Three-Dimensional Bravais-Lattice Types

From the book : Fundamentals of the Physics of Solids [Structure and Dynamics]


Symmetry-implied restrictions on the side lengths and angles of the Bravais cell are listed in Table 5.7. The name of the crystal system refers to the shape of the Bravais cell.



The seven crystal systems and the three types of centering could thus be expected to give rise to 21 types of centered lattices. In reality, only seven centered lattice types appear as centering does not always lead to new lattice types: in particular, in the triclinic system centering does not lead to a single new type. In other cases symmetries of the simple lattice are broken in the centered lattice. The new lattice types and the primitive ones are listed Table 5.8 for each crystal system.

The variety of notational conventions used for point groups exists for Bravais lattice types, too. In the Schoenflies notation the subscript of Γ   specifies the crystal system (t = triclinic, m = monoclinic, o = orthorhombic, q = tetragonal, rh =rhombohedral, h = hexagonal, c = cubic), and the superscript refers to the centering the Bravais cell (c = base-centered, v =body-centered, f = face-centered). Another convention uses a code of the form xY , where x is the international symbol for the crystal system (2nd column), and Y – one of the letters P, C (S), I,or F – specifies the centering type of the lattice.


The rhombohedral (trigonal) lattice is an exception: although it is a primitive lattice, its centering type is denoted by R.Thus hP stands for hexagonal, and hR for rhombohedral crystal system. The reader will understand in hindsight why the rhombohedral system appears as a nonprimitive type of the hexagonal system. A third notation uses a code of the form Yx, where Y , once again, refers to the centering type (Y = P, C, I, F, R), and x is the short international symbol for the point group of the Bravais lattice. E.g., for body-centered cubic lattices the notations Γv c ,cI,and Im¯ 3m are used equally. All three notations are given in parentheses at the listing of Bravais-lattice types.

The Hierarchy of Crystal Systems


The previous enumeration of crystal systems proceeded from the simplest point groups to those with more symmetry elements. By adopting the opposite approach, a certain natural hierarchy can be observed among crystal systems.

The most straightforward way to demonstrate this is to take a cubic (in two dimensions: square) or hexagonal crystal system, and to reduce the symmetry by appropriate deformations of the crystal.

Of all two-dimensional point lattices square and hexagonal ones possess the highest symmetries. Small deformations cannot take them into one another. A small deformation of the square lattice – stretching one of its sides – leads to a simple rectangular lattice, while stretching a square lattice along its diagonal leads to a centered rectangular lattice. The same lattice type is obtained when a hexagonal lattice is stretched or compressed along a mirror line or in a direction perpendicular to it. Square and hexagonal crystal systems are thus aid to be higher in the hierarchy than the rectangular one. Further (shearing) deformation of both types of rectangular lattices leads to oblique lattices. The hierarchy of two-dimensional crystal systems is summarized in Fig. 5.24.



In the three-dimensional case we shall start off with the most regular primitive cell, the cubic one. It can be deformed by pulling or pushing on two opposite faces: angles are left intact while a side is stretched or compressed. The result is a rectangular prism with a square base – an object with tetragonal symmetry.

By stretching the object in another direction, a general rectangular parallelepiped with orthorhombic symmetry is obtained. A further shearing deformation along one of the planes changes the inclination of the edge initially perpendicular to the shearing plane. The result is an object that possesses only monoclinic symmetry. Finally, another shearing in another plane leads to the most general triclinic parallelepiped. The objects obtained through the above sequence of deformations of a cube are shown in Fig. 5.25.

Trigonal and hexagonal crystal systems are not yet included in the above scheme. By stretching the cubic lattice along a space diagonal, a rhombohedral lattice is obtained. Another small deformation leads to a lattice that possesses the symmetries of the monoclinic crystal system. Small deformations of a cubic (regular) lattice cannot lead to a hexagonal lattice. On the other hand, deformations of a hexagonal lattice may lead to a lattice showing orthorhombic symmetry – more precisely, to a base-centered orthorhombic Bravais lattice.

Figure 5.26 gives a summary of this hierarchy of three-dimensional crystal systems.


Each time the primitive cell is deformed, a symmetry is broken. Therefore the symmetries of a crystal system ranked lower in the hierarchy form a subgroup of the symmetries of the system ranked higher. The same hierarchy could have been derived by starting with the cubic and hexagonal point groups (Oh and D6h), and choosing smaller and smaller subgroups. One must, however, exercise due care: although the symmetry group D3d of the rhombohedral
lattice is a subgroup of D6h, the symmetry group of the hexagonal lattice, the former should not be considered to lie under the latter in the hierarchy as no small deformation of the hexagonal lattice can lead to a rhombohedral one.

In the above presentation of the hierarchy of crystal systems we started off with a simple cubic lattice and arrived at lower-symmetry lattices through subsequent deformations. Analogously, one can start with a body- or face-centered cubic lattice and track down deformed lattices with lower symmetries. It is a straightforward matter to prove that whichever cubic Bravais lattice type is taken, a suitably chosen deformation will take it into one of the tetragonal lattice types. Further deformations will transform any tetragonal
lattice type into one of the orthorhombic lattices. The four orthorhombic lattice types are, in turn, deformed into either of the monoclinic lattice types.

Finally, a small deformation of either monoclinic lattice type results in a triclinic lattice. Similarly to the case of the simple cubic lattice, when a face or body-centered cubic lattice is stretched along the body diagonal, a rhombohedral lattice is obtained. As the hexagonal system has no centered type one can state that the hierarchy among crystal systems does not change when the various types within each system are taken into account. This hierarchy is of particular importance when the crystal classes are assigned to the crystal systems in the next section.