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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:
|
Cubic: |
a = b = c, |
α = β = γ = 90° |
|
Tetragonal: |
a = b ≠ c, |
α = β = γ = 90° |
|
Orthorhombic: |
a ≠ b ≠ c, |
α = β = γ = 90° |
|
Hexagonal: |
a = b ≠ c, |
α = β = 90°, γ = 120° |
|
Monoclinic: |
a ≠ b ≠ c, |
α =
γ = 90°, β ≠ 90° |
|
Triclinic: |
a ≠ b ≠ c, |
α ≠
β ≠
γ ≠
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:
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.
| |
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 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. |
| |
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
,
,
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. |
| |
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
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
a′r = 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.
References
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† |
|
1. |
Copper (Al)‡ |
Cu, Ag, Au, Al, Pt, Ni, Pb |
|
2. |
Iron (A2) |
Fe, Mo, W, Li, Na, K, Ba |
|
3. |
Magnesium (A3) |
Mg, Be, Gd, Rh, Zr; Zn§,
Cd§ |
|
4. |
Diamond (A4) |
C (diamond), Si, Ge, Sn (grey) |
|
5. |
Rock salt (sodium chloride) (B1) |
NaCl, and most alkali halides except Cs
compounds; |
|
|
|
MgO, CaO, SrO, FeO, CaS,
CaSe, AsSn, ThC |
|
6. |
Caesium chloride (B2) |
CsCl, CsBr, CsI, TlCl; many ordered
intermetallic |
|
|
|
compounds such as AgMg, AuMg,
BaCd |
|
7. |
Fluorite (calcium fluoride) (C1) |
CaF2, BaF2, UO2,
ThO2, K2O, UN2, AuAl2 |
|
8. |
Zinc blende (B3) |
ZnS, ZnSe, BeS, CdS, GaAs, BN |
|
9. |
Wurtzite (B4) |
ZnS, ZnSe, ZnO, BeO, CdS, GaN |
|
10. |
Rutile (C4) |
TiO2, SnO2, MnO2,
CrO2, MgF2, FeF2, MnF2 |
|
11. |
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 |
|
12. |
Calcite |
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. |
E.D.Atkins
|
|
