Chapter 3: The Structure of Crystalline Solids
ISSUES TO ADDRESS... • How do atoms assemble into solid structures? (for now, focus on metals) • How does the density of a material depend on its structure? • When do material properties vary with the sample (i.e., part) orientation? Instructor: Eng. Tamer Haddad
Chapter 3 - 1
Energy and Packing • Non dense, random packing
Energy typical neighbor bond length
typical neighbor bond energy
• Dense, ordered packing
r
Energy typical neighbor bond length
r
typical neighbor bond energy
Dense, ordered packed structures tend to have lower energies. Instructor: Eng. Tamer Haddad
Chapter 3 - 2
Materials and Packing Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers
crystalline SiO2 Adapted from Fig. 3.22(a),
Callister 7e.
Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline
Si
Oxygen
noncrystalline SiO2 Adapted from Fig. 3.22(b),
Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 3
Crystal Structure:• The manner in which the atoms, ions or molecules are spatially arranged.
• There is a large number of different crystal structures all having long range atomic order.
Atomic Hard Sphere Model When describing crystalline structures, atoms of solid spheres and well defined diameters Lattice : 3D array of spheres centers Instructor: Eng. Tamer Haddad
Chapter 3 - 4
Section 3.3 – Unit Cells Unit cell: It is often convenient to subdivide the structure into small repeat entities called unit cell. smallest repetitive volume which contains the complete lattice pattern of a crystal.
• The unit cell is the basic structural unit or building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within. Instructor: Eng. Tamer Haddad
Chapter 3 - 5
Section 3.4 – Metallic Crystal Structures • How can we stack metal atoms to minimize empty space? 2-dimensions
vs.
Instructor: Eng. Tamer Haddad
Chapter 3 - 6
Metallic Crystal Structures • Tend to be densely packed. • Reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. - Electron cloud shields cores from each other
• Have the simplest crystal structures.
We will examine three such structures...
Instructor: Eng. Tamer Haddad
Chapter 3 - 7
Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.
ex: Aluminum, Copper, Silver, and Gold.
Adapted from Fig. 3.1, Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 8
Face Centered Cubic Structure (FCC) a: cube edge length R: atomic radius 2a
Unit cell edge length for facecentered Cubic a = 2R√2
a Adapted from Fig. 3.1(a),
Callister 7e.
• Each corner atom is shared among eight unit cells, whereas a face-centered atom belongs to only two.. 4 atoms/unit cell may be assigned to a given unit cell: 6 face x 1/2 + 8 Instructor: Eng. Tamer Haddad corners x 1/8 Chapter 3 - 9
Two Important Characteristics of a crystal structure:
(I) Coordination Number Coordination number: For metals, each atom has the same number of nearestneighbor or touching atoms, which is the coordination number. • For FCC, Coordination # = 12
Instructor: Eng. Tamer Haddad
Chapter 3 - 10
(II) Atomic Packing Factor (APF) APF =
Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres • APF for a face-centered cubic structure = 0.74 maximum achievable APF
2a
a
atoms volume 4 3 p ( 2a/4) 4 unit cell atom 3 APF = volume 3 a unit cell Instructor: Eng. Tamer Haddad
Chapter 3 - 11
Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
ex: Chromium, iron and tungsten.
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 7e.
2 atoms/unit cell: 1 center + 8 corners x 1/8 Instructor: Eng. Tamer Haddad
Chapter 3 - 12
Atomic Packing Factor: BCC • APF for a body-centered cubic structure = 0.68 3a
a 2a
Adapted from Fig. 3.2(a), Callister 7e.
R
a
Close-packed directions: length = 4R = 3 a
atoms volume 4 3 p ( 3a/4) 2 unit cell atom 3 APF = volume 3 a unit cell Instructor: Eng. Tamer Haddad
Chapter 3 - 13
Hexagonal Close-Packed Structure (HCP) • Not all metals have unit cells with cubic symmetry; hexagonal unit cell are another common metallic crystal structure.
ex: Cadmium, magnesium, titanium, and zinc.
Instructor: Eng. Tamer Haddad
Chapter 3 - 14
Hexagonal Close-Packed Structure (HCP) 6 atoms/unit cell = one-sixth of each of the 12 top and bottom face corner atoms, one-half of each of the 2 center face atoms, and all 3 mid plane interior atoms.
• Coordination # = 12 • APF = 0.74 • c /a = 1.633 with deviations for some HCP metals.
Instructor: Eng. Tamer Haddad
Chapter 3 - 15
Theoretical Density Computations, r Density = r =
r =
where
Mass of Atoms in Unit Cell Total Volume of Unit Cell
nA VC NA
n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol Instructor: Eng. Tamer Haddad
Chapter 3 - 16
Theoretical Density, r • Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n=2
R atoms unit cell
r= volume unit cell
a = 4R/ 3 = 0.2887 nm
a 2 52.00
a3 6.023 x 1023
g mol
rtheoretical = 7.18 g/cm3
ractual
= 7.19 g/cm3
atoms mol
Instructor: Eng. Tamer Haddad
Chapter 3 - 17
Densities of Material Classes In general rmetals > rceramics > rpolymers 30 Why? Metals have...
Ceramics have... • less dense packing • often lighter elements
Polymers have...
r (g/cm3 )
• close-packing (metallic bonding) • often large atomic masses
• low packing density (often amorphous) • lighter elements (C,H,O)
Composites have... • intermediate values
Metals/ Alloys
20
Platinum Gold, W Tantalum
10
Silver, Mo Cu,Ni Steels Tin, Zinc
5 4 3 2
Titanium
Aluminum Magnesium
Graphite/ Ceramics/ Semicond
Composites/ fibers
Polymers
Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass -soda Concrete Silicon Graphite
1
PTFE Silicone PVC PET PC HDPE, PS PP, LDPE
0.5 0.4 0.3
Glass fibers GFRE* Carbon fibers CFRE* Aramid fibers AFRE*
Wood
Data from Table B1, Callister 7e.
Instructor: Eng. Tamer Haddad
Chapter 3 - 18
Section 3.6 – Polymorphism/Allotropy • For metals and non-metals, two or more distinct crystal structures for the same material (polymorphism/allotropy) due to temperature and pressure)
iron system liquid
carbon diamond, graphite
High Pressures
Ambient Conditions
BCC
1538ºC -Fe
FCC
1394ºC -Fe 912ºC
BCC
Instructor: Eng. Tamer Haddad
-Fe
Chapter 3 - 19
Crystalline and Non-crystalline Materials Single Crystals For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption,
Instructor: Eng. Tamer Haddad
Chapter 3 - 20
Polycrystalline Crystals Most crystalline solids are composed of a collection of many small crystals or grains. Ex. Solidification of polycrystalline specimen.
Grain boundary
Instructor: Eng. Tamer Haddad
Chapter 3 - 21
SUMMARY • Atoms may assemble into crystalline or amorphous structures. • Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. • We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). • Materials can be single crystals or polycrystalline.
Instructor: Eng. Tamer Haddad
Chapter 3 - 22
Question 3-10 Some hypothetical metal has the simple cubic crystal structure shown in Figure 3.23. If its atomic weight is 74.5 g/mol and the atomic radius is 0.145 nm, compute its density.
Instructor: Eng. Tamer Haddad
Chapter 3 - 23
