Crystals:
Recall Crystal = Bravais lattice + Basis.
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Bravais lattice = repeated set of points, R = n1a1 + n2 a2 + n3 a3
Basis = location of atoms “decorating” the lattice
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!
!
di
• Primitive cell: always has volume V = a1 ⋅ a2 × a3 . One option for the cell is
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the parallelepiped with edges a1 , a2 , a3 .
• Naming convention for cell dimensions and angles: Lengths a, b, c; angles, α
= (b-c angle) etc. as shown below. Lengths are the "lattice parameters".
• Cubic lattice parameters always a × a × a (= conventional cell dimensions);
hexagonal a × a × c, etc.
14 Bravais lattices
Including point group (lattice
decoration), 230 “Space Group”
Symmetry classes. Example,
hexagonal structures with Space
group #194:
Cubic primitive and conventional cells
BCC (body center)
FCC (face center)
Examples: Iron, Na
Examples: Cu, Al, Ni,
Silicon*, NaCl*, etc
* with basis
Close packed: face center cubic close-packed,
hexagonal close-packed
FCC: A Bravais lattice; atoms are all the same; colors show ABCABC stacking.
HCP: Atoms all the same but 2-atom basis. colors show ABAB stacking.
diamond&structure:
FCC&bravais,&basis&2
tetrahedral&bonding
other&carbon&forms
graphite
graphene
Silicon,&germanium
Not&bravais (even&though&
atoms&identical)
tetrahedral&bonds
nanotubes
“Buckyball”
GaAs (zincblende)
IIIIV&
semiconductors:
tetrahedral&like&
silicon,&can&view&
roughly&as&
covalent&bonded&
framework&with&
partial&charge&
transfer,&e.g.
“GaIqAs+q”
GaN
(wurtzite)
CsCl;
ionic
NaCl ;&ionic
These& materials:& completely& ionic&to&
good&approximation;&formation&
energy&=&Coulomb&energy&of&
assembled& charges.&
high$Tc
YBa2Cu3O7
K3C60 (superconductor)
FCC&structure
“AlQ3”
organic
semiconductor
Images4of4two$dimensional4tilings:444What&is&the&Bravais lattice?&&The&basis?&
Lattice4Planes
(200)&planes,& simple& cubic.
Parallel&equalIspaced& planes& intersect&all&
Bravais lattice&points.
(Proof&in&terms&of&reciprocal&lattice)
Indexing:&&•&atom&at&corner&of&cell,&edges&=&
lattice& vectors.&
•&choose&plane&nearest&origin.&
•&indices& are&integer&divisors&of&the&lengths& of&
cell&edges&intercepted& by&plane.&
•&Usual&notation&h,&k,&ℓ.
(Planes&more&than&needed& to&intersect& all&
Bravais lattice&points&if&indices& have&
common&denominator)
Lattice4Planes
(100)&planes,& FCC
Note,&lattice&planes& intersect&Bravais
lattice points&(not&necessarily& all&atoms)
But&note,&FCC&and&BCC&are&always&
indexed& as&if&they&were&simple& cubic&with&
a&basis&
Reciprocal Lattice:
Vectors form k-space Fourier components of Bravais (direct-space) lattice.
Plane waves have symmetry of lattice,
!
!
!
To construct: K! = hb + kb + "b
1
2
3
!
!
!
etc.; also
a ×a
b1 = 2π ! ! !
a1 ⋅ a2 × a3
2
3
• Then can show: Wavefronts of
e
! !
iK ⋅ R
e
! ! !
iK ⋅( r + R )
=e
! !
ai ⋅ b j = 2πδ ij
are Bragg planes.
! !
iK ⋅ r