Commit 40c3ece7 authored by Anton Akhmerov's avatar Anton Akhmerov
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parent a7a138d7
......@@ -13,7 +13,6 @@ In this lecture we will:
### Crystal classification
- **_Lattice_**
+ periodic pattern of *lattice points*, which all have an identical view
+ lattice points are not necessarily the same as atom positions
+ there can be multiple atoms per lattice point
......@@ -21,25 +20,21 @@ In this lecture we will:
+ multiple lattices with different point densities possible
- **_Lattice vectors_**
+ from lattice point to lattice point
+ $N$ vectors for $N$ dimensions
+ multiple combinations possible
+ not all combinations provide full coverage
- **_Unit cell_**
+ spanned by lattice vectors
+ has 4 corners in 2D, 8 corners in 3D
+ check if copying unit cell along lattice vectors gives full lattice
- **_Primitive unit cell_**
+ smallest possible $\rightarrow$ no identical points skipped
+ not always most practical choice
- **_Basis_**
+ only now we care about the contents (i.e. atoms)
+ gives element and position of atoms
+ properly count partial atoms $\rightarrow$ choose which belongs to unit cell
......@@ -52,12 +47,10 @@ In this lecture we will:
1. Choose origin (can be atom, not necessary)
2. Find other lattice points that are identical
3. Choose lattice vectors, either primitive (red) or not primitive (blue)
- lengths of lattice vectors and angle(s) between them fully define the crystal lattice
- for graphite: $|{\bf a}_1|=|{\bf a}_2|$ = 0.246 nm = 2.46 Å, $\gamma$ = 60$^{\circ}$
4. Specify basis
- using ${\bf a}_1$ and ${\bf a}_2$: C$(0,0)$, C$\left(\frac{2}{3},\frac{2}{3}\right)$
- using ${\bf a}_1$ and ${\bf a}_{2}'$: C$(0,0)$, C$\left(0,\frac{1}{3}\right)$, C$\left(\frac{1}{2},\frac{1}{2}\right)$, C$\left(\frac{1}{2},\frac{5}{6}\right)$
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