Chapter Outline

How do atoms arrange themselves to form solids?

• Fundamental concepts and language

• Unit cells
• Crystal structures
¾ Simple cubic
¾ Face-centered cubic
¾ Body-centered cubic
¾ Hexagonal close-packed
• Close packed crystal structures
• Density computations
• Types of solids
Single crystal
Polycrystalline
Amorphous

Chapter 3, Structure of solids CVLE224 Materials Science 1

 Types of Solids
Crystalline material: atoms self-organize in a periodic
array
Single crystal: atoms are in a repeating or periodic array
over the entire extent of the material
Polycrystalline material: comprised of many small
crystals or grains

Amorphous: disordered – lack of a systematic atomic

arrangement
Crystalline Amorphous

SiO2

Chapter 3, Structure of solids CVLE224 Materials Science 2

 Crystal structures
Why do atoms assemble into ordered structures (crystals)?
Energy of interatomic bond

Interatomic distance
0

Let’s consider nondirectional

bonding (like in metals)

Energy of the crystal < Energy of the amorphous solid

Chapter 3, Structure of solids CVLE224 Materials Science 3

 Crystal structure

To discuss crystalline structures it is useful to consider

atoms as being hard spheres with well-defined radii. In this
hard-sphere model, the shortest distance between two like
atoms is one diameter of the hard sphere.

2R
- hard-sphere model

We can also consider crystalline structure as a lattice of

points at atom/sphere centers.

Chapter 3, Structure of solids CVLE224 Materials Science 4

 Unit Cell
The unit cell is a structural unit or building block that can
describe the crystal structure. Repetition of the unit cell
generates the entire crystal.

Example: 2D honeycomb net can

be represented by translation of
two adjacent atoms that form a unit
cell for this 2D crystalline structure

Example of 3D crystalline structure:

Different choices of unit cells possible, we will consider

parallelepiped unit cell with highest level of symmetry

Chapter 3, Structure of solids CVLE224 Materials Science 5

 Metallic Crystal Structures

¾ Metals are usually (poly)crystalline; although formation

of amorphous metals is possible by rapid cooling

¾ As we learned in Chapter 2, the atomic bonding in metals

is non-directional ⇒ no restriction on numbers or
positions of nearest-neighbor atoms ⇒ large number of
nearest neighbors and dense atomic packing

¾ Atomic (hard sphere) radius, R, defined by ion core

radius - typically 0.1 - 0.2 nm

¾ The most common types of unit cells are

• faced-centered cubic (FCC)
• body-centered cubic (BCC)
• hexagonal close-packed (HCP).

Simple cubic (SC)is very rare

Chapter 3, Structure of solids CVLE224 Materials Science 6

Simple Cubic (SC) Crystal Structure p

RareGXHWRORZSDFNLQJGHQVLW\
2QO\3RSRORQLXPKDVWKLVVWUXFWXUH

$WRPVDUHORFDWHGDWHDFKRIWKHFRUQHUV
 Simple Cubic (SC) Crystal Structure p

¾ The hard spheres touch one another on the edge ⇒ the

cube edge length, a= 2R
¾ The coordination number, CN = the number of closest
neighbors to which an atom is bonded = number of
touching atoms, CN = 6
¾ Number of atoms per unit cell, n = 1. (For an atom that
is shared with m adjacent unit cells, we only count a
:fraction of the atom, 1/m). In Simple unit cell we have

corner atoms shared by eight cells: 8×1/8=1

¾ Atomic packing factor, APF = fraction of volume

occupied by hard spheres = (Sum of atomic volumes)/
(Volume of cell) = 0.52
Chapter 3, Structure of solids CVLE224 Materials Science 8
Face-Centered Cubic (FCC) Crystal Structure (I)
¾ Atoms are located at each of the corners and on the
centers of all the faces of cubic unit cell
¾ Cu, Al, Ag, Au have this crystal structure

Two representations
of the FCC unit cell

Chapter 3, Structure of solids CVLE224 Materials Science 7

 Face-Centered Cubic Crystal Structure (II)

¾ The hard spheres touch one another across a face

diagonal ⇒ the cube edge length, a= 2R√2
¾ The coordination number, CN = the number of closest
neighbors to which an atom is bonded = number of
touching atoms, CN = 12
¾ Number of atoms per unit cell, n = 4. (For an atom
that is shared with m adjacent unit cells, we only count a
fraction of the atom, 1/m). In FCC unit cell we have:
6 face atoms shared by two cells: 6×1/2 = 3
8 corner atoms shared by eight cells: 8×1/8 = 1
¾ Atomic packing factor, APF = fraction of volume
occupied by hard spheres = (Sum of atomic
volumes)/(Volume of cell) = 0.74 (maximum possible)
Chapter 3, Structure of solids CVLE224 Materials Science 8
 Face-Centered Cubic Crystal Structure (III)

Let’s calculate the atomic packing factor for FCC crystal

a = 2R 2
R

a
APF = (Sum of atomic volumes)/(Volume of unit cell)

4 3
Volume of 4 hard spheres in the unit cell: 4 × πR
3
Volume of the unit cell: a 3 = 16 R 3 2

16 3
APF = πR 16 R 3 2 = π 3 2 = 0.74
3
maximum possible packing of hard spheres

Chapter 3, Structure of solids CVLE224 Materials Science 9

 Face-Centered Cubic Crystal Structure (IV)
¾ Corner and face atoms in the unit cell are equivalent
¾ FCC crystal has APF of 0.74, the maximum packing for
a system equal-sized spheres ⇒ FCC is a close-packed
structure
¾ FCC can be represented by a stack of close-packed
planes (planes with highest density of atoms)

Chapter 3, Structure of solids CVLE224 Materials Science 10

 Body-Centered Cubic (BCC) Crystal Structure (I)
Atom at each corner and at center of cubic unit cell
Cr, α-Fe, Mo have this crystal structure

Chapter 3, Structure of solids CVLE224 Materials Science 11

 Body-Centered Cubic Crystal Structure (II)

¾ The hard spheres touch one another along cube diagonal

⇒ the cube edge length, a= 4R/√3
¾ The coordination number, CN = 8
¾ Number of atoms per unit cell, n = 2
Center atom (1) shared by no other cells: 1 x 1 = 1
8 corner atoms shared by eight cells: 8 x 1/8 = 1
¾ Atomic packing factor, APF = 0.68
¾ Corner and center atoms are equivalent

Chapter 3, Structure of solids CVLE224 Materials Science 12

 Hexagonal Close-Packed Crystal Structure (I)

¾ HCP is one more common structure of metallic crystals

¾ 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
¾ Cd, Mg, Zn, Ti have this crystal structure

Chapter 3, Structure of solids CVLE224 Materials Science 13

 Hexagonal Close-Packed Crystal Structure (II)

¾ The hard spheres touch one another on the edge ⇒ the

cube edge length, a= 2R
¾ Unit cell has two lattice parameters a and c. Ideal ratio
c/a = 1.633
¾ The coordination number, CN = 12 (same as in FCC)
¾ Number of atoms per unit cell, n = 6.
3 mid-plane atoms shared by no other cells: 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

¾ Atomic packing factor, APF = 0.74 (same as in FCC)

¾ All atoms are equivalent

Chapter 3, Structure of solids CVLE224 Materials Science 14

 Close-packed Structures (FCC and HCP)
¾ Both FCC and HCP crystal structures have atomic
packing factors of 0.74 (maximum possible value)
¾ Both FCC and HCP crystal structures may be generated
by the stacking of close-packed planes
¾ The difference between the two structures is in the
stacking sequence

HCP: ABABAB... FCC: ABCABCABC…

Chapter 3, Structure of solids CVLE224 Materials Science 15

 FCC: Stacking Sequence ABCABCABC...

Third plane is placed above the “holes” of the first plane

not covered by the second plane

Chapter 3, Structure of solids CVLE224 Materials Science 16

 HCP: Stacking Sequence ABABAB...

Third plane is placed directly above the first plane of atoms

Chapter 3, Structure of solids CVLE224 Materials Science 17

 Density Computations

Since the entire crystal can be generated by the repetition

of the unit cell, the density of a crystalline material, ρ = the
density of the unit cell = (atoms in the unit cell, n ) × (mass
of an atom, M) / (the volume of the cell, Vc)

Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)

Mass of an atom, M = Atomic weight, A, in amu (or g/mol)

is given in the periodic table. To translate mass from amu
to grams we have to divide the atomic weight in amu by
the Avogadro number NA = 6.023 × 1023 atoms/mol

The volume of the cell, Vc = a3 (FCC and BCC)

a = 2R√2 (FCC); a = 4R/√3 (BCC)
where R is the atomic radius

Thus, the formula for the density is: nA

ρ=
Vc N A

Atomic weight and atomic radius of many elements you

can find in the table at the back of the textbook front cover.

Chapter 3, Structure of solids CVLE224 Materials Science 18

 Polymorphism and Allotropy
Some materials may exist in more than one crystal
structure, this is called polymorphism. If the material is an
elemental solid, it is called allotropy.
An example of allotropy is carbon, which can exist as
diamond, graphite, and 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) and carbon
nanotube.
Chapter 3, Structure of solids CVLE224 Materials Science 19
 Single Crystals and Polycrystalline Materials

Single crystal: atoms are in a repeating or periodic array

over the entire extent of the material
Polycrystalline material: comprised of many small
crystals or grains. The grains have different
crystallographic orientation. There exist atomic mismatch
within the regions where grains meet. These regions are
called grain boundaries.

Grain Boundary

Chapter 3, Structure of solids CVLE224 Materials Science 20

 Polycrystalline Materials

Atomistic model of a nanocrystalline solid by Mo Li, JHU

Chapter 3, Structure of solids CVLE224 Materials Science 21

 Polycrystalline Materials

Simulation of annealing of a polycrystalline grain structure

from http://cmpweb.ameslab.gov/cmsn/microevolproj.html (link is dead)

Chapter 3, Structure of solids CVLE224 Materials Science 22

 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. This causes
anisotropy in the properties of crystals, for instance, the
deformation depends on the direction in which a stress is
applied.

In some polycrystalline materials, grain orientations are

random, so bulk material properties are isotropic

Some polycrystalline materials have grains with preferred

orientations (texture), so properties are dominated by those
relevant to the texture orientation and the material exhibits
anisotropic properties

Chapter 3, Structure of solids CVLE224 Materials Science 23

 Non-Crystalline (Amorphous) Solids

In amorphous solids, there is no long-range order. But

amorphous does not mean random, in many cases there is
some form of short-range order.

Schematic picture of
amorphous SiO2 structure

Amorphous structure from

simulations by E. H. Brandt

Chapter 3, Structure of solids CVLE224 Materials Science 24

 Summary

Make sure you understand language and concepts:

¾ Allotropy
¾ Amorphous
¾ Anisotropy
¾ Atomic packing factor (APF)
¾ Body-centered cubic (BCC)
¾ Coordination number
¾ Crystal structure
¾ Crystalline
¾ Face-centered cubic (FCC)
¾ Grain
¾ Grain boundary
¾ Hexagonal close-packed (HCP)
¾ Isotropic
¾ Lattice parameter
¾ Non-crystalline
¾ Polycrystalline
¾ Polymorphism
¾ Single crystal
¾ Unit cell

Chapter 3, Structure of solids CVLE224 Materials Science 25