Crystal Structure

On the nanometer level, crystal structures are symmetric arrangements of molecules (bound atoms) in three-dimensional space [19]. Driven purely by energy minimization, countless manifestations of symmetry are found in nature ranging from the arrange­ment of atoms in unit cells and water molecules in snowflakes to the facets of crystals such as quartz and diamond [20]. For a crystal constructed of identical molecules, the positions of all of the molecules in the structure can be predicted using four basic symmetry elements: (1) centers of symmetry; (2) two, three, four, or sixfold rotational axes; (3) mirror or reflection planes; or (4) combinations of a symmetry centers and rotational axes [21]. Combined with the constraint that space must be filled by the
resulting structural units, the symmetry elements can be used to construct structures that make up the seven basic crystal systems (cubic, hexagonal, rhombahedral, tetragonal, orthorhombic, monoclinic, and triclinic). Within the crystal systems, increasingly finer divisions of symmetry can be defined using Bravais lattices, crystal classes, or space groups (Table 7) [22]. A detailed description of how the symmetry elements relate to this hierarchy can be found in many texts on crystallography [19, 23], X-ray diffraction [21, 22], or mineralogy [20, 24]. As an aside, the convention is to name crystal structures after the mineral for which the positions of the atoms were first confirmed [16]. Thus, compounds showing face-centered cubic symmetry and belonging to the Fm3m space group are referred to as the rock salt structure, since NaCl was the first mineral proven to have this structure.

For oxide compounds, the particular crystal structure that is formed is related to the composition, the relative sizes of the atoms, and the tendency toward ionic or covalent bonding [16]. The composition of a pure crystalline material or more precisely the stoichiometry of the compound limits the possible crystal structures [13]. For example, a compound with a cation to oxygen ratio of 2:3 like Al2O3 cannot crystallize into the same type of structure as a compound with a cation to oxygen ratio of 1:1 like MgO [16]. The cation to oxygen ratio is constrained by the requirement that electrical neutrality be maintained. The ratio of the sizes of the cation (rc) to the radius of the oxygen anion (ra) also affects the types of structures that can form. As the size of the cation increases relative to oxygen, more oxygen ions can be packed around the cation center [16]. The possible coordination numbers and critical rc/ra ratios are given in Table 8 along with the resulting structure types [1]. Finally, the bond character also affects the crystal structure. For highly covalent crystals, the hybridization of the

Table 7 Hierarchical organization of crystal structures

Crystal system

Possible Bravais lattices

Crystal classes or point groups

Number of

space groups

Cubic

P, I, Fa

5

36

Hexagonal

P

7

27

Trigonal

P

5

25

Tetragonal

P, I

7

68

Orthorhombic

P, C, I, F

3

59

Monoclinic

P, C

3

13

Triclinic

P

2

2

7

14

32

230

aP primitive; C end centered; I body centered; F face centered

Table 8 Critical cation environments

to anion radius

ratios for stability

various coordination

rc/ra

Coordination

number

Configuration

Example

0 > rc/ra > 0.155

2

Linear

CO2

0.155° > rc/ra > 0.225

3

Triangle

O in rutile

0.225 > rc/ra > 0.414

4

Tetrahedron

Wurtzite

0.414 > rc/ra > 0.732

6

Octahedron

Rock salt

0.732 > rc/ra > 1.0

8

Cubic

Fluorite

1.0 > rc/ra

12

Cuboctahedron

A site in Perovskite

valence shell orbitals is often the determining factor in crystal structure [16]. For example, SiC has a radius ratio of 1:6, but it crystallizes into the wurtzite structure (tetrahedral coordination) because of the strong covalent nature of the bonds [13]. A number of methods exist to predict structures including radius ratios [16], Pauling’s rules [25], and Mooser-Pearson plots [13].

A majority of the important oxide ceramics fall into a few particular structure types. One omission from this review is the structure of silicates, which can be found in many ceramics [1, 26] or mineralogy [19, 20] texts. Silicate structures are composed of silicon-oxygen tetrahedral that form a variety of chain and network type structures depending on whether the tetrahedra share corners, edges, or faces. For most nonsilicate ceramics, the crystal structures are variations of either the face-centered cubic (FCC) lattice or a hexagonal close-packed (HCP) lattice with different cation and anion occupancies of the available sites [25]. Common structure names, examples of compounds with those structures, site occupancies, and coordination numbers are summarized in Tables 9 and 10 for FCC and HCP-based structures [13, 25]. The FCC – based structures are rock salt, fluorite, anti-fluorite, perovskite, and spinel. The HCP – based structures are wurtzite, rutile, and corundum.