Simple Cubic Unit Cell Basics

A simple cubic unit cell is defined by three mutually perpendicular unit vectors $ \hat{i}, \hat{j}, \hat{k} $ of equal length, typically set to the lattice constant $a$. Lattice directions are specified using Miller indices.

Lattice Vector Representation

The specified lattice vectors can be represented in terms of the unit vectors:

• The lattice vector $ [100] $ represents the direction along the x-axis. As a vector, $ \vec{a} = 1\hat{i} + 0\hat{j} + 0\hat{k} $.
• The lattice vector $ [111] $ represents the direction with equal components along all three axes. As a vector, $ \vec{b} = 1\hat{i} + 1\hat{j} + 1\hat{k} $.

Angle Calculation Using Dot Product

The angle $ \theta $ between two vectors $ \vec{a} $ and $ \vec{b} $ is determined using the dot product formula:

$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta $

Calculate the dot product $ \vec{a} \cdot \vec{b} $:

$ \vec{a} \cdot \vec{b} = (1\hat{i} + 0\hat{j} + 0\hat{k}) \cdot (1\hat{i} + 1\hat{j} + 1\hat{k}) = (1 \times 1) + (0 \times 1) + (0 \times 1) = 1 $

Calculate the magnitudes $ |\vec{a}| $ and $ |\vec{b}| $:

$ |\vec{a}| = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1 $

$ |\vec{b}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{1+1+1} = \sqrt{3} $

Substitute these values into the dot product formula:

$ 1 = (1)(\sqrt{3}) \cos \theta $

Solve for $ \cos \theta $:

$ \cos \theta = \frac{1}{\sqrt{3}} $

Find the angle $ \theta $:

$ \theta = \arccos\left(\frac{1}{\sqrt{3}}\right) \approx 54.7356^\circ $

Rounding to one decimal place, the angle is $ 54.7^\circ $.