<aside> 🦅 Periodicity(for a crystallin): Crystalline solids are periodic, it consists of regular arrays of identical basis which can consist of one or more atoms
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💃 Examples: of periodicity (ie translational symmetry)
For 2D lattices, a lattice translation vector, which covers all lattice points can be defined as
$$ \vec R_{n,m}=n\vec a+ m\vec b $$
where $n,m\in \Z$ and $\vec a$ and $\vec b$ are non-collinear lattice vectors with some angle $\phi$ between them
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/8fdfc061-ec07-47d2-bf8b-7e719dc0baf8/Bravais_lattice.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/8fdfc061-ec07-47d2-bf8b-7e719dc0baf8/Bravais_lattice.png" width="40px" /> Bravais lattice: distinct lattice type, there are five Bravais lattice in $2\rm D$
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| Lattice | no. of mirror planes | n-fold rotation |
|---|---|---|
| oblique | 0 | - |
| rectangular | 2 | 2 ($180\degree$) |
| square | 4 | 2 4 ($180\degree,90\degree$) |
| hexagonal | 6 | 2 3 6($\footnotesize{180\degree,120\degree,60\degree}$) |
| centred rectangular | 2 | 2 ($180\degree$) |
There can only be $2,3,4,6$-fold rotations that can be consistent with $2\rm D$ translational symmetry.
If we try something like $5$-fold we get:
$$ \text{lattice}+\text{basis}=\text{crystal structure} $$
To form a crystal structure we attach copies of the basis to points defined by translational vectors
💃 Example: consider graphene
Examples of primitive cells
🗒️ Note: unit cells typically: basis + 1 lattice point, though not always: rectangular with 2 of each
3D lattices are 2D lattices with extra points
1 Simple cubic
3 Face-centred cubic
5 Body tetragonal
7 End orthorhombic
9 face orthorhombic
11 Centrd monoclinic
2 Body centred cubic
4 Tetragonal
6 Orthorhombic
8 Body orthorhombic
10 Monoclinic
12 Triclinic
13 Rhombohedral
14 Hexagonal
💼 Case: lets focus on the cubic lattices
