Electron diffraction patterns provide information about the structure of crystalline materials by revealing the diffraction of electrons from crystal planes. Reflections in patterns represent the constructive interference of electrons from different crystal planes, with their position and intensity indicating the d-spacing of planes and occupation/symmetry of the structure. By indexing patterns, determining the d-spacings of reflections and relating them to crystal planes via Braggs law, the zone axis and unit cell parameters can be determined.
This is a tutorial on indexing diffraction patterns, deriving reflection conditions from SAED, derving point groups from CBED and combining both to find the space group. The slides contain exercises, the page to work on is at the end of the presentation and should be printed first to be able to measure on that page.
The document discusses techniques for measuring thin film thickness using a spectrophotometer. Spectrophotometers measure the interference patterns of light passing through thin films to determine optical thickness and refractive index. Key parameters like transmission, absorption coefficient, and wavelength are used to calculate film thickness and properties. Measurements are taken across the 300-900nm range to analyze interference spectra and optical band gaps of materials like SnO2 and ZnO.
Colour centres are point defects or defect clusters in crystal lattices that cause the material to change color. They occur when electrons or holes become trapped at defect sites. Common examples are the F-centre in alkali halides, which forms when an electron is trapped at a halide ion vacancy, and the H-centre and V-centre in alkali halides, which involve trapped holes. Defect clusters can also form through the interaction of multiple point defects, such as pairs or groups of F-centres. The defects cause color changes by absorbing visible light and exciting trapped electrons or holes to higher energy states.
Molecular Orbital Theory For Diatomic Species Yogesh Mhadgut
The Molecular orbital theory was put forward by Hund and Mulliken in 1930 and was later develop further by I. E-Lennard-Jones and Charles Coulson.
On the basis of valance bond theory in molecule atomic orbitals retain their identity but according to molecular orbital theory atomic orbitals combine and form new molecular orbitals
TEM Winterworkshop 2011: electron diffractionJoke Hadermann
1) The purpose of the lecture is to teach how to index and determine cell parameters from SAED patterns, determine possible space groups and point groups, and solve simple structures from PED patterns.
2) Example materials used are aluminum and rutile-type SnO2. The slides and exercises for determining unknown cell parameters from SAED patterns are provided.
3) The document provides a step-by-step example of determining the cell parameters of aluminum from its SAED patterns, starting with no prior knowledge of the material. The correct cell is determined to be cubic with a=b=c=4.06 Angstroms.
The document discusses estimating crystallite size using X-ray diffraction (XRD). It provides a brief history of XRD, introducing key concepts like the Scherrer equation published in 1918 relating crystallite size to peak broadening. It discusses factors that contribute to observed peak profiles, including instrumental broadening, crystallite size, microstrain, and others. It also covers considerations for accurately analyzing crystallite size such as deconvoluting instrumental and sample contributions, and effects of crystallite shape, size distribution, and the measurement technique.
The document provides information about x-ray diffraction training opportunities and classes at MIT, including safety training requirements. It describes x-ray diffraction data collection classes on November 6th and 13th that cover the Rigaku powder diffractometer and PANalytical X'Pert Pro respectively. It also lists several data analysis workshops in November and December focusing on programs like Jade and analyzing thin films and texture. The document then provides a basic overview of x-ray diffraction principles including Bragg's law and how diffraction provides information about crystalline structure.
- Grazing incidence X-ray diffraction (GIXRD) is a technique that allows analyzing thin film samples by varying the incident angle of the X-rays to change their penetration depth.
- GIXRD provides enhanced signals from thin film layers compared to conventional XRD and helps distinguish thin film peaks from substrate peaks. It can also be used to analyze phases, stress, and crystal structure as a function of depth.
- Examples showed how GIXRD allowed analyzing phase composition and residual stress at different depths in thin film solar cell structures and revealed surface treatment effects in a stainless steel sample.
This is a tutorial on indexing diffraction patterns, deriving reflection conditions from SAED, derving point groups from CBED and combining both to find the space group. The slides contain exercises, the page to work on is at the end of the presentation and should be printed first to be able to measure on that page.
The document discusses techniques for measuring thin film thickness using a spectrophotometer. Spectrophotometers measure the interference patterns of light passing through thin films to determine optical thickness and refractive index. Key parameters like transmission, absorption coefficient, and wavelength are used to calculate film thickness and properties. Measurements are taken across the 300-900nm range to analyze interference spectra and optical band gaps of materials like SnO2 and ZnO.
Colour centres are point defects or defect clusters in crystal lattices that cause the material to change color. They occur when electrons or holes become trapped at defect sites. Common examples are the F-centre in alkali halides, which forms when an electron is trapped at a halide ion vacancy, and the H-centre and V-centre in alkali halides, which involve trapped holes. Defect clusters can also form through the interaction of multiple point defects, such as pairs or groups of F-centres. The defects cause color changes by absorbing visible light and exciting trapped electrons or holes to higher energy states.
Molecular Orbital Theory For Diatomic Species Yogesh Mhadgut
The Molecular orbital theory was put forward by Hund and Mulliken in 1930 and was later develop further by I. E-Lennard-Jones and Charles Coulson.
On the basis of valance bond theory in molecule atomic orbitals retain their identity but according to molecular orbital theory atomic orbitals combine and form new molecular orbitals
TEM Winterworkshop 2011: electron diffractionJoke Hadermann
1) The purpose of the lecture is to teach how to index and determine cell parameters from SAED patterns, determine possible space groups and point groups, and solve simple structures from PED patterns.
2) Example materials used are aluminum and rutile-type SnO2. The slides and exercises for determining unknown cell parameters from SAED patterns are provided.
3) The document provides a step-by-step example of determining the cell parameters of aluminum from its SAED patterns, starting with no prior knowledge of the material. The correct cell is determined to be cubic with a=b=c=4.06 Angstroms.
The document discusses estimating crystallite size using X-ray diffraction (XRD). It provides a brief history of XRD, introducing key concepts like the Scherrer equation published in 1918 relating crystallite size to peak broadening. It discusses factors that contribute to observed peak profiles, including instrumental broadening, crystallite size, microstrain, and others. It also covers considerations for accurately analyzing crystallite size such as deconvoluting instrumental and sample contributions, and effects of crystallite shape, size distribution, and the measurement technique.
The document provides information about x-ray diffraction training opportunities and classes at MIT, including safety training requirements. It describes x-ray diffraction data collection classes on November 6th and 13th that cover the Rigaku powder diffractometer and PANalytical X'Pert Pro respectively. It also lists several data analysis workshops in November and December focusing on programs like Jade and analyzing thin films and texture. The document then provides a basic overview of x-ray diffraction principles including Bragg's law and how diffraction provides information about crystalline structure.
- Grazing incidence X-ray diffraction (GIXRD) is a technique that allows analyzing thin film samples by varying the incident angle of the X-rays to change their penetration depth.
- GIXRD provides enhanced signals from thin film layers compared to conventional XRD and helps distinguish thin film peaks from substrate peaks. It can also be used to analyze phases, stress, and crystal structure as a function of depth.
- Examples showed how GIXRD allowed analyzing phase composition and residual stress at different depths in thin film solar cell structures and revealed surface treatment effects in a stainless steel sample.
X-ray diffraction is a technique used to characterize nanomaterials by analyzing the diffraction patterns produced when X-rays interact with the crystal structure of a material. The document discusses the history, principles, instrumentation, and applications of XRD. It describes how XRD can be used to determine properties like crystallite size, dislocation density, strain, and identify crystalline phases by comparing to known standards. XRD provides a non-destructive way to analyze crystal structures with high accuracy and is suitable for both powder and thin film samples.
Solid state physics (schottkey and frenkel)abi sivaraj
This document discusses different types of lattice defects in crystals. It describes Schottky and Frenkel defects. A Schottky defect occurs in ionic crystals when equal numbers of oppositely charged ions leave their lattice sites, creating vacancies while maintaining overall charge neutrality. A Frenkel defect occurs when an atom moves from its lattice site to an interstitial site, producing a vacancy and interstitial. Schottky defects lower the crystal's density while Frenkel defects do not change density.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
This document provides an introduction to fundamentals of spectroscopy. It discusses various types of spectroscopy including UV-visible, infrared, Raman, and photoluminescence spectroscopy. The key topics covered are the electromagnetic spectrum, principles of absorption and emission of radiation, Beer-Lambert's law, instrumentation, applications of different spectroscopic techniques, and pioneers in the field of spectroscopy.
Molecular orbital theory (MOT) is an alternative model to valence bond theory that explains how atomic orbitals from different atoms combine to form molecular orbitals. The linear combination of atomic orbitals (LCAO) method considers the probability of finding electrons in atomic orbitals from different atoms. According to the LCAO method, molecular orbitals are formed from constructive and destructive interference of atomic orbitals. MOT can be used to explain bonding in homonuclear diatomic molecules like N2 and O2, heteronuclear diatomic molecules like CO and NO, and polyatomic molecules like CO2 and SF6. It can also describe bonding in octahedral transition metal complexes like hexaaquoferrate(II) ion
Frankel and Schottky defects occur in ionic crystals when ions leave their lattice sites. Frankel defects occur when an ion leaves its site and occupies an interstitial site, maintaining electrical neutrality and stoichiometry. Examples include AgCl and AgBr. Schottky defects occur when both cation and anion ions leave their lattice sites, creating vacancies that decrease the crystal's density. Examples include NaCl and KCl. The key difference between the defects is that Frankel defects do not change density while Schottky defects do due to vacancies formed by both ion types leaving lattice sites.
This document discusses using Orgel and Tanabe-Sugano diagrams to calculate crystal field splitting energies (10 Dq) and Racah parameters (B) for various d-block metal complexes. It provides examples of calculating these values for d1, d2, d7, and d9 systems using observed absorption bands and relationships between the transitions. The values obtained are then used to determine 10 Dq, B, and the nephelauxetic ratio β for complexes such as [Ti(H2O)6]3+, [V(H2O)6]3+, and [Co(H2O)6]2+.
The document provides an introduction to group theory and symmetry elements in chemistry. It defines symmetry elements like proper axes of rotation, planes of symmetry, centers of inversion, and improper axes of rotation. It explains how symmetry operations like rotations and reflections combine to form point groups, which describe the symmetry of molecules. Examples of common point groups like C2v, C3v, D4h, and D6h are given with molecular structures that belong to each group. The document outlines the process of assigning point groups to molecules and includes examples of molecules from various groups.
This document summarizes different types of defects in crystals. It classifies defects as zero-dimensional point defects, one-dimensional line defects, two-dimensional surface defects, or three-dimensional bulk defects. Point defects include vacancies, interstitials, Frenkel defects, and Schottky defects. Line defects include edge and screw dislocations. Surface defects include grain boundaries and twin boundaries. Bulk defects include precipitates, dispersants, inclusions, and voids. Defects can impact material properties and are sometimes deliberately introduced to improve properties.
This document provides an overview of group theory concepts. A group is a collection of elements that is closed under a binary operation, contains an identity element, and has inverse elements. Groups can be represented by multiplication tables. Symmetry operations within a point group can be classified into conjugacy classes based on their similarity transforms. Matrix representations allow symmetry operations to be modeled as transformations on object coordinates.
Miller indices specify directions and planes in crystal lattices using integer indices. They are represented by sets of integers in parentheses that indicate the intercepts of a plane or direction with the lattice's basis vectors. For planes, the intercepts are taken as reciprocals and represented by (hkl). Directions are represented by [hkl] and families of directions by <hkl>. Miller indices allow unambiguous identification of planes and directions that influence material properties like optical behavior and reactivity.
The document discusses different types of crystal defects including point defects, stoichiometric defects, and non-stoichiometric defects. Stoichiometric defects include Schottky and Frenkel defects which involve cation-anion pairs missing or cation dislocations. Non-stoichiometric defects result from deviations from the ideal ratio of cations to anions and include metal excess or deficiency defects involving anion or cation vacancies. Common examples of different defect types in various crystals are provided.
The document discusses various types of nuclear reactions. It defines nuclear reactions as processes where two nuclei or nuclear particles collide and produce different products than the initial particles. It describes several types of nuclear reactions including elastic and inelastic scattering, pickup and stripping reactions, compound nuclear reactions, radioactive capture, and photo disintegration. Elastic scattering involves the projectile and outgoing particles being the same, while inelastic scattering results in a loss of energy and particles scattered in different directions with different energies. Pickup reactions involve a gain of nucleons from the target, and stripping reactions involve one or more nucleons captured from the projectile. The document provides examples of each type of reaction.
Wilhelm Roentgen discovered x-rays in 1895. X-rays are generated when electrons collide with a metal target in a vacuum tube. When x-rays hit the planes of atoms in a crystal, they cause diffraction patterns that can be used to determine the crystal structure. X-ray diffraction analysis involves directing a beam of x-rays at a sample and measuring the angles and intensities of the diffracted beams to determine the sample's crystal structure, phase composition, and structural properties like lattice parameters. It is a non-destructive technique used to identify crystalline materials and phases.
This document discusses different types of crystal defects. It begins by defining an ideal crystal and explaining that real crystals contain defects due to deviations from a completely ordered atomic arrangement. Crystal defects are classified as point defects, line defects, planar defects, or bulk defects depending on their geometry. Point defects, which occur around a single atom, are further divided into vacancy defects, interstitial defects, Schottky defects, and Frenkel defects. Line defects include edge dislocations and screw dislocations. Planar defects involve grain boundaries and stacking faults, while bulk defects are voids, cracks, or impurity inclusions. The document provides examples and descriptions of each type of defect.
Solid State Synthesis of Mixed-Metal Oxidesanthonyhr
The document discusses solid-state synthesis of mixed metal oxides. It introduces solid state chemistry and synthesis, which involves producing solids by combining substances through high-temperature reactions. Specific aims are to synthesize new mixed metal oxide compounds with distinct properties for various applications. The methodology described involves using silica tubes lined with carbon and heating powdered metal reactants at high temperatures for 1-2 weeks to form novel crystals. Future work plans to carry out reactions combining tin, lead, antimony and bismuth, and synthesize new mixed metal oxides according to predicted products.
This document provides information about x-ray diffraction (XRD) training classes and basics of XRD. It outlines several safety and operation training sessions on specific dates for the x-ray diffraction equipment. It also provides a brief overview of key x-ray diffraction concepts including Bragg's law, diffraction patterns from single crystal and polycrystalline samples, diffraction geometries, phase identification using databases, and applications like determining crystal structure and strain.
Rietveld refinement is a widely used technique for determining crystal structures and quantifying crystalline materials from powder diffraction data. It works by minimizing the difference between observed and calculated diffraction patterns using least squares refinement. Key aspects include modeling the background, peak shape, unit cell parameters, atomic positions, and other structural details. Common software packages are used to perform the iterative refinement calculations.
5 10c exercise_index the aluminum tilt series_ex on saed do not know cell parsJoke Hadermann
This is a tutorial on how to index single crystal (electron) diffraction patterns when the cell parameters are not yet known. This follows after another tutorial for the case of known cell parameters, 5.10a.
This document discusses character tables and their use in determining irreducible representations. It provides examples of deriving character tables for the point groups C3v and C4v based on five theorems. These theorems relate the number of irreducible representations to the number of classes in the group, define characters as traces of matrices, require orthogonality of representations, and relate dimensions of representations to the group order. The document shows how characters in a table are used to determine basis functions that transform like specific irreducible representations. It also discusses decomposing reducible representations into sums of irreducible components.
X-ray diffraction is a technique used to characterize nanomaterials by analyzing the diffraction patterns produced when X-rays interact with the crystal structure of a material. The document discusses the history, principles, instrumentation, and applications of XRD. It describes how XRD can be used to determine properties like crystallite size, dislocation density, strain, and identify crystalline phases by comparing to known standards. XRD provides a non-destructive way to analyze crystal structures with high accuracy and is suitable for both powder and thin film samples.
Solid state physics (schottkey and frenkel)abi sivaraj
This document discusses different types of lattice defects in crystals. It describes Schottky and Frenkel defects. A Schottky defect occurs in ionic crystals when equal numbers of oppositely charged ions leave their lattice sites, creating vacancies while maintaining overall charge neutrality. A Frenkel defect occurs when an atom moves from its lattice site to an interstitial site, producing a vacancy and interstitial. Schottky defects lower the crystal's density while Frenkel defects do not change density.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
This document provides an introduction to fundamentals of spectroscopy. It discusses various types of spectroscopy including UV-visible, infrared, Raman, and photoluminescence spectroscopy. The key topics covered are the electromagnetic spectrum, principles of absorption and emission of radiation, Beer-Lambert's law, instrumentation, applications of different spectroscopic techniques, and pioneers in the field of spectroscopy.
Molecular orbital theory (MOT) is an alternative model to valence bond theory that explains how atomic orbitals from different atoms combine to form molecular orbitals. The linear combination of atomic orbitals (LCAO) method considers the probability of finding electrons in atomic orbitals from different atoms. According to the LCAO method, molecular orbitals are formed from constructive and destructive interference of atomic orbitals. MOT can be used to explain bonding in homonuclear diatomic molecules like N2 and O2, heteronuclear diatomic molecules like CO and NO, and polyatomic molecules like CO2 and SF6. It can also describe bonding in octahedral transition metal complexes like hexaaquoferrate(II) ion
Frankel and Schottky defects occur in ionic crystals when ions leave their lattice sites. Frankel defects occur when an ion leaves its site and occupies an interstitial site, maintaining electrical neutrality and stoichiometry. Examples include AgCl and AgBr. Schottky defects occur when both cation and anion ions leave their lattice sites, creating vacancies that decrease the crystal's density. Examples include NaCl and KCl. The key difference between the defects is that Frankel defects do not change density while Schottky defects do due to vacancies formed by both ion types leaving lattice sites.
This document discusses using Orgel and Tanabe-Sugano diagrams to calculate crystal field splitting energies (10 Dq) and Racah parameters (B) for various d-block metal complexes. It provides examples of calculating these values for d1, d2, d7, and d9 systems using observed absorption bands and relationships between the transitions. The values obtained are then used to determine 10 Dq, B, and the nephelauxetic ratio β for complexes such as [Ti(H2O)6]3+, [V(H2O)6]3+, and [Co(H2O)6]2+.
The document provides an introduction to group theory and symmetry elements in chemistry. It defines symmetry elements like proper axes of rotation, planes of symmetry, centers of inversion, and improper axes of rotation. It explains how symmetry operations like rotations and reflections combine to form point groups, which describe the symmetry of molecules. Examples of common point groups like C2v, C3v, D4h, and D6h are given with molecular structures that belong to each group. The document outlines the process of assigning point groups to molecules and includes examples of molecules from various groups.
This document summarizes different types of defects in crystals. It classifies defects as zero-dimensional point defects, one-dimensional line defects, two-dimensional surface defects, or three-dimensional bulk defects. Point defects include vacancies, interstitials, Frenkel defects, and Schottky defects. Line defects include edge and screw dislocations. Surface defects include grain boundaries and twin boundaries. Bulk defects include precipitates, dispersants, inclusions, and voids. Defects can impact material properties and are sometimes deliberately introduced to improve properties.
This document provides an overview of group theory concepts. A group is a collection of elements that is closed under a binary operation, contains an identity element, and has inverse elements. Groups can be represented by multiplication tables. Symmetry operations within a point group can be classified into conjugacy classes based on their similarity transforms. Matrix representations allow symmetry operations to be modeled as transformations on object coordinates.
Miller indices specify directions and planes in crystal lattices using integer indices. They are represented by sets of integers in parentheses that indicate the intercepts of a plane or direction with the lattice's basis vectors. For planes, the intercepts are taken as reciprocals and represented by (hkl). Directions are represented by [hkl] and families of directions by <hkl>. Miller indices allow unambiguous identification of planes and directions that influence material properties like optical behavior and reactivity.
The document discusses different types of crystal defects including point defects, stoichiometric defects, and non-stoichiometric defects. Stoichiometric defects include Schottky and Frenkel defects which involve cation-anion pairs missing or cation dislocations. Non-stoichiometric defects result from deviations from the ideal ratio of cations to anions and include metal excess or deficiency defects involving anion or cation vacancies. Common examples of different defect types in various crystals are provided.
The document discusses various types of nuclear reactions. It defines nuclear reactions as processes where two nuclei or nuclear particles collide and produce different products than the initial particles. It describes several types of nuclear reactions including elastic and inelastic scattering, pickup and stripping reactions, compound nuclear reactions, radioactive capture, and photo disintegration. Elastic scattering involves the projectile and outgoing particles being the same, while inelastic scattering results in a loss of energy and particles scattered in different directions with different energies. Pickup reactions involve a gain of nucleons from the target, and stripping reactions involve one or more nucleons captured from the projectile. The document provides examples of each type of reaction.
Wilhelm Roentgen discovered x-rays in 1895. X-rays are generated when electrons collide with a metal target in a vacuum tube. When x-rays hit the planes of atoms in a crystal, they cause diffraction patterns that can be used to determine the crystal structure. X-ray diffraction analysis involves directing a beam of x-rays at a sample and measuring the angles and intensities of the diffracted beams to determine the sample's crystal structure, phase composition, and structural properties like lattice parameters. It is a non-destructive technique used to identify crystalline materials and phases.
This document discusses different types of crystal defects. It begins by defining an ideal crystal and explaining that real crystals contain defects due to deviations from a completely ordered atomic arrangement. Crystal defects are classified as point defects, line defects, planar defects, or bulk defects depending on their geometry. Point defects, which occur around a single atom, are further divided into vacancy defects, interstitial defects, Schottky defects, and Frenkel defects. Line defects include edge dislocations and screw dislocations. Planar defects involve grain boundaries and stacking faults, while bulk defects are voids, cracks, or impurity inclusions. The document provides examples and descriptions of each type of defect.
Solid State Synthesis of Mixed-Metal Oxidesanthonyhr
The document discusses solid-state synthesis of mixed metal oxides. It introduces solid state chemistry and synthesis, which involves producing solids by combining substances through high-temperature reactions. Specific aims are to synthesize new mixed metal oxide compounds with distinct properties for various applications. The methodology described involves using silica tubes lined with carbon and heating powdered metal reactants at high temperatures for 1-2 weeks to form novel crystals. Future work plans to carry out reactions combining tin, lead, antimony and bismuth, and synthesize new mixed metal oxides according to predicted products.
This document provides information about x-ray diffraction (XRD) training classes and basics of XRD. It outlines several safety and operation training sessions on specific dates for the x-ray diffraction equipment. It also provides a brief overview of key x-ray diffraction concepts including Bragg's law, diffraction patterns from single crystal and polycrystalline samples, diffraction geometries, phase identification using databases, and applications like determining crystal structure and strain.
Rietveld refinement is a widely used technique for determining crystal structures and quantifying crystalline materials from powder diffraction data. It works by minimizing the difference between observed and calculated diffraction patterns using least squares refinement. Key aspects include modeling the background, peak shape, unit cell parameters, atomic positions, and other structural details. Common software packages are used to perform the iterative refinement calculations.
5 10c exercise_index the aluminum tilt series_ex on saed do not know cell parsJoke Hadermann
This is a tutorial on how to index single crystal (electron) diffraction patterns when the cell parameters are not yet known. This follows after another tutorial for the case of known cell parameters, 5.10a.
This document discusses character tables and their use in determining irreducible representations. It provides examples of deriving character tables for the point groups C3v and C4v based on five theorems. These theorems relate the number of irreducible representations to the number of classes in the group, define characters as traces of matrices, require orthogonality of representations, and relate dimensions of representations to the group order. The document shows how characters in a table are used to determine basis functions that transform like specific irreducible representations. It also discusses decomposing reducible representations into sums of irreducible components.
This document is the preface to the instructor's manual for Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion. It provides an overview of the contents of the manual, which contains solutions to the end-of-chapter problems from the textbook. The preface notes there are now 509 problems and the solutions range from straightforward to challenging. It stresses the solutions are only for instructors and should not be shared with students.
1) The document describes developing the element matrix equation for a beam element with a uniformly distributed load. It involves discretizing the beam into elements, deriving the governing differential equation and boundary conditions for a general element, determining the shape functions through interpolation, and assembling the element stiffness matrix.
2) The weighted integral form of the governing differential equation is used to derive the element matrix equation. Shape functions are obtained as Hermite cubic polynomials.
3) As an example, the document analyzes a beam with two elements and applies the derived element matrix equation and boundary conditions to solve for displacements and forces.
MTH101 - Calculus and Analytical Geometry- Lecture 44Bilal Ahmed
Virtual University
Course MTH101 - Calculus and Analytical Geometry
Lecture No 44
Instructor's Name: Dr. Faisal Shah Khan
Course Email: mth101@vu.edu.pk
The document contains information about the Post Graduate Common Entrance Test (PGCET) for the year 2017, to be held on July 1st from 2:30 pm to 4:30 pm. It provides details about the exam such as the date and time, subjects covered, number of questions, maximum marks, and duration. It also lists instructions for candidates regarding filling the answer sheet correctly and guidelines for answering questions. The exam will contain 75 multiple choice questions in the subject of electrical sciences, to be answered in 120 minutes. Candidates are advised to carefully read and follow all instructions provided.
The document provides instructions for a mock test paper on civil engineering. It consists of 65 multiple choice questions worth a total of 100 marks. Questions 1-25 and 56-60 carry 1 mark each, while questions 26-55 and 61-65 carry 2 marks each. Questions will cover both civil engineering topics and general aptitude. There are negative marks for incorrect answers. Calculators are allowed but no other aids. The purpose is to simulate exam conditions and help identify strengths and weaknesses to guide further preparation.
The document discusses topics in coordinate geometry including slopes, length of line segments, midpoints, and proofs. It provides formulas and examples for calculating slopes, lengths of lines, and midpoints. It also discusses using an analytical approach to prove geometric theorems through using coordinates, formulas, and algebraic manipulations rather than relying solely on diagrams.
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
This is a lecture on the hydraulics of gradually varied flow in open channels. It shows the profiles common in the open channels and some numerical examples using numerical integration.
CBSE XII MATHS SAMPLE PAPER BY KENDRIYA VIDYALAYA Gautham Rajesh
The document provides the blueprint for the Class XII maths exam, including the breakdown of questions by chapter and type (1 mark, 4 marks, 6 marks). It includes 13 chapters, with a total of 10 one-mark questions, 12 four-mark questions, and 7 six-mark questions. The document also provides a sample question paper following the same format, with Section A having 10 one-mark questions, Section B having 12 four-mark questions, and Section C having 7 six-mark questions. The question paper covers various topics like relations and functions, matrices, differentiation, integrals, differential equations, and probability.
In this work we discuss how to compute KLE with complexity O(k n log n), how to approximate large covariance matrices (in H-matrix format), how to use the Lanczos method.
We solve elliptic PDE with uncertain coefficients. We apply Karhunen-Loeve expansion to separate stochastic part from spatial part. The corresponding eigenvalue problem with covariance function is solved via the Hierarchical Matrix technique. We also demonstrate how low-rank tensor method can be applied for high-dimensional problems (e.g., to compute higher order statistical moments) . We provide explicit formulas to compute statistical moments of order k with linear complexity.
This document discusses integrals involving exponential functions. It shows that integrating the exponential function results in dividing the constant in the exponent. It evaluates the important definite integral from 0 to infinity of e^-ax, which equals 1/a. It also evaluates the double integral from -infinity to infinity of e^-a(x^2+y^2), which equals sqrt(pi/a). Taking derivatives of these integrals generates related integrals involving x and x^4 that are useful in kinetic theory of gases.
This first lecture describes what EMT is. Its history of evolution. Main personalities how discovered theories relating to this theory. Applications of EMT . Scalars and vectors and there algebra. Coordinate systems. Field, Coulombs law and electric field intensity.volume charge distribution, electric flux density, gauss's law and divergence
This document provides an overview of structural deflections and methods for calculating beam deflection. It discusses how deflections are an important part of structural analysis and excessive deflections can lead to failures. The deflection curve of a beam is described as the initially straight axis bending into a curve. Methods for determining the deflection curve including drawing shear and moment diagrams and identifying the concave upward and downward portions. Examples are provided to demonstrate calculating deflection curves for various beams. The double integration method for relating beam deflections to bending moments is described. Assumptions and limitations of the method are also stated. Further examples demonstrate applying the double integration and moment-area methods to calculate maximum deflections in beams.
This document provides information about determinants and their application to matrices and geometric shapes. It begins with definitions and examples of determinants of 2x2 and 3x3 matrices. The determinant of a matrix represents the area or volume of the geometric shape formed by the matrix's rows or columns. The document then provides an example of calculating the determinant of a 4x4 matrix using cofactor expansion. It includes two practice problems to calculate determinants. The final problems involve applying concepts like Kirchhoff's laws to solve for unknown currents in an electric circuit and traffic flows at intersections.
Similar to Electron diffraction: Tutorial with exercises and solutions (EMAT Workshop 2017) (20)
5 10b exercise_determine the diffraction symbol of ca_f2_ex on saedJoke Hadermann
These slides follow up on the slides with a tutorial on how to index ED patterns, 5 10a. From the result achieved in those slides, we now derive the reflection conditions and corresponding possible space groups.
Complementarity of advanced TEM to bulk diffraction techniquesJoke Hadermann
Lecture contains some examples of how advanced TEM techniques can help solve the structure if for some reason bulk diffraction techniques do not allow to propose a model.
Scheelite CGEW/MO for luminescence - Summary of the paperJoke Hadermann
This document summarizes a research paper that studied the incommensurate modulation and luminescence properties of CaGd2(1-x)Eu2x(MoO4)4(1-y)(WO4)4y phosphors. The researchers found that these materials exhibit incommensurate modulation of the cation ordering due to vacancies in the scheelite structure, which requires description in superspace. Replacing Mo6+ with W6+ switched the modulation from 3+2D to 3+1D, despite their similar sizes. Variations in Eu content changed luminescence intensity but not the modulation periodicity. The results contradict prior reports of simple ordered structures.
Mapping of chemical order in inorganic compoundsJoke Hadermann
Presentations of some of the possibilities of observing cation and anion order in perosvkite based structures in order to solve their structure or to solve other questions, when they could not be solved by bulk diffraction techniques. The examples include a (Pb,Bi)FeO3-d compound, a solid oxide fuel cell compound Sr(Nb,Co,Fe)O3-d and several brownmillerite related compounds, as well as a "relaxor ferromagnetic". This was an invited lecture given at the Spring meeting of the British Crystallographic Association in 2014.
New oxide structures using lone pairs cations as "chemical scissors"Joke Hadermann
Lone pair cations like Bi3+ and Pb2+ allow for asymmetric coordination environments which can be used to create crystallographic shear planes in perovskite structures. Shear planes involve shifting one part of the structure relative to the other. This results in a new class of anion-deficient perovskite structures called the AnBnO3n-2 homologous series. Unlike other homologous series, these structures have electrically and magnetically active interfaces between perovskite blocks, resulting in frustrated magnetic structures. The orientation and spacing of shear planes can be controlled through composition.
This lecture was given at the 2nd International School on Aperiodic Crystals in Bayreuth, Germany. It is an updated version of the lecture given on the 1st school, that can be found between my lectures as "TEM for incommensurately modulated materials".
Electron crystallography for lithium based battery materialsJoke Hadermann
This lecture was given at the IUCr (International Union of Crystallography) meeting in Madrid, 2011. Contents are focussed on the use of precession electron diffraction for functional materials, mainly lithium based battery materials, but also a perovskite was included, since a large part of the audience worked on that subject.
Solving the Structure of Li Ion Battery Materials with Precession Electron Di...Joke Hadermann
Very short summary of the paper "Solving the Structure of Li Ion Battery Materials with Precession
Electron Diffraction: Application to Li2CoPO4F", published in Chem. Mater.
Tem for incommensurately modulated materialsJoke Hadermann
This presentation is a teaching lecture given on the International School on Aperiodic Crystals and explains how to index electron diffraction patterns taken from incommensurately modulated materials, with exercises, and gives some examples of HAADF-STEM and HRTEM images on incommensurately modulated materials.
Modulated materials with electron diffractionJoke Hadermann
This lecture was given at the International School of Crystallography in Erice 2011, on the topic of Electron Crystallography. It explains the very basics of how to index commensurately and incommensurately modulated materials. It was meant for a 40 minute lecture.
Direct space structure solution from precession electron diffraction data: Re...Joke Hadermann
The presentation shows the main points from the publication "Direct space structure solution from precession electron diffraction data: Resolving heavy and light scatterers in Pb13Mn9O25" about how thee structure of Pb13Mn9O25 was solved using transmission electron microscopy.
Determining a structure with electron crystallography - Overview of the paper...Joke Hadermann
The route to a solved structure (in this case Pb13Mn9O25) on the basis of precession electron diffraction, combined with HAADF-STEM, HRTEM, EELS and EDX is shown.
Summary of the paper "Solving the Structure of Li Ion Battery Materials with Precession
Electron Diffraction: Application to Li2CoPO4F"
Cation Ordering In Tunnel Compounds Determined By TemJoke Hadermann
1) The document discusses tunnel manganites, which are compounds containing manganese oxide frameworks that form tunnel structures. These frameworks can accommodate various guest cations in their tunnels.
2) Several examples of tunnel manganite frameworks are described, including pyrolusite, ramsdellite, hollandite, and more complex examples. The document also discusses the types of guest cations that can occupy the tunnels.
3) New examples of tunnel manganites are presented, including SrMn3O6, CaMn3O6, and a todorokite compound with rock salt-type tunnel contents. Composite structural models are discussed that can be used to describe ordering of guest cations
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
Immersive Learning That Works: Research Grounding and Paths ForwardLeonel Morgado
We will metaverse into the essence of immersive learning, into its three dimensions and conceptual models. This approach encompasses elements from teaching methodologies to social involvement, through organizational concerns and technologies. Challenging the perception of learning as knowledge transfer, we introduce a 'Uses, Practices & Strategies' model operationalized by the 'Immersive Learning Brain' and ‘Immersion Cube’ frameworks. This approach offers a comprehensive guide through the intricacies of immersive educational experiences and spotlighting research frontiers, along the immersion dimensions of system, narrative, and agency. Our discourse extends to stakeholders beyond the academic sphere, addressing the interests of technologists, instructional designers, and policymakers. We span various contexts, from formal education to organizational transformation to the new horizon of an AI-pervasive society. This keynote aims to unite the iLRN community in a collaborative journey towards a future where immersive learning research and practice coalesce, paving the way for innovative educational research and practice landscapes.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
2. At the end of this lecture you should be able to
index SAED patterns if the cell parameters are known
determine the possible space groups from SAED
patterns
determine possible point groups from CBED patterns
Combine (3) and (4) to find thé space group.
3. Reflections: what do they represent? What
is their origin? What information can they
give us? cf. lectures Mark De Graef
4. Constructive vs. destructive interference
reflection – no reflection
position d-values of the planes
intensity occupation in the planes
both symmetry of the structure
21. *How?
You know the structure, thus all cell parameters.
Make a list of all reflections with hkl and their d-values.
• use excell to make the list yourself (e.g. cubic: )
• use free software like Powdercell
• ...
d hkl
8.00000 001
5.00000 010
4.23999 011, 01-1
4.00000 002
3.12348 012,01-2
3.00000 100
... ...
𝑑 =
𝑎
ℎ2 + 𝑘2 + 𝑙2
22. *How?
You know the structure, thus all cell parameters.
Make a list of all reflections with hkl and their d-values.
• use excell to make the list yourself (e.g. cubic: )
• use free software like Powdercell
• ...
d hkl
8.00000 001
5.00000 010
4.23999 011, 01-1
4.00000 002
3.12348 012,01-2
3.00000 100
... ...
𝑑 =
𝑎
ℎ2 + 𝑘2 + 𝑙2
23. 3Å
5Å
Experimentally: the other way around:
010
100
[001]
*
Let's agree to label the reflections
with the d-value instead of 1/(xÅ):
d hkl
8.00000 001
5.00000 010
4.23999 011, 01-1
4.00000 002
3.12348 012,01-2
3.00000 100
... ...
26. Exercise:
index the given patterns taken
from a CaF2 mineral
Solution slides (plus all other temporarily deleted slides)
are available after the lecture on
slideshare/johader
27. h k l d I F
1 1 1 3.15349 83.73 61.89
2 0 0 2.731 0.11 3.07
2 2 0 1.93111 100 96.55
3 1 1 1.64685 31.44 46.49
2 2 2 1.57674 0.2 6.81
4 0 0 1.3655 12.69 74.25
3 3 1 1.25307 11.35 38.65
4 2 0 1.22134 0.54 8.67
4 2 2 1.11493 23.75 61.87
5 1 1 1.05116 6.88 34.2
3 3 3 1.05116 2.29 34.2
You need this table made for CaF2
𝑑 =
𝑎
ℎ2 + 𝑘2 + 𝑙2
with a=5.4620 Å
SG. Fm3m
28. We are going to index these patterns:
Start with easiest:
highest symmetry or smallest interreflection distances
= usually lower zone indices (“main zones”)
(Online version:
workpage can be
found at the end.)
Tilt
series
32. probably this is <001>
(Cubic: [100], [010], [001] equivalent = <001>)
33. To do: measure the distances, compare to list d-hkl, index
consistently.
Step 1: Use the scalebar for the conversion
factor to 1/d-values.
Scalebar = R (in mm)
equal to 1/0.08 nm
R.d=L
L = R . 0.8Å = ?
Write down and
use this in the rest
of the exercise
Measure the scalebar
34. Step 2: measure the distance of two reflections, not on
the same line, calculate the corresponding d-value
Point 1
d
5.46 Å
3.15 Å
2.73 Å
Point 2
d
5.46 Å
3.15 Å
2.73 Å
1
2
To do: measure the distances, compare to list d-hkl, index
consistently.
35. Step 2: measure the distance of two reflections, not on
the same line, calculate the corresponding d-value
Point 1
d
5.46 Å
3.15 Å
2.73 Å
Point 2
d
5.46 Å
3.15 Å
2.73 Å
1
2
To do: measure the distances, compare to list d-hkl, index
consistently.
36. To do: measure the distances, compare to list d-hkl, index.
Step 3: look up in the table to which
reflection this corresponds
100
110
200
Point 1
d
Point 2
d
Point 1
hkl
Point 2
hkl
1
2
5.46 Å
3.15 Å
2.73 Å
5.46 Å
3.15 Å
2.73 Å
100
110
200
37. To do: measure the distances, compare to list d-hkl, index.
Step 3: look up in the table to which
reflection this corresponds
100
110
200
Point 1
d
Point 2
d
Point 1
hkl
Point 2
hkl
1
2
5.46 Å
3.15 Å
2.73 Å
5.46 Å
3.15 Å
2.73 Å
100
110
200
38. Keep in mind: d-values for all equivalent {hkl}!
100
010
1
2
If point 1 is 200 then point 2 is 020 or 002.
Choose and stick to your choice.
Righthanded system.
Step 4: make the indexation consistent
44. Next zone: with reflections closest to the central beam.
Because reflections far from the central beam:
lower d-values
larger amount of possible matches of hkl to this d
difficult to conclude which one is correct index!
1
3
5
45. Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
Point 1
d
2.57 Å
2.75 Å
3.15 Å
Point 2
d
1 2
2.57 Å
2.73 Å
3.15 Å
46. Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
Point 1
d
2.57 Å
2.75 Å
3.15 Å
Point 2
d
1 2
2.57 Å
2.73 Å
3.15 Å
47. Look up in the table to which reflection
this corresponds
110
200
111
110
200
111
Point 1
d = 3.15 Å
Point 2
d = 2.73 Å
hkl hkl
1 2
48. Look up in the table to which reflection
this corresponds
110
200
111
110
200
111
Point 1
d = 3.15 Å
Point 2
d = 2.73 Å
hkl hkl
1 2
49. Make the indexation in a consistent manner.
1 2
Point 2 should be indexed as
200
020
200
all are correct
-
50. Make the indexation in a consistent manner.
1 2
Point 2 should be indexed as
200
020
200
all are correct
-
51. Consistency:
This is a tilt series...
...so the common row needs to have the
same indices in all patterns
200 200
200 200
54. Consistency:
200111
111
-
1
200
3
1 and 3 have the same d-value
+
relation between 1 and 3 = vector 200
you need two indices such that
h1 k1 l1 = h3+2 k3 l3
(also possible 111 and 111, make a choice and stick to it for the following patterns)
- - - - -
83. Right upper zone:
Point 2
d
1.22 Å
1.11 Å
1.05 Å
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
We already know the
first point: 200.
200
2
84. Right upper zone:
Point 2
d
1.22 Å
1.11 Å
1.05 Å
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
We already know the
first point: 200.
200
2
85. Look up in the table to which reflection this corresponds:
We know already it is either 151 or 131 or 042 or 153
1.05 Å 151
131
042
Point 2
d
Point 2
hkl
200
2
86. Look up in the table to which reflection this corresponds:
We know already it is either 151 or 131 or 042 or 153
1.05 Å 151
131
042
Point 2
d
Point 2
hkl
200
2
87. If this were not a tilt series...
could have
been at
first sight
both 151
and 333...
Or in this particular case of 333: you would need to see 111 and 222 at 1/3 and 2/3 of the distance.
1.05 Å
Point 2
d
h k l d I F
1 1 1 3.15349 83.73 61.89
2 0 0 2.731 0.11 3.07
2 2 0 1.93111 100 96.55
3 1 1 1.64685 31.44 46.49
2 2 2 1.57674 0.2 6.81
4 0 0 1.3655 12.69 74.25
3 3 1 1.25307 11.35 38.65
4 2 0 1.22134 0.54 8.67
4 2 2 1.11493 23.75 61.87
5 1 1 1.05116 6.88 34.2
3 3 3 1.05116 2.29 34.2
88. If this were not a tilt series...
Point 2 could have been at first sight
both 115 and 333...
Again, consistency check:
Or in this particular case of 333: you would need to see 111 and 222 at 1/3 and 2/3 of the distance.
200
151
200
333
151
115 has same d-
value as 115
-
-
133 does not have
same d-value as 333
133
90. [001]
[015]-
[013]
-
[012]-
[035]-
[011]-
010
031 051
053
What if you don’t know the material and thus don't know
the reflection conditions?
You might have different reflection conditions.
You might have more possibilities for in-between
zones:
043
032
041021
[025]-
052 [014]-
[023]-
[034]-
When indexed correctly, the patterns in between
have to give you one of these as zone-index.
011
91. Last pattern, pattern bottom left:
Point 2
d
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
200
2
1.65 Å
1.58 Å
1.37 Å
92. Last pattern, pattern bottom left:
Point 2
d
Measure the distance of two reflections, not on the
same line, calculate the corresponding d-value
200
2
1.65 Å
1.58 Å
1.37 Å
93. Look up in the table to which reflection this corresponds.
We know already it is either: 151 or 131 or 042 or 153
1.65 Å 113
131
311
Point 2
d
Point 2
hkl
200
2
Make sure the vector addition is consistent.
94. Look up in the table to which reflection this corresponds.
We know already it is either: 151 or 131 or 042 or 153
1.65 Å 113
131
311
Point 2
d
Point 2
hkl
200
2
Make sure the vector addition is consistent.
At this point (table+vector
addition) also 113 is still
possible. However, when
considering the other zones
( next slides), you will see
that only 131 remains.
95. Calculate the zone-index = [013]
= in agreement with sections scheme (pdf p. 62-82)
-
200
131
062 200
131062
131
-
The indexation is indeed consistent
concerning vector addition for both.
Calculate the zone-index = [031]
= not in agreement with sections scheme
200
131
062 200
113026
113
-
-
CORRECT
WRONG
96. Make your analysis easier by not taking
ED patterns from separate crystals, but
taking different ED patterns from the
same crystallite, if possible.
=“Tilt series”
97. So now you have indexed these four patterns.
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-
98. ...indexed patterns give you info on phase,
orientation, cell parameters,...
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-
99. What if you do not know the cell
parameters beforehand?
Analyse the patterns try to propose basis vectors
(For example reflections closest to the central beam)
Same system as previous slides:
can you index all reflections?
If not, adapt your choice of
basis vectors and try again.
100. What if you need to know if
anything changed in the
symmetry?
Need to determine the space group
Determine the
extinction symbol
from the spot patterns
Determine the point
group from the CBED
patterns
101. Let's say we need to know if the CaF2 in the example
is still the Fm3m type CaF2.
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-
-
Part 1. You index the patterns using just the cell parameters.
Part 2. You determine the space group.
102. 1. Extinction symbol(SAED)
Space group?
P no reflection conditions
F h+k=2n, k+l=2n, h+l=2n
I h+k+l=2n
A/B/C k+l=2n/h+k=2n/h+k=2n
2. Point Group (CBED)
glide planes conditions on hk0/h0l/0kl
screw axes conditions on h00/0k0/00l
mirror planes, inversion
centre, rotation axes
no extra conditions
CBED
SAED
104. Reflection conditions can be looked up in tables in
International Tables for Crystallography Vol. A
Or using freeware such as Space Group Explorer
105. Be careful: forbidden reflections can be present
because of multiple diffraction
Incident electron wave
106. Can see this:
When reflection
conditions say this:
For example possible
010
100
020
100
111. Destroy double diffraction paths by tilting.
If becomes
If stays
then extinct, was due to DD
then not extinct.
112. Determine the general reflection condition from the patterns.
hkl:Centering Reflection condition
P no conditions for hkl
I hkl: h+k+l=2n
F hkl: h+k=2n, k+l=2n,
h+l=2n
A hkl: k+l=2n
B hkl: h+l=2n
C hkl: h+k=2n
113. hkl:
h+k+l=2n
h+k, k+l, h+l=2n
h+k=2n
Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-
(remember it is cubic
so h, k and l are
interchangeable)
114. hkl:
h+k+l=2n
h+k, k+l, h+l=2n
h+k=2n
Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
-
[013]
-
[011]
-
For these patterns
both would be
good....!?
Only disagreement
with hkl:h+k=2n is
in [001] where also
110 is absent.
However, this
could be due to a
special reflection
condition on hk0.
116. Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
[013]
-
[011]
-
200
042
hkl: h+k+l=2n
h+k, k+l, h+l=2n
h+k=2n
117. Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
[013]
-
[011]
-
200
042
hkl: h+k+l=2n
h+k, k+l, h+l=2n
h+k=2n
118. Determine the reflection conditions from the patterns.
200
042
hkl: h+k, k+l, h+l=2n
h+k=2n
242
x
x
x x
x
021 121
100
141
221
"hkl: h+k=2n"
should look like this:
119. It is possible to draw the wrong
conclusions if you do not have
enough zones!
120. Look up the matching extinction symbol
in the International Tables of
Crystallography.
?
123. Determine the reflection conditions from the patterns.
200
020
[001]
0kl: k=2n, l=2nhkl:h+k, k+l, h+l=2n
example is cubic,
so 0kl = hk0 = h0l
If it were 0kl:k+l=4n, k=2n, l=2n
it should look like:
200
020
[001]
040
400
220
240
440
420
124. Look up the matching extinction symbol
in the International Tables of
Crystallography.
?
125. Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
[013]
-
[011]
-
200
042
hhl: h+l=2n
h=2n,l=2n
hkl:h+k, k+l, h+l=2n
0kl: k+l=2n
126. Determine the reflection conditions from the patterns.
200
131
200
151
200111
200020
[001] [015]
[013]
-
[011]
-
200
042
hhl: h+l=2n
h=2n,l=2n
hkl:h+k, k+l, h+l=2n
0kl: k+l=2n
127. Determine the reflection conditions from the patterns.
200111
[011]
-
hkl:h+k, k+l, h+l=2n
0kl: k+l=2n
hhl: h+l=2n
311
511
222
422
111
-
h=2n,l=2n
128. Step 2: look up the matching extinction
symbol in the International Tables of
Crystallography.
?
?
129. 200 and 020 could be due to double
diffraction...
130. 200 and 020 could be due to double
diffraction...
Tilt around 200 until all other
reflections gone except h00 axis:
131. 200 and 020 could be due to double
diffraction...
Tilt around 200 until all other reflections
gone except h00 axis:
200 does not disappear
It is not double diffraction
00l: l=2n
not 00l: l=4n
132. Step 2: look up the matching extinction symbol in
the International Tables of Crystallography.
133. From the reflection conditions you get
the extinction symbol:
F - - -
Extinction symbol
134. From the reflection conditions you get
the extinction symbol:
F - - -
This still leaves 5 possible space groups
F23 Fm3 F432 F43m Fm3m
-
Extinction symbol
135. Point Group
From the reflection conditions you get
the extinction symbol:
F - - -
This still leaves 5 possible space groups
F23 Fm3 F432 F43m Fm3m
Only difference: rotation axes and mirror planes
cannot be derived from reflection conditions
need
-
Extinction symbol
136. Symmetry-relation in real space
𝑟′ = 𝑅. 𝑟 + 𝑡
R = rotation matrix
t=translation vector
Symmetry-relation amplitudes in reciprocal space
ℎ′ 𝑘′ 𝑙′ = ℎ𝑘𝑙 . 𝑅
137. Within one point group, there are the same
theoretical equivalences between the reflections
For example have the same equivalent reflections:
𝑃4 𝑃41 𝑃42 𝑃43 𝐼4 𝐼41
But not
𝑃4 𝐼4
However! Friedel's Law adds an inversion symmetry
On an experimental ED pattern all of the above have
the same equivalences between reflections!
Can discern only Laue symmetries.
138. Step 2: look up the matching extinction symbol in
the International Tables of Crystallography.
Need a different technique: CBED
141. For CBED you need sufficiently thick crystals:
t=10 nm t=20 nm t=30 nm
t=40 nm t=50 nm
CBED patterns
calculated using JEMS
142. Easiest* : "Eades method"
Obtain CBED patterns of high symmetry zones,
preferably showing HOLZ
Derive "diffraction group" of each zone
Look for the point group that can have all the
diffraction groups you found for the different zones
* purely personal opinion
143. You need two tables:
relation between symmetries in CBED patterns
and the diffraction groups
which point groups contain which diffraction groups
(both tables are included in these slides)
1
2
144. Table from: J.A. Eades,
Convergent beam
diffraction, in: Electron
Diffraction Techniques,
volume 1, ed. J. Cowley,
Oxford University Press,
1992
1
Each zone axis
CBED pattern
has its own
diffraction
group.
145. From: Atlas of Electron Diffraction, Jean-Paul Morniroli, freely available at
http://www.electron-diffraction.fr/Files/220_atlas_version_september_2015.pdf
Whole pattern, 3D symmetry
aka "whole pattern symmetry"
3D WP
Whole pattern, 2D symmetry
aka "whole pattern projection symmetry"
2D WPBright field, 3D symmetry
aka "bright field symmetry"
3D BF
no HOLZ lines? 2D BF = BF projection symm
146. Table from: J.A. Eades,
Convergent beam diffraction, in:
Electron Diffraction Techniques,
volume 1, ed. J. Cowley, Oxford
University Press, 1992
.
186. What would make a difference further?
For example:
-cell parameters
-look for a third zone etc.
(e.g. is there zone with 3 or 6 fold m-3m)
-SAED for reflection conditions
187. If you need
cell parameters a=b= 4.72 Å, c=3.16 Å
to be able to index all SAED patterns,
the point group is
4/mmm
m3m
188. If you need
cell parameters a=b= 4.72 Å, c=3.16 Å
to be able to index all SAED patterns,
the point group is
4/mmm
m3m
190. Combine with information about
reflection conditions from SAED patterns
020
200
110
020
011
002
hkl:
no conditions
h+k+l=2n
002
110
-
191. Combine with information about
reflection conditions from SAED patterns
020
200
110
020
011
002
hkl:
no conditions
h+k+l=2n
002
110
-
192. Combine with information about
reflection conditions from SAED patterns
020
200
110
020
011
002
hk0:
no conditions
h+k=2n
0kl:
no conditions
k+l=2n
k=2n or l=2n
002
110
-
hhl:
no conditions
l=2n
193. Combine with information about
reflection conditions from SAED patterns
020
200
110
020
011
002
hk0:
no conditions
h+k=2n
0kl:
no conditions
k+l=2n
k=2n or l=2n
002
110
-
hhl:
no conditions
l=2n
200. /1, 00 kk
Ewald sphere
Diffraction pattern
PED technique:
R. Vincent, P. Midgley,
Ultramicroscopy,1994, 53, 271
201.
202. Extract the intensities using
EDM (free)
EXTRAX (free)
PETS (free)
CRISP-EDT (not free)
...
203. • imperfect orientation
• not thin enough
• too small precession angle
• too large precession angle
• overlapping patterns (twins,
impurities, ...)
• …
Intensities will deviate from the correct values:
But good enough to solve structures.
Deviations have been proven to not destroy the possibility to solve structures. Even a rough division into weak - medium
- strong reflections is often good enough anyway [H. Klein&J.David, Acta Cryst. (2011). A67, 297–302]
204. Treat the data of the individual patterns:
• could apply Lorentz factor (no consensus whether
necessary)
𝐼′ = 𝐼. 𝑔 1 −
𝑔
2𝑅
2
(R=radius Laue circle)
• could symmetrize = replace reflections by average of
equivalent reflections (also gives Rsym)
• ...
205. Merge separate lists into one list
Scale factor based on common reflections
Possible pitfalls in merging:
• (wrong?) ratio of small amount of reflections is
imposed on all reflections
• merging is done in series, errors accumulate
• typical tilt series around main axis, but main axes
often stronger deviations by themselves
• should a weighing be used?
• ...
207. Manually + treatment of
patterns with PETS (free)
Automated in ADT (not
free)
You obtain a list of hkl
and intensities
PETS: Palatinus et al. Acta Cryst. (2015). A71, 235-24
208. Direct methods, or optimisation methods made
for single crystal data
SIR2011 (free)
Fox (free)
Endeavour (not free)
Superflip (free)
...
Same concerning refinement e.g. Jana (free)
Right structure càn come out (GIGO).
209. At the end of this lecture you should be able to
index SAED patterns if the cell parameters are known
determine the possible space groups from SAED
patterns
determine possible point groups from CBED patterns
Combine (3) and (4) to find thé space group.