Crystal Structures
A crystal is invariant under translation by a lattice vector.
The crystal lattice is a set of mathematical points defined by:
rn = n1a1 + n2a2 + n3a3 n1, n2, n3 integer
The vectors a1, a2, a3 define the unit cell. Atoms inside the unit cell are called thebasis.
= +
Crystal Symmetries
1) Translation Bravais Lattices (14 in 3D, 5 in 2D; fcc, bcc,…)
2) Point OperationsPoint Groups ( 32 in 3D, 10 in 2D; rotation, mirror)
1) + 2)Space Groups (230 in 3D, 17 in 2D; crystal structures)
Reciprocal lattice Diffractionpattern Fourier transform of the lattice
The reciprocal lattice consists of the vectors Ghkl defined by:
Ghkl = hg1 + kg2 + lg3 h, k, l integer (Miller indices)
g1= 2/Va2a3 for all permutations: 1,2,3 2,3,1 3,1,2 V = a1(a2a3)
The (hkl) define the lattice planes which produce a diffraction peak (Bragg, Laue).
Ghkl is perpendicular to the planes(hkl). Its length is given by theinverseof theplane spacing dhkl: |Ghkl| = 2/dhkl
Square brackets denote the direction [hkl] which is perpendicular to the planes (hkl).
Brillouin zone = Unit cell in reciprocal space (k-space).
Usually one chooses the Wigner-Seitz construction for the Brillouin zone, wherek=(000) is at the center and the surface planes are half way to the nearest reciprocal lattice points. More on that in Lecture 10.
Crystal Structure and Bonding
Covalent: Diamond; Binary compounds: Zincblende (cubic), Wurtzite (hexagonal)
Directional bonds determine the geometry, e.g. tetrahedral for sp3 bonding.
Surface: Re-bonding (= “reconstruction”) minimizes the density of broken bonds.
See Lecture 6 for the intricate structures formed at silicon surfaces.
Ionic: NaCl for ralkali rhalogen CsCl for ralkali rhalogen
Optimize the Coulomb (Madelung) energy for a given size of the ions.
Surface: Charge neutrality to avoid infinite Coulomb energy: NaCl(100)
Metallic, Molecular: fcc, hcp
Isotropic bonding favors close-packed structures (12 neighbors)
Two close-packed structures: face centered cubic (fcc) = abcabc stacking
hexagonal close packed (hcp) = ababab stacking
Surface: Also dense packing, i.e. fcc(111), hcp(0001) (four indices: a1, a2, a3, c)
Stacking of Close-Packed Layers
a abcaba
Low Index Planes for a Cubic Lattice
Covalent
IonicMetallic
NaCl structure:fcc structure:
CsCl structurebcc structure
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