Structures of Metals and Ceramics

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Structures of Metals and Ceramics

• Callister, 2000
• 5th Edition

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Outline of the Lecture

• Crystal Structures
• Unit Cell
• Metallic Crystal Structures
• Crystal Systems (Directions and Planes)
• Atomic Arrangements
• Linear and Planar Atomic Densities
• Noncrystalline Materials

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Crystal Structures

• Material classification can be made based on the
  regularity or
• irregularity of atom or ion arrangement with
  respect to each other.
• 
  Materials

Crystalline Material Atoms are situated in a
repeating or periodic array over large atomic
distances. All metals Ceramics Some of the
polymers
Noncrystalline Material (amorphous) Long range
atomic arrangement lacks in this type of
materials.
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Crystal Structures

• The crytalline structure of the materials range
  from simple to more complex and there are many
  different types of structures.
• The atoms or ions are thought as solid spheres
  with their sizes defined. This is called atomic
  hard sphere model. All atoms are identical in
  this model.
• Smallest repeating group is called UNIT CELL.
  Unit cells can be imagined as the building block
  of the crystal structure. Unit cells in general
  are paralelepipeds or prisms having three sets of
  parallel faces, one is drawn within the aggregate
  of spheres.

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Metallic Crystal Structure

• Atomic bonding is metallic, which is
  nondirectional. Therefore there are no
  restrictions as to the number and position of
  nearest neighbor atoms. For metals each sphere
  in crystal structure represents the ion core.
  There are three simple crystalline structure in
  metallic materials.
• Face centered cubic crystal structure (FCC)
• Body centered cubic crytal structure (BCC)
• Hexagonal Close-Packed Crystal Structure (HCP)

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• Face Centered Cubic Crystal Structure (FCC)

The ion cores touch one another across a face
diagonal. The cell comprises the volume of the
cube, which is generated from the centers of the
atoms at the corners. Each corner atom is shared
among eight unit cells. A face centered atom is
belongs to only two unit cells. Therefore, the
number of atoms per FCC (1/8)8(1/2)64
atoms/cell
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• Coordination number number of nearest neighbor
  or touching atoms.
• For FCC it is 12 atoms.
• Atomic packing factor (APF)
• APF0.74 for FCC. (see the example problem in
  textbook)
• Metals generally have high APF to maximize the
  shielding provided by
• electron cloud.

Volume of atoms/cell
Volume of the cell
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• Body Centered Cubic Crystal Structure (BCC)

There are atoms located at eight corners and a
single atom at the center. Center and corner
atoms touch one another along the cube diagonals
and the unit cell length (a) and atomic radius
(R) can be related through There are 2
atoms/BCC unit cell. Coordination number is 8 and
APF0.68.
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• Hexagonal Close-Packed Crystal Structure (HCP)

The top and bottom faces of the unit cell have
six atoms that form regular hexagons and a
single atom in the center. Another plane provides
three additional atoms is situated between top
and bottom planes. 1/6(12)1/2(2)3 6 atoms/HCP
cell Ideally c/a1.633, but for some metals this
ratio deviates from the ideal value. Coordination
number is 12 and APF0.74.
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• Theoretical density (mass/ volume) can then be
  calculated using the
• crystal structure of metallic solid material.

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Ceramic Crystal Structures

• Ceramics crytal structures are more complex since
  they are composed of different elements. Moreover
  the bonding in ceramics may range from purely
  ionic (nondirectional) to totally covalent
  (directional). We learned the calculation of
  ionic character of a covalent bond using the
  electronegativities and here are some examples
  for different materials

For totally ionic bonding Imagine crystalline
structure composed of positively and negatively
charged ions instead of atoms. The magnitude of
the electrical charge on each of the ions and
relative sizes of the cations and anions are
important factors for crystal structure. The
crystal must be electrically neutral (positive
and negative charges must be balanced). The
chemical formula reflects this
electroneutrality, such as, CaF2. Cations are
usually smaller than anions and rC/rA is less
than unity (Table 3.4). Cations and anions prefer
to have as many neighboring ions as possible.

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• Stable ceramic crystal is when those anions
  surrounding a cation are all in contact with that
  cation.

The coordination number (number of anion nearest
neighbors for a cation) depends on the
cation-anion (rC/rA) radius ratio. For a
specified coordination number, there is a
critical or minimum rC/rA ratio, which can be
calculated by geometrical methods. The Table
3.3 presents coordination numbers and nearest
neighbor geometries for various rC/rA ratios.
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• AX-Type Crystal Structures For ceramics having
  equal numbers of cations and anions (Acation,
  Xanion). Coordination number for both is 6. For
  example NaCl, MgO, MnS, FeO.
• Cesium Chloride structure Coordination number
  for cation and anion is 8.
• Zinc blende structure ZnS, ZnTe, SiC.
  Coordination number is 4. The bonding is covalent
  in this type structure.
• AmXp-Type Crystal Structures CaF2, UO2, PuO2,
  ThO2.
• AmBnXp Type Crystal Structures BaTiO3

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• It is also possible to calculate the theoretical
  density of ceramic material from the unit cell
  data.

Silicate Ceramics materials composed of Si and
O. There are many arrangements of SiO44-, which
is the basic unit for most of the silicate
ceramics. There are four O atoms bonded to a
single Si. Si-O bonds are strong, directional and
assumed to be covalent.
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• Carbon C can exist in various polymorphic (a
  material having more than one crystal structure
  such as FeBCC structure at room T, FCC structure
  at 9120C, it is also called allotropy for
  elemental solids) forms as well as amorphous
  state. Diamond and graphite are two different
  polymorphic forms of C.

Metastable C polymorph at room T and atmospheric
P.
Stable C polymorph at room T and atm. P.
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Crystal Systems

• The unit cell geometry x,y,z coordinate system
  is established with its origin at one of the unit
  cell corners and axes coincide with the edges of
  the paralelepiped extending from that corner, the
  origin.

There are six parameters to define the geometry
of the unit cell Three edge lengths a, b,
c Three interaxial angles a, ß, ? These
parameters are also called lattice parameters.
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• There are seven different possible combinations
  of the lattice parameters respresenting a unique
  crystal system.

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Note that cubic system is the most symmetric,
while trclinic is the least one. It is obvious
BCC and FCC are cubic systems, while HCP is a
hexagonal system.
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• It is often necessary to specify a particular
  crystallographic plane of atoms or direction.
  Labeling using indices helps us to define ceratin
  planes and directions. The basis for the
  estimation of index values is the unit cell.
• Crystallographic Directions
• The direction is a line between two points or a
  vector as shown below

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• Steps for defining a direction in a crystal
  system
• A vector is positioned such that it passes
  through the origin of the coordinate system. Then
  you can move the vector if you keep the
  parallelism.
• The length of the vector projection on each of
  the three axes is determined in terms of the unit
  cell dimensions (a, b, c).
• The three numbers are multiplied or divided by a
  common factor to reduce them to the smallest
  integer values.
• Three indices are enclosed in square brackets as
  uvw
• Remember to count for positive and negative
  coordinates based on the origin. When there is a
  negative index value, then show that by a bar
  over it, as

This vector has a component in y direction.
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Examples
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• In cubic crystals, all directions showed by the
  indices of

are equivalent. Therefore they can be grouped as
a family, which is shown as
Hexagonal Crystals There is a four-axis
(Miller-Bravais) coordinate system used for this
type of structures.
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• The three a1, a2, and a3 axes are placed withina
  single plane (basal plane) and at 120 angles to
  one another. The z axis is perpendicular to the
  selected basal plane.
• There will be four indices to define the
  direction as uvtw
• The first three indices are the projections of
  a1, a2, and a3 axes. Then convert from
  three-index system to four-index system as
  follows

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Crystallographic Planes

• Except hexagonal crystal system, crytallographic
  planes are specified using three Miller indices
  (hkl). Any two planes parallel to each other are
  equivalent and have same indices.
• The determination of the h,k, and l index numbers
  are as follows
• If the plane passes through the selected origin,
  then construct a new parallel plane or change the
  originto a corner of another unit cell.
• Plane intersects or parallels each of the axes
  the length of each axis is determined by using
  lattice parameters a,b, and c.
• Take the reciprocals of the lattice parameters.
  Therefore a plane that parallels an axis has a
  ZERO index. (1/infinityzero)
• You may then change these three numbers to the
  set of smallest integers using a common factor.
• Report the indices as (hkl).
• An intercept on the negative side of the origin
  is indicated by a bar over that index.

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• For cubic crystals Planes and directions having
  the same indices are perpendicular to one
  another.

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• We are interested in specification fo planes
  because atomic arrangement depends on the crystal
  structure.

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• A family of planes is formed by all those planes
  that are crystallographically equivalent, 100,
  111.
• 111

For tetragonal crystal structure, 100 family
contains
But
planes are not crystallographically equivalent.
Hexagonal Crystals Equivalent planes have the
same indices as directions (Figures 3.21 and
3.22). Four-index scheme is used (hkil) and the
index i is calculated by the sum of h and k
through i - (hk) This scheme defines the
orientation of a plane in a hexagonal crystal.
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Linear and Planar Atomic Densities

• Linear and planar atomic densities are one and
  two dimensional analogs of atomic packing
  factor. Linear density shows directional
  equivalency, i.e., equivalent directions have
  identical linear densities. Planar density shows
  planar equivalency. The following examples
  illustrate the determination of the linear and
  planar densities.

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Crystalline and Noncrystalline Materials

• Single Crystals When the periodic and repeated
  arrangement of atoms extends throughout the
  entirety of the specimen without interruption,
  the result is a single crystal. Single crystals
  may exist in nature but they may also be produced
  artificially.

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• Polycrystalline Materials are the materials made
  of a collection of small crystals or grains.
• A typical solidification of a polycrystalline
  specimen

Grain boundary
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• Anisotropy Some of the physical properties of
  single crystals may depend on the crystallograhic
  direction. For axample the elastic modulus,
  electrical conductivity of a single crystal may
  be different in 100 and 111 directions. This
  directionality of properties is termed
  anisotropy. This difference is usually due to
  variance of atomic or ionic spacings in different
  directions. The extend and magnitude of
  anisotropy are related with the symmetry of the
  crystal structure. The degree of anisotropy
  increases as the symmetry decreases.
• Isotropy Properties are independent of the
  crystallographic direction.
• Determination of Crystal Structures X-Ray
  Diffraction (XRD)
• Reading assignment for students.
• Noncrystalline Solids lack a systematic and
  regular arrangement of atoms over relatively
  large atomic distances.