Chapter 1 Introduction to Nanomaterial Science 1.1 Introduction Matter can be placed into broad categories according to size. Macroscopic matter is visible with the naked eye. Atoms and (most) molecules are microscopic with dimensions < 1nm. Mesoscopic particles, such as bacteria and cells that have dimensions on the order of micron(s), can be observed with optical microscopes. Falling into the gap between the microscopic and the mesoscopic is another class of matter, the nanoscopic particles. The size of nanoparticles is compared to that of other “small” particles in Figure, where the bacterium is huge in comparison . - 1 -
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Chapter 1
Introduction to Nanomaterial Science
1.1 Introduction
Matter can be placed into broad categories according to size. Macroscopic matter is
visible with the naked eye. Atoms and (most) molecules are microscopic with dimensions < 1nm.
Mesoscopic particles, such as bacteria and cells that have dimensions on the order of micron(s), can
be observed with optical microscopes. Falling into the gap between the microscopic and the
mesoscopic is another class of matter, the nanoscopic particles. The size of nanoparticles is
compared to that of other “small” particles in Figure, where the bacterium is huge in comparison .
Size comparisons of “small” particles
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In nanotechnology, a particle is defined as a small object that behaves as a whole unit
in terms of its transport and properties. Particles are further classified according to size : in terms of
diameter, coarse particles cover a range between 10,000 and 2,500 nanometers.
Fine particles are sized between 2,500 and 100 nanometers. Ultrafine particles, or
nanoparticles are sized between 100 and 1 nanometers. The reason for this double name of the same
object is that, during the 1970-80's, when the first thorough fundamental studies were running with
"nanoparticles" in the USA (by Granqvist and Buhrman) and Japan, they were called "ultrafine
particles" (UFP). However, during the 1990s before the National Nanotechnology Initiative was
launched in the USA, the new name, "nanoparticle" had become fashionable (see, for example the
same senior author's paper 20 years later addressing the same issue, lognormal distribution of sizes).
Nanoparticles may or may not exhibit size-related properties that differ significantly from those
observed in fine particles or bulk materials. Although the size of most molecules would fit into the
above outline, individual molecules are usually not referred to as nanoparticles.
Nanoclusters have at least one dimension between 1 and 10 nanometers and a narrow
size distribution. Nanopowders are agglomerates of ultra fine particles, nanoparticles, or
nanoclusters. Nanometer-sized single crystals, or single-domain ultra fine particles, are often
referred to as nanocrystals. Nanoparticle research is currently an area of intense scientific interest
due to a wide variety of potential applications in biomedical, optical and electronic fields. The
chemical processing and synthesis of high performance technological components for the private,
industrial, and military sectors requires the use of high purity ceramics, polymers, glass-ceramics
and material composites. In condensed bodies formed from fine powders, the irregular particle sizes
and shapes in a typical powder often lead to non-uniform packing morphologies that result in
packing density variations in the powder compact.
Nanoparticles are of great scientific interest as they are effectively a bridge between
bulk materials and atomic or molecular structures. A bulk material should have constant physical
properties regardless of its size, but at the nano-scale size-dependent properties are often observed.
Thus, the properties of materials change as their size approaches the nanoscale and as the percentage
of atoms at the surface of a material becomes significant. For bulk materials larger than one
micrometer (or micron), the percentage of atoms at the surface is insignificant in relation to the
number of atoms in the bulk of the material. The interesting and sometimes unexpected properties of
nanoparticles are therefore largely due to the large surface area of the material, which dominates the
contributions made by the small bulk of the material.
Nanoparticles often possess unexpected optical properties as they are small enough to
confine their electrons and produce quantum effects. For example gold nanoparticles appear deep red
to black in solution. Nanoparticles of usually yellow gold and gray silicon are red in color. Gold
nanoparticles melt at much lower temperatures (~300 °C for 2.5 nm size) than the gold slabs (1064
According to Siegel, Nanostructured materials are classified as Zero dimensional,
one dimensional, two dimensional, three dimensional nanostructures.
Fig. 3. Classification of Nanomaterials (a) 0D spheres and clusters, (b) 1D nanofibers,wires, and rods, (c) 2D films, plates, and networks, (d) 3D nanomaterials.
Nanomaterials are materials which are characterized by an ultra fine grain size (< 50
nm) or by a dimensionality limited to 50 nm. Nanomaterials can be created with various
modulation dimensionalities as defined by Richard W. Siegel: zero (atomic clusters, filaments and
cluster assemblies), one (multilayers), two (ultrafine-grained overlayers or buried layers), and three
(nanophase materials consisting of equiaxed nanometer sized
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grains) as shown in the above figure 3.
1.4 Properties Improvement
Nanotechnology offers an extremely wide range of applications in the fields of health,
medicines, electronics, optical communication and biological systems. Application of nanostructures
and nanomaterials are based on the following properties
Mechanical properties
The large amount of grain boundaries in bulk materials made of nanoparticles allows
extended grain boundary sliding leading to high plasticity.
Catalytic Properties
Due to their large surface, nanoparticles made of transition element oxides exhibit
interesting catalytic properties. In special cases, catalysis may be enhanced and more specific by
decorating these particles with gold or platinum clusters.
Magnetic Properties
In magnetic nanoparticles, the energy of magnetic anisotropy may be that small that
the vector of magnetization fluctuates thermally; this is called superparamagnetism. Such a material
is free of remanence, and coercitivity. Touching super paramagnetic particles are loosing this special
property by interaction, except the particles are kept at distance. Combining particles with high
energy of anisotropy with super paramagnetic ones leads to a new class of permanent magnetic
materials.
Optical Properties
Distributions of non-agglomerated nanoparticles in a polymer are used to tune the
index of refraction. Additionally, such a process may produce materials with non-linear optical
properties. Gold or CdSe nanoparticles in glass lead to red or orange coloration. Semi-conducting
nanoparticles and some oxide-polymer nanocomposites exhibit fluorescence showing blue shift with
decreasing particle size.
For many new applications, further new properties and materials are incorporated.
cells,sensors,steam electrolyzers,hydrogen seperation from hydrogen containing gas mixtures and
membrane reactors for the transport of protons.
It is well known that most properties of ceramic powders depend on their methods of
production. Solid state reactions are the most widely utilized process and are good for mass
producing cost-efficient powders, because the raw materials are simply calcinated to obtain the
products. However, it is difficult to obtain a homogeneous composition and dense, fine grained
sintered bodies because of poor dispersion by physical mixing. Therefore, wet chemical methods,
such as co-precipitation, sol-gel ,the pechini, glycine-nitrate and the citrate acid methods, were
studied. These methods provide a mixing of the element at an atomic scale which is known to
accelerate the reaction of the phase formation. The Pechini and glycene nitrate methods have been
used to obtain a variety of mixed metal oxides with precise stoichiometry.
This work deals with the preparation of nano powders of BaCeO3 perovskite by a new
combustion process , Self propogating auto igniting combustion technique. This method is more is
more advanced than any other methods now available since it does not require a prolonged heating
in air.
2.4 Review of Present Work
The strontium cerate nanopowder was characterized by various methods as shown
below.
First, x-ray diffraction (XRD) patterns of the strontium cerate nanopowder were
recorded using a Seifert XRD 3003 PTS diffractometer system using Cu Kα radiation (λ=0.15418
nm) in the range of 10° to 80° (2θ) to examine the crystallization and structural development of the
strontium cerate nanopowder. The XRD patterns of strontium cerate nanopowders calcined at 800°
C., 900° C., and 1000° C.. Strontium cerate nanopowder calcined at 800° C. exhibits a small
impurity phase due to the presence of strontium carbonate (SrCO3) in the nanopowder. Similarly,
strontium cerate nanopowder calcined at 900° C. exhibits a small impurity phase due to the presence
of cerium oxide (CeO2) in the nanopowder. However, the XRD pattern of strontium cerate
nanopowder calcined at 1000° C is consistent with the spectrum of pure strontium cerate, and no
peak attributable to possible impurities is observed. The sharp diffraction peaks show that the
synthesized strontium cerate nanoparticles have high crystallinity. The crystallite size of the
synthesized strontium cerate nanoparticles was calculated based on the major diffraction peaks using
the Debye-Scherrer .
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Second, chemical purity and stoichiometry of the nanopowders were tested by energy
dispersive X-ray spectroscopy (EDX). As illustrated by the EDX spectrum the atomic ratio of
strontium to cerium in the synthesized strontium cerate nanopowder was determined to be almost
equal to that of pure strontium cerate.
Third, the specific surface area of the synthesized strontium cerate nanoparticles was
determined based on Brunauer-Emmet-Teller (BET) analysis. The BET measurements were
performed on a Micromeritics Tristar 3000 analyzer. The difference in size for the two samples
calcined at different temperatures can be due to the higher purity of the samples calcined at 1000°
C., as explained above. The inconsistency between the crystalline size determined by the XRD and
BET methods can be due to conglomeration in the synthesized strontium cerate nanopowders.
Fourth, the morphology of the synthesized strontium cerate nanoparticles calcined at
1000° C at the higher magnification of 15,000 times. The morphology of the nanoparticles appears
to consist of large, highly dense crystals, whereas at the lower magnification of 7,500 times the
crystal morphology appears to be granular and porous with small particles.
Fifth, the morphology of the synthesized strontium cerate nanoparticles calcined at
1000° C . The strontium cerate nanoparticles appear to be agglomerated.
It is to be understood the implementations are not limited to the particular processes, devices, and/or
apparatus described which may, of course, vary.
Properties of Tb doped SrCeO3
The electrical conduction behavior of Tb doped SrCeO3 was investigated in different
gases at high temperatures. In air, oxygen or nitrogen SCTb shows small electronic-hole conduction
below 800°C and oxygen ionic conduction over 800°C with activation energy about 30 kJ/mol and
164–181 kJ/mol respectively. SCTb becomes a protonic conductor in hydrogen or methane in 500–
900°C, with the proton conductivity in the range of 10−3–10−2 S/cm, two to three orders of
magnitude higher than electronic or oxygen ionic conductivity of SCTb in air or oxygen. The
activation energy for protonic conduction in SCTb is 49 kJ/mol in methane and 54 kJ/mol in
hydrogen. The electrical conductivity of SCTb in water vapor-saturated nitrogen, air or oxygen is
higher than in corresponding gas without water vapor. Presence of water vapor does not affect the
electrical conduction of SCTb in hydrogen or methane. Gas permeation measurements show that
SCTb membrane is impermeable to hydrogen when the membrane is exposed to hydrogen or
methane upstream and nitrogen or oxygen downstream. These results confirm that SCTb is a pure
protonic conductor with very low electronic and oxygen ionic conductivity.
2.5 Synthesis Method
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2.5.1 Co-Precipitation from solution
Co-precipitation from solution is one of the methods used for the production of mixed oxides. This is a rapid technique has been developed for quantitatively concentrating several trace materials from aqueous solution. The metals are co-precipitated as dithiocarbomate chelats by adding an excess of another dissolved metal. The precipitated product is separated from the liquid by filtration, dried and thermally decomposed to the desired compound. Radio tracer experiments show that various materials can be co-precipitated under proper experimental conditions.
There are three main mechanisms of co-precipitation. Inclusion, occlusion and adsorption. An inclusion occurs when impurity occupies a lattice site in the crystal structure of the carrier resulting in a crystallographic defect; this can happen when the ionic radius and charge of the impurity are similar to those of the carrier.
An occlusion occurs when the adsorbed impurity get physically trapped inside a crystal as it grows. An adsorbate is an impurity that is weakly bound adsorbed to the surface of the precipitate.co-precipitation method is applicable in radio chemistry, acid mine drainage, radio nuclide migration, metal contaminant transport at industrial and defense sites, metal concentrations in aquatic systems etc.
2.5.2 Sol-gel method
Sol-gel method is a chemical technique widely used in the fields of material science and ceramic engineering. Such method is used primarily for the fabrication of the materials starting from a colloidal solution(sol) that acts as the precursor for an integrator network(or gel) or either discreet particles or network polymers. Sol-gel method has several multistep processes such as hydrolysis’ polymerization, drying and densification. A sudden increase in viscosity in the common feature in sol-gel processing indicating the on set of gel formation.
Sol-gel process is a multi step operation that involves calcinations for prolonged operation that involves calcination for prolonged duration at high temperatures for obtaining powders and processing of high volume of liquids with relatively low yield. The applications of sol-gel process are numerous. The important applications of sol-gel derived products are protective coatings, thin films and powders, nano scale powders, in opto mechanical area, for the production of hot mirrors and cold mirrors, lenses and dream splitters etc.
2.5.3 Hydrothermal synthesis
Hydrothermal synthesis is a process that utilizes single or heterogenous phase reactions, in aqueous media in evaluated temperature and pressure to crystalline ceramic materials directly from the solution. Upper limits of hydrothermal synthesis extended over 1000C and 500MPa pressure. Intensive research has led to a better understanding of hydrothermal chemistry which has significantly reduced the reaction time, temperature and pressure for hydrothermal crystallization of
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materials. The hydrothermal synthesis of ceramic powders processes two major advantages, the elimination or minimization of any high temperature calcination stage and use of relatively inexpensive raw materials.
One of the main applications of hydrothermal synthesis in the preparation of lithium polymer battery.
2.5.4 Emulsion synthesis
The emulsion synthesis is generally applicable for many ceramic powders or combinations of ceramic powders for which water soluble precursors are available. The aqueous solutions of the ceramic precursors are emulsified with an organic surfactant to provide a dispersion of aqueous droplets of nearly uniform size in the organic fluid. The emulsion process uses water-soluble precursors dispersed in the organic phase to produce spherical, uniform fine powders with minimized agglomeration at a relatively moderate coast.
The important applications of the emulsion synthesis are, small scale biodiesel production from the vegetable oil, small scale production of solid, liquid and gas fuel from wood, cogeneration of energy production using natural and fuel gas from biomass, fuel ethanol from sugar cane etc.
2.5.5 Combustion synthesis
Combustion synthesis is also known as self propagating high temperature synthesis, is a versatile material methods for the synthesis of a verity of solids. A highly exothermic reaction between the reactance producing a flame due to spontaneous combustion plays a prominent role in this method. From this method we get the desired product or its precursor is finally divided form. There are some essential conditions for combustion to take place. They are,
Initial mixture of reactance is highly dispersed and contains high chemical energy.
The powdered mixtures of the reactance (0.1 to 100nm particle size) are generally placed in an appropriate gas medium which favors an exothermic reaction.
The combustion temperature is anywhere between 1500 depending on the reaction.
Reaction times are very short since the desired product results soon after the combustion.
The application of combustion synthesis is important in the synthesis of americium based ceramics and combustion porous materials for bone replacement.
2.5.6 Solid state ceramic route method
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The ceramic is usually prepared by the conventional solid state ceramic route method
by ball milling the oxides of the metals, drying and calcininig at high temperature. The calcined
powder is again ball milled and pressed into suitable shapes and then sintered depending on the
composition of metals. The importance of sintering is, it is difficult to fully densify the powder
without the sintering. Note that the powder is prepared by the mixed oxide route.
Even sintering at high temperatures, it is unable to give good densification. So, it is
necessary to dope with any compound in its oxide form. During the doping process, the binding
agent like poly vinyl agent is added to the compound. By the doping process, the sintering
temperature can be reduced and necessary densification can be attained. The sintering after the
doping will give the density above the 90% of the theoretical density.
2.6. Characterization Techniques
The nanoparticles can be characterized by various techniques, which provide
important information for the understanding of their optical, morphological and structural features.
The most extensively used techniques can be categorized into the following:
(a) Ultraviolet-Visible (UV-Vis) spectroscopy
(b) FTIR spectroscopy
(c) X-ray diffraction (XRD)
2.6.1. UV-visible Spectroscopy
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Ultraviolet-visible (UV-vis) spectroscopy is widely utilized to quantitatively
characterize organic and inorganic nanosized molecules. It deals with the study of electronic
transitions between orbitals or bands of atoms, ions or molecules in gaseous, liquid and solid state.
A sample is irradiated with electromagnetic waves in the ultraviolet and visible ranges and the
absorbed light is analyzed through the resulting spectrum. It can be employed to identify the
constituents of a substance, determine their concentrations, and to identify functional groups in
molecules. Size dependent properties can be observed in a UV-vis spectrum, particularly in the nano
and atomic scales. These include peak broadening and shifts in the absorption wavelength. Many
electronic properties, such as the band gap of nanosized material, can also be determined by this
technique. For semiconductor nanocrystals, the absorption spectrum is broadened owing to quantum
confinement effects, and as their size reduces, there is no longer a distinct peak, rather there is a
band. Further more, semiconductor nanoparticles absorption peaks shifts towards smaller
wavelengths (higher energies) as their crystal size decreases.
The schematic representation of a typical UV-vis spectrometer is shown in the figure
2.1. The functioning of this instrument is relatively straightforward. A beam of light from a visible
and/or UV light source is separated into its component wavelengths by a prism or diffraction grating.
Each monochromatic (single wavelength) beam in turn is split into two equal intensity beams by a
half-mirrored device. One beam, the sample beam, passes through a small transparent container
(cuvette) containing a solution of the compound being studied in a transparent solvent. The other
beam, the reference beam, passes through an identical cuvette containing only the solvent. The
intensities of these light beams are then measured by electronic detectors and compared.
The intensity of the reference beam, which should have suffered little or no light absorption, is
defined as I0. The intensity of the sample beam is defined as I. Over a short period of time, the
spectrometer automatically scans all the component wavelengths in the manner described. The
ultraviolet (UV) region scanned is normally from 200 to 400 nm, and the visible portion is from 400
to 800 nm.
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Fig 2.1 Schematic representation of a dual beam UV-vis spectrophotometer
If the sample compound does not absorb light of a given wavelength, I = I0.
However, if the sample compound absorbs light then I is less than I0, and this difference may be
plotted on a graph versus wavelength. Absorption may be presented as transmittance (T = I/I0) or
absorbance (A= log I0/I). If no absorption has occurred, T = 1.0 and A= 0. The wavelength of
maximum absorbance is a characteristic value, designated as λmax. Different compounds may have
different absorption maxima and absorbance. Intensely absorbing compounds must be examined in
dilute solution, so that significant light energy is received by the detector, and this requires the use of
completely transparent (non-absorbing) solvents. Since the absorbance of a sample will be
proportional to its molar concentration in the sample cuvette, a corrected absorption value known as
the molar absorptivity is used when comparing the spectra of different compounds. This is defined
as: Molar Absorptivity, ε = A / l c (where A is the absorbance, c is the sample concentration in
moles/liter and l is the length of light path through the cuvette in cm) .
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2.6.2 Fourier Transform Infrared Spectroscopy.
FT-IR stands for Fourier Transform InfraRed, the preferred method of infrared
spectroscopy. In infrared spectroscopy, IR radiation is passed through a sample. Some of the infrared
radiation is absorbed by the sample and some of it is passed through (transmitted). The resulting
spectrum represents the molecular absorption and transmission, creating a molecular fingerprint of
the sample. Like a fingerprint no two unique molecular structures produce the same infrared
spectrum. This makes infrared spectroscopy useful for several types of analysis.
Infrared spectroscopy has been a workhorse technique for materials analysis in the
laboratory for over seventy years. An infrared spectrum represents a fingerprint of a sample with
absorption peaks which correspond to the frequencies of vibrations between the bonds of the atoms
making up the material. Because each different material is a unique combination of atoms, no two
compounds produce the exact same infrared spectrum. Therefore, infrared spectroscopy can result in
a positive identification (qualitative analysis) of every different kind of material. In addition, the size
of the peaks in the spectrum is a direct indication of the amount of material present. With modern
software algorithms, infrared is an excellent tool for quantitative analysis.
Fourier transform infrared spectroscopy is preferred over dispersive or filter methods
of infrared spectral analysis for several reasons:
• It is a non-destructive technique
• It provides a precise measurement method which requires no external calibration
• It can increase speed, collecting a scan every second
• It can increase sensitivity – one second scans can be co-added together to ratio out random noise
• It has greater optical throughput
• It is mechanically simple with only one moving part
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Fourier Transform Infrared (FT-IR) spectrometry was developed in order to
overcome the limitations encountered with dispersive instruments. The main difficulty was the slow
scanning process. A method for measuring all of the infrared frequencies simultaneously, rather than
individually, was needed. A solution was developed which employed a very simple optical device
called an interferometer. The interferometer produces a unique type of signal which has all of the
infrared frequencies “encoded” into it. The signal can be measured very quickly, usually on the order
of one second or so. Thus, the time element per sample is reduced to a matter of a few seconds rather
than several minutes.
Most interferometers employ a beamsplitter which takes the incoming infrared beam
and divides it into two optical beams. One beam reflects off of a flat mirror which is fixed in place.
The other beam reflects off of a flat mirror which is on a mechanism which allows this mirror to
move a very short distance (typically a few millimeters) away from the beamsplitter. The two beams
reflect off of their respective mirrors and are recombined when they meet back at the beamsplitter.
Because the path that one beam travels is a fixed length and the other is constantly changing as its
mirror moves, the signal which exits the interferometer is the result of these two beams “interfering”
with each other. The resulting signal is called an interferogram which has the unique property that
every data point (a function of the moving mirror position) which makes up the signal has
information about every infrared frequency which comes from the source. This means that as the
interferogram is measured, all frequencies are being measured simultaneously. Thus, the use of the
interferometer results in extremely fast measurements. Because the analyst requires a frequency
spectrum (a plot of the intensity at each individual frequency) in order to make identification, the
measured interferogram signal can not be interpreted directly. A means of “decoding” the individual
frequencies is required. This can be accomplished via a well-known mathematical technique called
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the Fourier transformation. This transformation is performed by the computer which then presents
the user with the desired spectral information for analysis.
Some of the major advantages of FT-IR over the dispersive technique include:
• Speed: Because all of the frequencies are measured simultaneously, most measurements by FT-IR
are made in a matter of seconds rather than several minutes. This is sometimes referred to as the
Felgett Advantage.
• Sensitivity : Sensitivity is dramatically improved with FT-IR for many reasons. The detectors
employed are much more sensitive, the optical throughput is much higher (referred to as the
Jacquinot Advantage) which results in much lower noise levels, and the fast scans enable the
coaddition of several scans in order to reduce the random measurement noise to any desired level
(referred to as signal averaging).
• Mechanical Simplicity: The moving mirror in the interferometer is the only continuously moving
part in the instrument. Thus, there is very little possibility of mechanical breakdown.
• Internally Calibrated: These instruments employ a HeNe laser as an internal wavelength
calibration standard (referred to as the Connes Advantage). These instruments are self-calibrating
and never need to be calibrated by the user.
These advantages, along with several others, make measurements made by FT-IR
extremely accurate and reproducible. Thus, it a very reliable technique for positive identification of
virtually any sample. The sensitivity benefits enable identification of even the smallest of
contaminants. This makes FT-IR an invaluable tool for quality control or quality assurance
applications whether it be batch-to-batch comparisons to quality standards or analysis of an
unknown contaminant. In addition, the sensitivity and accuracy of FT-IR detectors, along with a
wide variety of software algorithms, have dramatically increased the practical use of infrared for
quantitative analysis. Quantitative methods can be easily developed and calibrated and can be
incorporated into simple procedures for routine analysis. Thus, the Fourier Transform Infrared (FT-
IR) technique has brought significant practical advantages to infrared spectroscopy. It has made
possible the development of many new sampling techniques which were designed to tackle
challenging problems which were impossible by older technology. It has made the use of infrared
analysis virtually limitless.
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2.6.4 X-ray diffraction Studies
XRD involves monitoring the diffraction of X-rays after they interact with the
sample. It is a crystallographic technique used for identifying and quantifying various crystalline
phases present in solid materials and powders. In XRD the crystal structure as well as the size of the
grains and nanoparticles can be determined. When X-rays are directed at a regular crystalline
sample, a proportion of them are diffracted to produce a pattern. From such a pattern the crystal
phases can be identified by comparison to those of internationally recognized databases that contain
reference patterns. From the diffraction patterns, the uniqueness of nanocrystal structure, phase
purity, degree of crystallinity and unit cell parameters of the nanocrystalline materials can be
determined. X-ray diffraction technique is nondestructive and does not require elaborate sample
preparation, which partly explains the wide use of XRD methods in material characterization. In
crystallography, the solid to be characterized by XRD has a space lattice with an ordered three
dimensional distribution of atoms. These atoms form a series of parallel planes separated by a
distance d, which varies according to the nature of the material. For any crystal, planes have their
own specific d-spacing. When a monochromatic X-ray beam with wavelength λ is irradiated onto a
crystalline material with spacing d, at an angle θ, diffraction occurs only when the distance travelled
by the rays reflected from successive planes differs by an integer number n of wavelengths to
produce constructive interference. Such constructive interference patterns only occur when incident
angles fulfill the Bragg condition such that nλ = 2dsinθ. Condition for constructive interference is
given in figure 2.4.
Fig 2.4 Condition for constructive interference
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By varying the angle θ, the Bragg law condition is satisfied for different d-spacings in
polycrystalline materials. Plotting the angular positions versus intensities produces a diffraction
pattern, which is characteristic of the sample. When a mixture of different phases is present, the
resultant diffractogram is a superposition of the individual patterns. In a typical XRD pattern, the
diffracted intensities are plotted verses the detector angle 2θ. Each peak is then assigned a label
indicating the spacing of a crystal plane. Bragg’s law states the condition for sharp diffraction peaks
arising from crystals which are perfectly ordered. Actual diffraction peaks have a finite width
resulting from imperfections, either the irradiation source or the sample. As the crystallite
dimensions enter the nanoscale the peaks broaden with decreasing crystal size. The widths of the
diffraction peaks allow the determination of crystallite size. The size of the crystallites can be
determined using Debye–Scherrer equation:
where τ is the thickness of the nanocrystal, K is a constant which depends on the crystallite shape ,
λ = wavelength of X-rays, β is the full width at half maxima of the broadened peak .