15.11: Questions

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Quick questions

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: A.

What is a lattice ?

Answer: D.

How many Bravais lattices (3D) are there ?

Answer: C.

How many point groups (3D) are there ?

Answer: B.

What is a unit cell ?

Answer: 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
Diagram of plan view of diamond

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

\[\frac{\sqrt{3}}{16} \pi a^{3} \div a^{3} \approx 0.34\]

\[\Rightarrow 34 \%\]

a	A transforming matrix
b	A set of atoms
c	A group of symmetry elements
d	An infinite array of identical points repeated throughout space

a	24
b	32
c	12
d	19

a	A unit of volume
b	Any parallelepiped with lattice points at its corners
c	A parallelepiped containing only one lattice point
d	The angle between lattice vectors