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The Structure of Crystalline Solids
1. Crystal Structures
A crystalline material is one in which atoms are arranged in repeating arrays over large
atomic distances.
A noncrystalline material does not have long-range order, considered amorphous.
The crystal structure, or lattice structure, can affect material properties.
A unit cell is the smallest repeating unit in a crystalline material.
Atomic hard sphere model:
Three most common crystal structures in metals:
1. FCC: Face Centered Cubic; a cell of cubic geometry with atoms located at the corners
and centers of all cube faces.
2. BCC: Body Centered Cubic; cell of cubic geometry with atoms located at all eight
corners and one at the center.
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3. HCP: Hexagonal Close Pack; unit cell of hexagonal geometry with three layers of
atoms.
1. FCC : Copper , Aluminum, silver, gold
a = 2 R 2(1/2)
Coordination Number of Metals: Number of atoms touching any given atom.
For FCC coordination number = 12
Atomic packing factor (APF): the fraction of solid sphere volume in a unit cell
volume of atoms in unit cell
total unit cell volume
For FCC, APF=0.74.
2. Body Centered Cubic (BCC)
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The coordination number of a BCC crystal is 8.
The atomic packing factor is 0.68.
3. Hexagonal close-packed , HCP
On top/bottom planes, six atoms formed regular array at corners; another atom at center.
Triangular arrangement on middle plane.
Coordination number is 12 and atomic packing factor are 12 and 0.74. Same as FCC.
Example: Find a with respect to R, and the APF for a BCC metal having only one atomic
species.
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II. Density
Density is found from:
mass of atoms in cell
volume of cell
rho=n A/ Vc Na
Na - Avogadro’s Number (number of atoms/ mole)
A - atomic number of atoms (grams/mole)
n - number of atoms in unit cell
Example: Find the theoretical density of Chromium
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Crystalline and noncrystalline Materials:
There are seven kinds of crystal systems shown on page ____.
Next, step back from the single crystal and look at groupings of crystals. A material may
have different levels of crystalline.
a. Single crystal: entire substance formed from one large crystal
b. Polycrystalline: substance formed from multiple grains (small crystals)
c. Amorphous: No crystalline structure whatsoever. (polymers)
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Anisotropy: if physical properties vary in specific directions
Directions of crystals can make materials anisotropic.
Isotropic: when material properties don’t depend on direction.
Often true of amorphous materials. Why?
Crystallographic Directions, Planes and Miscellaneous Imperfections:
Crystallographic Directions
All crystal systems can be described by a generic unit cell that has six lattice parameters
a, b, c, andWe use the unit cell’s own geometry to describe directions within it.

In all, there are seven known crystal system geometries (of which cubic and hexagonal
are two).
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Example: How do we describe the vector shown below within the crystal?
1. Place the tail at the origin.
2a. Find the projection of each
component along the crystal
directions a,b and c.
2b. Determine the projected lengths
in terms of a,b & c respectively.
3. Divide the numbers by the
smallest nonzero value.
4. Place the three numbers w/o
commas inside square brackets.
Question: Will the same direction have the same three indices in the unit cell one step
above/below/right/left of the original?
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Crystallographic Planes
Crystallographic planes in a unit cell are represented by three Miller Indices (except in
hexagonal crystals).
Example: How do you describe a plane in a unit cell?
Steps:
1. Choose an origin that does not contact the plane.
2. Determine the distance to the intercept wrt a, b, and c respectively.
3. Take the reciprocals of the numbers found in 2.
4. Divide the three numbers by smallest nonzero value.
5. Place the indices in curved brackets wo commas.