Crystal Structures
A crystal is a solid whose atoms are arranged in a periodic pattern in three dimensions. A
point
lattice
can be constructed by taking three non-coplanar sets of parallel, equally spaced planes,
whose intersections form a set of lines which further intersect to form a set of points. When this
is done such that, for a given crystal system, each of the points has identical physical
surroundings, a point lattice is formed. A lattice point does not, however, need to correspond to a
particular atom location, and many systems are constructed with lattices in which each lattice
point is associated with two or more atoms. The lattice construction also gives a set of identical
unit cells
, which are parallelepipeds. Any crystal may be described with a number of different
point lattices, but the one commonly used generally gives the smallest and most symmetrical unit
cell.
Both the unit cell and the point lattice may be constructed by successive application of a set of
three
lattice vectors
(or ‘basis’ vectors)
a
,
b
and
c
, which define the edges of the unit cell when
drawn from a common origin. These vectors may be considered as unit vectors in a
crystallographic coordinate system (which need not be orthogonal). A point anywhere in the
lattice may then be specified in terms of the unit vectors as the
crystal coordinates
, (h
a
, k
b
, l
c
).
The magnitudes of
a
,
b
, and
c
, together with the angles between the vectors,
α
,
β
and
γ
are called
lattice parameters
(or lattice constants). 7 crystal systems and 14 Bravais lattices
Cubic systems have
a
=
b
=
c
and
α
=
β
=
γ
= 90
ο
.
(Simple cubic; body-centered; face-centered)
Tetragonal systems have
a
=
b
≠
c
and
α
=
β
=
γ
= 90
ο
.
(Simple; body-centered)
Orthorhombic systems have
a
≠
b
≠
c
and
α
=
β
=
γ
= 90
ο
.
(Simple; base-centered; body-centered; face-centered)
Rhombohedral (or ‘trigonal’) systems have
a
=
b
=
c
and
α
=
β
=
γ
≠
90
ο
.
Hexagonal systems have
a
=
b
≠
c
and
α
=
β
= 90
ο
and
γ
= 60
ο
.
Monoclinic systems have
a
≠
b
≠
c
and
α
=
β
= 90
ο
≠
γ
.
(Simple, base-centered)
Triclinic systems have
a
≠
b
≠
c
and
α
≠
β
≠
γ
≠
90
o
.

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