2. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Chapter Outline
How do atoms ARRANGE themselves
to form solids?
• Unit cells
• Crystal structures
Face-centered cubic
Body-centered cubic
Hexagonal close-packed
• Close packed crystal structures
• Density
• Types of solids
Single crystal
Polycrystalline
Amorphous
3.7–3.10 Crystallography – Not Covered / Not Tested
3.15 veidbeooos.Diffraction keds.heodleh.ocolem.– Not com
Covered / Not Tested
Learning objectives #5, #6 - Not Covered / Not Tested
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3. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Types of Solids
Crystalline material: periodic array
Single crystal:
periodic array over the entire extent of the material
Polycrystalline material: many small crystals or grains
Amorphous: lacks a systematic atomic arrangement
Crystalline Amorphous
SiO2 videos.edhole.com
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4. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Crystal structure
Consider atoms as hard spheres with a radius.
Shortest distance between two atoms is a diameter.
Crystal described by a lattice of points at center of
atoms
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5. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Unit Cell
The unit cell is building block for crystal.
Repetition of unit cell generates entire crystal.
Ex: 2D honeycomb represented by
translation of unit cell
Ex: 3D crystalline structure
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6. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Metallic Crystal Structures
Metals usually polycrystalline
amorphous metal possible by rapid cooling
Bonding in metals non-directional Þ
large number of nearest neighbors and dense
atomic packing
Atom (hard sphere) radius, R:
defined by ion core radius: ~0.1 - 0.2 nm
Most common unit cells
Faced-centered cubic (FCC)
Body-centered cubic (BCC)
Hexagonal close-packed (HCP).
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7. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Face-Centered Cubic (FCC) Crystal Structure (I)
Atoms located at corners and on centers of faces
Cu, Al, Ag, Au have this crystal structure
Two representations
of the FCC unit cell
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8. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Face-Centered Cubic Crystal Structure (II)
R
Hard spheres touch along diagonal
Þ the cube edge length, a= 2RÖ2
a
The coordination number, CN = number of closest
neighbors = number of touching atoms, CN = 12
Number of atoms per unit cell, n = 4.
FCC unit cell:
6 face atoms shared by two cells: 6 x 1/2 = 3
8 corner atoms shared by eight cells: 8 x 1/8 = 1
Atomic packing factor, APF
= fraction of volume occupied by hard spheres
= (Sum of atomic volumes)/(Volume of cell)
= 0.74 (maximum possible)
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9. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Atomic Packing Fraction
APF= Volume of Atoms/ Volume of Cell
Volume of Atoms = n (4p/3) R3
Volume of Cell = a3
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10. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Density Computations
r = mass/volume
= (atoms in the unit cell, n ) x
(mass of an atom, M) /
(the volume of the cell, Vc)
Atoms in the unit cell, n = 4 (FCC)
Mass of an atom, M = A/NA
A = Atomic weight (amu or g/mol)
Avogadro number NA = 6.023 ´ 1023 atoms/mol
The volume of the cell, Vc = a3 (FCC)
a = 2RÖ2 (FCC)
R = atomic radius
r = nA
c A V N
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11. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Density Computations
r = nA
c A V N
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12. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Body-Centered Cubic Crystal Structure (II)
a
Hard spheres touch along cube diagonal Þ
cube edge length, a= 4R/Ö3
The coordination number, CN = 8
Number of atoms per unit cell, n = 2
Center atom not shared: 1 x 1 = 1
8 corner atoms shared by eight cells: 8 x 1/8 = 1
Atomic packing factor, APF = 0.68
Corner and center atoms are equivalent
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13. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Hexagonal Close-Packed Crystal Structure (I)
Six atoms form regular hexagon surrounding one
atom in center
Another plane is situated halfway up unit cell
(c-axis) with 3 additional atoms situated at interstices
of hexagonal (close-packed) planes
Cd, Mg, Zn, Ti have this crystal structure
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14. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Hexagonal Close-Packed Crystal Structure (II)
Unit cell has two lattice parameters a and c.
Ideal ratio c/a = 1.633
The coordination number, CN = 12 (same as in FCC)
Number of atoms per unit cell, n = 6.
3 mid-plane atoms not shared: 3 x 1 = 3
12 hexagonal corner atoms shared by 6 cells:
12 x 1/6 = 2
2 top/bottom plane center atoms shared by 2 cells:
2 x 1/2 = 1
Atomic packing factor, APF = 0.74 (same as in FCC)
a
c
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15. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Density Computations Summarized
Density of a crystalline material,
r = density of the unit cell
= (atoms in the unit cell, n ) ´ (mass of an atom, M) /
(the volume of the cell, Vc)
Atoms in unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)
Mass of atom, M = Atomic weight, A, in amu (or g/mol).
Translate mass from amu to grams divide atomic
weight in amu by Avogadro number
NA = 6.023 ´ 1023 atoms/mol
Volume of the cell, Vc = a3 (FCC and BCC)
a = 2RÖ2 (FCC); a = 4R/Ö3 (BCC)
where R is the atomic radius
r = nA
c A V N
Atomic weight and atomic radius of elements are in the
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16. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Face-Centered Cubic Crystal Structure (III)
Corner and face atoms in unit cell are equivalent
FCC has APF of 0.74
Maximum packing Þ FCC is close-packed
structure
FCC can be represented by a stack of close-packed
planes (planes with highest density of atoms)
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17. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Close-packed Structures (FCC and HCP)
FCC and HCP: APF =0.74 (maximum possible value)
FCC and HCP may be generated by the stacking of
close-packed planes
Difference is in the stacking sequence
HCP: ABABAB... videos.edhole . c o m FCC: ABCABCABC…
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18. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
HCP: Stacking Sequence ABABAB...
Third plane placed directly above first plane of atoms
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19. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
FCC: Stacking Sequence ABCABCABC...
Third plane placed above “holes” of first plane not
covered by second plane
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20. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Polymorphism and Allotropy
Some materials can exist in more than one crystal
structure, Called polymorphism.
If material is an elemental solid: called allotropy.
Ex: of allotropy is carbon:
can exist as diamond, graphite, amorphous carbon.
Pure, solid carbon occurs in three crystalline forms –
diamond, graphite; and large, hollow fullerenes. Two kinds
of fullerenes are shown here: buckminsterfullerene
(buckyball) videos. aenddh coalreb.ocno nmanotube.
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21. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Single Crystals and Polycrystalline Materials
Single crystal: periodic array over entire material
Polycrystalline material: many small crystals (grains)
with varying orientations.
Atomic mismatch where grains meet (grain boundaries)
Grain Boundary
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22. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Polycrystalline Materials
Atomistic model of a nanocrystalline solid
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23. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Polycrystalline Materials
Simulation of annealing videos.edhole.com of a polycrystalline grain structure
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24. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Anisotropy
Different directions in a crystal have different packing.
For instance: atoms along the edge of FCC unit cell
are more separated than along the face diagonal.
Causes anisotropy in crystal properties
Deformation depends on direction of applied stress
If grain orientations are random bulk properties are
isotropic
Some polycrystalline materials have grains with
preferred orientations (texture): material exhibits
anisotropic properties
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25. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Non-Crystalline (Amorphous) Solids
Amorphous solids: no long-range order
Nearly random orientations of atoms
(Random orientation of nano-crystals can be
amorphous or polycrystalline)
Schematic Diagram
of Amorphous SiO2
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26. Introduction To Materials Science, Chapter 3, The structure of crystalline solids
Summary
Make sure you understand language and concepts:
Allotropy
Amorphous
Anisotropy
Atomic packing factor (APF)
Body-centered cubic (BCC)
Coordination number
Crystal structure
Crystalline
Face-centered cubic (FCC)
Grain
Grain boundary
Hexagonal close-packed (HCP)
Isotropic
Lattice parameter
Non-crystalline
Polycrystalline
Polymorphism
Single crystal
Unit cell
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Editor's Notes
Density calculation is emphasized in the tests (when you calculate density from structure – you use your understanding of structures).
We have discussed atomic structure, various types of the bonds between atoms, and the correlation between the two.
Now we are going to discuss the next level of structure of solid materials. The properties of solid materials not only depends on the bonding forces between atoms, but also depends on the internal structure of the solids and the arrangements of atoms in solid space.
The internal structure of solids determines to a great extent the properties of solids or how a particular solid material will perform or behave in a given application.
Give an example of simple cubic lattice.
Explain why one atom in honeycomb lattice can not be used as a unit cell.
Give an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
Give an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
Give an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
Give an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
The top figure shows a zone annealing simulation of a gas turbine blade alloy. The grain size in the initial, randomly oriented, microstructure is about ten microns. The figure shows the development of a creep resistant columnar grain structure, which results when the material is pulled through a highly localized hot zone (a narrow vertical region at the middle of the figure). Simulations allow determination of process parameters (pull speed, hot zone temperature) that optimize the grain aspect ratio (length and width).