SUMMARY
Lattices and lattice directions:
1. Lattice points: Point in a crystal with specific arrangement of atoms,
reproduced many times in a macroscopic crystal. The choice of the
lattice point within the unit cell is arbitrary.
2. Crystal basis: Arrangement of atoms within the unit cell.
3. Lattice vectors connect two lattice points. Primitive lattice vectors are
the shortest lattice vectors possible. Three of them span the lattice
space. All other lattice vectors can be expressed as a set of three
indices that tell the indices of the vector sum that reproduces a
particular lattice vector. The indices are integral for lattice vectors and
are non-integral for points within the unit cell. Square brackets are
used for a particular direction and carets are used for all equivalent
directions. [x y z] and <x y z>. Negative indices are given as
.
4. Unit cells are the space limited by a parallelepiped with edges that are
three, non-coplanar lattice vectors. The primitive unit cell contains only
one lattice point. The preferred primitive unit cell contains one lattice
point and has the shortest lattice vectors that are nearly equal. The
lattice vectors are chosen to have an obtuse angle between them, if
possible. The preferred setting often, but not always, has the symmetry
of greatest rank parallel with the c axis. (The monoclinic and trigonal-
rhombohedral lattices are exceptions.
5. Miller-Bravais notation is sometimes used for the hexagonal system,
i.e. [u v t w]. Because only two of the u v and t are needed to span the
basal plane, the third index is redundant and is given by u+v+t=0.
BIBLIOGRAPHY:
Bloss, F.D., Crystallography and Crystal Chemistry: An Introduction, 543 pp.,
Mineralogical Society of America, Washington, DC, 1994.
Kelly, A., and G.W. Groves,
Crystallography and Crystal Defects, 428 pp.,
Addison Wesley, N. Y., 1970.
Whittaker, E.J.W.,
Crystallography: An introduction for Earth Science (and other
solid-state) students
, 254 pp. pp., Pergamon Press, Oxford, 1981.