Structure of Crystalline Solids

1
Structure of Crystalline Solids

• Phenomenon
• electron diffraction pattern of a single crystal
  GaAs (gallium arsenide)

• surface of gold - (111) plane
• Atoms are periodically arranged in a crystalline
  solid

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Structure of Crystalline Solids

• Structure of a solid
• Structure of a building
• building blocks e. g. bricks
• how bricks are laid and bound
• Building blocks for a solid atoms
• How atoms are laid and bound
• Types of possible structures
• amorphous random packing of atoms
• e.g. glasses
• crystalline periodic packing of atoms (repeating
  patterns)
• e.g. most engineering metals and ceramics
• very important

• Structure of crystals
• Approximation
• atoms hard spheres of a certain radius
• Unit cell the repeating unit

Reading 3.1-3.6 (6th ed)
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Structure of Crystalline Solids

• Basic types of crystal structures
• Face-Centered Cubic (FCC)
• one atom at each corner of the cubic unit cell
• one atom at each face certer
• lattice 3D array of center positions of atoms

• 4 (equivalent) atoms are completely contained in
  a FCC unit cell
• each corner atom 1/8
• 8 x 1/8 1
• each face center atom 1/2
• 6 x 1/2 3

IMSE FCC Structure
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Structure of Crystalline Solids

• Face-Centered Cubic (FCC)
• geometric relationships
• volume of the unit cell

• total volume of atoms in a unit cell
• atomic packing factor (APF)

FCC is one of the closest packed structures. It
is still only 3 quarters filled by atoms.
Materials with FCC Al, Cu, Ni
a
Self Study Density calculation- Example 3.3
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Structure of Crystalline Solids

• Body-Centered Cubic (BCC)
• one atom at each corner of the cubic unit cell
• one atom at the body center

• 2 (equivalent) atoms are completely contained in
  a BCC unit cell
• each corner atom 1/8
• 8 x 1/8 1
• the body center atom 1

IMSE BCC Structure
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Structure of Crystalline Solids

• Body-Centered Cubic (BCC)
• geometric relationships
• volume of the unit cell

• total volume of atoms in a unit cell
• atomic packing factor (APF)

BCC is not as densely packed as FCC
Materials with BCC aFe, Cr, W
a
4R
Self Study Density calculation- tutorial question
?
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Structure of Crystalline Solids

• Hexagonal Close-Packed (HCP)
• unit cell is a hexagon
• c/a 1.633 (ideal value)

• 6 (equivalent) atoms are contained in a unit cell
• as close packed as FCC (APF 0.74)

IMSE HCP Structure
Materials with HCP Mg, a-Ti, Zn
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Structure of Crystalline Solids

• coordinates
• right hand system x, y, z
• use a as one unit

• Cubic crystal systems
• way of describing unit cells
• general
• a ? b ? g a ? b ? c
• Cubic
• a b g 90
• a b c

Reading 3.7-3.10 (excluding hexagonal)
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Structure of Crystalline Solids

• convention and procedure
• the indices, uvw, of a direction are simply the
  components of the equivalent vector (projected
  lengths of the direction on x, y and z)
• make uvw smallest integers
• put any minus sign on top
• use uvw to indicate a particular direction
• direction A
• components of vector 0, 1, 1/2
• smallest integers 0, 2, 1 (x 2, i.e. double the
  length)
• direction A 021
• direction B

• Indexing crystallographic directions
• a system of describing directions in crystals -
  Miller indices

111
B
1/2
A
021
100
110
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Structure of Crystalline Solids

• lt111gt includes
• homework fill in the rest of the lt110gt and lt111gt
  family members
• members of a family have the same arrangement of
  atoms (for a cubic system) although they are
  different directions

• family of directions
• lt100gt includes 100, 010 and 001 as well as
• lt110gt includes 110, 101, 011,
  

110
111
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Structure of Crystalline Solids

• convention and procedure
• the indices, hkl, of a plane are the reciprocals
  of the intercepts of the plane with x, y and z
• make hkl smallest integers
• put any minus sign on top
• use (hkl) to indicate a particular plane
• plane A
• intercepts x 1, y 1/2 and z 2/3
• reciprocals of intercepts 1, 2, 3/2
• smallest integers (reciprocals x 2) 2, 4, 3
• plane A (243)

• Indexing crystallographic planes
• a system of describing planes in crystals -
  Miller indices

the intercept on z is 8
(110)
(111)
2/3
A
1/2
(010)
the intercepts on x z are 8
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Structure of Crystalline Solids

• 111 includes
• homework fill in the rest of the 110 and 111
  family members
• members of a family have the same arrangement of
  atoms (for a cubic system) although they are
  different planes
• equivalent planes (they are exactly the same),
  e.g. (001) and

• family of planes
• 100 includes (100), (010) and (001)
• 110 includes (110), (101), (011),
  

(001)
(010)
(001)
(100)
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Structure of Crystalline Solids

• Packing of atoms in FCC and HCP
• close-packed plane (e.g. layer A)
• interstitial sites
• B e.g. the up right triangles
• C e.g. the upside down triangles
• stacking of close-packed planes
• first layer A
• second layer B
• third layer
• C ABCABC ... packing
• A ABAB packing

The third layer can be either C or A, but not
both on the same layer!
Reading 3.12 (6th ed)
14
Structure of Crystalline Solids

• ABC packing FCC

Viewing direction
Projection along the viewing direction
Video - Open Uni. T201/VC11
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Structure of Crystalline Solids

• AB packing HCP

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Structure of Crystalline Solids

• Some questions
• Why a material takes a certain structure, e.g. Al
  is FCC?
• So that the lowest potential energy is achieved
• Can a material takes more than one structure?
• For some materials, yes
• e.g. Fe is BCC (ferrite) at room temperature, but
  transforms to FCC (austenite) at 912C
• Why?
• Again, to lower the potential energy
  corresponding to conditions (e.g. temperature)

• Noncrystalline solids
• amorphous
• packing of atoms is random

Si
O
Crystalline (2D)
SiO2
Glass (2D)
Cristobalite
Reading 3.17 (6th ed)
17
Structure of Crystalline Solids - Summary

• Phenomenon
• Periodic diffraction patterns by X-ray or
  electrons
• Atomic images showing periodic patterns
• Basic structures
• FCC BCC
• Unit cells
• geometric relationships
• density
• packing factor

• Quantitative description
• Miller indices for directions and planes
• indexing directions or planes
• plotting directions or planes from indices
• Physical understanding
• Structures are determined by atomic structure and
  bonding
• minimum potential energy for the solid
• structural changes