Difração de Raios-X [7]

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First medical X-ray made by Wilhelm Röntgenof his wife Anna Bertha Ludwig's hand

Wilhelm Conrad Röntgen (1845 – 1923) was a German mechanical engineer and physicist, who,on 8 November 1895, produced and detected electro-magnetic radiation in a wavelength range known as X-rays or Röntgen rays, an achievement that earnedhim the first Nobel Prize in Physics in 1901.

Röntgen (1845 – 1923)

https://en.wikipedia.org/wiki/Wilhelm_Rontgen

Parts of the Diffractometer

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X-ray Tube: the source of X RaysIncident-beam optics: condition the X-ray beam before it hits the sampleThe goniometer: the platform that holds and moves the sample, optics, detector, and/or tubeThe sample & sample holderReceiving-side optics: condition the X-ray beam after it has encountered the sampleDetector: count the number of X-Rays scattered by the sample

X-Rays Diffractometer

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X-Rays Diffractometer

4>

Speed:- sample: θ- detector: 2θ

X-ray Generation

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XX--raysrays –– High energy*, highly penetrative High energy*, highly penetrative electromagnetic radiationelectromagnetic radiation

*E = *E = hchc//λλ λλ(X(X--rays)rays) = 0.02= 0.02--100100ÅÅ (avg. ~1 (avg. ~1 ÅÅ))λλ(visible light)(visible light) = 4000= 4000--72007200ÅÅ

XX--ray Vacuum Tuberay Vacuum Tube

Cathode (W)Cathode (W)–– electron electron generatorgenerator

Anode (Mo, Cu, Fe, Co, Cr) Anode (Mo, Cu, Fe, Co, Cr) ––electron target, electron target, XX--ray generatorray generator

X-ray Generation

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Electrons from the filament strike the target anode, producing characteristic radiation via the photoelectric effect.The anode material determines the wavelengths of characteristic radiation.While we would prefer a monochromatic source, the X-ray beam actually consists of several characteristic wavelengths of X rays.

X-ray Generation

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2.084920 Å2.08487ÅCr Kb0.632305 Å0.632288ÅMo Kb

2.293663 Å2.293606ÅCr Ka20.713609 Å0.713590ÅMo Ka2

2.289760 Å2.28970ÅCr Ka10.709319 Å0.709300ÅMo Ka1

ChromiumAnodes

MolybdenumAnodes

1.620830 Å1.62079ÅCo Kb1.392250 Å1.39220ÅCu Kb

1.792900 Å1.792850ÅCo Ka21.544426 Å1.54439ÅCu Ka2

1.789010 Å1.788965ÅCo Ka11.540598 Å1.54056ÅCu Ka1

Holzer et al.(1997)

Bearden(1967)

CobaltAnodes

Holzer et al.(1997)

Bearden(1967)

CopperAnodes

Wavelengths for X-Radiation are Sometimes Updated

Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect. Values from Bearden (1967) are reprinted in international Tables for X-Ray Crystallography and most XRD textbooks.Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)

Monochromators

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Diffraction from a monochromator crystal can be used to select one wavelength of radiation and provide energy discrimination.Most powder diffractometer monochromators only remove K-beta, W-contamination, and Brehmstralung radiation

Only HR-XRD monochromators or specialized powder monochromators remove K-alpha2 radiation as well.

A monochromator can be mounted between the tube and sample (incident-beam) or between the sample and detector (diffracted-beam)

An incident-beam monochromator only filters out unwanted wavelengths of radiation from the X-ray sourceA diffracted-beam monochromator will also remove fluoresced photons.A monochromator may eliminate 99% of K-beta and similar unwanted wavelengths of radiation. A diffracted-beam monochromator will provide the best signal-to-noise ratio, but data collection will take a longer time

Filters

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A material with an absorption edge between the K-alpha and K-beta wavelengths can be used as a beta filterThis is often the element just below the target material on the periodic table

For example, when using Cu radiation

Cu K-alpha = 1.541 ÅCu K-beta= 1.387 ÅThe Ni absorption edge= 1.488 Å

The Ni absorption of Cu radiation is:

50% of Cu K-alpha99% of Cu K-beta

Sup

pres

sion

Wavelength

Cu

Kα

W L

αC

u K

β

Ni filter

X-Rays Fluorescence

10>

Some atoms absorb incident X-rays and fluoresce them as X-rays of a different wavelength

The absorption of X-rays decreases the diffracted signalThe fluoresced X-rays increase the background noise

The increased background noise from fluoresced X-rays can be removed by using (i) a diffracted-beam monochromator or (ii) an energy sensitive detector.The most problematic materials are those two and three below the target material:

For Cu, the elements that fluoresce the most are Fe and Co

Mn Fe Co Ni Cu Zn GaV Cr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

Cs Ba L Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

K Ca Sc Ti Ge As Se Br Kr

F Ne

Cl Ar

N O

P S

B C

Si

Li Be

Na Mg

HeH

Fr Ra A

X-Rays Diffractometer

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Diffractometer

Bragg-Brentano geometry

(Primary monochromator)

(Secondary monochromator)

Bragg’s Law

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• For parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs when Bragg’s law is satisfied.

– In our diffractometers, the X-ray wavelength l is fixed.– A family of planes produces a diffraction peak only at a specific angle 2q.

• Additionally, the plane normal [hkl] must be parallel to the diffraction vector s

– Plane normal [hkl]: the direction perpendicular to a plane of atoms– Diffraction vector s: the vector that bisects the angle between the incident and

diffracted beam

θλ sin2 hkld= q q dhkl

dhkl

s[hkl]

Bragg’s Law

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2θ 2θ 2θ

For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams). Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.

s

[100] [110]s s

[200]

A polycrystalline sample should contain thousands of crystallites. Therefore, all possible diffraction peaks should be observed.

X-Rays Diffraction Patterns

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Position [°2Theta] (Copper (Cu))20 30 40 50

Counts

0

2000

4000

01000200030004000

0

2000

4000

SiO2 Glass

Quartz

Cristobalite

The three X-ray scattering patterns above were produced by three chemically identical forms SiO2 (silica).Crystalline materials like quartz and cristobalite produce X-ray diffraction patterns. Quartz and cristobalite have two different crystal structures, trigonal and tetragonal respectively.The amorphous glass does not have long-range atomic order and therefore produces only broad scattering features.

Quartz

Cristobalite

You can use XRD to determine

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Phase Composition of a SampleQuantitative Phase Analysis: determine the relative amounts of phases in a mixture by referencing the relative peak intensities

Unit cell lattice parameters and Bravais lattice symmetryIndex peak positionsLattice parameters can vary as a function of, and therefore give you information about, alloying, doping, solid solutions, strains, etc.

Residual Strain (macrostrain)Crystal Structure

By Rietveld refinement of the entire diffraction patternEpitaxy/Texture/OrientationCrystallite Size and Microstrain

Indicated by peak broadeningOther defects (stacking faults, etc.) can be measured by analysis of peak shapes and peak width

Interpretation of results

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hkl dhkl (Å) I (%)

{012} 3.4935 49.8{104} 2.5583 85.8{110} 2.3852 36.1{006} 2.1701 1.9{113} 2.0903 100.0{202} 1.9680 1.4

angle[2θ]

I [cts]

25.2000 372.0000

25.2400 460.0000

25.2800 576.0000

25.3200 752.0000

25.3600 1088.0000

25.4000 1488.0000

25.4400 1892.0000

25.4800 2104.0000

25.5200 1720.0000

25.5600 1216.0000

25.6000 732.0000

Reduced dI list

Position [°2Theta] (Copper (Cu))25 30 35 40 45

Counts

0

400

1600

3600DEMO08

• Diffraction data can be reduced to a list of peak positions and intensities– Each dhkl corresponds to a family of atomic planes {hkl}– individual planes cannot be resolved - this is a limitation of

powder diffraction versus single crystal diffraction

X-Rays Scattering

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Scattering by an Electron

),( 00 νλSets electron into oscillation

Emission in all directions

Scattered beams),( 00 νλCoherent

(definite phase relationship)

A

Polarization factorComes into being as we used unpolarized beam

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

221 2

42

4

20 θCos

cme

rIIP

X-Rays Scattering

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B Scattering by an Atom

Scattering by an atom ∝ [Atomic number, (path difference suffered by scattering from each e−, λ)]

Scattering by an atom ∝ [Z, (θ, λ)] Angle of scattering leads to path differencesIn the forward direction all scattered waves are in phase

electronan by scattered waveof Amplitudeatoman by scattered waveof Amplitude

Factor ScatteringAtomicf

=

=

f→

λθ )(Sin (Å−1) →

0.2 0.4 0.6 0.8 1.0

10

20

30

Schematic

λθ )(Sin

X-Rays Scattering

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C Scattering by the Unit cell (uc)

If atom B is different from atom A → the amplitudes must be weighed by the respectiveatomic scattering factors (f)The resultant amplitude of all the waves scattered by all the atoms in the uc gives the scattering factor for the unit cellThe unit cell scattering factor is called the Structure Factor (F)

Scattering by an unit cell = f(position of the atoms, atomic scattering factors)

electronan by scattered waveof Amplitudeucin atoms allby scattered waveof AmplitudeFactor StructureF ==

)](2[ zhykxhii feAeE ′+′+′== πϕ)(2 zhykxh ′+′+′= πϕIn complex notation

2FI ∝

)](2[

11

jjjj zhykxhin

jj

n

j

ij

hkln efefF ′+′+′

==∑∑ == πϕ

Structure factor is independent of the shape and size of the unit cell

Structure factor calculations

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nnie )1(−=π

)(2

θθθ

Cosee ii

=+ −

Atom at (0,0,0) and equivalent positions

)](2[ jjjj zhykxhij

ij efefF ′+′+′== πϕ fefefF hkhi === ⋅+⋅+⋅ 0)]000(2[ π

22 fF = ⇒ F is independent of the scattering plane (h k l)

ππ nini ee −=

1) ( −=πinodde

1) ( +=πinevene

Simple cubic (primitive):

Structure factor calculations

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Centered Orthorhombic C:Atom at (0,0,0) & (½, ½, 0) and equivalent positions

)](2[ jjjj zhykxhij

ij efefF ′+′+′== πϕ

]1[ )()]2

(2[0

)]021

21(2[)]000(2[

khikhi

hkhihkhi

efefef

efefF

++

⋅+⋅+⋅⋅+⋅+⋅

+=+=

+=

ππ

ππ

⇒ F is independent of the ‘l’ index

Real

]1[ )( khiefF ++= π

fF 2=

0=F

22 4 fF =

02 =F

Both even or both odd

Mixture of odd and even

e.g. (001), (110), (112); (021), (022), (023)

e.g. (100), (101), (102); (031), (032), (033)

(h + k) even

(h + k) odd

e.g. (100), (001), (111); (210), (032), (133)

Structure factor calculations

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Body-Centered Cubic:

Atom at (0,0,0) & (½, ½, ½) and equivalent positions

)](2[ jjjj zhykxhij

ij efefF ′+′+′== πϕ

]1[ )()]2

(2[0

)]21

21

21(2[)]000(2[

lkhilkhi

hkhihkhi

efefef

efefF

++++

⋅+⋅+⋅⋅+⋅+⋅

+=+=

+=

ππ

ππReal

]1[ )( lkhiefF +++= π

fF 2=

0=F

22 4 fF =

02 =F

(h + k + l) even

(h + k + l) odd

e.g. (110), (200), (211); (220), (022), (310)

Structure factor calculations

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Face Centered Cubic:Atom at (0,0,0) & (½, ½, 0) and equivalent positions

)](2[ jjjj zhykxhij

ij efefF ′+′+′== πϕ

]1[ )()()(

)]2

(2[)]2

(2[)]2

(2[)]0(2[

hlilkikhi

hlilkikhii

eeef

eeeefF

+++

+++

+++=

⎥⎦

⎤⎢⎣

⎡+++=

πππ

ππππ

Real

fF 4=

0=F

22 16 fF =

02 =F

(h, k, l) unmixed

(h, k, l) mixed

e.g. (111), (200), (220), (333), (420)

e.g. (100), (211); (210), (032), (033)

(½, ½, 0), (½, 0, ½), (0, ½, ½)

]1[ )()()( hlilkikhi eeefF +++ +++= πππ

Diffracted Intensity

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Ref.: D. B. Cullity & S. R. Stock, p.360 (2014).

Diffracted Intensity

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Structure Factor (F)

Multiplicity factor (p)

Polarization factor

Lorentz factor

Absorption factor

Temperature factor

Scattering from uc

Number of equivalent scattering planes

Effect of wave polarization

Combination of 3 geometric factors

Specimen absorption

Thermal diffuse scattering

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

θθ

θ 21

21

SinCos

SinfactorLorentz

( )( )θ21 2CosIP +=

Diffracted Intensity

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Multiplicity Factor:

Diffracted Intensity

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Polarization factorLorentz factor

( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

θθ

θ 21

21

SinCos

SinfactorLorentz( )( )θ21 2CosIP +=

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

θθθ

CosSinCosfactoronPolarizatiLorentz 2

2 21

0

5

10

15

20

25

30

0 20 40 60 80

Bragg Angle (θ, degrees)

Lor

entz

-Pol

ariz

atio

n fa

ctor

Application: Cubic Structures Determination

)( 222

22

lkhadhkl ++

=

θλθλ 2

22

42

sendsend hklhkl ⋅

=→⋅⋅=

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( )2222

22

4lkh

asen ++⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

=λθ

Cubic Structure factor:Primitive: N = 1, 2, 3, 4, 5, 6, 8, ...Body-Centered: N = 2, 4, 6, 8, 10, 12, ...Face-Centered: N = 3, 4, 8, 11, 12, 16, ...Diamond Cubic: N = 3, 8, 11, 16, 19, 24, ...

N

Application: Cubic Structures Determination

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Application: Cubic Structures Determination

30>

4,04918,00247,980,86851,19965268,735137,479

4,04876,67206,650,72391,01752758,3116,68

4,04916,33196,320,68750,97764656,015112,037

4,04855,33165,320,57920,86489849,55599,116

4,04804,00123,990,43450,71968741,23582,475

4,04803,67113,660,39830,68294739,1378,264

4,04702,6782,660,28980,56845432,5765,143

4,04601,3341,330,14500,39060522,3844,762

4,04451,0031,000,10880,3361519,2638,521

a (angstrom)

teste(N/N-min)N

sen^2(theta)/sem^2(theta-min)

sen^2(theta)theta-radtheta2thetapico #

Aluminium (Cu Kα radiation):

Difração de Raios-X

Notas de aula preparadas pelo Prof. Juno Gallego para a disciplina CARACTERIZAÇÃO MICROESTRUTURALDOS MATERIAIS. ® 2017. Permitida a impressão e divulgação. http://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/grupos/maprotec/educacional/

Cullity, B. D.; Stock, S. R. Elements of X-Ray Diffraction, 3rd edition. Pearson Education Limited, Essex, 2014.

Hammond, C. The Basics of Crystallography and Diffraction (3rd ed). Oxford University Press, Oxford, 2009.

Brandon, D.; Kaplan, W. D. Microstructural Characterization of Materials, 2nd edition. John Wiley & Sons Ltd, Chichester, 2008, 536p.

Goodhew, P. J.; Humphreys, J.; Beanland, R. Electron Microscopy andAnalysis. Taylor & Francis Inc.,New York, 2001.

https://www.doitpoms.ac.uk/tlplib/xray-diffraction/index.phphttp://www.xraydiffrac.com/http://www.pbs.org/wgbh/nova/photo51/

Bibliografia:

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