7. Crystal structure, Diffraction - TU Grazlampx.tugraz.at/~hadley/ss1/lectures18/apr10.pdfPb, Th...
Transcript of 7. Crystal structure, Diffraction - TU Grazlampx.tugraz.at/~hadley/ss1/lectures18/apr10.pdfPb, Th...
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Institute of Solid State PhysicsTechnische Universität Graz
7. Crystal structure,Diffraction
April 10, 2018
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face centered cubic, fcc
Bravais lattice
body centered cubic, bcc
simple cubic
basis
Bravais lattice Crystal
= +
Auguste Bravais
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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]
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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
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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
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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
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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
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Inequivalent atoms in the unit cell
An element can have two distinct positions
Diamond conventional unit cell
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Space group: 227point group: m3m
Basis:
Primitive lattice vectors:
Diamond
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zincblende
ZnSGaAsInP
space group 216F43m
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ZnSZnOCdSCdSeGaNAlN
Number 186
There are 2 polytypes of ZnS: zincblende and wurtzite
wurtzite
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NaCl
Number 225
Bravais: fcc
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CsCl
Number 221
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SrTiO3LiNbO3BaTiO3PbxZr1-xTiO3
Number 221simple cubic
perovskite
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YBa2Cu3O7
Number 47
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graphite
Polytypes of carbongraphite (hexagonal)carbon nanotubesdiamondrhobohedral graphitehexagonal diamond
Space group 194
4 inequivalent C atoms in the primitive unit cell
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atomic packing density
fcc, hcp = 0.74 random close pack = 0.64simple cubic = 0.52diamond = 0.34
343
3
/ 20.52
6LL
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Coordination number
Number of nearest neighbors an atom has in a crystal
sc 6
bcc 8 fcc 12
Graphene 3 diamond 4
hcp 12
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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
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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
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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
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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
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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:
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( ) 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)
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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: