MSE 605, Crystallography & Crystal Chemistry
Name: _______________________________
Homework 3
Due 24 September 2015
The numbers following each question give the approximate percentage of marks
allocated to that question. SHOW ALL YOUR WORK!
1. Consider melting temperature (Tm), tensile strength (σUTS), elastic modulus (E), and
thermal expansion coefficient (α) and describe how bonding affects each.
Reference an energy vs displacement diagram where appropriate.
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2. Briefly explain metallic bonding and describe how the structure and properties of
metallically-bonded materials result from the characteristics of metallic bonding.
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Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
3. For each of the compounds below:
a. Determine the cation and anion charges (ZC and ZA), effective ionic radii (rC and rA), and
coordinations (NC and NA). Then use Pauling’s Rules to calculate s, the electrostatic
bond valence strengths, in the given compounds. Finally use Pauling’s 1960 equation to
calculate the percent ionicity (P) of the bonds in each compound.
Hint:
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Use Shannon’s ionic radii data for rC and rA in sixfold coordination.
Note: Pauling originally used the univalent radii (rather than crystal radii or effective
ionic radii) to calculate radius ratios; however, the success rate of the predictions
does not depend strongly on which set of radii are used, as long as both the
cation and anion radii come from a set that was derived using consistent
assumptions.
χNa = 0.93,
χCd = 1.69
Compound
χCs = 0.79,
χCl = 3.16
ZC
ZA
χK = 0.82,
χBr = 2.96,
rC(VI)
χZn = 1.65,
χI =2.66,
rA(VI)
rC/rA
χTi = 1.54,
χS = 2.58,
NC
χMn = 1.55,
χO = 3.44
s
NA
P
NaCl
CsBr
KI
TiO2
ZnO
b. Comment on the s results for the different compounds
(e.g., what does higher s mean?).
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c. Carefully chosen examples can make the radius-ratio concept look like an accurate
predictive tool; however, it is often in error, particularly for complex structures. In fact,
ZnO is a good example of its limitations in that the coordinations which would be
predicted for it in this way are actually incorrect. Discuss reasons which might explain
this anomaly.
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Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
4. A model for the energy of an inert gas solid is given by the Lennard-Jones potential:
6
 σ 12
 r
 
σ 
E = 4ε   −    or E = ε  0
 r 
r  
 r


12



6
 r0  
− 2  
 r  
where -ε is the equilibrium lattice energy, σ is the spacing at which E = 0, r is the
interatomic spacing, and r0 is the equilibrium separation. Using Mathematica or
another mathematical package, do the following:
a. Plot the repulsive energy (first term), attractive energy (second term), and total
energy as a function of separation for Ar. The curves should be on the same plot.
Include only the range 3 Å ≤ r ≤ 5 Å and -0.1 eV ≤ E ≤ 0.2 eV.
Be sure to label all three curves.
b. Use data in the table below to calculate the Van der Waal's radius of Ar?
HINT: as E will not be > 0, σ gives a measure of how close two nonbonding
atoms can get.
15
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c. Plot the total energy as a function of interatomic separation for Ne, Ar, Kr, and Xe.
Include only the range 2.5 Å ≤ r ≤ 5 Å and -0.04 eV ≤ E ≤ 0.08 eV and be sure to
label all four curves.
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d. Find the equilibrium atomic spacing, r0, for each solid in the table and comment on
trends you notice.
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e = 1.6022 x 10-19 C
1 eV = 1.6022 x 10-22 kJ
Solid
ε (eV)
σ (Å)
Ne
0.0031
2.74
Ar
0.0104
3.40
Kr
0.0140
3.65
Xe
0.020
3.98
SHOW ALL YOUR WORK!
BONUS:
With reference to question 3c, what are the correct coordinations for ZnO?
Show how you calculated this value or cite the source where you found it.
Autumn 2015
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