Some simple structures

The structures described below are particularly simple and important types. Most of them are cubic, and most of them have all their atoms in special positions. The structure type is commonly named after the particular material described, except where otherwise specified. Lattice parameters and variable position parameters (if any) apply to this material. Other isostructural (isomorphous) materials (of which examples are given in the table) have slightly different numerical values for these parameters. A short qualitative comment is added to the formal description, which, however, is sufficient in itself to allow scale diagrams to be drawn and interatomic distances calculated.

Note that atoms are not separately listed if their coordinates can be derived from those listed by the use of lattice centring operators, as explained above. As a check, the total number of atoms of each kind in the unit is indicated.

1. Copper (‘Monatomic face-centred cubic’)

Cubic; all-face-centred lattice F
a = 361.5 pm
4 Cu at 0,0,0

       Each atom has 12 equidistant neighbours, in directions parallel to the face diagonal of the cube, at distances a/√2. The structure is a cubic close packing of equal spheres.
       An alternative description uses hexagonal axes of reference, with c_H along the diagonal of the cube and a_H along a face diagonal perpendicular to it; this unit cell is rhombohedrally centred and contains 3 Cu.   

2. Iron (‘Monatomic body-centred cubic’)

Cubic; body-centred lattice I
a = 286.6 pm
2 Fe at 0,0,0

      Each atom has 8 equidistant neighbours in directions parallel to the cube body diagonal, at distances √(3)a/2.

3. Magnesium (‘Hexagonal close-packed’)

Hexagonal; primitive lattice P

a = 320.9 pm, c = 521.0 pm
2Mg at ±(,,)
With an alternative choice of origin, Mg atoms are at 0,0,0 and ,,.

      Each atom has 6 equidistant neighbours in its own plane at a distance a, and two sets of 3 above and below. If c/a has the ideal value of 1.633, the whole array is in hexagonal close packing. The unit cell contains two close-packed layers, not related by a lattice vector (in contrast to cubic close packing, with three layers related by lattice vectors).

4. Diamond

Cubic; all-face-centred lattice F
 a = 356 pm
8 C at ±(,,)
With an alternative choice of origin, C atoms are at 0,0,0 and ,,.

      Each atom has 4 equidistant neighbours in directions parallel to the body diagonals of the cube, at distances √(3)a/4; but adjacent atoms have all their nearest-neighbour bonds oppositely directed. The structure could be derived by taking 8 body-centred cells, of side a/2, and leaving out half the atoms systematically. The C—C bonds are at the tetrahedral angle of 109.5°.

5. Rock salt, NaCl

Cubic; all face-centred lattice F
a = 563 pm
4 Na at 0, 0, 0
4 Cl at , 0, 0

With an alternative choice of origin, the position coordinates of Na and Cl can be interchanged.
        Each atom has 6 neighbours of the opposite kind, in directions parallel to the cube edges, at a distance a/2.

6. Caesium Chloride, CsCl

Cubic; primitive lattice P
 a = 411 pm
1 Cs at 0,0,0
1 Cl at , , 

With an alternative choice of origin, the position coordinates of Cs and Cl can be interchanged.
        Each atom has 8 neighbours of the opposite kind, in directions parallel to the cube body diagonal, at a distance of √(3)a/2.
        The structure is closely related to that of iron, to which it would reduce if Cs and Cl were replaced by indistinguishable atoms.

7. Fluorite, CaF₂

Cubic; all-face-centred lattice F
 a = 545 pm
4 Ca at 0, 0, 0
8 F at ±(, , )

       Each Ca atom has 8 equidistant F neighbours, at the corners of a cube; each F atom has 4 equidistant Ca neighbours, at the vertices of a regular tetrahedron. The Ca–F distance is √(3)a/4.

8. Zinc blende, ZnS

Cubic; all-face-centred lattice F
a = 542 pm
4 Zn at 0, 0, 0
4 S at , , 

        Each Zn atom is surrounded by 4 equidistant S atoms, at the corners of a regular tetrahedron, at a distance √(3)a/4; similarly, each S atom is surrounded tetrahedrally by 4 Zn atoms. All the Zn–S bonds lying parallel to a given body diagonal of the cube point in the same direction.
        The structure is closely related to that of diamond, to which it would reduce if Zn and S were replaced by indistinguishable atoms.

        Alternatively, the structure may be described using hexagonal axes of reference, chosen as for copper (q.v.); the unit cell is rhombohedrally centred, and contains 3 atoms of each kind, with Zn at 0, 0, 0 and S at 0, 0,  (or by reversing the sense of the c-axis and changing the origin these coordinates may be interchanged).
        Unlike any of the structures previously described, this has no centre of symmetry.

9.

Wurtzite, ZnS Hexagonal; primitive lattice P  a = 382 pm, c = 626 pm 2 Zn at , , 0; , ,  2 S at , , u; , ,  + u; with u         As in the zinc-blende structure, each Zn atom is surrounded by 4 S atoms at the corners of a tetrahedron, and each S similarly by 4 Zn atoms. The tetrahedra are, however, only regular if c/a and u have the ideal values of 1.633 and 0.375 respectively. The Zn—S bonds parallel to the c-axis all point in the same direction ; in contrast to the zinc-blende structure, this is a unique axis, and the symmetry is therefore polar.

The relation between the structures of wurtzite and zinc blende is the same as that between the structures of magnesium and copper, i.e. between the operations of hexagonal close packing and cubic close packing.      Intermediates between the wurtzite and zinc-blende structure, sometimes of considerable complexity (‘polytypes’), are found in a number of materials, including ZnS itself and SiC (carborundum).

10. Rutile, TiO₂

Tetragonal; primitive lattice P
a = 459.4 pm, c = 296.2 pm
2 Ti at 0, 0, 0 and , , 

  O at ±(u, u, 0) and ± ( + u,  − u, ), with u = 0.314

      Each Ti atom has as neighbours 6 O atoms, forming a nearly regular octahedron. Edges of the octahedra parallel to face-diagonals of the square base are shared with similar octahedra, forming chains parallel to the c-axis; octahedra in neighbouring chains share corners, making each O atom neighbour to 3 Ti atoms

11. Ideal perovskite: example, SrTiO₃

(‘Perovskite’ is the mineral name of CaTiO₃, whose actual structure, though it approximates to that of     SrTiO₃, is more complicated and of lower symmetry.)

Cubic; primitive lattice P
a = 390.5 pm
1 Ti at 0, 0, 0
1 Sr at , , 
3 O at , 0,0;   0, , 0;   0, 0, 

      Each Ti atom has 6 neighbouring O atoms, at a distance a/2, forming a regular octahedron. Each octahedron shares each corner with one similar octahedron to form a three-dimensional framework, with cavities holding the larger Sr atoms. Each Sr atom has 12 neighbouring O atoms at a distance √(2)a/2. Each O atom is linked to 2 Ti atoms and 4 Sr atoms.

        The ideal perovskite structure imposes a particular relation between ionic radii as a condition of cation–anion contact for both cations, and only occurs when this is nearly satisfied, as in SrTiO₃. With moderate misfit, related structures of lower symmetry are found, collectively described as ‘structures of the perovskite family’. Many materials possessing such structures at room temperature have a high-temperature form with the ideal perovskite structure.

12. Calcite, CaCO₃

Rhombohedrally centred hexagonal lattice, R
a_H = 499.0 pm, c_H = 1700 pm
6 Ca at 0, 0, 0 and 0, 0, 
6 C at 0, 0,  and 0, 0, 
18 O at x, 0, ; 0, x, ; , , ; , 0, ; 0, , ; x, x, ; with x = 0.257

         Each Ca atom has 6 equidistant O atoms at the vertices of an octahedron. Each O atom is shared between two octahedra, and also forms one corner of an equilateral triangle, perpendicular to the c-axis, with the carbon atom at its centre. The edges of the CO₃ triangle are shorter than any of the octahedron edges. CO₃ groups in successive layers perpendicular to the c-axis are rotated through 180°.

         An alternative description uses the edges of the primitive rhombohedron as axes of reference; then a_r = 636 pm, α = 46.08°.
        If the CO₃ groups were replaced by cylindrical discs, the unit cell would be half the height; the primitive rhombohedron would have a′_r = 404 pm, α′ = 76.17°.