The 14 3D Bravais lattices
The 14 3D Bravais lattices
Abstract and Keywords
This chapter constructs all the possible 3D translation sets compatible with the previously introduced 3D point groups, leading to the wellknown fourteen Bravais lattices. For each crystal system, the compatible lattices (both primitive and centred) are defined, together with the corresponding holohedry (lattice symmetry). The more complex centred lattices, such as the Rcentred trigonal lattice, receive particular attention. A reference is made to the different unit cell conventions (with a particular note about the monoclinic system), although this is too intricate a subject for such an introductory level textbook.
Keywords: holohedry, Bravais lattices, crystal system
9.1 Introduction
Since the reader of this book will most likely be familiar with the 14 Bravais lattices, the emphasis of this chapter will be on showing how these lattices are constructed and why no more (and no less) are needed. The procedure followed to derive the 14 Bravais lattices in 3D is closely related to that used in the 2D case. In fact, we can exploit the symmetry of the projections along symmetry directions, which, as we will see, must be one of the known 2D lattices.
In all but the trivial cases of the triclinic groups 1 and 1̄, where the three shortest independent translations are allowed to be at any angle with each other, there is at least one proper or improper axis of order 2 or higher. Following the procedure shown in Figure 3.4 , one can always construct one translation along and two translations perpendicular to the axis direction, neither necessarily minimal, and in general not orthogonal to each other. Pairs of such translations define a 2D lattice, which must be one of the five lattices defined above. The various cases are obtained by considering all the possible directions of the minimal translation vectors.
9.2 Construction of the 14 Bravais lattices
9.2.2 Monoclinic system
9.2.2.1 Classes 2, m and 2/m, holohedry 2/m, lattices P and C
The holohedry is easily derived by completing each of the classes with the inversion, which is necessarily a symmetry of the lattice. The lattice orthogonal to the two‐fold axis (the “unique” axis) is oblique, whereas the other two are rectangular, and can be either p or c. This is shown in Figure 9.1 (p.84)
There are two possibilities:

• The shortest translation not orthogonal to the unique axis is also parallel to it. This means that all the rectangular lattices containing the two‐fold axis are p. The translation along the two‐fold axis is chosen as the b basis vector (or c in the alternate setting). In this case, one can choose as a primitive (and also conventional) cell a prism with parallelogram basis. The edges of the latter are chosen as a and c (or a and c) basis vectors. The lattice is monoclinic primitive (symbol P).

• The shortest translation not orthogonal to the unique axis is in an arbitrary direction. Twice its projection on the two‐fold axis must however be a translation vector, and is chosen as the conventional b basis vector. Twice the projection on the plane perpendicular to the unique axis (the “basal plane”) is also a translation vector; the standard setting identifies this direction with the a‐axis. The a and b basis vectors define a c rectangular lattice containing the unique axis (the centering vector is the shortest translation not orthogonal to the unique axis). The c‐axis is chosen as a vector in the basal plane that is linearly independent of a. With these conventions, the lattice is monoclinic C‐centered (symbol C), because only the 2D lattice not containing the c‐axis is C‐centered. However, a plethora of other settings is possible, most of them duly reported in the ITC, making the monoclinic system the most intricate as far as conventions are concerned.
Problem 9.1: show that any monolinic lattice can be reduced to one of the two types C and P ‐ see Figure 9.2 .
9.2.3 Orthorhombic system
9.2.3.1 Classes 222, mm2 and mmm, holohedry mmm, lattices P, C, F and I.
The holohedry is easily derived again by completing each of the classes with the inversion. Once again, it is possible to find three translations (p.85)

• All of these 2D lattices are p. The primitive unit cell is a cuboid, and the lattice is “P‐orthorhombic”.

• One of the lattices orthogonal to a two‐fold axis is c. The resulting 3D lattice is A, B or C depending on the orthogonal direction, clearly a matter of convention. The C setting is the standard one, but A also occurs as a standard setting in the mm2 class.

• All three lattices orthogonal to two‐fold axes are c (it can be shown that if two of them are c, the third must also be c). The lattice is F, indicating that all faces are centered.

• One of the lattices bisecting two two‐fold axes is c. By using fractional coordinates, it can be shown that none of the other lattices can be c. In fact, if $\left[\frac{1}{2}\frac{1}{2}\frac{1}{2}\right]$ is a lattice point, then $\left[\frac{1}{2}\frac{1}{2}0\right]$ and permutations cannot be lattice points, otherwise (by difference) $\left[00\frac{1}{2}\right]$ would be a lattice point, contradicting the fact that [0 0 1] is the shortest translation
(p.86)along the z‐axis. The lattice is body‐centered orthorhombic (symbol I).
9.2.4 Tetragonal system
9.2.4.1 Classes 4, 4̄, 422, 4/m, 4mm, 4̄m2, 4̄mmm, holohedry 4/mmm, lattices P and I.
The basal‐plane lattice is square, and can only be p. The lattices orthogonal to the shortest in‐plane translations are rectangular, but can only be p. In fact, if $\left[\frac{1}{2}0\frac{1}{2}\right]$ existed, so would $\left[0\frac{1}{2}\frac{1}{2}\right]$ (by symmetry) and $\left[\frac{1}{2}\frac{1}{2}0\right]$ (by difference), contradicting the fact that [1 0 0] is the shortest in‐plane translation. This leaves only two possibilities:

• All of the 2D lattices are p. The primitive unit cell is a square‐based cuboid, and the lattice is P‐tetragonal.

• The rectangular lattices bisecting the in‐plane directions are c. The lattice is body‐centered tetragonal (symbol I).
Problem 9.2: show thatatetragonalF lattice can be reduced toI by an appropriate change of the unit cell.
9.2.5 Trigonal system
9.2.5.1 Classes 3, 3m1, 321, 3̄m1, lattices P and R.
This system is peculiar, in that each class can be supported by two lattices, P and R, with different holohedries.

• The P lattice is simply the 3D extension of the 2D hexagonal lattice by a translation along the z‐axis, and has holohedry 6/mmm. Here, the unit cell is a hexagonal prism.

• The R lattice is generated from an arbitrary translation by applying the point group 3̄m1, the equivalent points forming a trigonal antiprism. Differences between translations with the same z‐components are in‐plane translations, forming a hexagonal lattice (Figure 9.5 ). Sums of three symmetry‐equivalent translations with the same z‐component are translations along the z‐axis, so the z‐component of any translation must be an integral multiple of $\frac{1}{3}$ of a lattice vector. The primitive unit cell is a rhombohedron, i.e., a cube “stretched” along one of the body diagonals. The larger (three times the volume) hexagonal cell, formed by the in‐plane translations and by the shortest z‐axis translation, is used as a “conventional” cell in the “hexagonal” setting. Both rhombohedral and hexagonal settings are used and are listed in the ITC.
(p.87) 9.2.6 Hexagonal system
9.2.6.1 Classes 6, 6̄, 6̄m, 622, 6̄m2, 6mm, 6̄mmm, holohedry 6̄mmm, lattice P.
It is easy to see by subtraction of equivalent translations that the projections of a generic translation in the basal plane and perpendicular to it are also translations. The only allowed lattice is therefore P, and the unit cell is a hexagonal prism.
9.2.7 Cubic system
9.2.7.1 Classes 23, m3̄, 432, 4̄3m and 43̄m, holohedry 43̄m, lattices P, I and F.
The three lattices are generated by the shortest translations and its equivalents, which, by restriction, must be along one of the three inequivalent symmetry directions. In all cases, the cubic cell is used as the conventional cell, and the edges of the primitive cell are obtained by rotating the generating translation around a three‐fold axis (Figure 9.6 ).

• The shortest translation is along the primary direction (four‐fold axis). The lattice is primitive cubic (P), and the primitive unit cell is a cube.

• The shortest translation is along the secondary direction (threefold axis), and connects a corner with the center of the conventional cubic cell. The primitive cell is a rhombohedron with angles between edges $\alpha =109.47\xb0\left(\mathrm{cos}\alpha =\frac{1}{3}\right)$. The lattice is I − “body‐ centered cubic”.

• The shortest translation is along the tertiary direction (two‐fold axis along the face diagonals). The primitive cell is a rhombohedron with angles between edges α = 60°. The lattice is F – “face‐centered cubic”.
(p.88)