X-Ray Diffraction
(XRD)
AMAN JHA
Department of CEEES
DCRUST
Interaction of X-rays with matter
Incident X-rays
SPECIMEN Absorption (Heat)
Fluorescent X-rays
Electrons
Scattered X-rays
Compton recoil Photoelectrons
Coherent Incoherent (Compton modified)
From bound charges From loosely bound charges
Transmitted beam
The coherently scattered X-rays are the ones that are important from
XRD perspective.
Diffraction Basics
For electromagnetic radiation to be diffracted the spacing in the grating
should be of the same order as the wavelength
In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is
X-rays
Hence, X-rays can be used for the study of crystal structures
Neutrons and Electrons are also used for diffraction studies from materials.
Neutron diffraction is especially useful for studying the magnetic ordering
in materials
X-rays
Beam of electrons Target
A accelerating charge radiates electromagnetic radiation
XRD the first step
A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal.
The electrons oscillate under the influence of the incoming X-Rays and becomesecondary
sources of EM radiation.
The secondary radiation is in all directions.
The waves emitted by the electrons have the same frequency as the incoming X-rays
coherent.
The emission can undergo constructive or destructive interference.
Secondary
Incoming X-rays emission
Oscillating charge re-radiates In phase with
the incoming x-rays
Schematics
Sets nucleus into oscillation
Sets Electron cloud into oscillation
Small effect neglected
Crystalline materials are characterized by the orderly
periodic arrangements of atoms.
The (200) planes The (220) planes
of atoms in NaCl of atoms in NaCl
• The unit cell is the basic repeating unit that defines a crystal.
• 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.
The atoms in a crystal are a periodic array of coherent
scatterers and thus can diffract light.
• 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 light.
– The electrons interact with the oscillating electric field of the light
wave.
• Atoms in a crystal form a periodic array of coherent scatterers.
– The wavelength of Xrays 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
• XRays are also reflected, scattered incoherently, absorbed, refracted,
and transmitted when they interact with matter.
Diffraction occurs when light is scattered by a periodic array
with long-range order, producing constructive interference
at specific angles
•The electrons ineach atom coherently scatter light.
–We can regard each atom as a coherent point scatterer
–The strength with which an atom scatters light is proportional to the number of
electrons around the atom.
•The atoms in a crystal are arranged in a periodic array with long-range order and
thus can produce diffraction.
•The wavelength of Xrays are similar to the distance between atoms in a crystal.
Therefore, we use X-ray scattering to study atomic structure.
•The scattering of X-rays from atoms produces a diffraction pattern, which
contains information about the atomic arrangement within the crystal
•Amorphous materials like glass do not have a periodic array with long-range
order, so they do not produce a diffraction pattern. Their X-ray scattering pattern
features broad, poorly defined amorphous ‘humps’.
Crystalline materials are characterized by the long-range orderly
periodic arrangements of atoms.
•The unitcell is the basic repeating unit that defines the crystal
structure.
–The unit cell contains the symmetry elements required to uniquely define the
crystal structure.
–The unit cell might contain more than one molecule:
•for example, the quartz unit cell contains 3 complete molecules of SiO2.
–The crystal system describes the shape of the unit cell
–The lattice parameters describe the size of the unit cell
•The unit
cell repeats in all dimensions to fill space and produce the
macroscopic grains or crystals of the material
The diffraction peak position is a product of interplanar
spacing, as calculated by Bragg’s law
• Bragg’s law relates the diffraction angle, 2θ, to dhkl
–In most diffractometers, the X-ray wavelength λ is fixed.
– Consequently, a family of planes produces a diffraction peak only at a specific
angle 2θ.
• dhkl is a geometric function of the size and shape of the unit cell
– dhkl is the vector drawn from the origin to the plane (hkl) at a 90° angle.
– dhkl, the vector magnitude, is the distance between parallel planes of atoms
in the family (hkl)
– Therefore, we often consider that the position of the diffraction peaks are
determined by the distance between parallel planes of atoms.
Bragg’s law provides 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 Bragg’s law is satisfied.
– In our diffractometers, the X-ray wavelength λ is fixed.
– A family of planes produces a diffraction peak only at a specific angle 2θ.
• Additionally, the plane normal [hkl] must be parallel to the
diffraction vector s
– Plane normal [hkl]: the direction perpendicular to a plane of atoms
– Diffraction vector s: the vector that bisects the angle between theincident
and diffracted beam
You can use XRD to determine
• Phase Composition of a Sample
– Quantitative Phase Analysis: determine the relative amounts of phases in a
mixture by referencing the relative peak intensities
• Unit cell lattice parameters and Bravais lattice symmetry
– Index peak positions
– Lattice parameters can vary as a function of, and therefore give you information
about, alloying, doping, solid solutions, strains, etc.
• Residual Strain (macrostrain)
• Crystal Structure
– By Rietveld refinement of the entire diffraction pattern
• Epitaxy/Texture/Orientation
• Crystallite Size and Microstrain
– Indicated by peak broadening
– Other defects (stacking faults, etc.) can be measured by analysis of peak shapes
and peak width
Essential Parts of the Diffractometer
• X-ray Tube: the source of XRays
•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 XRays scattered by the sample
Information in a Diffraction Pattern
• Phase Identification
• Crystal Size
• Crystal Quality
• Texture (to some extent)
• Crystal Structure
How do we get X-rays?
• The cathode is heated by a heat
source to create an electron beam.
• The beam of electrons is then
accelerated by the high voltage
source, allowing them to collide
with the metal target (usually
Tungsten)
• X-rays are produced when the
electrons are suddenly decelerated
upon collision with the metal target
(Brehmsstrahlung)
• If the bombarding electrons have
sufficient energy, they can knock an
electron out of an inner shell of the
target metal atoms. Then electrons
from higher states drop down to fill
the vacancy, emitting x-ray photons
(characteristic x-rays)
X-ray production Spectrum
• The characteristic x-
rays, shown as two
sharp peaks in the
illustration occur
when vacancies are
produced in the n=1
or K-shell of the
atom
• The x-rays produced
by transitions from
the n=2 to n=1 levels
are called K-alpha x-
rays
• The x-rays produced
in the transition
from n=3 n=1 are
called K-beta x-rays.
