X-ray Diffraction - PowerPoint PPT Presentation

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X-ray Diffraction

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B = FWHM (full width at half max) or integral breadth. qB = Bragg Angle. Scherrer's Formula ... What is B? B = (2? High) (2? Low) B is the difference in ... – PowerPoint PPT presentation

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Title: X-ray Diffraction


1
X-ray Diffraction
  • The Basics
  • Followed by a few examples of
  • Data Analysis
  • by
  • Wesley Tennyson

NanoLab/NSF NUE/Bumm
2
X-ray Diffraction
  • Braggs Law
  • Lattice Constants
  • Laue Conditions
  • ? - 2? Scan
  • Scherrers Formula
  • Data Analysis Examples

3
Braggs Law
  • n? 2 d sin ?
  • Constructive interference only occurs for certain
    ?s correlating to a (hkl) plane, specifically
    when the path difference is equal to n
    wavelengths.

4
Bragg conditions
  • The diffraction condition can be written in
    vector form
  • 2kG 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

5
Lattice Constants
  • The distance between planes of atoms is
  • d(hkl) 2p / G
  • Since G can be written as
  • G 2p/a (hb1 kb2 lb3)
  • 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

6
Laue Conditions
  • a1?k 2p?1 a2?k 2p?2
  • a3?k 2p?3
  • Each of the above describes a cone in reciprocal
    space about the lattice vectors a1, a2, and a3.
  • the ?i are integers
  • When a reciprocal lattice point intersects this
    cone the diffraction condition is met, this is
    generally called the Ewald sphere.

7
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 ?, ?, and d
  • ? is known
  • ? is measured in the experiment (2?)
  • d is calculated
  • From the planes (hkl)
  • a is calculated

8
? - 2? Scan
The ? - 2? 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 ? - 2? Scan
9
? - 2? 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 (?i) Angle of reflection
    (?r)
  • This is done by moving the detector twice as fast
    in ? as the source. So, only where ?i ?r is the
    intensity of the reflect wave (counts of photons)
    measured.

10
? - 2? Scan
11
Smaller Crystals Produce Broader XRD Peaks
12
Scherrers 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

13
Scherrers Formula
  • What is B?
  • B (2? High) (2? Low)
  • B is the difference in angles at half max

Peak
2? high
2? low
Noise
14
When to Use Scherrers Formula
  • Crystallite size lt1000 Ã…
  • 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

Sherrers Formula References Corman, D.
Scherrers Formula Using XRD to Determine
Average Diameter of Nanocrystals.
15
Data Analysis
  • Plot the data (2? vs. Counts)
  • Determine the Bragg Angles for the peaks
  • Calculate d and a for each peak
  • Apply Scherrers Formula to the peaks

16
Bragg Example
17
Bragg Example
  • d ? / (2 Sin ?B) ? 1.54 ?
  • 1.54 ? / ( 2 Sin ( 38.3 / 2 ) )
  • 2.35 ?
  • Simple Right!

18
Scherrers Example
19
Scherrers Example
t 0.89? / (B Cos ?B) ? 1.54 ?
0.891.54 ? / ( 0.00174 Cos (98.25/ 2 ) )
1200 ? B (98.3 - 98.2)p/180 0.00174 Simple
Right!
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