1. CHARACTERIZATION OF THE STRUCTURE OF SOLIDS
Three main techniques:
X-ray diffraction
Electron diffraction
Neutron diffraction
Single crystal
Powder
Principles of x-ray diffraction
X-rays are passed through a crystalline material and the patterns produced give
information of size and shape of the unit cell
X-rays passing through a crystal will be bent at various angles: this process is called
diffraction
X-rays interact with electrons in matter, i.e. are scattered by the electron clouds of
atoms

2. Reflection (signal) only occurs when conditions for constructive interference between
the beams are met
These conditions are met when the difference in path length equals an integral number
of wavelengths, n. The final equation is the BRAGG’S LAW
Data are collected by using x-rays of a known wavelength. The position of the sample
is varied so that the angle of diffraction changes
When the angle is correct for diffraction a signal is recorded
With modern x-ray diffractometers the signals are converted into peaks
Intensity (a.u.)
2 degrees
(200)
(110)
(400)
(310)
(301)
(600)
(411)
(002)
(611)
(321)

3. TEST
NaCl is used to test diffractometers. The distance between a set of planes in
NaCl is pm. Using an x-ray source of 75 pm, at what diffraction angle
(2) should peaks be recorded for the first order of diffraction (n = 1) ?
Hint: To calculate the angle  from sin , the sin-1 function on the calculator
must be used

4. The lattice parameters a, b, c of a unit cell can then be calculated
The relationship between d and the lattice parameters can be determined
geometrically and depends on the crystal system
Crystal system
dhkl, lattice parameters and Miller indices
Cubic
Tetragonal
Orthorhombic
The expressions for the remaining crystal systems are more complex

5. THE POWDER TECHNIQUE
An x-ray beam diffracted from a lattice plane can be detected when the x-ray source, the sample and the detector are correctly oriented to give Bragg diffraction
A powder or polycrystalline sample contains an enormous number of small crystallites, which will adopt all possible orientations randomly
Thus for each possible diffraction angle there are crystals oriented correctly for Bragg diffraction
Each set of planes in a crystal
will give rise to a cone of diffraction
Each cone consists of a set of closely spaced dots each one of which represents a
diffraction from a single crystallite

6. Formation of a powder pattern
Single set of planes
Powder sample

7. Experimental Methods
To obtain x-ray diffraction data, the diffraction angles of the various cones, 2,
must be determined
The main techniques are: Debye-Scherrer camera (photographic film) or
powder diffractometer
Debye Scherrer Camera

8. Powder Diffractometer

9. The detector records the angles at which the families of lattice planes scatter (diffract) the x-ray beams and the intensities of the diffracted x-ray beams
The detector is scanned around the sample along a circle, in order to collect all the diffracted x-ray beams
The angular positions (2) and intensities of the diffracted peaks of radiation (reflections or peaks) produce a two dimensional pattern
Each reflection represents the x-ray beam diffracted by a family of lattice planes (hkl)
This pattern is characteristic of the material analysed (fingerprint)
Intensity
2 degrees
(200)
(110)
(400)
(310)
(301)
(600)
(411)
(002)
(611)
(321)

10. APPLICATIONS AND INTERPRETATION OF X-RAY POWDER DIFFRACTION DATA

11. Sample line broadening
*Strain effect
- variation in d
- introduced by defects, stacking fault, mistakes
- depends on 2θ

12. Scherrer equation
* Determination of size effect, neglecting strain (Scherrer, 1918)
*Thickness of a crystallite L = N dhkl
Lhkl = k λ / (β cosθ),
k: shape factor, typically taken as unity for β and 0.9 for FWHM

13. Identification of compounds
The powder diffractogram of a compound is its ‘fingerprint’ and can be used to identify the compound
Powder diffraction data from known compounds have been compiled into a database (PDF) by the Joint Committee on Powder Diffraction Standard, (JCPDS)
‘Search-match’ programs are used to compare experimental diffractograms with patterns of known compounds included in the database
This technique can be used in a variety of ways

14. PDF - Powder Diffraction File
A collection of patterns of inorganic and organic compounds
Data are added annually (2008 database contains 211,107 entries)

15. Example of Search-Match Routine

16. ? Outcomes of solid state reactions Product: SrCuO2?
Pattern for SrCuO2from database
Product: Sr2CuO3?
Pattern for Sr2CuO3from database

17. Phase purity
When a sample consists of a mixture of different compounds, the resultant diffractogram shows reflections from all compounds (multiphase pattern)
Sr2CuO2F2+
Sr2CuO2F2+ + impurity *

18. Determination of crystal class and lattice parameters
X-ray powder diffraction provides information on the crystal class of the unit cell (cubic, tetragonal, etc) and its parameters (a, b, c) for unknown compounds 1
Crystal class
comparison of the diffractogram of the unknown
compound with diffractograms of known compounds
(PDF database, calculated patterns) 2
Indexing
Assigning Miller indices to peaks 3
Determination of
lattice parameters
Bragg equation and lattice parameters
Cubic system

19. Selected data from the NaCl diffractogram 2 () h,k,l 27.47 111 31.82
PROBLEM
NaCl shows a cubic structure. Determine a (Å) and the missing Miller indices
( = Å). ? ?
Selected data from the NaCl diffractogram
2 ()
h,k,l
27.47
111
31.82 ?
45.62
56.47
222

20. a (Å)
Use at least two reflections and then average the results Å
(222)
Miller Indices A

22. F I P Systematic Absences Conditions for reflection
For body centred (I) and all-face centred (F) lattices restriction on reflections from
certain families of planes, (h,k,l) occur. This means that certain reflections do not appear in diffractograms due to ‘out-of-phase” diffraction
This phenomenon is known as systematic absences and it is used to identify
the type of unit cell of the analysed solid. There are no systematic absences for
primitive lattices (P)
Conditions for reflection F
i.e indices are all odd or all even I P
No conditions

23. to either the correct lattice(s).
Considering systematic absences, assign the following sets of Miller indices
to either the correct lattice(s).
Lattice Type
Miller Indices P I F
1 0 0 Y N
1 1 0
1 1 1
2 0 0
2 1 0
2 1 1
2 2 0
3 1 0
3 1 1 y

24. Autoindexing Generally indexing is achieved using a computer program.
This process is called ‘autoindexing’
Input:
Peak positions (ideally peaks)
Wavelength (usually = Å)
The uncertainty in the peak positions
Maximum allowable unit cell volume
Problems:
Impurities
Sample displacement
Peak overlap

25. Derivation of