Lattice spacingtypically

o1010 m 1

o1A

Max von Laue(1879-1960)

1914 Nobel prize

Laue 1912

Crystal diffraction

Today X-ray diffraction supplemented by electron and neutron diffration

Energies X-ray, electrons and neutrons wave-particle

hcE h

hcE

X-ray:o

1A E 12 k eV

Electrons:

Neutrons:

hp k

h hp 2mE

-31em 9.1 10 kg

o1A

E 150 eV

h hp 2mE

o1A -27

nm 1.6749 10 kg

E 0.08 eV

Typical Laue X-ray diffraction pattern

symmetry of the pattern

Laue X-ray diffractionYAlO3

c-axis normal to picture

symmetry of the crystal

Complementarity of the three types of radiation

X-ray diffraction Electron diffraction Neutron diffraction

•Photon energies 10keV-100keVlarge penetration depth

3D crystal structure

•scattering by electron density

best results for atoms with high Z

•Charged particle “strong” interaction with matter

low penetration depth

Study of: surfaces thin films

•Interaction with nuclei Improved efficiencyfor light atoms Inelastic scattering:phonons

•Magnetic moment interacts with moment of electrons

Magnetic scattering:Structure, magnons

Law describing the necessary condition for diffractionApplicable for photons, electrons and neutrons

2 sind n n: integer

Bragg’s lawCondition for efficient specular reflection 2 sinhkld n

(click for java applet)

Bragg Diffraction Law

Spacing dhkl between successive (hkl) planes

In cubic systems:222 lkh

adhkl

x

y

221102 ad

ad110

2110ad

Top view

dhkl for non cubic lattice later in the frameworkof the reciprocal lattice

Bragg’s law necessary condition

•structure factor

•atomic form factor

Intensity of particular(hkl) reflection

General theory of Diffraction

X-ray source

R

P

r

BR’

R’-r

X-ray source

R

P

r

BR’

R’-r

Plane wave incoming at P 0 ( )0

i R r i tP

kA A e

Scattered wave contribution from Pincoming at B

( )( ) i R rB P

kA A r e

Electron density at P0 ( ) ( )0 ( )i R r i ik

Bkt R rA A e r e

0 0( ) ( )

0 ( )i k R k R t i rk kA e r e

Total scattering from the entire volume:0( ) 3( ) k ki rA r e d r

0k k

23ri

23r)kk(i2

rde)r(

rde)r(A)(I 0

Diffraction experiment measures the intensity I of the scattered waves

where 0kk is the scattering vector

Diffracted intensity is the square of the Fourier transform of the electron density

In crystals )r( is periodic 1D example ,...,,,,n),nax()x( 3210

n

xna

i

n e)x(2

Fourier series expansion

0k k

k 1

af(x) a cos kx b sin kx2

2π periodic function decomposed into cos kx and sin kx

orikx

kk

f(x) c e

where

0

k k k

k k

ikx

a for k 021c (a ib ) for k 021 (a ib ) for k 02

1 f(x)e dx2

3dimensional case1dimensional case

n

xna

i

n e)x(2

G

rGiG e)r(

mana

i

n

xna

i

n

n

)max(na

i

n

ee

e)max(

22

2

with

122

22

)mnsin(imncosee mniman

ai

)x(e)max(n

xna

i

n 2

translational invariance of )r(

with respect to lattice vector

332211 anananrn

)r()rr( n

mrG n 2

Reciprocal lattice vectors

23ri rde)r()(I

Diffracted intensity is the square of the Fourier transform of the electron density

Remember:

G

rGiG e)r( periodic electron density

23r)G(i

GG rde)(I

with V

3r)G(i rdeV1)G( 22

G V)G(I

(click for information about -functions)

22G V)G(I

Scattering condition G is nothing but Bragg´s law !

mrG n 2 332211 anananrn

decomposition into so far unknown basis vectors321 glgkghG

with h, k, l integers

The reciprocal lattice

321g,g,gThe basis vectors of the reciprocal lattice are determined by:

)aa(aaaag

321

3211

2

)aa(aaaag

321

1322

2

)aa(aaaag

321

2133

2

igThese fulfill the condition ijji

ag 2

mrG n 2 holds, where321 glgkghG

Examples for reciprocal lattices

3 dimensions

2 dimensions

Important properties of the reciprocal lattice vectors321

glgkghGhkl

lies perpendicular to the lattice plane with Miller indices (hkl)hklG

simple example for the (111) plane in the cubic structure

),,a(a 001 ),a,(a 002

)a,,(a 003

),a,a(),a,(),,a(aa 0000021

)a,a,(aa 023

),a,a(aa 021

and)a,a,(aa 023

span the (111) lattice plane

vector )aa()aa( 2321 (111) plane

)aa()aa()aa()aa(

)aa()aa(

22213231

2321

)aa()aa()aa( 213213

0

111321Gggg

Distance dhkl between lattice planes (hkl) related to hklG according to

hklhkl d

G 2

d111

)G,a(cosa

d1111

111 1111

1111

GaGa

ijjiag 2

111

2Ga

111111

2G

d

G111

Equivalence between the scattering condition hklG

and Bragg´s law sindhkl2

lattice plane (hkl)

Ө

k0

k

-k0k

Ө

200

22 22 k)(coskkkkk

Elastic scattering: k=k0

212 cosk

22222 sincossincosk

1 2cos

sink2

sin4hkl

hkl dG

2

sindhkl2

Geometrical interpretation of the scattering condition hklG

k0

k

G2Ө

(000)

reciprocal lattice

Ewald construction

Crystal in random orientation not necessarily reflection rotation of the crystal

polychromatic radiation

(click for animation)

incoming monochromatic beam

Rotating crystal arrangement

Powder method / Debye Scherrer

determine unknown structure

Precise measurement of lattice constants

Laue method

transmission

reflection

Polychromatic X-rays

Orientation of crystal with known structure

The structure factor

22

V)G(IhklGhkl

Scattering condition ( Bragg’s law )necessary condition

Controls the actual intensity of the (hkl)-reflex

rde)r(V

rGi

cellc

hklhkl 31

hkl

hkl

hklG

rGi

G e)r(Remember: because crystal periodic )r()rr( n

Fourier-coefficients

rde)r(V

rGi

cellc

hklhkl 31

Majority of the electrons are centered in a small region around the atoms

core electrons

Scattering from valence electrons can be neglected

rde)r(eV

rGirGi

c

hklhkl

31

Atom in n-th unit cell is located at position r

rrr

atomic scattering factor fα

rGi

hklhklefF Structure factor

atomic scattering factor rde)r(f rGi hkl

3

Spherically symmetric

ddrdsinre)r(f cosrGi hkl 2

d)(cosdrdre)r( cosrGi hkl 2

where G , r’

rdrrG

rGsin)r( 24

sinkG 02

rdr

)/)(sinr))/)(sinrsin)r(f

2

444

atomic scattering factor

Maximum at Ө=0 (forward scattering)

rdr)r()(f

240 Z number of electrons/atom

(Click for calculations of

atomic scattering factors)

Structure factor of a lattice with basis

Structure factor of the bcc lattice: Conventional cell contains two atoms at

r1=(0,0,0)r2=(1/2,1/2,1/2)

Both atoms have the same atomic scattering factor f1 = f2 = f

Reciprocal unit cell: cube with cell side of 2π/a )l,k,h(a

Ghkl

2

)lkh(irGi

hkl efefF hkl

1

evenlkhforf

oddlkhforefF )lkh(i

hkl

2

01

( ) cos( ( )) sin( ( ))i h k le h k l i h k l

We observe e.g. diffraction peaks from (110), (200), (211) planes

but no peaks from (100), (111), (2,1,0) planes

•Shape and dimension of the unit cell can be deduced from Bragg peaks

•Content of the unit cell (basis) determined from intensities of reflections

If f1 = f2 peaks like (100), (111), (2,1,0) appearlike CsI

Similar situation in the case of fcc KCl/KBr: f(K+)=f(Cl-) = f(Br-)

KCl: Non-zero if all indices even

KBr: all fcc-peaks present