Advanced solid state physics WS2016 Bravais Lattice...Bravais Lattice There are 14 different basic...
Transcript of Advanced solid state physics WS2016 Bravais Lattice...Bravais Lattice There are 14 different basic...
Advanced solid state physics WS2016
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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
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Bravais Lattices
cubic
tetragonal
orthorhombic
rhombohedral
monoclinic
hexagonal triclinic
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Closed Packed Lattices: fcc and hcp
Layer Sequence
ABA:
hcp
ABC:
fcc
fcc
face centered
cubic
hcp
hexagonal
close packed
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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
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Crystal Structures of the Elements
Ch. Kittel, Einführung in die Festkörperphysik
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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
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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
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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!
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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)..
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Debye-Scherrer Method
Salt Crystal
same size atoms (KCl):
„sc“
different size atoms (KBr):
fcc
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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!
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Laue Method
• „white“ x-ray light
• oriented single crystal
Diffraction on NaCl (KCl?)
in direction <100>
Gerthsen Physik
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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!
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Protein crystal diffraction
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structure reconstruction
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record: 5 million atom image
virus capsid
Protein structure
databank:
www.rcsb.org
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quasi-crystals
Ho-Mg-Zn quasi-crystal
(Al-Pd-Mn) quasicrystal