Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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System Lengths &

Angles

Number ofLattices

Cubic a = b = c

α = β = γ = 90°3

Tetragonal a =b ≠ c

α = β = γ = 90°2

Orthorhombic a ≠ b ≠c

α = β = γ = 90°4

Rhombohedral a = b = c

α = β = γ ≠ 90°1

Monoclinic a ≠ b ≠c

α = γ = 90° ≠ β2

Triclinic a ≠ b ≠ c

α ≠ β ≠ γ1

Hexagonal a = b ≠ c

α = β = 90°γ = 120°

1

α

b

c

a

β

γ

Bravais Lattice

There are 14 different basic crystal lattices

Definition according to unit cell edge lengths and angles

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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Bravais Lattices

cubic

tetragonal

orthorhombic

rhombohedral

monoclinic

hexagonaltriclinic

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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Closed Packed Lattices: fcc and hcp

Layer Sequence

ABA:hcp

ABC:fcc

fccface centeredcubic

hcphexagonalclose packed

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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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

xy

z

a

4/

4/

4/

a

a

a

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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Crystal Structures of the Elements

Ch. Kittel, Einführung in die Festkörperphysik

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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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

hλde Broglie: 0,006 nm

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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X-ray diffraction: radiation source

X-ray tube

Voltage U =10 kV -100 kV

Spectrum of the radiation

Bremsstrahlungcharakteristic radiation

Copper-Anode, 38 kVeU

hchv ==

λ

minimalwavelength

Alternative:Synchrotron-radiation

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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Debye-Scherrer Method

• monochromatic source of light

• crystal powder (averaging overall orientations)

Crystals Film

Diffraction pattern of a fcc-crystal(MgO)

Structurfactor fcc: all indices are even or all odd !

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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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 SS2014Kapitel 1.1: Crystal structure

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Debye-Scherrer Method

Salt Crystal

same size atoms (KCl):„sc“

different size atoms (KBr): fcc

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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Debye-Scherrer Method

FeCo – Crystalordered: sc (with Basis)unordered: bcc

sc

bcc

Structure factor bcc:Sum of all indices must be even!

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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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 SS2014Kapitel 1.1: Crystal structure

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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 SS2014Kapitel 1.1: Crystal structure

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Protein crystal diffraction

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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structure reconstruction

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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record: 5 million atom image

virus capsid

Protein structure databank:www.rcsb.org

Advanced solid state physics SS2014Kapitel 1.1: Crystal structure

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quasi-crystals

Ho-Mg-Zn quasi-crystal

(Al-Pd-Mn) quasicrystal