Lattices and unit cells

A crystal is built up from regularly repeating ‘structural motifs’, which may be atoms, molecules, or groups of atoms, molecules, or ions. A space lattice is the pattern formed by points representing the locations of these motifs (Fig. 20.1). The space lattice is, in effect, an abstract scaffolding for the crystal structure. More formally, a space lattice is a three-dimensional, infinite array of points, each of which is sur rounded in an identical way by its neighbours, and which defines the basic structure of the crystal. In some cases there may be a structural motif centred on each lattice point, but that is not necessary. The crystal structure itself is obtained by associating with each lattice point an identical structural motif. The unit cell is an imaginary parallelepiped (parallel-sided figure) that contains one unit of the translationally repeating pattern (Fig. 20.2). A unit cell can be thought of as the fundamental region from which the entire crystal may be constructed by purely translational displacements (like bricks in a wall). A unit cell is commonly formed by joining neighbouring lattice points by straight lines (Fig. 20.3). Such unit cells are called primitive. It is sometimes more convenient to draw larger non primitive unit cells that also have lattice points at their centres or on pairs of opposite faces. An infinite number of different unit cells can describe the same lattice, but the one with sides that have the shortest lengths and that are most nearly perpendicular to one another is normally chosen. The lengths of the sides of a unit cell are denoted a, b, and c, and the angles between them are denoted α, β, and γ (Fig. 20.4). Unit cells are classified into seven crystal systems by noting the rotational symmetry elements they possess. A cubic unit cell, for example, has four threefold axes in a tetrahedral array (Fig. 20.5). A monoclinic unit cell has one twofold axis; the unique axis is by convention the b axis (Fig. 20.6). A triclinic unit cell has no rotational sym metry, and typically all three sides and angles are different (Fig. 20.7). Table 20.1 lists the essential symmetries, the elements that must be present for the unit cell to belong to a particular crystal system. There are only 14 distinct space lattices in three dimensions. These Bravais lattices are illustrated in Fig. 20.8. It is conventional to portray these lattices by primitive unit cells in some cases and by non-primitive unit cells in others. A primitive unit cell (with lattice points only at the corners) is denoted P. A body-centred unit cell (I) also has a lattice point at its centre. A face-centred unit cell (F) has lattice points at its corners and also at the centres of its six faces. A side-centred unit cell (A, B, or C) has lattice points at its corners and at the centres of two opposite faces. For simple structures, it is often convenient to choose an atom belonging to the structural motif, or the centre of a molecule, as the location of a lattice point or the vertex of a unit cell, but that is not a necessary requirement.

Fig. 20.1 Each lattice point specifies the location of a structural motif (for example, a molecule or a group of molecules). The crystal lattice is the array of lattice points; the crystal structure is the collection of structural motifs arranged according to the lattice.

Fig. 20.2 A unit cell is a parallel-sided (but not necessarily rectangular) figure from which the entire crystal structure can be constructed by using only translations (not reflections, rotations, or inversions).

Fig. 20.3 A unit cell can be chosen in a variety of ways, as shown here. It is conventional to choose the cell that represents the full symmetry of the lattice. In this rectangular lattice, the rectangular unit cell would normally be adopted.

Fig. 20.4 The notation for the sides and angles of a unit cell. Note that the angle α lies in the plane (b,c) and perpendicular to the axis a.

Fig. 20.5 A unit cell belonging to the cubic system has four threefold axes, denoted C3, arranged tetrahedrally. The insert shows the threefold symmetry.

Fig. 20.6 A unit belonging to the monoclinic system has a twofold axis (denoted C2 and shown in more detail in the insert).

Fig. 20.7 A triclinic unit cell has no axes of rotational symmetry.

Fig. 20.8 The fourteen Bravais lattices. The points are lattice points, and are not necessarily occupied by atoms. P denotes a primitive unit cell (R is used for a trigonal lattice), I a body centred unit cell, F a face-centred unit cell, and C (or A or B) a cell with lattice points on two opposite faces.

Fig. 20.9 Some of the planes that can be drawn through the points of a rectangular space lattice and their corresponding Miller indices (hkl): (a) (110), (b) (230), (c) (⁄10), and (d) (010).

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مواضيع ذات صلة

Surface pressure

Membrane formation

Micelle formation

The electrical double layer

Classification and preparation

The stability of proteins and nucleic acids

The structure of nucleic acids

Helices and sheets

Conformational energy

The Corey–Pauling rules

The different levels of structure

Viscosity