1

 The Basics
 Followed by a few examples of
 Data Analysis
 by
 Wesley Tennyson

2

 Bragg’s Law
 Lattice Constants
 Laue Conditions
 θ  2θ Scan
 Scherrer’s Formula
 Data Analysis Examples

3

 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.

4

 The diffraction condition can be written in vector form
 2k∙G + G^{2} = 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

5

 The distance between planes of atoms is
 d(hkl) = 2π / G
 Since G can be written as
 G = 2π/a (h*b_{1}+ k*b_{2 }+l*b_{3})
 Substitute in G
 d(hkl) = a / (h^{2} + k^{2} + l^{2})^{(1/2)}
 Or
 a = d * (h^{2} + k^{2} + l^{2})^{(1/2)}
 a is the spacing between nearest neighbors

6

 a_{1}∙∆k = 2πυ_{1 } a_{2}∙∆k
= 2πυ_{2}
 a_{3}∙∆k = 2πυ_{3}
 Each of the above describes a cone in reciprocal space about the lattice
vectors a_{1}, a_{2}, and a_{3}.
 When a reciprocal lattice point intersects this cone the diffraction
condition is met, this is generally called the Ewald sphere.

7

 When a diffraction condition is met there can be a reflected Xray
 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)

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9

 The incident Xrays may reflect in many directions but will only be
measured at one location so we will require that:
 Angle of incidence (θ_{i}) = Angle of reflection (θ_{r})
 This is done by moving the detector twice as fast in θ as the
source. So, only where θ_{i} = θ_{r} is the
intensity of the reflect wave (counts of photons) measured.

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 t = thickness of crystallite
 K = constant dependent on crystallite shape (0.89)
 l = xray wavelength
 B = FWHM (full width at half max) or integral breadth
 q_{B} = Bragg Angle

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 What is B?
 B = (2θ High) – (2θ Low)
 B is the difference in angles at half max

14

 Crystallite size <1000 Å
 Peak broadening by other factors
 Causes of broadening
 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

15

 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

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17

 d = λ / (2 Sin θ_{B}) λ = 1.54 Ǻ
 = 1.54 Ǻ / ( 2 * Sin (
38.3 / 2 ) )
 = 2.35 Ǻ
 Simple Right!

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