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Lesson 1

X-Rays & Diffraction

Nicola DöbelinRMS Foundation, Bettlach, Switzerland

January 14 – 16, 2015, Bern, Switzerland

Electromagnetic Spectrum

2

X rays:

Wavelength λ: 0.01 – 10 nm

Energy: 100 eV – 100 keV

Generation of X-radiation:

Shoot electrons on matter

Interatomic distances in crystals:

typically 0.15 – 0.4 nm

Interference phenomena only

for features ≈ λ

X-ray Tube

3

e‒

Filament (W)

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

Be window

Acceleration

VoltageV

acu

um

Filament

Current

X-ray Tube

4

e‒

Filament (W)

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

Point focus

Line focus

Generation of X-rays

5

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

6

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

Acceleration Voltage:

� Speed / Energy of Electrons

Bremsstrahlung

7

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 Generator Current:

� Number of Electrons

Characteristic Radiation

8

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

9

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

Old X-ray tubes

10

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

X-rays: Summary

11

• 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

12

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 ionization

2. Relaxation and emission of characteristic radiation

CuKα1

FeKα1

Fe atom

Elastic Scattering

13

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

14

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

Electrons of each atom become a source of

scattered radiation (spherical waves)

15

Interference pattern

n=2

+ =

Positive / constructive

interference

Amplification

+ =

Negative / destructive

interference

Extinction

16

Interference pattern

n=4n=8n=12n=25n=50

Bragg’s Law

17

n · λ = 2 · d · sin(θ)

d

λ

θθδ

2δ = n · λ δ = d · sin(θ)

d = (n · λ) / (2 · sin(θ))

Lattice Planes and Miller Indices

18

a

b

Definition:

A lattice plane is a plane

which intersects atoms of

a unit cell across the whole

3‐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.

Bragg’s Law

19

CuKα1 = 0.154056 nm

a = 0.2 nm

b = 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°

Single Crystal

20

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

2θ 2θ

2θ

Polycrystals, Powders

21

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

22

Diffraction at an angle 2θ° fromthe primary beam

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

Diffraction Cones

23

(120)

(100)

(010)

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

Powder sample:

Debye Ring

24

Gra

y V

alu

e

2θ Angle

Gra

y V

alu

e

2θ Angle

Powder Diffractometer

25

X-ray tube

Primary Beam

Powder

Sample

Diffraction Cones

«Secondary Beams»

X-ray Detectorscanning X-ray intensity

vs. 2θ angle

Powder Diffraction Pattern

26

10 20 30 40 50 60

0

500

1000

1500

2000In

ten

sity (

cts

)

Diffraction Angle (°2θ)

Lesson 2: All about powder diffractometers

Monochromatic X-radiation

27

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!

28

Monochromatic X-Radiation

Cu Radiation

Kβ absorption filter

filtered Radiation

Ideally: Isolate Kα1

Reality: Suppress Kβ and Bremsstrahlung

Wavelength (nm)

Inte

nsity

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Cu

Kα1

Kα2

Kβ

BremsstrahlungCu Radiation

Monochromator Crystal

(Graphite, d = 0.3352 nm)

CuKα1/2 Radiation

n · λ = 2 · d · sin(θ)

Cu RadiationEnergy dispersive

Detector

Digital filtering

Kβ-filtered Diffraction Pattern

29

27 28 29 30 31 32

0

100000

200000

300000

400000

500000

Inte

nsity

Diffraction Angle (°2θ)

CuKα1

CuKα2

LaB6 pattern, Cu radiation, Ni filter in primary beam (Bruker D8 Advance)

Kβ-filtered Diffraction Pattern

30

27 28 29 30 31 32

0

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)

LaB6 pattern, Cu radiation, Ni filter in primary beam (Bruker D8 Advance)

Graphite Monochromator

31

27 28 29 30 31 32

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Inte

nsity

Diffraction Angle (°2θ)

CuKα1

CuKα2

LaB6 pattern, Cu radiation, Graphite monochromator, secondary beam (PANalytical CubiX3)

Graphite Monochromator

32

27 28 29 30 31 32

0

100

200

300

400

500

Inte

nsity

Diffraction Angle (°2θ)

CuKα1 & CuKα2 duplet

CuKα Satellites

(= CuKα3)

LaB6 pattern, Cu radiation, Graphite monochromator, secondary beam (PANalytical CubiX3)

31 32 33 34 35 36

0

10000

20000

30000

40000

50000

Inte

nsity (

co

unts

)

Diffraction Angle (°2θ)

Energy-Dispersive Detector

33

Al2O3 pattern, Cu radiation, LynxEyeXE detector (Bruker D8 Advance)

CuKα1

CuKα2

31 32 33 34 35 36

0

1000

2000

3000

4000

5000

Inte

nsity (

co

un

ts)

Diffraction Angle (°2θ)

Energy-Dispersive Detector

34

CuKβ

CuKα1 & CuKα2 duplet

Edge of

Energy Window

Al2O3 pattern, Cu radiation, LynxEyeXE detector (Bruker D8 Advance)

31 32 33 34 35 36

0

5000

10000

15000

20000

25000

30000

Inte

nsity (

co

un

ts)

Diffraction Angle (°2θ)

Energy-Dispersive Detector + Kβ filter

35

CuKα1

CuKα2

50% loss of intensity(0.0125 mm Ni filter)

Al2O3 pattern, Cu radiation, LynxEyeXE detector (Bruker D8 Advance)

31 32 33 34 35 36

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Inte

nsity (

co

un

ts)

Diffraction Angle (°2θ)

Energy-Dispersive Detector + Kβ filter

36

CuKα1 & CuKα2 duplet

Al2O3 pattern, Cu radiation, LynxEyeXE detector (Bruker D8 Advance)

Edge of

Energy Window

50% loss of intensity(0.0125 mm Ni filter)

Monochromators

37

Kβ Filter:Selectively suppresses Kβand parts of Bremsstrahlung

Monochromator crystal andenergy dispersive detector:Suppress everything BUT Kα

Important difference forfluorescent samples

«Notch filter»

«Bandpass filter»

38

Ni-Filtered Cu Radiation, Fe Fluorescence

10 20 30 40 50 60 70 80

0

500

1000

1500

2000

Inte

nsity [cts

]

Diffraction Angle [°2θ] Cu radiation, Ni filter in 2nd beam

(PANalytical X’Pert Pro)

100% TiO2

95% TiO2

+ 5% NiFe2O4

90% TiO2

+ 10% NiFe2O4

75% TiO2

+ 25% NiFe2O4

50% TiO2

+ 50% NiFe2O4

25% TiO2

+ 75% NiFe2O4

Secondary-monochromated Cu-Radiation

39

10 20 30 40 50 60 70 80

0

200

400

600

800

1000

Inte

nsity [cts

]

Diffraction Angle [°2θ] TiO2 + NiFe2O4,

Cu radiation, 2nd graphite monochromator

(PANalytical CubiX3)

95% TiO2

+ 5% NiFe2O4

90% TiO2

+ 10% NiFe2O4

75% TiO2

+ 25% NiFe2O4

50% TiO2

+ 50% NiFe2O4

25% TiO2

+ 75% NiFe2O4

100% NiFe2O4

In case of Fluorescence, use:

- Secondary-beam monochromator

- Energy-dispersive Detector

- Different X-ray tube (Cr / Fe / Co)

Summary: Monochromators

40

Optical Element Effect on Spectrum Effect on Intensity

Kβ Filter Reduces Kβ peaks Moderate loss

Graphite Monochromator Eliminates Kβ peaks

Eliminates Fluorescence

Strong loss

Multi-bounce Monochromator Eliminates Kβ and Kα2

Eliminates Fluorescence

Massive loss

(mostly used on Synchrotrons)

Energy dispersive Detector Reduces Kβ peaks

Eliminates Fluorescence

No loss

Overview of Instruments

41

Lab Instrument Monochromator

RMS Foundation Bruker D8 Energy dispersive

Detector

Uni Bern Panalytical X’Pert Kβ-Filter

Uni Bern Panalytical CubiX Graphite

Monochromator

Bruker D8 Panalytical X’Pert Panalytical CubiX