1. Chapter 1: Crystal Structure

2. Chapter 1: Crystal Structure The Nobel “Booby” Prize!
See the “Ig Nobel” Prize discussed at:

3. The (Common) Phases of 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!

4. 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.

5. 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. 5

6. 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.

7. 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

8. At the end of this Chapter, you should:
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

9. 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

10. Elementary Crystallography
Solid Material Types
Crystalline
Polycrystalline
Amorphous
Single Crystals

11. Crystals are Everywhere!

12. More Crystals

13. 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)

14. 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.

15. 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

16. Solids Crystals: Amorphous solids: Short-range Order
Long-range Order
Amorphous solids:
~Short-range Order
No Long-range Order

17. 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”

18. Solids
Different arrangements of atoms (even the same atoms) give different properties
Single layer is graphene

19. 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.

20. Single Crystals Single Pyrite Solid Crystal Amorphous
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 Pyrite
Crystal
Amorphous
Solid
Single Crystals

21. 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

22. Polycrystalline Solids
In Polycrystalline Solids, the atomic order can vary from one domain to the next. The grains are usually 100 nm microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites.
Polycrystalline
Pyrite Grain

23. 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.

24. Space (Crystal) Lattice + Basis
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

25. 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.

26. 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.

27. Solid State Physics Crystals Can Diffract X-rays Quantum Mechanics
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.

28. 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.

29. (Scanning Tunneling Microscope)
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
(Scanning Tunneling Microscope)
Crystal Lattice
Structure of Platinum
Platinum

30. A Lattice is Defined as an Infinite Array of Points in Space
Crystal Lattice
Mathematically
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.
2 Dimensional Example  α a b C B E D O A y x

31. The simplest structural unit for a given solid is called the BASIS
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

32. Crystal Structure ≡ Lattice + 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

33. Crystal Structure = Space Lattice + Basis

34. 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

35. Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Crystal Structure 
Lattice 
Basis 

36. Crystalline Periodicity
Crystal Structure ≡ Lattice + Basis
For another example, see the figure.
Basis 
Crystal Structure 
Lattice 

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

38. Lattice with atoms at the corners
2 Dimensional Lattice
Lattice with atoms at the corners
of regular hexagons α a b C B E D O A y x O A C B F b G D x y a E H 38

39. Crystal Structure = Lattice + Basis
The atoms do not necessarily lie at lattice points!!
Crystal Structure = Lattice + Basis
Basis  
Crystal
Structure 39 39