x-ray diffraction

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Essential Parts of the Diffractometer

• X-ray Tube the source of X Rays
• Incident-beam optics condition the X-ray beam
  before it hits the sample
• The goniometer the platform that holds and moves
  the sample, optics, detector, and/or tube
• The sample sample holder
• Receiving-side optics condition the X-ray beam
  after it has encountered the sample
• Detector count the number of X Rays scattered by
  the sample

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Instrumentation

• Production of X-Rays
• Collimator
• Monochromator
• Filter
• Crystal monochromator
• Detector
• Photographic methods
• Counter methods

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The wavelength of X rays is determined by the
anode of the X-ray source.

• 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.

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Braggs law is a simplistic model to understand
what conditions are required for diffraction.

• For parallel planes of atoms, with a space dhkl
  between the planes, constructive interference
  only occurs when Braggs law is satisfied.
• In our diffractometers, the X-ray wavelength l is
  fixed.
• Consequently, a family of planes produces a
  diffraction peak only at a specific angle q.
• Additionally, the plane normal must be parallel
  to the diffraction vector
• Plane normal the direction perpendicular to a
  plane of atoms
• Diffraction vector the vector that bisects the
  angle between the incident and diffracted beam
• The space between diffracting planes of atoms
  determines peak positions.
• The peak intensity is determined by what atoms
  are in the diffracting plane.

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XRD-Methods

• Laue photographic method
• Braggs X-Ray spectrometer
• Rotating crystal method
• Powder method

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Laue photographic method

• In his first experiments, Max von Laue (Nobel
  Prize in Physics in 1914) used continuous
  radiation (with all possible wavelengths) to
  impact on a stationary crystal. With this
  procedure the crystal generates a set of
  diffracted beams that show the internal symmetry
  of the crystal. In these circumstances, and
  taking into account Bragg's Law, the experimental
  constants are the interplanar spacings d and the
  crystal position referred to the incident beam.
  The variables are the wavelength ? and the
  integer number n
• n ?  2 dhkl sin ?nh,nk,nl
• Thus, the diffraction pattern will contain (for
  the same spacing d) the diffracted beams
  corresponding to the first order of diffraction
  (n1) of a certain wavelength, the second order
  (n2) of half the wavelength (?/2), the third
  order (n3) with wavelength ?/3, etc. Therefore,
  the Laue diagram is simply a stereographic
  projection of the crystal

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The Laue method in transmission mode
The Laue method in reflection mode
Laue diagram of a crystal
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Braggs X-Ray spectrometer
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• When x-rays are scattered from a crystal lattice,
  peaks of scattered intensity are observed which
  correspond to the following conditions
• The angle of incidence angle of scattering.
• The pathlength difference is equal to an integer
  number of wavelengths.
• The condition for maximum intensity contained in
  Bragg's law above allow us to calculate details
  about the crystal structure, or if the crystal
  structure is known, to determine the wavelength
  of the x-rays incident upon the crystal.

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X-radiation for diffraction measurements is
produced by a sealed tube or rotating anode.

• Sealed X-ray tubes tend to operate at 1.8 to 3
  kW.
• Rotating anode X-ray tubes produce much more flux
  because they operate at 9 to 18 kW.
• A rotating anode spins the anode at 6000 rpm,
  helping to distribute heat over a larger area and
  therefore allowing the tube to be run at higher
  power without melting the target.
• Both sources generate X rays by striking the
  anode target wth an electron beam from a tungsten
  filament.
• The target must be water cooled.
• The target and filament must be contained in a
  vacuum.

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Rotating crystal method
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Most of our powder diffractometers use the
Bragg-Brentano parafocusing geometry.

• A point detector and sample are moved so that the
  detector is always at 2q and the sample surface
  is always at q to the incident X-ray beam.
• In the parafocusing arrangement, the incident-
  and diffracted-beam slits move on a circle that
  is centered on the sample. Divergent X rays from
  the source hit the sample at different points on
  its surface. During the diffraction process the X
  rays are refocused at the detector slit.
• This arrangement provides the best combination of
  intensity, peak shape, and angular resolution for
  the widest number of samples.

F the X-ray source DS the incident-beam
divergence-limiting slit SS the Soller slit
assembly S the sample RS the diffracted-beam
receiving slit C the monochromator crystal AS
the anti-scatter slit
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What is X-ray Powder Diffraction (XRD) X-ray
powder diffraction (XRD) is a rapid analytical
technique primarily used for phase identification
of a crystalline material and can provide
information on unit cell dimensions. The
analyzed material is finely ground, homogenized,
and average bulk composition is determined.
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• Fundamental Principles of X-ray Powder
  Diffraction (XRD)
• Max von Laue, in 1912, discovered that
  crystalline substances act as three-dimensional
  diffraction gratings for X-ray wavelengths
  similar to the spacing of planes in a crystal
  lattice.
• X-ray diffraction is now a common technique for
  the study of crystal structures and atomic
  spacing.
• X-ray diffraction is based on constructive
  interference of monochromatic X-rays and a
  crystalline sample.
• These X-rays are generated by a cathode ray
  tube, filtered to produce monochromatic
  radiation, collimated to concentrate, and
  directed toward the sample. The interaction of
  the incident rays with the sample produces
  constructive interference (and a diffracted ray)
  when conditions satisfy Bragg's Law (n?2d sin
  ?).

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• This law relates the wavelength of
  electromagnetic radiation to the diffraction
  angle and the lattice spacing in a crystalline
  sample.
• These diffracted X-rays are then detected,
  processed and counted.
• By scanning the sample through a range of
  2?angles, all possible diffraction directions of
  the lattice should be attained due to the random
  orientation of the powdered material.
• Conversion of the diffraction peaks to
  d-spacings allows identification of the mineral
  because each mineral has a set of unique
  d-spacings. Typically, this is achieved by
  comparison of d-spacings with standard reference
  patterns.

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• All diffraction methods are based on generation
  of X-rays in an X-ray tube. These X-rays are
  directed at the sample, and the diffracted rays
  are collected.
• A key component of all diffraction is the angle
  between the incident and diffracted rays. Powder
  and single crystal diffraction vary in
  instrumentation beyond this.

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Applications of XRD

• XRD is a nondestructive technique
• To identify crystalline phases and orientation
• To determine structural properties
• Lattice parameters (10-4Å), strain, grain size,
  expitaxy, phase composition, preferred
  orientation (Laue) order-disorder transformation,
  thermal expansion
• To measure thickness of thin films and
  multi-layers
• To determine atomic arrangement
• Detection limits 3 in a two phase mixture can
  be
• 0.1 with synchrotron radiation
• Spatial resolution normally none

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• Applications
• X-ray powder diffraction is most widely used for
  the identification of unknown crystalline
  materials (e.g. minerals, inorganic compounds).
  Determination of unknown solids is critical to
  studies in geology, environmental science,
  material science, engineering and biology. Other
  applications include
• characterization of crystalline materials
• identification of fine-grained minerals such as
  clays and mixed layer clays that are difficult to
  determine optically
• determination of unit cell dimensions
  measurement of sample purity

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• With specialized techniques, XRD can be used to
• determine crystal structures using Rietveld
  refinement
• determine of modal amounts of minerals
  (quantitative analysis)
• make textural measurements, such as the
  orientation of grains, in a polycrystalline
  sample
• characterize thin films samples by
• determining lattice mismatch between film and
  substrate and to inferring stress and strain
• determining dislocation density and quality of
  the film by rocking curve measurements
• measuring superlattices in multilayered
  epitaxial structures
• determining the thickness, roughness and density
  of the film using glancing incidence X-ray
  reflectivity measurements

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• Strengths and Limitations of X-ray Powder
  Diffraction (XRD)?
• Strengths
• Powerful and rapid (lt 20 min) technique for
  identification of an unknown mineral
• In most cases, it provides an unambiguous
  mineral determination
• Minimal sample preparation is required
• XRD units are widely available
• Data interpretation is relatively straight
  forward

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• Limitations
• Homogeneous and single phase material is best
  for identification of an unknown
• Must have access to a standard reference file of
  inorganic compounds (d-spacings, hkls)
• Requires tenths of a gram of material which must
  be ground into a powder
• For mixed materials, detection limit is 2 of
  sample
• For unit cell determinations, indexing of
  patterns for non-isometric crystal systems is
  complicated
• Peak overlay may occur and worsens for high
  angle 'reflections'

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