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EWALD SPHERE AND ITS CONSTRUCTION, what is k?, Reciprocal lattice vectors …
EWALD SPHERE AND ITS CONSTRUCTION
RECIPROCAL LATTICE
K.R
=2πn, n is an integer
Direct lattice position vectors ,
R
=n₁a₁ + n₂a₂ + n₃a₃
Reciprocal lattice vectors ,
K
=hb₁ +kb₂ + lb₃
The collection of all wave vectors, k that produce plane waves with a specific Bravais lattice's periodicity
A wave vector of a plane wave that has the periodicity of the direct lattice
k = 2π / λ
Wave vectors of plane waves that are unity at all direct lattice sites
X-RAY DIFFRACTION
Definition
phenomenon in which the atoms of a crystal, by virtue of their uniform spacing, cause an interference pattern of the waves present in an incident beam of X-rays.
Uses of XRD
to analyze physical properties such as phase composition, crystal structure and orientation of powder, solid and liquid samples
Applications of XRD
Measure the average spacings between
layers or rows of atoms
Determine the orientation of a single
crystal or grain
Measure the size, shape and internal
stress of small crystalline regions
Find the crystal structure of an unknown
material
XRD pattern of NaCl powder
How it occurs?
when incident X-rays constructively interfere with a crystalline solid such that Bragg's law is satisfied.
Types of XRD
Single crystal XRD
a crystal is mounted and centered within the X-ray beam.
Powder XRD
a polycrystalline sample is ground into a fine powder and mounted on a plate.
LAUE DIFFRACTION (RECIPROCAL SPACE)
A regular array of spots on a photographic emulsion resulting from X-ray scattered by certain groups of parallel atomic planes within a crystal
Laue method is useful mainly to determine the orientation of large single crystals
Principle of Laue Equation
Miller Indices
d = 1/(|h|)
Reciprocal Space
Laue Equation in one dimension
Laue Equation in three dimensions
Laue Equation
Laue Condition of Diffraction for a Lattice
G.R = 2\( \pi \)m for all R (lattice vector) and integral m
Discovered by
Max Theodor Felix von Laue (9 October 1879 - 24 April 1960), a German physicist who won the Nobel Prize in Physics in 1914
BRAGG'S LAW (REAL SPACE)
Equation
nλ = 2d sinθ
Important of Bragg's law
Bragg's law is useful for measuring wavelength and for determining the lattice spacing of crystals.
Applications of Bragg's law
In X-ray diffraction (XRD), the inter-planar spacing or d-spacing of a crystal is used for characterization and identification purposes.
In X-ray fluorescence spectroscopy (XRS), the crystals with known interplanar spacings are used to analyse the crystals in the spectrometer.
It is useful for conducting the measurements of the wavelength of different families of crystals.
Definition
A fundamental principle in physics that describes how X-rays interact with crystals to produce diffraction pattern.
Proposed by?
Lawrence Bragg and William Henry Bragg
When Bragg's equation has no solution?
When n=2 and λ>d. This is because sine of an angle can't be more than 1.
What does Bragg's law states?
When the X-ray is incident onto a crystal surface, its angle of incidence, θ, will reflect with the same angle of scattering, θ. And, when the path difference, d is equal to a whole number, n, of wavelength, λ, constructive interference will occur.
Ewald's sphere
is a sphere of radius 1/λ passing through the origin O of the reciprocal lattice.
:eight_pointed_black_star: Paul Peter Ewald, a
German physicist and crystallographer; 1888-1985
:eight_pointed_black_star:Reciprocal lattice/crystal is a map of the crystal in reciprocal space but it does not tell us which spots/reflections would be observed in an actual experiment.
:eight_pointed_black_star:The
Ewald sphere construction
selects those points which are actually observed in a diffraction experiment.
:eight_pointed_black_star:For each reciprocal lattice point that is located on the Ewald sphere of reflection,
the Bragg condition is satisfied
and diffraction arises.
:eight_pointed_black_star:the wavevector of the incident and diffracted x-ray beams
:eight_pointed_black_star:the diffraction angle for a given reflection
:eight_pointed_black_star:the reciprocal lattice of the crystal
1.Draw a circle with diameter 2/λ
2.Construct a triangle with the diameter as the hypotenuse and 1/d hkl as a side
3.The angle opposite the 1/d side is θ hkl
(from bragg's equation)
4.Now if we overlay 'real space' information on the Ewald sphere (mix real and reciprocal space information)
5.Assume the incident ray along AC and the diffracted ray along CP.Then automatically the crystal will have to be considered to be located at C with an orientation such that the d hkl planes bisect the angle OCP (∠OCP=2θ)
6.OP becomes the reciprocal space vector g hkl
:eight_pointed_black_star:When the ewald sphere (shown as circle in 2D below) touch the reciprocal lattice point.
:eight_pointed_black_star: That reflection is observed in an experiment (
41
reflection in the figure below)
:eight_pointed_black_star: The ewald sphere touches the reciprocal lattice (at point
41
)
:eight_pointed_black_star: Bragg's equation is satisfied for
41
:
In general, reciprocal lattice points do not lie on the sphere
To observe the reflection, then we must
:eight_pointed_black_star: Move the sphere
:eight_pointed_black_star: Move the crystal(rotate)
:eight_pointed_black_star:Change the size of the sphere
:eight_pointed_black_star:The only reflections that can be observed are those for which OH ≤ 2/λ (diameter of the Ewald sphere) and for which the reciprocal nodes lie within a sphere of centre O and radius 2/λ.
:eight_pointed_black_star:This sphere is called the limiting sphere (in yellow in Fig. 3)
:eight_pointed_black_star:Conversely, if the wavelength of the incident radiation is larger than the largest interplanar spacing dmax of the crystal, there is no reciprocal lattice node within the limiting sphere: no Bragg reflection can take place for λ > dmax.
what is
k
?
Reciprocal lattice vectors ,
K
Definition
Definition
Objectives
Conceived by
Demonstrate the relationship between
Ewald construction
Example
ΔK=Diffraction vector
In general
Limiting sphere
Reciprocal space
