Name: _____________________________________ Partner:_________________________________
Lab 5 – Symmetry & Defects in Crystalline Structures
Chem 125
Purpose and Motivation
Continue to test your structure-property hypotheses from Lab 2 by exploring the structures of solids!
Recall that so far on this structure-property journey, you have…
• observed real material behaviors – particularly ductile vs. brittle (copper, zinc, & glass) – and made
hypotheses about the underlying structures causing these properties (Labs 2-3).
• investigated how the interatomic potential relates to strength and stiffness of one bond between
two atoms (Lab 4).
• explored energy barriers present when moving a few atoms bonded in two dimensions (Review
Worksheet, Part 2).
This week, you will see how solid structures made of many atoms are really arranged at the atomic level,
still in just two dimensions. Next week’s lab will culminate our structural investigations, examining the
structures (and correlated properties) in three-dimensional solid networks.
While seeking the structure-property relationships responsible for solid properties, early scientists
discovered that solids are primarily characterized as one of two types: amorphous or crystalline. As
illustrated in Figure 1, amorphous solids lack long-range order on the atomic level; this is the case in glass
and most plastics. Crystalline solids, on the other hand, contain a highly organized, long-range framework
– a repetitive or periodic lattice.
The vast majority of technologically-important materials are crystalline solids, including all pure metals
(such as copper), alloys (like steel), semimetals (like silicon), salts & ceramics (like NaCl or Al2O3), many
molecular substances (lauric acid, sugars, ice), and some covalent network substances (like graphite and
diamond). Even viruses tend to form crystals! Thus, we will focus heavily on the structure-property
relationships in crystalline solids.
Amorphous
Crystalline
Polycrystalline
grain
boundary
Figure 1
Furthermore, it is exceedingly rare for real crystalline substances to be comprised of one giant, pure crystal
– they are instead made of many smaller pieces of crystal joined together at grain boundaries, forming a
polycrystalline network, as illustrated in Figure 1(c). In addition to grain boundaries, there are additional
defects or imperfections in most crystalline structures that also affect properties, including vacancies,
stacking faults, and dislocations. We will explore these defects in today’s lab. The growth of a polycrystal
is a complex process, but if you are interested please ask us and/or do some research online - it would make a
great Learning Journal entry!
© 2020 A. G. Caster & M.E. Eberhart. Department of Chemistry, Colorado School of Mines.
Revised 2020.03.02
The precise structures formed by crystalline substances determines many important material properties,
so your primary goal in today’s lab is learning how to determine and communicate crystalline
structures using the minimum amount of information. Take a look at Figure 2, which shows the atomic
structures inside of pure copper, zinc, and silicate glass (SiO2). How would you go about concisely
describing and comparing the positions of the atoms in these structures?
Zinc
Copper
Silicate Glass
Figure 2. Three-dimensional ball-and-stick representations of select solids. (Atom sizes not to scale.)
Clearly, we can greatly reduce the amount of information needed to compare copper and zinc by identifying
the symmetry elements present in these structures. This is made easier by beginning with 2D examples,
which you will do today, but we will eventually use the same formalism to discuss these real 3D crystals.
Because of the lack of symmetry, amorphous structures are much more difficult to describe – we don’t yet
have a good system for doing so! But by building an understanding of crystal packing and defect structure,
you will have the information needed to explain the behaviors of copper, zinc, and glass observed in Lab 2.
Warmup Puzzle (with your TA)
1. Letters have been grouped and each group represents a specific commonality. Can you identify that
commonality? You could even draw on them with a pencil to figure it out!
ABCDE
KMTU
VWY
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FGJL
PQR
HI
OX
NS
Z
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Part 1 – A Crystalline “Lattice Type” is Simply a Symmetry Class
Now apply these symmetry concepts to some extended patterns. A lattice is a periodic set of points upon
which a pattern (motif) can be “hung”, and the primitive unit cell is the smallest repeating parallelogram
on that lattice. (A conventional unit cell is a larger, but more popular and easier to identify, repeating
unit that occupies more space than a primitive cell.) We group lattices as being the same “type” or “class”
if they have same set of symmetry elements.
1. Choose any four nearest-neighbor points and connect them into a parallelogram; this is your primitive
unit cell. Identify symmetry elements (translation, mirror planes & rotations) for each unit cell.
Assume all dots are equivalent points. Label the following…
•
•
•
a capital “T” for translational symmetry.
a dashed LINE labeled “m” along each mirror plane.
the corresponding small, solid shapes on each rotation axis:
2-fold (180°) rotation axis
4-fold (90°) rotation axis
6-fold (60°) rotation axis
(A)
(B)
(C)
(D)
(E)
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2. Complete Table 1 below according to these specs. Reference Chapter 7 or the internet as needed.
• Check that the primitive unit cell (shaded), conventional unit cell (outlined), and
lattice vectors match what you completed in Question 4. Note: Any nearest-neighbor
parallelogram counts as a primitive unit cell, but preferred primitive cells shown here have
smallest, most equivalent lattice vector lengths & show all symmetry elements within one cell.
• Write the symmetry elements you identified on the previous page. For example, “T” for
translation, “2m” for 2 mirror planes, and to indicate a 4-fold rotation axis.
• From the randomized lists below, choose the correct name & unit cell parameters for
each lattice. A few examples are completed for you. You can use items more than once.
LATTICE TYPE
Oblique
Square
Hexagonal
Centered
Rectangular
Rectangular
VECTOR LENGTH
|a| = |b|
|a| ≠ |b|
arbitrary
Table 1: 2D Bravais Lattice Types & Characteristics
Lattice Type
Oblique
A
B
Centered
Rectangular
C
VECTOR ANGLE
𝜃=90°
𝜃=120°
𝜃=arbitrary
D
E
hexagonal
tetragonal
Lattice
Symmetry
Elements
Symmetry Class
T,
monoclinic
orthorhombic
| a| = | b|
Vector Lengths
Vector Angle(s)
T, 2m,
𝜽=arbitrary
3. Why is there no “centered square” lattice? Hint: Try drawing it. What are the symmetry elements?
4. In your own words…How does the letter-sorting warm-up relate to the concept of 2D Bravais Lattices?
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Part 2 – Adding a Motif to the Lattice
Lattices are not very interesting until we attach something to them – like atoms or tile patterns. The pattern
is called the motif. To reproduce the entire pattern, we only need to know the motif plus the lattice vectors
and angle.
5. For each pattern below, use the information from Part 1 to…
•
•
•
•
Identify the symmetry elements to determine the lattice type and write only its name on the
line (aka crystal class or Bravais Lattice). USE YOUR TABLE FROM THE PREVIOUS PAGE!!!
Mark the lattice points.
Shade in one primitive unit cell.
Outline one conventional unit cell (if there is one).
This can be challenging – follow these hints for the quickest success:
1. Find four IDENTICAL, nearest-neighbor points that make a parallelogram. Look around
each point in every direction to be sure the four points are truly identical.
2. Identify mirror planes and rotation axes of that parallelogram.
3. Compare the mirror planes & rotation axes to Table 1. If a pattern has translational
symmetry, but is NOT square, rectangular, centered rectangular, or hexagonal, it
must be OBLIQUE.**
______________________________
_____________________________
________________________________
___________________________
__________________________
**There is a sub-class (a “space group”) of the OBLIQUE class that has only translational symmetry.
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______________________________
________________________________
For the curious (optional)… Unusual Motifs
6. This beautiful pattern is known as a Penrose Tiling. Can you identify a symmetry class for it? Why or
why not? Feel free to do some web research!
7. An important and interesting class of materials are quasicrystals - the discovery of which lost Daniel
Shecthman his job but got him a Nobel Prize! Below is an image of one such quasicrystal, an Al-Pd-Mn
alloy. How do quasicrystals vary from the regular crystals we have been discussing so far? Again, web
research is a great idea!
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Part 3 - Breaking Symmetry: Crystalline Defects
We know that the symmetry in real crystals isn’t infinitely perfect; there are defects that form during
crystal growth. By observing the behavior of a single layer of spherical BBs, you can model some of the
ways that atoms come together to form 2D polycrystals, and identify some types of defects that occur.
8. Get a container of BBs and slowly pour them into your tray - without spilling! At first, add just a few BBs (about
10-20). Tilt the tray back and forth, imagining the BBs as atoms. What type of “atomic motions” are modeled
here? If this tray represents a pure substance in one single phase of matter (solid, liquid, gas,
supercritical fluid), what phase would it be?
9. Add enough BBs to the tray to cover >95% of the bottom. To demonstrate a 2D crystal, the atoms must be in a
single layer – a monolayer – where all atoms are in the same plane (none “popped up”). Add or remove BBs as
needed to make a single monolayer. To help you achieve this, you can carefully place a sheet of clear acrylic (a few
are available in each lab) on top of the atoms and press it gently flat; try to squish them into a monolayer without
significantly disturbing their positions.
(a) Now what phase of matter does your array most closely resemble (solid, liquid, or gas)?
(b) Is it polycrystalline or amorphous? Explain.
10. (a) Sketch what the arrangement looks like over the span of a few unit cells.
(a) What Bravais lattice type is this? _________________________________________
(b) Trace out one primitive unit cell onto your drawing.
(c) Draw straight lines through each of the closest-packed directions on your drawing.
(d) How many distinct closest-packed directions (CPDs) are there in this lattice type? CPDs = _____
(e) Why do you think the BBs naturally pack this way?
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11. Now, jiggle the tray just slightly for a few seconds while gently while holding the edges of the plastic sheet in
place, so that the atoms remain in a monolayer. DO NOT SHAKE VIGOROUSLY YET! Describe any change in
structure you observe. What type of atomic motion are you modeling here? (Hint: Atoms and molecules can
vibrate, rotate and translate.)
12. Identify the different types of defects in your tray. Make one sketch of each type in your and give a name to
each one, from the following list: vacancy, grain boundary, and stacking fault. Feel free to reference your
textbook and the internet. If you don’t see many defects, reset your BBs and try again.
13. Locate at least one grain boundary. If you see no grain boundaries, try jiggling the tray a little bit more, then reset
your tray of BBs and try again until you make some. While watching one particular grain boundary, tilt and jiggle
the tray VERY slightly in a couple of directions, always holding the plastic sheet gently in place. Describe what
happens to the boundary you are following as you tilt and gently jiggle the tray.
14. Locate at least one stacking fault. If you see no stacking faults, try jiggling the tray a little bit more, then reset your
tray of BBs and try again until you make some. While watching one particular stacking fault, tilt and jiggle the tray
VERY slightly in a couple of directions, always holding the plastic sheet gently in place. Describe what happens
to the stacking fault(s) you are following as you tilt and gently jiggle the tray.
15. Locate at least one vacancy. Repeat the instructions from step 6 or 7. Describe what happens to the vacancy
you are following as you tilt and gently jiggle the tray.
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16. Now obtain a small number (10-20) BBs of a different size. Add the new BBs to your existing batch. Shake the
tray around gently one more time, and describe any new types of defects that appear. What do we call these
defects?
17. Summarize: What happens to the types and numbers of defects as you jiggle the tray? What happens to
the crystallinity and lattice type?
18. What physical process (with a real material, that you did in Lab 2…) is analogous to you gently shaking
this tray of BBs?
19. Based on what you observed with these BBs… How do you think annealing a metal (heating to high
temperatures and then allowing the crystal to cool) changes the defect structure within the crystal?
20. List ways in which the BBs behave (a) similarly to and (b) differently from real atoms
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21. Based on what you’ve heard in this class, or prior knowledge, make an educated guess: Do real metal atoms
always pack into this type of lattice? Briefly explain what you know so far about packing in Cu vs. Zn vs.
other metals.
Before leaving lab…Clean Up!
Clean all transparencies thoroughly with wet paper towel, then DRY them before putting away.
Put all materials into their proper locations.
Post-Lab…
As usual, some more analysis questions are on Canvas as a post-lab quiz which you will be required to do
online. Please refer Canvas for the Link! If you finish lab soon, go through the questions on Canvas and
ask your TAs if you have any questions!
Time to start preparing for Report 1! Your TA & Canvas will provide details, but the due weeks are:
• Week of Oct 21: System Flow Diagram + Outline for Report 1
• Week of Oct 28: Report 1 Due (Structure-Property Relationships)
TA/Instructor Sign Out ______________ Date
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