Chapter 1: Crystal Structure

The Nobel “Booby” Prize! See the “Ig Nobel” Prize discussed at: http://improbable.com/ig/

Chapter 1: Crystal Structure

The (Common) Phases of Matter

“Condensed Matter” includes both of these. We’ll focus on Solids!

This doesn’t include Plasmas, but these are the “common” phases!!

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.

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• 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.

Liquids & Liquid Crystals

• 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.

Solids

Crystal StructureTopics

1. Periodic Arrays of Atoms2. Fundamental Types of Lattices3. Index System for Crystal Planes4. Simple Crystal Structures5. Direct Imaging of Crystal Structure6. Non-ideal Crystal Structures7. Crystal Structure Data

ObjectivesAt 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

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

Periodic Arrays of Atoms

Elementary Crystallography

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)

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.

The Three General Types of Solids

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 “Long-range Order”

• Amorphous solids do not have Long-Range 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 CrystalsSingle Pyrite

CrystalAmorphous

Solid

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.

PolycrystallinePyrite 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.

PolycrystallinePyrite 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.

CrystallographyCrystallography ≡ 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 PhysicsStarted in the early 20th Century when the fact that

Crystals Can Diffract X-rayswas discovered.

• Around that same time the new theory of

Quantum Mechanicswas 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.

CrystallographyA Basic Knowledge of Elementary

Crystallography is Essentialfor 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 LatticeCrystallography 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!

PlatinumPlatinum Surface(Scanning Tunneling

Microscope)

Crystal LatticeStructure of Platinum

MathematicallyA 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.

α

a

bCB ED

O

A

y

x

Crystal Lattice2 Dimensional Example

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

• In a crystalline material, the equilibrium positions of all the atoms form a crystal

Crystal Structure ≡ Lattice + Basis For example, see Fig.

Crystalline Periodicity

2 Atom Basis

Lattice

Crystal Structure

Crystal Structure ≡ Lattice + Basis For another example, see the figure.

Crystalline Periodicity

Lattice

Basis

Crystal Structure

Lattice

Crystal Structure

Crystalline PeriodicityCrystal Structure ≡ Lattice + Basis

For another example, see the figure.

Basis

A Two-Dimensional (Bravais) Lattice with Different Choices for the Basis

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E

HO A

CB

Fb G

D

x

y

a

α

a

bCB ED

O A

y

x

Lattice with atoms at the cornersof regular hexagons

2 Dimensional Lattice

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The atoms do not necessarily lie at lattice points!! Crystal Structure = Lattice + Basis

Basis

CrystalStructure