![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.](https://image.slidesharecdn.com/bravaislattices-250313114830-0a1d9f7e/85/In-mathematics-a-lattice-is-a-partially-ordered-19-320.jpg)
![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:](https://image.slidesharecdn.com/bravaislattices-250313114830-0a1d9f7e/85/In-mathematics-a-lattice-is-a-partially-ordered-20-320.jpg)
In mathematics, a lattice is a partially ordered
- 1. Presented by: Karung Phaisonreng Kom M.Sc applied geology (1st sem.), DoS in Earth Science University of Mysore, Mysore-06 Guided by: Dr. GS Gopalakhrishna Professor, DoS in Earth Science, University of Mysore, Mysore-06 University of Mysore Date 07-10-2011
- 2. Content Introduction Definition Bravais lattice in 2D and 3D Zone and zone laws Conclusion References
- 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.
- 13. Trigonal (1 lattice) The simple trigonal (or rhombohedral) is obtained by stretching a cube along one of its axis.
- 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.
- 17. Crystal System Possible Variations Axial Distances (edge lengths) Axial Angles Examples Cubic Primitive, Body centred, Face centred a = b = c = = = 90° α β γ NaCl, Zinc Blende, Cu Tetragonal Primitive, Body centred a = b ≠ c = = = 90° α β γ White tin, SnO2, TiO2, CaSO4 Orthorhombic Primitive, Body centred, Face centred, End centred a ≠ b ≠ c = = = 90° α β γ Rhombic Sulphur, KNO3, BaSO4 Hexagonal Primitive a = b ≠ c = = 90°, α β = 120° γ Graphite, ZnO, CdS Rhombohedral (trigonal) Primitive a = b = c = = ≠ 90° α β γ Calcite , Cinnabar Monoclinic Primitive, End centred a ≠ b ≠ c = = 90°, α γ ≠ 120° β Monoclinic Sulphur, Na2SO4.10H2O Triclinic Primitive a ≠ b ≠ c ≠ ≠ ≠ 90° α β γ 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