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| Diffraction for a particular radiation occurs when the slit width is lesser than the wavelength of radiation |
Light Diffraction
Diffraction was first studied in the case of light by Grimaldi and he coined the word "Diffraction". Posthumously the paper was published in the year 1665.
Double Slit Experiment - well known slit experiment presented by Young - This was helped in presenting wave theory of light against newton's particle theory. Interference produced from the diffracted rays giving constructive and destructive interference with light and dark areas.
In Youngs Experiment - The slit width should be less than the wavelength of the light.
In materials the atoms are periodically placed forming slits with a width of the order of angstrom. Therefore the in the electromagnetic spectrum the wavelength should be of order of angstrom and those are X-Rays.
Other sources like electrons or neutrons can also be used for diffraction when they are energized high enough such that they vibrate with wavelength of angstrom.
In Youngs Experiment - The slit width should be less than the wavelength of the light.
In materials the atoms are periodically placed forming slits with a width of the order of angstrom. Therefore the in the electromagnetic spectrum the wavelength should be of order of angstrom and those are X-Rays.
Other sources like electrons or neutrons can also be used for diffraction when they are energized high enough such that they vibrate with wavelength of angstrom.
Particle Diffraction
Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength
λ = h\p
where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles).
larger the
The geometric theories of diffraction of the three types of radiation—X rays, electrons, and neutrons—are identical, but the physical natures of the interaction with matter differ. It is this difference that determines the specifics and fields of application of each of these methods. X rays are scattered by the electron shells of atoms; neutrons are scattered by atomic nuclei (through short-range nuclear forces); and electrons are scattered by the electric potential of atoms.
X-ray
Routinely used to provide structural information on compounds and to identify samples
Used with both powder and single crystal samples
X-rays produced in the home lab or using synchrotrons
Can also be used to examine liquids and glasses
X-rays are scattered by electrons in atoms, the electron cloud is about the same size as the wavelength of the X-rays
The x-rays scattered from the different atoms interfere with one another either constructively or destructively
Follows Braggs Diffraction 2d sinθ= λ
For crystalline solids this interference pattern has sharp well defined peaks
The positions of the peaks are determined by the lattice for crystalline solid
d-spacing formulae
For a unit cell with orthogonal axes– (1 / d2hkl) = (h2/a2) + (k2/b2) + (l2/c2) (2 is square)
Hexagonal unit cells– (1 / d2hkl) = (4/3)([h2+ k2+ hk]/ a2) + (l2/c2) (2 is square)
Hexagonal unit cells– (1 / d2hkl) = (4/3)([h2+ k2+ hk]/ a2) + (l2/c2) (2 is square)
Energy and angle dispersive diffraction
An X-ray diffraction pattern is a measurement of X-ray intensity versus d-spacing
d-spacing, scattering angle and λare related by Bragg’s law 2d sinθ= λ
Energy dispersive diffraction -Fix 2θ and vary λ
Quick experiment with fixed sampling volume, but low resolution
Angle dispersive diffraction - Fix λand vary 2θ
High resolution but slow and sampling volume varies
Powder XRD
It is used routinely to assess the phase purity, crystallinity of materials, unit cell size and shape and crystal structure.
Each crystalline phase has a unique powder diffraction pattern
Measured powder patterns can be compared to a database for identification
Can distinguish between the same compound with different structures and different compounds with the same structure
Particle size
You can use the grain size calculator.
The width of the peaks in a powder pattern contain information about the crystallite size in the sample. From Scherrer equation L = (K λ/ βcosθ)
L - mean size of crystallites, K - constant roughly 1, depends on shape of crystallites, β- full width at half maximum in radians
(Width gives information about presence of microstrain also)
You can use the grain size calculator.
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| Grain size from XRD diffractogram can be obtained using Scherrer equation |
Indexing a powder pattern
The process of figuring out what Miller indices belong to each peak or “d-spacing” in a powder pattern is called indexing
During the indexing process the unit cell constants are also determined
Indexing can be done by hand or by computer
Phase identification can be done
by calculating the unit cell and then search the NIST crystal data database for known compounds with the same or similar unit cells
by comparing the measured pattern against the ICDD/JCPDS/COD data base. (fingerprinting identification of phases)
For multiphase compound, the relative amount of the phases can be determined by comparing the intensities of peaks in the sample with those from reference materials
Single crystal X-ray diffraction
A single crystal of about 0.1 mm in all dimensions is required
Several thousand “reflections” (intensities) are measured to solve an xtal structure
Uses the space group symmetry of the solid to aid the process
– a crystal contains an enormous number of atoms
– without using symmetry we would have an under determined problem
Production of x-rays
X-rays are produced in the laboratory by bombarding a metal target with high energy electrons
The high energy electrons knock electrons out of the core orbitals in target metal
These empty core orbitals are refilled by atomic transitions that lie in the x-ray region
The wavelength of x-rays produced depends on the element the target is made from
Electron diffraction–
primarily used for phase identification, and unit cell determination on small crystallites in the electron microscope, also used for gas phase samples
useful for very small particles of material
can give you unit cell and space group
Very high energy electrons are employed to examine small crystals of materials
Electrons interact strongly with matter– can only use thin samples to observe a diffraction pattern in transmission
Not good for solving crystal structures due to multiple scattering
Experiments are usually performed in a transmission electron microscope (TEM), or a scanning electron microscope (SEM) as electron backscatter diffraction.
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| The bluish glow from the central region of the crab nebula is due to synchrotron radiation. |
Synchrotron Radiation
It is the electromagnetic radiation emitted when charged particles are accelerated radially.
If the particle is non-relativistic, then the emission is called cyclotron emission while for relativistic particles its called as synchrotron emission
They have properties like high intensity, Plane polarized (linear and circular), intrinsically collimated, wide energy range and well defined time structure.
Neutrons
Neutron diffraction is a form of elastic scattering where the neutrons exiting the experiment have more or less the same energy as the incident neutrons.
Thermal neutrons have wavelengths near one angstrom and are therefore useful for interatomic interference studies.
Neutrons have two kinds of interactions
- short range nuclear interaction of the neutron with the atomic nucleus.
- interaction of the magnetic moment of the neutron with the spin and orbital magnetic moments of the atom
Neutrons Vs X-Rays
Neutrons: Particle wave, Has Mass, Spin 1/2, Has Magnetic dipole moment, Neutrons interact with the nucleus, Scattering power independent of 2θ
X-Rays: Electromagnetic wave, No mass, spin 1, no magnetic dipole moment, X-ray photons interact with the electrons, Scattering power falls off with 2θ
Neutrons interact with matter differently than x-rays. X-rays interact primarily with the electron cloud surrounding each atom. The contribution to the diffracted x-ray intensity is therefore larger for atoms with a large atomic number (Z) than it is for atoms with a small Z. On the other hand, neutrons interact directly with the nucleus of the atom.
Advantage over X-ray diffraction
- structure determination of composite crystals which contain both heavy and light atoms(hydrogen containing substances) because the scattering powers of light and heavy elements prove to be of the same order of magnitude.
- magnetic structure determination due to magnetic moment of the neutron
- A major difference with X-rays is that the scattering is mostly due to the tiny nuclei of the atoms. That means that there is no need for an atomic form factor to describe the shape of the electron cloud of the atom and the scattering power of an atom does not fall off with the scattering angle as it does for X-rays. Diffractograms therefore can show strong well defined diffraction peaks even at high angles. The superb high angle information means that the data can give very precise values for the atomic positions in the structure.
Neutron diffraction is a form of elastic scattering where the neutrons exiting the experiment have more or less the same energy as the incident neutrons.
Produced using a nuclear reactor or a spallation source.
Spallation neutrons are produced by bombarding a metal target with pulses of protons
Fission process: the energy of neutron is 1MeV which is very high to use for experiments. Therefore they are slowed down by cooling in water or carbon. Cooled neutrons are classified into three categories depending on their energies.
–hot neutrons: moderated at 2000°C, 0.1-0.5 eV, 0.3-1 Å, 10 000 m/s
–thermal neutrons: moderated at 40°C, 0.01-0.1 eV, 1-4 Å, 2000 m/s
–cold neutrons: moderated at -250°C, 0-0.01 eV, 0-30 Å, 200 m/s
Neutrons have two kinds of interactions
- short range nuclear interaction of the neutron with the atomic nucleus.
- interaction of the magnetic moment of the neutron with the spin and orbital magnetic moments of the atom
Neutrons Vs X-Rays
Neutrons: Particle wave, Has Mass, Spin 1/2, Has Magnetic dipole moment, Neutrons interact with the nucleus, Scattering power independent of 2θ
X-Rays: Electromagnetic wave, No mass, spin 1, no magnetic dipole moment, X-ray photons interact with the electrons, Scattering power falls off with 2θ
Neutrons interact with matter differently than x-rays. X-rays interact primarily with the electron cloud surrounding each atom. The contribution to the diffracted x-ray intensity is therefore larger for atoms with a large atomic number (Z) than it is for atoms with a small Z. On the other hand, neutrons interact directly with the nucleus of the atom.
Advantage over X-ray diffraction
- structure determination of composite crystals which contain both heavy and light atoms(hydrogen containing substances) because the scattering powers of light and heavy elements prove to be of the same order of magnitude.
- magnetic structure determination due to magnetic moment of the neutron
- A major difference with X-rays is that the scattering is mostly due to the tiny nuclei of the atoms. That means that there is no need for an atomic form factor to describe the shape of the electron cloud of the atom and the scattering power of an atom does not fall off with the scattering angle as it does for X-rays. Diffractograms therefore can show strong well defined diffraction peaks even at high angles. The superb high angle information means that the data can give very precise values for the atomic positions in the structure.
Neutrons have a magnetic moment and they are scattered by unpaired electron spins
Normal neutron scattering is scattering off the nucleus
Neutrons scattering patterns are sensitive to the arrangement of unpaired electron spins in a material
- If the unpaired spins on atoms and ions are ordered throughout the material scattering from these spins will contribute to the neutron diffraction pattern
The Rietveld method
Powder diffraction patterns can be curve fit so as to obtain structural information
– this process is called Rietveld refinement
– it is not a method of solving structures. It is a method for refining structures
Refine a model containing:
– structural parameters
– peak shape
– unit cell size
Small angle scattering
The arguments used in discussing the form factor for atoms can be applied to other systems– polymer particles, globular proteins, voids in samples
The low angle scattering from a sample of something like a colloid tells you about the size and shape of the particles
Software: Rietveld refinement can be done by FULLPROF (free software), the site contains various tutorials with examples. There are also numerous paid software's like Xpert High Score, Crystal Match, etc
https://ww2.chemistry.gatech.edu/class/6182/wilkinson/diffraction_methods.pdf
https://web.stanford.edu/group/glam/xlab/MatSci162_172/LectureNotes/04_Powder%20Diffraction,%20Powder%20Method.pdf
https://www.fhi-berlin.mpg.de/acnew/department/pages/teaching/pages/teaching__wintersemester__2008_2009/malte_behrens__x-ray_and_neutron_diffraction__081024.pdf
https://www.fhi-berlin.mpg.de/acnew/department/pages/teaching/pages/teaching__wintersemester__2008_2009/malte_behrens__x-ray_and_neutron_diffraction__081024.pdf
https://mrl.illinois.edu/sites/default/files/pdfs/Workshop08_X-ray_Handouts.pdf
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