MAS.020UF Introduction to Solid State Physics

Crystal structure

This page contains a list of definitions and concepts you should know for the exam.

Some common crystal structures you should know

• A crystal consists of a basis (the atoms of a primitive unit cell) repeated on one of the 14 Bravais lattices. You should be able to draw the conventional unit cell given the basis and the Bravais lattice as in this problem.
• Any sum of primitive lattice vectors $(\vec{a}_1,\vec{a}_2,\vec{a}_3)$ brings you from one Bravais lattice point to another.
• The volume of a unit cell is $\vec{a}_1\cdot(\vec{a}_2\times\vec{a}_3)$.
• A translation vector of the crystal is $\vec{T}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ where $l,m,n$ are integers.
• Different primitive unit cells can be defined for the same crystal. The primitive unit cell with the maximum symmetry is called the Wigner-Seitz cell. To construct the WIgner-Seitz cell, draw lines from a Bravais lattice point to all of its neighboring points and bisect these lines with planes. All of the points that can be reached from the initial point without crossing a plane are in the Wigner-Seitz cell.
• Drawing Wigner-Seitz cells in two dimensions
• Drawing Wigner-Seitz cells in three dimensions
• Miller indicies are used to define directions and planes in a crystal. You should be able to draw the arrangement of atoms at the surface of a crystal cut along a plane specified by Miller indices such as in this problem.
• Every crystal structure has a set of symmetries associated with it. These are specified by the point group and the space group. The space group uniquely determines the point group and the Bravais lattice.
• Every crystal can be associated with one of the 32 point groups. A point group is the collection of symmetries (rotations, reflections, inversion) that a crystal has where at least one point in the crystal does not move during the transformation. Point group symmetries can be represented by $3\times 3$ matrices. Some examples are:
  Rotation about the $x$-axis by
  Rotation about the $y$-axis by
  Rotation about the $z$-axis by
  Rotoinversion about the $x$-axis by an angle
  Rotoinversion about the $y$-axis by an angle
  Rotoinversion about the $z$-axis by an angle
  
  $$\text{Identity }E= \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$
  $$\text{Identity }E= \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{Inversion }i= \left[\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{reflection about }x-y\text{ plane } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{reflection about }x-z\text{ plane } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{reflection about }y-z\text{ plane } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotation by }\pi\text{ about the }x\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{rotation by }\frac{2\pi}{3}\text{ about the }x\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{2}\text{ about the }x\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{3}\text{ about the }x\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix}\right]$$ $$\text{rotation by }\pi\text{ about the }y\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{rotation by }\frac{2\pi}{3}\text{ about the }y\text{ axis } \left[\begin{matrix} -\frac{1}{2}& 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0\\ \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{2}\text{ about the }y\text{ axis } \left[\begin{matrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{3}\text{ about the }y\text{ axis } \left[\begin{matrix} \frac{1}{2} & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{matrix}\right]$$ $$\text{rotation by }\pi\text{ about the }z\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotation by }\frac{2\pi}{3}\text{ about the }z\text{ axis } \left[\begin{matrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0\\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0\\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{2}\text{ about the }z\text{ axis } \left[\begin{matrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotation by }\frac{\pi}{3}\text{ about the }z\text{ axis } \left[\begin{matrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotoinversion by }\pi\text{ about the }x\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{2\pi}{3}\text{ about the }x\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{\sqrt{3}}{2} \\ 0 & -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{2}\text{ about the }x\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{3}\text{ about the }x\text{ axis } \left[\begin{matrix} -1 & 0 & 0 \\ 0 & -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \end{matrix}\right]$$ $$\text{rotoinversion by }\pi\text{ about the }y\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{2\pi}{3}\text{ about the }y\text{ axis } \left[\begin{matrix} \frac{1}{2}& 0 & \frac{\sqrt{3}}{2} \\ 0 & -1 & 0\\ -\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{2}\text{ about the }y\text{ axis } \left[\begin{matrix} 0 & 0 & 1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{3}\text{ about the }y\text{ axis } \left[\begin{matrix} -\frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \\ 0 & -1 & 0 \\ -\frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \end{matrix}\right]$$ $$\text{rotoinversion by }\pi\text{ about the }z\text{ axis } \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{2\pi}{3}\text{ about the }z\text{ axis } \left[\begin{matrix} \frac{1}{2} & \frac{\sqrt{3}}{2} & 0\\ -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0\\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{2}\text{ about the }z\text{ axis } \left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$ $$\text{rotoinversion by }\frac{\pi}{3}\text{ about the }z\text{ axis } \left[\begin{matrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & -1 \end{matrix}\right]$$

Suggested Reading
Kittel Chapter 1: Crystal Structure or R. Gross und A. Marx: Kristallstrucktur 1.1 - 1.2

Resources
• Table of crystal classes in 3D
• International Tables for Crystallography is the authoritative reference for the classification of crystal symmetries.
• Inorganic Crystal Structure Database
• Materials project
• A python package for crystal calculations