📖 Definitions:
- Periodicity: crystal consists of a regular array of identical “structural units”
- Basis: the structural unit, can consist of 1 or more atoms (blue red and green object)
- Unit cell: Basis + the lattice vectors (shaded area)
- Lattice: An infinite array created by repeating the unit cell (whole picture)
- crystal: An infinite repetition of the lattice structure (whole picture)
The lattice vectors (yellow) which represent the position of the lattice points from a reference point can be written as
$$ \begin{aligned} \vec R&=n_1 \vec a_1+ n_2 \vec a_2 + n_3 \vec a_3 \, \quad &\text{in } {\rm 3D} \\ \vec R&=n_1 \vec a_1+ n_2 \vec a_2 \quad &\text{in } {\rm 2D} \end{aligned} $$
where $n_1,n_2,n_3\in \Z$ and $\vec a_1, \vec a_2, \vec a_3$ are primitive lattice vector
🗒️ Note: the main focus will be on $\rm 3D$ lattices
📖 Definitions:
- Primitive lattice vectors: A set of vectors that can be used to generate all the lattice points
- Bravais lattice: the simplest way points can be periodic
🗒️ Note: there are many different primitive vectors for one lattice (they all have the same area or volume)
💃 Example: we can re-write the lattice vector as
$$ \vec R=n_1(\vec a_1+ \vec a_2)+ (n_2-n_1) \vec a_2+n_3 \vec a_3 $$
which is a new set of primitive lattice vectors with $\vec a_1 + \vec a_2 ,\vec a_2,\vec a_3$
X-ray scattering from parallel planes
Properties:
We detect the scattered waves as a function of angle
The diffraction pattern is obtained by measuring the scattered intensity at different angles
Experimentally we use $\lambda \sim d\sim 0.1\,\rm nm$
Information can be gained by looking at the spikes of constructive interference (Bragg’s law)
$$ 2d\sin(\theta)=n\lambda $$
where $d$ is the distance between adjacent lattice planes, $\theta$ is the angle of incidence, $n\in \Z$, $\lambda$ wavelength of incident
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/4347c2ad-fb72-4991-bec6-b41f3d2592d1/Primitive_unit_cell.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/4347c2ad-fb72-4991-bec6-b41f3d2592d1/Primitive_unit_cell.png" width="40px" />
Primitive unit cell: Any region of space that contains only one lattice point and can be translated by lattice vectors $\vec R$ to fill the whole space without leaving gaps or forming overlaps
</aside>
💃 Example: The parallelepiped whose edges are the primitive vectors $\vec a_1, \vec a_2,\vec a_3$ is always a primitive unit cell.
The volume of a 3D primitive unit cell is
$$ V_\text{uc}=|\vec a_1 \cdot (\vec a_2 \times \vec a_3)| $$
