Chapter 1: Crystal Structure
Chapter 1: Crystal Structure
The Nobel “Booby” Prize!
See the “Ig Nobel” Prize discussed at: http://improbable.com/ig/
The (Common) Phases of Matter
Matter
Gases
Liquids &
Liquid Crystals
Solids
This doesn’t include Plasmas, but
these are the “common” phases!!
“Condensed Matter” includes both of these.
We’ll focus on Solids!
Gases
• Gases have atoms or molecules that do not
bond to one another in a range of pressure,
temperature & volume. Also, these
molecules have no particular order & they
move freely within a container.
Liquids & Liquid Crystals
• Similar to gases, Liquids have no atomic or
molecular order & they assume the shape of
their containers.
• Applying low levels of thermal energy can
easily break the existing weak bonds.
• Liquid Crystals have mobile
molecules, but a type of long
range order can exist; the
molecules have a permanent
dipole. Applying an electric field
rotates the dipole & establishes
order within the collection of
molecules.
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Solids
• Solids consist of atoms or molecules
undergoing thermal motion about their
equilibrium positions, which are at fixed
points in space.
• Solids can be crystalline, polycrystalline, or
amorphous.
• Solids (at a given temperature, pressure, volume)
have stronger interatomic bonds than liquids.
• So, Solids require more energy to break the
interatomic bonds than liquids.
Crystal Structure
Topics
1. Periodic Arrays of Atoms
2. Fundamental Types of Lattices
3. Index System for Crystal Planes
4. Simple Crystal Structures
5. Direct Imaging of Crystal Structure
6. Non-ideal Crystal Structures
7. Crystal Structure Data
Objectives
At the end of this Chapter, you should:
1. Be able to identify a unit cell in a
symmetrical pattern.
2. Know that (in 3 dimensions) there are
7 (& ONLY 7!!)
Possible unit cell shapes.
3. Be able to define cubic, tetragonal,
orthorhombic & hexagonal unit cell shapes
Periodic Arrays of Atoms
Experimental Evidence of periodic structures.
(See Kittel, Fig. 1.)
The external appearance of crystals gives some clues to this. Fig. 1
shows that when a crystal is cleaved, we can see that it is built up of
identical “building blocks”. Further, the early crystallographers noted
that the index numbers that define plane orientations are exact integers.
Cleaving a Crystal
Elementary Crystallography
Solid Material
Types
Crystalline
Single
Crystals
Polycrystalline
Amorphous
Crystals are Everywhere!
More Crystals
Early ideas
• Crystals are solid - but solids are not
necessarily crystalline
• Crystals have symmetry (Kepler!!!)
and long range order
• Spheres and small shapes can be
packed to produce regular shapes
(Hooke, Hauy)
The Three General Types of Solids
Single Crystal, Polycrystalline,
Amorphous
•Each type is characterized by the
size of the ordered region within the
material. An ordered region is a
spatial volume in which atoms or
molecules have a regular geometric
arrangement or periodicity.
All Solids!
• All solids have “resistance” to changes in
both shape and volume
• Solids can be Crystalline or Amorphous
• Crystals are solids that consist of a
periodic array of atoms, ions, or
molecules
– If this periodicity is preserved over “large”
(macroscopic) distances the solid has “Longrange Order”
• Amorphous solids do not have LongRange Order
– Short Range Order
Solids
• Crystals:
– Short-range Order
– Long-range Order
• Amorphous solids:
– ~Short-range Order
– No Long-range Order
Solids
• Different solids can have the
same geometrical
arrangements of atoms
– Properties are determined by
crystal structure, i.e. both crystal
lattice and basis are important
• Examples:
– Si, Diamond (C), GaAs, ZnSe have the
same geometry
– Si and C (Diamond) Form “Diamond
Structure”
– GaAs or ZnSe form a structure called
“Zinc Blende”
Solids
• Different arrangements of atoms (even the
same atoms) give different properties
Single layer is graphene
Crystalline Solids
• A Crystalline Solid is the solid form of a substance
in which the atoms or molecules are arranged in a
definite, repeating pattern in three dimensions.
• Single Crystals, ideally have a high degree of
order, or regular geometric periodicity,
throughout the entire volume of the material.
•
A Single Crystal has an atomic structure that
repeats periodically across its whole volume. Even
at infinite length scales, each atom is related to every
other equivalent atom in the structure by translational
symmetry.
Single Crystals
Single Pyrite Amorphous
Solid
Crystal
Polycrystalline Solids
• A Polycrystalline Solid is made up of an aggregate of
many small single crystals (crystallites or grains).
Polycrystalline materials have a high degree of order
over many atomic or molecular dimensions. These
ordered regions, or single crystal regions, vary in size &
orientation with respect to one another. These regions
are called grains (or domains) & are separated from one
another by grain boundaries.
Polycrystalline
Pyrite Grain
Polycrystalline Solids
• In Polycrystalline Solids, the atomic order
can vary from one domain to the next. The
grains are usually 100 nm - 100 microns in
diameter. Polycrystals with grains that are
< 10 nm in diameter are called nanocrystallites.
Polycrystalline
Pyrite Grain
Amorphous Solids
• Amorphous (Non-crystalline) Solids are
composed of randomly orientated atoms, ions, or
molecules that do not form defined patterns or lattice
structures. Amorphous materials have order only
within a few atomic or molecular dimensions. They
do not have any long-range order, but they have varying
degrees of short-range order. Examples of amorphous
material include amorphous silicon, plastics, & glasses.
Crystals
• The periodic array of atoms, ions, or molecules
that form the solid is called Crystal Structure
Crystal Structure =
Space (Crystal) Lattice + Basis
– Space (Crystal) Lattice is a regular periodic
arrangement of points in space, and is purely
mathematical abstraction
– Crystal Structure is formed by “putting” the
identical atoms (group of atoms) in the points
of the space lattice
– This group of atoms is the Basis
Departures From the “Perfect Crystal”
• A “Perfect Crystal” is an idealization that does not exist
in nature. In some ways, even a crystal surface is an
imperfection, because the periodicity is interrupted there.
• Each atom undergoes thermal vibrations around their
equilibrium positions for temperatures T > 0K. These can
also be viewed as “imperfections”.
• Real Crystals always have
foreign atoms (impurities),
missing atoms (vacancies), &
atoms in between lattice sites
(interstitials) where they should
not be. Each of these spoils the
perfect crystal structure.
Crystallography
Crystallography ≡ The branch of science that deals with
the geometric description of crystals & their internal
arrangements. It is the science of crystals & the math used to
describe them. It is a VERY OLD field which pre-dates Solid
State Physics by about a century! So (unfortunately, in some
ways) much of the terminology (& theory notation) of Solid State
Physics originated in crystallography. The purpose of Ch. 1 of
Kittel’s book is mainly to introduce this terminology to you.
Solid State Physics
Started in the early 20th Century when the fact that
Crystals Can Diffract X-rays
was discovered.
•Around that same time the new theory of
Quantum Mechanics
was being accepted & applied to various problems.
Some of the early problems it was applied to were
the explanation of observed X-ray diffraction
patterns for various crystals & (later) the behavior
of electrons in a crystalline solid.
Crystallography
A Basic Knowledge of Elementary
Crystallography is Essential
for Solid State Physicists!!!
• A crystal’s symmetry has a profound influence
on many of its properties.
• A crystal structure should be specified
completely, concisely & unambiguously.
• Structures are classified into different types
according to the symmetries they possess.
• In this course, we only consider solids with
“simple” structures.
Crystal Lattice
Crystallography focuses on the geometric properties of crystals. So, we imagine
each atom replaced by a mathematical point at the equilibrium position of that
atom. A Crystal Lattice (or a Crystal) ≡ An idealized description of the
geometry of a crystalline material. A Crystal ≡ A 3-dimensional periodic
array of atoms. Usually, we’ll only consider ideal crystals. “Ideal” means
one with no defects, as already mentioned. That is, no missing atoms, no atoms off
of the lattice sites where we expect them to be, no impurities,…Clearly, such an
ideal crystal never occurs in nature. Yet, 85-90% of experimental observations
on crystalline materials is accounted for by considering only ideal crystals!
Platinum Surface
Platinum
(Scanning Tunneling
Microscope)
Crystal Lattice
Structure of Platinum
Crystal Lattice
Mathematically 2 Dimensional Example
A Lattice is Defined
as an Infinite Array
of Points in Space
in which each point has
identical surroundings
to all others. The points
are arranged exactly
in a periodic manner.


y
B
α
b
C
D
A
O
a
x
E
Ideal Crystal ≡ An infinite periodic repetition
of identical structural units in space.
• The simplest structural unit we can imagine
is a Single Atom. This corresponds to a solid
made up of only one kind of atom ≡
An Elemental Solid.
• However, this structural unit could also be a
group of several atoms or even molecules.
The simplest structural unit for a given
solid is called the BASIS
• The structure of an Ideal Crystal can be
described in terms of a mathematical
construction called a Lattice.
A Lattice ≡
• A 3-dimensional periodic array of points in space.
For a particular solid, the smallest structural unit,
which when repeated for every point in the lattice
is called the Basis.
• The Crystal Structure is defined once both the
lattice & the basis are specified. That is
Crystal Structure ≡ Lattice + Basis
Crystals
Crystal Structure = Space Lattice + Basis
Crystalline Periodicity
• In a crystalline material, the equilibrium positions
of all the atoms form a crystal
Crystal Structure ≡ Lattice + Basis
For example, see Fig.
Lattice 
2 Atom Basis 
Crystal 
Structure
Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Crystal Structure
Lattice

Basis


Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Basis
Crystal Structure

Lattice


A Two-Dimensional (Bravais) Lattice with
Different Choices for the Basis
2 Dimensional Lattice
Lattice with atoms at the corners
of regular yhexagons
y
B
C
α
b
B
C
F
O
a
E
D
E
b
D
G
A
x
O
a
A
x
H
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The atoms do not necessarily lie at lattice points!!
Crystal Structure = Lattice + Basis
Basis


Crystal
Structure
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