Crystal Symmetry and the Unit Cell

Consider a cube as an example. Suppose the center of the cube is placed at the origin of its coordinate system and the symmetry operations that transform it into identity with itself are counted (Fig. 21.3). The x, y, and z coordinate axes are 4-fold axes of rotational symmetry, denoted by C4, because a cube that is rotated through a multiple of 90° (= 360°/4) about any one of these axes is indistinguishable from the original cube. Similarly, a cube has four 3-fold axes of rotational 

FIGURE 21.2 Three types of symmetry operation rotation about an n-fold axis, reflection in a plane, and inversion through a point. 

FIGURE 21.3 Symmetry operations acting on a cube. The various rotations of the cube are suggested by the curved arrows around the rotational axes. 

FIGURE 21.4 An ammonia molecule has a 3-fold axis of rotation and three mirror planes of symmetry. Each of the N —H bonds lies in a mirror plane. 

If the ammonia molecule is drawn as a pyramid with the nitrogen atom at the top (Fig. 21.4), then the only axis of rotational symmetry is a 3-fold axis passing downward through the N atom. Three mirror planes intersect at this 3-fold axis. 

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The largest protonated cluster of water molecules yet definitively characterized is the discrete unit lHi306l formed serendipitously when the cage compound [(CyHin)3(NH)2Cll Cl was crystallized from a 10% aqueous hydrochloric acid solution. The structure of the cage cation is shown in Fig. 14.14 and the unit cell contains 4 [C9H,8)3(NH)2aiCUHnOfiiai- The hydrated proton features a short. symmetrical O-H-0 bond at the centre of symmetry und 4 longer unsymmetrical O-H - 0 bonds to 4... 

In crystals of any material, the atoms present are always arranged in exactly the same way, over the whole extent of the solid, and exhibit long-range translational order. A crystal is conventionally described by its crystal structure, which comprises the unit cell, the symmetry of the unit cell, and a list of the positions of the atoms that lie in the unit cell. 

The minimum amount of information needed to specify a crystal structure is the unit cell type, that is, cubic, tetragonal, and so on, the unit cell parameters, and the positions of all of the atoms in the unit cell. The atomic contents of the unit cell are a simple multiple, Z, of the composition of the material. The value of Z is equal to the number of formula units of the solid in the unit cell. Atom positions are expressed in terms of three coordinates, x, y, and z. These are taken as fractions of a, b, and c, the unit cell sides, say and j. The x, y, and z coordinates are plotted with respect to the unit cell axes, not to a Cartesian set of axes. The space group describes the symmetry of the unit cell, and although it is not mandatory when specifying a structure, its use considerably shortens the list of atomic positions that must be specified in order to built the structure. 

The preparation of single crystals is difficult, but is successful in some case, so that we are well informed about the structures (1,6,17,21,22,23). The structures of the plastic phases are related to the well-known intermetallic phase Li3Bi, where the centres of the polycyclic P7 or Pn anions surround the positions of the Bi atoms in LisBi. The orientation of the polyanions is disordered (dynamically ). For these structures this orientation leads to a typical electron density distribution of a seemingly octahedral unit. In contrast the orientation of the anions is fixed for the crystalline phases. The symmetry of the unit cells as well as the distribution of cations and anions in these M3P7 and M3P11 type structures reflect the direct relationship to the structures of the plastic phases. 

Determine unit cell dimensions and symmetry information (the unit cell is the basic building block of the crystal). 

The helical hand of the individual stems in chiral but racemic polymers is a very severe and therefore critical criterion in the crystallization process. The constraints apply for each stem and are dictated by the symmetry of the unit-cell. Contrary to the stem length (which, if incorrect, can be healed or adjusted by later structural reorganization), helix chirality involves a flip of a coin type of decision - right- or left-handedness. 

Spectra of molecules in the crystalline state, i.e., of molecular crystals, are obtained from molecules which are at fixed positions (sites) in the lattice (Fig. 2.6-1C). Normal (first-order) infrared and Raman spectra can be seen as spectra of hyper molecules , the unit cells (Schneider 1974, Schneider et al., 197.5). As a consequence, any molecular vibration is split into as many components as there are molecules present in the unit cell. Their infrared and Raman activity is determined by the symmetry of the unit cell. In addition, the translational and rotational degrees of freedom of molecules at their sites are frozen to give rise to lattice vibrations translational vibrations of the molecules at their sites and rotational vibrations about their main inertial axes, so-called librations. 

Hartree-Fock calculations on molecules commonly exploit the symmetry of the molecular point group to simplify calculations such studies on perfectly ordered bulk crystalline solids are possible if one exploits the translational symmetry of the crystalline lattice (see Ashcroft and Mermin, 1976) as well as the local symmetry of the unit cell. From orbitals centered on various nuclei within the unit cell of the crystal Bloch orbitals are generated, as given by the formula (in one dimension) ... 

Systematically absent Bragg reflections Bragg reflections that have no intensity, because of translational components of any symmetry in the unit-cell contents and which have h, k, and I values that are systematic in terms of oddness or evenness. These absences depend only upon symmetry in the atomic arrangement in the crystal, and they can be used to derive the space group. For example, all reflections for which h + k is odd may be absent, showing that the unit cell is C-face centered. 

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