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Fenster schliessen

Which solids have simple one Atom Basis?

Noble Metals (Au, Ag, Hg, ...)
Alkali Metals (Li, Na, K, Rb, Cs, Fr)

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Crystal structure =

lattice + basis
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Definition of a Bravais lattice?

A(r) = A(r+T) with T = n1*a1 + n2*a2 + n3*a3 n1,2,3 are integer

  • The vectors are NOT unique.
  • There are 5 Bravais lattices in 2D and 14 in 3D 
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What is a primitive cell?

A primitive cell fills the whole space without holes or overlapping.

  • Parallelogram/-epipid based on the Bravais vectors. (not unique)
  • every other shape which fills the whole space without holes or overlapping.
  • Wigner-Seitz cell (rectangular or hexagon)

All primitive cells have the same Volume and have one lattice point. (Proof it)
(There are also non primitive cells which may have higher symmetry, body centered cube bcc)

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Honeycomb lattice

is no Bravais lattice.

Graphene has a honeycomb lattice (two hexagonal lattices )It can be discribed as a hexagonal  lattice with a basis of 2 atoms.

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Rotational symmetrie

There is not only Translational symmetrie but also rotational.

There are only 2, 3,4,6 fold rotation symmetries allowed (Proof it).

(5 is possible in quasi crystals)

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There are which and how many Bravais lattices?

  • oblique (least symmetry) phi != 90 |a1| != |a2|
  • rectangular phi = 90 |a1| != |a2| ( reflection and 180 Rotation)
  • square phi =90 |a1| = |a2| (90 Rotation)
  • hexagonal phi =60 |a1| = |a2| (60 Rotations)
  • centered rectangular rhombic