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Bas Nijholt
lectures
Commits
40c3ece7
Commit
40c3ece7
authored
Mar 31, 2018
by
Anton Akhmerov
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hmm...
parent
a7a138d7
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docs/lecture_5.md
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40c3ece7
...
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@@ 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 $
\r
ightarrow$ 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 $
\r
ightarrow$ 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: ${
\b
f a}_1={
\b
f a}_2$ = 0.246 nm = 2.46 Å, $
\g
amma$ = 60$^{
\c
irc}$
4.
Specify basis

using ${
\b
f a}_1$ and ${
\b
f a}_2$: C$(0,0)$, C$
\l
eft(
\f
rac{2}{3},
\f
rac{2}{3}
\r
ight)$

using ${
\b
f a}_1$ and ${
\b
f a}_{2}'$: C$(0,0)$, C$
\l
eft(0,
\f
rac{1}{3}
\r
ight)$, C$
\l
eft(
\f
rac{1}{2},
\f
rac{1}{2}
\r
ight)$, C$
\l
eft(
\f
rac{1}{2},
\f
rac{5}{6}
\r
ight)$
...
...
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