MSE 605, Crystallography & Crystal Chemistry
Name: _______________________________
Homework 2
Due 17 September 2015
The numbers following each question give the approximate percentage of marks
allocated to that question. SHOW ALL YOUR WORK!
1. Calculate the minimum stable radius ratio for cubic (eight-fold) coordination. Use
appropriate diagrams as necessary.
10
2. How many formula units are there in a unit cell of NaCl?
5
Autumn 2015
MSE 605, Crystallography & Crystal Chemistry
3. Use Evjen's summation method to calculate the Madelung constant of NaCl. Use sufficient
terms for a model including eight unit cells (2 x 2 x 2) and draw these cells out.
Autumn 2015
20
MSE 605, Crystallography & Crystal Chemistry
4. In cubic perovskite BaZrO3 (a = 4.193 Å) the atom positions are:
Ba: 0,0,0
Zr: ½,½,½
O: ½,½,0, ½,0,½ , 0,½,½
a. Use the bond lengths, r, with the parameters ro and b to calculate the bond valence
sums (effective charges) of all three ionic species according to:
nbonds
BV =
∑ exp[(r
0
− r)/b]
15
n =1
Ba2+ - O2-:
Zr2+ - O2-:
Zr4+ - O2-:
r0 = 2.285 Å
r0 = 2.34 Å
r0 = 1.937 Å
b = 0.37 Å
b = 0.37 Å
b = 0.37 Å
Hint: Make sure you include all the “nearest-neighbor” bonds (e.g., there are six of
them for oxygen). It may help to draw out more than one unit cell to see them all.
b. Which species are overbonded? Which are underbonded?
Autumn 2015
3
MSE 605, Crystallography & Crystal Chemistry
5. Consider a hard-sphere model of an hexagonal structure with an arbitrary ratio
δ = c/a. Assuming that spheres touch, three cases can be distinguished:
Case I: Spheres touch along <100>
Case II: Spheres touch along <243>
Case III: Spheres touch along <001>
(a) Calculate the packing fraction as a function of δ for each case.
(b) Using Excel or another graphical software tool, plot the packing fraction as a
function of δ. Use linear axes with 0 ≤ δ ≤ 5 and 0 ≤ PF ≤ 1 for your graph.
(c) Calculate the values of δ at which the transitions from one case to another occur
and indicate them on your graph?
(d) Calculate the packing fractions at the transitions
and indicate them on your graph?
(e) Based on your calculations/graph above, what is the maximum packing fraction
(corresponding to the ideal hcp structure)?
Use extra pages as necessary.
Autumn 2015
25
10
5
5
2