## Lattice and periodic system Actually, before this part, we should go and finish free energy surface. But let's just put it to the last and finish quantum related part first. ### Basis A group of vectors formed by some particles and can be used to represent any particles in this periodic system. ### Lattice The set of mathematical points to which the basis is attached. ### Primitive lattice For any two identical points (for the crystal) $r$, $r'$ satisfy $r'=r+n_1\vec{a_1}+n_2\vec{a_2}+n_3\vec{a_3}$, then $a_i$ is called *primitive* (translation vector) and forms a primitive lattice, $n_i$ are integers. Volume: $V=a_1\cdot a_2 \times a_3$ There are 5 Bravais lattices for two decades and 14 for 3D case. ### Wigner-Seitz cell The same method is used to define the first brilliant zone. This is here a method to define primitive cells. ### Space group It includes the symmetry operation possible to the cell. ### Defining a cell Typically requires $a, b, c, \alpha,\beta,\gamma$. $a$ along $x$, $b$ along $y$, $\alpha,\beta,\gamma$ define $c$ direction. ![[Drawing 2023-09-27 22.42.56.excalidraw.svg]] >[!Info] >See more on [Crystal structure](https://en.wikipedia.org/wiki/Crystal_structure) and [7.2.2: Lattice Structures in Crystalline Solids - Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/07%3A_The_Crystalline_Solid_State/7.02%3A_Formulas_and_Structures_of_Solids/7.2.02%3A_Lattice_Structures_in_Crystalline_Solids)