Chapter 1 Crystal Structures

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Chapter 1Crystal Structures
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Two Categories of Solid State Materials
Crystalline quartz, diamond.. Amorphous
glass, polymer..
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Ice crystals
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crylstals
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Lattice Points, Lattice and Unit Cell

• How to define lattice points, lattice and unit
  cell?

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LATTICE

• LATTICE An infinite array of points in space,
  in which each point has identical surroundings to
  all others.
• CRYSTAL STRUCTURE The periodic arrangement of
  atoms in the crystal.
• It can be described by associating with each
  lattice point a group of atoms called the MOTIF
  (BASIS)

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Notes for lattice points

• Don't mix up atoms with lattice points
• Lattice points are infinitesimal points in space
• Atoms are physical objects
• Lattice Points do not necessarily lie at the
  centre of atoms

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An example of 2D lattice
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An example of 3D lattice
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Unit cell A
repeat unit (or motif) of the regular
arrangements of a crystal is defined
as the smallest repeating unit which
shows the full symmetry of the crystal structure
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More than one ways
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How to assign a unit cell
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A cubic unit cell
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3 cubic unit cells
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Crystal system

• is governed by unit cell shape and symmetry

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The Interconversion of Trigonal Lattices
????????
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The seven crystal systems
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Symmetry
Space group point group translation
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Definition of symmetry elements

• --------------------------------------------------
  -----------
• Elements of symmetry
• ------------------------------------------------
• Symbol Description
  Symmetry operations
• --------------------------------------------------
  -------------------
• E Identity
  No change
• s Plane of symmetry
  Reflection through the plane
• i Center of symmetry
  Inversion through the center
• Cn Axis of symmetry
  Rotation about the axis by (360/n)o
• Sn Rotation-reflection
  Rotation about the axis by (360/n)o
• axis of symmetry
  followed by reflection through the
  
• 
  plane perpendicular to the
  axis
• --------------------------------------------------
  -------------------

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Center of symmetry, i
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Rotation operation, Cn
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Plane reflection , ?
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Matrix representation of symmetry operators
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Symmetry operation
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Symmetry elements
C3
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space group point group translation
Symmetry elements
Screw axes 21(//a), 21(//b), 41(//c) 42(//c), 31(//c) etc
Glide planes c-glide (- b), n-glide, d-glide etc
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21 screw axis // b-axis
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Glide plane
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Where are glide planes?
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Examples for 2D symmetry
http//www.clarku.edu/djoyce/wallpaper/seventeen.
html
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Examples of 2D symmetry
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General positions of Group 14 (P 21/c) unique
axis b
1 x,y,z identity
2 -x,y1/2,-z1/2 Screw axis
3 -x,-y,-z i
4 x,-y1/2,z1/2 Glide plane
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Multiplicity, Wyckoff Letter, Site Symmetry
4e 1 (x,y,z) (-x, ½ y,½ -z) (-x,-y,-z) (x,½ -y, ½ z)
2d i (½, 0, ½) (½, ½, 0)
2c i (0, 0, ½) (0, ½, 0)
2b i (½, 0, 0) (½, ½, ½)
2a i (0, 0, 0) (0, ½, ½)
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General positions of Group 15 (C 2/c) unique
axis b
1 x,y,z identity
2 -x,y,-z1/2 2-fold rotation
3 -x,-y,-z inversion
4 x,-y,z1/2 c-glide
5 x1/2,y1/2,z identity c-center
6 -x1/2,y1/2,-z1/2 2 c-center
7 -x1/2,-y1/2,-z i c-center
8 x1/2,-y1/2,z1/2 c-glide c-center
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P21/c in international table A
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P21/c in international table B
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Cn and ?
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Relation between cubic and tetragonal unit cell
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Lattice the manner of repetition of atoms, ions
or molecules in a crystal by an array of points
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Types of lattice

• Primitive lattice (P)
• - the lattice point only at corner
• Face centred lattice (F)
• - contains additional lattice points in the
  center of each face
• Side centred lattice (C)
• - contains extra lattice points on only one
  pair of opposite faces
• Body centred lattice (I)
• - contains lattice points at the corner of a
  cubic unit cell and body center

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Examples of F, C, and I lattices
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14 Possible Bravais lattices combination of
four types of lattice and seven crystal systems
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How to index crystal planes?
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Lattice planes and Miller indices
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Lattice planes
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Miller indices
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Assignment of Miller indices to a set of
planes1. Identify that plane which is adjacent
to the one that passes through the origin.2.
Find the intersection of this plane on the three
axes of the cell and write these intersections as
fractions of the cell edges. 3. Take
reciprocals of these fractions.Example fig.
10 (b) of previous page cut the
x axis at a/2, the y axis at band the z axis at
c/3 the reciprocals are
therefore, 1/2, 1, 1/3 Miller
index is ( 2 1 3 )
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Examples of Miller indices
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Miller Index and other indices

• (1 1 1), (2 1 0)
• 1 0 0 (1 0 0), (0 1 0), (0 0 1) .
• 2 1 0, -3 2 3
• lt1 0 0gt 1 0 0, 0 1 0, 0 0 1

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???
Assign the Miller indices for the crystal faces
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Descriptions of crystal structures

• The close packing approach
• The space-filling polyhedron approach

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Materials can be described as close packed

• Metal- ccp, hcp and bcc
• Alloy- CuAu (ccp), Cu(ccp), Au(ccp)
• Ionic structures - NaCl
• Covalent network structures (diamond)
• Molecular structures

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Close packed layer
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A NON-CLOSE-PACKED structure
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Close packed
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Two cp layers
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P sphere, O octahedral hole, T / T-
tetrahedral holes
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Three close packed layers in ccp sequence
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ccp
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ABCABC.... repeat gives Cubic Close-Packing (CCP)

• Unit cell showing the full symmetry of the
  arrangement is Face-Centered Cubic
• Cubic a b c, a b g 90
• 4 atoms in the unit cell (0, 0, 0) (0, 1/2, 1/2)
  (1/2, 0, 1/2) (1/2, 1/2, 0)

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hcp
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ABABAB.... repeat gives Hexagonal Close-Packing
(HCP)

• Unit cell showing the full symmetry of the
  arrangement is Hexagonal
• Hexagonal a b, c 1.63a, a b 90, g
  120
• 2 atoms in the unit cell (0, 0, 0) (2/3, 1/3,
  1/2)

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Coordination number in hcp and ccp structures
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hcp
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Face centred cubic unit cell of a ccp arrangement
of spheres
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Hexagonal unit cell of a hcp arrangement of
spheres
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Unit cell dimensions for a face centred unit cell
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Density of metal
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Tetrahedral sites
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Covalent network structures of silicates
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C60 and Al2Br6
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The space-filling approachCorners and edges
sharing
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Example of edge-sharing
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Example of edge-sharing
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Example of corner-sharing
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Corner-sharing of silicates