X-ray Diffraction The Basics
Followed by a few examples of
Data Analysis
by
Wesley Tennyson NanoLab/NSF NUE/Bumm

X-ray Diffraction :

NanoLab/NSF NUE/Bumm X-ray Diffraction Bragg’s Law
Lattice Constants
Laue Conditions
θ - 2θ Scan
Scherrer’s Formula
Data Analysis Examples

Bragg’s Law :

Bragg’s Law nλ = 2 d sin θ
Constructive interference only occurs for certain θ’s correlating to a (hkl) plane, specifically when the path difference is equal to n wavelengths.

Bragg condition’s :

NanoLab/NSF NUE/Bumm Bragg condition’s The diffraction condition can be written in vector form
2k∙G + G2 = 0
k - is the incident wave vector
k’ - is the reflected wave vector
G - is a reciprocal lattice vector such that where
G = ∆k = k - k’
the diffraction condition is met

Lattice Constants :

NanoLab/NSF NUE/Bumm Lattice Constants The distance between planes of atoms is
d(hkl) = 2π / |G|
Since G can be written as
G = 2π/a (h*b1+ k*b2 +l*b3)
Substitute in G
d(hkl) = a / (h2 + k2 + l2)(1/2)
Or
a = d * (h2 + k2 + l2)(1/2)
a is the spacing between nearest neighbors

Laue Conditions :

NanoLab/NSF NUE/Bumm Laue Conditions a1∙∆k = 2πυ1 a2∙∆k = 2πυ2
a3∙∆k = 2πυ3
Each of the above describes a cone in reciprocal space about the lattice vectors a1, a2, and a3.
the υi are integers
When a reciprocal lattice point intersects this cone the diffraction condition is met, this is generally called the Ewald sphere.

Summary of Bragg & Laue :

NanoLab/NSF NUE/Bumm Summary of Bragg & Laue When a diffraction condition is met there can be a reflected X-ray
Extra atoms in the basis can suppress reflections
Three variables λ, θ, and d
λ is known
θ is measured in the experiment (2θ)
d is calculated
From the planes (hkl)
a is calculated

θ - 2θ Scan :

NanoLab/NSF NUE/Bumm θ - 2θ Scan The θ - 2θ scan maintains these angles with the sample, detector and X-ray source Normal to surface Only planes of atoms that share this normal will be seen in the θ - 2θ Scan

θ - 2θ Scan :

NanoLab/NSF NUE/Bumm θ - 2θ Scan The incident X-rays may reflect in many directions but will only be measured at one location so we will require that:
Angle of incidence (θi) = Angle of reflection (θr)
This is done by moving the detector twice as fast in θ as the source. So, only where θi = θr is the intensity of the reflect wave (counts of photons) measured.

θ - 2θ Scan :

NanoLab/NSF NUE/Bumm θ - 2θ Scan

Smaller Crystals Produce Broader XRD Peaks :

Smaller Crystals Produce Broader XRD Peaks

Scherrer’s Formula :

t = thickness of crystallite
K = constant dependent on crystallite shape (0.89)
l = x-ray wavelength
B = FWHM (full width at half max) or integral breadth
qB = Bragg Angle Scherrer’s Formula

Scherrer’s Formula :

Scherrer’s Formula What is B?
B = (2θ High) – (2θ Low)
B is the difference in angles at half max 2θ high Noise 2θ low Peak

When to Use Scherrer’s Formula :

When to Use Scherrer’s Formula Crystallite size <1000 Å
Peak broadening by other factors
Causes of broadening
Size
Strain
Instrument
If breadth consistent for each peak then assured broadening due to crystallite size
K depends on definition of t and B
Within 20%-30% accuracy at best Sherrer’s Formula References
Corman, D. Scherrer’s Formula: Using XRD to Determine Average Diameter of Nanocrystals.

Data Analysis :

Data Analysis Plot the data (2θ vs. Counts)
Determine the Bragg Angles for the peaks
Calculate d and a for each peak
Apply Scherrer’s Formula to the peaks

Bragg Example :

Bragg Example

Bragg Example :

Bragg Example d = λ / (2 Sin θB) λ = 1.54 Ǻ
= 1.54 Ǻ / ( 2 * Sin ( 38.3 / 2 ) )
= 2.35 Ǻ
Simple Right!

Scherrer’s Example :

Scherrer’s Example

Scherrer’s Example :

Scherrer’s Example t = 0.89*λ / (B Cos θB) λ = 1.54 Ǻ
= 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) )
= 1200 Ǻ
B = (98.3 - 98.2)*π/180 = 0.00174
Simple Right!

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