Institute of Solid State PhysicsTechnische Universität Graz

7. Crystal structure,Diffraction

April 10, 2018

face centered cubic, fcc

Bravais lattice

body centered cubic, bcc

simple cubic

basis

Bravais lattice Crystal

= +

Auguste Bravais

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]

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

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

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

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

Inequivalent atoms in the unit cell

An element can have two distinct positions

Diamond conventional unit cell

Space group: 227point group: m3m

Basis:

Primitive lattice vectors:

Diamond

zincblende

ZnSGaAsInP

space group 216F43m

ZnSZnOCdSCdSeGaNAlN

Number 186

There are 2 polytypes of ZnS: zincblende and wurtzite

wurtzite

NaCl

Number 225

Bravais: fcc

CsCl

Number 221

SrTiO3LiNbO3BaTiO3PbxZr1-xTiO3

Number 221simple cubic

perovskite

YBa2Cu3O7

Number 47

graphite

Polytypes of carbongraphite (hexagonal)carbon nanotubesdiamondrhobohedral graphitehexagonal diamond

Space group 194

4 inequivalent C atoms in the primitive unit cell

atomic packing density

fcc, hcp = 0.74 random close pack = 0.64simple cubic = 0.52diamond = 0.34

343

3

/ 20.52

6LL

Coordination number

Number of nearest neighbors an atom has in a crystal

sc 6

bcc 8 fcc 12

Graphene 3 diamond 4

hcp 12

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

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

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

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

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:

( ) 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)

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: