Crystalline Structures
Edward A. Mottel
Department of Chemistry
Rose-Hulman Institute of Technology
Crystalline Solid State
 Study
of the arrangement of atoms/ions in
metallic and ionic crystals.
 Enables the calculation of the size of
atoms/ions.
 Allows the "environment" (the number of
atoms/ions and the types of atoms/ions)
surrounding each atom/ion to be studied.
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Physical Properties
Related to Solid Structure
 Density
Luster
 Hardness
 Electrical Properties
 Melting Point
 Magnetic Properties

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Distinctions Between Solid Structures
 Crystalline
versus Amorphous Solids
 Crystal Structure versus Crystal Morphology
 Equivalent Sphere versus Ionic Structures
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Unit Cells
b
a
b
a
g
c
Simple Cubic
a=b=c
a = b = g = 90°
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Triclinic
abc
a  b  g  90°
Crystal Structure Definitions
 Unit
cell
• cell dimensions
• unit cell length (a, b, c)
• cell angles (a, b, g)
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Crystal Structure Definitions
 Cell
•
•
•
•
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relationships
edge
face-diagonal
body-diagonal
cell volume
Crystal Structure Definitions
 Cell
•
•
•
•
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relationships
edge
face-diagonal
body-diagonal
cell volume
Crystal Structure Definitions
 Cell
•
•
•
•
relationships
edge
face-diagonal
body-diagonal
cell volume
If the cell edge is a, how long is the face diagonal?
If the cell edge is a, how long is the body diagonal?
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Crystal Structure Definitions
 Cell
•
•
•
•
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relationships
edge
face-diagonal
body-diagonal
cell volume
Crystal Lattice
The repeating pattern of the unit cell creates a lattice.
Find the
repeating
pattern
What is the unit cell composition?
The Unit Cell
Crystal Structure Definitions
 Begin
working on page J-4 of laboratory
manual
 Refer to page J-1 for definitions
• Unit Cell
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Solid State Model Kit
 Obtain
one kit for every 3-4 people.
 Sit in a circle so that everyone can see the
models as they are being built.
 Refer to pg. 5 of the Model Kit Manual (in the
box) for instructions
• how to select the plastic base to use.
• which holes of the base to use.
• how to build the models.
• how to take them apart.
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Equivalent Sphere Structures
Crystal Structure Definitions
 Read
pages J-1 through J-3.
 Start at page J-4 of laboratory manual; work
through page J-4 to J-7
 Work J-12 to J-13.
 Refer to page J-1 for definitions
• Net atoms per unit cell
• Percentage void space (packing efficiency)
• Coordination Number (CN)
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Questions to Consider
When Studying Solid State Structures
 How
do the atoms touch?
 What is the length of the edge of the unit cell in
terms of the atomic radii?
 How many net atoms are there per unit cell?
 What is the percentage void space in the
structure?
 Build the primitive (simple) unit cell (pg. 9, 102)
• Build the small cube, then the large cube
• Answer questions on page J-5 of lab manual
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Solid State Model Kit
Models to Build
 Primitive
(simple) unit cell (pg. 9 model kit)
• Build the small cube, then the large cube
• Page J-5 of lab manual
 Body-Centered Cubic (bcc) pg 18, J-6
 Face-Centered Cubic (fcc) pg 27, J-7
 Diamond (dia) pg 23
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Simple (Primitive) Cubic
 1)
6
 2) 2r
 3) 22 r
 4) 23 r
 5) 1 net atom per unit cell
 6) 47.6%
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Body-Centered Cubic
 1)
8
 2) 8
 3) 4r
 4) 4r/3
 5) 2 net atoms per unit cell
 6) 31.9%
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Face-Centered Cubic
 1)
12
 2) 4r
 3) 22 r
 4) 4 net atoms per unit cell
 5) 25.9%
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Equivalent Sphere Summary
Structure
Along which dimensions
of a cube do the atoms
touch
Length
simple (primitive)
cubic
edge
2r
body-centered
cubic
body-diagonal
4r
face-centered
cubic
face-diagonal
4r
Diamond
 Hardest
naturally occurring material
 Strong because
• strong covalent bonds
• 3-dimensional strength
 Same structure is formed by
• silicon and germanium
 Similar structure formed by several
semiconducting materials
• GaAs, GaP
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Diamond Structure
Atoms occur at
the corners of the
unit cell
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Diamond Structure
Atoms occur at
the face centers
of the unit cell
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Diamond Structure
Atoms occur at
four positions
within
the unit cell
The four positions
are at the centers
of alternate
smaller cubes
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Diamond Structure
Atoms occur at
four positions
within
the unit cell
The four positions
are at the centers
of alternate
smaller cubes
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Diamond Structure
Atoms occur at
the corners of the
unit cell
Atoms occur at
the face centers
of the unit cell
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Atoms occur at
four positions
within
the unit cell
The four positions
are at the centers
of alternate
smaller cubes
How many net carbon atoms are
there per unit cell?
Diamond Structure
How long is the
body-diagonal
in terms of r?
Some atoms
touch along
the
body-diagonal
The density of diamond is 3.51 g·cm-3.
What is the radius of a carbon atom?
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Graphite Structure
How many spheres can fit into the
following square (2-D)
1
7
7
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How many spheres can fit into the
following square (2-D)
49
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How many spheres can fit into the
following square (2-D)
52 + 8/2
Hexagonal packing is efficient.
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Solid State Model Kit
Models to Build
 Cubic
Closest Pack (ccp) pg 26, J-8
• Shaded area and hollow numbers
 Hexagonal Closest Pack (hcp) pg 24, 99, J-8
 Hole Calculation, Radius Ratio and
Coordination Number pg 93, J-11
• How is r/R calculated?
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Cesium Chloride
Two Different Kinds of Ions
One type of ion
occurs at the
corners of
a cube
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The other ion
occurs at the
center of
the cell
Sodium Chloride
A Different Structure with Two Kinds of Ions
One type of ion
occurs at
the corners
of a cube
the centers
of a each face
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The other ion
occurs at
the center of
the each edge
the center of
the cell
Questions to Consider
When Studying Solid State Structures
 How
do the ions touch?
 What is the length of the edge of the unit cell
in terms of the ionic radii?
 How many net ions are there per unit cell?
 What is the percentage void space in the
structure?
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Ionic Structures
 Cesium
Chloride (CsCl) pg 96
• Answer questions on page J-12 of lab
manual
 Rock Salt (NaCl) pg 33, 31
• Answer questions on page J-13 of lab
manual
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Cesium Chloride Structure
 i)
8, 8, 0
 ii) 8, 8, 0
 iii) cubic hole
 iv) 2r+ + 2r v) 1 cation
 vi) 1 anion
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 vii)
5.50/3 = 3.18 Å
 viii) 32.0 Å3
 ix) 30.3%
Rock Salt Structure
 i)
6, 6, 0
 ii) 6, 6, 0
 iii) octahedral hole
 iv) 2r+ + 2r v) 4 cations
 vi) 4 anions
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125 Å3
 viii) 38.0%
 vii)
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