3. Introduction
In geometry and crystallography, Bravais
Lattice, studied by AUGUSTE BRAVAIS(1850), is an
infinite array of discrete points generated by a set
of discrete translation operation describe by:
R= n1a1+ n2a2+ n3a3
where, n1, n2 and n3are any integers
a1, a2 and a3 are primitive
integers
R=vector
The 14 basic unit cells in 3 dimensions, called the
Bravais lattice.
4. Definition
Lattice:
In mathematics, a lattice is a partially ordered
set (also called a poset) in which any two elements have
a unique supremum (the elements' least upper bound;
called their join) and an infimum (greatest lower bound;
called their meet)
Space lattice:
Lattice of point onto which the atoms are hung.
Unit cell:
Building block, repeat in a regular way.
5. Bravais lattices in 2 dimensions
In each of 0-dimensional and 1-dimensional
space there is just one type of Bravais lattice.
In two dimensions, there are five Bravais
lattices. They are oblique, rectangular, centered
rectangular (rhombic), hexagonal, and square.
6. The five Bravais lattices
(oblique, rectangular, centered rectangular (rhombic), hexagonal,
and square).
7. Bravais lattices in 3 dimensions
Cubic (3 lattices)
The cubic system contains those Bravais lattices whose point
group is just the symmetry group of a cube. Three Bravais lattices with
nonequivalent space groups all have the cubic point group. They are the
simple cube, body-centered cubic, and face-centered cubic.
Simple cube
Body-centered cubic
Face-centered cubic
8. Tetragonal (2 lattices)
The simple tetragonal is made by pulling on two opposite
faces of the simple cubic and stretching it into a rectangular
prism with a square base, but a height not equal to the sides of
the square.
By similarly stretching the body-centered cubic one more
Bravais lattice of the tetragonal system is constructed,
the centered tetragonal.
Simple tetragonal Body centered tetragonal
9. Orthorhombic (4 lattices)
The simple orthorhombic is made by deforming the square
bases of the tetragonal into rectangles, producing an object with
mutually perpendicular sides of three unequal lengths.
The base orthorhombic is obtained by adding a lattice
point on two opposite sides of one object's face.
Simple orthorhombic Base orthorhombic
10. The body-centered orthorhombic is obtained by adding one lattice
point in the center of the object.
The face-centered orthorhombic is obtained by adding one lattice
point in the center of each of the object's faces.
11. Monoclinic (2 lattices)
The simple monoclinic is obtained by distorting the
rectangular faces perpendicular to one of the orthorhombic axis
into general parallelograms.
By similarly stretching the base-centered orthorhombic one
produces the base-centered monoclinic.
Simple monoclinic Base-centered monoclinic
12. Triclinic (1 lattice)
The destruction of the cube is completed by moving the
parallelograms of the orthorhombic so that no axis is
perpendicular to the other two.
The simple triclinic produced has no restrictions except
that pairs of opposite faces are parallel.
14. Hexagonal (1 lattice)
The hexagonal point group is the symmetry
group of a prism with a regular hexagon as base.
The simple hexagonal Bravais has the
hexagonal point group and is the only Bravais lattice
in the hexagonal system.
15.
16.
17. Crystal System Possible Variations
Axial Distances
(edge lengths)
Axial Angles Examples
Cubic
Primitive, Body
centred, Face
centred
a = b = c α = β = γ = 90°
NaCl, Zinc
Blende, Cu
Tetragonal
Primitive, Body
centred
a = b ≠ c α = β = γ = 90°
White
tin, SnO2, TiO2,
CaSO4
Orthorhombic
Primitive, Body
centred, Face
centred, End
centred
a ≠ b ≠ c α = β = γ = 90°
Rhombic
Sulphur, KNO3,
BaSO4
Hexagonal Primitive a = b ≠ c
α = β = 90°,
γ = 120°
Graphite, ZnO,
CdS
Rhombohedral
(trigonal)
Primitive a = b = c α = β = γ ≠ 90°
Calcite ,
Cinnabar
Monoclinic
Primitive, End
centred
a ≠ b ≠ c
α = γ = 90°,
β ≠ 120°
Monoclinic
Sulphur,
Na2SO4.10H2O
Triclinic Primitive a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° kyanite
18. Zone and zone laws
The faces on a crystal which are parallel to
themselves and parallel to common axis are said
to be form a zone and are called co-zonal or
tauto zonal faces. The corresponding axis is
called the zone axis.
Fig: A crystal of lead sulphate
Fig: Its stereographic projection to show the
co-zonal faces and the zone symbol in
brackets.
19. The zone axis need not always coincide with the
crystallographic axis like the general symbol for a face
is [hkl]. The general symbol [uvw] .
If any face (hkl) is parallel to an axis and
belongs to a particular zone (uvw) then it should
satisfy the condition is known as Weiss zone law.
The great circle appearing as circle arcs and
diameter represents the various zones. The Miller
index of the face (101) is obtained by the addition (111)
and (111). Likewise the index of the (111) face is
obtained by the addition of (100) and (001). This
relation is known as the Law of addition.
20. If [u'v'w'] & [u " v " w "] are the two zones, then
the face at the intersection point hkl is:
u' v' w' u' v' w'
u" v" w" u" v" w“
h= v' w" - v" w' ; k=w' u" -w" u' ; l= u' v" - u" v‘
In the same manner, if the indices of 2 parallel
faces are [h1k1l1] & [h2k2l2] then the zone symbol for the
zone [uvw] is:
21. Conclusion
The lattice types were first discovered in 1842 by
Frankenheim, who incorrectly determined that 15
lattices were possible.
Bravais lattice, any 14 possible lattices in 3-
dimensional configuration of points used to describe
the orderly arrangement of atoms
in a crystal.
Each point represent one or more atoms in the actual
crystal and if the points are connected by lines, a
crystal lattice is formed.
Zone law states that the intersection of 2 zones can
be a possible pole of a face. The miller indices of face
on a zone is a simple algebraic addition of the 2
22. Books
Leonid V. Azaroff, “Introduction to Solid”, tata
McGraw Hill publishing company ltd 1977, Pp-37-38.
Marjorie Senechal, Crystalline Symmetries, Pp-39-42.
WE Ford, a text of Mineralogy, Wiley Eastern Ltd. 1949.
Websites:
http://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf
http://en.wikipedia.org
http://phycomp.technion.ac.il/~sshaharr/intro.html
