# Introduction of Solid State Physics

## Chapter 1: Crystal structures

### Ch. 1-1: Periodic arrays of atoms: Crystal

digraph hierarchy {

nodesep=1.0
rankdir=LR                
node [fontname=Arial, color=skyblue,shape=Mrecord,style=filled]
edge [color=darkgreen]
solid->crystal
solid->"non-crystal (~nm)"
crystal->"single crystal (單晶): long-range order"
crystal->"polycrystal (多晶): short-range order"
}

Quasi-crystal:
A structure with long-range order with 5 or $\geqslant$ 7-fold symmetry, but without translational symmetry

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#### <font size=4>Lattice translation vectors</font>
- ideal crystal: infinite repetition of identical atoms or group of atoms <font color=red>basis</font>
- <font color=orange>lattice</font>: point to which basis is attached


A 3D lattice can be constructed by three translational vectors $\vec{a_{1}}$, $\vec{a_{2}}$ and $\vec{a_{3}}$

$\vec{T}=u_{1}\vec{a_{1}}+u_{2}\vec{a_{2}}+u_{3}\vec{a_{3}}$

$\{u_{1}, u_{2}, u_{3}\}\in$ integers

$\vec{T}:$ lattice translation vector

- If the origin is located at a given lattic point, $\vec{T}$ is also pointing at a lattice point.
- For a physical quantity $V(\vec{r})$ has the periodicity of the lattice then ==$V(\vec{r})=V(\vec{r}+\vec{T})$==

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#### <font size=4>Primitive cell</font>

$\vec{a_{1}}$, $\vec{a_{2}}$, $\vec{a_{3}}$ are said to be "primitive" vectors when the volume of parallelepiped is minimum.

==$V_{c}= | \vec{a_{1}} \cdot (\vec{a_{2}}\times \vec{a_{3}}) |$==

**primitive cell**: the parallelepiped formed by the 3 primitive vectors $\vec{a_{1}}$, $\vec{a_{2}}$, $\vec{a_{3}}$
- The smallest unit that constructs the lattice by repeating the cell.
- There is only not lattice point per primitive cell.
- Primitive cell is not uniquely defined $\rightarrow$ ($\vec{a_{1}}$, $\vec{a_{2}}$) ($\vec{a_{1}}'$, $\vec{a_{2}}'$) ($\vec{a_{1}}''$, $\vec{a_{1}}''$) are primitive

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#### <font size=4>Wigner-Seitz primitive cell</font>
A primitive cell not only has the smalled volume, but also keeps the symmetry of the crystal.

**Constructing WS cell**
1. Drawing lines from a given lattic point to all nearby lattice points.
2. At the mid point and normal to these lines, drwa new lines (or planes)
3. The smallest area (volume) enclosed is the WS cell.

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### Ch. 1-2: Fundamental types of lattice

#### <font size=4>Symmetry operation / Point symmetry group</font>
- Crystal lattice characterized by their symmetry
- They can be mapped into themselves by various symmetry operation: rotation, reflection, inversion

**Fundamental symmetry operation**
| symbol | element | operation |
| ------ | ------- | --------- |
| $E$ | identity | No operation |
| $\sigma$ | symmetry plane | Reflection in the plane |
| $i$ | inversion center | $(x,y,z)\rightarrow (-x,-y,-z)$ |
| $C_{n}$ | proper axis | Rotation (clockwise) by $\frac{2\pi}{n}$
| $S_{n}$ | improper axis | 1. Rotation by $\frac{2\pi}{n}$ <br> 2. Reflection in the plan $\perp$ to rotation axis |

:::info
In the crystal, there are only **8 independent** symmetry operations: $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $C_{6}$, $S_{4}$, $i$, $\sigma$
:::

All the symmetry operations of crystal lattices can be described by the combination of the 8 independent operations

**Point symmetry group**
- A group of symmetry operations that keep at least one point fixed

:::info
**<font size=4>Definition of a group:</font>**
A series of elements (operations)
$a$, $b$, $c$ .... can form a group $G=\{a,b,c...\}$ as long as the following properties are satisfied:
1. Closure (封閉性):
- If $a$, $b$ $\in G$, then $ab \in G$
2. Associativity (結合性):
- If $a$, $b$, $c$ $\in G$, then $(ab)c = a(bc)$
3. Identity (等同性):
- $E \in G$, such that $Ea=aE=a$
4. Inverse (反量性):
- If $a \in G$, then $a^{-1} \in G$ such that $a^{-1}a=aa^{-1}=E$
:::

$ex$:
$C_{3}$ group: $\{E, C_{3}^{1}, C_{3}^{2}\}$

$C_{3}^{1}$: rotation by $\frac{2\pi}{3}$

$C_{3}^{2}$: rotation by $2 \times \frac{2\pi}{3}$

#### <font size=4>Lattice types</font>
| lattice point group | elements | lattice types |
| ------------------- | -------- | ------------- |
| 1D | 2 | 1 |
| 2D | 10 | 5 |
| 3D | 32 | 14 |

**2D lattice types**
1. Square
- 
- $a_{1}=a_{2}$
- $\phi = 90^{\circ}$
2. Hexagonal
- 
- $a_{1}=a_{2}$
- $\phi = 120^{\circ}$
3. Rectangular
- 
- $a_{1}\neq a_{2}$
- $\phi = 90^{\circ}$
4. Centered Rectangular
- 
- $a_{1}\neq a_{2}$
- $\phi = 90^{\circ}$
5. Oblique
- 
- $a_{1}\neq a_{2}$
- $\phi \neq 90^{\circ}$