Version 1.0, August 31, 2001, Copyright, Hugh Jack 1993-2001
I.Crystal Structure Fundamentals
A.Crystal structure - atoms arranged in regular, repeating pattern over large distances
C.Lattice - 3-dimensional array of points arranged in a regular, repeating periodic structure
D.Unit cell - the smallest portion of a lattice which represents the symmetry and structure of
E.Equivalent sites - points in a lattice which are indistinguishable from other points in the
lattice - in a real crystal each equivalent site is associated with an identical atom or group of
1.Crystallographic axes - x, y, z
2.Crystallographic angles - a, b , g
II.Crystal Systems and Structures
a.1 equivalent site per unit cell
b.no crystals with only 1 atom per equivalent site
3.Face-Centered Cubic (FCC) lattice
a.4 equivalent sites per unit cell
b.common metals with 1 atom per equivalent site include - Al, Cu, Ni
4.Body-Centered Cubic (BCC) lattice
a.2 equivalent sites per unit cell
b.common metals with 1 atom per equivalent site include - Fe, Cr, W
1.a = b c, a = b = 90° , g = 120°
2.1 equivalent site per unit cell
3.Face-Centered Hexagonal (FCC)
a.2 atoms per equivalent site with close packed arrangement
b.common metals with HCP structure include - Zn, Mg, Ti
1.Set origin for crystal system
2.Points identified by translation from the origin
1.Move origin to tail of vector or draw parallel vector from origin
2.Determine translation to tip point
3.May also use tip minus tail to find translation
4.Convert indices obtained to lowest integer form
5.Place indices into brackets in form [uvw]
6.Negative indices indicated by a bar over the number
1.Find intercepts of plane with the three crystallographic axes
2.If plane includes the origin, move either the plane or the origin
3.Invert intercept values obtained
4.Convert reciprocals to lowest integer form
5.Place indices in parentheses in form (hkl)
6.Negative indices indicated by a bar over the number
D.Families of directions and planes
1.Families refer to directions or planes in a crystal which have exactly the same arrangement
and spacing of equivalent sites and atoms
2.Families of directions have the same atomic spacing
3.A family of directions is indicated by placing the indices for any one member of the family in
pointed brackets in the form <uvw>
4.Families of planes have the same planar arrangement of equivalent sites and atoms
5.A family of planes is indicated by placing the indices for any one member of the family in
a.Fraction (or %) of a vector which passes through atoms
b.Vector must pass through the center of the atom for the atoms to be counted
a.Fraction (or %) of the plane which is covered by the atoms spheres
b.Only atoms whose center lies on the plane are counted
a.Densest possible packing of identical size spheres
b.FCC and HCP structures for metals are close packed
c.Stacking of close packed layers of atoms
1.Single crystals - some materials can exist as large (macroscopic) single crystals
2.Polycrystalline - most materials exist as a set of contiguous small crystals
Grain boundary - interface between individual crystals
a.Isotropic materials have randomly oriented polycrystals - thus physical
properties are the same in all directions
b.Anisotropic materials have non-random orientations of the crystallographic
axes - thus physical properties vary with direction in the material
4.Amorphous materials - no long range orientation of the atoms to each other
5.Polymorphic and Allotropic Materials