Structure of Materials I
The solid state
The crystalline state
- Definitions and terms
- Space lattice and crystal systems
- Analytical description of the space lattice
Structures
- Single-element structures
- Alloys
- Ionic structures, Structure of inorganic glass
- Macromolecular structures
Crystal defects
1
Arrangement of atoms
The gaseous state
1019 atoms/cm3. no order.
free path length .
10-8..10-6 m (at 293 K and
atmospheric pressure).
W(r) is the probability to “meet”
an atom at a distance r from an
arbitrarily chosen origin
2
Arrangement of atoms
The liquid state
1022 atoms/cm3. short-range order.
position of atoms is a function of
time. atoms are “mobile”.
free path length is about the size of
an atom. particles are in contact and
move cooperatively.
at a distance r = 1 (atom diameter),
there is the largest probability to
meet another atom. at larger
distance, such correlation is missing.
there is just short-range order.
3
Arrangement of atoms
The glassy state
(solid amorphous state)
1022 atoms/cm3. short-range order.
position of atoms is not a function of
time. atoms are immobile. the
structure is frozen.
due to the locally inhomogeneous
structure is the density-probability
function less smooth than in case of
a liquid.
4
Arrangement of atoms
The crystalline state
1023 atoms/cm3. long-range order.
regular arrangement of atoms.
5
The solid state
An amorphous or non-crystalline solid is a solid that lacks the long-range order
that is characteristic of a crystal. In some older books, the term has been used
synonymously with glass.
amorphous
state
degree of order
crystalline
state
A crystal or crystalline solid is a solid material whose constituents
(such as atoms, molecules, or ions) are arranged in a highly ordered
microscopic structure, forming a crystal lattice that extends in all directions.
6
The solid state
degree of order
perfect glass
perfect crystal
inorganic glasses
organic glasses (polymers)
semicrystalline
polymers
metals
crystalline ceramics
short-range order exists
glass = class of materials
glass = state of structure
consist of an amorphous
and a crystalline part
defective crystal
polycrystal
A mesophase is a state of matter intermediate between liquid and solid.
mesophases
liquid crystal
plastic crystal
condis crystal
7
The solid state
Liquid (or melt)
solid amorphous
state
degree of order
solid crystalline
state
A given material can exist in crystalline, amorphous or intermediate
structures as function of the thermo-mechanical history, i.e., the
structure is controlled by the conditions of solidification/processing.
Crystallization requires
1. certain mobility of structural units (atoms/ions/molecules).
2. time for rearrangement of structural units (atoms/ions/molecules).
Consider amorphous metals, slowly crystallizing inorganic glass,
amorphous and semicrystalline poly(ethylene terephthalate), ….
8
The crystalline state: Crystal definition
A crystal is an infinitely extended three-dimensional periodic arrangement
of atoms, ions or molecules. Atoms, ions and molecules are considered as
structural motifs (crystallographic definition).
The quality of exhibiting properties with different values when measured along axes in different directions
A crystal is a homogeneous, anisotropic discontinuum.
A crystal is a naturally grown solid body/matter which is confined in
macroscopic shape/habit by defined planes (historic definition).
9
The crystalline state: Crystal definition
A crystal is an infinitely extended three-dimensional periodic arrangement
of atoms, ions or molecules. Atoms, ions and molecules are considered as
structural motifs (crystallographic definition).
NaCl crystal on aluminum support (black). The bright spots are the chloride ions.
www.iap.tuwien.ac.at/www/surface/STM_Gallery/nonmetals.html
10
The crystalline state: Crystal definition
A crystal is a homogeneous, anisotropic discontinuum.
What is the meaning of “homogeneous”?
In material science, the meaning of “homogeneous” is that
matter/system is uniform in composition (type/concentration of atoms)
and physical properties throughout. The antonym is heterogeneous.
Example:
Immiscible polymer blend
homogeneous
system
Example:
Semicrystalline polymers
composition 1
properties 1
composition 1
properties 1
composition 2
properties 2
composition 1
properties 2
heterogeneous
system
11
The crystalline state: Crystal definition
A crystal is a homogeneous, anisotropic discontinuum.
What is the meaning of “anisotropic”?
Anisotropy is synonymous for the dependence of a (given) property on
the direction within the crystal. The antonym is isotropic. Examples are
the thermal expansion, stiffness, thermal and electrical conductivity, ...
isotropic
system
anisotropic
system
12
The crystalline state: Crystal definition
A crystal is a homogeneous, anisotropic discontinuum.
Crystal habit of Halite (NaCl) and anisotropy of hardness
www.imp.ethz.ch/ABT/Mineral_1.pdf; www.natur-lexikon.com
13
The crystalline state: Mathematical representation/
description by a lattice
Crystal
= periodic, three-dimensional arrangement of
atoms, molecules or ions.
Space/point lattice
= mathematical abstraction of the crystal lattice
to a spatial periodic arrangement of points, with
the points corresponding lattice sites.
the kind of atoms, molecules, or ions is neglected.
Basis
= groups of atoms, molecules, or ions, which are
assigned to each lattice point.
Space/point lattice + Basis = Crystal
14
The crystalline state: Crystal lattice
All possible space/point-lattices can be classified by the length and direction of
three non-coplanar vectors (lattice constants and lattice angles), termed crystal
systems. Such defined body is the smallest repeating unit of the point lattice. It
represents the full symmetry of the crystal, and is termed unit cell.
There exists a total of seven different crystal systems with the lengths a, b, c,
and angles ,  and .
Unit cell and unit cell
parameters
15
The crystalline state:
The seven crystal
systems
16
The crystalline state: The seven crystal systems
cubic
rhombohedral
(trigonal)
tetragonal
orthorhombic
monoclinic
hexagonal
triclinic
17
The crystalline state: The 14 Bravais lattices
It was found by the French crystallographer BRAVAIS in 1849 that a primitive unit cell
(with points, representing the motifs/basis, only at the corners of the unit cell) is not
always advantageous for representation of the highest symmetry of a point lattice.
additionally occupied positions within an primitive unit cell
rhombohedral primitive
vs. face-centered cubic
rhombohedral primitive
vs. body-centered cubic
cubic primitive
The fcc and bcc unit cells show higher number of symmetry elements/operations.
18
The crystalline state:
The 14 Bravais
lattices
P
C
I
F
-
primitive
base-centered
body-centered
face-centered
triclinic
monoclinic
orthorhombic
rhombohedral
hexagonal
tetragonal
cubic
-
P
P, C
P, C, I, F
P
P
P, I
P, I, F
hcp (hexagonal close packing) is not a Bravais lattice because the two possible single lattice sites at half
height are not completely equivalent! Corner and inside atoms do not have the same neighborhood.
hcp can be explained by two‐atom‐motif with the additional atom at [[1/3 2/3 1/2]].
19
The crystalline state: Crystal systems of materials
cubic:
tetragonal:
hexagonal:
rhombohedral:
orthorhombic:
monoclinic:
triclinic:
/-Fe, Ta, Nb, W, Cr, V
(bcc)
Al, Cu, Ag, Au, Ni, -Fe, NaCl (fcc)
Po
(primitive)
Sn
Zn, Mg, Be, Ti, Cd, Co, Zr, Quartz
CaCO3 , Sb, Hg, PB-1
PE
PP, PA 6, cellulose
PET, PA 6.6, talc
Metals
Inorganic
Polymers
20
The crystalline state: Polymorphism
Polymorphism/Allotropy is the capability to exist in different crystal structures.
Many materials form different crystal structures depending on temperature and
pressure. The term ‘Polymorphism’ is applied to any material, the term ‘Allotropy’
is to be applied for elements only (e.g. the allotropy of Fe).
fcc with basis
[[000]] [[¼ ¼ ¼]]
The Allotropy of carbon
21
The crystalline state: Polymorphism
Polymorphism/Allotropy is the capability to exist in different crystal structures.
Many materials form different crystal structures depending on temperature and
pressure. The term ‘Polymorphism’ is applied to any material, the term ‘Allotropy’
is to be applied for elements only (e.g. the allotropy of Fe).
The Allotropy of iron
p = 1 bar
T (°C)
Liquid
Tm =1536
1392
911
R.T.
 – Fe
bcc
(body-centered cubic)
 – Fe
fcc
(face-centered cubic); a0 = 0.3638 nm
 – Fe
bcc
(body-centered cubic); a0 = 0.2895 nm
22
The crystalline state: Analytical description of the lattice
An analytical/mathematical description of the crystal or space lattice is needed
a)
b)
for quantification of the anisotropy of properties, and
for understanding of mechanisms of deformation (motion/displacement of
atoms, ions or molecules).
It includes addressing of points, directions, and planes, based on vector algebra.
23
The crystalline state: Analytical description of the lattice
Points
r = x a0 + y b0 + z c0
r = 1 a0 + 2 b0 + 1 c0
[[ 1 2 1 ]]
draw a vector r from the origin to
the lattice point of interest and
specify the position of the point in
fractions of the unit cell parameters
a0, b0, and c0
c
r = x a0 + y b0 + z c0
x, y, z are the coordinates of the
lattice point, and can be nonintegers.
b
a
identification of lattice points is by
double square brackets [[ x y z ]].
negative indices are indicated with
a bar on top of the number.
commas are not allowed.
24
The crystalline state: Analytical description of the lattice
Directions
c
c
shift through origin
nearest lattice point is [[ 1 0 1 ]]
direction is [ 1 0 1 ]
b
a
b
a
draw a line through the origin, being parallel to the given direction.
determine the coordinates u’, v’, w’ of any point on this line through the origin.
u’, v’, w’ can be non-integers. must be converted to a set of smallest integers u, v, w.
identification by single square brackets [ u v w ]. negative indices are indicated with a bar
on top of the number. commas are not allowed.
25
The crystalline state: Analytical description of the lattice
Planes
planes are specified by MILLER indices h, k, l.
parallel planes have identical indices.
if a plane passes through the origin, then a parallel plane must be constructed (or
a new origin must be defined).
the crystallographic plane intersects or parallels each of the three axes, and the
intersections m, n, o are determined in terms of the unit-cell parameters a0, b0, c0.
calculate the reciprocals 1/m, 1/n, 1/o. a plane which is parallel to an axis
intersects at infinity distance and the reciprocal is zero.
the set of reciprocals is changed to a set of smallest integers h, k, l.
identification of lattice directions is by single parenthesis ( h k l ).
negative indices are indicated as bar on top of the number.
commas are not allowed.
26
The crystalline state: Analytical description of the lattice
Planes
determine the intersections of the
plane with axes in fractions of the
unit cell parameters
m = 2 a0
n = 3 b0
p = 3 c0
c
calculate the reciprocals
1/m = 1/2
1/n = 1/3
1/p = 1/3
b
a
calculate the smallest integers
(multiply by 6)
h=3
k=2
l =2
(hk l) = (322)
27
The crystalline state: Analytical description of the lattice
Planes
c
c
b
a
c
b
a
m = 1 a0
n = ∞ b0
p = ∞ c0
1/m = 1/1
1/n = 1/∞
1/p = 1/∞
(hkl) = (100)
b
a
m = 1 a0 1/m = 1/1
n = 1 b0 1/n = 1/1
p = ∞ c0 1/p = 1/∞
(hkl) = (110)
Homework: Zones, Crystal Forms
m = 1 a0
n = 1 b0
p = 1 c0
1/m = 1/1
1/n = 1/1
1/p = 1/1
(hkl) = (111)
28
The crystalline state: Analytical description of the lattice
Interplanar spacing
The interplanar spacing d(hkl) is the shortest distance between two parallel planes.
c
Normal of lattice
plane (111), N (111)
interplanar distance for
an orthogonal system
d(hkl) 
b
1
h

a
 0
2

k
  

b

 0
2

 l
  

c

 0




2
a
d(hkl)
29
Single-element structures
Most of the chemical elements
crystallize in the following structures:
30 %
30 %
35 %
body-centered cubic
face-centered cubic
hexagonal close-packed
Characteristics of crystal structures
number of atoms per unit cell
n
atomic packing density
APD = n × V atom / V unit-cell
coordination number
C (number of neighbors at equal distance)
30
Single-element structures
(100)
a0 = 2r
c
Simple cubic
n
APD
C
b
=1
= 0.52
=6
Polonium (Po)
a
31
Single-element structures
(110)
Body-centered cubic
a0
c
n
APD
C
b
a
=2
= 0.68
=8
Tungsten (W)
Molybdenum (Mo)
Tantalum (Ta)
-Iron (-Fe)
a0 sqrt(2)
32
Single-element structures
(100)
a0
Face-centered cubic
c
n
APD
C
b
a
=4
= 0.74
= 12
a0
Silver (Ag)
Gold (Au)
Aluminum (Al)
Copper (Cu)
-Iron (-Fe)
most dense packing of spherical motifs
33
Single-element structures
(001)
Hexagonal close-packed
c
n
APD
C
=2
= 0.74
= 12
a0 = 2r
Magnesium (Mg)
Beryllium (Be)
Titanium (Ti)
b
6
1
2
7
8
a
3
9
5
4
most dense packing of spherical motifs
34
Single-element structures
face-centered cubic (fcc)
hexagonal close-packed (hdp)
A
c
B
b
c
A
C
B
A
A
a
b
a
most dense packing of spherical motifs
Stacking sequence in face-centered cubic and
hexagonal close-packed crystals
35
Single-element structures
most dense packing of spherical motifs
Stacking sequence in face-centered cubic (right) and
hexagonal close-packed crystals (left)
36