LINEAR DENSITYWhen planes slip over each other, slip

takes place in the direction of closest packing of atoms on the planes.

The linear density of a crystal direction [h k l] is determined as:

δ[h k l] = length# of atoms

2

ex: linear density of Al in [110] direction 

a = 0.405 nm

FCC: Linear Density• Linear Density of Atoms LD = 

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD

Adapted fromFig. 3.1(a),Callister & Rethwisch 8e.

BCC: Linear Density

Calculate the linear density for the following directions in terms of R:

a. [100]

b. [110]

c. [111]

PLANAR DENSITY When slip occurs under stress, it takes

place on the planes on which the atoms are most densely packed.

δ(hkl) = area# of atoms in a plane

Example: FCC unit cell

a2

(100)

z

y

x

δ(100)=4*1/4+1 a2

= a2

2

a=1

r24 δ(100)= 4r2

Quasicrystal A quasiperiodic crystal, or, in short, quasicrystal, is

a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. 

A material with sharp diffraction peaks with a forbidden symmetry by crystallography.

They have long-range positional order without periodic translational symmetry.

Shechtman was awarded the Nobel Prize in Chemistry in 2011 for his work on quasicrystals. “His discovery of quasicrystals revealed a new principle for packing of atoms and molecules.” 

HOW IS A QUASICRYSTAL DIFFERENT FROM A CRYSTAL?

ORDERED PERIODIC

QC ARE ORDERED

STRUCTURES WHICH ARE NOT PERIODIC

CRYSTALS

QC

AMORPHOUS

UNIVERSE

PARTICLES

ENERGYSPACE

FIELDS

GAS

BAND STRUCTURE

AMORPHOUS

ATOMIC NON-ATOMIC

STATE / VISCOSITY

SOLID LIQUIDLIQUID

CRYSTALS

QUASICRYSTALS CRYSTALSRATIONAL APPROXIMANTS

STRUCTURE

NANO-QUASICRYSTALS NANOCRYSTALS

SIZE

Where are quasicrystals in the scheme of things?

Quasicrystals

x

( )

1 -101 1

4

51

23

x

Al6Mn

1 m

Types of Quasicrystals

Two types: Quasiperiodic in Two Dimensions: This is also

referred to as polygonal or dihedral quasicrystals. It has sub elements namely octagonal, decagonal and dodecagonal. This has one periodic direction which lies perpendicular to the quasiperodic layers.

Quasiperiodic in Three Dimensions: This type has no periodic direction and icosahedral quasicrystals fall under this type.

New type: Icosahedral quasicrystals with broken symmetry fall in this category.

Characteristics

• hard and brittle• low surface energy

(non-stick)• high electrical

resistivity • high thermal

resistivity• high thermoelectric

power• …

Quasicrystalline coating method shows promise for cookware and prosthetic devices.

APPLICATIONS OF QUASICRYSTALSAPPLICATIONS OF QUASICRYSTALS

• WEAR RESISTANT COATING (Al-Cu-Fe-(Cr))WEAR RESISTANT COATING (Al-Cu-Fe-(Cr))• NON-STICK COATING (Al-Cu-Fe)NON-STICK COATING (Al-Cu-Fe)• THERMAL BARRIER COATING (Al-Co-Fe-Cr)THERMAL BARRIER COATING (Al-Co-Fe-Cr)• HIGH THERMOPOWER (Al-Pd-Mn)HIGH THERMOPOWER (Al-Pd-Mn)• IN POLYMER MATRIX COMPOSITES (Al-Cu-IN POLYMER MATRIX COMPOSITES (Al-Cu-

Fe)Fe)• SELECTIVE SOLAR ABSORBERS (Al-Cu-Fe-SELECTIVE SOLAR ABSORBERS (Al-Cu-Fe-

(Cr))(Cr))• HYDROGEN STORAGE (Ti-Zr-Ni)HYDROGEN STORAGE (Ti-Zr-Ni)

Application

Quasicrystals have been used in surgical instruments, LED lights and non stick frying pans. They have poor heat conductivity, which makes them good insulators.

Quasicrystals vary depending on their direction. One direction might conduct electricity easily - another

direction might not conduct electricity at all. Quasicrystalline cylinder liners and piston coatings in

motor-car engines would undoubtedly result in reduced air pollution and increased engine lifetimes.

Assignment 1

Chapter 2. Reciprocal Lattice

Issues that are addressed in this chapter include:

Bragg law Scattered wave amplitude Brillouin Zones Fourier analysis of the basis

• The set of all waves vectors k that yield plane wave with the periodicity of a given Bravais lattice.

Reciprocal lattice

• A diffraction pattern is not a direct representation of the crystal lattice

• The diffraction pattern is a representation of the reciprocal lattice

Reciprocal Lattice/Unit CellsWe will use a monoclinic unit cell to avoid orthogonal axes

We define a plane and consider some lattice planes

(001)

(100)

(002)

(101)

(101)

(102)