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
“Condensed Matter” includes both of these. We’ll focus on Solids!
This doesn’t include Plasmas, or Bose-Einstein condensates but these three are the “common” phases!!

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
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. 5

6. Liquid Crystals
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. 6

7. Solids Solids can be crystalline, polycrystalline, or amorphous.
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.

8. Crystallography & Crystalline Solids
The following material covers most of the topics in Ch. 1 (Crystal Structure) in the book by Kittel.
In the Supplemental book by Omar, the corresponding material is in Ch. 1, (Crystal Structures & Interatomic Forces), Sects. 1 – 7.
However, as will be true throughout the course, my discussion will use a large variety of sources other than those books.

9. 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

10. 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

11. Preliminary Remarks on Crystalline Solids
This material is an overview of the many types of crystalline solids & their crystal structures.
Mostly, this involves some simple mathematics of symmetry. However, it also involves learning some terminology & nomenclature.
Unfortunately, the first scientists to be interested in crystals were mostly geologists & crystallographers & not physicists. So, in order to "speak the language" of crystals, we must use the terminology that they developed. I sometimes find that inconvenient & abstract.

12. Despite this, understanding that language is necessary in order for us to be able discuss the PHYSICS of crystalline solids.
NOTE!! There are a huge number of websites that can be used to supplement the textbook discussions & my lectures. A list of some is posted on the Lecture Page.
We’ll spend 3 or 4 lectures on these topics. Some Lectures on this material are also posted on the  Lecture Page. Some of these may undergo some revisions as we proceed. A homework assignment on this material will be given out soon.

13. Periodic Arrays of Atoms (Kittel, Fig. 1.)
Experimental Evidence of periodic structures:
The external appearance of crystals gives some clues.
The figure shows that when a crystal is cleaved, we
can see that it is built up of identical “building blocks”.

14. Experimental Evidence of periodic structures.
The early crystallographers noted that
the index numbers that define plane
orientations are exact integers.
Cleaving a Crystal

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

16. Crystals are Everywhere!

17. More Crystals

18. Early ideas
All crystals are solids, but all solids are not crystalline! Crystals have symmetry (Kepler!!!) and
long range order
Spheres & small
shapes can be
packed to
produce regular
shapes (Hooke, Hauy)

19. 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.

20. 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, but they often have Short Range Order

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

22. Solids Examples: Si, Diamond (C), GaAs, ZnSe have the same geometry
Different solids can have the same geometrical arrangements of atoms
Their Properties are determined by crystal structure: Both crystal lattice & 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”

23. Solids
Different arrangements of atoms (even the same atoms) give different solid properties
2 very different solids made of only carbon (C) atoms!

24. 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.

25. Single Crystals Single Pyrite Solid Crystal Amorphous
A Single Crystal has a arrangement of atoms that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to each equivalent atom in the structure by translational symmetry.
Single Crystals
Single Pyrite
Crystal
Amorphous
Solid

26. 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

27. Polycrystalline Solids A polycrystal with grain boundaries
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.
A polycrystal with grain boundaries

28. Polycrystalline Solids with grains & grain boundaries:

29. Polycrystalline Solids
Many technologically important materials are
polycrystalline.
The figure is an
electron micrograph
of a Nb-Hf-W plate
with an electron
beam weld.
Each "grain" is a single crystal. If the grains are
randomly oriented, the overall component
properties are not directional.
Grain sizes typically range from 1 nm to 2 cm
(i.e., from a few to millions of atomic layers)

30. Polycrystalline Solids

31. Polycrystalline Solids
Photograph of a
Silicon Single Crystal.
Micrograph of a
polycrystalline stainless
steel sample showing
grains & grain
boundaries

32. 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.

33. Amorphous Solids
Amorphous (Non-crystalline) Solids 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.

34. Some examples from everyday life: 3. Window glass, 4. Cotton candy!
Amorphous Solids
Amorphous (Non-crystalline) Solids
have no regular long range order of
arrangement of atoms.
Some examples from everyday life:
1. Polymers, 2. Ceramics,
3. Window glass, 4. Cotton candy!
The two sub-states of amorphous solids
are the Rubbery and Glassy states

35. Amorphous Solids Amorphous (Non-crystalline) Solids
have no regular long range order of
arrangement of atoms.
Amorphous Solids can be prepared by
rapidly cooling molten material. Rapid
cooling minimizes time for the atoms to
pack into a more thermodynamically
favorable crystalline state.

36. Amorphous Solids
Illustration of the continuous random network structure of the atoms in an amorphous solid

37. Amorphous Solids
Amorphous materials - Materials, including glasses, that have no long-range order, or crystal structure.
Glasses - Solid, non-crystalline materials (typically derived from the molten state) that have only short-range atomic order.
Glass-ceramics - A family of materials typically derived from molten inorganic glasses & processed into crystalline materials with very fine grain size & improved mechanical properties.

38. Atomic arrangements in crystalline silicon and amorphous silicon
Atomic arrangements in crystalline silicon and amorphous silicon. (a) Amorphous silicon. (b) Crystalline silicon. Note the variation in the inter-atomic distance for amorphous silicon.

39. Space (Crystal) Lattice + Basis
Crystals
The periodic array of atoms, ions, or molecules that form the solid is called the Crystal Structure
Crystal Structure 
Space (Crystal) Lattice + Basis
The Space (Crystal) Lattice is a regular periodic arrangement of points in space, & is purely a mathematical abstraction
A Crystal Structure is formed by “putting” the identical atoms (group of atoms) on the points of the space lattice
This group of atoms is called the Basis

40. 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.
We also know that each atom undergoes thermal vibrations around their equilibrium positions for temperatures T > 0K. These can also be viewed as “imperfections”.

41. Departures From the “Perfect Crystal”
Real Crystals always have foreign atoms (impurities), missing atoms (vacancies), & atoms between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.

42. Crystallography Crystallography ≡ 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.

43. Crystallography Crystallography 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.

44. Crystals Can Diffract X-rays
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.

45. A Basic Knowledge of Elementary Crystallography is Essential
for Solid State Physicists!!!
A crystal’s symmetry has a profound influence on many (most!) 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.

46. (Scanning Tunneling Microscope)
Crystal Lattice
Crystallography focuses on the geometric properties
of crystals. So, we imagine each atom replaced by a
mathematical point at it’s equilibrium position.
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.
Platinum Surface
(Scanning Tunneling Microscope)
Platinum

47. (Scanning Tunneling Microscope)
Crystal Lattice
Usually, we’ll only consider ideal crystals. “Ideal” means
no defects. 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 crystals
is accounted for by considering only ideal crystals!
Platinum Surface
(Scanning Tunneling Microscope)
Crystal Lattice
Structure of Platinum

48. 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

49. 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

50. 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

51. Crystal Structure = Space Lattice + Basis

52. 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

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

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

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

56. 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 56

57. Crystal Structure = Lattice + Basis
The atoms do not necessarily lie at lattice points!!
Crystal Structure = Lattice + Basis
Basis  
Crystal
Structure 57 57