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An Introduction to Crystallography and Diffraction

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An Introduction to Crystallography and Diffraction Powered By Docstoc
					Geometrical Crystallography
  Part I: Two Dimensional Crystals

                S. Asgari
Dept. of Materials Science & Engineering
    Sharif University of Technology
                    Outline
 Motif, Lattice, Unit Cell

 Crystal System and Bravais Lattice

 Symmetry Operations in Two Dimensions

 The 10 Two Dimensional Point Groups

 The 17 Plane Groups
     Motif, Lattice, Unit Cell
Crystal: A regular arrangement of atoms or
 molecules repeated in one, two or three
 dimension(s).

            Crystal = Motif +Lattice

Motif: The repeat unit of a crystal.
Lattice: An imaginary network of points over
        which motifs are distributed.
      Motif, Lattice, Unit Cell
Unit Cell: A unit parallelogram of lattice which
          contains the Motif.




  Primitive Unit Cell : One lattice point per unit cell.
Two Dimensional Patterns
                       Lattice
An Important Property of Lattice:

The surroundings of all lattice points are identical.



  Is this a lattice?
Example: Structure of Graphite
 Each point represents a carbon atom.

 Two carbon atoms per lattice point.
          Lattice (Cont.)
Choice of Lattice (unit cell) is not unique.
Crystal System and Bravais Lattice

 Crystal System: Defines the shape of unit cell.

 Bravais Lattice : Defines the distribution of lattice
                    points within the unit cell.

  Number of crystal systems and Bravais lattices in 2-D and 3-D

                    Crystal System Bravais Lattice
 2-D Crystals               4              5
 3-D Crystals               7             14
                  2-D Crystals

Crystal systems:                      Bravais Lattices:

Oblique (a ≠ b, ≠ 60°, 90°)          Oblique P
Rectangular (a ≠ b, =90°)            Rectangular P
                                      Rectangular C
Square (a = b, =90°)
                                      Square P
Hexagonal (a = b, =60°)
                                      Hexagonal P

Note: (a = b, ≠ 60°, 90°) is sometimes called Rhombic or
Diamond but the suitable Bravais lattice is Rect. C.
2-D Bravais Lattices




Rectangular C and
Rhombic Lattices
       Symmetry Operations
 An object or figure has symmetry if some
 movement of the figure or operation on the figure
 bring it to a position identical to that before the
 operation.
 Symmetry operation of the first sort brings an
 object to its identical shape, e.g. translation,
 rotation and screw rotation.

 Symmetry operation of the second sort brings an
  object to its enantiomorph (mirror or inverted
  shape), e.g. reflection, inversion, glide reflection,
  roto-inversion.
          Rotational Symmetry


     Symbols

Hermann-Mauguin
   1, 2, 3, 4, 6


   Schoenflies
 C1, C2, C3, C4, C6
Limitation on Rotational Symmetry
     Due to Lattice Regularity
- An isolated object may have n-fold rotation axis.
- A lattice may only have 1, 2, 3, 4, 6 fold rotation
axis.

                          b = 2 . a.cos  = m . a
    Mirror Line


                      m
             H
                  m

m
Lattice Symmetry
 in 2-D Crystals

         Note:

  These are symmetry
  elements of lattice
  not crystal !