Transcript Document

X-Ray Diffraction
Dr. T. Ramlochan
March 2010
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Crystals
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A crystal is a solid material where the constituent atoms are arranged in
an orderly repeating pattern extending in all three spatial dimensions
CaSO4·2H2O
SrTiO3
Crystallography
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Crystals are divided into 7 lattice
systems → all crystalline materials
must fit in one of these unit cells
 lengths of edges (a, b, c) of unit cell
and the angles (α, β, γ) between
them are the lattice parameters
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The space group of a crystal is a
description of the symmetry of the
crystal → the unit cells do not just
repeat side-by-side
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Space groups in three dimensions are
made from combinations of different
symmetry operations (reflection,
rotation and improper rotation, the
screw axis and glide plane)
 230 unique space groups
Crystallography
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The atoms in a crystal lattice form planes (described by Miller indices) that
repeat
X-rays and diffraction
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X-rays were discovered in 1895 by Röntgen
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X-rays are electromagnetic radiation with wavelengths in the range of 0.52.5 Å
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As with visible light X-rays will
undergo diffraction when they
encounter an obstacle
 If the diffracting obstacle is on the
order of the size of the
wavelength, the propagating
waves will have interference due
to different waves having travelled
different path lengths
X-ray diffraction image of DNA by
Rosalind Franklin (1952)
X-rays and diffraction

Differences in the length of the path travelled lead to differences in
phase
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The introduction of phase differences produces a change in amplitude
→ summed amplitude of the waves can have any value
between zero and the sum of the individual amplitudes
Scattering of X-rays
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Atoms (or their electrons) will scatter X-rays in all directions
If atoms are arranged in space in a regular periodic fashion, as a
crystal, some of the scattered X-rays will undergo reinforcement in
certain directions and cancellation in other directions producing
diffracted beams
Diffraction is essentially reinforced scattering
Bragg’s Law

For a particular condition of scattering where the angle (θ) of the
incident beam and the ‘reflected’ X-rays are the same, the scattered Xrays will be completely in phase and undergo reinforcement if the path
difference is equal to a whole number of n wavelengths, such that
nλ = 2d sin θ

This was first identified by W.L. Bragg and is called Bragg’s Law
Bragg’s Law
For a fixed wavelength (λ) and value of d, there will be an angle theta (θ)
where diffraction (complete reinforcement) occurs

Diffractogram is a plot of the intensity of the diffracted X-rays vs. 2θ over
a range of angles
 Each peak represents a plane in the crystal lattice with a given
‘d-spacing’
 Basis for powder diffraction
27.470 [°]
56.691 [°]
45.615 [°]
31.817 [°]
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20
30
40
°2Theta
50
60
X-ray production
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X-ray are produced when electrically charged particles (e.g., electrons)
with sufficient kinetic energy give up some energy
 Non-characteristic (continuous) X-rays → electrons decelerated in an
electromagnetic field (Bremsstrahlung)
 Characteristic X-rays → if electrons have high enough kinetic energy
can knock electrons out of their shells
→ when an electron moves from an outer shell to an inner one it is
‘excited’ and releases excess energy directly as X-rays with
eV/wavelength characteristic of the atom released from
X-ray production

X-rays named according to shell being filled and number of shells
changed (e.g., K shell filled by L shell (Kα radiation) or M shell (Kß
radiation))
 Each peak represents a transition; more than one peak (‘family of
X-rays’); Kα (highest probability) is ~5 times stronger than Kß

Kα is a doublet (Kα1 and Kα2)
→ different spin states
Kα1 always about twice the
intensity of Kα2
For Cu
Kα1 1.540598 Å
Kα2 1.544426 Å
Kα 1.541874 Å
Kß 1.392250 Å
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X-ray production
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For XRD we want monochromatic X-rays (i.e., X-rays of a single wavelength
travelling in the same direction/plane)
 Can filter the beam by passing through a material with an absorption
edge between Kα and Kß wavelengths
 For Cu radiation use Ni filter → Kß reduced to 1/500; Kα reduced by 1/2
X-ray generation
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To generate X-rays → a) source of electrons, b) high accelerating
voltage, and c) a metal target
Use a water-cooled X-ray tube
 Evacuated glass tube with an anode (Cu target) and cathode
maintained at high negative potential (HT transformer)
 Filament is heated to emit electrons → accelerated towards target
 X-rays emitted through (X-ray-transparent) beryllium windows
X-ray diffractometer
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Diffractometer has two parts:
 Generator → to generate X-rays
 Goniometer → to scan sample through a range of angles
Diffraction optics/geometry
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X-rays diverge from source → pass through Soller slits and divergence
slits to define and collimate incident beam
Incident beam diffracted by ‘flat’ powder specimen
Diffracted beam passed through receiving slits
Secondary monochromator reduces background radiation from sample
X-rays collected by detector (proportional, Geiger, scintillation,
semiconductor)
Diffractograms
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Gives information about peak positions, intensity, and shape
C o unts
1000
500
C 3 S - A li te (M3 ) 6 8 .2 %
C 2 S - ß-B e li te 6.1 %
C 3 A - A lumi na te cub i c 6 .4 %
C 3 A - A lumi na te orthorhombic 2 .2 %
C 4 A F - B ro wnmi lle ri te 5 .3 %
Ma g ne si um o xi d e - P e ri clase 1 .2 %
C a lci um o xi d e - L i me 0 .5 %
P o ta ssi um sulfa te, b eta - A rcani te 0 .7 %
C a lci te 4 .5 %
Qua rtz 0 .0 %
Gypsum 1 .8 %
C a lci um sulfa te he mi hyd ra te 2 .3 %
A nhyd ri te 0 .4 %
P o rtla nd i te 0 .0 %
A p hthi tali te 0 .4 %
0
10
20
30
P o si ti o n [°2 The ta ]
20
0
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40
50
60