The document is a presentation on x-ray diffraction (XRD), covering both qualitative and quantitative analysis methods for crystalline materials. It details topics such as the generation of x-rays, Bragg's Law, various XRD methods, and specific analytical techniques like the Hanawalt method for qualitative analysis and various quantitative methods including the Rietveld method. Key points about the applications, limitations, and requirements for conducting XRD analysis are also highlighted.

1. X-Ray Diffraction –
Qualitative and Quantitative Analysis
Presenter: Syed Ali Afzal
Roll No.: MM-02/2017-18
Subject: AMCT
Presented to: Dr. Ali Dad Chandio
1

2. Presentation Layout
Introduction
i. General Uses
ii. X-Rays Generation
iii. Collimator
iv. Monochromator
v. Detection of X-Rays
Bragg’s Law
X-Ray Diffraction Methods
i. Laue’s Photographic Method
ii. Bragg’s X-Ray Spectrometer
Method
iii. Powder Crystal Method
2
Qualitative Analysis
i. Hanawalt Method
ii. Computer Based Search/Match
Program
Quantitative Analysis
i. Introduction
ii. Lattice Parameter Method
iii. The Absorption Method
iv. The Method of Standard
Additions
v. The I/Icorundum Method
vi. The Rietveld Method

3. 1. Introduction
3

4. i. General Uses
• Most useful in the characterization of crystalline
materials
• E.g. Metals, inter-metallic, ceramics, minerals, polymers and
plastics
• Used to identify phases, grain size, texture and crystal
imperfections
• Rapid and Non-destructive
• Qualitative and Quantitative analysis of crystalline
phases
• E.g. Coal ash, ceramic powder, corrosion products
etc.
• Characterization of solid-state phase transformations
• Lattice parameters and lattice type determinations
• Orientation of single crystal
• Limitations:
• Sample must be crystalline
• Identification requires existence of standard patterns
4

5. Example: Empirical formula: FeTiO3
Mixture of Two Phases (FeO and TiO2) Single phase mineral (FeTiO3)
5
Fe
Iron ion
Oxygen ion
Rocksalt
Rutile
Perovskite

6. ii. X-Rays Generation
• X-rays: Wavelength 10-10 to 10-18 m (0.1 to 100 Å)
• Only 0.3 to 2.5 Å is used for XRD
• High energy (50 KV) e- beam (W filament) on metal
target (Cu as target, Kα  1.5406 Å)
• Approx. 1% of total energy of e- beam is converted
into X-rays
• Rest of the e- beam energy is dissipated as heat
6

7. iii. Collimator
• Used for achieving a narrow beam of X-rays
• Filters a stream of rays so that only those travelling to a specified direction are allowed
through
7

8. iv. Monochromator
• The incident beam should consists of a single wavelength i.e. monochromatic
• Two methods to monochromatized output beam:
1. Filter of appropriate metal: A material that absorbs undesirable radiation but allows the
radiation of required wavelength to pass
E.g. Ni absorbs most of the radiations in Cu and transmits only Kα
2. Crystal monochromator: Use of a single crystal as diffractor to allow diffraction of only
desired wavelengths
E.g. Crystals of Graphite, Si, Ge and Quartz
8Fig 1. Characteristic X-ray spectra without a filter
Kα1
Kα2
Kβ
bremsstrahlung or white radiation

9. v. Detection of X-Rays
• Position of beams and their
intensities
• Photographic films and Counter
methods are usually used
 Photographic films:
• Special films used as flat or
cylindrical detectors  beam
appears as dark spot or line
• Darkening of spot is
proportional to beam intensity
• Useful when entire diffraction
pattern is desirable
• Easy to interpret position of
beams, difficult to use for
quantitative intensity data
• Time consuming  exposure of
several hours
9

10. v. Detection of X-Rays
 Counter Methods:
• Gieger-Muller counter
• Proportional counter
• Scintillation counter
• Solid-state semi-conductor detector
• Semi-conductor detector
10Scintillation counter

11. 2. Bragg’s Law
11

12. Bragg’s Law
• XRD is based on the constructive interference of monochromatic X-rays and the
crystalline sample
• The interaction of incident rays with the sample produces constructive interference
when conditions satisfy Bragg’s law.
• Path difference = 2d Sinθ
• For constructive interference: nλ=2d Sinθ
• If Bragg’s eq. not satisfied = no reflection
12

13. Bragg’s Law
13

14. 3. X-Ray Diffraction
Methods
14

15. i. Laue’s Photographic Method
Transmission Method
• Film placed behind crystal
• Adv: Orientation of single crystal, d-spacing
• Disadv: Sample must be thin enough for x-
rays to penetrate
Back Reflection Method
• Film is placed between x-ray source and
crystal
• Adv: Orientation of single crystal
• Disadv: Don’t provide phase identification
15

16. ii. Bragg’s X-Ray Spectrometer Method
• Bragg analyzed structures of Nacl, Kcl and ZnS
• Single plane generates several diffraction lines  diffraction pattern
16
XRD image of a single Alum crystal

17. iii. Powder Crystal Method
• XRPD  phase identification of crystalline materials + unit cell
dimensions
• Analyzed materials should be finely grounded and
homogenized
• Determine complex structure of metals and alloys
• Measurement of sample purity
17Diffraction from powder crystal

18. 4. Qualitative
Analysis
18

19. i. Hanawalt Method
• Qualitative Analysis:
Identification of a phase or phases in a specimen by comparison with “standard” patterns” (i.e.,
data collected or calculated by someone else), and relative estimation of proportions of
different phases in multiphase specimens by comparing peak intensities attributed to the
identified phases.
• Hanawalt Method:
• The Hanawalt Manual, lists standard phases from the JCPDS file, along with their eight
most intense d-spacings and intensities
• The d value of the strongest line on the pattern is used to determine which group is to be
consulted in the manual
• If the other six lines of one of these standard patterns match lines of similar relative
intensity in the unknown pattern, the standard selected is most likely a match for the
unknown
• To be more certain, the JCPDS data for the full pattern are then compared with the
unknown pattern; any lines from the unknown that do not match lines of the standard
may indicate the presence of a second phase and that the unknown pattern did not come
from a single phase
19

20. Example
20
Peak number 2θ (°) d (Å) I/I1
1 35.84 2.503 85
2 41.73 2.163 100
3 60.47 1.530 55
4 72.37 1.305 32
5 76.02 1.251 18
6 90.75 1.082 11
7 101.71 0.993 13
8 105.48 0.968 25
Fig 2. X-ray diffraction pattern of the unknown material
Table 1. Experimental Data for Identifying an Unknown Sample

21. Example
21
Table 2. Excerpt from the Hanawalt Search Manual
• TiC is the unknown specimen

22. ii. Computer Based Search/Match Programs
22
• The manual search needs to compare lines from the unknown pattern in various
permutations and combinations with the standard file; the most common computer
technique involves the opposite, that is, comparing the standards with the unknown
Fig 4. Different Phases produces different combination of peaksFig 3. Experimental XRD Data compared to Reference Pattern. Lines in
red represent reference pattern

23. ii. Computer Based Search/Match Programs
23
• The experimental data should contain all major peaks listed in reference pattern

24. ii. Computer Based Search/Match Programs
24
• Most diffraction pattern contains K-alpha 1 and K-alpha 2 peak doublets rather than just
single peaks
• Intensity of K alpha 1 = 2 (Intensity of K-alpha 2)

25. 5. Quantitative
Analysis
25

26. i. Introduction
• The determination of amounts of different phases in multi-phase
samples
• Determination of particular characteristics of single phases
including precise determination of crystal structure or crystallite
size and shape
• All quantitative analysis requires precise and accurate
determination of the diffraction pattern for a sample both in terms
of peak positions and intensities
• Many factors prevent the direct comparison of concentration with
peak intensity. The basic factor is the different x-ray absorption
properties of the substances in the sample
• The most common methods of Quantitative Analysis are:
26

27. ii. Lattice Parameter Method
• Applicable for continuous solid-solutions
• Accurate method of determining the chemical compositions by lattice parameters
• Determine composition of single phase, not amounts
• Determination of unit cell dimensions
27
Bragg's Equation
d*2 equation to calculate lattice parameters
Unit cell dimensions by diffraction peaks

28. ii. Lattice Parameter Method
• Microstructural Information
• Peak Broadening:
• Smaller crystallite size in nano-crystalline materials
• More stacking faults, microstrain and other defects in crystal
• An inhomogeneous composition in a solid solution
28

29. iii. The Absorption Method
• Requires the measurement of intensity from a diffraction peak in the mixture and from a pure
standard of material
• Ipure is the intensity of a peak from a pure phase
• I is the intensity of the same peak of the phase in mixture
• X is the weight fraction of the phase in the mixture
• (µ/ρ) is the mass-absorption coefficient of the phase
• (µ/ρ)m is the mass-absorption coefficient of the entire sample
• The mass-absorption coefficient of the sample and the phase under analysis must be known
(International Tables for X-Ray Crystallography)
• The accuracy of this technique depends strongly on consistent sample preparation and on
appropriate pure standards
29
Diffraction Equation for The Absorption Diffraction Method

30. iv. The Method of Standard Additions
• Also known as The Spiking Method or The Doping Method
• The peak intensity is first measured of the phase of interest then again measuring the
intensity after adding a small amount of this phase
• I1 is the intensity of a diffraction line from the sample
• I2 is the intensity of the same line after it has been spiked
• Co is the concentration of the phase of interest
• C1 is the amount of phase added to spike the sample
• Useful when only one phase is to be quantified
• Problems:
• well mixed powder,
• uniform crystallite size,
• production of an extensive database of diffraction patterns
30
Equation for The Method of Standard Additions

31. v. The I/Icorundum Method
• Perform rapid semi-quantitative analysis without standards
• Weight fraction Xs of phase a is calculated using:
• (Ia/Ic)unk is the intensity of the 100 peak of phase a divided by the intensity of the 100 peak of corundum in a
1:1 mixture of sample and corundum
• (Ia/Ic)JCPDS is the reference intensity ratio
31
Equation for I/Icorundum Method

32. vi. The Rietveld Method
• The Rietveld method refines user-selected parameters to minimize the
difference between an experimental pattern (observed data) and a model
based on the hypothesized crystal structure and instrumental parameters
(calculated pattern)
• Can refine information about a single crystal structure
• confirm/disprove a hypothetical crystal structure
• refine lattice parameters
• refine atomic positions, fractional occupancy, and thermal parameter
• Refine information about a single sample
• preferred orientation
• Refine information about a multiphase sample
• determine the relative amounts of each phase
32

33. vi. The Rietveld Method
• Advantages:
i. Differences between the experimental standard and the phase in the unknown
are minimized. Compositionally variable phases are varied and fit by the software.
ii. Pure-phase standards are not required for the analysis.
iii. Lattice parameters for each phase are automatically produced, allowing for the
evaluation of solid solution effects in the phase.
iv. The use of the whole pattern rather than a few select lines produces accuracy and
precision much better than traditional methods.
v. Preferred orientation effects are averaged over all of the crystallographic
directions, and may be modeled during the refinement.
33

34. 34
Thank you. Questions?