Indus University

IISHLS

Department of Chemistry

M.Sc. Analytical Chemistry

Semester: 3rd

Name of Subject: Qualitative Spectroscopic Techniques

Subject Code: MCH0321

Unit-4- X- ray Diffraction

 ❖Introduction:

→ X-ray diffraction is a tool for the investigation of the fine structure of matter.
→ This technique had its beginnings in von Laue's discovery in 1912 that crystals
diffract x-rays, the manner of the diffraction revealing the structure of the crystal.
→ At first, x-ray diffraction was used only for the determination of crystal structure.
Later on, however, other uses were developed, and today the method is applied,
not only to structure determination, but to such diverse problems as chemical
analysis and stress measurement, to the study of phase equilibria and the
measurement of particle size, to the determination of the orientation of one crystal
or the ensemble of orientations in a polycrystalline aggregate.
→ X-rays were discovered in 1895 by the German physicist Roentgen and were so
named because their nature was unknown at the time. Unlike ordinary light, these
rays were invisible, but they traveled in straight lines and affected photographic
film in the same way as light.
→ On the other hand, they were much more penetrating than light and could easily
pass through the human body, wood, quite thick pieces of metal, and other
"opaque" objects.
→ X-ray Diffraction in crystal was discovered by Max Von Laue. The wavelength
range is 0.01 to about 10 nm.
→ X-rays are short wavelength of electromagnetic radiation produced by the
deceleration of high energy electrons or by electronic transition of electrons
in the inner orbital of atoms.
→ The penetrating power of X rays depends on energy also, there are two types of
X-ray can be produced .
 1. Hard X-rays
•Have high frequency and more energy.

2. Soft X-rays
•Have less penetrating and have low energy

→ X-ray diffraction (XRD) is a non-contact and non-destructive technique used to

understand the crystalline phase, different polymeric forms and the
structural properties of materials.
→ The XRD pattern of a pure substance is therefore like a FINGERPRINT of that
substance.
→ XRD technique is based on the scattering of X-ray by crystals.
→ XRD technique is very important as it give the following information…

Measure the average spacing between layers of the rows of atoms

Determine the orientation of a single crystal

FInd the crystal structure of unknown material

Measure the size, shape and internal stress of small crystalline

region.

To determine the phase composition of the crystal

 ❖Generation of X-Rays

Fig. Schematic of filament X-ray tube

→ We have seen that x-rays are produced whenever high-speed electrons collide
with a metal target. Any x-ray tube must therefore contain

A source of electrons

A high accelerating voltage

A metal target

Vacuum: to control the number and speed of the accelerated electrons

independently.

Pyrex glass: connecting wires same coefficient of linear expansion as

glass.

Negative terminal of the x-ray tube is called cathode or filament. Along with filament 2 other
elements: connecting Along with filament 2 other elements: connecting wires and focusing cup
Filament made of tungsten wire 0.2 mm diameter coiled to form a vertical spiral 0.2 cm
diameter and 1 cm length
➢ Some basic terms

✓ Cathode

Filament

• Made of thin (0.2 mm) tungsten wire because tungsten:

o has a high atomic number (A 184, Z 74)
o is a good thermionic emitter (good at emitting electrons)
o can be manufactured into a thin wire
o has a very high melting temperature (3422°c)
• The size of the filament relates to the size of the focal spot. Some cathodes
have two filaments for broad and fine focusing

✓ Focusing cup

• Made of molybdenum as:

o high melting point
o poor thermionic emitter so electrons aren't released to interfere with
electron beam from filament
• Negatively charged to focus the electrons towards the anode and stop spatial
spreading

✓ Anode

• Target made of tungsten for same reasons as for filament

• Rhenium added to tungsten to prevent cracking of anode at high temperatures
and usage
• Set into an anode disk of molybdenum with stem
• Positively charged to attract electrons
• Set at angle to direct x-ray photon beam down towards patient. Usual angle is
5º - 15º

✓ Definitions

• Target, focus, focal point, focal spot: where electrons hit the anode
• Actual focal spot: physical area of the focal track that is impacted
• Focal track: portion of the anode the electrons bombard. On a rotating anode
this is a circular path
• Effective focal spot: the area of the focal spot that is projected out of a tube
→ A current is passed through the tungsten filament and heats it up.
→ As it is heated up the increased energy enables electrons to be released from the
filament through thermionic emission(i.e. Emission of electrons resulting from
the absorption of thermal energy).
→ Electron cloud surrounding the filament produced by thermionic emission is
termed “Edison effect”
→ The electrons are attracted towards the positively charged anode and hit the
tungsten target with a maximum energy determined by the tube potential
(voltage).
→ As the electrons bombard the target they interact via Bremsstrahlung and
characteristic interactions which result in the conversion of energy into heat
(99%) and x-ray photons (1%).
→ The x-ray photons are released in a beam with a range of energies (x-ray
spectrum) out of the window of the tube and form the basis for x-ray image
formation.
→ Since most of the kinetic energy of the electrons is converted into heat in the
target, the latter must be water-cooled to prevent its melting.
→ All x-ray tubes contain two electrodes, an anode (the metal target) maintained,
with few exceptions, at ground potential, and a cathode, maintained at a high
negative potential, normally of the order of 30,000 to 50,000 volts for diffraction
work.
→ X-ray tubes may be divided into two basic types, according to the way in which
electrons are provided: filament tubes, in which the source of electrons is a hot
filament, and gas tubes, in which electrons are produced by the ionization of a
small quantity of gas in the tube.
→ Filament tubes : They consist of an evacuated glass envelope which insulates
the anode at one end from the cathode at the other, the cathode being a tungsten
filament and the anode a water-cooled block of copper containing the desired
target metal as a small insert at one end.

→ One lead of the high-voltage transformer is connected to the filament and the
other to ground, the target being grounded by its own cooling water connection.
→ The filament is heated by a filament current of about 3 amp and emits electrons
which are rapidly drawn to the target by the high voltage across the tube.
→ Surrounding the filament is a small metal cup maintained at the same high
(negative) voltage as the filament: it therefore repels the electrons and tends to
focus them into a narrow region of the target, called the focal spot.
→ X-rays are emitted from the focal spot in all directions and escape from the tube
through two or more windows in the tube housing. Since these windows must be
vacuum tight and yet highly transparent to x-rays, they are usually made of
beryllium, aluminum, or mica.
→ X-rays are generated via interactions of the accelerated electrons with electrons
of tungsten nuclei within the tube anode. There are two types of X-ray generated:
characteristic radiation and bremsstrahlung radiation.

Fig. Characteristic radiation

Fig. Bremsstrahlung X-ray generation

 Fig. Characteristic Spectrum of X ray

→ When high velocity electrons will strike the metal target then it will produce x-
rays.
→ The continuous spectrum is caused by the rapid deceleration of electrons by the
target metal, the origin of the characteristic spectrum lies in the atoms of the
target material itself.
→ To understand this phenomenon , it is enough to consider an atom as consisting
of a central nucleus surrounded by electrons lying in various shell.
→ If one of the electrons bombarding the target has sufficient kinetic energy , it can
knock an electron out of the K shell leaving an atom in an excited high energy
state.
→ One of the outer electrons immediately falls into the vacancy in the K shell ,
emitting energy n the process and the atom is once again in its normal energy
state.
→ The energy emitted is in the form of radiation offer definite wavelength and is
infact characteristic of K radiation.
→ The K shell vacancy may be filled by an electron from any of the outer shell, thus
giving rise to a series of K lines. Kα and Kβ

❖ Basic Principle of X- ray Diffraction:

→ Wavelength of x-ray is 0.01 to 10 nm.

→ For analytical purpose , the wavelength of x-ray is 0.07 to 0.2 nm.

[A] X-ray Diffraction

→ This method is based on the scattering of x-rays by crystals.

→ By using this method, analyst can easily identify the crystal structure of any solid
compound with high degree of specify and accuracy.
→ XRD is highly important method as compared to other x- ray spectroscopy.

[B] X-ray Absorption

→ This is similar to absorption method in other region of electromagnetic spectra

like UV visible, IR spectroscopy.
→ It gives the information about the absorbing material.
→ In this method a beam of x-ray is passed through the sample and the fraction of
x-ray photons absorbed is consider to be a measure of the concentration of the
sample.
→ This method is least used as compared to other methods.
→ It is used only to detect imperfection in the internal structure , elemental
analysis , thickness measurement.
 [C] X-ray Fluorescence (Emission)

→ In this method, x-rays are generated within the sample and by measuring the
wavelength and intensity of the generated x-rays.
→ Analyst can perform qualitative as well as quantitative analysis.

❖ The Bragg’s Law for X-ray Diffraction

→ Bragg pointed out that the unlike reflection of ordinary light, reflection of X-ray
can take place only at a certain angle which are determined by the wavelength of
x-ray and distance between the plane of in the crystal.
→ Two geometrical facts are worth remembering:
→ (1) The incident beam, the normal to the reflecting plane, and the diffracted beam
are always coplanar.
→ (2) The angle between the diffracted beam and the transmitted beam is always
2. This is known as the diffraction angle, and it is this angle, rather than , which
is usually measured experimentally.
→ As previously stated, diffraction in general occurs only when the wavelength of
the wave motion is of the same order of magnitude as the repeat distance between
scattering centers. This requirement follows from the Bragg law.
→ Suppose a beam of x-ray falls on the crystal at glancing angle (tangential angle)
 , then some of these rays will reflected from the upper plate at same angle .
→ While some of these rays will be absorbed and get reflected from the successive
layer.
 Fig. The Bragg’s Law

 So, here angle will be XBY =  & YBB’ = 

 AB = A’X & CB = C’Y

 Path difference = XB’ + B’Y

→ Path difference is defined as an integral multiple of wavelength = n 

 n  = XB’ + B’Y

→ Now for ∆ XBB’ ; Now for ∆ YBB’ ;

Sin  = XB’ Sin  = YB’

BB BB’

 XB’ = BB’ Sin   YB’ = BB’ Sin 

 XB’ = d Sin   YB’ = d Sin 

  n  = XB’ + B’Y

 n  = d Sin  + d Sin 

n  = 2d Sin 
Where n = Order of diffraction
 = wavelength
d = Interplanar distance
 = Glancing angle

❖ Single Crystal X ray Diffraction

What is a crystal?

→ Atoms (molecules) pack together in a regular pattern to form a crystal.

→ The unit cell is the basic repeating unit that defines the crystal structure.
→ The unit cell contains the symmetry elements required to uniquely define the
crystal structure.
→ The unit cell might contain more than one molecule: For example, the quartz
unit cell contains 3 complete molecules of SiO2.
→ The crystal system describes the shape of the unit cell.
→ The lattice parameters describe the size of the unit cell.
→ The unit cell repeats in all dimensions to fill space and produce the
macroscopic grains or crystals of the material.
 Fig. Basic of Crystal structure

→ The crystal structure describes the atomic arrangement of a material.

→ The crystal structure determines the position and intensity of the diffraction peaks
in an X-ray scattering pattern.
→ Interatomic distances determine the positions of the diffraction peaks.
→ The atom types and positions determine the diffraction peak intensities.
→ Diffraction peak widths and shapes are mostly a function of instrument and
microstructural parameters.

→ To understand the atomic arrangement in the single crystal, single crystal X-ray
diffraction has been used to determine the crystal structure. When X-ray interacts
with the crystal, it will diffract into directions. The angles and intensities of these
diffracted X-rays will be measured to produce a 3D image of the density of
electron, which reflect the average positions of atoms in the crystal.
→ Further the length of chemical bonds between atoms, bond strength, their
disorder, and defects can be determined. The samples can be ranged from
minerals, alloys, vitamins, drugs to proteins.
→ When a crystal is mounted on a goniometer of the X-ray diffractometer, it will
gradually rotate. The electrons will be excited with X-rays, producing a
diffraction pattern (also known as reflection) according to the regularly spatial
arrangement. The 2D images will be collected due to the different rotations.
 Further these 2D images will be converted into a 3D model of the density of
electrons by the mathematical method of Fourier transforms.
→ The X-ray diffraction is known as a rapid, nondestructive analysis of
multicomponent mixtures without the need for extensive sample preparation.
This method has the ability to quickly analyze unknown materials and perform
materials characterization in the fields of metallurgy, mineralogy, forensic
science, archeology, condensed matter physics, and the biological and
pharmaceutical sciences. Using X-ray diffraction, series information can be
obtained, such as the phase identification, crystallinity, lattice parameters,
expansion tensors and bulk modulus, crystal structure refinement and
determination, size and strain broadening, and periodically arranged clusters.

➢ What is meant by Crystallography and why to study the structure of

crystalline solids?

→ Crystallography is the experimental science of determining the arrangement of

atoms in the crystalline solids. The properties of some materials are directly
related to their crystal structures.
→ For example, pure and undeformed magnesium and beryllium, having one crystal
structure, are much more brittle (i.e., fracture at lower degrees of deformation)
than pure and undeformed metals such as gold and silver that have yet another
crystal structure.
→ Furthermore, significant property differences exist between crystalline and non-
crystalline materials having the same composition. For example, non-crystalline
ceramics and polymers normally are optically transparent; the same materials in
crystalline (or semi-crystalline) form tend to be opaque or, at best, translucent.
→ An important reason to have an understanding of interatomic bonding in solids is
that, in some instances, the type of bond allows us to explain a material’s
properties. For example, consider carbon, which may exist as both graphite and
diamond. Whereas graphite is relatively soft and has a “greasy” feel to it,
diamond is the hardest known material. This dramatic disparity in properties is
directly attributable to a type of interatomic bonding found in graphite that does
not exist in diamond.
→ Thus, by studying the crystal structure and bonding nature of different materials,
we can investigate the reasons for the similar or dissimilar nature of the selected
materials in terms of different properties or different parameters.
→ The crystal structure and symmetry of a material play a vital role in determining
many of its physical properties, such as cleavage, electronic band structure, and
optical transparency, etc.
→ Based on the atomic arrangement in a substance, solids can be broadly classified
as either crystalline or non-crystalline.
→ In a crystalline solid, all the atoms are arranged in a periodic manner in all three
dimensions where as in a non-crystalline solid the atomic arrangement is random
or nonperiodic in nature.
→ A crystalline solid can either be a single crystalline or a polycrystalline. In the
case of single crystal, the entire solid consists of only one crystal and hence,
periodic arrangement of atoms continues throughout the entire material.
→ A polycrystalline material is an aggregate of many small crystals separated by
well-defined grain boundaries and hence periodic arrangement of atoms is limited
to small regions of the material called as grain boundaries.
 Fig. Two-dimensional representation of single crystal, polycrystalline
and non-crystalline material
→ The non-crystalline substances are also called as amorphous substances
materials. Single crystalline materials exhibit long range as well as short range
periodicities while long range periodicity is absent in case of poly-crystalline
materials and non-crystalline materials.
 ➢ X-ray Diffraction method

→ Diffraction can occur whenever the Bragg law n λ = 2d sinθ , is satisfied. This
equation puts very important condition on λ and θ for any given crystal.
→ With monochromatic radiation, an arbitrary setting of a single crystal in a beam
of x-ray will not in genera produce any diffracted beams.
→ Someway of satisfying the Bragg law must be devised , and this can be done by
continuously varying either λ or θ during the experiments.
→ The way in which these quantities are varied distinguish the three main
diffraction method.
 Method λ θ Applicable
for

Laue method Variable Fixed Single Crystal

Rotating Crystal Fixed Variable (in Single crystal
method Part)

Powder method Fixed Variable Crystalline or

polycrystalline
powder

❖ Von Laue method

→ The Laue method was the first diffraction method ever used. This method is used
to study the orientation of crystal and to verify crystal symmetry.
→ The Bragg angle θ is therefore fixed for every set of planes in the crystal, and
each set picks out and diffracts that particular wavelength which satisfies the
Bragg law for the particular values of d and involved.
→ Each diffracted beam thus has a different wavelength.
→ A small crystal (sample) is placed in the path of a narrow beam of X-rays from a
tungsten target at about 60 KV.
→ The x ray beam will pass through a pinhole collimator to get the sharp x-rays.
→ These sharp beams of x ray will go towards the sample to be examined.
→ These sharp x rays beam will pass through a crystal and gives

(1) Transmitted rays

(2) Diffracted rays (Transmission Laue method)
(3) Reflected rays (Back reflection Laue method)

→ There are two variations of the Laue method, depending on the relative positions
of source, crystal, and film (See the figure).

• Transmission Laue Method

→ The film is flat and placed perpendicular to the incident beam. The film in the
transmission Laue method (the original Laue method) is placed behind the
crystal so as to record the beams diffracted in the forward direction.
 Fig. Transmission Laue Method and Laue Pattern

→ This method is so called because the diffracted beams are partially transmitted
through the crystal.
→ Value of θ is calculated and relative spacing between the planes are estimated.

• Back-reflection Laue method

→ It is used for large and thick specimen, where the diffraction is difficult to
get.
→ In the back-reflection Laue method the film is placed between the crystal
and the x-ray source, the incident beam passing through a hole in the film, and
the beams diffracted in a backward direction are recorded.
 (a)Back Reflection Laue Method (b)Laue Pattern

→ In either method, the diffracted beams form an array of spots on the film as shown
in Fig.for a cubic crystal. This array of spots is commonly called a pattern, more
specifically, Laue pattern, but the term is not used in any strict sense and does
not imply any periodic arrangement of the spots.
→ Disadvantage of Back reflection method are
(1) A big crystal is required
(2) It is used to orient solid state experiments and to determine the
single crystal symmetry.
❖ Rotating crystal method

Fig. Rotating crystal method

→ X- ray is generated in the x-ray tube and x-ray beam is made monochromatic.
→ Monochromatic radiation will fall on the crystal fall on the crystal mounted
on a shaft which can be rotated at uniform angular rate.
→ Shaft will rotate the crystal at slow rate.
→ This cause the set of planes coming successively into their reflecting position.
→ Whenever it will follow the Bragg’s equation , in that condition reflection and
diffraction occurs.
→ Each plane will produce a spot inside the photographic film present in the
camera.
→ Photographic film will be fixed perpendicular to the incident x-rays beam
inside the cylindrical camera.
→ There are two methods available for photography.
 (a) Complete rotation method
→ Series of complete revolution takes place.
→ Each plane in the crystal diffracts x-ray four times during rotations.
→ These four beams are distributed in the form of rectangular pattern.
(b) Oscillation method
→ Crystal is oscillated through an angle of 150 to 200.
→ The photographic film is also moved accordingly. This method is used to
detect the size of unit cell in the crystal.

❖ Powder diffraction method

→ X-ray powder method is usually carried for polycrystalline materials.
→ The powder method of x-ray diffraction was devised independently in 1916
by Debye and Scherrer in Germany and in 1917 by Hull in the United States.
→ Basically, this method involves the diffraction of monochromatic x-rays by a
powder specimen. In this connection, "monochromatic" usually means the
strong K characteristic component of the general radiation.
→ x-ray tube operated above the K excitation potential of the target material.
"Powder" can mean either an actual, physical powder held together with a
suitable binder or any specimen in polycrystalline form.
→ The method is thus eminently suited for metallurgical work, since single
crystals are not always available to the metallurgist and such materials as
polycrystalline wire, sheet, rod, etc., may be examined nondestructively
without any special preparation.
→ There are three main powder methods in use, differentiated by the relative
position of the specimen and film:
→ (1) Debye-Scherrer method. The film is placed on the surface of a cylinder
and the specimen on the axis of the cylinder.
→ (2) Focusing method. The film, specimen, and x-ray source are all placed on
the surface of a cylinder.
→ (3) Pinhole method. The film is flat, perpendicular to the incident x-ray beam,
and located at any convenient distance from the specimen.
→ In all these methods, the diffracted beams lie on the surfaces of cones whose
axes lie along the incident beam or its extension; each cone of rays is
diffracted from a particular set of lattice planes.
→ In the Debye-Scherrer and focusing methods, only a narrow strip of film is
used and the recorded diffraction pattern consists of short lines formed by the
intersections of the cones of radiation with the film.
→ In the pinhole method, the whole cone intersects the film to form a circular
diffraction ring.
→ The powder photograph is obtained in the following way.
➢ Debye-Scherrer powder diffraction method
→ The given polycrystalline material is grinding to fine powder and this powder
can be taken either in a capillary tube made up of non-diffracting material or
is just struck on a hair with small quantity of binding material and fixed at the
center of cylindrical Debye-Scherrer camera as shown in Fig.
 Figure: Powder diffraction method

(a) Debye -Scherrer cylindrical camera

(b) Film mounted in camera
(c) Film on stretch-out

Fig. Debye-Scherrer powder diffraction setup and analysis

 Fig. Cross-section and pattern of powder diffraction method

→ A stripe of X-ray photographic film is arranged along the inner periphery of the
camera.
→ A beam of monochromatic X-rays is passed through the collimator to obtain a
narrow fine beam of X-rays.
→ This beam falls on the polycrystalline specimen and gets diffracted.
→ The specimen contains very large number of small crystallites oriented in random
directions.
→ So, all possible diffraction planes will be available for Bragg reflection to take
place.
→ Such reflections will take place from many sets of parallel planes lying at
different angles to the incident X-ray beam.
→ Also, each set of planes gives not only first-order reflections but also of higher
orders as well. Since all orientations are equally likely, the reflected rays will
form a cone whose axis lies along the direction of the incident beam and whose
 semi-vertical angle is equal to twice the glancing angle (θ), for that particular set
of planes.
→ For each set of planes and for each order, there will be such a cone of reflected
X-rays. There intersections with a photographic film sets with its plane normal
to the incident beam, form a series of concentric circular rings.
→ In this case, a part of the reflected cone is recorded on the film and it is a pair of
curves, the resulting pattern is shown in Fig.(c).
→ Diameter of these rings or corresponding curves is recorded on the film, and using
this the glancing angle and interplanar spacing of the crystalline substance can be
determined. Figure (b) shows the film mounted in the camera and the X-ray
powder pattern obtained.
→ The film on spread-out is shown in Fig (c). The distance between any two
corresponding curves on the film is indicated by the symbol S.
→ In case of cylindrical camera, the diffraction angle θ is proportional to S. Then,

Θ= S
4R
where R is representing the radius of the camera.
S is rhe distance between any two corresponding
curves on the film

If S1,S2,S3…..etc. are the distance between symmetrical lines on the stretched

film, then,

Θ1 = S1 , Θ2 = S2 , Θ3 = S3 …….
4R 4R 4R

Using these values of Θn in Bragg’s equation n  = 2 d sin Θn

Where n =1,2,3….= order of diffraction

d = Interplanar spacing
Θn = Angle of diffraction for nth order.
→ Each diffraction peak is actually a Debye diffraction cone produced by the tens
of thousands of randomly oriented crystallites in an ideal sample. A cone along
the sphere corresponds to a single Bragg angle 2theta
→ The linear diffraction pattern is formed as the detector scans along an arc that
intersects each Debye cone at a single point
→ Only a small fraction of scattered X-rays are observed by the detector.

→ In a Debye–Scherrer arrangement, after exposing a powder of a crystalline

material to monochromatic X-rays, the developed film strip will exhibit
diffraction patterns such as indicated in fig. 12. Each diffraction peak (dark line)
on the film strip corresponds to constructive interference at planes of a particular
interplanar spacing [d(hkl)]. The problem now consists of “indexing” the
individual lines – i.e., determining the Miller indices (hkl) for the diffraction
lines:
→ X-rays scatter from atoms in a material and therefore contain information about
the atomic arrangement
→ The three X-ray scattering patterns above were produced by three chemically
identical forms SiO2
→ Crystalline materials like quartz and Cristobalite produce X-ray diffraction
patterns –Quartz and Cristobalite have two different crystal structures –The Si
and O atoms are arranged differently, but both have long-range atomic order –
The difference in their crystal structure is reflected in their different diffraction
patterns.
→ Diffraction patterns are collected as absolute intensity vs 2θ, but are best reported
as relative intensity vs dhkl.
→ Example :

→ The peak position as 2θ depends on instrumental characteristics such as

wavelength.
▪ The peak position as dhkl is an intrinsic, instrument-independent,
material property.
▪ Bragg’s Law is used to convert observed 2θ positions to dhkl.
▪ The absolute intensity, i.e. the number of X rays observed in a given
peak, can vary due to instrumental and experimental parameters.
 ▪ The relative intensities of the diffraction peaks should be
instrument independent.
▪ To calculate relative intensity, divide the absolute intensity of every
peak by the absolute intensity of the most intense peak, and then
convert to a percentage.
▪ The most intense peak of a phase is therefore always called the
“100% peak”.
▪ Peak areas are much more reliable than peak heights as a measure
of intensity.
→ The wavelength of X rays are similar to the distance between atoms in a
crystal. Therefore, we use X-ray scattering to study atomic structure.
→ The scattering of X-rays from atoms produces a diffraction pattern, which
contains information about the atomic arrangement within the crystal.
→ The amorphous glass does not have long-range atomic order and therefore
produces only broad scattering features
→ Amorphous materials like glass do not have a periodic array with long-range
order, so they do not produce a diffraction pattern. Their X-ray scattering
pattern features broad, poorly defined amorphous ‘humps’.

❖ Analytical Information
▪ Phase Composition of a Sample
→ Quantitative Phase Analysis: determine the relative amounts of phases in a
mixture by referencing the relative peak intensities
▪ Unit cell lattice parameters and Bravais lattice symmetry
→ Index peak positions
→ Lattice parameters can vary as a function of, and therefore give you information
about, alloying, doping, solid solutions, strains, etc.
▪ Residual Strain (macrostrain)
▪ Crystal Structure
→ By Rietveld refinement of the entire diffraction pattern
▪ Epitaxy/Texture/Orientation
▪ Crystallite Size and Microstrain
→ Indicated by peak broadening
→ Other defects (stacking faults, etc.) can be measured by analysis of peak shapes
and peak width

➢ Phase Identification

→ The diffraction pattern for every phase is as unique as your fingerprint

→ Phases with the same chemical composition can have drastically different
diffraction patterns.
→ Use the position and relative intensity of a series of peaks to match experimental
data to the reference patterns in the database
The diffraction pattern of a mixture is a simple sum of the diffraction patterns
of each individual phase.

➢ Unit Cell Lattice Parameter Refinement

→ By accurately measuring peak positions over a long range of 2θ, you can
determine the unit cell lattice parameters of the phases in your sample
→ Alloying, substitutional doping, temperature and pressure, etc can create changes
in lattice parameters that you may want to quantify
→ Use many peaks over a long range of 2theta so that you can identify and correct
for systematic errors such as specimen displacement and zero shift
→ Measure peak positions with a peak search algorithm or profile fitting
→ Profile fitting is more accurate but more time consuming then numerically refine
the lattice parameters

➢ Crystallite Size and Microstrain

→ Crystallites smaller than ~120nm create broadening of diffraction peaks
• This peak broadening can be used to quantify the average crystallite size of
nanoparticles using the Scherrer equation
• Must know the contribution of peak width from the instrument by using a
calibration curve
→ Microstrain may also create peak broadening
• Analyzing the peak widths over a long range of 2theta using a Williamson-
Hull plot can let you separate microstrain and crystallite size

➢ Preferred Orientation (texture)

→ Preferred orientation of crystallites can create a systematic variation in diffraction
peak intensities
• Can qualitatively analyze using a 1D diffraction pattern
• A pole figure maps the intensity of a single peak as a function of tilt and
rotation of the sample
• This can be used to quantify the texture

➢ Analytical Information: Qualitative and Quantitative Analysis

→ A given substance always produces a characteristic diffraction pattern, whether

that substance is present in the pure state or as one constituent of a mixture of
substances.
→ This fact is the basis for the diffraction method of chemical analysis.
→ Qualitative analysis for a particular substance is accomplished by identification
of the pattern of that substance.
→ Quantitative analysis is also possible, because the intensities of the diffraction
lines due to one constituent of a mixture depend on the proportion of that
constituent in the specimen.
❖ Qualitative analysis
→ The particular advantage of diffraction analysis is that it discloses the presence
of a substance as that substance actually exists in the sample, and not in terms of
its constituent chemical elements.
→ For example, if a sample contains the compound AxBy, the diffraction method
will disclose the presence of AxBy as such, whereas ordinary chemical analysis
would show only the presence of elements A and B. Furthermore, if the sample
contained both AxBy and AxB2y, both of these compounds would be disclosed by
 the diffraction method, but chemical analysis would again indicate only the
presence of A and B.*
→ The powder pattern of a substance is characteristic of that substance and forms a
sort of fingerprint by which the substance may be identified.
→ If we had on hand a collection of diffraction patterns for a great many substances,
we could identify an unknown by preparing its diffraction pattern and then
locating in our file of known patterns one which matched the pattern of the
unknown exactly.
→ The collection of known patterns has to be fairly large, if it is to be at all useful,
and then pattern-by-pattern comparison in order to find a matching one becomes
out of the question.
→ What is needed is a system of classifying the known patterns so that the one which
matches the unknown can be located quickly. Such a system was devised by
Hanawalt in 1936. Any one powder pattern is characterized by a set of line
positions 26 and a set of relative line intensities I.
→ But the angular positions of the lines depend on the wavelength used, and a more
fundamental quantity is the spacing d of the lattice planes forming each line.
→ Hanawalt therefore decided to describe each pattern by listing the d and I values
of its diffraction lines, and to arrange the known patterns in decreasing values of
d for the strongest line in the pattern.
→ This arrangement made possible a search procedure which would quickly locate
the desired pattern. In addition, the problem of solving the pattern was avoided
and the method could be used even when the crystal structure of the substance
concerned was unknown.
→ 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 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
standards 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 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.

→ After the experimental values of d and I/l1are tabulated, the unknown can be
identified by the following procedure :

1) Locate the proper d1 group in the numerical index.

2) Read down the second column of d values to find the closest match to d2
. (In comparing experimental and tabulated d values, always allow for
the possibility that either set of values may be in error by 0.01A.)
3) After the closest match has been found for d1, d2 , and d3 , compare their
relative intensities with the tabulated values.
4) When good agreement has been found for the three strongest lines listed
in the index, locate the proper data card in the file, and compare the d
and I/I1 values of all the observed lines with those tabulated. When full
agreement is obtained, identification is complete.

• TiC is the unknown specimen

[Hanawalt Method]
→ Practical difficulties. In theory, the Hanawalt method should lead to the
positive identification of any substance whose diffraction pattern is included
in the card file.
→ In practice, various difficulties arise, and these are usually due either to
errors in the diffraction pattern of the unknown or to errors in the card
file.
→ Errors of the first kind, those affecting the observed positions and intensities
of the diffraction lines, have been discussed in various parts of this book and
need not be reexamined here.
→ There is, however, one point that deserves some emphasis and that concerns
the diffractometer. It must be remembered that the absorption factor for
this instrument is independent of the angle 2, whereas, in a Debye-
Scherrer camera, absorption decreases line intensity more at small than
at large angles; the result is that the low angle lines of most substances
appear stronger, relative to medium- or high-angle lines, on a
diffractometer chart than on a Debye-Scherrer photograph.
→ This fact should be kept in mind whenever a diffractometer pattern is
compared with one of the standard patterns in the ASTM file, because
practically all of the latter were obtained with a Debye-Scherrer camera.
→ On the other hand, it should not be concluded that successful use of the
Hanawalt method requires relative intensity measurements of extremely
high accuracy. It is enough, in most cases, to be able to list the lines in the
correct order of decreasing intensity.
→ Errors in the card file itself are generally more serious, since they may go
undetected by the investigator and lead to mistaken identifications.
→ Even a casual examination of the ASTM alphabetical index will disclose
numerous examples of substances represented in the file by two or more cards,
often with major differences in the three strongest lines listed.
→ Work is now in progress at the National Bureau of Standards to resolve such
ambiguities, correct other kinds of errors, and obtain new standard patterns.
→ The results of this work, which is all done with the diffractometer, are
published from time to time in NBS Circular 539, "Standard X-Ray
Diffraction Powder Patterns, and incorporated in card form in the most
recently issued sections of the ASTM file.
→ Whenever any doubt exists in the investigator's mind as to the validity of a
particular identification, he should prepare his own standard pattern. Thus, if
the unknown has been tentatively identified as substance X, the pattern of pure
X should be prepared under exactly the same experimental conditions used
for the pattern of the unknown.
→ Comparison of the two patterns will furnish positive proof, or disproof, of
identity. The Hanawalt method fails completely, of course, when the unknown
is a substance not listed in the card file, or when the unknown is a mixture and
the component to be identified is not present in sufficient quantity to yield a
good diffraction pattern.

❖ Quantitative Analysis

→ 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:
▪ Chemical analysis by parameter measurement.

Fig. Parameter v/s Composition Curve

→ The lattice parameter of a binary solid solution of B in A depends only on the
percentage of B in the alloy, as long as the solution is unsaturated.
→ This fact can be made the basis for chemical analysis by parameter
measurement. All that is needed is a parameter vs. composition curve, such as
curve be of Fig. , which can be established by measuring the lattice parameter
of a series of previously analyzed alloys.
→ This method has been used in diffusion studies to measure the change in
concentration of a solution with distance from the original interface.
→ Its accuracy depends entirely on the slope of the parameter-composition
curve.
→ In alpha brasses, which can contain from to about 40 percent zinc in copper,
an accuracy of 1 percent zinc can be achieved without difficulty.
→ This method is applicable only to binary alloys. In ternary solid solutions,
for example, the percentages of two components can be independently varied.
The result is that two ternary solutions of quite different compositions can
have the same lattice parameter.

▪ 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

→ To calculate unit cell lattice parameters from the diffraction peak

positions
 • Convert the observed peak positions, °2theta, into dhkl values
using Bragg’s Law:

• Determine the Miller indices (hkl) of the diffraction peaks from the
published reference pattern
– If you do not have access to a reference pattern that identifies (hkl) then you
will need to index the pattern to determine the (hkl)
• Use the d*2 equation to calculate the lattice parameters
– Most analysis programs contain an unit cell refinement algorithm for
numerically solving the lattice parameters
– These programs can also calculate and correct for peak position error due to
specimen displacement.

→ 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
 ▪ The Absorption Method
→ Requires the measurement of intensity from a diffraction peak in the
mixture and from a pure standard of material.

Diffraction Equation for The Absorption Diffraction Method

• 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

▪ 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.
→ Useful when only one phase is to be quantified.

Equation for The Method of Standard Additions

 • 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

▪ 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
❖ Applications in Pharmaceutical Industry and in Forensic
Science

❖ Pharmaceutical Industries

→ X-ray diffraction (XRD) can be used to unambiguously characterize the

composition of pharmaceuticals. An XRD-pattern is a direct result of the
crystal structures, which are present in the pharmaceutical under study. As
such, the parameters typically associated with crystal structure can be simply
accessed. For example, once an active drug has been isolated, an indexed X-
ray powder diffraction pattern is required to analyse the crystal structure,
secure a patent and protect the company’s investment.
→ For multi-component formulations, the actual percentages of the active
ingredients in the final dosage form can be accurately analysed in situ, along
with the percentage of any amorphous packing ingredients used.
→ XRD is the key technique for solid-state drug analysis, benefiting all stages
of drug development, testing and production.

❖ Forensic science

→ XRD is used mainly in contact trace analysis. Examples of contact traces are
paint flakes, hair, glass fragments, stains of any description and loose
powdered materials. Identification and comparison of trace quantities of
material can help in the conviction or exoneration of a person suspected of
involvement in a crime.
→ X-ray diffraction or XRD is one such technique which is nondestructive and
the sample requires minimum sample preparation prior to analysis. The only
requirement is that the sample should be homogeneous in nature so as to
provide uniform analysis results even if a small portion is analysed from a
bulk quantity. Samples commonly received for forensic testing commonly
include:

• Building materials such as cement, concrete, steel rods, bricks,etc

• Drugs of abuse and other banned substances

• Residues from site of arson such as kerosene oil, gasoline or cotton lints

• Explosive residues and splinters from sites

• Accident scene samples such as blood spots, paint chips,etc

• Assault samples such as torn garments, broken glass,cosmetic marks etc.

• Theft and robbery site samples which include gunshot residues, tools, murder
weapons, forged documents, blood residues,etc.

→ The samples can be analysed by XRD if it exhibits a degree of crystallinity

even if the remaining content is amorphous. The technique helps establish the
presence or absence of a particular material through comparison against a
reference XRD data base. The present article discusses some of the XRD
applications of common samples picked up from scene of crime.