being integers. 

Not all incommensurate structures are composite. It is possible to have incommensurate modulations in a structure composed of a single infinite building block, particularly if a weak cation fits rather loosely into a hole in a flexible framework. The polyhedra that compose the framework tend to twist to give the cation a distorted environment. These twists can often be described by a wave with a wavelength that may or may not be commensurate with the lattice translation of the crystal. If it is commensurate, the twisting is described as... 

Reaction of the triacetylide [ 1,3,5( = ) 3- 6 3] and the corresponding monodentate ligand (L) (L = isocyanide, phosphine, phosphite) gives [ 1,3,5 (C = CAuL) 3-C(lH3]. In the crystal structures of compounds with L — CN Bu and P(OMe)3 molecules related by lattice translations are connected through aurophilic interactions to form a polymer (Figure 2.73) [363]. 

Accordingly, glide planes are those planes which have the shortest b vectors a/2 <110> for fee, a/2 <111> for bcc, and a/3 <211.0) for hep lattices. Dislocations can split into so-called Shockley partials b = bx +b2, if b2>b +b. Since b and b2 are not translational vectors of the crystal lattice, they induce a stacking fault. The partial dislocation therefore bounds the stacking fault. 

---