from matplotlib import pyplot as plt 
import numpy as np
from math import pi
  

Solutions for lecture 9 exercises

Warm-up exercises

1. A primitive unit cell is a unit cell which contains a single lattice point. Because it only contains a lattice point, it is the unit cell with the smallest volume out of all unit cells.

2. Let us first consider the FCC lattice. Its primitive lattice vectors vectors are $$\begin{array}{rlrl}{\mathbf{a}}_1& =\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}}){\mathbf{a}}_2& =\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{z}}){\mathbf{a}}_3& =\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}}).\end{array}$$ With respect to the conventional unit cell, the basis in fractional coordinates is $◯(1/2,1/2,0)$, $◯(1/2,0,1/2)$, $◯(0,1/2,1/2)$ and $◯(0,0,0)$. With respect to the primitive unit cell, the basis is $◯(0,0,0)$. Let us now consider the BCC lattice. The primitive lattice vectors are $$\begin{array}{rlrl}{\mathbf{a}}_1& =\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}}+\hat{\mathbf{z}}){\mathbf{a}}_2& =a\hat{\mathbf{x}}{\mathbf{a}}_3& =a\hat{\mathbf{y}}.\end{array}$$ The basis of the conventional unit cell is $◯(0,0,0)$ and $◯(1/2,1/2,1/2)$. For the primitive unit cell the basis is $◯(0,0,0)$.

3. You would need at least two. A diatomic crystal could require more than two basis vectors if the proportion between the two elements is not one to one.

4. $$F=\frac{\pi }{6}$$

5. See lecture notes

Exercise 1: Diatomic crystal¶

1.

y = np.repeat(np.arange(0,8,2),4)
x = np.tile(np.arange(0,8,2),4)
plt.figure(figsize=(5,5))
plt.axis('off')

# WZ
plt.plot([5,5,7,7,5],[5,7,7,5,5], color='k',ls=':')
plt.annotate('WZ',(6,6.5),fontsize=14,ha='center')

# PUC1 
plt.plot([0,2,4,2,0],[4,6,6,4,4], color='k',ls=':')

# UPC2
plt.plot([6,4,2,4,6],[0,0,2,2,0], color='k',ls=':')

plt.plot(x,y,'ko', markersize=15)
plt.plot(x+1,y+1, 'o', markerfacecolor='none', markeredgecolor='k', markersize=15);

2. The area of the primitive unit cell is $A=a^2$. If the filled and empty circles are identical particle, the nearest neighbour distance becomes $a^{\ast }=\frac{a}{\sqrt{2}}$ and thus the area $A^{\ast }={a^{\ast }}^2=\frac{a^2}{2}=\frac{A}{2}$.

3. One set of primitive lattice vectors is $${\mathbf{a}}_{\mathbf{1}}=a\hat{\mathbf{x}},{\mathbf{a}}_{\mathbf{2}}=a\hat{\mathbf{y}}.$$ With respect to the primitive lattice vectors, the basis is $ \huge \bullet \normalsize(0,0), \quad \bigcirc(\frac{1}{2},\frac{1}{2}). $

4. The lattice is a cubic lattice. The basis of the crystal is

$ \huge \bullet \normalsize(0,0,0), \quad \bigcirc(\frac{1}{2},\frac{1}{2},\frac{1}{2}). $

An example of such a material is Cesium Chloride (CsCl)

5. We obtain the BCC lattice.

Example: Sodium (Na)

6. $$F=\frac{\sqrt{(}3)\pi }{8}$$

Exercise 2: Diamond lattice

1. The conevtional unit cell of diamond consists out of two shifted fcc lattices. One set of primitive lattice vectors is $$\begin{array}{rlrl}{\mathbf{a}}_{\mathbf{1}}& =\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{y}}){\mathbf{a}}_{\mathbf{2}}& =\frac{a}{2}(\hat{\mathbf{x}}+\hat{\mathbf{z}}){\mathbf{a}}_{\mathbf{3}}& =\frac{a}{2}(\hat{\mathbf{y}}+\hat{\mathbf{z}}).\end{array}$$ The volume of the primitive unit cell is $$V=|{\mathbf{a}}_{\mathbf{1}}\cdot ({\mathbf{a}}_{\mathbf{2}}\times {\mathbf{a}}_{\mathbf{3}})|=\frac{a^3}{4}$$

2. The primitive unit cell contains 2 atoms. With respect to the set of primitive lattcie vectors, the basis is $ \mathrm{C}(0,0,0)$ and $\mathrm{C}(\frac{1}{4},\frac{1}{4},\frac{1}{4})$.

3. One FCC lattice contains 4 atom. Because the diamond conventional unit cell contains two shifted FCC lattices, it will contain 8 atoms. The volume of the conventional unit cell is $V=a^3$.

4.

Visual hint

Consider the atom at (0.25,0.25,0.25) coordinates in the interactive diamond lattice image and look at its neighbours.

The distance between neirest neighbouring atoms is $d=\frac{\sqrt{3}a}{4}$

5. $$F=\frac{\sqrt{3}\pi }{16}$$

Exercise 3: Directions and Spacings of Miller planes

1. Miller plane is a plane that intersects an infinite number of lattice points. Miller indeces are a set of 3 integers which specify a set of parallel planes.

2.

Small hint

The $(hkl)$ plane intersects lattice at position vectors of $\frac{{\mathbf{a}}_{\mathbf{1}}}{h},\frac{{\mathbf{a}}_{\mathbf{2}}}{k},\frac{{\mathbf{a}}_{\mathbf{1}}}{l}$. Can you define two general vector inside the $(hkl)$ plane that span the plane?

3. The same hint as mentioned in the previous subquestion also applies here.

Subquestion 4

Big hint

There is always a neighbouring lattice plane which intersects the (0,0,0) lattice point.

Small hint

Do not forget to normalize your unit vector.