Lesson 1X-rays & Diffraction

Nicola DöbelinRMS Foundation, Bettlach, Switzerland

February 11 – 14, 2013, Riga, Latvia

Electromagnetic Spectrum

5

X rays: Wavelength λ: 0.01 – 10 nmEnergy: 100 eV – 100 keV

Generation of X-radiation:Shoot electrons on matter

Interatomic distances in crystals:typically 0.15 – 0.4 nm

Interference phenomena onlyfor features ≈ λ

Generation of X-rays

6

Accelerated electron impinges on matter:

Electron is deflected and decelerated by the atomic nucleus.(Inelastic scattering)

Deflected electron emits electromagnetic radiation.Wavelength depends on the loss of energy.

Bremsstrahlung (Deceleration radiation)

Bremsstrahlung

7

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20

20 kV, 20 mA

30 kV, 20 mA

40 kV, 20 mA

Continuous spectrum

0.25 0.30

Bremsstrahlung

8

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20

30 kV, 20 mA

Continuous spectrum

0.25 0.30

30 kV, 30 mA

30 kV, 40 mA

Characteristic Radiation

9

KLM

Eb (eV)0

76

122

933952

1097

8979

M4,5 (3d)

M2,3 (3p)

M1 (3s)

L3 (2p3/2)L1 (2p1/2)

L1 (2s)

K1 (1s)

Kα1 Kα2 Kβ

Cu

Wavelength of Kα1, Kα2, Kβ, Lα... are characteristic for the atomic species.

X-rays: Spectrum

10

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Cu

Kα1

Kα2

Kβ

Kα1

Kα2

Kβ

Mo

X-ray Tube

11

e‒

Filament

Target (Cu, Mo, Fe, Co, ...)

Be windowAccelerationVoltage

Vac

uum

FilamentCurrent

Generator settings: kV mA

Old X-ray tubes

12

Caution:Beryllium is toxic & carcinogenic!

- Never touch the windows!- Use appropriate covers!

Caution:Beryllium is toxic & carcinogenic!

- Never touch the windows!- Use appropriate covers!

Lifetime of a few years:

- Vacuum decreases� loss of intensity

- Tungsten from filament depositson target� contaminated spectrum (characteristicW spectrum starts to appear)

- Monitor the intensity

- Replace old tubes

Focal Point

13

Typical target size:Length: 10-12 mmWidth: 0.4-1.0 mm

Target

Point focus

Line focus

Take-off angle(typically 6°)

X-rays: Summary

14

• Generated in an X-ray tube

• Spectrum contains Bremsstrahlung (continuous) and characteristic radiation (Kα1, Kα2, Kβ) of target material

• Tube is characterized by:• Target material• Size and shape of target• Aceleration voltage and

current

Diffraction Basics

15

Interaction of X-rays with matter:

- Absorption (photoelectric effect, giving rise to fluorescence)- Elastic scattering (Thomson scattering)- Inelastic scattering (Compton scattering)

Absorption Photoelectric effect, Fluorescence

1. Absorption and ionization2. Relaxation and emission of characteristic radiation

CuKα1

FeKα1

Fe atom

Elastic Scattering

16

Electron oscillates in the electric field,emits photons of the same wavelength as

the incoming radiation (λs = λp).

λp

λs

CuKα1

CuKα1

Fe atom

Secondary wave is in phase (+ 180°) with primary wave.

Crystal Lattice

17

Crystal: Periodic arrangement of atoms/ions/molecules in 3 dimensions.

Electrons of each atom become a source ofscattered radiation (spherical waves)

xx.xx.xxxx Tagung18Image: http://www.forbes.com/

Positive interference (amplification)Negative interference (extinction)

More sources in ordered arrangement=

More distinct interference pattern

Bragg’s Law

19

n · λ = 2 · d · sin(θ)

d

λ

θθ 2θ

Diffracted beam looks like a «reflection», but it is scattered radiation

Bragg’s Law

20

CuKα1 = 0.154056 nma = 0.2 nmb = 0.5 nm

ba

b

a

2θ = 45.30°2θ = 17.72°

d = 0.2 nm

d = 0.5 nm

θ = 22.65°θ = 8.86°

Lattice Planes and Miller Indices

21

a

b

Definition:A lattice plane is a planewhich intersects atoms ofa unit cell across the whole3‐dimensional lattice.

d(100)

d(010)

d(110)

d(-210)

- Each lattice plane generates a diffraction peak.

- The 2θ angle of thepeak depends on the plane’s d-spacing.

- Diffraction peaks canbe labelled with the plane’s Miller index.

Single Crystal

22

A single crystal must be rotatedto bring each lattice plane indiffraction condition.

2θ 2θ2θ

Polycrystals, Powders

23

In an ideal powder everypossible orientation ofcrystals occurs.

In a random powderno orientation is preferred.

In an ideal powder all possible diffraction peaksare generated, regardless of sample orientation.

Diffraction Cones

24

Diffraction at an angle 2θ° fromthe primary beam

All possible rays form a cone = diffraction cone = Debye cone

Diffraction Cones

25

(120)

(100)

(010)

One Debye Cone for each lattice plane spacing (d value)

Powder sample:

Debye Ring

26

Gra

y V

alue

2θ Angle

Gra

y V

alue

2θ Angle

Powder Diffractometer

27

X-ray tube

Primary Beam

PowderSample

Diffraction Cones«Secondary Beams»

X-ray Detectorscanning X-ray intensity

vs. 2θ angle

Powder Diffraction Pattern

28

10 20 30 40 50 600

500

1000

1500

2000In

tens

ity (

cts)

Diffraction Angle (°2θ)

Lesson 2: All about powder diffractometers

Monochromatic X-radiation

29

n · λ = 2 · d · sin(θ)

http://fineartamerica.com

Diffraction angle θ depends on wavelength λ:

Polychromatic X-ray Beam

We need monochromatic X-radiation!

Monochromatic X-radiation

30

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Cu

Kα1

Kα2

Kβ

X-ray Beam from Tube

Bremsstrahlung

Kβ

Kα2

Kα1

Monochromator: Remove every wavelength but Kα1

Monochromator

31

X-radiation is absorbed by solid matter.

Absorption coefficient depends on wavelength.

There are steps (absorption edges) in the spectrum.

Wavelength (nm)

Abs

orpt

ion

Coe

ffic

ient

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Ni

K: 0.14879 nm

L-I: 1.22988 nm

Ni

«K» edge

Ni-Filter

32

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Cu

Kα1

Kα2

Kβ

Ni

Cu Radiation

Ni filter

Ni-Filter

33

Cu Radiation

Ni filter

Ni-filtered Cu Radiation

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Kα1

Kα2

Kβ

Ni

Kβ and Bremsstrahlung attenuated

No elimination of Kα2

Ni-filtered Diffraction Pattern

34

27 28 29 30 31 320

100000

200000

300000

400000

500000In

tens

ity

Diffraction Angle (°2θ)

CuKα1

CuKα2

Ni-filtered Diffraction Pattern

35

27 28 29 30 31 320

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Inte

nsity

Diffraction Angle (°2θ)

CuKβ

CuKα1 & CuKα2 duplet

Remaining Bremsstrahlung

Absorption Edge

Impurity

CuKα Satellites

(= CuKα3)

Ni Filter: Primary or Secondary Beam

36

BremsstrahlungKβ

Kα2

Kα1

Cu Radiation

Ni filter

Ni-filtered primary beam

Kα1

Kα2

Cu RadiationNi filter

Primary beam filter

Secondary beam filter

Kβ Filter

37

Target

Element

Kα1 (nm) Kα2 (nm) Kβ (nm) Kβ Filter Absorption

Edge λK (nm)

Cr 0.228975 0.229365 0.20849 V 0.2269

Fe 0.193631 0.194002 0.17567 Mn 0.1896

Co 0.178900 0.179289 0.16208 Fe 0.1744

Ni 0.165794 0.166178 0.15002 Co 0.1608

Cu 0.154059 0.154441 0.139225 Ni 0.1488

Mo 0.709317 0.713607 0.63230 Zr 0.6889

Ag 0.559422 0.563813 0.49708 Rh 0.5339

Birkholz, M. «Thin Film Analysis by X-ray Scattering», Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006.

Summary: K β Filter

38

Kβ Filter:

- Mostly eliminates Kβ

- Does not eliminate Kα2

- Moderate loss of intensity of Kα1 and Kα2

- Leaves an absorption edge in the foot of the diffraction peaks

- Attenuation of Kβ depends on thickness of filter foil

- Can be placed in the primary or secondary beam

Monochromator Crystal

39

Graphitesingle crystal

Emission Line Wavelength (nm) 2θ Bragg Diffraction

Condition (°)

CuKα1 0.154059 26.57

CuKα2 0.154441 26.64

CuKβ 0.139225 23.97

d (002) = 0.3352 nm

θ = 13.3°

n · λ = 2 · d · sin(θ)

2θ = 26.6°

Kβ and most of the Bremsstrahlung (BS)are not in diffraction condition � Extinction

Monochromator Crystal

40

BSKβ

Kα2

Kα1

Graphite Crystal

Kα1

Si / Ge CrystalsHigh-resolution monochromator

Graphitemonochromator

Kα1 Kα2

Graphite Monochromator

41

27 28 29 30 31 320

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000In

tens

ity

Diffraction Angle (°2θ)

CuKα1

CuKα2

Graphite Monochromator

42

27 28 29 30 31 320

100

200

300

400

500In

tens

ity

Diffraction Angle (°2θ)

CuKα1 & CuKα2 duplet

CuKα Satellites

(= CuKα3)

Monochromator Crystal

43

Monochromator Crystal:

- Completely eliminates Kβ

- Reduces background intensity

- Si / Ge eliminate Kα2, Graphite does not eliminate Kα2

- Severe loss of intensity of Kα1 (and Kα2)

- Graphite crystal can be placed in primary or secondary beam

- Si / Ge crystals are usually placed in the primary beam

- Monochromatic beam is polarized

Energy-Dispersive Detector

44

BS Kβ

Powder Sample

Detector

Kα2Kα1

Detector’s energy windowis set to Kα1/2, other

wavelengths are ignored(«digital filtering»)

Energy-Dispersive Detector

45

Energy-dispersive Detector:

- Completely eliminates Kβ

- Reduces background intensity

- Does not eliminate Kα2

- No loss of intensity of Kα1 and Kα2

Summary: Monochromators

46

• Monochromatic X-radiation is required for powder XRD

• Bremsstrahlung and Kβ must be eliminated from thetube’s spectrum

• 3 different types of monochromators:• Kβ filter (Cu tube + Ni filter, Mo tube + Zr filter)• Monochromator crystal• Energy-dispersive detector

• Most systems do not eliminate Kα2!

Overview of Instruments

47

Lab Instrument Monochromator

RTU Rigaku Ultima+ Graphite

Monochromator

RTU Panalytical X’Pert Ni-Filter

LU Bruker D8 Ni-Filter

Uppsala Uni Bruker D8 Ni-Filter

RTU Salaspils Bruker D8 Energy-dispersive

Detector

RMS (Uni Bern) Panalytical X’Pert Ni-Filter

RMS (Uni Bern) Panalytical CubiX Graphite

Monochromator