advanced solid state physics ws2016 bravais lattice...bravais lattice there are 14 different basic...
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Advanced solid state physics WS2016
1
System Lengths &
Angles
Number of
Lattices
Cubic a b c
a b g 90
3
Tetragonal a b c
a b g 90
2
Orthorhombic a b c
a b g 90
4
Rhombohedral a = b = c
a b g 90
1
Monoclinic a b c
a g 90 b
2
Triclinic a b c
a b g
1
Hexagonal a = b c
a b 90
g 120
1
a
b
c
a
b
g
Bravais Lattice
There are 14 different basic
crystal lattices
Definition according to
unit cell edge lengths and angles
Advanced solid state physics WS2016
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Bravais Lattices
cubic
tetragonal
orthorhombic
rhombohedral
monoclinic
hexagonal triclinic
Advanced solid state physics WS2016
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Closed Packed Lattices: fcc and hcp
Layer Sequence
ABA:
hcp
ABC:
fcc
fcc
face centered
cubic
hcp
hexagonal
close packed
Advanced solid state physics WS2016
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Diamond Lattice
fcc Lattice with Basis
(000) a, (1/4,1/4,1/4) a
View with bonds
Examples:
Diamond, Silicon, Germanium
x y
z
a
4/
4/
4/
a
a
a
Advanced solid state physics WS2016
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Crystal Structures of the Elements
Ch. Kittel, Einführung in die Festkörperphysik
Advanced solid state physics WS2016
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Diffraction with Waves (Light and Particles)
Diffraction of electrons and
X-rays from Silver
a) 36 keV - electrons
b) Copper-Kα-Radiation
λ = 0,154 nm.
p
hde Broglie: 0,006 nm
Advanced solid state physics WS2016
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X-ray diffraction: radiation source
X-ray tube
Voltage U =10 kV -100 kV
Spectrum of the radiation
Bremsstrahlung
charakteristic radiation
Copper-Anode, 38 kV eU
hchv
minimal
wavelength Alternative:
Synchrotron-
radiation
Advanced solid state physics WS2016
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Debye-Scherrer Method
• monochromatic source of light
• crystal powder (averaging over
all orientations)
Crystals Film
Diffraction pattern
of a fcc-crystal
(MgO)
Structurfactor fcc: all indices are even or all odd!
Advanced solid state physics WS2016
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Debye-Scherrer Method
(semi) sc-Crystal (KCl)
Visible Rings (inside out) :
(100), (110), (111), (200),
(210), (211), (220), (221)..
fcc-Crystal (Al)
Visible Rings (inside out) :
(111), (200), (220), (311), (222)..
Advanced solid state physics WS2016
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Debye-Scherrer Method
Salt Crystal
same size atoms (KCl):
„sc“
different size atoms (KBr):
fcc
Advanced solid state physics WS2016
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Debye-Scherrer Method
FeCo – Crystal
ordered: sc (with Basis)
unordered: bcc
sc
bcc
Structure factor bcc:
Sum of all indices must
be even!
Advanced solid state physics WS2016
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Laue Method
• „white“ x-ray light
• oriented single crystal
Diffraction on NaCl (KCl?)
in direction <100>
Gerthsen Physik
Advanced solid state physics WS2016
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Laue Method
Diffraction on Si
(fcc with basis)
forward direction <111>
3-fold symmetry!
Backward scattering in direction <111>
direction <100>
4-fold
symmetry!
Advanced solid state physics WS2016
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Protein crystal diffraction
Advanced solid state physics WS2016
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structure reconstruction
Advanced solid state physics WS2016
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record: 5 million atom image
virus capsid
Protein structure
databank:
www.rcsb.org
Advanced solid state physics WS2016
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quasi-crystals
Ho-Mg-Zn quasi-crystal
(Al-Pd-Mn) quasicrystal