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 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

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
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Bragg Example

• d ? / (2 Sin ?B) ? 1.54 ?
• 1.54 ? / ( 2 Sin ( 38.3 / 2 ) )
• 2.35 ?
• Simple Right!

18
Scherrers Example
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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!