Crystalline Structures

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Crystalline Structures

• Edward A. Mottel
• Integrated, First-Year Curriculum in Science,
  Engineering and Mathematics

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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 PropertiesRelated 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
Simple Cubic a b c a b g 90
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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 relationships
• edge
• face-diagonal
• body-diagonal
• cell volume

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Crystal Structure Definitions

• Cell relationships
• edge
• face-diagonal
• body-diagonal
• cell volume

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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 relationships
• edge
• face-diagonal
• body-diagonal
• cell volume

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The Unit CellCrystal 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 StructuresCrystal Structure
Definitions

• Start at page J-5 of laboratory manual work
  through page J-10
• 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 ConsiderWhen 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 KitModels to Build

• Primitive (simple) unit cell (pg. 9, 102)
• 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
• Cubic Closest Pack (ccp) pg 26, J-8
• Shaded area and hollow numbers
• Hexagonal Closest Pack (hcp) pg 24, 99, J-8

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Simple (Primitive) Cubic

• 1) 6
• 2) 2r
• 3) 2Ö2 r
• 4) 2Ö3 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/Ö3
• 5) 2 net atoms per unit cell
• 6) 31.9

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Face-Centered Cubic

• 1) 12
• 2) 4r
• 3) 2Ö2 r
• 4) 4 net atoms per unit cell
• 5) 25.9

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Equivalent Sphere Summary
Along which dimensions of a cube do the
atoms touch
Structure
Length
simple (primitive) cubic
edge
2 r
body-centered cubic
body-diagonal
4 r
face-centered cubic
face-diagonal
4 r
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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 four positions within the unit cell
Atoms occur at the corners of the unit cell
The four positions are at the centers of
alternate smaller cubes
Atoms occur at the face centers of the unit cell
How many net carbon atoms are there per unit cell?
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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 gcm3. What is the
radius of a carbon atom?
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Graphite Structure
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How many spheres can fit into the following
square (2-D)
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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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Experiment JJ Solid State IIElectrical
Resistance
metal
metal
semiconductor
semiconductor
superconductor
superconductor
Based on your observations, which type of
material corresponds to which line?
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Experiment JJ Solid State IIStructure

• Sodium nitrate crystal formation
• Bobby Pin heat treatment
• Copper wire work hardening

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Solid State Model KitModels 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 ChlorideTwo Different Kinds of Ions
One type of ion occurs at the corners of a cube
The other ion occurs at the center of the cell
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Sodium ChlorideA Different Structure with Two
Kinds of Ions
One type of ion occurs at
The other ion occurs at
the corners of a cube
the center of the each edge
the centers of a each face
the center of the cell
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Questions to ConsiderWhen 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

• vii) 5.50/Ö3 3.18 Å
• viii) 32.0 Å3
• ix) 30.3

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Rock Salt Structure

• i) 6, 6, 0
• ii) 6, 6, 0
• iii) octahedral hole
• iv) 2r 2r-
• v) 4 cations
• vi) 4 anions

• vii) 125 Å3
• viii) 38.0

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