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Bas Nijholt
lectures
Commits
40c3ece7
Commit
40c3ece7
authored
Mar 31, 2018
by
Anton Akhmerov
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a7a138d7
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docs/lecture_5.md
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40c3ece7
...
@@ 13,7 +13,6 @@ In this lecture we will:
...
@@ 13,7 +13,6 @@ In this lecture we will:
### Crystal classification
### Crystal classification

**_Lattice_**

**_Lattice_**
+
periodic pattern of
*lattice points*
, which all have an identical view
+
periodic pattern of
*lattice points*
, which all have an identical view
+
lattice points are not necessarily the same as atom positions
+
lattice points are not necessarily the same as atom positions
+
there can be multiple atoms per lattice point
+
there can be multiple atoms per lattice point
...
@@ 21,25 +20,21 @@ In this lecture we will:
...
@@ 21,25 +20,21 @@ In this lecture we will:
+
multiple lattices with different point densities possible
+
multiple lattices with different point densities possible

**_Lattice vectors_**

**_Lattice vectors_**
+
from lattice point to lattice point
+
from lattice point to lattice point
+
$N$ vectors for $N$ dimensions
+
$N$ vectors for $N$ dimensions
+
multiple combinations possible
+
multiple combinations possible
+
not all combinations provide full coverage
+
not all combinations provide full coverage

**_Unit cell_**

**_Unit cell_**
+
spanned by lattice vectors
+
spanned by lattice vectors
+
has 4 corners in 2D, 8 corners in 3D
+
has 4 corners in 2D, 8 corners in 3D
+
check if copying unit cell along lattice vectors gives full lattice
+
check if copying unit cell along lattice vectors gives full lattice

**_Primitive unit cell_**

**_Primitive unit cell_**
+
smallest possible $
\r
ightarrow$ no identical points skipped
+
smallest possible $
\r
ightarrow$ no identical points skipped
+
not always most practical choice
+
not always most practical choice

**_Basis_**

**_Basis_**
+
only now we care about the contents (i.e. atoms)
+
only now we care about the contents (i.e. atoms)
+
gives element and position of atoms
+
gives element and position of atoms
+
properly count partial atoms $
\r
ightarrow$ choose which belongs to unit cell
+
properly count partial atoms $
\r
ightarrow$ choose which belongs to unit cell
...
@@ 52,12 +47,10 @@ In this lecture we will:
...
@@ 52,12 +47,10 @@ In this lecture we will:
1.
Choose origin (can be atom, not necessary)
1.
Choose origin (can be atom, not necessary)
2.
Find other lattice points that are identical
2.
Find other lattice points that are identical
3.
Choose lattice vectors, either primitive (red) or not primitive (blue)
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

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$^{
\c
irc}$

for graphite: ${
\b
f a}_1={
\b
f a}_2$ = 0.246 nm = 2.46 Å, $
\g
amma$ = 60$^{
\c
irc}$
4.
Specify basis
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(
\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)$

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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