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Kai Sun
University of Michigan, Ann Arbor
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1D example
1D crystal 3 atoms/periodicity
Bravais lattice: 1. Choose one point in each unit cell. 2. Make sure that we pick the same point in
every unit cell. 3. These points form a lattice, which is
known as the Bravais lattice.
All sites are equivalent The vector connecting any two lattice sites
is a translation vector
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Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)
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Primitive cell in 2D
The Parallelegram defined by the two primitive lattice vectors are called a primitive cell.
the Area of a primitive cell: A = |𝑎 1 × 𝑎 2| Each primitive cell contains 1 site.
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Wigner–Seitz cell
the volume of a Wigner-Seitz cell is the same as a primitive cell each Wigner-Seitz cell contains 1 site (same as a primitive cell).
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2D Bravais lattices
http://en.wikipedia.org/wiki/Bravais_lattice
Green regions: conventional cells
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3D Bravais lattices
http://en.wikipedia.org/wiki/Bravais_lattice
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Cubic system
Conventional cells For a simple cubic lattice, a conventional cell = a primitive cell NOT true for body-centered or face-centered cubic lattices
How can we see it? sc: one conventional cell has one site (same as a primitive cell) bcc: one conventional cell has two sites (twice as large as a primitive cell) fcc: one conventional cell has four cites (1 conventional cell=4 primitive cells)
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Simple cubic
Lattice sites: 𝑎(𝑙 𝑥 + 𝑚 𝑦 + n 𝑧 )
Lattice point per conventional cell: 1 = 8 ×1
8
Volume (conventional cell): 𝑎3 Volume (primitive cell) : 𝑎3 Number of nearest neighbors: 6 Nearest neighbor distance: 𝑎 Number of second neighbors: 12
Second neighbor distance: 2𝑎
Packing fraction: 𝜋 6
≈ 0.524
Coordinates of the sites: (𝑙, 𝑛,𝑚) For the site 0,0,0 , 6 nearest neighbors: ±1,0,0 , 0, ±1,0 and 0,0, ±1 12 nest nearest neighbors: ±1,±1,0 , 0, ±1,±1 and (±1,0, ±1)
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Packing fraction
Packing fraction: We try to pack N spheres (hard, cannot deform).
The total volume of the spheres is 𝑁4 𝜋𝑅3
3
The volume these spheres occupy V > 𝑁4 𝜋𝑅3
3 (there are spacing)
Packing fraction=total volume of the spheres/total volume these spheres occupy
𝑃𝑎𝑐𝑘𝑖𝑛𝑔 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =𝑁4 𝜋
𝑅3
3
𝑉=
4 𝜋𝑅3
3
𝑉/𝑁=
4 𝜋𝑅3
3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑝𝑒𝑟 𝑠𝑖𝑡𝑒
=4 𝜋
𝑅3
3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑐𝑒𝑙𝑙
High packing fraction means the space is used more efficiently
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Simple cubic
Lattice sites: 𝑎(𝑙 𝑥 + 𝑚 𝑦 + n 𝑧 )
Lattice point per conventional cell: 1 = 8 ×1
8
Volume (conventional cell): 𝑎3 Volume (primitive cell) : 𝑎3 Number of nearest neighbors: 6 Nearest neighbor distance: 𝑎 Number of second neighbors: 12
Second neighbor distance: 2𝑎
Packing fraction: 𝜋 6
≈ 0.524
𝑃𝑎𝑐𝑘𝑖𝑛𝑔 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4 𝜋
𝑅3
3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑐𝑒𝑙𝑙
=4 𝜋
𝑅3
3
𝑎3 =4 𝜋
3(𝑅
𝑎)3=
4 𝜋
3(𝑎/2
𝑎)3=
𝜋
6≈ 0.524
About half (0.524=52.4%) of the space is really used by the sphere. The other half (0.476=47.6%) is empty.
Nearest distance= 2 R R= Nearest distance/2=𝑎/2
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bcc
Lattice sites
𝑎(𝑙 𝑥 + 𝑚 𝑦 + n 𝑧 ) and 𝑎[ 𝑙 +1
2𝑥 + 𝑚 +
1
2𝑦 + 𝑛 +
1
2𝑧 ]
Lattice point per conventional cell: 2 = 8 ×1
8+ 1
Volume (conventional cell): 𝑎3 Volume (primitive cell) : 𝑎3/2 Number of nearest neighbors: 8
Nearest neighbor distance: (𝑎
2)2+(
𝑎
2)2+(
𝑎
2)2=
3
2𝑎 ≈ 0.866𝑎
Number of second neighbors: 6 Second neighbor distance: 𝑎
Packing fraction: 3 8
𝜋 ≈ 0.680
Coordinates of the sites: (𝑙, 𝑛,𝑚) For the site 0,0,0 ,
8 nearest neighbors: ±1
2, ±
1
2, ±
1
2
6 nest nearest neighbors: ±1,0,0 , 0,±1,0 and (0,0, ±1)
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bcc packing fraction
Volume (primitive cell) : 𝑎3/2
Nearest neighbor distance: (𝑎
2)2+(
𝑎
2)2+(
𝑎
2)2=
3
2𝑎 ≈ 0.866𝑎
Packing fraction: 3 8
𝜋 ≈ 0.680
𝑃𝑎𝑐𝑘𝑖𝑛𝑔 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4 𝜋
𝑅3
3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑐𝑒𝑙𝑙
=4 𝜋
𝑅3
3
𝑎3/2=
8𝜋
3(𝑅
𝑎)3=
8𝜋
3(
34
𝑎
𝑎)3=
3𝜋
8≈ 0.680
About 68.0% of the space is really used by the sphere. About 32.0% of the space is empty.
Nearest distance= 2 R
R= Nearest distance/2=3
4𝑎
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fcc
Lattice sites
𝑎(𝑙 𝑥 + 𝑚 𝑦 + n 𝑧 ) 𝑎[ 𝑙 +1
2𝑥 + 𝑚 +
1
2𝑦 + 𝑛 𝑧 ]
𝑎[ 𝑙 +1
2𝑥 + 𝑚 𝑦 + 𝑛 +
1
2𝑧 ]
𝑎[𝑙𝑥 + 𝑚 +1
2𝑦 + 𝑛 +
1
2𝑧 ]
Lattice point per conventional cell: 4 = 8 ×1
8+ 6 ×
1
2= 1 + 3
Volume (conventional cell): 𝑎3 Volume (primitive cell) : 𝑎3/4 Number of nearest neighbors: 12
Nearest neighbor distance: (𝑎
2)2+(
𝑎
2)2+(0)2=
2
2𝑎 ≈ 0.707𝑎
Number of second neighbors: 6 Second neighbor distance: 𝑎
For the site 0,0,0 ,
12 nearest neighbors: ±1
2, ±
1
2, 0 , ±
1
2, 0, ±
1
2 and 0,±
1
2, ±
1
2
6 nest nearest neighbors: ±1,0,0 , 0,±1,0 and (0,0, ±1)
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fcc packing fraction
Volume (primitive cell) : 𝑎3/4
Nearest neighbor distance: (𝑎
2)2+(
𝑎
2)2+(0)2=
2
2𝑎 ≈ 0.707𝑎
𝑃𝑎𝑐𝑘𝑖𝑛𝑔 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =4 𝜋
𝑅3
3
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑐𝑒𝑙𝑙
=4 𝜋
𝑅3
3
𝑎3/4=
16𝜋
3(𝑅
𝑎)3=
16𝜋
3(
24
𝑎
𝑎)3=
2𝜋
6≈ 0.740
About 74.0% of the space is really used by the sphere. About 26.0% of the space is empty.
Nearest distance= 2 R
R= Nearest distance/2=2
4𝑎
0.740 is the highest packing fraction one can ever reach. This structure is called “close packing” There are other close packing structures (same packing fraction)
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Cubic system
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hexagonal
Primitive cell: a right prism based on a rhombus with an included angle of 120 degree. Volume (primitive cell):
𝑉𝑜𝑙 = 𝐴𝑟𝑒𝑎 × ℎ𝑒𝑖𝑔ℎ𝑡
= 21
2×
3
2𝑎1
2 𝑎3 =3
2𝑎2𝑐
2D planes formed by equilateral triangles Stack these planes on top of each other
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Examples
We choose three lattice vectors Three lattice vectors form a primitive or a conventional unit cell Length of these vectors are called: the lattice constants We can mark any unit cell by three integers: 𝑙𝑚𝑛
𝑡 = 𝑙𝑎 1 + 𝑚 𝑎 2 + 𝑛 𝑎 3 Coordinates of an atom: We can mark any atom in a unit cell by three real numbers: 𝑥𝑦𝑧. The location of this atom: 𝑥 𝑎 1 + 𝑦 𝑎 2 + 𝑧 𝑎 3 Notice that 0 ≤ 𝑥 < 1 and 0 ≤ 𝑦 < 1 and 0 ≤ 𝑧 < 1 Q: Why x cannot be 1? A: Due to the periodic structure. 1 is just 0 in the next unit cell
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Sodium Chloride structure
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Sodium Chloride structure
Face-centered cubic lattice Na+ ions form a face-centered cubic lattice Cl- ions are located between each two neighboring Na+ ions Equivalently, we can say that Cl- ions form a face-centered cubic lattice Na+ ions are located between each two neighboring Na+ ions
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Sodium Chloride structure
Primitive cells
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Cesium Chloride structure
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Cesium chloride structure
Simple cubic lattice Cs+ ions form a cubic lattice Cl- ions are located at the center of each cube Equivalently, we can say that Cl- ions form a cubic lattice Cs+ ions are located at the center of each cube Coordinates: Cs: 000
Cl: 1
2
1
2 1
2
Notice that this is a simple cubic lattice NOT a body centered cubic lattice For a bcc lattice, the center site is the
same as the corner sites Here, center sites and corner sites are
different
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Hexagonal Close-Packed Structure (hcp)
Let’s start from 2D (packing disks, instead of spheres). Q: What is the close-pack structure in 2D? A: The hexagonal lattice (these disks form equilateral triangles) Now 3D: Q: How to get 3D close packing? A: Stack 2D close packing structures on top of each other.
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Hexagonal Close-Packed Structure (hcp)
hcp: ABABAB… fcc: ABCABCABC…
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Diamond structure
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Carbon atoms can form 3 different crystals
Graphene (Nobel Prize carbon)
Diamond (money carbon/love carbon) Graphite (Pencil carbon)
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Not all lattices are Bravais lattices: examples the honeycomb lattice (graphene)
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Diamond lattice is NOT a Bravais Lattice either
Same story as in graphene: We can distinguish two different type of carbon sites (marked by different color) We need to combine two carbon sites (one black and one white) together as a (primitive) unit cell If we only look at the black (or white) sites, we found the Bravais lattice: fcc
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Cubic Zinc Sulfide Structure
Very similar to Diamond lattice Now, black and white sites are two different atoms fcc with two atoms in each primitive cell
Good choices for junctions
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Matter
Matters
gas/liquid:
Atoms/molecules can move around
solids:
Atoms/molecules cannot move
Crystals:
Atoms/molecules form a periodic
structure
Random solids:
Atoms/molecules form a random
structure