Crystal Symmetry Elements

$$\text{Fold Number}=\frac{360^{\circ}}{\text{Smallest Rotation Angle that makes the object look identical}}$$

不存在5-fold和高于6-fold的旋转轴：

以上是lecture1的内容，以下是lecture2：

Lattice

$$\text{LATTICE}+\text{MOTIF}=\text{CRYSTAL}$$

Rules for Choosing a Unit Cell

In crystallography, we pick a unit cell by following two rules (in order):

1. It must have the same symmetry as the whole crystal structure.
   • Example: If a crystal is cubic (symmetric like a cube), its unit cell must also be cubic.
2. It must contain the smallest number of lattice points.

Tips

• The crystal’s symmetry is determined by the whole atom pattern, not by the unit cell

• Higher fold symmetry is better

Primitive Unit Cell

我还是没完全理解这个概念，仅仅有一个点不是无敌了吗，无论要形成什么晶体都可以

Directions and Lines (2D)

2D晶格类型

共有5种，每种的关键区别在于边长关系|a|和|b|的夹角φ

1.Oblique lattice：|a|≠|b|，φ≠90°；对称性最低
2.Rectangular lattice：|a|≠|b|，φ=90°；具有矩形对称性
3.Centered Rectangular lattice：|a|≠|b|，φ=90°；在矩形晶格的中心额外有 1 个晶格点
4.Squared lattice：|a|=|b|，φ=90°；对称性高，含 4 次旋转轴
5.Hexagonal lattice：|a|=|b|，φ=120°；含 6 次旋转轴，是 2D 密堆积的典型晶格

2D Miller indices

Miller 指数不是 “截距比例”，而是 “单位晶胞内线的密度比例”，体现晶体周期性下”线的密集程度“

1.找截距
找一条不过原点的线，记录它在a轴和b轴的截距

2.转分数截距，截距/晶胞长度

3.取倒数，消去∞

4.简化，乘最小公倍数