15.11: Questions
- Page ID
- 31582
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You should be able to answer these questions without too much difficulty after studying this TLP. If not, then you should go through it again!
Which of these is not a lattice type?
| a | H |
| b | F |
| c | I |
| d | C |
- Answer
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A.
What is a lattice ?
- Answer
-
D.
How many Bravais lattices (3D) are there ?
- Answer
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C.
How many point groups (3D) are there ?
- Answer
-
B.
What is a unit cell ?
- Answer
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B.
Deeper questions
The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.
Construct a plan view of NaCl (sodium chloride).
NaCl has a face-centred cubic lattice. The motif is:
Cl @ (0,0,0);
Na @ (0,0,1/2);
Note 1: The motif coordinates are positions relative to each lattice point
Note 2: In a face centred cubic structure the lattice points are located at:
(0,0,0), (1/2,1/2,0), (1/2,0,1/2), (0,1/2,1/2)
- Answer
-
Your construction should look something like this
It should include:
A set of axes
The a and b lattice vectors
Two separate symbols for Cl and Na plus a key explaining the symbols
The fractional heights of the atoms.
Diamond has a face centred cubic lattice. Its motif is
C @ (0,0,0), (1/4,1/4,1/4)
Construct a plan view of the diamond unit cell.
Treating the carbon atoms as hard spheres calculate the packing efficiency of diamond.
Note: In a face centred cubic structure the lattice points are located at:
(0,0,0), (1/2,1/2,0), (1/2,0,1/2), (0,1/2,1/2)
- Answer
-
Your construction should look something like this
Packing Efficiency:
The volume of the unit cell is a3.
The volume of a single atom is\[\frac{4}{3} \pi r^{3}\]
There are 8 atoms per unit cell.
Note that the solid red line extends to a height of a/4. The total length of m is equal to 2r (the centre-centre distance of two touching atoms is 2r). The distance covered by m in the xy plane is .\[\frac{a}{2 \sqrt{2}}\]
We can see from this diagram that\[2 r=\frac{\sqrt{3}}{4} a\]
therefore
\[r=\frac{\sqrt{3}}{8} a\]
The volume of a single atom is
\[\frac{4}{3} \pi\left(\frac{\sqrt{3}}{8} a\right)^{3}=\frac{\sqrt{3}}{128} \pi a^{3}\]
The volume occupied by the 8 atoms is
\[\frac{\sqrt{3}}{16} \pi a^{3}\]
Therefore the packing efficiency is
\[\frac{\sqrt{3}}{16} \pi a^{3} \div a^{3} \approx 0.34\]
\[\Rightarrow 34 \%\]