Homework 3

PAUL REVERE: TOUGH AS NAILS
The Stuff of History + Materials Science and Solid State Chemistry
Homework 3
DUE FRIDAY, FEBRUARY 15, BY CLASS TIME (1 PM)
THE “STUFF” OF HISTORY
1. Read the description for the “Historical Research component of Project One on page 4 of the Assignments Supplement,
especially points 6, 7, and 8.
2. Read the description of the Paper one expectations, on pages 5-6 of the Assignments Supplement.
3. Give your nearest friend a “high five” and have a soda, preferably of the root beer or vanilla coke persuasions (diet
varieties are fine).
4. Produce a “statement of goals” for this paper. You can write this as a coherent paragraph (which might end up as a part
of your paper‘s introduction) or you can write it as a set of bullet points. Either way, make sure you cover the specific
points you hope to address in this paper, which should include the following:
a. Define and/or explain your modern artifact (in detail, including its cultural/social importance and its material
properties)
b. Define and/or explain your ancient counterpart (in detail, as above)
c. Come up with a hypothesis that connects the modern and ancient counterparts
d. Come up with a hypothesis that connects the item’s properties to its social and historical impacts
5. Give your nearest friend a “high five with a flipside” and enjoy a small to medium sized chocolaty treat (frozen versions
also acceptable).
6. Produce a tentative outline for this paper. Use roman numerals and sub-items (i.e., I.A.1…), and go to whatever degree
of detail that you can. Try to map out a coherent plan that takes you from an introduction through all of your material
and historical evidence/arguments, to a conclusion. This outline needs to explicitly achieve your goals as stated in item
three above.
7. This one is optional, though it is included because it will help you. Try to insert the evidence (historical and scientific)
you have gathered thus far into the outline. This evidence includes:
a. Quotes, citations, paraphrasings, or statistics taken from your ancient civilization textbook
b. Information from the Substance of Civilization
c. Information from other sources in the library (historical atlases…)
d. Information you researched in your answers to Exam One
e. Materials science research information
f. Materials science theory from your textbooks or other sources
This exercise should help you evaluate your ability to write paper one. You may discover several gaping holes in your
outline, in which case you can either change the emphasis of your paper or you can seek additional evidence.

M A T E R I A L S S C I E N C E
READINGS
o Crystallography. Callister Chapter 3 or Askeland Chapter 3 (Atomic and Ionic Arrangements). Read the entire
chapter, but don’t spend too much time studying all those crystal structures in Sections 3.7 and 3.8.
o Imperfections. Callister Chapter 4 or Askeland Chapter 4
o Check out the crystal structure models in the library, if you have time.
The key concepts in these sections are:
Crystallography & Structure of Solids (Chapter 3)
Single crystals, polycrystalline solids, and noncrystalline solids – What’s the difference?
Lattice, crystal structure, lattice parameters (angles and edge lengths), unit cell concepts – Know what the unit cell
represents, and be familiar with basic unit cell geometry.
Crystal geometry – Be familiar with the terminology for crystallographic direction indices and planar indices (Miller
indices), and be able to use direction indices and planar indices. For example, if you are asked a question about the
(101) in bcc, you should be able to sketch the unit cell and identify the plane.
Crystal structures. A lot of crystal structures are described in your textbook. You don’t need to memorize all
these structures, of course, but be familiar with the more common metal structures (face centered cubic and body
centered cubic). Be familiar with methods for determining the coordination number (number of nearest neighbors),
and the number of atoms per unit cell.
Atomic packing & density calculations – Why are these things important? They help us determine the properties of
a material. For example, metals with higher packing factors and high symmetry, such as face centered cubic, tend to
be much more ductile at lower temperatures than metals with lower packing factors or less symmetry.
o Atomic packing factor (APF)
o True density of a solid
o Planar atomic density
o Linear atomic density
Polymorphs and allotropes. These are just fancy terms that refer to multiple solid forms of compounds
(polymorphs) and elements (allotropes).
X-ray diffraction. This is a good topic to spend some time understanding. In fact, Bragg’s law is on my top ten list of
useful materials science equations. Take a look at how Bragg’s law is used in the determination of lattice parameters
and analyzing x-ray diffraction patterns for crystalline substances. Also, check out the equation that relates
interatomic spacing to the lattice parameter and Miller (planar) indices.
Imperfections in Solids (Chapter 4)
Defects topics tend to show up as parts of problems dealing with properties (e.g., mechanical or physical) and
crystallography. For example, solubility limits are related to the atomic structures and bonding, and they are important
in determining the strength or ductility of a material. The various types of defects can have different effects on
mechanical properties of different materials.
Differences in the scale and dimensions of 0-D (point), 1-D (line), 2-D (surface), and 3-D (volume) defects.
Point defects are quite important in determining lots of material properties, including electrical, chemical, thermal
and mechanical. Some examples:
o Vacancies. How does the equilibrium number of vacant sites in a solid change as a function of temperature?
o Impurities – substitutional (replace an atom in the crystal structure) and interstitial (squeeze an impurity into a
smaller space between the normal atomic positions in a structure).
o What factors that determine extent of solid-state solubility
Line defects are very important for determining mechanical properties…dislocations slip through a material in
response to applied stress. If we can stop the slip, we can strengthen and harden the material!
o Edge and screw dislocations
o Strain fields around an edge dislocation. How might these strained regions interact with other defects?
Surface defects – external surfaces, grain boundaries, twin boundaries
Volume defects – cracks, pores, voids, inclusions
Importance of defects in determining mechanical behavior of ceramic materials

PROBLEMS
Practice Problems
1. Askeland Problem 3-34. Beryllium has a hexagonal crystal structure, with a o = 0.22858 nm and c o = 0.35842 nm. The
atomic radius is 0.1143 nm, the density is 1.848 g/cm 3 , and the atomic weight is 9.01 g/mol. Determine
a. The number of atoms in each unit cell; and
b. The packing factor in the unit cell
2. Askeland Problem 3-78. Explain why most often polycrystalline materials exhibit isotropic properties.
3. Crystallographic planes. Sketch an fcc unit cell. Within the unit cell, sketch and label the following:
a. ( 0 0 1 ) b. 2 0 1
( ) c. [ 1 2 0]
4. Crystallographic planes and directions. Sketch a bcc unit cell. Within the unit cell, sketch and label the following:
a. [ 0 1 0 ] b. [ 2 2 1 ] c. ( 0 0 1 )
5. Nickel has the fcc structure and a lattice parameter of a = 0.3517 nm.
a. Determine the atomic radius in nm. Compare this value to that shown in the front of your textbook.
b. Determine the atomic volume in nm 3 (assume a spherical atom shape).
c. Determine the density in g/cm 3 and compare this answer to the Ni density value given in the table at the front
of your textbook.
6. Sketch an fcc unit cell.
a. Determine the planar density for the ( 1 1 1 ).
b. Determine the planar density for the ( 1 1 0 ).
c. Determine the linear density for the [ 0 1 1 ]
d. Determine the linear density for the [ 1 0 0 ].
e. Given the densities you calculated for (a)-(d), which planes and directions would provide the easiest path for
edge dislocation motion, or “slip”?
7. The zinc blende (ZnS) structure is shown below.
a. What is the coordination number for the Zn atoms?
b. How many atoms per unit cell in the ZnS structure?
8. Askeland 4-67. Why is most “gold” or “silver” jewelry made out of gold or silver alloyed with copper, i.e, what
advantages does copper offer?

9. Solid state solubility. Of the elements in the chart below, name those that would form each of the following
relationships with copper (non-metals only have atomic radii listed):
a. Substitutional solid solution with complete solubility
b. Substitutional solid solution of incomplete solubility
c. An interstitial solid solution
Element Atomic Radius (nm) Crystal Structure Electronegativity Valence
Cu 0.1278 FCC 1.9 +2
C 0.071
H 0.046
O 0.060
Ag 0.1445 FCC 1.9 +1
Al 0.1431 FCC 1.5 +3
Co 0.1253 HCP 1.8 +2
Cr 0.1249 BCC 1.6 +3
Fe 0.1241 BCC 1.8 +2
Ni 0.1246 FCC 1.8 +2
Pd 0.1376 FCC 2.2 +2
Pt 0.1387 FCC 2.2 +2
Zn 0.1332 HCP 1.6 +2
10. Vacancies. Calculate the fraction of atom sites that are vacant for pure lead at room temperature (25 °C) and at 310 °C
(just below its melting temperature). The activation energy for vacancy formation is about 53 kJ/mole. How do you
think the difference in the fraction of vacant sites will affect self-diffusion (diffusion of the atoms within their own crystal)?
11. Strain fields around dislocations. Sketch the distortion of a crystal lattice around an edge dislocation, and show the
preferred regions for (i) large substitutional atoms, (ii) small substitutional atoms, and (iii) interstitial atoms. HINT:
Determine the type of distortion (tension or compression) that the large and small substitutional atoms will have on a
perfect lattice, then think of how the distortion around a dislocation could best be balanced or relieved by the distortion
caused by the substitutional atoms.
Open-Ended Problems
1. An unforeseeable reaction has just destroyed the metal lining of your high temperature vacuum furnace. You can’t afford
to purchase a new lining, so you’ve decided to make a new lining in the Olin shop. The lining is basically a 6-inch metal
cube with a wall thickness of 0.125 inches. One problem: you don’t know the material used for the furnace lining. You
think you can safely assume it’s a pure metal, but other than that, you’re at a loss.
To determine the unknown metal, you run a standard x-ray diffraction scan, the results of which are shown in the
following figure. Crystallographic planes (hkl) are indexed on the pattern. The copper radiation used in the x-ray
diffraction test had a wavelength of 0.1542 nm.

a. Given the XRD data, what is the likely crystal structure of the unknown metal?
b. What is the lattice parameter of the unknown metal?
c. Based on your answers in parts (a) and (b), which of the following materials is the most likely match for the
unknown metal?
i. Tantalum
ii. Nickel
iii. Sodium
iv. Zirconium
v. Tungsten
vi. Chromium
d. Will you be able to fabricate the material in the Olin machine shop? Briefly explain your answer.
2. Noncrystalline SiO 2 is optically transparent. Polycrystalline SiO 2 is opaque or translucent. Single crystal SiO 2 is optically
transparent. Explain.