Lecture 02
Fundamental Properties of
Solids
14 Bravais Lattices divided into 7 Crystal Systems
A Symmetry based concept
Crystal System
1
Cubic
2
Tetragonal
3
We will take up these cases one by one
(hence do not worry!)
‘Translation’ based concept
Some guidelines apply
Shape of UC
Bravais Lattices
P
I
F
Cube



Square Prism (general height)




Orthorhombic Rectangular Prism (general height)
4
Hexagonal
5
Trigonal
6
Monoclinic
7
Triclinic
120 Rhombic Prism

Parallopiped (Equilateral, Equiangular)

Parallogramic Prism

Parallelepiped (general)

P
Primitive
I
Body Centred
F
Face Centred
C
A/B/C- Centred

C


3-Dimensional Unit Cells
Common Unit Cells with Cubic Symmetry
Simple Cubic
(SC)
Body Centered Cubic
(BCC)
Face Centered Cubic
(FCC)
1 atom/unit cell
2 atoms/unit cell
4 atoms/unit cell
(8 x 1/8 = 1)
(8 x 1/8 + 1 = 2)
(8 x 1/8 + 6 x 1/2 = 4)
coordination number 6 coordination number 8 coordination number 12
Base Centered Cubic
Atom/unit cell:
Coordination number:
Primitive & Conventional Unit Cells
Unıt Cell Types
Primitive
Conventional
(Non-primitive)
A single lattice point per cell
More than one lattice point per cell
The smallest area in 2 dimensions, or
The smallest volume in 3 dimensions
Volume (area) = integer multiple of
that for primitive cell
Simple Cubic (sc)
Body Centered Cubic (bcc)
Conventional Cell = Primitive cell
Conventional Cell ≠ Primitive cell
1
Cubic
P

Cube
I

F

C
I
P
a bc
Lattice point
      90
F
2
Tetragonal
Square Prism (general height)
P
I


F
C
I
P
a bc
      90
3
Orthorhombic Rectangular Prism (general height)
P
I
F
C




One convention
abc
I
P
Note the position of
‘a’ and ‘b’
abc
      90
F
C
P
4
Hexagonal
120 Rhombic Prism
a bc
    90,   120
A single unit cell (marked in blue)
along with a 3-unit cells forming a
hexagonal prism
Note: there is only one type of hexagonal
lattice (the primitive one)

I
F
C
P
5
Trigonal
Parallelepiped (Equilateral, Equiangular)
I

Rhombohedral
a bc
      90
Note the position of the origin
and of ‘a’, ‘b’ & ‘c’
Symmetry of Trigonal lattices
2
3
m
F
C
P
6
Monoclinic

Parallogramic Prism
One convention
abc
abc
    90  
Note the position of
‘a’, ‘b’ & ‘c’
I
F
C

P
7
Triclinic

Parallelepiped (general)
abc
   
I
F
C
Closed-packed structures
 There are an infinite number of ways
to organize spheres to maximize the
packing fraction.
The centres of
spheres at A, B,
and C positions
(from Kittel)
There are different ways you can pack spheres together.
This shows two ways, one by putting the spheres in an
ABAB… arrangement, the other with ACAC…. (or any
combination of the two works)
ENERGY AND PACKING
• Non dense, random packing
• Dense, regular packing
Dense, regular-packed structures tend to have
lower energy.
METALLIC CRYSTALS
• tend to be densely packed.
• have several 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.
• have the simplest crystal structures. 74 elements
have the simplest crystal structures – BCC, FCC
and HCP
We will look at three such structures...
SIMPLE CUBIC
STRUCTURE
(SC)
• Rare due to poor packing (only Polonium(Po)
has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
4
ATOMIC PACKING FACTOR
• APF for a simple cubic structure = 0.52
Adapted from Fig. 3.19,
Callister 6e.
5
FACE CENTERED CUBIC
STRUCTURE (FCC)
• Close packed directions are face diagonals.
--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
• Coordination # = 12
Adapted from Fig. 3.1(a),
Callister 6e.
6
ATOMIC PACKING FACTOR: FCC
• APF for a body-centered cubic structure = 0.74
a
Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
Adapted from
Fig. 3.1(a),
Callister 6e.
7
BODY CENTERED CUBIC
STRUCTURE (BCC)
• Close packed directions are cube diagonals.
--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
• Coordination # = 8
Adapted from Fig. 3.2,
Callister 6e.
8
ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
R
Adapted from
Fig. 3.2,
Unit cell contains:
1 + 8 x 1/8
= 2 atoms/unit cell
a
Callister 6e.
9
BCC
Chromium, iron, tungsten exhibit
bcc structure
Two atoms are associated with
each BCC unit cell
The coordination number for the
BCC is 8
the atomic packing factor for BCC
lower—0.68 versus 0.74 (FCC)
Packing Factor – FCC vs
BCC
HEXAGONAL CLOSE-PACKED STRUCTURE
(HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
A sites
B sites
A sites
Adapted from Fig. 3.3,
Callister 6e.
• Coordination # = 12
• APF = 0.74
10
Q. Most crystalline solids do not show primitive
cubic packing. Why is this the case?
Crystalline solids have atoms arranged such that the available
volume in the unit cell is used to the largest extent possible.
In other words, atoms are arranged in a unit cell such that the
ratio of volume occupied by the atoms to the total volume of
the unit cell is highest. The primitive cubic packing is not
efficient in terms of its packing density. Look at the image
below showing a primitive cubic packing
The primitive cubic packing only
allows a packing fraction of 52.4%.
This implies that the primitive cubic
packing has a large amount of unused
volume and is not the most efficient
packing configuration.
As a result, most materials do not
show primitive cubic packing
THEORETICAL DENSITY, / Density
Computations
Example: Copper
Data from Table):
• crystal structure = FCC: 4 atoms/unit cell
• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)
• atomic radius R = 0.128 nm (1 nm = 10 -7cm)
Result: theoretical Cu = 8.89 g/cm3
Compare to actual: Cu = 8.94 g/cm3
11
Characteristics of Selected Elements at
20C
Density
At. Weight
Element
Symbol (amu)
Aluminum
Al
26.98
Argon
Ar
39.95
Barium
Ba
137.33
Beryllium
Be
9.012
Boron
B
10.81
Bromine
Br
79.90
Cadmium
Cd
112.41
Calcium
Ca
40.08
Carbon
C
12.011
Cesium
Cs
132.91
Chlorine
Cl
35.45
Chromium Cr
52.00
Cobalt
Co
58.93
Copper
Cu
63.55
Flourine
F
19.00
Gallium
Ga
69.72
Germanium Ge
72.59
Gold
Au
196.97
Helium
He
4.003
Hydrogen
H
1.008
(g/cm3)
2.71
-----3.5
1.85
2.34
-----8.65
1.55
2.25
1.87
-----7.19
8.9
8.94
-----5.90
5.32
19.32
-----------
Atomic radius
(nm)
0.143
-----0.217
0.114
Adapted from
-----Table, "Charac-----teristics of
0.149 Selected
0.197 Elements",
inside front
0.071 cover,
0.265 Callister 6e.
-----0.125
0.125
0.128
-----0.122
0.122
0.144
----------12
Q. 1: The radius of a polonium atom is 168 picometers
(pm). α-polonium crystallizes in a primitive cubic unit cell,
with a cell edge length of 336 pm. Assuming an atomic mass
of 209 g/mol, calculate the theoretical density of polonium in
g/cc
Q. 2: Fe has an atomic mass of 55.85 g/mol and an atomic
radius of 124 pm. What is the side of the iron unit cell in pm?
What is the expected density of alpha iron in such a case,
assuming a BCC structure?
Q. 3: Steel is an alloy of iron and carbon. When low
carbon (predominantly iron) containing steel (such as
1018 carbon steel) is heated, it transforms from a
BCC to an FCC structure. Based on this information,
will you expect steel to contract or expand when it is
heated. Justify
Hexagonal Close Packed
Cell of an HCP lattice is visualized as a top
and bottom plane of 7 atoms, forming a
regular hexagon around a central atom.
In between these planes is a half-hexagon
of 3 atoms.
Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl
Hexagonal Close Packed
There are two lattice parameters in HCP,
a and c, representing the basal and height
parameters respectively. In the ideal
case, the c/a ratio is 1.633, however,
deviations do occur.
Coordination number for HCP are exactly
the same as those for FCC: 12
This is because they are both considered
close packed structures.
Hexagonal Close Packed (HCP) Structure:
(A Simple Hexagonal Bravais Lattice with a 2 Atom Basis)
The HCP lattice is not a Bravais
lattice, because the orientation of
the environment of a point varies
from layer to layer along the c-axis.
Structure of NaCl
Structure of Cesium
Chloride(CsCl)
Carbon structures
Zinic Sulfide Structure
Why are planes in a lattice important?
(A) Determining crystal structure
Diffraction methods directly measure the
distance between parallel planes of lattice
points. This information is used to
determine the lattice parameters in a
crystal and measure the angles between
lattice planes.
(B) Plastic deformation
Plastic (permanent) deformation in metals
occurs by the slip of atoms past each
other in the crystal. This slip tends to
occur preferentially along specific lattice
planes in the crystal. Which planes slip
depends on the crystal structure of the
material.
 (C) Transport Properties

In certain materials, the atomic
structure in certain planes causes the
transport of electrons and/or heat
to be particularly rapid in that plane,
and relatively slow away
from
the plane.
 Example: Graphite
 Conduction of heat is more rapid in
the sp2 covalently bonded lattice
planes than in the direction
perpendicular to those planes.
 Example: YBa2Cu3O7
superconductors
 Some lattice planes contain only Cu
and O. These planes conduct pairs of
electrons (called Cooper pairs) that
are responsible for superconductivity.
These superconductors are electrically
insulating in directions perpendicular
to the Cu-O lattice planes.
(GPa)
Unit cell
Assuming an ideal infinite crystal we define a unit cell by
c
Unit cell: a volume in space that
fills space entirely when translated
by all lattice vectors.
The obvious choice:
a parallelepiped defined by a, b, c,
three basis vectors with

the best a, b, c are as orthogonal
as possible
the cell is as symmetric as
possible (14 types)


a
A unit cell containing one lattice point is called primitive cell.
b