Introduction to X-Ray Diffraction

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

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)

Part 1. The X-ray Diffractometer

Primitive Crystallographers

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

Tube

Collimator

Tube

Crystal

Film

Von Laue’s CameraX-ray

SourcePin-hole

Optic Specimen

Film

Rosalind Franklin

Powder X-ray Diffraction

Tube

Powder

Film

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

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

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.

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

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

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-

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)

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”

The X-ray Diffractometer

X-ray Tube

monochromator

collimator

goniometer

Detector

specimen

shutter

Three-Circle Single-Crystal Diffractometer

⎟⎟⎠

⎞⎜⎜⎝

⎛==

32sin 54.73561 1-oχ

Part 2. The crystal.

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

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.

The Unit Cell

a

b

c

αβ

γ

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°

Reflection Planes in a 2D Lattice

b

a

(1.0)

(1,1)

(1,2)

(3,1)

Miller Indices (1/a 1/b)

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).

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

The Bragg-Brentano Geometry

Tube

measurement circle

focusing-circle

qq2

Detector

Sample

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.

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.

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.

From Signal to Intensity

Signal Raw

Lp correction

hkl

Scale Absorption Correction

Integration ABS correction

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

Reciprocal Space to Real Space and Back

Fourier Transform

( )∑ ∑∑∞ ∞

−∞=

∞

−++=h k l

hklhkl lzkyhxFV

xyz '2cos1)( απρPhase angle

( )[ ]∑ ++=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?

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

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)

….