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

2

Bravais Lattices

cubic

tetragonal

orthorhombic

rhombohedral

monoclinic

hexagonal triclinic

Advanced solid state physics WS2016

3

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

5

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

12

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