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Chapter: 3 Chemistry
    Section: 3.7 Properties of chemical bonds
        SubSection: 3.7.7 Crystal structures



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3.7.7 Crystal structures

A perfect infinite crystal possesses a lattice, an infinite set of points generated by three non-parallel vectors, such that each point is identical in itself and its surroundings.

With each lattice point may be associated a number of atoms. If their coordinates relative to the lattice point are given, together with the lengths and directions of the lattice vectors chosen to define the axes of reference, the complete structure is defined.

Position coordinates x, y, z are commonly expressed as fractions of the lattice parameters a, b, c; the parallelepiped defined by the lattice vectors a, b, c, is the unit cell.

Symmetry elements may be present imposing certain relations (i) between lattice parameters, (ii) between position coordinates of different atoms. Axes of reference are generally chosen in accordance with the symmetry. As a result of (i), and with a conventional choice of axes, crystals are classified into systems as follows:


a = b = c,

α = β = γ = 90°


a = bc,

α = β = γ = 90°



α = β = γ = 90°


a = bc,

α = β = 90°, γ = 120°



α = γ = 90°, β ≠ 90°



αβγ ≠ 90°

where α is the angle between b and c, and similarly for β and γ. Accidental equality of unit cell edges, and special values of interaxial angles not required by the symmetry, are disregarded in making this classification.

Atoms may be in general positions, or in special positions on point-symmetry elements. In the latter case, some or all of the position coordinates x, y, zare simple fractions; in the former, they are variable parameters whose values may change with temperature, pressure, or composition. It often happens, however, that atoms in general positions have, accidentally, parameters which are to a good approximation simple fractions.

If the translation vectors chosen as axes of reference generate all the points of the lattice, it is said to be primitive (P); otherwise it is centred. For centred lattices, if there is an atom at x, y, z, all translation-repeats of it within one unit cell can be derived by adding to x, y, z, components which are fractions of the lattice parameters. The kind of centring is summarized in the lattice centring operator; if, for example, this is written (0,0,0,0, , )+ it means that for every point at 0 + x, 0 + y, 0 + z, there is another identical point at 0 + x, + y, + z. The complete set of possibilities is as follows.

For any symmetry except hexagonal:

One-faced-centred lattice A [2 points],

(0,0,0;    0,, )+

Body-centred lattice I [2 points] (0,0,0;   , , ) +

All-face-centred lattice F [4 points]

(0,0,0;   0,, ;   , 0, ;   , , 0) +

      For hexagonal symmetry only:


Rhombohedrally centred lattice R [3 points],

(0,0,0;    , , ;    , , ) +

There are expressions similar to A for difference choices of axes, giving B- and C-face-centring. A- or B-face-centring is impossible in the tetragonal system; A-, B- or C- in the cubic. The same lattice could be named either P or C in the tetragonal and monoclinic systems, according to the choice of axes; similarly for I and F. A rhombohedrally centred lattice rotated through 180° from the conventional (obverse) orientation listed above is obtained by the operations (0,0,0;   , , ; , ,) +.

To describe a structure fully, we specify its system, its independent lattice parameters (thus choosing our axes of reference), its lattice type, and the position parameters of a set of atoms so chosen that (a) no two are separated by a lattice vector, (b) no atom not excluded by (a) is omitted. This description is complete and general, applicable to crystals of any system and any degree of complexity. Note that, if a letter or number specifying a position parameter is negative, it is conventional to write a bar over it, instead of a minus sign in front of it: thus , , stands for −x, −x, .

If the space group is known, the description can be abbreviated by listing position parameters for the asymmetric unit only, i.e. the set of atoms not related to each other by any symmetry element. This kind of abbreviation will not be used for the structures described below (see International Tables for X-ray Crystallography, Vol. 1, for details of its use).

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.



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 cH along the diagonal of the cube and aH along a face diagonal perpendicular to it; this unit cell is rhombohedrally centred and contains 3 Cu.  



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.



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).


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°.


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.



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.



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.



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.



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).



Rutile, TiO2
Tetragonal; primitive lattice P
a = 459.4 pm, c = 296.2 pm
2 Ti at 0, 0, 0 and , ,
4 O at ±(u, u, 0) and ± ( + u, u, ), with u = 0.31
      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.



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.



Calcite, CaCO3
Rhombohedrally centred hexagonal lattice, R
aH = 499.0 pm, cH = 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 ar = 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 ar = 404 pm, α′ = 76.17°.

Some interesting structures

An example of a solid state superconducting material is YBa2Cu3O7 which becomes superconducting below about 90 K. It can be made by slow cooling in an oxygen-rich atmosphere. The structure can be related to the cubic ideal perovskite (example 11). Imagine three of the cubic perovskite SrTiO3 unit cells stacked on top of each other. The stack of three centred Sr atoms are replaced with Ba, Y and Ba atoms in that order. Two oxygen atom sites are vacant and the unit cell sides perpendicular to the stacking change owing to the symmetry being broken and the unit cell becomes orthorhombic; primitive.

There are many intermetallic compounds whose formulae can be written as CeM2Si2 with M = Au, Ag, Cr, Cu,..., and with Si replaced by Ge. These compounds often exhibit antiferromagnetic ordering and, for example, the Ce atoms at the corners of the body centred lattice have spin up while the Ce atoms at the centre of the unit cell have spin down. Thus magnetic ordering means that the magnetic space group can be different from that of the atomic structure.


For a full account of crystallographic symmetry and notation, see International Tables for X-ray Crystallography,
  vol. I (ed. N. F. M. Henry and K. Lonsdale), Kynoch Press, Birmingham, England, for the International
  Union of Crystallography.
For a classified reference book of crystal structures, with lattice parameters and position parameters, see
  R. W. G. Wyckoff, Crystal Structures, 2nd edn, Interscience, New York.
For a description and discussion of numerous important structures and their symmetry, see H. D. Megaw,
  Crystal Structures—A Working Approach, Saunders, Philadelphia, Pa.
For a description and discussion of high temperature superconducting structures and their symmetry, see
  G. Burns and A. M. Glazer, Space Groups for Solid State Scientists, 2nd edn, Academic Press, London.

Table of materials

Name of structure type

Some representative examples


Copper (Al)

  Cu, Ag, Au, Al, Pt, Ni, Pb


Iron (A2)

  Fe, Mo, W, Li, Na, K, Ba


Magnesium (A3)

  Mg, Be, Gd, Rh, Zr; Zn§, Cd§


Diamond (A4)

  C (diamond), Si, Ge, Sn (grey)


Rock salt (sodium chloride) (B1)

  NaCl, and most alkali halides except Cs compounds;



       MgO, CaO, SrO, FeO, CaS, CaSe, AsSn, ThC


Caesium chloride (B2)

  CsCl, CsBr, CsI, TlCl; many ordered intermetallic



       compounds such as AgMg, AuMg, BaCd


Fluorite (calcium fluoride) (C1)

  CaF2, BaF2, UO2, ThO2, K2O, UN2, AuAl2


Zinc blende (B3)

  ZnS, ZnSe, BeS, CdS, GaAs, BN


Wurtzite (B4)

  ZnS, ZnSe, ZnO, BeO, CdS, GaN


Rutile (C4)

  TiO2, SnO2, MnO2, CrO2, MgF2, FeF2, MnF2


Ideal perovskite

  SrTiO3, KTaO3; BaZrO3, BaSnO3, DyMnO3, KMgF3,



      KFeF3; also high-temperature forms of BaTiO3,



      KNbO3, NaNbO3, LaAlO3; also, by omission of the 



      12-coordinated cation, ReO3, RhF3, MoF3



  CaCO3, MgCO3, FeCO3, NaNO3

            Unless otherwise specified, the structures listed are those found at room temperature.
            The type names (A1), (A2), etc. are those given to the structures in Vol. 1 of Strukturbericht (1920) and are still sometimes found in the literature.
           § These elements have axial ratios c/a much greater than the ideal value characteristic of close packing.



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