Practical Guide to Materials Characterization: Techniques and Applications 3527350713, 9783527350711

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Practical Guide to Materials Characterization

Practical Guide to Materials Characterization Techniques and Applications

Khalid Sultan

Author Dr. Khalid Sultan

Central University of Kashmir Department of Physics Ganderbal, Jammu and Kashmir 191131 Srinagar India Cover image

© marigold_88/Getty Images

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Contents List of Figures  x List of Tables  xvi Preface  xvii 1 1.1 1.2 1.3 1.4 1.5 1.6

Basics of Material Characterization Techniques  1 Introduction  1 Electromagnetic Spectrum  2 Fundamentals of Crystallography  4 Molecular Vibrations  10 Magnetism in Solids  10 Optical Properties of Solids  11 References  13

2 X-ray Diffraction  15 2.1 Introduction  15 2.2 Bragg’s Law  17 2.3 Von Laue Treatment: Laue’s Equation  18 2.4 Experimental Techniques  20 2.4.1 Laue Method  20 2.4.2 Rotating Crystal Method  20 2.4.3 Powder Method  21 2.5 Geometry and Instrumentation  21 Standard XRD Pattern  26 2.6 2.7 Applications  27 2.7.1 Orientation of Single Crystals  27 2.7.2 Structure of Polycrystalline Aggregates  29 2.7.3 XRD in the Pharmaceutical Field and Forensic Science  33 2.7.4 XRD in the Geological Field  34 2.8 Examples and Illustrations  34 2.8.1 XRD Data and Interpretation in the PrFe1–xMnxO3 System  34

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2.8.2 2.8.3

XRD Data and Interpretation in the La1–xCaxMnO3 System  37 XRD Analysis of EuFe1–xMnxO3 (x = 0.0, 0.3, 0.5)  38 References  40

3 3.1 3.2 3.3 3.4 3.5 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.7 3.8 3.8.1 3.8.2 3.8.3 3.8.4 3.9 3.9.1

Raman Spectroscopy  43 Introduction  43 Infrared and Raman Spectroscopy  44 Raman Spectra: Origin  45 Classical Theory of Raman Scattering  47 Quantum Theory of Raman Spectroscopy  48 Raman Spectrometer  50 Excitation Source  50 Sample Illumination  51 Wavelength Selector  51 Detection and Control System  52 Resonance Raman Spectroscopy  53 Special Techniques  55 High-pressure Raman Spectroscopy  55 Raman Microscopy  57 Surface-enhanced Raman Spectroscopy  59 Raman Spectroelectrochemistry  60 Applications and Illustrations  62 Raman Spectra of the PrFe1–xMnxO3 System at Different Concentrations of Mn Doped in Place of Fe  62 Raman Spectra and Measurements of the La1–xCaxMnO3 System (x = 0.0, 0.3, 0.5, and 0.7)  64 Temperature-dependent Raman Study of La1−xCaxMnO3 (x = 0.0 and 0.3)  66 References  71

3.9.2 3.9.3 4 4.1 4.1.1 4.1.2 4.1.3 4.1.3.1 4.1.3.2 4.1.3.3 4.1.3.4 4.1.3.5 4.1.4 4.1.4.1

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X-ray Spectroscopic Techniques  77 X-ray Absorption Spectroscopy  77 Introduction  77 Basic Principle of XAS  78 Experimental Aspects  83 Synchrotron Radiation  83 Experimental Setup  84 Transmission Mode  84 Fluorescence Mode  85 Electron Yield Mode  85 Examples and Analysis  86 X-ray Absorption Spectra of La1–xCaxMnO3 (x = 0.0, 0.3, 0.5, 0.7) Samples  86

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4.1.4.2 Electronic Structure of PrFe1–xMnxO3 by X-Ray Absorption Spectroscopy  88 4.2 X-ray Photoelectron Spectroscopy (XPS)  92 4.2.1 Introduction  92 4.2.2 Basic Principle  93 4.2.3 Energy Referencing  94 4.2.4 Instrumentation  96 4.2.5 XPS Spectra and Their Features  97 4.3 Auger Electron Spectroscopy  99 4.3.1 Introduction  99 4.3.2 Interactions of Electrons with Matter  100 4.3.3 Competition Between X-ray and Auger Electron Emissions  100 4.3.4 Auger Process  101 4.3.5 Kinetic Energy of the Auger Electron  103 4.3.6 Auger Spectra  104 4.3.7 Instrumentation  105 References  106



Magnetic Measurements  109 Introduction  109 Magnetization Measuring Instruments  111 Extraction Technique  112 Vibrating Sample Magnetometer  113 SQUID Magnetometer  114 Advantages and Disadvantage of a Vibrating Sample Magnetometer  115 Susceptibility Measurement  116 Examples and Illustrations  117 Magnetic Behavior Shown by Thin Films of the PrFe1–xMnxO3 System Deposited on Substrate Si (100)  117 Magnetic Behavior of the La1–xCaxMnO3 System Where x is the Concentration of Ca as Dopant and Equals 0.0, 0.3, 0.5, or 0.7  120 Magnetic Behavior of La1–xCaxMnO3 Thin Films Deposited on Si(100) with x = 0.0, 0.3, 0.5, and 0.7 Being the Concentrations of Ca  124 References  125

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3

Dielectric Measurements  129 Introduction  129 Polarization and Dielectric Constant  130 Electronic or Optical Polarization  131 Orientational Polarization  134 Atomic Polarizability  135 Interfacial Polarization  136 Mechanism for the Colossal Dielectric Response  137

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.3 5.4 5.5 5.5.1 5.5.2 5.5.3

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6.4 6.5 6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.7 6.8 6.9 6.10 6.11 6.11.1

Frequency Dependence of Polarizability  138 Classification of Dielectric Materials  138 Nonferroelectric Materials  138 Nonpolar Materials  139 Polar Materials  139 Dipolar Materials  139 Dielectric Dispersion: A Brief Discussion  140 Dielectric Loss and Relaxation  141 Complex Permittivity  141 Polarization Buildup  142 Jonscher’s Universal Law  145 Examples and Illustrations  148 Dielectric Behavior of PrFe1–xMnxO3 with x = 0.0, 0.1, 0.3, and 0.5 Being the Concentration of Mn Doped in a Pristine Compound  148 6.11.1.1 Frequency and Temperature Dependence of Dielectric Properties  148 6.11.2 Dielectric Properties of the EuFe1–xMnxO3 System in Which Different Concentrations of Mn Are Doped in EuFeO3  151 6.11.2.1 Dependence of Dielectric Behavior on Frequency  151 6.11.2.2 Dependence of Dielectric Behavior on Temperature  154 References  157 7 7.1 7.2 7.3 7.4 7.5 7.5.1 7.5.2 7.5.3 7.5.4 7.6 7.7 7.7.1 7.7.2 7.7.3

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Electron Microscopy  159 Introduction  159 Generation of an Electron Beam  159 Interaction of an Electron Beam with a Sample  160 Inelastic Scattering and Absorption  161 The Family of Electron Microscopes  164 The X-ray Microscope  165 The Transmission Electron Microscope  166 The Scanning Electron Microscope  172 The Scanning Transmission Electron Microscope  174 Atomic Force Microscopy  175 Examples and Illustrations  177 AFM Studies of PrFe1–xMnxO3 (x = 0.0, 0.1, 0.3, 0.5) Thin Films Grown on Si (100)  177 Atomic Force Microscopy in the Case of La1–xCaxMnO3 for (a) x = 0.0, (b) x = 0.3, (c) x = 0.5, and (d) x = 0.7  177 Morphological Studies and Elemental Analysis of EuFe1–xMnxO3 (x = 0.0, 0.3, and 0.5)  178 References  181

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8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.8.1 8.8.2 8.8.3 8.8.4 8.8.5

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Infrared Spectroscopy  183 Introduction  183 Instrumentation for FTIR  184 Fourier Transform  189 Electromagnetic Radiation  190 Infrared Absorption  190 Normal Modes of Vibration  191 Complicating Factors  192 Applications of IR Spectroscopy  192 Food Science  192 Chemistry in Clinical Practice  193 Plants  194 Disease Diagnosis  194 Environmental Applications  195 References  196

Index  199

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List of Figures Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 1.7 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 2.15 Figure 2.16 Figure 2.17

The electromagnetic spectrum.  3 A two-dimensional crystal.  5 Representation of r in terms of a , b, and c.  6 Example showing directions.  6 Two-dimensional Bravais lattice.  7 Simple cubic Bravais lattice.  8 Crystal systems.  8 Scattering of X-rays by an atom containing many electrons.  17 Diffraction of X-rays.  18 Scattering from two identical points, A and B.  19 Geometrical interpretation of the incident beam, scattered beam, reflecting plane, and normal.  19 Geometry of an X-ray diffractometer.  21 Arrangement of various components in a diffractometer.  23 A peak in the XRD pattern.  24 Arrangement of a diffractometer along with the monochromating crystal in the way of a diffracted beam.  24 XRD pattern of PFO.  26 Crystal rotation axes for the diffractometer.  27 Plotting method applied while determining orientation of the crystal.  28 Back-reflection patterns of recrystallized specimens.  30 Influence of the presence of strain on width and position.  32 XRD print of PrFe1–xMnxO3 samples at different concentrations of Mn along with an inset suggesting the shift of the (112) peak.  35 Lattice constants as a function of the Mn content in PrFe1–xMnxO3.  36 XRD pattern of La1–xCaxMnO3 compounds.  37 Lattice constants a, b / 2 , and c as a function of Ca concentration in La1–xCaxMnO3.  38

List of Figures

Figure 2.18 Rietveld treatment of the XRD pattern of EFMO at x = 0.0, 0.3, and 0.5.  39 Figure 3.1 Mechanism of infrared and Raman spectra.  46 Figure 3.2 Experimental setup for Raman scattering.  50 Figure 3.3 Energy levels in normal Raman, resonance Raman, and fluorescence.  54 Figure 3.4 Representation of energy levels in the resonance Raman process.  55 Figure 3.5 Depiction of the diamond anvil cell along with (a) details of the cell, (b) the side view, and (c) the front view.  56 Figure 3.6 An arrangement of scattering geometry using the laser diamond anvil cell experiment.  57 Figure 3.7 Schematic arrangement of the optical path for microprobe sampling.  58 Figure 3.8 Resonance Raman spectroelectrochemistry cell (controlled potentional electrolysis cell).  61 Figure 3.9 Resonance Raman spectroelectrochemistry cell (sandwich cell).  61 Figure 3.10 Raman spectra of PrFe1–xMnxO3 samples for x = 0.0, 0.1, 0.3, and 0.5.  63 Figure 3.11 Raman spectra of the La1–xCaxMnO3 system for (a) x = 0.0, (b) x = 0.3, (c) x = 0.5, (d) x = 0.7. The inset shows the B2g mode moving towards a higher wave number region due to increasing dopant concentration.  65 Figure 3.12 Raman spectra of LaMnO3 at various temperatures.  67 Figure 3.13 Raman spectra of La0.7Ca0.3MnO3 at various temperatures.  68 Figure 3.14 Variation of the intensity of some observed modes of LaMnO3 with temperature.  69 Figure 3.15 Variation of the intensity of some observed modes of LaMnO3 with temperature.  70 Figure 3.16 Variation of FWHM with temperature of various modes for (a) LaMnO3 and (b) La0.7Ca0.3MnO3.  70 Figure 4.1 Representation of a different sort of interaction, when X-rays are incident on the specimen.  78 Figure 4.2 Different phenomena when X-rays are absorbed.  79 Figure 4.3 Schematic of the absorption process.  79 Figure 4.4 Transition of an electron from the ground state going above the Fermi level.  80 Figure 4.5 Normalized absorption vs incident energy (eV).  81 Figure 4.6 Representation of the relative scale of energy that is involved in XAS.  82 Figure 4.7 Transitions that contribute to XAS edges.  82 Figure 4.8 Modern synchrotron facility basic design.  83 Figure 4.9 A modern XAS beamline’s typical components.  84

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List of Figures

Figure 4.10 The experimental setup for the various XAS detection techniques.  85 Figure 4.11 (a) Normalized O K edge XAS spectra of La1–xCaxMnO3 (x = 0.0, 0.3, 0.5, and 0.7) samples. (b) Difference spectra for O K edge XAS spectra of La1–xCaxMnO3 (x = 0.0, 0.3, 0.5, and 0.7) samples.  87 Figure 4.12 (a) Normalized Mn L3,2 edge XAS spectra of La1–xCaxMnO3 (x = 0.0, 0.3, 0.5, and 0.7) samples. (b) Difference spectra for Mn L3,2 XAS spectra of La1–xCaxMnO3 (x = 0.3, 0.5, and 0.7) samples.  88 Figure 4.13 NEXAFS spectra for the O K edge of PrFe1–xMnxO3 (x = 0.0, 0.1, 0.3, and 0.5) samples. It is observed that the spectral feature b gains weight as the Mn concentration is increased, which indicates increased hybridization of Mn 3d orbitals with the O 2p states.  89 Figure 4.14 Difference spectra for O K edge NEXAFS spectra of PrFe1–xMnxO3 (x = 0.1, 0.3, and 0.5) samples. It is seen from the difference spectrograph that the density of states increases with Mn doping (observed as increasing dips in all spectral peaks from a to f).  90 Figure 4.15 NEXAFS spectra for the Fe L3,2 edges of PrFe1–xMnxO3 (x = 0, 0.1, 0.3, and 0.5) samples. It is observed that Fe ions are in a 3+ state having octahedral symmetry for all these doping levels.  91 Figure 4.16 NEXAFS spectra of Mn L3,2 edges of PrFe1–xMnxO3 (x = 0.1, 0.3, and 0.5) samples. The multiplet features a2 (~ 638.0) and b2 (~ 639.1) in L3,2 spectra are attributed to mixed valence states of Mn.  92 Figure 4.17 A model atom’s XPS emission mechanism (top). The photoelectron is ejected as a result of an incoming photon. The relaxation process (bottom) for a model atom ejects an Auger electron.  94 Figure 4.18 The process that is involved in the BE and KE scale references.  95 Figure 4.19 XPS instrument schematic diagram.  96 Figure 4.20 Schematic diagram of an XPS spectrum.  98 Figure 4.21 (a) A wide scan 0 to 1000 eV binding energy range XPS spectrum of gold and (b) a 4f spin-orbit doublet at a higher resolution.  98 Figure 4.22 Variation of yield (Auger or X-ray) with atomic number.  101 Figure 4.23 (a) Knock-out of an electron from a core or L shell. (b) Electron jumps from a higher shell to a core level shell. (c) Emission of an Auger electron.  102 Figure 4.24 (a) Electron is knocked out from the core shell, i.e., the K shell. (b) Vacancy or hole in the core shell.  103 Figure 4.25 (a) Relaxation process occurs via transition of the electron from the L1 to the K shell. (b) Emission of the Auger electron from the L23 shell.  103 Figure 4.26 Distribution of energy of backscattered and secondary electrons produced via interaction of a nearly monochromatic electron beam with a sample.  105 Figure 4.27 Differentiated and direct spectrum of copper (Cu).  106 Figure 4.28 Auger electron spectroscopy instrument depicted schematically.  106

List of Figures

Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7

Figure 5.8

Figure 5.9 Figure 5.10 Figure 5.11 Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5

Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9 Figure 6.10

Figure 6.11 Figure 6.12 Figure 6.13

Arrangement for measuring magnetization of a ring.  110 Schematic arrangement for the measurement of a rod-shaped sample placed in a magnetic field produced by a solenoid.  113 Schematic arrangement of a VSM.  114 Schematic arrangement of a SQUID.  115 A substance in a field with a nonuniform nature.  116 Rod-shaped magnetic substances in an electromagnetic field.  117 ZFC and FC magnetization curves as a function of temperature at a magnetic field of 100 Oe in the PFMO system at different concentrations of Mn.  118 Magnetization in terms of ZFC and FC plots as a function of temperature at a magnetic field of 500 Oe for different concentrations of dopant.  120 Magnetization as a function of magnetic field at a temperature of 5 K for different concentrations of dopant.  121 Magnetization in terms of ZFC and FC plots as a function of temperature at a magnetic field of 100 Oe.  124 Field dependence of magnetization La1–xCaxMnO3 samples at 300 K and 10 K.  125 Polarization mechanism.  132 Electronic polarization of the dielectric.  132 Orientational polarization.  134 Frequency dependence of polarization.  138 Within the alternating nature of an electric field, the real and imaginary parts of the dielectric constant ε* = ε′ –jε′′ are such that the reference phasor is along Ic.  142 Buildup of polarization in materials having a polar dielectric nature.  143 Frequency dependence of the dielectric constant in the PFMO system for x = 0.0, 0.1, 0.3, and 0.5.  149 Dielectric constant as a function of temperature at different frequencies in the PFMO system.  150 Dielectric loss as a function of temperature at different frequencies in the PFMO system with x = 0.0, 0.1, 0.3, and 0.5.  151 Dielectric constant at different temperatures as a function of frequency and dielectric loss at room temperature as a function of frequency in EuFe1–xMnxO3 samples.  152 Frequency dependence of susceptibility in EuFe1–xMnxO3 samples.  153 Frequency dependence of conductivity in EuFe1–xMnxO3 samples.  154 Dielectric constant as a function of temperature at different frequencies in the EuFe1–xMnxO3 system for (a) x = 0.0, (b) x = 0.3, (c) x = 0.5, (d) the comparative behavior at 1 kHz.  155

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List of Figures

Figure 6.14 Temperature dependence of dielectric loss at different frequencies in the EuFe1–xMnxO3 system, such as (a) sample with x = 0.0, (b) sample with x = 0.3, (c) sample with x = 0.5, (d) comparative behavior at 1 kHz.  156 Figure 6.15 The AC conductivity at different frequencies as a function of temperature in the EuFe1–xMnxO3 system doped with different concentrations of Mn with (a) x = 0.0, (b) x = 0.3, (c) x = 0.5, (d) comparative behavior at 1 kHz.  157 Figure 7.1 Geometry of a thermionic triode electron gun.  160 Figure 7.2 Interaction of an X-ray with a sample, where S = secondary electrons, B = backscattered electrons, and D = diffracted electrons.  161 Figure 7.3 There are two ways in which an inner-shell-excited atom can relax with the knocking out of an electron from the K shell. (a) A characteristic (kα) X-ray is emitted and (b) an Auger electron (KLM) is ejected.  162 Figure 7.4 An X-ray spectrum excited from a specimen of molybdenum by a beam of 30 keV electrons. The K and L characteristic peaks are evident, superimposed on the Bremsstrahlung background.  163 Figure 7.5 Electron spectrum showing the relative abundance of secondary (S), Auger (A), and backscattered (B) electrons. The energy scale here is not continuous and E0 is typically >>50 eV.  164 Figure 7.6 The family of electron microscopes; X = X-ray detector and E = electron detector.  165 Figure 7.7 Schematic diagram of a scanning transmission X-ray microscope (STXM) attached to a synchrotron radiation source.  166 Figure 7.8 STXM images of a clay-stabilized oil-water emulsion. By changing the photon energy, different components of the emulsion become bright or dark and can be identified from their known X-ray absorption properties.  166 Figure 7.9 Early photograph of a horizontal two-stage electron microscope.  168 Figure 7.10 The first commercial TEM from the Siemens Company.  169 Figure 7.11 Two recent TEMs: (a) the JEOL grand ARM (300 KV, 63 pm resolution with aberration correction), (b) the FEI TITAN Themis, which can reach a 60 pm resolution at 300 KV if fitted with an aberration correction, (c) enclosed version of FEI TITAN.  170 Figure 7.12 TEM diffraction-contrast image (m ≈ 10,000) of polycrystalline aluminum. (M.J.Whelan, Oxford University.)  171 Figure 7.13 TEM image (right) of a stained specimen of mouse-liver tissue, corresponding approximately to the small square area within the light-microscope image on the left. (Bhatnagar, Biological Sciences Microscopy Unit, University of Alberta.)  171

List of Figures

Figure 7.14

Figure 7.15 Figure 7.16 Figure 7.17

Figure 7.18

Figure 7.19 Figure 7.20 Figure 7.21 Figure 7.22

Figure 7.23 Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4

A 3 MV HVEM constructed at the CNRS Laboratories in Toulouse and in operation during the 1970s. Courtesy of G. Dupouy (personal communication).  172 SEM at the RCA laboratories.  173 HITACHI-SU5000 SEM.  174 Photograph of the Chicago STEM and (bottom-left inset) an image of mercury atoms on a thin-carbon support film. Courtesy of Dr. Albert Crewe (personal communication).  175 Atomic columns in a strontium titanate crystal: (a) in a TEM using phase contrast and (b) using a STEM mode with a high-angle annular dark-field detector.  176 Elements of an AFM.  176 AFM image of the INAS islands on a GAAS substrate.  177 Shows 1 µm × 1 µm AFM images on Si (100) substrate for (a) x = 0.0, (b) x = 0.1, (c) x = 0.3, and (d) x = 0.5 PFMO samples.  178 Shows 1 µm × 1 µm AFM images on Si (100) substrate La1–xCaxMnO3 samples for (a) x = 0.0, (b) x = 0.3, (c) x = 0.5, and (d) x = 0.7.  179 The SEM and EDAX micrographs of EuFe1–xMnxO3 (x = 0, 0.3, and 0.5) sintered at 1300oC for 24 h.  180 Michelson interferometer.  185 Synchronization of the IR signal and the laser signal (Wikipedia).  186 Optical schematic of an FTIR spectrometer.  187 H2O and CO2 molecules.  191

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List of Tables Table 2.1 Table 2.2 Table 2.3 Table 3.1 Table 3.2 Table 3.3 Table 4.1 Table 6.1 Table 7.1 Table 7.2 Table 8.1 Table 8.2

Structural lattice parameters of the PrFe1–xMnxO3 system.  36 Various lattice parameters in the La1–xCaxMnO3 system.  38 Structural parameters and activation energy for EFMO.  40 Raman modes along with the corresponding atomic motion for PrFeO3.  64 The observed Raman modes with corresponding assignments.  65 Ionic radii of the constituent elements of La1–xCaxMnO3.  67 Atomic orbital nomenclature.  99 Dependence of various parameters on concentration x of Mn in PFMO.  151 Surface roughness and grain diameter.  179 EDAX of EuFe1–xMnxO3.  181 Degrees of freedom in linear and nonlinear molecules.  191 Infrared bands of major food components.  193

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Preface Characterization of materials is the measurement and determination of a material’s physical, chemical, mechanical, and microstructural properties. This technique provides the greater degree of awareness required to handle significant issues such as failure causes and process-related concerns, as well as allowing the manufacturer to make critical material decisions. The field of materials characterization is vast and diverse. Perhaps the best place to begin is at the beginning, with the first principle to consider being the depth to which characterization promotes the discovery of new materials: ● ●

●

Measuring a material’s property allows for experimental improvement; Taking unique measurements allows for distinction through improvement in specific areas; and Understanding the compositional and structural foundations of material attributes allows for rationally designed improvements.

Material characterizations is a crucial step to conduct before using the materials for any purpose. To ensure that the material under consideration can perform without failure during the life of the final product, it might be subjected to mechanical, thermal, chemical, optical, electrical, and other characterizations, depending on the purpose. This book focuses on the most extensively used experimental approaches for structural, morphological, and spectroscopic characterization of materials. One of the most important aspects of this book is the discussion of recent results in a wide range of experimental techniques and their application to the quantification of material properties. Furthermore, it covers the practical elements of the analytical techniques used to characterize a wide range of functional materials (both in bulk as well as thin film form) in a simple but thorough manner. For a wide range of readers, from beginners and graduate students to expert specialists in academia and industry, the book gives an overview of frequently used characterization approaches. One of my main aims in preparing this book was to put the basic characterizations used by material research students in the form of a single book. The book is divided into eight chapters.

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Preface

The first chapter gives the basic ideas of an electromagnetic spectrum, which is important as properties of materials are obtained using the interaction of light with matter. In addition, some fundamentals of crystallography, the magnetic materials, the molecular vibrations, and optical properties in materials have been defined. The second chapter is based on the one and foremost technique used in material sciences and is called the X-ray diffraction technique. After synthesis of any material, the first step is confirmation, which is obtained through the X-ray diffraction pattern. The basic theory, the experimental setup, along with some examples and applications have been included in this chapter. Chapter 3 concerns Raman spectroscopy. In addition to the X-ray diffraction technique, Raman spectroscopy may also be used for the identification of samples. In this chapter, basic theory, the instrumentation of Raman spectrometer, and illustrations are included. Chapter 4 discusses X-ray spectroscopic techniques. Three techniques, namely X-ray absorption spectroscopy, X-ray photoemission spectroscopy, and Auger electron spectroscopy have been explained along with the basic principle and experimental setup for each case. In Chapter 5 magnetism in solids has been incorporated along with the methods of magnetic measurements. Three important methods, like the extraction technique, vibrating sample magnetometer, and SQUID, are part of this chapter. Moreover, some examples for the measurement of magnetic properties, like temperature dependence of magnetization and field dependence of magnetization in some particular compounds, have been set up to give a clearer picturization of magnetic measurements. Chapter 6 refers to dielectric measurements. A fundamental description has been set up for dielectric properties before explaining the different trends of the dielectric constant and dielectric loss as a function of temperature and as a function of frequency. Alternating current conductivity is also part of this description. Chapter 7 discusses electron microscopy in material sciences where the scanning electron microscope, transmission electron microscope, scanning transmission electron microscopy, X-ray microscope, and atomic force microscopy are described in detail. Moreover, some examples of atomic force microscopy and scanning electron microscopy pertaining to some particular compounds form part of this chapter. Finally, the last chapter of book is on infrared spectroscopy. In this chapter the theory of infrared spectroscopy has been included along with instrumentation. Fourier transform infrared spectroscopy has also been explained. Lastly, some applications of infrared spectroscopy in our day-to-day life form part of this chapter.

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1 Basics of Material Characterization Techniques 1.1 Introduction In our daily life we come across different types of materials like metals, semiconductors, and ceramics. Scientific communities are continually exploring the nature of materials as well as their technological use, which requires understanding of some of the basic principles of physics and chemistry. The foremost thing about a material that is to be investigated is its structure and the related properties. The material is then fabricated to such a design that it results in exceeding the basic technological requirements. Checking the performance of the material is always a major focus in order to observe the behavior of the material under different conditions. The synthesis and characterization of materials play a crucial role for materials, research and nowadays a number of advanced instruments are available to understand the large range of mechanisms in materials. Since all characterization techniques are fundamental pillars in understanding the properties of a material, each characterization technique is based on some basic principles and some basic processes of physics. It is therefore worthwhile outlining the principles and processes that are being applied or used by different characterization techniques. Most characterization techniques involve the use of electromagnetic light, so information about the electromagnetic spectrum and related processes like reflection, refraction, absorption, transmittance, diffraction, interference, and dispersion are part of this chapter. Apart from this, the solid form of materials is the most stable form, with minimum free energy, and a detailed information about the crystallography is provided, which is beneficial for understanding the structural properties of materials using techniques like X-ray diffraction. These are followed up in subsequent chapters. Similarly, different kinds of molecular motion and vibrations are also present in solids that, using Raman spectroscopy and infrared spectroscopy, also provide information about the materials providing information about the different motion in solids that form the initial stage for obtaining information and data. Some techniques such as electron microscopy also make use of electrons. This chapter provides details about electron–matter interactions and related consequences providing the morphology, topography, elemental composition, etc. Apart from these structural studies, dielectric studies of materials involve terms like dielectric constant Practical Guide to Materials Characterization: Techniques and Applications, First Edition. Khalid Sultan. © 2023 Wiley-VCH GmbH. Published 2023 by Wiley-VCH GmbH.

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1  Basics of Material Characterization Techniques

and dielectric loss, which are also important parameters. The magnetic phenomenon involves magnetic terms and types of magnetism. All these terms are defined within this book.

1.2  Electromagnetic Spectrum The electromagnetic spectrum envelops electromagnetic waves with a wide range of frequencies and hence a wide range of wavelengths and energy. The frequency range is divided into various parts and electromagnetic waves in each part have different names. Going from the low-frequency end of the electromagnetic spectrum to the high-frequency end we have radio waves, microwaves, infrared, visible light, ultraviolet (UV), X-rays, and gamma rays. Each type has different characteristics of production, interaction, and applications. Gamma rays, X-rays, and high-energy UV are called ionizing radiations because their photons have energy to ionize an atom. Some of the frequencies are responsible for spectroscopy, which is also based on the interaction of electromagnetic waves with matter. Before the 1800s, the term “light” was interpreted by a general reader and even a specialized person as visible light. In the 1800s, it was found that light contains not only the visible part but something more as evidenced by William Herschel’s discovery of infrared light. The infrared region has three main parts, i.e., the far infrared, the mid-infrared, and the near infrared. The near-infrared region lies at the visible end of the electromagnetic spectrum. Similarly, in 1801 Johann Ritter identified the part of the spectrum that lies just beyond the violet end of visible light, which he termed deoxidizing rays. In the late ninetenth century, knowledge of these rays was well established and they were termed the UV rays. The UV region also consists of several parts, i.e., long-wave UV, medium-wave UV, and short-wave UV. James Clark Maxwell’s equations provided information about the existence of an infinite number of frequencies of electromagnetic waves and thus predicted the entire electromagnetic spectrum. Heinrich Rudolf Hertz was the first to generate radio waves and microwave radiation. The study of X-rays was first carried out by Wilhelm Röntgen in 1895. X-ray spectroscopy was developed by Karl Manne Siegbahn, who was then awarded the 1924 Nobel Prize in Physics for his work. The discovery of gamma rays was made by Paul Villard in 1900 during an investigation of radioactivity and was said to be electromagnetic radiation, with the shortest wavelength and hence the highest energy as well as frequency. All parts of the spectrum are important and, as a result of their different characteristics, are used in different spectroscopic techniques [1]. The electromagnetic spectrum is shown in Figure 1.1. Thus, based on the frequency, wavelength, and energy, which are fundamental factors in the spectrum, the electromagnetic radiations are divided into the following classes, regions, or bands: a) gamma radiation, b) X-ray radiation, c) UV radiation,

1.2  Electromagnetic Spectrum

Figure 1.1  The electromagnetic spectrum.

d) e) f) g)

visible radiation, infrared radiation, microwave radiation, and radio waves.

These regions are given in increasing order of wavelength. It should also be noted that there is no well-defined boundary between the regions and they fade into each other. Radio waves: These waves are released or received by antennas. The production of these waves involves the generation of an alternating current (AC) by a transmitter, which is an electric device available with an antenna. There is generation of oscillating electric and magnetic fields due to the oscillations of electrons in the antenna that lead to radiations that leave the antenna in the form of radio waves. Receiving these radio waves is associated with the coupling of oscillating electric and magnetic fields of waves with the electrons in the antenna. This causes the back-and-forth movement of electrons, thereby producing oscillating currents that are applied to radio receivers. These waves are used in the transmission of information in communication systems and in the Global Positioning System (GPS). Microwaves: These waves are absorbed by polar molecules in addition to being released and received by short antennas. These are considered as radio waves of short wavelength with the characteristic of being used in radars and satellites. Microwaves have the capability to penetrate a material in order to deposit the energy well inside the surface. As a result they are used in microwave ovens. Infrared radiation: The range of this frequency of radiation is approximately from 300 GHz to 400 THz. The far infrared part, ranging from 300 GHz to 300 THz,

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1  Basics of Material Characterization Techniques

lies towards the microwave edge and is absorbed by the rotational modes of gas, the molecular motions in liquid, and phonons in solids. These are strongly absorbed by the water available in the atmosphere of Earth so are opaque to them, but there are certain frequencies that are allowed and are under study by astronomers. The midinfrared part ranges from 30  THz to 120  THz. The hottest object or a blackbody radiator can emit radiations in this range while the skin of human beings also emits radiations of frequencies lying towards the lower end of this mid-region. As far as absorption of these radiations is concerned, they are absorbed by the vibration of molecules. The near-infrared part of this region ranges from 120 THz to 400 THz and lies towards the visible part of the electromagnetic spectrum. The higher frequencies are usually detected by photographic films and image sensors. Visible light: The range of this frequency of radiations is approximately from 400  THz to 790  THz. This part of the electromagnetic spectrum is visible to human eyes and hence is called visible light. It is released and absorbed by electrons in an atom or in a molecule when going from one energy level to another. A rainbow shows the visible part of the electromagnetic spectrum, consisting of seven colors. UV radiation: The wavelength range of these radiations of the electromagnetic spectrum is less than the visible part, but is considered to have the longest wavelength, with photons having such energy that they can ionize atoms. Shorter wavelength UV radiation and radiations having still smaller wavelengths, like X-rays and gamma rays, are known as ionizing radiations. UV also causes certain materials to glow, producing visible light, and the phenomenon is known as fluorescence. X-ray: This is radiation that can interact with matter through the Compton effect. X-rays can be hard X-rays and soft X-rays, differing by the kind of wavelength. Shorter wavelength X-rays are called hard X-rays and can easily pass through different materials along with some absorption. The one most important use of these radiations is in diagnostic imaging. In material sciences, the X-rays diffraction phenomena has a major role in the identification of different compounds and samples. As far as the production of X-rays is concerned, they are released when there is a sudden deceleration of fast-moving electrons while interacting with the target anode. Gamma rays: Gamma rays have the shortest wavelengths and were, discovered by Paul Ulrich Villard. These are considered to have the most energetic radiation, with no lower limit for their wavelength, and are usually used in the irradiation of food and seeds. In the field of astronomy these radiations have a role in the investigation of high-energy regions.

1.3  Fundamentals of Crystallography The state of matter in which there is a regular arrangement of atoms is referred to as solid and the regularity is expressed by symmetry elements. In modern times the regularity of atoms can be studied with the help of high-resolution transmission electron microscopy. The arrangement and pattern of atoms connected through various interatomic forces can be expressed in terms of a unit whose

1.3  Fundamentals of Crystallography

repetition gives rise to crystals. In crystals, the set of points surrounding any given point is identical to those of all other points and constitutes a lattice, where each point is called a lattice point. A lattice in a crystal describes a translational ­symmetry. In the case of three dimensions, the unit cell is in the form of a parallelepiped such that the origin lies at the corner of the unit cell and the three axes are represented in terms of sides of the unit parallelepiped. The three axes are ­connected with each other through angles α, β, and γ, and the minimum ­separation between two adjacent lattice points along the three axes is expressed in terms of lattice parameters x, y, and z. Consider a two-dimensional crystal having lattice parameters a and b such that the magnitudes of both a and b are different, as shown in Figure 1.2, where we observe that the mesh lines OB, O′B′, and O′′B′′ are parallel and thus constitute a set with regular spacing in between these mesh lines. Similarly, the same effect can be observed in mesh lines AB, A′B′, and A′′B′′. It has been determined that the spacing between parallel mesh lines depends not only on the lattice parameters a and b vectors but also on the angle between the a and b vectors. However, the case of the angle between the two sets depends on the ratio of magnitude of a and b. It should be noted that the value of the angle between the corresponding faces is constant as long as all the crystals are of the same substance [2]. In this analogy of lines in a mesh, in the planes of a crystal the faces of the crystal are parallel to the planes of the lattice and the lattice planes have a high density. In this case, we also have a set of parallel planes whose spacing depends on the lattice parameters and axial angles. However, the angle between various lattice planes depends on the ratio of the axial angles and lattice parameters. O

B

Oʹ

Bʹ

Oʺ

Bʺ

φ

A Aʹ Aʺ

a

b y

x

Figure 1.2  A two-dimensional crystal.

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1  Basics of Material Characterization Techniques

In a crystal the direction is expressed as a line. Let us assume two points P and P′ lying on a line with P at the origin, as shown in Figure 1.3. Using the concept of translation, the vector r joining P and P′ can be expressed in terms of a , b, and c vectors along the x, y, and z directions, respectively, as r = ua + vb + wc

(1.1)

The notation for direction is then represented by arrows and examples are shown in Figure 1.4. In cases where the value of any of the u, v, and w is negative, it is expressed in terms of bar. Moreover, P lies at the origin and in the case where u, v, and w are also integers, the P′ will lie on the lattice point such that the produced PP′ represents a row of lattice points and is called a rational line and accordingly we have a rational plane. To designate a plane let us consider that a plane from the set cuts the three Figure 1.3  Representation of r in terms of a , b , and c .

z

pʹ p

r wc

ua

y ua

vb x

z

[0

11

]

[121

y

[11

0]

x

Figure 1.4  Example showing directions.

]

1.3  Fundamentals of Crystallography

axes x, y, and z at A, B, and C respectively. Suppose that the origin lies at the lattice point and a , b, and c represent the separation between lattice points that are adjacent to each other. Here we define Miller indices (hkl) for a set of planes such that OA =

a h

OB =

b k

OC =

c l

The reason behind the introduction of Miller indices in crystal structures is that it simplifies calculations. Now the basic idea for a description pertains to the explanation of solids in terms of the Bravais lattice that pertains to the periodic arrangement in solids where a unit is repeated. In actual fact, these units can be an atom or group of atoms but the Bravais lattice depends on the geometry of the structure, whatever the unit is. A Bravais lattice can be understood in terms of the following definitions: ●

●

An infinite arrangement of lattice points is such that the array and orientation are the same from every point the arrangement is viewed. A Bravais lattice has all points such that the position vector R, for a1 , a2 , and a3 , does not lie on the same plane: R = n1a1 + n2a2 + n3a3

●

(1.2)

Here n1, n2, and n3 take integer values. This means that a point Σni ai can be obtained if moved in the direction of ai through ni steps, each of length ai.

The vector ai is called a primitive vector and is responsible for the generation of a lattice. In physics, it is said to span the lattice [3]. A two-dimensional Bravais lattice is shown in Figure 1.5, where P = a1 + 2a2 and Q = −a1 + a2 , and a simple cubic three-dimensional representation of a Bravais lattice is shown in Figure 1.6. Another concept in solid-state physics is of a reciprocal lattice. Let us try to understand this. If we have a set of points in a Bravais lattice and accordingly a plane wave of the form e ik⋅r , then all wave vectors K that produce periodicity of a given Bravais lattice are called a reciprocal lattice of a P Bravais lattice, such that e

Q a2 a1

Figure 1.5  Two-dimensional Bravais lattice.

iK ⋅(r + R)

= e iK ⋅r 

(1.3)

is satisfied for any r and all R in a Bravais lattice. There are seven types of crystal systems, which will be defined and are shown in Figure 1.7.

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1  Basics of Material Characterization Techniques

Figure 1.6  Simple cubic Bravais lattice.

a3

a1

a2

b

a α

β

c

γ simple cubic a=b=c α = β = γ = 90°

tetragonal a=b≠c α = β = γ = 90°

monoclinic a≠b≠c γ ≠ α = β = 90°

orthorhombic a≠b≠c α = β = γ = 90°

triclinic a≠b≠c α ≠ β ≠ γ ≠ 90°

rhombohedral a=b=c α = β = γ ≠ 90°

hexagonal a=b≠c α = β = 90° , γ = 120°

Figure 1.7  Crystal systems.

1) Cubic system: In this system all angles are equal to 90º and all of the axes are equal, such that we write a = b = c α = β = γ = 90o

1.3  Fundamentals of Crystallography

2) Tetragonal system: In this system all angles are equal to 90º and two axes are equal such that one axes is unequal to both of the equal axes. In this case we write a = b ≠ c α = β = γ = 90o 3) Orthorhombic system: In this system all three axes are at right angles but are unequal, such that a ≠ b ≠ c α = β = γ = 90o 4) Rhombohedral system: In this system all three axes are equal and no angle is equal to 90º, such that a=b=c α = β = γ ≠ 90o 5) Monoclinic system: In this system all three axes are not equal and two angles are equal to 90º and one angle is not equal to 90º, such that a ≠ b ≠ c α = γ = 90o β ≠ 90 o 6) Triclinic system: In this system all three axes are unequal in length and none of the angles between these axes is equal to 90º In other words, we write parameters a, b, and c as a ≠ b ≠ c No angle = 90° 7) Hexagonal system: In this system two angles are equal to 90º and one angle is equal to 120o. Similarly, the two axes are equal and one axis is unequal to both equal axes, such that a = b ≠ c α = β = 90o γ = 120o

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1.4  Molecular Vibrations A molecule consists of atoms that move relative to one another in such a way that the center of mass remains unaltered. The frequency of vibration is generally less than 1013 Hz. If a molecule is polyatomic in nature then the vibrational motion is described in terms of normal modes of vibration. These modes of vibration are not dependent on each other instead the different parts constituting a molecule are influenced by each normal mode of vibration. For a molecule that is nonlinear and has N atoms in total, the total number of normal modes of vibration are 3N – 6, but this number of modes becomes 3N – 5 if the molecules are linear. For a molecule with N atoms there should be 3N degrees of freedom corresponding to translation, vibration, and rotation. When the molecule is nonlinear in structure, it has three rotational degrees of freedom, while in the case of a linear molecule it has only two rotational degrees of freedom. Thus, the vibrational modes will be obtained by subtracting the translational and rotational degrees of freedom from 3N total degrees of freedom. Consider a diatomic molecule, which is always linear, to have only one possible mode of vibration that corresponds to the compressing and stretching of bonds joining the two atoms of the molecule. Let the molecule absorb an energy ΔE = hυ, which as a result causes it to become excited vibrationally. If one quantum of this energy is absorbed in the ground state, this will lead to fundamental vibration. Similarly, absorption of multiple quanta of energy causes excitement of first, second, and higher overtones. To probe the vibrational states of a molecule, a number of techniques can be used. The first and foremost technique is infrared spectroscopy. This is because the energy required for the vibrational transitions lies in the infrared region of the electromagnetic spectrum. However, Raman spectroscopy can also be applied in the study of vibrations.

1.5  Magnetism in Solids Magnetic materials in solids are very important because of their role in scientific development and industrial growth. The primary source of magnetism in materials is motion of electrons. Based on the behavior of these materials in response to an externally applied magnetic field, these materials can be classified into the following types: 1) Diamagnetic materials: When placed in an external magnetic field, these materials become magnetized in a direction opposite to that of the applied field. Diamagnetic substances do not possess any permanent magnetic moment but have the ability to expel the magnetic lines of an externally applied field. They have a temperature-independent negative value of magnetic susceptibility. 2) Paramagnetic materials: These substances possess a permanent magnetic moment. When they are placed in an external magnetic field the random

1.6  Optical Properties of Solids

magnetic moment becomes aligned in the direction of the magnetic field. In the absence of a magnetic moment, the material possesses a zero net magnetic moment. These materials have a positive but small value of magnetic susceptibility. 3) Ferromagnetic materials: These materials have magnetization even in the absence of an external magnetic field and that magnetization is called spontaneous magnetization. Spontaneous magnetization exists up to the temperature known as the Curie temperature. After the Curie temperature the substance becomes paramagnetic because a further increase in temperature causes thermal agitations, as a result of which there would be randomization in the magnetic moments. In ferromagnetic materials each atom acts as a tiny magnet pointing in the same direction. 4) Antiferromagnetic materials: These materials have magnetic moments of adjacent atoms that are antiparallel to each other. This behavior has been seen in mangenese oxide (MnO), in which the moments of adjacent atoms are antiparallel to each other in the absence of an external magnetic field and hence the net magnetization is zero. When a field is applied a small value of magnetization appears, which increases with an increase in temperature up to a specific temperature, called the Neel temperature. After this, magnetization decreases and the substance becomes paramagnetic. 5) Ferrimagnetic substances: These substances are the same as antiferromagnetic substances except that the adjacent antiparallel magnetic moments are unequal in magnitude. In other words, we may say that two sublattices have different values of magnetization. These materials also result in ferrites, which have high resistivity and anisotropic properties [4].

1.6  Optical Properties of Solids Understanding the optical properties of solids involves the investigation of the energy band structure, lattice vibrations, excitons, etc. Experimental study requires the observation of reflectivity, transmission, scattering, absorption, etc., and arrives at results for the dielectric constant and conductivity, for example. These properties have a direct dependence on the energy band structure [5]. Let us introduce complex optical conductivity and complex dielectric constant with the help of Maxwell’s equation and with the assumption of zero charge density, given by: ∇× H −

1 ∂D 4π = j c ∂t c

(1.4)

∇× E +

1 ∂B = 0 c ∂t

(1.5)

∇⋅ D = 0

(1.6)

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1  Basics of Material Characterization Techniques

∇⋅ B = 0 

(1.7)

Constitutive equations are given by D = εE B = µH j = σE 

(1.8)

From this equation the parameter ε defines the complex dielectric function. The field variables E and H can be given by ∇2 E =

εµ ∂ 2 E 4πσµ ∂E + 2  ∂t c 2 ∂t 2 c

∇2 H =

εµ ∂ 2 H 4π  σµ ∂H  + 2 ∂t c 2 ∂t 2 c

(1.9) (1.10)

In optical fields we define the solution of the equations for E in the form E = E0 e (

i K ⋅r − ω t )



(1.11)

where K is the propagation constant and ω is the frequency of light. The real part of K is the wave vector and the imaginary part represents the attenuation within the material. By substitution of this solution into the above equation for E , we obtain −K 2 = −

εµω 2 4π  iσµω  − c2 c2

(1.12)

For no loss, K becomes K0 and is a real quantity, such that K 0 =

ω εµ  c

(1.13)

For loss, we can express K as K 0 =

ω ε µ c complex

(1.14)

Here the complex dielectric constant is defined as εcomplex = ε + εcomplex =

4π  iσ = ε1 + iε2 ω

4π  iσ  εω  4π  i σ + = σ  4π  i  ω  ω complex

(1.15)

References

References 1 Ball, D.W. (2007). The electromagnetic spectrum: A history. Spectroscopy 22: 14. 2 Kelly, A. and Knowles, K.M. (2012). Crystallography and Crystal Defects. John Wiley & Sons. 3 Ashcroft, N. and Mermin, D. (1976). Solid State Physics. Harcourt College Publisher. 4 Globus, A., Packard, H., and Kagan, V. (1977). Distance between magnetic ions and fundamental properties in ferrites. Le Journal de Physique Colloques 38 (C1): C1–163. 5 Dressel Haus, M.S. (2018). Optical Properties of Solids in Solid State Physics. Springer.

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2 X-ray Diffraction 2.1 Introduction The most familiar method of determination of a crystal structure is the X-ray diffraction (XRD) technique. After the synthesis of a particular substance, the foremost step is to confirm that the prepared substance is the actual required substance. This is done through the use of XRD. The discovery of X-rays was accidental when, in 1895, Wilhelm C. Röntgen was doing work on cathode rays, and in 1901 he was awarded the Nobel Prize for his discovery [1]. It has been observed in solids that the interatomic distance is of the order of an angstrom. When investigating the concerned solid structure, we must have an electromagnetic spectrum with wavelengths of this order and thus energy in the order of thousands of electron volts. These energies are characteristic of X-rays [2] and are electromagnetic radiations that have higher energy and can pass through almost every object. A characteristic of X-rays originates when a metal target is bombarded with electrons of very high energy in order to remove the inner electron from an ion. To fill the vacancy, an electron from the higher energy level falls to the lower energy level, thereby releasing a distinctive X-ray. A complete description of the production of X-rays, properties of X-rays, the X-ray spectrum, factors governing the quality of X-rays, scattering of X-rays, etc., has been summarized by Weatherwax and Hamid [3, 4]. XRD is the most effective and nondestructive method that is applied for the characterization of crystalline substances. From investigation and analysis through XRD, we can obtain information about the structure of a material under ­investigation: phases, texture, grain size, strain, crystallinity, and many other parameters. XRD is the process in which the atoms of a sample, due to their unform spacing, lead to interference of a beam of X-rays. In this technique, XRD peaks are formed when there is a constructive interference of a monochromatic beam of X-rays that are scattered at particular angles from each set of lattice planes. The XRD pattern gives the fingerprint of the periodic arrangements in a sample [5]. In 1912 Laue found that crystalline solids behave as three-dimensional gratings when X-rays fall on them [6]. The X-rays originate in a cathode ray tube through the process of filament heating in order to release electrons, which are Practical Guide to Materials Characterization: Techniques and Applications, First Edition. Khalid Sultan. © 2023 Wiley-VCH GmbH. Published 2023 by Wiley-VCH GmbH.

16

2  X-ray Diffraction

accelerated towards the target and knock out inner electrons, thereby producing X-rays. The most commonly used target material is copper and the most common X-rays are Kα and Kβ. The X-rays are then filtered to obtain monochromatic radiation. The monochromatic radiation is then collimated and allowed to fall on the sample [5]. When the X-rays are diffracted from the sample they are detected and processed. XRD also allows the crystal structure to be modeled in order to obtain the unit cell [7]. The improvement and interest in XRD and particularly powder XRD are mainly due to the development of the Rietveld method [8]. The powder diffraction method was applied shortly after the discovery of XRD and the ­construction of the powder diffractometer, which is explained by Hull [9]. It was used to investigate simple substances like diamond and iron. Later on, this method was extended to include a wide spectrum of materials. The study of a non-cubic structure like the hexagonal structure of UCl3 and other uranium halides is explained by Zachariasen [10, 11]. In early ab initio approaches, two methods were used to obtain the structure from data, i.e., the direct method and the Patterson technique [12, 13]. After this there was development in the least squares methods of structure refinement and finally the Rietveld refinement [8]. Development of the Rietveld method was in response to need and was first performed with neutrons rather than X-rays. It has been reviewed by several researchers [14–16]. The applications of the Rietveld refinement using neutron data were then followed by its use in X-ray data [17, 18]. The Rietveld method using X-rays is now commonly used and is a most powerful tool, although it is not as precise as the method using neutrons [19]. The technique involves the phenomenon of diffraction, which is common in waves and is due to the interaction of light or waves with the object. It can be explained by considering an individual atom that has been interacting with X-rays. As soon as the X-rays fall on the atom, the electrons in that particular atom start oscillating about the mean position. When an electron decelerates, it causes the emission of X-rays. This process of absorption and emission of X-rays is called scattering. If we have a non-hydrogenic atom, then the same scattering can be considered from many electrons. This is shown in Figure 2.1, in which the atom contains many electrons. We will now look into the behavior of two waves that fall on the atom. Let us consider that one wave is scattered by electron A and another wave is scattered by electron B. The two scattered waves are in phase across the XX′ wavefront as the two have traveled the same distance and there is no path or phase difference. In this case, if we add these waves, the resultant wave will have the same wavelength and double the amplitude. If we now suppose another direction of scattered waves, as shown in Figure 2.1, the corresponding wavefront is represented by YY′. In this case there is a path difference and hence a phase difference. The condition is that if the path difference is not an integral multiple of the wavelength, then on adding these two waves the resultant wave will have less amplitude. In order to find the efficiency of an atom to scatter the waves, a quantity known as the atomic scattering factor [7] is defined, which is denoted by f such that

2.2  Bragg’s Law Y

A

D

Yʹ

X

B

C

Xʹ

Figure 2.1  Scattering of X-rays by an atom containing many electrons [7].

f =

amplitude of wave scattered by an atom amplitude of wavee scattered by one electron

This atomic scattering factor depends on the scattering angle as well as the wavelength of incident X-rays. Now if we extend these explanations to a group of atoms that are closely spaced, then it is obvious that there would be a contribution of scattered X-rays from each atom. The scattered waves from each atom may interfere constructively or destructively as the waves are in phase or out of phase respectively. In short, we can say that diffraction introduces scattering and interference and that diffraction is constructive interference of many waves that are scattered. The basic principle in the XRD technique is stated by Bragg’s law, which relates the three crucial parameters, namely the wavelength of incident radiation, lattice spacing, and the diffraction angle, in a particular crystal.

2.2  Bragg’s Law As explained earlier about the scattering of X-rays from many atoms, we find that if the waves scattered from different atoms can interfere constructively, then, accordingly, we have diffracted beams in particular and specific directions. These directions are dependent on the wavelength of the incident waves as well as the nature of the material under investigation and are best explained in terms of a law known as Bragg’s law [20]. Consider the scattering of waves from the atoms as shown in Figure 2.2, where it is supposed that an incident as well as a scattered beam make the same angle θ with the atomic planes. For the scattered waves to interfere constructively, they must be in phase and for this the path difference δ should be equal to the integral multiple of wavelength λ such that δ = nλ

(2.1)

We can write this equation in the form δ = DE + EC′ = 2EC′

(2.2)

δ = 2CE  sin θ

(2.3)

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18

2  X-ray Diffraction A

B

C

θ

D

Aʹ

E

θ

θ

Cʹ Bʹ

dʹ

B

Figure 2.2  Diffraction of X-rays.

where CE is the interplanar distance denoted by d such that δ = 2d sin θ

(2.4)

Comparing this with the equation δ = nλ , we have δ = 2d sin  θ

(2.5)

nλ = 2d sin θ

(2.6)

This is Bragg’s law and is the basic pillar in obtaining the diffraction pattern and hence various parameters of a crystal; n is the order of diffraction. The basic idea is clear from the equation as λ is known and θ can be obtained from experiment. Hence, this may lead to the calculation of the interplanar distance d. From this equation it is clear that the diffraction is possible only if λ < 2d, which provides the reason why optical waves cannot be used.

2.3  Von Laue Treatment: Laue’s Equation This is the more general method of treating the scattering of X-rays from individual atoms and then reuniting them to obtain the direction, having a diffraction maxima. Here we consider the scattering from two points A and B that are identical and at a distance r from each other. Suppose that n1 represents the unit vector in the direction of an incident beam, n2 represents the unit vector in the direction of a scattered beam and 2θ is the angle between n1 and n2 . From Figure 2.3 we can see that the path difference is given by Path difference = AM− BL = r .n1 − r .n2 = r .( n1 − n2 ) = r . N

(2.7)

where N = n1 − n2 is normal to the reflecting plane such that N = 2 sin  θ. The phase difference between rays is given by ϕ =

2π (r ⋅ N ) λ

(2.8)

2.3  Von Laue Treatment: Laue’s Equation n2 n1 Incident beam

r B

ed

er att

MA

sc

m

bea

L

Figure 2.3  Scattering from two identical points, A and B.

n2

n1

N = n1 − n2

θ θ Reflecting plane

Figure 2.4  Geometrical interpretation of the incident beam, scattered beam, reflecting plane, and normal.

Geometrical interpretations of the incident beam, scattered beam, reflecting plane, and normal are shown in Figure 2.4. It is to be noted that r may coincide with any of the three crystallographic axes. The condition that the rays will interfere constructively if and only if the phase difference is an integral multiple of 2π leads to the equations

2π (a ⋅ N ) = 2πh′ = 2πnh λ



2π (b ⋅ N ) = 2πk′ = 2πnk λ



2π (c ⋅ N ) = 2πl′ = 2πnl λ

(2.9)

where a, b, and c are the three crystallographic axes and h′, k′, and l′ are the three integers that differ from h, k, and l by the factor n. If we assume that A and B are nearest neighbors, then a, b, and c denote the interatomic distances along the respective crystallographic directions. Suppose that α, β, and γ are the angles that are formed by normal N with a, b, and c respectively. With this assumption a ⋅ N = aN cos α = 2 a  sin θ cos α  a ⋅ N = 2 a  sinθ cos α = h′λ = nhλ b ⋅ N = 2 b  sinθ cos β = k ′λ = nkλ c ⋅ N = 2 c  sinθ cos γ = l′λ = nlλ

(2.10)

19

20

2  X-ray Diffraction

These equations are known as Laue’s equations [21]. We know that in an orthogonal coordinate system we have the relationship for α, β, and γ such that 2 2 2 cos α + cos β + cos γ = 1

(2.11)

where cos α, cos β, and cos γ are the direction cosines of scattering normal. The three equations, numbered (2.10) and Equation (2.11), give us the value of α, β, γ, and θ if the value of h, k, l, and n are known. Thus, for the given plane, the equations numbered (2.10) determine the particular value of N and θ defining a scattering direction. For a fixed value of θ, cos α, cos β, and cos γ are proportional to h/a, k/b, and l/c, which is derived from the fact that the direction cosines normal to the arbitrary plane (hkl) are proportional to h/a, k/b, and l/c, and hence the arbitrary plane is the reflecting plane. The equation for the interplanar spacing for hkl planes is given as b c a d = cos α = cos β = cos γ  h k l

(2.12)

Combining with the equations in (2.10), gives 2d  sinθ = nλ 

(2.13)

where n denotes the order of reflection and is the greatest common factor among the integers h′, k′, and l′.

2.4  Experimental Techniques Satisfaction of Bragg’s law can be achieved through the variation of either θ or λ. The common techniques are (1) the Laue method, (2) the rotating crystal method, and (3) the powder method [21, 22].

2.4.1  Laue Method This method involves the process of holding the sample stationary and allowing a beam of continuous wavelengths of X-ray radiation to fall on the sample. There would be a number of planes of crystal selecting the proper wavelength so as to form a constructive interference and to obtain a geometric arrangement of bright spots. This method is usually preferred for the determination of orientation and symmetry of the single crystal. In this method, an X-ray beam is allowed to fall on the sample while it is lying at a fixed orientation. The beam has a range of wavelengths, and the  particular wavelength λ that satisfies Bragg’s law at the fixed position of the sample is automatically selected by the crystal.

2.4.2  Rotating Crystal Method This method involves the process of pointing a monochromatic beam of X-rays on to a single crystal sample rotating about an axis. Diffraction maxima are obtained if

2.5  Geometry and Instrumentation

and only if the sample orientation in relation to the incident beam satisfies Bragg’s law. The crystal is placed in the rotatory spindle and the photographic film lies on the inner side of a cylinder in such a way that the axis of the cylinder coincides with the axis of rotation. When a monochromatic beam having wavelength λ is allowed to fall on the crystal it is then rotated in order to obtain the diffraction condition. The diffraction pattern is then obtained on a screen.

2.4.3  Powder Method This method is used to determine the structure if the sample is in the form of a powder. The powder consists of a number of fine constituents. The constituents are oriented randomly and when a monochromatic beam falls on the sample there will always be orientations corresponding to the wavelength of the incident beam that satisfy Bragg’s law. In this way, a pattern of XRD can be obtained that can be used to determine various structural parameters of the materials under investigation.

2.5  Geometry and Instrumentation An X-ray diffractometer consists of three main components [23] that lie over the circumference of a circle, called the focusing circle (see Figure 2.5). These components are: ● ● ●

X-ray tube Sample holder Detector

An X-ray originates in the X-ray tube. In fact, X-ray spectra are released that have components like Kα and Kβ. Kα further consists of Kα1 and Kα2, where Kα1 has focusing circle detector x-ray source diffractometer circle

x-rays brought to focus here θ specimen

2θ

Figure 2.5  Geometry of an X-ray diffractometer [7].

21

22

2  X-ray Diffraction

a little bit shorter wavelength while the intensity is twice that of Kα2. In the production of X-rays, electrons are produced by heating the filament, which are then accelerated towards a target in order to produce X-rays. Copper is the most commonly used target for diffraction, having a Cu Kα radiation  of  1.5418 Å. These X-rays are allowed to fall on the sample after proper collimation. The sample and the detector are rotated in order to obtain the data. The data are obtained in such a way as to satisfy Bragg’s law, in order to get the constructive interference and thus peaks of intensity. Finally, a detector receives the signal, processes it, and transforms it to the count rate. The X-ray diffractometer is constructed to take into consideration the fact that the sample is rotating in the path on which X-rays are falling, at angle θ, and the detector is in a position on an arm to receive the diffracted rays. The detector can be rotated to an angle of 2θ. The component called a goniometer is applied to maintain the angle of the detector at a constant value and to rotate the sample [5]. The angle between the plane of the sample and the source of the X-ray is θ, known as Bragg’s angle, while the angle between the detector and the projection of the X-ray source is denoted as 2θ. Due to this reason, the diffraction patterns are also called θ–2θ scans. There is another component called the diffractometer circle, which is centered in the specimen with a source and detector at the circumference. The radius of this circle is fixed and the circle is also called the goniometer circle. However, the radius of the focusing circle is not fixed and can be increased as the angle 2θ is decreased. In a typical scan the range of the 2θ is from 30o to 140o, but we can go beyond these values depending on which crystal structure is under investigation. A full description of the components of the diffractometer (see Figure 2.6) is given: a) Source: As explained earlier, an electron is accelerated and directed towards the target inside an evacuated tube. The voltage and the current are variable and a particular voltage is applied depending on the target. The electron, having sufficient energy when it strikes the target, removes the electron from the K shell and as a result an electron from the outer shell falls to the K shell, thereby releasing the X-ray. The emitted radiation is also called fluorescent radiation. If the atomic number of the specimen of the sample is slightly less than the target, then the X-ray is absorbed. This causes a decrease in the intensity of the diffracted beam and an increase in the fluorescent radiation. Due to this situation the signal will be increased. To avoid fluorescence from the sample occuring, another type of radiation may be applied. In most diffractometers the wavelength of the X-ray is fixed [7]. b) Sample: The preparation of the sample is one of the basic requirements of the analysis and investigation. There are a number of books that deal with sample preparation [24–27]. In diffraction experiments, the crystalline powder is spread over a substrate. In actual practice, the specimen should consist of a number of small grains oriented in different directions. The mixing of the sample is also an important factor in the analysis, such that the specimen should represent the entire sample. In the case of θ–2θ scans, the source is kept fixed with respect to the sample. When the sample is in powder form there will always be grains oriented in such directions that satisfy Bragg’s law and allow the diffraction from a particular set of lattice planes. Each set of lattice planes in a crystal have spacings

2.5  Geometry and Instrumentation X-ray source receiving slit

Soller slits divergence slit

anti-scatter slit

Soller slits

to detector

specimen

Figure 2.6  Arrangement of various components in a diffractometer [5].

of d1, d2, d3,… and angles of θ1, θ2, θ3,…, where the angle θ is observed to increase as the spacing decreases so as to satisfy Bragg’s law. The intensity at each angle is recorded in order to form the XRD pattern. c) Optics: When coming out of the source an X-ray beam goes through a series of slits, called Soller slits, that collimate it. These slits are made up of heavy metals such as molybdenum. The purpose of a divergence slit is to define the extent of divergence of an incident beam. When a beam falls on the specimen it gets diffracted and again passes through another series of slits. The antiscatter slit decreases the background radiation and makes sure that X-rays reaching the detector are from the specimen. The beam finally passes through the receiving slit that converges and defines the extent of the convergence of the beam as it enters the detector. The arrangement of these elements is shown in Figure 2.6. An increase in the width of the slit will lead to a rise in the highest intensity of reflected rays in a pattern of diffraction, but it has some limitations, like the loss of resolution. However, the integrated intensity I, which is the peak area, does not depend on the slit width. An alteration of the slit width cannot change the ratio of integrated intensities corresponding to the two peaks, I1/I2, but may change the ratio of maximum intensities, M1/M2 (see Figure 2.7). Another set of Soller slits is also present after the receiving slit and before the monochromator (see Figure 2.8). The monochromator is used to obtain monochromatic radiation and to separate the unwanted radiation. In earlier times this process was carried out through the use of filters, but in modern times the monochromator used is monochromating graphite crystal, which is placed in the way of the diffracted beam in order to suppress the unwanted fluorescent radiation and pass only Kα radiation. Since the use of the diffractometer has emerged with time, there has therefore been a parallel improvement in the optics of the system. The improvements are directly based on the collimation of the beam. A method that was first proposed was to use optical fibers, also known as polycapillary fibers, which were used to transfer a large number of X-rays from the source to the specimen. These fibers are now available commercially. Another development was the use of multilayer mirrors for focusing and collimating the X-ray beam.

23

2  X-ray Diffraction

M

Intensity

24

FWHM

I

Diffraction angle 2θ

Figure 2.7  A peak in the XRD pattern [7]. curved graphite monochromator

anti-scatter slit

detector

Soller slits receiving slit

specimen

Figure 2.8  Arrangement of a diffractometer along with the monochromating crystal in the way of a diffracted beam [7].

d) Detector: An X-ray diffractometer can be associated with any of the three types of detectors, i.e., gas proportional detector, scintillation detector, and solid-state detector. The gas proportional detector has a noble gas inside a tube through the center of which passes a positively charged thin tungsten wire. As soon as the X-ray enters the tube it becomes absorbed, thereby causing the ejection of photoelectrons. The released electron is attracted towards the positively charged wire, thus giving birth to a charge pulse. When the initial X-ray enters the filled tube, it causes the production of a few hundreds of electrons and a small amount of

2.5  Geometry and Instrumentation

charge is created. The amplification can be achieved if the positive potential of the tungsten wire is high, such that the initial electrons produced are accelerated to collide with other atoms of the noble gas to produce further ionization and hence a large increase in charge. The collected charge is proportional to the energy of an incident X-ray photon and this proportionality is crucial in distinguishing X-ray photons having different energies. In the case of a scintillation detector, the X-rays cause a suitable crystal to exhibit fluorescence. A scintillation is produced as the incident photon is absorbed. The quantity of scintillation produced is directly related to the intensity of the X-ray, which is measured with the help of a photomultiplier. Moreover, the size of pulses is directly related to the energy of the absorbed photon. This detector is more efficient, but the energy resolution is not better than that of a proportional detector or a solid-state detector. Energy resolution is measured for the width of the peak, where a narrower peak means a higher resolution. A solid-state detector is a single crystal having an intrinsic silicon sandwiched between a p-type layer and an n-type layer. The importance of the intrinsic silicon in the detector is due to the best electrical resistivity that is too small at low temperatures and a small number of electrons having sufficient energy to move through the band gap. This feature is very important because of the fact that only electrons that are excited by the X-ray photon can pass through the band gap. Intrinsic silicon is usually impure and because of that it acts as a p-type semiconductor. To react to this, lithium (Li) is added to it with a much higher concentration on one side of the silicon crystal and a low concentration on the other side. This Li is in the state of Li+ such that free electrons are there in the crystal (Li→Li+ + e-). One side of the crystal with a high concentration of Li+ acts as an n-type layer while the other side acts as a p-type layer. In the middle, the crystal becomes intrinsic as the free electrons may recombine with the excess holes that were already present there. When the X-rays undergo interaction with silicon there would be excitation of electrons to the conduction band from the valence band and thereby creation of an electron–hole pair. On reversing the bias, the electron–hole pair can be isolated in order to determine the charge pulse. The number of electrons or holes produced are directly related to the energy of the X-ray. The number of electron hole pairs produced due to absorption of one photon is given by n =

energy  of  photon energy required to create one  pair

For example, an absorption of a Cu Kα photon having energy of 8.04 keV can create 2116 pairs. Successive photons produce slightly varying numbers, like 2110 or 2121. This difference causes variations of output pulse size and hence peak width. However, the energy resolution is better than a proportional detector or scintillation detector. The solid-state detector of silicon containing Li needs to be placed at the temperature of liquid nitrogen in order to minimize the thermal excitation of electrons at room temperature and to minimize the thermal diffusion of Li. It is said that a solid-state detector is more efficient over a wide range of energies. A germanium (Ge) detector can be used in the case of γ-rays and high-energy X-rays. Ge has the advantage that it is pure and there is no need for the Li to drift. Moreover, a Ge

25

2  X-ray Diffraction

detector has a high-energy resolution as well as a high signal-to-noise ratio as ­compared to the silicon detector. If we compare the absorption of the Cu Kα photon having energy of 8.04 keV, there may be the creation of 2772 pairs. The good values of resolution for a pure Ge detector are 145 eV and for a silicon (Li) detector are 165 eV. A better detector has a smaller resolution. Although a Ge detector has the best resolution, it does have disadvantages since silicon detectors are more efficient than Ge detectors in the range 1–20 keV. This is because the energies of the X-ray that are used in the diffraction process fall in this range.

2.6  Standard XRD Pattern

30

40

50 2θ°

Figure 2.9  XRD pattern of PFO [28].

60

(224) (400)

(132) (024) (312)

(114)

(220) (221)

(022) (202) (113)

(021)

(020)

(111)

(112)

As an example, let us consider the XRD plot of a compound like PrFeO3 (PFO) [28], as shown in Figure 2.9. As we can see in this pattern, there are a number of peaks, which are also called reflections. In an XRD pattern the intensity of the peak is plotted along the Y-axis and the measured diffraction angle 2θ along the X-axis. Each peak in the pattern represents the diffraction from a particular set of planes available in the sample under investigation. However, we can observe from the figure that the intensity of different peaks is different. This intensity depends on factors such as structure factor, incident intensity, and slit width. The position of the peaks is determined by the structure of the crystal and also the wavelength of the X-ray. It has been found that a crystal with less symmetry has a greater number of peaks [29]. The width of the peak, usually defined as full-width at half-maximum (FWHM) (see Figure 2.7), is applicable in the determination of parameters like crystallite size and strain. While taking the data in the diffraction experiment, the 2θ value is varied over a suitable range in most of the crystals, but if the crystal is totally unknown then the value of 2θ needs to be varied over the entire range. Nowadays there are certain kinds of software that allow the calculations of strain, crystallite size, lattice parameters, etc. There is still continuous development and advancement in software that aims to make life easier. The most important and crucial point to be noted is that the instrument must be aligned and calibrated in a proper fashion.

(110)

26

70

2.7 Applications

27

2.7 Applications XRD has applications in most fields of everyday life. Here a few have been included to give an idea of its importance.

2.7.1  Orientation of Single Crystals The properties of a polycrystalline crystal can be obtained through isolation of a single crystal. Single crystals are anisotropic so the orientation of the crystal has much importance in determining its properties such as, resistivity and yield strength. Crystals can be grown for research purposes or device purposes and accordingly we can produce quality crystals. Crystal orientation can be obtained through the use of the back-reflection Laue method, the transmission Laue method, and the diffractometer [30]. The process for a diffractometer follows. As the diffractometer makes use of monochromatic radiation, so a single crystal allows reflection if and only if the inclination of a set of planes to that of an incident beam is θ, which satisfies Bragg’s law. When the counter, which is fixed at angle 2θ, points out that reflection has occurred, then the inclination of reflecting planes to any chosen line or plane on the crystal surface is known from the position of the crystal. This needs two kinds of operation. First is the rotation of the crystal till a position is achieved that allows reflection and second is the location of the pole of the reflecting plane on a stereographic projection based on the known rotation angle. A diffractometer can be varied in different ways and is dependent on the type of goniometer used for holding and rotating the sample. The holder gives three possible and available rotation axes, as shown in Figure 2.10. One of the axes coincides with that of the axis of the diffractometer. The second axis AA′ is in the plane of the incident beam I, diffracted beam D, and the tangent to the specimen surface. The third axis BB′ is normal to the surface of the specimen. Let us consider determination of the orientation of a cubic crystal. For such a type of crystal the {111} planes are best reflectors and Aʹ there are four sets of these planes. The B β 2θ value for the reflection from {111} θ can be determined from the known I C value of spacing in {111} planes and θ the wavelength of incident radiation. A 2θ γ The counter can then be fixed at this D determined value of 2θ. After that, the N holder is to be rotated about the axis of the diffractometer such that the incliBʹ nation of its surface and the axis AA′ is counter equal to the incident beam as well as the diffracted beam. At this point the Figure 2.10  Crystal rotation axes for the diffractometer [30]. holder that carries the sample is to be

c02.indd 27

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28

2  X-ray Diffraction

fixed with respect to the diffractometer axis. Then, by making a rotation about the BB′ axis, the one edge of the specimen is made parallel to the diffractometer axis and this represents the initial position. Now the crystal is rotated about AA′ and BB′ until an inclination is achieved that leads to the observation of reflection on the counter. When the reflecting position is observed, the normal to one set of planes {111} coincides with CN, which is the line bisecting the angle made by incident beams with diffracted beams. The stereographically plotted poles of the diffracting planes are shown in Figure 2.11. The projection is done on a plane that is parallel to the specimen surface while the N–S axis of projection is parallel to the reference edge. If the crystal is rotated about the axis BB′ through an angle β, then the projection is also rotated about its center through angle β. The direction of CN can be represented by the pole N′. This is first at the center of the projection and moves through γ degrees along the radius if the crystal is rotated about the AA′ axis through γ degrees. Our purpose is to coincide N′ with a {111} pole and to point out the location of the latter on the projection. This can be done by varying γ for fixed β. The projection is covered point by point along a series of radii. It has been seen that it is sufficient to investigate only one quadrant as there would be at least one {111} pole in any one quadrant. If one pole gets located, the second is determined from the knowledge that it lies 70.5o from the first one. Two {111} poles are sufficient to fix the orientation of the crystal, but determining a third can be regarded as a check. Once the

N

β

PROJECTION

Nʹ

γ W

E

S

Figure 2.11  Plotting method applied while determining orientation of the crystal [30].

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2.7 Applications

orientation of the crystal is determined, it is then rotated in some other orientation, such as in the direction of the incident beam, for further X-ray examination in another direction and subsequent cutting along a particular selected plane.

2.7.2  Structure of Polycrystalline Aggregates In the previous section, we were concerned about the orientation of a single crystal, but in actual practice substances are produced in polycrystalline aggregate form consisting of many crystals. The properties corresponding to the “single-phase aggregate” are determined through the way the single crystals are put together in order to form a composite. The structure of an aggregate means the size, orientation, and quality of grains composing the aggregate. The properties of the multiphase aggregate depend on the properties corresponding to each phase and on the pattern in which the phases present as an aggregate. These properties are discussed separately. 1) Crystal size a) Grain size: The determination of grain size has an impact on the exploration of many other properties. In commercially available metals and alloys, the grain size is in the range from 1000 to 1 μm. These values are arbitrary and extreme and in a typical case the value lies in the range from 100 to 10 μm. XRD can yield information about the grain size. The transmission or a back-reflection pinhole photograph that is formed with the use of filters is better for this purpose. In the case of a back-reflection, the surface of the specimen is etched so as to remove any surface layer that is disturbed and may be present, as much of the diffracted radiation originates in the thin surface layer. There is a change in the pinhole photographs due to reduction of the grain size, as shown in Figure 2.12. The basic observation is to see the number of grains causing the diffraction and this number is proportional to a cross-sectional area of the beam of incidence as well as to the penetration depth (in the case of back-reflection) and specimen thickness (in the case of transmission). If the grain size is rough, only some of the crystals take part in diffraction and the photograph is made up of superimposed patterns due to the presence of white radiation. One pattern is due to diffraction from each crystal. This is shown in Figure 2.12 (a). If the grain size is quite fine, there would be an increase in the spots and the spots lying on the Debye rings would be stronger than others because they are due to the intense characteristic part of the incident radiation – this is shown in Figure 2.12 (b). If the grain size is still finer, the spots there would only be Debye rings, visible as shown in Figure 2.12 (c). A still finer grain size leads to smooth Debye rings, as shown in Figure 2.12 (d). There are a number of approaches used to estimate the grain size. Formulation of an equation relating the observed number of spots in the Debye ring and the grain size can be helpful, but this would involve a number of approximations and the result would probably not be accurate. Therefore, the best way would be to make an estimation by obtaining a sequence of specimens having known grain size numbers and then to obtain a sequence of standard photographs. A grain size number can then be obtained by comparing the diffraction pattern with any of the standard photographs. The comparison can only be made if and only if both are obtained under similar conditions. As soon as the

29

30

2  X-ray Diffraction (a)

(b)

(c)

(d)

Figure 2.12  Back-reflection patterns of recrystallized specimens [30].

grain size approaches a value between 10 and 1 μm, the Debye rings try to lose their spotty nature and become continuous. Between these values and 0.1 μm, there would be no change in the pattern due to diffraction. At about 0.1  μm, detectable broadening starts due to the reduced crystal size. Thus, in the range of 10 (or 1) μm to 0.1 μm XRD cannot be used for any change in grain size for a normal-sized incident beam. With the use of microbeam techniques, X-ray lines are spotty, even down to grain sizes smaller than those mentioned, thereby allowing for the observation and counting of spots from individual grains [31]. b) Particle size: For the size of an individual crystal less than 0.1 μm, particle size comes into play, which leads to broadening of the Debye rings. The strength of the broadening is given by the relationship [30] B =

0.9λ  t cos θ

(2.14)

2.7 Applications

where B represents the broadening corresponding to the diffraction line determined at half of its highest intensity and t represents the diameter of a crystal particle. Even if the crystal size is greater than 0.1 μm, all the diffraction lines are still associated with a measurable breadth. The breadth B in Equation (2.14) corresponds to the extra breadth due to only the particle size effect and B is necessarily zero if the particle size is greater than 0.1 μm. The main objective in determination of particle size is to find B from the measured breadth BM of the diffraction line. Warren’s method is the simplest of all methods applied for this purpose. The one that is not known is combined and mixed with a standard having a particle size greater than 0.1 μm and gives a diffraction line near the line of the unknown one to be determined. A diffraction pattern is obtained from the mixture using a Debye camera or diffractometer. The pattern consists of sharp lines for the standard case and broad lines for the unknown case. If BS is the breadth measured at half of the maximum intensity of the line for the standard case, then B is given by [32] B 2 = BM2 − BS2 

(2.15)

After inserting the determined value of B in Equation (2.14), we can obtain the particle size t. Apart from Warren’s method there are several other methods for determining particle size either involving the Fourier analysis of diffraction lines for the unknown as well as the standard or determining the integral breadth of a diffraction line instead of the breadth at half of the maximum intensity. This integral breadth is defined as the integrated intensity divided by the maximum intensity. In other words, it is the width corresponding to a rectangle having the same area and height as that of an observed line. It has emerged from various determinations that the difficulty in experimental measurement of particle size using line broadening rises with the size of the particle under investigation. In a broader sense we can say that crude measurement serves in the range 0–500  Å but the best technique is for the range 500–1000 Å. With the use of a camera technique for measurement, the upper maximum size measurable by line broadening is placed at 1000 Å while the diffractometer takes the limit to 2000 Å. Therefore, careful investigation is required. 2) Crystal quality Of the many imperfections, the one of most concern is strain, as it is the characteristic of the cold-worked state of metals and alloys. If there is an observance of some deformation in the polycrystalline portion of a metal, then there is a change in the shape of the grain. The change in the shape does not depend only on application of force but on the fact that each grain has contact with neighbors. Therefore, the grain is restrained by its neighbors. Thus, there is a presence of portions of bend and tension in a deformed grain in an aggregate. Such stresses are called residual stresses or microstresses as they vary from one grain to another. When stress is uniform over a very large distance, it is called a macrostress. The effect of strain on X-ray reflection is shown in Figure 2.13,

31

32

2  X-ray Diffraction

where (a) represents the unstrained grain with planes having equal spacing d0. If the grain is under the effect of a uniform strain that is also at right angles to the reflecting plane, then the spacing becomes larger than d0 and the corresponding change is shown in Figure 2.13 (b). In this case the diffraction line shifts towards the lower angle. In the case of a nonuniform strain, as shown in Figure 2.13 (c), the grain is bent such that on the top the spacing is greater than d0 while on the bottom the spacing is less than d0. This type of strain causes a number of sharp diffraction lines and the sum of these lines is represented by the broadened solid curve. In fact, only the broadened curve is obtained in experiments and a relation between nonuniformity of the strain and the broadening of the curve can be obtained if we differentiate using Bragg’s law. Therefore, we have the relation [30] b = ∆2θ = −2

∆d tan θ  d

(2.16)

In this equation b represents the extra broadening over the instrumental breadth of line because of fractional spacing Δd/d. The equation serves as the base that permits the variation in strain, Δd/d, that is to be calculated from the observed broadening. Since this value of Δd/d contains both tensile strain as well as compressive strain, it

(a)

CRYSTAL LATTICE

DIFFRACTION LINE

d0

NO STRAIN

(b)

UNIFORM STRAIN

(c)

NONUNIFORM STRAIN

2

Figure 2.13  Influence of the presence of strain on width and position [30].

2.7 Applications

needs to be divided by a factor of two in order to obtain tensile strain only or compressive strain only. The value of strain obtained is multiplied by the term E, called the elastic modulus, in order to get maximum stress. Whenever an annealed alloy or metal is cold worked, there would be broadening in the diffraction lines. There may be three main causes for the line broadening, i.e., nonuniform strain, small particle size, and stacking faults. However, there is no material to which all the broadening can be associated to the fine particle size. In actual practice, it is not possible to imagine cold work fragmenting the grain to an extent that may lead to particle size broadening without causing any strain. We can consider the processes of recovery, recrystallization, and grain growth. If the cold-worked metal or alloy is annealed, recovery takes place at a low temperature, recrystallization at a higher temperature, and grain growth at a still higher temperature. During the recovery the macrostress as well as the residual microstress are decreased in magnitude, but the hardness and strength are supposed to be very high. During the recrystallization process new grains are considered to be formed, eliminating the residual stress and abruptly decreasing the strength and hardness. The main cause of the broadening is nonuniform strain because of the residual microstress and we usually prefer to find the broad diffraction lines characteristic of cold-worked metal that partially sharpen during the process of recovery. As soon as the recrystallization process starts the lines begin to attain maximum sharpness and finally during grain growth the lines start to become spotty.

2.7.3  XRD in the Pharmaceutical Field and Forensic Science X-rays play a role in the formulation by giving the details of morphology, crystallinity, and quantity of components of the mixture. X-ray analysis is also applied in the characterization of compositions of pharmaceuticals. Since XRD is the direct consequence of the crystal structure that is available in the pharmaceutical product under investigation, so the various parameters associated with the crystal structure can be obtained. As an example, we can quote that when a drug is separated, an indexed pattern is used in the investigation of the structure and to lock a patent. For a formulation consisting of a multicomponent, the exact percentage of the active ingredients can be analyzed. Powder XRD continuously supports the upliftment in the wide varieties of the pharmaceutical field. It has applications as an active pharmaceutical ingredient (API) characterization as well as API identification in the process of manufacturing and developing drugs. Powder XRD is also helpful in the investigation of structures concerning variable hydrates. These are crystalline species having a nonstoichiometric amount of water within a channel in the lattice and the amount of water inside the variable hydrate is dependent on the relative humidity environment. A change in the unit cell size due to the availability of water can be seen through comparison with the powder XRD pattern obtained for species having different relative humidity environments. In some systems there is a rise in the specific lattice parameter that is directly dependent on the amount of water in the hydrated structure [33].

33

34

2  X-ray Diffraction

Powder XRD is the most efficient system because of its nondestructive nature and thus is the most favorable technique for the analysis of forensic specimens. An important example of this is the investigation of narcotics seizures, which are supposed to contain additional excipients. The addition of these leads to a profile that is used in identification. Some specimens contain excipients in the hydrated form or the polymorphic form and may be differentiated through the powder diffraction technique. The results obtained for cooperation between a police forensic laboratory and the academic institutes point to the possibility of applications of single crystal XRD as a supreme approach for identification of designer drugs in forensic fields. XRD in the forensic field is used for contact traces like hair, soils, strains, and glass fragments. The majority of the samples obtained and recovered in the forensic field are crystalline and XRD is the best-suited technique for the identification of these samples. Therefore, we can simply say that XRD can aid criminal investigations. XRD is also used in fabric examination as it limits the examination to one or two techniques, thereby requiring minimum expertise and time [34]. Apart from these XRD analyses of forensic specimens, document examination, textile fibers examination, pints and pigments examination, and explosive examination can also be carried out.

2.7.4  XRD in the Geological Field XRD is also the principal tool in geological investigations [35]. Each mineral is associated with a characteristic crystal structure and thus a characteristic XRD pattern. Thus, the XRD technique serves as the identification tool for geologists and nowadays they are totally dependent on it. The top layer of soil gives an idea about the properties of soil and this technique is useful for its quantification. The diffraction pattern obtained for an unknown can be obtained and compared with the data available in the library. The data base is helpful in quantitative as well as qualitative analysis of minerals. In addition to this, the XRD pattern also gives a clue to the extent of crystallinity, grain size, and strains. Rocks vary in composition and sampling of the rock has a crucial role in an investigation. The composition of minerals is subjected to variations because of factors like a hydrological change in temperature. Such variations are seen through the use of XRD and thus support the establishment of evolution. XRD data obtained also make the manufacture of jewellery on demand possible.

2.8  Examples and Illustrations 2.8.1  XRD Data and Interpretation in the PrFe1–xMnxO3 System The XRD print of the PrFe1–xMnxO3 system with x = 0.0, 0.1, 0.3, and 0.5 being the concentrations of the manganese (Mn) ion doped in PFO is shown in Figure 2.14. The pristine PFO has an orthorhombic structure and Pbnm as the associated space group. The Wyckoff position of Pr is 4c (x y 1/4), the Wyckoff position of Fe/Mn is

2.8  Examples and Illustrations

Figure 2.14  XRD print of PrFe1–xMnxO3 samples at different concentrations of Mn along with an inset suggesting the shift of the (112) peak [36].

4b (1/2 0 0), and the Wyckoff position of O2 is 8d (x y z). The XRD data predict that the compound has a single phase and the Pbnm space group is associated with the orthorhombic structure. Moreover, it has also been observed that there is no change in the structure or the symmetry when the sample is doped up to the Mn concentration of x = 0.5. The inset of Figure 2.14 shows that we also observe a shift in the peak in the direction of increasing 2θ as the concentration of the Mn ion is increased, thus depicting the presence of strain in the sample. This strain can be due to the occurrence of the Jahn‐Teller (JT) distortion phenomenon in the doped compound. The line width and the change in the different reflection peaks intensity are suggested by increasing the concentration of the Mn ion. As an illustration, consider the peak (020) that is not associated with the original structure and thus may be due to some changes in structure happening at high Mn ion contents. The structural parameters corresponding to different concentrations have been summarized in Table 2.1. From the variations in the lattice parameters with different concentrations of the Mn ion, it has been observed that the two lattice constants a and c reduce and one lattice constant b improves as the Mn content is increased. However, if we consider

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2  X-ray Diffraction

the volume of cells, there has been a small decreasing trend with a rise in the Mn amount. The interplanar distance changes slightly and we can also observe this from XRD patterns that the angle (θ) at which the peak (θ) occurs does not change too much when the sample is doped with Mn ion. The change in the lattice constants a, b, and c/√2 as a function of Mn content is shown in Figure 2.15. It is also suggested that up to the doping concentration of x = 0.5, the structural parameter c/√2  >  a and there is a decrease in a and c/√2 with an increase in the Mn ion. However, the change in the former is less than in the latter. Moreover, the compound PrFe1–xMnxO3 has been observed to demonstrate an O‐type (a