Physical Chemistry , 1st ed.

Now we consider systems where the angles between a,b, and care not all

90°. For this set we start with the least restricted case. If all distance and angle

parameters are different, the crystal is triclinic.If two of the angles are 90° and

one not, then the crystal is considered monoclinic.If all three unit cell dimen-

sions are the same (abc) and the angles are equal but not 90°, then the

crystal is described as trigonalor rhombohedral.And last, some crystals have a

sixfold symmetry. Such crystals are called hexagonal.

Finally, within some of the systems are subsystems that have additional

atoms or ions or molecules. These subsystems differ only by having corner

species in other parts of the unit cell, like the sides or the center of the unit cell.

Table 21.2 lists these details, and Figure 21.9 shows examples of the additional

possible cubic, tetragonal, orthorhombic, and monoclinic crystal structures.

These 14 possible crystal structures are called the Bravais lattices,after the

French scientist Auguste Bravais (1811–1863), who first described them in 1848.

The cubic crystal system, for example, is separated into three Bravais lattices

depending on whether the unit cell has species only at the corners (simpleor

primitive cubic); at the corners and the center of the unit cell (body-centered

cubic); or at the faces of the unit cell (face-centered cubic). Note that for the

body-centered cubic, the species (atom, ion, or molecule) in the center con-

tributes one full member to the stoichiometry of the cell, and the atoms, ions,

or molecules in the faces of the unit cell contribute ^12 of a member each. (Recall

that species at the corners contribute ^18 of a member each.) For face-centered

cubic unit cells, the facial species contribute, overall,^12  6 3 members to

the stoichiometry of the unit cell.

The tetragonal system can also have species in the center of the unit cell.

Orthorhombic crystals can have species in the center (body-centered), in all

faces (face-centered), only in opposing faces (end-centered), or only at the cor-

ners (primitive). Monoclinic unit cells can be either primitive or end-centered,

with species in one set of opposing faces. Triclinic, hexagonal, and trigonal unit

cells are just primitive.

Table 21.2 also lists another standard nomenclature for referring to the

types of Bravais lattice. Generally, the letter P is used to indicate a primitive

lattice, I is used for body-centered lattices, F for face-centered lattices, and C

for end-centered lattices.

21.3 Crystals and Unit Cells 735

a  b  c







a

b

c

Figure 21.8 Definition of unit cell parame-
ters. The lengths of the unit cell are labeled a,b,
and c, such that abc. The angles between
the unit cell dimensions are , , and ; they are
defined as shown.

Table 21.2 The fourteen Bravais lattices

Crystal system Bravais lattice Symbol(s)

Cubic Simple (or primitive) cubic sc, P

Body-centered cubic bcc, I

Face-centered cubic fcc, F

Tetragonal Simple (or primitive) tetragonal P

Face-centered tetragonal F

Orthorhombic Simple (or primitive) orthorhombic P

Body-centered orthorhombic I

Face-centered orthorhombic F

End- (or base-) centered orthorhombic C

Trigonala Simple (or primitive) trigonal P

Hexagonal Simple (or primitive) hexagonal P

Monoclinic Simple (or primitive) monoclinic P

End- (or base-) centered monoclinic C

Triclinic Simple (or primitive) triclinic P

aTrigonal is also known as rhombohedral in some texts.