L03A: Chapter 3 Structures of Metals

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L03A Chapter 3 Structures of Metals Ceramics

• The properties of a material depends on the
  arrangement of atoms within the solid.
• In a single crystal the atoms are in an ordered
  array called the structure. Single crystals are
  necessary for many applications and can be very
  large. For example, silicon crystals can be up to
  2 feet in diameter
• http//www.flickr.com/photos/davemessina/623130054
  9/
• A polycrystalline material consists of many
  crystals. Materials used for construction or
  fabrication are usually polycrystalline. For
  example
• http//www.cartech.com/news.aspx?id578
• In this chapter we examine typical crystal
  structures for metals, inorganic compounds, and
  carbon.
• You will see how to specify crystal planes and
  directions.
• You will learn how to calculate some properties
  of crystals from their structure, including the
  dependence on direction in their lattice.
• Review calculation of areas for squares and
  rectangles, and calculation of distances and
  areas for right triangles, e.g. at the Math
  Skills Review under Read, Study Practice at
  WileyPLUS.com.

Last revised January 12, 2014 by W.R. Wilcox,
Clarkson University
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Amorphous and crystalline materials

• A material is crystalline if the atoms display
  long-range order, i.e. the same repeating
  arrangement over-and-over.
• The atoms in some materials do not have
  long-range order. These are called amorphous or
  glassy. Most polymers are amorphous, but so are
  some ceramics, metals, and forms of carbon.

• Equilibrium structures are those with the minimum
  Gibbs energy G, although atomic movement in
  solids is so slow that equilibrium is often not
  reached at room temperature. (In thermodynamics
  youll see that G H TS)

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Hard-sphere model of crystals

• We may show the atoms as points or small spheres
  connected by lines, or we may show them as hard
  spheres of defined diameter in contact with one
  another.
• For a metal with a face-centered cubic lattice

Unit cell. When repeated, generates the entire
crystal.
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Metallic Crystal Structures

• Bonding is not directional
• Minimum energy when nearest-neighbor distances
  are small.
• The electron cloud shields the positive cores
  from one another.
• Metals have the simplest crystal structures.
• We will examine the three most common.
• Two of their unit cells are based on a cube

Virtual Materials Science and Engineering (VMSE)
? http//higheredbcs.wiley.com/legacy/college/cal
lister/1118061608/vmse/xtalc.htm
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Atomic Packing Factor (APF)
APF calculation for a simple cubic structure
The coordination number is the number of nearest
neighbors. What is it here?
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Body Centered Cubic Structure (BCC)
Examples Cr, W, Fe(?), Ta, Mo
Coordination number?
How many touch the one in the center?
Coordination number 8
Number of atoms per unit cell?
1 center 8 corners x 1/8 2
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Atomic Packing Factor for BCC
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Theoretical Density
where n number of atoms/unit cell
A atomic weight (g/mol) VC Volume of
unit cell a3 for cubic NA Avogadro
constant 6.022 x 1023 atoms/mol
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Example Theoretical Density of Chromium

• Cr is body-centered cubic
• A 52.00 g/mol
• R 0.125 nm
• n 2 atoms/unit cell

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Face Centered Cubic Structure (FCC)Examples Al,
Cu, Au, Pb, Ni, Pt, Ag
How many atoms in the unit cell touch the atom in
the center of the front face?
Atoms only touch along face diagonals.
How many additional atoms touch it in the unit
cell in front of this one?
Coordination number 8 4 12
How many atoms in one unit cell?
6 face x 1/2 8 corners x 1/8 4
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Atomic Packing Factor for FCC
This is the maximum achievable APF and is one of
two close-packed structures.
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Crystal Systems
a, b, and c are the lattice constants
Only for the cubic system are the angles all 90o
and the lattice constants all the same.
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Crystal structure

• Seven different possible geometries for the unit
  cell.
• There are 14 Bravais lattices, with each point
  representing the same atom or collection of atoms.

• Pure metals are usually FCC, BCC or HCP.
• Except for hexagonal, number of atoms per unit
  cell1/8 at corners1/2 at face centersAll of
  body centered

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Point Coordinates in a Lattice

• Point coordinates for the unit cell center are
• a/2, b/2, c/2 ? ½ ½ ½
• Point coordinates for unit cell corner are a, b,
  c ?111
• Translation by an integer multiple of lattice
  constants reaches an identical position in
  another unit cell

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Miller Indices for Crystallographic Directions
examples 1, 0, ½ gt 2, 0, 1 gt 201
VMSE with examples, problems, exercises
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Linear Density of Atoms (LD)
Number of atoms

• LD  

Length of direction vector
example linear density of Al in 110
direction  FCC with a 0.405 nm
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Miller Indices for Crystallographic Planes

• Reciprocals of the three axial intercepts for a
  plane, cleared of fractions common multiples.
• All parallel planes have the same Miller indices.
• Algorithm (procedure)
• 1. If the plane passes through the origin,
  translate so it does not.
• 2. Read off the intercepts of the plane with the
  axes in increments of the lattice constants (a,
  b, c). For example, 1, 2, 2
• 3. Take reciprocals of those intercepts. If it
  is parallel to an axis so that it doesnt
  intersect it, the reciprocal is 0. For
  example, 1, ½, ½
• 4. Convert the numbers to the smallest possible
  integer values. For example, 2, 1, 1
• 5. Enclose those numbers in parentheses, with no
  commas. For example (211).
• 6. As with directions, a bar over a number
  indicates it is negative.
• VMSE with illustrations, problems, exercises
• Families of equivalent planes. For a cubic
  structure, for example

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Three Low-index Planes
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Crystallographic Plane Examples
4. Miller Indices (110)
4. Miller Indices (100)
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Planar Density or Packing

• Atoms per unit area
• Very important for mechanical strength and for
  chemical properties.
• Essential step is to sketch the plane of
  interest, and then use geometry to relate lattice
  constant to atomic radius.
• For example, iron foil can be used as a catalyst.
  The atomic packing of the exposed plane is
  important.
• Draw (100) and (111) crystallographic planes
• b) Calculate the planar density for each of
  these planes.

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Planar Density of (100) ?-Iron (Ferrite)

• For T lt 912?C the equilibrium structure of iron
  is BCC.

Radius R 0.1241 nm
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Planar Density of (111) Ferrite
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Close-packed planes and structures

• 111 in FCC have metal atoms as close together
  as possible. Called close-packed.
• So FCC structure of metals is also sometimes
  called cubic close packed.
• In VMSE http//higheredbcs.wiley.com/legacy/colle
  ge/callister/1118061608/vmse/xtalclose.htm
• Watch close-packed 111 planes added to build
  FCC http//departments.kings.edu/chemlab/animatio
  n/clospack.html ABCABCABC , where A, B and C are
  three possible positions of atoms.
• In L-03B we look at another close-packed
  structure for metals built of the same planes,
  but in a different order, ABABAB.