9_crystal_structure.md

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import matplotlib.pyplot as plt

import numpy as np
from math import sqrt,pi

from common import draw_classic_axes, configure_plotting

import plotly.offline as py
from plotly.subplots import make_subplots
import plotly.graph_objs as go

configure_plotting()

py.init_notebook_mode(connected=True)

Lecture 9 – Crystal structure

based on chapter 12 of the book

!!! success "Expected prior knowledge"

Before the start of this lecture, you should be able to:

- use elementary vector calculus

!!! summary "Learning goals"

After this lecture you will be able to:

- Describe any crystal using crystallographic terminology, and interpret this terminology
- Compute the volume filling fraction given a crystal structure
- Determine the primitive, conventional, and Wigner-Seitz unit cells of a given lattice
- Determine the Miller planes of a given lattice

??? info "Lecture video"

<iframe width="100%" height="315" src="https://www.youtube-nocookie.com/embed/zl8htOplU1s" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Crystal classification

In the past few lectures, we derived some very important physical quantities for phonons and electrons, such as the effective mass, the dispersion relation. These systems we considered were mainly 1D. But most solids, such as crystals, are 3D structures. Describing 3D system is not much harder than describing 1D systems. It does however, require a new language and framework in order to fully describe such structures. Therefore the upcoming two lectures will focus on developing, understanding, and applying this language and framework.

Lattices and unit cells

Most of solid state physics deals with crystals, which are periodic multi-atomic structures. To describe such periodic structures, we need a simple framework. Such a framework is given by the concept of a lattice:

A lattice is an infinite set of points defined by an integer sums of a set of linearly independent primitive lattice vectors(will be explained later on).

Which for a 3D system translates to:

\mathbf{R}_{\left[n_{1} n_{2} n_{3}\right]}=n_{1} \mathbf{a}_{1}+n_{2} \mathbf{a}_{2}+n_{3} \mathbf{a}_{3}, \quad \mathrm{for } \:\: n_{1}, n_{2}, n_{3} \in \mathbb{Z}.

With \mathbf{a}_i being the primitive lattice vectors. For a n dimensional system, we have to define n linearly independent primitive lattice vectors in order to be able to map out the entire lattice.

This definition is pretty rigorous. But there exist multiple equivalent definitions of a lattice. A more informal definition is:

A lattice is a set of points where the environment of each point is the same.