Crystal symmetries infinite lattices

In contrast to discrete molecules, crystals have a lattice structure exhibiting three-dimensional periodicity. As a result, we need to consider additional symmetry elements that apply to an infinitely extended object, namely the translations, screw axes, and glide planes. Chapters 9 and 10 introduce the concept and nomenclature of space groups and their application in describing the structures of crystals, as well as a survey of the basic inorganic crystalline materials. 

In an infinite lattice with point dipoles placed at the nodes—a first model of a molecular crystal—the translational symmetry allows us to simplify equation... 

The four unit cells shown in Figure 1.21 have the same symmetry (a twofold rotation axis, which is perpendicular to the plane of the projection and passes through the center of each unit cell), but they have different shapes and areas (volumes in three dimensions). Furthermore, the two unit cells located on top of Figure 1.21 do not contain lattice points inside the unit cell, while each of the remaining two has an additional lattice point in the middle. We note that all unit cells depicted in Figure 1.21 satisfy the rule for the monoclinic crystal system established in Table /.//. It is quite obvious, that more unit cells can be selected in Figure 1.21, and an infinite number of choices is possible in the infinite lattice, all in agreement with Table 1.11. 

A simple translation is represented by the operation (E, T yJ, where E is the identity (represented by a unit matrix), and = wa + i b + wc, a vector of the translation lattice. Translational symmetry allows us to arrange a large number of identical molecules or atoms in such a way that they are all strictly equivalent provided that the crystal is infinitely large (or large compared to the shortest translations). The atoms or molecules on the surface of a crystal are equivalent to each other if they form a two-dimensional translation lattice, i.e. planar faces parallel to specific lattice planes in the three-dimensional lattice (Bravais law. Section 1.4.2). 

On the other hand, translation, as an operation by which the unit cell expands to the infinite crystal (infinite lattice + its basis/motif), was not yet integrated into the possible combinations of symmetry. This inclusion, with the analysis of the geometric consequences for the crystalline characterization, is aimed for the present sections. 

This description of the excited state does however not respect the translational symmetry of the crystal and an extra step has to be taken to obtain a more complete description. First, we change from discrete point indexation (1,2,... ij,..., N)to a more convenient representation based on the distance between two lattice sites. Figure 6.16 shows how the discrete labeling of lattice sites can be replaced by a representation based on the distance r between these through the vectors r. Although slightly more abstract, this choice is more versatile for an extended system with, in principle, infinite lattice sites and translational symmetry. 

A crystal is comprised of an infinite 3-D lattice of repeating units, of which the smallest building block is known as the asymmetric unit. When acted upon by crystal symmetry operations such as rotation axes or mirror planes (see Section 2.3.2), the asymmetric unit is duplicated to produce the contents of a unit cell (Figure 2.7). For any crystal lattice, it is possible to define an infinite number of possible unit cells (Figure 2.8). However, by convention, this unit is chosen to be a repeatable unit that... 

Of course, the main feature of a crystal, and what distinguishes it from other solids, is that it contains order. Conceptually, we describe crystals as being formed from a perfectly ordered array of atoms or molecules. This order has a helpful consequence by describing a small portion of the structure and the symmetry of crystal we can map out the atomic positions of an infinite lattice. The power of this approach is beguiling. It allows us to map all of the atoms in a crystal, which may be metres in... 

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

This strategy has been successfully applied to infinite periodic 1-D chains of Li atoms [28], through the first symmetry-broken application of the ab-initio UHF version of the Torino s CRYSTAL package [29]. The results of this work and of further treatments of 2-D lattices of Li and even Mg (Lepetit and coworkers, to be published) all confirm the validity of the intersticial picture. This is a case where the symmetry-broken HF solutions have led to a completely new picture of the electronic assembly. 

A technical problem occurs when one attempts to apply this approach to study a surface. The calculations described for the bulk crystal assume perfect symmetry and a solid of infinite extent often described in terms of cyclic or periodic boundary conditions. However, for a surface, the translational symmetry is broken, and the usual expansions in Fourier series used for the bulk are not appropriate. For the bulk, a few atoms form a basis which is attached to a lattice cell, and this cell is... 

A study of the external symmetry of crystals naturally leads to the idea that a single crystal is a three-dimensional periodic structure i.e., it is built of a basic structural unit that is repeated with regular periodicity in three-dimensional space. Such an infinite periodic structure can be conveniently and completely described in terms of a lattice (or space lattice), which consists of a set of points (mathematical points that are dimensionless) that have identical environments. 

We mentioned earlier that the true symmetry of the unit cell may not simply be manifested macroscopically upon infinite translation in three dimensions. Buerger has illustrated this with the mineral nepheline, (Na,K)AlSiC>4 (Buerger, 1978). The true symmetry of the nepheline crystal lattice, the symmetry of the unit cell, consists merely of a sixfold rotation axis (class 6) as would be exhibited by a hexagonal prism with nonequivalent halves. That is, there is no mirror plane perpendicular to the rotation axis. However, the absence of this mirror plane is obviously not macroscopically visible in the hexagonal prism form development of nepheline, implying a higher apparent symmetry (6/mmm). 

Symmetry operations which involve shifts, can apply only to regularly repeating infinite patterns, like crystal structmes. A repeated application of such an operation brings the structme not to the original position, but to a different one, separated from the original by an integer number of lattice translations. There are two types of such ( translational ) symmetry elements (see Table 2), besides primitive lattice translations a, b, c. 

The symmetry operations must be compatible with infinite translational repeats in a crystal lattice. 

Although degenerate orbitals in an isolated ion are partially lifted in a molecule and in a solid, the degeneracy often partially remains in a crystal lattice with high symmetry. In a molecule, this degeneracy is lifted by ion distortions, i.e. the Jahn-Teller distortion. However, in a crystal lattice, an infinite number of degrees of... 

Other possible unit cells with the same volume (an infinite number, in fact) could be constructed, and each could generate the macroscopic crystal by repeated elementary translations, but only those shown in Figure 21.6 possess the symmetry elements of their crystal systems. Figure 21.7 illustrates a few of the infinite number of cells that can be constructed for a two-dimensional rectangular lattice. Only the rectangular cell B in the figure has three 2-fold rotation axes and two mirror planes. Although the other cells all have the same area, each of them has only one 2-fold axis and no mirror planes they are therefore not acceptable unit cells. 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

---