Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf ·...

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Introduction to X-Ray Diffraction

Transcript of Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf ·...

Page 1: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Introduction to X-Ray Diffraction

Page 2: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Structure

H10-10m

Angstrom (Å)

Visible Light 4x10-7-9x10-7m4000-9000 Å

LeptonsX-rays 0.1 to 14 Å

FermionsNeutronsElectrons

Molecular, Extended, Quasi, etc

Page 3: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Interaction between X-ray and Matter

wavelength λο

intensity Io

• incoherent scatteringλC (Compton-Scattering)

• coherent scattering (XRD)λο (Bragg´s-scattering)

• Absorption (XANSE etc)Beer´s law I = Io*e-µd

• Fluorescence (XRF)λ> λo

• Photoelectrons (XPS)

Page 4: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Part 1. The X-ray Diffractometer

Primitive Crystallographers

Page 5: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Laue’s Experiment in 1912 Single Crystal X-ray Diffraction

Tube

Collimator

Tube

Crystal

Film

Page 6: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Von Laue’s CameraX-ray

SourcePin-hole

Optic Specimen

Film

Rosalind Franklin

Page 7: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Powder X-ray Diffraction

Tube

Powder

Film

Page 8: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Max von Laue put forward the conditions for scattering maxima, the Laue equations:

a(cosα-cosα0)=hλb(cosβ-cosβ0)=kλc(cosγ-cosγ0)=lλ

Max Theodor Felix von Laue

Page 9: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

W. H. Bragg and W. Lawrence Bragg

W.H. Bragg (father) and William Lawrence.Bragg (son) developed a simple relation for scattering angles, now call Bragg’s law.

θλ

sin2 ⋅⋅

=nd

Page 10: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Bragg’s Description

The incident beam will be scattered at all scattering centers, which lay on lattice planes.The beam scattered at different lattice planes must be scattered coherent, to give an maximum in intensity.The angle between incident beam and the lattice planes is called θ.The angle between incident and scattered beam is 2θ .The angle 2θ of maximum intensity is called the Bragg angle.

Page 11: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Bragg’s Law

A powder sample results in cones with high intensity of scattered beam. Above conditions result in the Bragg equation

or

θλ sin2 ⋅⋅=⋅ dn

θλ

sin2 ⋅⋅

=nd

Page 12: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Essential Parts of the X-ray Diffractometer

1. X-ray Tube: the source of X Rays2. Incident-beam optics: condition the X-ray beam

before it hits the sample3. The goniometer: the platform that holds and

moves the sample, optics, detector, and/or tube4. The sample & sample holder5. (Receiving-side optics: condition the X-ray beam

after it has encountered the sample) powder6. Detector: count the number of X Rays scattered

by the sample

Page 13: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The Generating of X-rays Emission Spectrum of aMolybdenum X-Ray Tube

Bremsstrahlung = continuous spectra(near misses, bending radiation)

characteristic radiation = line spectra(direct collisions)

50,000 Volts

0.040 amps

Vacuum

M

K

L

Kα1 Kα2 Kβ1 Kβ2

e-

2000 wattsChilled water

e-

Page 14: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Monochromator

Type 1. FilterNi foil will filter CuY foil will filter Mo(high flux)

Type 2. CrystalGraphite (cheap)Ge (better, expensive)(lower flux)

Type 3. X-ray MirrorsThin layers (~10A)of Heavy Metals(very expensive)(high flux)

Page 15: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

DetectorsCCD/Phosphor

– Fast, High DQE, Inexpensive– High Background, Hot Spots– Resolution inverse of phosphor thickness

IMAGE PLATE– Wide Area, Adaptable– High Background, Slow– Resolution based on grain size

MWPC/Micro Gap (VANTEC-2000)– No Detector Background, Fast– Limited Area, Expensive– Resolution limited to wire separation

Pixel Area Detector (PAD)– Silicon Strip Technology, very adaptable– Background, very expensive– Resolution based on individual “chips”

Page 16: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The X-ray Diffractometer

X-ray Tube

monochromator

collimator

goniometer

Detector

specimen

shutter

Page 17: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Three-Circle Single-Crystal Diffractometer

⎟⎟⎠

⎞⎜⎜⎝

⎛==

32sin 54.73561 1-oχ

Page 18: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Part 2. The crystal.

To view an atom (structure)with X-raysyou must find a way to hold it in place!

Page 19: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The Crystal

A crystal is a solid material whose constituent atoms (molecules and ions) are arranged in an orderly repeating pattern extending in all three spatial dimensions.

Page 20: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The Unit Cell

a

b

c

αβ

γ

Page 21: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Crystal SystemsCrystal systems Axes system

cubic a = b = c , α = β = γ = 90°

Tetragonal a = b ≠ c , α = β = γ = 90°

Hexagonal a = b ≠ c , α = β = 90°, γ = 120°

Rhomboedric a = b = c , α = β = γ ≠ 90°

Orthorhombic a ≠ b ≠ c , α = β = γ = 90°

Monoclinic a ≠ b ≠ c , α = γ = 90° , β ≠ 90°

Triclinic a ≠ b ≠ c , α ≠ γ ≠ β°

Page 22: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Reflection Planes in a 2D Lattice

b

a

(1.0)

(1,1)

(1,2)

(3,1)

Miller Indices (1/a 1/b)

Page 23: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Three Dimensions

The (200) planes of atoms in NaCl

The (220) planes of atoms in NaCl

Parallel planes of atoms intersecting the unit cell are used to define directions and distances in the crystal.

These crystallographic planes are identified by Miller indices (hkl).

Page 24: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Relationship between d-value and the Lattice Constants

λ = 2 d s i n θ Bragg´s lawThe wavelength is knownTheta is the half value of the peak positiond will be calculated

1/d2= (h2 + k2)/a2 + l2/c2 Equation for the determination of the d-value of a tetragonal unit cell

h,k and l are the Miller indices of the peaksa and c are lattice parameter of the elementary cellif a and c are known it is possible to calculate the peak position if the peak position is known it is possible to calculate the lattice parameter

Page 25: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The Bragg-Brentano Geometry

Tube

measurement circle

focusing-circle

qq2

Detector

Sample

Page 26: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Powder Pattern and Structure

The d-spacings of lattice planes depend on the size of the elementary cell and determine the position of the peaks.The intensity of each peak is caused by the crystallographic structure, the position of the atoms within the elementary cell and their thermal vibration.The line width and shape of the peaks may be derived from conditions of measuring and properties - like particle size - of the sample material.

Page 27: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

D8 ADVANCE Bragg-Brentano Diffractometer

A scintillation counter may be used as detector instead of film to yield exact intensity data.Using automated goniometers step by step scattered intensity may be measured and stored digitally.The digitised intensity may be very detailed discussed by programs.More powerful methods may be used to determine lots of information about the specimen.

Page 28: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The atoms in a crystal are a periodic array of coherent scatterers and thus can diffract X-rays.

Diffraction occurs when each object in a periodic array scatters radiation coherently, producing concerted constructive interference at specific angles.

The electrons in an atom coherently scatter X-rays. The electrons interact with the oscillating electric field of the X-ray.

Atoms in a crystal form a periodic array of coherent scatterers.The wavelength of X rays are similar to the distance between atoms.Diffraction from different planes of atoms produces a diffraction

pattern, which contains information about the atomic arrangement within the crystal

X-Rays are also reflected, scattered incoherently, absorbed, refracted, and transmitted when they interact with matter.

Page 29: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

From Signal to Intensity

Signal Raw

Lp correction

hkl

Scale Absorption Correction

Integration ABS correction

Page 30: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Intensity and hkl

• I (hkl) == Reflections (hkl file = *.hkl)– hkl => where the atoms are

• hkl ~ d = Position in reciprocal space

– I => what the atoms are• Intensity ~ number of scattering electrons

I(hkl) = K|F(hkl)|2 ( )[ ]∑ ++=i

iiii lzkyhxfhklF π2exp)(observed

Calculated

Electron density = ρ(xyz)

Fourier Transform

Page 31: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Reciprocal Space to Real Space and Back

Fourier Transform

( )∑ ∑∑∞ ∞

−∞=

−++=h k l

hklhkl lzkyhxFV

xyz '2cos1)( απρPhase angle

Page 32: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions
Page 33: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

( )[ ]∑ ++=i

iiiical lzkyhxfhklF π2exp)(2)()( obshklFKhklI =

atom (i)Calc. structure factor

for atom (i)

Minimize D (modify x,y,z +

Thermal p’s)

Model building employing known Intensities

Structure SolutionPhase Refinement

Fourier

L.S./Fourier( )222∑ −=hkl

calcobshkl kFFwD

( )( )

2/1

22

222

2⎟⎟⎟⎟

⎜⎜⎜⎜

⎛ −=

hklobs

hklcalcobs

Fw

kFFwwR p

DpD

ΔΔ

∂∂ ~

( )∑ ∑∑∞ ∞

−∞=

−++=h k l

hklhkl lzkyhxFV

xyz '2cos1)( απρ

( )∑ ∑∑∞ ∞

−∞=

−++=h k l

hklhkl lzkyhxFV

xyz '2cos1)( απρ

Guess?

Page 34: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

Summary

• Crystal – periodicity – Reduce N X 1020 → N problem

• Lattice/Symmetry describes the periodicity– Visual and Mathematical

• Scattered X‐rays describe the atomicity – Intensity  ~ e‐ density [ρ(xyz)]

• A Model is fit (L.S.) to the ρ(xyz) map.– Minimize the difference between the observed F(hkl) and the 

calculated F(hkl).

• Report the Model

)(~)( hklIhklF

Page 35: Introduction to X-Ray Diffractionxray.tamu.edu/practicals/class_2009/0practicals_intro.pdf · Powder X-ray Diffraction. Tube Powder. Film. Max von Laue put forward the conditions

The Model

• Unit Cell/Symmetry/Coordinates_symmetry_cell_setting             Monoclinic

_symmetry_space_group_name_H‐M    P2(1)/c

_cell_length_a                     11.379(11)

_cell_length_b                     39.43(3)

_cell_length_c                    9.824(8)

_cell_angle_alpha                  90.00

_cell_angle_beta                   109.87(3)

_cell_angle_gamma                  90.00

ATOM  Element X Y Z Thermal P.

F1A  F  ‐0.4863(6)  0.09196(14)  1.2353(7)  0.084(2)

F2A  F  0.4873(6)  0.07626(13)  0.6898(7)  0.072(2)

O1A  O  0.3271(8)  0.03134(15)  1.1028(9)  0.067(3)

O2A  O  ‐0.0537(7)  0.03345(15)  1.2981(8)  0.057(2)

….