Institute of Solid State PhysicsTechnische Universität Graz
7. Crystal structure,Diffraction
April 10, 2018
face centered cubic, fcc
Bravais lattice
body centered cubic, bcc
simple cubic
basis
Bravais lattice Crystal
= +
Auguste Bravais
fcc
Al, Cu, Ni, Sr, Rh, Pd, Ag, Ce, Tb, Ir, Pt, Au, Pb, Th
Primitive Vectors:
Basis Vector:
Number 225
1
2
3
ˆ ˆ2 2
ˆ ˆ2 2
ˆ ˆ2 2
a aa y z
a aa x z
a aa x y
1 (0,0,0)B A
B
C
[111]
bcc
W Na K V CrFe Rb Nb Mo Cs Ba EuTa
Primitive Vectors:
Number 229
1
2
3
ˆ ˆ ˆ2 2 2
ˆ ˆ ˆ2 2 2
ˆ ˆ ˆ2 2 2
a a aa x y z
a a aa x y z
a a aa x y z
Basis Vector: 1 (0,0,0)B
hcp
Mg, Be, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Gd, Tb, Dy, Ho, Er, Tm, Lu, Hf, Re, Os, Tl
Space group 194
Hexagonal Bravais latticeBasis vectors:
2 1 11 2 3 3 2(0,0,0) ( , , )B B
AB
C
Hexagonal unit cellCrystallographic unit cell
Asymmetric unit
Unit cellcell 5.09000 6.74800 4.52300 90.000 90.000 90.000 natom 3 Fe1 26 0.18600 0.06300 0.32800 Fe2 26 0.03600 0.25000 0.85200 C 6 0.89000 0.25000 0.45000 rgnr 62 Cohenite (Cementite) Fe3 C
Asymmetric unitGenerated by PowderCell
Powder cell
http://www.bam.de/de/service/publikationen/powder_cell.htmhttps://www.ccdc.cam.ac.uk/Community/csd-community/freemercury/
Mercury
PowderCell
Standard data file: *.cif
Inequivalent atoms in the unit cell
An element can have two distinct positions
Diamond conventional unit cell
Space group: 227point group: m3m
Basis:
Primitive lattice vectors:
Diamond
zincblende
ZnSGaAsInP
space group 216F43m
ZnSZnOCdSCdSeGaNAlN
Number 186
There are 2 polytypes of ZnS: zincblende and wurtzite
wurtzite
NaCl
Number 225
Bravais: fcc
CsCl
Number 221
SrTiO3LiNbO3BaTiO3PbxZr1-xTiO3
Number 221simple cubic
perovskite
YBa2Cu3O7
Number 47
graphite
Polytypes of carbongraphite (hexagonal)carbon nanotubesdiamondrhobohedral graphitehexagonal diamond
Space group 194
4 inequivalent C atoms in the primitive unit cell
atomic packing density
fcc, hcp = 0.74 random close pack = 0.64simple cubic = 0.52diamond = 0.34
343
3
/ 20.52
6LL
Coordination number
Number of nearest neighbors an atom has in a crystal
sc 6
bcc 8 fcc 12
Graphene 3 diamond 4
hcp 12
Crystal structure determination
STM FIM TEM Scanning tunneling microscope
Field ion microscope Transmission electron microscope
Usually x-ray diffraction is used to determine the crystal structure
Institute of Solid State Physics
Crystal diffraction (Beugung) Technische Universität Graz
light photonssound phononselectron waves electronsneutron waves neutronspositron waves positronsplasma waves plasmons
Everything moves like a wave but exchanges energy and momentum as a particle
Expanding a 1-d function in a Fourier series
xAny periodic function can be represented as a Fourier series.
f(x)
01
( ) cos(2 / ) sin(2 / )p pp
f x f c px a s px a
a
multiply by cos(2p'x/a) and integrate over a period.
0 0
( ) cos 2 / cos 2 / cos 2 /2
a ap
p
acf x px a dx c px a px a dx
0
2 ( ) cos 2 /a
pc f x px a dxa
Expanding a 1-d function in a Fourier series
xAny periodic function can be represented as a Fourier series.
f(x)
01
( ) cos(2 / ) sin(2 / )p pp
f x f c px a s px a
a
2 pGa
cos sin2 2
ix ix ix ixe e e ex xi
( ) iGxG
G
f x f e
*
G Gf fFor real functions:
2 2p p
G
c sf i
reciprocal lattice vector
Institute of Solid State Physics
Fourier series in 1-D, 2-D, or 3-D Technische Universität Graz
In two or three dimensions, a periodic function can be thought of as a pattern repeated on a Bravais lattice. It can be written as a Fourier series
( ) iG rG
G
f r f e
Reciprocal lattice vectors (depend on the Bravais lattice)
Structure factors (complex numbers)
2 - ... 1,0,1,... G vb v ba
In 1-D:
( ) iG rG
G
f r f e
Any periodic function can be written as a Fourier series
Structure factor
Reciprocal lattice vector G
vi integers
1 1 2 2 3 3G b b b
2 3 3 1 1 2
1 2 31 2 3 1 2 3 1 2 3
2 , 2 , 2a a a a a ab b ba a a a a a a a a
kx
ky
Reciprocal lattice (Reziprokes Gitter)
Reciprocal space (Reziproker Raum)k-space (k-Raum)
p k
2k
k-space is the space of all wave-vectors.
A k-vector points in the direction a wave is propagating.
wavelength: momentum:
kx
kz
ky
G
exp( ) cos( ) sin( )x y z x y ziG r G x G y G z i G x G y G z Plane wave: