Introduction to
X-Ray Diffraction
Structure
Molecular, Extended, Quasi, etc
10-10m
Angstrom (Å)
H
Visible Light 4x10-7-9x10-7m
4000-9000 Å
Leptons
X-rays 0.1 to 14 Å
Fermions
Neutrons
Electrons
Interaction between X-ray and Matter
• incoherent scattering
λ
C
(Compton-Scattering)
• coherent scattering (XRD)
wavelength λο
λο (Bragg´s-scattering)
• Absorption (XANSE etc)
intensity Io
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
Tube
Crystal
Collimator
Film
Von Laue’s Camera
X-ray
Source
Pin-hole
Optic
Specimen
Film
Rosalind Franklin
Powder X-ray Diffraction
Film
Tube
Powder
Max Theodor Felix von Laue
Max von Laue put forward
the conditions for scattering
maxima, the Laue equations:
a(cosα-cosα0)=hλ
b(cosβ-cosβ0)=kλ
c(cosγ-cosγ0)=lλ
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.
n⋅λ
d=
2 ⋅ sin θ
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
n ⋅ λ = 2 ⋅ d ⋅ sin θ
„ or
d =
n ⋅λ
2 ⋅ sin θ
Essential Parts of the X-ray Diffractometer
1. X-ray Tube: the source of X Rays
2. Incident-beam optics: condition the X-ray beam
before it hits the sample
3. The goniometer: the platform that holds and
moves the sample, optics, detector, and/or tube
4. The sample & sample holder
5. (Receiving-side optics: condition the X-ray beam
after it has encountered the sample) powder
6. Detector: count the number of X Rays scattered
by the sample
The Generating of X-rays
Emission Spectrum of a
Molybdenum
X-Ray Tube
0.040 amps
Vacuum
e-
50,000 Volts
2000 watts
Chilled water
characteristic radiation = line spectra
(direct collisions)
M
e-
L
K Kα1 Kα2
Kβ1
Kβ2
Bremsstrahlung = continuous spectra
(near misses, bending radiation)
Monochromator
Type 1. Filter
Ni foil will filter Cu
Y foil will filter Mo
(high flux)
Type 2.
Crystal
Graphite (cheap)
Ge (better, expensive)
(lower flux)
Type 3.
X-ray Mirrors
Thin layers (~10A)
of Heavy Metals
(very expensive)
(high flux)
Detectors
„ CCD/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
monochromator
specimen
Detector
collimator
shutter
goniometer
X-ray Tube
Three-Circle Single-Crystal
Diffractometer
⎛ 2⎞
⎟
χ = 54.73561 = sin ⎜⎜
⎟
⎝ 3⎠
o
-1
Part 2. The crystal.
To view an atom (structure)with X-rays
you 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
c
a
α
β
γ
b
Crystal Systems
Crystal 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
(3,1)
Miller Indices (1/a 1/b)
(1,2)
b
(1.0)
(1,1)
a
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 in θ
Bragg´s law
„ The wavelength is known
„ Theta is the half value of the peak position
„ d will be calculated
2
2
2
2
2
1/d = (h + k )/a + l /c
2
Equation for the determination of
the d-value of a tetragonal unit cell
„ h,k and l are the Miller indices of the peaks
„ a and c are lattice parameter of the elementary cell
„ if 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
Detector
Tube
q
focusingcircle
Sample
measurement circle
2q
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.
ƒDiffraction occurs when each object in a periodic array scatters
radiation coherently, producing concerted constructive
interference at specific angles.
ƒThe electrons in an atom coherently scatter X-rays.
ƒThe electrons interact with the oscillating electric field of the X-ray.
ƒAtoms in a crystal form a periodic array of coherent scatterers.
ƒThe wavelength of X rays are similar to the distance between atoms.
ƒDiffraction from different planes of atoms produces a diffraction
pattern, which contains information about the atomic arrangement
within the crystal
ƒX-Rays are also reflected, scattered incoherently, absorbed,
refracted, and transmitted when they interact with matter.
From Signal to Intensity
Integration
Signal
Lp correction
ABS correction
Raw
hkl
Scale
Absorption
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
Calculated
I(hkl) = K|F(hkl)|2
F (hkl ) = ∑ f i exp[2π (hxi + kyi + lzi )]
i
observed
Fourier Transform
Electron density = ρ(xyz)
Reciprocal Space to Real Space and Back
Fourier Transform
(
Phase angle
1 ∞ ∞ ∞
'
ρ ( xyz) = ∑ ∑∑ Fhkl cos 2π hx + ky + lz − α hkl
V h k =−∞ l
)
Model building employing known Intensities
Structure Solution
Phase Refinement
Fourier
ρ ( xyz) =
(
Guess?
1 ∞ ∞ ∞
∑ ∑∑ Fhkl cos 2π hx + ky + lz − α hkl'
V h k =−∞ l
)
atom (i)
Calc. structure factor
for atom (i)
I (hkl ) = K F (hkl ) obs
F (hkl ) cal = ∑ f i exp[2π (hxi + kyi + lzi )]
2
i
ρ ( xyz) =
D =
∑w
hkl
(F
2
obs
hkl
(
2
⎛
− kFcalc
w
F
⎜∑
obs
hkl
wR 2 = ⎜
2 2
⎜
w
F
∑
obs
⎜
hkl
⎝
(
)
)
2 2
1/ 2
⎞
⎟
⎟
⎟
⎟
⎠
− kF calc
)
2 2
Minimize D
(modify x,y,z +
Thermal p’s)
(
1 ∞ ∞ ∞
∑ ∑∑ Fhkl cos 2π hx + ky + lz − α hkl'
V h k =−∞ l
L.S./Fourier
∂D ΔD
~
∂p Δp
)
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)]
F ( hkl ) ~
I ( hkl )
• 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
The Model
• Unit Cell/Symmetry/Coordinates
_symmetry_cell_setting _symmetry_space_group_name_H‐M _cell_length_a _cell_length_b _cell_length_c _cell_angle_alpha _cell_angle_beta _cell_angle_gamma ATOM Element X
Y
Monoclinic
P2(1)/c
11.379(11)
39.43(3)
9.824(8)
90.00
109.87(3)
90.00
Z
Thermal P.
F1A F F2A F O1A O O2A O ….
1.2353(7) 0.6898(7) 1.1028(9) 1.2981(8) 0.084(2)
0.072(2)
0.067(3)
0.057(2)
‐0.4863(6) 0.4873(6) 0.3271(8) ‐0.0537(7) 0.09196(14) 0.07626(13) 0.03134(15) 0.03345(15)