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X-ray Diffraction

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

NanoLab/NSF NUE/Bumm

X-ray Diffraction

- Braggs Law
- Lattice Constants
- Laue Conditions
- ? - 2? Scan
- Scherrers Formula
- Data Analysis Examples

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.

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

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

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.

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

? - 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

? - 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.

? - 2? Scan

Smaller Crystals Produce Broader XRD Peaks

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

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

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.

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

Bragg Example

Bragg Example

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

Scherrers Example

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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